Statement
I hereby declare that this submission is my own work and to the best of my knowledge
it contains no material previously published or written by another person, nor material
which to a substantial extent has been accepted for the award of any other degree or
diploma at UNSW or any other educational institution, except where due
acknowledgement is made in the thesis. Any contribution made to the research bycolleagues, with whom I have worked at UNSW or elsewhere, during my candidature,
is fully acknowledged.
I also declare that the intellectual content of this thesis is the product of my own
work, except to the extent that assistance from others in the project’s design andconception or in style, presentation and linguistic expression is acknowledged.

(Parham Mokhtari)(date)

i

ii

“It seemed to me a superlative thing — to know the explanation of
everything, why it comes to be, why it perishes, why it is.
So spoke Socrates nearly 2500 years ago...”Robyn Williams,
in Foreword of “More Big Questions,Paul Davies in conversation with Phillip Adams” (Piper Films, 1998)

iii

iv

Dedicatedto my mother,to my father,

Pari

Parvizand to my sister,
Tara

v

vi

Acknowledgements
During the course of my research endeavours which have ultimately led to thecompletion of this doctoral dissertation, I have had the extremely good fortune of being
associated with a number of people whose support I herein wish to acknowledge.
I extend my gratitude first and foremost to Dr Frantz Clermont, whom I have been
proud to consider my research supervisor, professional colleague, and personal friend.
Ever since our intellectually explosive Australian summer of 1993/94 when the seeds ofthe present research work were sown, he has consummately transcended his supervisoryrole, and provided intense and almost brotherly guidance at every stage of the research
process. Our uncountable and immeasurably inspiring, free-spirited discussions, haveenriched my experience as a doctoral candidate, and fuelled my own passion for the
Science in human speech. I am boundlessly grateful to Frantz for having imparted his
scholarly outlook, his visionary thinking, and his exacting attitude in scientific research.
May he, my scientific mentor, find frequent satisfaction in this permanent record of ouroften challenging but always exciting and inspired journey.
I amicably thank Dr Michael (Spike) Barlow, for his selfless friendship, both on
and off the tennis court; and for his numerous and insightful comments and suggestions
both informally and at the Speech Group meetings held at the School. My sincere
thanks also extend to Dr Santha Sampath, for her moral support and encouragement
during the entire duration of my candidature, and for her involvement and suggestions
at the Speech Group meetings at which my research work was the topic of discussion.
I gratefully thank the following people, both past and present at the School: Dr
Michael Wagner for having taken me on board as a doctoral candidate early in 1993,
for his initial suggestions of pursuing research in the area of speaker characterisation,and for his many hours of involvement and guidance during approximately the first yearof my candidature; Dr Hiroaki Oasa for his early involvement, interest, and suggestions;Cmdr. Clive Cooper for his amiable and inspiring, but all too brief presence, especiallyat those three-way meetings with Frantz at the Gorman House café during 1994; Mrvii

Andy Quaine for his moral support and guidance during a particularly traumatic period
in 1995; Dr Xin Yao for sharing some of his expertise in numerical optimisation
methods; Mrs Willma Nelowkin, Mr Phillip Clark, Mr Jeff Howard, Mr Wen Ung, Mr
Joe Holloway, Mr Aaron Mihe, Mr Crispin Russell and Mr Colin Stevens for theirtechnical support in computer-related matters; Mrs Eileen Trot, Mrs Pam Gianakakis
and Ms Alison McMaster for their invaluable, secretarial assistance; and Dr Charles
Newton, Head of the School, for his invariable support.
I thank Ms Peta Kennedy and Ms Diane Cook of the Postgraduate Office of the
Academy, for their kind assistance in matters relating to University administration; andthe University College itself, for granting me the financial support of a Postgraduate
Research Scholarship.
My special thanks are extended to Ms Marija Tabain and Ms Catherine Watson,
whose kind invitation and encouragement to present a seminar at the Speech, Hearingand Language Research Centre at Macquarie University in Sydney (in July 1997) did
significantly help to crystallise my thoughts on some of the work relevant to Chapter 5.
I warmly thank my dear friends Mr Bruce Taloni and Mr Brendan Cooper; thelatter for his candid friendship and encouragement even at a distance, and the former
whose frequent enquiries as to my state of health and that of the “baby” thesis did help
me to retain an overall sense of balance. In that regard, I also extend my heartfelt thanks
to Mr Ehsan Keyhani, Mrs Mehrassa Keyhani, Dr Hashem Etminan, Mrs Mina Etminan,
Dr Fereidoon Ghasemi, Mrs Mina Ghasemi, and to Mr Shahrokh Vaziri, whose
invariable encouragements, warm conviviality, and refined culture are deeply impressedin my mind.
Finally, I wish to thank my dear family, to whom this work is dedicated. Their
unfailing and unwavering love and multi-faceted support through times of crisis and joy
alike, have been the mainstay of my passage not only through this period of doctoral
research, but more inclusively, throughout my years. The strength I carry with mebeyond these formative years, is firmly rooted in their singular belief in my ability tosucceed. I hope that in this volume they will find at least a small part of the fruits of
their labours.

viiiAbstract
The acoustic speech signal naturally contains information pertaining to both the
linguistic message and the identity of the speaker. Separation of these two, primarysources of variability has been of considerable concern to speech scientists for many
years, yet we still lack a model of the speech communication process that can accountfor a wide variety of speakers even for a restricted set of vowels of a given dialect of
spoken English.
Basic research was therefore undertaken to gain a better understanding of theseparate influences and of the interactions between phonetic and speaker-specificattributes of vowel sounds. First, a new methodology for performing computer vowel
recognition was developed, which has the ability to reveal the contrastive influence ofvowel-speaker interactions across different regions of the frequency spectrum. The
results obtained from applying this methodology have: (i) yielded significant insights
into the long-standing problem of phonetic-speaker dichotomy; and (ii) prompted asearch for an even more fundamental explanation in terms of the physical properties ofthe speech production mechanism.
To this end, a new functional representation of vocal-tract shapes was derived,
which depends directly on resonance parameters while retaining the uniquenessproperties of the Linear-Prediction model of speech production. This hybrid modelling
approach was used together with a new, articulatory method of speaker normalisation,
to quantify speaker differences in vocal-tract shapes, and thus to define physicalcorrelates of the phonetic-speaker interactions which were earlier shown to adversely
affect vowel recognition accuracy in certain frequency bands.
In sum, this research work embraces two major domains of computer speechscience and technology — namely, speech acoustics and speech production. Beyond
advancing knowledge in speaker characterisation, it ultimately has implications in thestill unresolved problem of speech or speaker recognition by computer.
ix

x

Contents
Acknowledgements . . . . . . . . . . . . . . . . . . .

vii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . ix
Contents . . . . . . . . . . . . . . . . . . . . . . . . . xi
1 General Introduction . . . . . . . . . . . . . . . . . . . 1
1.1 Research Context: The Speech-Speaker Dichotomy . . . . . . . . .

1

1.1.1 On the Sources of Speech Variability . . . . . . . . . . . .

2

1.1.2 On the Sources of Speaker Variability . . . . . . . . . . . .

3

1.2 Research Scope . . . . . . . . . . . . . . . . . . . . .

5

1.3 Thesis Outline

6. . . . . . . . . . . . . . . . . . . . .

2 Speech-Speaker Dichotomy: An Interpretive Review of the Literature . 9
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . .

92.1.1 An Historical Account of the Dichotomy . . . . . . . . . . .

92.1.2 Structural Profile of Literature . . . . . . . . . . . . .

12

2.2 Acoustic-Auditory Evidence and Influential Role . . . . . . . . .

13

2.2.1 Human Auditory-Perceptual Evidence of Spectral Dichotomy . . .

13

2.2.2 Overcoming the Dichotomy by Auditorily-motivated Spectral Emphasis .

16

2.2.2.1 Frequency-scale Transformations .

. . . . . . . . . . .

2.2.2.2 Importance of Relative Spectral Measures .

16. . . . . . . .

19

2.2.3 Emerging Perspective . . . . . . . . . . . . . . . .

21

2.3 Acoustic-Phonetic Evidence and Approaches . . . . . . . . . .

24

2.3.1 Spectral Manifestations and Descriptive Parameters . . . . . .

25. . . . . . . . . .

25. . . . . . . . . . . . . . .

27

2.3.2 Consequences in Vowel/Speaker Recognition by Machine . . . .

30

2.3.1.1 Phonetic Quality of Spoken Vowels .
2.3.1.2 Speaker Individuality .

2.3.2.1 Speaker Recognition .

. . . . . . . . . . . . . . .

30

2.3.2.2 Vowel Recognition .

. . . . . . . . . . . . . . .

34

xi

2.3.3 Approaches to Overcoming the Dichotomy . . . . . . . . .
. . . . . . . . .

38
39. . . . . . . . . . . . . .

42

2.4 Acoustic-Articulatory Evidence and Approaches . . . . . . . . .

46

2.4.1 The Speaker Factor: Sources of Variability and Consequences . . .

47

2.3.3.1 Towards Robustness in Parameterisation .
2.3.3.2 Speaker Normalisation .

2.4.1.1 Theoretical Framework .

. . . . . . . . . . . . . .

2.4.1.2 Most Common Empirical Observations .

47. . . . . . . . .

48

2.4.2 Persistent Approaches to Speaker Normalisation . . . . . . .

51

2.4.3 Speaker Normalisation Beyond Vocal-Tract Length . . . . . .

55. . . . . . . . . . . . .

55. . . . .

62

. . . . . . . . . . . . . . . . .

69

3 Speech Materials and Acoustic Parameterisation . . . . . .

73

3.1 Introduction . . . . . . . . . . . . . . . . . . . . .

73

3.2 Speech Materials and Speaker Sets

. . . . . . . . . . . . .

73

3.3 Acoustic Data used to Unfold the Dichotomy . . . . . . . . . .

81

3.3.1 Vocalic Steady-State Detection . . . . . . . . . . . . .

81

3.3.2 Formant Estimation . . . . . . . . . . . . . . . . .

86

2.4.3.1 Under-exploited Evidence .

2.4.3.2 Articulatory Parameterisation and the Inverse Problem

2.5 Concluding Discussion

3.3.2.1 Goodness Criteria .

. . . . . . . . . . . . . . . .

87. . . . . . .

87

3.3.2.3 IFV-related Problems and Corrections. . . . . . . . . .

89

3.3.2.4 LP-related Problems and Corrections .

. . . . . . . . . .

92

3.3.3 Cepstral Analysis . . . . . . . . . . . . . . . . . .

97

3.3.2.2 Formant-tracking Method and Evaluation Aids .

3.4 Acoustic Data used to Validate the Dichotomy. . . . . . . . .

98

3.4.1 Formant Estimation . . . . . . . . . . . . . . . . .

99

3.4.1.1 Amplitude-Section Approach .

. . . . . . . . . . . .

99. . . . . . . . . . . . . .

100

3.4.2 Simplified Cepstrum . . . . . . . . . . . . . . . . .

101

3.5 Summarising Perspective . . . . . . . . . . . . . . . . .

104

4 Acoustic-Phonetic Analysis of Speech-Speaker Dichotomy . . .

107

4.1 Introduction . . . . . . . . . . . . . . . . . . . . .

107

4.2 Methodology for Unfolding the Dichotomy . . . . . . . . . . .

109

4.2.1 Approach . . . . . . . . . . . . . . . . . . . .

110

3.4.1.2 Spectrogram Approach .

xii

4.2.2 Spectral Representations . . . . . . . . . . . . . . .
4.2.3 Classification Methods and Distance Formulations . . . . . . .

112
114

. . . . .

116

. . . .

121

4.3 The Dichotomy Unfolded . . . . . . . . . . . . . . . . .

135

4.3.1 Behaviour of Classification Accuracy Using Linear Classifier . . .

135

4.3.2 Behaviour of Classification Accuracy Using Quadratic Classifier . .

137

4.2.3.1 Classifier based on Hyper-Planar Decision Boundaries

4.2.3.2 Classifier based on Hyper-Quadratic Decision Boundaries

4.4 The Dichotomy Explained. . . . . . . . . . . . . . . .

4.4.1 Spectral Regions of Vowel-Speaker Dichotomy

146. . . . . . .

146

4.4.2 Acoustic-Phonetic Explanation of the Dichotomy . . . . . . .

151

4.4.3 Dependence on Spectral Representation . . . . . . . . . .

155

4.4.4 Dependence on Speaker Homogeneity . . . . . . . . . . .

162

4.4.5 Dependence on Idiolectal Differences . . . . . . . . . . .

174

4.5 Concluding Summary . . . . . . . . . . . . . . . . . .

183

5 Vocal-Tract Shape Parameterisation and Estimation . . . . .

187

5.1 Introduction . . . . . . . . . . . . . . . . . . . . .

187

5.2 Rationale . . . . . . . . . . . . . . . . . . . . . .

189

5.2.1 Resonance-Based Parameterisation of Area-Functions. . . . .

189

5.2.2 Inherent Uniqueness in LP-Derived Area-Functions . . . . . .

195

5.3 Parameters of Unique LP-derived Area-Functions. . . . . . . .

201

5.3.1 Dependence of LP-derived VT-Shapes on Formant Frequencies . .

203. . . . . . . . . . . . . .

203. . . . . . . . . . . . . . .

207

5.3.2 Dependence of LP-derived VT-Shapes on Formant Bandwidths . .

212

5.3.1.1 Partial Theoretical Proof
5.3.1.2 Empirical Validation .

5.3.2.1 Theoretical Motivation .

. . . . . . . . . . . . . .

212

5.3.2.2 Empirical Justifications .

. . . . . . . . . . . . . .

214

5.4 Method of Area-Function Parameterisation and Estimation. . . . .

221

5.4.1 Description . . . . . . . . . . . . . . . . . . . .

221

5.4.2 Evaluation of Proposed Area-Function Parameterisation . . . . .

227

5.4.2.1 Inter-parameter correlations .

. . . . . . . . . . . . .

5.4.2.2 Representation of Directly Measured Area-Functions .

227. . . . .

229

5.5 Evaluation of Estimation Method . . . . . . . . . . . . . .

234

5.5.1 Inter-repetition Alignment of Area-Functionsxiii. . . . . . . .

236

5.5.2 Re-estimation of Directly Measured Area-Functions . . . . . .
5.5.2.1 Model-matched Conditions .

. . . . . . . . . . . . .

239
239

5.5.2.2 More Realistic Conditions .

. . . . . . . . . . . . .

246

5.5.2.3 Model-based Formant Correction .

. . . . . . . . . . .

249

5.5.3 Estimation of Area-Functions from Measured Formants . . . . .

253

5.5.3.1 Formant-induced Variability of LP Area-Functions. . . . . .

254

5.5.3.2 Closed-glottis Correction of Formant Bandwidths .

. . . . . .

260

5.6 Concluding Summary . . . . . . . . . . . . . . . . . .

264

6 An Articulatory Explanation of Speech-Speaker Dichotomy . .

269

6.1 Introduction . . . . . . . . . . . . . . . . . . . . .

269

6.2 Articulatory Approach. . . . . . . . . . . . . . . . . .

270

6.3 Methodology for Articulatory Speaker Normalisation . . . . . . .

273

6.3.1 Speaker Normalisation of VT Fixed Structure . . . . . . . .

273

6.3.2 Speaker Normalisation of Articulatory Setting . . . . . . . .

276

6.3.3 Speaker Normalisation of Vowel-Specific Articulatory Strategy . .

281

6.4 Articulatory Explanation of the Dichotomy . . . . . . . . . . .

283

6.4.1 Contributions of VT Structure to the Dichotomy . . . . . . .

284

6.4.2 Contributions of Articulatory Setting to the Dichotomy . . . . .

293

6.4.3 Contributions of Articulatory Strategy to the Dichotomy. . . . .

303

6.5 The Dichotomy Undone . . . . . . . . . . . . . . . . .

321

6.5.1 Articulatory Consequences. . . . . . . . . . . . . .

323. . . . . . . . . . . .

328

6.6 Concluding Summary . . . . . . . . . . . . . . . . . .

333

7 Concluding Discussion . . . . . . . . . . . . . . . . .

337

7.1 Acoustic-Phonetic Perspective . . . . . . . . . . . . . . .

337

7.1.1 Experimental Methods . . . . . . . . . . . . . . . .

337

6.5.2 Acoustic-Phonetic Consequences

7.1.1.1 A Posteriori Selection of Frequency Sub-bands .

. . . . . . .

7.1.1.2 Recognition Framework for Diagnosing the Dichotomy .

337. . . .

338

7.1.2 Spectral Manifestations of the Dichotomy and Implications . . . .

340

7.2 Acoustic-Articulatory Perspective . . . . . . . . . . . . . .

343

7.2.1 Experimental Methods . . . . . . . . . . . . . . . .

343

7.2.1.1 Vocal-tract Shape Parameterisation .

. . . . . . . . . .

343

7.2.1.2 Articulatory Speaker Normalisation .

. . . . . . . . . .

346

xiv

7.2.2 Articulatory Explanation of the Dichotomy . . . . . . . . .

347

A Formant-tracking Analysis Conditions and Sequence Charts .

353

B Mathematical Derivation of the SM Model . . . . . . . . .

363

C A Lossy Vocal-Tract Acoustic Model . . . . . . . . . . .

369

D The LP Method of Inversion “Stepped Down” . . . . . . .

375

E Representation of Directly Measured Area-Functions . . . .

383

F Re-estimation of Directly Measured Area-Functions . . . . .

391

G Closed-glottis Correction of Formant Bandwidths . . . . . .

399

Bibliography . . . . . . . . . . . . . . . . . . . . . .

403

xv

xvi

Chapter 1
General Introduction

1.1 Research Context: The Speech-Speaker Dichotomy
If there is a single objective which epitomizes over half a century of international
research efforts concerned with the processes of human speech communication, it is tofind order and regularity in the variability which exists therein. Owing to their intrinsic
dimensionality and complexity, the sources of variability which pervade the threedomains of speech production, acoustics, and perception, still pose a challenge tospeech scientists and technologists alike. Indeed, whether it is to contribute toward theultimate goal of a unified theory of speech communication, or toward the application ofsuch knowledge in spoken language systems or in aids for the handicapped, variability is
still a significant and an eminent research problem.
Human speech communication naturally implies auditory perception of an acoustic
speech signal, which itself is produced by the vocal apparatus of a speaker. The acoustic
speech signal therefore carries not only the linguistic message which the speaker wishes
to convey, but also a host of para-linguistic information about the speaker’s identity,
gender, emotions, state of health, and other characteristics which may be perceived by
the listener. The kernel of the variability problem is therefore made up of two main
components: the speech or the linguistic message itself, and the idiosyncrasies of the
speaker who produces it. Furthermore, the influences of these two components are so
intricately intertwined in the acoustic speech signal, that a complete definition of theirseparate manifestations continues to be elusive. The inherent difficulties of this longstanding problem are better appreciated, and an investigative approach more easily
formulated, by considering the potential sources of variability which underlie the twocomponents of what we henceforth will refer to as the speech-speaker dichotomy or,1interchangeably, as the dichotomy in short.

1.1.1 On the Sources of Speech Variability
Sources of variability which are normally attributed to the speech component of the
dichotomy are broadly categorised in terms of either prosodic or segmental propertiesof the acoustic speech stream. The former include signal energy (or stress), syllable
duration (or rhythm), and fundamental frequency (or intonation) which characterise
utterances across syllables, words, and sentences; they embody generally polysegmental aspects of speech. By contrast, segmental properties are known to more
directly embody the phonetic or the linguistic aspects of the spoken message, and aredescribed in terms of the individual phonemes, which themselves are generally believed
to be the building blocks (or phonetic units) of spoken syllables, words, and phrases.
The segmental units of the speech stream are, in turn, broadly categorised as either
consonants or vowels, depending on the degree to which the airstream through the
vocal-tract is obstructed during the production of those speech sounds. Consonants,
which include stops, fricatives, and approximants, involve a relatively greater degree ofobstruction than vocalic sounds; they can therefore give rise to non-periodic, noise-like,or sudden bursts of acoustic energy, caused either by air turbulence at a very narrow
constriction, or by a release following complete closure in the vocal-tract. By contrast,
spoken vowels are produced with relatively more open vocal-tract configurations and,more importantly, their articulatory, acoustic, and perceptual properties are relatively
better apprehended. In particular, the acoustic properties of spoken vowels are firmly
rooted in the spectral domain, where they have been studied mainly in terms of thecompact, articulatorily- and perceptually-relevant description afforded by the first few
formants (or acoustic resonances of the vocal-tract).
The importance of vowels, and our relatively superior understanding of theirproperties, can be explained partly by their high rate of occurrence in spoken language(Denes (1963) has estimated that the vowel phonemes occur over 30% of the time inspoken English). In addition, they exhibit relatively stable spectral characteristics andconsistently high levels of acoustic energy, and are therefore regarded as “islands ofreliability” (Lea et al., 1975) in the acoustic speech stream, particularly in context of2

stressed syllables. Even so, the segmental properties of spoken vowels are known to be
influenced by potential sources of variability which include speaking rate, linguistic
stress, and coarticulation with preceding and following phonemes. Traditionally, thesetypes of variability are controlled by embedding the vowel (generically represented by
the symbol V) in the quasi-neutral phonetic context afforded by the formation of /hVd/words; consistency in stress and speaking rate is then achieved when those monosyllabic
words are read aloud (in citation form) from lists, in which the order of the words are
randomised to reduce inter-repetition bias.

1.1.2 On the Sources of Speaker Variability
It is often taken for granted that any of the types of variability discussed above could
potentially arise from, and could therefore be used to characterise, either inter- or intraspeaker variations. Whilst the former refers to the speaker differences which are impliedin the speech-speaker dichotomy, the latter refers to the repeatability of speech sounds
by a single speaker on different occasions; variability of this kind might be induced in
the short-term by varying emotions, in the medium-term by different states of health,and in the longer-term by the consequences of ageing. In this regard, Nolan (1983,p.59) aptly puts forward the view that each speaker’s vocal apparatus determines“merely the range within which variation in a particular parameter is constrained to takeplace”, with the assertion that there is “no acoustic feature which escapes the plasticityof the vocal tract.”
Leaving aside the possibilities of intended disguise or mimicry, or indeed of
extrinsically-induced variations in speech production beyond what might be regarded asa speaker’s normal manner of speaking, it is generally accepted that a given speaker(particularly, a phonetically naive speaker) will not be able to exactly reproduce a given
set of speech sounds, even in an emotionally neutral, contemporaneous setting, andeven after accounting for measurement variability. In this regard, however, Broad(1972) has noted that the magnitude of the resulting, so-called inter-repetition variation(of measured formant frequencies) is of the same order as the human auditoryperceptual tolerance to perturbations in those acoustic parameters of vocalic speechsounds. Indeed, it is generally assumed that under controlled conditions (i.e., in the3absence of the types of systematic variability discussed above, or attempts at disguise or
mimicry), the variability of acoustic parameters measured in any given vowel of a single
speaker is considerably less than the variability measured in the same vowel of two or
more speakers; consequently, the phonetic equivalence (Broad, 1976) of the measured
parameter values is more easily defined in the single-speaker case.
Perhaps the most well-known consequence of the greater magnitude of inter- thanintra-speaker variability, is the often dramatic deterioration in the performance of an
automatic speech recogniser, when used, without proper training, by more than one
speaker. The extent of that deterioration — a presumed consequence of the dichotomy
problem — depends partly on the nature and the degree of the differences between the
speakers involved. In order to cope with this problem, state-of-the-art automatic speech
recognition (ASR) systems often use statistically-based algorithms for speaker
adaptation of either the training data or the model parameters (e.g. Young, 1996).
However, at best (as we shall see in Chapter 2), such algorithms account only
incompletely for the basic sources of inter-speaker variability; their effectiveness istherefore ultimately curtailed by under-exploitation of our limited but expandingknowledge of the fundamental components of speaker variability.
In this vein, it may be of some advantage to note the obvious fact that the acoustic
speech signal originates from the vocal apparatus of the speaker. Differences between
the acoustic realisations of phonetically-equivalent speech sounds uttered by different
speakers can therefore be related to the size, shape, and idiosyncratic usage of theirarticulatory organs. The two commonly accepted, broad categories which describeinter-speaker differences at an articulatory level, are known as organic and learned,
respectively (Garvin and Ladefoged, 1963). Whilst the former refers to anatomicaldifferences in the size and shape of the speakers’ articulatory organs such as the tongue,teeth, laryngeal and other vocal-tract fixed structures, the latter refers to the
behavioural differences in their individual usage of the moveable articulators during
speech production.
The most substantial types of organic differences are undoubtedly those which arefound between the vocal-tract anatomical structures of men, women, and children.
Analogously, the most substantial types of learned differences are those which
4distinguish between the dialects or idiolects of a given language. Whilst both of thesetypes of differences serve to differentiate between groups of speakers, within eachgroup (for example, adult male speakers of a particular dialect of English) there exist
individual or intrinsic, inter-speaker differences which, although generally smaller inmagnitude than group differences, can still give rise to substantial acoustic variability.
Indeed, as Nordström (1977, p.81) has pointed out in his interpretation of a wide rangeof vowel formant data, “one does not need to go beyond comparisons within the same
speaker category (male speakers of the same dialect) to find immense variability.”
An important consideration in this regard is the degree of homogeneity (orconversely, heterogeneity) of a given speaker group. Speakers who possess similar,
vocal-tract anatomical and behavioural characteristics, might be expected to exhibit afair degree of homogeneity in their measured acoustic data, and are therefore less likely
to be problematic from an ASR point of view. By contrast, a more heterogeneousgroup of speakers might be expected to exhibit larger amounts of acoustic, interspeaker variability, which may arise from a combination of organic and learned,
articulatory differences; interestingly, whilst the acoustic manifestations of thesedifferences might detrimentally affect the performance of an ASR system, they are
generally of distinct advantage in the converse task of automatic speaker recognition.

1.2 Research Scope
It is well-known, and our introductory exposition of the problem has made quite clear,that the speech-speaker dichotomy is the kernel of a vast and intricate network ofvariability in the processes of human speech communication. Whilst the two primary
components of variability manifest in the acoustic speech signal can be attributed,
respectively, to the linguistic message (the speech) and the communicator of thatmessage (the speaker), each component implies a multitude of potential sources ofvariability, whose intertwined influences have rendered a complete decoupling ofspeech and speaker formidable. Clearly, one cannot hope to undertake in a single study,
an investigation of the speech-speaker problem in its entirety. We must therefore
narrow the scope of our research, by imposing limits on the potential “determinants ofacoustic parameter values” (Broad, 1982).
5

As noted earlier, our understanding of the vowel sounds of English, accumulated
over decades of research, perhaps exceeds that of any other class of speech sounds.
Indeed, our greater knowledge of the properties of spoken vowels is reflected not onlyin the acoustic domain where their spectral properties are best known in terms of theformant parameters, but also in the articulatory domain, where a number of models
have been proposed which account for the physiologically and acoustically relevant
properties of vowel articulation. The research scope of this thesis will therefore belimited to the segmental properties of the spoken (non-nasalised) vowels of English.
Our emphasis on the importance of relating the acoustic manifestations of the speechspeaker dichotomy to its physical causes, will become apparent in later chapters where
we shall adopt a vocal-tract modelling approach to provide such physical explanations
for the vowel sounds considered.
As for the speaker component of the dichotomy, we shall restrict our attention todifferences amongst adult, male speakers with no known speech impairments. The
potential sources of inter-speaker variability will include intrinsic and idiolectal
differences, which may be related to a combination of vocal-tract anatomical andbehavioural idiosyncrasies. Furthermore, by eliciting natural productions of vowel
sounds in a contemporaneous, laboratory environment, we forsake the wide range ofintra-speaker variability which has been attributed to emotions, states of health, ageing,
and either intended disguise or socially-induced variations. As a result, the full range ofintra-speaker variability in vowel production is kept to a minimum, and a relatively
invariant norm is established for each vowel of each speaker, from which may proceed asystematic investigation of purely phonetic and inter-speaker influences.

1.3 Thesis Outline
As outlined below, the results of our acoustic-phonetic and articulatory investigationsof the speech-speaker dichotomy are presented in two core chapters (4 and 6). Those
chapters are preceded by an interpretive review of the relevant literature (Chapter 2)
and a complete description of our speech data (Chapter 3); they are connected by our
detailed treatment of the problem of vocal-tract shape parameterisation and estimation(Chapter 5); and they are followed by a concluding discussion (Chapter 7).
6

In Chapter 2 we address the question of how we are justified in pursuing aproblem which has long been central to speech science and technology. Past attempts tofind regularity in the intertwined influences of speech and speaker variability, whether
qualitatively suggested or quantitatively shown, are recalled and placed in context.
Previous methods which have been proposed to overcome the influences of speechspeaker interactions are then reviewed, together with an overall perspective on theliterature concerning the speech-speaker dichotomy. In short, we will argue in Chapter
2 that despite our relatively more comprehensive knowledge of both the acoustic andarticulatory properties of spoken vowels, the dichotomy problem manifested in those
speech sounds still warrants further research.
In Chapter 3 we present and justify the range of speech materials and methods ofacoustic parameterisation used in our investigation of the speech-speaker dichotomy. In
particular, we describe three different datasets of vowels, recorded in the timehonoured /hVd/ context by adult, male speakers of Australian and American English.
Differences in the degree of phonetic and speaker complexity of those datasets, afford
the opportunity of examining the dependence of the dichotomy both on the degree of
speaker homogeneity, and on the presence or absence of idiolectal speaker differences.
We also recount the methods of acoustic parameterisation used to carefully extract theformants and the linear prediction (LP) cepstra at the steady-state of each vocalic
nucleus.
Those sets of parameters are then used in Chapter 4 to investigate the vowelspeaker dichotomy in the acoustic-phonetic domain. In particular, the spectral
manifestations of vowel-speaker interactions are unfolded by way of vowel recognition
experiments, designed specifically to lay bare the influences of phonetic and speakervariabilities along the spectral continuum. An indispensable tool in our methodology for
unfolding the dichotomy, is a new cepstral distance measure (Clermont and Mokhtari,1994) which facilitates the selection of frequency-bands in machine classification ofvowels. The interpretively superior formant parameters then provide an acousticphonetic explanation of the observed spectral manifestations of the dichotomy. We
conclude our acoustic-phonetic investigations by addressing the questions raised earlier,which concern speaker homogeneity and the influence of idiolectal speaker differences.
7

In search of a more complete elucidation of the problem of speech-speakerdichotomy, we then extend our investigations into the domain of speech production.
However, a methodological framework for investigating the physical correlates of the
speech-speaker dichotomy would be impractical if it were to rely on direct articulatory
measurements, which are in general difficult to acquire. In Chapter 5 we therefore
reconsider the well-known inverse problem of mapping the geometry of the humanvocal-tract from the acoustic speech signal. In particular, we derive a new functional
representation of vocal-tract shapes, which depends directly on resonance parameters(formants) while retaining the uniqueness properties of the well-known linear prediction(LP) method of inversion.
The vocal-tract shapes thus obtained from the formant data, which themselves aremeasured in Chapter 3 and used in Chapter 4 to unfold and explain the acousticphonetic manifestations of the dichotomy, then afford a physical explanation of the
dichotomy in Chapter 6. Towards that end, we describe a new method of articulatory
speaker normalisation, which facilitates a tripartite decomposition of the articulatory
sources of inter-speaker variability. Our investigative loop is then closed via resynthesisof acoustic parameters from the speaker-normalised vocal-tract shapes, thus allowing a
direct interpretation of the articulatory sources of speaker variability in terms of ouracoustic-phonetic methodology of cepstrum-based vowel recognition and formantbased explanation of the dichotomy.
In sum, Chapters 4, 5, and 6 collectively provide more complete insights into, andthus advance our understanding of the speech-speaker problem in the two domains ofspeech acoustics and speech production. In Chapter 7 we then draw conclusions
regarding the various contributions which arise from the research work described in thisthesis.

8

Chapter 2
Speech-Speaker Dichotomy:
An Interpretive Review of the Literature

2.1 Introduction
The speech-speaker dichotomy is not a new problem. Its long history is first given due
consideration in Section 2.1.1, where we will recall recurrent references to the opposing
and complementary roles attributed to the lower and the higher spectral regions of
spoken vowels in particular. This historical profile will also highlight that previous
contributions towards elucidating the intertwined influences of phonetic and speakerspecific sources of variability in the segmental properties of spoken vowels, span the
acoustic as well as the articulatory and auditory domains of speech communication.
Consequently, in Section 2.1.2, we will propose to re-examine the dichotomy problem
by attempting a tripartite review of the literature, from which will emerge ourcontention that previous works do not provide a cohesive perspective on the problem
and that further elucidation is warranted.

2.1.1 An Historical Account of the Dichotomy
Scattered amongst the vast assemblage of previous contributions, are a number ofisolated statements of a largely qualitative nature, which nevertheless betray anunderlying consensus regarding the likely acoustic correlates of the dichotomy.
One of the earliest of such quotes, dates back to a decade before the invention of
the sound spectrograph in 1946. Using very laborious techniques to estimate theresonance frequencies of sung vowels, Lewis (1936, p.97) very cautiously offered thefollowing conjecture:“It may be that the typical quality of a vowel is determined mainly by the two resonatorsof lowest frequency, with individual voice differences resulting from the action of the9other resonators...”Following relatively more comprehensive spectral analyses of the vowels “Ah” and “O”
sung by six trained, male singers, Lewis and Tuthill (1940, p.156) were then able toconclude“...that individual uniqueness in voice is significantly related to the operation of one ormore high frequency resonators which are relatively invariable. ... The two low frequency
resonators are undoubtedly crucial in the determination of vowel character, while the highfrequency resonators probably contribute only to less basic tonal qualities.”

More substantial evidence to support this view, had to await the invention of thesound spectrograph (Koenig et al., 1946). Indeed, this experimental tool paved the way
for less laborious, more large-scale acoustic-phonetic studies, which firmly establishedthe importance of the formant frequencies (or resonance frequencies of the vocal tract)1
in acoustic-level descriptions of spoken vowels. Spectrographic analyses were soon
complemented by auditory-perceptual experiments, using the so-called “pattern
playback” machine which synthesised speech sounds from hand-painted spectrograms.
Equipped with considerable expertise in both spectrographic analyses and perceptual
experiments, Delattre (1951, p.872) was able to assert the following:“No doubt that formant 3 is much less responsible than formants 1 and 2 for the linguisticcolor of vowels. Formant 3 is mainly to be considered as one of the many higher
resonances... . Being the lowest of these, it has the most perceptible effect. ... as a wholethese high resonances above formant 2 have very little effect on color, they mostly add
intelligibility without changing the color and are probably responsible for voice quality.”

However, perhaps the most direct interpretation of the formants remains firmly
rooted in the articulatory domain. For example, transcending the inherent complexity of
the relations between formants and articulatory configurations assumed during the
production of voiced speech sounds, Peterson (1959, p.151) noted that“...the lower formant frequencies depend more upon gross cavity sizes and constrictions
than upon exact shapes. The adjustment of the higher formant frequencies in generaldepends progressively more upon specific cavity sizes and shapes.”

More specifically, in his classic work on the acoustic theory of speech production,
1

Hereafter denoted in symbolic form as F1 , F2 , F3 , ... in ascending order of frequency.

10Fant (1960, p.48) begins his analytical treatment of the formant-based constraints on the
spectra of voiced speech sounds thus:“The relative importance of the separate formants of voiced sounds decreases withincreasing order above F2. F1 and F2 are the main determinants of vowel quality. F3 andF4 contribute significantly to the phonetic quality of front vowels, but in back vowels
they are of minor importance only. F3 and F4 ... also provide certain information on
personal voice characteristics.”

Further evidence of the relative importance of the formants is given by Bernard(1967, p.103) who, in the course of his painstaking, spectrographic measurements ofthe first three formant frequencies of vocalic speech sounds recorded by 170 speakers
of Australian English, was compelled to disregard the third formant in determining theso-called onset, steady-state or target, and offglide of those monophthongal anddiphthongal sounds:“It seems clear that the first and second formants together make a much greater
contribution to the specification of phonetic colour than the third formant and accordingly
the third formant was not considered when deciding at which points to makemeasurements.”

Nor has this common view failed to persist in more recent times. For example, Saito
and Nakata (1985, p.35) offer the following general observations, confirmed by formant
measurements of vowels recorded by speakers of Japanese:“Characteristics of a vowel depend mainly on formants in the low frequency range, thatis, the first and second formants. ... The higher-order formant frequencies show arelatively small range of variation between vowels that depends mostly on the speaker.”More specifically in the context of synthesis of voiced sounds using vocal-tract
analogues, Linggard (1985, p.15) notes that“...in the frequency range of interest, upto 5kHz, there are only four or five formants. In
general terms, two of these are necessary to specify vowel quality, a third is required toestablish speaker identity, and the fourth/fifth may be added to give natural voice
quality.”

A similar point of view is offered by Ladefoged (1993, p.211):“...the position of the fourth and higher formants in most vowels is indicative of a11

speaker’s voice quality rather than the linguistic aspects of the sounds.”

All the statements quoted above are in general agreement; together, they provide acommon conception — accumulated over 50 years of speech research and spanning all
three domains of speech production, acoustics, and perception — of the spectral
correlates of phonetic quality, as distinct from para-linguistic attributes which have beenalternately referred to as “voice quality”, “tonal quality”, “personal characteristics”,“speaker identity”, or “individual uniqueness” in spoken vowels. Indeed, the common
view on the acoustic correlate of the vowel-speaker dichotomy is that of a bipartite
separation of vowel and speaker influences along the frequency scale — phonetic
variability is widely regarded to be manifest in the low spectral regions which containthe first two formants, while speaker-specific influences are believed to be manifest
mainly in the higher spectral regions which contain the third and higher formants.
Moreover, these spectral correlates are of importance both perceptually andarticulatorily.

2.1.2 Structural Profile of Literature
As stated earlier, the literature relevant to the dichotomy problem does encompass thethree major domains of auditory-perception, acoustics, and production of speech. Not
unexpectedly, these three bodies of literature contribute unequally to elucidating the
problem; in particular, it will transpire that the acoustic literature plays the more
substantial role. Indeed, it is the acoustic manifestations of the dichotomy to which theperception and articulatory literature constantly return. Hence, our interpretive reviewof the acoustic-auditory, the acoustic-phonetic, and the acoustic-articulatory literature,
in search of evidence and insights into the speech-speaker dichotomy.
Nor is this particular order of presentation purely incidental. As we shall see inSection 2.2, the perception literature not only has contributed to our understanding of
the dichotomy, but it has also exerted an influential role on the spectral representations
commonly used in acoustic analyses and in machine recognition of speech and speaker.
By contrast, the more substantial, acoustic literature reviewed in Section 2.3, provides
relatively more explicit evidence of the spectral manifestations of phonetic quality,
speaker individuality, and the consequences of their interactions in recognition by12machine. The latter has provided the impetus for speaker normalisation as an approachto overcome the dichotomy — but has this approach yielded any fundamental insights
regarding the dichotomy problem itself? In Section 2.4 we then review the articulatory
literature, which provides perhaps the most sparse evidence of the speech-speakerdichotomy. Although the articulatory domain is likely to hold the greatest potential foryielding fundamental insights into the nature of the dichotomy, exploitation of thatdomain has admittedly been thwarted by methodological obstacles. In particular,
nonuniqueness in acoustic-to-articulatory mapping, and the limitations in methods ofarticulatory parameterisation, have together impeded progress towards more complete
descriptions of speaker differences in articulatory terms, which might then be expected
to yield more informed approaches to overcome the dichotomy by speakernormalisation. Finally, in Section 2.5 we provide a concluding perspective on thevarious bodies of literature reviewed, which will sum up the weaknesses and relative
states of incompleteness of previous works concerned with the speech-speakerdichotomy, and thus point towards our own contributions presented in the following
chapters.

2.2 Acoustic-Auditory Evidence and Influential RoleThe literature on human auditory-perception of speech provides quantitative evidence
which does substantiate the statements quoted in Section 2.1. Indeed, we will first recall
evidence to support the role of the higher spectral regions in providing perceptual cuesof speaker identity, and of the low spectral regions in providing the primary cues toperceived phonetic quality of spoken vowels. Equally importantly, we will also attempt
to reconstruct the influential role which the perception literature has played in shaping
the popular approaches to parameterisation of speech in the acoustic domain.

2.2.1 Human Auditory-Perceptual Evidence of Spectral Dichotomy
The perceptual studies which have shed light on the speaker-related potency of thehigher spectral regions, have invariably adopted a speaker identification approach
whereby a panel of listeners reasonably familiar with the voices of the speakers
recorded, are asked to identify the speaker when presented with auditory stimuli

13

comprising the (acoustically manipulated) recorded vowel sounds. One of the earliest ofsuch studies is that of Peters (1954, cited in Hecker, 1971, p.40), who manipulated
recorded speech sounds by passing them through single octave-band filters; theaccuracy of listeners’ identification of the speaker was found to be highest for theoctave-band spanning the range from 1.2 to 2.4 kHz.
Similar results were reported by Compton (1963) and Dukiewicz (1970) using
filtered versions of the vowel / i / recorded, respectively, by 9 speakers of American
English and 5 male speakers of Polish. For example, Compton used various low-passand high-pass filters with cut-off frequencies of 255Hz, 510Hz, and 1020Hz; listeners’
accuracy in speaker identification (found to be significantly greater than chance even for
stimuli of duration 25msec) was adversely affected only under the low-pass conditions,clearly indicating perceptual preference for spectral information above 1020Hz. Alongthe same lines, Dukiewicz used single band-pass filters spanning the range from 128Hz
to 8192Hz in single-octave, double-octave, and triple-octave pass-bands; and foundlisteners’ accuracy in speaker identification of the stimuli (recorded with monotonous
pitch, and with a duration of approximately 1sec) to be the highest in the single-octave
band from 2048Hz to 4096Hz, in the double-octave band from 2048Hz to 8192Hz,
and in the triple-octave band from 1024Hz to 8192Hz. Although the results for twoother vowels / a / and / u / reported by Dukiewicz were less consistent, significant
improvements in accuracy were obtained when the single-octave range 4096Hz to
8192Hz was extended downwards to 2048Hz.
More direct and perhaps more conclusive results were reported recently by
Kitamura and Akagi (1994, 1996), who were able to claim that “voice quality can becontrolled” simply by replacing the higher frequency band of one speaker with that ofanother speaker. Their auditory stimuli consisted of 5 vowels recorded by speakers ofJapanese; and then resynthesised (with normalised pitch and energy contours) using an
FFT cepstrum-based analysis-resynthesis method, which consisted of joining one
speaker’s low frequency range with another speaker’s higher frequency range for thesame vowel. Listeners’ accuracy in speaker identification was found to be a monotonic
function of the cross-over frequency defining the boundary between the low and thehigher spectral ranges. In their earlier study, Kitamura and Akagi found the listeners’
14preference for one speaker over the other to occur at about 2340Hz; in their later study(in which there is no indication of any difference in methodology compared with theearlier study) they found the cross-over frequency to occur at 1740Hz.
Consistent with the implied speaker-specificity of the higher spectral regions,Kuwabara and Takagi (1991) found human perception of speaker individuality to be
most sensitive to percentage changes in F3 alone. Their result was obtained using two
male, Japanese speakers’ recordings of the all-sonorant nonsense word / aoiue /, thespectral characteristics of which were modified using an all-pole analysis-resynthesis
method whereby the frequency and bandwidth of spectral poles corresponding toindividual formant peaks could be directly, and independently manipulated.
Although the studies reviewed thus far have been more directly concerned with
the importance of spectral information in providing perceptual cues to speaker identity,
one could almost infer from the more recent, Japanese studies in particular, that theauditory percept of vowel identity is more strongly linked with the low spectral regions.
For example, Kitamura and Akagi’s (1994) method of spectral swapping between
speakers did no harm to their listener’s identification of the phonetic identity of thevowel stimuli.
Much of the earlier literature on human auditory-perception of speech does also
provide a solid foundation for the perceptual salience of the low spectral regions which
encompass the first two formants, as the primary carriers of phonetic information in
spoken vowel sounds. For example, perceptual experiments carried out by Delattre etal. (1952) using the famous “pattern playback” machine, showed that two formants (orspectral prominences) are sufficient for the auditory perception of 16 cardinal vowels.
In particular, they found that a single formant is sufficient for the perception of backvowels, while two formants are required for categorical perception of front vowels. The
perceptual role of F3 was then relegated to that of supporting the spectral prominence
formed by the high F2 in front vowels.
Evidence to support the phonetic importance of the low spectral regions is also
provided in the classic studies of Pols et al. (1969) and Singh and Woods (1971). In
those studies, which were primarily concerned with the perceptual dimensionality of the
Dutch and American English vowel spaces, respectively, the steady-state portions of15

recorded vowels of a single speaker were presented (in randomised triads and pairs,
respectively) to listeners who were asked to rate the degree of dissimilarity in those
speech sounds. Multidimensional scaling analyses of the perceptual similarity matrices
then revealed that the two principal dimensions of their listeners’ auditory vowel space
were highly correlated with the first two formant frequencies (Pols et al., 1969), andalso with the articulatorily-motivated features “height” and “advancement” (Singh andWoods, 1971).

2.2.2 Overcoming the Dichotomy by Auditorily-motivated Spectral Emphasis
2.2.2.1 Frequency-scale Transformations
The auditory-perceptual findings reviewed in the previous section provide quantitative
evidence (i) that at least on a per-speaker basis, the perceptual structure of spoken
vowel sounds depends primarily on the spectral information in the low-frequency range
which encompasses the first two formants, (ii) and that the perception of speakerindividuality relies heavily on the spectral information carried in the higher-frequency
ranges which encompass the third and higher formants. It therefore follows that one
method of overcoming the dichotomy is to suppress the speaker-specific informationwhilst retaining the phonetic information, by re-scaling the frequency axis such that thehigher frequency range is de-emphasised (by contraction along the axis).
In this vein, it is of interest to note that the human auditory system already appears
to have such a mechanism. Indeed, it is well-known (Békésy, 1960) that acousticmechanical stimulation of the basilar membrane (in the coiled structure of the cochlea,
situated in the inner ear) maps approximately equal length-intervals onto a non-linear
frequency scale which is approximately logarithmic for frequencies above about 500Hz,
thus effectively de-emphasising the higher spectral ranges by a reduction in frequency
resolution. An empirically-determined approximation of the non-linear frequency scaleof the auditory system was given in tabular form by Zwicker (1961), and later inanalytical form by Zwicker and Terhardt (1980); indeed, according to the so-called
critical band (or Bark) scale, the bandwidth of each critical band within which spectral
integration occurs at the auditory level, increases with the centre-frequency of the band.

16Perhaps the most influential role of perceptually-motivated frequency-scaletransformations as exemplified in critical band processing of the acoustic speech signal,
is in machine recognition of speech. For example, Dautrich et al. (1983) obtained higher
accuracies in an isolated word recognition task, using band-pass filters spaced along a
critical band (frequency) scale which was uniform below, and highly nonlinear above
1500Hz. Similarly, Davis and Mermelstein (1980) obtained superior performance inmonosyllabic word recognition by machine, using cepstral parameters derived on thecomparable mel-frequency scale. In particular, Davis and Mermelstein (1980, p.364)
concluded that the mel-frequency cepstral coefficients (MFCC’s) “allow better
suppression of insignificant spectral variation in the higher frequency bands.” In light ofthe evidence reviewed earlier regarding the contrastive roles of the lower and the higher
spectral regions, it is not at all suprising that MFCC’s should perform so well in
automatic speech recognition (ASR), and that they have consequently been adopted as
the acoustic parameters of choice in most state-of-the-art ASR systems (e.g. Young,1996).
Nor has the well-known linear-prediction (LP) method of speech analysis (Markel
and Gray, 1976) been spared from auditorily-motivated methods of spectral emphasis.
For example, Strube (1980) derived a computationally efficient and flexible method oftransforming LP parameters such that the all-pole spectrum is represented on a warped(or a non-linear) frequency scale. He then showed that an LP-based method of analysisresynthesis on the critical band (or Bark) scale yielded superior intelligibility of Germanmonosyllabic words. Consistent with the findings of Delattre et al. (1952) reviewed
earlier (in Section 2.2.1), Strube found the perceptual superiority of the frequencywarped method to be particularly prominent for a 5th-order model (which can yield
only up to two spectral peaks), intelligibility scores for which were comparable to anormal (linear frequency scale) LP model of twice that order.
Perhaps the most celebrated importation of auditory processing methods in the LPmodel, is the perceptual linear predictive (PLP) analysis of speech (Hermansky, 1990).
Prior to autoregressive modelling, the short-term power spectrum is warped onto a
Bark scale along the frequency axis, submitted to spectral reduction by critical bandprocessing, and its amplitude is equalised in order to account for the non-equal
17perceptual sensitivity of loudness at different frequencies. A relatively low-order LP
analysis is then afforded by the considerable reduction in spectral resolution; thus,
across a spectral range which normally would encompass the first four formants, a 5thorder PLP model yields a gross spectral shape which contains either one peak (for mostback vowels) or only two peaks (for mid- to front vowels). Similarly to ourinterpretation of the results of Strube (1980), the phonetic relevance of the one- or twopeak spectral representation is consistent with the perceptual findings of Delattre et al.
(1952) reviewed earlier.
In addition, however, it is Hermansky’s interpretation of the results of speakerdependent and speaker-independent, phoneme and word recognition accuracy obtainedusing PLP (and LP) models of increasing order, which is of particular concern in our
search for insights regarding the speech-speaker dichotomy. Not only did the PLP
model generally outperform the conventional LP model in those automatic speech
recognition tasks, it also exhibited two different types of behaviour of machinerecognition accuracy as a function of the model order: in speaker-dependent
experiments, accuracy was found to increase nearly asymptotically with the order of themodel; by contrast, in inter-speaker experiments the highest accuracy was obtained fora 5th-order PLP model.
Consequently, Hermansky (1990, p.1743) was able to conjecture “that the
linguistically relevant speaker-independent cues lie in the gross shape of the auditory
spectrum” which is captured “by the one or two spectral peaks of the 5th-order PLP
model”, and that the “finer details of the auditory spectrum, modeled by additional
poles of the higher-order PLP models, carry more speaker-dependent information.” Insum, Hermansky states that the PLP model“...allows for the effective suppression of the speaker-dependent information bychoosing the particular model order.”Thus, the phonetic component of the acoustic speech signal is apparently emphasised by
the perceptually-based frequency-scale transformation and the subsequent reduction ofspectral information to only two peaks in the 5th-order PLP spectrum. Furthermore,
Hermansky has shown that the second spectral peak of the 5th-order PLP analysis is18

consistent with the so-called effective second formant F2′ which is of considerable
significance in the perception of two-formant vowel sounds (Fant and Risberg, 1962).
2.2.2.2 Importance of Relative Spectral Measures
Whilst the studies reviewed thus far in this section do lend support to the contrastingroles of the low and the higher spectral regions of spoken vowel sounds, there is longstanding evidence to suggest that a mere re-scaling of the frequency axis is generally
insufficient to account for the gross spectral differences manifest between groups of
speakers such as adult males, adult females, and children. In this vein, Chiba andKajiyama (1958, pp.193-194) attempted to explain the human auditory-perceptual
ability to recognise the phonetic identity of vowels spoken by a wide variety of speakers
spanning gender and age groups, by the “space pattern theory”:“A vowel is characterized by its relative formants, provided the centres of the formants
are situated within certain frequency regions fixed for a given vowel...”.

More specifically in terms of the physical mechanisms of the auditory system, Potter
and Steinberg (1950, p.812) suggested that“...within limits, a certain spatial pattern of stimulation on the basilar membrane may be
identified as a given sound regardless of position along the membrane.”

Ladefoged and Broadbent (1957) arrived at the same general conclusion byobserving that listeners’ preference for the phonetic identity of a vowel embedded in aphrase, is influenced by the absolute values of the formants in the parts of the phrase
preceding the test vowel. Thus, they put forward the suggestion that whilst the phonetic
information is encoded in the relative positions of the formants, speaker-specificinformation may be encoded in their absolute values.
Indeed, it is well known (e.g., Potter and Steinberg, 1950; Peterson and Barney,1952) that although the formants of a given vowel (presumed phonetically equivalent)
spoken by an adult male, an adult female, and a child, do not have the same absolute
positions along the frequency axis (no matter what the scale), they do appear to have
similar relative locations. In particular, Potter and Steinberg’s (1950) measurements ofthe formant frequencies of 10 vowels spoken by 25 speakers of American English

19

including children, adult females and males, revealed a tendency for the formant
distributions of some vowels to elongate along lines passing approximately through theorigin. Potter and Steinberg (1950, p.816) then suggested that “different speakers tend
to hold ratios of the formant frequencies constant for a given sound.” To illustrate thispoint, they showed that the ratios of adjacent formant frequencies in mels ( M 2 M 1 andM 3 M 2 ) “appear to be fairly constant over the range of speakers” for each vowel, andthat the ratio M 3 M 2 in particular “is significantly different for the two groups, front
and back vowels.”
A relatively more recent, but similar approach to overcoming gender and agegroup differences in vowel formant frequencies, is Syrdal and Gopal’s (1986) proposalof a so-called “inherent or speaker-independent” normalisation method whereby the
first three formant frequencies (and the fundamental frequency F0 ) are first transformed
onto a Bark scale, then three new dimensions are defined by taking the Bark-scaledifferences F1 − F0 , F2 − F1 , and F3 − F2 . Applying this transformation to the wellknown set of formant frequencies of 10 vowels recorded by 33 adult male, 26 adultfemale, and 15 child speakers of American English (Peterson and Barney, 1952), Syrdal
and Gopal (1986, p.1095) found that it “greatly reduced between-speaker variability”.
Similarly to Potter and Steinberg (1950), they also found that the Bark-difference
F3 − F2 dimension “clearly distinguished” between the front and back vowels of the 76
speakers (as did Clermont and Broad (1995), using the steady-state vowel formant
frequencies of 4 adult, male speakers of Australian English).
By contrast with the preceding, formant-based approaches, Hermansky (1990) has
demonstrated that the additional spectral processing involved in obtaining a 5th-order
PLP spectrum, is able to reduce the significant differences in the underlying formant
structure of the vowels spoken by an adult male and a child, and thus yield two-peaked
spectra which reflect essentially the phonetic or the linguistic content of the spoken
sounds. However, whilst Hermansky’s (1990) illustration of 5th-order PLP-based
spectrograms of an utterance spoken by an adult male and a child speaker shows the
PLP’s effectiveness in suppressing speaker-group differences and enhancing the
linguistic message, other types of auditorily-based whole-spectrum representations havenot been as successful in overcoming the dichotomy without the help of a subsequent
20

normalisation step. For example, Bladon et al. (1983) have shown that a reasonable
first-order approximation to the normalisation of the quasi-auditory spectra of vowels
spoken by adult males and females, is to shift the spectra of female speakers downward
in frequency by 1Bark, thus accounting for the generally higher-valued vowel formant
distributions of female compared with male speakers.
In a similar vein, Nearey’s (1978) so-called Constant Log Interval Hypothesis(CLIH) suggests that a given speaker’s vowel formant data can be normalised withrespect to some reference data simply by an additive constant on log(F1 ) and log(F2 ) .
Nearey’s results of experiments similar to those of Ladefoged and Broadbent (1957)
using synthetic vowel stimuli were found to support that hypothesis, indicating that ashift in the perceived identity of a test vowel could be altered by the absolute formant
values of a single, carrier vowel. The CLIH was also tested on the 76-speaker vowel
formant data of Peterson and Barney (1952), and was found to greatly reduce thedifferences between the three groups of speakers. However, in light of the perceptual
evidence reviewed earlier regarding the speaker-specificity of the higher formants,
Nearey (1978) might have gained further valuable insights by extending his
experimental methodology to include formants higher than the second. Clearly, a
constant additive term applied to formant frequencies in the logarithmic domain, is
equivalent to a constant multiplicative factor applied to the formants on a linear (Hz)scale (indeed, Nearey’s Constant Ratio Hypothesis, or CRH); and the greatest effect ofsuch a normalisation method is therefore on the higher, rather than the lower formants.

2.2.3 Emerging Perspective
Our foregoing review was driven primarily by the contributions of the auditoryperception literature in elucidating the problem of speech-speaker dichotomy. Indeed,
there is sufficient evidence to suggest that in human perception of spoken vowel
sounds, phonetic and speaker-specific information is not evenly distributed throughout
the entire spectral range. Rather, they appear to be perceptually manifest in terms of abipartite separation of the frequency axis into a predominantly phonetic, low spectral
range which encompasses the first two formants or spectral prominences, and a
predominantly speaker-specific, higher spectral range which encompasses the third and21higher formants.
In addition, it emerges that in attempting to overcome the speech-speakerdichotomy by perceptually-motivated spectral emphases which are intended to suppress
the speaker component while enhancing the phonetic component, the perception
literature has played an influential role in shaping the commonly adopted methods ofacoustic parameterisation of speech. For example, the critical band concept (Zwicker,1961) which stems from pioneering research on the acoustic-mechanical properties of
the auditory system (Békésy, 1960), has been applied with some success in machinerecognition of speech; the mel-frequency cepstrum enjoys a popularity today which isclearly founded in those early, auditory-perceptual findings. Indeed, the auditorilymotivated, non-linear transformations of the frequency scale, are entirely consistent
with the perceptual evidence which appears to place greater phonetic emphasis on the
low spectral regions of spoken vowel sounds.
Nevertheless, the relatively large spectral differences in the spoken vowels of adultmale, adult female, and child speakers, have been found to require rather more elaborate
methods of spectral emphasis. For example, relative formant measures (Potter andSteinberg, 1950; Syrdal and Gopal, 1986), formant scaling (Nearey, 1978), and shiftingof a quasi-auditory spectral representation along the Bark scale (Bladon et al., 1983),
are just three of the perceptually-motivated methods which have been proposed toovercome those types of speaker-group differences. On the other hand, the one- or
two-peaked, 5th-order PLP spectrum is purported to already suppress such speakergroup differences, and thereby yield a predominantly phonetic spectral representation(Hermansky, 1990).
Quite clearly, spectral representation is an important issue which permeates
throughout the auditory-perception literature. In particular, each of the wide range ofstudies reviewed thus far, can be categorised on the basis of their preference for either acontinuous (whole-spectrum) or a discrete (formant) spectral representation. On the
one hand, the formants which have proved to be indispensable in acoustic-phonetics,
are more directly interpreted as the resonance frequencies of the vocal-tract during
speech production. On the other hand, there are conflicting points of view as to theperceptual saliency of those parameters, and a whole-spectrum representation is often
22

regarded as more akin to the human auditory representation of speech sounds.
Indeed, there has been much debate as to the appropriateness of formant andwhole-spectrum representations of speech, and the debate is far from being entirely
settled. One of the central issues in the continuing debate, is the ability of either theformant or a whole-spectrum representation to adequately account for the human
auditory-perceptual judgement of phonetic distances between vowel sounds. For
example, Bladon (1982) argues that a distance measure based on a quasi-auditory
whole-spectrum representation is better able to predict the nearly equal, perceptual
phonetic distances between the pairs of front vowels / i /-/ e / and / i /-/ Q /, whilst aformant-based distance would clearly yield a much smaller distance for the former
compared with the latter pair of vowels.
Bladon (1982) also argues in favour of a whole-spectrum approach, on grounds ofthe general indeterminacy of formants. In support of this view, Klatt (1982) notes thatwhilst formant-based approaches to speech recognition by machine can be quite errorprone due to gross errors in formant-tracking, such gross errors do not usually occur inhuman auditory-perception of speech. Consequently, Klatt (1982, p.1280) hypothesisesthat maybe “humans do not detect and label formant peaks when making phonetic
judgements.”
Further support for this hypothesis is found, for example, in the classic study ofChistovich and Lublinskaya (1979), which suggests that the auditory system performs
spectral integration of formant peaks when the centre-frequencies of those peaks arecloser than about 3.5 Bark (or critical bands). Thus, the F2 and the F3 of certain front
vowels appear to be auditorily represented as a single, merged spectral prominence.
Both the second spectral peak of Hermansky’s (1990) 5th-order PLP model, andDelattre et al.’s (1952) observations regarding their listeners’ preference for somewhat
higher second formants in two-formant vowel stimuli, are consistent both with theauditory integration theory and with the concept of an effective second formant F2′ .
The above considerations collectively raise doubts as to the perceptual
compatibility of acoustic-phonetic distance measures based simply on raw formant
frequencies. However, Klatt (1982) has shown that the sensitivity of perceptual
phonetic quality judgements to acoustic changes in vowel sounds is largest for23percentage shifts in the formant frequencies (the first two formant frequencies inparticular), compared with changes in other spectral-shape parameters such as formant
bandwidths, relative formant amplitudes, and overall spectral tilt. As argued both byKlatt (1982, 1986) and by Bladon and Lindblom (1981), distance measures based on
quasi-auditory, whole-spectra should therefore benefit from enhancement of theformant peaks, and suppression of other, phonetically less relevant spectral variations.
Indeed, the literature does suggest that while the formant frequencies play a central rolein the auditory-perceptual characterisation of spoken vowel sounds, their perceptual
relevance is conditioned by the combined and interactive influences of the formant
peaks in the whole-spectrum representation of those sounds.
We have begun our search for evidence of the speech-speaker dichotomy in thissection by considering the acoustic-auditory literature. On the weight of the evidence
found in previous works, it is clear that the perceptual characterisation of the dichotomy
is enlightening and important, to the point of having influenced modern approaches toacoustic parameterisation and spectral representation of speech. However, it is also
understood that perceptual evidence alone would provide only a partial understandingof the dichotomy in the acoustic-phonetic domain, to which we now turn for more
substantial, and perhaps more direct insights.

2.3 Acoustic-Phonetic Evidence and Approaches
In our survey of the acoustic-phonetic literature, we will first search (in Section 2.3.1)
for evidence of the spectral manifestations of the speech-speaker dichotomy, whilst also
attending to the range of acoustic parameters used to describe those manifestations. We
will then examine (in Section 2.3.2) the consequences of the dichotomy in the two,
converse tasks of automatic speaker and speech recognition, from which we expect togain further insights into the spectral manifestations and parameterisation issues
regarding the dichotomy. The latter task in particular will be shown (in Section 2.3.3)
to have prompted various approaches to overcome the dichotomy, either by way ofmore robust spectral parameterisation, or by more explicit methods of speakernormalisation or adaptation in speech recognition by machine.

24

2.3.1 Spectral Manifestations and Descriptive Parameters
In this section we seek to gain a better understanding of the spectral manifestations of
the speech-speaker dichotomy in spoken vowel sounds, by reviewing first the basic
literature which has established the standard view on their phonetic quality, and thenthe more dispersed but substantial literature concerned with the manifestations of
speaker individuality in those sounds.
2.3.1.1 Phonetic Quality of Spoken Vowels
Although basic concepts in acoustic-phonetics had been known from many years earlier,the invention of the sound spectrograph (Koenig et al., 1946) made it possible toperform acoustic analyses of speech more easily and more extensively than ever before.
This development spawned a number of pioneering studies concerned with acoustic
descriptions of spoken language; and by far the most widely studied sounds have since
been the non-nasalised vowels with which we are here concerned.
From the very first series of spectrographic observations, the regularity in thepatterns of dark bands which mark the spectral regions of high acoustic energy acrossthe duration of spoken vowels, words, and phrases, confirmed that the phonetic content
of those utterances could be deciphered by what is now regarded as the art and thescience of spectrogram reading. In this context, Kopp and Green (1946, p.87) notedthat “similarities in the oral use of English are more numerous than differences, despite
variations in vocal quality and sectional pronunciation differences.” Denoting the darkbands or formants as “bars”, Kopp and Green (1946, p.82) made the following generalobservations regarding an acoustic-phonetic description of the vowel space afforded byspectrograms of one speaker’s recordings of several diphthongs:“Bar 1 is limited in movement to a small area at the bottom of the pattern, and bar 3 isboth limited in the area of its movement and is frequently weak or absent from thepattern. Because bar 2 is changed in position and shape more than either of the otherbars, and because of the consistent manner in which the second bars of all sounds areconnected, the position of bar 2 is considered to be the principal zone or the hub for all
sounds.”Thus was established the importance of the second formant frequency F2 in describing

25the principal acoustic dimensions of vocalic speech sounds.
More extensive spectrographic measurements of 10 steady-state vowels recordedby an American English phonetician, then led Potter and Steinberg (1950, p.814) tostate that the “first and second formant positions show the greatest movement fromvowel to vowel.” Indeed, the vowel formant distribution of that speaker plotted on theF1 F2 plane was clearly found to be “such as to identify each vowel distinctly.” Whilsteven more extensive measurements of vowel formant frequencies have shown that there
exists significant overlap amongst vowel classes when data of a large or heterogeneousgroup of speakers are pooled (e.g., Peterson and Barney, 1952; Bernard, 1967, cf.
Clermont, 1996; Hillenbrand et al., 1995), these and many other acoustic-phonetic
studies have since established the standard view that the F1 F2 plane is the mostdescriptive, two-dimensional representation of the phonetic quality of spoken vowel
sounds.
It follows therefore, that the bulk of phonetic information in those speech sounds
should be contained in the low spectral regions which encompass the first two formants.
In this vein, Pols et al. (1973) have shown that the maximally vowel-discriminating
plane found by rotation of the first two dimensions of a principal components analysisof the 13 -octave band spectra (spanning the frequency range from 100Hz to 10kHz) of
12 Dutch vowels recorded by 50 adult male speakers, is highly correlated with the plane
defined by log(F1 ) and log(F2 ) measured from the same data. This classic result attests
both to the phonetic significance of the first two formant frequencies of spoken vowels,
and to that of the low spectral regions wherein they reside.
By contrast with the major role of F1 and F2 in describing the phonetic quality of
spoken vowels, the higher formants (and by extension, the higher spectral regions)would appear to play only a minor or subsidiary role. For example, the third and fourthformants are often regarded as complementary to the high F2 in shaping andemphasising the spectral prominence of high front vowels; retroflexion, as in theAmerican English vowel / ¼ /, is known to substantially lower the F3 . On the other
hand, these higher resonances and the spectral regions in which they are contained,
appear to manifest a great deal of speaker-specific information, as our following reviewof the acoustic-phonetic literature on speaker individuality will bear out.
26

2.3.1.2 Speaker Individuality
Indeed, the speaker-specificity of the higher formants has been observed in numerous
studies, and in different research contexts. For example, in a spectrographic study of 7
vocalic sounds including diphthongs and diphthongised monophthongs recorded byspeakers of two main varieties (or idiolects) of Australian English, Burgess (1969)
observed several, idiolect-specific trends in the behaviour of F3 in those speech sounds,
thus implying that the third formant might provide a reliable cue for the speaker’s
idiolect. In a different spectrographic study of spoken texts recorded by 5 speakers andby two well-known imitators who attempted to imitate the voices of those 5 speakers,
Endres et al. (1971, p.1845) observed that the formant structure in the vowels / iá / and/ a / of the original speakers and of the imitators “do not agree, especially in the higherfrequency bands” above about 2kHz.
More quantitative, analyses of variance of the formant frequencies of 12 Dutch
vowels recorded by 50 adult, male speakers, clearly showed (Pols et al., 1973, Table II,
p.1095) the speaker variance in log(F3 ) to be much higher than that in either log(F1 )or log(F2 ) (23% of the total variance in the data, compared with 5% and 2%,
respectively); by contrast, the vowel variance in the same data was observed to behave
conversely (43%, compared with 86% and 94%, respectively). Similarly, analyses ofvariance of the first four formant frequencies measured in 5 vowels contained in 4
words, recorded several times over a duration of 5 years by 9 adult, male speakers ofJapanese, led Saito and Itakura (1983, p.150) to observe that the variances of the“speaker factor” in “the higher formant frequencies are higher than those of the lower
ones.”
Evidence of the speaker-specificity of the higher spectral regions also appears instudies adopting a whole-spectrum approach to parameterisation of the acoustic speech
signal. For example, histograms of the so-called “average absolute difference” between
22 speakers’ long-term spectral correlation matrices obtained using a 35-band filterbank spanning the range from 270Hz to 10kHz, was found by Li and Hughes (1974,
p.835) to have a higher distribution when the frequencies below 2200Hz were deemphasised with a + 6 dB/oct filter; thus, they concluded that “low-frequency

27

deemphasis helps to retain more intertalker differences.” In a study of the long-term
averaged spectra of a phonetically-balanced list of 25 spondee words recorded by 20
male, 18 female, and 2 child speakers of Finnish, Kiukaanniemi et al. (1982, p.23)
found, for each group of speakers separately, “surprisingly large individual variations in
the distribution of speech power especially at frequencies 1000-3000Hz.”
Using more contemporary methods of speech signal analysis to obtain the longterm average, cepstrally-smoothed spectra (by truncation of the FFT cepstrum to 11
terms) of 2 words recorded by 9 adult, male speakers of Japanese, Furui and Akagi(1985, p.6) conclude that“...the individuality of the spectral envelope derived from the cepstrum is almost
completely represented by the difference of spectral levels between 2.5 and 3.5 kHz.”

Similar conclusions emerge from Kitamura and Akagi’s (1994, Figure 3, p.1185)
profiles of the vowel and speaker components of spectral variance, computed for 5
vowels recorded by 10 male speakers of Japanese. Representing each vowel spectrumwith 60 FFT cepstral coefficients (and using an auditorily-motivated frequency-scale),they confirmed their perceptual findings (discussed earlier, in Section 2.2.1) that whilst“vowel characteristics” are concentrated mainly in the lower spectral region from about
603Hz to 2212Hz, “speaker individualities” are to be found mainly above 2212Hz.
Whilst the studies reviewed thus far have provided evidence concerning thespectral correlates of speaker individuality in spoken utterances which contain vocalic
sounds, the question arises whether those spectral correlates are related to, or might bepredicted by, the greater speaker-specificity of certain phonetic (or vowel) subspaces.
In this regard, Stevens et al. (1968) provide an indication of the importance of thephonetic feature “front-back” for vowels. In particular, identification of speakers byvisual inspection of voice spectrograms was found to be more reliable for utterances
containing stressed front vowels, than for those containing stressed back vowels.
Consistent with the emerging view on the spectral correlates of speaker individuality,
Stevens et al. (1968, p.1602) then suggest that “the quality of a talker’s voice may be
judged, both visually and aurally, on the basis of high-frequency components, say in therange 2000-3000Hz.”

28

The speaker-related potency of front vowels is also borne out in the formant
modelling study of Sato et al. (1982). Applying the bi-planar model of Broad and
Wakita (1977) to the first three formant frequencies of 5 vowels extracted fromsentences recorded by each of 4 adult, male speakers of Japanese, they obtained larger,
inter-speaker differences in the coefficients of the planar equations for the front vowels
than for the back vowels. On the other hand, van den Heuval and Rietveld (1992)
obtained a higher “ratio of speaker specificity” (RSS) for the vowel / a / compared withthe vowels / i / and / u /, in 24 /CVC / nonsense words recorded by 5 male and 5 female
speakers of Dutch. However, their whole-spectrum representation (LP cepstrum) andcomputation of spectral distance using dynamic time warping (DTW), did not yieldinsights regarding the contributions of different spectral regions to the RSS.
On the basis of the accruing evidence regarding both the spectral correlates andthe phonetic subspaces of greater speaker potency, the acoustic-phonetic literature on
speaker individuality seems to implicate the higher spectral regions which encompass
not only F3 and the higher resonances, but also the F2 of front vowels. By contrast,
the more commonly accepted view on the acoustic correlates of phonetic information in
spoken vowels, as reviewed earlier, clearly points to the importance of the low spectral
regions which encompass the entire range of both F1 and F2 . Furthermore, these
implications are also consistent with the auditory-perceptual findings reviewed earlier(in Section 2.2), which imply that whilst as many as two formants are necessary for theperception of vowel sounds, the higher spectral range which appears to carry
perceptually important information on speaker individuality, may also include the F2 offront vowels.
Together, these conclusions then raise the question of whether the phonetic andthe speaker-specific influences shown to be manifest predominantly in the low and in
the higher spectral regions, respectively, might not overlap and impinge on each otherto yield the phenomenon of vowel-speaker interactions. In this vein, it is worth notingthat the acoustic-phonetic studies reviewed in this section have adopted mainly direct
methods of observation such as analyses of variance, in order to quantify separately
either the phonetic or the speaker component of spectral variability. By contrast, wemight expect the effects of vowel-speaker interactions to be more explicitly manifest,
29and therefore better described, in terms of their consequences in vowel and speakerrecognition by machine.

2.3.2 Consequences in Vowel/Speaker Recognition by Machine
The evidence reviewed in the previous section concerning the spectral manifestations
and acoustic-phonetic correlates of vowel quality and of speaker individuality, to a large
extent foreshadow the consequences of the speech-speaker dichotomy in automatic
vowel and speaker recognition. Indeed, those spectral regions or spectral parameterswhich have been directly observed to manifest the bulk of either speaker individuality orphonetic quality, can be expected to also yield the highest accuracies in automatic
speaker and vowel recognition, respectively. In addition, these converse tasks offer the
potential to observe the spectral manifestations of the vowel-speaker dichotomy as
follows: in speaker recognition by machine emerges the question of the mostappropriate vowel subspaces; in automatic vowel recognition, vowel-speaker
interactions may be induced by performing recognition on an inter-speaker (or aspeaker-independent) basis. In the following two sections, therefore, further insights
into the phenomenon of speech-speaker dichotomy are sought by reviewing the relevant
literature on automatic speaker and vowel recognition, respectively.
2.3.2.1 Speaker Recognition
One of the earliest studies to consider the relative importance of formants in speakerrecognition by machine, appears to be that of Carré (1971). Speaker identification was
attempted using pairwise combinations of the first three formant frequencies extracted
from 2 repetitions of a given, dynamic vocalic utterance (described as containing threesyllables within a duration of about 500msec extracted for formant analysis) recordedby 10 male speakers of French. Formant trajectories on the F2 F3 plane were found to
be more effective in discriminating between the speakers, than those on the F1 F2 plane.
The effectiveness of different formants of different vowels in speaker identification
by machine has since received attention in several studies. For example, Sambur (1975)
found that the values of the second, third, and fourth formant frequencies measured in
the steady-state of certain vowels contained in sentences recorded over a period of 3.530

years by 21 adult, male speakers of American English, were amongst the mostimportant acoustic parameters for automatic identification of those speakers. For thefront vowels / i /, / I / and / Q /, the best combination of high inter-speaker variability andlow intra-speaker variability was found in F2 and F4 , whilst for the back vowel / u /,the most effective parameter was found to be F3 . The first four formant frequencies of
11 vowels recorded in /hVd/ context by 1 female and 9 male, adult speakers of British
English were assessed for their speaker discriminating power by Paliwal (1984a), whofound the best performance in both F2 and F3 of the mid- to front vowels / ¶ / and / I /.
More recently, Mella (1994) found that on average, the performance of F3 in automatic
speaker identification is superior to that of F1 and F2 . Speaker identification performed
using the first three formant frequencies in 4 repetitions of 7 vowels in /pVr/ and /bVr/
context in sentences recorded by 10 male speakers of French (of the Lorraine region),showed that for F2 , the front vowels / i /, / e /, / E /, and the central vowel / § / yielded
the highest accuracies, whilst for F3 , the back vowel / u / performed the best. Similarly,
Cooper and Clermont (1994) showed that the F2 of front vowels and the F3 of backvowels yielded the highest accuracies in speaker identification, using 5 repetitions of/CVd/ monosyllables recorded by 4 adult, male speakers of Australian English. Theseauthors also reported that speaker-identification experiments performed on Peterson
and Barney’s (1952) formant data of 33 adult, male speakers of American English,yielded the highest accuracy (96%) using only F2 and F3 of 8 of the 10 vowels, having
excluded the two back vowels / A / and / U /.
Evidently, the contribution of formants in the upper spectral range (including F2
of front vowels) appears to be greater than that of the formants in the low spectral
range. However, formant extraction is admittedly a non-trivial task; not surprisingly,
therefore, the majority of studies in automatic speaker recognition have opted for thecomputationally less expensive and more robust, whole-spectrum approach to acoustic
parameterisation. Unfortunately, this approach is often taken to implicitly condone theuse of the entire available spectral range (i.e., up to half the sampling frequency of the
speech signal), without further regard for the relative effectiveness of different spectral
regions, as shown in the numerous studies reviewed thus far in this chapter.
Nevertheless, previous works which have adopted a whole-spectrum approach in31

speaker recognition, do offer insights regarding the effectiveness of different phonemes.
For example, Kashyap (1976) used a rather limited set of phonemes (four front
vowels, one high back vowel, and two nasal consonants) extracted from a list of words
recorded by 4 speakers, and obtained the highest inter-speaker distances (aimed atbeing used more effectively in automatic speaker recognition) for the front vowels / I /and / E /. A more comprehensive study of the relative importance of 24 different
phonemes (Höfker, 1976, cited by Jesorsky, 1978, p.117), found that amongst thevowels, the front vowels / i /, / e /, / E /, and / I / yielded the highest accuracies in theidentification of 12 speakers. Using 12th-order LP cepstra and FFT mel-frequency
cepstra measured in 6 sentences recorded (at a sampling frequency of 8kHz) over the
telephone by 125 speakers, Eatock and Mason (1994) conducted an even morecomprehensive study of the effectiveness of 75 different phonemes in a speakerverification task based on vector quantisation (VQ), and obtained a rank-ordering of
those phonemes, the top four of which were the nasal / N /, the low vowel / A /, and thetwo high front vowels / i / and / I /. By comparison, Liou and Mammone (1995) used
12th-order mel-frequency cepstral coefficients, the signal energy, and the first andsecond-order differences of those parameters (across time) measured for 30 speakers,
and found the statistical F-ratio of speaker discriminability yielded by a neural tree
network (NTN), to be the highest for the front vowel / e /, the retroflex / ¼ /, and thediphthong / aI / which does involve a series of front-vowel configurations (Clermont,
1991).
In contrast with those studies which have adopted a whole-spectrum approachwithout attempting to seek further insights regarding the spectral ranges of importance,
van den Heuvel et al. (1993) performed speaker discriminant analysis using the output
of each of 19 frequency sub-bands spanning the frequency range up to 8kHz on a Bark
scale. Their speech material consisted of the vocalic steady-state of 24 isolated /CVC /nonsense words recorded by 15 adult, male speakers of Dutch. Most importantly,
despite their use of a filter-bank, their interpretations of the recognition results were
offered in terms of the speakers’ formant frequency ranges. In particular, they found thebest speaker discrimination in the frequency bands around F3 of / a /, and around F2and F3 of / I /. Interestingly, these results qualify the earlier finding of van den Heuval
32and Rietveld (1992) regarding the higher ratio of speaker specificity (RSS) obtained forthe vowel / a / of their speakers of Dutch (as reviewed in Section 2.3.1.2). Although vanden Heuvel and Rietveld did not venture in their earlier study to spectrally decompose
their cepstrum-based, and thus whole-spectrum representation, it is indeed likely thatthe higher spectral regions containing F3 made the largest contribution to the high RSS
obtained for that vowel.
By contrast, the following two studies were able to confirm the relative
importance of the higher spectral regions in speaker recognition by machine using a
cepstrum-based approach, albeit at the cost of cepstral re-analysis in each frequency
band of interest. First, Hayakawa and Itakura (1994) performed automatic speakerrecognition using cepstral coefficients obtained by selective LP analysis (Makhoul,
1975b) within various frequency sub-bands (in the range up to 16kHz) of 5 repetitionsof 5 words, recorded every 3 months over a duration of 1 year by 5 male speakers ofJapanese. Whilst they could not therefore guarantee an equivalent accuracy of spectral
representation across the different frequency bands, they were able to conclude that theband spanning 4 − 10 kHz carries as much speaker-specific information as the lowfrequency range extending up to 4kHz. A methodologically more consistent method of
re-analysis was adopted recently by Lin et al. (1996), who obtained 20th-order FFT
cepstral coefficients in two, equal-interval frequency ranges spanning 0 − 5 kHz and3 − 8 kHz, respectively. Text-independent, VQ-based speaker identification using

sentences recorded by 24 male and 14 female speakers then yielded a lower error-rate
for the higher spectral range ( 1.1 %) than for the lower spectral range ( 5.8 %).
Prompted perhaps mainly by a desire to gain higher levels of accuracy in automatic
speaker recognition, recent studies clearly reflect an increasing interest in the relative
importance of different spectral regions and phonetic subspaces. Further examples
include some recent attempts at separate analyses and weighted recombination of FFTbased frequency sub-bands (e.g., Besacier and Bonastre, 1997; Auckenthaler and
Mason, 1997), and at least one attempt to incorporate an empirically-determined
phonetic weighting in speaker verification (e.g. Liou and Mammone, 1995). However,the speech-speaker dichotomy does have serious implications in speaker-independent,
speech recognition which need to be elucidated as well. To this end, therefore, a careful
33

review of studies of vowel recognition in particular, is attempted in the next section.
2.3.2.2 Vowel Recognition
Whilst in automatic speaker recognition there is the question of determining the mostspeaker-specific acoustic-phonetic subspaces, the converse problem of speakerindependent vowel recognition by machine offers the potential of observing the
manifestations of vowel-speaker interactions, albeit indirectly via recognition accuracy.
For example, it is well known that in general, a speech (or vowel) recognition system
will perform worse on an inter-speaker, than on an intra-speaker basis (e.g., Rabiner
and Juang, 1993, p.285). However, it is quite clear from our earlier review of theliterature on phonetic quality, speaker individuality, and automatic speaker recognition,that vowel recognition accuracy will depend not only on the presence of inter-speakervariability, but also on the spectral range or the spectral parameters selected.
Indeed, Paliwal (1984b, p.104) obtained results which confirm the importance ofthe low spectral range in vowel recognition by machine. In particular, he found spectral
pre-emphasis to be detrimental to vowel recognition on an intra-speaker basis, andattributed the deterioration in recognition accuracy to the “undue weight on highfrequency components” which do not carry reliable information pertaining to vowel
identity.
Using an LP-based whole-spectrum representation, Ainsworth and Foster (1985)
obtained results which more directly confirm the implications of vowel-speaker
interactions in recognition. They performed vowel classification experiments both on an
intra- and an inter-speaker basis, using 10 repetitions of 11 vowels recorded in /hVd/
context by 4 male and 4 female speakers of English. Each vocalic steady-state was first
represented by a 128-point spectrum up to 5kHz (obtained by Fourier transformationof the 10th-order LP autoregressive coefficients). Vowel classification was thenperformed using first the entire spectral range, then only the spectral points up to about
3200Hz. Whilst intra-speaker accuracy suffered a 3% drop (to 93%) as a result of the
reduction in spectral range, inter-speaker vowel classification accuracy was found toincrease by 3% (to 43%). These results not only confirm the relative phonetic
unimportance of the spectral range between 3200Hz and 5000Hz, but also demonstrate34

its speaker-related, detrimental influence in vowel recognition.
As that particular spectral range includes perhaps only the high end of the third
formant distribution of most adult female speakers (e.g. Peterson and Barney, 1952),
Ainsworth and Foster’s (1985) result is more likely indicative of the phonetic
unimportance and speaker-specificity of mainly the fourth and fifth formant frequency
ranges. However, Arslan and Hansen’s (1997) results do bear on the frequency rangesof the lower formants. They performed speaker-independent, isolated word recognition
using a hidden Markov model (HMM), in each of 16 frequency bands spaced uniformly
across the spectral range up to 4kHz, and with a dataset of 48 adult male speakers,
including equal numbers of native speakers of American English and speakers of
Turkish-, Mandarin-, and German-accented English. Speech recognition accuracy was
found to be highest in the frequency pass-bands below about 1800Hz; by contrast,
accuracy was found to be much lower in the pass-bands above about 1800Hz. On theother hand, the accuracy of classification of the speakers’ accents was found to behighest in the so-called mid-range frequencies spanning 1500 − 2500 Hz.
In addition to their whole-spectrum approach using a filter-bank analysis, Arslan
and Hansen (1997) also performed speech recognition and accent classification using
each of the first four formant frequencies separately. In speaker-independent speech
recognition, F2 followed by F1 were found to yield the highest scores; by contrast, F2
and F3 yielded the best performance in accent classification. Although accent
classification involves characteristics of groups of speakers rather than individuals, theseresults are nevertheless consistent with those reviewed earlier, which together
substantiate the contrastive roles of the low and the higher spectral regions in regard to
the speech-speaker dichotomy.
In contrast to Arslan and Hansen’s (1997) use of each formant separately inrecognition, most previous works have used the formants in crescendo (i.e. first only

F1 , then F1 and F2 together, then the first three formants, and so on). In light of theevidence reviewed earlier, this approach implicitly acknowledges the phonetic
importance of the first two formants; in addition, it may afford the potential ofobserving the influences of vowel-speaker interactions on recognition accuracy, as firstthe phonetic, and then the presumably more speaker-specific formants are recruited in35the recognition process. Indeed, somewhat analogously to the whole-spectrum resultsof Ainsworth and Foster (1985) reviewed above, one might expect vowel classification
accuracy to be detrimentally affected upon recruitment of F3 and higher formants.
Table 2.1 shows a summary of the results reported in eight different studies.
Summarised in each column from left to right, following the authors and the year of
publication, are the vowel and speaker materials used, the type of decision boundaries
used in the classifier, and the frequency scale of the formants. In the last two columns
are listed the inter-speaker vowel classification accuracies obtained using only the firsttwo, then the first three formants.
Consistent with our initial expectations, recruitment of F3 is reported to affectinter-speaker vowel classification accuracy detrimentally (by about 3.5 % on average),but in only three of the studies listed (Assmann et al., 1982; Chung et al., 1988; andKumar, 1996). By contrast, the remaining five studies report an improvement inaccuracy (by about 8.3 % on average). Apparently therefore, the effectiveness of thethird formant in manifesting vowel-speaker interactions, is not consistently observablein this type of recognition experiment. Nor can this discrepancy in the role of the third
formant be explained entirely in terms of methodological differences — both rises andfalls in accuracy are reported using both linear and logarithmic frequency scales, bothlinear and quadratic classifiers, and with both smaller and larger populations of speakers
of different languages or dialects.
It is perhaps not surprising that this apparent discrepancy should not have been
noted in previous studies on vowel recognition, which have been primarily concernedwith accuracy and not with explaining the speech-speaker dichotomy problem. It would
appear, however, that spectral representation can play a determining role in revealing
manifestations of the dichotomy in recognition. On the one hand, the inconsistency inthe contribution of the third formant noted above, suggests that formant-based spectral
representations in vowel recognition may undermine the possibility of gaining insights
into the dichotomy. On the other hand, quite a number of studies reviewed earlier
suggest that a whole-spectrum representation is likely to more consistently exhibit the
detrimental consequences of speaker-induced variability in the higher spectral regions,on inter-speaker vowel classification accuracy.
36

VowelRecognition
Accuracy (%)
Previous Study

VowelsSpeakers

Classifier

Frequency
scale

F1F2

F1F2F3

Welch &
Wimpress(1961)

10
AmericanEnglish33 male &26 female(Peterson &†Barney, 1952)

Quadratic

Hz

87

91

50 maleQuadraticlog(Hz)

71.3

75.3

5 male &5 female(Mahalanobis)ln(Hz)

8077

20 maleQuadratic

Hz

92.4

92.2

Hz

74.9

83.6

log(Hz)

75.2

83.6

Bark

76.1

83.6

Quadratic

Hz

60.5

73.0

Quadratic

Hz

68.2

81.0

(Mahalanobis)

Hz

82.8

72.0

Quadratic
Quadratic
Quadratic
Quadratic

Hz
Hz
Hz
Hz

84.4
68.2
80.6
72.7

83.6
64.0
76.4
71.5

Chung et al.
(1988)

12
Dutch
10
Canadian
English
8
KoreanHillenbrand &Gayvert(1993)

10
AmericanEnglishZahorian &Jagharghi(1993)

11
AmericanEnglish12
AmericanEnglishPols et al. (1973)
Assmann et al.
(1982)

Hillenbrand
et al. (1995)
Kumar (1996)

11
AustralianEnglishLinear

33 male,26 female, &15 child(Peterson &Barney, 1952)
10 male,10 female, &10 child
45 male,48 female, &46 child
36 male(of 3 idiolectal‡
groups)14 “Broad”
11 “General”
11 “CultivatedQuadratic

LinearTable 2.1: Summary of speech materials, methods, and inter-speaker (or speaker-independent)
automatic vowel classification accuracies obtained in 8 previous acoustic-phonetic studies, using thefirst two, then the first three formant frequencies. Accuracies in bold-font indicate detrimental effect of†
the third formant. [ Only those vowels which were correctly identified unanimously by a panel of 26‡
listeners; Unpublished results of Kumar’s (1996) Master’s thesis.]

In an attempt to reconcile these two behaviours, it seems important to reconsider
the whole-spectrum versus formants issue raised earlier in Section 2.2.3, from the pointof view of recognition. First, by invoking basic principles of pattern recognition, it canbe argued that if the ratio of the amount of training data to the dimensionality of thefeature set is sufficiently high, and provided also that the probabilistic assumptions of
the classifier are reasonably met in the given data, then an additional, independent
parameter may either improve, or at worst leave unaffected the classification accuracy
37

yielded by a nearly optimal classifier (Duda and Hart, 1973, p.67), even if that extra
parameter ( F3 ) were to exhibit potentially confounding, inter-speaker variations.
Indeed, this argument finds support in the very low feature-set dimensionality of only 2
or 3 formants, and in the often reasonable assumption of a normal distribution offormant data on a per-vowel basis (as required by a quadratic classifier).
It is also clear that in a recognition framework, the formant parameters are treatedas independent, mutually orthogonal dimensions. By contrast, a whole-spectrum
representation used in recognition naturally embodies formant-peak interactions which
occur both on an intra-frame (i.e., merging of proximate formant peaks) and on aninter-frame basis (e.g., inter-vowel or inter-speaker overlap of different formant peaks).
In addition, both the perceptual and acoustic-phonetic literature reviewed earlier
implicate the speaker-specificity of not only the third and higher formants, but also theF2 of front vowels, which cannot simply be excluded from a vowel recognition
experiment using the first two formants. It is therefore reasonable to speculate that thedichotomy should be more clearly manifest in vowel recognition using a wholespectrum approach, rather than the discrete, formant-based representation which would
appear not only to discard important formant-peak interactions, but also to limit thedegree to which vowel and speaker influences might be decoupled.

2.3.3 Approaches to Overcoming the Dichotomy
Having reviewed the relevant literature on the spectral manifestations of the speechspeaker dichotomy and its consequences in vowel and speaker recognition by machine,
we now direct our attention to acoustic-phonetic approaches intended to overcome thedichotomy. In contrast to the frequency-scale transformations and relative spectral
measures which (as reviewed in Section 2.2.2) are motivated by properties of human
auditory-perception, the approaches to be described in the following two Sections are
driven mainly by the criterion of higher accuracies in speaker-independent speech
recognition by machine, and do not have a perceptual basis. In Section 2.3.3.1, we
consider one of the more important trends in gaining robustness of spectral parameters,
hence higher accuracies, in recognition. In Section 2.3.3.2 we then consider some of themore explicit methods of speaker normalisation and adaptation. A recurrent question is38

whether any of these approaches have yielded, irrespective of their success in dealing
with or normalising speaker differences, any new insights into the dichotomy problem.
2.3.3.1 Towards Robustness in Parameterisation
We deduced from the literature reviewed in the previous section, that the formants may
not be the best spectral parameters for elucidating the speech-speaker dichotomy in avowel recognition framework. However, this does not imply that the formants are notuseful parameters in vowel recognition; on the contrary, their potential in that context(and more generally in speech recognition) has often been advocated (e.g., Broad,
1972; Zahorian and Jagharghi, 1993; Holmes et al., 1997). Unfortunately, the formants
do suffer the serious limitation of being difficult to measure robustly — a limitation
which has prevented a wider use of those parameters in speech recognition systems. It
is therefore a fortunate coincidence that the more robustly measured, whole-spectrum
parameters such as the cepstrum, do hold the potential of elucidating the dichotomy
problem via recognition.
Independently of its potential ability to yield insights into the dichotomy problem,the cepstrum has indeed enjoyed tremendous popularity and relative success both inspeech recognition (e.g., Davis and Mermelstein, 1980; Paliwal and Rao, 1982; Young,1996) and in speaker recognition (e.g., Luck, 1969; Atal, 1974; Furui, 1981; Mammone
et al., 1996). The demand for higher accuracies, however, has inevitably led tocontinued efforts aimed at improving the robustness of the cepstrum in either of thosetasks. As noted in Section 2.2, the perception literature has been particularly influential
in this regard, motivating the transformation of the cepstrum onto the phonetically morerobust, mel-frequency scale. An equally important method for improving the robustnessof the cepstrum in recognition, is that of liftering (or weighting of the cepstrum in thequefrency domain), the objective of which is aptly summarised by Rabiner and Juang(1993, p.169):“... to control the noninformation-bearing cepstral variabilities for reliable discriminationof sounds.”

