AN ADAPTIVE ALGORITHM FOR MEL-CEPSTRALANALYSIS OF SPEECH
Toshiaki Fukadat, Keiichi Tokudatt, Takao Kobayashzjttand Satoshi Imaittt
thformation Systems Research Center, Canon Inc., Kawasaki, 211 JAPAN (toshiakiOcis.canon.co.jp)
ttDepartment of Electrical and Electronic Engineering, Tokyo Institute of Technology, Tokyo, 152 JAPANtttPrecision and Intelligence Laboratory, Tokyo Institute of Technology, Yokohama, 227 JAPANABSTRACTS P E C T R A LE S T I M A T I O NB A S E DO N
MEL-CEPSTRAL REPRESENTATION
2.1 S p e c t r a lM o d e l and C r i t e r i o nWe represent the model spectrum H ( e J w )by the M-thorder mel-cepstral coefficients ? ( m )as follows:2.

Thispaperdescribes
a mel-cepstralanalysismethod
and its adaptive algorithm. In the proposed method,
weapply the criterion used in the unbiased estimation of log
spectrum to the spectral model represented by themelcepstral coefficients. To solve the non-linear minimization
problem involved in the method, we give an iterative algorithm whose convergence is guaranteed. Furthermore, we
derive an adaptive algorithm for the mel-cepstral analysis
by introducing an instantaneous estimate
for gradient ofthecriterion.Theadaptivemel-cepstralanalysissystem
is implemented with an IIR adaptive filter
which has anexponential transfer function, and whose stability is guaranteed. We also present examplesof speech analysis andresults of a n isolated word recognition experiment.
1.

M~ ( mz'-")~ ( z=)exp

(1)

m=O

whereThe phase characteristic
i-' = e--3' is given byof the all-pass transfer function(I - a 2 ) s i n w(1 a ~ ) c o s w- 2 0 'For example,for a samplingfrequency of lOkHz, LJ is agood approximation to the me1 scale based on subjective
pitch evaluations when a = 0.35[4].
To obtainanunbiasedestimate,weuse the following
criterion[2] and minimize it with respect to {?(m)}$o.

G = tan-'

INTRODUCTION

The spectrum represented by the mel-cepstral coefficients have frequency resolution similar to that of the human ear which hashighresolution
at low frequencies[l].
As a result, mel-cepstral coefficients are useful for speechsynthesis and recognition. For obtaining mel-cepstral coefficients, several methods have been proposed. For example, the mel-cepstralcoefficients are obtained from the LPC
coefficients by using the technique of spectral resampling.
No strict method, however, is proposed in which the spectral model is represented by mel-cepstral coefficients and aspectral criterion is minimized.
In this paper, we propose a mel-cepstral analysis method
anditsadaptivealgorithm.Inthemel-cepstralanalysis
method, the model spectrum is represented by the M-thorder mel-cepstral coefficients and the criterion used in theunbiased estimation of log spectrum[2] is minimized withrespect to the mel-cepstral coefficients. The minimization
problem is solved efficiently by an iterative technique using the FFT, recursion formulas, and a fast algorithm that
requires O ( M Z )arithmetic operations. We can show thatthe convergence is quadratic and typically a few iterations
are sufficient to obtain the solution.
Furthermore, we present an adaptive algorithm for themel-cepstralanalysis.Toderivetheadaptivealgorithm,
we introduce an instantaneous estimatefor the gradient of
the criterion in a similar manner of the LMS algorithm[3].
The adaptive analysis system is implemented with an IIR
adaptive filter which has the structure of the MLSA filter[4] and whose stabilityis guaranteed. The adaptive analysis
system requires O ( M ) operations per sample to implement
the M-th order mel-cepstral analysis.
We show examples
of analysis for synthetic and speech signal.To evaluate theproposed methods, an isolated word recognition experiment
was carried out.

