rceps

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, $$F_s=7418Hz$$.

load mtlb

% 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;

% 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)")

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);

% 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)")