PHASE RECONSTRUCTION FROM AMPLITUDE SPECTROGRAMSBASED ON VON-MISES-DISTRIBUTION DEEP NEURAL NETWORK
Shinnosuke Takamichi† , Yuki Saito† , Norihiro Takamune† , Daichi Kitamura‡ , and Hiroshi Saruwatari††

Graduate School of Information Science and Technology, The University of Tokyo, Japan.
‡
Department of Electrical and Computer Engineering, National Institute of Technology,
Kagawa College, Japan.
ABSTRACT

1. INTRODUCTION
A variety of audio signal processing and machine learning-based
methods, such as audio source separation and speech enhancement,
involve processing the amplitude spectrogram obtained via shortterm Fourier transform (STFT). Also, statistical parametric text-tospeech synthesis [1] is shifting from vocoder-based (source filter
model-based) to STFT-based frameworks [2, 3, 4]. To produce synthetic speech, we require the corresponding phase spectrogram, but
it is often unavailable. The Griffin-Lim method [5] is a well-known
example that iteratively estimates of the phase spectrogram through
STFT and inverse STFT while fixing the amplitude spectrogram.
This signal-processing-based method has high portability withouta priori training but causes unnatural artifacts in synthetic speech.
Therefore, this paper proposes building a trainable phase reconstruction method using generative models.
A deep neural network (DNN) is a powerful generative model.
There are two types of distributions: non-parametric [6, 7, 8, 9] andparametric ones. This paper addresses the latter. There are many
types of the parametric approaches related to the Gaussian distribution, e.g., isotropic multivariate Gaussian [10] and temporal-deltaconstrained Gaussian [11, 12] (a.k.a., trajectory DNN in statisticalparametric speech synthesis). The loss function (e.g., mean squared
error) for training DNNs is often derived to minimize the negative
log-likelihood of the distribution. The straightforward way to predict a phase spectrogram from an amplitude spectrogram is to use

Targetphase
Phase loss
Freq. 1(von Mises distribution)…

Index Terms— speech analysis, phase reconstruction, deepneural network, von Mises distribution, group delay

Predicted
phase…

This paper presents a deep neural network (DNN)-based phasereconstruction from amplitude spectrograms. In audio signal andspeech processing, the amplitude spectrogram is often used for processing, and the corresponding phase spectrogram is reconstructedfrom the amplitude spectrogram on the basis of the Griffin-Lim
method. However, the Griffin-Lim method causes unnatural artifacts in synthetic speech. Addressing this problem, we introduce thevon-Mises-distribution DNN for phase reconstruction. The DNN isa generative model having the von Mises distribution that can model
distributions of a periodic variable such as a phase, and the model
parameters of the DNN are estimated on the basis of the maximum
likelihood criterion. Furthermore, we propose a group-delay loss
for DNN training to make the predicted group delay close to a natural group delay. The experimental results demonstrate that 1) the
trained DNN can predict group delay accurately more than phases
themselves, and 2) our phase reconstruction methods achieve betterspeech quality than the conventional Griffin-Lim method.

Amplitude
Freq. 1
Freq. F

Group-delay
loss

DNN

Freq. F

Group-delay
calculation

Fig. 1. Overview of proposed phase reconstruction method. This figure shows frame-by-frame phase prediction rather than multi-frameor sequence-wise prediction for clear illustration.
these models. However, the Gaussian distribution is inappropriate
for modeling distributions of a phase that is a periodic variable witha period of 2π.
This paper proposes phase reconstruction from amplitude spectrograms based on the von-Mises-distribution DNN. The von Mises
distribution is a probability distribution on the circle [13], which issuitable for modeling periodic variables. The von-Mises-distribution
DNN is a generative model that has the von Mises distribution asa conditional probability distribution. The von-Mises-distribution
shallow neural network was originally proposed by Nabney et al.
[14], and this paper utilizes it for predicting the phase spectrogramfrom the amplitude spectrogram. The loss function for DNN training, named the phase loss (see Fig. 1), is defined by minimizing likelihoods of the von Mises distribution. Also, we propose another loss
function named group-delay loss, which has a stronger connection to
the amplitude spectrum [15]. The group-delay loss is used to makegroup delay of the predicted phase close to that of the target phase.
Since the group delay and group-delay loss are differentiable by thephase, the DNN can be trained by a standard backpropagation algorithm. We conduct objective and subjective evaluations to evaluate
the effectiveness of the proposed methods. The results demonstratethat 1) the trained DNN can predict group delay accurately more thanphases themselves, and 2) our phase reconstruction method outperforms the conventional Griffin-Lim method in terms of quality ofsynthesized speech.
2. GRIFFIN-LIM PHASE RECONSTRUCTION
This section briefly describes the conventional Griffin-Lim method[5]. The Griffin-Lim method is a signal-processing-based iterative algorithm to reconstruct a phase spectrogram from theamplitude spectrogram. Let x = [x1 , · · · , xt , · · · , xT ] andy = [y 1 , · · · , y t , · · · , y T ] be amplitude and phase spectrograms, respectively. xt = [xt,0 , · · · , xt,f , · · · , xt,F ] and y t =[yt,0 , · · · , yt,f , · · · , yt,F ] are the amplitude and phase at frame t,
respectively. f is the frequency index, and F corresponds to the
978-1-5386-8151-0/18/$31.00 c 2018 IEEE

