Complex cepstral analysis
This example uses
cceps to show an echo. Generate a sine of frequency 45 Hz, sampled at 100 Hz. Add an echo with half the amplitude and 0.2 s later. Compute the complex cepstrum of the signal. Notice the echo at 0.2 s.
Fs = 100; t = 0:1/Fs:1.27; s1 = sin(2*pi*45*t); s2 = s1 + 0.5*[zeros(1,20) s1(1:108)]; c = cceps(s2); plot(t,c) xlabel('Time (s)') title('Complex cepstrum')
x— Input signal
Input signal, specified as a real vector. By the application of a linear phase term, the input is altered to have no phase discontinuity at ±π radians. That is, it is circularly shifted (after zero padding) by some samples, if necessary, to have zero phase at π radians.
n— Length of zero-padded signal
Length of zero-padded signal, specified as a positive real integer.
xhat— Complex cepstrum
Complex cepstrum, returned as a vector.
nd— Number of samples
Number of samples of circular delay added to
x, returned as a
positive real scalar.
xhat1— Second complex cepstrum
Second complex cepstrum, returned as a vector.
computed using an alternative factorization algorithm specified in the references  and . This method can be applied only to finite-duration signals. See the Algorithm
section below for a comparison of the Fourier and factorization methods of computing the
Cepstral analysis is a nonlinear signal processing technique that is applied most commonly
in speech processing and homomorphic filtering .
is an implementation of algorithm 7.1 in . A lengthy Fortran
program reduces to these three lines of MATLAB® code, which compose the core of
h = fft(x); logh = log(abs(h)) + sqrt(-1)*rcunwrap(angle(h)); y = real(ifft(logh));
rcunwrap in the above code segment is a special version of
unwrap that subtracts a straight line from the phase.
rcunwrap is a local function within
cceps and is not
available for use from the MATLAB command line.
The following table lists the pros and cons of the Fourier and factorization algorithms.
|Fourier||Can be used for any signal.||Requires phase unwrapping. Output is aliased.|
|Factorization||Does not require phase unwrapping. No aliasing||Can be used only for short duration signals. Input signal must have an all-zero Z-transform with no zeros on the unit circle.|
In general, you cannot use the results of these two algorithms to verify each other. You can use them to verify each other only when the first element of the input data is positive, the Z-transform of the data sequence has only zeros, all of these zeros are inside the unit circle, and the input data sequence is long (or padded with zeros).
 Oppenheim, Alan V., Ronald W. Schafer, and John R. Buck. Discrete-Time Signal Processing. Upper Saddle River, NJ: Prentice Hall, 1999, pp. 788–789.
 Steiglitz, K., and B. Dickinson. “Computation of the Complex Cepstrum by Factorization of the Z-transform.” Proceedings of the 1977 IEEE® International Conference on Acoustics, Speech and Signal Processing, pp. 723–726.
 Digital Signal Processing Committee of the IEEE Acoustics, Speech, and Signal Processing Society, eds. Programs for Digital Signal Processing. New York: IEEE Press, 1979.