PARAMETER GENERATION ALGORITHM CONSIDERING MODULATION SPECTRUM
FOR HMM-BASED SPEECH SYNTHESIS
Shinnosuke Takamichi†,‡ , Tomoki Toda† , Alan W. Black‡ , and Satoshi Nakamura†† Graduate School of Information Science, Nara Institute of Science and Technology (NAIST), Japan‡ Language Technologies Institute, Carnegie Mellon University (CMU), U.S.A
Email: shinnosuke-t@is.naist.jp
ABSTRACTThis paper proposes a novel parameter generation algorithm forhigh-quality speech generation in Hidden Markov Model (HMM)based speech synthesis. One of the biggest issues causing significant
quality degradation is the over-smoothing effect often observed ingenerated parameter trajectories. Global Variance (GV) is known
as a feature well correlated with the over-smoothing effect and ametric on the GV of the generated parameters is effectively used as apenalty term in the conventional parameter generation. However, thequality of the synthetic speech is far from that of the natural speech.
Recently, we have found that a Modulation Spectrum (MS) of thegenerated parameters, which is also regarded as an extension of theGV, is more sensitively correlated with the over-smoothing effectthan the GV. This paper incorporates a metric on the MS as a newpenalty term in the proposed parameter generation algorithm. The
experimental results demonstrate that the proposed parameter generation algorithm considering the MS yields significant improvements
in synthetic speech quality compared to the conventional parameter
generation algorithm considering the GV.
Index Terms— HMM-based speech synthesis, over-smoothing,global variance, modulation spectrum, parameter generation
1. INTRODUCTION
Statistical parametric speech synthesis based on Hidden Markov
Models (HMMs) [1] is an effective framework for generating diverse types of synthetic speech. Speech parameters, i.e., spectral
and excitation features and HMM-state duration are simultaneously
modeled with context-dependent HMMs in a unified framework[2]. In synthesis, the speech parameter trajectories are generated
by maximizing the likelihood of the HMMs [3]. This approachallows us not only to produce smoothly varying speech parameter
trajectories under a small footprint [4] but also to apply several techniques for flexibly controlling synthetic speech [5, 6, 7] to various
speech-based systems [8, 9].
One of the critical problems in HMM-based speech synthesisis that the parameter trajectories generated from the HMMs are often over-smoothed. This phenomenon causes significant degradationof the perceptual quality and makes synthetic speech sound muffled[10]. To address this over-smoothing problem, Toda and Tokuda[11] proposed a parameter generation algorithm considering Global
Variance (GV). The GV is defined as the 2nd order moment of theparameter trajectory. The GV of the generated trajectories is often
smaller than that of natural speech parameter trajectories. They integrated a metric on the GV into the parameter generation algorithm tokeep the GV close to natural one, and they reported that the speech
quality was significantly improved. This conventional method has
been widely studied [12, 13, 14] because of not only its practical

978-1-4673-6997-8/15/$31.00 ©2015 IEEE

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advantage for improving synthetic speech quality but also its mathematically tractable formulation. However, the synthetic speech quality still sounds unnatural compared to the natural one.
Recently, Takamichi et al., [15] have found that a Modulation
Spectrum (MS) of the generated parameter trajectory is more sensitively correlated with the over-smoothing effect than the GV. Because the MS is regarded as an extension of the GV, it enables toquantify more characteristics of the parameter trajectory than theGV. [15] has reported compensating the MS of the generated trajectories using a simple post-filter process. However, this framework based on the post-filtering possibly causes adverse effects due
to completely ignoring the HMM likelihood. Moreover, it is notstraightforward to further apply various extensions, such as model
adaptation [6] and joint optimization in training [16], to this framework as it is not well formulated mathematically.
This paper proposes a speech parameter generation algorithm
considering the MS. The proposed algorithm generates the parameter trajectories by maximizing a novel objective function consistingof the HMM and MS likelihoods. The MS likelihood works as apenalty term to make the MS of the generated parameters close tothat of natural ones. The proposed algorithm is applied to both spectral and F0 components. The experimental results demonstrate thatthe proposed algorithm significantly outperforms the conventionalalgorithm considering the GV.
2. CONVENTIONAL PARAMETER GENERATIONALGORITHM CONSIDERING GLOBAL VARIANCE
Context-dependent HMMs are trained using natural speech parameters. In synthesis, after constructing a sentence HMMs corresponding to input text to be synthesized, the HMM state sequenceq = [q1 , · · · , qt , · · · , qT ]⊤ is determined , where qt is state index at
frame t. Then, the parameter trajectory is generated by maximizing
the objective function consisting of both HMM and GV likelihood
as follows:()ω(1)
ĉ = argmax log N W c; µq , Σq v N (v (c) ; µv , Σv ) ,
c

