MEL-GENERALIZED CEPSTRAL ANALYSIS— A UNIFIED APPROACH TO SPEECH SPECTRAL ESTIMATION
Keiichi Tokuda†, Takao Kobayashi††, Takashi Masuko†† and Satoshi Imai†††Department of Electrical and Electronic Engineering, Tokyo Institute of Technology, Tokyo, 152 Japan††Precision and Intelligence Laboratory, Tokyo Institute of Technology, Yokohama, 227 Japan

ABSTRACT'' α=0

The generalized cepstral analysis method is viewed as a uniﬁed approach to the cepstral method and the linear prediction
method, in which the model spectrum varies continuously fromall-pole to cepstral according to the value of a parameter γ. Since
the human ear has high resolution at low frequencies, introducing
similar characteristics to the model spectrum, we can represent
speech spectrum more eﬃciently. From this point of view, this paper proposes a spectral estimation method which uses the spectral
model represented by mel-generalized cepstral coeﬃcients. The
eﬀectiveness of mel-generalized cepstral analysis is demonstrated
by an experiment of HMM-based isolated word recognition.

I.

$

Generalized
Cepstral Analysis [4]#Mel-Generalized
Cepstral Analysis [8], ([10])γ = −1

Linear Prediction[1], [2]"#WarpedLinear Prediction[11], ([12])γ=0Unbiased Estimationof Log Spectrum[9], ([3])"&&

INTRODUCTION$| α | < 1, −1 ≤ γ ≤ 0!

Mel-Cepstral
Analysis [13]!
%%(a) Block analysis (frame basis)
Linear prediction [1], [2] is a generally accepted method for
obtaining all-pole representation of speech. However, in somecases, spectral zeros are important and a more general modelingprocedure is required. Although cepstral modeling can represent
poles and zeros with equal weights, the cepstral method [3] witha small number of cepstral coeﬃcients overestimates the bandwidths of the formants. To overcome this problem, we proposed
the generalized cepstral analysis method [4], [5]. The generalized
cepstral coeﬃcients [6] are identical with the cepstral and AR coeﬃcients when a parameter γ equals 0 and −1, respectively. Thus,
utilizing the generalized cepstral representation, we can vary themodel spectrum continuously from the all-pole spectrum to that
represented by the cepstrum according to the value of γ.
Since the human ear has high resolution at low frequencies, introducing the similar characteristics to the model spectrum, we
can represent speech spectrum more eﬃciently. The spectrum
represented by the mel-generalized cepstrum [7], i.e., frequencytransformed generalized cepstrum, has frequency resolution similar to that of the human ear with an appropriate choice of thevalue of a parameter α. Hence, it is expected that the melgeneralized cepstral coeﬃcients are useful for speech spectral representation.
From the above point of view, this paper proposes a melgeneralized cepstral analysis method [8], in which we apply thecriterion used in the UELS (unbiased estimation of log spectrum)[9] to the spectral model based on the mel-generalized cepstral
representation. It can be shown that the minimization of the criterion is equivalent to the minimization of the mean square of thelinear prediction error. As a result, the proposed method can beviewed as a uniﬁed approach to speech spectral analysis, which includes several speech analysis methods (see Fig. 1). The detailed
discussion is given in Section VI.
Although the method involves a non-linear minimization problem, it can easily be solved by an iterative algorithm [8]. The
convergence is quadratic and typically a few iterations are suﬃcient to obtain the solution. We can also show that the stabilityof the obtained model solution is guaranteed [8].
Finally, we show some simulation results of natural speech analysis to demonstrate that the characteristic of the obtained spectrum varies according to the values of α and γ. It is shown that'$| α | < 1, −1 ≤ γ ≤ 0

' α=0

$

Adaptive Generalized
Cepstral Analysis [14]#Adaptive Mel-Generalized
Cepstral Analysis

γ = −1

Adaptive
Linear Prediction[15], ([16])"#Adaptive Warped
Linear Prediction([12], [17])γ=0Adaptive Cepstral
Analysis [18]

"
&
&

!

