Question

4. Using Matlab: 4.1. Plot | H(n the following cases: (the frequency range: 0-20 KHz) a. a 0.2 0.5 ms. b. α-0.8 C.α-0.2 c-0.1 ms. c 0.5 ms. 4.2. Consider a signal whose Fourier Transform is given by: 50000ω Plot the transfer function l x(o) l and the output l Y(o) l in each of the above cases (stated in part 4.1) 4.3. Find the Inverse Fourier Transorms of| X(oand Y(o) , and generate the audio signals x and y

Answer 1

clear all;
close all;

alpha = input("Enter alpha value :\n");

c_ms = input("Enter c value in ms :\n");

c = c_ms*0.001; %convert c to seconds

fmin = 0;

fmax = 20000;

w = 2*pi*fmin:0.01:2*pi*fmax;

%%%%%%%%%%%%%%% solution to 4.1 %%%%%%%%%%%%%%%%%%

Hw = sqrt((1+alpha^2) + 2*alpha*cos(w*c));

figure;plot(w, Hw)
titletext1=['|H(w)| vs w for alpha = ', num2str(alpha),' and c = ', num2str(c_ms),' ms'];
title(titletext1);
xlabel('w=2*pi*f (rad/s) ----> ');
ylabel('|H(w)|');
xlim([min(w) max(w)]);

%%%%%%%%%%%%%%%    Solution to part 4.2   %%%%%%%%%%%%%%%%%%%%

Xw = (50000*w)./(2*pi*2000+w).^2;

figure;plot(w,Xw);
title('X(w) vs w ');
xlabel('w=2*pi*f (rad/s) ----> ');
ylabel('X(w)');
xlim([min(w) max(w)]);

Yw = Hw.*Xw;

figure;plot(w,Yw);
titletext2=['Y(w) vs w for alpha = ', num2str(alpha),' and c = ', num2str(c_ms),' ms'];
title(titletext2);
xlabel('w=2*pi*f (rad/s) ----> ');
ylabel('Y(w)');
xlim([min(w) max(w)]);

%%%%%%%%%%%%%%%%     Solution to part 4.3    %%%%%%%%%%%%%

xt = abs(ifftshift(ifft(Xw))); % time domain signal from frequency domain signal
audiowrite('x.wav',xt,2*fmax); % save x(t) in .wav format

yt = abs(ifftshift(ifft(Yw)));

% different filename for different alpha and c
audio_title = ['y_alpha_',num2str(alpha),'_c_', num2str(c_ms),'.wav'];
audiowrite(audio_title,yt,2*fmax); % save y(t) in .wav format

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Command Window > magni tude_spectrum Enter alpha value 0.2 Enter c value in ms 0.5 >magni tude_spectrum Enter alpha value: 0.8 Enter c value in ms: 0.5 magni tude_spectrum Enter alpha value: 0.2 Enter c value in ms 0.1

Plots of |H(w)| for difference values of alpha and c:

Figure 1 (on gpuwks) File Edit View Insert Iools Desktop Window Help [H(w)l vs w for alpha-0.2 and c = 0.5 ms 1.2 1.15 1.1 1.05 0.95 0.9 0.85 0.8 10 0 4 12 × 10 w=2*pi*f (rad/s)-->

Figure 1 (on gpuwks) File Edit View Insert Iools Desktop Window Help [H(w)l vs w for alpha-0.8 and c = 0.5 ms 1.8 1.6 1.4 1.2 3 0.8 0.6 0.4 0.2 10 0 4 12 × 10 w=2*pi*f (rad/s)-->

Figure 1 (on gpuwks) File Edit View Insert Iools Desktop Window Help [H(w)l vs w for alpha = 0.2 and c = 0.1 ms 1.2 1.15 1.1 1.05 0.95 0.9 0.85 0.8 12 10 4 0 × 10 w=2*pi*f (rad/s)-->

Plot of X(w) [independent of alpha and c] :

Figure 2 (on gpuwks) File Edit View Insert Iools Desktop Window Help X(w) vs w 0.9 0.8 0.7 0.6 30.5 0.4 0.3 0.2 0.1 10 0 4 W-2 pi*f (rad/s) -->

Plots of Y(w) for difference values of alpha and c:

Figure 3 (on gpuwks) File Edit View Insert Iools Desktop Window Help Y(w) vs w for alpha = 0.2 and c 0.5 ms 1.2 0.8 0.6 0.4 0.2 0 10 12 4 w=2*pi*f (rad/s)--> ×104

Figure 3 (on gpuwks) File Edit View Insert Iools Desktop Window Help Y(w) vs w for alpha = 0.8 and c 0.5 ms 1.8 1.6 1.4 1.2 0.8 0.6 0.4 0.2 0 10 12 0 4 w=2*pi*f (rad/s)--> × 10

Figure 3 (on gpuwks) File Edit View Insert Iools Desktop Window Help Y(w) vs w for alpha = 0.2 and c 0.1 ms 1.2 r 0.8 0.6 0.4 0.2 0 10 12 0 4 w=2*pi*f (rad/s)--> ×104