Proc. Interspeech, pp. 42-45, 2010.

Phase Equalization-Based Autoregressive Model of Speech Signals
Sadao Hiroya, Takemi Mochida
NTT Communication Science Laboratories, NTT Corporation, Japanhiroya@idea.brl.ntt.co.jp, mochida@idea.brl.ntt.co.jp

Abstract

3585This paper presents a novel method for estimating a vocal-tract
spectrum from speech signals, based on a modeling of excitation signals of voiced speech. A formulation of linear prediction
coding with impulse train is derived and applied to the phaseequalized speech signals, which are converted from the original
speech signals by phase equalization. Preliminary results showthat the proposed method improves the robustness of the estimation of a vocal-tract spectrum and the quality of re-synthesized
speech compared with the conventional method. This techniquewill be useful for speech coding, speech synthesis, and real-time
speech conversion.
Index Terms: LPC, vocal-tract spectrum, phase equalization(A)

1. Introduction
Linear prediction coding (LPC) (or the autoregressive model:AR) is a fundamental technique for the estimation of a vocaltract spectrum from speech signals and has been widely used
for speech synthesis and speech coding. However, the estimatedvocal-tract spectrum of voiced speech is affected by harmonics, because the model assumes Gaussian noise as the excitation signals even for voiced speech. This causes degradationof the re-synthesized speech quality. To overcome the problem, methods based on an adjustment of the analysis window or
on modeling of excitation signals for voiced speech have been
proposed. The former refers to a pitch synchronous analysis[1] and a glottal closure interval analysis [2], but the problems
still remain: the analysis window is not long enough and estimating glottal closure intervals is difficult. One of the latter isdiscrete all-pole (DAP) modeling [3] which assumes a periodic
impulse excitation in LPC for voiced speech, but the assumption is not satisfied for natural speech. Another is LPC with aglottal source hidden Markov model (HMM) [4]. The method
is robust but has high computational complexity because theglottal source HMM requires the estimation of many parameters (e.g. the mean and the covariance of fifteen HMM states
for each vowel) in order to represent phase characteristics ofspeech signals. To reduce the computational complexity of robust estimation of vocal-tract spectrum using LPC, modifying
phase characteristics of natural speech to be fitted into a simple
periodic impulse excitation model would be beneficial.
Moriya and Honda have proposed a method for compensating phase characteristics of speech signals using a matchedfilter, called phase equalization [5]. They found both the speech
spectrum and the quality of the phase-equalized speech is almost equivalent to those of the original speech, respectively:
humans cannot distinguish changes in short-time phase characteristics of speech signals. Their method is convenient compared with a phase compensation using an all-pass filter [6].
The phase-equalized speech signals can be considered to be theoutput of the LPC filter whose input is the impulse train spaced

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(B)

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1472

(C)

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1989

(D)

-1989

G(t0)

G(t1)

G(t2)

G(t3)

t0

t1

t2

t3

G(tI)

(E)
Gaussian noisetI

Figure 1: Examples of the English vowel /i/. (A) Original speech
signals; (B) phase-equalized speech signals; (C) LPC residual
signals; (D) phase-equalized LPC residual signals; (E) excitation signal model for PEAR.

at the pitch period. Thus, it is expected that a formulation of
LPC with an impulse train would be easy, and then the computational complexity would be reduced.
In this paper, we propose a phase equalization-based autoregressive (PEAR) model of speech signals and show that a robust vocal-tract spectrum is obtained from the phase-equalized
speech signals using PEAR.

2. Phase equalization
In phase equalization, the idea is to convert the phase characteristics of the original speech signals to the minimum phase. Thisis done by applying an adaptively matched filter and converting
the LPC residual signals to a nearly zero phase [5]. In the voiced
speech frame, the LPC residual signals e(t) are considered to be
the impulse train of the pitch period:e(t) = s(t) −

p
∑
i=1

a(i)s(t − i),(1)

Proc. Interspeech, pp. 42-45, 2010.

60[Fig. 1 (D)]. Then, the phase-equalized speech signals are approximately the minimum-phase sequence [Fig. 1 (B)]. Figure 2
shows an example of speech spectra of the original speech signals and phase-equalized speech signals. We can see that the
spectrum of the phase-equalized speech signals is almost thesame as that of the original ones.

50

Amplitude (dB)

40

30

20

3. Proposed method

10

Phase equalization has been used to optimize the excitation signals of voiced speech for low-bit rate speech coding [5, 7, 8, 9],
but not to estimate the vocal-tract spectrum of voiced speech. In
this section, we describe our method for estimating a vocal-tract
spectrum from the phase-equalized speech signals, based on themodeling of excitation signals of voiced speech.

