The CTFT of yt can be written in terms of its magnitude and

The CTFT of y(t) can be written in terms of its magnitude and phase as Y(j omega) = |Y(j omega)|e^j phi(omega), where phi(omega) = Y(j omega). For many signals, either the phase or the magnitude alone can be used to construct a useful approximation of the signal y(t). For instance, consider the signals y_2(t) and y_3(t) with CTFTs Y_2(j omega) = |Y(j omega)| and Y_3(j omega) = e^j phi (omega). Demonstrate analytically that y_2(t) and y_3(t) will be real signals whenever y(t) is real. Construct a vector Y2 equal to the magnitude of Y. Store in y2 the inverse Fourier transform of Y2. Play this vector using sound. Construct a vector Y3 which has the same phase as Y, but has magnitude equal to one for each frequency. Store in y3 the inverse Fourier transform of Y3. Play this vector using sound. Based upon the two signals you just heard, which component of the Fourier transform would you say is most crucial for representing an audio signal: the magnitude or the phase?

Solution

load splat;
y=y(1:8192);
N=8192;
fs=8192;
sound(y,fs);
Y=fftshift(fft(y));
%%part a
w=[-pi:2*pi/N:pi-pi/N]*fs;
plot(w,abs(Y));
y=ifft(fftshift(Y));
y=real(y);
%%part b
Y1=conj(Y);
y1=ifft(fftshift(Y1));
y1=real(y1);
sound(y1,fs);
%ifft(Y1(jw))=y(-t);thats is the complex conjucative of fft(y(t))
%willproduce the image version(y(-t)) y(t).

%% oriented subject.

%%part d
Y2=abs(Y);
y2=ifft(fftshift(Y2));
y2=real(y2);
sound(y2,fs);
%%Part E
Y3=angle(Y);
y3=ifft(fftshift(Y3));
y3=real(y3);
sound(y3,fs);
% when we paly the magitude based signal we can hear the sound which
% represt the original signal.But when we taking the phase based signal we
% cant hear anything. which says the magnitude of the CTFT will hold the
% crucial information of original signal.
%% Thanking you. if you have doubts u can contact me %%whatsaap +919790819924