STATISTICAL F0 PREDICTION FOR ELECTROLARYNGEAL SPEECH ENHANCEMENTCONSIDERING GENERATIVE PROCESS OF F0 CONTOURS WITHIN PRODUCT OF
EXPERTS FRAMEWORKKou Tanaka1 , Hirokazu Kameoka2 , Tomoki Toda3 , and Satoshi Nakamura1
1

Graduate School of Information Science, Nara Institute of Science and Technology, Japan
2
NTT Communication Science Laboratories, NTT Corporation, Japan
3
Information Technology Center, Nagoya University, Japan{ko-t, s-nakamura}@is.naist.jp, kameoka.hirokazu@lab.ntt.co.jp, tomoki@icts.nagoya-u.ac.jp

ABSTRACT
We have previously proposed a statistical fundamental frequency (F0 ) prediction method that makes it possible topredict the underlying F0 contour of electrolaryngeal (EL)speech from its spectral feature sequence. Although thismethod was shown to contribute to improving the naturalness
of EL speech as a whole, the predicted F0 contour was still
unnatural compared with that in normal speech. One possible solution to improve the naturalness of the predicted F0
contours would be to take account of the physical mechanismof vocal phonation. Recently a statistical model of voice F0
contours was formulated by constructing a stochastic counterpart of the Fujisaki model, a well-founded mathematicalmodel representing the control mechanism of vocal fold vibration. This paper proposes a Product-of-Experts model toincorporate this generative model of voice F0 contours into
the statistical F0 prediction model. Based on the constructed
model, we derive algorithms for parameter training and F0
prediction. Experimental results revealed that the proposed
method successfully outperformed our previously proposed
method in terms of the naturalness of the predicted F0 contours.
Index Terms— Electrolaryngeal speech enhancement,
F0 prediction, Generative model, Product of Experts
1. INTRODUCTION
Speech is one of the most common tools in human communication. Since speech is produced by the vocal apparatus,
the produced sounds are physically constrained by the conditions of human body. Unfortunately, there are many people with disabilities that prevent them from producing speech
freely, leading to communication barriers. Those who are unable to produce speech freely involve laryngectomees, whohave undergone an operation to remove the larynx including
the vocal folds for such reasons as injury or laryngeal cancer. The ability by these people to generate excitation sounds
is severely impaired because they no longer have their vocal
folds. One alternative means of producing voice for these patients involves the use of electrolaryngeal (EL) speech, which
is produced by using the excitation signals mechanically generated from an electrolarynx. EL speech is reasonably intelligible, but somewhat unnatural particularly due to the mechanical sounding of the excitation signals.
To address this problem, we have previously proposed
methods that aim to convert EL speech to normal-sounding
speech, by predicting the fundamental frequency (F0 ) contour from the spectrum sequence of the EL speech based on
This work was supported in part of JSPS KAKENHI 26280060.

978-1-4799-9988-0/16/$31.00 ©2016 IEEE

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Gaussian Mixture Models (GMMs) followed by synthesizing
the speech waveforms according to the predicted acoustic parameters [1–3]. These methods were shown to contribute toimproving the naturalness of EL speech [1, 2] and also preserving its intelligibility [3]. However, the F0 contours predicted using these methods still sounded unnatural comparedwith that in normal speech. This was because the predicted F0
contours were not necessarily guaranteed to satisfy the physical constraint of the actual control mechanism of the thyroid
cartilage, even though they were optimal in a statistical sense.
In this regard, these methods still had a plenty of room for
improvement. One possible solution to improve the naturalness of the F0 contours of the converted speech would be toincorporate a generative model of voice F0 contours into thestatistical F0 prediction model to take account of the physicalmechanism of vocal phonation.
One of the authors previously proposed a statistical modelof voice F0 contours [4–6], formulated by constructing astochastic counterpart of the Fujisaki model [7], a wellfounded mathematical model representing the control mechanism of vocal fold vibration. The Fujisaki model [7] assumes
that an F0 contour on a logarithmic scale is the superposition of a phrase component, an accent component and a basevalue. The phrase and accent components are considered to
be associated with mutually independent types of movementof the thyroid cartilage with different degrees of freedom andmuscular reaction times. The model proposed in [4–6] hasmade it possible to estimate the underlying parameters of theFujisaki model that best explain the given F0 contour, byusing powerful statistical inference techniques.
To incorporate the generative F0 contour model into thestatistical F0 prediction framework, this paper proposes a
Product-of-Experts (PoE) model [8] combining the abovementioned two models. Since the PoE model is obtained bymultiplying the densities of different models, it usually becomes complicated due to the renormalization term. To avoid
this, we introduce a latent trajectory model proposed in [9] toreformulate the prediction model so that it can be smoothly
combined with the generative F0 contour model.
2. GMM-BASED STATISTICAL F0 PREDICTION
We briefly review our statistical F0 prediction method [1–3],
which exploits the idea of statistical voice conversion techniques [10, 11]. The aim of this method is to predict F0 contours from the spectral parameters of EL speech. As withvoice conversion methods, it consists of training and prediction processes.
In the training process, the parameters λG of the joint
probability density p((x[k]T , o[k]T )T |λG ) described as a
Gaussian mixture model (GMM) are trained, where T de-