.

rr+{exp R ( w ) - R ( w ) - 1) dw

whereR(w) = logIN(w)-log

1 H ( P ) I*(4)(5)

and I N ( w )is the modified periodogram of a weakly stationary process z(n) with a time window of length N . To takethe gain factor A' outside from H ( r ) , we rewrite (1)as
M~ ( z=)expb(m)+ " , ( z ) = I C . ~ ( r )

(6)

m=O

where

Ii' = exp b(0)

(7)

(8)
m=l

andE.(nz)=Since H ( r ) is a minimum phase system, we can show thatis equivthe minimization of E with respect to {Z(m)}E=oalent to that of

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with respect tob = [ b ( l ) , b ( 2 ) , . .. , b(M)IT.
(12)
The gain factor 11'that minimizes E is obtained by setting

d E / d I i = 0:

K=G(13)+where emin is the minimized value of E .
2.2

Solution for theMinimizationProblem

Since E is convex with respect to b [5], the minimization problem of (11) can be solved by the Newton-Raphson
method. For thei-thresultb('), solving a set of linear
equations++f

we have the values

Ab(') = [Ab(')(l),Ab(')(2),. . . ,A d i ) ( M ) I T(15)
where H is the Hessian matrix H = @ € / a babT. Then thenext result is obtained as follows:b('+1)= b(') + Ab(').
(16)
The gradient V Eis given by
Ve = -2F = -2[?(1), ?(2), . . . ,
(17)
where

and the Hessian matrix H is given byM

H = 2{h(i?~')}i,j=1

Fig. 2. Block diagram of the adaptive mel-cepstralanalysis.
and em(n)is the output of the filter am(z) (see Fig. 1).
To derive anadaptivealgorithm,
we introducean instantaneousestimateinasimilarmannerof the LMS
algorithm[3](19)whereIn this paper, to suppress fluctuation of b, we estimate Ve
using an exponential window as follows:
71

(20)

Since the matrix H is a symmetric Toeplitz plus Hankel
matrix, (14) can be solved using a fast recursive algorithm[6] which requires O ( M 2 ) arithmeticoperations. Coefficients h ( i , j ) and F(m) can be calculated efficiently using
the FFT and recursion formulas[5]. We can obtain an initial guess b(O) from the FFT cepstrum using a recursion
formula[5]. The convergence is quadratic because the Hessian matrix is positive definite. We have found that typically a few iterations are sufficient to obtain the solution.

ADAPTIVEMEL-CEPSTRALANALYSISALGORITHM
3.1 DerivationofAdaptiveAlgorithm
Replacing H in (14) with the unit matrix,we can derive3.

the method of steepest descent from the Newton-Fbphsonb(') the next result
method. That is, from the i-th resultb('+') is given by

where p is the adaptation step size.
Assuming that the time window length N is sufficiently
large, we can interpret (11) as the mean square of e(n):E = E [e2(n)](22)
where e(n) is the output of the inverse filter l / D ( t ) driven
by ~ ( n )According.
to the assumption, the gradient givenin (17) becomesO€("f= -2(1- 7)

Tn-'e(i) e t ):=-m- T ~ E ( ' + - ' )- 2(1 - r ) e ( n ) e t ) , 0 5 T < 1.(26)With this estimate of the gradient, we can specify an adaptive algorithm based on the method of steepest descent:
the coefficients vector b(") a t time n is updated asb("+') = b(")- p ( n ) Te('+).

(27)

When the gain of the signal z ( n ) is time-varying, p is
normalized as follows:
ap(n) = O<a<l(28)M ~ ( n ')where d n ) is an estimate of E at time nnE('*)= (1 - A )X"-' e2(i)I=--- Adn-')+ (1 - X ) e 2 ( n ) ,0 5 X < 1. (29)
Using (13), we can get an estimateof I< at timen from e(").
The block diagram of the adaptive mel-cepstral analysisis
we will discuss
depicted in Fig. 2. In thenextsection,a realization method of the exponential transfer function

1/D(z).
The mel-cepstralcoefficients { E ( r n ) } ~ = = , can be obtained
from I< and b using ( 7 ) and (9). Notethatthe
above
algorithm is equivalent to the adaptive cepstral
analysisalgorithm[7] when cr = 0.

Realization of ExponentialTransferFunction
Although the transfer function l / D ( r ) is not a rational
function,the MLSA filter[4] canapproximate l/D(z) inFig. 2 with sufficient accuracy. The complex exponential

3.2

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Input&...-+I

output-2dIIII11234 1 5 2Frequency (kHz)(a) Basicfilter F ( z )LOOutplut(a) LPC analysis

3I4
Frequency (kHz)( b ) M e l - c e p s t r aaln a l y s i sFig. 4. Spectral estimates for synthetic signal(MWhen F ( z ) is expressed asI I

I+F ( z ) = Fl(2) F Z ( Z )(35)
the exponential transfer function is approximated in a cascade form
I/D(z) = exp(-F(z)) = exp(-FI(z)) .exp(-Fz(z))- R L ( - F I ( ~ ).)R L ( - F ~ ( ~ ) )(36)
as shown in Fig. 3(c). Ifmay F1(ejw) , m a y F2(eJW) < max F ( e j w )(37)

JII(b) R L ( - F ( z ) )N ~ / D ( z ) L = 4

WIIw

1IWI

M

F ? ( r )=TABLE I Coefficients of R4(w)1b ( m )@,,,(z).