Frequency [%]Nyquist frequency. Both xt,f and yt,f are real-valued variables,
but only yt,f is a variable with a period of 2π. The Griffin-Lim
method randomly initializes y first. 1) Then it takes inverse STFT
to obtain a waveform from x and y. 2) The method takes STFT
to re-obtain y from the waveform. 3) We substitute the original
x for the re-obtained x and then go back to step 1). These inverse
STFT and STFT are iteratively performed until they have converged.
The method can reconstruct the phase spectrogram consistent withthe given amplitude spectrogram but makes some artifacts in thesynthesized speech, e.g., extra reverberation and phasiness owing to
inappropriate initialization of y.

40
36.2
35
32.5
30
25
20
14.4
15
12.9
10
5
2.31
1.50
0.17 0.01
0 0.00 0.03
 2 3 4 5 6
-4 -3 -2 -
0Phase [rad]Fig. 2. Histogram of predicted phases. The target phases have arange of [0, 2π], but the predicted phases have a range of [−4π, 6π].
All frames and frequency bins of the evaluation data in Section 4.1
are represented in this figure.

3. PHASE RECONSTRUCTION BASED ON
VON-MISES-DISTRIBUTION DNNSThis section introduces the von-Mises-distribution DNN and proposes loss functions for the DNN training.

ω = πf /F is the angular frequency. If the filter has P complexvalued poles zp = rp exp(jωp ) (p = 1, ..., P ), we have log A(ω) =PP P∞ nQlog P
n=1 rp /n cos n(ω − ωp ), wherep=1
p=1 Ap (ω) = 2
Ap (ω) is the amplitude of the single-pole model for the specific p-thpole [15]. Then, the group delay can be given by
Z π
P X
∞
X
dφ(ω)
−
=c
log Ap (ω) cos(nω)dω,n cos n(ω − ωp )dω−π
p=1 n=1

3.1. von Mises distribution
The von Mises distribution P (vM) (·) [13] is a probability distribution for a periodic variable yt,f , given as
exp (κ cos (yt,f − µ))P (vM) (yt,f ; µ, κ) =,(1)
2πI0 (κ)
where µ is the mean direction of the distribution (analogous to themean of the Gaussian distribution), κ is the shape parameter (analogous to the precision of the Gaussian distribution), and I0 (·) is themodified Bessel function of the first kind of order 0. The negative
log likelihood given y t is:− log P (vM) (y t ; µ, κ) ∝ −FXcos (yt,f − µ) + Const.,(4)
where c is a constant value; this shows the strong correlation between
the group delay and the amplitude spectrum, which motivates us toutilize the group delay as a regularization term.
In this paper, we approximate the group delay at frame t andfrequency bin f with the following equation:∆yt,f = − (yt,f +1 − yt,f ) .
(5)
Since ∆yt,f is also a periodic variable, the group-delay loss is defined as similar to Eq. (3):

(2)

f =0

where Const. is a value constant to µ. Not only yt,f but also a maximum likelihood estimate of µ has a period of 2π.
3.2. DNN training

Lgd (y t , ŷ t ) =We train a DNN that has the von Mises distribution as a conditional
probability distribution. The mean direction is predicted from x at
each frame and frequency. Here, let G (·) be the DNN. The predicted phase (mean direction) ŷ = [ŷ 1 , · · · , ŷ t , · · · , ŷ T ] is given
as ŷ = G (x). The following sections propose two loss functions
for estimating model parameters of G (·): phase loss Lph (y t , ŷ t )and group-delay loss Lgd (y t , ŷ t ).

− cos (yt,f − ŷt,f ) .