In line with Rabiner and Juang’s prescription, are the perceptually-motivatedarguments put forward by proponents of the whole-spectrum approach (e.g., Klatt,39

1982, 1986; Bladon and Lindblom, 1981, as discussed in Section 2.2.3), who advocate
spectral enhancement of the formant peaks for improved vowel recognition inparticular. Klatt’s (1982) proposal of a spectral slope measure based on an auditory
filter-bank, was indeed an important first step towards enhancing formant peaks inspectral distance computation. However, a major advance in that regard had alreadybeen made by Yegnanarayana (1978), who first showed that the negative derivative of
the LP phase spectrum (NDPS, also known as the group delay spectrum) has several
desirable properties, including not only the flattening of the phonetically unimportant
spectral slope and the enhancement of formant peaks (see also Fuchi and Ohta, 1978),but also the additive (rather than multiplicative) influence of formant peaks in thatspectral representation. Most importantly, Yegnanarayana showed that the NDPS isequivalent to the so-called quefrency-weighted cepstrum (QCEP, also known as theroot power sum), and that its phonetically relevant properties are therefore embodied
both in the QCEP and, by extension, in the cepstral distance measure based on the
QCEP (Yegnanarayana and Reddy, 1979).
The QCEP has since been shown to perform better than the unweighted LP
cepstrum in speaker-dependent vowel recognition by machine (Paliwal, 1982), and in
speaker-dependent isolated word recognition in the presence of noise (Hanson and
Wakita, 1987). More specifically in the context of the speech-speaker dichotomy,
Tohkura (1986) has shown that a cepstral lifter based on the reciprocal of theempirically-determined variance in each cepstral coefficient, is not only very similar to
the QCEP, but also performs much better than the unweighted cepstrum in speakerindependent isolated word recognition. Indeed, Tohkura (1986, p.761) concluded thatthe relative de-weighting of the low-order cepstral coefficients “tended to equalize theperformance of the recognizer across different talkers.”
However, the best performance was reported by Tohkura (1986) with the numberof cepstral coefficients approximately equal to the LP analysis order; the higher-ordercoefficients were found to be detrimental to recognition accuracy. Indeed, the role ofthe higher cepstral coefficients had earlier been questioned by Davis and Mermelstein(1980, p.364) who found, using unweighted, mel-frequency cepstra in monosyllabic
word recognition, that the first six coefficients (based on a sampling frequency of40

10kHz) already yielded high accuracies, and that the “importance of the higher
cepstrum coefficients appears to depend on the speaker.” The importance of spectral
smoothing (by cepstral truncation) in order to reduce the sensitivity of distancemeasures used in speech recognition, is now well-established, as indicated for example
by Shikano and Itakura (1992, p.422): “... broadening of the peaks is indispensable,
since just a slight deviation in formant frequencies between two patterns would result ina large distance value.”
A compromise between the desirable, high-quefrency emphasis of the QCEP, andthe efficacy of spectral smoothing by cepstral truncation, was proposed by Itakura andUmezaki (1987) in the form of a Gaussian lifter, and by Juang et al. (1987) in the formof a raised-sinusoidal, band-pass lifter. The former was evaluated in speaker-dependent,
isolated word recognition, and found to yield the highest recognition accuracies with anequivalent frequency resolution (up to 5kHz) of about 300Hz. The latter, band-passlifter was evaluated in both vowel and isolated digit recognition on an inter-speakerbasis, and was found to yield the highest accuracies compared with several competing,
cepstrum-based distance measures. Juang et al. (1987, p.949) then attributed thesuccess of the band-pass liftering approach, to the effective suppression of the following
two sources of variability: that of higher quefrency terms which determine the moredetailed, higher-resolution components of the spectral shape, are “inherent artifacts of
the analysis procedure”; and that of low-quefrency terms which affect mainly the grossspectral shape and tilt, are “primarily due to variations in transmission, speakercharacteristics, and vocal efforts ...”.
Admittedly, cepstral liftering achieves a wholesale modification of the entire shapeof the spectrum by a combination of enhancement and smoothing of spectral peaks, theeffect of which is difficult if not impossible to predict in specific frequency sub-bands. Inview of the evidence accumulated in the auditory-perception and acoustic-phonetic
literature reviewed earlier in this chapter, attempts to control the so-called“noninformation-bearing variabilities” in the spectrum by cepstral liftering are therefore,
at best, indirect approaches to dealing with the speech-speaker dichotomy. Indeed,whilst the importance of cepstral liftering is itself acknowledgment of the dichotomy
problem, that method does not allow direct control of any particular frequency sub41

band. Nor are current cepstral distances capable of yielding similarity measures inselectable frequency sub-bands. Further research, which more directly addresses the
problem of speech-speaker dichotomy by allowing the flexibility of selecting spectral
ranges in either cepstrum derivation or cepstral distance computation, is clearly
warranted!
2.3.3.2 Speaker Normalisation
In contrast to the indirect approach of cepstral liftering to overcome the dichotomy, theharmful influences of inter-speaker variability in speech recognition have been more
explicitly acknowledged, and more directly dealt with, by so-called methods of speakernormalisation. As with the dual approach to spectral parameterisation evident in thevarious bodies of literature reviewed thus far, methods of speaker normalisation may
also be categorised as adopting either a formant or a whole-spectrum approach. A
relevant question, then, is to what extent either of these direct approaches haveacknowledged or further elucidated the acoustic-phonetic manifestations of the speechspeaker dichotomy.
Whilst the following three methods of speaker normalisation are theoreticallyapplicable to any number of formants, they were originally intended for only the first
two. Gerstman (1968) linearly mapped the maximum range of each speaker’s F1 andF2 separately onto a normalised scale from 0 to 999. Using the classic dataset ofPeterson and Barney (1952), Gerstman (1968, p.79) reported “no significant F1 or F2
differences” in the per-vowel mean formant values computed for each of the threegroups of speakers (adult male, adult female, and child) after normalisation. Vowel
classification performed by Gerstman using the normalised formant values, their sums,
and their differences (with F3 used solely to identify the retroflex vowel), then yielded
misclassifications (approximately 12%) for only those vowel tokens which had also
failed of unanimous correct identification by a panel of 26 listeners. The first two
formants were also the focus of Lobanov’s (1971) normalisation of the overall meanand standard-deviation of each speaker’s vowel formant distribution. Applied to thesteady-state vowel formants of 5 adult, male speakers of Russian, his method yielded ahigher index of adjacent-vowel separability, compared either with Gerstman’s range42

normalisation, or with a linear scaling to normalise the speakers’ maximum formant
values. More recently, Di Benedetto and Liénard (1992) proposed an “isomorphic
transformation” of each speaker’s F1 F2 vowel formant distribution, onto a normalised
plane where each vowel token retains its relative distances from three reference vowels(chosen as the three point vowels / i /, / A /, and / u /). Applied to the Peterson and Barney(1952) dataset of all 76 speakers’ formants, the method was found to yield degrees of
pairwise vowel discrimination comparable to those yielded by a modified version of
Gerstman’s method whereby only the three point vowels are used in the rangenormalisation. However, by definition, this method untenably assumes that thosereference vowels are phonetically, precisely equivalent across speakers. An equally
drastic assumption of all three methods reviewed above, has led to their blatantdisregard of the speaker-related influences of formants higher than the second, whichmight potentially have an important role in speaker normalisation.
By contrast, Assmann et al. (1982) applied Nearey’s (1978) CLIH method (seeSection 2.2.2.2) to the first three formant frequencies of 10 vowels recorded by 5 maleand 5 female speakers of Canadian English. As expected, they obtained higher vowel
classification accuracies with the speaker-normalised, than with the original formants.
However, in contrast to the drop in accuracy obtained originally upon recruiting thethird formant (from 80% to 77%, as listed earlier in Table 2.1), the speaker-normalised
data yielded a small rise in accuracy from 91% to 93%. Whilst these results indirectly
confirm the speaker-related potency of the third formant in particular, apparently
contradictory results were obtained by Pols et al. (1973), who had earlier applied asimilar method of speaker normalisation (i.e., translation of each speaker’s, logarithmic
vowel formant data to a common, overall mean value) to the first three formants of 12
vowels recorded by 50 male speakers of Dutch. Indeed, further attesting to theinconsistency of formant-based vowel recognition results, their speaker-normalised data
yielded a relatively smaller rise than did their original data, upon recruiting the third
formant (from 78.3 % to 80.5 %, in contrast with the larger rise from 71.3 % to 75.3 %as listed earlier in Table 2.1).
The apparently inconsistent behaviour of formants in vowel recognition as typified
in those studies, and the general lack of insights regarding the contribution of acoustic43

phonetic subspaces or of higher formants, raises the question of whether a wholespectrum approach to speaker normalisation is more likely, or indeed has previously
been used, to provide such insights into the speech-speaker dichotomy. One of thesimplest approaches to speaker normalisation using whole-spectrum parameters, is thatof long-term mean subtraction (which has also been used with the aim of removing
frequency-dependent recording or transmission-channel distortions in automatic speakerrecognition, as proposed for example by Atal (1974) using the LP cepstrum). A similar
method was indeed used by Plomp et al. (1967) in order to remove the vowel-averaged
speaker component in each of their

13-octave frequency bands spanning 100Hz to

10kHz; later, the same method applied by Klein et al. (1970), Pols et al. (1973), andvan Nierop et al. (1973) to the principal-components space of those filter-bank spectra,
was found to be quite effective in speaker normalisation of Dutch vowel acoustic data.
However, it would appear that further insights into the spectral regions or phonetic
subspaces most affected by mean spectrum (or cepstrum) normalisation, have not been
forthcoming in the literature.
In contrast to the above method which assumes that the speaker component of theacoustic speech signal is primarily contained in the overall mean spectral parameters,
frequency warping (Matsumoto and Wakita, 1978) is a more general and potentially amore powerful method of speaker normalisation. By analogy with the popular methodof dynamic time warping (DTW, see Sakoe and Chiba, 1978) used in ASR, dynamic
frequency warping (DFW) is intended to optimally align two given spectra, withconstraints on the warping path applied in a dynamic programming framework.
Paliwal and Ainsworth (1985) showed specific examples of the effectiveness of
DFW in matching the LP-derived spectrum of a given vowel of one speaker, with thatof another speaker. However, they also found that the same method could equally wellmatch the spectra of two different, but neighbouring vowels. Not surprisingly therefore,
the use of DFW in inter-speaker vowel classification using 4 male and 4 female
speakers’ data, yielded worse results on average.
A similar degradation in inter-speaker vowel classification performance wasreported by Matsumoto and Wakita (1986) using vowel data recorded by 12 male and
12 female speakers of American English. However, DFW was found to improve44

classification accuracy when the overall spectral slope was concurrently speakernormalised. Indeed, Sejnoha and Mermelstein (1983) had also found that DFW applied
to mel-frequency filter-bank vowel spectra of 7 male and 6 female speakers, improved
both within- and across-gender, vowel classification accuracy, when applied together
with a speaker-dependent compensation of overall spectral tilt. These results suggestthat DFW should perhaps yield improved performance in vowel classification using the
NDPS representation (as embodied in the QCEP), which inherently suppresses theinfluence of the overall spectral slope.
Regarding the type of DFW best suited to improving vowel classification
accuracy, Sejnoha and Mermelstein (1983) found the “best frequency warp paths ... to
be nonlinear and strongly dependent on the speech segment.” By contrast, Matsumotoand Wakita (1986) found that the largest contribution to improved accuracy was
yielded by a linear frequency warp (where the frequency locations of spectral peaks orformants are effectively scaled by a common factor). However, acoustic-phonetic
insights into the dichotomy were offered only by Matsumoto and Wakita, who reported
that the (linear) frequency warp improved inter-speaker classification of mainly the midto high front vowels.
Insights of a similar kind were offered by Suomi (1984), who also adopted awhole-spectrum approach to speaker normalisation. Superimposing the genderdependent mean, critical-band spectra (in perceptually-motivated frequency andloudness units of Bark and phons, respectively) of 8 vowels recorded by 8 male and 8
female speakers of Finnish, Suomi (1984, p.204) first observed that “the differences
appear phoneme-specific, particularly in the upper part of the spectrum.” However,Suomi (1984, p.205) then obtained (cf. Bladon et al., 1983, as reviewed earlier in
Section 2.2.2.2) “a close match ... in the upper bands by shifting the male spectra one
Bark upwards and three phons downwards relative to the female spectra.” Following
normalisation, the accuracy of classifying the female speakers’ vowels based on themale speakers’ training data was found to improve from 70.7 % to 88.1 %. Prior to
speaker normalisation, those gender differences had been found to detrimentally affectthe classification accuracy of mainly the mid- to front vowels.
In view of these results, and indeed of the much earlier studies of Potter and45

Steinberg (1950) and Peterson and Barney (1952), it is not at all surprising that Umesh
et al. (1996) recently found that the optimum, non-linear frequency warp which would
allow gender-normalisation of the warped spectra by only a linear translation, was infact very similar to the perceptually-motivated, mel-frequency scale. Thus, in addition
to its inherent emphasis of the low spectral range, the popular mel-frequency cepstrum
is expected to hold the potential of facilitating normalisation of inter-gender differencesin spoken vowels. However, it seems clear that spectral warping approaches are aimed
at alleviating the effects of and not at explaining the speech-speaker dichotomy.
Nor is one further equipped to unfold the dichotomy problem, by turning tomethods of speaker adaptation such as those advocated by Furui (1989; 1991; 1992).
While they embody the appealing idea of incrementally improving recognition
performance, and have therefore the potential of securing technological advances, it isfair to say that the problem is thus acknowledged but not explained. No moreelucidating are those common approaches adopted in ASR, which simply rely on vast
amounts of data in the hope of capturing sufficient variability amongst speakers.

2.4 Acoustic-Articulatory Evidence and Approaches
We have reviewed in Sections 2.2 and 2.3, respectively, the relevant parts of theauditory-perceptual and acoustic-phonetic bodies of literature which bear on the
problem of speech-speaker dichotomy. As intimated earlier (both in Section 2.1 and in
Chapter 1), an even more fundamental perspective on the dichotomy problem is
expected if the intertwined sources of variability are examined in the speech production
domain. In that context, we first consider (in Section 2.4.1) previous attempts totheoretically formalise, and to empirically describe the articulatory sources of interspeaker variability and their acoustic-phonetic consequences. We then review (inSection 2.4.2) the most persistent approaches to overcoming the dichotomy byarticulatorily-motivated methods of speaker normalisation. We shall then argue (inSection 2.4.3) that if more complete articulatory insights are to be gained, then there isa need to advance beyond those persistent approaches, and to more inclusively accountfor the articulatory sources of speaker variability.

46

2.4.1 The Speaker Factor: Sources of Variability and Consequences
In contrast to the physical correlates of the phonetic quality of spoken vowels, whichhave long been the subject of investigation both directly by way of articulatory
measurements and indirectly in terms of articulatory modelling which has helped toestablish the basic relations between speech production and acoustics (e.g., Chiba andKajiyama, 1958; Fant, 1960; Flanagan, 1972), the literature on the physical correlates of
speaker differences in spoken vowels is considerably more sparse. Nevertheless, in thefollowing two sections, we first review some prominent attempts to formalise thepotential articulatory sources of speaker variability in a theoretical or a descriptive
framework, and then we re-examine some of the most common, directly measured andacoustically inferred observations of speaker differences in the production of spoken
vowels.
2.4.1.1 Theoretical Framework
As briefly mentioned in our introductory chapter, Garvin and Ladefoged (1963)
qualitatively categorised the physical sources of inter-speaker variability in terms oforganic and learned differences. Whilst organic differences refer to the size and shapeof speakers’ fixed vocal-tract anatomies, learned differences imply idiosyncracies inboth the static and dynamic aspects of articulation during speech production. However,this bipartite categorisation is not a mutually exclusive one — as pointed out for
example by Stevens (1971), the so-called learned characteristics may partly depend on
the speaker’s vocal-tract anatomical structure and the constraints which it imposes onboth the static postures and the dynamic gestures of articulators.
Nolan (1983), who first points out that the literature on speaker recognition inparticular had for some twenty years ventured no further than Garvin and Ladefoged’s
bipartite categorisation of speaker differences (an observation which is almost equallyvalid to this day), then himself proceeds to “reject” that framework on grounds of its
oversimplification of the speaker variability problem. Nolan’s criticism of the organic
versus learned framework is founded partly on the non-trivial influence of so-calledorganic attributes, on both learned (speaker-related) and linguistic (or phonetic) aspects47

of speech production; and partly also on the complexity of the so-called learned
attributes, which themselves are conditioned by such factors as communicative intent.
Ultimately, Nolan advances the notion that a speaker’s fixed vocal-tract anatomy
merely sets limits on the range of possible articulatory implementations available in the
acoustic transmission of a given linguistic message; furthermore, that there should bemade a more explicit distinction between the long-term qualities and the short-term or
realisational rules, in describing the so-called learned speaker differences.
In fact, Laver (1980) had earlier expounded a detailed framework for the phonetic
description of voice quality (including both laryngeal and supralaryngeal structures andarticulators), which Nolan (1983) then used in his own investigations of the acoustic
correlates of long-term segmental properties. Although Laver’s descriptive framework
is initially motivated by voice quality distinctions at an auditory-perceptual level, it isdefined entirely in terms of the so-called articulatory settings, such as “raised-larynx
voice”, “labiodentalised voice”, or “palatalised voice” (to name only three). Settings arethus described in relation to an overall “neutral” articulatory configuration and, whilsteach setting is liable to exert different amounts of influence (both at an articulatory andan acoustic level) on different types of speech segments, it does afford an articulatory
description of a speaker’s long-term voice quality.
Unfortunately, it would appear that neither Laver’s (1980) descriptive framework,
nor Nolan’s (1983) cautions against the inadequacies of the organic-learned framework,
have had much direct influence in the literature concerned with the speech-speakerdichotomy. As the overall anatomical size of a speaker is perhaps the most clearlyvisible and easily accessible of individual characteristics, it can almost be expected thatgross differences of the organic type should have dominated previous approaches to
describing speaker variability in speech production. Indeed, this is borne out in the next
section, in which we review the most common, empirical observations of thearticulatory sources and acoustic consequences of speaker differences in the static,
segmental properties of spoken vowels.
2.4.1.2 Most Common Empirical Observations
As noted above, perhaps the most obvious difference between speakers is in the overall
48

size of their vocal-tract. For example, it is well known that on average, the vocal-tractof an adult male is larger than that of an adult female, which in turn is larger than that ofa child. The most acoustically significant, physical measurement relating to vocal-tract
size, is the total length of the vocal-tract airway from the glottis to the lips (e.g. Fant,
1960). In this vein, if one assumes that different speakers adopt very similar articulatory
configurations for a given vowel, and that the only source of variability is in the total
length of their vocal-tract, then acoustic theory predicts that the corresponding formant
frequencies will be scaled inversely to the ratio of the speakers’ vocal-tract lengths.
Indeed, the acoustic manifestations of vocal-tract length-related, organic differences
between speaker groups (Garvin and Ladefoged, 1963), are the progressively higher,
average vowel formant frequencies of men, women, and children, as portrayed forexample in the classic study of Peterson and Barney (1952). In this vein, it has beenshown that the formant frequencies of (presumed) phonetically equivalent vowels
spoken by a wide range of speakers, can be modelled to a first approximation by lines of
constant formant ratio (e.g. Peterson, 1952; 1961).
Nevertheless, the accuracy of that basic, linear relation in measured acoustic data,
has generally been observed to be greater in regard to the so-called individual (Garvinand Ladefoged, 1963), organic speaker differences (e.g., differences between speakersof the same gender). For example, Stevens and House (1963) were able to show areasonable agreement between the percentage differences in 3 adult, male speakers’
mean vocal-tract lengths measured from mid-sagittal X-ray images, and the percentage
differences in those speakers’ first and second formant frequencies averaged over 8
vowels. Later, Stevens (1971, p.216) obtained a reasonably linear relation (as predictedby theory) between those same, three speakers’ average vocal-tract lengths, and theirmean values of F3 , arguing that“... since the third formant does not change markedly from vowel to vowel, ... it provides
a more precise indication of average vocal-tract length than does the first or second
formant.”

Implicit in that statement is the greater speaker-specificity of the third formant, by
virtue of its more direct relation to the vocal-tract length of the speaker.
By contrast, the relation between the vocal-tract lengths (and of the formants) of49

speakers of different gender and age groups, has been shown to be more complicated.
Despite some recent evidence to the contrary (e.g. Yang and Kasuya, 1995, 1996), midsagittal X-ray images have revealed that the lengths of the oral and pharyngeal parts ofthe vocal-tract (approximately the distance from the lips to the back wall of thepharynx, and from the soft palate to the glottis, respectively) are, in general, scaled
differently from speaker to speaker, particularly across gender and age groups. For
example, Chiba and Kajiyama (1958) found that, although the total length of the vocaltract of an 8 year-old girl was 70% of the total vocal-tract length of a 26 year-old male,the young girl’s oral and pharyngeal cavity lengths were 77% and 64% of thecorresponding cavity lengths of the adult male. Similar, physiological findings werereported by Goldstein (1979) in her modelling study of the childhood and adolescent
growth curves of vocal-tract anatomical distances. Furthermore, the acoustic
consequences of inter-gender differences of that type were studied in detail by Fant
(1966, 1975a), who used formant frequency measurements from speakers of eight
different languages to show the importance of so-called non-uniform formant scaling,
and to explain the non-uniformity in terms of female speakers’ relatively smaller ratio ofpharyngeal to oral cavity lengths, compared to adult males. Both the acoustic andarticulatory evidence thus support Chiba and Kajiyama’s (1958, p.193) generalstatement, that“... the length of the mouth cavity does not vary so much with different individuals as thatof the pharynx cavity.”

Not incompatible with that statement, is the long-held view that the speakerspecific dimensions of the so-called “larynx tube” which extends above the glottis, may
provide a phonetically-independent, acoustic cue which is characteristic of the speaker.
If such a structure, typically about 2 − 3 cm in length, were assumed to be acoustically
uncoupled from the rest of the pharyngeal cavity (as is approximately the case when therelatively narrow larynx tube opens to a much wider pharynx), then it may indeed beresponsible for the relatively fixed, higher formant observed (e.g. Ladefoged, 1993) inthe vicinity of 3kHz in some speakers’ spectrograms of vocalic speech sounds. Thishypothesis was put forward more than sixty years ago by Bartholomew (1934), and has

50since been revisited in describing potential articulatory sources of inter-speakervariability (Stevens, 1971), in providing reliable cues to speaker identification by
spectrograms (Stevens et al., 1968), and in explaining the phenomenon of the “singer’s
formant” (Sundberg, 1974, 1995). From an acoustic point of view, the phonetic
independence and the potential speaker-specificity of the so-called larynx resonance isconsistent with its typical frequency location being higher than the normal range of thefirst two formants. However, robust identification of such a fixed, higher formant,
which admittedly has been more clearly observed in sung (rather than spoken) vowels,
can generally be expected to be problematic.
The emphasis on speaker differences in vocal-tract fixed anatomy has recently
been reaffirmed, from both an acoustic and an articulatory point of view. Owren andBachorowski (1997) reported an accuracy of 99% in automatic gender identification
using only the fundamental frequency and a formant-based estimate of the vocal-tract
length, in the first (mid-front) vowel of the utterance “Test n Test” recorded by 50 maleand 75 female speakers. On the other hand, direct, physical measurements were used by
Honda et al. (1997) to study the effects of the size and shape of speakers’ orofacial
structure on their production of spoken vowels. Whilst the “lower facial height or thesize of the mandibular symphysis” was found to be inversely correlated with the firstformant frequency, the speaker-dependent “spread of the second formant” was found to
be correlated with the “oral cavity aspect ratio and mandibular symphysis shape”.
In sum, the most common empirical observations of the physical sources of interspeaker variability and their acoustic manifestations in the static, segmental propertiesof spoken vowels, clearly emphasise the importance of vocal-tract fixed anatomical
structures. A relevant question then arises whether previous, articulatorily-motivatedapproaches to overcoming the speech-speaker dichotomy by speaker normalisation,
also bear the influence of this persistent emphasis on speaker differences in vocal-tract
fixed structure. Our review of the relevant literature in the following section will shedsome light on this issue.

2.4.2 Persistent Approaches to Speaker Normalisation
Analogously to the acoustic-auditory and acoustic-phonetic approaches to overcoming
51the speech-speaker dichotomy (as reviewed in Sections 2.2.2 and 2.3.3, respectively),
the speaker normalisation problem has also been approached from an acousticarticulatory point of view. Similarly to those approaches reviewed earlier, a primaryobjective of importing acoustic-articulatory principles in speaker normalisation has been
to overcome the detrimental influences of inter-speaker variability, and thereby gain
higher accuracies in speaker-independent, automatic speech recognition. Previous
methods which have aimed to account for articulatory sources of speaker differences inspeech recognition, have either explicitly estimated certain articulatory parameters fromacoustic measurements, or only implicitly accounted for the acoustic consequences of
speaker-induced variations in those articulatory parameters.
One of the earliest of the explicit approaches is the study by Wakita (1977), whoused the linear-prediction (LP) model to first estimate the instantaneous vocal-tract
length (VTL) for any given frame of vocalic speech data (the method will be describedin Chapter 5). Both the frequencies and bandwidths of the LP poles corresponding to
the formants of the measured frame of speech, were then scaled by the ratio of theestimated VTL and a fixed, reference length of 17cm (a typical length of an adultmale’s vocal-tract). Whilst normalisation of both the speaker and phonetic variation in
VTL led to an overall shrinkage of the formant space of the 9 vowels of the 14 maleand 12 female speakers of American English used in that study, the VTL-related
speaker normalisation of the first three formant frequencies yielded a nearly consistentincrease in the ratio of inter- to intra-vowel variance between pairs of neighbouringvowels, and led to an improvement from 78.9 % to 84.4 % in inter-speaker vowel
classification accuracy using a Bayes’ classifier.
One of the advantages of Wakita’s (1977) method of speaker normalisation,
particularly in the context of recognition, is that it makes no a priori assumption about
the phonetic identity of the analysis frame. However, if the phonetic identity can beassumed to be known, a more physically meaningful approach (Ishizaki, 1978a; Broad
and Wakita, 1978) is to first estimate the VTL of all the vowels of each speaker, thennormalise each speaker’s average VTL to a fixed reference value. This approach retains
each speaker’s inter-vowel distribution of vocal-tract length, and thereby avoids thepotentially harmful (and unrealistic) shrinkage of the entire vowel formant space as52noted by Wakita (1977). It is also interesting to note that this method of speakernormalisation is similar to Pols et al.’s (1973) translation of speakers’ logarithmic
formant frequencies to a common mean value (as reviewed earlier in Section 2.3.3.2), ifthat translation were constrained to equal intervals along each dimension.
Indeed, uniform speaker normalisation of mean vocal-tract length entails a linear
expansion or contraction of each speaker’s vocal-tract area-functions2 by a single,
speaker-dependent scaling factor. Broad and Wakita (1978, p.85) computed the scaling
factor for each of their 26 speakers, using all 9 vowels available in a reference dataset,and noting only the phonetic equivalence of the “canonical allophones” of those vowel
phonemes to the corresponding vowels in each speaker’s data. On the other hand,
Ishizaki (1978a, p.67) recognised the potential speaker-specificity of “lip protrusion andthe up-and-down motion of the larynx” in certain rounded vowels, and therefore used
only the three unrounded vowels / i /, / e /, and / A / to obtain a more realistic, and perhaps
a more purely organic or anatomically-based estimate of the differences between 8
adult, male speakers of Japanese.
In contrast to the explicit estimation of VTL prior to speaker normalisation, anumber of studies have shown that improvements in speaker-independent speech
recognition accuracy can be gained by allowing certain, vocal-tract length-relatedtransformations of the acoustic (usually whole-spectrum) parameters. For example,
Golibersuch (1983) used per-speaker histograms of the first three formant frequencies(measured from about 40sec of speech recorded by each speaker), to determine theoptimum linear frequency warp between pairs of speakers’ filter-bank spectra (spanningcentre-frequencies from 350Hz to 3400Hz, on a logarithmic scale). In a speakerindependent, isolated word recognition task using each of 4 male speakers’ data in turnfor training, and 10 other male and 6 female speakers for testing, the linear frequency
warp was found to improve recognition accuracy from about 32% to 42%, on average;as the training speakers were all male, the largest improvements in accuracy wereobtained for the female speakers. In this vein, Matsumoto and Wakita (1986) also
reported improvements in inter-gender vowel recognition accuracy (cf. Section 2.3.3.2)
2

The profile of cross-sectional area as a function of the distance along the vocal-tract airway from theglottis to the terminating plane at the lips.

53

using linear frequency warping. However, they avoided the error-prone task of formant
estimation, by using a suitably constrained dynamic programming framework to
determine the optimum, linear fequency warp path; in particular, their frequency
warping function was constrained to allow variations in vocal-tract length between
13cm and 21cm.
More recent approaches to articulatorily-motivated speaker normalisation, haveattempted to obtain higher accuracies in speaker-independent speech recognition by
allowing for VTL-related transformations of the cepstrum parameters. For example,
Claes et al. (1997) derived a transformation of the mel-frequency cepstrum which
linearly warps the original frequency axis, based on speaker-group differences in the
median of the third formant frequency. Thus implicitly accounting for VTL-relateddifferences between adult and child speakers, they obtained a reduction in the word
error rate yielded by a continuous-density HMM-based, continuous word recogniser,from 4.3 % to 1.9 %.
On the general assumption (similar to that of Matsumoto and Wakita, 1986) thatthe formant parameters themselves cannot be reliably estimated, Lin and Che (1995),
and very recently Lee and Rose (1998), proposed to maximise speech recognition
accuracy by performing recognition on several, differently computed cepstrum vectors,
based on a finite number of degrees of linear expansion or compression of FFT-based
spectra. Lin and Che (1995) reported a monotonic improvement in isolated word
recognition accuracy using 18 mel-frequency cepstral coefficients in a discrete-density
HMM recogniser trained on the data of 20 male speakers, when the effective halfsampling frequency of the test spectra of 10 female speakers was increased in threesteps, from the original 6.0 kHz up to 7.8 kHz (thus compressing those spectra, andeffectively lengthening the female speakers’ vocal-tract lengths by a factor of 1.3 ).
They also obtained a monotonic improvement in recognition accuracy when therecogniser was trained on 20 female speakers’ data and tested on 10 male speakers’
data, as the effective half-sampling frequency of the latter spectra was decreased in two
steps, from 7.2 kHz down to 6.0 kHz (thus expanding those spectra, and effectively
shortening the male speakers’ vocal-tract lengths by a factor of about 0.83 ). In thislatter case, Lin and Che (1995, p.203) pertinently noted that “use of less spectral54information can lead to a higher word recognition accuracy...”, even — one should note
— with a powerful HMM recogniser trained on presumably sufficient amounts of data.
In this vein, Lee and Rose (1998) also reported improved performance in an
HMM-based, speaker-independent (mixed gender) speech recognition task, using acontinuous-digits dataset recorded over the telephone. Mel-frequency cepstra of 12thorder were computed from spectra which were originally represented by 22 mel-spaced
filter-bands up to about 4kHz, but subsequently allowed to be compressed or expanded
along the frequency axis, with 13, equally-spaced linear warping factors between 0.88
and 1.12 . The speaker-dependent warping factors were optimised both in training andtesting the recogniser, and the average factor for the male and female speaker groups
was found to be 1.00 and 0.94 , respectively.

2.4.3 Speaker Normalisation Beyond Vocal-Tract Length
Clearly, the literature on articulatorily-motivated speaker normalisation in speech
recognition, is dominated by either direct or indirect approaches to account for speakerdifferences in vocal-tract length. This is perhaps not surprising, given that the so-calledorganic differences between speaker groups such as adult males, adult females, andchildren, are the most significant and the most readily interpretable, both physiologicallyand acoustically. However, the overriding emphasis on vocal-tract length has impeded
progress towards a better understanding of the so-called learned differences between
speakers which, as we shall see in the next section, is far from insignificant.
2.4.3.1 Under-exploited Evidence
Indeed, the existing theoretical frameworks for describing speaker differences in vowel
production (as reviewed in Section 2.4.1.1) already claim that a minimal description of
those differences should include both vocal-tract anatomy and articulatory behaviour.
Amongst the studies which provide empirical evidence to support this view, are thosewhich are concerned with elucidating the relative significance of uniform and nonuniform scaling of vocal-tract length. For example, in advocating non-uniform scaling
between adult male and female speakers, Fant (1975a, p.13) claims that it can halve thespeaker-group variance remaining in the first three formant frequencies after uniform55

length scaling; thus, “it can be expected that non-uniform normalization brings out
dialectal differences between speakers more clearly than uniform scaling.” Similarly,
Nordström (1977, p.91) concludes from his vocal-tract modelling study of male-femalechild differences, that “anatomical differences ... only explain part of the formant
differences”, and that it is “probable that the vocal tract form varies between men andwomen/children.” The speaker differences which Fant grossly categorises as “dialectal”,
and which Nordström refers to as “vocal tract form”, are indeed the so-called learned
differences in articulatory behaviour, the acoustic consequences of which can be at leastas significant as those induced by anatomical or organic differences.
In this vein, some of the acoustic implications of learned speaker differences werenoted by Saito and Itakura (1983, p.151), whose contribution to elucidating speakerindividuality in vowel formant data was reviewed earlier in Section 2.3.1.2. In
particular, they noted that“...the normalization of frequency spectral differences by the use of the ratios between theformant frequencies ..., or corrections based on the vocal tract length ..., will beinsufficient, because variations in the ratio of formant frequencies are not always in aradial direction, but in a circular one...”Similarly, in his bi-planar modelling of the first three formant frequencies of 30 vowels
recorded by each of 3 male and 3 female, adult (and phonetically-trained) speakers ofAmerican English, Broad (1981, p.1428) concluded thus:“The significant inter-speaker variations in the orientations of the planes demonstrates astatistically detectable departure from uniform scaling.”

Applying the same, bi-planar model to the formants of 5 vowels recorded by 4 adult,
male speakers of Japanese, Sato et al. (1982) made similar observations, and arrived at
a very similar conclusion (as one might have predicted, given that differences in vocaltract length are perhaps of secondary importance between speakers of the same genderand age group). Analogously in terms of a whole-spectrum representation, we havealready reviewed (in Section 2.3.3.2) those studies in which non-linear (rather thanlinear) frequency warping was found to more completely normalise the spectral
differences between speakers (e.g., Sejnoha and Mermelstein, 1983; Paliwal and
Ainsworth, 1985; Matsumoto and Wakita, 1986).
56

All of this evidence, which arises from acoustic and vocal-tract modelling studies,clearly points beyond vocal-tract length (or more generally, vocal-tract anatomical size)
as an articulatory source of speaker differences. In this vein, one wonders whether
direct articulatory measurements have at all been used to shed light on the learned typesof speaker differences which lie beyond vocal-tract length. Admittedly, this question
itself presupposes a methodology for comparing articulatory data of different speakers(who in general have different vocal-tract anatomical sizes and shapes), such as to bringout those vocal-tract shape-related differences which might be attributed to articulatory
behaviour rather than anatomy.
One particularly insightful approach to inter-speaker comparison of vocal-tract
shapes was devised by Liljencrants (1971), who computed inter-speaker correlations
between corresponding pairs of Fourier coefficients used to model two adult, malespeakers’ tongue shape profiles during vowel production. The profiles themselves wereobtained from mid-sagittal X-ray images, by sampling at intervals of about 5mm along
the length of the vocal-tract airway, the distance to the tongue contour with respect to a
speaker-dependent coordinate system. Although only the “DC term” and the first two
Fourier components were extracted by Liljencrants, he found that the speakers’ tongue
shapes were highly correlated in their first Fourier components ( r = 0.895 and 0.957
for the cosine and sine terms, respectively), and far less correlated in their second
Fourier components ( r = 0.365 and 0.637 , respectively). Thus, Liljencrants (1971,p.16) concluded that whilst the “gross shapes are similar for the two subjects uttering
similar vowels”, there appears to be “a more subject dependent fine structure.” Apartfrom Mermelstein’s (1967, p.1292) qualitative assertions regarding the phonetic
importance of the lower formants and therefore of the low-frequency components ofvocal-tract shapes, this is perhaps the first and only, direct evidence to support
Peterson’s (1959) general statement quoted earlier (in Section 2.1.1), which itself also
implicates the speaker-specificity of the higher formants.
Empirical observations of speaker differences in directly measured vocal-tract
shapes (whether mid-sagittal traces or area-functions obtained therefrom), are indeed
scarse in the literature. Another notable exception is Harshman et al.’s (1977) threeway factor analysis (so-called PARAFAC) of tongue shapes obtained from mid-sagittal
57

tracings of 10 vowels produced by 5 speakers of American English. A distinct
advantage of the PARAFAC analysis adopted in that study, is its explicit modelling ofboth vowel and speaker-related statistical variations along each tongue contour;
furthermore, as opposed to principal components analysis, the factors yielded by
PARAFAC are not constrained to be orthogonal. Regarding the principal dimensions ofvowel variation, two factors were found which were then described, respectively, as“front raising” and “back raising” of the tongue (Ladefoged et al., 1977, p.21); thevowel distribution on the corresponding factor plane was found to be very similar to the
traditional vowel chart (albeit slightly rotated), with opposing and extreme vowel
configurations of / o / and / i /, and of / A / and / u /, respectively, along each dimension.
Although the PARAFAC analysis also yielded factors describing the speaker variationin the tongue-shape data, Harshman et al. (1977) did not offer new insights from aspeaker-related point of view, other than to point out a correlation between the first
speaker factor and the speakers’ oral cavity lengths, and between the second speakerfactor and the ratio of the speakers’ pharynx to oral cavity lengths. Thus, although dataon detailed tongue shapes were available, those authors did not venture beyond
considerations of gross, anatomical speaker differences, which indeed appear to havebeen the dominant influence in the literature.
In line with the notion of a more speaker-specific, pharyngeal cavity length,
Hermansky and Broad’s (1989) hypothesis represents a significant refinement of thestructurally-motivated partitioning of the vocal-tract into oral and pharyngeal sections.
In particular, they advanced that the phonetic and the speaker-specific attributes of
vocal-tract articulatory configurations can be associated, respectively, with the front
and the back cavities which are divided by the main place of lingual constriction. Direct
evidence to support this hypothesis was shown in tracings of mid-sagittal X-ray imagesof vocal-tract vowel configurations of an adult male and a child, which revealed that
although the child’s vocal-tract was much smaller and had a proportionally shorter
pharynx, the size and shape of the child’s vocal-tract “front cavity” (anterior to the mainplace of lingual constriction) was comparable to that of the adult male speaker, both fora front and a back vowel. The acoustic consequence of this hypothesis wasdemonstrated by showing that the second spectral prominence of the 5th-order PLP
58

model (Hermansky, 1990) is highly correlated with the resonance of the front cavity(assumed to be decoupled from the back cavity) of a vocal-tract model. Hence, thephonetic salience of the two-peaked PLP spectrum (as discussed earlier in Section
2.2.2) was re-interpreted articulatorily, in terms of its ability to capture the effective
second formant F2′ , or the phonetically important front cavity resonance. However, itremains to be tested whether their hypothesis of speaker differences being manifest
mainly in the shape and size of the vocal-tract cavity behind the main place of lingual
constriction, is equally tenable for speakers having similar vocal-tract anatomical sizes
and proportions (e.g., for speakers of the same gender and age group).
Two further exceptions to the dominant influence of vocal-tract structure, include
the magnetic resonance imaging (MRI) studies of Baer et al. (1991) and of Narayanan
et al. (1997), who superimposed the MRI-measured, vocal-tract area-functions for eachof 4 point vowels (/ i /, / Q /, / A / and / u /) sustained by two male speakers, and for eachof 3 point vowels (/ i /, / a / and / u /) sustained by two male and two female speakers,
respectively. Regarding speaker differences, Baer et al. (1991, p.812) noted the largest
amount of variability in the vowel / Q /, for which one speaker’s area-function was“generally wider” and showed “more peaks and dips” than that of the other speaker; thelip opening of one speaker was also systematically found to be larger than that of theother speaker. However, whilst one speaker’s entire area-function for / u / was found tobe “shifted to the left” with respect to that of the other speaker, this difference wascautiously attributed to a systematic measurement error in the position of thepharyngeal starting point, at which all area-functions were aligned. By contrast,
Narayanan et al. (1997) aligned their MRI-measured area-functions of two male andtwo female speakers at the lips, thereby allowing the position of the glottal end to varyon each graph, according to the inter-speaker differences in vocal-tract length for eachvowel. Speaker differences were then qualitatively noted in the anterior region (or front
cavity) for the vowels / u / and / a /, and in the back cavity for the vowels / u / and / i /.
These observations led Narayanan et al. (1997, p.1008) to suggest that speakerdifferences are most pronounced in vocal-tract regions of large cross-sectional area
(i.e., in vocal-tract cavities, rather than at places of constriction), since in those regions,“individual anatomical differences are allowed to show prominently.”
59However, perhaps the most detailed, articulatory study of speaker differences, wasrecently conducted by Högberg (1995), who compared the vocal-tract area-functions of
10 non-nasalised vowels obtained from X-ray images of one adult male and one adultfemale speaker of French. A particularly noteworthy, methodological advance relative
to previous studies, which have clearly stumbled on the problem of how best to alignand thereby quantitatively compare area-functions or vocal-tract shapes of different
speakers, was Högberg’s assumption that at least on a per-vowel basis, the purely
anatomical differences between the two speakers could be normalised by separately
scaling the lengths of the oral and pharyngeal parts of the vocal-tract. Although
Högberg did acknowledge the potential contribution of lip protrusion and larynx heightto speaker differences in vocal-tract length (similarly to Ishizaki, 1978a, as reviewed
earlier in Section 2.4.1.2), the two speakers were found to exhibit similar degrees of lip
rounding and larynx raising from vowel to vowel. Thus having scaled down the lengthsof the male speaker’s area-functions (by an overall average of 17%), Högberg (1995,p.45) found the largest amount of speaker differences in “articulation strategies”, forthe vowels “of the most neutral character”, namely / E /, / § /, and / O /. In addition, hefound that the female speaker “used more extreme articulations” for the vowels / i / and/ a /. Averaged over all 10 vowels, the male-to-female ratios of vocal-tract areas were
found to be largest in the lower pharynx, and at the lips. Also somewhat consistent with
Hermansky and Broad’s (1989) hypothesis reviewed earlier, Högberg (1995, p.41)
found the largest inter-vowel variation of those ratios in the oral cavity, thus “implyingthat the differences in the mouth are quite vowel dependent.”
Whilst the studies reviewed above do provide some direct articulatory evidence
which underscores the importance of speaker differences beyond vocal-tract length,
they also attest to the difficulties of quantitatively comparing articulatory data acrossspeakers in a meaningful way. For example, three different methods of accounting fororganic differences prior to direct comparisons of vocal-tract shapes across speakers,
are the anatomically-based, speaker-dependent, mid-sagittal reference grid used by
Harshman et al. (1977); the uniform vocal-tract length normalisation effectively carriedout by Liljencrants (1971); and the inter-gender, non-uniform length normalisation usedby Högberg (1995). On the other hand, the organic versus learned distinction has been
60

completely ignored in those studies where the original vocal-tract area-functions ofdifferent speakers are simply aligned either at the glottis (Baer et al., 1991) or at the lips
(Narayanan et al., 1997). Indeed, despite the potential guidance of existing theoreticalframeworks (as reviewed in Section 2.4.1.1), there does not appear to be a consistentor a generally accepted method of separately accounting for the so-called learned,
articulatory sources of speaker variability. It is therefore conceivable that the
exploitation of such direct articulatory evidence as reviewed above, has been partly
inhibited by the very absence of an accepted methodology for adequately describing andaccounting for the various sources of speaker differences in vowel production.
In this vein, the valiant efforts of Payan and Perrier (1993) and of Perrier et al.
(1995) to incorporate acoustic-articulatory knowledge in speaker normalisation ofvowel formant data, are to be applauded. Their proposed method is founded on the
acoustic-articulatory principle that in certain vowels for which the vocal-tract cavities
can be assumed to be acoustically, nearly uncoupled, there exist strong formant-cavity
affiliations. Using Fant’s (1960) well-known, model-based formant nomograms, Perrier
and his colleagues first infer the formant-cavity affiliations for a given speaker’s
recordings of the point vowels / i /, / a /, / u /, and various vocalic transitions such as / iy /,/ iu /, and / ua /, during which sudden changes in formant-cavity affiliations are known tooccur. They then obtain so-called normalisation coefficients which map, on a pervowel basis, the effective lengths of the speaker’s vocal-tract cavities (determined by
the measured frequencies of the formants which are affiliated with those cavities) tothose of a standardised, mid-sagittal articulatory model (Maeda, 1979; 1988). Having
thus computed the normalisation coefficients for the point vowels, those of intermediate
vowels are found by interpolation, and the entire, vowel formant space of the speaker
can then be mapped onto that of the articulatory model. Thus, rather than attempt toseparate the organic and the learned components of vowel production, Payan andPerrier (1993, p.418) acknowledge that“...the consequences of anatomical and gestural differences on the size of back and front
cavities could be variable, depending on the location of the articulation in the vocal-tract.”

In view of the acoustic-articulatory knowledge involved in their method of speaker61

normalisation, it is surprising that Perrier et al. (1995) then found it to perform only
slightly better on F2 , and actually worse on F1 of measured vowel data, compared withthe mainly heuristic, acoustic-phonetic methods reviewed earlier in Section 2.3.3.2 (i.e.,
Gerstman, 1968; Lobanov, 1971; Di Benedetto and Liénard, 1992). On the other hand,in view of the significant amount of literature reviewed in this chapter which points tothe speaker-specificity of the third and higher formants, it is also surprising that Perrier
et al. (1995) did not evaluate their method of speaker normalisation on F3 , which
nevertheless played a crucial role in determining formant-cavity affiliations.
Furthermore, whilst their method of normalisation is significantly more informed thanprevious approaches, the large amount of expert knowledge (and often subjective
judgement) which is required in order to infer formant-cavity affiliations from measured
formants alone, may perhaps be overly prohibitive in applying the method to more
extensive datasets.
As aptly stated by Payan and Perrier (1993, p.417), a method of speakernormalisation “is efficient if it can account for the causes of variability.” However, thesimplicity of this statement belies the lack of cohesion and incompleteness of previous
approaches to overcome the speech-speaker dichotomy by speaker normalisation,
whether perceptually motivated, statistically-based, or vocal-tract length-related. In thisvein, it would seem that a more inclusive approach to accounting for the physicalsources of speaker variability, must involve both vocal-tract length- and shape-related
articulatory parameters, and must be guided by a sufficiently complete, theoreticalframework for describing speaker differences in the speech production domain.
2.4.3.2 Articulatory Parameterisation and the Inverse Problem
If a more complete description of the vocal-tract shape is to be used either to overcome
the speech-speaker dichotomy in speech recognition, or more generally to define
physical correlates of speaker differences in an acoustic dataset of spoken language,
then direct articulatory measurements, whether by non-intrusive methods such as X-ray,
MRI, or ultra-sound imaging, or by more intrusive methods such as electro-magnetic
mid-sagittal articulometry, are clearly impractical; nor are those methods immune tovarious sources of measurement error. For example, the mapping from two-dimensional62

(usually mid-sagittal) images to a more complete, three-dimensional representation(usually the area-function) is still an active research problem; the three-dimensional
reconstruction afforded by MRI still requires the speaker to sustain the speech soundfor many seconds; magnetic tracking of small pellets attached to the tongue or the lips
can only provide localised information on specific articulators.
A more generally applicable, and certainly a more practical approach, would be toestimate articulatory parameters from the acoustics of speech. However, acoustic-toarticulatory mapping (or the so-called speech inverse problem) is itself an unresolved
and controversial area of research (e.g. Schroeter and Sondhi, 1994) — even for thesubset of vocalic sounds, the acoustic and articulatory properties of which are relatively
better understood. Indeed, the most significant problem in estimating articulatory
parameters from acoustic parameters, is that in general, the inverse mapping is nonunique (as epitomized for example by the ventriloquist, who is able to articulate speechsounds without the normal movements of the lips and the jaw).
Some limited conclusions on the extent of the non-uniqueness problem can be
drawn from studies of human articulatory activity during speech production. A numberof such studies (e.g., Lindblom et al., 1979; Gay et al., 1981) have shown that when
speakers are physically inhibited from assuming their normal articulatory configuration
for a given vowel (e.g., by inhibiting either lip or jaw movement), they are able tocompensate for the acoustic perturbation, by recruiting the help of other articulators(e.g. the tongue). In particular, Gay et al. (1981) inferred from X-ray images, that when
speakers perform such articulatory compensation in unnaturally perturbed conditions,they attempt to attain a vocal-tract area-function which is appropriate for the given
vowel (especially at the acoustically more sensitive place of maximum constriction).
The phenomenon of articulatory compensation has also been observed in normal(or uninhibited) speech. For example, Hughes and Abbs (1976) found that in repetitionsof the vowels / i /, / Q / and / E / in /hVbVb/ context, a speaker achieves a nearly invariant
degree of vertical lip opening near the centre of each acoustic vowel nucleus, but withhighly varying, vertical positions of the lower lip, the jaw, and the upper lip. Those
three articulators were thereby shown to exhibit “motor equivalence”, insofar as theirindividual “target” positions are subordinate to the more acoustically important, vocal63

tract “goal” of an appropriate lip aperture for the given vowel. Such “trading relations”
between articulators were also observed by Maeda (1991), who first used his own
articulatory model (Maeda, 1979) to determine the components of the mid-sagittal
profile pertaining to the positions of the tongue-dorsum and the jaw, at the “target” ofthe vowels / i / and / a / recorded in various consonantal contexts by two female speakers
of French. He then found, for each of those vowels, a positive correlation between
tongue fronting (or backing) and jaw opening (or closing); and determined that bysubtracting the articulatory variation due to this correlation, it was possible to reducethe variability in those articulatory parameters to a degree which is comparable to thatof the first two formants measured in the corresponding acoustic data.
Such findings of compensatory articulation support the view that a moreacoustically-relevant representation is afforded by the entire vocal-tract area-function,
than by a model of the mid-sagittal profile which includes the positions of individual
articulators. In addition to the potential exacerbation of the non-uniqueness problem,
such mid-sagittal articulatory models also suffer the plight of speaker-dependence.
Indeed, as rightly stated by Hogden et al. (1996, p.1821):“...many of the ... vocal tract models used to study the inverse mapping problem areessentially models of a single speaker.”

Nowhere is this more obvious than in mid-sagittal models (e.g., Mermelstein, 1973;
Coker, 1976; Sorokin, 1992) which necessarily include both moveable articulators (e.g.,
the tongue and the lips) and the vocal-tract hard-structure (e.g., the outline of the palate
and the rear pharyngeal wall). In this vein, there have been some recent attempts tonormalise certain anatomical dimensions of mid-sagittal articulatory models to those ofa particular speaker, prior to using that speaker’s acoustic data in the inverse mapping(e.g., McGowan, 1997; Mathieu and Laprie, 1997). Whilst this procedure may beeffective in decoupling the so-called organic and learned components of vocal-tract
shapes, it still relies partly on the availability of a mid-sagittal X-ray or MR image ofeach speaker, without which the possibilities of articulatory compensation (in this case,between moveable articulators and the vocal-tract hard-structure) might potentially
undermine the uniqueness of the normalisation itself. On the other hand, explicit

64

modelling of the vocal-tract area-function can provide a speaker-independent
framework for acoustic-to-articulatory mapping, while still being reasonably amenable
to the decoupling of anatomical and behavioural characteristics.
Nevertheless, some studies of human articulatory behaviour imply that even thearea-function is not immune to the non-uniqueness problem. For example, Perkell et al.
(1993) found that some speakers exhibit a (negative) correlation between tongue-bodyraising and lip rounding in the vowel / u / recorded in a wide range of phonetic contexts.
Owing to the relative independence of those two articulators, and their spatially distal
influence on the shape of the vocal-tract airway, variations in the measured positions of
the tongue and lip pellets were assumed to imply changes in the vocal-tract areafunction; however, the degree of acoustic compensation was determined using only F2 .
By contrast, Savariaux et al. (1995) offered a more complete set of formant
measurements in their acoustic and articulatory study of 11 speakers’ lingualcompensations to an inhibiting lip tube, in the production of the vowel / u /. Six of the
speakers were found to partially compensate for the unnaturally wide lip-opening, andone speaker was found to compensate nearly completely, by retracting the tongue body
towards a more velo-pharyngeal (than velo-palatal) constriction. Consistent with thephonetic importance of the low formant space (as reviewed earlier in this chapter),
Savariaux et al. (1995) used only F1 and F2 as a measure of their speakers’ success inacoustic compensation; by contrast, their speakers were generally not as successful incompensating for the effect of the inhibiting lip tube on F3 .
The human-based studies of articulatory behaviour reviewed above, together
substantiate the reality of the non-uniqueness problem. However, model-based
experiments afford more precise control on the variability of articulatory and acoustic
parameters, and potentially more direct insights into the fundamental nature of the nonuniqueness. In this vein, one of the most celebrated studies of the non-uniquenessproblem is that of Atal et al. (1978), who used one of the most “realistic” vocal-tract
acoustic models then available (i.e., one which includes acoustic energy losses arisingfrom viscosity, heat conduction, radiation at the lips, yielding vocal-tract walls, and amean glottal resistance to airflow). Their well-known illustrations of so-called “fibers”
in articulatory parameter space, have since been repeatedly cited as evidence that65

essentially the same set of acoustic parameters can be generated using a realistic vocaltract model, with a continuously-varying combination of articulatory parameter-values.
However, in describing vocal-tract area-functions either in terms of their individual
section-areas, or even in terms of a more economical parameterisation adapted from
Stevens and House’s (1955) three-parameter model, Atal et al. (1978) used a greaternumber of articulatory than acoustic parameters; consequently, as they had earlier
shown mathematically, non-uniqueness is guaranteed by the inevitable creation of a“null-space”. For example, Atal et al. (1978, p.1547) note that “the mouth opening forvowel / i / can vary considerably without affecting the formant frequencies”, and they
indeed give evidence that the first three formant frequencies (and also the first two
formant bandwidths) are relatively invariant for the wide range of mouth openings
considered (approximately 0.3 cm2 to 12cm2). However, their evidence also clearlyshows that for the same range of articulatory parameter variation, the fourth formant
frequency varies from 3973Hz down to 3672Hz, and the third formant bandwidthvaries from 102Hz to a completely unrealistic 2439Hz (cf. Atal et al., 1978, Figure 13,
p.1548); furthermore, it was found that such “differences in the formant bandwidths andthe higher formant frequencies were perceptible to expert listeners.” These observations
underscore the importance not only of constraining model-generated vocal-tract shapes
within realistic ranges of variation (as might appear to be more directly possible using a
mid-sagittal articulatory model), but also of ensuring that the non-uniqueness problem is
not unnecessarily exacerbated by inappropriate articulatory parameterisation which
could lead to under-specification of the acoustic-to-articulatory mapping.
However, in addition to the issue of articulatory parameterisation, the literaturedoes also highlight the importance of the type of acoustic vocal-tract model. Indeed, it
has been repeatedly shown that, regardless of the articulatory parameterisation, the nonuniqueness problem is almost inevitable when using a fully lossy, so-called “realistic”
vocal-tract acoustic model. For example, contrary to the methods adopted by Atal et al.
(1978), Charpentier (1984) intentionally over-specified the inverse mapping by using agreater number of acoustic parameters (the first five formant frequencies and the first
three formant amplitudes) than articulatory parameters (the six parameters of Flanagan
et al.’s (1980) area-function model). Yet, Charpentier’s (1984) detailed analyses, using66

a vocal-tract acoustic model which included all standard losses except at the glottis,
clearly showed the rampant non-linearities and potential ambiguities of the inverse
mapping. Approaching the problem from a more basic angle, Fant (1980) considered asingle-resonance, single-section acoustic tube, and showed that whilst the length of the
tube could be determined by the frequency of the single resonance (using the wellknown quarter-wavelength formula), its cross-sectional area was a non-unique functionof the resonance bandwidth, owing to the complementary effects of lip radiation loss
and internal surface losses included in the model.
Partly as a result of the potential multiplicity of positions of individual articulatorsin yielding a nearly invariant area-function (as implied by certain types of articulatorycompensation), and perhaps more fundamentally owing to the intrinsic non-uniquenessof so-called realistic articulatory and vocal-tract acoustic models, stringent constraints
have had to be applied in acoustic-to-articulatory mapping (e.g., McGowan, 1994;
Gupta and Schroeter, 1993; Sorokin, 1992; Schroeter and Sondhi, 1992; Parthasarathy
and Coker, 1992). Indeed, in order to combat the theoretical and practical inevitabilityof non-uniqueness, previous methods have relied on pre-determined codebooks basedon vector quantisation of articulatory space (Larar et al., 1988); static and dynamic
constraints on articulatory parameters imposed in a dynamic programming framework(Schroeter and Sondhi, 1989); pitch-synchronous analysis to increase the reliability ofmeasured acoustic parameters, and to jointly estimate the parameters of a glottal model(Gupta and Schroeter, 1993); some acoustic-phonetic knowledge in order to ensure areasonable, initial articulatory configuration, thereby avoiding local optima in ananalysis-by-synthesis framework (Lin and Fant, 1989); and linear interpolation between
pre-determined (logarithmic) area-functions of three extreme articulatory configurationsof a given speaker (Butler and Wakita, 1982).
Clearly, both human-based and model-based studies imply that non-uniqueness isinherent to the “realistic” vocal-tract3. However, non-uniqueness at the acoustic-toarea-function level has also been shown to exist when the vocal-tract is treated as a3

Indeed, a further complicating factor is the potential non-uniqueness believed to be caused bycompensatory relations between the supralaryngeal vocal-tract and the glottis, especially those whichmight affect the formant bandwidths (e.g. Schroeter and Sondhi, 1994).

67

completely lossless acoustic tube. For example, in their pioneering investigations of theacoustic properties of the so-called three-parameter model (which specifies the place
and the degree of lingual constriction, and the degree of lip opening), Stevens andHouse (1955) noted the existence of two separate articulatory configurations yielding
the same set of first three formant frequencies for the vowel / u /. Not surprisingly, those
two configurations are similar to the human-based, articulatory compensation strategies
observed more recently by Perkell et al. (1993) and Savariaux et al. (1995).
However, perhaps the most general formulation of the non-uniqueness propertiesof a completely lossless vocal-tract model, was derived by Schroeder (1967) andMermelstein (1967). Briefly, those authors found that if the logarithmic area-function ismodelled in terms of the coefficients of the cosine-series, then for reasonably small
perturbations about a uniform area-function, a change in each odd-indexed coefficient
will induce a proportional change in a unique formant frequency; by contrast,
perturbation of any of the even-indexed coefficients will leave the formants unaffected.
This fundamental result provides the first, direct link between acoustic and vocal-tract
shape parameters. Equally importantly, it provides a theoretically-grounded explanationof the non-uniqueness which is inherent to any, completely lossless model of the vocaltract — in particular, symmetric perturbations of an area-function (where the axis ofsymmetry is at the mid-length of the area-function) are acoustically inconsequential;
furthermore, there exists an infinite range of such, acoustically-inconsequential vocaltract shape-perturbations, at every possible value of the entire vocal-tract length.
To summarise, the literature clearly indicates that acoustic-to-articulatory mappingwith either a fully lossy (i.e., a “realistic”) or a completely lossless vocal-tract acoustic
model, is prone to non-uniqueness; moreover, this inherent non-uniqueness is prone toexacerbation by inappropriate choice of articulatory parameters. Assuming that theentire shape of the vocal-tract (as embodied in the area-function) can be parameterisedin an acoustically relevant way, the question then arises whether there exists any
particular combination of vocal-tract losses which might render the non-uniquenessproblem somewhat less problematic. In fact, it is well known (Atal, 1970; Wakita,1973) that for the linear-prediction (LP) vocal-tract model which has only a single

68source of energy dissipation (either at the glottis or at the lips), uniqueness4 istheoretically guaranteed!
If, as implied at the start of this section, a more complete description of the lengthand shape of the vocal-tract can be expected to yield more informed methods ofovercoming the speech-speaker dichotomy by speaker normalisation, then perhaps thebest approach would be to adopt a minimal and orthogonal parameterisation of the
vocal-tract area-function, and entrust to the uniqueness properties of the LP model thetask of acoustic-to-articulatory mapping. However, whilst the LP-based method ofarea-function estimation does resolve the non-uniqueness problem, it remains
problematic from several other points of view. Indeed, since its inception nearly three
decades ago, it has suffered considerable neglect, primarily on grounds of its presumed
incapability to yield anything but pseudo-area-functions which are generally regarded as
unreliable and too far removed from “real” vocal-tract shapes (e.g., Sondhi, 1979).
Nevertheless, we may yet remain hopeful as perhaps Broad and Shoup (1975, p.254)
were, when they made the following statements in regard to the work of Atal (1970)
and Wakita (1973), which“...suggests that the combination of formant frequency and formant bandwidth
information can give fairly reasonable estimates of vocal tract shapes. These results arevery important for understanding the behavior of the acoustic speech wave in terms ofunderlying articulatory processes, although this possibility has not yet been exploited very
extensively.”

Dare we hope to find the LP model useful in advancing our understanding of the
speech-speaker dichotomy from a speech production point of view?

2.5 Concluding Perspective
Our interpretive review of the auditory-perception, acoustic-phonetic, and articulatory
literature on the long-standing problem of speech-speaker dichotomy has, as far as weare aware, for the first time assembled together the numerous but often disparate works
which provide evidence of both the separate and intertwined influences of phonetic andspeaker-specific attributes of spoken vowels. Indeed, despite this large and growing
4

Strictly, uniqueness of the discrete-sectioned, relative area-function of known length.

69

body of evidence which is held together perhaps by the frequent return to the acousticdomain descriptions of vowel and speaker variability, there is a distinct lack of cohesion
and an immaturity which is reflected both in the diversity of previous approaches, and intheir still incomplete explanation of the speech-speaker dichotomy.
In the acoustic-auditory literature there is ample evidence to suggest that humanperception of vowel identity and of speaker identity is inclined towards favouring,
respectively, the information contained in the low and in the higher parts of thefrequency spectrum. In that context, the perception literature has played an influential
role in shaping current approaches to spectral parameterisation; particularly theauditorily-motivated methods of frequency scale-transformation and relative spectral
measures, which are partially successful in suppressing the speaker-specific andenhancing the phonetic information in spoken vowel sounds. In addition, the perception
literature has clearly fuelled the ongoing debate of formant versus whole-spectrum
representation, and thereby almost inadvertently served as a catalyst for the widespread
appropriation of whole-spectrum parameters such as the mel-frequency cepstrum, inmachine analysis and recognition of speech.
However, the most substantial body of evidence concerning the vowel-speaker
duality of the spectral continuum, is found in the acoustic-phonetic literature. Indeed,
the well-known, classic studies which have firmly established the phonetic importanceof the low spectral range encompassing the first two formants of spoken vowel sounds,
are complemented by numerous, but relatively unacknowledged works which provide
direct evidence of the speaker individuality contained in the higher spectral regions
which include and extend beyond F2 of front vowels. Indirect, but equally compelling
evidence is provided in studies of speaker recognition by machine which, consistently
with the studies on human auditory-perception, report higher accuracies when using thehigher spectral regions or the higher formants of mid- to front vowels in particular.
While studies on speaker recognition and speaker individuality have thus afforded
insights into the acoustic-phonetic subspaces of relatively greater speaker potency, thevowel recognition paradigm appears to hold the potential of more completely revealing
the spectral manifestations of vowel-speaker interactions. However, the literature on
automatic vowel recognition suggests that the detrimental influences of inter-speaker70

variability on recognition accuracy may not be consistently observed using the formants.
Indeed, apart from the sheer popularity and demonstrated superiority of the cepstrum instate-of-the-art ASR, the whole-spectrum representation itself would appear to embody
more completely (and with greater auditory-perceptual relevance) the types of formantpeak interactions which define the very essence of vowel-speaker interactions.
Nevertheless, previous attempts to overcome the speech-speaker dichotomy bysuppressing speaker-related influences, have failed to exploit the full potential of thewhole-spectrum representation which is commonly used in ASR. For example, thepopular method of cepstral liftering does not afford direct control of frequency subbands; nor have the more direct methods of speaker normalisation, whether based on
the whole-spectrum or the formants, addressed the fundamental issue of the apparent
phonetic-speaker duality of the low and the higher spectral ranges. Clearly, a new
approach to cepstral distance computation is called for which overcomes the limitationof operating over the entire available spectral range.
Even so, it emerges from the acoustic-articulatory literature that a more informed,
and therefore a more effective method of overcoming the speech-speaker dichotomy byspeaker normalisation, is potentially afforded by transcending the acoustics of speech,
and accounting for the physical sources of speaker differences. However, whilst
reasonably effective, articulatorily-motivated methods of speaker normalisation havepreviously been proposed, they almost invariably account for only the gross differences
in vocal-tract length. Although the overall size of the vocal-tract is admittedly the mostobvious and the most easily estimated, physical source of variation between adult male,adult female, and child speakers, it is also well-known that such anatomical differences
can only partly explain the wide range of inter-speaker variations in the acoustics ofspeech; even more so for speakers of the same gender and age group, who may possess
vocal-tracts of comparable size, and who may therefore exhibit mainly learned (whether
inter- or intra-idiolectal), rather than organic differences.
Indeed, importation of acoustic-articulatory principles in speaker normalisation, is
still in its infancy. On the one hand, a method of speaker normalisation which relies on
direct articulatory measurements would in general be too prohibitive and impractical.
On the other hand, the inherent non-uniqueness of acoustic-to-articulatory mapping is a71

major stumbling block which continues to divert research efforts away from thepotential use of estimated articulatory parameters. However, even if the non-uniquenessproblem were alleviated by using the inherently unique, LP vocal-tract acoustic model,
there still remains the non-trivial problem of adopting a suitable parameterisation of the
vocal-tract area-function, such that the uniqueness properties of the LP model areretained, and more importantly, that the estimated vocal-tract shapes may bequantitatively compared in a physically meaningful way across different speakers. In
that context, adherence to a sufficiently complete, theoretical framework for describing
speaker differences in articulatory terms, is highly desirable.
Quite clearly, our interpretive review of the literature has shown that whilst an
acoustic-phonetic approach to elucidating the speech-speaker dichotomy is necessary, it
is not sufficient. Our acoustic-phonetic investigations in Chapter 4 will therefore becomplemented by an acoustic-articulatory investigation of the dichotomy in Chapter 6.
However, the latter presupposes an appropriate method of vocal-tract shape estimationand parameterisation, which we do consider in Chapter 5. In order to stage the scene,
the phonetic and the speaker complexity of the spoken vowel data used in this thesis,
and the acoustic parameterisation of those data, are described in the following chapter.

72

Chapter 3
Speech Materials and Acoustic Parameterisation

3.1 Introduction
In this chapter we describe the speech materials and the methods of acoustic
parameterisation which have yielded the data used in the remainder of this thesis. As
foreshadowed in our analysis of the literature (in Chapter 2), the very definition of our
proposed study of speech-speaker dichotomy raises the issue of both the speech andspeaker components of the implied datasets, which is given careful consideration in
Section 3.2. Our review of the literature (in Chapter 2) also highlighted the distinction
between the so-called discrete representation of the acoustics of speech in terms of theformant parameters, and the so-called continuous or whole-spectrum representation
afforded by the cepstrum. In view of the numerous, salient features of both types ofspectral representation, our study of speech-speaker dichotomy will rely heavily on bothformants and cepstra. Sections 3.3 and 3.4 are therefore devoted to describing themethods of acoustic parameterisation of the datasets used to investigate the speechspeaker dichotomy. A summarising perspective is offered in Section 3.5, which points
to the subsequent chapters of this thesis by drawing attention to the usage therein of theacoustic speech data described in this chapter.

3.2 Speech Materials and Speaker Sets
An acoustic-phonetic study of speech-speaker dichotomy would ideally require an
experimental set of spoken language data which embody the two key attributes ofphonetic complexity and speaker heterogeneity. The former refers to a sufficient
coverage of the acoustic-phonetic space of each speaker, while the latter implies
sufficient differences between speakers, as to render speech-speaker interactions

73

measurable by experimental acoustic-phonetic methods.
The phonetic aspect of the present study was established in our analysis of theliterature (in Chapter 2), from which it emerges that even for the relatively better
understood acoustic-phonetic properties of vowel sounds, there still exist fundamental
gaps in our knowledge of the speaker-specific aspects of those properties. It is wellknown, however, that the acoustic-phonetic properties of vowels vary not only with the
speaker, but also according to intra-speaker variability, coarticulation with neighbouring
speech segments, and prosodically-induced segmental variations such as those arisingfrom the presence or absence of linguistic stress. The monosyllabic /hVd/ context has
traditionally been used for studying vowel sounds, firstly because it offers the formationof common words in English, while at the same time nearly nullifying coarticulatory
influences from the preceding consonant, since the articulatory configuration adopted
during the /h/ normally approximates that of the following vowel. In addition, thereading aloud of /hVd/ words in isolated-citation form tends to encourage wellarticulated vocalic configurations or gestures; and randomisation of the words in eachlist helps to maintain consistency in prosodic variables such as intonation and linguistic
stress.
The speaker aspect of the present study was also established in our review of theliterature (in Chapter 2), where attention was drawn to the distinct lack of emphasis inprevious works, in regard to explaining the types of differences that exist between
speakers of a given language and of the same gender, be they idiolectal differences orwithin-dialect, so-called intrinsic differences. In this vein, it is by far more problematicto secure sufficient speaker heterogeneity, than it is to secure phonetic complexity,
especially when one is limited to sampling only the adult male population of speakers ofa particular variety of spoken English (e.g., Australian English, or American English).
On the one hand, most of the current technology in automatic speech recognition(ASR) is based on a data-driven approach whereby large amounts of training speech
data are obtained from as large as possible a population of speakers of the target
language, in order to help increase the robustness of ASR systems against speakervariation, at least on a gender-specific basis. On the other hand, the larger the
population of speakers, the greater are the chances that differences between speakers
74may be obscured or blurred into a more homogeneous continuum of variations.
Heterogeneity therefore does not necessarily imply great numbers of speakers. Indeed,
acoustic data of just a few speakers may suffice to clearly observe certain
manifestations of vowel-speaker interactions, provided only that those speakers are
sufficiently different from one another, either in a physiological or an articulatorybehavioural sense.
With these issues in mind, we now proceed to describe the three sets of speechmaterials on which our acoustic-phonetic and articulatory study of vowel-speakerdichotomy is based. The flow-diagram in Figure 3.1 summarises the nature of the
speech materials and speaker populations which comprise each of our three datasets,
and gives an overview of the pre-processing steps taken (and described in the
succeeding sections of this chapter) to acquire the formant and cepstrum parameters,
which together form the main body of acoustic data used in the remainder of this thesis.
The first dataset (Clermont, 1991), used primarily to unfold the phenomenon ofvowel-speaker dichotomy, comprises eleven non-nasalised vowels in /hVd/ context,
recorded five times contemporaneously by each of four adult male, native speakers1 ofAustralian English. The dataset, which is a subset of a larger corpus of /CVd/monosyllables recorded in citation form by the said four speakers, contains thefollowing vowels in syllable-stressed position: V = / i / (as in heed), / I / (as in hid), / E /(as in head), / Q / (as in had), / a / (as in hard), /  / (as in hod), /  / (as in hoard), / U / (asin hood), / á / (as in who’d), / « / (as in hud), and / ¶ / (as in herd). Five repetitions of themonosyllabic words in randomised order, were presented to each speaker on acomputer screen, at a comfortable rate of about 3.7 seconds per word. The recordings
were made in an acoustically-treated booth, with a light-weight electret microphone(flat frequency response from 50 Hz to 20 kHz) “clipped on the speaker’s shirt, roughly
10 cm below the neck” (Clermont, 1991), and the speech waveforms were quantised to

12 bits and sampled at Fs = 10 kHz. A semi-automatic, acoustic prosodic method wasthen used to break up the recorded speech stream into each individual (monosyllabic)
word, with leading and trailing intervals of silence discarded. Although the above

1

Henceforth referred to as speakers A, B, C, and D.

75

Recorded/ hVd/
monosyllables
Section 3.2
FC dataset(Clermont, 1991)
4 adult, male speakers ofAustralian English;
11 vowels; 5 repetitions in
isolated-citation.

PB dataset(Peterson & Barney, 1952)
32 adult, male speakers ofAmerican English;
10 vowels; 2 repetitions in
isolated-citation.

JB dataset(Bernard, 1967; Clermont, 1996)
36 adult, male speakers ofAustralian English(14 Broad, 11 General, 11 Cultivated);
11 vowels; once in isolated-citation,and once in sentential context.

Section 3.4.1
Section 3.3.1
Vocalic steady-state detection:
cepstrum-based method ofInter-Frame Variance (IFV).

Vowel target located byvisual inspection ofspectrogram.

Vowel target located byvisual inspection ofspectrogram.

Section 3.3.2
Automatic formanttracking based on LP
spectral-matching anddynamic programming(Clermont, 1991).

Section 3.4.1.1

Section 3.4.1.2

Formant-frequency estimation
by weighted-average ofspectral harmonics(cf. Potter & Steinberg, 1950).

Formant frequencies
measured on spectrogram.

Modification of LP
analysis conditions andtemporal location of
vocalic steady-state

Section 3.4.2
Synthetic
formant bandwidths:
Bn, n=1,2,3(3 rounds of examinations).

Section 3.3.3
Cepstral analysis of
vocalic steady-state

LP inverse-routines:
fb2a & a2cp.

(default LP analysis conditions).

{Fn, Bn}(FC),n=1,2,3,4.

{ck}(FC),
k=1,2,..,14.

{ck}(PB),{Fn}(PB),
n=1,2,3. k=1,2,..,14.

{ck}(JB), {Fn}(JB),
k=1,2,..,14. n=1,2,3.

Figure 3.1: Speech data map of the thesis. Flow-chart summarising the three sets of speech materials,
and the methods of analysis used to obtain the formant and cepstrum parameters, which comprise theacoustic data of this study. To the top-left of each module is enumerated the section of this chapter inwhich the respective bodies of data or methods of processing are discussed.
76

description suffices for our present purpose, further details regarding the speechmaterials and the methods of data collection and recording can be found in Clermont(1991, pp.75-108).
Although one could argue that the FC dataset (as hereafter referred to) detailed
above leaves much to be desired by way of sampling a sizeable population of adult malespeakers of Australian English, it nevertheless embodies a number of salient properties
which render it suitable as a good starting point for investigating the question of vowelspeaker dichotomy. Firstly, it comprises a well-balanced and representative sampling of
the acoustic vowel-space of each speaker, with the seven most steady-state, consecutive
frames taken (as described later, in Section 3.3) from the vocalic nucleus of each of thefive repetitions of each monosyllable. Admittedly, longer-term intra-speaker variability
is not captured in the present dataset, but nor has this type of variability ever been sowell-defined in the literature that its absence might be taken to undermine our intended
explanations of acoustic-phonetic and articulatory differences between the four
speakers. Secondly, the relatively small number of speakers does lend itself to an
exploratory framework whereby the intricacies of vowel-speaker interactions might beaccounted for and explained within reasonable bounds of computational complexity.
Thirdly, the four speakers of the FC dataset have been determined by informal auditoryperceptual judgements (Clermont, 1991), to span a range of idiolectal variations which
encompass the so-called General and Cultivated varieties of Australian English, thereby
embodying presumably sufficient amounts of speaker differences to induce the vowelspeaker dichotomy. As stated earlier, the four-speaker FC dataset will therefore be used
to unfold the dichotomy, first in the acoustic-phonetic (in Chapter 4) and then in thearticulatory (in Chapter 6) domain.
With a view to extending our initial investigations to a dataset which contains alarger number of speakers and in addition, offers an advantage in terms of formant
measurements having already been made, we therefore consider next the seminal studyof Peterson and Barney (1952), whose published vowel formant (averaged) data of 33
adult male, 28 adult female, and 15 child speakers are well-known, and have been
widely used in the literature. The speech materials of this dataset comprise tenmonosyllabic /hVd/ words, read twice in randomised order by each of the said 76
77

speakers of American English. The ten vowels placed in syllable-stressed position are as
follows: V = / i / (as in heed), / I / (as in hid), / E / (as in head), / Q / (as in had), / A / (as inhod), /  / (as in hawed), / U / (as in hood), / u / (as in who’d), / « / (as in hud), and / ¼ / (asin heard). The first three formant frequencies were originally measured (as describedlater, in Section 3.4.1) at a temporal location which was judged within each vowel
nucleus to be the most steady-state.
Although only the per-vowel averaged formant frequencies for each of the threegroups of speakers were originally published (Peterson and Barney, 1952, Table II,
p.183), the full body of vowel formant data was recently restored and made publicly
available by Watrous (1991), based on careful comparisons of three existing versions of
the data. As the electronic location of the data given by Watrous (1991) no longer
appeared to hold the data-file at the time that we wished to acquire it (January 1994),
we obtained an electronic copy of the UNIX compressed tar file via anonymous ftp
from “thumper.bellcore.com” in directory “pub”, where it had been deposited by
Spiegel (“comp.speech” Internet newsgroup, 1994).
Only a subset of Peterson and Barney’s data will be used in the present study, andcomprises the first three formant frequencies of the steady-state portion of each of two(contemporaneous) repetitions of the ten vocalic nuclei in /hVd/ context, spoken by 32of the adult male speakers of American English. The data for male speaker number 2
were not used, because the first two formants ( F1 and F2 ) of his vowel /  / (in both
repetitions) are listed as having identical values, presumably owing to inseparable ormerged formant peaks in the measured harmonic spectra. Evidence in support of thisobservation can be found in Figure 8 of Peterson and Barney (1952, p.182), in which
the second repetition of all 76 speakers’ formants are plotted on the F1 F2 plane. That
figure clearly shows a number of data points for the vowels /  / and / A /, which lie along
the F1 = F2 line. The restored version of the formant data suggests that the /  / shownat F1 = F2 = 560 Hz belongs to the second repetition of adult male speaker number 2.
Whilst from a perceptual point of view the merged formant peaks may be completely
acceptable (indeed, annotations which accompany the restored data suggest that both
repetitions of that speaker’s /h d/ were unanimously identified correctly by a panel of
26 listeners), specification of a vowel with identical first and second formant
78frequencies would clearly present difficulties in an acoustic-phonetic, and particularly inan articulatory investigation; hence, the formant data of speaker number 2 weredisregarded.
It is not immediately clear as to the degree of (acoustic-phonetic or articulatory)
heterogeneity of the group of 32 male speakers of the PB dataset (as hereafter referredto). Indeed, as discussed earlier, the larger the number of speakers drawn from the same
variety of a given language, the greater are the chances of introducing homogeneity inthe group as a whole. Unfortunately, we remain today with only an indication of thediversity of the PB speakers. Referring to all 76 speakers, Peterson and Barney (1952,
p.177) state that only two of them “were born outside the United States”, while “a few
others spoke a foreign language before learning English”. It is unknown whether any of
those are amongst the subset of 32 male speakers. However, it is also stated that, incomparison with the adult female and child speakers, “the male speakers represented amuch broader regional sampling of the United States; the majority of them spoke
General American.” Certainly, the PB dataset is well-known and often-used in theliterature; it does comprise formant measurements which are otherwise difficult toacquire accurately and consistently from a large group of speakers; and it would appear
to embody a healthy mix of regional American-English dialects. We shall therefore use
the subset of 32 adult male speakers primarily to validate the vowel-speaker dichotomy,in the acoustic-phonetic domain (in Chapter 4).
The presence of either dialectal (PB dataset) or idiolectal (FC dataset) variations
in each of the two groups of speakers reviewed thus far, would presumably override therelatively more subtle differences which exist between speakers of the same dialect ofAmerican English, or of the same idiolect of Australian English, respectively. Vowelspeaker interactions arising from these latter, so-called intrinsic types of speakerdifferences might only be clearly observed by using a dataset in which the dialectal
tendency of each speaker is known, and can therefore be taken into account. Indeed,
Bernard’s (1967) pioneering study of Australian English provides exactly that, witheach speaker having been categorised (by auditory impressions rendered by Bernard
during an informal interview) as belonging to one of the three idiolects or varieties
known as Broad, General, and Cultivated. Furthermore, the JB dataset (as hereafter79referred to) offers the convenience of formant frequency measurements, analogously to
the PB dataset.
Whilst the original study involved a total of 170 adult male, native speakers ofAustralian English, subsequent publications by Bernard (1970, 1989) have only dealt
with per-vowel averages of the formant data for each of the three idiolectal groups.
Clermont’s (1996) recent resurrection of Bernard’s original formant measurements hasrevealed, however, that a complete subset of the first three formant frequencies
measured in the steady-state of each vocalic nucleus (the same eleven vowels that werelisted earlier in connection with the FC dataset) of /hVd/ monosyllabic words recorded
once in isolated-citation and once in sentential context, could be obtained only byretaining 36 of the speakers. Fortunately, this subset includes a fairly even distributionof speakers of each idiolect, with 14 Broad, 11 General, and 11 Cultivated speakers.
Moreover, Clermont (1996) has shown that the differences between the original dataset
and the retained subset in terms of per-vowel formant averages computed for each of
the three idiolects, are “quite tolerable”, which indicates that the 36-speaker subset may
indeed be considered a representative sampling of the original 170 speakers, both as awhole and in terms of each idiolectal group.
In contrast with the PB dataset where the dialectal distribution of the speakers is
known only qualitatively at best, the idiolectal grouping of the 36-speaker JB dataset is
known precisely. The JB dataset will therefore be used in Chapter 4 to further validate
the phenomenon of vowel-speaker dichotomy, particularly from the point of view of
elucidating vowel-speaker interactions induced by either idiolectal or intrinsic speakerdifferences.
The remainder of this chapter is devoted to a complete description of the methodsof acoustic parameterisation used to construct the three bodies of acoustic data, as
outlined in Figure 3.1. Section 3.3 describes the acoustic parameterisation of the more
phonetically-emphasised (FC) dataset, which will be used to unfold the phenomenon ofvowel-speaker dichotomy. Section 3.4 then describes the acoustic parameterisation of
the two formant-based (PB and JB) datasets, which will permit validation of the
dichotomy using not only a greater number of speakers, but also exploiting the contrast
between the effects of idiolectal and intrinsic speaker differences clearly defined.
80

3.3 Acoustic Data used to Unfold the Dichotomy
As described in the preceding section, the speech materials used to first unfold theacoustic-phonetic manifestations of vowel-speaker interactions, were selected on thebasis of embodying sufficient phonetic complexity and speaker heterogeneity, whilst
ensuring that the involved and exacting task of acoustic-phonetic parameterisation of
the entire dataset could still be carried out within a reasonable period of time. As shown
earlier in Figure 3.1, the FC dataset of eleven non-nasalised vowels in /hVd/ context,
recorded five times by four adult male, native speakers of Australian English, is firstsubjected to an automatic procedure (described in Section 3.3.1) to arrive at an initial
estimate of the temporal location of the most steady-state portion of each vocalic
nucleus. A semi-supervised approach (described in Section 3.3.2) is then adopted tomaintain consistency in estimating the first four formants of the vocalic steady-states,
using an automatic formant-tracker (Clermont, 1991) based on Linear Prediction (LP)
spectral-matching and dynamic programming. Finally, the well-known LP cepstrum
parameters are extracted (as described in Section 3.3.3) from the same, steady-state
frames of vocalic speech data, in preparation for the acoustic-phonetic investigations to
be described in Chapter 4.

3.3.1 Vocalic Steady-State Detection
An implicit assumption in our proposed study of vowel-speaker interactions, as noted
earlier in Section 3.2, is the selection of appropriate speech materials such that
influences external to vowel and speaker variations are controlled or minimised. Thus,
for example, the time-honoured /hVd/ monosyllabic context was selected to increase
the chances of consistently stressed acoustic realisations of well-articulated vocalicgestures. Closely associated with the notion of well-articulated vowel production, is theconcept of the so-called vowel target which the vocal-tract articulators presumably
succeed in attaining some time after the initial consonant /h/ but before the subsequent
articulatory gesture which leads into the final consonant /d/. The acoustic manifestationof that target articulatory configuration is the so-called vocalic steady-state during
which there is, by definition, relatively little spectral change. Indeed, it is in the steady-81state portion of each vowel nucleus that we aim to make our measurements of formant
and cepstrum parameters, which will then be taken to represent that speaker’s acoustic
target for the given vowel.
Towards facilitating our identification of the steady-state portion of each vowel
nucleus (of which there are 5 repetitions of 11 vowels from 4 speakers, i.e., a total of
220 vowel targets to identify), we note the power and simplicity of the procedure usedby Broad and Wakita (1977) to locate five consecutive, steady-state frames insmoothed formant trajectories which they obtained from acoustic recordings ofsustained vowels. Their operational definition of steady-state was simply “minimum
interframe formant frequency variation”, which they found by first computing the interframe variance in the formant frequencies of each combination of five consecutive
frames across the entire utterance, and then locating the temporal location of the global
minimum.
However, if the formants themselves have not yet been estimated, the question
then arises whether the criterion of minimum inter-frame formant frequency variation
cannot be replaced by a more general criterion of minimum inter-frame spectral
variation. In this vein, we note that the Euclidean distance between a pair of LP cepstra
is equivalent to the root-mean-square (rms) distance between the corresponding pair of
cepstrally-smoothed, LP log-magnitude spectra (Gray and Markel, 1976). We cantherefore still embrace the spirit of Broad and Wakita’s (1977) steady-state detection
algorithm, while adopting a cepstrum-based criterion of minimum spectral variation,
which uses a far more robust acoustic parameterisation than the formants. Furthermore,whilst the LP cepstrum can be used indiscriminately to represent the short-time LP
spectrum of both sonorant and obstruent segments of the speech stream, the spectrally
most steady-state region of an /hVd/ utterance can be expected to lie within the vocalic
nucleus. Consequently, the inter-frame cepstral variance can be computed in all groupsof any fixed number of consecutive frames throughout the entire duration of each /hVd/utterance, without prior segmentation of the vocalic nuclei.
In addition to a whole-spectrum approach thus advocated mainly for practicalreasons, we also need a spectral representation, and just as importantly a cepstral
distance measure, which tends to emphasise the formant peaks and to de-emphasise the82spectral valleys in between those peaks. To this end, we note the “formant-sensitivity”
property (Clermont, 1991, p.114) of the Negative Derivative of the LP Phase Spectrum(NDPS), as captured in the so-called NDPS cepstral distance measure proposed by
Yegnanarayana and Reddy (1979) and shown as follows:
2

d

2
QCEP

2
1 π  dφ
1 NCC
dφ ′ 
(c, c ′) = ∫0  −  −  −   dθ ≈ ∑ k(ck − ck′ ) ,2 k =1π  dθ
dθ 

[

]

(3.1)

where φ and φ ′ are the LP phase spectra of two frames of speech which are to becompared, c k and c k′ are the kth components of the corresponding pair of cepstral
vectors ( c and c ′ , respectively), θ is the normalised frequency variable, and NCC isthe number of cepstral coefficients (Gray and Markel (1976) have shown that theapproximate equality in Equation 3.1 tends to an exact equality as NCC → ∞ , but thata value equal to the LP order of analysis M yields a sufficiently high correlation between
the two distances evaluated over a large number of speech frames). Also known as the
root-power-sum (RPS) or the quefrency-weighted cepstrum (QCEP), the equivalenceof the kc k sequence to the LP-NDPS (also known as the group delay spectrum), andthe remarkable “formant-enhancement” properties (Clermont, 1991, p.111) of thelatter, were first shown by Yegnanarayana (1978) and by Fuchi and Ohta (1978). The
so-called NDPS or quefrency-weighted cepstral distance measure in Equation 3.1
inherits those properties, and is therefore equivalent to a spectral distance measure withenhanced sensitivity to variations in the formant frequencies. In addition, the cepstral
distance measure on the right-hand side of Equation 3.1 is computationally far more
economical than the exact NDPS distance measure (shown in the middle of that
Equation).
Our algorithmic procedure for estimating the temporal location of the vocalicsteady-state in each /hVd/ utterance, is therefore described by the following steps:
1. Using the autocorrelation method of LP analysis (Makhoul, 1975a; Markel andGray, 1976), compute M = 14 th-order LP cepstrum in each frame of length25.6 msec (speech waveform Hamming-windowed, and pre-emphasised using thedigital filter 1 − 0.98z −1 ), with a frame-advance of 5.0 msec, yielding a total of N
frames across the entire /hVd/ utterance.

