INTERSPEECH 2016
September 8–12, 2016, San Francisco, USA

Majorisation-minimisation based optimisation of the composite autoregressive
system with application to glottal inverse filtering
Lauri Juvela1,2 , Hirokazu Kameoka2,3 , Manu Airaksinen1 , Junichi Yamagishi4 , Paavo Alku1
1

Aalto University, Department of Signal Processing and Acoustics, Finland
2
University of Tokyo, Japan
3
Nippon Telegraph and Telephone Corporation, Japan
4
National Institute of Informatics, Japan{lauri.juvela,manu.airaksinen,paavo.alku}@aalto.fikameoka.hirokazu@lab.ntt.co.jp, jyamagis@nii.ac.jp

Abstract

method resulted in significant improvement for female voice in[7]. However, the pre-requisite for all such modelling is reliable
glottal inverse filtering, and since QCP requires accurate glottal
closure instant (GCI) estimates, its performance is likely to degrade with a breathy voice or noisy speech. A review on other
GIF methods and glottal excitation parameterisations used formethod evaluation is given in [8].

The composite autoregressive system can be used to estimate a speech source-filter decomposition in a rigorous manner, thus having potential use in glottal inverse filtering. By
introducing a suitable prior, spectral tilt can be introduced
into the source component estimation to better correspond tohuman voice production. However, the current expectationmaximisation based composite autoregressive model optimisation leaves room for improvement in terms of speed. Inspiredby majorisation-minimisation techniques used for nonnegative
matrix factorisation, this work derives new update rules for themodel, resulting in faster convergence compared to the original
approach. Additionally, we present a new glottal inverse filtering method based on the composite autoregressive system andcompare it with inverse filtering methods currently used in glottal excitation modelling for parametric speech synthesis. These
initial results show that the proposed method performs comparatively well, sometimes outperforming the reference methods.
Index Terms: Composite autoregressive system, majorisationminimisation, glottal inverse filtering

The Composite Autoregressive system (CAR) [9] provides
a convenient statistical approach for estimating the source-filter
decomposition from speech by modelling the signal as a linear combination of various source-filter pair templates that are
estimated from the data. However, the current expectationmaximisation optimisation method is time-consuming and canbe improved upon. Optimising the CAR model is equivalent
to minimising the reconstruction error of the autoregressive
source-filter model, i.e., the Itakura-Saito divergence [10]. This,
with the structure of the CAR system, connects the model tonon-negative matrix factorisation (NMF) with I-S divergence[11], further motivating us to apply a majorisation-minimisation
based approach similar to I-S NMF presented in [12, 13].
Additionally, this paper proposes a glottal inverse filtering
technique based on the CAR system. The conventional frameby-frame analysis sometimes suffers from noisy data or thesparse harmonics in high-pitched voices. As the vocal tract remains relatively stationary over several frames, we hypothesisethat optimisation over a longer time period gives benefits overthe conventional approach. To add flexibility to the CAR optimisation process, we introduce a generalised Gamma distribution prior into the model, which induces sparsity in the representation and enables adding frequency dependent prior information for the source model. We set the source spectral priors
using the Liljencrants-Fant (LF) [14] glottal model in order toembed spectral tilt in the estimated source templates. The resulting inverse filtering method is evaluated by using synthetic
speech data and calculating various objective error measures
based on glottal flow parameterisations [8]. These measures arethen compared with those of the established GIF methods IAIF
and QCP currently used for in glottal vocoding.

1. Introduction
The source-filter model for speech production states that aspeech signal can be decomposed to a glottal source excitation
and a vocal tract filter. However, the solution for the decomposition problem is not unique and evaluation of different solutions is difficult since glottal volume velocity ground truth is
not available without intrusive measurements. Various glottal
inverse filtering (GIF) techniques use some specific knowledge
from the speech production theory to aid in the estimation of
the source-filter decomposition. Techniques such as the iterative adaptive inverse filtering (IAIF) [1] model the spectral tilt
present in the voice source with a low order all-pole filter andestimate it in cascade with the vocal tract filter. Other popular
type of techniques, such as closed phase covariance analysis [2]and quasi-closed phase (QCP) [3] analysis, use weighted linear
prediction to focus on the glottal closed phased of the speech
cycle, attenuating the source contribution in the spectral estimate.
Glottal inverse filtering has applications in excitation modelling for parametric speech synthesis, as originally proposedin [4] using IAIF. Recently, the overall quality and flexibility
for varying phonation types in synthesis has been improved bymodelling the glottal excitation waveforms with Deep Neural
Networks [5, 6]. Moreover, use of the more advanced QCPCopyright © 2016 ISCA