ICASSP 2016

Fig. 1. Original Fujisaki model [7].
notes transposition, and x[k] and o[k] denote a source feature
and a target feature at time frame k, respectively. The corresponding joint feature vectors can be obtained by performing
automatic frame alignment with Dynamic Time Warping. As
a source feature, the spectral segment feature of EL speech
is extracted on a frame-by-frame basis from the mel-cepstra
at multiple frames around the current frame k [12]. The target feature o[k] = (y[k], ∆y[k])T consists of the static anddelta (time derivative) components of the log-scaled F0 valuey[k], extracted on a frame-by-frame basis from the target
normal speech. Note that to improve prediction accuracy, weinterpolate unvoiced frames of F0 patterns by using splineinterpolation and remove micro-prosody [13].
In the prediction process, given the spectral segment sequence x = (x[1]T , . . . , x[K]T )T of EL speech, the most
likely F0 sequence y = (y[1], . . . , y[K])T can be obtained asfollows:ŷ = argmax p(o|x, λG ) subject to o = W y ,
(1)
y

T

T T

where o = (o[1] , . . . , o[K] ) denotes the joint static anddynamic feature vector sequence, W is a constant matrix thattransforms the static feature vector sequence y to o. Namely,
each row of W consists of the coefficients of an identity mapping operator or time differential operator. p(o|x, λG ) is the
GMM with the trained parameters, which we approximate asp(o|x, λG )=∑p(o|x, m, λG )p(m|x, λG )m

≃ p(o|x, m̂, λG )p(m̂|x, λG ),(2)with m̂ = argmaxm p(m|x, λG ). Here, m = (m1 , . . . , mK )indicates a sequence of mixture indices. p(o|x, m, λG ) is(o|x)
given as the product of p(o[k]|x[k], mk , λG ) = N (o[k]; emk ,
(o|x)
(o|x)
(o|x)
D mk ) over k where emk and D mk are the mean vector
and covariance matrix of mk -th mixture component, respectively. Thanks to this approximation, the solution to Eq. (1)is given explicitly as follows:−1−1ŷ = (W T D (o|x) W )−1 W T D (o|x)m̂ em̂ ,(o|x)

where em̂
(o|x)

(o|x)

(3)
(o|x)

is a stacked vector of the mean vectors em̂1 , . . . ,(o|x)

em̂K and D m̂is a block diagonal matrix where each block

(o|x)
(o|x)
is D m̂1 , . . . , D m̂K .

3. GENERATIVE MODEL OF VOICE F0 CONTOURS
The generative model of F0 contours proposed in [4–6] is astochastic counterpart of a discrete-time version of the Fujisaki model [7].
The Fujisaki model (shown in Fig. 1) assumes that a logscaled F0 contour y(t) is the superposition of a phrase component yp (t), an accent component ya (t) and a base valueµb . The phrase and accent components are assumed to be
the outputs of different second-order critically damped filters,
excited with Dirac deltas up (t) (phrase commands) and rectangular pulses ua (t) (accent commands), respectively. Here,