0

−10
0

1000

2000

3000

4000
5000
Frequency (Hz)

6000

7000

8000

Figure 2: Speech spectrum of the original speech signals (thinline) and the phase-equalized speech signals (thick line) of the
English vowel /i/.

where s(t) is the original speech signals, a(i) is the LPC coefficients, and p is the dimension of the LPC coefficients. However,the LPC residual signals for natural speech are not a zero-phase[Fig. 1 (C)]. So the impulse train is reconstructed from the filter
output using the M + 1 tap FIR filter h(t) as follows. Provided
one pulse exists at a known position t0 in the frame for the sakeof simplicity, the modeled input is represented as δ(t − t0 ) andthe reconstructed input g(t) is expressed as∑

M/2g(t) =h(τ )e(t − τ ).

h

∑
t

)2

∑

M/2

h(τ )e(t − τ ) − δ(t − t0 ).

(3)

τ =−M/2

If the autocorrelation function of e is a delta function for thetime delay up to M + 1, then
v
/u
u M/2
u ∑
e(t0 + τ )2 .
(4)h(t) = e(t0 − t) t
τ =−M/2

That is, the LPC residual is converted into a positive impulse
train through the FIR filter whose coefficients are the values of
the LPC residual itself, which is reversed at a reference positionin the time domain. To reduce abrupt change in the equalizing
filter response, the FIR filter coefficients are low-pass filtered[7]. For the obtained h, the phase-equalized speech signals x
are computed by∑

t6=t0 ,··· ,tI

t=t0 ,··· ,tI

where G(t) for t = t0 , · · · , tI is the impulse amplitude, σ is thecovariance, and I + 1 is the number of impulses in the frame.
The phase-equalized LPC residual signals f are calculated fromthe phase-equalized speech signals x like in Eq. (1):f (t) = x(t) −The optimum filter coefficients h are derived by minimizing themean squared error between g and δ(t − t0 ) in the frame:argminThe phase-equalized speech signals are considered to be theoutput of the LPC filter whose input is the impulse train corresponding to pitch mark t0 , · · · , tI and the Gaussian noise elsewhere in the frame [Fig. 1 (E)]. Thus, we consider minimizing
the following function:∑∑
σ −1 f (t)2 +σ −1 (f (t) − G(t))2 ,
(6)

(2)

τ =−M/2(

3.1. PEARh(τ )s(t − τ ).

(5)

τ =−M/2

Here, the number of filter taps M + 1 matches the pitch period
in the frame. The positions of pitch mark t0 , t1 , · · · in the frame
are detected on the basis of the LPC residual signals as in [7].
Figure 1 shows an example of the results of phase equalization. The phase-equalized LPC residual signals show very
sharp pitch spikes at the instant corresponding to the pitch markâ(i)x(t − i).

(7)

i=1

The LPC coefficients â are calculated by solving the following
simultaneous equations:



R(0)
. . . R(p − 1)â(1)
 R(1)

. . . R(p − 2)   â(2) 




..
.
.
.
..
..

  .. .
R(p − 1) . . .
R(0)
â(p)


I
∑
x(ti − 1)G(ti )  R(1) −


i=0


I


∑
 R(2) −x(ti − 2)G(ti ) 


,
=
(8)
i=0




.
.


.




I
∑


R(p) −
x(ti − p)G(ti )i=0where R is an autocorrelation function of the windowed phaseequalized speech signals x:

M/2x(t) =

p∑

R(q) =

L−1∑

x(t)x(t + q),

(9)

t=0

where L is the window length. As Eq. (8) is a Toeplitz matrix,
we can use the Levinson algorithm to efficiently solve the equations [10]. For given LPC coefficients, the optimum impulse
amplitude is obtained from the phase-equalized LPC residual
signals like in Eq. (7): G(t) = f (t) for t = t0 , · · · , tI . Therefore, we first calculate the phase-equalized LPC residual signals

Proc. Interspeech, pp. 42-45, 2010.

(A) Phase-equalized speech signals

Phase equalization

Autocorrelation / Cross-correlation

200
150

x(t i -p)x(t i )autocorrelation

100x(t i -p)G(t i )cross-correlation

Original speech signals s

50
0Conventional LPC & Inverse filtering

−50
−100
−150

LPC residual signals e

Pitch mark analysis

Phase equalization

Pitch mark ti

Phase-equalized speech signals x

Amplitude analysis

−200
0

5

10

15

20

25

30

35

40

Lag p (sample)(B) Original speech signals
Autocorrelation / Cross-correlation

200
150

s(t i -p)s(t i )autocorrelation

100s(t i -p)G(t i )cross-correlation

500Impulse amplitude G

−50
−100

PEAR

−150
−200
0

5

10

15

20

25

30

35

40

Lag p (sample)