Adaptive Mel-Cepstral
Analysis [13]!
%%(b) Adaptive analysis (sample by sample basis)Fig. 1. A uniﬁed view of speech spectral analysis.
(References in parentheses are closely related methods.)
we can improve the performance of speech recognition by choosing
the values of α and γ.

II.

SPECTRAL MODEL AND CRITERION

The generalized logarithmic function [6] is a natural generalization of the logarithmic function:
(wγ − 1) /γ, 0 < | γ | ≤ 1.
(1)sγ (w) =log w,
γ=0
The cepstrum c(m) of a real sequence x(n) is deﬁned as the inverse
Fourier transform of the logarithmic spectrum, while the melgeneralized cepstrum cα,γ (m) is deﬁned as the inverse Fouriertransform of the generalized logarithmic spectrum calculated ona warped frequency scale βα (ω):∞
sγ X(ejω ) =cα,γ (m) e−jβα (ω) m
m=−∞

1

(2)

TABLE I

Spectral representation based on mel-generalized
cepstrum. (equation (5)).

x(n)
1/D(z)

γ = −1γ=0γ=1−1 ≤ γ ≤ 1α=0|α| < 1

all-pole
cepstralall-zerogeneralized cepstral

warped all-polemel-cepstral
warped all-zeromel-generalized cepstral

where X(ejω ) is the Fourier transform of x(n). The warped frequency scale βα (ω) is deﬁned as the phase response of an all-passsystemz −1 − α 
= e−jβα (ω) , | α | < 1(3)
Ψα (z) =−1
1 − α z z=ejω
where
βα (ω) = tan−1(1 − α2 ) sin ω.
(1 + α2 ) cos ω − 2α(4)The phase response βα (ω) gives a good approximation to auditory
frequency scales with an appropriate choice of α [12], [19].
In this paper, we assume that a speech spectrum H(ejω ) canbe modeled by the M + 1 mel-generalized cepstral coeﬃcients asfollows:

 M

m
c
(m)
Ψ
(z)
H(z) = s−1
α,γ
γ
α
m=0

 
1/γ
M



m

 1+γ
cα,γ (m) Ψα (z), 0 < |γ | ≤ 1

=

m=0

M





cα,γ (m) Ψm
 exp
α (z),

γ=0

m=0

From (5), it is seen that for (α, γ) = (0, −1) the model spectrum
takes the form of the all-pole representation and for (α, γ) =(0, 0) the model spectrum is identical with the spectrum represented by the cepstrum. The relation between the form of themodel spectrum and the values of (α, γ) is summarized in Table I.
T
We determine c = [cα,γ (0), cα,γ (1), . . . , cα,γ (M )] in such a
way that the following spectral criterion derived in the UELS [9]is minimized:
E=
where

1
2π

π
−π

{exp R(ω) − R(ω) − 1} dω2R(ω) = log IN (ω) − log  H(ejω ) 

(6)

(7)

and IN (ω) is the modiﬁed periodogram of a weakly stationary
process x(n) with a window w(n) whose length is N :2
 N −1−jωn 
w(n) x(n) e
IN (ω) = 



n=0

N
−1


w2 (n).

(8)n=0

When (α, γ) = (0, 0) , the proposed method is identical with
the UELS, because in the UELS the estimator of the log spectrum is represented by the cepstrum. Furthermore, (6) has thesame form as the spectral criterion in the maximum likelihood estimation of Gaussian stationary AR process [1]. Therefore, when(α, γ) = (0, −1), the proposed method is identical with the linear
prediction method [1], [2]. As shown in Fig. 1, some other spectral
estimation methods are also included in the proposed method as
special cases.
Since E is convex with respect to c when −1 ≤ γ ≤ 0,
∇E =