WL
I

I=1

exp w 2: R L ( w )='(30)

Thus l/D(z) is approximated as follows:
RL(-F(~N) ) exp(-F(z)) = 1 / D ( z )where F ( z ) is defined by(31)+1

LA L , ,(-w)'
k 1Mm=lThe filter structure of F ( z ) is showninFig.
3(a). Figure3(b) shows the block diagram of the MLSA filter& ( - F ( z ) ) for the case of L = 4.
When we use the coefficients A4,1 showninTable
I,& ( - F ( z ) ) is stable and becomes a minimum phase system under the condition
F(eJ"') 5 6.2.
(33)
Furthermore, we canshow
thattheapproximationerror l o g l / D ( e j W )- l o g R 4 ( - F ( e j W ) ) doesnotexceed
0.24dB[8] under the condition
F ( e j w ) 5 4.5.
(34)

11IIII(40)Wfor speech sounds, RL(-Fl(z)) . R L (-F~(z))
approximatesthe exponential transfer functionl / D ( z )with sufficient accuracy and becomes a stable system.
To implementtheM-thorderadaptivemel-cepstral
analysis, the analysis system requires O ( M ) operations per
sample.Thusitcanbeimplementedwithonecurrently
available DSP.

function exp w is approximated by a rational function

A L , Iw

(39)

m=?

Since we empirically found thatm i t v I F 1 ( e J Y ) I , m a s I F ? ( e j w )<4.5
I+1,

it is expected that RL(-F1(ejw)) RL(-F2(eJW)) approximates l/D(ej") more accurately than RL(-F(e'")).
In the following experiments, we let
Fl(2) = b ( l ) Q l ( Z )(38)(c) Two-stage cascade structure&(-F1(z)). & ( - F z ( Z ) ) 2 1 / D ( z )Fig. 3. Implementation of l / D ( z ) .

1= 12).

4. E X P E R I M E N T A L RESULTS
In the following experiments, o was set to 0.35 and M
wasset to 12 except that it wasset to 15 in 4.3. In theadaptive algorithm, u , X, and T were set to 0.12, 0.98, and0.92, respectively.

4.1 Analysis of S y n t h e t i c S i g n a l
Figure 4 shows the spectral estimates for synthetic signal compared with the LPC analysis. The signal
was generated by driving an ARMA filter by a pulse train with unit
variance. It is seen that mel-cepstral analysis can estimate
resonances and anti-resonanceswhich the LPC analysis cannot estimate accurately.
Figure 5(a) shows the convergence characteristics of theadaptive mel-cepstral analysis for the same synthetic signal. The spectral estimates obtained from the mel-cepstralcoefficients at the 400th and the 800th iterations are givenin Figs. 5(b) and (c), respectively.FromFig. 5 , it is seenthat the adaptive mel-cepstral analysis has fast and stable
convergence characteristics.

Analysis of N a t u r a lS p e e c hFigure G shows the result of a natural speech analysis.
The signal shown in Fig. G(a) is a natural Japanese speech
4.21-139

M = 12.;‘aI0. 3 5200

00I2000400

400(a)Waveform

600(ms)600(ms)

J

600

800

N u m b e r o ift e r a t i o n s(a)-40m-true

4Iestimate

true

200
400
( b ) Mei-cepstraicoefficienls

-1NI

-2d1

2

3

4
5
Frequency (kHz)( b ) 400 i t e r a t i o n s

12

3

45U”