(6)

The group-delay loss makes ∆ŷt,f close to ∆yt,f . Because Eq. (5)for ŷ t is a linear transformation of y t , the backpropagation algorithm can be used as the same as in Section 3.2.1.
3.2.3. Multi-task learning
On the basis of the multi-task learning formulation, the DNN can betrained using both phase loss and group-delay loss. The loss function
L (y t , ŷ t ) is:
L (y t , ŷ t ) = Lph (y t , ŷ t ) + αLgd (y t , ŷ t ) ,(7)
where α is the weight of the secondary task (group-delay loss). Note
that, since ranges of Eq. (3) and Eq. (6) are the same, no scale
normalization factor is required.

The phase loss function is derived from Eq. (2) as follows:FX− cos (∆yt,f − ∆ŷt,f ) .

f =0

3.2.1. Phase loss

Lph (y t , ŷ t ) =

F
X

(3)

f =0

Model parameters of G (·) are iteratively updated by backpropagation algorithm to minimize this loss function. The minimum pointof ŷt,f is periodic, i.e., ŷt,f = yt,f ± 2πN , where N is an arbitrary
integer value.

3.3. Discussion
The generalized version of the von Mises distribution is the generalized cardioid distribution [17] given as

3.2.2. Group-delay loss

P (GC) (yt,f ; µ, κ, ψ) =Group delay of speech has a high potential in speech processing,such as speech and speaker recognition [15, 16]. The group delay is defined as the negative derivative of phase by frequency. In
general, speech production via a human vocal tract is well modeled as an all-pole filter defined by A(ω) exp(jφ(ω)), where A(ω)
and φ(ω) are the amplitude and phase functions, respectively, and(cosh (κψ))1/ψ (1 + tanh (κψ) cos (yt,f − µ))1/ψ
, (8)2πP1/ψ (cosh (κψ))
where P1/ψ is the associated Legendre function of the first kind ofdegree 1/ψ and order 0. The von Mises distribution is the special
case (ψ → 0) of this distribution. Also, this distribution is equivalent
2

0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
− 0.05

Cosine dist.
(phase)to the cardioid distribution and the wrapped Cauchy distribution forψ = 1 and ψ = −1, respectively. The negative log likelihood forµ for the cardioid and wrapped Cauchy distributions are equal to
Eq. (2). Therefore, DNNs with these two distributions are trained in
the same manner. One possible way to extend our work is to model
phases using this generalized cardioid distribution. Other possible
ways are to use an asymmetric distribution [18] and mixture model.
As described in Section 3.2.1, the phase loss is minimized
when ŷt,f = yt,f ± 2πN for arbitrary N . Therefore, von-Misesdistribution DNNs suffer from an exploding value of ŷt,f . To
investigate this, Fig. 2 plots a histogram of the predicted phase
spectrograms. We can see that the predicted phase spectrograms
have a wider range than the target phase ([0, 2π]), but the value does
not explode.

max
medianmin

Cosine dist.
(group delay)

0.8

4. EXPERIMENTAL EVALUATION
4.1. Experimental setup

0.6
0.4
0.2
0.0
− 0.2

2

4 8
PH

2 4 8
PH+GD

2

4 8 [kHz]
GD

Fig. 3. Box plots of cosine distances between target and predicted
phases (upper) and group delays (lower). The box indicates the first
and third quartiles.

Evaluations were performed using the JSUT corpus [19], a freeJapanese speech corpus uttered by a female speaker. We used 5,000
utterances (approx. 6 hours) of the subset BASIC5000 for training
and 300 utterances of the subset ONOMATOPEE300 for evaluation.
Speech signals were downsampled at a rate of 16 kHz. The window
length, shift length, and FFT length were set to 400 samples (25 ms),
80 samples (5 ms), and 512 samples, respectively. The Hamming
window was used. Features fed to a DNN were the joint vectorsof the log amplitude spectra at current and ±2 frames, and they
were normalized to have zero-mean unit-variance. The DNN architectures were Feed-Forward networks that included 3 × 1024-unit
gated linear unit [20] hidden layers. The activation of the output
layer was a linear function. We empirically explored DNN architectures within Feed-Forward networks and found that the gated linear
unit is significantly better than a rectified linear unit (ReLU) [21] or
LeakyReLU [22] hidden units. The DNN parameters were randomly
initialized and the AdaGrad algorithm [23] with its learning rate set
to 0.001 was used for the optimization algorithm.
We compared the conventional Griffin-Lim and three proposed
phase reconstruction methods.
PH only phase loss (Eq. (3))
GD only group-delay loss (Eq. (6))
PH+GD multi-task learning (Eq. (7))
In the Griffin-Lim method, phase spectrograms were randomly initialized. The number of iterations was set to 100. The weight α for
multi-task learning was set to 0.1. In the proposed methods, phasesin the low frequency band are first estimated by DNNs, and those in
the remaining frequency bins are randomly generated. We used threesettings of frequency bands of the predicted phase spectrograms: 0-2(96 dim.), 0-4 (128 dim.), and 0-8 kHz (257 dim.). After predicting
the phase spectrogram by DNNs, we further applied the Griffin-Lim
method to refine the phase. The number of iterations of the phaserefinement was 100.