Figure 3: Examples of the English vowel /i/. (A) Autocorrelation function for the phase-equalized speech signals (thick line)
and cross-correlation function between the impulse train andthe phase-equalized speech signals (thin line) for the lag. (B)Autocorrelation function for the original speech signals (thickline) and cross-correlation function between the impulse train
and the original speech signals (thin line) for the lag.

as the initial impulse amplitude and then determine the LPC coefficients and the amplitude iteratively. If G(t) = 0 for allt, e.g. the unvoiced speech, then Eq. (8) is equivalent to theconventional LPC: the autocorrelation method. By substituting
Eqs. (4) and (5) into Eq. (9) under the assumption that the autocorrelation function of e is a delta function for the time delay
up to M + 1, R is an autocorrelation∑function of the windowed
L−1
original speech signals s: R(q) =t=0 s(t)s(t + q). This
supports the result of Fig. 2 by the Wiener-Khinchin theorem.
The differences between PEAR and conventional
LPC are∑the cross-correlation terms of the Eq. (8): Ii=0 x(ti − q)G(ti )(q = 1, · · · , p). We compare∑ the function with the autocorrelation one at the pitch mark: Ii=0 x(ti −q)x(ti ) (q = 1, · · · , p).
Figure 3(A) shows the cross-correlation function between theimpulse train G and the phase-equalized speech signals x, andthe autocorrelation function of the phase-equalized speech signals x at the pitch mark. The cross-correlation function is similar to the autocorrelation function. This indicates that PEAR
can efficiently reduce the effect of impulse at the pitch mark (i.e.
the effect of the harmonics in a vocal-tract spectrum) from thephase-equalized speech signals. Moreover, notice that, though
PEAR worked well for the phase-equalized speech, it did notwork well for the original speech input because LPC residual
signals of the original speech are not considered to be the impulse train. Figure 3(B) shows the cross-correlation function
between the impulse train G and the original speech signals s,
and the autocorrelation function of the original speech signals
s at the pitch mark. The cross-correlation function is different
from the autocorrelation function of the original speech signals.
3.2. Algorithms
Figure 4 shows the algorithms of the proposed method. The
LPC residual signals are calculated by using the conventional
LPC inverse filtering technique. Then, for voiced speech, thephase-equalized speech signals and the pitch mark are obtainedby using the LPC residual signals. LPC inverse filtering fromthe phase-equalized speech signals gives the initial impulse am-

Phase-equalized speech signals x Pitch mark ti

Impulse amplitude GPEAR
LPC coefficients ^aImpulse amplitude G

Figure 4: Algorithms for phase equalization and PEAR.

plitude.
For the pitch mark and the initial impulse amplitude, we
determine the LPC (PEAR) coefficients. And then the LPC coefficients and the impulse amplitude are determined iteratively.
3.3. TANDEM window
Even when PEAR is applied to estimate the vocal-tract spectrum, the obtained spectrum is not temporally stable. Kawahara has found that the temporally stable power spectrum of aperiodic signal can be calculated as the average of two power
spectra by using a pair of time windows temporally separated
for half of the fundamental period, called a TANDEM window[11]. According to the Wiener-Khinchin theorem, the power
spectrum is the Fourier transform of the corresponding autocorrelation function. Thus, we can apply the TANDEM window
with the PEAR as follows: We use the average of two autocorrelation function and the average of two terms of x×G in Eq. (8)for the temporally separated windows. Theoretically, the TANDEM window works well when the pitch period is constant in
the frame, but the assumption is not satisfied for natural speech.
Therefore, to cope with a variable pitch period T0 , · · · , TI in
the frame, we propose the following shift value:
/ I
I
∑
∑
wi /Ti 2
wi /Ti2 ,
(10)
i=0

i=0

where w is a Gaussian weight function. The analysis window is
a Blackman window with a 3.5-fold pitch period [11].

4. Experiments
We evaluated the proposed method using natural speech. The
speech signals were recorded at a sampling rate of 16 kHz.
Twenty LPC coefficients were obtained with a 4-ms frame shift.
No pre-emphasis was used.
4.1. Effect of PEAR
Figure 5 shows the vocal-tract spectrum of the English vowel /i/for the conventional LPC and PEAR (without a TANDEM win-Proc. Interspeech, pp. 42-45, 2010.