The paper is structured as follows: in the theoretical section 2, we overview the composite autoregressive system, deriving new update rules (section 2.1) and a glottal inverse filtering method (section 2.2), and finally explaining the role ofpriors (section 2.3). In the experimental section, we define ourtest signals (section 3.1), examine the convergence rate of the
proposed optimisation algorithm (section 3.2) and compare theproposed GIF method with established methods (section 3.3).

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http://dx.doi.org/10.21437/Interspeech.2016-735

form update rules satisfying the non-negativity requirement for
activations Uni,j and source spectral templates Fki . Minimising
the I-S divergence DIS directly is difficult, but local optima canbe found iteratively using the auxiliary function approach: first
construct an upper bound auxiliary function GIS , where the majorising function is equal to DIS at exactly one point and can be
minimised in closed form. After this, minimise the auxiliary
function and update the model parameters, which is guaranteed
to decrease the original objective function DIS for the next iteration.
The generalised Gamma distribution (with parameters
η, d, p) [15] provides a flexible way to apply various desirable
properties to the model, as described in section 2.3. Here weshow the derivation for the activations Uni,j and note that thesource spectrum templates Fki can be derived in a similar manner. The generalised Gamma prior for U is given byUpp(U ) ∝ U d−1 exp −
,
(6)
η
Up
+ const.
(7)
log(p(U )) = (d − 1) log(U ) −ηFigure 1: Figure illustrating the concept of the composite autoregressive system: signal spectrogram is modelled as a linear
combination of I × J source-filter pairs compounded from I
source and J filter templates.

2. Composite autoregressive system

Since U p is concave at p < 1, an upper bound is given by thetangent at point V , where V is an auxiliary variable.

The composite autoregressive system models the signal as aweighted sum of source–filter pair spectra compounded fromI source and J filter templates. Each of these components attime instant n has the distribution Xni,j ∼ NC (0, Λi,jn ), where
i,j
i,j
Λi,j
n = diag(λ1,n , · · · , λK,n ) and
λi,j
k,n =

Uni,j Fki
2j
|A (ej2πk/K )|= Uni,j Fki Hkj ,Aj (z) = 1 − aj1 z −1 − · · · − ajP z −P ,U p ≤ pV p−1 (U − V ) + V p + const.
Then the prior log-probability is bounded up to constant bylog(p(U )) ≤ (d − 1) log(U ) −

(1)
(2)

i,j

X

Λi,j
n = diag(φ1,n , . . . , φK,n ),

1 p−1
1
pV
(U − V ) + V pηη(9)Disregarding the constant terms, the posterior upper boundfunction is given by subtracting the prior upper bound from thelikelihood-based GIS presented in [12, 13]:
"
#
i,j 2
X X Yk,n (ξk,n
X Fki Uni,j H j
)
k
GIS =
+
αk,n
Fki Uni,j Hkj
i,j
i,j
k,n

X
−
(d − 1) log(Uni,j )where Fki corresponds to the vocal source spectrum at k:th frequency bin, Hkj = 1/|Aj (z)|2 is an all-pole filter, and Uni,j
is a non-negative weighting coefficient. The observed complex
spectrogram Yn is then given by the sum of these components
X i,j
Yn =
Xn ∼ N (0, Φn ),
(3)
Φn =

(8)

n,i,j

(4)


1 i,j p
1
i,j p−1i,ji,j(Un − Vn ) + (Vn ) ,− p(Vn )
η
η

i,j

P
where φk,n = i,j λi,jk,n . The model structure is illustrated inFig. 1. The original EM-based optimisation algorithm is presented in [9], while this paper introduces a new majorisationminimisation algorithm presented in section 2.1. By using either optimisation approach, maximising the probability density
function of Yn with respect to φk,n amounts to minimising the
Itakura-Saito divergence, which is given up to constant byX  Yk,n
DIS (Y , Φ) =+ log(φk,n ) .
(5)
φk,n

(10)

i,j
where (αk,n , ξk,n
, Vni,j ) are the auxiliary variables and theequality for the upper bound holds only when
i,j
ξk,n
=

Uni,j Fki Hkj
,
φk,n

(11)

αk,n = φk,n ,(12)Vni,j = Uni,j .