83

2. Initialise frame-group counter to n = 1 .
3. Compute Inter-Frame Variance (IFV) in the group of NSSF = 7 potential steadystate frames starting at frame n, as follows:
IFV(n) =1 NSSF 2∑ dQCEP (c(n + ssf − 1), c(n)),
NSSF ssf =1

(3.2)

2
where c(n) is the cepstral vector at the nth frame, d QCEP( ) is the NDPS cepstral

distance defined earlier in Equation 3.1, and:c (n) =

1 NSSF∑ c(n + ssf − 1) .
NSSF ssf =1(3.3)

4. Increment the frame-group counter n by 1; go to step 3 while n ≤ N − NSSF + 1 .
5. Determine the index n of the global minimum in the IFV profile; store the entire
IFV profile for future reference.
A fine example of a correctly located vocalic steady-state is shown in Figure
3.2(a), which displays the acoustic speech waveform in the middle graph, the spectralpoles obtained by LP analysis of each frame in the lower graph, and, time-aligned above
those plots, the computed IFV profile across the entire duration of the first repetition of/hUd/ recorded by speaker C. Although the IFV profile appears to be quite noisy,
especially during the apparently noise-ridden parts of the waveform before theaspiration of the /h/ and also well after the /d/ release, there is clearly a global minimum
which identifies the start of the seven-frame vocalic steady-state to occur shortly after
the onset of voicing (after just one or two glottal periods, as shown by the left verticalmarker superimposed on the waveform). As shown in the spectrogram-like display of
the LP poles, the spectral variation within the automatically identified steady-state isindeed quite small compared with the rest of the utterance, and the formant structure
which is apparent in that region provides visual confirmation of a correctly identified
vowel target, just before the rapid transition towards the /d/ consonant.
Of all 220 /hVd/ utterances thus analysed, the temporal location of the global
minimum in the IFV profile was found to lie outside the vowel nucleus (as judged byvisual observation of the speech waveform and the time-aligned display of LP poles) for
only four utterances of one speaker (A). The vowel target was incorrectly detected

84

IFV
Amplitude
Frequency (Hz)

(a)

5.5
4.5
3.5
2.5
1.5
0.5

5000
4500
4000
3500
3000
2500
2000
1500
1000
500
0
0

0.1

0.2

0.3

0.4
Time (msec)

0.5

0.6

0.7

0.8

IFV

4.5
3.5
2.5
1.5

Frequency (Hz)(b)

Amplitude

0.5

5000
4500
4000
3500
3000
2500
2000
1500
1000
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0
0

0.1

0.2

0.3

0.4

0.5

Time (msec)

Figure 3.2: Panel (a): a good choice of vocalic steady-state location, in the first repetition of / hUd/
recorded by speaker C. Panel (b): an erroneous choice of vocalic steady-state location, in the fifth
repetition of / h«d/ recorded by speaker A. Time-aligned in each panel, from top to bottom: InterFrame Variance (IFV) profile obtained by the algorithm described in Section 3.3.1 (arrow points to theglobal minimum); speech waveform, with vertical dashed lines demarcating the entire, automaticallyselected steady-state; spectrogram-like display of LP poles (length of each vertical stroke proportional
to the pole bandwidth) obtained with default analysis conditions (see IFV algorithm in Section 3.3.1),
with vertical dashed lines demarcating the seven frames automatically selected as the steady-state.
85

once before the onset of voicing (during the /h/ aspiration), once after the /d/ release,and twice during the /d/ closure. In the first case (the fifth repetition of /h«d/ recordedby speaker A), the spectrally most steady-state sequence of seven consecutive frames
was found by the algorithm to lie before the onset of voicing, during the /h/ aspiration.
As shown in Figure 3.2(b), there is indeed less inter-frame variation of the LP poles inthe selected frames than in the vowel nucleus itself, where the first formant appears to
split into two closely-spaced poles, and where the third and fourth formants are not at
all clearly represented. Nevertheless, the second-ranked minimum in the IFV profile
does fall inside the vowel nucleus, and might therefore serve as a more likely candidate
for the vocalic steady-state in this utterance. Similar observations were made in the fifth
repetition of /hUd/ (steady-state detected just after the /d/ release) and in the third andfourth repetitions of /hUd/ (steady-state detected during the /d/ closure) recorded byspeaker A. In all four cases, the potential steady-state location was manually correctedby selecting the next candidate minimum in the IFV curve lying within the vocalic
nucleus.
Although the completely unsupervised algorithm described above did appear to
identify very likely, candidate steady-state locations for the great majority of the /hVd/utterances recorded by our four speakers, we emphasise that it can only be regarded asproviding an objective guide as to the vicinity of the steady-state within each vowel
nucleus. As it was expected that manual intervention would be necessary to ensure
correct identification of each vowel target, we made no further attempt to improve on
the reliability and precision of the very simple, steady-state detection algorithm. Indeed,a more precise, final decision on the location of the vocalic steady-state awaits theformant estimation stage (described in the following section), where we also take steps
to ensure that the first four formants are measured with as much care and consistency as
possible, across the seven consecutive frames which best characterise the vowel target.

3.3.2 Formant Estimation
As will become plainly evident in the remaining chapters of this thesis, both theacoustic-phonetic and articulatory interpretations of vowel-speaker dichotomy rely
crucially on the formant parameters. One of our main objectives in acoustic
86

parameterisation of the dataset destined to unfold the dichotomy, was therefore tosecure the best possible formants in the steady-state region of each of the 220 vowel
nuclei recorded by our four speakers. As discussed in the preceding section, thecompletely unsupervised algorithm described therein was used to obtain an objective,
initial estimate of the temporal location of the seven most steady-state, consecutive
frames in each /hVd/ utterance. Only four of those estimates were found to lie(incorrectly) outside the vowel nucleus, thus requiring manual intervention to select a
local minimum in the IFV profile within the vowel nucleus. Notwithstanding theapparent success of our automatic algorithm, more extensive manual intervention was
unavoidably required in order to overcome both IFV- and LP-related problems in
obtaining the best formants, as described in Sections 3.3.2.3 and 3.3.2.4, respectively.
3.3.2.1 Goodness Criteria
It is first necessary to define “best possible formants” in the context of this study. Ourdefinition is based on the requirement of formants which best characterise the vowel
target of each /hVd/ monosyllable recorded. That is, we require the trajectories of thefirst four formant frequencies ( F1 , F2 , F3 , and F4 ) to be as well-defined as possible
throughout the selected, seven consecutive frames. However, “well-defined” is here
taken to imply not only smooth and nearly-horizontal formant frequency trajectories,
but also an avoidance of excessively wide formant bandwidths. In addition, whilst theutmost care is taken to preserve the natural occurrence of intra-speaker variability
across the five repetitions of each utterance, our acoustic-phonetic knowledge is alsobrought to bear on the task of ensuring inter-repetition consistency in the per-vowel
formant estimates of each speaker.
3.3.2.2 Formant-tracking Method and Evaluation Aids
Having thus defined our goals in regard to formant measurements, we next describe themethods used to attain them. First, the task was carried out on the data of one speakerat a time, by two, acoustic-phonetically-trained examiners (the present author and his
doctoral supervisor Frantz Clermont). The formant-tracker itself (Clermont, 1991) is anentirely automatic, analysis-by-synthesis method based on spectral matching and87

dynamic programming. Briefly, it selects the first four formants of each frame fromamongst all combinations of the poles yielded by LP analysis of that frame, then applies
a formant-sensitive spectral distance measure (the NDPS cepstral distance measuredescribed in the preceding section) to construct a distance matrix which spans theselected frames of data and finally yields, by the method of dynamic-programming, an
optimum path which defines the formant trajectories across those frames. Given thetemporal constraints imposed by the formant-tracker, each steady-state was therefore
padded with a preceding and a following group of seven frames (a total of 21 frames inall) in order to secure greater continuity in the formant-tracks of the seven central
frames.
Formant trajectories yielded by the tracker, were examined by superimposing them
onto the corresponding, spectrogram-like display of LP poles (examples of which wereshown earlier in Figure 3.2), together with a pair of vertical lines demarcating the sevencentral, potentially steady-state frames. Also displayed on the same screen were timealigned plots of the corresponding speech waveform and of the energy profile across theentire utterance, both of which proved useful in visually confirming the location of thevowel nucleus. Our per-speaker examinations proceeded one vowel at a time, often
displaying the aforementioned graphical plots for two or three repetitions side by side,
thus increasing the chances of retaining in the examiners’ minds, a semblance of crossrepetition consistency in assessing the formant-tracks and the relative temporal
locations of the selected steady-states. Also quite useful in this regard, is the so-called“formant sequence chart” (cf. Potter and Steinberg, 1950), where all four formant
frequencies are plotted (with frequency on the ordinate) in sequence from left to right,
across the seven frames, five repetitions, and eleven vowels of each speaker. We found
the formant sequence chart useful not only for confirming the vowel-to-vowel relationsof the formant frequencies (whereby gross errors in formant labelling can be identified),but, more importantly, for examining their variability across the seven steady-state
frames and especially across the five repetitions of each vowel. Particularly variable
formant trajectories, or those which appeared to be markedly out of line in comparison
with the neighbouring repetitions of the same vowel, could thus be rapidly identified
and marked for a more careful re-examination of the pole-display and superimposed88formant trajectories of each individual utterance.
The vocalic steady-state location and the measured formant-tracks were visually
examined as described above, and corrections administered in three consecutive passes.
Each pass was preceded by automatic estimation of the formant trajectories across 21
frames centred about those seven frames which were selected at the previous round of
examinations as the steady-state of each /hVd/ utterance recorded by that speaker. The
formant-tracker was first applied to the steady-state of each vowel nucleus asdetermined by the global minimum in the corresponding IFV curve (with the four
corrections as described Section 3.3.1). The first pass of examinations then prompted
either a more suitable choice of steady-state location, a more appropriate set of LP
analysis conditions, or both. Once a new set of instructions were formulated concerning
the 21 frames and the LP analysis conditions for each utterance, the formant-tracker
was then re-run to produce a new set of formant estimates. These were examined using
the tools and displays described earlier, a fresh set of instructions were prepared for theformant-tracker based on the latest corrections to steady-state location and analysis
conditions, and the entire process was repeated a third and final time, to yield thesteady-state locations and formant trajectories ultimately to be used in the remainder ofthis study.
3.3.2.3 IFV-related Problems and Corrections
The difficulties encountered at each examination of estimated formant data, can bebroadly categorised as occurring either as a result of inter-repetition inconsistency, or
simply as a result of one or more of the first four formants not being sufficiently welldefined by the LP poles. Included in the former category are those vowel nuclei whichwere found to carry acoustic-phonetic cues of diphthongisation (typically / i / and / á /which, as also noted by Bernard (1967, 1989), tend to acquire an “onglide” for somespeakers of Australian English, thus rendering the nominal monophthongs perceptuallycloser to / i / and / á /, respectively). If spectral variation persists throughout adiphthongised vowel nucleus, then admittedly that part of the nucleus determined by the
IFV algorithm (described in Section 3.3.1) to have the least spectral change, will notalways coincide with the segment which might be selected by a trained phonetician as89the steady-state or vowel target of the nominal monophthong. A case in point is the
fourth repetition of /h id/ recorded by speaker B. As shown in Figure 3.3(a), the global
minimum in the IFV profile occurs soon after the onset of voicing. However, the profilein the vicinity of the global minimum is quite shallow, and is followed towards the endof the vowel nucleus by comparably small inter-frame variances which are only local
minima. The corresponding display of LP poles shows that the automatically selected
seven frames (demarcated in Figure 3.3(a) by the left-most pair of vertical, dashed lines)
lie in what Bernard (1967, 1989) might refer to as the “onglide”, where in this case thesecond formant is still rising towards its target for the nominal monophthong. Hence, inorder to more accurately identify the intended vowel target, and thus also to retain agreater degree of inter-repetition consistency, it was necessary to manually correct thetemporal location of the vocalic steady-state by choosing a more suitable location (asindicated in Figure 3.3(a) by the right-most pair of vertical, solid lines), displaced by atotal of 27 frames (or 135 msec) towards the end of the vowel nucleus, just prior to theso-called offglide into the final /d/ consonant. On the other hand, Figure 3.3(b) shows acorrectly located vocalic steady-state in the second repetition of /h ád/ recorded byspeaker D. Despite the nearly constant rise of the second formant throughout the vowel
nucleus, all four formant frequencies measured in the automatically-selected steadystate region demarcated by the vertical dashed lines, are consistent across repetitions forthis speaker.
By far the majority of problematic cases requiring manual intervention, however,were due to ill-defined formants, even when the temporal location of the automaticallydetected steady-state would otherwise have been acceptable. After the third and final
round of examinations, the location of the steady-state had been manually shifted,
owing either to an incorrectly identified vowel target, or to ill-defined formants at whatmight have passed as a suitable vowel target, in 159 (or 72 %) of the 220 utterances.
Although this figure might seem to undermine the IFV method of vocalic steady-state
detection, the power of that simple method in yielding a good initial estimate of thevowel target is better appreciated by noting that the average shift in steady-state
location was only 26 msec, which is less than half of the maximum length of eachseven-frame window ( 55.6 msec).
90

IFV
Amplitude
Frequency (Hz)

(b)

5000
4500
4000
3500
3000
2500
2000
1500
1000
500
0
0.10

0.15

0.20

0.25

0.30
Time (msec)

0.35

0.40

0.45

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0.15

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0.25

0.30
Time (msec)

0.35

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5
4
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2
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0

Amplitude

IFV

Frequency (Hz)

(a)

5
4
3
2
1
0

5000
4500
4000
3500
3000
2500
2000
1500
1000
500
0
0.10

Figure 3.3: Panel (a): steady-state automatically selected in the vocalic “onglide”, in the fourth
repetition of / hid/ recorded by speaker B. Panel (b): a good choice of vocalic steady-state location, forthe second repetition of / h ád/ recorded by speaker D. Time-aligned in each panel, from top tobottom: Inter-Frame Variance (IFV) profile obtained by the algorithm described in Section 3.3.1(arrow points to the global minimum); speech waveform, with vertical dashed lines demarcating theentire, automatically-selected steady-state; spectrogram-like display of LP poles (length of each verticalstroke proportional to the pole bandwidth) obtained with default analysis conditions (see IFV algorithm
in Section 3.3.1), with vertical dashed lines demarcating the seven frames automatically selected as thesteady-state. In panel (a), the manually corrected steady-state is demarcated by vertical solid lines.

913.3.2.4 LP-related Problems and Corrections
As stated earlier, manual intervention entailed not only selecting a more suitable steadystate position within the vowel nucleus, but also modifying the LP analysis conditions
used by the automatic formant-tracker. The default analysis conditions were the same as
those used earlier to generate the IFV profiles, i.e., LP order of analysis M = 14 , preemphasis coefficient PR = 0.98 , analysis frame-length FL = 25.6 msec, a Hammingwindow of the same length applied to each frame prior to analysis, and the frameadvance FA = 5.0 msec. However, for the sake of obtaining the “best possible
formants” as defined earlier, it was found necessary either to increase or to decrease theorder of analysis M, and often to select a higher, a lower, or a frame-by-frame adaptive(Gray and Markel, 1974) pre-emphasis PR.
Not surprisingly, the rarely-documented fourth formant proved particularly
troublesome in this regard, almost independently of vowel or speaker. An example inpassing was shown previously in Figure 3.3(a) where, in order to secure a more welldefined fourth formant trajectory in the manually corrected steady-state region, it was
found necessary to reduce the LP order of analysis to M = 12 and to increase the preemphasis to PR = 0.99 . As a result of ill-defined formants alone, the LP analysis
conditions required modification in 165 (or 75 %) of the 220 vowel nuclei.
The beneficial effect of increasing the LP order of analysis is illustrated in Figure
3.4, which shows the IFV profile, the speech waveform, and two displays of LP poles,
for the fourth repetition of /hEd/ recorded by speaker D. The pole-display shown
directly below the waveform was obtained with the default LP analysis conditions, andthe vertical lines indicate the steady-state region which for this utterance was correctly
identified by the global minimum in the IFV profile. However, at the default value
M = 14 there appear spurious poles of wide bandwidth which tend to overlap with thesecond and the third formants, and, more importantly, the fourth formant (atapproximately 3500 Hz) is appreciably unsteady and ill-defined. By contrast, an
increase in the LP order of analysis to M = 18 , as shown in the pole-display at thebottom of Figure 3.4, provides a far better definition of the fourth formant (as seen by
the more nearly horizontal trajectory and the smaller bandwidths) and, presumably by

92

IFV
Frequency (Hz)Frequency (Hz)Amplitude

5
4
3
2
1
0

5000
4500
4000
3500
3000
2500
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1500
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500
0
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4500
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0
0.10

0.15

0.20

0.25

0.30
Time (msec)

0.35

0.40

0.45

0.50

Figure 3.4: An example of the beneficial effect of increasing the LP order of analysis to obtain morewell-defined formants (F4 in particular), in the fourth repetition of / hEd/ recorded by speaker D. Top
two graphs: Inter-Frame Variance (IFV) profile obtained by the algorithm described in Section 3.3.1(arrow points to the global minimum); speech waveform, with vertical dashed lines demarcating theentire, automatically-selected steady-state. Bottom two graphs: spectrogram-like display of LP poles(length of each vertical stroke proportional to the pole bandwidth) obtained with default analysis
conditions (above), and with the LP order of analysis increased to M= 18 (below), (vertical dashedlines demarcate the automatically selected, seven steady-state frames).

93

virtue of the two additional, conjugate pole pairs available in the analysis, yields
spurious poles which are somewhat less erratic in behaviour and better separated fromthe true formants.
An illustration of the beneficial effect of reducing the LP order of analysis isprovided by the fifth repetition of /h«d/ recorded by speaker C. As shown in Figure 3.5,
the IFV algorithm selected a very likely steady-state location (in fact, it was manuallyadvanced by just one frame in order to secure slightly less variable formant trajectories).
However, it is not immediately apparent from the LP analysis performed at M = 14(pole-display shown directly below the speech waveform), whether or not the second
formant is represented by either of the two, closely-spaced and broader-bandwidthpoles at approximately 1 kHz. By contrast, a reduction in the order of analysis to
M = 13 (pole-display shown at the bottom of Figure 3.5) yields a more acceptable,

narrower-bandwidth second-formant trajectory which lies in between the two sets ofpoles obtained previously, and which, more importantly, is more consistent than eitherof the two previous sets of poles, with the steady-state second formant as measured in
the first four repetitions of the back vowel in /h«d/ recorded by the same speaker.
In general, an ill-defined formant was manifested on the pole-display either as aformant which appeared to be “split” into two poles of broader bandwidth (as justdescribed and illustrated in Figure 3.5), a formant which simply disappeared for one ormore consecutive frames, or a formant trajectory which appeared to be relatively
variable or “noisy” (mainly in its frequency-location from frame to frame, but also in its
bandwidth). As described above, an increase in the LP order of analysis M often helped
to better define certain formants with the aid of additional spectral poles, whilst a
decrease in M often resolved the ambiguity when a single formant was erroneously
represented by more than one spectral pole. In addition, an increase or a decrease in thedefault pre-emphasis PR = 0.98 (or in some cases, an adaptive pre-emphasis) often
helped to better define formants by virtue of the greater or lesser degree, respectively,of overall spectral tilt which, as indeed was found, tended to have more influence on thepoles in the higher frequency-bands.
However, when a formant-trajectory still showed signs of being variable or“noisy” despite changes in LP analysis conditions as described above, then one of two
94

IFVFrequency (Hz)Frequency (Hz)Amplitude

3.5
3.0
2.5
2.0
1.5
1.0
0.5
0

5000
4500
4000
3500
3000
2500
2000
1500
1000
500
0
5000
4500
4000
3500
3000
2500
2000
1500
1000
500
0
0.15

0.20

0.25

0.30

0.35
Time (msec)

0.40

0.45

0.50

0.55

Figure 3.5: An example of the beneficial effect of decreasing the LP order of analysis to obtain morewell-defined formants (F2 in particular), in the fifth repetition of / h«d/ recorded by speaker C. Top twographs: Inter-Frame Variance (IFV) profile obtained by the algorithm described in Section 3.3.1(arrow points to the global minimum); speech waveform, with vertical dashed lines demarcating theentire, automatically-selected steady-state. Bottom two graphs: spectrogram-like display of LP poles(length of each vertical stroke proportional to the pole bandwidth) obtained with default analysis
conditions (above), and with the LP order of analysis reduced to M=13 (below), (vertical dashed lines
demarcate the automatically selected, seven steady-state frames).

95

remaining options were chosen to remedy the situation. First, the so-called (DP-2)
smoothing function of the formant-tracker (Clermont, 1991) was put into effect,
whereby the frequencies of each of the first three, tracked formants would be perturbed
in steps of ±30 Hz, and the quefrency-weighted cepstral distance measure (Equation
3.1) used to construct a much larger, dynamic-programming distance matrix from whichsmoother formant trajectories might be selected, owing to the greater number ofpossibilities and also to a more stringent continuity constraint imposed on thosetrajectories. This option of the formant-tracker was indeed used a total of 7 times on

F1 , 16 times on F2 , and 25 times on F3 , confirming our expectations that the higher
formants might tend to be more prone to measurement errors induced by frame-toframe variability in the frequency-locations of LP poles.
Failing the above procedure, our last resort was to decrease the frame-advance inorder to allow all seven, consecutive frames to fit within the longest span of acceptable,
steady-state formant trajectories that could be found in the vowel nucleus. Remarkably,
this option was selected for only one of our four speakers (speaker A), whose fourthformant trajectories proved particularly elusive. Only three of that speaker’s vowels(those in /hd/, /hád/, and /h¶d/) were found not to require a reduction in the frameadvance in any of the five repetitions. On the other hand, all five repetitions of his two
front vowels in /hEd/ and /hQd/, four of the five repetitions of his two front vowels in/hid/ and /hId/, and four repetitions of his back vowel in /hUd/, were found to require a
reduced frame-advance, often in addition to changes in LP analysis order and preemphasis. However, in order to retain a reasonable lower limit on the length of eachseven-frame steady-state, the frame-advance was never decreased below FA = 1.0 msec
(thus retaining a minimum steady-state duration of 31.6 msec).
Whilst the main criterion for assessing the goodness of measured formants
traditionally refers to their frequencies (as exemplified here by the use of the NDPS
cepstral distance measure in the formant-tracker), the formant bandwidths have also
played an indirect role in that assessment, by visual examination of the LP pole displays
in order to avoid excessively large formant bandwidths. A single exception to the ruleoccurred for the first repetition of /h ád/ recorded by speaker A, where the formanttracker yielded a fourth formant which was found to be inconsistent (by nearly 500 Hz)
96with the fourth formant frequencies obtained for the other repetitions of the same vowel
recorded by the same speaker. Examination of the LP pole display confirmed that the
tracked, “spurious” pole lying in between the third and fourth formants, was indeed of aslightly narrower bandwidth than the more consistent, true fourth formant.
Interestingly, the tracker was induced into tracking what we considered (as a result ofcross-repetition comparisons) the true fourth formant, only after changing the NDPSbased cepstral distance measure to an unweighted one, which is equivalent to the rms
logarithmic-magnitude distance between cepstrally-smoothed LP spectra. Thus, thereliability of bandwidth measurements cannot be expected to rival that of the formant
frequencies, and their use in either an acoustic-phonetic or an articulatory investigationof vowel-speaker dichotomy will require further, careful examinations which we duly
consider in subsequent chapters.
In sum, the methods described in this section were used to ensure the best possible
conditions for estimation of the first four formants, thus securing a reliable, acousticphonetic description of the eleven nominal monophthongs recorded by our four
speakers of Australian English. The formant sequence charts in Appendix A provide an
overview of the formant frequency data thus obtained for each speaker after all threerounds of examinations. Above all, they confirm the inter-repetition consistency in eachformant frequency measured in the vocalic steady-state of our four speakers’ recordingsof /hVd/ monosyllables. Also included in Appendix A are tables which describe our
final choice of LP analysis conditions used to obtain the formant measurements for eachutterance.

3.3.3 Cepstral Analysis
The methods of speech data analysis described in Sections 3.3.1 and 3.3.2 were used,
respectively, to obtain an objective estimate of the temporal location of the vocalicsteady-state in each of the 220 /hVd/ utterances recorded, and to adjust those estimateson the basis of acquiring the best possible trajectories of the first four formants whichcharacterise each vowel target. Whilst the formants do provide a discrete and readilyinterpretable acoustic-phonetic parameterisation of vocalic speech data, the cepstrum
offers a whole-spectrum representation which, as argued in Chapter 2, is likely to yield97

key insights regarding manifestations of vowel-speaker interactions along the spectral
continuum. To this end, the cepstrum was obtained by the autocorrelation method of
LP analysis (Makhoul, 1975a; Markel and Gray, 1976) of the 220 utterances, in each of
the seven consecutive frames which comprise the vocalic steady-state located by themethods described in the previous section.
As explained in the preceding section, the LP analysis conditions used by theautomatic formant-tracker were determined manually for each utterance, such that thebest set of formants could be selected from amongst the LP poles. Thus, the preemphasis coefficient PR and the order of analysis M were manually varied whennecessary, in order to ensure that the first four formants in the vocalic steady-state wereboth well-defined (as observed on the spectrogram-like display of LP poles) andreasonably consistent across repetitions (as observed on the formant sequence chart
which was prepared prior to each round of examinations). However, it is not usual
practice, either in speech or speaker recognition systems or in the research laboratory,
to obtain the cepstrum with analysis conditions which vary from one utterance toanother for the sake of measuring the best formants. In order to maintain consistency inspectral representation across vowels, speakers, and repetitions of utterances, thedefault LP analysis conditions ( M = 14 , PR = 0.98 ) were therefore used to obtain 14
cepstral coefficients {c1 , c 2 ,..., c14 } per steady-state frame.

3.4 Acoustic Data used to Validate the Dichotomy
In contrast with the acoustic data selected to first unfold the nature of vowel-speaker
interactions, the two sets of validating data (PB and JB) embody a larger population of
speakers, at the expense of a slightly reduced phonetic complexity. Indeed, thesedatasets comprise effectively two repetitions of vowels in /hVd/ context, spoken by 32
adult male speakers of American English, and 36 adult male speakers of AustralianEnglish, respectively. Furthermore, they each are in existence today (so far as thepresent study is concerned) in terms of a single formant-pattern measured at the steadystate of each vocalic nucleus. As summarised earlier in Figure 3.1, the two studies(Peterson and Barney (1952) and Bernard (1967), respectively) differ slightly in their
method of formant frequency estimation; those methods will be reviewed in Section 3.4.1.
98

Subsequently, in Section 3.4.2, we will describe our method of obtaining so-called
simplified LP cepstrum parameters from the available formant data, such that ourinvestigations of vowel-speaker interactions manifest in the FC dataset can be paralleledusing the PB and JB datasets, despite the unavailability of the actual speech waveformsof the latter.

3.4.1 Formant Estimation
In the following two sections we briefly review the methods used by Peterson andBarney (1952) and by Bernard (1967), respectively, to obtain the invaluable sets ofvocalic steady-state formant data which play a validating role in subsequent chapters of
this thesis.
3.4.1.1 Amplitude-Section Approach
The approach adopted by Peterson and Barney (1952) to estimate the frequency
locations of the first three formants of the recorded American English vowel sounds,
involved first the identification of the “practically steady state” interval of each vowel,
by visual examination of a broad-band spectrogram prepared (using a soundspectrograph) for each monosyllabic /hVd/ word. Lacking a more complete descriptionof their procedure for selecting the steady-state portion of each vowel nucleus, we only
know that they selected an interval between the influence of the initial /h/ (which
presumably is quite negligible anyway) and the influence of the final /d/ consonant.
Although they do not refer to any difficulties associated with non-steady or
diphthongised vowel nuclei, their sample spectrograms shown (Peterson and Barney,
1952, Figure 2, p.176) for one of the female speakers (not used in our present study)
suggest that they may indeed have encountered apparently transitional vowel nuclei,
with which they must have dealt in a partly subjective but presumably systematicmanner.
A so-called amplitude section (or harmonic magnitude-spectrum) was obtained at
the selected steady-state portion of each vowel nucleus, and calibrated both on its
vertical (dB) and horizontal (Hz) axes. Guided by the method proposed by Potter andSteinberg (1950), the frequency of each formant was then obtained by computing an99

amplitude-weighted average of the main harmonic components which make up thatformant peak, as follows:

∑w f
F=
,
∑w
i i

i

(3.4)

i

i

where fi is the frequency of the i th spectral harmonic component, and wi is its
amplitude normalised with respect to the amplitude of the dominant component. It isinteresting to note that Potter and Steinberg’s (1950) approximation of the formant
centre-frequency according to Equation 3.4, arose from their speculations regarding thehuman perceptual mechanisms in dealing with formant-like concentrations of spectralenergy, rather than from any consideration of how those spectral components areshaped by the human speech production mechanism.
The main problems associated with the amplitude-section approach to formant
estimation, include the often-encountered asymmetry in the amplitudes of spectral
harmonic components of a formant, and the insufficient density of spectral harmoniccomponents required to resolve a formant. The former problem was effectively dealt
with by computing the weighted-average of spectral components (Equation 3.4), thus
overcoming the ambiguity which might exist in having to select only a single
component. The latter problem mainly occurred with high fundamental frequencies ofvoicing, as would be expected of the majority of adult female and child speakers in thegroup; one might therefore safely assume that the problem was not as prevalent for thesubset of adult male speakers whose data will be used in our present study.
3.4.1.2 Spectrogram Approach
In contrast with the amplitude-section approach just reviewed, Bernard’s (1967)
formant measurements relied mainly on spectrograms of the /hVd/ words, which heprepared using a model Kay spectrograph. Thus, his visual acuity, manual dexterity, andacoustic-phonetic knowledge were all brought to bear on the task of tracing the first
three formant trajectories by hand, and estimating their frequencies at the onset, target,and offset of each vocalic nucleus.
In that context, Bernard (1967, 1989) distinguishes at least two types of acoustic
100realisations of the nominal monophthongs recorded. In the first type, the formant
trajectories (those of F1 and F2 in particular) were found to conform with the ideal
notion of a single, “well-defined target” (Bernard, 1989, p.191), located within thetemporal region demarcated on the left by the onset of the vocalic nucleus (with orwithout a so-called onglide), and on the right by the start of the so-called offglide where
the effect of the final /d/ consonant starts to bend particularly the F2 trajectory towards
an offset value. In those cases, the measured formant-pattern of interest in our present
study is that which Bernard took at a position mid-way through the steady-state of what
he appropriately calls a “one-target” sound. In the second, so-called “two-target” typeof sounds (which were found to occur predominantly for / i / and / u /), he recorded
formant measurements taken at the onset, the offset, and at the centres of each of twotargets thus identified. For convenience, “only the F-patterns he estimated at the second
target were taken as representative of the monophthongs realised with diphthongal
quality” (Clermont, 1996, p.146), and thence used in our present study.

3.4.2 Simplified Cepstrum
If we are to parallel our forthcoming unfolding of vowel-speaker interactions withsimilar investigations using the validating datasets (PB and JB), we therefore require (inaddition to the formants) an acoustic parameterisation of those datasets in terms ofwhole-spectrum information which is carried by the cepstrum. However, as the original
speech waveforms of the PB and JB datasets are either unavailable or would require
inordinate efforts to acquire and segment in the same manner as in the original studies,
we chose to make full use of the available formant data by creating from themcorresponding sets of LP cepstra. By analogy with the notion of “simplified spectrum”
put forward by Clermont (1991, p.110) in the context of his spectral-matching approach
to formant estimation, an LP cepstrum thus obtained, may also be referred to as
simplified because it embodies spectral-shape information derived only from (a limited
number of) formants, and is therefore free of the influence of so-called spurious poles
which usually accompany the true formant poles yielded by LP analysis of a frame ofvoiced speech data.
One way of obtaining the so-called simplified LP cepstrum from formant101

measurements is first to convert the formants to LP autoregressive coefficients using
the reverse conversion algorithm “FB2A” (ILS, 1983), which recursively computes thecoefficients of the LP inverse polynomial by expanding the product of terms eachcorresponding to a resonance pole (Markel and Gray, 1976, p.7). The autoregressive
coefficients can then be converted to LP cepstrum coefficients using the well-known
recursive algorithm “A2CP” (Markel and Gray, 1976, p.230). The data processing
involved in this method remains within the confines of the LP model, merelytransforming between various LP parametric representations (i.e., from the subset offormant poles, to the autoregressive coefficients, to the cepstrum) of a simplified, allpole spectrum.
An alternative but equivalent method of transforming formants to cepstra is thefollowing, direct transformation (Itahashi, 1984, 1988):
kπBn−
 2 kπFn 2c k = ∑ e Fs cos,k n
 Fs 

(3.5)

where Fn and Bn are, respectively, the frequency and bandwidth of the nth pole, Fs isthe assumed sampling frequency, and c k is the kth cepstral coefficient. This formulation,
which can be derived from the very definition of the cepstrum in terms of a minimumphase, all-pole model (Oppenheim et al., 1968), provides some useful insights whichmight be gained only indirectly from the LP-based recursive routines. In particular, thedecaying-exponential factor in Equation 3.5 suggests that the contribution of a broadbandwidth pole to any of the cepstral coefficients is proportionately diminished incomparison with the contribution of a narrow-bandwidth pole with the same centrefrequency. Hence, the so-called spurious poles which are usually of much wider
bandwidth than their true formant counterparts, already have a much smaller influenceon the cepstrum obtained by LP analysis of a frame of voiced speech having clear
formant structure.
Nevertheless, a so-called simplified cepstrum obtained using either Equation 3.5 or
the LP recursive routines, is completely stripped of influence not only from the spurious
poles, but also from the somewhat difficult-to-estimate higher formants which may still
lie within the frequency-band extending to the half-sampling frequency.

102

In this regard, it may be noted that, using either of the two methods, the required
parameter Fs can be chosen rather arbitrarily, subject only to the theoretical constraintthat it exceed twice the frequency of the highest formant considered. However, in orderto maintain consistency with the LP cepstra obtained from the FC dataset (as describedearlier in Section 3.3.3), the sampling frequency is fixed at Fs = 10 kHz when mapping
the three available formants of the PB and JB datasets to corresponding sets of cepstra.
It may also be noted that formant bandwidths are required in the computation of
the simplified cepstrum (using either of the two methods), as might intuitively be
expected when the spectral shape is to be specified, and not just the frequency locationsof the formant peaks. However, neither of the two classic datasets (PB and JB) include
formant bandwidths, which are in any case notoriously difficult to estimate reliably. In
this regard, Bogert’s (1953) early attempt at measuring the formant bandwidths of 100of the PB vowels recorded by the 33 male speakers, is certainly a landmark effort. On
the harmonic amplitude spectra used by Peterson and Barney (1952) to obtain formantfrequency estimates (as described in Section 3.4.1.1), Bogert carefully drew a free-handcurve enveloping the amplitudes of the harmonic components of each formant peak, andthereby obtained an estimate of the bandwidth by measuring the width of the curve at
3dB below its peak amplitude.
By replacing Bogert’s free-hand curves with a number of template curves ofdifferent bandwidths obtained from an analytic relation, Dunn (1961) was able toimprove on the accuracy of the bandwidth estimates, which he obtained on 400 samplesof the PB vowels, comprising both repetitions of all ten /hVd/ words recorded by 20 of
the male speakers. Notwithstanding the numerous sources of measurement error which
include the sparse density of spectral harmonic components in each amplitude section,
he was able to identify a likely trend. In a classic plot of per-vowel formant bandwidthversus formant frequency averaged over both repetitions and all 20 speakers, Dunn
(1961, Fig.2, p.1739) suggested a piecewise-linear relationship on a logarithmic scale,with one line for each of the first three formants.
One way of supplementing the available formant frequencies of the PB and JB
data with “synthetic” bandwidths prior to computing the cepstra, might therefore be toformulate an equation for each of the three linear relations suggested by Dunn, and thus103

to determine the bandwidth of each formant on the basis of its centre-frequency. An
alternative approach might be to adopt a fixed value for each of the first three formant
bandwidths (regardless of formant frequency), based on the overall mean values ofabout 50 Hz, 64 Hz, and 115 Hz found by Dunn (1961) for the subset of the PB data.
(Note that Fant’s (1972) well-known equations which express each of the first threeformant bandwidths in terms of combinations of the formant frequencies, and likewise
Hawks and Miller’s (1995) piecewise-polynomial expression which relates the
bandwidth of a formant to its centre-frequency, are based on closed-glottis bandwidth
data. In contrast, bandwidths measured by means of spectrographic analyses (Bogert,
1953; Dunn, 1961) or by LP analysis (the present study, as described earlier in Section
3.3.2) are based on frames of speech data which typically encompass at least two
periods of the glottal cycle, and thus are presumably averaged over both open andclosed glottis conditions).
Our final choice regarding the synthetic bandwidths required to convert the PB
and JB sets of formant data into simplified cepstra, depends to a large extent on thenature of vowel-speaker interactions to be unfolded and later validated using those
cepstra in Chapter 4, and on the robustness of our acoustic-phonetic findings in the faceof varying formant bandwidth values. It would therefore seem pertinent to first assess
the robustness of the acoustic-phonetic manifestations of vowel-speaker interactions
using simplified cepstra generated from the FC dataset, for which we do have theoriginal LP cepstra as a baseline for comparison. Indeed, it would be rather fortuitous
to find those acoustic-phonetic manifestations relatively invariant despite the radical
simplification of the cepstrum (and hence the spectrum) to carry information contained
only in the first three formant frequencies.

3.5 Summarising Perspective
In this chapter we described the three sets of recorded /hVd/ monosyllabic speech data,
and the methods used to parameterise those data in terms of formants and cepstra
measured at the vocalic steady-state (or acoustic vowel target). In the summarising
perspective which follows, we shall briefly preview the role of the speech and speakermaterials, and of the acoustic parameters derived from those datasets, in terms of their104

usage in the forthcoming chapters of this thesis.
As summarised earlier in Figure 3.1, the FC dataset (Clermont, 1991) includes five
repetitions of eleven vowels recorded by four adult, male speakers of AustralianEnglish, who have been determined (at least qualitatively, from a subjective, auditoryperceptual point of view) to form a sufficiently heterogeneous speaker group. The
acoustic parameters measured from that dataset will therefore be used in Chapter 4 to
unfold the phenomenon of vowel-speaker dichotomy. By contrast, the well-known andreadily available PB dataset (Peterson and Barney, 1952) includes acoustic vowel dataof 32 adult, male speakers of American English. Although it is limited to only two
repetitions of each vowel per speaker, the more populous dataset offers the possibilityof validating the phenomenon of dichotomy first unfolded using the data of only four
speakers. However, as the dialectal heterogeneity of the PB dataset is known only
qualitatively, we also consider the recently resurrected (Clermont, 1996) subset of data
(Bernard, 1967) which includes effectively two repetitions of eleven vowels recorded by
14 Broad, 11 General, and 11 Cultivated, adult, male speakers of Australian English.
Indeed, whilst the JB dataset will also play a validating role in Chapter 4, it provides theunique opportunity of separating the effects of idiolectal and so-called intrinsic speakerdifferences.
Acoustic parameterisation of the three datasets just reviewed, has yielded both adiscrete and a continuous spectral representation in terms of the formant and the
cepstral parameters, respectively. The salient features of those two types of spectral
representation will be duly exploited in the following three chapters. First, as argued in
our analysis of the literature (in Chapter 2), a whole-spectrum representation may lend
itself more effectively to an investigation of the frequency-band dependence of vowelspeaker interactions. The LP cepstra obtained for the FC dataset, and later thesimplified LP cepstra derived from the PB and JB formant data, will therefore play avital role in Chapter 4, in unfolding and validating, respectively, the phenomenon ofvowel-speaker interactions along the spectral continuum. The interpretive superiority of
the formants will then be exploited in Chapter 4, in order to provide an acousticphonetic explanation of the phenomenon. Furthermore, the articulatory significance ofthe formants will be reasserted first in Chapter 5, where we present a new method of105

vocal-tract shape parameterisation and estimation which maps the geometry of the
vocal-tract from the formants, and then in Chapter 6, where we use those estimatedvocal-tract shapes to provide an articulatory explanation of the phenomenon of vowelspeaker interactions, and of the speaker differences which underly those interactions.

106

Chapter 4
Acoustic-Phonetic Analysis of Speech-Speaker Dichotomy

4.1 Introduction
Our review of the literature (in Chapter 2) has firmly established the importance andcurrency of the long-standing problem of speech-speaker dichotomy. Indeed, even for
the relatively better understood, vocalic sounds of spoken language, we are still lacking
a unified account of the separate influences of phonetic and speaker-specific attributes
which are intertwined in the acoustic speech signal. A better understanding of thenature of those separate influences, and of their interactions, would have implications inboth speech and speaker recognition by computer, and would also pave the way formore versatile, multi-speaker speech synthesis.
In this vein, the relatively more extensive literature on spoken vowel sounds isclearly replete with the time-honoured view that the most important, phonetic
information is carried by the first two formants (or resonances of the vocal-tract) F1
and F2 . Not only are those parameters generally considered the best acoustic-phonetic
descriptors of vowels, they are also strongly correlated with the traditional articulatory
description of vowels in terms of the two dimensions tongue height and advancement,
respectively. By contrast, there is mounting evidence in the literature that the third andhigher formants, or indeed the spectral regions which encompass those higher
resonances, carry more speaker-specific information, and therefore manifest greater
amounts of inter-speaker variability.
If certain spectral regions of vowel sounds indeed contain predominantly either
phonetic or speaker-specific information, then it would seem possible to observe thisphenomenon in its duality, if machine classification of these sounds were attempted on
an intra- and an inter-speaker basis. However, the issue of spectral manifestations of107vowel and speaker variability, raises the equally-important question of spectral
representation. Despite the inherent difficulties in measuring the formant frequencies,
they still hold a place of pride in acoustic-phonetics, and are perhaps the mostarticulatorily relevant, acoustic parameters. On the other hand, the cepstrum is easily
obtained from the acoustic speech signal, and is perhaps better suited to an investigationof the frequency-band dependence of the vowel-speaker dichotomy, owing to its wholespectrum representation. However, new methods of cepstrum and cepstral distancecomputation are required in order to enable the selection of frequency sub-bands in
cepstrum-based vowel classification. In Section 4.2 we elaborate on the methodologicalaspects of our acoustic-phonetic investigations, which indeed use the LP cepstrum invowel classification experiments designed to contrast the manifestations of phonetic andspeaker influences along the spectral continuum.
In Section 4.3 we then apply our methodology to unfold the vowel-speakerdichotomy using the FC dataset described in Chapter 3. In our attempt to unfold thephenomenon with a greater certainty of having suppressed external or artefactual
influences, we shall compare the results obtained using classifiers with hyper-planar (inSection 4.3.1) and hyper-quadratic (in Section 4.3.2) decision boundaries. However, inorder to retain only the intrinsic manifestations of the vowel-speaker dichotomy, wemust also address the related issues of data sufficiency and cepstrum dimensionality towhich the latter, more sophisticated classifier is particularly sensitive.
Having thus unfolded the phenomenon of dichotomy using the ensemble ofmethods centred around the cepstrum, in Section 4.4 we then turn to the interpretively
more superior formant parameters to provide an acoustic-phonetic explanation of the
dichotomy. Indeed, the question of which vocalic, acoustic-phonetic subspaces are themost heavily influenced by vowel-speaker interactions, is best approached byinterpreting the cepstrum-based classification results in terms of the speakers’ vowel
formant distribution.
Whilst the FC dataset provides sufficient phonetic richness and speaker differences
to unfold the dichotomy, the question remains whether phonetic-speaker interactions
will be equally manifest in a more populous, and perhaps more homogeneous dataset.
As explained in Chapter 3, the PB dataset allows the possibility (in Section 4.4.4) of108

validating the dichotomy from the point of view of speaker homogeneity. However, inorder to be confident in applying our methodology to the so-called simplified cepstra
generated from the formants of the PB dataset, we first examine (in Section 4.4.3) the
dependence of the dichotomy on the simplified spectral representation using the FC
dataset.
As the FC and PB datasets are known to contain either idiolectal or dialectal
speaker differences, the further question arises whether the dichotomy is wholly reliant
on that type of speaker heterogeneity, or whether it is also manifest when only withinidiolect, or so-called intrinsic speaker differences are present. In Section 4.4.5 weaddress that question using the JB dataset (as described in Chapter 3), in terms of(perceived) idiolectal homogeneity amongst adult, male speakers of Australian English.
Finally, in Section 4.5 we conclude with a summary of our acoustic-phonetic
findings in regard to the vowel-speaker dichotomy, and point forward to ourforthcoming, articulatory investigations.

4.2 Methodology for Unfolding the Dichotomy
As described in Chapter 3, the dataset which is destined to unfold the phenomenon ofvowel-speaker dichotomy comprises five repetitions of eleven vowels recorded in
citation-form /hVd/ context by four adult, male, native speakers of Australian English(Clermont, 1991). The seven most steady-state, consecutive frames were carefully
identified within each vowel nucleus according to criteria aimed at obtaining the best
possible trajectories of the first four formants. Those seven frames were then reanalysed to obtain the 14th-order LP cepstrum parameters which we shall use to unfold
the dichotomy, with the aid of the ensemble of methods herein described.
In Section 4.2.1 we provide an outline of our general approach, which consists ofusing the LP cepstra in vowel classification experiments designed to elicit theconsequences of vowel and speaker influences across the spectral continuum. In
Section 4.2.2 we provide justifications for considering the formant-enhanced spectral
representation afforded by the quefrency-weighted cepstrum (QCEP), and motivate theuse of simplified cepstra derived from the FC formant data. In Section 4.2.3 wedescribe the two methods of vowel classification used in this chapter, which yield
109

hyper-planar and hyper-quadratic decision boundaries, respectively. In order toaccommodate frequency sub-band selection in the classification experiments, newformulations of cepstral distance computation and cepstrum transformation are derived,
which offer the flexibility of observing the behaviour of classification accuracy as afunction of an increasing spectral range.

4.2.1 Approach
In formulating an approach to unfold the vowel-speaker dichotomy, we must address
two principal requirements. By definition, we require to contrast the acoustic
manifestations of phonetic and inter-speaker differences. At the same time, we require
to interrogate the frequency-band dependence of those differences. As presently
described, our approach to unfolding the acoustic manifestations of vowel-speaker
interactions, hinges on automatic (computer) vowel classification experiments designed
specifically to lay bare those interactions along the spectral continuum.
First, the inherent duality of the vowel-speaker problem is heeded by contrasting
the performance of vowel classification carried out alternately on an intra-speaker (orspeaker-dependent), and on an inter-speaker (or speaker-independent) basis. As thephonetic dimension is present in both types of experiment, the vowel classification task
is of equal relevance. However, whilst the former type of experiment is oblivious tointer-speaker differences and therefore embodies principally the phonetic dimension
unimpeded, the latter type of experiment explicitly introduces the speaker dimension,and thus involves the more error-prone task of phonetic discrimination in the presenceof inter-speaker variability. In that regard, it is generally accepted that, at least fornormal, adult male speakers of sufficiently similar dialects of a given language, theacoustic-phonetic manifestations of inter-speaker differences are generally of a smaller
magnitude than those of inter-vowel differences for a given speaker. Variability induced
by inter-speaker differences can therefore be regarded as contaminating the acoustic
vowel space, thereby degrading the performance of vowel classification according to
the degree of manifestation of those differences. The contrast in performance of vowel
classification first in the absence, and then in the presence of inter-speaker variability,
can therefore be expected to shed some light on the relative strength of vowel and110speaker influences.
However, the question of the frequency-band dependence of those influences
remains unanswered pending a method to contrast classification performance in various
spectral regions. In that context, our analysis of the literature (in Chapter 2) has alreadysuggested the potentially contrasting roles of the lower spectral regions which
encompass the normal range of phonetic variation in the first two formants, and the lessphonetically important (and perhaps more speaker-specific) higher spectral regions
which contain the third and higher formants. If the acoustic manifestations of speakerdifferences are indeed more potent in the so-called higher than in the lower spectral
regions, one might expect to observe their effect on vowel classification more clearly byinitially considering only the lower, phonetically-relevant frequency bands, then using
progressively greater amounts of spectral information recruited from the higher
frequency bands.
Sadly, the traditional approach to frequency-band selection either by using filterbanks or by re-analysing or re-sampling the speech waveform, leaves much to be
desired in terms of both flexibility and consistency in spectral representation. Indeed,
the popular cepstral distance measures used in speech processing and recognition
systems have to date been formulated with the inflexible constraint of comparing framesof speech over the entire available spectral range extending to half the sampling
frequency. We shall attempt to remedy this deficiency in Section 4.2.3, where we
propose a new transformation of LP cepstra, and a new formulation of a cepstral
distance measure, both of which are expressed parametrically in terms of a lower and an
upper spectral limit which demarcate any selected frequency band within the available
spectral range. This new approach to cepstrum derivation and cepstral distancecomputation does provide the flexibility of performing vowel classification using
spectral information contained in selected frequency sub-bands, without the restrictions
normally imposed by a fixed bank of bandpass filters, and without the need to re-analyseor re-sample the speech waveform.
To recapitulate, the acoustic manifestations of vowel-speaker dichotomy will be
unfolded by means of vowel classification experiments, which at once aim to resolve theseparate influences of phonetic and speaker-specific attributes of spoken vowels, and to111

determine the frequency-band dependence of those attributes and their interactions. In
particular, the contrast between the performance of intra- and inter-speaker vowel
classification will provide a basis for assessing the relative degree of acoustic
manifestation of vowel and speaker influences. Determination of the frequency-band
dependence of those influences will be facilitated by new formulations of cepstrum
derivation and cepstral distance computation, which allow a more flexible approach toscanning the available spectral range incrementally from low to higher frequency bands.

4.2.2 Spectral Representations
An important step which precedes the classification process and is well-known to play amajor role in its performance, is the acoustic parameterisation which yields a particular
type of spectral representation to be used in the classifier. In this vein, the cepstrum iswidely regarded as one of the foremost parameters in state-of-the-art spoken language
systems. Not only is the LP cepstrum obtained from the speech waveform robustly andwith relatively low computational cost, it has also been found to outperform many
alternative spectral representations in both speech and speaker recognition tasks. In
Chapter 3 we gave a complete description of our methods of obtaining LP cepstra in
the vocalic steady-state of /hVd/ monosyllables recorded by four speakers of AustralianEnglish (the FC dataset). In view of our planned experiments of vowel classification
using those cepstra, we now re-examine some of our assumptions regarding analysis
conditions and type of cepstral representation.
First, despite the variable analysis conditions adopted by necessity during theformant estimation procedure, the LP analysis order and pre-emphasis coefficient were
subsequently fixed ( M = 14 , PR = 0.98 ) when obtaining the cepstrum in each frame, inorder to secure consistency in spectral representation across repetitions, vowels, andspeakers. A fixed order of analysis maintains consistency in the accuracy of spectralshape information carried by the LP cepstrum, since potentially the same number ofpoles (a maximum of seven complex-conjugate poles) are made available to model the
short-time spectral envelope of each vocalic frame. The order of analysis was chosen on
the basis of guidelines put forward by Markel and Gray (1976, pp.154-156), whosuggested that for voiced speech, a value equal to the sampling frequency in kHz would112

offer most likely a sufficient number of poles to model the expected number of formant
peaks in the spectrum, and that an additional number of terms (between two and five)should be added to model remaining spectral-shape characteristics, such as the spectral
slope which results from the combined effects of glottal and lip radiation. Thus, we
added four to the value of Fs = 10 kHz to arrive at a reasonable order of analysis
M = 14 .

The next question concerns the dimensionality of the cepstrum itself. By
definition, an exact representation of the LP spectrum is only obtained by having
recourse to an infinite number of cepstral coefficients. Truncation of the cepstrum to a
finite number of coefficients yields a smoothed representation of the all-pole spectrum.
Nevertheless, Gray and Markel (1976) have shown that distances yielded by a
cepstrum-based distance measure have a significantly high correlation ( 0.98 ) with rootmean-square (rms) distances between LP log-magnitude spectra computed over a large
number of frames of speech, when the number of cepstral coefficients retained (NCC) isequal to the analysis order M. This finding is of particular importance in recognition
systems where the dimensionality of the feature space is required to be as small as
possible, because it implies that the low-order cepstral coefficients retain the essential
components of spectral-shape information. Indeed, state-of-the-art speech and speakerrecognition systems do take advantage of this property and generally use only the loworder cepstrum (often supplemented with derivative parameters such as the deltacepstrum). We are therefore justified in using the first NCC = M = 14 cepstralcoefficients {c1 , c 2 ,..., c14 } , as indicated earlier in Section 3.3.3, to represent each frameof vocalic speech data in our vowel classification experiments.
As also indicated in Chapter 3, the Negative Derivative of the LP Phase Spectrum(NDPS) offers an enhanced form of spectral representation which emphasises theformant peaks and minimises the interactions between neighbouring peaks. Those
properties are captured by the root-power-sum (RPS) or the quefrency-weighted
cepstrum (QCEP), where each cepstral coefficient is simply multiplied by its index (orquefrency) to yield the sequence kc k . Whilst quefrency-weighting of LP cepstralcoefficients has also been shown to improve their performance in vowel classification(Paliwal, 1982), our primary motivation for adopting the formant-enhanced spectral
113representation afforded by the QCEP is to accentuate the acoustic-phonetic (andtherefore perhaps the articulatory) relevance of the already quite powerful LP cepstrum.
In this vein, it is also of relevance to consider the simplified cepstrum (cf. Section
3.4.2), which carries only formant information, and is therefore cleansed of theinfluence of spurious and other spectral-shaping poles. In Chapter 3 we discussed thesimplified cepstrum in the context of generating cepstral data for the PB and JB
datasets which currently exist only in terms of the first three formant frequencies. Prior
to performing vowel classification experiments on those datasets, it would indeed seempertinent to first validate the results obtained with the original cepstra of the FC dataset,by using simplified cepstra generated from the measured formants.

4.2.3 Classification Methods and Distance Formulations
We next describe the vowel classification process itself, which uses the quefrencyweighted LP cepstrum (QCEP) parameters discussed previously. We proceed from a
discussion of data-partitioning strategies, to the classification rule, and finally to the two
methods of classification with their respective entourages of cepstral distanceformulations and methodological peculiarities.
A pattern-recognition approach is adopted, whereby the available data are divided
into a training set which is used to form prototypes for each class (vowel), and a test
set which is then used to arrive at an estimate of the probability of error in classification.
There exist many such data-partitioning schemes, an excellent review of which is given
by Toussaint (1974). Perhaps the simplest is the so-called re-substitution (or R) method
where all of the available data are used for both training and testing. However, theory
confirms intuition that the R-method provides an overly optimistic estimate ofclassification error, which therefore approaches the so-called true error-rate frombelow. On the other hand, estimates of misclassification yielded by the so-called holdout(or H) method have been shown to be overly pessimistic (thus approaching the trueerror-rate from above), as the available data are divided only once into mutually
exclusive sets for training and testing, thereby stretching the capability of the classifier
to generalise beyond the training data. A more reliable version of the H-method consistsof averaging the estimates of misclassification obtained by randomly partitioning the114data several times into pairs of equal-sized training and test sets. However, theaveraged estimate is known to be still overly pessimistic.
A less biased approach is given by the leave-one-out (or U) method, where asingle pattern sample is withheld from the training data and used for testing, and theprocess is repeated over all such data-partitions to arrive at an average error-rate. A
closely related and more general form of the U-method is the so-called rotation (or Π)
method, where a small subset (greater than or equal to a single sample, but where theratio of total samples to left-out samples is an integer) of the available data is withheld
from training and used for testing, and the error-rate is again determined by rotating thedata and averaging the results.
In view of the reduction in bias achieved by the U-method and the Π-method,which therefore more closely approach the true rate of misclassification, we shall adopt
a variant of these approaches to data-partitioning. In particular, our requirement tocontrast the performance of vowel classification on an intra- and an inter-speaker basis,
leads us to adopt a “leave-one-out” approach where the data of one repetition at a time
are withheld in intra-speaker experiments, and the data of one speaker at a time arewithheld in inter-speaker experiments. Thus for the FC dataset, intra-speaker vowel
classification accuracy is averaged over five experiments per speaker (before being
averaged over all four speakers’ results), whilst inter-speaker vowel classification
accuracy is averaged over four experiments. This would appear to be the best approachto obtaining minimally-biased estimates of classification accuracy, while at the sametime conforming to our methodological requirements aimed at unfolding the vowelspeaker dichotomy.
Having decided on a method of data-partitioning, we now turn to the classifier
itself. In order to place the vowel-speaker dichotomy on a more solid footing, we
consider two classifiers which differ mainly in the level of complexity used to represent
each vowel prototype. In particular, the simpler of the two classifiers computes vowel
prototypes using only first-order statistics (the mean) of the training data, whilst themore sophisticated classifier uses both first- and second-order statistics (the mean andcovariance). As a result, the two classifiers may be distinguished on the basis of theirdecision boundaries: they use effectively hyper-planar and hyper-quadratic decision115

boundaries, respectively.
A common feature of both classifiers is the well-known minimum-distanceclassification rule, whereby distances are computed from a given test sample to each of
the class prototypes, and the test sample is then assigned to that class from whose
prototype it was found to have the minimum distance. Implicit in this approach toclassification is the requirement of a distance measure. Indeed, our plan to investigate
the frequency-band dependence of vowel and speaker influences does require a method
to compute the distance between two given frames of speech, not only across the entire
available spectral range as implied in cepstral distance measures available to date, but
within any selected frequency sub-band which lies in that range.
In the following two sections, we therefore describe the two classifiers which welater use to unfold the vowel-speaker dichotomy. In Section 4.2.3.1 we describe theclassifier which uses hyper-planar decision boundaries, together with our new,parametric cepstral distance measure which allows selection of any frequency sub-band
without requiring to re-compute the cepstrum. In Section 4.2.3.2 we then describe theclassifier which uses hyper-quadratic decision boundaries, together with our new
method of cepstrum computation which models the spectrum within any selected
frequency sub-band, and which therefore allows interpretation of classification results interms of those selected spectral regions.
4.2.3.1 Classifier based on Hyper-Planar Decision Boundaries
The classifier first adopted to unfold the dichotomy, uses the training data to compute asingle, per-class prototype which is the mean (or centroid) of the LP cepstrum vectors
belonging to that class (vowel). For N v vowels and S training samples per vowel, the ithvowel prototype is therefore computed as follows:

ci =

1 S
∑ c il ,S l =1i = {1, 2, ..., N v },(4.1)

where c denotes an LP cepstrum vector with NCC = M = 14 elements. Each cepstrum
vector in the test data is then assigned to that class whose prototype is the “closest”(i.e., yields the smallest distance), and a correct or a false classification is tallied
according to whether or not, respectively, the assigned class is the same as that of the116sample being tested. When the minimum-distance classification rule is thus employedwith vowel prototypes constructed from only first-order statistics, the resulting decisionboundaries which separate the classes are well-known to be hyper-planar (Duda andHart, 1973; Tou and Gonzalez, 1974). (For a two-class problem, the hyper-plane issimply the perpendicular bisector of the line segment joining the two class means.)
The distance measure used to compare a given test sample with each vowel
prototype, is based on the formant-enhanced NDPS representation given by the QCEP
(cf. Section 4.2.2). We have already discussed the advantages of this spectral
representation in Chapter 3, where we capitalised on the formant-sensitivity property of
Yegnanarayana and Reddy’s (1979) NDPS cepstral distance measure in our definitionof inter-frame variance, used successfully to detect the vocalic steady-states of therecorded utterances which comprise the FC dataset. However, as with all cepstral
distances proposed to date, the measure of spectral similarity is integrated effectively
over the entire available spectral range which extends to half the sampling frequency(cf. Equation 3.1 in Chapter 3). In order to perform vowel classification using spectral
information only within selected frequency sub-bands, direct selection of a sub-band in
the distance measure itself would be of tremendous advantage both in terms ofcomputational efficiency and consistency in spectral representation, as it would
circumvent both a return to the speech waveform and a re-analysis with potentiallydifferent conditions.
Indeed, we have proposed a parametric cepstral distance measure (Clermont andMokhtari, 1994) which satisfies the criteria just outlined, and thus affords greater
flexibility in our search for the frequency-band dependence of the vowel-speakerdichotomy. The mathematical derivation of the distance measure begins by replacing the
fixed limits of the integral in Equation 3.1, with the normalised-frequency variables θ 1and θ 2 which represent, respectively, the lower and the upper limit of a selected
frequency band within the entire spectral range [0, π] . As the NDPS itself is expressedin terms of a discrete cosine transform of the quefrency-weighted LP cepstrum:−dφ(e jθ )dθNCC≈ ∑ kc k cos(kθ) ,
k =1

117

(4.2)

(where the approximation tends to an exact equality as NCC → ∞ ), modification of thelimits of integration in Equation 3.1 and substitution of Equation 4.2 yields thefollowing expression:
2

d

2
QCEP

θ 2  NCC1k (c k − c k′ )cos(kθ) dθ .
(c, c ′, θ1 , θ 2 ) =∑∫θ(θ 2 − θ1 ) 1  k =1


(4.3)

Owing to the parametric limits of integration, Parseval’s theorem is rendered generally
inapplicable, and Equation 4.3 is consequently decomposed then integrated, as follows:
2d QCEP(θ1 , θ 2 ) =[]1
A(θ , θ ) + B(θ1 , θ 2 ) ,(θ 2 − θ1 ) 1 2(4.4)

where:

A(θ1 , θ 2 ) = ∫θ2 NCCθ1∑ [k (c − c ′ )cos(kθ)] dθ ,
2

k

(4.5)

k

k =1

 NCC −1 NCC

B(θ1 , θ 2 ) = ∫ 2 ∑ ∑ kl(c k − c k′ )(cl − cl′)cos(kθ) cos(lθ) dθ ,θ1
 k =1 l = k +1

θ2

(4.6)

and where it is understood that the distance is computed between the two cepstral
vectors c and c ′ . The results of carrying out the integrations in Equation 4.5 and 4.6,
which themselves are the additive components of the parametric distance in Equation
4.4, are summarised as follows:
NCC

[

]

A(θ1 , θ 2 ) = ∑ α k k (c k − c k′ ) ,
k =1

2

(4.7)

where:

α k (θ1 , θ 2 ) =(θ 2 − θ1 ) + sin(2kθ 2 ) − sin(2kθ1 ) ,
2

4k

(4.8)

and
NCC −1 NCC

B(θ1 , θ 2 ) = ∑∑ β [kl(c − c ′ )(c − c ′)] ,
kl

k

k

l

l

(4.9)

k =1 l = k +1

where:

β kl (θ1 , θ 2 ) =sin((k − l )θ 2 ) − sin((k − l )θ 1 )(k − l )118+

sin((k + l )θ 2 − sin((k + l )θ1 ))(k + l ).(4.10)Equations 4.4, and 4.7 through 4.10 describe a new, parametric cepstral distancemeasure (PCD) which at once takes advantage of the formant-enhancement property of
the NDPS, and allows selection of any frequency sub-band [θ1 , θ 2 ] within the available
spectral range.
Our formulation of a frequency-band selective cepstral distance has a number ofsalient features. First, it is computationally more efficient than having to re-analyse thespeech waveform for every selection of a new frequency band. Second, it is more
flexible than the traditional filter-bank approach to frequency sub-band analysis, as the
only constraint on allowed values of the parameters which demarcate the selected
spectral region is that they lie within the available spectral range extending to half thesampling frequency, i.e., 0 ≤ θ1 < θ 2 ≤ π . Third, consistency in spectral representationis strictly maintained, as the mean-square difference is computed between the same two,
cepstrally-smoothed NDPS, irrespective of the selected frequency band. This lastproperty contrasts, for example, with the inevitably inconsistent spectral representation
which would result either by re-sampling the speech waveform and re-analysing over a
low-pass or a band-pass frequency range with the same or a different LP order ofanalysis, or by applying selective linear-prediction analysis (Makhoul, 1975) over thedesired frequency band. By contrast with those methods, our parametric cepstral
distance measure simply re-uses the cepstrum, which itself is computed only once,
defined up to half the sampling frequency, and obtained with a fixed LP order ofanalysis M.
In Figure 4.1 we illustrate the PCD as it operates on selected frequency sub-bandsof a pair of speech spectra measured in frames 60msec apart, in the diphthong nucleusof the monosyllabic word “hoy” recorded by speaker D of the FC dataset. The exact
NDPS of those two frames are shown superimposed in the top graph, and reveal
substantial differences in the frequency of the third formant F3 , and in the frequency of
the fifth formant F5 . By contrast, the first, second, and fourth formants appear to have
only smaller differences in their centre-frequencies, but larger differences in theirbandwidths, which are known to be approximately (inversely) correlated with theheights of the formant peaks in the NDPS representation. As the PCD operates on the
cepstrally-smoothed (rather than the exact) spectra, those are shown superimposed in
119

35
30

Exact NDPS (rad/rad)

25
20
15
10
5
0
-5
-10
0

500

1000 1500 2000 2500 3000 3500 4000 4500 5000

0500

1000 1500 2000 2500 3000 3500 4000 4500 5000

Cepstrally smoothed NDPS (rad/rad)

8
6
4
2
0
-2
-4
-6

Magnitude difference in NDPS (rad/rad)

-8

30
Exact
Cepstrally smoothed

25
20
15
10
5
0
0

500

1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency (Hz)Distances between NDPS (rad/rad)

11
10
9
8
7
6
5
4
3
2

Exact
PCD

1
0
0.1-0.7

0.9-1.5 1.6-2.2
2.4-3.0
3.3-3.9
Selected frequency sub-bands (kHz)

4.0-4.6

Figure 4.1: Illustration of parametric cepstral distance measure (PCD), applied to a pair of speech
spectra measured in frames 60msec apart, in the diphthong nucleus of the monosyllabic word “hoy”
recorded by speaker D of the FC dataset. Top graph: the two, exact NDPS, obtained via FFT operations
on the LP autoregressive coefficients. Second graph from top: the two, cepstrally-smoothed NDPS,
obtained by cosine-expansion of the truncated cepstral series (NCC= M= 14). Third graph from top: themagnitude of the differences between the pairs of exact and cepstrally-smoothed NDPS. Bottom graph:profiles of distances computed within selected spectral ranges — Euclidean distance between exact
NDPS (diamond symbols joined by solid lines); PCD (plus symbols joined by dashed lines).

120the second graph from the top. The effect of cepstral truncation to NCC = M = 14
terms is clearly seen to normalise the differences observed in the formant bandwidths,
and to retain the differences where they were observed to occur between the formant
centre-frequencies. This observation is confirmed in the third graph from the top, whichshows superimposed the magnitude of the differences between the exact pair of NDPS
(solid curve), and between the cepstrally-smoothed pair of NDPS (dashed curve) —clearly, the former is more sensitive to the bandwidth-related difference in the range ofthe second formant. Finally, in the bottom graph we show the Euclidean distancescomputed within selected frequency sub-bands between the exact pair of NDPS
(diamond symbols joined by solid lines), and the distances computed using the PCD in
the same, selected frequency sub-bands (plus symbols joined by dashed lines). The large
discrepancy between the two distances computed in the third sub-band ( 1.6 − 2.2 kHz)
clearly illustrates the much lower sensitivity of the PCD to differences in formant
bandwidths. Indeed, the largest distances yielded by the PCD occur in the two
frequency sub-bands which encompass the third formants (in the range 2.4 − 3.0 kHz)and the fifth formants (in the range 4.0 − 4.6 kHz), respectively, which earlier wereobserved to exhibit the largest differences in their centre-frequencies.
The salient features of the PCD illustrated above therefore allow us to confidently
pursue the question of frequency-band dependence of vowel and speaker influences, byappropriate selection of parameter values in our cepstral distance measure used invowel classification experiments. In particular, our approach to scanning the available
spectral range will be to fix the lower spectral limit at zero Hz ( θ 1 = 0 ), and to increase
the upper spectral limit up to the half-sampling frequency ( θ 2 = π ) in steps of 20 Hz. A
complete set of intra- and inter-speaker vowel classification experiments will be
performed at each step, and the accuracy profiles then plotted with the value of θ 2 on
the abscissa. It is indeed the behaviour of classification accuracy as a function of theupper spectral limit with which we are primarily concerned in Section 4.3, as we unfold
the vowel-speaker dichotomy.
4.2.3.2 Classifier based on Hyper-Quadratic Decision Boundaries
In order to gain more confidence in the behaviour of classification accuracy obtained121

using the classifier based on hyper-planar decision boundaries, we also consider a moresophisticated classifier which is founded on the principles of Bayes decision theory.
Both first- and second-order statistics are brought to bear in the computation of eachvowel prototype, which comprises the mean and the covariance of the LP cepstra ofthat vowel in the training data. Assuming N v vowels and S training samples per vowel,
the ith vowel prototype is therefore specified by the mean cepstral vector c i given
earlier in Equation 4.1, and by the following estimate of the NCC × NCC , symmetriccovariance matrix:

1 S
T
C i = ∑ (c il − c i )(c il − c i ) ,S l =1i = {1, 2, ..., N v } ,(4.11)

where ( ) denotes the transpose of the vector or matrix enclosed in the parentheses.
T

The classifier discussed in the previous section makes no a priori assumptions
about the distribution of the data, and uses effectively hyper-planar decision boundaries
determined solely on the basis of the class means. On the other hand, it is well-known
that if each class can be assumed to have a multivariate normal distribution which iscompletely specified by its mean vector and covariance matrix, Bayes decision theory
offers an analytically tractable solution in terms of hyper-quadratic decision functions.
Whilst under those ideal conditions the Bayes classifier is known to be optimal on anaverage basis (Tou and Gonzalez, 1974, p.121), only sub-optimal performance may beachieved in practice, owing to the necessarily finite amount of available training data,
and to the likely departures of the per-class distributions from being strictly Gaussian.
If, however, the vowel-speaker dichotomy first unfolded with the linear classifier isconfirmed using the more powerful, statistical approach, we may then assume withgreater confidence that the observed phenomenon is intrinsic in nature, and not anartefact of the classification methodology.
Although the usual formulation of the Bayes classification rule assigns a test
sample to that class which yields the maximum (logarithmic) probability, negation ofthat probability will re-cast it in terms of a measure of distance which one seeks tominimise. Adopting the latter approach for the sake of consistency with the minimumdistance classification rule used in the previous section, the distance measure used to122compare a given test sample c with the ith vowel prototype ( c i , C i ) is given by thefollowing expression:[]di (c) = 12 (c − c i ) C i−1 (c − c i ) + 12 ln C i − ln( pi ) ,T

where ( ) denotes the matrix inverse,−1(4.12)

denotes the matrix determinant, and pi isthe a priori probability of the ith vowel. As the data-partitioning schemes describedearlier yield equal numbers of samples per vowel in both the training and test sets, the apriori probabilities are therefore assumed to be vowel-independent and equal to thereciprocal of the number of classes, as follows:

pi =

1
,
Nv

i = {1, 2, ..., N v }.

(4.13)

Hence, the last term of the distance measure in Equation 4.12 can be ignored, as it has
no effect on the classification results. We note in passing, that if the covariance matricesof all classes are assumed to be equal (or if a single, pooled covariance matrix is used),
Equation 4.12 reduces to the well-known Mahalanobis distance measure. Moreover, if
the covariance matrices are replaced by an identity matrix, Equation 4.12 reduces to theunweighted, linear decision function described in the previous section.
Thus far we have assumed that the distance measure (Equation 4.12) operates on
the original, quefrency-weighted cepstrum which represents the NDPS over the entire
available spectral range extending to half the sampling frequency. However, as
explained earlier, we would also like to perform vowel classification experiments where
the distance between test and prototype cepstra is computed within selected frequency
sub-bands. Owing to the use of the inverse and determinant of the covariance matrix,
Equation 4.12 cannot easily be reformulated (as we did for the Euclidean distancemeasure in the previous section) to operate on a selected frequency sub-band while stillusing the original cepstra. To overcome this problem, we propose a new formulation of
the cepstrum itself, whereby the original, cepstrally-smoothed NDPS is re-modelledover any selected frequency sub-band, and the unmodified Bayes distance measure canthen be used while retaining consistency in spectral representation.
We begin our mathematical derivation of the partial, quefrency-weighted
cepstrum (P-QCEP) by defining, analogously to Equation 4.2, the cepstrally-smoothed123

negative derivative of the LP phase spectrum X (θ) , which is obtained by the following
cosine expansion of only the first NCC terms of the kc k sequence (QCEP):
NCC

X(θ) = ∑ kc k cos(kθ) .

(4.14)

k =1

Those first NCC, non-zero terms of the QCEP can therefore be retrieved from the
cepstrally-smoothed NDPS, by the following, inverse cosine transformation:
kc k =

1 , k = 0
.
γ =
2
0
k
>
,


γ π
X (θ) cos(kθ)dθ ,π ∫0(4.15)In order to obtain a new cepstral sequence c~k which models the cepstrally-smoothed
NDPS over a selected frequency sub-band [θ1 , θ 2 ] , it is only required to modify thelimits of integration in Equation 4.15 accordingly. Substitution of Equation 4.14 into
the modified, inverse cosine transform then yields the following expression:c~k (θ 1 , θ 2 ) =θ 2 NCCγlc l cos(lθ) cos(kθ ′ )dθ ,(θ 2 − θ1 ) ∫θ1 ∑l =1(4.16)

where γ is defined as in Equation 4.15, and the new, normalised frequency variable θ ′is defined by the following transformation:θ′ =π(θ − θ1 )(θ 2 − θ1 ),(4.17)

which linearly maps the selected frequency interval θ ∈[θ1 , θ 2 ] to θ ′ ∈[0, π] .
Similarly to our decomposition of Equation 4.3 when deriving the parametriccepstral distance measure in the previous section, Equation 4.16 is first decomposed
then integrated, as follows:c~k (θ 1 , θ 2 ) =γA (θ , θ ) + Bk (θ1 , θ 2 ) ,(θ 2 − θ1 ) k 1 2[](4.18)

where:NCC

θ2

Ak (θ1 , θ 2 ) = cos(λθ 1 )∑ lcl ∫ cos(lθ) cos(λθ)dθ ,
l =1

NCC

θ1

θ2

Bk (θ1 , θ 2 ) = sin(λθ1 )∑ lcl ∫ cos(lθ)sin(λθ)dθ ,l =1

and where:124

θ1(4.19)(4.20)

λ=kπ.
(θ 2 − θ1 )(4.21)

Two distinct cases arise in Equations 4.19 and 4.20, depending on whether or not theindex l is equal to λ . When l = λ , the two trigonometric functions in each integral
have the same argument, and the following expressions are obtained:

Ak (θ1 , θ 2 ) =sin(2lθ 2 ) − sin(2lθ1 )

1 NCC
lcl cos(lθ1 )(θ 2 − θ1 ) +,∑
2 l =12l
(4.22)

Bk (θ1 , θ 2 ) = cos(2lθ1 ) − cos(2lθ 2 )1 NCC
lc l sin(lθ 1 ).
∑l
2 l =12


(4.23)

When l ≠ λ , the integral of Equation 4.19 is carried out to yield the following
expression:
NCC
1
Ak (θ1 , θ 2 ) = cos(λθ 1 )∑ lc l x kl (θ 1 , θ 2 ) ,2
l =1(4.24)

where:

x kl (θ1 , θ 2 ) =sin((l − λ )θ 2 ) − sin((l − λ )θ1 )l−λ+sin((l + λ)θ 2 ) − sin((l + λ )θ1 )l+λ, (4.25)

and the integral of Equation 4.20 is carried out to yield the following expression:
NCC
1
Bk (θ1 , θ 2 ) = sin(λθ1 )∑ lcl y kl (θ1 , θ 2 ) ,2
l =1(4.26)

where:

y kl (θ 1 , θ 2 ) =cos((l − λ )θ 2 ) − cos((l − λ )θ1 )l−λ−cos((l + λ )θ 2 ) − cos((l + λ )θ 1 )l+λ. (4.27)

The two sets of Equations 4.22 and 4.23 ( l = λ ), and 4.24 through 4.27 ( l ≠ λ ),respectively, are substituted back into Equation 4.18, and after some algebraic
simplification we arrive at the following expression for the P-QCEP:γ
c~k (θ 1 , θ 2 ) = ∑ lcl ζ kl (θ 1 , θ 2 ) ,2 l =1
NCC

where:

125(4.28)

sin(2lθ 2 − lθ1 ) − sin(lθ1 ),cos(lθ1 ) +
θ
θ
2
l
−
(
)

2
1
ζ kl (θ 1 , θ 2 ) = k−1) sin(lθ 2 ) − sin(lθ1 )(2l(θ 2 − θ1 )
,
2
2

l
θ
−
θ
−
k
π
[
]
(
)
2
1


[

]

l=λ
.

(4.29)l≠λ

It is easy to verify that when θ 1 = 0 and θ 2 = π (i.e., when the entire spectral range isselected), Equations 4.28 and 4.29 simply yield the QCEP without modification.
We note in passing that the derivation of the partial cepstrum (P-CEP), which
represents the cepstrally-smoothed LP log magnitude spectrum (LMS) over a selected
frequency sub-band, would yield the same result as above, except for the quefrencyweight l which would be removed from Equation 4.28. However, as explained
previously, the NDPS offers a formant-enhanced spectral representation which we
therefore adopt consistently in our vowel classification experiments.
The P-QCEP just derived, and summarised in Equations 4.28 and 4.29, captures
the shape of the cepstrally-smoothed NDPS within any selected frequency sub-band.
Most importantly, our formulation of the P-QCEP circumvents re-analysis of the speech
waveform, and merely defines it as a transformation of the cepstrum. Consequently, it
maintains consistency in spectral representation, which other methods (such as selective
LP analysis) do not. This point is aptly illustrated in Figure 4.2, where the P-QCEP iscomputed in two different frequency sub-bands of a single frame of speech (a vocalicsteady-state frame taken from /h ád/ recorded by speaker B), and used to re-computethe partial NDPS (via the discrete cosine transform in Equation 4.14) in those selected
sub-bands. Clearly, the spectral representation is consistent with that of the original
QCEP, and the selected portions of the cepstrally-smoothed NDPS are faithfully
reproduced without excessive computational cost.
A relevant issue which arises in regard to the practical use of the P-QCEP, is thenumber of coefficients to retain. It is particularly important to address the question of
P-QCEP dimensionality in the context of classification experiments using the quadratic
distance measure in Equation 4.12, as the inadvertent inclusion of redundant parameters(i.e., those which do not sufficiently contribute to modelling the spectral shape in a
selected sub-band) will no doubt tend to ill-condition the per-class covariance matrices.
This, in turn, would lead to instability in the computation of the inverse and the
126

40

exact
cepstrally-smoothed

35
30

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25
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original, cepstrally-smoothed
partial

8

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4

NDPS (rad/rad)NDPS (rad/rad)

8

500

2
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original, cepstrally-smoothed
partial

2
0
-2

-4

-4

-6

-6

-8

3500

-8
500

1000
1500
Frequency (Hz)

2000

3500
4000
Frequency (Hz)Figure 4.2: Illustration of partial (quefrency-weighted) cepstrum (P-QCEP), computed within two,
selected spectral regions (lower-left graph: 100Hz to 2150Hz; lower-right graph: 3050Hz to 4350Hz)
for a vocalic steady-state frame taken from / h ád/ recorded by speaker B of the FC dataset. In the top
graph are shown superimposed the exact (dashed-dotted curve) and the cepstrally-smoothed (solid
curve), negative derivative of LP phase spectra (NDPS), obtained with M= NCC= 14. As shown in the
two lower graphs, cosine-expansion (using Equation 4.14) of the P-QCEP derived from the original
cepstrum (using Equations 4.28 and 4.29) faithfully models the cepstrally-smoothed NDPS over therespective, selected frequency sub-bands.

127

determinant of those matrices, which then would yield meaningless or biased
classification results. Naturally, over the entire spectral range [0, π] , one needs no morethan the original NCC coefficients to exactly represent the cepstrally-smoothed NDPS.
However, when only a portion of the entire spectrum is selected, the dimensionality of
the P-QCEP is complicated in at least two ways.
First, as the mean spectral level in a frequency sub-band of the NDPS (or indeedof the LMS) is not necessarily equal to zero, the usual assumption c 0 = 0 does notcarry over to the P-QCEP (nor to the P-CEP). Consequently, the zeroth P-QCEP
coefficient c~ must also be computed in order to retain the relative spectral amplitudes
0

across frames, vowels, and speakers. Whilst this may appear to increase thedimensionality of the P-QCEP compared with the original cepstrum, a second, more
serious complication occurs which tends to counteract that effect.
Indeed, selection of any frequency sub-band will inevitably reduce the number ofspectral peaks and valleys encompassed, and is therefore expected to lead to a
reduction in the number of P-QCEP coefficients required to model that portion of thespectral shape. Of course, the validity of that assumption will depend to a certain extent
on the particular spectral range selected, and will also vary from one speech sound toanother (e.g., the frequency band [500, 1000] Hz may contain two formant peaks for a
mid-back vowel, but only part of a spectral valley for a high-front vowel, thus requiring
perhaps a greater number of coefficients for the former than for the latter). However, it
seems reasonable to assume that higher-order coefficients (beyond the original NCC)
are not required to model a sub-band of the cepstrally-smoothed NDPS which itself isbased on a truncated cepstral series. Ideally, one might hope to methodically reduce thenumber of P-QCEP coefficients retained, for example by selecting the number ofcoefficients in proportion to the range of the selected frequency sub-band.
In order to confirm the general validity of our assumptions, and perhaps to arrive
at a reasonable method of determining the appropriate dimensionality of the P-QCEP,
an empirical investigation was therefore carried out, in which the mean and standard
deviation σ of each P-QCEP coefficient was computed over the entire FC dataset of
vocalic steady-states, as a function of the upper limit θ 2 of the selected spectral range[0, θ 2 ] . Each of the 15 graphs in Figure 4.3 shows the mean and ±1σ error-bar for a
128

10
5

c~0 2

0
-5
-10
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5000

10

c~1

5
0
-5
-10

10

c~2

5
0
-5
-10

Upper limit θ 2 of spectral range [0, θ 2 ] Hz

Figure 4.3: Mean (diamond symbols) and ±1σ (error-bars) in each partial quefrency-weighted
cepstrum (P-QCEP) coefficient from the 0th to the 14th (continued on the next two pages), computed
over all the FC dataset (7 steady-state frames, 5 repetitions, 11 vowels in / hVd/ context recorded by 4
adult, male speakers of Australian English), as a function of the upper limit θ2 of the selected
frequency sub-band [0, θ2], incremented in steps of 20Hz. Also shown in each graph is a vertical(dashed) line which indicates: (i) for the zeroth coefficient, the maximum spectral range, and (ii) for
the 1st through 14th coefficients, the minimum spectral range of acceptable variation in that coefficient,
lest it becomes statistically redundant (as determined by a nominal threshold of 0.4 in the standarddeviation) and must therefore be excluded from the parameter set used in the classifier with quadratic
decision functions (Equation 4.12). As discussed in the text, these graphs suggest a method ofrecruiting progressively higher-order P-QCEP coefficients in proportion to the selected spectral range.
The suggested rule is implemented in our classification experiments by a step-wise linear function in
the number of coefficients, from 2 at the narrowest spectral range, to all 14 coefficients at full spectral
range. N.B.: the zeroth P-QCEP coefficient is weighted by the square-root of two before computing
statistics and performing classification experiments, in order to account for its differential contribution
in a weighted-Euclidean cepstral distance measure (as in the first term of Equation 4.12).

129

10
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c~03
-5
-10
0

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10
5

c~04
-5
-10

10
5

c~05
-5
-10

10
5

c~06
-5
-10

10
5

c~07
-5
-10

10
5

c~08
-5
-10

Upper limit θ 2 of spectral range [0, θ 2 ] Hz

Figure 4.3: (continued from previous page).

130

10

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c~10

5
0
-5
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10

c~11

5
0
-5
-10

10

c~12

5
0
-5
-10

10

c~13

5
0
-5
-10

10

c~14

5
0
-5
-10

Upper limit θ 2 of spectral range [0, θ 2 ] Hz

Figure 4.3: (continued from previous page).

131

single P-QCEP coefficient, obtained in steps of 20 Hz across the available spectral
range. Clearly, our assumptions regarding the statistical redundancy of higher-ordercoefficients in narrower spectral ranges, is confirmed by the smaller standard-deviationsin those parameters for low values of θ 2 . Furthermore, the minimum spectral range
required for the variations in each coefficient to gain statistical significance (asdetermined qualitatively on each graph, in terms of the spectral range at which there
appears a marked increase in σ ) clearly shows a gradual, upward progression as afunction of the P-QCEP coefficient index. A nominal threshold of 0.4 on the minimum
standard-deviation for any given coefficient to be deemed non-redundant, yields anearly linear progression in the number of P-QCEP coefficients, from only 1 coefficientat about 100 Hz, to all 14 coefficients at full spectral range. On the other hand, both themean and standard-deviation of the parameter c~ tend to zero as θ approaches half02

the sampling frequency, indicating that the zeroth coefficient should not be used in thequadratic classifier when the selected frequency band is nearly the full spectral range.
These observations are summarised in Figure 4.4, where the highest allowed index
of P-QCEP coefficients is indicated by a step-wise function of an increasing spectral
range (solid lines); the required exclusion of the zeroth P-QCEP is indicated by thesingle step at 3870Hz, shown in the bottom right corner of the graph. Our empiricalresults therefore confirm the simple but reasonable assumption that on average, thenumber of P-QCEP coefficients retained (compared with the original NCC) be
approximately determined by the ratio of the selected and the entire spectral range. For
example, if the frequency sub-band extending to 2500Hz is selected, then it isreasonable to assume that on average, the first seven P-QCEP coefficients (in additionto c~ ) will suffice to faithfully represent a cepstrally-smoothed NDPS which originally0

is defined over a 5kHz range with 14 cepstral coefficients. Our results also suggest amuch coarser step-size for θ 2 than proposed for the classification experiments
described in the previous section. In order to avoid marked discontinuities in the
behaviour of classification accuracy when an increment in θ 2 is accompanied byrecruitment of a higher-order P-QCEP coefficient, the upper limit of the spectral range
will be increased in steps of 200 Hz when using the quadratic classifier (in contrast withthe step-size of 20 Hz proposed for the linear classifier). The vertical (dashed) lines in
132

14
13
12

Index of P-QCEP coefficient

11
10
9
8
7
6
5
4
3
2
1
0
0

1000

2000
3000
4000
5000
Upper limit of spectral range (Hz)Figure 4.4: Summary of allowed P-QCEP coefficients as a function of upper spectral limit (increased
in steps of 200Hz, as indicated by the dashed, vertical lines), based on the empirical results of P-QCEP
mean and standard deviation as shown in Figure 4.3.

Figure 4.4 span the allowed number of P-QCEP coefficients to be used at each of the
25 steps across the entire spectral range.
Thus far we have discussed the issue of P-QCEP dimensionality only from aspectral representation point of view. However, we must also be wary of a phenomenon
which has been brought to light mainly in the pattern-recognition literature, and which
has come to be known as the “curse of dimensionality” (e.g., Meisel, 1972, p.13). In
essence, a lower limit is imposed on the ratio of the number of training samples perclass (i.e., the number of P-QCEP vectors per vowel) and the dimensionality of theparameter set (i.e., the number of coefficients in each P-QCEP vector). If that ratio isless than unity, the sample covariance matrix will be less than full rank, andcomputation of its inverse will require a method such as Singular Value Decomposition(SVD) to eliminate the inevitable nullity. Even if the ratio is greater than unity and theparameters themselves are reasonably uncorrelated and non-redundant, there is still thedanger of obtaining heavily biased or nonsensical classification results, unless thetraining sample population is sufficiently greater than the intrinsic dimensionality of the133sample vectors. It has been suggested, for example, to observe a minimum ratio of
between 3 and 5 (Foley, 1972; Sarma and Venugopal, 1977). A more conservative
estimate is given by Piper (1992), who suggests a minimum number of training samples
per class an order of magnitude greater than the dimensionality of the parameter set.
In our speaker-independent experiments the number of training samples per vowel
is equal to 105 (7 frames, 5 repetitions, and 3 speakers in each round), which should
perhaps be sufficiently large to avoid the problem of dimensionality (as the ratio is at
worst equal to 105 ÷ 14 = 7.5 ). However, we may expect the problem to be manifest in
our speaker-dependent experiments, where the number of training samples per vowel isreduced to only 28 (7 frames and 4 repetitions in each round, per speaker), and where
the ratio is therefore as low as 2. One way of dealing with the problem is to impose an
upper limit on the number of P-QCEP coefficients presented to the quadratic classifier.
Unfortunately, by reducing the number of coefficients in order to counterbalance theeffect of a small training sample size, we would also compromise on the accuracy ofspectral representation.
Although our inter-speaker experiments may be more immune to the “curse ofdimensionality”, we must also be wary of sub-optimal results caused by a violation of
the assumption of normality. Whilst on a speaker-dependent basis the per-vowel
distribution of the cepstrum or of the P-QCEP coefficients can perhaps be assumed tobe reasonably multivariate Gaussian, this assumption is likely tenuous when the acoustic
parameters of several speakers are pooled. On the other hand, if the number of speakers
is sufficiently large, the per-vowel distribution of their acoustic parameters may tend tobe sufficiently homogeneous that the assumption of normality is upheld. However, ourinter-speaker experiments involve training the classifier on only three speakers’ data andtesting it on the acoustic parameters of the remaining speaker. If the speakers are
sufficiently consistent across repetitions of the same vowel, and sufficiently different
from each other, then the assumption of normality which underpins the quadratic
classifier may be violated, and the yielded classification accuracies therefore suboptimal. It remains to be seen whether any of the potential difficulties outlined above
will tend to undermine the phenomenon of vowel-speaker dichotomy, to which we now
direct our attention.
1344.3 The Dichotomy Unfolded
We shall now use the methodology developed in the previous section, to unfold thevowel-speaker dichotomy by way of vowel classification experiments performed using
the FC dataset. The behaviour of intra- and inter-speaker vowel classification accuracy
as a function of an increasing upper spectral limit, is first observed in Section 4.3.1
using the classifier with hyper-planar decision boundaries (described in Section 4.2.3.1).
The validity of those observations is then tested in Section 4.3.2 using the classifier withhyper-quadratic decision boundaries which, as previously foreshadowed, must also be
addressed in terms of its sensitivity to cepstrum dimensionality in relation to the amountof training data.

4.3.1 Behaviour of Classification Accuracy using Linear Classifier
Vowel classification experiments were first performed using the classifier described in
Section 4.2.3.1, where each vowel prototype is characterised by the mean of thequefrency-weighted cepstral vectors (QCEP) for that vowel in the training data, and theclassifier therefore uses hyper-planar decision boundaries. Our parametric cepstral
distance measure (PCD) was used to perform a complete set of experiments at eachselected frequency sub-band [0, θ 2 ] , and the upper limit θ 2 of the spectral range wasincremented in steps of 20 Hz.
First, a set of speaker-dependent experiments were performed, where one
repetition at a time was left out of the training set and used as test data, and the results
were averaged over those five experiments per speaker. The solid curve in Figure 4.5
shows the behaviour of classification accuracy averaged over all four speakers, as afunction of an increasing upper spectral limit. Classification accuracy rises gradually to
91.6 % as the spectral range is increased to 1600 Hz, beyond which it changes very

little. The best performance of 92.6 % is achieved at a spectral range of 2320 Hz(which encompasses nearly all of the first two formants of our four speakers, as weshall see later in Section 4.4), and the accuracy tapers off to 90.2 % at full range( 5000 Hz). Our intra-speaker classification curve is therefore in complete agreementwith the well-established acoustic-phonetic theory (discussed in Chapter 2) that the first

135

100

Vowel classification accuracy (%)

90
80
70
60
50
40
30

Intra-speaker
Inter-speaker

20
10
0

500

1000

1500 2000 2500 3000 3500 4000 4500 5000
Upper limit of spectral range (Hz)Figure 4.5: Intra- and inter-speaker vowel classification accuracy (solid curve and dashed curve,
respectively) obtained with a linear classifier, as a function of the upper limit θ2 of the spectral range[0,θ2]. Dataset (FC): 14th-order LP cepstra of 7 vocalic steady-state frames in 5 repetitions of 11 vowels
recorded in / hVd/ context by 4 adult, male speakers of Australian English.

two formant peaks provide the most discriminating acoustic cues in the spectra of
spoken vowels for any single speaker.
Next, a set of speaker-independent experiments were performed, where the
cepstra of one speaker at a time were left out of the training set and used as test data.
The dashed curve in Figure 4.5 shows the behaviour of classification accuracy averaged
over all four experiments. Classification accuracy rises to a peak of 74.4 % as the
spectral range is increased to 1680 Hz, then deteriorates as the higher spectral regions
are cumulatively recruited. The downward trend in accuracy appears to modulate
through several falls and smaller rises (the most significant of which occurs at about
3 kHz), finally resting on 66.0 % at full spectral range.

Consistent with our expectations regarding the effects of speaker differences onvowel classification, accuracies are unfailingly higher on an intra- than an inter-speakerbasis. However, a more striking contrast is apparent in the behaviour of the two curves
as a function of the spectral range. By comparison with the inter-speaker curve, the136behaviour of intra-speaker vowel classification accuracy might almost be viewed asnearly asymptotic, the asymptote of a little over 90 % having been reached just beyond
1500 Hz, and only minor changes in performance appearing across the higher spectral

regions. By contrast, the inter-speaker curve has an unmistakable peak at 1680 Hz,
marking the spectral boundary below which lies apparently useful phonetic information,
and above which the consequences of speaker differences appear to dominate. Indeed,
the detrimental influence of inter-speaker variability is perhaps most pronounced in theseries of falls in accuracy from 74.4 % at 1680 Hz, to 66.9 % at 2980 Hz — acumulative drop of 7.5 % across a 1300 Hz spectral region. The small shift in the intraspeaker curve from 91.9 % to 91.2 % across the same spectral region is, by comparison,an order of magnitude less significant.
This contrast in the behaviour of intra- and inter-speaker vowel classification
accuracy does suggest a dichotomous relation between the spectral manifestations ofvowel and speaker influences, which is not at all in disagreement with the extensive but
largely unheeded body of literature reviewed in Chapter 2. On the contrary, ourexperimental results at once confirm the well-established phonetic importance of the
low spectral regions of spoken vowels, and provide compelling evidence in support ofthe speaker potency of the higher spectral regions, particularly the frequency-band
which extends from about 1.6 to 3.0 kHz. The vowel-speaker dichotomy is thus definedin terms of the two, juxtaposed spectral regions which we have identified by way ofvowel classification experiments designed to reveal the consequences of vowel andspeaker influences along the spectral continuum. An intriguing question then arises,
whether these accuracy curves can be interpreted as embodying spectral regions ofprimary phonetic and speaker influence corresponding to particular formant ranges.
However, before we venture to pursue that question (in Section 4.4), we first attempt
to validate our findings using a more sophisticated classifier.

