Phase of 2D Rect Fourier Transform

I have been able to create a 3D rectangular pulse and to evaluate the fft of it, but when it comes to the phase it looks like it’s wrong shifted. The phase should only admitt values 0,pi (or -pi) instead is multiplied by a linear shift. I guess it's because matlab is receiving the pulse not centered in the way it is wanted. Is there a way to fix it, to have a correct phase’s graph? Thanks!

close all
clear
clc
npoints=512;
perc=1;
dt=6*1E-7/(npoints*perc);  % Tempo di campionamento
df=1/(npoints*dt); % Frequenza di campionamento

t(1)=0;
f(1)=0;
for k=2:npoints/2
    t(k)=(k-1)*dt;
    t(npoints-k+2)=-t(k);
    f(k)=(k-1)*df;
    f(npoints-k+2)=-f(k);
end
t(npoints/2+1)=t(npoints/2+2)-dt;
f(npoints/2+1)=f(npoints/2+2)-df;

ts=ifftshift(t);
fs=fftshift(f);

figure
[X,Y]=meshgrid(ts,ts);

D = npoints/2;        % to indicate origin at the center of the function
a = 100;          % change it to enlarge or reduce the pulse
y = repmat(1:npoints,npoints,1);
x = y';
rect = zeros(npoints);
rect(D-a:D+a-1,D-a:D+a-1) = ones(2*a);
rect=(rect);
surf(X,Y,rect);
shading interp
axis tight
title ('Rect 3D');
rect=ifftshift(rect);
figure, surf(X,Y,rect);
shading interp
axis tight
title ('Rect 3D shifted');

R = fft2((rect));
R = fftshift(R);
[X,Y]=meshgrid(fs,fs);
figure;
surf(X,Y,abs(R)); 
shading interp
axis tight
title('Fourier Transform of Rectangular function');
%plot real part
figure;
surf(X,Y,real(R)); 
shading interp
axis tight
title('Real part');

Rm=abs(R);
imm=imag(R);
re=real(R);
re(abs(re) < 1e-12) = 0;
imm(abs(imm) < 1e-12) = 0;
phase=atan2(imm, re);

%plot phase
[X,Y]=meshgrid(fs,fs);
figure
surf(X,Y,phase);
shading flat
axis tight
title ('Phase of the rect');

[vittorio todisco]

I got closer to the solution by replacing

R = fft2((rect));

with:

R = (fft(ifftshift(fft(rect).')).');

and

phase=atan2(imm, re);

with

R=re+imm;
phase=angle(R);

The graph that I get shows only value between 0 and pi, and the linear phase shift is removed. But if I see the graph from the top I see some diagonal lines... is the graph correct?

[David Goodmanson]

Hi vittorio,

a bit hard to say. If the matrix you are fft'ing has the same pattern as this (here n = 8, pulse width = 5)

ans =

     1     1     1     0     0     0     1     1
     1     1     1     0     0     0     1     1
     1     1     1     0     0     0     1     1
     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0
     1     1     1     0     0     0     1     1
     1     1     1     0     0     0     1     1

with the upper corner of ones being the largest, then the fft should be real and the phase plot does whatever it does. Incidentally,

R=re+imm;    phase=angle(R);

is not correct because imm is real, so it needs a factor of i.

[vittorio todisco]

Hi David, thanks for your answer.

I solved the problem, the solution if you or someone is interested, is available here:

https://dsp.stackexchange.com/questions/41556/matlab-phase-of-2d-rect-fourier-transform

Basically I forgot to multiply for the imaginary part, just as you said. And I recreated the rect pulse using the Matlab function rectpulse.