(13)

Differentiating this with respect to Uni,j yields a full second degree equation, from which we choose the solutionpbU + b2U + 4aU cU
Uni,j ←,(14)
2aU
X Fki H jpkaU =+ (Uni,j )p−1 ,
(15)
φk,n
η

k,n

2.1. Auxiliary function approach
In this section we derive a majorisation-minimisation optimisation algorithm for the composite autoregressive system using
the auxiliary function approach. This is done by minimising the
Itakura-Saito divergence between the signal and model spectrograms. The procedure is similar to deriving the multiplicative
update rules for NMF as done in [12, 13]. In addition, we apply
a generalised Gamma prior to the activations and source spectral components to better control the behaviour of these partsof the model. Under certain restrictions this results in closedkbU = K(d − 1),
X Yk,n Fki H j (Uni,j )2
k
cU =
,
φ2k,n
k

969

(16)
(17)

bH = N J(d − 1),
X Yk,n Uni,j H j (Fki )2
k
cH =
,
φ2k,n
n,j

Power (dB)

where positivity is guaranteed by constraining d ≥ 1. Similarly,
we obtain the update rule for FkipbH + b2H + 4aH cH
i
,
(18)
Fk ←
2aH
X Uni,j H jpk+(Fki )p−1 ,
(19)
aH =
φ
η
k,n
k,i
n,j
(20)
(21)

Frequency

where ηk,i is frequency dependent to accommodate for the LF
prior. Here also p and d are chosen differently from Uni,j prioras the priors have different desirable properties. Using a uniform prior simplifies the update rule to one resembling the multiplicative NMF update rule [12, 13] with IS-divergence:
v
uP
i,j j
Yk,n Un Hk
u
u n,j
2
φ
k,n
Fki ← Fki u
.
(22)
t P
i,j j
Un Hk
n,j

Figure 2: Examples of LF pulse spectra used for the priors of
Fki . Using different LF priors for the individual source components allows varying degrees of spectral tilt.
2.3. Setting the prior parameters
The generalised Gamma parameters can be chosen to achievedifferent goals for the activations and source templates. For theactivations Uni,j , the main requirement for the prior is to induce
sparsity. The power parameter p can be set anywhere between
zero and one, with smaller values providing more sparsity andp = 1 corresponding to the inverse Gamma prior used in [9]. In
this work we used p = 0.7.
For the source templates Fki , the desired properties for theprior are slightly different. First, to encourage spectral tilt, we
set ηki , which is proportional to the mean of the distribution, tocorrespond to a LF spectral prior in the update equations (18)and (19). The spectra were created by sampling the LF parameter space evenly and using the expression for LF spectral envelope given in [16]. Examples of such spectra are shown in
Fig. 2. Second, to avoid potential numerical problems occurring with very small values of Fki , we set d slightly above one,since the d-dependent term in Eq. (10) effectively acts as a logarithmic barrier function, which drives the error towards infinity
as Fki tends to zero. This type of barrier function has been used
explicitly for positive matrix factorisation in [17].

φk,nFinally for the filter templates, we follow a similar procedureand enforce the all-pole constraint by solving the filter coefficients from the normal equations
 j
 j   j
r0
r1j
. . . rPj −1
a1
r1
j
j
j
j
j
 r1



r
r
a
0
P −2   2  r2 
=
(23)
 .