4.3.2 Behaviour of Classification Accuracy using Quadratic Classifier
If the vowel-speaker dichotomy unfolded in the previous section is indeed an intrinsic
phenomenon of speech, one might expect to observe a similar contrast in the behaviourof intra- and inter-speaker vowel classification accuracy obtained using a different137

classifier. To test this hypothesis, we used the more sophisticated, quadratic classifier
described in Section 4.2.3.2, where each vowel prototype is characterised by the mean
vector and covariance matrix of the P-QCEP vectors for that vowel in the training data.
The P-QCEP were computed from the original, 14 th-order LP cepstra, and a complete
set of experiments were performed at each selected frequency sub-band [0, θ 2 ] , where
the upper limit θ 2 of the spectral range was incremented in steps of 200 Hz. The
number of P-QCEP coefficients retained at each selected spectral range, was dictatedby our earlier, empirical findings regarding the statistical redundancy of each coefficient,as summarised in Figure 4.4. Indeed, whilst it may be tempting to select the subset of PQCEP coefficients which yield the highest classification accuracy at each spectral range,
our earlier results provide sufficient statistical evidence to contend and refute theadoption of that method.
The intra- and inter-speaker curves of vowel classification accuracy thus obtained,
are superimposed in Figure 4.6, together with the earlier results yielded by the linear
classifier. The behaviour of vowel classification accuracy averaged over the four
separate, speaker-dependent experiments is shown by the diamond symbols (joined withshort-dashed lines). Although the curve appears to be more noisy (owing to the stepwise recruitment of P-QCEP coefficients) especially at the lower frequency sub-bands,
the overall trend is quite similar to the intra-speaker curve generated earlier with thelinear classifier (solid line). Classification accuracy rises to 81.6 % as the spectral range
is increased to 1600 Hz, and the performance does not significantly improve (nordegrade) as the spectral range is increased further to about 3 kHz.
By contrast, the inter-speaker curve obtained using the quadratic classifier (shownin Figure 4.6 by the plus symbols joined with dash-dotted lines) rises to a peak of
74.9 % as the spectral range is increased to 1600 Hz, and monotonically drops to66.2 % as the spectral range is increased further to 2800 Hz. The overall behaviour inclassification accuracy is quite similar to the inter-speaker curve generated earlier withthe linear classifier (dashed line), up to about 3.5 kHz. Although both curves tend torise a little as the spectral range is increased just beyond 3 kHz, the quadratic classifierappears not to follow the subsequent, downward trend in accuracy exhibited by thelinear classifier as the upper limit is increased to full spectral range. Nevertheless, the
138

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Vowel classification accuracy (%)

90
80
70
60
50
40
30
Intra-speaker (linear classifier)Inter-speaker (linear classifier)Intra-speaker (quadratic classifier)Inter-speaker (quadratic classifier)

20
10
0

500

1000

1500 2000 2500 3000 3500 4000 4500 5000
Upper limit of spectral range (Hz)Figure 4.6: Intra- and inter-speaker vowel classification accuracy obtained with a linear classifier(solid and dashed curves, respectively, replicated from Figure 4.5) and with a quadratic classifier(diamond and plus symbols, respectively), as a function of the upper limit θ2 of the spectral range[0,θ2]. Dataset (FC): 14th-order LP cepstra of 7 vocalic steady-state frames in 5 repetitions of 11 vowels
recorded in / hVd/ context by 4 adult, male speakers of Australian English. The number of P-QCEP
coefficients used in the quadratic classifier at each spectral range, was dictated by the step-wisefunction plotted in Figure 4.4.

accuracy at full spectral range ( 72.4 %) is still lower than the accuracy achieved at the
aforementioned peak ( 74.9 %) with about half the number of P-QCEP coefficients. The
quadratic classifier therefore does confirm the speaker potency of the spectral midrange, defined earlier as approximately [1.6,3.0] kHz.
Notwithstanding this confirmation of the vowel-speaker dichotomy as portrayed in
the overall behaviour of the accuracy curves yielded by the quadratic classifier, there
appears a considerable discrepancy in the levels of accuracy attained by the twoclassifiers, in both the intra- and inter-speaker experiments. Contrary to the naive
expectation that hyper-quadratic decision boundaries should perform no worse thanhyper-planar decision boundaries, the intra-speaker curve yielded by the quadratic
classifier is consistently lower by about 10 % when the spectral range is extended
beyond 1000 Hz. Furthermore, it appears to taper off more rapidly as the spectral range

139

is increased beyond 4000 Hz, and attains only 77.1 % at full spectral range. The levelsof accuracy shown along the inter-speaker curve are also questionable, especially as theupper limit of the spectral range is increased from 1800 Hz to 2800 Hz, where thequadratic classifier performs consistently worse than the linear classifier.
This apparently sub-optimal performance was foreshadowed in Section 4.2.3.2
where, specifically in connection with the quadratic classifier which relies to a greater
extent on the statistical properties of the data at hand, we cautioned against an
excessive number of P-QCEP coefficients compared with the training sample size —thus to avoid the so-called “curse of dimensionality”. In this vein, Foley (1972) has
shown that for a two-class problem with multivariate normal distributions, theperformance of a quadratic classifier only begins to satisfactorily approach that of the
optimum (Bayes) discriminant when the ratio of the number of training samples perclass to the parameter-set dimensionality is greater than or equal to 3. Given a range of
P-QCEP coefficients extending up to 14, and only 28 training samples per vowel (7frames in each of 4 repetitions for the intra-speaker experiments), we may therefore
infer that the sub-optimal accuracies yielded by the quadratic classifier in the intraspeaker experiments are indeed a consequence of the “curse of dimensionality”. On theother hand, we had earlier predicted that the inter-speaker experiments might beimmune to the “curse of dimensionality”, owing to a sufficiently large training sample
size. If the sub-optimal accuracies yielded by the quadratic classifier and shown along
the inter-speaker curve in Figure 4.6 are rather a consequence of an ill-satisfied
assumption of normality, they cannot be expected to improve simply by providing agreater number of training samples indiscriminately.
In order to test our hypotheses without compromising on the accuracy of spectral
representation (i.e., without having to reduce the number of P-QCEP coefficients used
at each selected spectral range), we turn to the more comprehensive version of the FC
dataset which comprises 11 time-normalised frames in each of 9 vowel nuclei (the twovowels /  / and / U / are not included) recorded 5 times in /CVd/ context by the same,
four speakers of Australian English, with the leading consonants C = /h, b, d, g, p, t, k/.
As described in Chapter 3, the centre frames 5, 6, and 7 are used, on the basis that they
are individually the three time-normalised frames which most closely match (in a140cepstral distance sense) the average steady-state frame of the corresponding vowel in/hVd/ context. Although this leads to a reduction in the number of frames per vocalic
nucleus from seven to only three, the seven-fold increase in the number of leading
consonants yields an overall advantage of three times the number of training (and test)
samples over the smaller subset of /hVd/ vocalic steady-state data.
The top graph in Figure 4.7 shows the intra- and inter-speaker profiles of vowel
classification accuracy obtained with the linear and the quadratic classifiers, using all of
the /CVd/ data as outlined above. Our first observation is in regard to the phenomenonof vowel-speaker dichotomy itself, which does not seem to have suffered from theabsence of the two, extreme back vowels, nor from the more relaxed assumptions
regarding the exact location of the steady-state in each vowel nucleus. Despite thosedifferences and the three-fold expansion of the data, the relative behaviour of the intraand inter-speaker curves retains the imprint of vowel-speaker dichotomy which
contrasts the low and the higher spectral regions.
More importantly in connection with our concerns regarding the relative
performance of the two classifiers, the quadratic classifier yields an intra-speaker curve
which is more nearly asymptotic than the one shown earlier in Figure 4.6, and whichclearly equals or outperforms the linear classifier over all the spectral ranges considered.
It would appear therefore, that the previously sub-optimal performance of the quadratic
classifier, which degraded even further as the spectral range was extended and a greaternumber of coefficients recruited, can indeed be explained by an insufficient number oftraining samples per vowel, which has been increased from 28 to 84 in the present,
intra-speaker experiments. By contrast, the relative level of accuracy obtained by thetwo classifiers in the inter-speaker experiments has not dramatically changed incomparison with the earlier, intertwined curves shown in Figure 4.6. This may imply
that the previous training sample size of 105 was perhaps sufficient to avoid the “curseof dimensionality”, and that a different explanation of the sub-optimal performance ismore likely.
In order to further substantiate our conclusions regarding the effect of training
sample size, we explicitly reduce the amount of data by using only the first, third, andfifth repetitions, those being perhaps the most independent group of three repetitions
141

100

(a)

Vowel classification accuracy (%)

90
80
70
60
50
40
Intra-speaker (linear classifier)Inter-speaker (linear classifier)Intra-speaker (quadratic classifier)Inter-speaker (quadratic classifier)

30
20
0

500

1000 1500 2000 2500 3000 3500 4000 4500 5000

100

(b)

Vowel classification accuracy (%)

90
80
70
60
50
40
Intra-speaker (linear classifier)Inter-speaker (linear classifier)Intra-speaker (quadratic classifier)Inter-speaker (quadratic classifier)

30
20
0

500

1000 1500 2000 2500 3000 3500 4000 4500 5000

100

(c)

Vowel classification accuracy (%)

90
80
70
60
50
40
Intra-speaker (linear classifier)Inter-speaker (linear classifier)Intra-speaker (quadratic classifier)Inter-speaker (quadratic classifier)

30
20
0

500

1000 1500 2000 2500 3000 3500 4000 4500 5000
Upper limit of spectral range (Hz)Figure 4.7: Intra- and inter-speaker vowel classification accuracy obtained with a linear classifier(solid and dashed curves, respectively) and with a quadratic classifier (diamond and plus symbols,
respectively), as a function of the upper limit θ2 of the spectral range [0,θ2]. Dataset (FC): 14th-order
LP cepstra of the centre frames 5, 6, and 7 of 11 time-normalised frames in 5 repetitions of 9 vowels
recorded in / CVd/ context by 4 adult, male speakers of Australian English. The number of P-QCEP
coefficients used in the quadratic classifier at each spectral range, was dictated by the step-wisefunction plotted in Figure 4.4. Panel (a): all data used in experiments (number of training samples per
vowel TSPV: intra-speaker = 84; inter-speaker = 315). Panel (b): data reduced to the three repetitions
1, 3, and 5 (TSPV: intra-speaker = 42; inter-speaker = 189). Panel (c): data reduced to the two leading
consonants / h, b/ (TSPV: intra-speaker = 24; inter-speaker = 90).

142

amongst the five. Having thus reduced the number of training samples per vowel from84 to 42 in the intra-speaker experiments, the resulting accuracy curves in Figure 4.7(b)clearly show an overall degradation in the performance of the quadratic classifier
relative to that of the linear classifier. By contrast, their relative performance in theinter-speaker experiments does not appear to have been altered by the reduction in the
training sample size from 315 to 189.
An even further reduction of data is achieved (independently of the previous
reduction in the number of repetitions) by considering the utterances with only the firsttwo leading consonants C = /h, b/. Having thus reduced the per-vowel training sample
size to only 24, the intra-speaker curves shown in Figure 4.7(c) indicate an even morestriking degradation in the performance of the quadratic classifier, which continues thetrend through Figures 4.7(a), (b), and (c), of an increasingly sub-optimal performanceas the training sample size is nearly halved at each step. Our hypothesis is further
supported by the accuracy curves shown in Figure 4.7(c), on the one hand by thegradually degenerating performance of the quadratic classifier as the spectral range isincreased beyond about 3 kHz and the number of P-QCEP coefficients approaches the
maximum of 14, and on the other hand by the generally superior performance of thequadratic classifier over the linear classifier in the very low spectral ranges up to about1200 Hz where the number of P-QCEP coefficients is sufficiently small to avoid the“curse of dimensionality”.
Presumably owing to the smaller amount of consonant-induced variability in theselected centre frames of the vocalic nuclei, the inter-speaker curves shown in Figure
4.7(c) are slightly higher in accuracy compared with the earlier results using all seven
leading consonants; those curves, however, are still intertwined and their relative levelsof accuracy almost unaffected by the further reduction in training size to 90 samples per
vowel. Indeed, even in Figure 4.7(a) where the inter-speaker experiments have recourse
to 315 training samples per vowel, the quadratic classifier performs worse than thelinear classifier over several spectral ranges. In order to test our hypothesis that the suboptimal performance is caused by a weakening assumption of normality when the dataof three, spectrally dissimilar speakers are pooled to train the classifier, we therefore
reverse our method of data-partitioning such that the P-QCEP of only a single speaker
143

100

Vowel classification accuracy (%)

90
80
70
60
50
40
Intra-speaker (linear classifier)Inter-speaker (linear classifier)Intra-speaker (quadratic classifier)Inter-speaker (quadratic classifier)

30
20
0

500

1000

1500 2000 2500 3000 3500
Upper limit of spectral range (Hz)

4000

4500

5000

Figure 4.8: Intra- and inter-speaker vowel classification accuracy obtained with a linear classifier(solid and dashed curves, respectively) and with a quadratic classifier (diamond and plus symbols,
respectively), as a function of the upper limit θ2 of the spectral range [0,θ2]. Dataset (FC): 14th-order
LP cepstra of the centre frames 5, 6, and 7 of 11 time-normalised frames in 5 repetitions of 9 vowels
recorded in / CVd/ context by 4 adult, male speakers of Australian English. The number of P-QCEP
coefficients used in the quadratic classifier at each spectral range was dictated by the step-wise function
plotted in Figure 4.4. The intra-speaker curves are replicated from Figure 4.7(a), where for eachspeaker, one repetition at a time was left out of the training data and used as test data. The interspeaker curves are here generated by training each classifier on one speaker’s data at a time, and using
the remaining three speakers’ data as the test set — thus better satisfying the quadratic classifier’s
underlying assumption of multivariate normality in the training parameter set.

at a time are used for training, the data of the remaining three speakers are used to test
the classifier, and the results are averaged over the four separate experiments.
Notwithstanding the inevitably pessimistic bias in the classification accuracies thus
obtained, if the per-vowel distribution of the training data is indeed more nearly
multivariate Gaussian on a speaker-dependent basis as we suspect, then the quadratic
classifier should consistently perform better than the linear classifier.
In Figure 4.8 are shown the accuracy curves obtained by using the complete set of/CVd/ data in order to secure a sufficient training sample size (for the intra-speaker
experiments in particular) and, in addition, by training the inter-speaker experiments on
only one speaker at a time in order to better satisfy the quadratic classifier’s underlying

144

assumption of multivariate normality. Indeed, having finally met the criteria in regard toboth dimensionality and normality, the quadratic classifier consistently performs noworse than the linear classifier, as expected both intuitively and theoretically.
Furthermore, the phenomenon of vowel-speaker dichotomy is strongly conveyed inboth sets of intra- and inter-speaker curves. The two, intra-speaker curves of vowel
classification accuracy almost reach what might be regarded as their respective,
asymptotic values as the spectral range is extended to a little over 1.5 kHz, and at full
spectral range the linear and quadratic classifiers attain 92.8 % and 94.7 %,respectively. By contrast, the modified method of data-partitioning used with both thelinear and quadratic classifier in the inter-speaker experiments, yields accuracy curves
which attain a peak of 67.7 % at 1680 Hz and 73.4 % at 1800 Hz, respectively. As the
spectral range is increased further to 3000 Hz, classification accuracy drops to only
58.4 % and 59.5 %, respectively, and at full spectral range the two curves attain only
55.7 % and 60.2 %, respectively.

The results presented in this section have not only confirmed the phenomenon ofvowel-speaker dichotomy which we first unfolded in the previous section using thelinear classifier, but have also upheld some inherent limitations of classification methods
which rely on second- or higher-order statistics. In particular, our intra-speaker
experiments have drawn attention to the importance of providing a sufficiently large
number of training samples per class, compared with the dimensionality of each sample.
Indeed, the “curse of dimensionality” was regarded by Rosenberg and Sambur (1975,
p.173) as rendering the use of potentially more powerful, quadratic discriminantfunctions “impractical”, owing to the “sample size required to obtain reliable estimatesof covariance matrices”. On the other hand, our inter-speaker experiments haveunderscored the frailty of embracing larger amounts of training data indiscriminately,
without regard to the underlying assumptions of the classifier. In view of the inherent
difficulties associated with the use of the quadratic classifier as evinced in this section,
we shall retain only its emphatic confirmation of the vowel-speaker dichotomy, andhenceforth revert to the more simple but stable classifier which uses hyper-planar
decision boundaries.

1454.4 The Dichotomy ExplainedThus far we have unfolded the phenomenon of vowel-speaker dichotomy by means ofvowel classification experiments using the LP cepstrum. While those accuracy curves
do provide compelling evidence of the dichotomous nature of vowel-speaker
interactions across the spectral continuum, in this section we seek to explain the
dichotomy in greater detail, and thereby gain a more insightful perspective on thephenomenon.

4.4.1 Spectral Regions of Vowel-Speaker Dichotomy
It is quite clear from our review of the literature (in Chapter 2) that the first two
formant frequencies, and hence the spectral regions which include those two formants,
carry the most important, acoustic information required to discriminate the vowels of
any single speaker. From a vowel classification point of view, one would therefore
expect the accuracies obtained in a speaker-dependent task to be relatively independentof the spectral regions which extend beyond the second formant. Indeed, the nearlyasymptotic behaviour of our intra-speaker accuracy curve (shown earlier in Figure 4.5)
already confirms that implication.
However, a more informative perspective is offered in Figure 4.9 where, adjacentto the F1 F2 plane, are plotted those portions of the intra-speaker accuracy curve which
encompass the F1 and the F2 ranges, respectively, of the four speakers’ formant
distribution. Classification accuracy rises to 65.7 % as the upper limit of the spectral
range is extended across the F1 range, to the lowest F2 (565Hz) of the formant
distribution, and continues to rise gradually to 91.4 % as the spectral range is further
extended to the mid- F2 of the four speakers’ vowel formant distribution (hereafter
defined as the mean F2 of the quasi-neutral vowel in /h¶d/, at 1585Hz). By contrast,
the inclusion of spectral information contained in the regions beyond the mid- F2 , results
in a nearly asymptotic increase of accuracy to 92.1 % at the highest F2 (2391Hz),
followed by a slight decrease to 90.2 % at full range (5000Hz).
As noted earlier in Section 4.3.1, the highest accuracy ( 92.6 %) is achieved at aspectral range of 2320Hz. The close proximity of that frequency value to the upper146

limit of the F2 distribution renders our results in complete agreement with thetraditional notion that the F1 F2 plane provides the best separation of vowels on an
intra-speaker basis. However, our results further suggest that, at least for the data at
hand, the most important spectral regions for intra-speaker vowel discrimination arethose which extend to the mid- F2 of the formant distribution. Indeed, the horizontal(dashed) line in Figure 4.9, which cuts through the centre of the ellipse drawn aroundthe four speakers’ formant distribution for / ¶ / (the mid- F2 defined earlier at 1585Hz),
also intersects the intra-speaker accuracy curve at what might be regarded as its “knee”.
By contrast with the gradual improvement in accuracy as the spectral range is extended
to the mid- F2 , classification performance is relatively independent of the spectral
regions which include those parts of the formant distribution lying above the mid- F2 .
By analogy with Figure 4.9, a more enlightening perspective on the behaviour of
our inter-speaker accuracy curve (first shown in Figure 4.5) is gained by plotting therelevant portions of that curve adjacent to the four speakers’ F2 F3 plane, as shown in
Figure 4.10. Classification accuracy rises to a maximum of 74.4 % as the spectral range
is increased up to 1680Hz, which is only 95Hz higher than the mid- F2 defined earlier.
As shown by the vertical and horizontal (dashed) lines in Figure 4.10, the peak inaccuracy is attained at a spectral range which is yet to embrace the majority of the F2of the four speakers’ front vowels, nor any of their F3 distribution. Indeed, theinclusion of spectral information contained in those regions beyond 1680Hz, results in adrop in accuracy to 70.0 % at the highest F3 (3240Hz), and to 66.0 % at full range(5000Hz). As noted earlier in Section 4.3.1, the detrimental influence of inter-speakervariability is most strongly manifest in the cascade of falls in accuracy from the peak to
only 66.9 % at 2980Hz. Figure 4.10 clearly emphasises the acoustic-phonetic relevanceof that higher spectral region, and shows that the cumulative drop of 7.5 % in accuracy
occurs as the spectral range is extended to include the F2 of the speakers’ front vowels,and nearly their entire F3 distribution.
If the dichotomy observed above is indeed a result of inter-speaker differencescausing confusion in vowel classification, then one might expect a large concentrationof speaker variance in those spectral regions where vowel classification accuracy has
been observed to decrease. The methodology adopted to decompose the vowel and147

F2 (Hz)
2500

/i/

/I/

Upper limit of spectral range (Hz)

/E/
/Q/

2000

/ á /
/«/

1500
/¶/

1000

/a/
//

/U/
//

40

50

60

70

80

90

100

500
200

Vowel classification accuracy (%)

300

400

500

600

700

800

900

F1 (Hz)100

Vowel classification accuracy (%)

Speaker A
Speaker B
Speaker C
Speaker D

90
80
70
60
50
40
200

300

400
500
600
700
Upper limit of spectral range (Hz)

800

900

Figure 4.9: F1F2 vowel space of all 4 male speakers (FC dataset), with a 2σ ellipse drawn aroundeach vowel cluster. Adjacent to the abscissa and ordinate are plotted the portions of the intra-speakeraccuracy curve (from Figure 4.5) which span the F1 and the F2 ranges, respectively. The horizontal(dashed) line cuts through the centre of the ellipse for the quasi-neutral vowel / ¶ /; the mid-F2 thus
defined, also intersects the accuracy curve at what might be regarded as its “knee”.

148

F3 (Hz)
3500
/i/
//
/I/

Upper limit of spectral range (Hz)

/a/

3000

/«/

//

/E/

2500
/Q/
/¶/

2000

/ á /

Speaker A
Speaker B
Speaker C
Speaker D

/U/

35

40

45

50

55

60

65

70

75

80

1500
500

Vowel classification accuracy (%)

1000

1500

2000

80

Vowel classification accuracy (%)

2500

F2 (Hz)

75
70
65
60
55
50
45
40
35
500

1000
1500
2000
Upper limit of spectral range (Hz)2500

Figure 4.10: F2F3 vowel space of all 4 male speakers (FC dataset), with a 2σ ellipse drawn aroundeach vowel cluster. Adjacent to the abscissa and ordinate are plotted the portions of the inter-speakeraccuracy curve (from Figure 4.5) which span the F2 and the F3 ranges, respectively. The vertical andhorizontal (dashed) lines intersect the accuracy curve at its peak (at 1680Hz), and cut across theformant plane in order to emphasise the acoustic-phonetic relevance of the spectral regions of primaryphonetic or speaker influence.

149

16
Vowel variance
Speaker variance
14

12

10

8

6

4

2

0
301-1585
1585-2391 1680-3240(minF1 → mid-F2) (mid-F2 → maxF2) (peak → maxF3)

3108-5000(minF4 → full-range)

Selected spectral range (Hz)Figure 4.11: Profile of speaker variance and vowel variance in the FC dataset, computed using theparametric cepstral distance measure (PCD, see Section 4.2.3.1) in selected frequency sub-bands. The
four sub-bands were selected on the basis of the formant ranges of the 4 speakers’ vowel data which, asshown in Figures 4.9 and 4.10, correspond closely with the prominent features of the accuracy curves.
The first sub-band extends from the lowest F1 to the mid-F2 (which is close to the ‘knee’ of the intraspeaker accuracy curve). The second sub-band extends from the mid-F2 to the highest F2. The third
sub-band extends from the peak in the inter-speaker accuracy curve (which is only 95Hz higher thanthe mid-F2) to the highest F3. The fourth sub-band extends from the lowest F4 to the full spectral range.

speaker contributions to the total variance is similar to that which Pols et al. (1973) andvan Nierop et al. (1973) used in their frequency analysis of Dutch vowels. In particular,
the vowel contribution is the variance in the 11 vowel centroids found by averaging
over the data of all four speakers per vowel, while the speaker contribution is thevariance in the 4 speaker centroids found by averaging over the data of all vowels per
speaker. In our variance computations, however, we use the parametric cepstral
distance measure (PCD, derived in Section 4.2.3.1), which allows greater flexibility
than has hitherto been possible in the selection and analysis of frequency sub-bands.
Figure 4.11 shows the amount of speaker variance (plus symbols joined with solidlines) and vowel variance (diamond symbols joined with dashed lines) computed in eachof four selected frequency sub-bands, which themselves were chosen on the basis of our150

speakers’ vowel formant distribution, and the corresponding, prominent features of theaccuracy curves shown in Figures 4.9 and 4.10. The largest amount of vowel variance isindeed found in the two lowest frequency sub-bands which together span the entire

F1 F2 range. In particular, the spectral region which we earlier found to have the mostsignificant contribution to classification accuracy on an intra-speaker basis (that which
extends to the mid- F2 at 1585Hz), is here shown to have both the largest vowel
variance and the smallest speaker variance. By contrast, vowel variance sharply drops
and speaker variance rises to a maximum in the very spectral region where inter-speakervowel classification accuracy was earlier observed to decrease, from its peak at
1680 Hz to the highest F3 at 3240Hz. In the highest frequency sub-band which extends

from the lowest F4 (3108Hz) to the full range (5000Hz), vowel variance reduces evenfurther to a level which is comparable to speaker variance in the same spectral range.
Whilst our analyses of variance can only provide an overall measure of spectral
dispersion, and therefore cannot be expected to yield a more detailed explanation of theaccuracy curves which themselves were determined by multi-speaker, vowel-to-vowel
interactions, the profiles of spectral variance shown in Figure 4.11 do confirm the vowel
and speaker-specific regions which we first unfolded by way of classification
experiments. Indeed, the variance profiles computed over the broad spectral regions
defined by the formant ranges which were also found to correspond with prominent
features of the accuracy curves, do provide evidence for a high concentration of vowel
variance in the low spectral regions, and for a high concentration of speaker variance
together with a decreasing vowel variance in the higher spectral regions. Subtle
differences in methodology and spectral representation notwithstanding, ourconclusions regarding the spectral regions of primary vowel and speaker influence are inagreement with the spectral variance analyses reported for female speakers of Dutch(van Nierop et al., 1973, Fig.1), and more recently for male speakers of Japanese(Kitamura and Akagi, 1994, Fig.3).

4.4.2 Acoustic-Phonetic Explanation of the Dichotomy
Thus far in our explanation of the vowel-speaker dichotomy, we have considered only
the gross features of the accuracy curves, which we related first to the speakers’ vowel
151formant distribution, then to the relative amount of vowel and speaker variance in broadfrequency sub-bands. In particular, we have observed that vowel-speaker interactions
are more strongly manifest in the spectral regions spanning the high- F2 and the F3 ofthe speakers’ vowel formant distribution. Indeed, Figure 4.10 clearly indicates thatinter-speaker vowel classification accuracy begins to drop as the spectral range is
extended to include the F2 of the speakers’ front vowels and the F3 of their backvowels, the overall effect of which was identified earlier as the phenomenon of
dichotomy. This apparent consistency between certain formant ranges and prominent
features of the classification accuracy curves encourages a more detailed phonetic
analysis of the vowel misclassifications which occur across the higher spectral regions,with a view towards identifying the vocalic subspaces which have caused thedichotomy.
To this end, first recall from our description (in Section 4.2.3) of the datapartitioning method adopted in our vowel classification experiments, that the interspeaker accuracy curve is the mean of four curves, each generated by training theclassifier on the data of three speakers, and testing on the data of the remaining, fourth
speaker. Each of those four, per-speaker accuracy curves can be further decomposed interms of the contribution to accuracy of each vowel tested. We therefore examined eachof those 44 accuracy curves (11 vowels × 4 speakers), in order to determine the vowel
and speaker contributions to the drop in accuracy observed across the higher spectral
regions, and thus providing a more detailed acoustic-phonetic interpretation of themisclassifications which have caused the dichotomy.
The results of this manual decomposition of the inter-speaker accuracy curve areshown in Table 4.1, where the entry in the ith row and jth column lists: (i) the vowel(s)confused with the ith vowel of the jth speaker, (ii) the formant(s) responsible for the
confusion, and (iii) the degree of contribution to the dichotomy as measured by thepercentage drop in accuracy (to the nearest 5 %) in each individual curve. Our aim
being to provide an acoustic-phonetic explanation of the main drop in the inter-speakeraccuracy curve across the higher spectral regions, only those misclassifications which
directly contribute to that overall drop across the mid- F2 and F3 ranges are considered.
The formant(s) deemed responsible for the confusions in each case, are determined by152

Speaker
Vowel
/i/

A

B

C

D

/ , á / (F2&F3, 10%)/ I / (F3, 50%)

/ I / (F2, 15%; F3, 40%)

/ I / (F2, 30%)/I/

/ i / (F2, 5%)/ E / (F3, 10%)/ i / (F2, 10%; F3, 5%)/ E / (F2, 80%)

/ E / (F3, 30%)/E/

/ i / (F2, 100%)/Q// E, U / (F2, 10%)/ E / (F3, 15%)

/a/
/«/

/ « / (F3, 90%)
/ a / (F3, 20%)

//
//

/ a / (F3, 5%)/ a / (F3, 40%)

/ a / (F3, 15%)

/ Q / (F2&F3, 5%)/ U / (F3, 5%)/ U / (F3, 35%)

/ Q,  / (F3, 30%)

/ U / (F3, 5%)/ I, E / (F2&F3, 15%)

/ Q / (F3, 15%)/ ¶ / (F3, 10%)/U// ¶ / (F3, 15%)/ á // ¶ / (F3, 10%)/ , Q / (F3, 20%)/ ¶ / (F3, 15%)/ ¶ / (F2, 10%)/ E / (F3, 20%)

/ i / (F2&F3,10%)/ i / (F3,15%)/¶// á / (F2, 25%)/ á / (F2, F3, 60%)

/ i, « / (F2&F3, 10%)/  / (F3, 60%)
/ E, Q / (F3, 10%)

Table 4.1: Acoustic-phonetic decomposition of the dichotomy in inter-speaker vowel classification
behaviour (Figure 4.5), in terms of the vowel misclassifications that contribute to the drop in accuracy
across the higher spectral regions which encompass the high-F2 and the F3 of the speakers’ vowel
formant distribution. ‘F2&F3’ indicates misclassifications caused by overlapping F2 and F3 ranges.

relating the spectral region of the drop in accuracy, to the speakers’ F1 F2 and F2 F3formant distributions shown, respectively, in Figures 4.9 and 4.10.
Misclassifications are observed to arise in those higher spectral regions, from
confusions: (i) between the F3 of back vowels and the F2 of front vowels, (ii) amongst
the F3 of front vowels, (iii) amongst the F3 of back vowels, and (iv) amongst the F2 offront vowels, in this increasing order of detrimental effects. An example of the firstcategory of misclassifications is the drop of about 15 % for the back vowel /  / ofspeaker D as the upper limit of the spectral range is increased from about 2.1 kHz to
2.3 kHz, owing to confusions with the front vowels / I / and / E /. Referring to the F2 F3formant distribution, we infer that the confusions are most likely caused by the F3 ofthe back vowel overlapping with the F2 of the (other speakers’) front vowels.
Confusions amongst the F3 of front vowels include the drop in accuracy of about153

50 % in / i / of speaker B across the spectral region from about 2.7 kHz to 2.9 kHz,owing to misclassifications with the neighbouring vowel / I /. Examples of confusions
amongst the F3 of back vowels include the drop of 40 % in /  / of speaker B owing tomisclassifications with the vowel / a /, and the drop of 90 % in / a / of speaker D causedby misclassifications with the highly overlapping vowel / « /. In addition to theconfusions between and amongst clearly front or back vowels, are the misclassificationsof the quasi-neutral vowel / ¶ / of speakers B and C, which occur respectively across theF3 range with /  /, and across both the F2 and F3 ranges with / á /.
However, the largest single contribution to the dichotomy occurs across the
spectral region from about 1.8 kHz to 2.4 kHz, where the accuracy curve for the vowel/ E / of speaker A drops by 100 % owing to confusions with / i / in the F2 range of thosefront vowels. Similarly across the same spectral region, classification accuracy of thefront vowel / I / of speaker D drops by 80 % owing to confusions with the neighbouringvowel / E / in the F2 range.
The vowel confusions shown in Table 4.1 and discussed above, are a directconsequence of vowel-speaker interactions in the higher spectral regions, and as such,they embody the specific types of speaker differences that are manifest in our spoken
vowel data. In this context, it is pertinent to recall from our review of the literature (inChapter 2) that Sambur (1975) found the F2 of front vowels and the F3 of the backvowel / u / to be the most speaker-discriminating formant parameters for those spoken
vowels of American English. More recently, Mella’s (1994) study of French spoken
vowels has also shown that F3 of / u / and F2 of front vowels are the best formant
parameters for speaker identification. These findings are further supported by van den
Heuvel et al.’s (1993) study which showed that for the Dutch vowels / i / and / a /, thelargest amount of speaker variability is contained in the spectral regions near the F2 andthe F3 of those vowels, respectively. Whilst these studies were concerned either withspeaker identification or with direct measures of inter-speaker variability on a per-vowel
basis, they support our own conclusions regarding the relative speaker-specificity ofdifferent regions of the acoustic-phonetic vowel space, which we have here determinedby decomposing the vowel-speaker interactions which caused a drop in inter-speakervowel classification accuracy across the higher spectral regions.
154

4.4.3 Dependence on Spectral Representation
Thus far we have unfolded, and provided an acoustic-phonetic explanation of thevowel-speaker dichotomy, by way of vowel classification experiments performed using
either the original, 14th-order LP (quefrency-weighted) cepstrum, or the partial but
equivalent spectral representation afforded by the P-QCEP. In both cases, the cepstrum,and hence the spectral shape of each vocalic frame of speech, is determined by a
combination of all the poles (which include typically six or seven complex-conjugate
pairs) yielded by LP analysis. Amongst those poles are usually found the formants, orresonances of the vocal tract which, owing to their relatively smaller bandwidths,
generally have the strongest influence in shaping the prominent, local peaks in the
spectral envelopes of spoken vowels.
In view of the well-known articulatory relevance of the formants, an importantquestion is therefore whether our methodology of unfolding the vowel-speakerdichotomy can be applied with equal success using so-called simplified cepstra which,as defined earlier (in Section 3.4.2), contain only formant information. This question isof particular relevance to our forthcoming validation of the dichotomy in Sections 4.4.4
and 4.4.5 using the PB and JB datasets, which contain only the centre-frequencies ofthe first three formants of each vowel token. It is of equally vital concern to establish
the dependence of the vowel-speaker dichotomy on the formant parameters, in view ofour forthcoming articulatory explanation in Chapter 6. In this section, we therefore test
the robustness of the dichotomy by performing vowel classification experiments using
simplified cepstra computed from the formants which, as described in Chapter 3, were
carefully measured in each of the seven, vocalic steady-state frames of the recorded/hVd/ monosyllables which make up the FC dataset.
If we are to maintain consistency with the number of acoustic parameters affordedby the PB and JB datasets, then we must restrict ourselves to only the first three,measured formant frequencies, and use fixed formant bandwidths. Rather than use
arbitrary bandwidths, however, we take advantage of our formant measurements and fix
the bandwidths to their respective mean values computed over all eleven vowels andfour speakers ( B1 = 99 Hz, B2 = 128 Hz, and B3 = 218 Hz). Similarly in the following

155

section where the PB dataset will be used to validate the dichotomy, we are fortunate inhaving recourse to Dunn’s (1961) measurements of the mean formant bandwidths
computed over a subset of 20 of the adult male speakers.
With only three formants, the order of the LP polynomial is perforce M = 6 , andthus only the first six, simplified cepstral coefficients are required to fully specify themodel. In view of the analysis conditions used to derive the original LP cepstra( NCC = M = 14 ), an important question then arises regarding the number of simplified
cepstral coefficients required to secure a spectral representation as close as possible tothe original. This question is easily answered by computing the distance between
corresponding pairs of original (14th-order) and simplified cepstra, for a range of valuesof NCC selected in generating the latter. The solid curve (joining the diamond symbols)in Figure 4.12 shows the mean of those distances computed over all 1540 frames of the
FC dataset, using the PCD selected over the full spectral range [0, 5000] Hz. Despite
the much lower LP order of M = 6 , the global minimum of that curve is obtained when
the number of simplified cepstral coefficients is equal to the original NCC = 14 .
As the simplified cepstra thus generated do not contain any poles above thehighest F3 (3240Hz), an inevitable mismatch occurs between the original and thesimplified cepstra in the highest available spectral regions which encompass the fourthand possibly the fifth formants. Distances obtained using the PCD would therefore be
expected to be smaller, if the selected spectral range is restricted to exclude thosehighest regions. Indeed, the dashed curve (joining the plus symbols) in Figure 4.12
shows that when the selected spectral range is limited to extend only to the highest F3 ,smaller mean distances are obtained at values of NCC not far from the global minimum,
while the minimum itself remains at NCC = 14 . It is also interesting to note from thecurves in Figure 4.12, that as the simplified spectral representation is either smoothed orenhanced beyond a certain level (in particular, when NCC ≤ 7 or NCC ≥ 25 ,respectively), the mean distances computed over the narrower spectral range begin toexceed those computed over the full spectral range. This result is to be expected if theexcessive blurring or sharpening, respectively, of the formant peaks in the simplified
spectra can be assumed to yield distances which are, on average, larger than thoseobtained over the mismatched spectral regions lying above the highest F3 .
156

24
Full spectral range [0,5000]HzSpectral range [0,3240]Hz

22
20

Mean PCD (rad/rad)

18
16
14
12
10
8
6
4
2
0
2

4

6

8
10 12 14 16 18 20 22 24
Number of simplified cepstral coefficients NCC

26

28

Figure 4.12: Mean distances computed over the entire FC dataset, between original cepstra(NCC= M= 14) and corresponding, simplified cepstra (M= 6, NCC= 2, 3, ... , 28) generated from the first
three, measured formant frequencies, with bandwidths fixed to mean values (B1 = 99Hz, B2 = 128Hz,
B3 = 218Hz), and sampling frequency Fs = 10kHz. Quefrency-weighted cepstral distances werecomputed using the PCD, first over the full spectral range [0, 5000]Hz (diamond symbols joined withsolid lines), then over the spectral range [0, 3240]Hz which extends only to the highest F3 (plussymbols joined with dashed lines).

In sum, the mean cepstral distances shown in Figure 4.12 strongly suggest that if
the simplified cepstra are generated using the original sampling frequency Fs = 10 kHz,
then the closest spectral representation to the original is obtained by retaining the same
number of simplified cepstral coefficients NCC = 14 , despite the much lower LP orderof M = 6 which inevitably results from using only the first three formants.
Nevertheless, even with NCC = 14 and the PCD computed over the reduced
spectral range, a non-negligible, mean distance of about 4.7 rad/rad is obtained, whichis due to at least four main differences between the original and the simplified spectral
representations. The first and perhaps most obvious difference is that the simplified
cepstra are explicitly cleansed of the influence of spurious poles. Whilst the NDPS
spectral representation is relatively insensitive to real poles (e.g., Yegnanarayana, 1978)
and to complex-conjugate, spurious poles which are usually of much wider bandwidth157

than their true formant counterparts, part of the mean distance noted above may be dueto the existence of narrower-bandwidth spurious poles in the original cepstra, especiallythose which tend to distort the spectral valleys between formant peaks.
The cause of the second type of difference lies in our use of fixed formant
bandwidths in generating the simplified cepstra. Although the degree of mismatch isperhaps lessened by using the means of the measured bandwidths, and although the
NDPS-based cepstral distance measure has been observed to be relatively less sensitive
to bandwidth variations than to differences in formant centre-frequency (Clermont andMokhtari, 1994), the amplitude of each formant peak is known to be inversely
proportional toits bandwidth in the NDPS spectral representation (e.g.,

Yegnanarayana, 1978), and large differences in bandwidth might therefore contribute toat least a portion of the mean distance noted above.
Perhaps a more consequential difference between the original and the simplified
cepstra, is due to the fixed LP analysis conditions used to obtain the former, which inmany cases differed from the analysis conditions finally adopted in measuring theformants themselves (as explained in Chapter 3). It can therefore be expected that theformant peaks in the simplified spectral representation are not always as clearly definedin the original cepstrum. This might indeed contribute a major portion of the mean
distance noted above.
The fourth and perhaps least consequential difference, is caused by the higher
parts of the speakers’ F3 range overlapping with the lower parts of their F4 range.
Indeed, the lowest F4 was measured at 3108Hz, which implies an overlapping spectral
region of width 132Hz. Any F4 peaks which are present in the original spectra might
contribute to larger distances within that spectral range.
Bearing in mind these potential differences in spectral representation, a new set of
intra- and inter-speaker vowel classification experiments were performed as a functionof an increasing spectral range, using the linear classifier and methods of datapartitioning and cepstral distance computation described earlier in Section 4.2.3, butusing the simplified cepstra with NCC = 14 . The resulting accuracy curves arecompared in Figure 4.13 with those obtained earlier using the original cepstra. Although
the intra-speaker curves are quite similar in their overall behaviour, the simplified
158

100
Intra-speaker

Vowel classification accuracy (%)

90
80
70
60

Inter-speaker
50
40
30

Original cepstra
Simplified cepstra

20
10
0

500

1000

1500 2000 2500 3000 3500
Upper limit of spectral range (Hz)

4000

4500

5000

Figure 4.13: Intra- and inter-speaker vowel classification accuracy yielded by a linear classifier, as afunction of the upper limit θ2 of the spectral range [0,θ2], using either the original cepstra (solid
curves) or the simplified cepstra (dashed curves) generated from the first three, measured formant
frequencies, with bandwidths fixed to mean values (B1 = 99Hz, B2 = 128Hz, B3 = 218Hz), sampling
frequency Fs = 10kHz, and NCC= 14. Dataset (FC): LP cepstra of 7 vocalic steady-state frames in 5
repetitions of 11 vowels recorded in / hVd/ context by 4 adult, male speakers of Australian English.
cepstra yield a more nearly-asymptotic curve, which more closely matches our
expectations regarding the well-known tendency of the F3 and higher spectral regions
to provide only minor improvements to vowel discrimination on an intra-speaker basis.
By contrast, the inter-speaker curve yielded by the simplified cepstra isconsistently higher in accuracy than that obtained using the original cepstra. Clearly, theremoval of spurious poles, the more consistent definition of formant peaks, and theeffective normalisation of formant bandwidths in the simplified spectral representation,lead to an improved performance of vowel discrimination on an inter-speaker basis.
Nevertheless, that improvement appears to be manifest across the entire spectral range,
and the phenomenon of vowel-speaker dichotomy which is signalled by the drop ininter-speaker vowel classification accuracy across the high- F2 and F3 spectral regions,is therefore retained. Indeed, classification accuracy rises to a peak of 81.9 % as the
spectral range is extended to 1860Hz, and subsequently drops to 74.7 % at 2980Hz.

159

Speaker
Vowel
/i/

A

/I/

/ i / (F2, 30%)

B

C

D

/ I / (F3, 20%)

/ I / (F2, 55%;F3, 15%)

/ I / (F2, 10%)/ E / (F3, 25%)

/ i / (F2, 5%)/ E / (F2, 15%)/ á / (F2&F3, 5%)/E// i / (F2, 100%)

/Q/
/a/

/ « / (F3, 100%)/«// a / (F3, 5%)/ a / (F3, 10%)//

/ a / (F3, 5%)/ a / (F3, 5%)/ a / (F3, 20%)

/ U / (F3, 70%)/ « / (F3, 5%)/ U / (F3, 45%)

/ Q / (F2&F3, 5%)

//
/U/

/ E / (F2&F3, 5%)/  / (F3, 45%)/  / (F3, 55%)/  / (F3, 20%)/ á // I / (F3, 25%)/ ¶ / (F2, 15%)/ ¶ / (F3, 15%)/¶// á / (F2, 5%)/ á / (F2, F3, 15%)

/ Q / (F3, 5%)

Table 4.2: Acoustic-phonetic decomposition of the dichotomy in inter-speaker vowel classification
behaviour for the FC dataset using simplified cepstra derived from the first three formant frequencies
with bandwidths fixed to mean values (B1 = 99Hz, B2 = 128Hz, B3 = 218Hz), in terms of the vowel
misclassifications that contribute to the drop in accuracy across the higher spectral regions which
encompass the high-F2 and the F3 of the speakers’ vowel formant distribution. ‘F2&F3’ indicatesmisclassifications caused by overlapping F2 and F3 ranges.

Although the frequency location of the peak in accuracy is nearly 200Hz higher than thepeak obtained using the original cepstra, a comparison with the four speakers’ F1 F2and F2 F3 vowel formant distributions shown earlier in Figures 4.9 and 4.10 reveals thatthe turning point in the accuracy curve still occurs as the spectral range is extended toinclude the F2 of front vowels and the F3 of back vowels.
In view of the general similarities observed in the behaviour of vowel classification
accuracy across the available spectral range using either simplified or original cepstra,
the question then arises whether the drop in the inter-speaker accuracy curve across thehigher spectral regions is caused by speaker-induced vowel confusions similar to thosewhich we listed earlier in Table 4.1. The results of our vowel-speaker decomposition of
the inter-speaker curve yielded by the simplified cepstra, shown in Table 4.2, indeed
confirm that the majority of so-called dichotomous vowel confusions which occur

160across those higher spectral regions, are caused by the same combinations of formants,
vowels and speakers found earlier. In particular, the two most significant contributions
to the dichotomy still occur, respectively, across the F2 range as the vowel / E / ofspeaker A is confused with / i /, and across the F3 range as the vowel / a / of speaker D
is confused with / « /.
Despite those and other similarities, the magnitude of the percentage drops inaccuracy shown in the entries of Table 4.2 are often different to those listed earlier inTable 4.1, and certain vowel misclassifications appear to replace others listed earlier.
For example, there are three relatively large drops in accuracy which involve the F3
range of the vowel / U /, none of which were observed earlier using the original cepstra:a drop of about 70% occurs as /  / of speaker D is confused with / U /; a drop of 55%
occurs as / U / of speaker B is confused with /  /; and a drop of 45% occurs as / U / ofspeaker A is confused with /  /. In the high front vowels, a redistribution of the drops inaccuracy caused by misclassifications of / i / as / I / occurs, which deemphasises thecontribution of the F3 range of speaker B, and emphasises instead the F2 range of
speaker C. Also, misclassifications of the quasi-neutral vowel / ¶ / of those two speakers
either no longer occur, or are much reduced in effect.
These examples serve to illustrate the inaccuracy of the naive assumption, that theLP cepstrum measured with fixed analysis conditions in the steady-state of vocalic
nuclei will clearly and consistently reflect the formant structure of those speech sounds.
Even the formant-enhancement property of the NDPS cannot entirely compensate forthe slight aberrations in the frequency-location of formant peaks, their bandwidths, andthe largely unknown and undocumented effects of spurious poles yielded by the usual
methods of LP analysis. As a result, our detailed decomposition of the individual vowel
confusions contributing to the dichotomy has revealed certain differences comparedwith the dichotomous confusions obtained using the original cepstra, and thosedifferences can be attributed directly to the types of differences in spectral
representation discussed earlier. Indeed, one might even expect that the dichotomous
misclassifications which occur using simplified cepstra, are either more readily, or more
directly related to the speakers’ formant ranges, and that a more accurate, acousticphonetic explanation of the dichotomy will therefore follow from the simplified spectral161

representation.
In this light, it is encouraging to note that the vowel-speaker decomposition of the
dichotomy shown in Table 4.2 confirms our previous results which implicate confusions
primarily amongst the F2 of front vowels, amongst the F3 of back vowels, and amongst
the F3 of front vowels. It is indeed this broad acoustic-phonetic explanation of the
dichotomy which is to be of long-lasting value in our interpretation of vowel-speaker
interactions, and which now permits us to consider using simplified cepstra to further
explore the phenomenon of dichotomy with the PB and JB datasets.

4.4.4 Dependence on Speaker Homogeneity
The clearly dichotomous, inter-speaker accuracy curves presented thus far, have been
generated using a population of speakers who differ quite significantly in the spectral
properties of their vowels. An intriguing question then arises, whether vowel-speaker
interactions will yield a similar dichotomous behaviour in inter-speaker classification ofvowels spoken by a more densely populated, and perhaps more homogeneous group of
speakers. To answer this question, we now turn to the PB dataset which, as describedin Chapter 3, comprises two repetitions of ten vowels spoken in /hVd/ context by 32
adult, male speakers of American English. Although very little is known about the
dialectal composition of those speakers, we have argued (in Chapter 3) that thehomogeneity of the group as a whole is probably enhanced by the sheer number of
speakers. If that assumption is indeed correct, then the PB dataset offers the possibilityof assessing the dependence of the vowel-speaker dichotomy on the degree of speakerhomogeneity.
In order to perform vowel classification experiments using the methodology
presented earlier, we first need to convert the available formant data into cepstra. In thiscontext, it is encouraging to recall our results of the previous section which confirmedthat the dichotomous behaviour of inter-speaker vowel classification accuracy is indeed
preserved when the spectral representation is simplified to contain only formant
information. We therefore proceed to generate simplified cepstra from the first threeformant frequencies of the PB dataset, using the LP method described in Section 3.4.2.
Analogously to our methodology used in the previous section, the formant bandwidths
162

100
Speaker Homogeneity

Vowel classification accuracy (%)

90
80
70
60

Speaker Heterogeneity

50
40
8 sheepiest speakers
26 sheepiest speakers
all 32 speakers
24 goatiest speakers
6 goatiest speakers

30
20
0

500

1000

1500 2000 2500 3000 3500
Upper limit of spectral range (Hz)

4000

4500

5000

Figure 4.14: Trend towards a more clear dichotomy or a more nearly asymptotic behaviour in interspeaker vowel classification accuracy, as a function of an increasing or a decreasing proportion,
respectively, of “goat” to “sheep” speakers. Dataset (PB): 2 repetitions of 10 vowels spoken by 32 adult,
male speakers of American English; simplified cepstra were generated from the first three formant
frequencies, with sampling frequency Fs = 10kHz, NCC= 14, and formant bandwidths fixed to meanvalues (B1 = 50Hz, B2 = 65Hz, and B3 = 114Hz) as measured by Dunn (1961).
are fixed to their respective mean values, as measured by Dunn (1961) for a subset of
20 of the adult male speakers ( B1 = 50 Hz, B2 = 64 Hz, and B3 = 115 Hz). Our resultsof the previous section further suggest that NCC = 14 simplified cepstral coefficientsare required with the sampling frequency set to Fs = 10 kHz, despite the lower LPmodel order of M = 6 .
The solid curve in Figure 4.14, representing inter-speaker vowel classification
accuracy as a function of upper spectral limit, exhibits a maximum of 85.2 % byincluding spectral information up to 1900Hz, then drops to 83.1 % for the spectral
range extending to 2300Hz. Although this small dip is reminiscent of our earlier interspeaker curves obtained for the FC dataset, the drop of only 1.3 % from the peak
(1900Hz) to full range (5000Hz) suggests that a potentially dichotomous behaviour has
been blurred as a result of averaging across the individual accuracy curves of all 32
speakers.

163

If the spectral dichotomy observed earlier in Figure 4.13 is indeed an intrinsic
phenomenon, then one could speculate that the blurring effect illustrated by the solid
curve in Figure 4.14 has occurred as a result of an imbalance towards a greater degreeof homogeneity amongst the speakers, and that any significant amount of speakerdifferences which have caused the small dip in accuracy in the higher spectral regions isdue to only a small subset of the 32 speakers, the “goats” as might be referred to. The
more clearly dichotomous, accuracy curves obtained earlier for our four very dissimilar
speakers of Australian English may therefore be used as typical evidence for identifying
the so-called “goat” speakers amongst the 32. Indeed, our “leave-one-speaker-out”
approach to data-partitioning allows us to identify those individual classification curves
which show strong evidence of a spectral dichotomy, and which therefore would
characterise the “goat” speakers. By contrast, the more nearly asymptotic individual
accuracy curves would identify the spectrally more homogeneous, “sheep” speakers.
This criterion can be quantified in terms of a measure of the degree of dichotomy,
by computing the difference between the highest accuracy in each individualclassification curve, and the accuracy attained at full spectral range. The score thus
obtained is equal to zero for nearly-asymptotic curves where the maximum accuracy isachieved at full-range. By contrast, a more dichotomous classification curve with aclearly defined peak followed by a drop in accuracy, would yield a higher score, up to a
maximum of 100%. The scores thus obtained for each speaker are listed in the second
column of Table 4.3, and the individual accuracy curves of the two speakers with theequal highest score of 20% (speakers 16 and 20) are shown superimposed with theoverall accuracy curve in Figure 4.15(a).
Our examination of the 32 speakers’ individual accuracy curves has revealed thatin a large number of cases, the accuracy peaks then drops, but subsequently rises again
as the spectral range is extended further to 5000Hz. A mild example is the individual
curve of speaker 16 shown in Figure 4.15(a); more emphatic examples are the accuracy
curves of speakers 1 and 22, shown superimposed with the overall accuracy curve in
Figure 4.15(b). In those cases, the measure of dichotomy defined earlier does not
capture the significant drop in accuracy which occurs across a frequency sub-band
between the peak and full-range. It is therefore necessary to compute the extent of that164

Speaker
16
20
26
9
5
23
12
15
8
22
25
28
33
19
29
1
11
14
18
27
32
3
6
13
7
17
24
30
4
10
21
31

max – full range
(%)
20.0
20.0
15.0
10.0
10.0
10.0
10.0
0.0
0.0
0.0
5.0
10.0
10.0
5.0
5.0
0.0
5.0
0.0
0.0
0.0
0.0
5.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0

largest drop
(%)
25.0
15.0
15.0
15.0
10.0
5.0
5.0
0.0
5.0
10.0
5.0
10.0
10.0
10.0
10.0
10.0
5.0
5.0
0.0
10.0
10.0
5.0
0.0
5.0
0.0
0.0
0.0
0.0
5.0
5.0
0.0
0.0

error at full range
(%)
45.0
30.0
30.0
25.0
25.0
30.0
25.0
40.0
30.0
20.0
20.0
10.0
10.0
10.0
10.0
10.0
10.0
15.0
20.0
10.0
10.0
5.0
15.0
10.0
10.0
10.0
10.0
10.0
0.0
0.0
5.0
5.0

Mean
(%)
30.0
21.7
20.0
16.7
15.0
15.0
13.3
13.3
11.7
10.0
10.0
10.0
10.0
8.3
8.3
6.7
6.7
6.7
6.7
6.7
6.7
5.0
5.0
5.0
3.3
3.3
3.3
3.3
1.7
1.7
1.7
1.7

Table 4.3: Rank-ordered list of the 32 adult, male speakers of the PB dataset (speakers numbered
1,3,4,5,...,33 in Column 1) in terms of the following three attributes of their individual classification
curves: Column 2: the difference between the maximum accuracy and that which is attained at full
spectral range (5000Hz); Column 3: the largest drop in accuracy across the higher spectral regions;
Column 4: the classification error at full spectral range. The rank-ordering is based on the mean ofthose three scores, as listed in Column 5 (speakers who have the same mean score are listed according
to their speaker number). Highlighted in bold-font at the top of the table are the 6 “goatiest” speakers
with the highest mean scores (15% or greater); highlighted in bold-font at the bottom of the table are
the 8 “sheepiest” speakers with the lowest mean scores (less than 5%).

largest, continuous drop in accuracy, and thus to take into account the potency of thehigher spectral regions without being bound by the value at full-range. Those scores arelisted for each speaker in the third column of Table 4.3.
Our selection of “goat” and “sheep” speakers would not be complete without also

165

100

(a)

Vowel classification accuracy (%)

90
80
70
60
50
40
30

all 32 speakers
speaker 16
speaker 20

20
10
0

500

1000 1500 2000 2500 3000 3500 4000 4500 5000

100

(b)

Vowel classification accuracy (%)

90
80
70
60
50
40
30

all 32 speakers
speaker 1
speaker 22

20
10
0

500

1000 1500 2000 2500 3000 3500 4000 4500 5000

100

(c)

Vowel classification accuracy (%)

90
80
70
60
50
40
30

all 32 speakers
speaker 4
speaker 8
speaker 15

20
10
0

500

1000 1500 2000 2500 3000 3500 4000 4500 5000
Upper limit of spectral range (Hz)Figure 4.15: A few individual speakers’ accuracy curves superimposed with the overall inter-speakercurve obtained using all 32 speakers of the PB dataset. Panel (a): the two most clearly dichotomous,
individual accuracy curves (“goat” speakers 16 and 20). Panel (b): a drop in accuracy underscores thepotency of the higher spectral regions, despite a further rise in accuracy towards full spectral range(speakers 1 and 22). Panel (c): relatively low accuracies are indicative of outliers (speakers 8 and 15)
notwithstanding the absence of a spectral dichotomy; by contrast, a nearly asymptotic curve which
attains 100% at full spectral range is indicative of an extreme “sheep” (speaker 4).

166considering what might be regarded as a more conventional definition of the former as
outliers in the group, and of the latter as representative or typical speakers in the group.
If the measure of dissimilarity which is implicit in this definition can be taken in the
broadest sense to operate over the entire available spectral range, then it can bequantified in terms of the classification error at full-range (i.e., the difference between
100% and the classification accuracy at 5000Hz). Those errors are listed for eachspeaker’s individual accuracy curve in the fourth column of Table 4.3; the greater theerror (or the lower the accuracy at full-range), the more “outlier” (hence the “goatier”)
the speaker. We have already observed two examples of this type of contrast in Figures4.15(a) and (b), where the lower accuracy at full-range for speakers 16 and 22,
respectively, indicate that they are the more “outlier” of each pair of speakers.
However, two extreme cases are the individual curves of speakers 8 and 15 shownsuperimposed in Figure 4.15(c): although they do not show any evidence of a spectral
dichotomy (as might be signalled by a drop in accuracy over the higher spectral
regions), the relatively low accuracies observed both across the entire spectral range
and at 5000Hz suggest that those two speakers are “outliers” in the group. By contrast,
the individual accuracy curve of speaker 4 (shown in the same Figure) is amongst theleast dichotomous, and attains 100% at full-range.
To summarise, we propose three measures to quantify the degree of “goatiness”
or “sheepiness” of each speaker in terms of the following attributes of their individualclassification curves: (i) the difference between the maximum accuracy and that which isattained at full spectral range, (ii) the largest drop in accuracy across the higher spectral
regions, and (iii) the classification error at full-range. Linear coefficients of correlation
computed between those scores listed in Table 4.3 in columns 2 and 3 ( r = 0.75 ),columns 2 and 4 ( r = 0.58 ), and columns 3 and 4 ( r = 0.44 ), indicate that they eachwould contribute non-redundantly to a combined measure of “goatiness”. Although in
general an optimally weighted combination of those scores (each of which can take on avalue between 0% and 100%) may be found to be superior, we have found that theirmean (listed in the fifth column of Table 4.3) will suffice to rank-order the 32 speakers
in terms of their quantified degree of “goatiness” (higher mean score) or “sheepiness”(lower mean score).
167

Indeed, the list of speakers in Table 4.3 is sorted in decreasing order of the final,
mean score. That rank-ordered list is then used to categorise the speakers (as indicatedby the broken lines in Table 4.3) into the following, three groups: (i) the 6 “goatiest”
speakers (with the highest mean scores, and the most clearly dichotomous, individual
accuracy curves), (ii) the 8 “sheepiest” speakers (with the lowest mean scores, and themost nearly asymptotic, individual accuracy curves), and (iii) the remaining 18 speakers
who are neither extreme “goats” nor extreme “sheep”.
In order to assess the influence of speaker homogeneity on the phenomenon ofvowel-speaker dichotomy, we performed a new set of inter-speaker vowel classification
experiments using the generated, simplified cepstra of the speakers taken from various
combinations of those three groups. First, the 8 “sheepiest” speakers (listed at thebottom of Table 4.3) are treated as a separate group, and their overall curve ofclassification accuracy is shown by the triple-dashed line in Figure 4.14. Next, the 18
speakers listed in the middle of Table 4.3 are added to the group, and the resultingclassification curve for the 26 “sheepiest” speakers is shown by the long-dashed line in
Figure 4.14. Those 18 speakers are then pooled with the 6 speakers listed at the top of
Table 4.3, and the resulting accuracy curve for the 24 “goatiest” speakers is shown by
the short-dashed line in Figure 4.14. Finally, the 6 “goatiest” speakers are treated as aseparate group, and their overall classification curve is shown by the dash-dotted line in
Figure 4.14.
Those curves of inter-speaker classification accuracy shown superimposed withthe original, solid curve in Figure 4.14, clearly show a trend towards higher accuracies
and a more nearly asymptotic behaviour as the population is made to comprise a higher
proportion of “sheep” speakers; and a trend towards a clearer dichotomy as the
proportion of “goat” speakers is increased. This striking progression in the behaviour of
the accuracy curves can therefore be taken to suggest that even amongst themselves,the “sheepiest” speakers are spectrally a more homogeneous group, and the “goatiest”
speakers are spectrally a more heterogeneous group. Before assessing the validity ofthat assumption more directly, we seek to gain a more informative perspective on the
acoustic-phonetic relevance of the spectral regions over which inter-speaker vowel
classification accuracy is observed to either rise or fall. In particular, the relevant
168portions of the two accuracy curves obtained, respectively, for the 6 “goatiest” and the
8 “sheepiest” speakers, are replotted in Figure 4.16 adjacent to the F1 F2 vowel formant
space of the two groups of speakers, and in Figure 4.17 adjacent to their F2 F3 vowel
space.
As shown in Figure 4.16, the two classification curves rise in accuracy as theupper limit of the spectral range is increased across the speakers’ entire F1 distribution.
The two curves then begin to diverge as the spectral range is extended beyond about
1kHz — the curve for the 6 “goat” speakers dips slightly then rises to a peak of 79.2 %at 1480Hz, while the curve for the 8 “sheep” speakers continues to rise to 86.3 % at thesame range. However, the most striking difference in the behaviour of the two curves
occurs as the spectral range is extended beyond 1480Hz to include the F2 of the
speakers’ front vowels, whereupon the nearly asymptotic behaviour of the “sheep”
speakers’ accuracy curve contrasts with a marked drop in the “goat” speakers’ accuracy
curve. As shown in Figure 4.17, the clearly detrimental influence of speaker differencesin the higher spectral regions continues to elicit a downward trend in the accuracy curveof the “goat” speakers across their entire F3 range, to only 65.8 % at 3500Hz. By
contrast, the accuracies obtained for the “sheep” speakers change very little over the
spectral ranges which encompass their high- F2 and F3 , attaining 88.8 % at 3500Hz.
(As shown previously in Figure 4.14, the accuracy displayed on each curve at full
spectral range is the same as that which is attained at 3500Hz, thus confirming that thesimplified spectral representation contains no formants higher than the F3 .)The sharp contrast in the behaviour of the accuracy curves for the “goat” and the“sheep” speakers does suggest, as noted earlier, that those two groups of speakers
should differ markedly in their degree of spectral homogeneity. In so far as thesimplified cepstrum carries, by definition, spectral information pertaining only to theformants, it is reasonable to expect a reflection of those differences in homogeneity, inthe speakers’ vowel formant space. Indeed, the F1 F2 and F2 F3 vowel formant
distributions shown in Figures 4.16 and 4.17 both for the 6 “goatiest” speakers (solid
ellipses) and for the 8 “sheepiest” speakers (dashed ellipses) would seem to offer amore direct interpretation of the acoustic-phonetic implications of the presumed
heterogeneity and homogeneity, respectively, of those two groups of speakers. Clearly,
169

F2 (Hz)
2500

/i/

/I/

Upper limit of spectral range (Hz)

/E/

/¼/

1500

1000

40

50

60

70

80

90

/«/

/u/

/A/

/U/

500
100

/Q/

2000

200

Vowel classification accuracy (%)

300

400

//

500

600

700

900

F1 (Hz)100

Vowel classification accuracy (%)

800

90

6 ’goat’ speakers
8 ’sheep’ speakers

80
70
60
50
40
200

300

400
500
600
700
Upper limit of spectral range (Hz)

800

900

Figure 4.16: F1F2 vowel space of the 6 “goatiest” and the 8 “sheepiest” male speakers (PB dataset),with a 2σ ellipse drawn around each vowel cluster (solid and dashed lines, respectively). Adjacent tothe abscissa and ordinate are plotted the portions of the inter-speaker accuracy curves (from Figure4.14) which span the F1 and the F2 ranges, respectively. The horizontal (heavy dashed) line cuts
through the formant plane at 1480Hz, which corresponds to the peak of the clearly dichotomous
accuracy curve obtained for the “goat” speakers.

170

F3 (Hz)
3500

Upper limit of spectral range (Hz)

/i/

3000

//

/A/
/I/
/«/

2500
/E/
/Q/

2000
/u/
/U/
/¼/

40

50

60

70

80

90

100

1500
500

Vowel classification accuracy (%)

1000

1500

2000

100

Vowel classification accuracy (%)

2500

F2 (Hz)

90
80
70
60
6 ’goat’ speakers
8 ’sheep’ speakers

50
40
500

1000
1500
2000
Upper limit of spectral range (Hz)2500

Figure 4.17: F2F3 vowel space of the 6 “goatiest” and the 8 “sheepiest” male speakers (PB dataset),with a 2σ ellipse drawn around each vowel cluster (solid and dashed lines, respectively). Adjacent tothe abscissa and ordinate are plotted the portions of the inter-speaker accuracy curves (from Figure4.14) which span the F2 and the F3 ranges, respectively. The vertical and horizontal (heavy dashed)lines intersect the clearly dichotomous accuracy curve at its peak (at 1480Hz), and by cutting acrossthe formant plane they emphasise the acoustic-phonetic relevance of the spectral regions of primaryphonetic or speaker influence.

171

Mean (Hz)

800
700
600
500
400
300
200

S.D. (Hz)

250
200
150
100
50
0

F1

/i/

/I/

/E/

/Q/

/«/

/A/

//

/U/

/u/

/¼/

/i/

/I/

/E/

/Q/

/«/

/A/

//

/U/

/u/

/¼/

/i/

/I/

/E/

/Q/

/«/

/A/

//

/U/

/u/

/¼/

Mean (Hz)

2500
2000
1500
1000
500

S.D. (Hz)

F2

250
200
150
100
50
0

Mean (Hz)

3500
3000
2500
2000
1500

S.D. (Hz)

F3

250
200
150
100
50
0

Figure 4.18: Mean (upper graph) and standard deviation (lower graph) in the first three formant
frequencies F1, F2, and F3 for each vowel, computed for the 6 “goatiest” speakers (diamond symbols
joined by solid lines) and for the 8 “sheepiest” speakers (plus symbols joined by dashed lines) of the PB
dataset, as determined by the rank-ordered list in Table 4.3.

172

the two groups differ not only in the orientation of their per-vowel ellipses, but also inthe size of those ellipses — as might be expected of a more homogeneous group, the“sheep” speakers are characterised by a generally smaller dispersion of formant values
per vowel. Perhaps the two most noticeable differences in the orientation of the ellipses
are for the mid-front vowels / I / and / E / which, for the “sheep” speakers, have a muchnarrower and well-separated distribution along the F2 dimension; by contrast, the“goat” speakers appear to have a much greater degree of overlap amongst their front
vowels along the F2 dimension, and in particular in the F2 F3 plane.
Differences in the formant distributions of the two groups of speakers can befurther quantified by a comparison of their per-vowel mean and standard deviation ineach of the first three formant frequencies. As shown in Figure 4.18, the profiles of pervowel mean formant frequency are nearly indistinguishable across the two groups of
speakers; the most consistent trend appears to be the slightly lower mean F2 (by 45Hz
on average) of the “goat” speakers, in all of the vowels except / ¼ /. By contrast, theprofiles of standard deviation shown in Figure 4.18 do indicate substantial differences,which confirm our earlier observations regarding the “sheep” speakers’ relatively
smaller vowel ellipses. While the standard deviations in F1 are comparably low for bothgroups of speakers, the “goat” speakers have a consistently larger standard deviation inF2 across all the vowels (by 60Hz on average), and a consistently larger standard
deviation in F3 of all the back vowels (by 33Hz on average).
Our measurements of formant dispersion are thus consistent with the notion ofinter-speaker variability causing the drop in vowel classification accuracy across thehigher spectral regions, which clearly exhibit a more potent manifestation of vowelspeaker interactions for the “goat” speakers. Even greater insights are gained bydecomposing the vowel and “goat” speaker contributions to that drop in accuracy, theresults of which are listed in Table 4.4. For example, the high front vowel / i / of speaker
16 (who was ranked the “goatiest” speaker in Table 4.3, and whose individual accuracy
curve was shown earlier in Figure 4.15(a) to be the most clearly dichotomous in thecontext of all 32 speakers) is confused with / I / in the F3 range, and the front vowel/ Q / of speaker 20 (ranked second in Table 4.3) is confused in the F2 range with the F3of the back vowel / « /. However, Table 4.4 reveals that the main contributions to the173

Speaker
Vowel
/i/

5

9

16

20

26

/ E / (F2)

/ i / (F2)

/ I / (F3)/I/

/ E / (F2)

/ i / (F2)/E/

/ I / (F2)

/ Q / (F2)/Q/

/ E / (F2)/ « / (F2&F3)

/«/

23

/ U / (F3)

/ U / (F3)

/A/
//
/U/

/ u / (F3)/u//  / (F3)/¼/

Table 4.4: Acoustic-phonetic decomposition of the dichotomy in inter-speaker vowel classification
behaviour observed (dash-dotted curve in Figure 4.14) for the 6 “goatiest” of the 32 adult, malespeakers of the PB dataset, in terms of the vowel misclassifications that contribute to the drop in
accuracy across the higher spectral regions which encompass the high-F2 and the F3 of the 6 speakers’
vowel formant distribution. ‘F2&F3’ indicates misclassifications caused by overlapping F2 and F3
ranges.

drop in accuracy across the higher spectral regions are caused by confusions amongst
the F2 of front vowels, and amongst the F3 of back vowels.
In this vein, it is remarkable to note the general similarity of the misclassifications
listed here for the 6 “goatiest” speakers of the PB American English dataset, to thoselisted earlier (in Tables 4.1 and 4.2) for our four speakers of Australian English.
Admittedly, the persistent trends in acoustic-phonetic manifestations of the dichotomy
in two different datasets of spoken English, reinforce our definition of vowel-speaker
interactions, and further underscore the speaker-related potency of the higher spectral
regions which encompass the high- F2 and the F3 of spoken vowel sounds.

4.4.5 Dependence on Idiolectal Differences
Thus far we have unfolded and provided an acoustic-phonetic explanation of thephenomenon of vowel-speaker dichotomy using two datasets of vowels recorded byspeakers known to be of mixed idiolectal or dialectal backgrounds. To what extent do
the observed manifestations of the dichotomy depend on the types of speakerdifferences which exist between idiolects (of Australian English, as in the FC dataset) or
between dialects (of American English, as in the PB dataset)? The JB dataset offers a174unique opportunity to address that question, in terms of the idiolectal variations foundbetween speakers of Australian English.
However, as the speaker population of the JB dataset is comparable in size to thatof the PB dataset, one can almost predict that the dichotomy manifest in the former
would be as “blurred” as that obtained in our investigations at the start of the previous
section. Although our method of speaker rank-ordering could then be recalled to
identify the “goat” speakers and thereby obtain a clearer dichotomy, we herein
disregard the issue of speaker homogeneity as it was elucidated in the previous section,
and assume that a blurred dichotomy is sufficient to indicate the presence of vowelspeaker interactions. Indeed, in this section we seek to gain insights regarding theidiolectal dependence of the dichotomy, despite potential blurring of the phenomenon
which might be caused by other factors (such as the number of speakers) contributing
to speaker homogeneity.
As described in Chapter 3, the JB dataset comprises formants measured in thesteady-state of effectively two repetitions of 11 vowels, spoken in /hVd/ context by 36
speakers of Australian English. Bernard’s (1967) own auditory judgments during an
informal interview with each speaker, form the basis of our idiolectal labelling of 14
Broad (numbered 1 through 14), 11 General (numbered 15 through 25), and 11
Cultivated (numbered 26 through 36) speakers of Australian English. Speakers withineach of those three groups can therefore be regarded as idiolectally homogeneous, atleast as determined from a perceptual point of view by an expert phonetician.
In preparation for our vowel classification experiments, we convert the formant
frequencies to LP simplified cepstra using the method outlined in Section 3.4.2, with
NCC = 14 , Fs = 10 kHz, and assuming fixed formant bandwidths equal to the meanvalues measured earlier for the same set of 11 vowels recorded by our four speakers ofAustralian English ( B1 = 99 Hz, B2 = 128 Hz, and B3 = 218 Hz). The dependence ofthe vowel-speaker dichotomy on idiolectal speaker differences is then assessed, byperforming inter-speaker vowel classification as a function of an increasing upper
spectral limit, in three, distinct experiments. First, idiolect-independent vowel
classification is performed by training the classifier on the data of 35 of the speakers at atime, and testing on the data of the remaining speaker. Next, inter-idiolect vowel
175

80

Vowel classification accuracy (%)

70

60

50

40

30

idiolect-independentinter-idiolectintra-idiolect

20
0

500

1000

1500 2000 2500 3000 3500
Upper limit of spectral range (Hz)

4000

4500

5000

Figure 4.19: Dichotomy in idiolect-independent, inter-idiolect, and intra-idiolect vowel
classification behaviour on an inter-speaker basis, using the JB dataset of 14 “Broad”, 11 “General”,
and 11 “Cultivated”, adult male speakers of Australian English.
classification is performed where, rather than leave out one speaker at a time, thetraining set comprises the data of all the speakers in two idiolectal groups, and theclassifier is tested using the data of all the speakers in the remaining, third idiolectal
group. Finally, intra-idiolect vowel classification is performed by adopting the leaveone-speaker-out approach within each of the three idiolectal groups at a time.
The solid curve in Figure 4.19 shows the behaviour of idiolect-independent, interspeaker vowel classification accuracy using the data of all 36 speakers. It exhibits apeak of 73.9 % at 1700Hz, followed by a drop in accuracy to 70.7 % at 2480Hz, and asmaller rise to 72.3 % at full spectral range. As predicted, this behaviour is quite similar
to the blurred dichotomy observed earlier (in Figure 4.14) for the 32 speakers of the PB
dataset, and therefore suggests that the effect of a more clear dichotomy has been
diminished by averaging over a large number of speakers.
Admittedly a more extreme form of eliciting the effects of idiolectal speakerdifferences, is to explicitly acknowledge the three separate groups of speakers byperforming inter-idiolect vowel classification as described above. The mean of the three

176

80

Vowel classification accuracy (%)

70

60

50

40

30

Broad
General
Cultivated

20
0

500

1000

1500 2000 2500 3000 3500
Upper limit of spectral range (Hz)

4000

4500

5000

Figure 4.20: Idiolectal decomposition of the mean, inter-idiolect vowel classification accuracy curve(dash-dotted line in Figure 4.19), in terms of the three accuracy curves obtained by training on two
idiolects and testing the data of all the speakers in the third idiolect (as noted in the legend).
accuracy curves thus obtained is shown in Figure 4.19 by the dash-dotted curve, which
exhibits a peak of 73.7 % at 1720Hz, followed by a drop in accuracy to 70.3 % at
2540Hz and a small rise to 71.7 % at full spectral range. This behaviour is very similar
to that of the idiolect-independent accuracy curve shown in the same graph anddiscussed earlier. Indeed, if vowel-speaker interactions are assumed to be manifest evenin a blurred dichotomy, then those curves together confirm the phenomenon in thepresence of idiolectal speaker differences.
Further insights are gained by decomposing the mean, inter-idiolect accuracy
curve in terms of its three constituent curves, as shown superimposed in Figure 4.20.
Although the three curves exhibit comparable degrees of dichotomous behaviour (i.e.,
similar drops in accuracy across the higher spectral regions), the highest accuracies are
attained along the entire curve for the speakers of the General idiolect, and the lowest
accuracies are achieved for the Broad speakers. These results certainly confirm thefindings of Kumar (1996) who performed inter-idiolect vowel classification on the samedataset using the formant frequencies, and found that when a Bayes classifier is trained177using the data of speakers in the remaining two idiolects, the best performance isachieved for the General speakers, and the worst performance for the Broad speakers.
This is not entirely unexpected as it seems reasonable to assume that the General
speakers exhibit acoustic-phonetic characteristics which are, almost by definition,intermediate between the other two, more extreme idiolects; and that the Broad
speakers have characteristics which are on the whole more distant, and perhaps marked
with greater variation compared with the other two idiolectal groups. Indeed, thisinterpretation of the relative levels of the three accuracy curves also increases ourconfidence in Bernard’s (1967) idiolectal categorisation of the speakers, which itself
was based purely on auditory-perceptual judgments rendered during an informalinterview with each speaker.
The idiolect-independent and inter-idiolect vowel classification accuracy curvespresented thus far, are consistent with our earlier results using the FC and the PB
datasets, and show that the vowel-speaker dichotomy is indeed manifest in the presenceof idiolectal speaker differences. We now address the question of whether the
dichotomy is preserved despite the absence of idiolectal differences, by performinginter-speaker vowel classification experiments separately for the speakers within eachidiolectal group. The dashed curve in Figure 4.19 is the mean of the three, idiolectdependent or intra-idiolect accuracy curves thus obtained. It exhibits a rise in accuracy
to a peak of 73.7 % at 1640Hz, followed by a drop in accuracy to 69.7 % at 2700Hz
and a small rise to 71.2 % at full spectral range. The remarkable similarity of the
behaviour of this curve to that of the idiolect-independent (solid) and the inter-idiolect(dash-dotted) curves, immediately suggests that the phenomenon of vowel-speakerdichotomy does not necessarily rely on idiolectal differences between speakers. In fact,
if we were to rank-order the three curves shown in Figure 4.19 in terms of their overall
degree of dichotomy, we should rank the intra-idiolect curve highest, on the basis of its
slightly larger drop in accuracy across the higher spectral regions (a drop of 4.0 % as
the upper limit of the spectral range is extended from 1640Hz to 2700Hz), and its
overall lower accuracy at full spectral range. Although this comparison reveals only
minor contrasts in the behaviour of the three accuracy curves, it does suggest that the
detrimental influence of within-idiolect, or intrinsic speaker differences on vowel
178

80

Vowel classification accuracy (%)

70

60

50

40

30

Broad
General
Cultivated

20
0

500

1000

1500 2000 2500 3000 3500
Upper limit of spectral range (Hz)

4000

4500

5000

Figure 4.21: Idiolectal decomposition of the mean, intra-idiolect vowel classification accuracy curve(dashed line in Figure 4.19), in terms of the three accuracy curves obtained by performing interspeaker vowel classification using only the speakers within each idiolectal group.
classification accuracy in the higher spectral regions, is no less significant than theinfluence of speaker differences between idiolects.
This implication is upheld by the dichotomous behaviour of the three individual,
intra-idiolect accuracy curves superimposed in Figure 4.21, all of which exhibit a dropin accuracy across the higher spectral regions. Similarly to the three, inter-idiolectcurves shown earlier in Figure 4.20, the curve pertaining to the speakers of the Broad
idiolect is consistently the lowest in accuracy; and the curve obtained for the General
speakers is highest in accuracy both across the higher spectral regions and at full
spectral range. However, the highest accuracy itself is attained along the Cultivatedspeakers’ curve, which exhibits a peak of 78.1 % at 1640Hz, followed by a large dropin accuracy to 71.5 % at 2520Hz. By comparison, the accuracy curve obtained for the
speakers of the General idiolect exhibits a less dramatic drop, from a peak of 77.3 % at
1720Hz, to 74.4 % at 2680Hz.
Whilst the dichotomous behaviour of all the accuracy curves generated thus far
using the JB dataset does show that the detrimental influence of speaker differences in179

Vowel/i/“Broad”

Idiolectal group (Australian English)“General”“Cultivated”/ I / (F2, 10%; F3, 15%)

/ I / (F3, 10%)/ I / (F2, 5%)/ i / (F2, 15%)

/ i / (F2, 40%)

/ E / (F2, F3, 35%)

/ E / (F2, 15%)

/ E / (F2, 5%)/ I / (F2, 10%)/ I / (F2, 20%)

/ E / (F2, 15%; F3, 5%)

/I/
/E/

/ I / (F2, 5%)/ á / (F2, 5%)/Q// E / (F2, 5%)

/a/
/«/

/ a / (F3, 20%)//

/ a / (F3, 5%)/ a / (F2, 5%)/  / (F3, 5%)/ U / (F3, 5%)/// U / (F3, 5%)/ U / (F3, 10%)/U// / (F3, 5%)/ / (F3, 20%)/ á // ¶ / (F2, 15%)/¶// á / (F2, 15%)

/ Q / (F2&F3, 10%)/ á / (F3, 5%)

Table 4.5: Acoustic-phonetic decomposition of the dichotomy in inter-idiolect vowel classification
behaviour (dash-dotted curve in Figure 4.19), in terms of the vowel misclassifications that contribute to
the drop in accuracy across the higher spectral regions which encompass the high-F2 and the F3 of the
JB speakers’ vowel formant distribution. ‘F2&F3’ indicates misclassifications caused by overlapping F2
and F3 ranges.

the higher spectral regions is independent of (perceived) idiolectal differences, we arecompelled to ask whether those detrimental influences are caused by similar types ofvowel confusions. Our acoustic-phonetic decomposition of the contributions to thedrop in accuracy in the inter-idiolect curve (the dash-dotted curve in Figure 4.19,
whose three constituent curves were shown in Figure 4.20) is listed in Table 4.5. As
highlighted in that Table, the two largest contributions are caused, respectively, byconfusions of the General speakers’ front vowel / I / with / i / in the F2 range of thosevowels, and by confusions of the Cultivated speakers’ front vowel / I / with / E / in theF2 and F3 ranges. The two largest contributions amongst the back vowels are caused,
respectively, by confusions of the General speakers’ / « / with / a /, and by confusions of
their / U / with /  /, both occurring in the F3 range of those vowels.
For the sake of comparison, the results of our acoustic-phonetic decomposition of
the vowel and speaker contributions to the drop in accuracy observed in the higher
spectral regions for each of the three intra-idiolect curves (in Figure 4.21), are listed in180

Vowel
/i/

1

2

3

4

/I/

/i/
(F2)

/E/
(F2)

/i/
(F2)

/i/
(F3)

/E/

/Q/
(F2)

/Q/

/«,¶/
(F2)

/a/
/«/

/a/
(F2)

/E/
(F2)
/i/
(F3)
/ á /
(F2)

/i/
(F2)

/«/
(F3)
/a/
(F2)

/U/
(F3)
//
(F3)

/U/
/¶/
(F2)

/i/
(F2)

/I/
(F2)

/a/
(F3)

/a/
(F2)

/a/
(F3+)

/U/
(F3)

//

/¶/

Speaker (of “Broad” Idiolect)
6
7
8
9
10

/E/
(F3+)
/«/
(F3)

//

/ á /

5

11

12

/E/
(F2)

/I/
(F2)

/I/
(F2)

/«/
(F3)
//
(F3+)

/«/
(F3)

13

14

/i/
(F2,
F3)

/i/
(F2)

/I/
(F2,
F2&F3)

/«/
(F3)
/a/
(F2,
F3)

//
(F3)
/U/
(F3)

/U/
(F3)

//
(F3)
/ á /
(F3)

//
(F3)
/¶/
(F2)

/ á /(F3+)/ á /(F2)/ á /
(F2)

/¶/
(F2)

/ á /(F3)

Table 4.6(a): Acoustic-phonetic decomposition of the dichotomy in intra-idiolect, inter-speakervowel classification behaviour for the 14 “Broad” speakers of the JB dataset (dashed curve in Figure4.21), in terms of the vowel misclassifications that contribute to the drop in accuracy across the higher
spectral regions which encompass the high-F2 and the F3 of those speakers’ vowel formant distribution.
‘F2&F3’ indicates misclassifications caused by overlapping F2 and F3 ranges; ‘F3+ ’ indicatesmisclassifications which occur in spectral regions higher than the speakers’ F3 range.

Tables 4.6(a), (b), and (c), for the 14 Broad, 11 General, and 11 Cultivated speakers,
respectively. Consistent with the less dramatic drop in accuracy noted earlier for the
General speakers’ accuracy curve in Figure 4.21, there appear to be relatively fewer
misclassifications listed in Table 4.6(b) than in Tables 4.6(a) and (c). Table 4.6(b) is
particularly sparse in confusions amongst the General speakers’ back vowels —confusions which appear profusely for both the Broad and Cultivated speakers. Despite
the relatively large contributions to the dichotomy in inter-idiolect vowel classificationby the General speakers, they appear to have only a relatively smaller contribution to
the dichotomous behaviour of the intra-idiolect accuracy curve.
These idiolect-specific characteristics notwithstanding, the vowel confusions listed
in Tables 4.5 and 4.6 convey much the same trend observed earlier both for the FC

181

Vowel
/i/

15

16

17

18

Speaker (of “General” Idiolect)
19
20
21
22

/I/
(F3)
/i/
(F3)
/I/
(F2)

/I/
/I/
(F2,F3)

/E/

/E/
(F2)

/E/
(F2)

23

24

25

/I/
(F2,F3)
/E/
(F2)
/I/
(F2)

/Q/
/a/
/a/
(F3)

/«/
//

/U/
(F3)

//
/U/
/ á /

/E/
(F2)

/¶/
(F3)

/¶/

Table 4.6(b): Acoustic-phonetic decomposition of the dichotomy in intra-idiolect, inter-speakervowel classification behaviour for the 11 “General” speakers of the JB dataset (dotted curve in Figure4.21), in terms of the vowel misclassifications that contribute to the drop in accuracy across the higher
spectral regions which encompass the high-F2 and the F3 of those speakers’ vowel formant distribution.

Vowel
/i/

26

27

/I/

/i/
(F2)
/I/
(F2)

/I/
(F2,F3)/E/(F2)/ á /
(F2)

/E/

28

Speaker (of “Cultivated” Idiolect)
29
30
31
32
33

34

/I/
(F3)
/E/
(F2)

/E/
(F2)

/i/
(F2)

/E/
(F2)

/I/
(F2)

35

36

/I/
(F2)
/i/
(F2)
/I/
(F2)

/I/
(F2)
/i/
(F2)

/E/
(F2,F3)

/Q/
/a/
/«/

/a/
(F3)

/a/
(F3)

/a/
(F3)

//
//

/U/
(F3)

/U/
(F3)

/Q/
(F2&F3)

/U/
(F3)
//
(F3+)
/¶/
(F3)

/U/
/ á /
/¶/

/I/
(F2)

/ á /(F3)/ á /(F2)

Table 4.6(c): Acoustic-phonetic decomposition of the dichotomy in intra-idiolect, inter-speakervowel classification behaviour for the 11 “Cultivated” speakers of the JB dataset (dash-dotted curve in
Figure 4.21), in terms of the vowel misclassifications that contribute to the drop in accuracy across thehigher spectral regions which encompass the high-F2 and the F3 of those speakers’ vowel formant
distribution. ‘F2&F3’ indicates misclassifications caused by overlapping F2 and F3 ranges; ‘F3+ ’
indicates misclassifications which occur in spectral regions higher than the speakers’ F3 range.
182

dataset (in Tables 4.1 and 4.2) and for the “goat” speakers of the PB dataset (Table 4.4)
— contributions to the drop in accuracy across the higher spectral regions arise mainly
from confusions amongst the F2 of front (and mid-) vowels, and amongst the F3 ofback vowels. Those manifestations of vowel-speaker interactions have been shown inthis section to appear in their basic form, irrespective of (perceived) idiolectal speakerdifferences. That the dichotomy should thus be manifest even for idiolectallyhomogeneous groups of speakers, is yet further validation of its fundamental nature as abasic phenomenon of speech.

