INTERSPEECH 2013

Beyond bandlimited sampling of speech spectral envelope
imposed by the harmonic structure of voiced sounds
Hideki Kawahara1 , Masanori Morise2 , Tomoki Toda3 , Ryuichi Nisimura1 , Toshio Irino1
1

Department of Design Information Sciences, Wakayama University, Japan
2
Faculty of Engineering, University of Yamanashi, Japan
3
Nara Advanced Institute of Science and Technology, Japankawahara@sys.wakayama-u.ac.jp

Abstract

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A new spectral envelope estimation procedure is proposed to recover details beyond band limitation imposed by the Shannon’ssampling theory when interpreting periodic excitation of voiced
sounds as the sampling operation in the frequency domain. The
proposed procedure is a hybrid of STRAIGHT, a F0-adaptive
spectral envelope estimation and the auto regressive model parameter estimation. Wavelet analyses of these spectral models
on the frequency domain enabled objective evaluation of thisrecovery procedure. The proposed procedure provides betterspeech quality especially when parameter manipulation is introduced.
Index Terms: Speech analysis, envelope spectrum, sampling
theory, speech modiﬁcation, transfer function

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1. Introduction
Flexible modiﬁcation of speech sounds requires spectral envelope modeling as well as the excitation source modeling. Channel VOCODER [1], linear predictive coding (LPC) [2, 3, 4],
cepstrum-based method [5, 6, 7], sinusoidal models [8, 9] andSTRAIGHT [10, 11] are representative examples of such spectral models. Estimation of relevant spectral envelopes fromvoiced sounds is not simple, because vocal tract transfer functions consist of components which violate the band limitation
requirement. This requirement is imposed by the equivalent
spectral sampling of the periodic excitation of voiced sounds
when represented in the frequency domain. The violation ofband limitation is due to the fact that vocal tract transfer functions are represented by rational functions having a cosine series
as the denominator (in discrete time systems). Because the cosine series is in denominator, even though the number of termsof the cosine series is ﬁnite, Fourier series representation of thetransfer function has inﬁnite number of terms. However, after
over a half century from introduction of Shanonn’s sampling
theory [12], new views on sampling problem emerge and provide basis to alleviate (or evade) this band-limitation barrier byusing side information (in our case F0 and knowledge about vocal tract transfer function) [13, 14].
This article reformulates our previous proposals [15, 16] on
spectral shape compensation for improving manipulated speechsound quality and proposes an optimized design of a hybrid procedure of STRAIGHT and LPC.
A vocal tract can be modeled using an one dimensional nonuniform acoustic tube [17, 18]. This approximation yields anall-pole transfer function shown in the upper plot of Fig. 1. The
squared absolute value of the denominator of the transfer function of a non-uniform acoustic tube is a cosine series with real

Copyright © 2013 ISCA

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Figure 1: Transfer function example of one dimensional acoustic tube modeling of Japanese vowel /a/ (upper plot) and pseudocolor map of the magnitude of its continuous wavelet transformusing Morlet wavelet (lower plot).
valued coefﬁcients. Logarithmic conversion of the denominator
has sharp negative peaks and they correspond to sharp formant
peaks of the transfer function. Attributes of these peaks (formants) have strong contribution to perceptual identity and quality of speech sounds. Acoustic measurements of the vocal tract
transfer function using real humans [19] and three dimensional
replica [20] veriﬁed that the peak shapes are actually very sharp.
Continuous wavelet analysis using Morlet wavelet [21] isintroduced to clarify the band-unlimited nature of vocal tract
transfer functions. The mother wavelet ψ(f ) used here has thefollowing form.
„
ψ(f ) =

2k0

ejk0 f − e− 2«f2

e− 2 ,(1)

where f represents the frequency axis of Fig. 1. It is a Gaussian
wavelet with a small correction factor to satisfy the admissibility condition. The wave number coefﬁcient k0 = 4 is used here.

