Change phase of a signal in frequency domain (MatLab)

Question

I posted this question on dsp.stackexchange, and was informed that it was more relevant for stackoverflow as it is primarily a programming question:

I am attempting to write a code which allows me to change the phase of a signal in the frequency domain. However, my output isn't exactly correct, so something must be wrong. For a simple example assume that we have the function y = sin(2*pi*t) and want to implement a phase shift of -pi/2. My code looks as follows:

clear all
close all

N = 64; %number of samples
fs = 10; %sampling frequency
ts = 1/fs; %sample interval
tmax = (N-1)*ts;
t = 0:ts:tmax;
y = sin(2*pi*t);

figure
plot(t,y)

% We do the FT
f = -fs/2:fs/(N-1):fs/2;
Y = fftshift(fft(y));

% Magnitude spectrum
figure
plot(f,abs(Y));

phase = angle(Y);

% Phase spectrum
figure
plot(f,phase)

Y = ifftshift(Y)

% Attempt at phase shift
Y = Y.*exp(-i*2*pi*f*pi/2);

% Inverse FT
u = ifft(Y);

figure
plot(t,real(u))

All plots look OK except for the final plot which looks as follows:

This looks almost correct but not quite. If anyone can give me some pointers as to how my code can be corrected in order to fix this, I would greatly appreciate it! I have a feeling my mistake has something to do with the line Y = Y.*exp(-i*2*pi*f*pi/2);, but I'm not sure how to fix it.

Answer 1

I can't really get into the Fourier analysis details (because I do not really know them), but I can offer a working solution with some hints:

First of all, You should express Your wave in imaginary terms, i.e.:

y = exp(1i*2*pi*t);

And what's even more crucial, You have to truly shift only the phase, without messing with the whole spectrum:

% Attempt at phase shift
Y = abs(Y).*exp(1i*angle(Y)-1i*pi/4); % -pi/4 shift

You should note that the shift isn't related to frequency anymore, which I guess makes sense. Finally You can plot the results:

figure
plot(t,real(u),'k')
hold on
plot(t,real(y),'r')

real(y) is actually a cosine function, and You started with sine, but hopefully You get the idea. For pi/4 shift i got something like this (started with red, finished with black):