4.5 Concluding Summary
In this chapter we have approached the long-standing problem of speech-speakerdichotomy from a novel perspective, which involves contrasting the behaviour of vowel
classification accuracy as a function of an increasing upper spectral limit, first on an
intra-speaker basis where the phonetic dimension is unimpeded by speaker differences,then on an inter-speaker basis where phonetic and speaker-specific influences were
expected to interact. A minimum-distance, linear classifier was used to first unfold thedichotomy using the phonetically-rich FC dataset of vocalic steady-states recorded in/hVd/ context by four adult, male speakers of Australian English. Progressive
recruitment of higher-frequency spectral information in vowel classification, wasachieved by way of a new, parametric cepstral distance measure (PCD), which allows
selection of any frequency sub-band within the available spectral range. The dichotomy
was then validated using a more sophisticated, quadratic classifier, for which frequency
sub-band selectivity was achieved by a recursive transformation of the cepstrum toobtain a so-called partial (quefrency-weighted) cepstrum (P-QCEP). Owing to thequadratic classifier’s sensitivity to data dimensionality, its role was limited to validating
the general, contrastive behaviour of the accuracy curves, whereas all subsequent
analyses were carried out using the less sensitive linear classifier.
The dichotomy itself is embodied in the contrast between the nearly-asymptotic
behaviour of intra-speaker vowel classification accuracy, and the dichotomous
behaviour of inter-speaker accuracy across the spectral continuum. Whilst the former
underscores the importance of the low spectral regions for the purposes of vowel
183

discrimination, the latter evinces the speaker-related potency of the higher spectral
regions where classification accuracy is observed to drop. The acoustic-phonetic
relevance of those spectral regions of primary phonetic and speaker influence wasdetermined by referring to the speakers’ vowel formant distribution, which first
confirmed the well-known phonetic salience of the spectral regions which encompass
the two lowest formants or resonances of the vocal tract F1 and F2 . By contrast, adetailed acoustic-phonetic decomposition of the vowel and speaker contributions to thedrop in inter-speaker vowel classification accuracy across the higher spectral regions,clearly implicated speaker-induced confusions mainly amongst the F2 of front vowels,
and amongst the F3 of back vowels. Apart from clearly substantiating previous, largely
unacknowledged works regarding the relative speaker-specificity of those acousticphonetic, vocalic subspaces, we have provided a new methodology for examining
vowel-speaker interactions in the acoustic-phonetic domain.
Prior to using the PB and JB datasets to validate the dichotomy, its dependence onspectral representation was investigated by performing vowel classification using
simplified cepstra generated from the carefully measured formants of the FC dataset.
The simplified spectral representation was shown to preserve the dichotomy, and toyield the same, general trend of speaker-induced vowel confusions in the higher spectral
regions. Indeed, one might expect the explicit removal of the so-called spurious poles,
together with the effective normalisation of formant bandwidths, to have yielded a more
direct, and perhaps a more truthful, acoustic-phonetic explanation of the vowel-speakerdichotomy in terms of the speakers’ formant ranges.
Simplified LP cepstra were then created from the formants of the PB dataset, andused both to validate the dichotomy and to explore its dependence on speakerhomogeneity. Indeed, the relatively larger, and evidently more homogeneous populationof 32 adult, male speakers, only yielded a so-called blurred dichotomy in the behaviourof inter-speaker vowel classification accuracy across the spectral continuum. A new
method of rank-ordering the speakers in terms of their degree of “sheepiness” or“goatiness” with respect to the given population, then led to a progression of accuracy
curves, from the most nearly asymptotic (for the “sheepiest” speakers), to the mostclearly dichotomous (for the “goatiest” speakers). The spectral manifestation of the184dichotomy was thus clearly shown to depend on the degree of heterogeneity amongst
the speakers. Moreover, a comparison of the per-vowel formant statistics of the “goat”
and the “sheep” speakers showed the heterogeneity of the former to be manifest mainly
in terms of a larger dispersion in the F2 of all vowels, and in the F3 of the back vowels.
Differences between the American English (PB) and our Australian English (FC)dataset notwithstanding, an acoustic-phonetic decomposition of the drop in accuracy
observed across the higher spectral regions for the “goatiest” (PB) speakers
corroborated the speaker-induced vowel confusions noted earlier amongst the F2 offront vowels and amongst the F3 of back vowels, thus raising our confidence in thebasic nature of the vowel-speaker dichotomy.
Finally, the JB dataset of 14 Broad, 11 General, and 11 Cultivated, adult malespeakers of Australian English, was used to determine the dependence of the dichotomy
on idiolectal speaker differences. Our results first confirmed the overall levels of
accuracy expected of each idiolectal group, thus raising our confidence in the idiolectal
labelling of each speaker, which was based on purely auditory-perceptual impressions.
The behaviour of inter-speaker vowel classification accuracy across the spectral
continuum was then shown to be equally dichotomous, regardless of the presence orabsence of (perceived) idiolectal speaker differences. Moreover, an acoustic-phonetic
decomposition of the vowel and speaker contributions to the drop in accuracy observed
across the higher spectral regions, confirmed our earlier findings using the FC and the
PB datasets, implicating the speaker-related potency of the F2 of front (and mid-)vowels and the F3 of back vowels.
The consistency of our acoustic-phonetic explanation of the dichotomy acrossthree datasets of spoken English vowels, urges further investigations, as set out in ourIntroduction (in Chapter 1), aimed at providing a physical, or articulatory explanationof the phenomenon. This relatively uncharted area of speech research deserves carefulconsiderations, unhampered by computational burdens imposed by excessively large
amounts of data of a large number of speakers. Our articulatory explanation will
therefore focus on the four-speaker, Australian English (FC) dataset already used in thischapter to unfold the dichotomy.
Lacking direct articulatory measurements of those speakers, we shall attempt a
185physical explanation in terms of vocal-tract shapes estimated from the measured
acoustic parameters. However, as reviewed in Chapter 2, acoustic-to-articulatory
mapping is itself a highly problematic area of research which has not yet converged to acommonly-accepted solution. The following chapter is therefore devoted to gaining a
fresh outlook on the problem of vocal-tract shape parameterisation and estimation,
driven by specific requirements which emerge from our current, acoustic-domain
investigations. In Chapter 6 we then apply the area-function parameterisation andestimation methods developed in Chapter 5, and thereby examine the phenomenon ofvowel-speaker dichotomy from a speech production point of view.

186Chapter 5
Vocal-Tract Shape Parameterisation and Estimation

5.1 Introduction
The spectral manifestations of the interactions between phonetic and speaker-specificattributes of steady-state vowels of spoken Australian and American English, were
unveiled in the preceding chapter, by way of classification experiments based upon thelinear-prediction (LP) cepstrum. The flexibility afforded by our parametric cepstral
distance measure (PCD, derived in Section 4.2.3.1) indeed complemented the wholespectrum representation of the LP cepstrum, and allowed direct examination of thespectral ranges which predominantly contain either phonetic or speaker-specificinfluences. A more detailed examination of the vowel-speaker dichotomy was thenperformed by having recourse to the spectral resonance (or formant) frequencies.
Although the task of formant-estimation is considerably more problematic than that of
obtaining the LP cepstrum, the interpretive superiority of the formants led to a morerevealing, acoustic-phonetic explanation of the dichotomy. As set out in ourIntroduction (Chapter 1), we now seek an even more fundamental interpretation of thevowel-speaker interactions which have thus far been elucidated only in the acoustic
domain, by extending our investigations into the domain of speech production. A firststep towards this end is taken here by considering the intertwined problems of vocaltract shape estimation and parameterisation.
As suggested in our literature review (Chapter 2), difficulties in acquiring directlymeasured physiological data have to a large extent prohibited a detailed, or a large-scale
investigation of the articulatory sources of speaker variability. Acoustic-to-articulatory
mapping, on the other hand, is potentially a more efficient and practical method, which
may therefore facilitate investigations of the articulatory correlates of the many types of187

variability in the acoustic speech signal. Sadly, the prospect of utilising estimated (rather
than directly measured) articulatory data seems to have suffered from want of a“perfect” solution to the so-called inverse problem, which should be capable ofrecovering, with a large degree of confidence, the actual vocal tract configurations ofany given speaker. Our review of the relevant literature (in Section 2.4.3.2) has alreadyunderscored the crucial role of both the articulatory parameterisation and the vocaltract acoustic model in determining the intrinsic nonuniqueness of the inverse mapping.
In our pursuit of an articulatory explanation of the phenomenon of dichotomy, we
therefore seek a method of acoustic-to-articulatory mapping which employs a vocaltract acoustic model with the least degree of intrinsic nonuniqueness.
In addition, we seek a vocal-tract shape parameterisation which not only inherits
the uniqueness properties of the chosen vocal-tract acoustic model, but also permits a
direct relation with the acoustic parameters used in the previous section to provide an
acoustic-phonetic explanation of the dichotomy. In this vein, it is hardly surprising thatthe formants, which were ideally suited to providing an acoustic-phonetic explanation,
are also the acoustic parameters which have been traditionally, and remain to this day,the most directly associated with the physical properties of the vocal-tract. It follows
that, in order to obtain an articulatory interpretation of the dichotomy analogous to theacoustic-phonetic explanation offered in Chapter 4, it would be of considerable benefit
to adopt a vocal-tract shape parameterisation which is related as directly as possible tothe formants.
Our aim in this chapter is therefore to develop an approach to the estimation of
vocal-tract area-functions from the acoustic speech waveform of non-nasalised vowels,
which satisfies two main criteria: (1) by necessity, the model adopted for the inverse
mapping must be capable of yielding inherently unique vocal-tract shapes; and (2) thearea-function parameterisation should be expressible directly as a function of theacoustic resonances of the vocal-tract.
In Section 5.2 we address these two issues of resonance-based parameterisation
and uniqueness, and thereby provide a bipartite rationale for our proposed approach.
Theoretical and empirical results are brought to bear in Section 5.3, on the problem ofdetermining the resonance-based parameters of unique, LP-derived area-functions. In
188

Section 5.4 we then describe our hybrid method of area-function estimation, and use it
to evaluate our proposed method of vocal-tract shape parameterisation. In Section 5.5
we first introduce a new method of quantifying inter-repetition variability amongst
vocal-tract shapes, which then allows an evaluation of our hybrid method of inversion.
We conclude in Section 5.6 with a summary of the contributions arising from our workin vocal-tract shape estimation and parameterisation, and with a discussion of its
significance in the context of our articulatory explanation of the dichotomy which is tofollow in Chapter 6.

5.2 Rationale
Our criteria for an appropriate method of acoustic-to-articulatory mapping, as statedabove, incite a twofold rationale concerning parameterisation and uniqueness of
estimated vocal-tract shapes, which we therefore consider, respectively, in the next two
sections.

5.2.1 Resonance-Based Parameterisation of Area-Functions
The seminal work of Schroeder and Mermelstein (1965), subsequently expanded byeach author separately (Schroeder, 1967; Mermelstein, 1967), is an acousticallymotivated and theoretically-derived parameterisation of the vocal-tract area-function. In
contrast with the many physiologically-based models of the vocal-tract which haveappeared in the literature, the parameterisation derived by these authors is rooted in the
acoustic, and more specifically in the resonance, properties of the vocal-tract. It istherefore re-examined here in some detail, with a view to highlighting its strengths andlimitations, and placing it in the context of our forthcoming, articulatory investigation of
the vowel-speaker dichotomy.
In order to better appreciate the implications of the modelling paradigm pioneeredby those two authors, we have included in Appendix B, a mathematical re-derivation ofwhat we hereafter will refer to as the Schroeder-Mermelstein (SM) model. Indeed, as
re-derived therein, the complete model is summarised in the following two equations:mπx ln A( x ) = ln A0 + ∑ am cos
,
 L 
m =1
M

189

(5.1)



which defines the parameterisation of the vocal-tract (logarithmic) area-function A(x) oflength L, in terms of the coefficients a m of its Fourier cosine series; andδFn= − 1 2 a2 n −1 ,
Fn

(5.2)

which defines the theoretical relation between the acoustic- and the articulatory-domain
parameters. In particular, as illustrated in the nomograms on the left in Figure 5.1,
antisymmetric perturbation of the area-function of a uniform, completely losslessacoustic tube according to each of the odd-indexed shape parameters, does induce aquasi-linear perturbation of a unique formant frequency about its neutral value. By
contrast, as shown in the nomograms on the right in Figure 5.1, perturbation of aneven-indexed parameter a2 n has no first-order1 effect on the formant frequencies. The
SM model thereby establishes an acoustically-important distinction between symmetricand antisymmetric perturbations of a uniform area-function (where the symmetry isdefined with respect to the mid-point along the length of the area-function).
Not long after its inception, the SM model’s potential in yielding articulatory
insights into acoustic speech phenomena was demonstrated by Broad (1972, p.418),
who combined Equations 5.1 and 5.2, then differentiated the result with respect to time
to obtain a “measure of the average acoustically detectable movement throughout the
vocal tract.” The coarse, phonetic segmentation (of an all-sonorant utterance) thus
afforded, was then refined by integrating the expression only across a selected length ofthe vocal-tract, thereby obtaining a segmentation based on the acoustically-inferredvelocity profile of, for example, the tongue dorsum. The SM model was also advocatedby Broad and Wakita (1977), who showed that it aptly predicts the F2 -dependence of
vocal-tract shapes, in terms of the distinction between front and back places of
constriction; similarly, the top-left panel in Figure 5.1 illustrates the F1 -dependence,
which is aptly predicted in terms of the open versus closed distinction.
However, despite the interpretive potential of the SM model, it is known to suffer
the limitation of nonuniqueness. In particular, while the antisymmetric-shapeparameters a 2n −1 are each associated with a unique formant frequency Fn , the1

By “first-order”, we refer not only to the magnitude of the induced perturbation, but also to its
monotonicity (or quasi-linearity).

191

symmetric-shape parameters a 2n remain ambiguous in the inverse mapping. In this vein,
Borg (1946) is credited (by Schroeder, 1967; Mermelstein, 1967; Schroeter andSondhi, 1994) to have proven that not a single, but a doubly-infinite set of eigenvalues,
which correspond to two different sets of boundary conditions, is both necessary andsufficient to uniquely determine the state of a completely lossless resonant system. The
lip impedance function, which is defined as the driving-point impedance of the vocaltract as seen from the lips, is particularly useful in this context, because it embodies
knowledge of both sets of eigenvalues; in terms of the complex frequency variables = jω , it is expressed as follows:∞∏ (s − ω ) ,Z ( s) =∏ (s − ω )
in

∞

n =1

n =1

n

(5.3)

( c-c )nwhere the zeros correspond to the well-known formants, and the poles of the lip
impedance function correspond to the resonances under “closed-closed” (c-c) boundary
conditions (implying an infinite acoustic impedance at both ends of the acoustic tube).
Analogously to Equation B.13 (in Appendix B), that second set of eigenvalues is givenby the following expression:
2

λ

(c-c)
n

2
ω (c-c)nπ= n 2 =   , Lcn = 1, 2, 3, ... ,(5.4)

and the “c-c” resonance frequencies themselves interleave with the formant frequencies,
as follows:ω 1 < ω 1(c-c) < ω 2 < ω (c-c)< ω 3 < ... .
2

(5.5)

Both Schroeder (1967) and Mermelstein (1967) have pointed out that the poles ofthe lip impedance function are related to the even-indexed parameters a2 n of the SMmodel in the same way that the formants are related to the odd-indexed parameters (inEquation 5.2). However, those poles can only be measured using a special apparatus
known as a “lip impedance tube” (Schroeder, 1967; Gopinath and Sondhi, 1970) which
has a mouthpiece sealed to the speaker’s lips, and which requires articulation ofsustained vowels without phonation. The nonuniqueness problem then stems from thefact that the acoustic speech signal contains resonance information pertaining to only
one of the required, two sets of boundary conditions — namely, the formants (or the192

zeros of the lip impedance function, but not its poles); hence, the symmetriccomponents of vocal-tract shapes remain undetermined in the inverse mapping, unlessadditional constraints are imposed.
Indeed, Mermelstein (1967) imposed the constraint of antisymmetry (which is,after all, a corollary of the SM model itself), simply by setting the undetermined, evenindexed parameters to zero. While this turned out to be a reasonable constraint for anumber of vocalic configurations, it led to a poor re-estimate of the X-ray measured
area-function (Fant, 1960) for the Russian vowel / u / in particular, which has a strong
symmetric component. More recently, Yehia and Itakura (1994, 1996) proposed toestimate the values of the first eight parameters of the SM model, using only the first
three formant frequencies; in particular, the ambiguity associated with the values of thefive shape-parameters a2 , a4 , a6 , a7 , and a8 , was resolved by a combination of staticand dynamic constraints on the estimated shapes themselves. Although, as reviewed
much earlier (in Chapter 2), the nonuniqueness problem is exacerbated when thenumber of vocal-tract shape parameters exceeds the number of acoustic parameters, theSM model is inherently nonunique if only the recorded speech signal is available.
In that context, it is important to emphasise that the SM model embodies thenonuniqueness of a completely lossless vocal-tract, perhaps in its most fundamental
form. Whilst other, more physiologically-relevant parameterisations of the vocal-tract
may afford more direct constraints on individual articulators such as the tongue body,the jaw, and the lips, the SM model captures the essential, acoustically-relevant
manifestations of nonuniqueness in the entire shape of a completely lossless vocal-tract.
It is also interesting to note that the acoustic-theoretic principles on which the SMmodel is founded, are equally the basis of the more recent, Distinctive Regions andModes (DRM) model of Mrayati et al. (1988). Admittedly, the DRM model has alreadybeen used more extensively than the SM model: for example, in /V-V/ and /V-C-V/
trajectory modelling (Carré and Mrayati, 1991; Carré et al., 1992), in automatic
recognition of place of articulation of plosives (Soquet and Saerens, 1994), in automatic
recognition of vowels and /V-V/ sequences (Candille and Meloni, 1995), in real-timearticulatory speech synthesis (Hill et al., 1995), and in acoustic-to-articulatory mapping
itself (Richards et al., 1995). However, as the DRM model is founded on the same193

acoustic principles (of a completely lossless acoustic tube) which are embodied in the
SM model, it does share the nonuniqueness properties of the latter, albeit without someof the distinct advantages of the SM model’s parameter-set.
Indeed, one of the advantages of the SM model, is its inherently smooth
representation of vocal-tract area-functions, which therefore allows (cf. Equation 5.1)
computation of the cross-sectional area at any position x along the length of the vocaltract. The degree of smoothness itself is determined by the number of formants used in
the inversion; thus, as conjectured by Mermelstein (1967), the low-order formants canbe used to obtain a bandlimited version of an area-function. In that vein, the efficiencyof the SM model is manifest in its minimal set of parameters, which describe the vocaltract shape in terms of an orthogonal set of basis functions (the cosine terms inEquation 5.1).
However, one of the disadvantages of the SM model2, is that the estimated vocaltract shape depends on the assumed vocal-tract length (VTL); Mermelstein (1967) has
already illustrated the potential sensitivity of the shape-length dependence. Based on
Zue’s (1969) thesis, Paige and Zue (1970) then proposed to optimise VTL such that theresulting shape is least eccentric compared with a uniform area-function. Within theanalytical framework of the SM model, they obtained a quadratic expression for theerror criterion with respect to vocal-tract length, thus firmly establishing the existenceof a unique, global minimum for any given area-function. Furthermore, they were able
to re-estimate the vocal-tract lengths of Fant’s (1960) six Russian vowels, to within thetolerance prescribed by Fant’s own, X-ray measurements ( ± 5 %). This combination oftheoretical elegance and empirical support has encouraged further use of the underlying
principle, which has since been referred to as “minimal articulatory antagonism”(Lindblom and Sundberg, 1971), “minimal articulatory difference” (Bonder, 1983b), or“minimum of muscle work” (Sorokin, 1992).
To summarise our preceding discussion, it is apparent that the great strength of
the SM model lies in its functional representation of vocal-tract shapes in terms oforthogonal parameters, one-half of which are acoustic-phonetically significant by virtue

2

Indeed, of any articulatory model which retains the vocal-tract length as an explicit parameter.

194

of their quasi-linear, one-to-one mapping with the formant frequencies. However, theabsence of resonance information (in the acoustic speech signal) pertaining to therequired, second set of vocal-tract boundary conditions, implies that only theantisymmetric shape components can be inferred in practice. We therefore direct ourattention in the next section, to the linear prediction (LP) vocal-tract model, which does
inherently resolve the nonuniqueness problem without recourse to further, articulatory
constraints.

5.2.2 Inherent Uniqueness in LP-derived Area-Functions
Indeed, as briefly reviewed in Chapter 2, the LP model does theoretically guarantee
uniqueness in acoustic-to-articulatory mapping. In view of the inherent nonuniquenessof the completely lossless vocal-tract model reviewed in the previous section, we are
therefore compelled to ask how the LP model secures uniqueness in vocal-tract shapes.
In search of an answer to that question, we first consider the uniqueness issue as dealt
with by the model’s two main proponents, namely Atal and Wakita.
Atal (1970) is widely credited (e.g., by Markel and Gray, 1976) for being the firstto derive an acoustic tube model of the vocal-tract directly from the speech waveform,
and to show that the formant frequencies and bandwidths are sufficient to uniquelydetermine the area-function in terms of a finite number of equal-length sections. In
Atal’s version of the LP vocal-tract model, the formant bandwidths are brought about
by a single, frequency-independent source of loss at the lip-end of an otherwise losslessacoustic tube. Invoking electrical-circuit analysis theory, Atal then proved (see, forexample, Strube, 1977) that a discrete, M-section area-function is uniquely obtained
through knowledge of the first M, discrete-time samples of the autocorrelation functionof the pressure developed across the resistive lip termination, in response to a unit
volume-velocity impulse at the glottis.
Atal and Hanauer (1971, Appendix F) then showed that an Mth-order, all-poletransfer function with all its poles inside the unit circle in the z-plane, can always beinterpreted as that of a lossless acoustic tube with M equal-length sections, terminatedby a unit acoustic resistance at the lips. Furthermore, they defined a procedure fordetermining the area-function of such an acoustic tube, using the covariance method of195

LP analysis. However, as pointed out by Markel and Gray (1976, p.79), they could
make no claim “as to the applicability of the method for estimating vocal tract areafunctions”, owing to the potential instability of the covariance method of analysis,
which may cause poles to lie on or outside the unit circle in the z-plane.
On the other hand, Wakita (1973) showed that the inverse filter obtained by the
autocorrelation method of LP analysis, whose stability is theoretically guaranteed(Markel and Gray, 1976, p.103), is an equivalent representation of a lossless acoustic
tube model with a resistive termination at the glottal end, and a short-circuit at the lips.
Furthermore, he demonstrated that plausible vocal-tract shapes can be obtained directlyfrom the speech waveform, provided certain analysis conditions (such as appropriate
pre-emphasis) are applied.
Wakita and Gray (1975) then shed some light on the uniqueness of area-functions
estimated by that method. In particular, they derived an expression for the lip
impedance function of the M-section acoustic-tube model, both for a lossless and alossy glottal termination. First, consistently with Borg’s (1946) proof reviewed in theprevious section, Wakita and Gray (1975, p.579) showed that for a lossless model, thenumerator and the denominator polynomials of the lip impedance function are
independent of each other, such that “both of them are needed for the determination ofa unique tube shape”; indeed, the roots of the two polynomials thus obtained are merely
the resonance frequencies which satisfy, respectively, the two different sets of boundary
conditions. By contrast, upon reinstatement of the lossy, resistive glottal termination,
the two polynomials were found to be inter-dependent; it was shown that either of
those polynomials can be obtained from the other, simply by changing the sign of all of
the reflection coefficients. Indeed, while for the completely lossless model the two setsof eigenvalues (the roots of the numerator and denominator polynomials, respectively)
are independent and real-valued (i.e., they have frequencies only), for the LP modelwith a resistive glottal termination they are dependent and complex-valued (i.e., they
have both frequencies and bandwidths).
Whilst the formant bandwidths are thus clearly implicated in both Atal’s andWakita’s explanations of the uniqueness of LP-derived area-functions, we sadly remain
ignorant of the vocal-tract shape-related manifestations of uniqueness. In particular,
196

despite the implications clearly foreshadowed by the SM model, the literatureapparently offers no insights regarding the roles of the antisymmetric and the symmetriccomponents of LP-derived vocal-tract shapes, and their possible relations with theformant frequencies and bandwidths — we only know that by specifying the first M 2formants, a discrete or step-wise, M-section area-function is obtained uniquely.
In that regard, it is important to note that the shape of an LP-derived areafunction (similarly to that of a completely lossless model, as reviewed in the previous
section) depends on its length. The principle of minimal articulatory effort which wasapplied first by Paige and Zue (1970), was then adapted to the LP method of inversion
by Wakita (1977), who proposed to determine the vocal-tract length by minimising theeccentricity of the discrete LP area-function with respect to a uniform tube. Indeed,
Wakita was able to re-estimate the lengths of five of Fant’s (1960) area-functions of six
Russian vowels, with errors of the order of ± 5 % using the first four formant
frequencies and bandwidths. Furthermore, he obtained reasonable estimates of the
vocal-tract length of vowels recorded by several, adult male and female speakers ofAmerican English.
However, owing to the fact that the effective half-sampling frequency is inversely
proportional to the length of each vocal-tract section, there remains the analysis artefactof an upper limit on allowed vocal-tract length (Wakita, 1977):

Lmax =

cM
,
4 Fhi

(5.6)

where Fhi is the frequency of the highest formant considered, M is the number ofsections, and c = 35300 cm/sec is the velocity of sound propagation in the vocal-tract.
If for a given set of formant frequencies and bandwidths, there cannot be found an
optimum vocal-tract length less than Lmax , Wakita offers no alternative but to accept
that upper limit (and the vocal-tract shape thus obtained).
Another limitation which is common to both the completely lossless and the LPbased inversion method, concerns the absolute value of estimated vocal-tract areas. For
a completely lossless acoustic tube, the formants are independent of the value of thearea scaling factor A0 (cf. Equation 5.1); similarly for the LP acoustic tube model,
which is completely specified by its reflection coefficients. Theoretically, A0 should not197be so large as to violate the assumption of acoustic plane-wave propagation in the tract,and it should not be so small as to imply frication at the place of constriction for avowel; physiologically, constraints on tongue-body volume suggest that A0 should nothave an excessive, phonetic range of variation. In practice, the area scaling factor isusually determined such that either the area at the glottal end, or the average area, or
the vocal-tract volume, remains fixed.
Whilst both the LP and the SM model are thus capable of yielding only
normalised3, or relative vocal-tract area-functions, there appears to be no evidence inthe literature to suggest that this limitation presents an obstacle in using such estimatedshapes to gain physiological insights into acoustic speech phenomena. On the contrary,a very small number of studies have even attempted to gain such insights using the LP
method of inversion — these include Crichton and Fallside’s (1974) experimental
system for deaf speech training; Gath and Yair’s (1988) successful identification of
lingual “tremor” in the sustained sound / l / recorded by Parkinsonian subjects; andHansen and Womack’s (1996) plausible descriptions of the articulatory differences inthe vowel / e /, in the word “help” recorded in neutral and angry states of emotional
stress. Although the estimated shapes cannot be claimed to be exactly those producedby the speakers, the plausibility of the insights gained in such studies does suggest thatthe consequences of the LP model’s well-known limitations (as described, e.g., by
Wakita, 1979; Sondhi, 1979) may not be as severe as the literature generally portrays.
In that context, perhaps the most misleading impression conveyed in the literatureon LP-based inversion, is that area-functions can be estimated directly by LP analysis of
the speech waveform, after applying appropriate pre-emphasis. Indeed, Wakita (1973,
p.422) does suggest that even a simple pre-emphasis of 6 dB/octave is “essential”, inorder to approximately equalise the combined spectral-slope effects of the glottal
source and lip radiation (Fant, 1960), and thereby reduce the possibility of obtaining“unusual tract shapes”. More elaborate methods of counteracting the influences ofsource and radiation characteristics by flattening the spectral slope and enhancing the3

Throughout this chapter, “normalised area-function” implies that the scale on the ordinate isdimensionless, owing to the fact that the area scaling factor (A0 in Equation 5.1) is acousticallyinconsequential; this should not be confused with speaker normalisation of area-functions, to which werefer in Chapter 6.

198

formant peaks, include the adaptive inverse filtering approach developed by Nakajima etal. (1973), the adaptive enhancing filter proposed by Tanaka and Nakajima (1975), andFuchi’s (1977) use of the negative derivative of the LP phase spectrum (NDPS) to
obtain an idealised, “stop/pass bands” spectrum. According to Sondhi (1979), “the only
tenable conclusion is that the area recovery is very strongly dependent on the assumed
source and radiation characteristics.”
Apart from the fact that the influences of source and radiation characteristics on
the formants can sometimes be appreciable, it is rarely acknowledged that LP analysis
usually yields a set of poles, amongst which may be found not only the formants, butalso the so-called spurious poles. Whilst the LP spectrum may not be adversely affectedby the presence of spurious poles (especially those of wide bandwidth), it is fair toassume that the shape of the LP area-function itself might be significantly affected — asfar as the LP vocal-tract model is concerned, a spurious pole which lies in between two“true” formants is just another resonance of the acoustic tube. Implicit in Wakita’s (andAtal’s) descriptions of the LP-based method of inversion, is therefore the retention of
only those poles which can be considered as the “true” formants.
However, the requirement of formants (estimation of which is itself non-trivial)
does not resolve the issue of the relevance of measured formants to the LP model.
Indeed, one of the major criticisms of the LP model is that it “lumps” all the sources ofloss into a single, resistive termination (at the glottal end in Wakita’s model). By
contrast, the formants measured from the acoustic speech waveform presumably include
the effects of all losses which naturally occur in the human vocal-tract. For example(Fant, 1960; Flanagan, 1972), the glottal inductance tends to raise the centrefrequencies of formants which are more strongly affiliated with the pharyngeal cavity;
the viscous and heat-conduction losses which are manifest along the surface of the
vocal-tract walls, contribute mainly to the bandwidths of the higher formants; vocaltract wall-vibrations tend to increase both the bandwidths and centre-frequencies ofmainly the lower formants; and the lip-radiation impedance tends to increase thebandwidths of mainly the higher formants, and also to lower the centre-frequencies offormants which are more strongly affiliated with the oral cavity.
Wakita (1979) proposed to compensate for the LP model’s lack of sufficiently
199

realistic losses, by way of a “formant frequency conversion chart” based on formants
synthesised with a more “realistic” (i.e., a more lossy) vocal-tract model. Similarly,
Hafer and Coker (1975) had earlier attempted to correct measured formant frequencies
prior to inversion, in order “to account for inductive yielding walls of the vocal tract”.
However, it remains questionable to what extent such model-based formant correction
procedures are applicable to real, measured data; and how viable it is to formulate asimilar correction procedure for formant bandwidths (which are, after all, moresignificantly affected than the formant frequencies by the LP model’s simplified
assumptions in regard to vocal-tract losses).
Despite these limitations, the LP-based method of inversion is at an advantage
compared with many other such methods, owing to the fact that the LP model is an
acoustic analysis model, whereby conversion of formants into an area-function is direct(or non-iterative) and computationally inexpensive. However, as suggested by Broad
and Shoup (1975, quoted in Section 2.4.3.2), the potential of the LP vocal-tract model
to yield physiological insights into acoustic speech phenomena has to date been
shamefully under-exploited. Perhaps a determining factor in the overwhelming neglectof the LP model over the past two decades, is its very coarse, step-wise representationof area-functions, from which it is admittedly difficult to extract crucial articulatory
landmarks such as the place of constriction. More importantly, the coarse
representation inhibits quantitative comparisons of area-functions of potentiallydifferent lengths, whether of different vowels or different speakers.
Those types of problems might conceivably be resolved by smoothing the discrete
LP area-function, and thereby obtaining a continuous and parameterised representation.
Indeed, Wakita and Gray (1975, Figure 4) used a Chebyshev polynomial approximation(also used earlier by Nakajima et al., 1973) to illustrate the LP-derived area-functions offive American English vowels. Although Wakita and Gray (1975, p.578) thus obtained“visually satisfying results” with the places of constriction more precisely defined, their
particular choice of smoothing function was admittedly “somewhat arbitrary”.
On the other hand, as reviewed in the previous section, the SM model does afford
a smooth representation of vocal-tract shapes, and more importantly, a resonance-based
parameterisation. Whilst it may therefore seem advantageous to use the SM model to200

parameterise LP-derived area-functions, it is not immediately obvious whether the
acoustically-relevant components of LP-derived vocal-tract shapes are equivalent tothose of a completely lossless vocal-tract on which the SM model is based. In the next
section we seek to answer that question, by attempting to identify the parameters of
unique, LP-derived area-functions.

5.3 Parameters of Unique LP-derived Area-Functions
In Section 5.1 we defined the two main criteria which our method of inversion would
need to satisfy — namely, uniqueness and formant-based parameterisation. Our
bipartite rationale then addressed these issues by invoking two classic, but sadly overneglected models, which do promise to fulfil our requirements. First, we reviewed the
SM model, which does indeed provide an acoustically-meaningful parameterisation of
the vocal-tract area-function. However, its underlying assumption of a completely
lossless vocal-tract, implies that a unique area-function can only be obtained if both thezeros (formants) and the poles of the lip impedance function are known. This
fundamental limitation then led us to consider the LP vocal-tract model, which is the
only one to guarantee uniqueness of area-functions estimated using acoustic
parameters, all of which can be measured from the recorded speech signal. However,whilst both the formant frequencies and the bandwidths are implicated in the LPmodel’s uniqueness, it has never been shown how LP-derived vocal-tract shapes exploit
the information contained in the formants, and of how those shapes might best be
parameterised in an acoustically meaningful way.
As far as we are aware, Wakita and Gray (1975) were the first and only
researchers to attempt to relate the LP model with the earlier work of Schroeder (1967)
and Mermelstein (1967). Indeed, they suggested that the LP and the SM models could
be used interchangeably to obtain a unique vocal-tract shape, once the appropriate
acoustic parameters had been identified by LP analysis. In particular, they proposed thatthe frequencies of the zeros and poles of the lip impedance function (as required by the
SM inversion method) could be found after LP analysis of the speech waveform, bysetting the glottal reflection coefficient to unity in order to render the LP vocal-tract
model completely lossless, then finding the required zeros of the lip impedance function201

by solving for the roots of the modified LP polynomial, and similarly the required poles,
after changing the sign of all of the reflection coefficients. It is interesting to note that
the LP line spectrum pair (LSP) parameters (Itakura, 1975), which are often regarded
merely as an alternative set of LP acoustic parameters, are indeed the inter-leaving
frequencies of the zeros and poles of the lip impedance function of a completely lossless
LP vocal-tract model, and are therefore precisely the necessary acoustic parameterswhich Wakita and Gray (1975) suggested for obtaining a unique vocal-tract shape using
the SM inversion method.
Does this imply that the poles of the lip impedance function are contained in the
acoustic speech signal after all, and that the SM model can therefore be used to obtain
unique vocal-tract shapes without resort to the lip impedance-tube method proposed bySchroeder (1967)? To answer that question, it is important to note that Wakita andGray’s (1975) proposed method of uniting the LP and the SM models, relies on postanalysis modification of the LP glottal reflection coefficient; subsequent conversion to
LSP frequencies does not change the shape of the LP area-function, which itself is
uniquely determined by the frequencies and bandwidths of the LP poles yielded in theoriginal analysis. The poles of the lip impedance function thus obtained, are therefore
only indirectly inferred from the LP area-function, rather than directly measured fromthe acoustic speech signal.
It emerges from the preceding discussion, that our knowledge of the uniquenessproperties of LP-derived vocal-tract shapes is still rather limited. Although in theirrespective formulations of the LP-based inversion method, Atal and Wakita certainly
provide evidence of the uniqueness of LP-derived area-functions, they stop short ofdescribing how the components of estimated vocal-tract shapes themselves are involvedin the uniqueness. By contrast, the SM model lends direct insights into thosefundamental components of vocal-tract shapes which are of greatest relevance fromacoustic-phonetic, perceptual, and articulatory points of view. Notwithstanding thedifferences between the respective vocal-tract models, it would therefore seemappropriate to use the SM model as a basis for identifying the parameters of unique LPderived area-functions.
As reviewed in the previous section, the underlying difference between the SM
202model and the LP acoustic-tube model is that the latter includes a resistive element at
the glottal end. The acoustic consequences of a purely resistive glottal termination havebeen studied both theoretically (e.g., Fant, 1960; Flanagan, 1972) and empirically (e.g.,
Badin and Fant, 1984). As a result, it is well-established that a resistive glottal
termination influences the formant frequencies far less than the bandwidths;
furthermore, that the resulting losses are frequency-independent. We might therefore
expect that the LP model shares the distinctive articulatory-acoustic relations embodied
in the SM model — in particular, that it inherits the property of a quasi-linear relation
between formant frequencies and the corresponding, antisymmetric shape componentsof the logarithmic area-function. We address this issue in Section 5.3.1, by presenting
new results which shed light on the formant frequency-dependence of LP-derived vocaltract shapes. In Section 5.3.2 we then address the question of fundamental concern,
whether the formant bandwidths, which provide the crucial second-half of acoustic
information required to secure uniqueness of LP-derived vocal-tract shapes, are at all
related to the undetermined, symmetric components of those shapes.

5.3.1 Dependence of LP-derived VT-Shapes on Formant Frequencies
Our first step towards marrying the SM model and the LP model, is to prove theirequivalence in regard to the dependence of vocal-tract shapes on the formant
frequencies. This proof is first considered from a theoretical point of view (in Section
5.3.1.1), with the simplifying assumption of a two-section vocal-tract area-function. It
is then extended (in Section 5.3.1.2) with empirical results which substantiate ourhypothesis of the equivalence of the two models, and provide the most extensive
validation of the SM model ever to appear.
5.3.1.1 Partial Theoretical Proof
One of the distinguishing features of the completely lossless SM model reviewed in
Section 5.2.1, is its continuous representation of vocal-tract area functions, in terms ofthe variable x. Hence, each parameter of the SM model specifies a component of the
vocal-tract shape, which is independent of the number of sections used to implement themodel in practice (as would be required, for example, to synthesise the acoustic

203resonances of a given area-function using numerical procedures). The LP model, on theother hand, not only represents the area-function in terms of a discrete number oflossless sections M (which, in the absence of real-valued poles, is equal to twice thenumber of LP poles effectively used), but is also characterised by a resistive (i.e., alossy) termination at one end of the vocal-tract. In order to confirm our hypothesis that
LP-derived vocal-tract shapes are governed by the same basic principles that define theSM model, it is therefore necessary first to recast the main result of the SM model interms of a discrete-sectioned, lossless acoustic tube.
Bonder (1983a) has shown that Webster’s Horn Equation (Equation B.3 or B.4,
in Appendix B) can be used to derive an analytical solution for the resonances of alossless acoustic tube with up to 10 equal-length sections. For simplicity, we willconsider a single-resonance acoustic tube which, according to the LP model, comprises
only two sections of equal length L / 2 . The simplified version of Bonder’s (1983c) socalled “n-tube formula” then reduces to the following expression (cf. also Fant, 1960,
p.65; Flanagan, 1972, p.70):

πLF1  A1
tan 2 ,= c  A2

(5.7)

where A1 and A2 are the cross-sectional areas of the front and the back vocal-tract
sections, respectively, and F1 is the frequency of the single resonance.
According to the SM model (Equation 5.2), the most efficient way to perturb F1
from its neutral value is to perturb the uniform area-function according to theantisymmetric shape parameter a1 . For a two-section acoustic tube, Equation 5.1 thenyields the following area-perturbations which would induce a positive perturbation in
the first formant frequency:
Am = e − a1 cos( ( 2 m −1)π / 4) ,m = 1, 2 ,(5.8)

where it is assumed that the area of each section takes on the value given by Equation
5.1 at the centre of that section (i.e., at x = 3 L / 4 and x = L / 4 for the front and theback sections, respectively), and that A0 = 1 . Equations 5.8 and 5.7 then together yield
the following expression for the single resonance-frequency F1 in terms of the vocaltract shape parameter a1 :

204

F1 =

c
tan −1 e − a1 2 .
πL

(5.9)

This last result, which is the homologue of Equation 5.2 for n = 1 , can be
regarded as an exact version of the SM model for a lossless two-tube. It is exact in thatit correctly predicts the value of the single resonance frequency as it would be obtainedby a sufficiently accurate numerical procedure to solve Webster’s Horn Equation for a
two-section area-function, independently of the degree of eccentricity of that areafunction with respect to a uniform tube. Assuming c = 35300 cm/sec and L = 17.65 cm
as in our earlier illustration of the SM model (Figure 5.1), the solid curve in Figure 5.2
illustrates the result just derived, showing the predicted variations in F1 about its
neutral value of 500 Hz, for area-function perturbations over a fairly wide range

a1 ∈[−2,+2] . The exactness of Equation 5.9 is indeed evidenced by the way in which itcorrectly predicts the F1 asymptotes of 0Hz and 1000Hz, respectively, for large
positive and large negative perturbations of a1 . By contrast, the linearity of the SMmodel (Equation 5.2 for n = 1 , shown here by the dotted curve) is strictly only valid for
relatively small perturbations about a uniform area-function.
Having established the exact form of the SM model for a completely lossless areafunction with two sections, we now proceed to derive an analogous expression for the
LP model. According to Wakita & Gray’s (1975) formulation, the single resonance of a
two-tube LP-model is given by the complex-conjugate roots of the following
numerator-polynomial of the lip impedance function:z 2 + µ 1 (1 + µ 2 )z + µ 2 = 0 ,(5.10)

where µ 1 = ( A1 − A2 ) / ( A1 + A2 ) is the reflection coefficient which describes the shapeof the two-section area-function, µ 2 = ρc / Ag(LP) is the glottal reflection coefficientwhich determines the effective (LP) glottal opening area Ag(LP) and hence the losses
introduced by the glottal resistance, and z = e j2πLF / c . Solving for the roots of thequadratic Equation 5.10, the single resonance frequency is then expressed in terms ofthe two reflection coefficients, as follows:
F1 = 4µ 2 − µ 12 (1 + µ 2 ) 2 c,tan −1 
2 πL−µ 1 (1 + µ 2 )



205

(5.11)

1000
Exact SM 2-tube modelLP 2-tube model, µ2 = {1.0, 0.7, 0.4}
Linear SM model

900

Formant Frequency F1 (Hz)

800
700

µ2=0.4600µ2=0.7500µ2=1.0

400
300
200
100
0
-2

-1.5

-1

-0.5

0

a1

0.5

1

1.5

2

Figure 5.2: Theoretical behaviour of the resonance frequency F1 of a 2-tube, as a function of the
shape-perturbation parameter a1. Solid curve: exact relation (Equation 5.9) derived from a completely
lossless 2-tube. Dashed curves: exact relation (Equations 5.11 and 5.12) derived from a 2-section LP
model, shown here for three values of the glottal reflection coefficient µ2={1.0, 0.7, 0.4}. Dotted curve:
linear relation predicted by the SM model (Equation 5.2).

where the range of the inverse-tangent function is taken as [0, π] by adding π to theresult whenever the argument is negative-valued, in order to ensure that only the polelying in the top half of the z-plane is considered.
Finally, to obtain an expression analogous to Equation 5.9, we need to derive therelation between the reflection coefficient µ 1 and the area-function parameter a1 .
Making use of Equation 5.8 and the definition of the reflection coefficient in terms of
the individual areas, we obtain the following:a µ 1 = − tanh 1  .
 2

(5.12)

Substitution of Equation 5.12 into Equation 5.11 then yields the desired relation which
is analogous to the earlier-derived Equation 5.9, with the extra flexibility afforded by
the resistive loss-element at the glottal end of the two-tube LP-model — an infinite
glottal impedance (i.e. a closed glottis) can be obtained by setting µ 2 = 1 , while finite
and increasing losses are introduced as µ 2 decreases towards zero.

206

The dashed curves in Figure 5.2 illustrate the LP-based equations just derived, forthree different values of the glottal reflection coefficient µ 2 = {1.0, 0.7, 0.4} . As
expected, when µ 2 = 1 (completely lossless, two-tube LP-model) the dashed curve
coincides exactly with the solid curve, which was obtained earlier from the exact
version of the two-tube SM model (Equation 5.9). The dashed curve for µ 2 = 0.7
shows a very similar behaviour of the resonance frequency, indicating that the inclusionof a moderate amount of loss at the glottal end does not significantly alter the relation
between F1 and the shape of a two-tube. However, it is interesting to note that as theglottal resistance is made to play a more significant role (dashed curve for µ 2 = 0.4 ),the behaviour of F1 in the two-tube LP-model begins to approach the quasi-linear
behaviour (dotted curve) predicted by the original SM model (Equation 5.2), which isbased on a completely lossless acoustic-tube, and which itself makes no assumptions as
to the number of vocal-tract sections.
We should emphasise, however, that the range over which a1 has been perturbed
in Figure 5.2, is much wider than would normally be expected for vocalic configurations(for example, Mermelstein’s (1967) analysis of Fant’s (1960) six Russian vowels
yielded values for a1 spanning approximately [−1, 1] , i.e., about half of the range
shown here). It is also quite clear in Figure 5.2 that the small but systematic differences
between the theoretically-derived models, are only of significance for such large
perturbations as are shown at the positive and the negative extremes along the abscissa.
Within the more restricted range of perturbations (e.g., [−1, 1] ) which are typical of
vocal-tract shapes for vowels, Figure 5.2 therefore confirms that the behaviour of theresonance frequency F1 as a function of the shape-perturbation parameter a1 isessentially common to both models. Our results concerning the two-section vocal-tract,
thereby support a partial theoretical proof of the validity of the SM model within the LPmodelling framework.
5.3.1.2 Empirical Validation
We now aim to further substantiate the partial theoretical proof presented thus far, byextending our validation to vocal-tract shapes with more than two sections. Two
methods are used to illustrate empirically the relation between formant frequencies and207

antisymmetric (and symmetric) perturbations of an LP area-function. In particular, weshow that this relationship is essentially the same as that predicted by the SM model for
a completely lossless acoustic tube.
The first method is similar to our earlier illustration of the SM model (Figure 5.1,
in Section 5.2.1), and involves nomograms depicting formant frequency variations as afunction of perturbations in a single SM-model parameter at a time. However, incontrast to the earlier results which were generated using a completely lossless vocaltract model, the resonances of computed vocal-tract shapes are here synthesised using
the LP vocal-tract model, which not only involves a discrete number of equal-length
sections (strictly equal to twice the number of formants), but also includes a resistive
termination at the glottal end. Hence, for each perturbation of an SM-model parameter

an , an area-function with M equal-length sections is first computed, with the area of
each section taking on the value given by Equation 5.1 at the centre of that section, andassuming A0 = 1 . The discrete and normalised area-function Am , m = 1,..., M is thenconverted to a set of reflection coefficients µ m , m = 1,..., M − 1 , and the glottal (theMth) reflection coefficient is chosen such that 0 < µ M ≤ 1 . A well-known recursivealgorithm (Markel and Gray, 1976) is then used to transform the set of M reflection
coefficients to a set of ( M + 1) LP autoregressive coefficients, which defines the Mthorder polynomial of the LP inverse filter. Finally, the M / 2 formant frequencies are
obtained by solving for the roots of the LP inverse-filter polynomial4.
Figure 5.3 shows a set of nomograms obtained for LP area-functions of M = 8
sections, resulting in the synthesis of only the first four formants. As in Figure 5.1, thepanels on the left-hand side depict formant-frequency nomograms for perturbations ofeach of the first three odd-indexed parameters ( a1 , a3 , and a5 ) separately, while thoseon the right-hand side show perturbations of each of the first three even-indexedparameters ( a2 , a4 , and a6 ). Also shown in each panel are the sets of superimposed,eight-section LP area-functions, each with a fixed total length L = 17.65 cm, whose
shapes are determined according to the single parameter being perturbed about its
neutral (zero) value. Rather than the usual, step-wise depiction of LP area-functions,
4This well-known sequence of procedures for obtaining the formants of a given, discrete area-function,
will henceforth be referred to as “LP synthesis”.

208

2
1.8

1.6

1.6

1.4

1.4

Normalised Area

Normalised Area

2
1.8

1.2
1
0.8
0.6
0.4

1
0.8
0.6
0.4

0.2

0.2

0

0
0

2

4

6
8
10
12
14
Distance from glottis (cm)

16

18

0

4000

4000

3500

3500

Formant Frequency (Hz)Formant Frequency (Hz)

1.2

3000
2500
2000
1500
1000
500

2

4

6
8
10
12
14
Distance from glottis (cm)

0

0.1

0.2

0.3

0.4

16

18

0.4

0.5

16

18

0.4

0.5

2000
1500
1000

0

0.1

0.2

0.3

a2

2

2

1.8

1.8

1.6

1.6

1.4

1.4

Normalised Area

Normalised Area

0.5

2500

0
-0.5 -0.4 -0.3 -0.2 -0.1

0.5

1.2
1
0.8
0.6
0.4

1.2
1
0.8
0.6
0.4

0.2

0.2

0

0
0

2

4

6
8
10
12
14
Distance from glottis (cm)

16

18

0

4000

4000

3500

3500

Formant Frequency (Hz)Formant Frequency (Hz)

0.4

3000

a1

3000
2500
2000
1500
1000
500

2

4

6
8
10
12
14
Distance from glottis (cm)

3000
2500
2000
1500
1000
500

0
-0.5 -0.4 -0.3 -0.2 -0.1

0

0.1

0.2

0.3

0.4

0
-0.5 -0.4 -0.3 -0.2 -0.1

0.5

a3

0

0.1

0.2

0.3

a4

2

2

1.8

1.8

1.6

1.6

1.4

1.4

Normalised Area

Normalised Area

18

500

0
-0.5 -0.4 -0.3 -0.2 -0.1

1.2
1
0.8
0.6
0.4

1.2
1
0.8
0.6
0.4

0.2

0.2

0

0
0

2

4

6
8
10
12
14
Distance from glottis (cm)

16

18

0

4000

4000

3500

3500

Formant Frequency (Hz)Formant Frequency (Hz)

16

3000
2500
2000
1500
1000
500
0
-0.5 -0.4 -0.3 -0.2 -0.1

2

4

6
8
10
12
14
Distance from glottis (cm)

3000
2500
2000
1500
1000
500

0

0.1

0.2

0.3

0.4

0.5

a5

0
-0.5 -0.4 -0.3 -0.2 -0.1

0

0.1

0.2

0.3

a6

Figure 5.3: Nomograms depicting changes in the first four formant frequencies, as a function of
perturbations in each of the first six, vocal-tract shape-parameters of the SM model. Each M= 8-section
area-function is of length L= 17.65cm, and the area scaling factor is A0 = 1.0; the formants aresynthesised using the LP vocal-tract acoustic model, with the glottal reflection coefficient µ M = 1.0(solid curves), 0.7 (dashed curves), and 0.4 (dotted curves).

209we emphasise their shapes by plotting a piecewise-linear representation obtained byjoining the section-centres; these area-functions are therefore merely a more coarselysampled version of those shown earlier in Figure 5.1.
For each formant in Figure 5.3, are shown three curves which have been obtained,
respectively, using a completely lossless implementation of the LP vocal-tract model( µ M = 1 , shown by the solid curve), and two other configurations with increasing
amounts of loss ( µ M = 0.7 , shown by the dashed curve; and µ M = 0.4 , shown by thedotted curve). It is evident that in most cases, the dashed and the solid curves areindistinguishable — only in a few cases, mainly in the higher formants and for extreme
values of perturbation, are the dotted curves visually separable from those pertaining toa more tightly-closed glottis. Figure 5.3 thus confirms that the resistive glottal
termination has only a very small influence on the LP-synthesised formant frequencies.
In particular, as the equivalent glottal opening area is increased by decreasing the valueof µ M , a slightly steeper (more negative) slope is obtained for the first-order relation
between each Fn and the corresponding shape parameter a2 n−1 . Nevertheless, thegeneral behaviour of the formant-frequency nomograms is similar to that observed in
Figure 5.1, with each of the odd-indexed Fourier cosine coefficients claiming first-ordercontrol of a unique formant frequency, and with the even-indexed parameters exerting
only second-order influence by comparison. We therefore conclude that these results
are in agreement with the theoretical predictions of formant frequency behaviour basedon the SM model (Equation 5.2).
In our second, empirical validation of the equivalence of the SM and LP models,
we aim to reveal the strength and the inherent structure of the relations between shapeand resonance parameters, by considering a more extensive range of shape-parameter
perturbations about a neutral tube. Retaining the eight parameters ( an , n = 1,...,8 )which are implied by the SM model (Equation 5.1) for an 8-section area-function, a
total of 6561 different vocal-tract shapes are generated by all possible combinations ofternary perturbations of each parameter ( −0.1, 0.0, + 0.1 ). The area-functions arefound using Equation 5.1, and assuming a fixed vocal-tract length L = 17.65 cm and anormalised area-scaling factor A0 = 1 ; a completely lossless acoustic tube is assumed( µ 8 = 1 ), and the first four formant frequencies are obtained using the LP forward
210

a1

a3

a5

a7

F1(rel)-0.996-0.001

0.000

0.000

F2(rel)-0.001-0.991-0.004-0.001F3(rel)0.000-0.004-0.986-0.010F4(rel)0.000-0.001-0.007-0.997

Table 5.1: Linear correlation coefficients (Pearson’s r) between each of the first four (relative)
formant frequencies LP-synthesised from 6561 perturbations of an 8-section, uniform area-function,and each of the first four odd-indexed VT-shape parameters from which those area-functions weregenerated.

routines. Finally, the coefficient of correlation (or Pearson’s r) is computed across all
6561 configurations, between the shape parameters and the relative formant frequencies
Fn(rel) = ( Fn − Fn(neut) ) / Fn(neut) (i.e., each formant frequency normalised with respect to its

neutral value Fn(neut) = (2n − 1)c / 4 L for the given vocal-tract length, as implied in
Equation 5.2).
Table 5.1 shows the correlations thus obtained. As expected, the table is heavily
dominated by the entries along the main diagonal. Whilst the SM model theoreticallyestablishes the negatively-sloped and quasi-linear relation between the nth relative
formant frequency and the corresponding shape parameter a2 n−1 , the high, negative
correlations along the main diagonal of Table 5.1 summarise the strength of thatlinearity. In addition, the relatively insignificant correlations in all of the off-diagonal
entries of Table 5.1, support the notion of a predominantly one-to-one mapping
between each odd-indexed Fourier cosine parameter and the corresponding formant
frequency. As far as we are aware, this evidence provides the most extensive, empiricalvalidation of the SM model ever reported. More importantly, our results further
validate the equivalence of the SM and the lossless LP model5.
Indeed, the theoretical and empirical evidence presented in this section supports
our hypothesis that the LP model and the SM model are isomorphic in regard to thefundamental relation between formant frequencies and the antisymmetric components of
the logarithmic area-function. This mutual congruence in the basic acoustic properties
5

A complementary proof of the equivalence of those two models, is to show that the LP method of
inversion can be “stepped down” such as to yield area-functions which would have been obtained using
the SM model, starting from the same set of formant frequencies. The interested reader is referred toAppendix D.

211

of the two models can therefore be used to advantage, by adopting a parameterisationof LP-derived area-functions in terms of the Fourier cosine coefficients of the SMmodel (cf. Equation 5.1). Insofar as this type of parameterisation allows a smooth andcontinuous representation of vocal-tract shapes, it immediately resolves the difficulties
of accurately identifying crucial articulatory landmarks such as the place of constriction,and suggests ways to approach the problem of quantitatively comparing two or more
LP-derived area-functions (elaborated in Section 5.5.1). However, the key strength of
the SM-model parameterisation lies in its acoustically-meaningful description of vocaltract shapes, which has hitherto never been attempted within the LP modelling
framework.

5.3.2 Dependence of LP-derived VT-Shapes on Formant Bandwidths
Thus far, we have been fortunate to be able to recall the SM model and unfold it as far
as elucidating the acoustic determinants of the antisymmetric components of LP-derived
vocal-tract shapes. However, despite the acknowledged importance of the formant
bandwidths in securing uniqueness in the LP-based method of inversion, it has neverbeen explained how these acoustic parameters contribute to the shape of LP-derived
area-functions. In the next two sections, we therefore consider this problem from atheoretical and an empirical point of view, respectively.
5.3.2.1 Theoretical Motivation
In this section we present a theoretical derivation of the relation between the formant
bandwidths and the glottal reflection coefficient of the LP vocal-tract model. Whilst asimilar derivation was presented briefly by Kasuya and Wakita (1979), we hereinpresent a more detailed derivation which leads to an expression involving the mean
bandwidth, and which then allows us to foreshadow the bandwidth-dependence of LPderived vocal-tract shapes.
According to the well-known recursive algorithms for converting between thepolynomial (autoregressive) coefficients and the reflection coefficients (e.g., Markel andGray, 1976), the glottal reflection coefficient µ M is equal to the Mth coefficient of theinverse-filter polynomial (see for example Equation 5.10 which shows that, for a212second-order polynomial in z, the constant term is equal to the highest-indexedreflection coefficient µ 2 ). In the same vein, it is relevant to recall Wakita’s (1973)
statement that, as the LP inverse-filter polynomial has only real or complex-conjugateroots, the coefficient of its highest-order term ( z − M ) is determined only by the
bandwidths of those roots. Therefore it is not surprising that the bandwidths should
determine the value of the glottal reflection coefficient, which itself defines the effective
glottal opening area AM +1 , and hence the amount of losses dissipated through thecharacteristic resistance of the glottal termination.
Indeed, a well-known theorem of algebra (e.g., Chrystal, 1964, p.432) states that,for a monic polynomial (such as that of the LP inverse-filter), the constant term (or theMth autoregressive coefficient, which is equal to the glottal reflection coefficient µ M ) is
proportional to the product of all M roots of the polynomial, as follows:µ M = (−1)

M

M

∏z .

(5.13)

m

m =1

Without loss of generality, the LP order of analysis M can be assumed to be even,
thereby reducing the constant of proportionality to positive unity. Further, since theroots of the LP polynomial occur in complex-conjugate pairs, the product of all of thecomplex roots is equal to the product of their magnitudes, such that Equation 5.13 canbe rewritten as follows:Mµ M = ∏ zM .

(5.14)

m =1

It is also well known that the bandwidth of the mth root of the LP inverse-filter
polynomial is proportional to the natural-logarithm of the magnitude of that root in thez-plane, and is given by the following:
Bm = −

Fs
ln z m ,π(5.15)

where Fs is the sampling frequency. The sum of the bandwidths of all M roots cantherefore be expressed as follows:
M

Fs

 M



∑ B = − π ln ∏ z  .
m

m =1

m

m =1

213

(5.16)

Substituting for the product of the magnitude of the roots from Equation 5.14,
and noting also that Fs = cM / 2 L (which is a property common to all vocal-tract
models with a discrete number of sections, such as the LP model), Equation 5.16 canfinally be rewritten as follows:B=−cln(µ M ) ,2πL

(5.17)

where B denotes the mean resonance bandwidth. According to this last result, themean bandwidth of the M / 2 independent poles of an Mth-order LP analysis, is
proportional to the logarithm of the glottal reflection coefficient, and is inversely
proportional to the assumed vocal-tract length.
More importantly, Equation 5.17 states that, for a fixed vocal-tract length, theglottal reflection coefficient is the sole determinant of the mean bandwidth of theformants generated by an LP area-function. This implicates the remaining reflection
coefficients ( µ 1 ,..., µ M −1 ) which define the shape of the vocal-tract, in determining thevalue of the individual formant bandwidths relative to the mean bandwidth B for the
given LP area-function. In the next section we demonstrate the truth of this implication,
and as a result, empirically derive a new set of vocal-tract shape parameters which
relate distinctively to the relative formant bandwidths, analogously to the SM model(Equation 5.2) which relates each odd-indexed Fourier cosine coefficient to a unique,relative formant frequency.
5.3.2.2 Empirical Justifications
With this clear objective in mind, we now endeavour to explore the dependence of LPderived vocal-tract shapes on the formant bandwidths. In particular, we embrace thetheoretically-motivated framework of the SM model, and seek to gain insights into the
bandwidth-dependence of shapes perturbed about a neutral tube. However, rather thanventure a hypothesis on the type of perturbation which might be necessary to induce a
distinctive change in only a single formant bandwidth, we use the LP inverse method to
obtain the vocal-tract shapes corresponding to such acoustic perturbations.
In order to obtain vocal-tract shapes perturbed from the neutral configurationaccording to variations in a single bandwidth at a time, we first need to identify the214formant frequencies and bandwidths of a uniform LP area-function. A cursory
examination of the LP-based nomograms presented earlier (in Figure 5.3) reveals thatthe neutral-tube formant frequencies of the LP model are basically the same as those of
a completely lossless uniform tube of the same length L, as given by the quarterwavelength formula; i.e., Fn = (2 n − 1)c / 4 L , regardless of the value of the glottalreflection coefficient. As the resistive glottal termination of the LP acoustic-tube model
induces energy losses which are frequency-independent, the values of the M / 2
bandwidths of a uniform tube represented by M equal-length sections are identical tothe mean bandwidth, for which we have derived an exact relation (Equation 5.17). For
example, an M-section, uniform LP area-function of total length L = 17.65 cm andglottal reflection coefficient µ M = 0.7 , has the following set of formant parameters:

Fn = (2 n − 1)500 Hz and Bn ≈ 113.5 Hz, for n = 1,..., M / 2 .
The bandwidth-dependence of LP-derived vocal-tract shapes can now be assessedby perturbing a selected bandwidth, while keeping all other formant parameters (i.e., thefrequencies and the remaining bandwidths) fixed at their respective, neutral values.
Assuming a nominal, neutral bandwidth of 100 Hz, each formant bandwidth is chosen in
turn, and perturbed two steps below (50 Hz and 80 Hz) and two steps above (125 Hz
and 200 Hz) the neutral value. The LP inverse method is then used to obtain a vocaltract shape for each set of formant parameters, by the well-known transformations
through LP autoregressive coefficients and LP reflection coefficients. As our aim is todiscern an underlying pattern in the vocal-tract shapes obtained in this way, the firstseven formants are specified in order to secure M = 14 vocal-tract sections, which does
permit a satisfactory spatial resolution for visually identifying the general form of thearea-function perturbation.
Figure 5.4 shows the 14-section, LP-derived vocal-tract shapes obtained by theprocedure just described, for changes in each of the first three formant bandwidths. It is
immediately apparent that perturbation of only the nth formant bandwidth, induces anapproximately sinusoidal perturbation of the area-function, with 2n − 1 half-cycles
within the length of the vocal-tract from the glottis to the lips. Recall from Section 5.2.2that the area scaling factor A0 is acoustically inconsequential in both the completely
lossless and the LP vocal-tract model; hence, the area-functions are plotted in such a
215

B1
2.5

2

1.5

1

0.5

0
0

2

4

6

8

10

12

14

16

10

12

14

16

10

12

14

16

B2
1.4

1.2

1

0.8

0.6
0

2

4

6

8

B3
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0

2

4

6

8

Distance from glottis (cm)Figure 5.4: Bandwidth-dependence of LP-derived vocal-tract shapes. Each formant bandwidth Bn isperturbed over five values {50, 80, 100, 125, 200}Hz, with all other bandwidths fixed at the nominal,
neutral value 100Hz; the formant frequencies are fixed at neutral values Fn = (2n-1)c / 4L, and the vocaltract length is fixed at L=17.65cm. The top, middle, and bottom graphs show the LP-derived areafunctions for perturbations in B1, B2 and B3, respectively. Seven formants are used in order to obtain
14-section LP area-functions (solid lines); at about the mid-length of each area-function, the largest
through smallest area corresponds to the increase in bandwidth from 50Hz to 200Hz as listed above.
Dashed curves: envelopes of the 1st, 3rd, and 5th sinusoidal components for the given L, superimposedon the step-wise area-functions in order to emphasise the functional form of the shape perturbations.

216

way as to visually emphasise the sinusoidal component, by setting the mean logarithmic
area to zero in the bottom two graphs, and by setting a constant area for the firstsection in the top graph.
The significance of the LP-based result portrayed in Figure 5.4, is perhaps best
appreciated in the context of the SM model. In Section 5.3.1 we established a link
between the SM model and the LP model, by showing that the latter does inherit thefundamental relation between antisymmetric shape components and perturbations in theformant frequencies. In Figures 5.1 and 5.3 we then showed that for both the SM andLP models, respectively, the formant frequencies are relatively independent of thesymmetric shape components. Remarkably, Figure 5.4 reveals that when a completely
lossless vocal-tract model is augmented with a resistive termination at the glottal end(as in Wakita’s formulation of the LP model), symmetric (sinusoidal) perturbations of
the uniform area-function are associated with distinctive variations in the resulting
formant bandwidths. Our results therefore elucidate the bandwidth-dependence of LPderived area-functions, and furthermore suggest a functional form of the required,symmetric shape perturbations (as shown in Figure 5.4 by the superimposed, dashedcurves), in terms of the odd-indexed coefficients of the Fourier sine (rather than the
SM model’s original Fourier cosine) series of the logarithmic area-function.
Having thus identified the shape components which fundamentally relate to theindividual bandwidths, a more complete assessment of the implied, shape-resonancerelations is afforded by nomograms prepared using the LP model. Considering only the
first four formants, LP area-functions are each represented by M = 8 sections of equal
length. Each of the shape parameters (i.e., the odd-indexed Fourier cosine coefficientsa2 n−1 and the newly-implicated, odd-indexed Fourier sine coefficients, henceforthdenoted as b2 n−1 ) is perturbed across the range [−0.8,+0.8] in steps of 0.05, while
keeping all other parameters fixed at zero value. Either discrete cosine expansion (forthe a2 n−1 ) or discrete sine expansion (for the b2 n−1 ) is then used to obtain 8-section
area-functions of fixed length L = 17.65 cm, from which the first four formant
frequencies and bandwidths are LP-synthesised (assuming a nominal value for theglottal reflection coefficient µ M = 0.7 which, according to Equation 5.17, yields a
reasonably realistic, mean bandwidth of B ≈ 113.5 Hz).
217

750

750
a{1}
a{3}
a{5}
a{7}

700
650

F1 (Hz)

650

600

600

550

550

500

500

450

450

400

400

350

350

300
-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

2200
a{1}
a{3}
a{5}
a{7}

1800

1400

1400

1200

1200

1000

1000

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

3400
a{1}
a{3}
a{5}
a{7}

3000

800
-0.8

2600

2600

2400

2400

2200

2200

2000

2000

1800

1800
-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1600
-0.8

4000

4000

3800

3800

3600

3600

3400

3400

3200

3200

3000

2400
2200
-0.8

0.2

0.4

0.6

0.8

b{1}
b{3}
b{5}
b{7}

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

b{1}
b{3}
b{5}
b{7}

3000
2800

2600

0

3200

2800

2800

-0.2

3400

3200

F4 (Hz)

-0.4

1800
1600

1600
-0.8

-0.6

2000

1600

800
-0.8

F3 (Hz)

300
-0.8
2200

2000

F2 (Hz)

b{1}
b{3}
b{5}
b{7}

700

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0.6

0.8

3000
a{1}
a{3}
a{5}
a{7}

-0.6

2800

b{1}
b{3}
b{5}
b{7}

2600
2400
-0.4

-0.2

0

0.2

0.4

0.6

0.8

a2 n−1

2200
-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

b2 n−1Figure 5.5(a): Formant-frequency nomograms generated by perturbing a uniform, 8-section LP areafunction, according to each of the first four, odd-indexed Fourier cosine (∆a2n-1, left graphs) and sine(∆b2n-1, right graphs) shape-components, for n= 1 (diamond symbols joined by green lines), n= 2 (plussymbols joind by blue lines), n= 3 (square symbols joined by red lines), and n= 4 (cross symbols joinedby black lines).

218

220
200
180
160

B1 (Hz)

220
a{1}
a{3}
a{5}
a{7}

200

160

140

140

120

120

100

100

80

80

60

60

40
-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

300
a{1}
a{3}
a{5}
a{7}

200

150

150

100

100

50

50

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

350
a{1}
a{3}
a{5}
a{7}

250

0
-0.8

0

0.2

0.4

0.6

0.8

b{1}
b{3}
b{5}
b{7}

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

200

150

150

100

100

50

50

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

b{1}
b{3}
b{5}
b{7}

250

200

0.8

0
-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

350
a{1}
a{3}
a{5}
a{7}

300
250

250
200

150

150

100

100

50

50

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

b{1}
b{3}
b{5}
b{7}

300

200

0
-0.8

-0.2

300

350

B4 (Hz)

-0.4

350

300

0
-0.8

-0.6

250

200

0
-0.8

B3 (Hz)

40
-0.8
300

250

B2 (Hz)

b{1}
b{3}
b{5}
b{7}

180

0.8

a2 n−1

0
-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

b2 n−1Figure 5.5(b): Formant-bandwidth nomograms generated by perturbing a uniform, 8-section LP
area-function, according to each of the first four, odd-indexed Fourier cosine (∆a2n-1, left graphs) andsine (∆b2n-1, right graphs) shape-components, for n= 1 (diamond symbols joined by green lines), n= 2
(plus symbols joind by blue lines), n= 3 (square symbols joined by red lines), and n= 4 (cross symbols
joined by black lines).

219

Figure 5.5(a) shows the influence of each of the first four parameters a2 n−1 (leftgraphs) and b2 n−1 (right graphs) on each of the first four formant frequencies. It is quite
clear that the first-order influence of each parameter a2 n−1 , is a negatively-sloped,
quasi-linear variation in the corresponding formant frequency Fn . By comparison, theinfluence of each of the remaining Fourier-cosine shape parameters on that formant
frequency, as indeed of each of the Fourier-sine shape parameters shown in the righthand graphs, is relatively minor. The largest of these second-order effects appear in theF3 nomograms for perturbations in a7 (left graph) and in b7 (right graph), which
describe the antisymmetric and the symmetric shape-component, respectively, of highest
spatial resolution for the 8-section LP model. An interesting phenomenon is observed in
the bottom-left graph of Figure 5.5(a), where there appears a discontinuity in the
nomogram for F4 , as a7 is reduced below about −0.5 . This is merely an artefact of the
8-section vocal-tract model, which prohibits measurement of formants higher than theeffective upper spectral limit (or half-sampling frequency), equal to 4kHz for the given
vocal-tract length. Notwithstanding such analysis artefacts, the nomograms in Figure5.5(a) further support our earlier evidence concerning the distinctive influence ofantisymmetric vocal-tract shape perturbations on the formant frequencies. Furthermore,they confirm that symmetric shape perturbations caused by variations in the oddindexed Fourier sine components (which we earlier associated with the formant
bandwidths), indeed have relatively little effect on LP-synthesised formant frequencies.
The analogous set of formant bandwidth nomograms for the 8-section LP model isshown in Figure 5.5(b), where the graphs on the left and on the right again pertain to
perturbations in the parameters a2 n−1 and b2 n−1 , respectively. These nomograms do
support our earlier hypothesis of a distinctive relation between each odd-indexed
Fourier sine parameter b2 n−1 and the corresponding bandwidth Bn . In particular, theyshow that the relation is negatively-sloped and quasi-linear. However, a comparison of
Figure 5.5(b) with Figure 5.5(a) reveals that the so-called second-order influences on
the bandwidths are relatively greater than those on the formant frequencies. For
example, the variations in B1 caused by perturbations in a3 appear to be even larger inmagnitude than the so-called first-order effect of b1 on that bandwidth. Nevertheless,similarly to the first-order formant-frequency nomograms shown on the left in Figure220

5.5(a), the first-order nomograms shown on the right in Figure 5.5(b) are eachcharacterised by a more strongly negative slope than any of the other curves for thesame bandwidth.
Our earlier, theoretical motivations (in Section 5.3.2.1), together with theempirical evidence just presented, strongly suggest that the odd-indexed Fourier sinecomponents of LP-derived vocal-tract shapes are distinctively related to the formant
bandwidths, whose very existence secures uniqueness in the LP-based method of
inversion. Our results thereby elucidate the uniqueness property of LP-derived areafunctions, for the first time in terms of their underlying shape components. Together
with the earlier results presented in Section 5.3.1, we have indeed identified the intrinsic
parameters of unique LP area-functions.

5.4 Method of Area-Function Parameterisation and Estimation
Our results presented in the previous section, were concerned with the formant
dependence of LP-derived vocal-tract shapes. In particular, in Section 5.3.1 weconfirmed the validity of the SM model within the LP modelling framework, and in
Section 5.3.2 we then provided a theoretical motivation and empirical justifications
regarding the components of LP-derived area-functions which depend on the formant
bandwidths. In order to fulfill our requirements as set out in Section 5.1, in the next
section we unite the newly-identified parameters of unique, LP-derived area-functions,and thereby propose a hybrid, LP-SM method of inversion. In Section 5.4.2 we thenuse the inversion method to evaluate our proposed parameterisation of vocal-tract
shapes.

5.4.1 Description
From our theoretical and empirical results presented in Section 5.3, follows our
proposed extension of the SM model (cf. Equation 5.1) to include two important sets ofacoustic parameters. In particular, our results concerning the formant frequencydependence of LP-derived area-functions suggest that the asymmetric shapecomponents a2 n−1 of the original SM model be indeed retained; and our results
concerning the bandwidth-dependence of LP-derived area-functions suggest that the221symmetric shape components, originally represented (Schroeder, 1967; Mermelstein,1967) by the even-indexed Fourier cosine coefficients a2 n , be replaced with the oddindexed Fourier sine coefficients b2 n−1 . Our proposed method of area-function
parameterisation is therefore given by the following expression:(2n − 1)πx (2n − 1)πx 
ln A( x ) = ln A0 + ∑ a2 n−1 cos + ∑ b2 n −1 sin
,




L
L
n =1
n =1
M /2

M/2

(5.18)

where M is assumed to be even, without loss of generality.
Perhaps the most distinctive feature of this new parameterisation, is that itprovides a mathematically complete and minimal description of vocal-tract shapes (upto the desired degree of smoothness M). Indeed, the sine and cosine terms in Equation
5.18 form a mutually orthogonal set of basis functions which describe the vocal-tract
shape in terms of the acoustically-relevant, spatial components. Furthermore, thecompleteness afforded by this set of basis functions does confirm both the necessity andthe sufficiency of the formant frequencies and bandwidths.
As the principle of orthogonality precludes compensatory relations amongst ourchosen parameters, the uniqueness property which is inherent to the LP vocal-tract
model is thereby retained. However, whilst it is easily shown that all of thetrigonometric terms in Equation 5.18 are mutually orthogonal, it is important to notethat the symmetric components do have a non-zero mean, spatial level, and that theb2 n−1 coefficients are therefore not strictly orthogonal to the area scaling factor A0 .
This implies that in parameterising a given area-function according to Equation 5.18,
the sine coefficients can only be uniquely determined if a consistent method is adopted
to first determine the area scaling factor.
Towards this end, our proposed method takes advantage of the boundary
conditions of each of the odd-indexed Fourier sine terms which, by definition, have zero
value both at x = 0 and at x = L . In particular, we first determine the antisymmetriccomponents of the given, logarithmic area-function (by odd-indexed, discrete cosine
transformation), and subtract them from the original vocal-tract shape in order to yielda primarily symmetric area function. (If the antisymmetric shape components could bedetermined completely by identifying an infinite number of odd-indexed Fourier cosine222coefficients, then the area function obtained after subtraction would be purely
symmetric). The area scaling factor is then determined such that ln A0 equals the meanof the logarithmic areas at the glottal and at the lip ends. Having thus defined A0 , thesymmetric shape parameters b2 n−1 are found uniquely by odd-indexed, discrete sinetransformation. This simple and consistent method also ensures that the first M / 2 of
the a2 n−1 and the b2 n−1 parameters found for a given value of M, are exactly replicatedwhen a higher value of M is used to parameterise the same area-function.
The block diagram of Figure 5.6 shows how our proposed method of areafunction parameterisation is used within the LP-based inversion method. Central to themethod is the well-known set of recursive routines to convert, in turn, from formant
frequencies and bandwidths (and the effective LP sampling frequency, as determined by
the number of formants M / 2 and the assumed vocal-tract length L), first to LP
autoregressive coefficients, then to LP reflection coefficients, and finally to a step-wise,
LP area-function. The raw, M-section LP area-function thus obtained for a particularvalue of the vocal-tract length, is then parameterised according to Equation 5.18 (asdescribed in the preceding paragraph) in order to determine the shape parameters{a 2 n−1 , b2 n−1 } n = 1,..., M / 2 . This entire procedure is repeated for a range of vocaltract lengths, and that length is finally chosen which minimises the eccentricity of the
vocal-tract shape (the so-called minimal articulatory distance, or MAD) with respect toa uniform area-function. Analogously to Paige and Zue’s (1970) derivation which was
based on the original SM model, it can easily be shown (e.g., using Parseval’s theorem)that the MAD criterion to be minimised when using our parameterisation method(Equation 5.18), is given by the following expression:

MAD 2 =1 L
1 M /2 2
2
=
ln
A
(
x
)
dx
a2 n −1 + b22n −1
[
]
∑
∫
0
2 n =1L[](5.19)

Consistently with the SM model, logarithmic areas are used in the articulatory distancemeasure in order to place greater emphasis on differences in the acoustically-salient
locations of constriction along the vocal tract.
Although our method of area-function estimation illustrated in Figure 5.6 is similar
to that proposed by Wakita (1977), who was indeed the first to combine the LP-based223

LP Inversion Method
Target Formants{Fn,Bn}, n=1,...,M/2

fb2a

a2µ

µ2areaDiscrete
LP Area-Function
Ai, i=1,...,M

Fs=cM/2L

Area-Function
Parameterisation(Equation 5.18)

L
Optimisation of L
to minimise εVT-Shape Parameters{a2n-1, b2n-1}, n=1,...,M/2

Computation of
MAD criterion

εFigure 5.6: Block-diagram of proposed area-function estimation method, valid for L < Lmax.

LP Inversion Method
Target Formants{Fn,Bn}, n=1,...,M/2

a2µ

fb2a

µ2areaDiscrete
LP Area-Function
Ai, i=1,...,M+2

Area-Function
Parameterisation(Equation 5.18)

Fs=c(M+2)/2L
Fn, Bnn=(M/2)+1

L

Initialise a higher formant:
F(M/2)+1 = c(M+1)/4L
B(M/2)+1 = (B1+...+BM/2)/(M/2)
Optimisation of L
to minimise εεOptimisation ofF(M/2)+1 and B(M/2)+1
until aM+1 = bM+1 = 0

VT-Shape Parameters{a2n-1, b2n-1}, n=1,...,(M/2)+1

Computation of
MAD criterion

Figure 5.7: Block-diagram of proposed area-function estimation method, to be used when an
optimum vocal-tract length L < Lmax could not be found using the method of Figure 5.6.

224

inversion method with the vocal-tract length optimisation criterion of Paige and Zue(1970), our proposed method does extend the work of these authors. First, therestrictively coarse, step-wise representation of the vocal-tract shape yielded by the LPmodel, is replaced by a smooth outline from which crucial articulatory landmarks suchas the place of constriction can be determined more accurately. Note that the degree ofsmoothness depends on the number of shape parameters used (cf. Equation 5.18), andultimately, on the number of formants used in the inversion. We shall return to thisimportant aspect of the proposed model in Section 5.4.2.2, where we evaluate therepresentational fidelity of the vocal-tract shape parameters, as a function of the degreeof smoothness M.
As a result of the smooth representation, the computation of the MAD criterion
used to optimise the vocal-tract length is no longer restricted to the entire interval[0, L] . Indeed, if so desired, the integral in Equation 5.19 can be numerically computed
across any selected interval [ x1 , x 2 ] along the length of the vocal-tract, and different,
physiologically-motivated sections of the area-function can thus be weighted differently.
In that context, Yehia and Itakura (1994) determined from area-functions measured on
mid-sagittal X-ray images of a single speaker of Japanese, that the two regions near the
mid-length of the vocal-tract and close to the glottis, respectively, are less likely thanother regions to exhibit large phonetic variations, and should therefore be given extra
weight in the computation of MAD with respect to that speaker’s average areafunction.
Our proposed method of inversion also overcomes the limitation of an upperbound Lmax on allowed vocal-tract lengths which, as discussed in Section 5.2.2, isinherent to all vocal-tract models which have only a limited number of sections. If the
MAD criterion is not found to have a minimum for a vocal-tract length less than Lmax
(which itself depends on the value of the highest formant frequency; cf. Equation 5.6),
Wakita (1977) offers no alternative but to accept this upper limit as the final solution.
However, it is interesting to note that Zue (1969) had already addressed this issue in his
own inversion method, which was based on a completely lossless acoustic-tube model,
and which therefore made use of the zeros (formants) and poles of the lip impedance
function. Indeed, Zue (1969) found that the frequency of the highest singularity often
225exceeded the upper spectral limit (i.e., the effective half-sampling frequency), whichitself is determined by the number of sections and by the length of the vocal-tract. In
order to overcome this limitation while retaining a fixed number of vocal-tract sections,
he augmented the low-order poles and zeros with a number of higher-order singularities
whose frequencies were chosen to be those of a uniform tube of the same length. Thissolution to the problem is particularly attractive, since it is founded on the theoreticalresult that the higher-order singularities of a non-uniform area-function converge tothose of a neutral tube of the same length. It can also be shown (simply by substituting
the well-known quarter-wavelength formula for the highest given formant frequency,
into Equation 5.6) that if the highest singularity is equal to its neutral value for a given
vocal-tract length, then that length is always less than the upper limit Lmax, and the
problem is thus resolved.
Our own solution to this problem in the context of the LP-based inversion methodof Figure 5.6, is indeed founded on the principle of adding a higher formant to the given
list of target formants. However, we transcend the simple assumption of a neutral
higher formant, by invoking the very property of our resonance-based parameterisation
which predicts that the additional, higher formant will primarily influence the vocal-tract
shape components of correspondingly higher resolution. Whilst a neutral value for thehigher formant only aspires towards affecting the resulting vocal-tract shape as little aspossible, our method of area-function parameterisation can be used to explicitly enforce
this outcome.
Figure 5.7 shows the block-diagram of our method of area-function estimation
which is used in case the earlier method outlined in Figure 5.6 fails to minimise theMAD criterion for a vocal-tract length less than Lmax. At each value of L higher thanthat upper limit, a single, higher formant is introduced, with a frequency which is
initially set to its neutral value for the given L, and with a bandwidth which is initially
set to the mean of the target bandwidths (following our results of Section 5.3.2.1). As aresult of the additional formant, the effective sampling frequency is increased by

c / L Hz; the number of vocal-tract sections is increased by 2; and the discrete, LP areafunction is then parameterised with two more shape parameters ( a M +1 and bM +1 ) than
had previously been used. However, in order to maintain consistency with the vocal226

tract shapes obtained at the original (lower) shape-resolution, we insist that the higher
formant must not itself contribute to additional, higher-resolution components of theresulting vocal-tract shape. In particular, as shown by the inner loop in Figure 5.7, weiteratively optimise both the frequency and bandwidth of the additional, higher formant(using steepest-descent optimisation), such that the resulting, parameterised LP areafunction retains its original degree of smoothness, i.e., F( M / 2 )+1 and B( M / 2) +1 are jointly
optimised until a M+1 = 0 and bM +1 = 0 , to within a specified tolerance — only then do
we proceed to compute the MAD criterion for the area-function obtained at thatparticular vocal-tract length. As indicated by the outer loop in Figure 5.7, this entire
procedure is repeated in search of an optimum vocal-tract length which minimises the
MAD criterion, and which then yields the final, parameterised area-function.

5.4.2 Evaluation of Proposed Area-Function Parameterisation
Whilst our method of area-function parameterisation (Equation 5.18) follows naturally
from our earlier derivation of the parameters of unique LP-derived area-functions, it
still warrants an evaluation of its effectiveness. We shall therefore evaluate our
proposed parameterisation, first (in Section 5.4.2.1) by using our hybrid method of
inversion to obtain the correlations between synthetic formant and estimated shapeparameters; and then (in Section 5.4.2.2) by testing the ability of our shape parameters
to capture the spatial characteristics of directly-measured area-functions obtained fromthe literature.
5.4.2.1 Inter-parameter Correlations
We herein aim to evaluate the acoustic-phonetic relevance of our proposed set of vocaltract shape parameters, in the context of the inversion method described in the previous
section. In particular, we would like to confirm that the statistical correlation between
each formant parameter and the corresponding shape parameter, is indeed as significant
as suggested in our results of Section 5.3. This hypothesis is tested by first synthesisingformant values from a large number of different area-functions, then applying our
hybrid LP-SM method of inversion to re-estimate the shape parameters and evaluate
their correlation with the synthetic formants.

227

First, area-functions are generated by all combinations of ternary perturbations( −0.1 , 0.0 , and +0.1 ) of the first four cosine parameters ( a2 n −1 , n = 1,...,4 ) and of thefirst four sine parameters ( b2 n−1 , n = 1,...,4 ), thus yielding a total of 6561 distinct
shapes, with a fixed vocal-tract length L = 17.65 cm and a normalised area scaling
factor A0 = 1.0 . Assuming a nominal value for the glottal reflection coefficient( µ 8 = 0.7 , for which the mean of the first four bandwidths is predicted by Equation
5.17 to be B ≈ 113.5 Hz), the formant frequencies and bandwidths of all 6561 perturbed
vocal-tract shapes are LP-synthesised. Those synthetic formants are then used to reestimate the area-functions, using our hybrid LP-SM method of inversion.
Consistently with the SM model (and similarly to our earlier investigation in
Section 5.3.1.2), correlations between vocal-tract shape and resonance parameters arecomputed using the relative formant frequencies

Fn(rel) = ( Fn − F (neut) ) / F (neut) ;consistently with our results of Section 5.3.2 where we found that the shape of the LP
area-function determines only the formant bandwidth pattern about the mean value,correlations are computed using the relative bandwidths Bn(rel) = ( Bn − B ) / B . The first
four, relative formant frequencies and bandwidths are then paired with each of the eight
vocal-tract shape parameters in turn, and the coefficient of correlation (or Pearson’s r)
is computed across all 6561 configurations.
Indeed, the high correlations along the main diagonal of Table 5.2 do provide
support for a strongly linear relation between each relative acoustic resonance and thecorresponding vocal-tract shape parameter. The weakest of these correlations ( −0.917 )summarises the strength of the linear relation between the relative second formant
bandwidth and the corresponding shape parameter b3 , while the strongest correlation( −0.996 ) holds between the relative F4 and the corresponding shape parameter a7 . By
comparison, the most significant, off-diagonal correlation ( +0.550 ) is that which
describes the strength of the relation between the relative third formant bandwidth andthe highest-indexed Fourier sine coefficient b7 . As already foreshadowed in ournomograms of Figure 5.5, the magnitude of the off-diagonal elements in Table 5.2
suggest that the influence of certain asymmetric shape parameters a2 n−1 on the formant
bandwidths, is not entirely insignificant. However, as we have already discussed, thefirst-order effect retains its distinctiveness owing to the negative sign of induced
228

a1

a3

a5

a7

b1

b3

b5

b7

F1(rel)-0.995

0.143

0.170

0.213

0.245

0.092

-0.005-0.175F2(rel)0.150-0.990

0.173

0.225

0.249

0.119

-0.012-0.189F3(rel)

0.174

0.178

-0.983

0.241

0.274

0.129

0.004

-0.224F4(rel)

0.216

0.229

0.249

-0.996

0.334

0.154

0.008

-0.274B1(rel)

0.124

0.528

0.333

0.113

-0.922-0.165

0.174

0.377

B2(rel)-0.066

0.041

0.461

0.135

-0.199-0.917

0.299

0.464

B3(rel)-0.020-0.243-0.062

0.263

0.236

0.327

-0.9250.550

B4(rel) -0.007 -0.114 -0.426 -0.357 0.432 0.421 0.449 -0.938
Table 5.2: Linear correlation coefficients (Pearson’s r) between each of the first four, relative formant
frequencies and bandwidths LP-synthesised from 6561 perturbations of an 8-section, uniform areafunction (with the glottal reflection coefficient fixed at a nominal value µ 8 = 0.7), and each of the firsteight VT-shape parameters re-estimated from those formant data using the hybrid LP-SM method of
inversion described in Section 5.4.1.

perturbations in the acoustic parameters, which is here reflected in the strongly negative
correlations along the main diagonal of Table 5.2.
These correlations do lend strong support to our proposed method of vocal-tract
shape parameterisation. Indeed, our proposed modification and extension of the SMmodel is here justified quantitatively, by the strength of the one-to-one mapping
between the odd-indexed Fourier sine parameters and the LP formant bandwidths. It isalso interesting to note that the strong correlations obtained between the first-order
pairs of formant and shape parameters, do indirectly underscore the appropriateness of
the MAD criterion which was used in the inversion.
5.4.2.2 Representation of Directly Measured Area Functions
Our evaluation thus far has established the quasi-linear, one-to-one mapping between
vocal-tract shape parameters and the LP-based formant frequencies and bandwidths.
Equally relevant to this, acoustically-motivated evaluation of the parameterisation, is toconsider its accuracy in merely representing area-functions obtained from direct
measurements of the human vocal-tract. In that context, it is important to first evaluate
the inherent smoothing of directly-measured area-functions, and then to assess therelative contribution of each parameter in representing those vocal-tract shapes.

229

The vocal-tract shapes which we shall use for this purpose, are taken frompublished works in which the authors have tabulated the area-functions measured using
either X-ray imaging or magnetic resonance imaging (MRI) techniques. Whilst suchdirectly-measured data are themselves prone to various, non-negligible sources of error(e.g., the necessary assumptions on the lateral width of vocal-tract sections when
mapping mid-sagittal cross-dimensions to cross-sectional areas), it is generally acceptedthat they do provide, simply by virtue of their having been acquired by direct
observation, the most accurate information that one might currently obtain on the entire
shape of the human vocal-tract during vowel production.
The six, X-ray measured area-functions of Russian vowels given by Fant (1960),
are perhaps the most celebrated and widely used data on vocalic area-functions, and formany years remained the only, relatively complete and easily-accessible of such data.
More recent, MRI methods apparently have none of the potential hazards to which
subjects are exposed during X-radiography, and in addition they can be used toreconstruct a volumetric profile of the vocal-tract airway. Nevertheless, they still sufferfrom a number of disadvantages, which include the requirement of repeatedly sustained
articulations due to the long image-acquisition time, and the poor detection of bone
structures such as the teeth, due to their low concentrations of hydrogen.
Notwithstanding these limitations, a number of recent studies have used MRI to obtain
invaluable area-function measurements which they have also generously published. Baeret al. (1991) provide area-functions of the four point vowels for one adult, male speakerof American English and one adult, male speaker of British English; Yang and Kasuya(1994) list area-functions of five vowels produced by two adult speakers (one male andone female) and one child, male speaker of Japanese; and Story et al. (1996) tabulate
the area functions of 11 sustained vowels of a single, male speaker of American English.
In addition, Beautemps et al. (1995) list the area-functions of the three cardinal vowels
which were initially obtained from mid-sagittal X-ray images of a single, male speakerof French, and which were then refined by an optimisation procedure (applied to themapping from cross-dimensions to areas) in order to minimise the error between themeasured and resynthesised formants. In total, these five studies offer 33 directlymeasured, vocalic area-functions produced by six adult, male speakers of four different
230languages, including two dialects of English.
As the three area-functions of French vowels given by Beautemps et al. (1995) are
originally listed with unequal section lengths, we first re-sample them by linear
interpolation of the logarithmic areas in order to obtain equal-length sections not longer
than the smallest original section-length (approximately 0.3 cm). Similarly, Yang andKasuya’s (1994) five area-functions of Japanese vowels are re-sampled at equal-lengthintervals no greater than 0.5 cm, owing to their treatment of the area at the lip opening
as a separate section of unspecified length, which we here assume to have a nominal
value of 0.01 cm. Each of the 33 directly-measured (and possibly re-sampled) areafunctions is then re-scaled (or normalised) to a mean logarithmic value of zero,consistently with the acoustical insignificance of the area-scaling factor in the LP vocaltract model.
According to the well-known Nyquist sampling theorem, the number of
independent parameters used to model a given area-function must not exceed the total
number of vocal-tract sections. Each of the 33, re-scaled area-functions is therefore
parameterised (as described in Section 5.4.1) with M equal to the highest, even integer
no greater than the number of vocal-tract sections. The area-functions with the fewest
number of sections (18) are those given by Baer et al. (1991); consequently, each of the33, re-scaled and parameterised area-functions is re-expanded according to Equation
5.18, with M / 2 ranging from 0 to 9, the former yielding a uniform area-function with
only an area-scaling factor, and the latter yielding a smoothed vocal-tract shape with thefirst 18 Fourier-series terms. A quantitative measure of the goodness of fit at each
degree of smoothness (integer-value of M / 2 ), is obtained by computing the rootmean-square (rms) error between each logarithmic area-function and its smoothedversion, computed at the centre of each original, equal-length section.
The diamond symbols (joined by solid lines) in Figure 5.8 show the average rms
error in smoothing the 33 directly-measured area functions, at each value of M / 2 . As
expected, the rms error is a monotonically decreasing function of the number (M) of
shape parameters used to represent each area-function. Our results clearly show that thefirst two, sine and cosine terms are the most important spatial components of directlymeasured, vocalic area-functions, and that only small, incremental improvements are231gained by including terms at ( M / 2) = 3 and higher. This observation agrees with theconclusions drawn by Liljencrants (1971), who found it necessary to use only the firsttwo (even-indexed) Fourier sine and cosine terms to represent X-ray measured, midsagittal tongue-contours mapped onto a Cartesian coordinate system.
As the degree of spatial resolution M / 2 is identical to the number of formants
used in the inversion, the diamond symbols in Figure 5.8 provide an optimistic
benchmark for the rms error to be expected when using the inversion method. For
example, if we assume that only the first four formants are available in general, the solid
curve in Figure 5.8 suggests an average, lower-bound rms error of about 0.22 whencomparing estimated vocal-tract shapes with so-called exact, or directly-measured areafunctions. Errors in excess of this benchmark could then be ascribed to the estimation
method itself, rather than the parameterisation.
In order to assess the relative contribution of antisymmetric and symmetriccomponents of directly-measured area functions, the rms errors were also computedwith respect to the smoothed vocal-tract shapes obtained by re-expanding only the
Fourier cosine terms, and only the Fourier sine terms, respectively, at each value ofM / 2 . The averaged results shown in Figure 5.8, clearly illustrate the nearly equal
contribution of antisymmetric components (plus symbols, joined by dashed lines) andsymmetric components (square symbols, joined by dotted lines). These results alsosuggest that on average, the first two cosine components ( a1 and a3 , whose
contributions are shown along the dashed lines as M is increased from 0 to 1, and from
1 to 2, respectively) and the first two sine components ( b1 and b3 , whose contributions
are shown along the dotted lines as M is increased from 0 to 1, and from 1 to 2,
respectively) are the most important spatial components of the 33 area-functions.
The results shown in Figure 5.8 and discussed above, were concerned with therms errors averaged over all of the 33 area-functions. However, as foreshadowed bySchroeder (1967) and Mermelstein (1967), individual vocal-tract shapes of certain
vowels can be strongly associated with either symmetric or antisymmetric components.
For example, Schroeder and Mermelstein found that the area-function of the Russianvowel /u/ given by Fant (1960), is very poorly represented by antisymmetric shapecomponents alone, and can not therefore be adequately estimated by the SM model,
232

1.2

antisymmetric + symmetricantisymmetric only
symmetric only1

rms error

0.8

0.6

0.4

0.2

0
0

1

2

3

4
5
6
Spatial resolution (M/2)

7

8

9

Figure 5.8: Average root-mean-square (rms) error in representing 33 directly-measured areafunctions obtained from the literature (see text). Each area-function is first parameterised according toEquation 5.18, with the total number of shape-parameters (a2n-1 and b2n-1) equal to the highest eveninteger not greater than the number of equal-length vocal-tract sections. Representations atincreasingly higher degrees of spatial resolution (M/2 in Equation 5.18) are then obtained bytruncating the parameters at M/2={0,1,2,...,9}, and the rms error of fit is computed between thesmoothed and the original, logarithmic area-functions. Diamond symbols: average rms error in
representation of each area-function using both antisymmetric (a2n-1) and symmetric (b2n-1) shapecomponents. Plus symbols: average rms error in representation using only antisymmetric (a2n-1) shapecomponents. Square sumbols: average rms error in representation using only symmetric (b2n-1) shapecomponents.
using only the formant frequencies. Their findings are confirmed by our own results(shown in Appendix E) of the individual rms profiles for each area-function, whichindicate that mid- to high-, back vowels (e.g., Fant’s (1960) Russian /u/; Baer et al.’s(1991) British English /u/ and American English /u/; Yang and Kasuya’s (1994)
Japanese /u/ and /o/; Beautemps et al.’s (1995) French /u/; and Story et al.’s (1996)
American English /o/, /U/, and /u/) most consistently exhibit vocal-tract shapesymmetry, owing presumably to the location of the linguo-velar constriction at
approximately midway along the length of the vocal-tract.
By contrast, the two point vowels /i/ and /a/, which are usually associated withextremes of lingual articulation in the high-front and in the low-back parts of the vocal
tract, are the most consistently antisymmetric of the directly-measured area-functions233

(except for Beautemps et al.’s (1995) French /a/, which admittedly has a more
symmetric shape, owing to the much smaller and decreasing areas towards the lip end).
From our quantitative evidence may be inferred the physiological point of view thatasymmetry in vocal-tract shapes implies conservation of the mass of the tongue body —for example, when the tongue moves forwards and upwards to form a constriction for ahigh-front vowel, the pharyngeal cavity is endowed with a proportionately larger
volume; and when it moves backwards and downwards to form a constriction for a
low-back vowel, larger areas are naturally attained in the front, oral cavity.
Our results can also be interpreted from an acoustical point of view, as aconsequence of the close relationships which we have shown to exist between
antisymmetric shape components and the formant frequencies, and between symmetricshape components and the formant bandwidths. Indeed, the relative dominance of thefirst two Fourier-series shape components, does confirm the relative importance of thefirst two formants in vowel production. The significant contribution of the twoantisymmetric shape parameters a1 and a3 in particular, provides indirect proof(gleaned here only from results of shape parameterisation) that F1 and F2 are indeed
the most important acoustic determinants of the phonetic identity of vowels. At thesame time, however, the significance of the symmetric shape parameters in representing
directly-measured area-functions, is consistent with their importance in securing
uniqueness in LP-derived vocal-tract shapes.