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Fig. 2. Command function modeling with HMM.
it must be noted that the phrase and accent commands do not
usually overlap each other. The base value is a constant value
related to the lower bound of the speaker’s F0 , below which
no regular vocal fold vibration can be maintained. The log F0contour, y(t), is thus expressed asy(t) = yp (t) + ya (t) + µb ,
(4)
where
yp (t) = gp (t) ∗ up (t),(5)
ya (t) = ga (t) ∗ ua (t).
(6)
Here, ∗ denotes convolution over time. gp (t) and ga (t) are
the impulse responses of the two second-order systems, which
are known to be almost constant within an utterance as wellas across utterances for a particular speaker.
A key idea in the proposed model [4–6] is that the sequence of the phrase and accent command pair (i.e., the underlying parameters of the Fujisaki model) is modeled as apath-restricted hidden Markov model (HMM) with Gaussian
emission densities (shown in Fig. 2) so that estimating thestate transition of the HMM directly amounts to estimating
the Fujisaki-model parameters.
We hereafter use k to indicate the discrete time index.
Given a state sequence s = (s1 , . . . , sK ) of the above HMM,
the conditional distributions of the phrase command sequence
up = (up [1], . . . , up [K])T and the accent command sequence ua = (ua [1], . . . , ua [K])T are given asp(up |s, λF ) = N (up ; µp , Σp ),(7)
p(ua |s, λF ) = N (ua ; µa , Σa ),(8)
respectively, where λF denotes the parameters of the HMM.
µp and µa denote the mean sequences of the state emission
densities and Σp and Σa are diagonal matrices whose diagonal elements correspond to the variances of the state emission densities. From Eqs. (5) and (6), the relationships between y p = (yp [1], . . . , yp [K])T and up and between y a =(ya [1], . . . , ya [K])T and ua can be written as
Gp up = y p ,(9)
Ga ua = y a ,(10)where Gp and Ga are Toeplitz matrices where each row is a
shifted copy of the convolution kernels gp [1], . . . , gp [K] andga [1], . . . , ga [K]. By using ub to denote the baseline component, the log F0 sequence y is given as y = y p +y a +ub +n
where n is an additive noise component corresponding to micro prosody. If we assume that n follows a Gaussian distribution with mean 0 and covariance Γ, the conditional distribuT
T T
tion of y given u = (uT
p , ua , ub ) is defined asp(y|u) = N (y; Gp up + Ga ua + ub , Γ).

(11)

We further assume that ub follows a Gaussian distribution
with mean µb 1 and covariance Σb . Then, from Eqs. (7), (8)and (11), the conditional distribution of y given s is given asp(y|s, ΛF )=∫p(y|u)p(u|s, λF )du= N (y; µF , ΣF ),(12)

where µF = Gp µp + Ga µa + µb 1 and ΣF = Gp Σp GT
p +Ga Σa GT
+
Σ
+
Γ.
b
a
4. PROPOSED MODEL4.1. Product-of-Experts Strategy

probabilities given a state sequence s. We consider constructing a PoE model by combining Eqs. (15) and (16) followed by marginalization, rather than by simply combining
the marginal distributions p(y, x|λG ) and p(y|λF ), whichmakes the parameter training and F0 prediction problems
excessively hard. To do so, we first combine the densitiesof p(y|o) and p(y|u) to obtain p(y|o, u). Since both ofthese distributions are Gaussians, the product of their distributions can be easily obtained by completing the square of
the exponentp(y|o, u)∝ N (y; Ho, V ) · N (y; Gu, Γ)= N (y; µy|o,u , Σy|o,u ),PoE [8] is a general technique to model a complicated distribution of data by combining relatively simpler distributions(experts). Since the distribution is obtained by multiplying
the densities of the experts, the way the experts are combined
is somewhat similar to an “and” operation. In this section, weconstruct a PoE model by treating the two models introducedin the previous sections as the experts.
Training a PoE model by maximizing the likelihood of thedata usually becomes difficult since it is hard even to approximate the derivatives of the renormalization term. By contrast,
we propose an elegant formulation that allows the use of the
EM algorithm for both parameter training and F0 prediction.
To do so, we first introduce a latent trajectory model proposedin [9] to reformulate the GMM-based statistical F0 prediction
model, which plays a key role in making this possible.
4.2. Latent-Trajectory-GMM-Based F0 Prediction
We reformulate the GMM-based statistical F0 prediction
model presented in Sec. 2 by employing the idea proposedin [9]. Instead of treating o as a function of y, we treat oas a latent variable to be marginalized out, that is related
to y through a soft constraint o ≃ W y. The relationshipo ≃ W y can be expressed through the conditional distribution p(y|o){}
p(y|o)∝ exp − 12 (W y − o)T Λ(W y − o)(13)
= N (y; Ho, V ),(14)
where H = (W T ΛW )−1 W T Λ and V = (W T ΛW )−1 .
Λ is a constant positive definite matrix that can be set arbitrarily. As with Sec. 2, the joint distribution p(x, o|λG ) ismodeled as a GMM. Namely, given mixture indices m, theconditional distribution p(x, o|m, ΛG ) is defined as a Gaussian distribution. Thus, the joint distribution of y, x, o, andm can be described using the distributions defined abovep(y, x, o, m|λG ) = p(y|o)p(x, o|m, λG )p(m|λG ),(15)
where p(m|λG ) is the product of mixture weights of the
GMM. By marginalizing o and m out, we can readily obtainthe joint distribution p(y, x|λG ), which can be used as acriterion to train λG and predict optimal y in a consistentmanner, unlike the method presented in Sec. 2. We use thismodel to construct our PoE model in the next subsection.

