A speech recording includes an echo caused by reflection off a wall. Use autocorrelation 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 an unbiased estimate of the signal autocorrelation. Select and plot the section that corresponds to lags greater than zero.

[Rmm,lags] = xcorr(mtEcho,'unbiased');
Rmm = Rmm(lags>0);
lags = lags(lags>0);
figure
plot(lags/Fs,Rmm)
xlabel('Lag (s)')

The autocorrelation has a sharp peak at the lag at which the echo arrives. Cancel the echo by filtering the signal through an IIR system whose output,
, obeys
.

[~,dl] = findpeaks(Rmm,lags,'MinPeakHeight',0.22);
mtNew = filter(1,[1 zeros(1,dl-1) alpha],mtEcho);

Plot the filtered signal and compare to the original.

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)