∂E
=0
∂c

ε = E[e2 (n)] → min
Fig. 2. Interpretation of the proposed spectral analysis as theleast mean square of the linear prediction error e(n).
gives the global minimum [8]. When γ = −1, (9) can be solved
directly, because (9) becomes a set of linear equations. Especially,
when (α, γ), = (0, −1), (9) corresponds to the normal equationsin the linear prediction method. In other cases, to solve the minimization problem, we have given an iterative algorithm [8] whose
convergence is quadratic. We have found that typically a few iterations are suﬃcient to obtain the solution. Furthermore, the stability of the model solution H(z) is guaranteed when −1 ≤ γ ≤ 0[8].
Assuming that D(z) is the gain-normalized version of H(z), i.e.,
the 0th impulse responses of D(z) and 1/D(z) are constrained
to unity, the inverse ﬁlter output e(n) shown in Fig. 2 is theprediction error [4]. It can be shown that minimizing E withrespect to H(z) is equivalent to minimizing(10)
ε = E e2 (n)with respect to D(z) [8]. Thus, the proposed method can beinterpreted as the least mean square of the linear prediction error
as shown in Fig. 2.

III.
. (5)

(9)

e(n)
-

EXAMPLE OF SPEECH ANALYSIS

Fig. 3 shows the estimated spectra for a Japanese sentence”naNbudewa” uttered by a male. A sampling frequency of 10kHz
was used. The signal was windowed by a 25.6ms Blackman window with a 5ms shift and then analyzed using the mel-generalized
cepstral analysis method with M = 12 and several values of(α, γ). The spectra for (α, γ) = (0, −1), (0, 0), and (0.35, 0)
are identical with those obtained by the linear prediction method(LP), the UELS (UELS), and the mel-cepstral analysis method(MCEP), respectively.
From the ﬁgure, it is seen that the obtained spectra have highresolution at low frequencies when α = 0.35. For vowels (e.g.,/e/), it is clear that the resonances are represented accurately
when γ is close to −1. On the other hand, for nasal cases (seeportion of /N/), it is seen that a more accurate representationof the anti-resonances is provided as γ approaches 0 at the expense of increasing the bandwidths of the resonances. From thesefacts, we surmise that every phoneme has its own optimal valuesof (α, γ, M ). However, the parameters (α, γ, M ) are generally
chosen to have constant values. Thus, in the case where the proposed method is applied to speech recognition, it is a reasonable
way to choose the values of (α, γ, M ) which maximize the speech
recognition accuracy.

IV.

PARAMETER FOR SPEECHRECOGNITION

We use the mel-generalized cepstrum as parameter for speech
recognition. However, the values of (α, γ, M ) might be diﬀerent from those used in spectral estimation. In the following, we
assume that the speech signal is analyzed with (α1 , γ1 , M1 ) andthen the obtained mel-generalized cepstral coeﬃcients are transformed to those with (α2 , γ2 , M2 ), which are given by

 ∞

m
H(z) = s−1
c
(m)
Ψ
(z)
.
(11)
α
,γ
γ2
α2
2 2
m=0

The coeﬃcients cα2 ,γ2 (m) can be obtained from cα1 ,γ1 (m) using
the mel-generalized logarithmic transformation [20], which consists of recursion formulas for the frequency transformation [21]aFrequency(kHz) Frequency(kHz) Frequency(kHz)

0
1
2
3
4
5
40(dB)01234540(dB)
0
1
2
3
4
5

N
200

b u

d

e
w
400
(a) Waveform

a

n
(ms)

(α, γ , M) = (0, -1, 12)(α, γ , M) = (0, -1/3, 12)

LP

GCEP

(α, γ , M) = (0, 0, 12)UELS
40(dB)(b) Spectral estimates ( α = 0)

Fig. 3.