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25- 29 August 2013, Lyon, France

The lower plot of Fig. 1 shows the absolute value of the
wavelet transform Wψ,H (a, f ). The vertical axis represents
the repetition interval of the scaled version of the carrier signal
ejk0 f of each Morlet wavelet. The vertical axis has the dimension of time (from bottom to top, the values roughly correspond
to 1, 2, 4, 8 and 16 ms). The pseudo color mapping represents
the absolute value of the wavelet transform of the differentiatedlogarithmic conversion of the vocal tract transfer function H(f )and the numbers indicated on the color bar show the magnitudein terms of dB.
«
„
Z ∞
1
λ − f d ln |H(λ)|ψ∗
Wψ,H (a, f ) = pdλ,adλ|a| −∞(2)
where ∗ represents complex conjugate and 1/a is shown on
the vertical axis of Fig. 1. This representation illustrates thathigher components violating band limitation requirement imposed by the spectral sampling due to periodic excitation arelocated around these sharp peaks.

Figure 2: Schematic diagram of the proposed method. Circled
numbers represent corresponding steps in 1.2 of the text.
quality, several types of excitation models are proposed to overcome this difﬁculty of LPC [25, 26].
1.2. Outline of the proposed method
By combining merit of LPC with STRAIGHT (also with enhancement based on consistent sampling and numerical tuning),
the true spectral envelope is likely to be recovered in the following steps. Figure 2 illustrates a procedure to implement thisidea.
In the ﬁrst step, temporally stable power spectrum (TANDEM spectrum) is calculated. In the next step, periodic spectral
variations due to periodic excitation is selectively suppressedby using an F0-adaptive triangular smoother having width of
2f0 on the frequency axis. In the third step, over-smoothing
due to smearing caused by time windowing and the triangularsmoother are compensated by the digital ﬁlter on the frequency
axis designed based on consistent sampling theory and numerically tuned afterwards. Let name the spectral envelope obtainedat this point as STRAIGHT spectrum. (It is a power spectrum.)
In the fourth step, LPC spectral envelope is estimated from autocorrelation coefﬁcients calculated from the inverse Fouriertransform of the STRAIGHT spectrum. In the ﬁfth step, thesmeared LPC spectrum is calculated by smoothing it using thefrequency domain representation of the time window and thetriangular smoother. In the sixth step, the correction spectrum
is yielded as the LPC spectrum divided by the smeared LPC
spectrum. In the ﬁnal step, the extended STRAIGHT spectrum is calculated by multiplying the correction spectrum to the
STRAIGHT spectrum.
This is a reformulation and extension of our proposals [15,16]. The next section starts from brief descriptions of TANDEM
and STRAIGHT, followed by detailed explanations of the proposed method with numerical examples.

1.1. Spectral smearing by modeling
Spectrum modeling using a set of band-pass ﬁlters, such as
channel VOCODER [1], smears these sharp peaks, because
spectral shape resolution is limited by the bandwidth of the
band-pass ﬁlters. Cepstrum liftering [5, 6, 7] for removing periodic variations representing harmonic structure also smears
these sharp peaks, because it is a low-pass ﬁltering on the frequency axis for removing components varying ﬁner than 2f0
period. For example, using excitation of 100 Hz pulse train,details shown in upper than 200 Hz (repetition interval on thevertical axis) region in the bottom plot of Fig. 1 are removed.
Spectral smoothing operations used in STRAIGHT (bothlegacy-STRAIGHT [10] and TANDEM-STRAIGHT [11, 16])
also smear these details, while they are not strictly band-limited,
rapidly changing components on the frequency axis are attenuated. These smoothing operations are error-tolerant1 implementation [10] of piece-wise stair-case interpolation andpiece-wise linear interpolation for TANDEM-STRAIGHT andlegacy-STRAIGHT respectively. One of our previous proposals [16] reduces this attenuation effect by numerically adjusting amount of enhancement introduced to implement the digitalcompensation ﬁlter for consistent sampling [13, 22].
All these procedures also suffer from spectral smearing due
to time windowing when they are implemented using short-term
Fourier transform [23, 24]. Frequency domain representation ofthese time windowing functions provides impulse responses of
low-pass ﬁlters on the frequency axis.
LPC introduces other smearing. Formulation of LPC by
Itakura [2] is originally a maximum likelihood estimation of autoregressive model parameters by assuming speech as a weakly
stable stochastic process. Since the model structure, in this case,agrees with the physically correct transfer function, the spectralenvelope calculated from the estimated parameters has properly
sharp peaks. It is not band-limited. However, since the actual
excitation signal of speech sounds is not an independent Gaussian random noise, the true speech spectral envelope (although
it is not directly observable) has peaks and dips due to local(short range) correlations. All these details are smeared out inthe spectral envelope derived from LPC analyses. Since smearing of these details signiﬁcantly deteriorates synthesized speech

