•  Explain why the auto-correlation function of y(n) has peaks at the time instants   n=0, n=N,  and  n=-N.

•  Fix the delay at 1000 samples and choose the amplitude of the echo to be 0.5. Which sound signal among the available choices (Train, Chirp, Splat) leads to the largest peak in the normalized auto-correlation function at  N=1000?  Listen to the sounds with echo in all cases. Which case has the most pronounced echo?

• MATLAB PROJECT. You need to use Matlab to solve this part. First download the file echo_ex (you can do this by holding down the SHIFT key and by clicking the highlighted file name with the LEFT button of your mouse. This will save the file to a directory on your computer. Make sure to save the file to the directory in which you are running Matlab).

Then go to the Matlab command window and load a file that contains two signals with echoes by typing:


load echo_ex

           Now listen to both signals by typing:

soundsc('Exercise1');
soundsc('Exercise2');

and also plot their normalized auto-correlation functions using the commands:

plot(-6999:6999,xcorr(Exercise1,'coeff'));

grid;

figure;

plot(-10000:10000,xcorr(Exercise2,'coeff'));

grid;
 

• Use the above results to remove the echo from the signal Exercise2  via inverse filtering.  The amplitude of the echo is 0.75. You can do this by using the time-domain method explained in the OVERVIEW section of this experiment. The following is a suggested code for your Matlab program. Make sure to listen to sounds before and after echo removal!
  
  

  N=2000; % delay in samples

  y=Exercise2; % Signal with echo

  x(1:N)=y(1:N); % First N samples of x and y agree

  for i=N+1:length(y)

  x(i)=y(i)-0.75*x(i-N); % Inverse filtering

  end

  subplot(211)

  plot(y); % Plot signal with echo

  title('SIGNAL WITH ECHO');

  subplot(212)

  plot(x); % Plot signal without echo

  title('SIGNAL WITHOUT ECHO');

  soundsc(y); % Listen to signal with echo

  soundsc(x); % Listen to signal without echo
   

•  Assume, by mistake, you end up using the wrong value of N (say by estimating it wrongly). Repeat the above inverse filtering procedure by using N=500. Listen again to the sounds before and after echo removal. What do you hear?