µy|o,u = (VΣy|o,u = (V

−1
−1

−1 −1

+Γ

)

−1 −1

+Γ

)

(V

−1

(17)
−1

Ho + Γ

Gu),

(18)
(19)

,

T
T T
where G = [Gp Ga I] and u = (uT
p , ua , ub ) .

From Eqs.
(15), (16) and (17), the joint distribution of y, x, o, u, m ands can be constructed asp(y, x, o, u, m, s|λG , λF )(20)
= p(y|o, u)p(x, o|m, λG )p(u|s, λF ) p(m|λG )p(s|λF ).
|{z}p(y,x,o,u|m,s,λG ,λF )This can be used as the complete data likelihood for parameter
training and F0 prediction as explained later. By marginalizing o and u out, we can readily obtain the joint distribution p(y, x, m, s|λG , λF ), which can be used as a criterion
to train λG and λF and predict optimal y in a consistent manner.
Since both p(x, o|m, λG ) and p(u|s, λF ) are Gaussians,
let us write them as ()
p(x, o|m, λG )= N

[ ] [ ] []x
µx
P xx P xo −1
;
,
,
o
µo
P ox P oop(u|s, λF )= N (u; µu , P −1
u ).

(21)
(22)

Then, from Eqs. (17), (21) and (22), it can be shown thatp(y, x, o, u|m, s, λG , λF ) is given asp(y, x, o, u|m, s, λG , λF ) y 
[] [x
A b +A bA= N  o ; A11 b1 + A12 b2 , A11
21
21 1
22 2
u


]
A12 
,
A22

(23)

where

−1
−1
O −V −1 H −Γ−1 G[]  Σy|o,u
A11 A12O
P xx −P xoO , (24)
A21 A22 =  −H T V −T −P ox P ooO−GT Γ−T

O

O


O
[ ]
b1
P
µ
−
P
µ
xo o 
 xx xb2 = P oo µo − P ox µx ,P u µu

Pu(25)

by completing the square of the exponent. Note that A11 ,A12 , A21 and A22 can be written explicitly using the blockwise inversion formula.
4.4. Parameter Training and F0 Prediction

4.3. Deriving PoE
In the same way as Eq. (15), we write the model presented inSec. 3 in the form of a joint distributionp(y, u, s|λF ) = p(y|u)p(u|s, λF )p(s|λF ),(16)
where p(u|s, λF ) is given as the product of state emission
densities and p(s|λF ) the product of the state transition

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The problems of parameter training and F0 prediction can beformulated as the following optimization problems:{λ̂G , λ̂F , m̂, ŝ} = argmax log p(ỹ, x̃, m, s|λG , λF ),(26)

λG ,λF ,m,s{ŷ, m̂, ŝ} = argmax log p(y, x̃, m, s|λ̂G , λ̂F ).
y,m,s

(27)

+ log p(m|λ̂G ) + log p(s|λ̂F ),(28)

where θ = {y, m, s}. From Eq. (23), we obtain][[ ] [ ]
o
y′
′
′
,
m
,
q
= A21 b1 + A22 b2 +
E u
x̃
([ ]
)
[ ]
ō
y′
A21 A−111x̃ − A11 b1 − A12 b2 =: ū , (29)E][[ ] [ ]T [ ][ ] [ ]T
ō ō
o o
y′
′
′
,
m
,
q
= A22 − A21 A−1
11 A12 + ūū , (30)u ux̃which are the values to be computed at the “E-step” by substituting θ into θ′ . At the “M-step”, we compute{y, m, s} ← argmax Q(θ, θ′ ).