5.5 Evaluation of Estimation Method
Evaluation of any proposed method of inversion has always been a major weakness instudies concerned with acoustic-to-articulatory mapping. Whilst confidence is usuallyraised if acoustic and articulatory data are simultaneously measured, existing sources oferror in directly measured articulatory data (some of which were discussed earlier, in
Section 5.4.2.2) do suggest that they should not be given carte blanche as the ultimate
reference. On the other hand, there seems to have been some over-reliance on indirect
evaluations, such as perceptual or spectrographic fidelity of resynthesised speech — it
has rightly been asserted (e.g., Sondhi, 1979) that, owing mainly to the problem of nonuniqueness, resynthesis alone is not sufficient for evaluating estimated articulatory data.
234

In this vein, perceptual judgements are probably even more inconclusive, owing to thesuggested “many-to-one relationship between articulatory/acoustic and perceptual
targets” (Gay et al., 1991, p.445) for vowel sounds.
These limitations notwithstanding, directly-measured area functions are admittedly
the best reference that we have regarding data on real vocal-tract shapes. However,owing to technical obstacles, simultaneously-measured acoustic data are rare, and evenwhen they are documented, the formant bandwidths are invariably missing. It would
therefore seem expedient to evaluate our method of inversion first using acoustic dataobtained by synthesis which, although admittedly biased towards a more favourable
assessment, does afford a systematic definition of the various sources of error. To thisend, in Section 5.5.2 we will take as a point of departure the 33 published, directlymeasured vocal-tract area-functions which we used earlier (in Section 5.4.2.2), andimpress certain assumptions which enable us to evaluate our hybrid LP-SM method of
inversion first under model-matched (i.e., LP-matched), then under presumably more
realistic conditions.
In Section 5.5.3 we then proceed to estimate vocal-tract shapes from themeasured formant data of our four adult, male speakers of Australian English. In
particular, we will first use our measurements of the first four formant frequencies andbandwidths, to quantify the formant-induced variability of LP-estimated area-functions,and thus arrive at a prescription for dealing with the large amounts of variability
normally encountered in measured bandwidths. We will then consider the well-known,
but ill-treated problem of the incompatibility of measured formant bandwidths withthose ideally required by the LP vocal-tract model.
However, a necessary requirement in the forthcoming evaluation is to be able toquantify the variability in estimated vocal-tract shapes. As area-functions are generallyof different lengths, and as variations in the horizontal position of the lips and in thevertical position of the larynx preclude them from being considered as fixed articulatory
landmarks, the problem of area-function alignment is highly non-trivial; nor has it beenaddressed to a sufficient degree in the literature. In Section 5.5.1 we therefore propose
a method of aligning parameterised area-functions, which we shall then use both in our
subsequent evaluation of our estimation method, and also later in our articulatory235

investigation of the speech-speaker dichotomy in Chapter 6.

5.5.1 Inter-repetition Alignment of Area-Functions
As already discussed in our review of the literature (in Chapter 2), the problem of
directly comparing two or more vocal-tract shapes and quantifying their differences, hasrarely been addressed in previous studies. A major obstacle which has impeded progressin that regard, is presumably the absence of an accepted method of aligning vocal-tractarea-functions (whether directly measured or estimated from acoustics) such as torender physically meaningful comparisons. The problem is exacerbated by the wellknown tendency for the position of the lips and of the larynx to vary from one
articulatory configuration to another, thus prohibiting the use of either end of areafunctions as an invariant positional reference. As the overall lengths of different areafunctions are also different in general, one is faced with the non-trivial problem of how
best to align area-functions prior to quantifying their shape-related differences.
One might suppose that if certain, fixed articulatory landmarks could be locatedalong each area function, then they might be aligned by piecewise-linear expansion orcontraction along the length dimension, anchored at those landmarks. Unfortunately,
this attractive solution is not only unrealistic in the context of area-function estimation,
but would also be quite difficult and laborious to achieve with a high degree ofprecision even with direct articulatory measurements. Previous approaches to thisproblem include Harshman et al.’s (1977) elaborate definition of a speaker-specific,
mid-sagittal reference grid; Wood’s (1979) alignment of area functions (measured from
X-ray films) by anchoring them at the single point of reference defined by the positionof the central incisors; and Högberg’s (1995) inter-gender normalisation of the total
lengths of X-ray measured area functions, by a piecewise-linear rescaling which relieson the definition of the boundary between the oral and the pharyngeal parts of the vocal
tract. Clearly, such methods cannot be applied to estimated vocal-tract shapes, whichlack the convenience of physiological landmarks.
However, if one requires to quantify the variability, or the dispersion in a group of
area-functions which are variants of the same, basic vocal-tract shape, then one might

236

Glottal-end ofreference VT-shapeLip-end ofreference VT-shape
Reference VT-shape

Test VT-shape

l rlx

l rlx
VT length-axis

Figure 5.9: Schematised illustration of our method of aligning similarly-shaped vocal-tract areafunctions. The longest area-function is selected as the so-called reference shape, and the test shape isthen shifted along the length-axis in steps of ∆x, within the limits set by the relaxation intervals oflength lrlx at both ends of the reference shape. At each step, the root-mean-square (rms) error between
the two shapes (logarithmic area-functions) is computed within their mutually-overlapping regions,and the two area-functions are finally aligned at that position which yielded the minimum rms error.

take advantage of the fact that those shapes are assumed to share a common set ofarticulatory characteristics. This condition may arise, for example, when area-functions
are to be compared across a number of analysis frames and repetitions of a given vowelof a single speaker. One way of overcoming the lack of a fixed frame of reference, isthen to allow each area function a positional degree of freedom along the length axis,
subject to the constraint of a minimal amount of variation about the typical, or
prototype vocal-tract shape which they all are assumed to embody. The problem ofaligning the given set of (presumed) phonetically equivalent area functions, thus reduces
to a minimisation of the root-mean-square (rms) dispersion of the vocal-tract shapes
relative to each other.
As initially proposed and demonstrated by Clermont (1991; 1993) who
successfully time-aligned repetitions of formant-contours of diphthongs, the easiest
approach (shown schematically in Figure 5.9) is to slide each shape (formant-contour
or, in the present study, vocal-tract area-function) along the length axis, until the rms
difference is minimised with respect to a so-called reference shape, which itself is237

chosen to be the longest one of the set. As the shapes are, by definition, not too
dissimilar, a global minimum in the rms error can be found by sliding each, so-called testarea-function, only within certain limits which are defined by so-called relaxationintervals placed at both ends of the reference area-function.
In practice, this is achieved by first expanding the Fourier series terms in Equation
5.18 in order to obtain the smoothed versions of the parameterised area-functions
themselves, sampled at small, equal-length intervals ∆x . The area-function with thelongest length is then selected as a reference, and a sufficient number of sections isallowed at either end to accommodate the relaxation intervals, each of length l rlx . Eachof the test area-functions is then shifted within these outer limits, and at each step therms distance is computed between the test and reference, logarithmic area-functions(consistently with the SM model), over that part of the overall length which is common
to both (the mutually-overlapping region, as hereafter referred to). The global
minimum in the rms error is thus found to within an accuracy of ± 12 ∆x for each areafunction.
As a result of the procedure outlined above, each of the parameterised vocal-tract
shapes is endowed with an extra parameter which identifies its position along the x-axis,
relative to a fixed frame of reference defined by the longest area-function in that set. A
so-called prototype area-function can then be found simply by computing the average ofthe (logarithmic) areas at every interval ∆x . The total length of the prototype isdetermined by the mean length of the area-functions in the set; and its end-points arelocated equi-distantly about the mid-point of the overall mutually-overlapping (MOL)
region. Finally, the prototype area-function thus found, can be parameterised according
to Equation 5.18, and regarded as the basic vocal-tract shape of which the given set ofarea-functions were assumed to be variants.
As we shall see in Chapter 6, the procedure just outlined will prove to be very
useful in reducing the multi-frame and multi-repetition area-functions, to yield a single,
prototype vocal-tract shape for each vowel of each speaker. In the following sections,
we shall use it primarily to quantify the variability in vocal-tract shapes estimated undervarious conditions, by retaining the minimum rms error yielded after each alignment.

238

5.5.2 Re-estimation of Directly Measured Area-Functions
The first part of our evaluation concerns re-estimation of vocal-tract area-functions
which have already been directly measured using X-ray imaging or MRI techniques —in particular, the 33 area-functions obtained from the literature and used earlier in
Section 5.4.2.2. The formant data required as input to our hybrid LP-SM method of
inversion, will be synthesised first under model-matched conditions (in Section 5.5.2.1),
then under more realistic conditions (in Section 5.5.2.2). Our method of inter-repetitionalignment will be used to quantify the differences between the original and re-estimatedarea-functions, thereby assessing the detrimental influences of using model-mismatchedacoustic data. We then appraise (in Section 5.5.2.3) the effectiveness of using suchmodel-based results in correcting the (presumed) “more realistic” formant data prior to
inversion.
5.5.2.1 Model-matched Conditions
The LP model which lies at the heart of our inversion method, makes certain unrealistic
assumptions (as discussed earlier in Section 5.2.2), the two most drastic of which are
the following: (i) the number of equal-length vocal-tract sections is equal to twice thenumber of formants considered; and (ii) the only source of losses in the vocal-tract is afrequency-independent, resistive element at the glottal end. We herein conform to thefirst of these assumptions, by representing each of the 33 directly-measured areafunctions in terms of 8 equal-length sections, obtained by parameterisation (with M = 8
in Equation 5.18) and re-expansion of the areas at the centre of each section (i.e.,
effectively by interpolating the smoothed area-function at equal-length intervals).
Assuming a nominal value µ 8 = 0.8 for the glottal reflection coefficient, the first four
formant frequencies and bandwidths of each of the smoothed, 8-section area-functions
are then synthesised using a vocal-tract acoustic-simulation model (see Appendix C for
details) with all losses removed except for the glottal resistance. The synthetic formants
thus obtained under model-matched conditions (and listed in Table 5.3), are then used
to re-estimate each of the area-functions, using our hybrid LP-SM method of inversion
described earlier in Section 5.4.1.

239

Vowel/i/

Formant Frequency (Hz)
F1
F2
F3
F4
232.4 2271.5 3125.1 3909.7

Formant Bandwidth (Hz)
B1
B2
B3
B4
20.2
70.0
35.4
178.1

/e/

440.5

2025.22952.23912.1

19.6

55.9

107.3

121.2

/A/

708.3

1232.32606.93858.7

128.7

55.8

32.8

77.7

/o/

564.7

1028.82457.93435.1

85.2

22.9

19.5

143.5

/u/

298.2

665.6

2362.93397.2

45.6

24.4

19.0

168.3

/Þ/

294.6

1897.32367.43310.3

34.6

19.8

86.2

123.3

/i/

271.6

2144.92899.33946.8

23.4

68.6

67.2

142.3

Baer et al. (1991)

/Q/

641.7

1529.32475.24019.7

96.6

99.0

36.1

70.0

(Speaker TB)

/A/

524.3

1165.82483.43864.2

128.1

36.2

29.7

92.3

/u/

302.4

1335.72288.53399.3

76.9

9.2

112.1

74.7

/i/

282.9

2230.13157.03826.3

24.3

74.0

82.5

120.7

Baer et al. (1991)

/Q/

694.0

1656.93116.34264.0

67.5

64.7

40.3

145.8

(Speaker PN)

/A/

600.5

1337.63002.74099.9

164.9

43.3

27.6

82.5

/u/

318.3

1806.52495.83514.5

24.3

10.1

60.5

177.8

/i/

282.4

2103.72646.63530.4

42.4

75.9

79.1

77.5

Yang & Kasuya

/e/

541.7

1844.92506.83598.4

48.8

49.5

75.2

112.1

(1994)/a/

710.9

1211.52583.33571.6

91.0

49.2

63.3

79.6

/o/

578.9

1067.22607.53702.2

37.8

11.2

26.6

200.3

/u/

415.0

1581.22378.93390.3

69.5

22.6

90.6

100.4

Beautemps et al.

/i/

232.9

2013.02977.63741.7

21.7

74.4

69.6

120.9

(1995)/a/522.4

1338.72465.13578.8

35.4

75.5

42.6

124.2

/u/

362.5

1030.42312.93589.4

34.3

10.1

166.9

56.2

/i/

236.4

2441.23345.23836.9

8.8

61.2

60.9

169.6

/I/

501.3

2189.52868.73749.6

7.4

40.7

86.7

164.3

/E/

670.4

2196.32910.83996.8

8.6

48.7

87.2

171.2

/Q/

761.7

1971.92755.53746.9

8.2

40.4

98.5

152.7

Story et al.

/«/

754.3

1379.82656.03533.5

17.4

41.2

78.3

148.2

(1996)/A/

835.7

1232.02811.53734.3

48.1

29.7

19.9

187.0

//

673.7

1156.12780.43417.8

13.8

46.2

74.4

152.0

/o/

477.3

1019.72540.13647.8

17.5

7.9

92.2

169.4

/U/

487.2

1003.82675.73588.3

28.0

38.3

72.1

148.8

/u/

281.0

1395.82421.33526.5

12.2

0.8

77.4

184.4

/¼/

570.3

1662.82970.93569.3

19.4

78.2

98.9

90.0

Study

Fant (1960)

Table 5.3: Formant frequencies and bandwidths of 33 directly-measured area-functions, synthesisedusing a transmission-line analog of the vocal-tract (see Appendix B) with a glottal resistance as the
only source of loss. Each area-function is first parameterised with M = 8 shape parameters, which arethen used in Equation 5.18 to obtain 8 equal-length sections. The glottal reflection coefficient is fixed
at a nominal value µ 8 = 0.8.

240As the LP-based inversion is theoretically reversible, the 8-section area-functions
yielded by constraining the inversion to use the original vocal-tract lengths are almost
identical to the original, 8-section area-functions. Indeed, the average rms error
obtained after alignment (with ∆x = 0.05 cm and l rlx = 2.0 cm), was found to be only
0.04 . However, the MAD criterion can only be tested by allowing it to determine theoptimum length of each area-function under these, so-called model-matched conditions.
In particular, the method depicted in Figure 5.6 is first used in search of aminimum in the MAD criterion, for a range of vocal-tract lengths starting from 12cm
and incrementing in steps of 0.4 cm up to the limit Lmax, which is determined by thehighest (the fourth) formant frequency (cf. Equation 5.6 in Section 5.2.2). A second
search is then carried out over a much smaller range of vocal-tract lengths whichbrackets the identified minimum in the MAD criterion, to determine the optimum length
at a finer resolution of 0.05 cm. Of the 33 area-functions considered, only two (Fant’s(1960) Russian /u/ and Story et al.’s (1996) American English /u/) were found not tohave a minimum in the MAD criterion for a length less than their respective Lmax. The
optimum lengths of these area-functions were therefore obtained using the method
depicted in Figure 5.7, with an additional, fifth formant used to overcome the Lmax
limitation (as described in Section 5.4.1).
In Figure 5.10 we illustrate typical curves of MAD versus L obtained in finding an
optimum vocal-tract length, shown here only for the six area-functions of Fant (1960).
Our results are indeed comparable to the curves obtained by Paige and Zue (1970) for
the same area-functions, even though their method of inversion was somewhat different
from ours. Similarly to their results, Figure 5.10 indicates that the high-front vowel / i /is the most eccentric (relative to a uniform tube) of the six vocal-tract shapes, even after
optimisation of the vocal-tract length to minimise eccentricity; the optimised areafunction of the mid-front vowel / e / is not only the shortest, but also the least eccentricof the six vocal-tract shapes; the re-estimated area-function of the lip-rounded vowel/ u / is found to be the longest. However, contrary to the results of Paige and Zue (1970,
Table III) and of Wakita (1977, Figure 6), who both slightly underestimated the lengthof this high-back vowel, our LP-based method clearly overestimates its length, yielding
the largest, positive error ( 13.6 %) amongst the re-estimated lengths of the 33 area241

1.6

1.4

/i/

1.2

MAD

1

0.8

/o/

/A/

/u/

0.6

/Þ/
0.4

/e/

0.2
15

16

17

18
19
20
Vocal-tract length (cm)

21

22

23

Figure 5.10: Curves of MAD versus L, obtained in determining the optimum vocal-tract length ofeach of Fant’s (1960) six Russian vowels. The vocal-tract shapes are re-estimated using the method
given in Section 5.4.1 (cf. Figures 5.6 and 5.7), with the MAD criterion defined in Equation 5.19. The
first four, synthetic formant frequencies and bandwidths (cf. Table 5.3) are used as input to theinversion method.

Re-estimated vocal-tract length (cm)

23
22
21
20
19
18
Fant (1960)
Baer et al. (1991) - TB
Baer et al. (1991) - PN
Yang and Kasuya (1994)
Beautemps et al. (1995)
Story et al. (1996)
line of best fit (r = 0.82)

17
16
15
14
14

15

16

17
18
19
20
21
Original vocal-tract length (cm)22

23

Figures 5.11: Scatter-plot of re-estimated versus original vocal-tract lengths, for the 33 directlymeasured area-functions obtained from the literature. The vocal-tract shapes are re-estimated from thefirst four formant frequencies and bandwidths (cf. Table 5.3) synthesised under model-matched
conditions. Solid line: 45-degree line drawn for reference only. Dashed line: result of a linear
regression analysis (correlation coefficient 0.82; cf. Equation 5.20).
242

functions.
The scatter-plot in Figure 5.11 shows the relation between the re-estimated andoriginal vocal-tract lengths for all 33 area-functions considered. The distribution of thescatter suggests that our method of area-function estimation tends to underestimate thelengths of originally shorter area-functions, and to overestimate the lengths of originallylonger area-functions, thus exaggerating the normal variation in vocal-tract length fromvowel to vowel. Indeed, a linear regression analysis of the 33 pairs of vocal-tract
lengths, yields a correlation coefficient of 0.82 , and the following relation:
Lre-estimated = 1.46 Loriginal − 8.06 ,(5.20)

which is shown by the dashed line in Figure 5.11. It is interesting to note that the line ofbest fit intersects the solid line (which represents an ideal condition where the reestimated and the original lengths are identical) at about L = 17.5 cm, which is known
to be approximately the average vocal-tract length of an adult male.
The obvious exaggeration of inter-vowel variations in vocal-tract length
notwithstanding, it is important to note that the vocal-tract shapes themselves are reestimated remarkably well. Indeed, as illustrated by the dashed curves superimposed on
the graphs in Appendix F, articulatory features such as the place of lingual constriction,and the place of labial constriction for the lip-rounded vowels, are preserved fairly
accurately. For example, whilst the total length of the re-estimated area-function of
Fant’s (1960) / u / is clearly overestimated by the largest amount (as discussed earlier),
the re-estimated vocal-tract shape itself preserves (after alignment) not only the placesof linguo-velar and labial constriction at approximately 7- 8 cm and 19cm from theoriginal position of the glottis, respectively, but also retains the shape and volume of theback and the front cavities. The overestimated length appears to be therein manifested
mainly by an exaggeration in the amount of lip-protrusion, and also partly by the
position of the larynx, which is effectively lowered by 0.55 cm compared with original.
Similarly, although the re-estimated length of Story et al.’s (1996) area-function for / A /is underestimated by the largest amount ( −15.2 %), its shape retains the main place ofconstriction at approximately 6 - 6.5 cm from the glottis, and appears to be adversely
affected mainly towards the front part of the oral cavity.

243Fant (1960)

Yang and Kasuya (1994)

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

/i/

/e/

/A/

/o/

/u/

/Þ/

/i/

Baer et al. (1991) -- speaker TB

/e/

/a/

/u/

/o/

Baer et al. (1991) -- speaker PN

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

/i/

/Q/

/A/

/u/

/i/

Beautemps et al. (1995)

/A/

/u/

Story et al. (1996)

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

/Q/

0

/i/

/a/

/u/

/i/ /I/ /E/ /Q/ /«/ /A/ // /o//U//u//¼/

Vowel

Figure 5.12: Root-mean-square (rms) errors computed using the alignment procedure (cf. Section
5.5.1), between vocal-tract shapes re-estimated from the first four formants synthesised under various
conditions, and the shapes which represent each original area-function with the same degree ofsmoothness (M/2)=4, for each of 33 directly-measured area-functions obtained from the literature.
Diamond symbols, joined by solid lines: 8-section area-functions; LP synthesis (cf. Table 5.3). Plussymbols, joined by dashed lines: original section-lengths and areas; a lossy vocal-tract model (cf. Table
5.4).

244

In drawing conclusions regarding the appropriateness of the MAD criterion for
estimating the vocal-tract length, we are compelled, in light of the evidence discussed
above, not only to compare the re-estimated and the original lengths, as earlier studies
have done, but also to quantify the dissimilarity between the original and the reestimated vocal-tract shapes. As discussed earlier in Section 5.2.2, quantification ofdifferences between raw, step-wise LP area-functions of different lengths is non-trivial(and indeed has apparently never been attempted!), owing partly to the fact that theoriginal and the re-estimated section boundaries would no longer coincide. By contrast,
our method of area-function parameterisation yields a smoothed representation of eacharea-function, and the alignment method described in Section 5.5.1 can then be used toquantify the differences between original and re-estimated vocal-tract shapes.
The rms distances thus computed after alignment of each re-estimated vocal-tract
shape and the smoothed, parameterised version of the corresponding, original areafunction, are shown by the diamond symbols (joined by solid lines) in Figure 5.12.
Whilst they all are, as expected, numerically larger than the mean rms distance ( 0.04 )computed earlier for the vocal-tract shapes re-estimated using their original lengths, themean of the 33 distances ( 0.12 ) is still lower than the mean rms error ( 0.22 ) incurred in
representing those area-functions using the same number of shape parameters (asobtained in Section 5.4.2.2). The largest, anterior shift required to align the pairs of
vocal-tract shapes was found to be 0.65 cm, for Story et al.’s (1996) American English/ A /, the length of which was the most underestimated; the rms distance computed forthat area-function ( 0.45 ) is also the largest, and is mainly attributable to the inaccurate
re-estimation of the size of the oral cavity and of the lip opening area. The largestalignment-shift towards the glottal end was found to be −0.55 cm, for Fant’s (1960)
Russian / u /, the length of which was the most overestimated. The remaining vocal-tract
shapes were found to require shifts along the length-axis by amounts intermediate
between those two extremes, and they generally also yielded lower, rms distances. It is
encouraging, therefore, to note that the close resemblance of the re-estimated shapes tothe original area-functions (as shown in the graphs of Appendix F), is quantitatively
confirmed by the rms errors yielded by our automatic method of alignment.
The results presented above, collectively provide quantitative support for the245inversion method used under model-matched conditions, and with only a finite numberof formants. Thus, they also lend credence to the MAD criterion, by which the vocaltract length (and effectively the vocal-tract shape) is determined. However, in reality,
the formant frequencies and bandwidths measured from the acoustic speech signal are
not “matched” with the LP vocal-tract model. We therefore continue our evaluation of
the inversion method in the next section, using somewhat more realistic, “modelmismatched” conditions.
5.5.2.2 More Realistic Conditions
In order to introduce more realistic conditions in our evaluation of the inversion
method, we herein relax the constraints arising from two critical assumptions underlying
the LP vocal-tract model referred to in the previous section. The first of thoseassumptions concerns the rarely acknowledged fact, that the formants synthesised using
any type of transmission-line analog of the vocal-tract, depend to a certain extent on thenumber of sections used to represent a given area-function. Indeed, it was to avoid
mismatch in synthesised formants which led Wakita (1977, 1979) to first reduce Fant’s(1960) area-functions to 8 sections prior to using the LP method (with four formants)
to re-estimate those vocal-tract shapes. As it is well known (e.g., Fant, 1960) that amore accurate acoustic simulation of the vocal-tract is afforded by using a greaternumber of sections, we introduce more realistic conditions (and thus a model-mismatch)by synthesising the formants using the original section-lengths and areas of the 33 areafunctions used previously.
The second, crucial assumption concerns the distribution of acoustic energy losses
in the vocal-tract. More realistic conditions are simulated by using a transmission-line
analog which does include not only the glottal resistance (once again assuming anominal value of 0.8 for the glottal reflection coefficient), but also a glottal inductance;
a viscosity factor, heat-conduction and wall-vibration losses at each section; and aradiation impedance at the lips (see Appendix C for a more detailed description). The
first four formant frequencies and bandwidths thus synthesised (and listed in Table 5.4)
are then used to re-estimate each of the 33 area-functions using our hybrid LP-SM
method of inversion.
246

Vowel/i/

Formant Frequency (Hz)
F1
F2
F3
F4
293.1 2311.5 3106.9 3804.9

Formant Bandwidth (Hz)
B1
B2
B3
B4
87.9
81.7
290.6
192.5

/e/

461.1

1997.62859.13670.1

56.3

105.5

245.6

559.3

/A/

686.6

1108.42480.33625.2

167.6

136.4

103.7

214.4

/o/

546.4

888.7

2399.83536.2

116.9

80.3

55.8

146.8

/u/

301.3

631.3

2393.13774.6

89.4

52.2

42.7

101.9

/Þ/

339.8

1524.32361.33407.8

72.6

120.7

68.0

99.9

/i/

322.3

2185.32852.23963.9

66.6

69.4

110.5

148.9

Baer et al. (1991)

/Q/

643.8

1442.52278.63645.5

88.7

114.7

153.3

386.9

(Speaker TB)

/A/

540.2

1089.32457.13777.3

104.8

65.7

72.4

201.1

/u/

341.4

1097.02374.93536.3

99.2

48.8

64.7

65.7

/i/

333.2

2293.12997.63727.1

67.6

77.6

349.2

191.5

Baer et al. (1991)

/Q/

695.5

1595.02916.93815.7

62.2

88.5

390.2

580.6

(Speaker PN)

/A/

622.8

1269.62894.54034.7

154.1

87.5

153.1

219.4

/u/

351.5

1465.42454.93570.0

51.5

62.6

50.6

73.1

/i/

329.3

2082.52481.03240.9

82.0

114.4

165.6

205.3

Yang & Kasuya

/e/

545.2

1598.82280.33437.9

63.1

95.1

107.3

117.1

(1994)/a/

670.4

1080.92515.53438.9

76.3

81.8

86.5

123.3

/o/

508.4

858.7

2598.73728.7

46.3

44.5

57.0

66.4

/u/

431.0

1252.32341.43421.8

93.7

63.0

88.5

98.6

Beautemps et al.

/i/

292.9

2070.43054.03720.7

87.0

85.8

138.8

205.9

(1995)/a/518.7

1304.92380.63548.5

60.1

90.4

77.8

136.1

/u/

358.8

867.4

2339.33444.9

77.9

47.9

118.3

67.1

/i/

290.1

2521.13196.23949.2

76.7

89.5

340.9

257.0

/I/

501.4

2031.62602.83594.8

40.5

95.1

153.5

244.1

/E/

648.5

2009.02612.63595.0

42.8

133.5

230.6

525.6

/Q/

730.9

1840.22570.23470.0

41.3

126.3

229.3

345.0

Story et al.

/«/

692.2

1302.12626.43371.6

55.3

107.2

178.2

391.8

(1996)/A/

808.9

1147.92807.13550.6

126.5

124.3

155.5

532.2

//

632.6

1069.7

2370.13007.6

77.0

182.9

723.5

107.3

/o/

403.7

868.4

2521.03802.8

58.7

49.8

86.8

134.1

/U/

407.3

943.1

2741.33817.6

63.9

63.1

77.6

104.5

/u/

315.1

1126.02484.83787.1

67.7

55.7

76.4

193.4

/¼/

535.2

1572.92274.13072.9

56.2

121.8

164.5

279.1

Study

Fant (1960)

Table 5.4: Formant frequencies and bandwidths of 33 directly-measured area-functions, synthesisedusing a transmission-line analog of the vocal-tract (see Appendix C) which includes glottal impedance,
viscosity, heat-conduction, wall-vibration, and lip-radiation losses. The original, published sectionlengths and areas are used, and the glottal reflection coefficient is fixed at a nominal value µ glott = 0.8.

247

The re-estimated area-functions are shown in Appendix F (dotted curves),
superimposed on each of the original area-functions (step-wise, solid lines) from which
the formants were synthesised, and the area-functions (dashed curves) re-estimated in
the previous section under the model-matched conditions. It is immediately apparent
from these graphs, that despite the deliberate mismatch in the number of sections and in
the vocal-tract acoustic model used to synthesise the formants, the re-estimated shapes
still capture most of the essential, articulatory features such as constriction locations
and cavity sizes, often with remarkable fidelity. Indeed, there do not appear to be any
catastrophic errors, as for example a front vowel having a back place of constriction.
On the other hand, a number of area-functions are not so well re-estimated. The
worst cases are the area-functions of the vowel / u /, for which a comparison of Tables
5.3 and 5.4 reveals not only a change in the bandwidth pattern, but also a drop, often
more than 200Hz, in the second formant frequency; this has been found to be caused
primarily by the mismatch in the number of vocal-tract sections. However, this does notappear to affect Fant’s (1960) Russian / u / which already has quite a backward place of
lingual articulation; rather, it affects mainly the remaining, more fronted configurationsof / u /, and similarly the fronted / Þ / of Fant (1960). It is interesting to note that some ofthese area-functions are amongst those found earlier (in Section 5.4.2.2) to have a
distinctively symmetric shape.
Our qualitative observations of the superimposed area-functions are confirmed in
Figure 5.12, where the rms errors obtained after alignment are shown by the plus
symbols (joined with dashed lines), superimposed with those obtained earlier. Indeed,
the largest increases in rms error compared with the previous, model-matched results,
occur for / u / of Story et al. (1996) and of Baer et al.’s (1991) speaker PN. Not
unexpectedly, rms errors generally increase as a result of the model mismatch.
However, in a number of cases the rms error is only slightly worse than before (Fant’s(1960) / u /, Yang and Kasuya’s (1994) / u / and / e /, and Story et al.’s (1996) / « /, / A /,and / ¼ /); and most intriguingly, the rms error obtained for Yang and Kasuya’s / A / isactually lower than that obtained under model-matched conditions.
The results discussed above can be summarised in terms of the mean of the rms
errors for all 33 area-functions ( 0.37 ) which, although greater than the mean rms error248

( 0.22 ) found in representing those area-functions with the same number of parameters,
might still be regarded favourably, considering the large number of fairly accurately reestimated vocal-tract shapes shown in Appendix F. Indeed, not only do those
superimposed area-functions show little evidence of gross errors, they also demonstratethat the inversion method is likely to succeed in capturing the main articulatory featuresof the majority of the vocalic area-functions considered, to within the spatial resolution
afforded by the first M = 8 shape parameters, and using only the first four formant
frequencies and bandwidths.
5.5.2.3 Model-based Formant Correction
Clearly, re-estimation of directly-measured area-functions using formants synthesised
under more realistic conditions than afforded by the LP vocal-tract model, generally
yields worse results. In particular, we have found (in Section 5.5.2.2) that the mean rms
error between original and re-estimated vocal-tract shapes under model-mismatched
conditions, is about three times larger than that obtained (in Section 5.5.2.1) undermodel-matched conditions. Nearly two decades ago, Wakita (1979) proposed tocombat this problem by using a so-called “formant conversion chart” to compensate forthe effects on the formant frequencies, of the LP model’s incompleteness in regard to
vocal-tract losses. However, his conversion chart apparently does not include theeffects of using different numbers of vocal-tract sections; nor has there appeared aconversion chart for the formant bandwidths.
A more complete version of Wakita’s conversion chart is afforded by the formants
which we have already synthesised from the 33 directly-measured area-functions, firstunder model-matched, then under more realistic conditions. In particular, the 4 formant
frequencies listed earlier in Tables 5.3 and 5.4 are plotted along the ordinate and theabscissa, respectively, in Figure 5.13(a); similarly the 4 formant bandwidths, which areplotted in Figure 5.13(b). The solid curve shown superimposed in Figure 5.13(a), is a
cubic-polynomial fit to the 132 pairs of data points in the logarithmic-frequency
domain, and is described as follows:

(

)

[(

ln Fn(LP) = 0.0494 ln Fn(FL))] − 1.14[ln(F )] + 9.71ln(F )− 21.8
3

(FL)
n

249

2

(FL)
n

(5.21)

Formant Frequency (kHz) -- LP-based VT model with 8 sections
Formant Bandwidth (Hz) -- LP-based VT model with 8 sections

(a)

5.0
F1
F2
F3
F4

4.0
3.0
2.0

1.0

0.5

0.2
0.2

0.5
1.0
2.0
3.0
Formant Frequency (kHz) -- fully lossy VT model

4.0

5.0

1000
B1
B2
B3
B4
100

10

1
0.5
0.5

1

10
100
Formant Bandwidth (Hz) -- fully lossy VT model

1000

Figure 5.13: Formant frequency (a) and bandwidth (b) conversion charts (more complete version of
the frequency chart first proposed by Wakita, 1979). The first four formants of each of the 33 directlymeasured area-functions obtained from the literature, are synthesised using first the LP vocal-tract
model with the glottal reflection coefficient fixed at a nominal value µ glott = 0.8 (ordinate, see Table5.3), then a lossy transmission-line analog of the vocal-tract (abscissa, see Table 5.4). The solid curvein(b)(a) is the best cubic fit to the set of logarithmic-frequency data, and is described by Equation 5.21 in
the text.

250

As pointed out by Wakita (1979), the wall-impedance effects in the low-frequency, F1
range appear to cause the most substantial deviation from the 45º straight (dashed) line.
In addition, our more complete, formant frequency conversion chart highlights theextent of model mismatch caused by the combination of vocal-tract losses and numberof sections, which appear mainly to affect the F2 range.
In contrast with the formant frequencies, our formant bandwidth conversion chart
shown in Figure 5.13(b) does not suggest any regularity which might be exploited byfitting the data with a function. On the contrary, the data appear to be spread acrossregions of the chart which lie far away from the ideal, 45º line. Taken one bandwidth ata time, the B1 and the B2 data points (diamond and plus symbols, respectively) do
appear to form an elongated cluster which may be fit with a straight line of large,
positive slope; the B4 data points (cross symbols) might be considered to form a cluster
which is elongated horizontally, and which might therefore be fit with a straight line
having very little variation in the LP-based values along the ordinate. On the other hand,
the B3 data points (square symbols) appear to be scattered across a large region of thechart, with no apparent regularity. Clearly, formant bandwidth correction is not as
amenable a task as formant frequency correction.
Before becoming too despondent over these problematic results, it is worth notingthat they are purely model-based (i.e., they are based entirely on formants synthesisedusing either the LP or a more lossy vocal-tract model). In this vein, we have alreadyquestioned (in Section 5.2.2) the viability of using model-based correction procedures,
when it remains unknown to what degree those models can be trusted to be accurate or
realistic. Indeed, the fidelity of even a fully-lossy vocal-tract model which does include
many of the acoustic and aerodynamic properties missing from the LP model, has notbeen firmly established. Perhaps we should therefore not expect model-based formant
correction procedures, even if they could be formulated, to significantly andconsistently improve the plausibility of LP-derived vocal-tract shapes by rendering themsomehow more “realistic” than they already are.
In that context, it is highly instructive to enquire whether those five studies fromwhich we have taken the 33 directly-measured area-functions, have also reported results
concerning the discrepancy between measured and re-synthesised formant data. Those251

studies have indeed published two sets of formant frequency values based, respectively,on acoustic measurements (from recordings made either at the time of the articulatory
measurements or, more commonly, on a separate occasion) and vocal-tract simulations
using the measured area-functions themselves. Fant (1960, Table 2.31-1, p.109) used adigital computer (BESK) implementation of a 20-section vocal-tract model, and foundthat the “average deviation of calculated data from spoken data is of the order of 5 percent in F2 and F3 and 10 per cent in F1 ”; Baer et al. (1991, Table III, p.813)
synthesised waveforms from their measured area-functions using a variant of Kelly and
Lochbaum’s (1963) model which includes non-ideal terminations at the lips and at theglottis, and found that the “formant frequencies of the subjects’ utterances differ
significantly from the computed resonances of the area functions”; Yang and Kasuya(1994, Table 4, p.626) used Sondhi and Schroeter’s (1987) hybrid time-frequency
domain synthesiser which incorporates source-tract interactions and a number of vocaltract losses, and found differences of up to 7.2 % in the formant frequencies of theirmale subject; Beautemps et al. (1995, Table 2, p.35) used a lossless vocal-tract modelwith the wall-vibration and lip-radiation taken into account by a correction to theformant frequencies and a correction to the length of the lip-section, respectively, anddespite explicit optimisation of their directly-measured area-functions to match themeasured and model-synthesised formants, they still obtained errors of up to 10 % in
the formant frequencies; Story et al. (1996, Table IV, p.549) used a “wave-reflection
analog vocal tract model” which includes viscosity, wall-vibration, and lip-radiation
losses, in addition to an acoustic side-branch to model the piriform sinuses, and stillfound deviations from measured formant frequencies up to nearly 15 % in F1 , up to
16 % in F2 , and up to nearly 20 % in F3 .

In row (a) of Table 5.5 we list the rms difference (in Hz) between the measured
and synthetic values for each of the first three formant frequencies published in those
five studies, and the rms difference in the fourth formant frequencies published by Yang
and Kasuya (1994) and by Beautemps et al. (1995). By comparison, row (b) lists therms differences (in Hz) between the formants synthesised first under model-matched,then (presumed) more realistic conditions (those listed earlier in Tables 5.3 and 5.4,
respectively). Both sets of results suggest rms errors on the order of 10% — clearly
252

F1

F2

F3

F4

B1

B2

B3

B4

(a)

53.5

183.7

251.2

145.6

-

-

-

-

(b)

40.0

160.6

185.5

228.0

40.4

54.5

166.4

173.4

Table 5.5: Root-mean-square (rms) difference (in Hz) in each of the first four formant frequencies
and bandwidths, either synthesised or measured, and pertaining to the 33 directly-measured areafunctions obtained from the literature. (a): rms difference between measured formant frequencies andthose synthesised from the directly-measured area-functions, as published in the five studies referred toin the text (cf. Section 5.5.2.3). (b): rms difference between formants first LP-synthesised with an 8section representation (cf. Table 5.3), then synthesised using the original section-lengths and areas,
with a lossy transmission-line analog (cf. Table 5.4).

greater than the 3 − 5 % perceptual difference limen suggested by Flanagan (1955).
However, it is even more interesting to note that for the first three formant frequencies,
the rms errors listed in row (a) of Table 5.5 (those inferred from the studies cited
above) are in fact larger than those obtained (and listed in row (b) of Table 5.5) in ourevaluation of the discrepancies between the LP and a more realistic vocal-tract model!
That result alone is indicative of the futility of formant correction procedures
based on model data. This assertion is indeed confirmed by an experiment in which we
attempted to correct the formants listed in Table 5.4, according to the correction chartsof Figure 5.13. In particular, we used the cubic-polynomial of Equation 5.21 to correct
the formant frequencies, and four separate, linear functions to correct the formant
bandwidths. Perhaps not surprisingly, the mean rms error yielded for the re-estimatedarea-functions ( 0.33 ), was found to be only slightly lower than that obtained originally
under model-mismatched conditions ( 0.37 ). In addition to confirming the futility ofmodel-based formant correction, these results also suggest that the mismatched
conditions simulated in the previous section may indeed have been exaggerated, by
virtue of our over-reliance on model-generated (or synthetic) formant data. In the next
section we therefore consider the inversion problem using real (or measured) formants.

5.5.3 Estimation of Area-Functions from Measured Formants
If we are to use our hybrid LP-SM method of inversion to estimate area-functions from
real, measured acoustic data, then we are compelled to alleviate the following problems
which would otherwise potentially undermine the interpretability of our results. The

253

first problem (considered in Section 5.5.3.1) concerns the sheer variability which is
known to plague measurements of formant bandwidths in particular. The second
problem (considered in Section 5.5.3.2) concerns the potential mismatch between
measured formant bandwidths, and those more appropriate to the LP vocal-tract model.
Although both of these problems have long been acknowledged, we still lack aquantitative assessment of, and an accepted method of dealing with, their potentiallyharmful influences on estimated area-functions.
5.5.3.1 Formant-induced Variability of LP Area-Functions
Whilst formant-tracking has traditionally been concerned with reliably measuring theformant frequencies, we have shown in this chapter that the formant bandwidths do playa vitally supporting role in ensuring uniqueness in the LP-based method of inversion.
Indeed, the distinctive relation which we have shown to exist between each formant
bandwidth and the corresponding, symmetric component of the LP-derived, logarithmic
area-function, is sufficient to suspect that large amounts of inter-frame or interrepetition variation in bandwidth measurements might induce significant variability inthe estimated vocal-tract shapes. As this would clearly be an undesirable input to ourforthcoming, articulatory interpretation of the phenomenon of dichotomy, it is ofutmost importance to quantify the extent to which measurement variability in either
formant frequencies or bandwidths are likely to influence the estimated vocal-tract
shapes.
To that end, a preliminary step is taken by computing the relative sensitivity of ourshape parameters to variations in the formant frequencies and bandwidths. This isachieved by fitting the distribution of each of the four formant frequencies and the four
bandwidths generated (in Section 5.4.2.1) from the 6561 area-functions perturbed
about the neutral-tube, using a linear combination of all eight, re-estimated shapeparameters, as shown by the following, multiple-linear regression formulae:
M/2

M/2

m =1

m =1

M/2

M/2

m =1

m =1

Fn(rel) = ∑ α (nF, m) a2 m −1 + ∑ β (nF, m) b2 m −1 ,Bn(rel) = ∑ α (nB,m) a2 m −1 + ∑ β (nB, m) b2 m −1 ,254

n = 1,..., M / 2 ,(5.22)

n = 1,..., M / 2 ,(5.23)

a1

a3

a5

a7

b1

b3

b5

b7

F1(rel)-0.535-0.040-0.040-0.044-0.015-0.017-0.0060.018

F2(rel)-0.041-0.516-0.050-0.053-0.022-0.009-0.011

0.019

F3(rel)-0.048-0.053-0.479-0.071-0.029-0.014-0.0070.020

F4(rel)-0.019-0.021-0.026-0.380-0.011-0.006-0.0020.007

B1(rel)

0.228

0.954

0.590

0.197

-0.776

0.032

0.064

0.082

B2(rel)-0.164

0.073

0.968

0.255

0.106

-1.867

0.493

0.618

B3(rel)-0.340-0.954-0.494

0.273

0.089

0.643

-2.1841.469

0.276 -0.073 -1.064 -0.726
0.581
1.191
1.627 -2.170
B4(rel)Table 5.6: Coefficients of hyperplanar model, obtained from the 6561 pairs of acoustic/ vocal-tract
shape data generated in Section 5.4.2.1, for each of the first four, relative formant frequencies andbandwidths, in terms of the first eight shape parameters (cf. Equations 5.22 and 5.23). Boldface figurescorrespond to the so-called first-order parameter relations which were earlier shown (in Table 5.2) tohave high coefficients of correlation (|r| > 0.9).

where the superscript “(rel)” refers to the relative formant parameters, as defined earlier
in Section 5.4.2.1. The resulting set of M = 8 hyperplanes (each of which passes
through the origin), are fully described by the following direction-cosines with respect
to each shape parameter: α (nF,m) , β (nF, m) , α (nB, m) , β (nB, m) , n = 1,..., M , m = 1,..., M / 2 . The
coefficients themselves are found such that each hyperplane fits the 6561 data points ina least-mean-squares sense, using a generalised linear least-squares algorithm (Press etal., 1988, p.537) based on singular value decomposition (SVD).
Table 5.6 lists the coefficients of sensitivity thus obtained. In particular, we draw
attention to those coefficients along the main diagonal, which quantify the sensitivity ineach of the so-called first-order parameter relations, shown earlier to have the mostsignificant coefficients of correlation (see Table 5.2, in Section 5.4.2.1). Indeed,similarly to our earlier results, the coefficients of sensitivity exhibit dominant andnegative values along the main diagonal, corresponding to the first-order relations
between Fn(rel) ↔ a 2 n−1 , and between Bn(rel) ↔ b2 n−1 . The factor − 1 2 which is predicted
theoretically by the SM model (cf. Equation 5.2), is determined empirically by themultiple-linear regression analysis as α 1( ,F1) = −0.535 , α (2F, 2) = −0.516 , α (3F,3) = −0.479 ,and α (4F, 4) = −0.380 , respectively, for each of the first four formant frequencies. The
analogous coefficients of sensitivity for each of the first four formant bandwidths, areβ 1( ,B1) = −0.776 , β (2B, 2) = −1.867 , β (3B, 3) = −2.184 , and β (4B, 4) = −2.170 , respectively. These255

increasingly larger-valued, negative coefficients, were already foreshadowed by theslopes of the relevant nomograms shown earlier in Figure 5.5(b).
In order to obtain a first approximation of the sensitivity of each shape parameterwith respect to a unit perturbation in the corresponding, relative formant parameter, we
assume that the matrix found above is diagonally dominant, and simply compute thereciprocal of each of those hyperplane coefficients listed along the main diagonal. The
inverse-sensitivities thus obtained, are −1.87 , −1.94 , −2.09 , and −2.63 , respectively,
for each of the first four, asymmetric shape parameters a2 n−1 , and −1.29 , −0.54 ,−0.46 , and −0.46 , respectively, for each of the first four, symmetric shape parametersb2 n−1 . The results of this simple calculation imply that vocal-tract shapes obtained byour hybrid LP-SM method of inversion, are less sensitive to unit variations in therelative bandwidths, than they are to unit variations in the relative formant frequencies.
In view of the notorious difficulties of measuring formant bandwidths robustly, thispreliminary analysis provides some hope that the bandwidth-induced variations in vocaltract shapes may not be so disastrous as to obscure important information pertaining,for example, to the phonetic identity of the vowel being considered.
We now turn to the formant data of our four adult, male speakers of AustralianEnglish, in order to better assess the detrimental influences of formant variations on
estimated vocal-tract shapes. As described in Chapter 3, we used both our phonetic
knowledge and the formant-enhancement properties of the NDPS, in order to ensure
consistent and relatively noise-free estimates of the formant frequencies in particular.
However, the formant bandwidths can be expected to have relatively larger amounts ofboth inter-frame and inter-repetition variability, which may therefore induce a greater
amount of variability in LP-derived vocal-tract shapes, despite the lower sensitivity of
those shapes to the bandwidths than to the formant frequencies (as found above). In
order to assess the relative influence of formant frequency and bandwidth variations on
estimated vocal-tract shapes, the mean and standard-deviation ( σ ) of each formant
frequency and bandwidth were first computed on a per-vowel, per-speaker basis, acrossthe 7 steady-state frames and 5 repetitions. The following two sets of formant data
were then prepared for each vowel of each speaker: (i) with the 4 bandwidths fixed attheir mean values, the 4 formant frequencies are set equal to ±1σ about their respective256

mean values, in all possible ( 2 4 = 16 ) combinations; and (ii) with the 4 formant
frequencies fixed at their mean values, the 4 bandwidths are set equal to ±1σ about
their respective mean values, in all possible ( 2 4 = 16 ) combinations. For each of those
two sets of 16 formant-patterns, our hybrid method of inversion was then used toestimate the corresponding groups of 16 area-functions, which in turn were subjected tothe alignment procedure described in Section 5.5.1.
Three representative examples of the area-functions thus obtained are shown in
Figure 5.14, where the two groups of 16, aligned vocal-tract shapes are superimposedin each graph. In particular, it is fairly consistently observed that the formant frequencyperturbed vocal-tract shapes (the green area-functions) cluster together more closelythan the bandwidth-perturbed shapes (the red area-functions). This immediately
confirms our suspicions raised in the previous paragraph, that despite the smaller
sensitivities of vocal-tract shape parameters to variations in bandwidths, the measured,inter-frame and inter-repetition variations in the bandwidths are sufficiently large(compared with the measured variations in the formant frequencies) as to induce agreater amount of variability in the estimated shapes.
The results are summarised in Figure 5.15, which shows the per-vowel, mean rms
dispersion (averaged over the 4 speakers) in the groups of estimated vocal-tract shapes.
These results first confirm that the dispersion in vocal-tract shapes caused by ±1σvariations in measured formant bandwidths (shown by the plus symbols joined bydashed lines), is indeed consistently larger than the shape-dispersions induced by ±1σvariations in the formant frequencies (diamond symbols joined by solid lines).
Nevertheless, this quantitative evidence confirms our observations in Figure 5.14, thatat least for the formant data at hand, the bandwidth-induced articulatory dispersions are
not so significantly larger than the formant frequency-induced dispersions, that theyshould cause catastrophic distortions of the estimated vocal-tract shapes. Indeed, themean, formant frequency-induced dispersions for the two back vowels in “hod” and“hoard”, are numerically within the same range as the mean, bandwidth-induced
dispersions for some of the front vowels. This finding is all the more remarkable since,
as discussed earlier, the measured formant bandwidths were not subject to the rigorous
constraints imposed by our methodology for extracting the steady-state formant
257

Normalised Area

4

/i/

3
2
1
0

Normalised Area

0

2

4

6

8

10

12

14

16

18

9
8
7
6
5
4
3
2
1
0

/ U/

0

2

4

6

8

10

12

14

16

18

20

22

Normalised Area

4

24

/a/

3
2
1
0
0

2

4

6
8
10
12
14
Distance from furthest glottal position (cm)16

18

Figure 5.14: Illustration of the vocal-tract shape dispersions induced by all combinations of ±1σ
perturbations in each of the first four, measured formant frequencies (green area-functions) andbandwidths (red area-functions) of the FC dataset. Top panel: vowel / i / of speaker C. Middle panel:
vowel / U/ of Speaker D. Bottom panel: vowel / a / of Speaker B. All area-functions are estimated using
our hybrid LP-SM method of inversion. As exemplified in these three graphs, it is consistently
observed in all 11 vowels of all 4 speakers of the FC dataset, that the bandwidth-induced shapedispersions are larger than the formant-induced shape dispersions.

258

0.3
Formant BANDWIDTH-induced dispersion
Formant FREQUENCY-induced dispersion
Consequences of bandwidth correction (see Section 5.5.3.2)

mean rms dispersion

0.25

0.2

0.15

0.1

0.05

0

heed

hid

head

had

hard

hod

hoard hood who’d hud

herd

Figures 5.15: Dispersions about the prototype vocal-tract shape for each of the eleven vowels, foundusing the alignment procedure described in Section 5.5.1, and averaged over the four speakers.
Diamond symbols (joined by solid lines): average dispersions induced by all combinations of ±1σ
perturbations in each of the first four formant frequencies. Plus symbols (joined by dashed lines):
average dispersions induced by all combinations of ±1σ perturbations in each of the first four formant
bandwidths. The mean and standard-deviation for each formant parameter was computed separately foreach vowel of each speaker, across the 7 steady-state frames and 5 repetitions.
Square symbols (joined by dotted lines): rms differences between pairs of vocal-tract shapes estimatedfrom the mean formants of the FC dataset, before and after bandwidth correction (see Section 5.5.3.2).

frequencies; nor did we use a pitch-synchronous acoustic analysis which might have
reduced the bandwidth measurement variability caused by the varying position of theanalysis window relative to the peaks of glottal excitation in the speech waveform.
In sum, our results indicate that bandwidth-induced dispersions in vocal-tract
shape (mean rms error of 0.181 ) are larger than formant frequency-induced dispersions(mean rms error of 0.097 ) by approximately a factor of two. It is interesting to notethat by comparison, the normalised standard-deviation of the bandwidths themselves(mean value of 0.163 ) is larger than that of the formant frequencies (mean value of
0.048 ) by a factor greater than three6.

Although studies concerned with the expected measurement variability in formant
bandwidths are quite rare, Markel and Gray (1976, p.178) do offer the following
6

The smaller factor obtained for the dispersions in vocal-tract shapes than for the formants themselves,is partly explained by the coefficients of inverse-sensitivity found earlier.

259

generalisation:“...linear prediction bandwidth estimates may be in error by as much as a factor of 2.5.”

In light of this and the evidence summarised in Figure 5.15, we must acknowledge thepotentially harmful influence of the presumably non-phonetic variations in formant
bandwidths, and prescribe a method of dealing with those variations. As the bandwidths
do clearly play a role in determining the so-called second-half of shape components
which are otherwise missing in the completely lossless model, their phonetic variation(i.e., from vowel to vowel) should perhaps be retained. Towards this end, we havedetermined to use only the mean of each bandwidth on a per-vowel basis, computed
across frames, repetitions, and speakers. This prescription for controlling the nonphonetic variability in the formant bandwidths is all the more pertinent that our vowel
classification experiments using simplified cepstra have already shown (in Chapter 4)that the acoustic-phonetic phenomenon of speech-speaker dichotomy is independent ofthese acoustic parameters.
5.5.3.2 Closed-glottis Correction of Formant Bandwidths
Whilst the sheer variability in measured formant bandwidths can be controlled by using
only their per-vowel mean values as recommended above, a serious question remains as
to the relevance of those mean bandwidths to the LP vocal-tract model. The potential
mismatch stems from the fact that while “true” or measured formant bandwidths reflect
all types of losses in the human vocal tract, such as those caused by the glottal source,
acoustic viscosity, heat-conduction, wall-vibration, and lip radiation, the LP vocal-tract
model has only a single, frequency-independent source of acoustic energy loss locatedat the glottal end. Hence, as stated by Fant (1980, p.85):“The bandwidths we need for the inverse LPC-based transforms are the bandwidths of a
production model which has losses at the glottis only and lacks the cavity wall shunt.”

However, as we have earlier argued (in Section 5.5.2.3), model-based formant
correction procedures are not easily formulated, nor do they appear to be particularly
effective. In this vein, Fant’s (1980, p.85) very next, more general statement is perhaps
a more appropriate guide to pursuing the problem:

260

“From the true formant frequencies and bandwidths we thus have to make a best guess ofwhat bandwidths the LPC model would generate.”

If model-based data are to be avoided, then we are naturally drawn towards measured
bandwidth data, in search of a clue as to the necessary “best guess”.
In that regard, it is relevant to note that any measured bandwidth is known to beapproximately an additive combination of its individual components (in Hz), whichthemselves are determined by the configuration-dependent contribution from each of
the separate sources of loss in the vocal-tract (e.g., Fant, 1960; Flanagan, 1972).
Bandwidths measured from the acoustic speech signal are thus the total sum of theirindividual components arising from glottal, vocal-tract internal, lip-radiation and otherlosses; by contrast, the LP vocal-tract model “would generate” bandwidths from only
the glottal resistance. The question then arises whether one can approximately
compensate for the losses which are missing in the LP model, by subtracting thosecomponents of each bandwidth which might have been measured had there been nosource of loss at the speaker’s glottis.
Such measurements of so-called “closed-glottis bandwidths” have indeed been
made on human speakers (Fant, 1962; Fujimura and Lindqvist, 1971), by applying a
sweep-tone excitation signal transcutaneously at the throat while the speaker maintains
a static articulatory configuration with closed glottis, and by measuring the acoustic
response near the speaker’s lips. Using those empirical data, Fant (1972) then brought
acoustic theory to bear on the problem of deriving a formula for each of the first threebandwidths, in terms of the first three formant frequencies and an average, fourthformant frequency. More recently, Hawks and Miller (1995) used those empirical data
to derive a more general equation which is able to predict the bandwidth of any of theformant frequencies up to nearly 5kHz, given only the centre-frequency of that formant.
If we can assume that the per-vowel, mean values of the measured bandwidths
embody all of the natural sources of loss which arise for the given vocalic configuration,and that Hawks and Miller’s (1995) equation embodies the relation between frequencies
and bandwidths of closed-glottis formants, then we might expect that by subtracting
thus estimated closed-glottis bandwidths from the measured bandwidths, we would
obtain an approximation of the per-vowel, mean glottis-only bandwidths. It is indeed
261

400
350

Formant Bandwidth (Hz)

300
250
200
150
100
50
0
0

500

1000

1500
2000
2500
Formant Frequency (Hz)

3000

3500

4000

Figures 5.16: Illustration of the viability of closed-glottis bandwidth correction. Solid curve: Hawks
and Miller’s (1995) empirically-derived equation relating the closed-glottis bandwidth of a formant
with its centre-frequency (assuming their default conditions for an adult male speaker). Diamondsymbols: the per-vowel mean of each of the first four formants computed over the 7 steady-state frames,
5 repetitions, and 4 adult, male speakers of the FC dataset.
the glottis-only bandwidths which the LP vocal-tract model ideally would require inorder to yield area-functions which (presumably) more closely resemble those actually
produced by a given speaker. Clearly, it is difficult to test this particular hypothesis
without simultaneously-measured acoustic and vocal-tract shape data; however, we
may at least test the viability of the procedure by determining whether the subtraction
would yield admissible (i.e., positive and non-zero) bandwidths.
To that end, in Figure 5.16 is shown Hawks and Miller’s (1995) bandwidthversus-frequency curve (assuming their default conditions for adult male speakers),
together with the mean frequencies and bandwidths of each of the first four formants of
the 11 vowels of our adult, male speakers of the FC dataset (diamond symbols). This
Figure shows that our measured bandwidth data do follow the expected trend of higher
values for higher formant frequencies. More importantly, it clearly shows that our meandata do consistently lie above the curve, and that the difference (or the vertical distance)
between each mean bandwidth (or data point) and the corresponding, closed-glottis
bandwidth (along the curve), would indeed yield a positive and non-zero, “glottis-only”

262

bandwidth7.
Whilst the corrected bandwidths (listed in Appendix G, together with themeasured, mean formant frequencies and bandwidths for each vowel) do confirm theviability of the procedure, it would also be interesting to know how, and to what extent,these bandwidth corrections influence the estimated vocal-tract shapes themselves.
Shown in each of the 11 panels in Appendix G, are superimposed the aligned areafunctions estimated from our mean formant data before (dashed line) and after (solidline) bandwidth correction. Those graphs show that the consequences of our proposed
method of bandwidth correction are not only harmless, but may even be consideredbeneficial on the whole. Differences between the pairs of vocal-tract shapes appear tobe the largest for the quasi-neutral vowel / ¶ / (which is also somewhat shorter after
bandwidth correction), the fronted / á / (which develops a more well-definedconstriction), and the back vowel /  / (whose front cavity expansion reduces in volume
to a somewhat more realistic level). It is also interesting to note that after bandwidthcorrection, the two, high-back vowels /  / and / U / are no longer found to require theaddition of a fifth, synthetic formant in order for the inversion method to determine an
optimum vocal-tract length (cf. Section 5.4.1).
Our qualitative observations are quantitatively confirmed in terms of the rms
differences between the pairs of (logarithmic) area-functions, shown by the squaresymbols (joined by dotted lines) in Figure 5.15 (see the end of the previous section).
That graph clearly indicates that the shape-related consequences of our proposed
method of bandwidth correction, are less severe than either the formant bandwidth- or
frequency-induced dispersions found in the previous section. Indeed, the overall meanrms error ( 0.03 ) is even less than that found earlier for the 33 directly-measured areafunctions re-estimated under model-matched conditions ( 0.04 , see Section 5.5.2.1).
It is highly encouraging that the consequences of formant bandwidth correction do
not appear to be as drastic as the literature would generally seem to suggest. However,
we must concede that our proposed method of determining the mean, “glottis-only”

7

It is important to note that such, “glottis-only” bandwidths would represent only an average of thequasi-periodic bandwidth values, owing to our speech analysis frame-lengths typically extending overmore than one glottal cycle.

263

bandwidths which ought to be better matched with the LP vocal-tract model, must be
more conclusively put to the test, perhaps with the help of simultaneously-measured
acoustic and vocal-tract shape data. Until such time, we may rest assured that at least
for the formant data at hand (the FC dataset), the proposed correction applied to the
per-vowel mean bandwidths, should not endanger the phonetic interpretability of theestimated area-functions, nor their articulatory manifestations of the speech-speakerdichotomy, which we wish to investigate in the forthcoming chapter.

5.6 Concluding Summary
Our aim in this chapter was to find a method of mapping the geometry of the humanvocal-tract from the acoustic speech signal, in order to facilitate an extension of ouracoustic-phonetic investigations of the speech-speaker dichotomy (in Chapter 4) into
the domain of speech production. As outlined in Section 5.1, we sought a method of
inversion which would satisfy the following two criteria: uniqueness and formant-based
parameterisation. These criteria served as guideposts in our bipartite rationale (inSection 5.2), where we reviewed the strengths and the limitations of the SchroederMermelstein (SM) model and the linear-prediction (LP) vocal-tract model.
The main strength of the SM model is its description of vocal-tract area-functionsin terms of orthogonal parameters which relate quasi-linearly and uniquely to theresonance frequencies of the acoustic-tube. Its main limitation in the context of areafunction estimation, is that only one-half of the required information is available in the
acoustic speech signal (i.e., only the formant frequencies). Hence, the estimated vocaltract shapes are ambiguous in their symmetric components, unless further constraints
are introduced.
The LP model, on the other hand, is well-known to yield a unique vocal-tract
shape (for a given vocal-tract length), and is notable for its mathematical tractability andlow computational complexity in comparison with other inversion methods which havebeen proposed. However, while the formant bandwidths are implicated as the necessary
second-half of acoustic information required to obtain a unique LP area-function, it has
never been explained how they contribute to the shape-related uniqueness of those areafunctions. Furthermore, the LP-based method of inversion suffers from a number of264

limitations, including the analysis artefact of an upper limit Lmax on allowed vocal-tract
length, and the coarse, step-wise representation which inhibits comparison of areafunctions of different length.
With the aim of elucidating the uniqueness property of the LP inversion method,and perhaps overcoming some of its inherent limitations, we then used the SM model asa probe to examine the formant-dependence of LP-derived vocal-tract shapes. In
Section 5.3 we first validated the SM model within the LP modelling framework, andthereby showed that those two models are indeed isomorphic in regard to the quasilinear relation between the formant frequencies and the antisymmetric (cosinusoidal)
perturbations of the uniform (logarithmic) area-function. We then derived a theoretical
proof of the dependence of the mean formant bandwidth on the glottal reflection
coefficient, and thereby provided the motivation for regarding the relative bandwidths
more pertinent to shape components of LP-derived area-functions. Remarkably, ourempirical investigations then revealed those bandwidth-related shape components to be
the symmetric (sinusoidal) perturbations of the uniform (logarithmic) area-function.
Given the acknowledged (but to date largely unexplained) importance of formant
bandwidths in securing uniqueness in LP-derived area-functions, our proposed methodof vocal-tract shape parameterisation (given in Section 5.4) is therefore an important
contribution to our understanding of the uniqueness property of the LP inverse method.
More importantly, it overcomes the inherent discreteness of LP-derived vocal-tract
shapes, and provides a smooth representation by way of a compact and orthogonal setof parameters which also lend themselves to an acoustic-phonetic interpretation.
Furthermore, the artificial limitation of Lmax is easily overcome by invoking the very
property of the SM model which explicitly relates higher-resolution shape components
with the higher formants. In particular, the vocal-tract shape is estimated to a specified
degree of spatial resolution, while using a sufficient number of vocal-tract sections toensure that the highest available formant still lies within the spectral range of the vocaltract model.
In Section 5.4.2 we first used our hybrid LP-SM method of inversion to evaluate
our proposed method of area-function parameterisation. In particular, we obtained highcoefficients of correlation between each of the formant frequencies and bandwidths
265synthesised from 6561 different area-functions, and the corresponding shape-parameters
re-estimated from those formant data. The acoustic relevance of our shape parameters
thus confirmed, their physical relevance was assessed by representing 33 directly
measured, human vocal-tract area-functions obtained from the literature. On average,
the most important shape-components of those area-functions were found to be the firsttwo pairs of both antisymmetric and symmetric components. This evidence confirmed
the phonetic importance of the first two formant frequencies of vowels; however, it also
confirmed the importance of the bandwidths in describing unique vocal-tract shapes.
The hybrid method of inversion itself was finally evaluated in Section 5.5, with thehelp of a new method for aligning and quantifying the difference between similarlyshaped area-functions. We first re-estimated the 33 directly measured area-functions
obtained from the literature, using the first four formant frequencies and bandwidths
synthesised under model-matched conditions (i.e., a vocal-tract model with a glottal
resistance as the only source of loss, and the number of vocal-tract sections equal totwice the number of formants). Whilst the phonetic range of variation in vocal-tract
length was found to be exaggerated upon re-estimation, the vocal-tract shapes werefairly accurately re-estimated, with a mean rms error of 0.12 . More realistic conditions
were then imposed, by re-estimating the area-functions from formants synthesised using
a more lossy vocal-tract model, and by retaining the original section-lengths and areas.
The mean rms error between original and re-estimated shapes under model-mismatched
conditions, was found to be 0.37 .
In the hope of finding a reasonably accurate method of compensating for the LPmodel’s simplified assumptions in regard to vocal-tract losses and number of sections,
we presented a more comprehensive version of the “formant frequency conversion
chart” first proposed by Wakita (1979). Whilst we were able to fit the model-generated
formant data with a third-order polynomial function in logarithmic frequencies, theanalogous conversion chart for formant bandwidths did not suggest the possibility offinding a single functional description. Indeed, the futility of approaching the formant
correction problem from a purely model-based point of view, was further reinforced by
our finding that on average, errors between formant frequencies measured from the
speech waveform and those re-synthesised from the directly-measured area-functions266

are in fact larger than the corresponding errors between the formant frequencies
synthesised, respectively, under model-matched and model-mismatched conditions.
In Section 5.5.3 we then turned to our measured vowel formant data (of the FC
dataset). As a prelude to examining formant-induced variability using those real data,
we first established that LP-derived vocal-tract shapes are actually less sensitive to unit
perturbations in the relative formant bandwidths, than to unit perturbations in therelative formant frequencies. Despite this lower sensitivity, the inter-frame and interrepetition variability in our measured bandwidth data was large enough to induce
dispersions in estimated vocal-tract shapes, approximately twice that induced byvariability in the formant frequencies. In view of our forthcoming, articulatory
investigation of the speech-speaker dichotomy which we had earlier explained in termsof the formant frequencies (in Chapter 4), our prescription was therefore to control theinter-frame, inter-repetition, and the inter-speaker variations in each formant bandwidth(while retaining their phonetic variation, the importance of which was indirectlysuggested by our results in Section 5.4.2.2), by reducing the bandwidths to their meanvalues computed on a per-vowel basis.
Whilst the measurement variability in formant bandwidths was thus controlled,
their relevance to the LP vocal-tract model was still questionable. Our approach todealing with this problem was then to recognise that the bandwidths required by the LPmodel are those which would arise from a vocal-tract having only a resistive glottal
termination. We therefore used Hawks and Miller’s (1995) empirically-derived equation
to determine the so-called “closed-glottis” bandwidths of our mean formant data, thensubtracted those from the original values in order to approximate the required, glottisonly bandwidths. Although a more conclusive proof of this procedure would perhaps
require simultaneously-measured acoustic and area-function data, our results do
support its viability, and furthermore indicate only minor differences in the vocal-tract
shapes re-estimated after bandwidth correction.
Finally, the detailed evaluation presented in Section 5.5 raises our confidence inLP-derived and parameterised vocal-tract shapes obtained by the hybrid LP-SM method
described in Section 5.4, which itself secures both uniqueness and resonance-based
parameterisation of estimated vocal-tract area-functions. In the following chapter we267therefore proceed to use our method of inversion and area-function parameterisation,with a view toward an investigation of the speech-speaker dichotomy in the speech
production domain.

268

Chapter 6
An Articulatory Explanation of Speech-Speaker Dichotomy

6.1 Introduction
From the very start, the central theme of our thesis has been the long-standing problemof speech-speaker dichotomy. In Chapter 4 we introduced a new methodology for
examining the interactions between phonetic and speaker-specific attributes in spoken
vowel sounds, and thereby unfolded and explained the vowel-speaker dichotomy in the
acoustic-phonetic domain. The remarkably systematic and orderly manifestations ofvowel-speaker interactions across the spectral continuum, and the readily interpretable,
acoustic-phonetic relevance of those spectral regions of either phonetic or speakerinfluence, then heightened our initial resolve (stated in Chapter 1) to gain an even more
fundamental interpretation of the dichotomy, from a speech production point of view.
Towards that end, in Chapter 5 we addressed the problem of mapping the geometry of
the human vocal-tract from the acoustic speech signal. The formant-based
parameterisation of LP-derived area-functions yielded by our extended version of the
Schroeder-Mermelstein (SM) model, then provided the necessary tools to proceed withan articulatory investigation of the vowel-speaker dichotomy, with sufficient confidence
in the uniqueness and plausibility of our estimated vocal-tract area-functions.
In this chapter we therefore seek an articulatory explanation of the vowel-speakerdichotomy, and for this purpose we shall use the FC dataset of /hVd/ monosyllables
recorded by our four, adult male speakers of Australian English. We begin in thefollowing section with a discussion of our overall philosophy on describing speakerdifferences in the articulatory domain. In particular, we shall define three potential,
articulatory sources of speaker variability associated with the supralaryngeal vocaltract, which will then serve as a descriptive framework for explaining the vowel-speaker269

interactions, whose manifestations we had previously observed only in the acousticphonetic domain.
As we shall see, a significant component of our articulatory approach will involve
speaker normalisation of the estimated vocal-tract area-functions. In Section 6.3 wetherefore describe our methodology for speaker normalisation of each of the threearticulatory sources of variability. In Section 6.4 we then apply our methodology toprovide a three-part articulatory explanation of the dichotomy. The final proof of ourarticulatory explanation will then be revealed in Section 6.5, where we examine both thearticulatory and acoustic-phonetic consequences of complete speaker normalisation. We
conclude in Section 6.6 with a summary of our contributions.

6.2 Articulatory Approach
The vowel-speaker dichotomy which we first unfolded and explained in the acousticphonetic domain (in Chapter 4) is, by definition, a consequence of the interactions
between phonetic and speaker-specific attributes of the spoken vowel sounds
considered. Insofar as the vocal-tract area-functions corresponding to those vocalic
data were estimated (in Chapter 5) from the formant parameters which earlier afforded
an acoustic-phonetic explanation of the dichotomy, we should therefore expect them tocarry a similarly potent blend of phonetic and speaker-specific attributes. Indeed, to theextent that those estimated area-functions represent articulatory manifestations of boththe phonetic and inter-speaker variabilities which exist in the original, acoustic data,
they embody the very elements required to provide a physical explanation of the
dichotomy.
Perhaps the simplest approach to an articulatory explanation, is to observe and toquantify the differences between our four speakers’ area-functions — particularly thosewhich correspond to the pairs of vowels, confusions amongst which were found (inChapter 4) to be the main contributors to the dichotomy. However, the apparent
simplicity of this direct approach belies the methodological complexities in comparingarea-functions of different vowels and different speakers, and in formulating plausible
articulatory interpretations of speaker differences, without the benefit of an adequate
framework for describing those differences. Alas, nor does the literature seem to have270

embraced any of the pioneering efforts (e.g., Garvin and Ladefoged, 1963; Stevens,
1971) or the more substantial attempts (e.g., Laver, 1980; Nolan, 1983) to provide sucha descriptive framework, which might then facilitate our investigation of the articulatory
correlates of the vowel-speaker dichotomy.
In this vein, our review of the literature (in Chapter 2) has already shown thatdifferences in overall vocal-tract length, particularly between speakers of different
genders, have received by far the greatest amount of attention, sadly to the almost
complete exclusion of all other, potential articulatory sources of speaker variability.
Whilst the size of the vocal-tract (and hence its length) is perhaps the most important,physical source of variation between male and female speakers, or indeed between adultand child speakers, rarely has there been an attempt to explain more subtle differencesin articulatory behaviour, as might be expected amongst our four, adult male speakersof the FC dataset used to unfold the dichotomy. Those differences, whether of an
idiolectal or more intrinsic nature, arise from what has been broadly termed acquired orlearned articulatory behaviour, as distinct from the so-called organic or anatomicaldifferences which may be accounted for mainly in terms of vocal-tract size (Garvin and
Ladefoged, 1963; Stevens, 1971).
In that context, we have also reviewed (in Chapter 2) a number of studies whichhave attempted to provide a more complete account of the potential sources of socalled learned articulatory differences between speakers, from which emerges adistinction between long-term and so-called realisational differences (Nolan, 1983).
Although the term “articulatory setting” was coined by Honikman (1964) mainly inregard to cross-linguistic differences in the long-term utilisation of the articulatoryorgans, Laver’s (1980) more comprehensive, descriptive framework of settings issufficiently general to allow for speaker differences at the two broad levels which
Garvin and Ladefoged (1963) have called group (e.g., across languages, dialects, or
genders) and individual (e.g., within genders and idiolects). By contrast, differences inrealisation or articulatory strategy have received attention mainly at the level of groupsof speakers, such as dialectal or socio-linguistic variations in the realisation ofphonemes, as accounted for by phoneticians and linguists.
From the foregoing discussion emerges an integrated perspective on the potential
271

articulatory (supralaryngeal) sources of speaker differences, three of which stand out asessential components in any, more extensive or elaborate, descriptive framework. Thosebasic components, or articulatory features as hereafter referred to, describe speakerdifferences in the overall size of the fixed vocal-tract structure, in the long-term bias or
articulatory setting, and in the residual, phoneme- (or vowel-) specific articulatory
strategy. These three features comprise what might be regarded as a minimal, yetsufficiently descriptive framework, which indeed will form the basis of our articulatory
explanation of the vowel-speaker dichotomy.
If our explanation is to be tripartite as suggested, it will require isolation of the
speaker differences carried by each of the three articulatory features in turn, followed byan assessment of their individual contributions to the overall, acoustic-domain
manifestation of the dichotomy. Naturally, the process of speaker normalisation whichis thereby called for, is best applied directly to the estimated area-functions themselves,
and the normalised acoustic data then obtained by an appropriate method of articulatory
synthesis. For example, the formant parameters synthesised from all the area-functions
after speaker normalisation of vocal-tract structure and articulatory setting, will define
an acoustic-phonetic space where only vowel-specific, articulatory strategy-related
speaker differences are present. Simplified cepstra generated from those formants, canthen be used in inter-speaker vowel classification experiments of the type conducted in
Chapter 4, and the contributions of speaker differences in articulatory strategy to thedichotomy thus determined.
In order to maintain consistency with our LP-based method of acoustic-toarticulatory mapping, the LP model is also used for synthesising the formant parameters
after each stage of speaker normalisation of the vocal-tract area-functions. Furthermore,
the number of vocal-tract sections for each area-function (and hence the order M of theLP synthesis model), is the same as the number of sections used to estimate that areafunction (i.e., equal to either 8 or 10, depending on whether an artificial, fifth formant
was required in order to overcome the Lmax limitation, as described in Chapter 5). Thismethod guarantees that the original formants (those which were used as inputs to theinverse mapping) are perfectly resynthesised from the unmodified area-functions. The
formants synthesised after normalisation of pairwise articulatory sources of inter272

speaker variability, can therefore be compared with the original formants, and theacoustic-phonetic consequences of the area-function modifications thus quantified
within a consistent methodological framework.
Our approach to an articulatory explanation of the vowel-speaker dichotomy istherefore based on the concept of systematically depriving the phenomenon of certain,
articulatorily-defined sources of inter-speaker variability, thereby forcing a reduction in
the degree of spectral manifestations of vowel-speaker interactions — a reduction,
however, with pre-defined physical correlates. This approach involves the successive
steps of articulatory speaker normalisation, synthesis of normalised acoustic data, interspeaker vowel classification as a function of an increasing upper spectral limit, and an
acoustic-phonetic decomposition of the behaviour of the accuracy curve thus generated.
Starting with the estimated area-functions of our four speakers of Australian English(FC dataset), those steps will be carried out three times in Section 6.4, each timenormalising inter-speaker variations in two of the three articulatory features describedabove, thus providing a threefold articulatory decomposition of the vowel-speakerdichotomy. Towards that end, we first describe (in Section 6.3) our methodology forspeaker normalisation of vocal-tract structure, articulatory setting, and vowel-specificarticulatory strategy.

6.3 Methodology for Articulatory Speaker Normalisation
A prerequisite to speaker normalisation of the three articulatory features described in
the preceding section, is a definition of each of those features in terms of quantifiable
aspects of estimated vocal-tract area-functions. In the following three sections, we shalltherefore motivate and justify our assumptions in regard to the definition of each of thearticulatory features in turn, and thus describe our collective methodology for speakernormalisation of vocal-tract fixed structure, articulatory setting, and vowel-specificarticulatory strategy.

6.3.1 Speaker Normalisation of VT Fixed Structure
In our search for a definition of the overall size of each speaker’s vocal-tract anatomy
or fixed structure, we are naturally drawn away from shape-related aspects of area-273

functions, and focus, almost by default, on their length. Indeed, as mentioned in theprevious section and discussed in greater detail in our review of the literature (inChapter 2), vocal-tract length is the physical parameter most widely studied in regard to
speaker differences, with a clear emphasis on the more radical differences between men,
women, and child speakers, and an equally clear (although often implicit) connection
with the overall size of the vocal-tract structure. However, as the vocal-tract length ofany given speaker changes from one vowel configuration to another, the question arises
whether the lengths pertaining to certain articulatory configurations are likely to be
more relevant than others, to the size of that speaker’s fixed vocal-tract anatomy.
Although the length along the path of the vocal-tract airway depends to a certain
extent on the posture and position of the tongue, the most significant contributors to
the phonetic variation in vocal-tract length are the position of the lips and of the larynx,
which vary typically over a range of up to 1 or 2cm each (e.g., Perkell, 1969). The
clearly lip-rounded vowels included in our (FC) dataset of Australian English, are theback vowels /  / and / U /, and the fronted vowel / á /, all of which therefore havecharacteristically longer overall lengths (as confirmed in Chapter 5); the back vowel /  /is also known to be accompanied by lip rounding (e.g., Singh and Singh, 1976;
Ladefoged, 1993). At the same time, it is well-documented (e.g., Lindblom andSundberg, 1971; Ewan and Krones, 1974; Riordan, 1977; Högberg, 1995) that thelarynx is usually lowered in all those vowels, concomitant with lip-rounding. Indeed, in
an attempt to gain a more faithful measure of the “articulatory organ size” in terms of
vocal-tract lengths estimated from vocalic speech data recorded by speakers ofJapanese, Ishizaki (1978b, p.1050) argues thus: “Since the pronunciations of / u / and/ o / are usually accompanied with lip protrusions and glottal movement downward,
vowels / i /, / e / and / a / are chosen ...”. On the other hand, there is additional evidence inthe literature (e.g., Ewan and Krones, 1974; Riordan, 1977; Högberg, 1995) where it
has been observed that the larynx is raised above its average position, in the mid- tohigh front vowels such as / i /, / I /, and / E /, for which the lips are usually either spread or
neutral.
By virtue of the significant potential for variations in lip protrusion/spreading andlarynx lowering/raising in those seven vowels, inter-speaker variability in the overall274

lengths of their area-functions are potentially compounded by differences both inanatomical size and articulatory behaviour. It would seem, therefore, that a moreaccurate (or less biased) estimate of the size of a speaker’s fixed vocal-tract anatomy isprovided by the lengths of only the non-rounded, mid- to low vowels, which
presumably are least susceptible to extreme positions of the lips and of the larynx. In
this connection, it is interesting to note Lindblom and Sundberg’s (1971) suggestion,
based on vocal-tract simulation experiments, that a good estimate of the vocal-tract
length is provided by F3 of open vowels, which would include our / Q /, / a /, and / « /. In
addition to being articulated more centrally and therefore being least constricted, thetwo vowels / « / and / ¶ / are also known to be “neutral for the feature rounding” (Singhand Singh, 1976, p.57). Our measure of the speaker-dependent vocal-tract fixed
structure is therefore defined as the mean length of the four vowels / Q /, / a /, / « /, and/ ¶ /. Thus for the sth speaker, the mean, structure-related vocal-tract length is obtained
as follows:()

Ls = 14 Ls/ Q / + Ls/ a / + Ls/ « / + Ls/ ¶ / ,(6.1)

where the mean length Ls/ • / for each of the four vowels is computed over all 5
repetitions and 7 frames.
Having thus defined the first of our three articulatory features, we come now to a
description of its normalisation across speakers. In that context, recall from our reviewof the literature (in Chapter 2) that the differences in vocal-tract lengths of male andfemale speakers have been shown (Chiba and Kajiyama, 1958; Högberg, 1995) to be
unevenly distributed across the length of the vocal-tract — differences in the length ofthe pharyngeal region are generally greater than differences in the length of the oral
region. However, as all four of our speakers of Australian English are male, we may
assume that their pharynx-to-oral length ratios are not as disparate as might be expectedof inter-gender comparisons. Consequently, the vocal-tract fixed structure is speakernormalised by uniform scaling of the vocal-tract lengths of all area-functions, as
follows:′ = k s Lsvrf ,
Lsvrf

(6.2)

where the prime superscript denotes the normalised length, and the indices refer to the275

v th vowel, the r th repetition, and the f t h frame. The speaker-dependent length-scaling
factor itself is defined by the following expression:
L (ref)
,
ks =
Ls

(6.3)

where the reference length is simply the mean, structure-related length of all N s = 4
speakers, as follows:
L

(ref)

1 Ns
=
∑ Ls .
N s s =1

(6.4)

It is conceivable that if the area-functions of both male and female speakers were
included, then a non-uniform length normalisation might have to be considered. It is not
at all clear, however, where the boundary between the oral and pharyngeal parts of
estimated area-functions would be identified. On the other hand, the studies ofNordström (1977) and Yang and Kasuya (1995, 1996) have cast considerable doubt on
the significance of non-uniform length scaling, and suggest that uniform scaling is
indeed relatively more important, both from acoustic and perceptual points of view.
After speaker normalisation as described in Equation 6.2, each speaker’s mean
vocal-tract length computed over the selected subset of four vowels will, by definition,
be equal to the average, reference length L (ref) , while inter-vowel, inter-repetition andinter-frame variations in length are naturally retained. Those length-normalised areafunctions can then be assumed to have been cast in a common, reference vocal-tract
anatomy or fixed structure, and the remaining speaker differences therefore related only
to articulatory behaviour, which we consider next in terms of setting and vowel-specificstrategy.

6.3.2 Speaker Normalisation of Articulatory Setting
Articulatory setting is the term preferred by Laver (1980) in his descriptive framework
for “voice quality”, which has been described as “a quasi-permanent quality running
through all the sound that issues from [a speaker’s] mouth” (Abercrombie, 1967, p.91).
Although the term was first used in reference to “the gross oral posture ... requisite as aframework for the ... merging and integrating of the isolated sounds ... of a language”

276

(Honikman, 1964, p.73), it has also been described to be useful “for extralinguistic
purposes, as a phonetic component of voice quality identifying the individual speaker”(Laver, 1980, p.3). It is in the latter context that we adopt the concept of articulatory
setting as a useful component or feature of speaker idiosyncrasy in vowel production.
To the extent that an articulatory setting “does not imply simply the particular
articulations of the individual speech sounds ..., but ... their common, rather than their
distinguishing components” (Honikman, 1964, p.73), or indeed that it is “polysegmental, ... a property of a stretch greater than a single segment” (Laver, 1980, p.3),
it may therefore be computationally defined as an articulatory feature averaged over all
the available, segmental data of a speaker. Analogously to our definition (in theprevious section) of a speaker’s fixed vocal-tract structure in terms of the mean lengthof their estimated area-functions, a speaker’s articulatory setting can be defined as themean shape of their area-functions. However, in contrast to our definition of structure
in terms of the mean length computed over only a subset of the vowels, setting is more
appropriately defined as the mean vocal-tract shape computed over all the vowels.
This working definition of articulatory setting is by no means complete — inLaver’s (1980) descriptive terminology, it can take into account only latitudinal,
supralaryngeal settings, and therefore ignores the so-called longitudinal, supralaryngeal
settings which comprise long-term idiosyncrasies in labial protrusion/labiodentalisationand larynx height (precisely those aspects of vowel articulation which we sought toexclude in our speaker normalisation of vocal-tract structure). However, to the extent
that our definition of setting includes the shape-related (or latitudinal) aspects of labial
idiosyncrasies, speaker-specific behaviour in the longitudinal positions of the lips and of
the larynx will later be absorbed into the vocal-tract length component of per-vowel
articulatory strategy (in Section 6.3.3). Whilst the so-called laryngeal, velopharyngeal,
and overall muscular tension settings are also ignored, they are in any case irrelevant toour study of the vocal-tract area-functions of non-nasalised vowels. Our present
definition of articulatory setting in terms of mean vocal-tract shape thus retains theessential components of Laver’s detailed descriptive framework, while adhering to areasonable degree of both conceptual and computational simplicity.
That simplicity, however, is threatened by a serious question which arises in277

regard to the required averaging of area-functions across all the vowels of a given
speaker. As the area-functions are generally of different lengths, and do not possess any
fixed articulatory landmarks except the glottal and lip ends which mark the acousticallyrelevant extreme end-points, there appears to be no obvious way in which they might bealigned along the length-axis prior to averaging or any other type of comparison. In thiscontext, recall that our method of alignment described in Chapter 5 can only accountfor variations in length across frames and repetitions, and is therefore applicable only ona per-vowel basis.
Our discussion in the previous section regarding the expected phonetic variations
in larynx height, clearly rules out the highly simplified assumption (which is so often
implicit in the literature) that vocal-tract area-functions can be aligned at the glottal end.
At the same time, the obvious contributions of labial posture to the overall length of the
vocal-tract cast some doubts on the robustness of aligning area-functions at the lip end.
The problem of comparing area-functions of different vowels without the benefit of a
fixed articulatory frame of reference, would thus appear to be insurmountable, and may
indeed have partially discouraged the pursuit of articulatory explanations of basic
speech phenomena using inverse methods based on area-functions.
One clue to a possible resolution of this problem lies in the observation (in theliterature) that there exists a correlation between lip protrusion and larynx lowering.
The significance of the correlation is such that, for example, Shirai and Honda (1977,
p.284) assume as part of the control of their articulatory model, “that the position of theglottis varies with the lip parameter” in a simple, linear relationship (with a coefficientof 0.3 , which implies a smaller range of variation in glottal height compared with lip
protrusion). There has also been put forward an acoustic-theoretic explanation of theobserved correlation: in his model-based study of the acoustical significance ofarticulatory maneuvers of [ y ], Wood (1986, p.391) concludes that “larynx depression iscomplementary to lip rounding and restores spectral sensitivity to palatal andpharyngeal tongue movements otherwise disturbed by the labial activity.”
Acoustic theory also provides a second clue as to the possible location of a fixedarea-function landmark. In particular, the Schroeder-Mermelstein (SM) model, which
formed the basis of our parameterisation (in Chapter 5) of vocal-tract shapes estimated278

by the linear-prediction (LP) model, clearly asserts that for variations in the formant
frequencies only, there exists a single location at the centre of an area-function (i.e., at
x = 12 L cm from either end) where the cross-sectional area is invariant (see Figure 5.1).
This node, which also marks a point of antisymmetry in the behaviour of formant
frequency-sensitivity functions of a uniform tube, might therefore be considered atheoretically-motivated landmark (as aptly pointed out by Fuchi and Ohta, 1979) withconsiderable, acoustic-phonetic significance.
Both the physiological and acoustic-theoretic evidence considered, support thenotion of aligning estimated vocal-tract area-functions at approximately their midlength. Indeed, the emergent concept of centre-alignment is perhaps the least biased
with respect to both the glottal and labial ends of area-functions which, by virtue of
their amounts of overhang being determined by the vocal-tract length, are thus allowed
to characterise phonetically-relevant variations in larynx lowering and lip protrusion.
Whilst a somewhat off-centre alignment may be more appropriate in order to enforce asmaller range of positions in larynx height compared with that of lip protrusion (aspreferred by Shirai and Honda, 1977), there is also evidence of some speakers who
adopt quite comparable ranges of phonetic variation in the positions of their larynx andlips (e.g., Perkell, 1969; Högberg, 1995). Insofar as the comparative range of positionalvariation may well be speaker-dependent, it may be impossible to precisely define apoint of reference which can then be applied with certainty to all estimated areafunctions; or indeed to determine speaker-specific points of reference on the basis ofestimated area-functions alone. In the face of these difficulties, we therefore retain the
more tractable concept of centre-alignment, which may adequately serve our purpose
without introducing undue computational complexity and methodological uncertainty.
Our method of computation and speaker normalisation of articulatory setting istherefore described as follows:
1. Speaker-normalise the vocal-tract fixed structure by scaling the lengths of allarea-functions (as described in Section 6.3.1). All area-functions are then assumed
to have originated from a common (or reference) vocal-tract anatomy.
2. Align the structure-normalised, per-vowel area-functions of each speaker across279

the 7 frames and 5 repetitions (as described in Section 5.5.1.2), thus minimising
the root-mean-square (rms) difference in vocal-tract shapes amongst each groupof 35 area-functions. The resolution along the length-axis for these and all
subsequent computations, is ∆x = 0.05 cm per section.
3. Compute the length of the mutually-overlapping (MOL) region for the 35 aligned
area-functions of each vowel of each speaker, defined as the length over which all
35 area-functions exist (i.e., excluding the glottal and labial ends where somearea-functions may extend beyond others).
4. Retaining the alignment performed in step 2, align the groups of 35 area-functions
across vowels and speakers, at the mid-length of their respective MOL regions
found in step 3.
5. Compute the mean vocal-tract shape per speaker, by averaging the area-functions(in the logarithmic-area domain) across all frames, repetitions, and vowels. At theglottal and labial ends where longer area-functions may extend beyond the shorter
ones, averaging is still performed (at each ∆x ) albeit over the reduced number ofarea-functions.
6. Compute the overall mean vocal-tract shape, by averaging the per-speaker mean
area-functions (in the logarithmic-area domain) across all speakers. Overhanging
area-functions at the glottal and labial ends are dealt with as described in step 5.
7. At each vocal-tract section (of length ∆x ) and for each speaker separately,
compute the ratio of overall mean area and per-speaker mean area (provided thelatter is defined at that section). Analogously to Equation 6.3, these ratios are thespeaker-specific area-scaling factors required to normalise the area-functions ofeach speaker with respect to articulatory setting as defined earlier.
8. For each individual area-function of each speaker (aligned as in step 4), multiply
the area at each vocal-tract section (of length ∆x ) by the area-scaling factor found
for that speaker in step 7.
To summarise the computational procedure described above, all the area-functions are
speaker-normalised with respect to vocal-tract structure, aligned to minimise interframe and inter-repetition variations in shape, centre-aligned across vowels and280

speakers, and speaker-normalised with respect to articulatory setting. By definition, theaverage vocal-tract shape per speaker will then be identical to the overall mean shapecomputed in step 6. To the extent that the alignment of each area-function as computedin step 4 is retained in all subsequent steps, any discontinuities in the mean areas foundin steps 5 and 6 (as a consequence of averaging over a reduced number of areafunctions towards the glottal and labial ends) are also retained in subsequent steps,
without adverse effect.

