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, $$F_s=7418Hz$$.

loadmtlb% To hear, type soundsc(mtlb,Fs)

Model the echo by adding to the recording a copy of the signal delayed by $$\Delta$$samples and attenuated by a known factor $$\alpha$$: $$y(n)=x(n)+\alpha x(n-\Delta )$$. 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, $$w$$, obeys $$w(n)+\alpha w(n-\Delta )=y(n)$$. Plot the filtered signal and compare it to the original.

dl = locs(2)-1; mtNew = filter(1,[1 zeros(1,dl-1) alpha],mtEcho); 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)