40

30

5. Discussion
F1

F2

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Amplitude (dB)

10

0

−10

−20

−30

−40
0

500

1000

1500

2000
Frequency (Hz)

2500

3000

3500

4000

Figure 5: Vocal-tract spectrum of the English vowel /i/ using
conventional LPC (thin line) and PEAR (thick line). Speech
spectrum (dashed line).
(A) PEAR without TANDEM window

As we noted in Sec. 3.1, PEAR requires the iterative estimationof the LPC coefficients and the impulse amplitude. However,
we found that the initial impulse amplitude gives a good result
for estimating the LPC coefficients, so the proposed algorithms
may have less computational complexity than other algorithms.
A real-time speech conversion technique is important forinvestigating human speech production mechanisms. For example, F1 perturbation studies of vowels using the techniquehave revealed that auditory feedback essentially plays a role inhuman speech production [11,12]. However, technique utilized
by the conventional LPC is sometimes problematic: F1 perturbation for female speaker makes noise in vowel signals due
to incorrect vocal-tract estimation. Thus, the proposed method
would be useful for solving the problem.

6. Conclusions
We presented a novel vocal-tract spectrum estimation method
and showed that it is superior to conventional LPC in terms ofrobust estimation of vocal-tract spectrum and the quality of resynthesized speech.

7. Acknowledgements
The authors thank Drs. H. Gomi, H. Kameoka, and Y. Kamamoto for many useful and helpful discussions.
(B) PEAR with TANDEM window

Figure 6: Example of the LPC spectrum sequence of PEAR (A)without and (B) with TANDEM window.

dow) with a 30-ms Hamming window. A lag window (100 Hz)
for the conventional LPC was used. The number of iterations
for PEAR was five. The average fundamental frequency was
258 Hz. We can see the vocal-tract spectrum of the conventional LPC was largely affected by the harmonics for the firstformant frequency (F1) compared with that of PEAR.
We compared the re-synthesized speech signals of PEAR
with those of the conventional LPC. For this purpose, theimpulse amplitude was determined to minimize frequencyweighted mean square error between the phase-equalized
speech signals and the synthesized speech signals [7]. The
signal-to-noise ratios (S/Ns) were 9.26 dB for PEAR and 8.55
dB for the conventional LPC. Moreover, an informal listening
test showed the speech re-synthesized using PEAR is superior
to that re-synthesized using the conventional LPC.
4.2. Effect of TANDEM window
Figure 6 shows the LPC spectrum sequence of PEAR with andwithout a TANDEM window of the Japanese word /udemae/.
With the TANDEM window, the temporal continuity of the
vocal-tract spectrum was slightly improved, and thus the quality of the re-synthesized speech and S/N (9.72 dB) was also
improved.

8. References[1]

Mathews, M.V., Miller, J.E., and David, E.E., “Pitch synchronous
analysis of voiced sounds,” J. Acoust. Soc. Am., 179, 1961.

[2]

Lu, J., Murakami, H., and Kasuya, H., “Estimation of vocal tract
transfer functions using multi-closure intervals linear prediction,”
IEICE Trans. Fundamental., 1011–1014, 1990.

[3]

El-Jaroudi, A. and Makhoul, J., “Discrete all-pole modeling,”
IEEE Trans. Signal Processing, 411-423, 1991.

[4]

Sasou, A. and Tanaka, K., “Glottal excitation modeling using
HMM with application to robust analysis of speech,” Proc. ICSLP, 704–707, 2000.

[5]

Moriya, T. and Honda, M., “Speech coder using phase equalization and vector quantization,” Proc. ICASSP, 1701–1704, 1986.

[6]

Funaki, K., Miyanaga, Y., and Tochinai, K., “Recursive ARMAX
speech analysis based on a glottal source model with phase compensation,” Signal Processing, 279–295, 1999.

[7]

Honda, M., “Speech coding using waveform matching based on
LPC residual phase equalization,” Proc. ICASSP, 213–216, 1990.

[8]

Stachurski, J. and McCree, A., “A 4kb/s hybrid MELP/CELP
coder with alignment phase encoding and zero-phase equalization,” Proc. ICASSP, 1379–1382, 2000.

[9]

Hiwasaki, Y., Mano, K., and Kaneko, T., “An LPC vocorder
based on phase-equalized pitch waveform,” Speech Communication, 277–290, 2003.

[10] Golub, G.H. and van Loan, C.F., “Matrix computations (3rd ed.),”Johns Hopkins University Press, 1996.
[11] Kawahara, H., Morise, M., Takahashi, T., Nisimura, R., andBanno, H., “TANDEM-STRAIGHT: A temporally stable power
spectral representation for periodic signals and applications tointerference-free spectrum,” Proc. ICASSP., 3933–3936, 2008.
[12] Purcell, D.W. and Munhall, K.G., “Compensation following realtime manipulation of formants in isolated vowels,” J. Acoust. Soc.
Am., 2288–2297, 2006.
[13] Villacorta, V.M., Perkell, J.S., and Guenther, F.H., “Sensorimotor
adaptation to feedback perturbations of vowel acoustics and its
relation to perception,” J. Acoust. Soc. Am., 2306–2319, 2007.