6.3.3 Speaker Normalisation of Vowel-Specific Articulatory Strategy
Thus far we have described our procedure for speaker normalisation of two articulatory
features, both of which relate to long-term aspects of vowel production. First the vocaltract structural differences between speakers are identified and normalised, then thevowel-averaged bias in vocal-tract shape or articulatory setting. The speaker differences
which remain after normalisation of structure and setting, can therefore be attributed to
idiosyncratic behaviour in articulatory strategy on a per-vowel basis.
As with the long-term features, vowel-specific articulatory strategy comprises twocomponents — one which relates to longitudinal variations in articulatory behaviour,such as the amount of lip protrusion, larynx lowering or raising, and the posture of the
tongue and other articulators as they influence the length of the path along the vocaltract airway; and one which relates to latitudinal variations, such as the degree of labialconstriction, and the posture of the tongue and other articulators as they determine theshape of (or cross-sectional areas along) the vocal-tract airway. We therefore define theformer in terms of speaker differences in mean vocal-tract length per vowel, and thelatter in terms of differences in mean vocal-tract shape per vowel. Assuming structuraland setting-related differences to have been speaker-normalised, the residual differencesin mean vocal-tract length and shape for each vowel separately then make up what wehere define as articulatory strategy.
Our algorithmic procedure for the computation and speaker normalisation of the
shape-related component of vowel-specific articulatory strategy is described as follows:
1. Starting with the structure- and setting-normalised, completely aligned area-281

functions obtained after step 8 in Section 6.3.2, compute the per-vowel mean
vocal-tract shape for each speaker, by averaging (in the logarithmic-area domain,
and with ∆x = 0.05 cm per section) over the 7 frames and 5 repetitions.
2. Compute the mean vocal-tract shape for each vowel, by averaging the perspeaker mean area-functions obtained in step 1 above (in the logarithmic-areadomain) across all speakers.
3. For each vowel separately, at each vocal-tract section (of length ∆x ) and for eachspeaker, compute the ratio of overall mean area and per-speaker mean area
(provided the latter is defined at that section). Similarly to step 7 in Section 6.3.2,
these ratios are the speaker-specific area-scaling factors required to normalise thearea-functions of each speaker with respect to articulatory strategy, on a pervowel basis.
4. For each individual area-function of each speaker, multiply the area at each vocaltract section (of length ∆x ) by the area-scaling factor found for that speaker andfor that vowel in step 3 above.
Following the steps described above, the per-vowel average vocal-tract shape of eachspeaker will, by definition, be identical to the overall mean shape computed for thatvowel in step 2.
The length-related component of articulatory strategy is then speaker normalisedby uniform length-scaling of all area-functions on a per-vowel basis, as follows:′′ = k sv Lsvrf
′ ,
Lsvrf

(6.5)

where the prime superscript denotes the structure-normalised length obtained earlier in
Section 6.3.1, and the double-prime superscript denotes the structure- and strategynormalised length. The length-scaling factor in Equation 6.5 is both speaker- andvowel-dependent, as defined by the following expression:
k sv =

Lv′ (ref)
,
Lsv′

(6.6)

where the denominator is the mean, structure-normalised length for each vowel of eachspeaker, and the numerator is simply the mean, structure-normalised length of all
N s = 4 speakers for the vth vowel, as follows:
282

Lv′ (ref) =1 Ns∑ Lsv′ .
N s s =1

(6.7)

It is important to note that the speaker normalisation of length-related differences inarticulatory strategy as described above, is carried out without altering the relativealignment of area-functions along the length-axis. Indeed, the alignment secured in
Section 6.3.2 is retained, by scaling (i.e., either lengthening or shortening) each areafunction in equal proportions about the fixed point of reference which is defined at thecentre (or mid-length) of the MOL regions found previously.

6.4 Articulatory Explanation of the Dichotomy
As outlined in Section 6.2, our approach to an articulatory explanation of the vowelspeaker dichotomy consists of a tripartite decomposition of the speaker differences
contained in the estimated vocal-tract area-functions, followed by resynthesis of theformant parameters, and a return to the acoustic-phonetic methods of analysis andexplanatory framework developed in Chapter 4. In this way, speaker differences in thethree articulatory features (defined in the previous Section) may be assessed both interms of their acoustic-phonetic correlates, and the spectral manifestations of vowelspeaker interactions which they each induce.
If the algorithmic procedures outlined in the previous Section are performed in theorder that we presented them, the resulting area-functions will have been completely
speaker normalised with respect to vocal-tract structure, articulatory setting, andvowel-specific articulatory strategy. However, as argued in the exposition of ourarticulatory approach (in Section 6.2), we require to isolate the speaker differences
carried in each of those three articulatory features in turn, in order to assess theirindividual contributions to the overall phenomenon of vowel-speaker dichotomy. In
principle, this can be achieved after having performed complete speaker normalisation,by methodically re-introducing the speaker differences found in each of the three
features in turn. Of course, this process of speaker un-normalisation is not necessary for
isolating the differences in vowel-specific articulatory strategy (as we shall require in
Section 6.4.3), because the procedures described in Sections 6.3.1 and 6.3.2 alreadyyield the desired (structure- and setting-normalised) area-functions. On the other hand,283

speaker differences in vocal-tract structure (as required in Section 6.4.1) must be reintroduced in the completely-normalised data, by dividing (rather than multiplying) the
length of each area-function by the appropriate, per-speaker scaling factor k s defined in
Section 6.3.1. Similarly, speaker un-normalisation of articulatory setting (as required in
Section 6.4.2) is achieved by dividing (rather than multiplying) the area of each section(of length ∆x ) of each area-function by the appropriate, per-speaker area-scaling factor
for that vocal-tract section, as found in step 7 of the procedure described in Section
6.3.2. In order to ensure the section-by-section relevance of those area-scaling factors,care must be taken to perform this latter operation prior to speaker normalisation of thelength component of vowel-specific articulatory strategy, as described in Section 6.3.3.
In the following three Sections we thereby examine the contributions of speakerdifferences in vocal-tract structure, articulatory setting, and vowel-specific articulatory
strategy, to the vowel-speaker dichotomy.

6.4.1 Contributions of VT Structure to the Dichotomy
Table 6.1 lists the mean of our speakers’ estimated vocal-tract lengths, averaged overthe 7 frames and 5 repetitions per vowel. In the last column are listed the speakeraveraged lengths for each vowel, which confirm our earlier observations (in Chapter 5)
regarding the general validity of the phonetic variations in estimated vocal-tract length.
Indeed, the four vowels with the longest mean lengths (/ U /, /  /, / á /, and /  /, in orderof decreasing length) are precisely those which are known to be articulated with lip
rounding and larynx lowering, and the vowels with the shortest mean lengths (/ Q /, / E /,/ I /, and / i /, in order of increasing length) are the four front vowels which are known to
be articulated either with an open oral tract or with spread lips and raised larynx.
The third row from the bottom in Table 6.1 lists the mean lengths per speaker,
computed over all eleven vowels. The overall mean of 17.51 cm agrees well with thecommonly accepted value for the typical vocal-tract length of an adult male. However,in line with our earlier discussions regarding the potential contributions of lip rounding
and larynx height to variations in vocal-tract length, the range of 0.87 cm in the overallmean lengths across the four speakers presumably bears the influence of bothanatomical and behavioural differences.
284

Speaker
Vowel
/i/

A
15.76

B
16.14

C
16.61

D
16.26

Mean
16.19

/I/

15.39

16.09

16.33

16.37

16.05

/E/

14.74

16.20

16.00

16.21

15.79

/

14.94

15.30

16.08

15.27

15.40

16.38

17.00

17.39

16.65

16.85

16.89

16.82

17.65

17.14

17.12

//

17.26

18.28

17.59

17.86

17.74

//

19.37

19.62

20.08

21.02

20.02

/U/

20.88

19.81

21.22

22.33

21.06

/ á /

19.06

18.69

19.81

20.35

19.47

/ /

16.84

16.22

16.76

17.63

16.86

17.05

17.29

17.77

17.92

17.51

16.26

16.33

16.97

16.67

16.56

1.018

1.014

0.976

0.993

Q/
/a/
/«/

¶

Overall
Mean
SubMeanks

Table 6.1: Mean estimated VT-lengths (cm), and length-scaling factors required to speaker normalise
VT structure. Shown for each of the 11 vowels, are the mean VT-lengths computed over all 7 frames
and 5 repetitions per speaker. The per-vowel mean VT-lengths are listed in the last column on theright, and the per-speaker mean VT-lengths are listed in the row labelled “Overall Mean”. By contrast,
the row labelled “Sub-Mean” lists the mean VT-lengths computed only over the subset of four vowels
deemed to be least affected by lip protrusion/retraction and larynx raising/lowering (/ Q, a, « , ¶ / , asargued in Section 6.3.1), and which therefore yield the per-speaker length-normalisation factors (ks,listed in the bottom row) required for speaker normalisation of VT fixed structure.

Indeed, as argued in Section 6.3.1, there is ample evidence in the literature tosuggest that speaker differences in vocal-tract fixed structure are best described interms of the mean length of only the non-rounded, mid- to low vowels / Q /, / a /, / « /,and / ¶ /, owing to their relatively low susceptibility to extreme and variable degrees oflip protrusion and larynx lowering. The mean vocal-tract lengths computed over that
subset of four vowels are therefore shown in the second row from the bottom (labelled“Sub-Mean” and highlighted in bold-font) in Table 6.1. Whilst our arguments forselecting those four vowels were motivated purely on the basis of direct physiologicalobservations reported in the literature, it is encouraging to note that the computed submean lengths are consistently about 1cm shorter than the overall mean lengths, thus
lending further credibility to our assumption of having largely excluded the influences of285

lip rounding and larynx lowering, and thereby arriving at a more effective measure of
the speaker differences in vocal-tract fixed structure.
Listed in the bottom row of Table 6.1 are the uniform length-scaling factors k s ,which were computed according to Equation 6.3, assuming normalisation of eachspeaker’s sub-mean length to the mean, reference value of L (ref) = 16.56 cm. These
length-scaling factors and the sub-mean lengths from which they were derived, together
suggest that differences in the size of our four speakers’ vocal-tract fixed structures areindeed fairly small, amounting to a range of only 0.71 cm from the shortest ( 16.26 cm
for speaker A) to the longest ( 16.97 cm for speaker C). By comparison, the estimatedvocal-tract lengths averaged over the vowels / i /, / e /, and / a / spoken by eight adult,
male speakers of Japanese was found by Ishizaki (1978a) to range from 16.7 cm to18.4 cm.

As indicated by the k s values, normalisation of the speaker differences in vocaltract structure entails uniform lengthening of all the area-functions of speakers A and B
by 1.8 % and 1.4 %, respectively, and uniform shortening of all the area-functions of
speakers C and D by 2.4 % and 0.7 %, respectively. Whilst those length-scaling factors
are admittedly quite small compared with the commonly accepted average difference ofabout 15-20% between the vocal-tract lengths of adult males and females (as found,e.g., in the X-ray studies of Chiba and Kajiyama (1941) and Högberg (1995), or in
Fant’s (1966) physiological interpretation of inter-gender differences in vowel formant
distributions), nor is there any reason to expect larger differences in the vocal-tract
anatomical sizes of our four speakers, as they are all adult males. However, a more
relevant issue in regard to our articulatory explanation of the vowel-speaker dichotomy,
is the extent to which those vocal-tract structural differences play a part in the spectral
manifestation of the dichotomy as unfolded in Chapter 4. In particular, we seek thecontributions of speaker differences in vocal-tract structure (as revealed in Table 6.1) tothe overall drop in inter-speaker vowel classification accuracy observed across thehigher spectral regions.
In order to perform the relevant vowel classification experiments, we require
acoustic data which contain the speaker differences in vocal-tract fixed structure alone.
Towards that end, we first carry out speaker normalisation of all three articulatory286

features (structure, setting, and strategy) as described in Section 6.3; then re-introduce
the speaker differences in vocal-tract structure alone, by uniform un-normalisation of
the area-function lengths of each speaker, according to the scaling factors listed inTable 6.1. The formants of each of the resulting area-functions are then obtained by LP
synthesis, using either 8 or 10 vocal-tract sections depending on the number usedoriginally to estimate that area-function (as explained earlier, in Section 6.2). In order tomaintain consistency with our earlier assumptions regarding simplified cepstra
generated for the FC dataset and used in vowel classification experiments (cf. Section
4.4.3), only the first three, synthesised formant frequencies are therefore used tocompute NCC = 14 simplified LP cepstral coefficients (as described in Section 3.4.2),
assuming the formant bandwidths to be fixed at their respective, original mean values
computed over all eleven vowels and four speakers ( B1 = 99 Hz, B2 = 128 Hz, andB3 = 218 Hz).

The resulting behaviour of inter-speaker vowel classification accuracy across theavailable spectral range is shown by the solid curve in Figure 6.1, superimposed withboth the inter- and intra-speaker accuracy curves obtained earlier using the original,
simplified cepstra (cf. Figure 4.13). As expected, the inter-speaker accuracies obtainedusing the simplified cepstra generated from the partially speaker-normalised articulatory
data are consistently higher than those obtained using the original, simplified cepstra.
For example, at full spectral range (5000Hz), the acoustic data which bear the influenceof only the vocal-tract structural differences between the speakers yield 85.6 % —clearly higher than the original 77.5 %, and understandably lower than the 90.8 %attained by the intra-speaker curve.
Surprisingly, as the upper spectral limit is extended across the region from1380Hz to 1900Hz, the accuracies slightly exceed those obtained along the intraspeaker curve. On face value, this might be taken to imply that speaker differences invocal-tract fixed structure are slightly beneficial to vowel classification across thatfrequency sub-band. However, the method of data-partitioning used in the inter-speaker
experiment yields a larger number of training samples per vowel, which itself may leadto an improved performance in classification (as found in connection with the quadratic
classifier in Chapter 4), even more so if the vocal-tract structural differences amongst
287

our four speakers are insufficient to counteract the presumed improvement in accuracy.
In this vein, it is interesting to note that the higher accuracies occur across a spectral
range where vowel-speaker interactions are yet to manifest their full detrimental
influence.
Indeed, the solid curve of vowel classification accuracy shown in Figure 6.1
clearly indicates that even those small differences in vocal-tract structure (as identified
in Table 6.1) are sufficient to induce vowel-speaker interactions across the higher
spectral regions, which then cause the drop in classification accuracy from a peak of
89.4 % at 1580Hz, to 85.3 % as the spectral range is extended further to 3140Hz. As

the detrimental influence of inter-speaker variability is caused only by vocal-tract
structural differences between the speakers, the drop of 4.1 % in accuracy is naturally
smaller than the drop of 7.2 % observed in the original, inter-speaker curve.
Nevertheless, the contrast between the dichotomous behaviour on the one hand, and thenearly-asymptotic, intra-speaker curve on the other, does clearly underscore thecontributions of vocal-tract fixed structure to the vowel-speaker dichotomy.
Analogously to our acoustic-phonetic explanation of the dichotomy in Chapter 4,
a more informative perspective on the contributions of vocal-tract structure to thedichotomy is gained by plotting the relevant portions of the inter-speaker accuracy
curve (the solid curve in Figure 6.1) adjacent to the abscissa and ordinate of the F1 F2and the F2 F3 vowel formant distributions used to generate the simplified cepstra.
Indeed, the acoustic-phonetic relevance of the spectral regions of primary phonetic
influence is clearly indicated in Figure 6.2, by the continuous rise in vowel classification
accuracy as the upper spectral limit is extended across the entire F1 range, and acrossthe F2 range of the back vowels. The peak in classification accuracy at 1580Hz(marked by the horizontal, dashed line in Figure 6.2, and by the horizontal and vertical,
dashed lines in Figure 6.3) coincides almost perfectly with the mid- F2 of our four
speakers’ vowel space, as defined earlier in terms of the mean F2 of the quasi-neutral
vowel / ¶ /. The acoustic-phonetic relevance of the vowel-speaker interactions inducedby speaker differences in vocal-tract structure is then revealed in both Figures 6.2 and
6.3, by the gradual drop in classification accuracy as the spectral range is extended
further to include first the F2 of the front vowels, and later the F3 of both the back and
288

100

Vowel classification accuracy (%)

90
80
70
60
50
40
Original (intra- and inter-speaker)
Contributions of VT Structure (inter-speaker)
30
20
0

500

1000

1500 2000 2500 3000 3500
Upper limit of spectral range (Hz)

4000

4500

5000

Figure 6.1: Contributions of speaker differences in vocal-tract fixed structure, to the dichotomous
behaviour in inter-speaker vowel classification accuracy (FC dataset). The upper and lower, dashedcurves are the intra- and inter-speaker accuracy curves, respectively, obtained in Chapter 4 (see Figure4.13) using simplified cepstra generated from the first three, measured formant frequencies, withbandwidths fixed to mean values (B1 = 99Hz, B2 = 128Hz, B3 = 218Hz), sampling frequency Fs = 10kHz,
and NCC= 14. The solid curve shows the behaviour of inter-speaker vowel classification accuracy
obtained using simplified cepstra generated from the first three formant frequencies synthesised fromthe vocal-tract area-functions, after speaker normalisation of articulatory setting and vowel-specificarticulatory strategy (with formant bandwidths fixed at the mean values listed above).

the front vowels.
Having thus confirmed the persistence of the dichotomy and its broad acousticphonetic explanation in the presence of only vocal-tract anatomical differences between
the speakers, it seems natural to enquire about the more detailed, acoustic-phonetic
manifestations of those differences. In view of our definition of vocal-tract structure interms of mean length, the acoustic-phonetic correlates of the related speaker differences
are easily predicted. In particular, uniform scaling of the vocal-tract length of any areafunction induces a uniform (but inverse) scaling of the formant frequencies. Moreover,equal percentage changes in the formant frequencies give rise to larger absolute changes(in Hz) in the higher than in the lower formants. Consequently, the individual vowel
formant clusters are expected to exhibit relatively larger, inter-speaker variations along289

F2 (Hz)
2500

/i/

/I/

Upper limit of spectral range (Hz)

/E/
/Q/

2000
/ á /

/«/

1500
/¶/

/a/

1000
//
/U/

Speaker A
Speaker B
Speaker C
Speaker D

//

50

60

70

80

90

100

500
200

Vowel classification accuracy (%)

300

400

500

600

700

900

F1 (Hz)100

Vowel classification accuracy (%)

800

90
80
70
60
50
200

300

400
500
600
700
Upper limit of spectral range (Hz)

800

900

Figure 6.2: F1F2 vowel space of all 4 male speakers (FC dataset), synthesised from their estimatedarea-functions after speaker normalisation, retaining differences in vocal-tract structure only. Eachvowel cluster is shown with a 2σ ellipse. Adjacent to the abscissa and ordinate are plotted the portionsof the inter-speaker accuracy curve (solid line in Figure 6.1) which span the F1 and the F2 ranges,
respectively. The horizontal (dashed) line intersects the accuracy curve at its peak (at 1580Hz), andcuts across the formant plane in order to emphasise the acoustic-phonetic relevance of the spectral
regions of primary phonetic or speaker (VT structure) influence.

290

F3 (Hz)
3500

Upper limit of spectral range (Hz)

/i/
/I/

3000
//

//

/a/

/«/

/E/

2500

/Q/
/¶/
/U/

/ á /

2000
Speaker A
Speaker B
Speaker C
Speaker D

50

60

70

80

90

100

1500
500

Vowel classification accuracy (%)

1000

1500

2000

100

Vowel classification accuracy (%)

2500

F2 (Hz)

90
80
70
60
50
500

1000
1500
2000
Upper limit of spectral range (Hz)2500

Figure 6.3: F2F3 vowel space of all 4 male speakers (FC dataset), synthesised from their estimatedarea-functions after speaker normalisation, retaining differences in vocal-tract structure only. Eachvowel cluster is shown with a 2σ ellipse. Adjacent to the abscissa and ordinate are plotted the portionsof the inter-speaker accuracy curve (solid line in Figure 6.1) which span the F2 and the F3 ranges,
respectively. The vertical and horizontal (dashed) lines intersect the accuracy curve at its peak (at1580Hz), and cut across the formant plane in order to emphasise the acoustic-phonetic relevance of thespectral regions of primary phonetic or speaker (VT structure) influence.

291

imaginary lines running through the origin, and the elongation along those radial lines is
expected to be more pronounced in the F2 F3 than in the F1 F2 plane.
These predictions are indeed borne out by the vowel formant distributions shownin Figures 6.2 and 6.3. For example, although the vocal-tract structural differencesamongst our four speakers were earlier shown to be relatively small, speaker C (whowas found to have the longest mean length) clearly occupies the lower-left region of theellipse drawn around each vowel cluster. Also, in comparison with the original formant
distribution (cf. Figures 4.9 and 4.10), the ellipses are narrower in the direction
perpendicular to imaginary, radial lines, and their elongation parallel to those radial lines
is most clearly evident for the group of front vowels in the F2 F3 plane. Indeed, speakernormalisation of articulatory setting and vowel-specific articulatory strategy has not
only reduced the overall size of each ellipse (while retaining a small degree of radialelongation), it has almost completely transformed the original, fan-shaped distributionof the front vowels in the F2 F3 plane, and replaced it with a more compact distribution
which is elongated in the direction of increasing F2 and F3 . As the direction ofelongation of the entire group of front vowels in that plane nearly coincides with theprincipal axis of each individual ellipse, confusions amongst the F2 and the F3 ofneighbouring front vowels might be expected to contribute significantly to the drop inclassification accuracy observed across the higher spectral regions.
Indeed, an acoustic-phonetic decomposition of the vowel misclassifications which
contribute to the drop in accuracy across those higher spectral regions, reveals that thecontributions arising from confusions amongst the F2 and amongst the F3 of the front
vowels, are more significant than those arising from confusions amongst the F3 of theback vowels. As shown in Table 6.2, the two largest contributions are caused by thedrop of 40% in the F2 range and 60% in the F3 range of the front vowel / E / ofspeaker A due to confusions with / I /, and the drop of 25% in the F2 range and 30% in
the F3 range of the front vowel / i / of speaker C, also due to confusions with theneighbouring vowel / I /. Notwithstanding some minor differences in the exact nature of
the vowel confusions, our acoustic-phonetic decomposition of the drop in accuracy inthe original inter-speaker curve (see Table 4.2) showed that amongst the front vowels,/ E / of speaker A and / i / of speaker C were indeed the main contributors to the292Speaker
Vowel
/i/

A

/I/

/ i / (F2, 15%)/E/

/ I / (F2,40%; F3,60%)B/ i / (F2, 20%)

/Q/
/a/

C

D

/ I / (F2,25%; F3,30%)/ I / (F2, 5%)/ E / (F2, 10%; F3, 15%)

/ E / (F3, 5%)/ ¶ / (F3, 15%)/ « / (F3, 25%)/ « / (F3, 5%)/«// « / (F3, 10%)
/ a / (F3, 30%)

/ a / (F3, 5%)

//
//
/U/

/ U / (F3, 20%)/  / (F3, 5%)/  / (F3, 25%)/ á //¶/

Table 6.2: Acoustic-phonetic decomposition of the dichotomy in inter-speaker vowel classification
behaviour (solid curve in Figure 6.1), showing the contributions of speaker differences in vocal-tract
structure, in terms of the vowel misclassifications that contribute to the drop in accuracy across thehigher spectral regions which encompass the high-F2 and the F3 of the speakers’ vowel formant
distribution.

dichotomy, even when the data had not yet been subjected to speaker normalisation.
Our present results therefore suggest that the main contributions of vocal-tract
structure to the overall dichotomy are the confusions amongst the F2 and amongst theF3 of the front vowels of speakers A and C, whom we have earlier identified,
respectively, with the shortest and the longest, structure-related vocal-tract lengths
amongst the four adult, male speakers of the FC dataset.

6.4.2 Contributions of Articulatory Setting to the Dichotomy
In Section 6.3.2 we described the steps required to compute each speaker’s overallmean vocal-tract shape, or articulatory setting. Briefly, the structure-normalised areafunctions of each speaker are first aligned to minimise inter-frame and inter-repetition
variations in shape, then centre-aligned across vowels (and speakers) at the mid-lengthof the MOL region determined for each group of frame- and repetition-aligned areafunctions. The mean vocal-tract shape is then computed for each speaker, by averaging(in the logarithmic-area domain) across all frames, repetitions, and vowels, at every
vocal-tract section (of length ∆x = 0.05 cm) where there exists an aligned area-function.
As also pointed out in Section 6.3.2, any discontinuities at the glottal and labial ends of293

the mean area-functions thus found are retained in subsequent computations, without
adverse effect. However, for purely aesthetic reasons, those raw, mean vocal-tract
shapes are first re-parameterised across their entire lengths, and only the resulting,smoothed area-functions are here displayed.
Figure 6.4 shows superimposed the four, mean vocal-tract shapes thus obtained.
The origin on the abscissa is defined as the overall mean position of the glottis, whichlies a distance equal to half the overall mean length (cf. Table 6.1) to the left of thecentral landmark at which all groups of area-functions are aligned; the vertical, dottedline on the right marks the overall mean position of the radiating plane at the lips, whichitself is 17.51 cm from the mean position of the glottis. Each speaker’s mean vocal-tract
shape extends beyond those average glottis and lip markers, as far as there exists any
individual, aligned area-function of that speaker.
The mean vocal-tract shapes shown superimposed in Figure 6.4 do reveal longterm idiosyncracies which may be interpreted in terms of articulatory settings adopted
by each of our four speakers. First, it appears that the bulk of speaker differences inarticulatory setting are manifest in approximately the front half of the vocal-tract.
Indeed, the mean vocal-tract shapes are quite similar up to about 6cm from the averageglottal position, which corresponds perhaps to a large part of the pharynx. Although theroot of the tongue, the lower part of the tongue body, and even the muscles of thepharynx walls are capable of either constricting or expanding the cross-sectional areasin that part of the vocal-tract on a long-term basis, thereby giving rise to so-called“pharyngeal settings” (Laver, 1980), Stevens and House’s (1955, p.485) observationsof X-ray data presented in the literature “suggest ... that the first few centimeters of the
tract above the glottis change through only a restricted range of cross-sectional areas incomparison with other portions of the tract.” Consequently, that part of the vocal-tract(which includes the so-called larynx tube) is often modelled with a fixed cross-sectional
area profile and a fixed length of about 2cm (e.g., Stevens and House, 1955; Fant,
1960; Flanagan et al., 1980; Carré and Mrayati, 1991). To the extent that any
significance may be given to an anatomical (as opposed to a behavioural) interpretationof that part of the vocal-tract, the apparent similarity in our speakers’ mean areafunctions up to about 6cm from the glottis is therefore consistent with our findings in294

6Speaker A
Speaker B
Speaker C
Speaker D

5

Normalised Area

4

3

2

1

0
-2

0

2

4
6
8
10
12
14
16
Distance from mean glottal position (cm)18

20

Figure 6.4: Speaker differences in articulatory setting. Shown superimposed are the 4 speakers’ mean
VT-shapes computed over all their vowels after speaker normalisation of VT structure, cross-frameand -repetition alignment, and centre-alignment of all area-functions across vowels and speakers. The
vertical, dotted lines on the left and the right indicate, respectively, the overall mean position of theglottis (which defines the origin on the abscissa) and of the lips (at 17.51cm). The setting areafunctions extend beyond those lines as far as there exists any single, aligned area-function of thatspeaker.
the previous Section, of the fairly similar vocal-tract anatomical sizes of these four
speakers.
By contrast, the middle and front parts of the mean vocal-tract shapes shown in
Figure 6.4 suggest a wider range of speaker differences in articulatory setting in theupper pharynx and oral parts of the vocal-tract. As the shapes embody, by definition,
articulatory properties which are shared by all the vowels of each speaker, they arenaturally less eccentric (or less constricted) compared with most of the individual areafunctions of which they are the average. Nevertheless, if we consider the acousticallymore salient, local minima in the cross-sectional area profiles, there appear two main
articulators responsible for the locations of long-term constriction — the tongue itself,
which determines the shape of the settings near the mid-length of the vocal-tract; andthe lips, which determine the (latitudinal) settings towards the very front of the vocaltract.
295

The location of the mean lingual constriction alone suggests a separation of thefour speakers into three categories: speaker D exhibits the most anterior location atabout 10.1 cm from the mean position of the glottis; speaker B exhibits the most
posterior location at about 7.5 cm from the mean glottal position; and speakers A and C
share a location intermediate between those two extremes, at about 9.1 cm from theglottis. Whilst the hard palate, the soft palate, and other landmarks along the length ofthe vocal-tract cannot be precisely located on our area-functions, we may neverthelessventure to label the observed lingual settings in terms of a standard vocal-tract anatomyof an adult male speaker, as depicted for example by Perkell (1969, p.6, Figure 2.3).
Indeed, the same reference vocal-tract outline was also used by Mrayati and Carré(1988, p.224, Figure 9) to illustrate the physical relevance of the vocal-tract sections of
their DRM model. If the total length along the vocal-tract airway from the glottis to thelips is assumed to be 17.5 cm (as found in Table 6.1), then the three distinct locations ofconstriction observed in Figure 6.4 might be interpreted as long-term tendencies
towards uvularised (speaker B), velarised (speakers A and C), and palato-velarised(speaker D) lingual articulations, respectively. Whilst these terms are consistent bothwith the physiological theory of phonetics put forward by Peterson and Shoup (1966)
and with Laver’s (1980) framework for describing latitudinal, supralaryngeal settings
which involve the tongue body, we should nevertheless proceed with caution, and avoid
placing undue emphasis on such finely graded articulatory labels in an absolute sense.
Our estimated articulatory data should rather be interpreted in terms of the overall
trends of speaker differences in vocal-tract shape.
Apart from his relatively more fronted, average place of lingual constriction,
Speaker D is further separated from the other speakers on the basis of his long-term(latitudinal) lip posture, which clearly appears more constricted. Although Laver (1980)
provides quite a detailed ensemble of descriptors for latitudinal, labial settings, many of
those detailed, physiological specifications are inapplicable to vocal-tract areafunctions, which cannot by themselves discriminate, for example, between the verticaland horizontal dimensions of constriction or expansion. Speaker D might therefore be
regarded simply as having a labial setting which involves a more pronounced degree oflip rounding in comparison with the other speakers.
296

In order to ascertain whether the speaker differences in articulatory setting thus far
described are likely to have any contributions at all to the vowel-speaker dichotomy, we
require acoustic data which contain only those setting-related differences. To this end,
the original area-functions are again completely speaker-normalised with respect to all
three articulatory features (structure, setting, and strategy) as described in Section 6.3,
and each section of the resulting area-functions of each speaker is then divided by theappropriate area-scaling factor for that speaker (as found in step 7 in Section 6.3.2),
thereby re-introducing the speaker differences in articulatory setting. The first three,LP-synthesised formant frequencies are then used with our fixed formant bandwidths togenerate NCC = 14 cepstral coefficients by the same methods described in the previous
Section; and inter-speaker vowel classification experiments are repeated to yield
another accuracy profile as a function of increasing spectral range.
The resulting behaviour of classification accuracy is shown by the solid curve in
Figure 6.5, superimposed with the original intra- and inter-speaker accuracy curves firstobtained in Chapter 4 (using simplified cepstra) and shown again in the previous
Section (cf. Figure 6.1). At nearly every value of the upper spectral limit, the accuracies
attained after retaining only speaker differences in articulatory setting are intermediate
to those obtained in the original intra- and inter-speaker experiments. Indeed, theaccuracy of 85.5 % attained at full spectral range (5000Hz) is nearly identical to thefull-range accuracy obtained in the previous Section when only the speaker differences
in vocal-tract structure were retained. The detrimental influence of speaker differences
in vocal-tract structure and articulatory setting (as we have defined them) are therefore
equal in magnitude when we consider the entire spectral range.
More importantly in terms of the contributions of articulatory setting to thedichotomy, the solid curve in Figure 6.5 does exhibit a quasi-dichotomous behaviour.
Classification accuracy rises to a peak of 84.9 % at 1600Hz, then drops to 81.5 % as
the spectral range is extended further to 2500Hz. Whilst this drop of 3.4 % is slightly
less than the drop of 4.1 % found (in the previous Section) to be precipitated byspeaker differences in vocal-tract structure, it does occur across a narrower spectral
range. Classification accuracy then begins to rise as the spectral range is extended
beyond 2500Hz, and attains a full-range value which is slightly higher than its earlier
297

100

Vowel classification accuracy (%)

90
80
70
60
50
40
Original (intra- and inter-speaker)
Contributions of Articulatory Setting (inter-speaker)
30
20
0

500

1000

1500 2000 2500 3000 3500
Upper limit of spectral range (Hz)

4000

4500

5000

Figure 6.5: Contributions of speaker differences in articulatory setting, to the dichotomous behaviour
in inter-speaker vowel classification accuracy (FC dataset). The upper and lower, dashed curves are theintra- and inter-speaker accuracy curves, respectively, obtained in Chapter 4 (see Figure 4.13) using
simplified cepstra generated from the first three, measured formant frequencies, with bandwidths fixed
to mean values (B1 = 99Hz, B2 = 128Hz, B3 = 218Hz), sampling frequency Fs = 10kHz, and NCC= 14.
The solid curve shows the behaviour of inter-speaker vowel classification accuracy obtained using
simplified cepstra generated from the first three formant frequencies synthesised from the vocal-tractarea-functions, after speaker normalisation of vocal-tract fixed structure and vowel-specific articulatory
strategy (with formant bandwidths fixed at the mean values listed above).

peak at 1600Hz.
An acoustic-phonetic interpretation of the quasi-dichotomous behaviour of theaccuracy curve is again gained by plotting the relevant portions of that curve adjacentto the abscissa and ordinate of the F1 F2 and the F2 F3 vowel formant distributions fromwhich the simplified cepstra were generated. As shown in Figure 6.6, classification
accuracy rises across the entire F1 range, and continues to rise as the spectral range is
extended across the F2 of the speakers’ back vowels. The peak in accuracy occurs atvery nearly the mid- F2 of the formant distribution, as defined earlier in terms of themean F2 of the quasi-neutral vowel / ¶ /. Classification accuracy then drops as the
spectral range is extended across the entire F2 of the speakers’ front vowels. Whilst the
local minimum along the accuracy curve at 2500Hz indeed occurs beyond the highest298

F2 , in Figure 6.7 it is shown to occur near the middle of the speakers’ F3 range. The
subsequent improvement in classification accuracy is therefore attributed partly to thehigh- F3 range of the speakers’ back vowels, but perhaps more strongly to the entire F3
range of their four front vowels / i /, / I /, / E /, and / Q /, which are admittedly somewhat
better separated than the back vowels along the F3 dimension.
Before we decompose the speaker-induced vowel misclassifications which havecaused the drop in accuracy across the high F2 and the low F3 ranges, we should
perhaps attempt to gain a better understanding of the acoustic-phonetic consequencesof the speaker differences in setting, which we have described thus far only inarticulatory terms. Some of the acoustic correlates of the lingual and labial settings
described earlier in connection with Figure 6.4, were indeed discussed by Laver (1980).
For example, a slightly retracted tongue body (as in the uvularised setting of speaker B)
ought to lower the F2 , and the implied contraction in the uvular region should tend toraise the F3 of back vowels (Fant, 1975b). On the other hand, a relatively more raised
lingual bias (as in the palato-velarised setting of speaker D) is generally expected toraise the F2 , particularly in the front vowels where the acoustic sensitivity in the palatalregion is enhanced by the local constriction. By contrast, lip rounding (as also exhibitedby speaker D) generally lowers all formant frequencies, with a more pronounced effecton higher formants (Lindblom and Sundberg, 1971); it should therefore lower the F3 inparticular, and tend to counteract any rise in F2 of back vowels caused by concomitantpalato-velarisation.
The formant distributions shown in Figures 6.6 and 6.7 allow us to verify thesepredictions of the acoustic correlates, as the formants were synthesised from the areafunctions following the speaker normalisation procedure described earlier, which retains
speaker differences in articulatory setting alone. The F2 F3 plane (Figure 6.7) inparticular highlights the setting-related idiosyncrasies of speakers B and D as discussed
above. For example, the formant distribution of speaker B appears to form an upper
layer in F3 , especially in the back vowels (as predicted by the uvular contraction).
Whilst it is tempting to attribute the lower F2 of that speaker’s high front vowels to thepresumed tongue body retraction, recall from Figure 6.4 that speaker B has a moreconstricted setting just anterior to the vocal-tract region where those vowels are likely
299

F2 (Hz)
2500

/i/
/I/

Upper limit of spectral range (Hz)

/E/
/Q/

2000
/ á /

1500

/«/
/¶/

/a/

1000
//
/U/

Speaker A
Speaker B
Speaker C
Speaker D

//

50

60

70

80

90

100

500
200

Vowel classification accuracy (%)

300

400

500

600

700

900

F1 (Hz)100

Vowel classification accuracy (%)

800

90
80
70
60
50
200

300

400
500
600
700
Upper limit of spectral range (Hz)

800

900

Figure 6.6: F1F2 vowel space of all 4 male speakers (FC dataset), synthesised from their estimatedarea-functions after speaker normalisation, retaining differences in articulatory setting only. Eachvowel cluster is shown with a 2σ ellipse. Adjacent to the abscissa and ordinate are plotted the portionsof the inter-speaker accuracy curve (solid line in Figure 6.5) which span the F1 and the F2 ranges,
respectively. The horizontal, heavy-dashed line intersects the accuracy curve at its peak (at 1600Hz);
the horizontal, light-dashed line intersects the curve at its local minimum (at 2500Hz). Those lines cut
across the formant plane in order to emphasise the acoustic-phonetic relevance of the spectral regionsof primary phonetic or speaker (articulatory setting) influence.

300

F3 (Hz)
3500

Upper limit of spectral range (Hz)

/i/

3000
//

/a/
/«/

//
/I/

2500

/E/
/Q/
/¶/
/ á //U/

2000
Speaker A
Speaker B
Speaker C
Speaker D

50

60

70

80

90

100

1500
500

Vowel classification accuracy (%)

1000

1500

2000

100

Vowel classification accuracy (%)

2500

F2 (Hz)

90
80
70
60
50
500

1000
1500
2000
Upper limit of spectral range (Hz)2500

Figure 6.7: F2F3 vowel space of all 4 male speakers (FC dataset), synthesised from their estimatedarea-functions after speaker normalisation, retaining differences in articulatory setting only. Eachvowel cluster is shown with a 2σ ellipse. Adjacent to the abscissa and ordinate are plotted the portionsof the inter-speaker accuracy curve (solid line in Figure 6.5) which span the F2 and the F3 ranges,
respectively. The vertical and horizontal, heavy-dashed lines intersect the accuracy curve at its peak (at1600Hz); the vertical and horizontal, light-dashed lines intersect the curve at its local minimum (at2500Hz). Those lines cut across the formant plane in order to emphasise the acoustic-phonetic
relevance of the spectral regions of primary phonetic or speaker (articulatory setting) influence.

301

to have their main place of linguo-palatal constriction (from about 12cm to 15cm fromthe glottis). The lowering of F2 and the concurrent raising of F3 observed in Figure 6.7
is therefore explained by the effective lengthening of the back cavity and shortening of
the front cavity, respectively, which would arise from the resulting, more anterior placeof constriction in those high front vowels.
By contrast, the formant distribution of speaker D appears to form a lower layer in
F3 , as predicted by the greater degree of lip rounding observed in Figure 6.4. At thesame time, the presumed tendency to raise the tongue body indeed causes a higher F2
in the front vowels, especially in /Q / (see also the F F distribution in Figure 6.6),
12whose constriction near the palato-velar region of the vocal-tract (Peterson and Shoup,
1966, p.45, Figure 2) would tend to be further constricted by the palato-velarisationsuggested in Figure 6.4 for speaker D.
In comparison with the F2 F3 formant distribution synthesised from the areafunctions containing speaker differences only in vocal-tract structure (cf. Figure 6.3 inthe previous Section), each of the front vowel clusters shown in Figure 6.7 is elongatedin a direction perpendicular to radial lines, thus clearly reflecting the speaker differencesin long-term vocal-tract shape rather than length. In view of these clearly identifiable
acoustic-phonetic differences which are also justified articulatorily, the question thenarises to what extent the speaker differences described above induce the drop inclassification accuracy observed across the high F2 and the low F3 ranges.
An acoustic-phonetic decomposition of the vowel misclassifications which occur
in those higher spectral regions is shown in Table 6.3. The largest contributionsiI

amongst the front vowels are the confusions of the vowel / / of speaker B with / /(owing to the low F2 of the former, which we explained in terms of a more frontedI

place of lingual constriction), and of the vowels / / and /E

Q / of speaker D with / i / and/ /, respectively (owing to the higher F2 , which we explained in terms of a palatovelarised setting, or a more raised tongue body). The largest contribution amongst theaback vowels arises from confusions of the vowel / / of speaker C in the F3 range with«the highly overlapping vowel / /. Amongst the more central vowels, the consistently
high F3 values of speaker B cause confusions of his vowel /¶á / with the quasi-neutral/ / — confusions which had also occurred when the original simplified cepstra were302

Speaker
Vowel
/i/

A

B

C

/ I / (F2, 15%)

/ I / (F2, 40%)

/ I / (F2, 10%)/I/

/ E / (F2, 25%)

/ E / (F2, 15%)

/ i / (F2, 10%)/E/D
/ i / (F2, 30%)
/ I / (F2, 20%)
/ i / (F2, 5%)

/Q/
/a/

/ E / (F2, 25%)/  / (F3, 5%)/«// a / (F3, 5%)/// a / (F3, 15%)/// « / (F3, 35%)/ « / (F3, 5%)/ a / (F3, 10%)

/ a / (F3, 5%)/ U / (F3, 5%)/U// á // ¶ / (F3, 20%)/¶/

Table 6.3: Acoustic-phonetic decomposition of the dichotomy in inter-speaker vowel classification
behaviour (solid curve in Figure 6.5), showing the contributions of speaker differences in articulatory
setting, in terms of the vowel misclassifications that contribute to the drop in accuracy across thehigher spectral regions which encompass the high-F2 and the low-F3 of the speakers’ vowel formant
distribution (from the peak at 1600Hz to the local minimum at 2500Hz).

used (as listed in Table 4.2).
Clearly, this acoustic-phonetic decomposition confirms our earlier observations
that only the F2 of the front vowels and the F3 of the back (and central) vowels are
responsible for the drop in accuracy which is induced by speaker differences inarticulatory setting, and which was observed to occur between 1600Hz and 2500Hz.
An acoustic-phonetic decomposition of the rise in accuracy beyond 2500Hz also
confirms our earlier predictions concerning the role of setting-related differences in theF3 of the speakers’ front vowels — improvements in classification accuracy occur
mainly in the F3 range of the vowel / i / of speaker B and the vowels / I /, / E /, and / Q / ofspeaker D. This is in clear contrast with the results of the inter-speaker experiments
performed in the previous Section, where it was found that both the F2 and the F3 offront vowels were the main contributors to the dichotomy induced by speakerdifferences in vocal-tract structure.

6.4.3 Contributions of Articulatory Strategy to the Dichotomy
Thus far we have accounted for the two, long-term articulatory sources of inter-speakervariability in vowel production, namely structure, which we defined in terms of the303

mean vocal-tract length of the mid- to low vowels, and setting, which we defined interms of the mean vocal-tract shape computed over all the vowels following two stagesof area-function alignment. The contributions of vocal-tract structure to the dichotomy
were found to arise mainly in the F2 and the F3 of the front vowels of speakers A andC, who themselves were found to have the smallest and the largest vocal-tract
anatomical sizes, respectively, amongst our four speakers. By contrast, thecontributions of articulatory setting to the dichotomy were found to arise mainly in theF2 of the front vowels and the F3 of the back and central vowels of the two speakers B
and D, who had earlier been found to have vocal-tract anatomical sizes close to theaverage of our four speakers, and whose articulatory settings were then described interms of long-term tendencies, respectively, towards uvularisation concomitant with amore forward place of constriction in front vowels, and towards palato-velarisation (orraising of the tongue body) concomitant with lip rounding.
Upon speaker normalisation of both structure and setting, the area-functions are
assumed to be associated with a common, or reference vocal-tract fixed anatomy, andto have a common, or reference long-term articulatory bias. The remaining, or residual
speaker differences can then be described in terms of both the lengths and shapes of thenormalised area-functions on a per-vowel basis, and thus collectively regarded as
idiosyncratic behaviour in vowel-specific articulatory strategy. Insofar as thearticulators are capable of altering the effective length along the vocal-tract airway fromthe glottis to the radiating plane at the lips, residual differences in length can arise fromidiosyncratic articulatory behaviour in the amount of lip protrusion and larynx loweringor raising, and perhaps to a lesser extent in the posture and position of the tongue. On
the other hand, residual differences in shape are determined by the so-called latitudinalcomponents of inter-speaker articulatory variability, which are measured perpendicular
to the direction of airflow, and which involve the degrees of lingual and labialconstriction, and the areas of the vocal-tract cavities and constrictions formed by the
tongue and the opposing tract walls.
These residual differences in articulatory strategy are shown in Figure 6.8, where
each of the eleven panels displays superimposed the per-vowel mean (or prototype)area-functions of our four speakers, as computed in step 1 of the algorithmic procedure304

described in Section 6.3.3. The lengths of the displayed area-functions have thus been
subjected to speaker normalisation of vocal-tract structure as described in Section 6.3.1
and later quantified in Table 6.1. Their shapes have likewise been subjected to speakernormalisation of articulatory setting, as described in Section 6.3.2 and discussed in theprevious Section in connection with Figure 6.4. As in the latter, the left and rightvertical (dotted) lines in each panel of Figure 6.8 indicate, respectively, the overall
average position of the glottis (which defines the origin on the abscissa) and of the lip
termination (at 17.51 cm).
Whilst the length-related speaker differences in articulatory strategy are conveyed
graphically in Figure 6.8, they are also quantified in Table 6.4, which lists the mean,
structure-normalised vocal-tract length of each speaker, on a per-vowel basis. In thelast column are listed the speaker-averaged mean lengths for each vowel (cf. Equation
6.7 in Section 6.3.3), which then allow computation of the length-scaling factors k sv
(according to Equation 6.6) required for speaker normalisation of the length componentof vowel-specific articulatory strategy (cf. Equation 6.5).
A brief examination of each group of superimposed area-functions in Figure 6.8
reveals relatively smaller amounts of inter-speaker variation in the lower parts of the
vocal-tract which include the first 4 to 6cm above the glottis. A similar observation wasmade in connection with the speakers’ vowel-averaged vocal-tract shapes or settings (inthe previous Section), in which context we offered an explanation in terms of therelative inflexibility of the lower part of the pharynx and the so-called larynx tube. It istherefore remarkable that the structure- and setting-normalised area-functions (in Figure6.8) should also have relatively invariant shapes across speakers on a per-vowel basis,in that part of the vocal-tract which is more likely than other parts to be associated withthe speakers’ fixed anatomy. In this regard, it could well be argued that the plausibilityof our estimated area-functions is partially supported by the absence of unrealisticallylarge areas or expanded cavities in that lowest part of the vocal-tract which, as we havenoted, is often modelled explicitly with a fixed length and cross-sectional area profile.
Rather, the bulk of speaker differences in articulatory strategy appear to be
manifest in the vocal-tract regions which correspond to the entire front or oral cavity,
and approximately the upper half of the pharynx. Speaker variations occur in the size of305Normalised Area

6
Speaker A
Speaker B
5 Speaker C
Speaker D
4

/i/

3
2
1
0
-2

0

2

4

6

8

10

12

14

16

18

20

Normalised Area

6
Speaker A
Speaker B
5 Speaker C
Speaker D
4

/I/

3
2
1
0
-2

0

2

4

6

8

10

12

14

16

18

20

Normalised Area

6
Speaker A
Speaker B
5 Speaker C
Speaker D
4

/E/

3
2
1
0
-2

0

2

4
6
8
10
12
14
Distance from mean glottal position (cm)

16

18

20

Figure 6.8: Speaker differences in vowel-specific articulatory strategy (4 pages of graphs). Shownsuperimposed for each of the 11 vowels, are the 4 speakers’ prototype area-functions obtained after
speaker normalisation of VT structure and articulatory setting. The vertical, dotted lines shown in eachgraph on the left and the right indicate, respectively, the overall mean position of the glottis (which
defines the origin on the abscissa) and of the lips (at 17.51cm). Area-functions are aligned at the centreof the mutually overlapping region per vowel per speaker (as described in Section 6.3.2).

306

Normalised Area

6
Speaker A
Speaker B
5 Speaker C
Speaker D
4

/ á

3
2
1
0
-2

0

2

4

6

8

10

12

14

16

18

20

Normalised Area

6
Speaker A
Speaker B
5 Speaker C
Speaker D
4

/¶/

3
2
1
0
-2

0

2

4

6

8

10

12

14

16

18

20

Normalised Area

6
Speaker A
Speaker B
5 Speaker C
Speaker D
4

/Q/

3
2
1
0
-2

0

2

4
6
8
10
12
14
Distance from mean glottal position (cm)Figure 6.8: (continued from previous page).

307

16

18

20

12
Speaker A
Speaker B
11 Speaker C
Speaker D
10

//

9

Normalised Area

8
7
6
5
4
3
2
1
0
-2

0

2

4

6

8

10

12

14

16

18

20

8

Normalised Area

Speaker A
Speaker B
7 Speaker C
Speaker D
6

/U/

5
4
3
2
1
0
-2

0

2

4
6
8
10
12
14
Distance from mean glottal position (cm)Figure 6.8: (continued from previous page).

308

16

18

20

12
Speaker A
Speaker B
11 Speaker C
Speaker D
10

/ /

9

Normalised Area

8
7
6
5
4
3
2
1
0
-2

0

2

4

6

8

10

12

14

16

18

20

8

Normalised Area

Speaker A
Speaker B
7 Speaker C
Speaker D
6

/a/

5
4
3
2
1
0
-2

0

2

4

6

8

10

12

14

16

18

20

Normalised Area

6
Speaker A
Speaker B
5 Speaker C
Speaker D
4

/«/

3
2
1
0
-2

0

2

4
6
8
10
12
14
Distance from mean glottal position (cm)Figure 6.8: (continued from previous page).
309

16

18

20

expanded regions, in the degrees and locations of primary (or secondary) constriction,and in the amount of lip rounding and protrusion. For example, in the two, high front
vowels / i / and / I /, speaker D appears to have a distinctly more forward place ofprimary lingual constriction, while that of speaker B is both longer and more retracted.
In those vowels and also in the fronted / á /, speaker A appears to have a greater
volume in the back cavity expansion compared with the other speakers. In the mid-front
vowel / E /, speaker D has quite a distinct vocal-tract configuration in the oral cavity,with a more fronted place but reduced degree of constriction; on the other hand,
speaker A exhibits a more voluminous front cavity, a greater degree of constriction, andan overall shorter vocal-tract length (as quantified in Table 6.4). In the lip-rounded backvowels /  /, / U /, and /  /, speaker D exhibits a more anterior place of lingual
constriction, a greater amount of lip rounding, and larger areas in the front-cavity
expansion; on the other hand, speaker B exhibits a more retracted place of constriction
which also appears to extend over a longer part of the vocal-tract. Whilst the areafunctions of all the back vowels suggest a fairly large amount of speaker variability inthe shape and volume of the expanded, front cavity, it is doubtful that these types ofphysical differences will have a proportional impact in the acoustic domain, as theformant sensitivity to changes in cross-sectional area generally reduces with increasingarea.
Indeed, it remains to be determined whether any of the speaker differences inarticulatory strategy shown in Figure 6.8 and listed in Table 6.4, will contribute to thevowel-speaker dichtomy which is spectrally manifest in the behaviour of inter-speakervowel classification accuracy. In order to perform the necessary vowel classification
experiments, we require acoustic data which contain speaker differences in articulatory
strategy alone. Consistently with the methodology adopted in Chapter 4 and in theprevious two Sections, the first three formant frequencies LP-synthesised from thestructure- and setting-normalised area-functions are therefore used with fixed formant
bandwidths, to generate NCC = 14 simplified cepstral coefficients. Those cepstra arethen used to perform inter-speaker vowel classification experiments with the same
increasing spectral range, as described in Section 4.2.3.1.
The resulting accuracy curve is shown in Figure 6.9 (solid curve), superimposed
310

Speaker
Vowel

A

B

C

D

Mean

/i/

VTL
ksv

16.04
1.009

16.36
0.990

16.21
0.999

16.15
1.003

16.19

/I/

VTL
ksv

15.67
1.024

16.32
0.984

15.94
1.007

16.27
0.987

16.05

/E/

VTL
ksv

15.00
1.052

16.42
0.961

15.61
1.011

16.10
0.981

15.79

/Q/

VTL
ksv

15.22
1.012

15.51
0.993

15.69
0.981

15.17
1.015

15.40

/a/

VTL
ksv

16.68
1.010

17.23
0.978

16.97
0.993

16.54
1.019

16.85

/«/

VTL
ksv

17.20
0.996

17.05
1.004

17.22
0.994

17.03
1.006

17.12

//

VTL
ksv

17.57
1.010

18.53
0.958

17.16
1.034

17.74
1.001

17.74

//

VTL
ksv

19.73
1.015

19.89
1.007

19.60
1.022

20.88
0.959

20.02

/U/

VTL
ksv

21.26
0.991

20.08
1.048

20.71
1.017

22.18
0.950

21.06

/ á /

VTL
ksv

19.40
1.004

18.94
1.028

19.33
1.007

20.21
0.963

19.47

/¶/

VTL
ksv

17.15
0.984

16.45
1.025

16.36
1.031

17.51
0.963

16.86

Table 6.4: Mean VT-lengths (cm) obtained after speaker normalisation of VT structure, and thefactors required to speaker normalise length-related differences in vowel-specific articulatory strategy.
Shown for each of the 11 vowels, are the mean VT-lengths computed over all 7 frames and 5
repetitions per speaker, after normalisation of VT structure (cf. Table 6.1). The per-vowel mean VTlengths are listed in the right-most column, and the per-speaker length-normalisation factors (ksv)
required to speaker-normalise length-related differences in articulatory strategy, are listed in the rows
directly below the lengths for each vowel. Shading indicates a significant contribution of the lengthrelated differences in strategy to the vowel-speaker dichotomy (see text for detailed discussion).
with both the inter- and intra-speaker accuracy curves obtained using the original,
simplified cepstra. Evidently, the speaker differences in vowel-specific articulatory
strategy do significantly contribute to the vowel-speaker dichotomy. Indeed, of thethree accuracy curves generated thus far in this Chapter and shown by the solid curves
in Figures 6.1, 6.5, and 6.9, this latter curve is clearly the closest to the original, interspeaker accuracy curve at every step along the spectral continuum.
At full spectral range (5000Hz), the accuracy of 78.1 % is only marginally better
than that of 77.5 % obtained using the original, simplified cepstra. This indicates that

311

100

Vowel classification accuracy (%)

90
80
70
60
50
40
Original (intra- and inter-speaker)
Contributions of Articulatory Strategy (inter-speaker)
30
20
0

500

1000

1500 2000 2500 3000 3500
Upper limit of spectral range (Hz)

4000

4500

5000

Figure 6.9: Contributions of speaker differences in vowel-specific articulatory strategy, to thedichotomous behaviour in inter-speaker vowel classification accuracy (FC dataset). The upper andlower, dashed curves are the intra- and inter-speaker accuracy curves, respectively, obtained in Chapter
4 (see Figure 4.13) using simplified cepstra generated from the first three, measured formant
frequencies, with bandwidths fixed to mean values (B1 = 99Hz, B2 = 128Hz, B3 = 218Hz), sampling
frequency Fs = 10kHz, and NCC= 14. The solid curve shows the behaviour of inter-speaker vowel
classification accuracy obtained using simplified cepstra generated from the first three formant
frequencies synthesised from the vocal-tract area-functions, after speaker normalisation of vocal-tract
fixed structure and articulatory setting (with formant bandwidths fixed at the mean values listed
above).

the detrimental influence of speaker differences in articulatory strategy (as defined in
our modelling methodology) is, on the whole, nearly identical in magnitude to that of
the three articulatory sources of speaker variability combined. By comparison, theaccuracies attained at full spectral range in Figures 6.1 and 6.5 were about 8% higher,
indicating that the overall detrimental influences of speaker differences in either
structure or setting alone, are relatively lower in magnitude.
As far as vowel-speaker interactions are concerned in the higher spectral regions,a closer examination of Figure 6.9 reveals a slightly lower peak in accuracy, and a lesspronounced drop in accuracy across the higher spectral regions. In particular,classification accuracy rises to a maximum of 80.8 % as the spectral range is extended
to 1780Hz, dips slightly to 79.9 % at 1960Hz and rises again to 80.8 % at 2160Hz,
312

then drops to 75.8 % as the spectral range is extended further to 3140Hz. The mostextensive drop in accuracy therefore amounts to 5.0 % which, although less than theoriginal drop of 7.2 %, is still larger than the drop in accuracy attributed to structureand setting-related speaker differences ( 4.1 % and 3.4 %, respectively). Amongst thethree, articulatory sources of inter-speaker variability, vowel-specific articulatory
strategy therefore appears to claim the largest contribution to the vowel-speakerdichotomy.
Whilst the sheer extent of the drop in accuracy across the higher spectral regions
is a good indication of the magnitude of the contribution to the dichotomy, theappearance of a double-peak in accuracy which spans nearly 400Hz (from 1780Hz to
2160Hz) raises an important question regarding the acoustic-phonetic relevance of therespective spectral regions over which the presumed influences of vowel-speaker
interactions play either a small, or a more substantial role. Naturally, that question isbest addressed by plotting, as we are now accustomed to do, the relevant portions of
the inter-speaker accuracy curve adjacent to the abscissa and ordinate of the synthesised

F1 F2 (in Figure 6.10) and F2 F3 (in Figure 6.11) vowel formant distributions whichwere earlier used to generate the simplified cepstra.
In Figure 6.10, classification accuracy is shown to steadily rise across the four
speakers’ entire F1 range. That steady rise in accuracy is briefly interrupted in the low-

F2 range, by a local peak and a small drop in accuracy which is caused by confusions
amongst the F2 of the highly overlapping vowels / a / and / « / — interestingly, a similar
dip in accuracy was observed in the original inter-speaker curve, and also in the two
accuracy curves shown earlier in this Chapter. However, as the spectral range is
extended further, classification accuracy rises substantially to the first of the two global
peaks, at approximately the low F2 of the three, high front vowels. Indeed, as shown inboth Figures 6.10 and 6.11, the spectral region between 1780Hz and 2160Hz where thesmall, speaker-induced drop in accuracy is matched by an equal rise back to the peak of
80.8 %, encompasses the F2 of mainly the three front vowels / i /, / I /, and / E /. By

comparison, a more substantial and persistent deterioration in classification performanceoccurs as the spectral range is extended beyond the second peak (at 2160Hz), andacross nearly the entire F3 distribution.
313

F2 (Hz)
2500

/I/

/i/

Upper limit of spectral range (Hz)

/E/

2000

/Q/

1500

/«/
/¶/

/ á /

/a/

1000
//
/U/
//

30

40

50

60

70

80

90

100

500
200

Vowel classification accuracy (%)

300

400

500

600

700

800

900

F1 (Hz)100

Vowel classification accuracy (%)

Speaker A
Speaker B
Speaker C
Speaker D

90
80
70
60
50
40
30
200

300

400
500
600
700
Upper limit of spectral range (Hz)

800

900

Figure 6.10: F1F2 vowel space of all 4 male speakers (FC dataset), synthesised from their estimatedarea-functions after speaker normalisation, retaining differences in vowel-specific articulatory strategy
only. Each vowel cluster is shown with a 2σ ellipse. Adjacent to the abscissa and ordinate are plotted
the portions of the inter-speaker accuracy curve (solid line in Figure 6.9) which span the F1 and the F2
ranges, respectively. The horizontal, dashed lines intersect the accuracy curve at its two peaks (at1780Hz and 2160Hz, respectively), and cut across the formant plane in order to emphasise theacoustic-phonetic relevance of the spectral regions of primary phonetic or speaker (vowel-specificarticulatory strategy) influence.

314

F3 (Hz)
3500/i/

Upper limit of spectral range (Hz)

//

3000
/a/
//

/«/

/Q/

2500

/I/
/E/
/¶/
/ á /

2000/U/

Speaker A
Speaker B
Speaker C
Speaker D

30

40

50

60

70

80

90

100

1500
500

Vowel classification accuracy (%)

1000

1500

2000

100

Vowel classification accuracy (%)

2500

F2 (Hz)

90
80
70
60
50
40
30
500

1000
1500
2000
Upper limit of spectral range (Hz)2500

Figure 6.11: F2F3 vowel space of all 4 male speakers (FC dataset), synthesised from their estimatedarea-functions after speaker normalisation, retaining differences in vowel-specific articulatory strategy
only. Each vowel cluster is shown with a 2σ ellipse. Adjacent to the abscissa and ordinate are plotted
the portions of the inter-speaker accuracy curve (solid line in Figure 6.9) which span the F2 and the F3
ranges, respectively. The vertical and horizontal, dashed lines intersect the accuracy curve at its two
peaks (at 1780Hz and 2160Hz, respectively), and cut across the formant plane in order to emphasise
the acoustic-phonetic relevance of the spectral regions of primary phonetic or speaker (vowel-specificarticulatory strategy) influence.

315

Speaker
Vowel
/i/

A

B

C

/ I / (F3, 100%)/ I / (F2, 20%)D

/ E / (F2, 5%)

/I/
/E/

/ i / (F2, 100%)/ i / (F2, 20%)

/ i / (F2, 25%)

/ E / (F3, 60%)

/ E / (F3, 10%)/ á / (F2, 20%)

/ I / (F3, 5%)

/Q/
/a/

/ « / (F3, 80%)/«// « / (F3, 10%)
/ a / (F3, 15%)

/ a / (F3, 5%)/ U / (F3, 80%)

//
//
/U/

/  / (F3, 15%)/  / (F3, 75%)/ á // ¶ / (F2, 35%)/¶/

/ E / (F2, 5%)

Table 6.5: Acoustic-phonetic decomposition of the dichotomy in inter-speaker vowel classification
behaviour (solid curve in Figure 6.9), showing the contributions of speaker differences in vowelspecific articulatory strategy, in terms of the vowel misclassifications that contribute to the drop in
accuracy across the higher spectral regions which encompass the high-F2 and the F3 of the speakers’
vowel formant distribution.

These observations are confirmed by an acoustic-phonetic decomposition (inTable 6.5) of the vowel misclassifications which contribute to the drop in accuracy
across the higher spectral regions (including the region between the two global peaks).
Significant contributions to the drop in accuracy are found to occur as a result ofconfusions mainly amongst the F3 of the back vowels and amongst the F3 of the front
vowels. However, contributing confusions are also found to occur amongst the F2 ofthe front vowels, despite the concurrent improvement in classification performanceacross that spectral range as observed earlier. The most significant of the dichotomous
vowel confusions listed in Table 6.5 are next discussed with reference to the formant
distributions of Figures 6.10 and 6.11, and with reference to the articulatory
manifestations of speaker differences in vowel-specific strategy shown earlier in thearea-functions of Figure 6.8 and the vocal-tract lengths listed in Table 6.4.
Confusions amongst the F2 of the front vowels are indeed listed in Table 6.5 for
all four speakers, but none more significant than the 100% drop in accuracy caused by
confusions of the vowel / E / of speaker A with the vowel / i /. Precisely the same vowel316confusion was reported earlier using the original acoustic data (both the measured andthe simplified cepstra, cf. Tables 4.1 and 4.2, respectively), but not in Tables 6.2 and6.3 where the contributions of structure and setting were identified. We may therefore
assume that this major contribution to the dichotomy is attributed entirely to speakerdifferences in articulatory strategy. As shown in Figure 6.8, those differences in strategy
are nevertheless rather subtle. In particular, it would appear that the articulatory
strategy adopted by speaker A to produce the vowel / E / involves a shorter vocal-tract
length (see also the shaded entry in Table 6.4) and a slightly greater degree of linguopalatal constriction, compared with the other speakers. The higher F2 values observedin Figures 6.10 and 6.11 for the vowel / E / of speaker A (higher values which are also
clearly evident in the original formant distributions shown in Chapter 4, cf. Figure 4.9),
are therefore explained by a combination of potentially three articulatory strategies: amore raised larynx, a greater amount of lip spreading, and a closer approximation of the
tongue to the hard palate.
A less dramatic contribution to the dichotomy which also occurs in the F2 rangeof the front vowels, is caused by confusions of the vowel / I / of speaker D with thefronted / á /. An acoustic-phonetic explanation is provided in Figure 6.10, which shows
clearly lower values of F2 for / I / of speaker D — so low that they encroach upon theellipse drawn around the formant cluster of the neighbouring, lip-rounded vowel (asalso observed in the original formant distribution, in Figure 4.9). Although not itself
implicated in the observed misclassifications, that speaker’s F3 distribution for / I / isshown in Figure 6.11 (and indeed in Figure 4.10) to be relatively high. In this vein, it iswell known that the second and third formants of high front vowels are strongly
affiliated, respectively, with the back and the front vocal-tract cavities which areseparated by the linguo-palatal constriction (e.g., Fant, 1960; Lindblom and Sundberg,
1971). The lower F2 and higher F3 observed in the high front vowels of speaker D
might therefore be brought about by an anterior shift in the place of constriction, which
would lengthen the back cavity (i.e., lower F2 ) and simultaneously shorten the front
cavity (i.e., raise F3 ). This is indeed the articulatory explanation emergent from Figure
6.8, which clearly suggests a relatively more forward place of constriction in the highfront vowels of that speaker. As a consequence of this idiosyncratic articulatory317

behaviour, vowel-speaker interactions are induced in the acoustic domain which cause
the observed confusions in the F2 range with the neighbouring, lip-rounded vowel.
Table 6.5 also reveals confusions in the F2 range of the fronted vowel / á / ofspeaker D, with the more central vowel / ¶ /. A lower distribution of F2 in the former isagain implicated in the formant plots of Figures 6.10 and 6.11, and the articulatory
explanation suggested in Figure 6.8 is similar to that discussed above — speaker D
appears to prefer a relatively more anterior place of constriction. However, there is also
evidence of a longer vocal-tract length (as quantified in the shaded entry in Table 6.4).
The implied lowering of the larynx would further lengthen the back cavity and thus help
to lower the F2 ; by contrast, the equally implied lip protrusion would help to maintain anearly constant front-cavity length, thereby counteracting the potential rise in F3 causedby the more fronted place of constriction. Indeed, this is confirmed in the formant plotof Figure 6.11, where it is shown that the F3 distribution for / á / of speaker D is close
to the average for that vowel.
This latter explanation is of particular significance, as it also partly explains therise in accuracy as the spectral range is extended to the second, global peak at 2160Hz.
The vowel / á / of speaker D is indeed one of the main contributors to that rise inaccuracy, owing to a gradual correction of the misclassifications discussed above, as theupper spectral limit is extended across the low- F3 range. However, a comparison of
Figure 6.11 with the F2 F3 vowel formant distributions shown in the previous two
Sections and in Chapter 4, indicates that while the originally lowered F2 of thatspeaker’s / á / is entirely explained by his idiosyncratic behaviour in articulatory
strategy, there is also originally a lowered F3 , which we have attributed to his
articulatory setting (the greater degree of lip rounding on a vowel-averaged or longterm basis, as discussed in the previous Section). Whilst that lowered F3 was not foundby itself to cause a drop in classification accuracy, its nullification by speakernormalisation of articulatory setting (as observed in Figure 6.11) has caused a rise inaccuracy, which occurs only after the strategy-induced drop in the F2 range.
This example highlights an important aspect of our methodology: when potential
articulatory sources of speaker variability are considered in isolation, they may induce
certain vowel-speaker interactions (either to the detriment or to the benefit of vowel318

classification accuracy) which had not transpired and were therefore latent in theoriginal data, where the sources of variability are combined and themselves allowed tointeract. Another example listed in Table 6.5 is a contribution to the strategy-induced
dichotomy in the F3 range of the vowel / a / of speaker A, which is confused with / « /.
Admittedly, those two vowels are nearly indistinguishable in both the F1 F2 and theF2 F3 planes. However, this particular confusion which contributes an 80% drop in
accuracy across the higher spectral regions, had not occurred until the speakerdifferences in articulatory strategy were explicitly isolated.
The two significant contributions to the dichotomy in the F3 range of the backvowels, which had also occurred using the original, simplified cepstra (cf. Table 4.2),
are listed in Table 6.5 as confusions of the vowel /  / of speaker D with / U /, and of thevowel / U / of speaker B with /  /. The formant distributions in Figures 6.10 and 6.11
offer an acoustic-phonetic explanation of the former in terms of a lower F1 , F2 , andespecially F3 in /  / of speaker D, and of the latter in terms of a slightly lower F2 but
much higher F3 in / U / of speaker B.
The superimposed area-functions for /  / in Figure 6.8 certainly indicate a range of
speaker differences not only in the length and volume of the oral cavity which is formedby the retracted and lowered tongue body, but also in the place of lingual constriction.
In particular, speaker D appears to have the most expanded oral cavity, a main place of
lingual constriction which is displaced further down the pharynx compared with that of
the other speakers (at about 5.8 cm from the glottis), and a more fronted, secondary
lingual constriction (at about 10.3 cm from the glottis). Attesting to the articulatory
plausibility of this double-constriction, are directly measured area-functions of similar,
low back vowels reported by Wood (1979, p.26, Figure 1: / A /) for a speaker of British
English, and by Story et al. (1996, p.545, Figure 6: /  /) for a speaker of American
English. According to the SM model (cf. Figure 5.1), the F3 of a uniform acoustic tube
can be lowered by alternately expanding and constricting the areas at multiples of
1
5

L cm from the lips. Thus, given the overall mean length L = 17.74 cm for /  / (as listed

earlier in Table 6.1), the half-wavelength of the F3 shape-component is 15 L ≈ 3.5 cm.
Indeed, as shown in Figure 6.8, the more expanded front cavity and the secondary
lingual constriction exhibited by the area-function for /  / of speaker D do occur,319

respectively, at about 3.5 cm and 7cm from the mean position of the lips. Those shaperelated idiosyncrasies in articulatory strategy therefore explain the low values of F3
observed for /  / of speaker D.
The superimposed area-functions for / U / in Figure 6.8 suggest that speaker B has
a more retracted place of constriction, and one which extends over a longer part of the
vocal-tract. In addition, his vocal-tract length for that vowel is markedly shorter (seealso the shaded entry in Table 6.4). As discussed previously in connection with thisspeaker’s articulatory setting (which we cautiously labelled uvularisation), a contractionof the vocal-tract areas in the uvular region will tend to raise the F3 of back vowels.
The idiosyncratic behaviour in articulatory strategy adopted by speaker B in his
production of / U /, which indeed appears to involve more constricted areas in the upper
pharynx in addition to an overall shorter vocal-tract length (implying a less loweredlarynx and less protruded lips, as shown in Figure 6.8), therefore explains the higher
values of F3 which have caused confusions with /  /, and hence the contribution to thedrop in accuracy observed in the higher spectral regions.
Finally, significant contributions to the drop in classification accuracy across themid- to high range of our speakers’ F3 distribution, are listed in Table 6.5 as arisingfrom confusions amongst the F3 of the high front vowels of speaker B. In particular, adrop of 100% in accuracy occurs as that speaker’s / i / is confused with / I /, and a dropof 60% occurs as his vowel / I / is confused with / E /. Whilst both of these vowel
confusions were also found to occur in Chapter 4 (using either the original or thesimplified cepstra, cf. Tables 4.2 and 4.3, respectively), their contributions to thedichotomy as shown in Table 6.5 are considerably greater. This implication of the
speaker differences in articulatory strategy playing a major role in the observed vowel
confusions, is further underscored by the complete absence of those confusions in ourearlier acoustic-phonetic decompositions in Tables 6.2 and 6.3. The acoustic-phonetic
manifestations of those differences in strategy as shown in Figure 6.11, indicate a lower
distribution of F3 in the high front vowels of speaker B, which clearly explain the
confusions observed in the F3 range with the neighbouring vowels. The articulatory
explanation, as noted earlier in connection with Figure 6.8, is a more retracted place of
linguo-palatal constriction in those high front vowels of speaker B — as the third320

resonance is mainly affiliated with the vocal-tract cavity in front of the constriction, amore retracted place of constriction would lengthen that front cavity, and hence lower
the F3 as observed in the vowel formant distributions.
In sum, our acoustic-phonetic decomposition of the strategy-induced drop ininter-speaker vowel classification accuracy observed across the higher spectral regions,has revealed that the largest contributions arise from confusions amongst the F3 of theback vowels of speakers B and D, amongst the F3 of the high front vowels of speakerB, and to a lesser extent amongst the F2 of the mid-front vowel of speaker A. Thesevowel confusions are caused by idiosyncrasies in vowel-specific articulatory strategy,
which are summarised (together with their observed acoustic-phonetic consequences)
as follows: a less lowered larynx in the back vowels and a generally more retracted
place of lingual constriction for speaker B (causing a higher F3 in the high back vowels,
and a lower F3 in the high front vowels); a larger front-cavity expansion in the backvowels and a generally more forward (even if secondary) place of constriction for
speaker D (causing a lower F2 in the front and central vowels, and a lower F3 in theback vowels); a more raised larynx and a greater degree of linguo-palatal constriction in
the mid- to high front vowels of speaker A (causing a higher F2 in those front vowels).
By comparison, as shown by the vowel formant distributions, the area-functions, andthe list of vowel misclassifications, speaker C is the most representative (or the leastextreme) of the four speakers of the FC dataset.

6.5 The Dichotomy Undone
The methodology used in the previous Section to obtain a tripartite articulatory
explanation of the vowel-speaker dichotomy, relied on speaker normalisation of only
two of the three articulatory features in turn, thereby retaining the speaker differences inthe third or remaining feature. If our explanations of the dichotomy have indeed
revealed the influences of three separate and complementary, articulatory sources ofinter-speaker variability, then it might be expected that speaker normalisation of all
three components would completely remove the inter-speaker variations in our dataset,and retain the influences of only the inherent, phonetic variability contained therein.
Thus completely deprived of the inter-speaker component of variability, the vowel321

EstimatedVT Area-Functions(FC dataset)

Section 6.3
Speaker normalisation of
VT Fixed Structure

Speaker normalisation of
Articulatory Setting

Speaker normalisation ofVowel-specific
Articulatory Strategy

Section 6.5.1
Completely speaker-normalised
VT Area-Functions

LP Synthesis
Fixedformant bandwidths:
B1 = 99Hz
B2 = 128Hz
B3 = 218Hz

Section 6.5.2
Formant frequencies:{F1, F2, F3}

Sampling frequency:
Fs = 10kHz

LP inverse-routines:
fb2a & a2cp

{ck},
k=1,2,..,14.
Section 4.2.3.1

Inter-speaker
Vowel Classification
Section 6.5.2
Behaviour of accuracy curve
across the spectral continuum

Figure 6.12: Undoing the vowel-speaker dichotomy by complete articulatory speaker normalisation.
Flow-chart of the computational procedure adopted to speaker-normalise the estimated vocal-tract areafunctions, LP-synthesise formant frequencies, generate simplified cepstra, and perform inter-speakervowel classification as a function of an increasing spectral range. Enumerated at the top-left ofcomputational modules or output data modules, are the Sections in this or in previous Chapters where
the respective algorithmic procedures or generated data are discussed.

322

speaker interactions which we have observed and explained in both the acousticphonetic and articulatory domains should cease to be manifest, and the remaining,phonetic component might then be regarded as prototypical of the speakers contained in
the given dataset.
In Figure 6.12 is shown a flow-chart of the computational procedure adopted toreveal the articulatory and acoustic-phonetic consequences of undoing the vowelspeaker dichotomy as suggested above. First, our four speakers’ estimated vocal-tractarea-functions are subjected to all three stages of articulatory speaker normalisation(structure, setting, and strategy) as described in Section 6.3. In Section 6.5.1 we briefly
examine those completely speaker-normalised area-functions, and discuss their merit inrepresenting prototypical vocal-tract shapes of the Australian English monophthongal
vowels which comprise the phonetic dimension of our FC dataset. Those speakernormalised area-functions are then input to the subsequent stages of LP synthesis offormants, of generation of simplified cepstra, and of inter-speaker vowel classification
as a function of an increasing spectral range. Both the synthesised formant distribution
and the resulting behaviour of classification accuracy across the spectral continuum are
therefore examined in Section 6.5.2, where we finally bring to light the acousticphonetic consequences of our articulatory method of complete speaker normalisation.

6.5.1 Articulatory Consequences
As described above, the area-functions obtained after complete speaker normalisationof vocal-tract structure, articulatory setting, and vowel-specific articulatory strategy,
contain only the phonetic component of the original variability, and are therefore
expected to be immune to vowel-speaker interactions. However, they are also expected
to contain the original, intra-speaker variations which are retained throughout theprocess of speaker normalisation. The per-vowel mean of the completely speakernormalised area-functions (computed across frames, repetitions, and speakers, as in step
2 of the strategy-normalisation procedure described in Section 6.3.3), may therefore be
regarded as a prototypical vocal-tract configuration for the steady-state of that vowel
recorded in /hVd/ context by our four, adult male speakers of Australian English.
In Figure 6.13 are shown the per-vowel prototype area-functions thus computed,323

Normalised Area

4

/i/
/I/
/E/
/Q/

3
2
1
0
-2

0

2

4

6

8

10

12

14

16

Normalised Area

4

18

20

/ á
/
/ /

3
2
1
0
-2

0

2

4

6

8

10

12

14

16

7

20

//
/U/

6

Normalised Area

18

5
4
3
2
1
0
-2

0

2

4

6

8

10

12

14

16

7

20

//
/a/
/«/

6

Normalised Area

18

5
4
3
2
1
0
-2

0

2

4
6
8
10
12
14
Distance from mean glottal position (cm)

16

18

20

Figure 6.13: Prototype vocal-tract area-functions of the 11 Australian English vowels (FC dataset),
as obtained in step 2 of the procedure described in Section 6.3.3. The vertical, dotted lines on the left
and the right indicate, respectively, the overall mean position of the glottis (which defines the origin on
the abscissa) and of the lips (at 17.51cm). Area-functions are aligned at the centre of the mutually
overlapping (MOL) region per vowel per speaker (as described in Section 6.3.2).
324

grouped approximately according to their main place of constriction. If these areafunctions are indeed to be trusted as prototypical of the Australian English vowels
included in our FC dataset, then they should embody at least certain basic properties
which are normally expected of vocal-tract configurations for those or similar vowels.
For example, the individual lengths of the prototype area-functions are expected toconfirm what we know about the phonetic relevance of lip rounding and larynx
lowering. The prototype vocal-tract lengths which appear in Figure 6.13 and whichwere also listed earlier in the last column of Table 6.4, indeed confirm that the liprounded and larynx-lowered vowels are, in order of decreasing length, the back vowels/ U / and /  /, the fronted / á /, and to a lesser extent the back vowel /  /. By comparison,
the lengths of the central to low vowels / ¶ /, / « /, and / a / are fairly close to the average,
and thus confirm our arguments (in Section 6.3.1) which led to the selection of a vowel
subset more likely to convey speaker differences in vocal-tract fixed structure. In that
context, whilst the low front vowel / Q / (which we also included in that vowel subset)
appears in Figure 6.13 to have the most inward-displaced glottal end (albeit marginally),
it is difficult to separate the influences of our methods of inter-repetition alignment andcentre-alignment in that regard. However, in regard to vocal-tract length, our prototype
area-functions of the four front vowels are not surprisingly the shortest, and differ only
by a slight progression of increasing length with vowel height.
As mentioned above, the four groups of area-functions in Figure 6.13 are arranged
approximately according to their main place of constriction along the length of the
vocal-tract. In the top graph are shown superimposed the four front vowels, with theirmain places of lingual constriction clearly located in the palatal region, between 10cm
and 13cm from the overall mean position of the glottis. Consistent with our knowledgeof the constrictive properties of these vowels, their place and degree of linguo-palatal
constriction appear to be more retracted and more expanded, respectively, in the orderfrom the high-front / i / to the low-front / Q /. In addition, there appear alternative or
secondary places of constriction in the pharyngeal part of the vocal-tract for the lowerof the front vowels, as often remarked or portrayed in the articulatory literature (cf. forexample the MRI-measured area-functions for / Q / shown in Baer et al., 1991, and in
Story et al., 1996).
325

In the second graph from the top are shown superimposed the prototype areafunctions of the very fronted / á / and the quasi-neutral / ¶ /. In accordance with its
relative quasi-neutrality, the latter area-function is indeed the least constricted along its
entire length. However, it does appear to have a tendency towards a fronted lingual
posture, which indeed confirms the slightly fronted position of its formant distribution
in the F1 F2 plane (e.g., Figure 4.9), with a lower F1 and a higher F2 compared withthe standard reference of Fn = (2n − 1)500 Hz.
Of all the area-functions shown in Figure 6.13, the prototype for the vowel / á /appears to be the most fronted, with its main place of constriction located in perhaps
the most anterior part of the hard palate. It is encouraging to note that Liljencrants’(1971) projections of a Swedish speaker’s X-ray measured mid-sagittal tongue contours
into a Cartesian coordinate system, also suggest a more fronted place of constriction for
the vowel /  / than for / i /. Whilst that vowel may indeed be articulated with extreme
frontness in Australian English (cf. its characteristically low F3 in, e.g., Figure 4.10),
there does appear to be a tendency for the raised tongue body to continue its
approximation to the roof of the mouth along nearly the entire length of the palate. In
this vein, recall from Figure 6.8 that two of our speakers (B and C) appeared to havetheir main place of constriction for / á / at about the same location as for / i / along thehard palate (at about 12cm from the overall mean position of the glottis); consistent
with Wood’s (1986) articulatory and acoustic findings on the / i /-/ y / contrast, speakers
B and C thus appeared to distinguish / i / from / á / by their degrees of lip rounding andlarynx lowering. By contrast, speakers A and D were shown in Figure 6.8 to have amore fronted place of constriction for the lip-rounded vowel. Our prototype areafunction for / á / portrayed in Figure 6.13 with (the shape-inferred articulatory features)rounded lips and lowered larynx, thus clearly bears the influence of the articulatory
strategies of all four speakers.
In the third graph from the top (in Figure 6.13) are shown superimposed theprototype area-functions for the two, high back vowels /  / and / U /. The velarised
articulations of these two vowels are confirmed by the location of the lingual
constriction at about 8cm to 9cm from the overall mean position of the glottis.
Consistent with the more extreme position of the formant cluster for /  / in the F1 F2
326

plane (e.g., Figure 4.9), it is shown in Figure 6.13 with a greater degree and a longer
place of constriction. Both vowels are shown correctly with large amounts of lip
rounding and larynx lowering.
In the bottom graph are shown superimposed the remaining three, prototype areafunctions for /  /, / a /, and / « /, which are appropriately portrayed as having the mostopen vocal-tract configurations. Moreover, their places of constriction are correctly
located in the pharyngeal part of the vocal-tract, at about 5 to 7cm from the mean
position of the glottis. By comparison with the front vowel / Q / which was shown
earlier to involve a secondary, low place of constriction, these three back vowels havethe lowest places of constriction in the pharynx, with an upward progression in place(and a more subtle progression towards a lesser degree of constriction) from /  / to / « /.
In accordance with the principle of conservation of the tongue body mass, the sameorder of progression is also observed in the area of the most expanded part of the oralcavity.
Our presentation in Figure 6.13 of the purely phonetic range of variation
embodied in prototype area-functions obtained after complete speaker normalisation,
was partly inspired by a similar presentation of X-ray measured area-functions by Wood
(1979), who thereby demonstrated the existence of four, so-called preferred locationsof constriction for spoken vowels. Analogously to his main contention in that work, ourprototype vocal-tract shapes are broadly categorised into three (rather than four) mainplaces of constriction, as follows1: (i) in the palatal region for the fronted vowels suchas / i /, (ii) in the velar region for the high back vowels such as /  /, and (iii) in thepharyngeal region for the low vowels such as / a /. The distinction drawn by Wood
(1979) between the places of constriction for /  / (further back) and / U / (more anterior)does not appear in our prototype area-functions — nor is it suggested, however, in theformant distributions of those two vowels, which appear (e.g., in Figure 4.9) partiallyoverlapping in the extreme high-back region of the F1 F2 plane. Nevertheless, thelingual constriction of our prototype for the more extreme (or tense) vowel /  / does
1

These three categories are represented in Figure 6.13 by the area-functions in the top and in thebottom two panels; / á / is grouped together with the fronted configurations in category (i), and / ¶ / isdisregarded in this context owing to its quasi-neutrality.

327

appear to extend further back into the upper pharynx compared with the area-function
for / U /, thus partly agreeing with Wood (1979), despite the differences in the phonetic
quality of the vowels spoken by his subjects (among them a speaker of Southern British
English) and our speakers of Australian English.
In the final analysis, we shall perhaps never be justified in claiming that areafunctions estimated by acoustic-to-articulatory mapping are exactly the vocal-tract areafunctions actually produced by the given speaker. In this vein, however, it is wellknown that direct methods of area-function measurement such as X-radiography or
MRI are also prone to various, non-negligible sources of error. Although in Chapter 5
we evaluated our hybrid method of area-function parameterisation and estimation
mainly from a theoretical point of view, a more lasting proof of viability is that which
emerges from articulatory interpretations of the variability contained in those estimatedarea-functions. In this Section, we have provided further evidence of the adequacy of
our inverse method in yielding sufficiently realistic area-functions, in terms of the more
widely documented and certainly better understood, phonetic dimension of variability in
our spoken vowel data.

6.5.2 Acoustic-Phonetic Consequences
As argued earlier, normalisation of the speaker differences in vocal-tract structure,articulatory setting, and vowel-specific articulatory strategy, is expected to yield areafunctions which retain variability only along the phonetic and intra-speaker dimensions.
Consequently, the vowel-speaker interactions which have been the focus of our study,
should cease to exist, and their manifestations in the acoustic-phonetic domain effaced.
In the previous Section we discussed the articulatory consequences of complete speakernormalisation, in terms of the phonetic relevance of the per-vowel prototype areafunctions thus obtained. We now examine its acoustic-phonetic consequences,
particularly in terms of the spectral manifestations of the vowel-speaker dichotomy.
As outlined earlier in Figure 6.12, the formants of the completely speakernormalised area-functions are first obtained by LP synthesis. NCC = 14 simplified
cepstral coefficients are then generated from the first three formant frequencies together
with the fixed formant bandwidths which we have adopted throughout this chapter.
328

100

Vowel classification accuracy (%)

90
80
70
60
50
40
Original (intra- and inter-speaker)
Completely speaker-normalised (inter-speaker)
30
20
0

500

1000

1500 2000 2500 3000 3500
Upper limit of spectral range (Hz)

4000

4500

5000

Figure 6.14: The dichotomy undone by complete articulatory speaker normalisation (FC dataset).
The upper and lower, dashed curves are the intra- and inter-speaker accuracy curves, respectively,
obtained in Chapter 4 (see Figure 4.13) using simplified cepstra generated from the first three,measured formant frequencies, with bandwidths fixed to mean values (B1 = 99Hz, B2 = 128Hz,
B3 = 218Hz), sampling frequency Fs = 10kHz, and NCC= 14. The solid curve shows the behaviour ofinter-speaker vowel classification accuracy obtained using simplified cepstra generated from the first
three formant frequencies synthesised from the vocal-tract area-functions, after speaker normalisationof vocal-tract fixed structure, articulatory setting, and vowel-specific articulatory strategy (withformant bandwidths fixed at the mean values listed above).

Applying the same methodology as described in Section 4.2.3.1, those cepstra are thenused to obtain our final profile of inter-speaker, vowel classification accuracy as afunction of increasing spectral range.
The behaviour of the resulting accuracy curve is shown in Figure 6.14 (solid line),superimposed with the intra- and inter-speaker accuracy curves (dashed lines) whichwere obtained in Chapter 4 using the original, simplified cepstra. The solid curve rises
to 89.9 % as the spectral range is increased to 1580Hz, beyond which it changes very
little; the best performance of 91.7 % is achieved at full spectral range (5000Hz).
Indeed, the nearly-asymptotic behaviour of the solid curve is quite comparable to thatof the original, intra-speaker curve, thus confirming that the inter-speaker variations in
our spoken vowel data have been effectively normalised, and only within-speakervariations retained. Unquestionably, the vowel-speaker dichotomy has been undone!
329

F2 (Hz)
2500

/i/
/I/

Upper limit of spectral range (Hz)

/E/
/Q/

2000
/ á /

/«/

1500
/¶/

/a/

1000
//
/U/

Speaker A
Speaker B
Speaker C
Speaker D

//

50

60

70

80

90

100

500
200

Vowel classification accuracy (%)

300

400

500

600

700

900

F1 (Hz)100

Vowel classification accuracy (%)

800

90
80
70
60
50
200

300

400
500
600
700
Upper limit of spectral range (Hz)

800

900

Figure 6.15: F1F2 vowel space of all 4 male speakers (FC dataset), synthesised from their estimatedarea-functions after complete speaker normalisation of vocal-tract structure, articulatory setting, andvowel-specific articulatory strategy. Each vowel cluster is shown with a 2σ ellipse. Adjacent to theabscissa and ordinate are plotted the portions of the inter-speaker accuracy curve (solid line in Figure6.14) which span the F1 and the F2 ranges, respectively. The horizontal (dashed) line cuts through thecentre of the ellipse for the quasi-neutral vowel / ¶ /; the mid-F2 thus defined (at about 1590Hz), alsointersects the accuracy curve at what might be regarded as its “knee”.