[
]
⊤
⊤ ⊤
where c = c⊤
is a speech parameter vector
1 , · · · , ct , · · · , cT
sequence of T frames, ct = [ct (1) , · · · , ct (d) , · · · , ct (D)]⊤ is a
D-dimensional parameter vector at frame t, d is a dimensional index,
W is the weighting matrix for calculating the dynamic features [3],and N (·; µ, Σ) denotes Gaussian distribution of a mean vector µ[]⊤ ⊤and a covariance matrix Σ. µq = µ⊤and Σq =q1 , · · · , µqT
diag [Σq1 , · · · , ΣqT ] are the mean vector and the covariance matrix,
where µqt and Σqt are a mean vector and a covariance matrix of thestate qt , respectively. v (c) = [v (1) , · · · , v (d) , · · · , v (D)]⊤ is

ICASSP 2015

the GV vector [11], where the d-th component v (d) is given by:
v (d)

=

c̄ (d)

=

T
1 ∑

T t=1

(ct (d) − c̄ (d))2 ,1 ∑cτ (d),T τ =1

(2)

T

(3)

where µv and Σv are the mean vector and the covariance matrixof the GV, respectively. ωv is a weight of the GV likelihood. The
maximization of the objective function is performed using the gradient method. To determine the initial parameter trajectories, theparameter trajectories analytically determined by maximizing only
the HMM likelihood are linearly transformed to maximize the GV
likelihood [11]. Then, they are iteratively updated by maximizing
the objective function.

We employ the steepest descent algorithm to iteratively update thegenerated parameter trajectory as follows:∂Lsĉ(i+1) = ĉ(i) + α,(11)∂c c=ĉ(i)where α is the step size parameter, and i is an iteration index. The
first derivative of Ls is calculated as
(
)
∂Ls
⊤ −1
= ωs −W ⊤ Σ−1
q W c + W Σq µq ,
∂c
[
]
⊤
⊤
⊤ ⊤
+ s′1 , · · · , s′t , · · · , s′T
(12)
s′t

=

[st (1) , · · · , st (d) , · · · , st (D)]⊤ ,st (d)=(s (c) − µs )⊤ ps(d) f t (d) ,

(13)
(14)
⊤

f t (d) =[ft,d (0) , · · · , ft,d (m) , · · · , ft,d (M − 1)] ,(15)ft,d (m) = −2 (Rd,m cos kt + Id,m sin kt) .
(16)

3. PROPOSED PARAMETER GENERATION ALGORITHMCONSIDERING MODULATION SPECTRUM
3.1. Modulation Spectrum Definition

3.3. Initialization

Whereas the GV represents the temporal scaling of the parameter
trajectory as a scalar value in each feature dimension, the MS explicitly represents the temporal fluctuation as a vector, and its sum
over all modulation frequency except bias is equal to the GV [15].
In this paper, we define the MS s (c) of the parameter trajectory c
as follows:[]⊤(4)s (c) =s (1)⊤ , · · · , s (d)⊤ , · · · , s (D)⊤ ,For initialization, we can basically use the same idea as used in theconventional algorithm, i.e., first generating the parameter trajectoryby maximizing only the HMM likelihood and then it is further transformed to maximize the other likelihood. To transform the parameter trajectory so that the MS likelihood is increased, we use theMS-based post-filter [15], which is given by)
σd,m (
sd (m)′ = ′(17)
sd (m) − µ′d,m + µd,m ,σd,m[sd (0) , · · · , sd (m) , · · · , sd (M − 1)]⊤ ,
(5)
2
2
Rd,m
+ Id,m,(6)( T)2 ( T)2∑∑
ct (d) cos kt+
ct (d) sin kt ,(7)where µd,m and σd,m are a mean and a standard deviation of sd (m),′respectively. µ′d,m and σd,m
are those of MS of generated trajecto′ries. We in advance estimate µ′d,m and σd,musing the generated trajectories included in the training data. Finally, the initial parameter
trajectory c(0) is determined using the filtered MS and the original
phase components of the parameter trajectory before the filtering.

s (d)sd (m)

