Real cepstrum and minimum phase reconstruction
[y,ym] = rceps(x)
The real cepstrum is the inverse Fourier transform of the real logarithm of the magnitude of the Fourier transform of a sequence.
rceps(x) returns the real cepstrum
of the real sequence
x. The real cepstrum is a
[y,ym] = rceps(x) returns
both the real cepstrum
y and a minimum phase reconstructed
ym of the input sequence.
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, .
load mtlb % To hear, type soundsc(mtlb,Fs)
Model the echo by adding to the recording a copy of the signal delayed by samples and attenuated by a known factor : . 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; subplot(2,1,1) plot(t,[orig echo]) legend('Original','Echo') subplot(2,1,2) plot(t,mtEcho) legend('Total') xlabel('Time (s)')
% To hear, type soundsc(mtEcho,Fs)
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)')
Cancel the echo by filtering the signal through an IIR system whose output, , obeys . Plot the filtered signal and compare it to the original.
dl = locs(2)-1; mtNew = filter(1,[1 zeros(1,dl-1) alpha],mtEcho); subplot(2,1,1) plot(t,orig) legend('Original') subplot(2,1,2) plot(t,mtNew) legend('Filtered') xlabel('Time (s)')
% To hear, type soundsc(mtNew,Fs)
rceps is an implementation of algorithm 7.2
in , 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))));
 Oppenheim, Alan V., and Ronald W. Schafer. Digital Signal Processing, Englewood Cliffs, NJ, Prentice-Hall, 1975.
 Programs for Digital Signal Processing, IEEE Press, New York, 1979.