2. The proposed method
The proposed method shares the following TANDEM and
STRAIGHT procedures. A brief introduction to these procedures and examples are presented here.
2.1. TANDEMThe TANDEM procedure eliminates temporal variation due toperiodicity by F0-adaptive window design and F0-adaptive averaging [27, 11]. Let P (ω, t) represent the power spectrum
around time t of a windowed voiced speech and T0 = 1/f0
represent the fundamental period. Then, TANDEM spectrum
PT (ω, t), which does not have temporally varying component

1 Error here refers fundamental frequency estimation error and background noise as additive power spectrum error.

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Figure 4: STRAIGHT spectrum (blue line) and TANDEM
spectrum (red line) (upper plot) and wavelet analysis resultof STRAIGHT spectrum using gC (τ ) and g2 (τ ) lifters (lowerplot). The horizontal orange blob and sharp cyan spikes foundin Fig. 3 are suppressed. However, large sharp peaks corresponding to formants are not salient due to smoothing.
consistent sampling, followed by the numerical tuning to reduce
spectral smearing around formant peaks [16], based on a set ofsimulations using vocal tract area functions of eleven English
vowels [28] and glottal waveform model [29] with random parameter perturbations.
«
„
2πτ
.
(5)
gC (τ ) = q̃0 + 2q̃1 cos
T0

due to excitation periodicity is calculated by the following equation.
P (ω, t − T40 ) + P (ω, t + T40 )PT (ω, t) =(3)
2
This procedure yields temporally stable representation, but
the spectrum still has periodic variations due to excitation periodicity. The variation is a single cosine and multiplied to power
spectral envelope. However, the variation represented in the
log-power spectrum has harmonic components in the cepstrum
domain, because of the logarithmic nonlinearity. The wavelet
transform shows this nonlinear effects and periodic variation
due to periodic excitation clearly as shown in Fig. 3.

The lifter gA (τ ), A ∈ {1, 2} is F0 adaptively designed to eliminate periodic variations due to the harmonic structure. The lifter
g1 (τ ) is used in TANDEM-STRAIGHT implementation.
sin(πf0 τ )= F [h1 (ω)],πf0 τ
j0 |ω| ≥ ω20.
h1 (ω) =1
otherwise
ω0
g1 (τ ) =2.2. STRAIGHT (with compensation and enhancement)
These interfering variations due to periodicity are selectively
eliminated by the F0-adaptive smoothing in STRAIGHT.
Smearing caused by the over-smoothing mentioned before iscompensated by the digital ﬁlter in the frequency domain. This
digital ﬁlter is also utilized to enhance spectral details for improving perceptual quality of the manipulated and resynthesized
speech. Figure 4 shows STRAIGHT spectrum and its wavelet
analysis results.
Whole procedures are approximately implemented as liftering in the cepstrum domain [22]. The following equation
represents the whole process.
PT ST (ω) = exp (F [gA (τ )gC (τ )CT (τ )]) ,

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Figure 3: TANDEM spectrum (upper plot) and its wavelet
analysis result (lower plot). A segment around vowel /i/ in a
Japanese utterance /arigatou/ (“thank you” in English) spoken
by a male is used. The horizontal orange blob in the lower plotcorresponds to the harmonic structure (f0 = 145 Hz) and cyan
periodic spikes correspond to the negative periodic dips in theupper plot.

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(6)
(7)

The lifter g2 (τ ) is used in legacy-STRAIGHT implementation.
„«2sin(πf0 τ )g2 (τ ) == F[h2 (ω)],
(8)
πf0 τ
(
“ 0 ˛ ˛” |ω| ≥ ω0h2 (ω) =(9)˛ ˛1otherwise
1 − ˛ ωω0 ˛
ω02.3. LPC from STRAIGHT spectrum(4)

TANDEM spectrum and STRAIGHT spectrum are derived
from the corresponding power spectra and can be converted toautocorrelation coefﬁcients rk , (k = 0, 1, . . . , N ) by inverse
Fourier transform. A Toeplitz matrix R and an autocorrelation
vector r, composed from the autocorrelation coefﬁcients rk are

where CT (τ ) represents the cepstrum calculated from the TANDEM spectrum. The symbol F represents Fourier transform.
The lifter gC (τ ) is the approximate implementation of the compensation digital ﬁlter, which is initially designed based on the36

used to estimate the LPC coefﬁcients an , (n = 1, 2, . . . , p),
where p is the assumed number of poles. Let a represent theprediction coefﬁcient vector consisting of ak . The following
equation yields the least square solution and is the maximum
likelihood estimate of the underlying autoregressive process,when the process is excited by a Gaussian white noise.
a = R−1 r.

