Question

Use MATLAB to: Design an Echo Filter. The output should be the original signal followed by three echoes with attenuations of 10%, 15 % and 25% respectively and the delays should be 0.9 sec, 1.4 sec, and 1.8 sec respectively for each echo. Hint: Based on this information the original signal appears at the output with the same amplitude, the first echo has amplitude 0.9 times the amplitude of the original (10% attenuation), the second echo has amplitude 0.85 (15% attenuation) and the third echo has amplitude of 0.75 (25% attenuation). 1) 2) Use a sampling frequency of F-8192. Load the signal Splat (see class MATLAB demo video) ii) Provide an expression for the unit sample response h[n] in terms of delayed unit samples. Answer: h[n] Hint: For h[n], the delays should be given in samples (discrete time) not in seconds. Note that discrete time in samples and continuous time in seconds are related by tnTs, where T-1/F iv) Provide an expression of the system function H(z) for your echo filter, Answer: H(z) = Hint: For H(z), the delays should be given in samples (discrete time) not in seconds. Provide the discrete time filter difference equation. Answer: y[n]= Hint: For the difference equation, the delays for xIn] should be given in samples (discrete time) not in v) seconds. vi Provide time domain and frequency domain plots of your signal before and after filtering. vi) Use the sound command to hear the signal before and after the indicated audio effect. vii Turn in a formal report including, this paper with the answer blanks filled, team member names, introduction, conclusion, MATLAB code, plots and all expressions required in previous parts.

Answer 1

-7373 0 2521714

********* Matlab Code for the other parts ************

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% echo filters and effects on ausio signals

clear;
load splat.mat;                 % loading the audio files
x=y;                            % x = input
Ts = 1/Fs;                      % sampling period
t = 0:Ts:(length(x)/Fs)-Ts;          % time vector

z = tf('z',Ts);
H = 1 + (0.1*z^(-7373)) + (0.15*z^(-11470)) + (0.25*z^(-14746));    % filter transfer function
y = lsim(H,x);                                                      % filter output

%%%%%%%%%%%%% Time domain plots %%%%%%%%%%%%%%
subplot(2,1,1)
plot(t,x);
xlabel('Time (s)');
ylabel('X');
title('Time domain plots of the original signal and with the echo');

subplot(2,1,2)
plot(t,y);
xlabel('Time (s)');
ylabel('Y');

%%%%%%%%%%%%% Freq domain plots %%%%%%%%%%%%%%

Y1 = fft(x);
L = length(x);
% Compute the two-sided spectrum P2.
% Then compute the single-sided spectrum P1 based on P2 and the even-valued signal length L.
P2 = abs(Y1/L);
P1 = P2(1:L/2+1);
P1(2:end-1) = 2*P1(2:end-1);
% Define the frequency domain f and plot the single-sided amplitude spectrum P1.
f = Fs*(0:(L/2))/L;
figure;
subplot(2,1,1)
plot(f,P1)
title('Single-Sided Amplitude Spectrum of X(t) and Y(t)')
xlabel('f (Hz)')
ylabel('|Y1(f)|')

Y2 = fft(y);
L = length(y);
% Compute the two-sided spectrum P2.
% Then compute the single-sided spectrum P1 based on P2 and the even-valued signal length L.
P2 = abs(Y2/L);
P1 = P2(1:L/2+1);
P1(2:end-1) = 2*P1(2:end-1);
% Define the frequency domain f and plot the single-sided amplitude spectrum P1.
f = Fs*(0:(L/2))/L;
subplot(2,1,2)
plot(f,P1)
xlabel('f (Hz)')
ylabel('|Y2(f)|')

%%%%%%%%%%%%%%%% Hearing the sounds before and after the echo effect %%%%%%%%%%%

sound(x,Fs);
pause(3);
sound(y,Fs);

******* End of Code *********

Output:

Single-Sided Amplitude Spectrum of X(t) and Y(t) 0.01 0.005 0 500 1000 1500 2000 2500 3000 3500 4000 4500 f (Hz) 0.01 0.005 0 500 1000 1500 2000 2500 3000 3500 4000 4500 f (Hz)