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rceps

Real cepstrum and minimum-phase reconstruction

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Syntax

[y,ym] = rceps(x)

Description

[y,ym] = rceps(x) returns both the real cepstrum y and a minimum phase reconstructed version ym of the input sequence.

example

Examples

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Open Live Script

A speech recording includes an echo caused by reflection off a wall. Use the real cepstrum to filter it out.

In the recording, a person says the word MATLAB®. Load the data and the sample rate, Fs=7418Hz.

load mtlb

% To hear, type soundsc(mtlb,Fs)

Model the echo by adding to the recording a copy of the signal delayed by Δ samples and attenuated by a known factor α: y(n)=x(n)+αx(n-Δ). Specify a time lag of 0.23 s and an attenuation factor of 0.5.

timelag = 0.23;
delta = round(Fs*timelag);
alpha = 0.5;

orig = [mtlb;zeros(delta,1)];
echo = [zeros(delta,1);mtlb]*alpha;

mtEcho = orig + echo;

Plot the original, the echo, and the resulting signal.

t = (0:length(mtEcho)-1)/Fs;

% To hear, type soundsc(mtEcho,Fs)

subplot(2,1,1)
plot(t,[orig echo])
legend("Original","Echo")

subplot(2,1,2)
plot(t,mtEcho)
legend("Total")
xlabel("Time (s)")

Figure contains 2 axes objects. Axes object 1 contains 2 objects of type line. These objects represent Original, Echo. Axes object 2 with xlabel Time (s) contains an object of type line. This object represents Total.

Compute the real cepstrum of the signal. Plot the cepstrum and annotate its maxima. The cepstrum has a sharp peak at the time at which the echo starts to arrive.

c = rceps(mtEcho);

[px,locs] = findpeaks(c,Threshold=0.2,MinPeakDistance=0.2);

clf
plot(t,c,t(locs),px,"o")
xlabel("Time (s)")

Figure contains an axes object. The axes object with xlabel Time (s) contains 2 objects of type line. One or more of the lines displays its values using only markers

Cancel the echo by filtering the signal through an IIR system whose output w obeys w(n)+αw(n-Δ)=y(n). Plot the filtered signal and compare it to the original.

dl = locs(2)-1;

mtNew = filter(1,[1 zeros(1,dl-1) alpha],mtEcho);

% To hear, type soundsc(mtNew,Fs)

subplot(2,1,1)
plot(t,orig)
legend("Original")

subplot(2,1,2)
plot(t,mtNew)
legend("Filtered")
xlabel("Time (s)")

Figure contains 2 axes objects. Axes object 1 contains an object of type line. This object represents Original. Axes object 2 with xlabel Time (s) contains an object of type line. This object represents Filtered.

Input Arguments

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Input signal, specified as a real vector.

Output Arguments

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Real cepstrum, returned as a vector.

Minimum phase real cepstrum, returned as a vector.

Algorithms

The real cepstrum is the inverse Fourier transform of the real logarithm of the magnitude of the Fourier transform of a sequence.

Note

rceps only works on real data.

rceps is an implementation of algorithm 7.2 in [2], that is,

y = real(ifft(log(abs(fft(x)))));

Appropriate windowing in the cepstral domain forms the reconstructed minimum-phase signal:

w = [1;2*ones(n/2-1,1);ones(1-rem(n,2),1);zeros(n/2-1,1)];
ym = real(ifft(exp(fft(w.*y))));

References

[1] Oppenheim, Alan V., and Ronald W. Schafer. Digital Signal Processing, Englewood Cliffs, NJ, Prentice-Hall, 1975.

[2] Programs for Digital Signal Processing, IEEE Press, New York, 1979.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

Introduced before R2006a

See Also

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