Frequency (kHz)(c) 800 iteratinos

u40(d8) (c)Spectralestimates(Mel-cepstralanalysis)Fig. 5. Convergence characteristics of the adaptive
mel-cepstral analysis algorithm.
/naNbudewa/sampledat
10kHz. Figure6(b) shows themel-cepstral coefficient ? ( I ) versus iteration number. Log
magnitude spectra shown in Figs. G(c) and (d) are obtained
from the mel-cepstral coefficients {2(rn)}z=o
at intervalsof 5ms. From Fig. 6, it is seen that the proposed analysis
methods have high resolution a t low frequencies in spiteof a smallorder of analysis(M = 12), and the adaptivealgorithm has sufficiently fast convergence characteristics
for speech analysis.
A p p l i c a t i o nt oS p e e c hR e c o g n i t i o nTo evaluate the performance of the mel-cepstral analysis method(MC) and the adaptive mel-cepstral analysis
method(AMC),anisolatedwordrecognitionesperiment
was carried out. We selected a vocabulary of highly confusing 20 Japanese city names uttered twice by five malespeakers. The recognition wasaccomplishedusing
theDTW(DynamicTimeWarping)andtheEuclidean
melcepstral distance calculated from the mel-cepstralcoefficients a t intervals of 10ms. For comparison, we used themel-cepstral coefficients(M = 15) calculated from the LPC
coefficients(M = 12) usingtherecursionformula(LPC).
Table I1 shows theresults for therecognitionrates.
Weachieved higher recognition rates using MC and AMC thanthat using LPC.
4.3

( d ) Spectralestimates(Adaptivemei-cepstraianalysis)Fig. G. An example of analysis for natural speech.

TABLE I1Recognition

rates for severalanalysis
methods.

Analysis

AMC

intraspeaker
interspeaker
total
90.0 %
91.8 %
98 %
91.0 %

91.2 %
93.2 %
92.4 %REFERENCES
G. Fant : “Speech sound and features,” MIT Press, Cambridge (1973).
S. Imai and C. Furuichi : “Unbiased estimator of log spect r u m and its application to speech signalprocessing,” inProc. 1989 EURASIP, Sep.1988,pp.203-206.
B. Widrow and S. D. Stearns : AdaptiveSignal Processing,Englewood Cliffs, N.J.: Prentice-Hall, 1985.
S. Imai, K. Sumita a n d C. Furuichi : “Me1 log spectrum
approximation (ML SA) filter for speech synthesis,” Trans.
IECE, vol.JGG-A,pp.122-129,Feb.1983.
K. Tokuda, T. Kobayashi, T. Fukada, H. Saito and S. Imai :“Spectral estimation of speech based on mel-cepstral representation,” Trans. IEICE, vol. Ji4-A, pp.1240-1248,Aug.
1991.
I. Gohberg and I. Koltracht : “Efficient algorithm for
Toeplitz plus Hankel matrices,’’ Integral Equations andOperator Theory, vol. 12,,pp.136-142, 1989.
K. Tokuda, T. Kobayashl, Shoji Shiomoto a n d S. Imai
: “Adaptive filtering based o n cepstral representation Adaptive cepstral analysis of speech,” in Proc. ICASSP
90, Apr. 1990,pp.377-380.
T. Kobayashi and S. Imai : “Complex Chebyshev approximation for IIR digital filters using an iterative WLS technique,” in Proc. ICASSP 90, A p r . 1990,pp.1321-1324.
T.Fukada, K. Tokuda, T. Kobayashi and S. Imai : “A studyof speech coding based on the adaptive mel-cepstral analysis,’’ IEICE Technical Report, SP91-73, Oct. 1991, pp.53GO.
K . Tokuda, T. Kobayashi, T. Fukada and S. Imai : “Adaptive mel-cepstral analysis of speech,” Trans. IEICE, vol.
J74-A, pp.1249-1256,Aug.1991.

5. C O N C L U S I O N
In this paper, we have presented the mel-cepstral analysis method and its adaptive algorithm. The mel-cepstralanalysis is efficient for the estimationof spectra which haveresonancesandanti-resonancesat
low frequencies. The
adaptive mel-cepstral analysis system is implemented withan IIR adaptive filterwhich has an exponential transfer
function and whose stability is guaranteed. It is shown thatthe adaptive algorithm requires O ( M ) operations per sample to obtain the M-th order mel-cepstral coefficients andhasfast convergence properties. The effectiveness of the
proposed methods was also shown by the some experimental results. Potential applicationof the adaptive analysis
to speech coding[9] is currently investigated.

ACKNOWLEDGMENT
Acknowledgment is made to the Director of Information Systems Research Centerof Canon Incorporated forpermission to publish this paper.

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