.
. ,
..
..
 ..
.
.   ..   .. rPj
ajP
rPj −1 rPj −2 . . .
r0j
where
 v
u P Y U i,j F i 

k,n n
k 


 u

u n,i
φ2
k,n
j
−1
ju
.
r1,...,P = DFT
Hk t P
i,j i
Un Fk






n,i φk,n(24)

Despite the filter convergence is not directly guaranteedwith this update rule, we observe convergence in practice at afaster rate than with the original EM algorithm (see Fig. 3).

3. Experiments
3.1. Test signals

2.2. Inverse filtering based on the CAR model

For evaluating glottal inverse filtering methods, a test signal
with the ground truth for source and filter should be available,
effectively forcing the use of synthetic data. Instead of using theconventional fixed filters and fundamental frequency (f0 ) for
test data, we attempt to create more realistic test signals by using simple analysis-synthesis framework: first extract the LPC
spectral envelope and f0 from real continuous speech, then use
the pitch information to create an LF model based excitation
signal, and finally combine these to synthesise the test signals.
To achieve this, we used a modified version of the GlottHMM
vocoder [4], where the GIF was replaced with plain LPC analysis, and the glottal excitation pulses were created using the antialiased LF model presented in [18]. This excitation signal was
considered as the ground truth for the vocal source in the GIF
evaluation. The speech database used for the experiment consists of Finnish sustained vowel utterances in neutral, breathy,
and pressed phonation styles, totalling 9.5 minutes in 792 utterances. In analysis-synthesis, the LF parameters were chosenfrom three sets of parameters in accordance with the phonationstyle, using parameters from the software presented in [18]. A
noise component was added to the voiced excitation in based on

The expected value of individual model components is given byhiλi,jUni,j Fki Hkjk,ni,j
E Ŷk,n= Yk,n= Yk,n
,
φk,n
φk,n

(25)

i,j
where Yk,n ∈ C. The source components Ŝk,ncan be estimatedby removing the all-pole filter contribution Hkj and the source
estimate is the obtained by summing the source components:

Ŝk,n =

X
i,j

i,j
Ŝk,n
=

X
i,j

Yk,n

Uni,j Fki
.
φk,n

(26)

The reference methods operate in a framework where thesource estimate is obtained in time domain by inverting the estimated vocal tract filter. In order to conform to this, the obtained
source spectrogram is cancelled from the signal spectrogram toobtain vocal tract spectrum estimates at each frame, and finallyall-pole filters are fit onto these before inverse filtering. Alternatively the complex valued source spectrogram can be converted
directly to time domain with inverse short-time Fourier transform.

970

100.8

EM
MM
MM-LF

100.7

Table 1: Objective error measures for GIF: MSE and H1H1
are given as average absolute errors and the rest as averagerelative errors. Smaller error measures indicate better performance. I source templates and J filter templates were used for
CAR, and a total number of N voiced frames contribute to theerror measure.

Itakura-Saito divergence

100.6
100.5
100.4
100.3
100.2
100.1

Neutral phonation (I = 5, J = 3, N = 26593)
MSE
H1H2 HRF NAQ QOQ
IAIF
6.31e-04
2.05
0.36
0.13
0.18
QCP
7.92e-04
2.03
0.79
0.14
0.28
CAR-MM
8.35e-04
1.76
0.28
0.23
0.24
CAR-MM-LF 8.18e-04
1.74
0.49
0.16
0.26

100
10−0.110−0.2
0

10

20

30

40

50

60 70 80 90 100 110 120 130 140 150
Iteration index

Figure 3: The average Itakura-Saito divergence as a functionof iteration index illustrates the convergence rates of the different optimisation algorithms. Majorisation-minimisation based
methods (MM) and (MM-LF) tend to converge faster than thebaseline expectation maximisation based method (EM).

Breathy phonation (I = 5, J = 3, N = 32281)
MSE
H1H2 HRF NAQ QOQ
IAIF
4.95e-04
4.91
0.39
0.07
0.11
QCP
9.69e-04
2.44
0.71
0.17
0.24
CAR-MM
4.82e-04
3.36
0.37
0.07
0.10
CAR-MM-LF 5.36e-04
4.06
0.43
0.07
0.12

the harmonic-to-noise ratio similarly to [4].