330

F3 (Hz)
3500

Upper limit of spectral range (Hz)

/i/
/I/

3000
/a/
//

/«/

//

/E/

2500

/Q/
/¶/
/U/

/ á /

2000
Speaker A
Speaker B
Speaker C
Speaker D

50

60

70

80

90

100

1500
500

Vowel classification accuracy (%)

1000

1500

2000

100

Vowel classification accuracy (%)

2500

F2 (Hz)

90
80
70
60
50
500

1000
1500
2000
Upper limit of spectral range (Hz)2500

Figure 6.16: F2F3 vowel space of all 4 male speakers (FC dataset), synthesised from their estimatedarea-functions after complete speaker normalisation of vocal-tract structure, articulatory setting, andvowel-specific articulatory strategy. Each vowel cluster is shown with a 2σ ellipse. Adjacent to theabscissa and ordinate are plotted the portions of the inter-speaker accuracy curve (solid line in Figure6.14) which span the F2 and the F3 ranges, respectively. The vertical (dashed) line cuts through thecentre of the ellipse for the quasi-neutral vowel / ¶ /; the mid-F2 thus defined (at about 1590Hz), alsointersects the accuracy curve at what might be regarded as its “knee”.

331Articulatorily and acoustically, we have therefore succeeded in normalising thedata of our four speakers to those of a single, prototype speaker, whose vocal-tractorganic and articulatory behavioural characteristics are, by definition, the average of ourfour speakers’ characteristics. Owing to the four-fold increase in the amount of(effectively) single-speaker data, it is likely that the resulting, within-speaker variations
represent an even greater degree of homogeneity compared with the original, nonnormalised data of each speaker separately. In this context, we note that as the spectral
range is extended beyond 1340Hz, the accuracies along the solid curve in Figure 6.14
are consistently higher than those obtained along the original, intra-speaker curve; atfull spectral range (5000Hz), the normalised data yield an improvement of 0.9 %.
Indeed, this small but consistent improvement is explained by the following two factors:(i) as noted above, the speaker-normalised data afford a greater richness of (effectively)
within-speaker variability; (ii) in addition, our method of data-partitioning yields agreater number of per-vowel training samples in inter-speaker experiments.
The reduction of inter-speaker variability and the consequent increase in thedensity of per-vowel variations, are perhaps best portrayed by the F1 F2 and F2 F3vowel formant distributions which were earlier used to generate the simplified cepstra.
As shown in Figures 6.15 and 6.16, the vowel formant clusters are indeed much smaller
than those depicted in the original formant distributions (cf. Figures 4.9 and 4.10). As
the means of the clusters are hardly affected by the speaker normalisation procedure,
the overall vowel separability is therefore enhanced, both in the formant space, and in
the cepstral domain as indicated earlier by the higher accuracies obtained. Moreover,the density of each vowel formant cluster is clearly much higher, owing to the four-foldincrease in the sheer number of data points which represent effectively the frames andrepetitions of our single, prototype speaker.
Adjacent to the abscissa and the ordinate of the vowel formant planes in Figures
6.15 and 6.16 are plotted those portions of the inter-speaker accuracy curve (solid line
in Figure 6.14) which span the respective formant ranges. Similarly to the behaviour ofthe original intra-speaker curve (cf. Figure 4.9), classification accuracy rises across theentire F1 range, and continues to rise as the spectral range is extended further acrossthe low F2 of our four speakers’ articulatorily-normalised formant distribution. At the332

mid- F2 (as defined by the mean F2 of the quasi-neutral vowel / ¶ /, which is equal to
1591Hz in the present, resynthesised data), classification accuracy nearly reaches its
asymptotic value of a little over 90%, and improves only marginally as the spectral
range is extended across the high F2 and the entire F3 range.
In further support of our conclusions in Chapter 4 regarding the intra-speakervowel classification results, the behaviour of our final, inter-speaker accuracy curve(shown in Figures 6.14, 6.15 and 6.16) first confirms the phonetic importance of the
low spectral regions which extend up to the mid- F2 of the vowel formant distribution.
It also confirms that the spectral regions higher than the mid- F2 , where we haveobserved the manifestations of speaker differences to be the most potent, contribute
relatively little to vowel discriminability even in the absence of the detrimental influenceof inter-speaker variations. Perhaps more importantly, the behaviour of our final, interspeaker accuracy curve attests to the effectiveness of our articulatory method of
speaker normalisation, which itself yielded (in Section 6.4) a three-part, physicalexplanation of the vowel-speaker dichotomy. This undoing of the dichotomy bycomplete speaker normalisation of the estimated vocal-tract area-functions, is indeed
our final proof both of the speaker-related potency of the higher spectral regions, and of
the viability of our tripartite definition and normalisation of the articulatory sources of
speaker differences in spoken vowels.

6.6 Concluding Summary
In this Chapter we presented a new methodology for speaker normalisation of vocaltract area-functions, which first facilitated a physical explanation of the vowel-speakerdichotomy in terms of three articulatory (supralaryngeal) sources of inter-speakervariability. Those sources of speaker differences were defined as follows: (i) vocal-tract
fixed structure, determined by the mean vocal-tract length computed over the mid- to
low vowels; (ii) articulatory setting, determined by the mean vocal-tract shapecomputed over all the vowels; and (iii) vowel-specific articulatory strategy, determinedby the mean vocal-tract length and shape computed on a per-vowel basis. Our
articulatory approach to explain the vowel-speaker dichotomy then consisted of:normalising the speaker differences in two of the three articulatory features in turn;333

Vowel formant range
F2 of front

Structure
SettingääStrategy
OriginaläF3 of backäää

F3 of frontäää

Contribution todichotomy (%)

4.1
3.4
5.0
7.2

Table 6.6: Contributions of vocal-tract structure, articulatory setting, and vowel-specific articulatory
strategy, to the vowel-speaker dichotomy, in terms of: (i) the vowel formant ranges of our four adult,
male speakers of Australian English (FC dataset) across which were observed significant contributions
to the drop in inter-speaker vowel classification accuracy, and (ii) the largest extent of that drop in
accuracy across the higher spectral regions. Also shown for comparison are the formant ranges and thetotal drop in accuracy obtained (in Chapter 4) using the original, simplified cepstra.
synthesising the first three formant frequencies of the resulting, normalised areafunctions; generating simplified cepstra from those synthetic formants; performing interspeaker vowel classification experiments as in Chapter 4; and finally carrying out an
acoustic-phonetic decomposition of the drop in classification accuracy observed acrossthe higher spectral regions.
Indeed, all three classification accuracy curves thus obtained were found to exhibit
a drop in accuracy across the higher spectral regions. However, systematic differenceswere found not only in the magnitude of the largest drop in each accuracy curve, butalso in the spectral range across which the most significant vowel confusions occurred.
Table 6.6 summarises the contributions of each of the three articulatory features to thedichotomy, as found in this Chapter for the FC dataset. The baseline for comparison isthe behaviour of the inter-speaker vowel classification accuracy curve obtained in
Chapter 4 using the original, simplified cepstra. Our acoustic-phonetic decomposition ofthat accuracy curve revealed significant vowel confusions across the high F2 and theentire F3 range, yielding an overall drop in accuracy of 7.2 %. By contrast, each of thethree articulatory sources of speaker variability were found (in Section 6.4) to havecontributions to the dichotomy mainly across two of the three, vowel formant ranges.
For example, the strategy-induced vowel-speaker interactions which yielded the largest
overall drop of 5.0 % in accuracy, were manifest mainly across the entire F3 range. The

334

Speaker

A
Structure

B

C

Smallest (16.26 cm)

Largest (16.97 cm)

distribution,
especially in F2 andF3 of front vowels

distribution,
especially in F2 andF3 of front vowels¬ higher formant
Velarised¬ lower formant

Uvularised¬ high F in backVelarised3

vowelsSetting

D

More anterior
constriction along
hard-palate¬ low F and highPalato-velarised
high F2 in lowfront vowels
Lip-rounded
low F3 in backvowels¬¬2

F3 in high-front
vowelsMore raised larynx& greater degree of
constriction in midto high-front vowels

Strategy¬ high F in mid2

to high-front
vowelsLess lowered larynx
in back vowels, &generally more
retracted place ofconstriction¬ high F in highback vowels¬ low F in high33front vowels

Larger front-cavity
expansion in backvowels, & generally
more anterior placeof constriction¬ low F in front& central vowels¬ low F in back2

3

vowelsTable 6.7: Articulatory interpretations and acoustic-phonetic consequences of the speaker differences
in vocal-tract structure, articulatory setting, and vowel-specific articulatory strategy, which were foundin Section 6.4 to contribute to the vowel-speaker dichotomy in the FC dataset.

next largest contribution of 4.1 % was caused by structure-induced, inter-speaker vowel
confusions mainly across the F2 and the F3 range of the front vowels. By contrast, thesmallest contribution of 3.4 % was caused by setting-related speaker differences mainlyin the F2 range of the front and the F3 range of the back vowels.
Nor did we find the articulatory sources of dichotomous vowel confusions to beuniformly distributed across all four speakers. Table 6.7 summarises our interpretationsof the speaker idiosyncracies in each of the three articulatory features, together with
their most pronounced, acoustic-phonetic consequences which were found to contributeto the dichotomy. For example, the main contribution of speaker C to the dichotomy
was found to occur in the F2 and the F3 range of the front vowels, and was explained
almost exclusively in terms of his relatively larger vocal-tract anatomy. Although thevowel confusions across the same formant ranges for speaker A were similarly
explained by his relatively smaller vocal-tract anatomy amongst our four speakers, his335

contribution to the dichotomy in the F2 range of the mid- to high-front vowels was, inaddition, explained by idiosyncratic behaviour in articulatory strategy in those vowels.
By contrast, speakers B and D, who were found to have vocal-tract anatomical sizes
close to the average of our four speakers, were then found to contribute almost
exclusively in their setting- and strategy-related idiosyncracies in vowel production.
As stated repeatedly, we cannot simply assume (nor do we claim) that ourestimated vocal-tract area-functions are those which the speakers actually produced
during their recording sessions. On the contrary, numerous sources of error are known
to exist along the path from the recorded acoustic speech signal, through formant
measurements and the many simplifying assumptions of the inverse method and its
underlying vocal-tract model, to the estimated area-functions themselves. Whilst thosesources of inaccuracy were considered in Chapter 5 mainly from a vocal-tract modelling
point of view, we reserved a more conclusive judgement on their significance, pending
further analyses of the area-functions, aimed at examining and interpreting those data inthe context of an actual speech phenomenon. In this Chapter we have indeed achieved
precisely what we set out to achieve in Chapter 1, namely to gain some articulatory
insights into the phenomenon of vowel-speaker dichotomy, at least as it is manifest in
the given dataset of spoken vowels. The credibility of our articulatory explanation can
only uphold the plausibility of our estimated area-functions, and lead us to advocate
further use of the hybrid LP-SM method of area-function parameterisation andestimation, in order to better understand some basic speech phenomena which still await
exploration or elucidation from a speech production point of view.

336Chapter 7
Concluding Discussion
As stated in the opening paragraph of this thesis, one of the most eminent and longstanding problems in understanding and describing the processes of human speechcommunication, whether in production, acoustics, or perception, is that of variability.
As also explained in our Introduction (Chapter 1), the kernel of the variability problem
is the speech-speaker dichotomy, which has indeed been the central theme of thepresent work. In our interpretive review of the literature (in Chapter 2), we assembled
the evidence presented and the approaches used in previous studies which bear, either
directly or indirectly, on the dichotomy problem. In the four intervening, core chapters,
we then described our own methods and investigative results concerning the dichotomy
manifest in the steady-state, segmental properties of spoken vowels. In the following,
concluding Sections 7.1 and 7.2 of this thesis, we endeavour to consolidate our main
contributions, whilst discussing implications and raising questions which point to future
work, from both an acoustic-phonetic and an acoustic-articulatory perspective.

7.1 Acoustic-Phonetic Perspective
Our acoustic-phonetic investigations of the speech-speaker dichotomy described in
Chapter 4, contribute both a more complete ensemble of experimental methods (Section
7.1.1) and a more unified description of the spectral manifestations of the dichotomy(Section 7.1.2) than have been offered to date.

7.1.1 Experimental Methods
7.1.1.1 A Posteriori Selection of Frequency Sub-bands
Perhaps the most important concept which has permeated throughout the present work,is that the phonetic and the speaker-specific components of acoustic variability are not337

equally distributed along the entire frequency range. It was not without some surprise,
therefore, that we previously noted the absence from the literature (despite its long
history of support for that concept), of a convenient and computationally efficientmethod of accessing spectral information within any available frequency sub-band.
Indeed, as reviewed in Chapter 2, previous studies which have at all shed light on thefrequency-band dependence of phonetic and speaker variability, have either been stuckwith the inflexibility of low-pass, high-pass, or banks of band-pass filters; or relied on
re-partitioning of FFT spectra; or resorted to more computationally intensive, reanalyses of the acoustic speech signal at each, new frequency sub-band.
In that regard, our parametric cepstral distance measure (PCD) and partial,
quefrency-weighted cepstrum (P-QCEP), both derived in Chapter 4, present asignificant advance towards greater flexibility and computational simplicity in the
powerful and popular, cepstral comparison of speech sounds. Indeed, our parametricformulations of cepstral distance computation and cepstrum derivation, respectively,
have greatly facilitated our investigations of the frequency-band dependence ofphonetic-speaker interactions. In particular, it is the a posteriori selection of any
frequency sub-band within the available spectral range, using only the low-order
cepstral coefficients, which renders those two methods potentially useful in a wide
range of applications including the recognition framework which we have adopted
throughout our investigation of the dichotomy.
7.1.1.2 Recognition Framework for Diagnosing the Dichotomy
One important aspect of our literature review exposed in Chapter 2, has been to identify
experimental probes into the dichotomy and its acoustic-phonetic manifestations. In thisregard, we therein suggested and, in subsequent chapters, demonstrated that vowelspeaker interactions are suitably implicated in machine recognition of vowels on a
speaker-independent basis, where both phonetic and inter-speaker variabilities (and theirinteractions) have a direct influence on the level of accuracy attained within eachselected frequency sub-band. Our experimental approach to the problem of speechspeaker dichotomy manifest in spoken vowels, was accordingly founded in a vowel

338

recognition framework.
However, we have also shown in Chapter 4 that the recognition methodology
therein adopted is able to yield a comprehensive diagnosis of the dichotomy, which isunprecedented. First, the PCD and the P-QCEP (used with the hyper-planar and thehyper-quadratic classifiers, respectively) afford a more complete profile of thefrequency-band dependence of vowel-speaker interactions. In particular, our wholespectrum approach avoids the inherent problems of the discrete, formant-based spectral
representation (as discussed in Section 2.3.2.2), and yields arbitrarily-detailed profilesof vowel classification accuracy as a function of an increasing upper spectral limit.
Second, our method of contrasting the accuracy profiles obtained on an intra- and aninter-speaker basis, leaves very little doubt that the observed differences are caused byinteractions between phonetic and inter-speaker variabilities.
No less important are our post-recognition interpretation and decomposition of
the generated accuracy curves. First, an acoustic-phonetic explanation of the grossfeatures of the accuracy profiles was afforded by plotting the relevant portions of thosecurves adjacent to the speakers’ F1 F2 and F2 F3 vowel formant distributions. Second,the overall mean, inter-speaker accuracy curve was decomposed in terms of its
constituent, per-vowel and per-speaker accuracy curves, thus facilitating a rankordering of the vowels and speakers according to their individual contributions to thedichotomy.
It would be tempting at this point to advance that the necessity of building up an
entire, acoustic-phonetic methodology, is itself indicative of the relative immaturity of
the current outlook on the long-standing problem of speech-speaker dichotomy — evenfor the most well-known class of speech sounds (vowels) recorded by the most widely
studied group of speakers (adult males). For the least, however, we should like toexpress the firm view that the ensemble of methods summarised above, indeed, yield along-awaited procedure for diagnosing the speech-speaker dichotomy manifest in any
segmental, acoustic dataset of spoken language (with the caveat that if non-vocalic
sounds are involved, the formant-based interpretation would need to be relinquished fora more suitable one).

339

7.1.2 Spectral Manifestations of the Dichotomy and Implications
The methodology recapitulated in the previous section, was used in Chapter 4 first to
unfold, then to provide a detailed, acoustic-phonetic explanation of the dichotomy
manifest in the steady-state vowels recorded in /hVd/ context by four, adult malespeakers of Australian English. The nearly asymptotic behaviour of the intra-speakeraccuracy curve, confirmed the well-known phonetic importance of the low spectral
range encompassing the first two formants. In particular, intra-speaker vowel
classification accuracy was found to be relatively independent of the spectral regions
higher than the mid- F2 of the speakers’ vowel formant distribution. By contrast, thebehaviour of the inter-speaker accuracy curve exposed the speaker-related potency of
those spectral regions higher than the mid- F2 . Indeed, the detrimental influences ofinter-speaker variability were clearly manifest by the drop in accuracy, caused by
misclassifications mainly amongst the F2 of front vowels, amongst the F3 of backvowels, and amongst the F3 of front vowels, in this decreasing order of detrimental
effects. It is interesting to note that this bipartite, acoustic-phonetic interpretation of the
manifestations of vowel-speaker interactions in terms of the front-back vowel
distinction, meshes quite well with Broad’s (1981) insightful, multi-speaker extension of
Broad and Wakita’s (1977) bi-planar model of the first three formant frequencies of
spoken vowels.
Our confidence in the intrinsic nature of the observed phenomenon was thenreinforced, by obtaining essentially the same behaviour of vowel classification accuracy
using first a more powerful classifier; then so-called simplified cepstra; and finally, twodifferent datasets containing larger numbers of speakers of American and AustralianEnglish, respectively.
First, the more statistically-reliant, hyper-quadratic classifier accentuated theimportance of using sufficient amounts of training data in order to avoid the so-called“curse of dimensionality”. Indeed, it is well known that the severity of that problem
increases both with the number of parameters used, and with the statistical complexityof the classifier; hence, the highly data-driven nature of state-of-the-art ASR systems.
Our approach to avoiding this problem was to acknowledge the intrinsically smaller340

dimensionality of the partial cepstrum, and therefore retain proportionally fewer PQCEP coefficients in classification experiments performed in frequency sub-bands. The
drop in inter-speaker vowel classification accuracy thus obtained across the higher
spectral regions was found to persist, thereby lending further support for the intrinsic
nature of the dichotomy.
The dichotomy also persisted upon using so-called simplified cepstra generated
from only the first three formant frequencies (with fixed bandwidths). Whilst this result
certainly paved the way for using the formant-based, but more populous, PB and JB
datasets, at a deeper level it partially addressed the important question of the influenceof spectral representation on the manifestations of vowel-speaker interactions. In
particular, it provided strong evidence that, consistently with findings reported in theauditory-perception literature, the frequency locations of the formant peaks are indeed
the most important features of the whole-spectrum representation of spoken vowels. In
addition, it proved the pertinence of our formant-based interpretations of the grossfeatures of the accuracy profiles.
In that context, it is relevant to recall that the NDPS representation consistently
used in all our classification experiments, does already enhance the formant peaks andsuppress non-phonetic influences such as the overall spectral tilt. However, an
intriguing question now arises as to the dependence of the dichotomy on the number ofquefrency-weighted cepstral coefficients retained, or equivalently, the degree ofsmoothing applied to that formant-enhanced spectral representation. On the one hand,
given the inconsistency of the (discrete) formant representation itself in revealing the
detrimental influences of phonetic-speaker interactions in speaker-independent vowel
recognition experiments reported in the literature (as discussed at length in Section
2.3.2.2), there is good reason to suspect that the clearly dichotomous behaviour of our
accuracy curves obtained using 14th-order cepstra might begin to blur if the cepstral
distance measure is rendered more sensitive by increasing the number of cepstralcoefficients, and thereby sharpening the formant peaks of the smoothed NDPS. On theother hand, given the reported, phonetic relevance of the low-order LP and PLP models(e.g., Strube, 1980; Hermansky, 1990), further smoothing of the NDPS by truncation of
the cepstrum to less than 14 coefficients may be expected, even without any
341

perceptually-motivated pre-processing, to yield levels of accuracy comparable to thoseattained at the peak of our inter-speaker accuracy curve near the mid- F2 of the
speakers’ vowel formant distribution. Further work is warranted, however, in order togain more detailed insights into the dependence of the dichotomy on spectral
smoothing.
We also sought to evaluate the dependence of the dichotomy on speakerhomogeneity, using the PB dataset of vowels recorded by 32 adult, male speakers ofAmerican English. In sum, we found that a spectrally more homogeneous group of
speakers is more likely to exhibit a blurred dichotomy; and conversely, that a spectrally
more heterogeneous group is more likely to exhibit a clearly dichotomous accuracy
curve (where spectral homogeneity can be quantified, for example, by per-vowel, interspeaker variation in each formant frequency). Whilst a quantitative measure of eachspeaker’s degree of “sheepiness” or “goatiness”, respectively, was therefore defined on
the basis of the gross features of the speakers’ individual accuracy curves, a more
informative index or rank-ordering criterion might also take into account the moredetailed, phonetic decomposition of each speaker’s accuracy curve.
Finally, the JB dataset of 14 Broad, 11 General, and 11 Cultivated, adult malespeakers of Australian English was used to investigate the dependence of the dichotomy
on idiolectal speaker differences. Briefly, the dichotomous behaviour of the interspeaker accuracy curve was found to be independent of the speakers’ (perceived)
idiolectal homogeneity — regardless of the presence or absence of idiolectal
differences, we were still able to explain the drop in accuracy across the higher spectral
regions in terms of speaker-induced confusions amongst the F2 of front (and mid-)vowels, and amongst the F3 of back vowels. This consistency in the acoustic-phonetic
decomposition of the dichotomous vowel confusions across three different datasets ofspoken vowels, does attest to the basic nature of the observed phenomenon. It also
encourages further investigations using our acoustic-phonetic methodology, applied todatasets of vowels recorded by speakers of different dialects, and of different languages.
Admittedly, the implications of the dichotomy for ASR would be more completely
assessed by extending our investigations to include other classes of speech sounds than
the steady-state vowels of English. However, at least for the spoken vowels considered342

(the relative importance of which is well established), our results provide a far more
solid basis than has been offered to date, for the very old concept of phonetic andspeaker-specific influences being more strongly manifest, respectively, in the low and in
the higher ranges along the spectral continuum. Moreover, our recognition results
obtained before and after speaker normalisation leave little doubt that speakerindependent ASR systems stand to gain by focusing on the more phonetically relevant,lower spectral regions of spoken vowels, and by exploiting the higher spectral regionsof those sounds for speaker adaptation. Conversely, for automatic speaker recognition,
our results raise the strong possibility that classification accuracy would be improved byfocusing on the higher spectral regions of vocalic speech sounds. Whilst the benefits ofauditorily-motivated frequency scales already attest to the importance of emphasising
the lower and de-emphasising the higher spectral regions, the PCD and the P-QCEP
arguably afford more direct control of frequency sub-bands and, a fortiori, of thespectral influences of phonetic and speaker variabilities.

7.2 Acoustic-Articulatory Perspective
Analogously to the bipartite structure of the previous section, our acoustic-articulatory
investigations of the speech-speaker dichotomy in Chapters 5 and 6 contribute both amethodology for acquiring and analysing vocal-tract shapes (Section 7.2.1), and a firstattempt at using those estimated shapes for a more complete description than has beenoffered to date, of the likely physical correlates of the speaker differences implicated in
the acoustic-phonetic phenomenon of dichotomy (Section 7.2.2).

7.2.1 Experimental Methods
7.2.1.1 Vocal-tract Shape Parameterisation
As discussed at length in Section 2.4.3.2 and later in Chapter 5, acoustic-to-articulatory
mapping is in general plagued by the problem of non-uniqueness, which itself depends
crucially on the articulatory parameterisation and the acoustic vocal-tract model used.
In regard to the latter, it has long been known (Atal, 1970; Wakita, 1973) that the linear
prediction (LP) vocal-tract model is the only one for which we know that uniqueness(within certain limits) is an inherent property. However, the LP method of inversion is343

generally regarded with disdain (e.g., Schroeter and Sondhi, 1994), partly owing to its
presumed incapability of yielding sufficiently plausible or realistic vocal-tract shapes;
but also partly because of its highly discretised representation of the vocal-tract areafunction, which prohibits physiologically meaningful comparisons of vocal-tract shapesof different lengths, across different vowels and different speakers.
In Chapter 5 we addressed these issues at a fundamental level, by invoking thecompletely lossless, Schroeder-Mermelstein (SM) model to provide a new explanationof the uniqueness property of the LP model, and ultimately to parameterise the latter interms of our extended version of the SM model. In particular, we first showed that the
LP model shares with the SM model, the fundamental, quasi-unique relation between
the formant frequencies and the odd-indexed coefficients of the cosine series of thelogarithmic area-function. Our extension of the SM model then emerged from ourdiscovery of an analogous, quasi-unique relation in the LP model, between the formant
bandwidths and the odd-indexed coefficients of the sine series of the logarithmic areafunction. Our resulting parameterisation of the discrete-sectioned LP area-function thus
provides the first ever, fundamentally shape-related explanation of the LP model’sinherent uniqueness; at the same time, it affords an acoustically meaningful, smooth
representation of LP-derived vocal-tract shapes, which are then more amenable tocross-vowel and cross-speaker comparisons.
An important pending question regarding the hybrid LP-SM method of inversion,is the generality of the MAD criterion used to estimate the vocal-tract length (VTL). In
that regard, one of the advantages of more physiologically-oriented articulatory models(e.g., Mermelstein, 1973; Coker, 1976) is that they obviate the necessity of VTL as an
explicit parameter. However, apart from the prohibitive, speaker-dependence of thosemodels, they have also been found to require both static and dynamic constraints toovercome their inherent non-uniqueness. In that context, the principle of minimal
articulatory effort which was first applied by Zue (1969) as a static constraint in areafunction estimation, has since been widely used (in various formulations, and under
different pseudonyms) with reasonable effect. Indeed, our results also suggest thatreasonably plausible lengths (and concomitantly, reasonably plausible vocal-tract
shapes) are yielded by that constraint, provided a sufficient number of acoustic344

parameters (e.g., 4 formants) are used. Further work along the lines pursued recently(e.g., Sorokin’s (1992) more physiologically-oriented definition of minimum “muscle
work”, or Yehia and Itakura’s (1994) use of the speaker’s average vocal-tract shape asa reference area-function in place of the uniform tube), may unveil even more realistic
alternatives to, or interpretations of the MAD criterion.
Another outstanding issue is the choice of formant bandwidths which, as we havebeen able to establish theoretically and empirically, play a vitally supporting role inensuring uniqueness of LP-derived vocal-tract shapes. Indeed, those acoustic
parameters provide the crucial second-half of shape components, which otherwise
remain ambiguous in the completely lossless model on which the original SM
parameterisation is based. Owing to the notorious unreliability of bandwidths measured
from the acoustic speech signal, we first averaged the values of each bandwidth acrossrepetitions and speakers, while retaining their phonetic variation. We thenacknowledged the fact that the LP vocal-tract model has only a single source of
acoustic energy loss lumped at the glottal end, by subtracting from those averaged
bandwidths, the so-called closed-glottis bandwidths predicted by Hawks and Miller’s(1995) equations derived from empirical data. The plausibility of our thus estimatedarea-functions encourages us to speculate that a more sophisticated procedure fordetermining the most appropriate formant bandwidths may not be necessary. However,further work is called for if our method is to be applied (at least in spirit) to datasets
such as PB and JB, which comprise only formant frequencies.
Another promising avenue for further research, would be to derive an articulatory
distance formula which yields the rms distance between area-functions, using theparameters of our extended SM model. As the relation between our vocal-tract shapeparameters and the logarithmic area-function (i.e., the cosinusoidal series expansion) is
quite similar to the relation between the cepstral coefficients and the log-magnitude
spectrum (or similarly, the relation between the QCEP and the NDPS), the derivation ofsuch a distance measure should follow closely our derivation (in Chapter 4) of the PCD.
Although there is the additional complexity that area-functions will potentially havedifferent lengths, a parametric articulatory distance measure (PAD) may afford a
reduced computational load and a greater flexibility in quantifying shape-related345

differences within physiologically-motivated, sub-regions of the vocal-tract.
7.2.1.2 Articulatory Speaker Normalisation
Our review (in Chapter 2) of previous methods proposed to overcome the speechspeaker dichotomy by speaker normalisation, showed that on the one hand, purely
acoustic-phonetic approaches have rarely afforded insights into the underlying causes ofthe dichotomy; and that on the other hand, articulatorily-motivated methods have rarelytranscended the persistent trend of dealing with only the gross differences in vocal-tract
length. One of the outstanding deficiencies in previous works has clearly been theunwillingness to adopt a sufficiently complete, theoretical framework for describing
speaker differences in physical terms. For example, it has long been suggested that thephysical sources of inter-speaker variability are both organic (or anatomical) andlearned (or behavioural) in nature, yet by far the majority of previous studies concernedwith importing articulatory principles in speaker normalisation, have cared to accountfor only the former. In a similar vein, whilst the concept of long-term average acoustic
parameters has been vociferously embraced in automatic speaker recognition and otherapplications, it has rarely been connected with the old and neglected concept of longterm articulatory setting.
Our articulatory method of speaker normalisation expounded in Chapter 6, doestranscend the incompleteness of prevalent approaches, by acknowledging three basic
sources of speaker differences — namely, vocal-tract fixed structure, long-term
articulatory setting, and vowel-specific articulatory strategy. In order to quantify and to
speaker-normalise any combination of these articulatory sources of speaker variation,
they were each given an operational definition in terms of the estimated area-functionsof any given speaker. Thus, structure was defined as the mean vocal-tract length,computed over only the non-rounded, mid- to low vowels, which are presumably least
influenced by speaker differences in extreme laryngeal and labial postures; setting was
defined as the mean vocal-tract shape, computed over all the vowels after speakernormalisation of structure; and strategy was defined as the mean vocal-tract shape andlength computed on a per-vowel basis, after speaker normalisation of both structure andsetting. Each one of these three articulatory features was then retained for cross346

speaker comparison, by normalisation of the remaining two, while normalisation of all
three features completely removed the corresponding articulatory sources of interspeaker variation.
However, a methodological obstacle in comparing any pair of area-functions, orindeed in computing an average vocal-tract shape as required in the normalisation
procedure, is that of lengthwise alignment. As area-functions are generally of different
lengths, and as there appears to be no reliable way of identifying on them fixed
anatomical landmarks, it is not at all surprising that the literature fails to provide acommonly accepted method of alignment. In Chapter 6 we therefore proposed toconform with the acoustic theory on which the SM model is founded, and centre-alignour estimated area-functions, thereby imposing the least amount of bias to the labial andglottal ends. Although the mean vocal-tract shapes and the comparisons thereof were
found to be quite plausible, our method of alignment may be rendered even more
realistic by acknowledging that phonetic variations in the inferred, articulatory featuresof larynx height and lip protrusion, are in general of different magnitude; and that thedifference in magnitude itself is potentially speaker-dependent.

7.2.2 Articulatory Explanation of the Dichotomy
An articulatory explanation of the speech-speaker dichotomy was offered in Chapter 6,
the completeness of which was confirmed in the penultimate section therein. Indeed,whilst the articulatory completeness of our normalisation method is intuitively clear
from the description of the method itself, its acoustic-phonetic completeness was only
confirmed by the nearly asymptotic, inter-speaker accuracy profile obtained after
normalisation of all three articulatory components of inter-speaker variability. That
result provided the final proof of the effectiveness of our articulatory method of
normalisation, and at the same time, it confirmed the validity of our recognition
framework as a powerful diagnostic of the speech-speaker dichotomy.
However, the articulatory explanation itself was obtained en route to that ultimate
proof, whereby the individual contributions to the drop in accuracy across the higher
spectral regions were identified for the FC dataset of four adult, male speakers ofAustralian English. Indeed, we were able to identify contributions to the dichotomy
347

from two of the speakers’ differences in vocal-tract structure; the other two speakers’
idiosyncratic articulatory settings; and three of the speakers’ idiosyncratic articulatory
strategies. Speaker differences in vocal-tract fixed structure were found to induce
misclassifications mainly amongst the F2 and amongst the F3 of front vowels. Not
surprisingly, these were caused by the two speakers who were found to have theshortest and the longest, structure-related vocal-tract lengths. Perhaps less anticipated,is the fact that the admittedly marginal difference between those speakers’ mean,
structural lengths of 16.26 cm and 16.97 cm, respectively, is apparently sufficient tocontribute to the observed drop in vowel classification accuracy across the higher
spectral regions. The other two speakers were then found to have dichotomy-inducing
idiosyncrasies in long-term articulatory setting, with one speaker preferring a slightly
uvularised setting together with a more anterior place of constriction for high-front
vowels, and the other speaker preferring a palato-velarised and slightly lip-roundedsetting.

These long-term behavioural idiosyncrasies were

found

to

inducemisclassifications mainly amongst the F2 of front vowels and amongst the F3 of backvowels. However, perhaps not unexpectedly for our dataset of four, adult malespeakers, the largest contributions to the dichotomy were found to be caused bydifferences in vowel-specific articulatory strategy, including vocal-tract shape-inferredvariations (i) in larynx height of both front and back vowels, (ii) in the maximum area ofthe front cavity expansion of back vowels, and (iii) in the place of constriction along thelength of the vocal-tract. These dimensions of speaker variation in strategy were foundto induce misclassifications mainly amongst the F3 of both front and back vowels.
In view of the clearly marked differences in both articulatory setting and strategy
observed in Chapter 6 for speaker D, it is interesting to consider whether thosearticulatory-behavioural differences can be regarded as embodying at least some of the
distinguishing characteristics of spoken Australian English. Informal auditory
impressions (Clermont, 1991) indeed suggest that, of the four speakers in the FC
dataset, speaker D tends mostly towards the so-called Cultivated variety, while theother three speakers tend towards the General variety of Australian English. Therefore,
a question of practical importance is whether the perceptually-determined idiolectal
labelling of the speakers in the JB dataset (used to validate the dichotomy in Chapter 4),348

can be confirmed articulatorily; and by extension, whether our acoustic-articulatory
methodology might then be useful in idiolectal labelling of speakers based on thecharacteristics of their inferred vocal-tract shapes.
Naturally, a more global question arises to what extent our articulatory
explanations could be generalised across different speaker datasets. Even considering
only those languages which have an inventory of steady-state vowels similar to that ofEnglish, we feel compelled to suggest that in general, different groups of speakers will
most probably exhibit different types of articulatory variations. For example, dialectal or
idiolectal differences may be caused by different types of vowel-specific articulatory
strategies, while individual differences in “voice quality” (in the sense of Laver, 1980)
may be caused by various types of articulatory settings, and more diverse groups of
speakers which include adult males, adult females, and children, would most certainly
exhibit a much wider range of vocal-tract structural differences. However, within thelimitations of our study, we have determined that if a speaker dataset is endowed withthe types of physical variations described in Chapter 6 and summarised above, then adichotomous behaviour in inter-speaker vowel classification accuracy is indeed very
likely.
In this vein, we are admittedly rather fortunate in having had recourse to such an
exemplary dataset of four speakers, whose physical differences in vowel production
appear to be so uniformly distributed as to allow an unbiased investigation of theacoustic-phonetic consequences of variations in structure, setting, and strategy. It isindeed clearly important to acknowledge (as we have found) that vowel-speaker
interactions in the higher spectral regions can be induced by speaker differences in any
one of those three, articulatory sources of variability. Furthermore, our results suggestthat each of those three types of physical idiosyncrasies may have a distinctive,
acoustic-phonetic signature which can be decoded on the basis of the spectral range ofgreatest detrimental influence. Thus, for example, we might expect that a group of
speakers with a particularly wide range of variation in vocal-tract size, would manifest
the largest drop in inter-speaker vowel classification accuracy across the spectral ranges
encompassing the F2 and the F3 of their front vowels.
Dare we now hope for a more “universal” explanation of the speech-speaker349

dichotomy than we have been able to offer? In this vein, Peterson’s (1959) visionary
statement quoted much earlier in Section 2.1.1, concerning the contrastive influences of“gross” and more “exact” vocal-tract shapes on the lower and the higher formants,
respectively (indeed, a statement which clearly foreshadows the SM model to appear
more than half a decade later), is perhaps the most direct, and the most “universal”
explanation yet, of the contrastive roles of phonetic and speaker influences in spoken
vowel sounds. Whilst an even more “universal” interpretation will require an extensionof our methods in order to cope with greater complexity in both the phonetic andspeaker components of variability, our present results do contribute a far more detailed
substantiation of that explanation than has ever been attempted. Nor do they preclude
the possibility of a more general interpretation, which may lead towards the definitionof the principal, articulatory components of inter-speaker variation in vowel
production.
Indeed, perhaps the most distinguishing characteristic of our articulatory methodof speaker normalisation, is its chief purpose not as a method of normalisation per se,
but as an integral component of an acoustic and articulatory, explanatory framework!
In this vein, we are reminded of Payan and Perrier’s (1993, p.417) statement which wequoted earlier in Section 2.4.3.1, that a method of speaker normalisation “is efficient if
it can account for the causes of variability.” We have of course argued that such
efficiency is not to be expected from purely acoustic-domain methods which primarily
aim to increase recognition accuracy using the entire available spectral range; nor fromacoustic-domain methods informed only by auditory-perceptual criteria. Rather, it isbest attained by returning to the acoustic domain only after having explicitly accounted
for the articulatory sources of speaker variability.
In that light, our acoustic-articulatory methodology does represent a significant
departure from previous works, as we have succeeded in actually using estimated vocaltract area-functions for a wider purpose than the mere investigation of the inverse
problem itself. While we cannot claim to have rendered either “perfect” or the “mostrealistic” vocal-tract shapes, we were clearly able to advance understanding of the
dichotomy problem in an acoustic-articulatory framework where direct physicalmeasurements were not involved. Therefore, notwithstanding the need for further
350

evaluation of our method of inversion and area-function parameterisation, the research
work reported in this thesis lends support to the following, emerging contention.
Namely, that the imperfections of acoustic-to-articulatory mapping (in particular thoseascribed to the over-neglected LP model) do not and should not impede progress
towards a better understanding of the articulatory bases of phonetic and speakervariability in the acoustics of speech.

351

352

Appendix A
Formant-tracking Analysis Conditions and Sequence Charts
In this appendix are shown, for each of the four (adult, male) speakersof the FC dataset (cf. Chapter 3), a table which summarises theformant-tracking problems encountered and LP analysis conditions
finally adopted, and a figure (“formant sequence chart”, as used byPotter and Steinberg, 1950) which shows the entire, steady-statevowel formant-frequency data finally extracted from the vocalic nucleiof the recorded, /hVd/ monosyllabic words.

353

Repetition
I

II

III

IV

V

Summary? F3? F4, F3
M18
FA10
? F4, F3
M13
FA15
? F4, F3
M13
FA10
? F4, F1
adPR
FA25
? F4

? F3, F4
M18
FA13
? F3, F4

? F3, F4
M18
FA25
? F3, F4

? F3, F4

? F3

FA25
? F4
adPR
FA20
? F4, F1, F2

FA25
? F4

heed
hid
? F4

head

FA10
? F4
PR100
FA10
? F4, F1
M15
FA10
? F4
M11, PR80
FA20
? F3
M15
(?F1,F2:sts)(?F2,F4:sts)

? F4
FA↓
? F4

FA20
? F4
M15

FA30
? F4
M15
FA12
? F4
M11, PR80
FA20? F4, F3, F2
M16
FA35
? F4
M12

? F2, F4
M12
DP2: F1, F2
? F4
M16

? F4
M12, PR96
FA30

? F4
M12
FA30

? F4
M12, PR90
FA20

? F4

? F4
M13

? F4
M15

herd

? F4
M15

? F4, F3
M12, PR99
FA14? F4
M16, PR90(?F2,F4:sts)? F4, F3
M12, PR100
FA18
? F4
M17, PR90

? F4

hud

had
hard

FA10
? F4, F3
M17
FA15
? F4
M12, adPR

FA10? F3, F2, F4? F3, F4

hod
hoard
hood

FA20
? F4
M15
? F4
PR90
FA40
? F4 (spur)

who’dDP1:uwCep? F4? F3, F4
adPR
? F4
M17, adPR

FA10
? F4
M12, PR80
FA20? F4, F3

? F2
M15

FA↓
? F4

? F3

? F4

Table A.1: Formant-tracking LP analysis conditions for speaker A. The first column lists theorthographic representations of the / hVd/ monosyllabic words, the vowel nuclei in which the formant
measurements were taken. The second through sixth columns list, for each of the 5 repetitions of eachvowel, the ill-tracked formant(s) and the LP analysis conditions finally chosen to secure “good” (seeSection 3.3.2.1) formants. (?: problematic formant-track(s); M: LP analysis order; PR: coefficient of
pre-emphasis multiplied by 100; adPR: frame-by-frame adaptive pre-emphasis; FA: frame advance inmsec multiplied by 10; DP1:uwCep refers to the unweighted cepstral distance measure used in theformant tracker; DP2: refers to the use of a formant-trajectory smoothing function in the formant
tracker; spur: refers to a spurious-pole trajectory; sts: refers to a steady-state detection problem). An
empty entry implies no formant tracking problems using the default analysis conditions (see Section
3.3.2). The last column lists a summary, where applicable, of the consistently encountered (i.e., acrossall 5 repetitions) problem(s) and required correction(s) for each vowel.

354

herd
0

500

1000

1500

2000

2500

3000

3500

4000

heed

hid

head

had

hard

hod

hoard

hood

who’d

hud

F4
F3
F2
F1
4500

Figure A.1: Formant-frequency sequence chart for speaker A. Along the abscissa are listed theorthographic representations of the / hVd/ monosyllabic words, the vowel nuclei in which the formant
measurements were taken. For each vowel, there are 35 data points per formant, corresponding to the 7
steady-state frames in each of the 5 repetitions.

355

Repetition
I

II

III

IV

V

Summary? F2, F3, F4
M9, PR88? F3, F2, F4
M10? F4(spl), F2
M12, PR99? F4(spl), F2
M11, PR99

hid? F4(spl)
M10? F4(spl), F3
M9? F3, F4, F2
M16
DP2: F3? F4(spl)
M10? F4, F2

heed

? F4(spl)
M10? F4(spl)
M12? F4(spl)
M↓

head

? F4
M15, PR95? F1, F3
M13

? F4 (..F5)
M13? F4 (..F5)
M12
DP2: F1..3

had

? F3
M13

? F4
M13

? F4, F2
M13
DP2: F2? F3, F4
M12

? F4,F2,(F1)M12
DP2: F1..3? F4(spl), F2
M11, PR80
DP2: F2? F2, F3, F4
M13
DP2: F1..3? F3, F4
M13

hard
hod

? F4, F3
M12

? F3, F4
M12(?F4:sts)hood? F4(spl), F3
M9? F4(spl)
M12

who’d

? F2
M13, PR95

? F4
M13

herd

? F3,F4,(F2)M11, PR90? F2, F4
M13

hoard

hud

? F1, F2
adPR

? F1..4
M17, adPR
DP2: F2,F3? F4, F2
M12
DP2: F2,F3

? F4
M15

? F4, F3
M12
DP2: F3? F2, F4
M12, PR95? F4, F2, F3
M12, PR100
DP2: F2,F3? F3 (..F4)
M13

? F2
M12
DP2: F2
? F2

? F3
M13
? F4
M15, adPR? F3, F4
M13

? F4
M↓

M↓

? F4, F2, F3
M15, PR95
DP2: F2? F2, F3, F4
M13

Table A.2: Formant-tracking LP analysis conditions for speaker B. The first column lists theorthographic representations of the / hVd/ monosyllabic words, the vowel nuclei in which the formant
measurements were taken. The second through sixth columns list, for each of the 5 repetitions of eachvowel, the ill-tracked formant(s) and the LP analysis conditions finally chosen to secure “good” (seeSection 3.3.2.1) formants. (?: problematic formant-track(s); M: LP analysis order; PR: coefficient of
pre-emphasis multiplied by 100; adPR: frame-by-frame adaptive pre-emphasis; FA: frame advance inmsec multiplied by 10; DP2: refers to the use of a formant-trajectory smoothing function in the formant
tracker; spl: refers to a “split” formant trajectory; sts: refers to a steady-state detection problem). An
empty entry implies no formant tracking problems using the default analysis conditions (see Section
3.3.2). The last column lists a summary, where applicable, of the consistently encountered (i.e., acrossall 5 repetitions) problem(s) and required correction(s) for each vowel.

356

herd
0

500

1000

1500

2000

2500

3000

3500

4000

heed

hid

head

had

hard

hod

hoard

hood

who’d

hud

F4
F3
F2
F1
4500

Figure A.2: Formant-frequency sequence chart for speaker B. Along the abscissa are listed theorthographic representations of the / hVd/ monosyllabic words, the vowel nuclei in which the formant
measurements were taken. For each vowel, there are 35 data points per formant, corresponding to the 7
steady-state frames in each of the 5 repetitions.

357

Repetition
I

II

III

IV

V

Summary? F3, F4
M16, adPR? F4, F3, F2
M13, adPR

? F3
M13

? F3
M15, adPR? F3
M15, adPR

? F3

heed
hid

? F1, F3
M15

? F4
M15

? F2, F3
M15
DP2: F3? F3, F4
M18
DP2: F3? F4
M18, adPR

? F4
M15

head

? F4
M15

had

? F4
M15, adPR

IRV: F3, F4

hard

? F3
M13

? F3
M13

? F2, F4
M12
DP2: F2? F2, F4
M13

? F1..4
M13? F3, F4
M13

? F3, F4
M15

? F3, F4
M13

? F4, F1, F3
M17, PR90? F3, F4

hod
hoard

? F3
M13

? F2, F4
M13

? F1, F3
M13

? F3
adPR

? F4
M13

? F1, F2, F3
M13
DP2: F1..3? F3, F4
M13

? F3, F4
M13, adPR? F2, F4
M15

? F3, F4
M13

? F4

who’d

? F3, F4
M16, adPR
DP2: F3? F1, F2, F4
M13

? F3

hood

? F3, F1
M13, adPR
DP2: F3?F2(spl)F1,3
M12? F2, F4
M13

? F2(spl), F3
M13

hud
herd

? F2, F3
M13, adPR
DP2: F3? F2, F3
M12

? F4
M12

M↑

IRV: F3? F2, F4
M13

M↓

? F2
M13

Table A.3: Formant-tracking LP analysis conditions for speaker C. The first column lists theorthographic representations of the / hVd/ monosyllabic words, the vowel nuclei in which the formant
measurements were taken. The second through sixth columns list, for each of the 5 repetitions of eachvowel, the ill-tracked formant(s) and the LP analysis conditions finally chosen to secure “good” (seeSection 3.3.2.1) formants. (?: problematic formant-track(s); M: LP analysis order; PR: coefficient of
pre-emphasis multiplied by 100; adPR: frame-by-frame adaptive pre-emphasis; FA: frame advance inmsec multiplied by 10; DP2: refers to the use of a formant-trajectory smoothing function in the formant
tracker; spl: refers to a “split” formant trajectory). An empty entry implies no formant tracking
problems using the default analysis conditions (see Section 3.3.2). The last column lists a summary,where applicable, of the consistently encountered (i.e., across all 5 repetitions) problem(s) and required
correction(s) for each vowel. (IRV: refers to the apparently intrinsic presence of inter-repetition
variability.)

358

herd
0

500

1000

1500

2000

2500

3000

3500

4000

heed

hid

head

had

hard

hod

hoard

hood

who’d

hud

F4
F3
F2
F1
4500

Figure A.3: Formant-frequency sequence chart for speaker C. Along the abscissa are listed theorthographic representations of the / hVd/ monosyllabic words, the vowel nuclei in which the formant
measurements were taken. For each vowel, there are 35 data points per formant, corresponding to the 7
steady-state frames in each of the 5 repetitions.

359

Repetition
I

II

III

IV

V

Summary? F3, F4
M13

? F3
M13

? F3, F4
M15

? F3

hid

? F3
M13

? F3
M13

? F3, F4
M17
DP2: F3? F3, F4
M18, adPR

head

? F4
M15

? F4
M15, adPR? F4, F3
M15, adPR

? F4
M18

? F4
M18

? F4
M↑

? F3, F4
M15

? F4, F3
M15

? F3
M13, adPR

? F3

had

? F3
adPR

? F3
IRV: F4

hard? F3
M12, adPR? F3
M12, adPR? F3, F4
M12, adPR? F3
M↓, adPR

hod? F3, F1, F2
M15, adPR? F3
M11, PR96? F2, F3, F1
M13, adPR
DP2: F1..3
? F3
M15

heed

hoard
hood

? F4, F1,2,3
M15, adPR
DP2: F1..3? F3, F4
M13, adPR
DP2: F3

who’dhud? F3, F4
M12, PR96
DP2: F3

DP2: F3?F3(spl)F1,2
M13, adPR? F4
M16, adPR? F4
M15, PR96? F3, F4
M16

? F3
M15

? F3, F4
M13
DP2: F3? F2, F3
M13

? F4, F3
M13
DP2: F3? F3, F4
M12, PR93
DP2: F3? F3, F4
M17

? F2
M13, adPR
DP2: F2,F3? F1, F3, F4
M13
DP2: F3? F3, F4

? F4
M16
? F1, F4
M12

? F3
M12

M↓

? F4
M13

herd

Table A.4: Formant-tracking LP analysis conditions for speaker D. The first column lists theorthographic representations of the / hVd/ monosyllabic words, the vowel nuclei in which the formant
measurements were taken. The second through sixth columns list, for each of the 5 repetitions of eachvowel, the ill-tracked formant(s) and the LP analysis conditions finally chosen to secure “good” (seeSection 3.3.2.1) formants. (?: problematic formant-track(s); M: LP analysis order; PR: coefficient of
pre-emphasis multiplied by 100; adPR: frame-by-frame adaptive pre-emphasis; FA: frame advance inmsec multiplied by 10; DP2: refers to the use of a formant-trajectory smoothing function in the formant
tracker; spl: refers to a “split” formant trajectory). An empty entry implies no formant tracking
problems using the default analysis conditions (see Section 3.3.2). The last column lists a summary,where applicable, of the consistently encountered (i.e., across all 5 repetitions) problem(s) and required
correction(s) for each vowel. (IRV: refers to the apparently intrinsic presence of inter-repetition
variability.)

360

herd
0

500

1000

1500

2000

2500

3000

3500

4000

heed

hid

head

had

hard

hod

hoard

hood

who’d

hud

F4
F3
F2
F1
4500

Figure A.4: Formant-frequency sequence chart for speaker D. Along the abscissa are listed theorthographic representations of the / hVd/ monosyllabic words, the vowel nuclei in which the formant
measurements were taken. For each vowel, there are 35 data points per formant, corresponding to the 7
steady-state frames in each of the 5 repetitions.

361

362

Appendix B
Mathematical Derivation of the SM Model
For the sake of completeness, we herein offer a complete rederivation of the SM vocal-tract model (cf. Chapter 5) which was firstpresented by Schroeder and Mermelstein (1965) and later expandedby each author separately (Schroeder, 1967; Mermelstein, 1967).
Whilst we follow mainly Schroeder’s (1967) mathematical derivationin the latter half of this Appendix, we start with some basic
assumptions and properties of a completely lossless acoustic tube,
which help to clarify certain aspects of the nonuniqueness problem
referred to in Chapter 5. The final result of the SM model itself isgiven in Equations B.22 and B.20, which are re-stated as Equations
5.1 and 5.2, respectively, in Section 5.2.1.

363

364

A static configuration of the supralaryngeal vocal-tract is usually modelled as anacoustic tube with a cross-sectional area-function A(x) which varies along its length (thex-axis) from the glottis (at x = 0 ) to the lips (at x = L ). Only planar (i.e., onedimensional) wave propagation is considered, which is a valid assumption forfrequencies below about 4 kHz, where acoustic wavelengths are larger than a typicalcross-sectional dimension. If the tube is further regarded as completely lossless andrigid-walled, then the spatio-temporal behaviour of acoustic pressure p(x,t) and volume
velocity u(x,t) within the vocal-tract are governed by the following two, well-known
equations (e.g., Morse and Ingard, 1968, p.243) of momentum (Equation B.1) andcontinuity of mass (Equation B.2):

ρ ∂u( x, t )∂p( x, t )=−∂x
A( x ) ∂t(B.1)∂u( x, t )A( x ) ∂p( x, t )
,
=− 2
∂x
ρc
∂t

(B.2)

where ρ is the air density and c is the velocity of sound propagation through the air
inside the acoustic tube. These differential equations are combined to yield the wellknown Webster’s Horn Equation, which can be expressed either in terms of acousticpressure:∂ ∂p( x, t )  A( x ) ∂ 2 p( x, t ), A( x )
= 2
∂x 
∂x 
c
∂t 2

(B.3)

or in terms of volume velocity:∂  1 ∂u( x, t ) 1 ∂ 2 u( x, t ).
=∂x  A( x ) ∂x  c 2 A( x ) ∂t 2

(B.4)

Without loss of generality, the time-dependence of acoustic pressure and volume
velocity is taken to be complex sinusoidal, as follows:p( x, t ) = p( x )e jωt ,(B.5)

u( x, t ) = u( x )e jωt ,(B.6)

where j is the imaginary unit, and ω is the angular frequency. If the vocal-tract areafunction is then assumed to be that of a uniform tube (such that A( x ) = A0 , someconstant), Equation B.3 reduces (by also eliminating the time-dependence using

365Equation B.5) to the simple one-dimensional wave equation for acoustic pressure:d 2 p( x )ω2=−p( x ) ,
dx 2
c2

(B.7)

and Equation B.4 likewise reduces (using Equation B.6) to the simple one-dimensional
wave equation for volume velocity:d 2 u( x )ω2=−u( x ) .
dx 2
c2

(B.8)

Simplified boundary conditions are defined by first regarding the glottis as aperfect volume-velocity source with infinite impedance, such that:u(0) = 0,

dp(0)
= 0.
dx

(B.9)

Ignoring end corrections, the open lip-end of the vocal-tract is then assumed to havezero impedance, such that:p( L) = 0,

du( L)= 0.
dx(B.10)

With these boundary conditions, the solutions to Equations B.7 and B.8 are therefore
characterised by the following eigenvectors for acoustic pressure and volume velocity,respectively:(2n − 1)πx pn ( x ) = cos
,


2L

un ( x ) =1(2n − 1)πx 
sin
,


2L
Z0

n = 1, 2, 3, ...

n = 1, 2, 3, ... ,(B.11)(B.12)

and by the eigenvalues:ω 2n  (2n − 1)π  2
λn = 2 =  , 2L 
cn = 1, 2, 3, ... ,(B.13)

where n denotes the index of the mode of resonance, and Z 0 = ρc / A0 is thecharacteristic impedance of the uniform acoustic tube.
Mermelstein (1967) then proceeds to invoke first-order perturbation theory toyield an expression for δλ n in terms of an assumed form of perturbation of the uniform
area-function. Deeper insights are gained, however, by re-tracing Schroeder’s (1967)
derivation, as it continues to build on the acoustic properties of the vocal-tract model.
366

In particular, Schroeder’s derivation is founded on the physical property that a
perturbation δA( x ) of the uniform area-function induces a change in the total energy

En of the nth mode of resonance, as follows:LδEn = − ∫ Pn ( x )δA( x )dx ,0(B.14)

where Pn ( x ) is the spatial distribution of acoustic radiation pressure for the nthresonance mode, and is given by the difference between the potential energy per unit
volume ( pn2 ( x ) / 2ρc 2 ) and the kinetic energy per unit volume ( Z 02 un2 ( x ) / 2ρc 2 ). Using
Equations B.11 and B.12, the resultant difference in energy densities is given by thefollowing:

Pn ( x ) =1(2 n − 1)πx 
cos
.
2


2ρc
L

(B.15)The total energy density of the nth resonance mode, on the other hand, is the sum of thepotential and kinetic energy densities for that mode. Using Equations B.11 and B.12,
the following relation is obtained:

L
,
2cZ 0

En =

(B.16)

which, when combined with Equation B.15, yields the following result for acoustic
radiation pressure:Pn ( x ) =En(2n − 1)πx 
cos
.


A0 L
L

(B.17)

Returning to Equation B.14, Schroeder (1967) then suggests an expression for thearea-function perturbation in terms of a Fourier cosine series which, in view of theform of Pn ( x ) in Equation B.17, secures the principle of orthogonality in computing
the integral, and thereby eliminates all but one term, per resonance mode. Thus,
assuming:∞mπx δA( x ) = A0 ∑ a m cos
,

L 
m =1

(B.18)and inserting Equations B.17 and B.18 into Equation B.14, orthogonality of thecomponents of the cosine series leads to the following, rather elegant solution:

δEn = − 1 2 En a2 n −1 ,367

(B.19)

which states that a perturbation of the uniform area-function by the single cosine term

a2 n−1 will, to first approximation, change the total energy of only the nth mode ofresonance.
The final step invokes a theorem by Ehrenfest which is “independent of the type ofwave equation” (Schroeder, 1967, p.1003), and which states that, for small, adiabatic
perturbations of an undamped linear oscillator, the relative change in the total energy ofthe nth resonance mode is equal to the relative shift in the frequency of that mode. Thismeans that Equation B.19 can be re-written in terms of the nth resonance (or formant)
frequency Fn = ω n / 2 π , as follows:δFn= − 1 2 a2 n −1 .
Fn

(B.20)The essence of the model is captured in this last result, which directly relates eachformant frequency to a unique vocal-tract shape parameter.
The final form of the vocal-tract shape parameterisation which led to this result, isobtained by first observing that for small perturbations, the natural-logarithm of thearea-function can be approximated as follows:ln[A( x )] = ln[A0 + δA( x )] ≈ ln( A0 ) +δA( x ).
A0

(B.21)

As suggested by Schroeder (1967), a logarithmic representation guarantees positivevalued areas, and in addition, it linearly separates the constant scale-factor from theperturbation term. This is clearly seen by inserting δA( x ) from Equation B.18 into
Equation B.21, to obtain the following:mπx ln A( x ) = ln A0 + ∑ am cos
,
 L 
m =1
M

(B.22)

where, in practice, the summation is limited to a finite number of terms M.
The complete model, which we refer to as the Schroeder-Mermelstein (SM)model, is summarised in Equation B.22 (reproduced as Equation 5.1 in Chapter 5)
which defines the vocal-tract shape parameterisation, and in Equation B.20 (reproducedas Equation 5.2 in Chapter 5) which defines the relation between the acoustic- and thearticulatory-domain parameters.

368

Appendix C
A Lossy Vocal-Tract Acoustic Model
This appendix describes the lossy vocal-tract acoustic model used in
Chapter 5 (first in Section 5.2.1 where we disregarded all the losses,
then in Section 5.5.2.2 where the losses were used) to synthesise theformants of given area-functions (in particular, the formant
frequencies shown in the nomograms in Figure 5.1, and the formant
frequencies and bandwidths listed in Table 5.4). The model is atransmission-line analog of the vocal-tract, suitable for the synthesisof the steady-state formants of static vocal-tract configurations
corresponding to non-nasalised, vocalic speech sounds, with the
vocal-tract excited at the glottis and with the radiating plane at the
lips.

369

370

mth vocal-tract section

glottis

jωLs

Rs

ug

Zg

Yw

lips
jωLs

-jωCp

Gp

lossy

Zl

lossless

Figure C.1: Electrical transmission-line analog of the vocal-tract.

Our implementation of a lossy vocal-tract acoustic model and the associated method offormant synthesis is based primarily on the numerical method described by Atal et al.
(1978, pp.1539-1542), together with other, well-known references such as Fant (1960),
Flanagan (1972), Wakita and Fant (1978), Badin and Fant (1984), and Ohmura (1993).
In particular, as shown in Figure C.1, an M-section vocal-tract area-function isrepresented as a serial concatenation of M electrical transmission-line sections,terminated at one end by a constant current source and a (time-invariant) glottal
impedance, and at the other end by a lip impedance across which acoustic energy isradiated. The relation between the acoustic pressure p (or electrical voltage) andvolume velocity u (or electrical current) at the input side (the glottal end, denoted by
the subscript g) and at the output side (the lip end, denoted by the subscript l) of theentire network, is then given by the following expression: p g ( s )   Α Β   p l ( s ) u ( s)  =   u ( s)  ,Γ∆g l 


(C.1)

where s = σ + jω is the complex-frequency variable, and the so-called, overall
transmission matrix is given as follows:Α Β  1
 Γ ∆  = 1 Z

  g

371

0 M α m1 ∏m =1 γ mβm .
δ m 

(C.2)

In Equation C.2 above, the first matrix on the right-hand side is the transmission
matrix of the glottal impedance, which itself is defined as follows: 12νd g l g2 0.875  ωρd g ,+Z g = R g + jωL g = 
2 ρPs  + j 3 A  A 
A
g
g
g


 

(C.3)

where ν = 1.86 × 10 −4 dyne.sec.cm-2 is the coefficient of viscosity, ρ = 1.14 × 10 −3 g.cm-3
is the density of air, d g = 0.3 cm and l g = 1.8 cm are the assumed depth (or vertical
thickness) and length of the (quasi-stationary) glottal slit, Ps = 8.0 cm H2O is theassumed pressure drop across the glottis; and the glottal opening area itself is computedby first determining the equivalent glottal resistance of the given area-function (withglottal-end section-area A(0) , glottal reflection coefficient µ glott , and the speed ofsound in the vocal-tract airway c = 35300 cm/sec) as follows:Rg =ρc (1 + µ glott ),A(0 ) (1 − µ glott )(C.4)

then equating the result with the first term on the right-hand side of Equation C.3 andnumerically solving for Ag (which is a smooth and monotonically decreasing functionof R g ).
The second matrix on the right-hand side of Equation C.2 is the transmission
matrix of the mth vocal-tract section, and is given as follows:

α m
γ
 m

Rs
βm   1
cosh (sl m c )( ρc S m ) sinh(sl m c ) =
G + Y 1 + R (G + Y ) δm   pcosh(sl m c )w
spw  m ( S m ρc ) sinh (sl m c )
(C.5)

In Equation C.5 above, the first matrix on the right-hand side is the transmission matrixof the lossy component of the mth section, while the second matrix is that of its losslesscomponent (see Figure C.1). The frequency-dependent elements of the former comprisethe following, series resistance (which represents the viscosity-induced loss contributedat the mth section):l S
R s = m 2m
Am

ρνω
,
2

(C.6)

the following, parallel conductance (which represents the heat-conduction loss
contributed at the mth section):
372

Gp =

(η − 1)l m S m
ρc 2

λω
,
2ξρ

(C.7)

and the following, parallel admittance (which represents the loss contributed by thereciprocal of the wall-vibration impedance at the mth section):−1

 c  2  ρ  
ω 
 
  Bw 0 + j
Yw =  .
2π 
 Fw   2πl m Am  


(C.8)

In Equations C.5 through C.8, the variables and constants are defined as follows: Am
and l m are the cross-sectional area (cm2) and the length (cm), respectively, of the mth
vocal-tract section; S m = SF × 2 πAm cm is its circumference (where Fant (1972)
suggests that the so-called “shape factor” be set to the value SF = 2 , whichcorresponds to an elliptical cross-sectional shape); η = 1.4 is the adiabatic gas constant;

λ = 5.5 × 10 −5 cal.cm-1sec-1deg-1 is the coefficientofheat-conduction

of

air;

ξ = 2.4 × 10 −1 cal.g-1deg-1 is the specific heat of air; and the closed-tract formant
frequency and bandwidth values of Fw = 190 Hz and Bw 0 = 80 Hz are approximately
those used by Badin and Fant (1984).
Returning now to Equation C.1 and substituting for the acoustic pressure at the lip
end pl (s ) = Z l u l (s ) in terms of the lip impedance Z l , the following expression is
obtained for the volume-velocity transfer function of the entire vocal-tract model:
H (s ) =u l (s )1=.
u g (s ) ΓZ l + ∆(C.9)

The lip impedance is assumed to be that of a piston in an infinite plane baffle, and iscomputed as follows (Flanagan, 1972; Ohmura, 1993):

ρc  Q (−1) q +1  ω
Zl =∑A(L )  q =1 q!(q + 1)!  c
A(L ) 
π 

2q

 2ω
4 Q
(−1) q−j ∑π q =0 (2q + 1)!!(2q + 3)!!  c

2 q +1,
(C.10)

A(L ) π where A(L ) is the cross-sectional area at the lip end, q!!= 1 × 3 × 5 × ... × (q − 2) × q , andthe number of terms Q = 8 in order to allow sufficient accuracy for frequencies up to
about 5kHz.

373

The resonances (or formants) of a given area-function are then simply the poles ofthe volume-velocity transfer function defined in Equation C.9 (where Γ and ∆ arefound by performing the matrix multiplication in Equation C.2), and are located in thesecond quadrant of the s-plane. In practice, they are found by searching for the zeros of
the reciprocal of the transfer function in that quadrant, in increasing order of frequency

ω . Atal et al. (1978) describe a search algorithm which we have found to be effective
in many cases, but which still requires manual intervention in order to ensure that aformant has not been erroneously skipped. Having determined (and perhaps corrected)
the first few formants s n = σ n + jω n , n = 1, 2, ... (where σ n ≤ 0 and 0 < ω n < ω n+1 ) in
the s-plane, the nth formant frequency is then given by:
Fn = ω n 2π ,(C.11)

Bn = − σ n π .

(C.12)and its bandwidth is given by:

The method of synthesis described above is particularly flexible, as it allows
selective inclusion of vocal-tract losses. For example, if any particular loss element is
not required, then its equivalent resistance is simply set equal to zero. Thus, in ourillustration of the formant frequency nomograms in Figure 5.1, we used a completely
lossless implementation of the above procedure, by appropriately nullifying all of theresistances in the transmission-line analog. By contrast, to obtain the formant
frequencies and bandwidths listed in Table 5.4, we used the completely lossy version of
the vocal-tract acoustic model.

374

Appendix D
The LP Method of Inversion “Stepped Down”
In this appendix, we describe and illustrate one way of “steppingdown” the LP method of inversion, such that it yields the same vocaltract area-function which the completely lossless, SM model would
yield using the same set of formant frequencies, and assuming
Mermelstein’s (1967) constraint of vocal-tract shape antisymmetry.
This supplements our empirical evidence presented in Section 5.3.1.2,
concerning the fundamental equivalence of the LP model and the SMmodel, in regard to their quasi-linear and one-to-one relation between
each formant frequency and the corresponding, antisymmetric vocaltract shape parameter.

375

376

As stated in Section 5.3.1.2, a complementary, empirical validation of the fundamental
similarity of the SM and the LP vocal-tract models, is to show that the LP model can beused to derive the same vocal-tract shapes which are yielded by the completely lossless,
SM model, starting from the same set of formant frequencies. To this end, we first notethat the essential difference in the way in which the two models secure uniqueness inestimated area-functions, can be stated in terms of the acoustic parameters required as
input to the respective method of inversion. Indeed, whilst the SM model requires theformant frequencies and the frequencies of the poles of the lip impedance function, theLP model requires the formant frequencies and the formant bandwidths. Initially, one
may be tempted to try to simulate the completely lossless SM model, simply by setting
the formant bandwidths to zero in the LP model. However, this is almost equivalent tounderspecifying the inverse problem, and in any case results in errors when converting
the LP autoregressive coefficients to reflection coefficients using the well-known
recursive algorithm (Markel and Gray, 1976). A different constraint must therefore beapplied to the formant bandwidths which are used as input to the LP method of
inversion, while ensuring that their values are positive, finite, and non-zero.
In order to resolve the nonuniqueness problem using the SM method of inversion,
Mermelstein (1967) imposed a constraint of antisymmetry on allowed vocal-tract
shapes (by allowing only the odd-indexed cosine parameters to be non-zero). If we areto coerce the LP model to yield the same vocal-tract shapes that would be obtained by
the SM model under these conditions, it is therefore necessary to assume that only thetarget formant frequencies are available, and that the values of the formant bandwidths
are to be determined such as to obtain a purely antisymmetrical vocal-tract shape. The
block diagram shown in Figure D.1 shows our solution to this problem. Since only theformant frequencies are assumed to be relevant, the input formant bandwidths are
initially set to arbitrary values. The LP inverse routines are then used (either with afixed vocal-tract length, or by applying Wakita’s (1977) method as briefly discussed in
Section 5.2.2) to obtain a discrete-sectioned, normalised LP area-function, theantisymmetric components of which are found by inverse discrete cosine transformation(IDCT) of the logarithmic area-function, while its symmetric components are ignored
by setting the even-indexed parameters of the SM model to zero.
377

L
Fs=cM/2L

LP Inversion Method

Initialise Formants:
Target {Fn},Nominal {Bn},n=1,...,M/2

fb2a

a2µ

µ2areaDiscrete
LP Area-Function
Ai, i=1,...,M

Update Formants
at jth iteration{Bn*}j+-{Fn*}j
+

+

+

-

* j-1

Target {Fn}{Fn }Retain only antisymmetric
VT-Shape Parameters{a2n-1}, n=1,...,M/2

LP
Synthesis

LP (lossless)
Synthesis

Area-Function
Parameterisation(Equation 5.1)SetµΜ = 1Area-Function
Expansion(Equation 5.1)
Antisymmetric
LP Area-Function
Ai*, i=1,...,M

Figure D.1: Block-diagram of area-function estimation method based on the LP model stepped down
to a completely lossless model, with the constraint of logarithmic area-function antisymmetry.

Having thus imposed the constraint of antisymmetry on the logarithmic areafunction, the LP synthesis routines are used to obtain a new set of formant bandwidths.
The arbitrary bandwidths initially used as input to the LP inverse method are thenreplaced with the bandwidths just synthesised from the antisymmetric area-function, anda new vocal-tract shape is estimated using the LP method of inversion. This process isrepeated until the bandwidths are sufficiently similar from one iteration to the next.
However, as shown in Figure D.1, the formant frequencies also need to be iteratively
updated, in order to account mainly for the second-order effect of discarding thesymmetric components, but also to counteract the small, systematic influence of a finite
LP glottal resistance (as discussed in Chapter 5, in regard to the nomograms in Figure5.3). This is achieved by LP-resynthesising the formant frequencies of the area-function
which is forced to be antisymmetric at every iteration, with the glottal reflection
coefficient µ M = 1 to simulate complete closure of the acoustic tube at the glottal end.
The difference between these formant frequencies and the original, target values thendefines the incremental change at each successive iteration.
378

Convergence is finally declared when the magnitude of all required adjustments toformant frequencies and bandwidths does not exceed a certain limit (typically 1.0 Hz).
The vocal-tract shape yielded by the LP inverse method at the last iteration is then
expected to be almost perfectly antisymmetric (to within numerical accuracies
determined by the condition for convergence), with a completely lossless LP resynthesis
yielding the target formant frequencies. An interesting by-product of this optimisation
scheme is, in answer to the question posed earlier, the identification of the formant
bandwidths required for the LP model to yield an antisymmetric area-function.
Figure D.2 illustrates our results obtained by applying this method to the formant
frequency data computed by Mermelstein (1967, Table II, p.1288), who used acompletely lossless vocal-tract model to obtain the resonances of the six Russian vowelsof Fant (1960), from the quantised area-functions (the so-called “exact areas”)
measured by Fant from mid-sagittal X-ray images. As only the first three formant
frequencies are given by Mermelstein, we use an LP model with M = 6 vocal-tract
sections, and the three formant bandwidths are initialised each to a nominal value of
30Hz. In order to avoid artifactual influences on the vocal-tract shapes that might arise
from also estimating their lengths, these were fixed to Fant’s original, X-ray measured
values. The optimisation process described above was then found to converge in no
more than 5 iterations for each of the six Russian vowels, yielding the final,
antisymmetric LP area-function shown in each panel of Figure D.2 together with thefinal values of the formant frequencies and bandwidths required as input to the LP
inverse routine.
An interesting, preliminary observation in regard to the formant bandwidths
required to finally secure an antisymmetric area-function, is that although they are
individually perturbed from their arbitrary initial values, the mean formant bandwidth of
30Hz is preserved for each of the six vocalic configurations tested. This observationsuggests that the relative bandwidths, rather than their absolute values, are more
relevant to the LP vocal-tract shape. Indeed, this already foreshadows our theoreticaland empirical results presented in Section 5.3.2, where we take up this important issue
in greater detail.
How do the vocal-tract shapes yielded by our iterative, LP-based method,379

compare with those determined by the SM model? Superimposed on each of the six,
LP-derived area-functions shown in Figure D.2, is the antisymmetric vocal-tract shapeobtained by applying Mermelstein’s (1967, Appendix B) approach to optimise only the
odd-indexed parameters of the SM model, using the same, target formant frequencies
which we used in our LP-based approach. In deciding the number of sections withwhich to represent each area-function, we note that Mermelstein (1967, Appendix A)
himself chose section-lengths of 0.5 cm in order for the representation “to be valid up toa frequency of 4kHz”. Whilst a greater number of sections would more faithfully
represent an area-function having many details and discontinuities, both the LP and the
SM model imply that the theoretical minimum of M = 2 LFs / c sections ensures thatthe vocal-tract shape is acoustically valid up to a frequency of Fs / 2 Hz (with an upper
limit of about 4kHz imposed by the assumption of planar acoustic-wave propagation).
In view of the inherent smoothness of area-functions generated by the SM model (cf.
Equation 5.1), and in order to maintain consistency with the LP method, the theoreticalminimum of 6 vocal-tract sections (which corresponds to a frequency range
encompassing the first three formants) was therefore used to implement the SM methodof inversion.
The estimated, 6-section area-functions are shown in Figure D.2 by the diamondsymbols, plotted at the centre of each section; dashed curves are used to emphasise theessentially smooth and continuous representation of vocal-tract shapes afforded by the
SM model. Notwithstanding minor differences in implementational details, our smootharea-functions are quite similar to those obtained by Mermelstein (1967, Figure 3).
Furthermore, the close proximity of the diamond symbols to the corresponding sectionsof the LP-derived area-functions, indicate that the latter are essentially a discretesectioned representation of the smooth area-functions obtained by the SM model. These
results demonstrate the success of our optimisation scheme of Figure D.1, in stepping
down the LP method of inversion such that it yields area-functions otherwise attainable
only by using a completely lossless vocal-tract model. Most importantly, ourtransformation of the LP model was founded on the very principle of symmetric versus
antisymmetric shape perturbations, which itself is the basis of the SM model.

380

/i/
7

Initial
224.0

Final
(5 iter’s)
223.9

2256.02256.43166.03192.6

30.0

3.9

30.0

5.5

30.0

80.6

6

F1
F2
F3
B1
B2
B3

5
4
3
2
1
0
0

2

4

6

8

10

12

14

16

18

/e/
2.5

Initial
421.0

Final
(4 iter’s)
420.7

1971.01971.82908.02909.0

30.0

17.3

30.0

22.8

30.0

49.9

2

F1
F2
F3
B1
B2
B3

1.5

1

0.5

0
0

2

4

6

8

10

12

14

16

18

/Þ/
3

Initial
283.0

Final
(3 iter’s)
282.8

1733.01733.52319.02319.0

30.0

23.0

30.0

28.4

30.0

38.6

2.5

F1
F2
F3
B1
B2
B3

2

1.5

1

0.5

0
0

2

4

6

8

10

12

14

16

18

20

Distance from glottis (cm)Figure D.2: Vocal-tract area-functions estimated from the first three formant frequencies given by
Mermelstein (1967, Table II) for Fant’s (1960) six Russian vowels. Solid lines: 6-section areafunctions obtained by Mermelstein’s (1967) optimisation method with the constraint a2n=0, using acompletely lossless vocal-tract model. Diamond symbols: 6-section area-functions represented at theirsection-centres, and obtained by the stepped-down LP inversion method shown in Figure D.1. Dashedcurves: smoothed version of the 6-section, LP-derived area-functions shown by the diamond symbols,
obtained by expanding the estimated vocal-tract shape parameters according to Equation 5.1. Table toright of each graph: first three formant frequencies and bandwidths used to initialise the stepped-down
LP inversion method (frequencies taken from Mermelstein (1967), bandwidths set to nominal value
30Hz), and those yielded by our method (as input to the LP inverse routines) at the final iteration prior
to convergence.

381

/u/
7

Initial
250.0

Final
(5 iter’s)
249.1

601.0

599.1

2308.02308.1

30.0

57.7

30.0

28.2

30.0

4.1

6

F1
F2
F3
B1
B2
B3

5
4
3
2
1
0
0

2

4

6

8

10

12

14

16

18

20

/o/
3.5

Initial
523.0

Final
(5 iter’s)
523.4

911.0

909.7

2350.02350.0

30.0

45.0

30.0

35.1

30.0

9.8

3

F1
F2
F3
B1
B2
B3

2.5
2
1.5
1
0.5
0
0

2

4

6

8

10

12

14

16

18

20

/A/
3.5

Initial
669.0

Final
(4 iter’s)
669.2

1139.01137.62487.02487.5

30.0

40.7

30.0

36.2

30.0

13.0

3

F1
F2
F3
B1
B2
B3

2.5
2
1.5
1
0.5
0
0

2

4

6

8

10

12

14

16

18

Distance from glottis (cm)Figure D.2: (continued from previous page).

382

Appendix E
Representation of Directly Measured Area-Functions
In this appendix are shown, for each of the 33, directly measured
area-functions obtained from the literature (cf. Section 5.4.2.2), theprofile of root-mean-square (rms) error as a function of the spatial
resolution M 2 used to represent those area-functions, using our
method of parameterisation (cf. Equation 5.18). In each figure, thediamond symbols (joined by solid lines) show the rms errors obtainedin representing that area-function with both symmetric andantisymmetric shape components; the plus symbols (joined by dashedlines) show the rms errors using only the antisymmetric shapecomponents; and the square symbols (joined by dotted lines) show
the rms errors using only the symmetric shape components. The
overall mean of each of those three, rms error profiles (across all 33
area-functions), was shown in Figure 5.8.
As discussed in Section 5.4.2.2, the rms errors obtained using
only antisymmetric shape components are consistently larger for themid- to high-, back vowels which have a place of constriction near the
mid-length of the vocal-tract (e.g., Fant’s (1960) Russian /u/; Baer etal.’s (1991) British English /u/ and American English /u/; Yang andKasuya’s (1994) Japanese /u/ and /o/; Beautemps et al.’s (1995)
French /u/; and Story et al.’s (1996) American English /o/, /U/, and
/u/).

383

/i/

/u/

2.4

2.4

2.2

2.2

2

2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0
0

1

2

3

4

5

6

7

8

9

0

1

2

3

/e/

4

5

6

7

8

9

6

7

8

9

6

7

8

9

/o/

2.4

2.4

2.2

2.2

2

2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0
0

1

2

3

4

5

6

7

8

9

0

1

2

3

/Þ/

4

5

/A/

2.4

2.4

2.2

2.2

2

2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0
0

1

2

3

4

5

6

7

8

9

0

1

2

3

4

5

Spatial resolution (M/2)

Figure E.1: Profiles of rms error in representing the area-functions of the 6 Russian vowels of
Fant (1960).

384

/i/

/u/

2.4

2.4

2.2

2.2

2

2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0
0

1

2

3

4

5

6

7

8

9

0

1

2

3

/Q/

4

5

6

7

8

9

6

7

8

9

/A/

2.4

2.4

2.2

2.2

2

2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0
0

1

2

3

4

5

6

7

8

9

0

1

2

3

4

5

Spatial resolution (M/2)

Figure E.2: Profiles of rms error in representing the area-functions of the 4 British English
vowels (Speaker PN) of Baer et al. (1991).

385

/i/

/u/

2.4

2.4

2.2

2.2

2

2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0
0

1

2

3

4

5

6

7

8

9

0

1

2

3

/Q/

4

5

6

7

8

9

6

7

8

9

/A/

2.4

2.4

2.2

2.2

2

2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0
0

1

2

3

4

5

6

7

8

9

0

1

2

3

4

5

Spatial resolution (M/2)

Figure E.3: Profiles of rms error in representing the area-functions of the 4 American English
vowels (Speaker TB) of Baer et al. (1991).

386

/i/

/u/

2.4

2.4

2.2

2.2

2

2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0
0

1

2

3

4

5

6

7

8

9

0

1

2

3

4

/e/

5

6

7

8

9

6

7

8

9

/o/

2.4

2.4

2.2

2.2

2

2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0
0

1

2

3

4

5

6

7

8

9

0

1

2

3

4

5

/a/
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0

1

2

3

4

5

6

7

8

9

Spatial resolution (M/2)
Figure E.4: Profiles of rms error in representing the area-functions of the 5 Japanese vowels(adult, male speaker) of Yang and Kasuya (1994).

387

/i/
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0

1

2

3

4

5

6

7

8

9

5

6

7

8

9

5

6

7

8

9

/u/
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0

1

2

3

4

/a/
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0

1

2

3

4

Spatial resolution (M/2)

Figure E.5: Profiles of rms error in representing the area-functions of the 3 French vowels of
Beautemps et al. (1995).

388

/i/

/Q/

2.4

2.4

2.2

2.2

2

2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0
0

1

2

3

4

5

6

7

8

9

0

1

2

3

/I/

4

5

6

7

8

9

6

7

8

9

6

7

8

9

/«/

2.4

2.4

2.2

2.2

2

2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0
0

1

2

3

4

5

6

7

8

9

0

1

2

3

/E/

4

5

/A/

2.4

2.4

2.2

2.2

2

2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0
0

1

2

3

4

5

6

7

8

9

0

1

2

3

4

5

Spatial resolution (M/2)

Figure E.6: Profiles of rms error in representing the area-functions of the 11 American English
vowels of Story et al. (1996). (Continued on the following page.)

389

//

/u/

2.4

2.4

2.2

2.2

2

2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0
0

1

2

3

4

5

6

7

8

9

0

1

2

3

4

/o/

5

6

7

8

9

6

7

8

9

/¼/

2.4

2.4

2.2

2.2

2

2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0
0

1

2

3

4

5

6

7

8

9

0

1

2

3

4

/U/
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0

1

2

3

4

5

6

7

Spatial resolution (M/2)

Figure E.6: (continued from previous page).

390

8

9

5

Appendix F
Re-estimation of Directly Measured Area-Functions
This appendix pertains to our discussions in Section 5.5.2, concerning
re-estimation of the 33, directly measured area-functions (obtained
from the literature, cf. Section 5.4.2.2), using our hybrid LP-SM
method of inversion. In each panel is shown the original, step-wisearea-function1 (solid line), the smooth area-function re-estimatedunder model-matched conditions (dashed curve), and the smootharea-function re-estimated under model-mismatched conditions(dotted curve). The latter two area-functions are optimally alignedwith respect to the original area-function, according to our method ofalignment described in Section 5.5.1.
Model-matched conditions (cf. Section 5.5.2.1) imply that thefirst four formant frequencies and bandwidths used as inputs to theinversion (cf. Table 5.3), were synthesised using an 8-section
representation of the original area-function, and an LP vocal-tract
model with a nominal, glottal reflection coefficient of µ 8 = 0.8 .
Model-mismatched conditions (cf. Section 5.5.2.2) imply thatthe first four formant frequencies and bandwidths used as inputs tothe inversion (cf. Table 5.4), were synthesised using the original
section lengths and areas, and with a vocal-tract acoustic model (seeAppendix C) which includes glottal impedance ( µ glott = 0.8 ),viscosity, heat-conduction, wall-vibration, and lip-radiation losses.

1

As explained in Chapter 5, all area-functions are first “normalised” such that the area scaling factor
in Equation 5.1 is equal to unity (A0 = 1). Hence the dimensionless ordinate on each graph, and thelabel “Normalised Area”, which should not be confused with the speaker normalisation of areafunctions referred to in Chapter 6.

391

/i/

/u/

4.5

8

4

7

3.5

6

3

5

2.5
4
2
3

1.5

2

1

1

0.5
0

0
-2

0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

/e/

10

12

14

16

18

20

22

14

16

18

20

22

14

16

18

20

22

/o/

2.5

6
5

2

4
1.5
3
1
2
0.5

1

0

0
-2

0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

/Þ/

10

12

/A/

3.5

5
4.5

3

4
2.5

3.5
3

2

2.5
1.5

2
1.5

1

1
0.5

0.5

0

0
-2

0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10

12

Distance from glottis (cm)Figure F.1: Original and re-estimated area-functions of the 6 Russian vowels of Fant (1960).

392

/i/

/u/

4.5

3.5

4

3

3.5
2.5
3
2.5

2

2

1.5

1.5
1
1
0.5

0.5
0

0
-2

0

2

4

6

8

10

12

14

16

18

-2

0

2

4

6

/Q/

8

10

12

14

16

18

20

12

14

16

18

20

/A/

4.5

3

4
2.5
3.5
3

2

2.5
1.5
2
1.5

1

1
0.5
0.5
0

0
-2

0

2

4

6

8

10

12

14

16

18

-2

0

2

4

6

8

10

Distance from glottis (cm)Figure F.2: Original and re-estimated area-functions of the 4 British English vowels (SpeakerPN) of Baer et al. (1991).

393

/i/

/u/

4

2.5

3.5
2
3
2.5

1.5

2
1

1.5
1

0.5
0.5
0

0
-2

0

2

4

6

8

10

12

14

16

18

0

2

4

6

8

/Q/

10

12

14

16

18

20

22

14

16

18

20

22

/A/

3.5

4

3

3.5
3

2.5

2.5
2
2
1.5
1.5
1

1

0.5

0.5

0

0
-2

0

2

4

6

8

10

12

14

16

18

0

2

4

6

8

10

12

Distance from glottis (cm)Figure F.3: Original and re-estimated area-functions of the 4 American English vowels(Speaker TB) of Baer et al. (1991).

394

/i/

/u/

3.5

2.5

3
2
2.5
1.5

2
1.5

1

1
0.5
0.5
0

0
0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

/e/

10

12

14

16

18

20

22

14

16

18

20

22

/o/

3.5

8

3

7
6

2.5

5
2
4
1.5
3
1

2

0.5

1

0

0
0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10

12

/a/
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0

2

4

6

8

10

12

14

16

18

20

22

Distance from glottis (cm)Figure F.4: Original and re-estimated area-functions of the 5 Japanese vowels (adult, malespeaker) of Yang and Kasuya (1994).

395

/i/
3.5
3
2.5
2
1.5
1
0.5
0
0

2

4

6

8

10

12

14

16

18

20

22

12

14

16

18

20

22

12

14

16

18

20

22

/u/
4
3.5
3
2.5
2
1.5
1
0.5
0
0

2

4

6

8

10

/a/
2.5

2

1.5

1

0.5

0
0

2

4

6

8

10

Distance from glottis (cm)Figure F.5: Original and re-estimated area-functions of the 3 French vowels of Beautemps etal. (1995).

396

/i/

/Q/

5

3

4.5
2.5

4
3.5

2

3
2.5

1.5

2
1

1.5
1

0.5

0.5
0

0
0

2

4

6

8

10

12

14

16

18

0

2

4

6

/I/

8

10

12

14

16

18

12

14

16

18

12

14

16

18

/«/

3

6

2.5

5

2

4

1.5

3

1

2

0.5

1

0

0
0

2

4

6

8

10

12

14

16

18

0

2

4

6

/E/

8

10

/A/

3

12

2.5

10

2

8

1.5

6

1

4

0.5

2

0

0
0

2

4

6

8

10

12

14

16

18

0

2

4

6

8

10

Distance from glottis (cm)Figure F.6: Original and re-estimated area-functions of the 11 American English vowels of
Story et al. (1996). (Continued on the following page.)

397

//

/u/

7

4.5
4

6

3.5
5
3
4

2.5

3

2
1.5

2
1
1

0.5

0

0
0

2

4

6

8

10

12

14

16

18

20

22

0

2

4

6

8

/o/

10

12

14

16

18

20

22

14

16

18

20

22

/¼/

7

3

6

2.5

5
2
4
1.5
3
1
2
0.5

1
0

0
0

2

4

6

8

10

12

14

16

18

20

22

0

2

4

6

14

16

18

20

8

/U/
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0

2

4

6

8

10

12

Distance from glottis (cm)Figure F.6: (continued from previous page).

398

22

10

12

Appendix G
Closed-glottis Correction of Formant Bandwidths
This appendix pertains to our discussions in Section 5.5.3.2,
concerning correction of measured formant bandwidths in order toobtain the “glottis-only” bandwidths ideally required by the LP vocaltract model. Table G.1 lists the per-vowel mean formant frequencies
and bandwidths computed over all 7 steady-state frames, 5 repetitions
and 4 speakers of the FC dataset; it also lists the corrected, or socalled “glottis-only” mean bandwidths, obtained by subtracting the“closed-glottis” value predicted by Hawks and Miller’s (1995)
equation. Figure G.1 then shows the influence of the proposed
bandwidth correction on LP-derived area-functions, estimated using
the mean formant data both before and after correction.

399

F1

F2

F3

F4

B1

B2

B3

B4

(Hz)

(Hz)

(Hz)

(Hz)

(Hz)

(Hz)

(Hz)

(Hz)

/i/

345

2175

2968

3582

77
31

126
55

293
166

268
85

/I/

367

2099

2803

3541

81
38

121
53

241
128

213
34

/E/

479

2019

2711

3542

104
67

139
75

224
118

337
158

/Q/

654

1798

2589

3641

126
86

124
69

213
116

329
141

/a/

741

1213

2595

3550

145
104

159
116

234
136

335
156

/ /

618

959

2521

3616

107
68

111
69

178
85

270
84

/ /

422

712

2563

3400

75
37

89
49

195
100

239
74

/ U/

397

857

2381

3325

94
54

145
104

339
256

358
200

/ á/

341

1660

2292

3344

57
11

99
48

104
26

200
40

/ «/

724

1237

2599

3518

133
93

174
131

260
162

317
141

/ ¶/

465

1585

2491

3591

84
47

120
71

113
23

201
18

Table G.1: Per-vowel mean formant frequencies and bandwidths, computed across all 7
steady-state frames, 5 repetitions, and 4 speakers of the FC dataset. The second row of figures
for each vowel lists the mean formant bandwidths obtained after subtracting the so-called
closed-glottis bandwidth predicted by Hawks and Miller’s (1995) equation. These mean formant
values (both before and after bandwidth correction) are used to estimate the area-functions
shown in Figure G.1.

400

/i/

/ á /

4

4

3

3

2

2

1

1

0

0
0

2

4

6

8

10

12

14

16

18

0

2

4

6

8

/I/

10

12

14

16

18

20

12

14

16

18

20

14

16

18

20

/¶/

4
3
3

2
2

1

1

0

0
0

2

4

6

8

10

12

14

16

18

0

2

4

6

8

/E/

10

/Q/
4

3
3

2
2

1

1

0

0
0

2

4

6

8

10

12

14

16

18

0

2

4

6

8

10

12

Distance from glottis (cm)Figure G.1: Influence of closed-glottis bandwidth correction on LP-derived area-functions.
Shown in each of the 11 panels (continued on next page) are the area-functions estimated fromthe per-vowel mean formant data (FC dataset) listed in Table G.1, before (dashed line) andafter (solid line) correction of the bandwidths by subtraction of the closed-glottis bandwidth
obtained by Hawks and Miller’s (1995) equation. Each pair of area-functions thus obtained, isaligned using the method described in Section 5.5.1.

401

//

//

7
10
6

9
8

5
7
4

6
5

3
4
2

3
2

1
1
0

0
0

2

4

6

8

10

12

14

16

18

20

22

0

2

4

6

8

/U/

10

12

14

16

18

20

12

14

16

18

20

/a/

10

6

9
5

8
7

4

6
5

3

4
2

3
2

1

1
0

0
0

2

4

6

8

10

12

14

16

18

20

22

0

2

4

6

14

16

18

20

/«/
5

4

3

2

1

0
0

2

4

6

8

10

12

Distance from glottis (cm)Figure G.1: (continued from previous page).

402

8

10

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