Signal Processing 13 (1987) 79-83 79 North-Holland

S H O R T C O M M U N I C A T I O N

O N T H E C O M P U T A T I O N O F C O M P L E X C E P S T R U M

T H R O U G H D I F F E R E N T I A L C E P S T R U M

G.R. REDDY and V.V. RAO

Department of Electrical Engineering, Indian Institute of Technology, Madras 600 036, India

Received 21 June 1986 Revised 4 November 1986

Abstract. A relation between complex cepstrum and differential cepstrum is derived. The problem of aliasing associated with the computation of complex cepstrum through differential cepstrum is demonstrated with an example.

Zusammenfassung. Es wird der Zusammenhang zwischen dem komplexen Cepstrum und dem differentiellen Cepstrum hergeleitet. Anhand eines Beispieles wird das Problem der Uberfaltung erl~iutert, das mit der Bestimmung des komplexen Cepstrums mit Hilfe des differentiellen Cepstrums verbunden ist.

R6sum6. Une relation entre le cepstre complexe et le cepstre diff6rentiel est d6riv6e. Le probl~me de recouvrement associ6 avec le calcul du cepstre complexe ~ l'aide du cepstre diff6rentiel est d6montr6 avec un exemple.

Keywords. Complex cepstrum, differential cepstrum.

Introduction

Complex cepstrum has wide applications in the field of seismic signal processing [7], image enhancement

[2] and other related fields. The advantage of the cepstral processing is that the process of convolution

of two signals in the time domain is reduced to mere addition in the cepstral domain. But the main

disadvantage of the cepstrum is the ambiguity in its phase relationship and the necessity for a complicated

phase-unwrapping algorithm [6]. Polydoros and Fam [3] have introduced the differential cepstrum as a

new tool for homomorphic deconvolution without the need for phase unwrapping. The concept of

differential cepstrum has been extended to multidimensional signals [4, 5]. In this paper we bring out the relation between the complex cepstrum and differential cepstrum. We also present through an example

the problem of aliasing in computing the complex cepstrum through the differential cepstrum as compared

to cepstral computation via phase-unwrapping methodology.

Relation between complex cepstrum and differential cepstrum

The differential cepstrum, Xd(n), of a sequence x ( n ) is defined as

t " Z - , [ d X ( z ) / d z ] ~d(n)= Z- [X,,(z)]= L X(z) 3"

0165-1684/87/$3.50 O 1987, Elsevier Science Publishers B.V. (North-Holland)

(1)

8 0 G.R. Reddy, V. V. Rao / On the computation o f complex cepstrum

The Fourier transform representation of (1) is

F_~F-e-i'°F[nx(n)]] ~(n)= L ~ J" (2)

As there is no explicit logarithmic operation involved in the computation of ~d(n) as given by (2), the need for phase unwrapping is avoided. The relation between ~d(n) and the complex cepstrum £(n) is derived as follows: From (1)

)(d(z) d l n X ( z ) dX(z) - dz - d----~ ' (3)

zXd(z)= ~ -n~(n)z-". (4)

Taking the inverse z-transform on both sides of (4) we get the complex cepstrum

.~(n)=-~d(n+l)/n fo rn¢O. (5)

Because of the normalization process involved in its definition, the differential cepstrum does not retain the constant scale factor information in the sequence x(n), hence £(0) which is equal to the average value of lnlX(w) I has to be computed separately. Bednar and Watt [1] proposed a method for computing the complex cepstrum which is equivalent to the method discussed above. Through some simulated examples they have shown that the complex cepstrum computed either through phase unwrapping or by using their method will give identical results. In the examples they considered the complex cepstral values are decaying fast enough and are almost zero beyond ~N, where N is the data length. The FFT size 2N used in their examples is long enough not to give rise to any aliasing problem even if the complex cepstrum is computed through the differential cepstrum. But, in general, it is observed that the complex cepstrum computed through the differential cepstrum using DFT is more prone to aliasing problems as we demonstrate here with an example.

The aliasing problem in computing complex cepstrum through differential cepstrum using DFT

When differential cepstrum 3~d(?l ) is computed using DFT of size N, we can get a periodic differential cepstrum, ~dp(n), where

oo

~dp(n) = Y~ £a(n+kN). (6) k = - - o o

For a sequence x(n) with Z-transform

rn i m o

Azr l-I (1--ak z-l) I] (1--bkZ) k = l k = l V f ~

,qL ~ Z ] -- p, Po

I] (1--CkZ-') 1] (1-dkz) k = l k = l

(7)

Signa l P rocess ing

G.R. Red@, V.V. Rao / On the computation of complex cepstrum 81

where lakl, [bkl, Ick[ and Idkl are all less than unity, the differential cepstrum is [3];

n--1 ~ n - 1 - - a k , n > 1 ,

k = l

- r, = l ,

/ Po mo

E dk "+ l - E b~ "+1, n < l . k k = l k = l

(8)

It should be noted that ~d(n), unlike complex cepstrum, does not decay with an envelope of 1/n. This indeed can give rise to aliasing problem when it is computed with fixed size DFT. Hence the periodic cepstrum, ~p(n), will also be aliased. In a way when the complex cepstrum is computed through differential cepstrum using DFT, the periodic complex cepstrum ~p(n) in terms of actual complex cepstral values

~(n) is

~p(n) = - (n + k N ) ~ ( n + k N ) , (9) /I k =-,x~

whereas, if it is computed using the phase-unwrapping algorithm,

oo

~p(n)= ~ ~ ( n + k N ) , (10) k=--oo

Hence the complex cepstrum computed using (9) will always be more aliased than the one computed using (10).