0
1
2
3
4
5

0
1
2
3
4
5

b u

d

e
w
400
(a) Waveforma(ms)(α, γ , M) = (0.35, -1, 12)

WLP

40(dB) (α, γ , M) = (0.35, -1/3, 12)

MGCEP

40(dB)
0
1
2
3
4
5

N
200

MCEP
40(dB) (α, γ , M) = (0.35, 0, 12)(b) Spectral estimates ( α = 0.35)

Spectral estimates of natural speech for several values of (α, γ, M ).

and the generalized logarithmic transformation. The algorithm
is shown in Table II. Although parameters for speech recognition should be independent on input gain, the values of cα2 ,γ2 (m)depend on the input gain except for γ2 = 0. To avoid this, wenormalize cα2 ,γ2 (m) as

 ∞

−1

m
cα2 ,γ2 (m) Ψα2 (z)(12)H(z) = Kα2 sγ2m=1

where cα2 ,γ2 (m) is the normalized mel-generalized cepstrum. The
coeﬃcients cα2 ,γ2 (m) are also obtained in the recursion shown in
Table II.
The LPC-cepstrum [22], which is widely used in speech recognition, is identical with the coeﬃcients obtained by the proposed
method with (α1 , γ1 ) = (0, −1) and (α, γ2 ) = (0, 0). We can also
derive several known recursion formulas as special cases of themel-generalized logarithmic transformation [20].

V.

a

0
Frequency(kHz) Frequency(kHz) Frequency(kHz)n0

SPEECH RECOGNITIONEXPERIMENT

A preliminary evaluation of the mel-generalized analysis in word
recognition was carried out using the ATR 5240 Japanese word
data base. Only data from one speaker (speaker MAU) was usedwith just one realization of each word. Training was conductedon the even words, while testing was conducted on the odd words
except 2 words including a phoneme which does not appear in the
training words. We used 32 diﬀerent phoneme models from thestandard ATR labeling scheme, plus an additional silence model,i.e., 33 models in all. The grammar used was a simple 2618 word
lexicon, mapping phoneme sequences to words. The type of HMM
used was a continuous gaussian mixture density model with no
explicit duration modeling. All models were 5-state, 3-mixture
left to right models with no skips. The ﬁrst and last states were
non-emitting.
All feature vectors comprised of 13 mel-cepstral coeﬃcients and
13 delta mel-cepstral coeﬃcients. Both cepstral and delta cepstral coeﬃcients included 0th coeﬃcients. The signal was windowed by a 25.6ms Blackman window with a 5ms shift. The
12th order mel-generalized cepstral analysis was applied with
(α1 , γ1 , M1 ) = (0, −1, 12), (0.35, −1/3, 12), (0.35, 0, 12), where
the analyses with (α1 , γ1 , M1 ) = (0, −1, 12) and (0.35, 0, 12) areequivalent to the linear prediction method and the mel-cepstral

analysis method, respectively. The obtained mel-generalized cepstral coeﬃcients were transformed into 13 mel-cepstral coeﬃcients
i.e., (α2 , γ2 , M2 ) = (0.35, 0, 12).
The recognition results are shown in Fig. 4. As shown in theﬁgure, the mel-generalized cepstral analysis with (α1 , γ1 , M1 ) =(0.35, −1, 12) shows the best result. In this experiment, we tried
only 3 sets of (α1 , γ1 , M1 ) for spectral estimation and one set of(α2 , γ2 , M2 ) for recognition parameter. More experiments will
give the optimal set of parameters.

VI.