Pressed phonation (I = 5, J = 3, N = 26774)
MSE
H1H2 HRF NAQ QOQ
IAIF
9.27e-04
1.75
0.72
0.14
0.20
QCP
8.51e-04
2.03
1.23
0.20
0.30
CAR-MM
8.26e-04
1.68
0.47
0.18
0.24
CAR-MM-LF
8.18e-04
1.74
0.49
0.16
0.26

3.2. Comparison of optimisation methods
The convergence rates of three CAR system optimisation methods were examined briefly: first, the original EM-based method(EM) [9], second, a MM-based method with uniform source
prior (MM) using Eq. (22), and third, a MM-based method with
LF based source prior (MM-LF). Since optimising the CAR
system amounts to minimising the I-S divergence, we examine
the divergence as a function of the iteration index to compare theconvergence rates. For this experiment, we used the sustained
vowel data described in section 3.1. The time-normalised average I-S divergence curves are presented in Fig. 3. Both MM and
MM-LF converge faster than the original EM-based method,with MM reaching slightly lower error at convergence.

perform reasonably well for neutral and pressed phonation, andtend to outperform the reference methods for breathy phonation.

4. Discussion and Conclusion
This paper applied the composite autoregressive system to glottal inverse filtering analysis of speech production. The optimisation process was improved with an auxiliary function -based
method for the composite autoregressive system, leading to afaster convergence rate. Generalised gamma prior was added
to the optimisation to add sparsity to the component activations
and spectral tilt to the source spectral templates.
The proposed glottal inverse filtering technique based on
the CAR system was studied experimentally with synthetic data
derived from natural sustained vowel utterances using the LF
model for glottal excitation. The method was compared withexisting inverse filtering techniques IAIF and QCP using objective error measures based on glottal parameterisations. For
breathy phonation style, the proposed method tends to outperform the reference methods, while also performing reasonably
well for other test data. These initial experiments show promise
for application in glottal vocoding, but the proposed method is
not as such applicable to data with variable linguistic content.
Instead of examining the full utterance at once, the proposed
method should be modified to work on relatively stationary segments of the signal and compound these results afterwards.
Future work includes tuning the model parameters and finding the optimal amount of source and filter templates to be
used for inverse filtering. The proposed inverse filtering methodshould also be refined by decoupling the all-pole filter gains
from the activations. Additionally, the source prior used for
CAR-MM-LF in this work is rather soft and did not produce
much difference to CAR-MM. Setting sharp harmonic source
priors instead to enforce source spectral tilt should be studied.

3.3. Glottal inverse filtering
The performance of the proposed GIF method was comparedwith the established methods IAIF [1] and QCP [3]. The objective evaluation of the GIF was carried out by comparing
the errors in the following glottal flow parameterisations: themean squared error (MSE) between the ground truth and estimated glottal flow derivatives; H1H2 [19], which measures thedifference between the first and second harmonics in dB; theHarmonic Richness Factor (HRF) [20], which characterises theamount spectral tilt; the Normalised Amplitude Quotient (NAQ)[21], which describes the relative length of the glottal closing
phase; and Quasi-Open Quotient (QOQ) [22], measuring therelative length of the glottal open phase. Glottal inverse filtering was performed for the test sets described in section 3.1 andthe waveforms were normalised by first scaling the glottal flow
derivative to match ground truth energy and then shifting the
integrated glottal flow minimum to zero. After this, the error
measures were evaluated in voiced frames.
The reference methods were used with their default settingsfor 16 kHz sample rate, and SEDREAMS [23] GCI detection
was used for QCP. The CAR optimisation was run for 150 iterations for both proposed inverse filtering methods: CAR-MM
without source prior, and CAR-MM-LF with LF-based prior.
Table 1 presents the average errors, grouped by the phonationstyle classes: neutral (top), breathy (middle), and pressed (bottom). The presented error measures are mean absolute error for
MSE and H1H2 (dB), and mean relative absolute error for HRF,
NAQ and QOQ, with smaller error measures signifying better
performance. The results show that the proposed methods both

5. Acknowledgements
This work was supported by the Academy of Finland (proj.
no. 256961 and 284671), and the European Union TEAMMUNDUS scholarship (TEAM1400081).

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