fft2 results don't match the analytical Fourier transform results

Hello everyone,

I am trying to implement fft2 and compare the results with the analytical Fourier trasnform. For that I have chosen a constant rectangular field with constant value (the constant c = 1) which its analytical Fourier trasnform already exist. However the results of my fft2 and analytival Fourier trasnform do not match each other. Can you help me to find out my mistake?

clear
clc

% Define the dimensions of the square field
Lx = 2; % length of the rectangle
Lz = 1; % width of the rectangle
x = 0:0.005:1; % x vector
z = 0:0.005:1; % z vector
xgrd = length(x);
zgrd = length(z);
kxgrd = length(x);
kzgrd = length(z);
dkx = 1/(x(2)-x(1)); % Sampling frequency in x
dkz = 1/(z(2)-z(1)); % smapling frequency in z
kxval = (-length(x)/2:length(x)/2-1)*(dkx/length(x)); % Frequency vector
kzval = (-length(z)/2:length(z)/2-1)*(dkz/length(z));
% Define the size of the rectangular field and the constant value
c = 1; % constant value

rect = zeros(xgrd,zgrd); % initialize the rectangular field to all zeros
rect(xgrd/4+1:3*xgrd/4,zgrd/4+1:3*zgrd/4) = c; % set the central region to the constant value

% Evaluate the analytical Fourier transform on a grid of u and v values
for indkx = 1:kxgrd
    for indkz = 1:kzgrd
        F_analytical(indkx,indkz) = Lx*Lz*sinc(pi*Lx*kxval(indkx))*sinc(pi*Lz*kzval(indkz)); % must match the fft2 results
    end
end

func = rect; % cosine signal
% Compute the Fourier transform
func_wo_shift = fft2(func);
func_shift = fftshift(fft2(func));

Answer 1

[Paul]

Hi Seyedalireza,

The code below get closer, maybe all the way there, by addressing a few issues

% Define the dimensions of the square field
Lx = 2; % length of the rectangle
Lz = 1; % width of the rectangle

I multiplied by Lx and Lz to get the correct physical dimensions

x = Lx*(0:0.005:1); % x vector
z = Lz*(0:0.005:1); % z vector

Define dx and dz for use below

dx = (x(2)-x(1));
dz = (z(2)-z(1));
xgrd = length(x);
zgrd = length(z);
kxgrd = length(x);
kzgrd = length(z);

Defined in terms of dx and dz

dkx = 1/dx; % Sampling frequency in x
dkz = 1/dz; % smapling frequency in z

The next two lines are incorrect, I believe.

kxval = (-length(x)/2:length(x)/2-1)*(dkx/length(x)); % Frequency vector
kzval = (-length(z)/2:length(z)/2-1)*(dkz/length(z));

The middle of the frequency vector should be zero, which is important in this problem because that's where all the action occurs. But 0 doesn't show up.

kxval(99:103)

ans = 1×5

-1.2438 -0.7463 -0.2488 0.2488 0.7463

With numel(x) = numel(z) = 201, the frequency vectors should be

kxval = (-100:100)/201*dkx;
kzval = (-100:100)/201*dkz;
kxval(99:103)

ans = 1×5

-0.9950 -0.4975 0 0.4975 0.9950

% Define the size of the rectangular field and the constant value
c = 1; % constant value
rect = zeros(xgrd,zgrd); % initialize the rectangular field to all zeros
rect(xgrd/4+1:3*xgrd/4,zgrd/4+1:3*zgrd/4) = c; % set the central region to the constant value

Warning: Integer operands are required for colon operator when used as index.

Warning: Integer operands are required for colon operator when used as index.

That warning is troublesome, but it looks like rect comes out the way you want.

figure
pcolor(x,z,rect.'),shading interp
xlabel('x'),ylabel('Z')

I think this is the correct expression for F_analytical, assuming we're only interested in its amplitude and not its phase. Note that sinc is defined as sinc(x) = sin(pi*x)/(pi*x);

% Evaluate the analytical Fourier transform on a grid of u and v values
for indkx = 1:kxgrd
    for indkz = 1:kzgrd
%        F_analytical(indkx,indkz) = Lx*Lz*sinc(pi*Lx*kxval(indkx))*sinc(pi*Lz*kzval(indkz)); % must match the fft2 results
        F_analytical(indkx,indkz) = Lx/2*Lz/2*sinc(Lx/2*kxval(indkx))*sinc(Lz/2*kzval(indkz)); % must match the fft2 results
    end
end
func = rect; % cosine signal
% Compute the Fourier transform
func_wo_shift = fft2(func);
func_shift = fftshift(fft2(func));

figure
pcolor(kxval,kzval,abs(F_analytical.')),colorbar
xlabel('Kx'); ylabel('Kz')
axis([-10 10 -10 10])

I'm going to assume that F_analytical is supposed to be the Continous Time Fourier Transform. If so, the result from fft2 has to be multiplied by dx*dz for comparison

figure
pcolor(kxval,kzval,abs(dx*dz*func_shift.')),colorbar
xlabel('Kx');ylabel('Kz')
axis([-10 10 -10 10])

[Seyedalireza Abootorabi]

thank you for the very clear answer.

Answer 2

[Benjamin Thompson]

Your problem is 201 by 201 samples in size? Making the size a power of 2. So use 256.

[Seyedalireza Abootorabi]

I don't know if I understood but my problem is with the difference in magnitude sizes and the lack of perfect similirity between the shapes.

[Benjamin Thompson]

Yes, and the difference is probably due to fft2 padding your data out to 256 because it is required to operate on data sizes that are a power of 2, while your analytical method has not such restriction. The different data sizes causes the results of fft2 to be changed.

[Paul]

Based on the doc page fft2, it doesn't pad the input unless specified via the optional second and third input arguments.