Example

The problem of aliasing has been demonstrated through a simulation example. The signal considered is a damped sinusoid of 100 Hz with a damping constant of 0.1 and sampled at a frequency of 10 kHz. This basic signal is convolved with three impulses at n = 0, 23 and 46 with impulse strengths 1.0, 1.6 and 0.8 respectively. This wavelet is shown in Fig. l(a). Fig. l(b) is the complex cepstrum computed with an FFT size of 256 through Tribolet's phase-unwrapping algorithm [6]. The complex cepstrum computed through the differential cepstrum with N = 256 and N = 512 are shown in Figs. l(c) and l(d) respectively. Comparing Figs. l(c) and l(d) it can be seen that there is more aliasing-iaa the case of Fig. l(c). Further, comparing Figs. l(d) and l(b), we find that there is still a certain amount of aliasing in the case of Fig. l(d), though increasing N from 256 to 512 has considerably reduced the aliasing error. To see more clearly the effect of aliasing, the time domain wavelet has been reconstructed using the complex cepstral values of Figs. l(b)-(d). The signal reconstructed from the cepstral values of Fig. l(b) gives exactly the original time domain wavelet as in Fig. l(a). The wavelets reconstructed from the cepstral values of Figs. l(c) and l(d) are shown in Figs. 2(a) and 2(b) respectively. These figures clearly demonstrate the problem of aliasing in computing a complex cepstrum through a differential cepstrum and the reduction of the aliasing error as the DFT length is increased.

Vol. 13, No. 1, July 1987

82 G.R. Reddy, V. V. Rao / On the computation of complex cepstrum

x(n) /'~ (a)

0 . . . . . " I I I I ~ I I I ~ n 255

1.0 ^

x(n)

(b)

i ., : ,, ~ :~ . ~, , , . . . . r , • , i i , ~n 255

3'0

x(n)

0

x(n)

0 "~ : y ~-I-;._~nlr : v ; ~ ; v " "" '" '~5'51

- t . O

1.0

~(n)

i (d)

• I i I i 1 1 • I I I I . i I . I i I t ~k

Fig. 1. Complex cepstrum computation of a wavelet. (a) Original time domain wavelet. (b) Complex cepstrum com- puted through the phase-unwrapping methodology with NFFr=256. (C) Complex cepstrum computed through the differential cepstrum with NFF r = 256. (d) Complex cepstrum computed though the differential cepstrum with NF~ = 512.

4.0

x(n)

(b)

I I i ; I-,"P ~, I I I- I-I-

---o,- n 511

Fig. 2. Time domain wavelet reconstruction using the complex cepstrum. (a) Reconstructed signal using the cepstral values of Fig. l(c). (b) Reconstructed signal using the cepstral values

of Fig. l(d).

C o n c l u s i o n s

The computa t ion o f complex cepstrum without the need o f phase unwrapping th rough differential

cepstrum is brought out. It has been demonst ra ted that aliasing is a major problem when complex cepstrum

is computed through differential cepstrum using the DFT. One may have to pay the price in terms of

using an FFT of greater length for the benefit o f avoiding the phase unwrapping.

Signal Processing

G.R. Reddy, V. V. Rao / On the computation of complex cepstrum 83

References

[1] J.B. Bednar and T.L. Watt, "Calculating the complex cep- strum without phase unwrapping or integration", IEEE Trans. Acoust., Speech, Signal Process., Vol. ASSP-33, August 1985, pp. 1014-1017.

[2] A.V. Oppenheim and R.W. Schafer, Digital Signal Process- ing, Prentice-Hall, Englewood Cliffs, NJ, 1975.

[3] A. Polydoros and A.T. Fam, "The differential cepstrum: Definition and properties", Proc. IEEE Internat. Conf. on Circuits and Systems, 1981, pp. 77-80.

[4] D.R. Reddy and R. Unbehauen, "The N-dimensional differential cepstrum", Proc. lnternat. Conf. on Digital Signal Processing, 1984, pp. 145-148.

[5] D.R. Reddy and R. Unbehauen, "The two dimensional differential cepstrum", IEEE Trans. Acoust., Speech, Signal Process., Vol. ASSP-33, October 1985, pp. 1335-1337.

[6] J.M. Tribolet, "A new phase unwrapping algorithm", IEEE Trans. Acoust., Speech, Signal Process., Vol. ASSP-25, April 1977, pp. 170-177.

[7] J.M. Tribolet, Seismic Applications of Homomorphic Signal Processing, Prentice-Hall, Englewood Cliffs, N J, 1979.

Vol. I3, No. 1, July 1987