Table 1. Results of preference tests: conventional Griffin-Lim
method vs. proposed methods. Bold indicates preferred method thathas a p-value smaller than 0.05
Method A
Scoresp-value Method B
Griffin-Lim 0.497 vs. 0.503
0.871
PH (2 kHz)
Griffin-Lim 0.280 vs. 0.720 < 10−9 PH (4 kHz)
Griffin-Lim 0.277 vs. 0.723 < 10−9 PH (8 kHz)
Griffin-Lim 0.453 vs. 0.547
0.022
PH+GD (2 kHz)
Griffin-Lim 0.233 vs. 0.767 < 10−9 PH+GD (4 kHz)
Griffin-Lim 0.247 vs. 0.753 < 10−9 PH+GD (8 kHz)
Griffin-Lim 0.447 vs. 0.553
0.009
GD (2 kHz)
Griffin-Lim 0.463 vs. 0.537
0.073
GD (4 kHz)
Griffin-Lim 0.490 vs. 0.510
0.619
GD (8 kHz)

hand, when using only group-delay loss (“GD”), group delay can bepredicted more accurately than phase for all settings of frequency
bands. This result shows us that the Feed-Forward DNN can predict group delay better than phases themselves. Finally, combinedphase loss and group-delay loss (“PH+GD”) predicts phase moreaccurately than “GD” and group delay more accurately than “PH”
for all settings of frequency bands. Therefore, we can demonstrate
the effectiveness of the proposed multi-task training.
4.3. Comparison of Griffin-Lim and proposed methods
To evaluate the effectiveness of the proposed methods, we compared
the quality of synthetic speech of Griffin-Lim and proposed methods. Preference AB tests (listening tests) on speech quality were
performed in our crowdsourcing evaluation system. 30 listeners participated in each test. Approximately $0.46 were paid to each listener. Each listener preferred better-quality speech and answered forten pairs of samples, i.e., 300 answers were obtained in each evaluation. Speech samples of pairs of methods were randomly presented
to the listeners. These settings are also used not only here but also in
the following sections.
Table 1 lists the results. The proposed methods outperform theconventional Griffin-Lim method in all settings of loss functions andfrequency bands. In particular, “PH+GD” always has significantly
better scores than the conventional Griffin-Lim method. These re-4.2. Prediction accuracy
We evaluated prediction accuracies of phases and group delay. Fig.
3 shows cosine distances between target and predicted phases andgroup delays. The distances were averaged over all frames and frequency bins of the predicted phases (0-2, 0-4, or 0-8 kHz).
The prediction accuracy of “PH (2 kHz)” ranges from 0.15 to
0.31, and the distribution seems symmetric. Also, the accuracy becomes smaller as the frequency band widens (“PH (4, 8 kHz)”). This
result is natural because phases at higher frequency bins are easilychanged by the temporal position of frame analysis. On the other3

Log spectral convergence

Cosine dist.
(phase)

0.30
0.25
max

0.20

median

0.15

min

0.10
0.05

Cosine dist.
(group delay)

0.8
0.6

0.7
0.8
0.9

Random

1.0
1.1
1.2
1.3
0

PH+GD (4 kHz)20
40
60
Iteration index

PH (4 kHz)
80

100

Fig. 5. Log spectral convergence by phase refinements. When thevalue is −∞, perfect reconstruction through STFT and inverse STFT
is achieved. This is the result of one of the evaluation datasets, but
the same tendency was observed in all evaluation datasets.

