cceps

Complex cepstral analysis

Syntax

`xhat = cceps(x)[xhat,nd] = cceps(x)[xhat,nd,xhat1] = cceps(x)[...] = cceps(x,n)`

Description

Cepstral analysis is a nonlinear signal processing technique that is applied most commonly in speech processing and homomorphic filtering [1].

 Note   `cceps` only works on real data.

`xhat = cceps(x)` returns the complex cepstrum of the real data sequence `x` using the Fourier transform. The input is altered, by the application of a linear phase term, 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.

`[xhat,nd] = cceps(x)` returns the number of samples `nd` of (circular) delay added to `x` prior to finding the complex cepstrum.

`[xhat,nd,xhat1] = cceps(x)` returns a second complex cepstrum, `xhat1`, computed using an alternative factorization algorithm [1][2]. 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 complex cepstrum.

`[...] = cceps(x,n)` zero pads `x` to length `n` and returns the length `n` complex cepstrum of `x`.

Examples

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Using cceps to show an echo

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') ```

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Algorithms

`cceps` is an implementation of algorithm 7.1 in [3]. A lengthy Fortran program reduces to these three lines of MATLAB® code, which compose the core of `cceps`:

```h = fft(x); logh = log(abs(h)) + sqrt(-1)*rcunwrap(angle(h)); y = real(ifft(logh)); ```
 Note   `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.

AlgorithmProsCons
FourierCan be used for any signal.Requires phase unwrapping. Output is aliased.
FactorizationDoes not require phase unwrapping. No aliasingCan 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).

References

[1] Oppenheim, Alan V., Ronald W. Schafer, and John R. Buck. Discrete-Time Signal Processing. Upper Saddle River, NJ: Prentice Hall, 1999, pp. 788–789.

[2] 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.

[3] Digital Signal Processing Committee of the IEEE Acoustics, Speech, and Signal Processing Society, eds. Programs for Digital Signal Processing. New York: IEEE Press, 1979.

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