DISCUSSION

Several speech spectral estimation methods which use spectral
representation other than conventional AR or ARMA representation have been proposed [10], [13], [4], [9], [12], with the intentionof introducing the characteristics of the human auditory sensation. Based on the proposed method, we can treat them within aframework as shown in Fig 1.
On the other hand, there exist models for simulating the human auditory system, e.g., [23], [24], [25]. Although the proposed
method also takes the characteristics of human auditory sensation into consideration, it is essentially diﬀerent from the models
of auditory system; the proposed method is an approach whichapplies a cost function, i.e., (6) or (10), to a mathematically welldeﬁned model, i.e., (5). Consequently, in the proposed method,
the transfer function of the synthesis ﬁlter is clearly deﬁned as
the spectral model of (5). As a result, based on the proposed
method, we can ﬁnd many applications of the proposed analysis
method in a similar manner of the linear prediction method; analysis/synthesis of speech [7], [26], derivation of adaptive analysis
algorithms [18], [13], [14] (Fig. 1(b)), speech coding [11], [27], etc.
The spectral representation obtained as the result of the PLP
analysis [10] is similar to that obtained by the mel-generalized
cepstral analysis. The diﬀerence between the two methods canbe summarized as follows. The PLP method ﬁrst applies auditorylike modiﬁcation to the power spectrum of speech and thenapplies the linear prediction method to the modiﬁed spectrum.
As a result, the PLP method obtains a spectral representation
similar to the mel-generalized cepstral representation of (5) with
(α1 , γ1 , M1 ) = (0.47, 0, 5) (except for the equal loudness preemphasis in the PLP analysis). On the other hand, the proposed
method ﬁrst assumes the mel-generalized cepstral model of (5)and then determines the mel-generalized cepstral coeﬃcients in

TABLE II

Mel-generalized logarithmic transformation: calculation of cα2 ,γ2 (m) from cα1 ,γ1 (m).

α = (α2 − α1 )/(1 − α1 α2 )

(i−1)

 cα1 ,γ1 (−i) + α cα2 ,γ1 (0),(i−1)
2 (i−1)(i)(1 − α ) cα2 ,γ1 (0) + α cα2 ,γ1 (1),cα2 ,γ1 (m) = c(i−1) (m − 1) + α c(i−1) (m) − c(i)Kα2 = s−1γ1α2 ,γ1m=0, i = −M1 , . . . , −1, 0 (14)
 m=1


α2 ,γ1
α2 ,γ1 (m − 1) , m = 2, 3, . . . , M2(0)cα2 ,γ1 (m) = c(0)m = 1, 2, . . . , M2(15)
α2 ,γ1 (m)/ 1 + γ1 cα2 ,γ1 (0) ,c(0)α2 ,γ1 (0),cα2 ,γ2 (m) = cα2 ,γ1 (m) +

m−1

k=1

cα2 ,γ2 (0) = sγ2 (Kα2 ) ,Linear Prediction(α 1, γ 1, M 1)=(0, -1, 12)
Mel-Generalized
Cepstral Analysis(α 1, γ 1, M 1)=(0.35, -1/3, 12)
Mel-Cepstral
Analysis(α 1, γ 1, M 1)=(0.35, 0, 12)k 
γ2 cα2 ,γ1 (k) cα2 ,γ2 (m − k) − γ1 cα2 ,γ2 (k) cα2 ,γ1 (m − k) ,m
cα2 ,γ2 (m) = cα2 ,γ2 (m) (1 + γ2 cα2 ,γ2 (0)) ,

77.5%

80.7%

79.6%
75
80
Recogniton accuracy (%)

Fig. 4.

Recognition accuracy for several values of (α1 , γ1 , M1 ).

such a way that the criterion of (6) or (10) is minimized. Therefore, the solution of the proposed analysis is unique and optimal in
the sense of the least mean square of the prediction error (Fig. 2).
The motivation of [28] is similar to that of the proposed method.
However, [28] does not clearly separate spectral representation,
spectral criterion, and parameterization for speech recognition.

VII.

(13)





CONCLUSION

In this paper, we have described a new spectral estimation
method based on the mel-generalized cepstral representation,
which can be regarded as a uniﬁed approach to speech spectral
estimation. We have demonstrated that since the characteristicsof the obtained spectrum varies according to the values of α andγ, we can improve the performance of a speech recognition system
by choosing the values of α and γ. Investigation of the optimal
set of (α1 , γ1 , M1 ) and (α2 , γ2 , M2 ) for speech recognition is our
future work.

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