(31)

y,m,s

The update equations are omitted owing to space limitations.
5. EXPERIMENTAL EVALUATION
5.1. Experimental Conditions

Correlation
coefficients

0.8
GMM-based
Fujisaki-model-based1
ProposedFujisaki-model-based2

0.7
0.6
0.5
2

4

6

8

10

12

14

16

18

20

Number of iteration times

Fig. 3. F0 correlation coefficients between predicted F0 patterns from EL speech and extracted F0 patterns from normal
speech.
5Confidence interval (95%)
Mean Opinion Score

where ỹ and x̃ denote the observed F0 contour extracted
from normal speech and the observed spectral sequence extracted from non-larynx speech. Both of these problems canbe solved using the EM algorithm by treating o and u as
latent variables. Owing to space limitations, here we only
derive an algorithm for solving Eq. (27).
The likelihood of y, m and s given the complete data{x̃, o, u} is given by Eq. (20). By taking the conditional
expectation of log p(y, x̃, o, u|m, s, λ̂G , λ̂F ) with respect too and u given x̃, y = y ′ , m = m′ and s = s′ and thenadding log p(m|λ̂G )p(s|λ̂F ), we obtain an auxiliary function[]
Q(θ, θ′ ) =Eo,u|x̃,y′ ,m′ ,s′ log p(y, x̃, o, u|m, s, λ̂G , λ̂F )4
3
2
1
GMM-based

Fujisaki-model-based1

Proposed

Fig. 4. Result of opinion test on naturalness.
• GMM-based: Predict F0 contours with the GMMbased method.
• Fujisaki-model-based1: Fit the Fujisaki model to thepredicted F0 contours obtained with the GMM-based
method.
• Proposed: Predict F0 contours with an approximatedversion of the proposed method, in which the E-stepis replaced with the GMM-based and Fujisaki-modelbased2 methods.
• Fujisaki-model-based2: Fit the Fujisaki model to thepredicted F0 contours obtained with Proposed.
5.2. Experimental Results

We conducted objective and subjective evaluation experiments to evaluate the performance of the proposed method.
For the objective evaluation, we evaluated the F0 correlation
coefficients between the predicted and target F0 contours. We
also subjectively evaluated the naturalness of the F0 contourof converted speech.
The source speech was EL speech uttered by one malelaryngectomee, and the target speech was normal speech uttered by a professional female speaker. Each speaker utteredabout 50 sentences in the ATR phonetically balanced sentence
set [15]. We conducted a 5-fold cross validation test in which
40 utterance pairs were used for training, and the remaining
10 utterance pairs were used for evaluation. The sampling frequency was set at 16 kHz. The settings of HMM in the generative F0 contour model were the same as reported in [4–6].
For initialization of the mixture index sequence m andstate sequence s, we performed the conventional GMM-based
and Fujisaki-model-based methods. Note that the Fujisakimodel-based method refers to a post-processing method thatconsists of first applying the GMM-based method and thenfitting the Fujisaki model to the predicted F0 contour using
the method of [5, 6], which is similar to a post-processing
method for HMM-based speech synthesis [16]. Note that inthis experiment, we implemented a simplified and approximated version of the proposed method, in which the E-stepprocedure is replaced with the conventional Fujisaki-modelbased and GMM-based methods. Therefore, the convergenceof the algorithm implemented for the current evaluation is notstrictly guaranteed. The speech used for evaluation were synthesized using STRAIGHT [17] given the mel-cepstrum sequence and F0 contour. The methods selected for comparison
were:

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As Fig. 3 shows, Proposed obtained the highest prediction
accuracy because of eq. (18) meaning an “and” operationfor eq. (11) and (14). Therefore, we found that it is effectiveness to construct a PoE model. Futhermore, since
Fujisaki-model-based2 has higher correration coefficients
than Fujisaki-model-based1, we found that the predicted F0
contours by Proposed were given good influences by considering not only the GMM-based method but also the Fujisaki
model. Note that we used the predicted F0 contours obtainedwith the GMM-based method as the input for the Fujisakimodel-based1 method in this experiment. To make a morefair comparison, it would be necessary to modify the Fujisakimodel-based method so as not to depend on the GMM-based
method. In additional, as for proposed, there is no large difference of correration coefficients in each iteration. To makeexactly evaluation, we have to conduct the PoE model not
replaced E-step with the conventional methods.
As Fig. 4 shows, Proposed outperformed the conventional
methods, GMM-based and Fujisaki-model-based1. This result is reasonable since Proposed obtained the highest prediction accuracy as in Fig. 3.
6. CONCLUSIONSIn this paper, to improve F0 prediction performance in electrolaryngeal speech enhancement, we proposed a Product-ofExperts model that combined two conventional methods, astatistical F0 prediction method and a statistical F0 contour
modeling method based on its generative process. Experimental results revealed that the proposed method successfully
outperformed our previously proposed method in terms of thenaturalness of the predicted F0 contours.

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