=
=
=

t=0

t=0

where 2M is a length of Discrete Fourier Transform (DFT). k =−πm/M is a modulation frequency. The MS likelihood is definedas N (s (c) ; µs , Σs ) where µs and Σs are a DM -by-1 mean vector
and a DM -by-DM
covariance matrix,] respectively. Σ−1is reps
[
(1)
(D)
(d)
resented as ps , · · · , p(d)
,
·
·
·
,
p
where
p
is
DM
-by-M
s
s
s
matrix of which columns correspond to s (d). In this paper, the MS
is calculated utterance by utterance and its mean vector and covariance matrix are calculated from the whole utterances of the training
data.
3.2. Generation Algorithm
We define an objective function Ls maximized in the proposed parameter generation algorithm as follows:(8)ĉ = argmax Ls ,c()ωs
Ls = log N W c; µq , Σq
N (s (c) ; µs , Σs ) ,(9)()
1 ⊤ ⊤ −1= ωs − c W Σq W c + c⊤ W ⊤ Σ−1
q µq
2
1
⊤
−1
− s (c)⊤ Σ−1(10)s s (c) + s (c) Σs µs + K̄,2
where ωs denotes the MS weight for controlling balance between
the HMM and MS likelihoods, which is set to the ratio between thenumber of dimensions, i.e., ωs = M/ (3T ) in this paper. K̄ isa constant value to c. Because the objective function is given by
a quartic form like that in the conventional algorithm considering
the GV, it is hard to analytically solve its maximization problem.

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3.4. Application to F0 Contour Generation
The proposed parameter generation is also applied to the F0 components modeled with Multi-Space probability Distribution (MSD)HMM [17]. In the F0 parameter generation, after unvoiced/voiced
determination, F0 values at only voiced frames are generated fromthe corresponding probability density functions while setting the precision matrices (inverse matrix of the covariance matrix) at the unvoiced/voiced boundaries to zero matrices to allow discontinuoustransitions. Therefore, the MS is calculated from the concatenatedvoiced frames in this paper. Moreover, we reform ct (d) of Eq. (7)as ct (d) − c̄ (d) as pointed out in [15]. In this implementation, theMS is directly affected by the discontinuous transitions at the unvoiced/voiced boundaries. This causes some adverse effects in theMS-based post-filtering process. To avoid this, we adopt the initialization method used in the conventional GV-based algorithm rather
than the MS-based post-filtering in this paper.
3.5. Discussion
Though we can also integrate the GV term into the proposed objective function, i.e., a product of the HMM, GV, and MS likelihoods,
the proposed objective function effectively recovers the GV likelihood as well without it because the MS involves the GV as we described. Figures 1 and 2 illustrate an example of the GV and the MSof the generated parameter trajectories. “HMM,” “GV,” and “MS”
indicate the results of the generated parameter trajectories using the
traditional algorithm without the GV [3], the conventional algorithm

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Fig. 1. Examples of GV of generated mel-ceptral coefficients.

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Fig. 3. HMM likelihood for the Fig. 4. HMM likelihood for thegenerated spectral parameters. generated F0 contours.

Fig. 2. Examples of MS of generated the 9-th mel-ceptral coefficient.

with the GV [11], and the proposed algorithm with the MS, respectively. “nat” indicates those of natural speech parameter trajectories.
We can find that the proposed generation algorithm well recovers
not only the MS but also the GV. On the other hand, “GV” cannot
recover the MS appropriately. It makes the MS slightly larger but
the resulting MS is still very different from the natural one. This isbecause the GV can model only an average value of the MS components over modulation frequencies.
The footprint of the synthesis system using the proposed algorithm is slightly larger compared to that using the conventionalalgorithm with the GV because the MS is M D-dimensional vector whereas the GV is D-dimensional vector. We may reduce thefootprint by considering only low modulation frequency components
which has more dominant effect on speech perception [18].
Finally, the MS-based post-filter [15] tends to generate the overtransformed trajectory and synthesize over-emphasized speech because it completely ignores the criterion like the HMM likelihood.
On the other hand, the proposed algorithm can effectively generate
the naturally-fluctuated parameter trajectory by jointly maximizing
both the HMM and MS likelihoods. In addition, as the proposed
algorithm is well formulated mathematically, it is straightforward
to apply various techniques, such as context dependent modeling,
adaptation, and joint optimization in training.
4. EXPERIMENTAL EVALUATION
4.1. Experimental Condition
We trained a context-dependent phoneme Hidden Semi-Markov
Model (HSMM) [19] for an English male speaker ”RMS” and an
English female speaker ”SLT” from the CMU ARCTIC database[20]. We used 593 sentences from subset A for training and 100 sentences from subset B for evaluation. Speech signals were sampled
at 16 kHz. The shift length was set to 5 ms. The 0th-through-24th
mel-cepstral coefficients were extracted as a spectral parameter and