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This Gaussian assumption does not hold for voiced speech
and the estimates may consist of bias due to model mismatch.
However, this provides the best available estimate and the function form of the transfer function around the spectral peaks
agrees with the physically relevant model (although parametervalues are biased). The next step is designed to use this physically correct shape to compensate for smearing caused by oversmoothing introduced by the time windowing and periodicity
suppression of STRAIGHT procedure.
The power spectral representation of LPC spectral envelope
PA (ω) is deﬁned by the following equation.
=

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where ωs = 2πfs represents the sampling angular frequency.
Let name PA (ω) the LPC spectrum in Fig. 2.

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Figure 5: The extended STRAIGHT spectrum (read thick line)
is overlaid on STRAIGHT spectrum (blue thin line) and TANDEM spectrum (green thin line) (upper plot) and wavelet analysis result of extended STRAIGHT spectrum is also shown(lower plot). Sharp peaks are recovered in the upper plot andsharp cyan peaks represent recovered band-unlimited components.

2.4. Spectral correction
Applying spectral smoothing using the frequency domain representation of time windowing function followed by cepstral liftering deﬁned by Eq. 4 yields smeared version of the LPC spectrum PSim (ω). The correction spectrum is deﬁned by the ratioof the LPC spectrum and the smeared LPC spectrum. The ﬁnal extended STRAIGHT spectrum PE (ω) is calculated by thefollowing equation.
PA (ω)PT ST (ω).
PSim (ω)

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˛1 − Pp an exp(−j2πω/ωs )˛2n=1

PE (ω) =500

0repetition interval on f−axis (Hz)PA (ω)−50
0method [36] to calculate less biased autocorrelation is an interesting possibility.
One of the authors proposed to use statistical approach forrecovering smeared spectral peaks by making use of distributed
information from the other parts of utterances [37]. Since thecurrent proposed method is frame-based, it is interesting to integrate with such statistical approaches. These are interesting
issues for further study.
It is important to note that the actual speech production
process involves nonlinear interactions between vocal fold vibration and resulted acoustic impedance variations within one
pitch period [38]. These detailed effects should be tested subjectively, since temporal aspects of masking also signiﬁcantly
affects perceived SNR and timbre [39].

(12)

The coefﬁcient PA (ω)/PSim (ω) is the correction spectrum in
Fig. 2. As shown in Fig. 5, the difference between STRAIGHT
spectrum and the extended STRAIGHT spectrum is visible onlyaround formant peaks. The modiﬁcation introduced by the correction spectrum is highly localized. This localized correction
enables to combine merit of non-parametric spectrum model(STRAIGHT) and parametric spectrum model (LPC spectrum)
without introducing wrong side-effects. Preliminary listening
tests of the proposed method indicated improvement in perceptual quality and intelligibility of the manipulated speech. Manipulated speech samples are linked to our web page [30].

4. Conclusion
A new spectral envelope recovery procedure is introduced as a
hybrid of STRAIGHT and LPC to overcome band-limitation
imposed by spectral sampling due to periodic excitation invoiced sounds. Parameter optimization based on objective as
well as subjective tests are underway.

3. Discussion
LPC coefﬁcients generally suffer from estimation bias due toperiodic excitation of voiced sounds [31]. The proposed spectral correction also suffers from this bias. Less biased methods [31, 32, 33, 34] should be tested for the LPC spectrum calculation process in the proposed method. The proposed procedure uses slightly excessive number of coefﬁcients based on
preliminary tests. An automatic determination algorithm shouldbe implemented. Since, various representations of LPC relatedparameter are mutually converted uniquely [35], they are essentially identical to auto-correlation parameter (excluding covariance method). In this respect, applying generalized cepstrum

5. Acknowledgements
The authors appreciate Keisuke Saito of our group for testing
preliminary implementation of our proposal. They also appreciate Tomoyasu Nakano and Masataka Goto for their inspiring
discussions and proposals on spectral estimation [40]. Noboru
Morimoto made the authors revisit wavelet analysis. This work
is supported by the Aid for Scientiﬁc Research No.24300073,
23700221 and 24650085 from JSPS.

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