0.4
0.2
0.0
0.2

0.6

2

4 8
PH

2 4 8
PH+GD

2

4 8 [kHz]GD

Table 3. Results of preference tests: effects of group-delay loss.
Bold indicates preferred method that has a p-value smaller than 0.05
Method A
Scoresp-value Method B
PH (2 kHz) 0.487 vs. 0.513
0.514
PH+GD (2 kHz)PH (4 kHz) 0.486 vs. 0.514
0.500
PH+GD (4 kHz)PH (8 kHz) 0.545 vs. 0.455
0.031
PH+GD (8 kHz)

Fig. 4. Box plots of cosine distances between predicted and refinedphases (upper) and group delays (lower). The box indicates the first
and third quartiles.
Table 2. Results of preference tests: proposed methods with different frequency bands. Bold indicates preferred method that has ap-value smaller than 0.05
Method A
Scoresp-value Method B
PH (2 kHz) 0.270 vs. 0.730 < 10−9 PH (4 kHz)PH (4 kHz) 0.507 vs. 0.493
0.744
PH (8 kHz)
PH+GD (2 kHz) 0.223 vs. 0.777 < 10−9 PH+GD (4 kHz)
PH+GD (4 kHz) 0.493 vs. 0.507
0.744
PH+GD (8 kHz)GD (2 kHz) 0.513 vs. 0.487
0.514
GD (4 kHz)GD (4 kHz) 0.567 vs. 0.433
0.001
GD (8 kHz)has different tendencies: speech quality for 0-8 kHz frequency bands
was significantly worse than that for 0-4 kHz. We will investigate
the reason for this.
4.6. Effect of group-delay loss
We investigated the effectiveness of group-delay loss compared withphase loss. First, we evaluated convergence of phase refinements.
Fig. 5 shows the log spectral convergence [25] of “PH (4 kHz)” and“PH+GD (4 kHz).” For comparison, results of randomized initialization of phases are also shown as “Random.” The proposed methods(“PH (4 kHz)” and “PH+GD (4 kHz)”) have a smaller value for the
spectral convergence than “Random.” Also, “PH+GD (4 kHz)” hasa smaller value than “PH (4 kHz)”, i.e., “PH+GD (4 kHz)” is theclosest to the perfect reconstruction.
In addition, we compared speech quality of “PH” and “PH+GD.”
Table 3 lists the results of the comparison with frequency bands of 02, 0-4, and 0-8 kHz. “PH” is preferred more in 0-8 kHz with statistical significance, but “PH+GD” is preferred more in 0-2 and 0-4 kHz(without statistical significance). Therefore, we can demonstrate theeffectiveness of group-delay loss in speech quality.

sults demonstrate the effectiveness of the proposed methods.
4.4. Effect of phase refinements
As explained in Section 4.1, after phases were predicted by DNNs,
they were refined by the Griffin-Lim method. Here, we evaluatedeffects of the phase refinements. Fig. 4 shows cosine distances between predicted and refined phases and group delays. Tendencies
were almost the same as those in Fig. 3. When phase loss is usedin training (“PH” and “PH+GD”), phase information is comparably preserved. Similarly, when group-delay loss is used in training (“GD” and “PH+GD”), the group-delay information is comparably preserved. In the preliminary evaluation, we compared speech
quality of refined and unrefined phases (i.e., phases predicted by
DNNs were directly used for finally synthesized speech). The results
demonstrated that unrefined phases had significantly worse speech
quality than refined phases.

5. CONCLUSION
This paper presented DNN-based phase reconstruction from an amplitude spectrogram. Based on maximum likelihood estimation of
the von Mises distribution, we introduced two loss functions for
DNN training: phase loss and group-delay loss. We demonstrated1) the trained DNNs can predict group delay more accurately thanphases, and 2) our proposed phase reconstruction methods achieve
better speech quality than the conventional Griffin-Lim phase reconstruction method. For future work, we will implement other probability distributions for periodic variables, propose other approaches
for phase refinements, and integrate our method with text-to-speechsynthesis that generates amplitude spectrograms [4].
Acknowledgments: Part of this work was supported by SECOM
Science and Technology Foundation, and JSPS KAKENHI Grant
Number 18K18100.

4.5. Effect of frequency bands
We compared effects of frequency bands of predicted phases within
one loss function (“PH”, “PH+GD”, or “GD”). Table 2 shows theresults of preference AB tests on speech quality. In “PH” and“PH+GD,” speech quality for 0-4 kHz frequency bands is significantly better than that for 0-2 kHz, and is comparable with that for
0-8 kHz. These results suggest that at least 0-4 kHz frequency bands
needs to be predicted but the spectrograms in the higher frequency
bands may be excited by random phases (this tendency is similar tothe harmonics plus noise model [24]). A curious point is that “GD”
4

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