4212

log-scaled F0 and 5 band-aperiodicity [21, 22] were extracted as
excitation parameters. The STRAIGHT analysis-synthesis system[23] was employed for parameter extraction and waveform generation. The feature vector consisted of spectral and excitation
parameters and their delta and delta-delta features. 5-state left-toright HSMMs were used. The DFT length to calculate the MS was
set to 8192. Diagonal covariance matrices were used in the HSMM,
the GV probability density function, and the MS probability density
function.
We compared the following parameter trajectories, HMM: parameters generated by the traditional generation [3], GV: parameters
generated by the conventional algorithm considering the GV [11],MS: parameters generated by the proposed algorithm considering
the MS, and nat: natural speech parameters. To confirm the effect of
the generation algorithms, we did not compare them to parameters
filtered by the MS-based post-filter [15]. In order to avoid slightly
artificial sounds caused by enhancing the high modulation frequency
components of “MS” speech, we applied 50 Hz-cutoff low pass filter
to the generated parameter trajectories used in the subjective evaluation1 . We first conducted an objective evaluation by the likelihoods
used in the conventional and proposed generation algorithms. Then,
we conducted a subjective evaluation on speech quality. The “GV”
was used in the aperiodic component to generate synthetic speech.
4.2. Objective Evaluation Using Likelihood
The generation algorithms were evaluated by the HMM, GV, and MS
likelihoods for the generated trajectories. Additionally, we estimatedthe log-MS log sd (m) probability density function and also calculated its likelihood to deeply discuss the results. Figures 3–10 show
4 types of likelihoods for spectral and F0 components, respectively.
Note that these results were averaged between two speakers.
HMM and GV likelihoods: We can find in Fig. 3 and Fig. 4that the HMM likelihoods for both spectral and F0 components inthe proposed algorithm (“MS”) are lower than the traditional algorithm (“HMM”) and the conventional algorithm (“GV”), but they
are still reasonably higher than those of natural speech parameter
trajectories (“nat”). For the GV likelihoods shown in Fig. 5 andFig. 6, “MS” can effectively recover the GV likelihood as in the
1 As a result of our preliminary subjective test, we have found that there
was no significant quality difference between analysis-synthesized speechsamples with/without the MS components over 50 Hz.

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Fig. 11. Preference scores on speech quality with 95% confidence
interval.

Fig. 7. MS likelihood for the gen- Fig. 8. MS likelihood for the generated spectral parameters.
erated F0 contours.
“GV.” These results demonstrate that the proposed algorithm wellpreserves the conventional criteria.
MS and log-MS likelihoods: In the spectral component, it isobserved in Fig. 7 that the MS likelihood of “MS” is larger thanthat of “HMM” and “GV.” In the F0 component shown in Fig. 8,
the MS likelihood of “MS” is larger than that of the “GV.” From thecomparisons between “MS” and “GV,” which use the same initial
parameter trajectories but different objective functions, it is shownthat the proposed objective function is effective for recovering the
MS likelihood. However, the MS likelihood of “HMM” is higher
than that of “MS.” Moreover, in both spectral and F0 components,
the MS likelihoods of “HMM” are higher than those of “nat.” These
results are hard to interpret. To analyze them, the Fig. 9 and Fig. 10
illustrate the log-MS likelihoods. We can see more reasonable resultsin these figures. These results imply that a probability density of the
MS is well modeled by the Gaussian distribution on the logarithm
domain. Nevertheless, we confirmed there was no perceptual quality
difference between the MS modeling and the log-MS modeling in
the proposed parameter generation algorithm.
4.3. Subjective Evaluation
We conducted a preference test (AB test) on speech quality by 8
listeners. The pair of synthetic speech of “GV” and “MS” was pre-4213

sented to listeners in a random order. Listeners were asked whichsample sounded better in terms of speech quality.
The result of the preference test for two speakers is illustrated
in Fig. 11. We can see that the score of “MS” increases over the“GV.” This result shows that the proposed algorithm can generatebetter quality of synthetic speech than the conventional algorithm
considering the GV.
5. CONCLUSION
In this paper, we have integrated a Modulation Spectrum (MS) into
the parameter generation algorithm in HMM-based speech synthesis. The parameter trajectories are generated by jointly maximizing the HMM and MS likelihoods. The experimental results havedemonstrated that the proposed algorithm can recover the MS while
preserving the conventional HMM and Global Variance (GV) likelihoods. Moreover, the result of perceptual evaluation on speech quality has shown that the proposed algorithm yields significantly better
quality than the conventional algorithm considering the GV. As future work, we will conduct more investigations, such as a comparison to the MS-based post-filter [15] and application to continuous
F0 modeling [24]. Moreover, we will incorporate the MS into thetraining phase like the GV-constrained trajectory training [16].
Acknowledgements: Part of this work was supported by JSPS
KAKENHI Grant Number 26280060 and Grant-in-Aid for JSPS
Fellows Grant Number 26 · 10354, and part of this work was executed under ”JSPS Strategic Young Researcher Overseas Visits
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