MATLAB: Derivative using FFT properties

derivativefft

Hi, I am trying to make a calculation by taking the first deriviative without success, I've tried the next couple of methods:
    %%%the data is a seismic shot gather
    %%%size(data_TX,1)=1001 (number of time samples)
    %%%siz (data_TX,2)=256     (number of increments)
        data_FK=fft2(data_TX);
    % setting the the time and spacial freq vectors
        fq=linspace(0,1/dx,size(data_TX,1)-1/2*dt);
        kx=linspace(0,1/dy,size(data_TX,2)-1/2*dx);
    %1 method
        [FQ,KX]=meshgrid(ifftshift(fq),ifftshift(kx));
        Dt_data_FK=2i*pi*FQ.*data_FK;
        Dx_data_FK=2i*pi*KX.*data_FK;
    %in this case I need to transpose the matrices
        FQ=FQ';
        KX=KX';
        Dt_data_FK=2i*pi*FQ.*data_FK;
        Dx_data_FK=2i*pi*KX.*data_FK;
        Dt_data_TX=ifft2(Dt_data_FK); 
        Dx_data_TX=ifft2(Dx_data_FK); %I get in this case a complex data set ?
    %2nd method
        data_FK=fftshift(fft2(data_TX));
        Dt_data_FK=2*pi*1i*diag(fq)*data_FK;
        Dx_data_FK=2*pi*1i*data_FK*diag(kx);
        Dt_data_TX=ifft2(ifftshift(Dt_data_FK),'symmetric');
        Dx_data_TX=ifft2(ifftshift(Dt_data_TX),'symmetric');
note: I need to make the calculation for a cube CUBE(k,j,i), k=1001 time samples, j=256 recievers,i=256 shots.
I tried at first to use fftn but all ways produced me with wrong results. the theory is right but i'm not sure exactly how to take the derivative.

Best Answer

Based on the website below:
http://www.cl.cam.ac.uk/teaching/2000/ContMaths/JGD-notes/node11.html
I believe you would do something similar to the example below:
Nx = 64; % Number of samples collected along first dimension

Ny = 32; % Number of samples collected along second dimension

dx = 1;  % x increment (i.e., Spacing between each column)

dy = .1; % y increment (i.e., Spacing between each row)

x = 0 : dx : (Nx-1)*dx; % x-distance

y = 0 : dy : (Ny-1)*dy; % y-distance

data_spacedomain = randn(Ny,Nx); % random 2D matrix

Nyq_kx = 1/(2*dx); % Nyquist of data in first dimension

Nyq_ky = 1/(2*dy); % Nyquist of data in second dimension

dkx = 1/(Nx*dx);   % x-Wavenumber increment

dky = 1/(Ny*dy);   % y-Wavenumber increment

kx = -Nyq_kx : dkx : Nyq_kx-dkx; % x-wavenumber

ky = -Nyq_ky : dky : Nyq_ky-dky; % y-wavenumber

data_wavenumberdomain = zeros(size(data_spacedomain)); % initialize data
% Compute 2D Discrete Fourier Transform
for i1 = 1 : Nx
for j1 = 1 : Ny
for i2 = 1 : Nx
for j2 = 1 : Ny
data_wavenumberdomain(j1,i1) = data_wavenumberdomain(j1,i1) + ...
(2i*pi*ky(j1))*(2i*pi*kx(i1))* ...
(data_spacedomain(j2,i2)*exp(-1i*(2*pi)*(kx(i1)*x(i2)+ky(j1)*y(j2))));
end
end
end
end
differentiated_data = ifft2(ifftshift(data_wavenumberdomain));
I have a program for performing this on a 1D dataset that you might leverage (in your own way) to produce the same effect on 2D datasets - see my answer in the post below:
http://www.mathworks.com/matlabcentral/answers/21700-finding-the-velocity-from-displacement
[EDIT 6/22]
Practically speaking, if you were going to compute the derivative of a 2D image, where your dimensions are either space-space or time-time, then you would need to use fft2, ifftshift, and meshgrid:
Nx = 64; % Number of samples collected along first dimension
Ny = 32; % Number of samples collected along second dimension
dx = 1;  % x increment (i.e., Spacing between each column)
dy = .1; % y increment (i.e., Spacing between each row)
x = 0 : dx : (Nx-1)*dx; % x-distance
y = 0 : dy : (Ny-1)*dy; % y-distance
data_spacedomain = randn(Ny,Nx); % random 2D matrix
Nyq_kx = 1/(2*dx); % Nyquist of data in first dimension
Nyq_ky = 1/(2*dy); % Nyquist of data in second dimension
dkx = 1/(Nx*dx);   % x-Wavenumber increment
dky = 1/(Ny*dy);   % y-Wavenumber increment
kx = -Nyq_kx : dkx : Nyq_kx-dkx; % x-wavenumber
ky = -Nyq_ky : dky : Nyq_ky-dky; % y-wavenumber
% Compute 2D FFT
data_wavenumberdomain = fft2(data_spacedomain);
% Compute grid of wavenumbers
[KX, KY] = meshgrid(ifftshift(kx),ifftshift(ky));
% Compute 2D derivative
data_wavenumberdomain_differentiated = (2i*pi)^2*KX.*KY.*data_wavenumberdomain; 
% Convert back to space domain
data_spacedomain_differentiated = ifft2(data_wavenumberdomain_differentiated );
If you only want to take the partial derivative of the the "image" with respect to one of your dimensions, then all you would do is:
d_MAT_x = 2i*pi*KX.*data_wavenumberdomain;
d_MAT_y = 2i*pi*KY.*data_wavenumberdomain;
Then just transform back into time domain using ifft2
[EDIT 6/27]
1. "Nx" and "Ny" are the number of columns and number of rows, respectively. Thus, you should have:
[KX,FQ]=meshgrid(ifftshift(kx),ifftshift(fq));
2. Your "fq" and "kx" may be defined wrong (I said in an earlier comment that if the number of rows or columns is an odd number then we need to do something different). Also, it seems strange that you would define "fq" in terms of "dx" and "dt"... and "kx" in terms of "dy" and "dx". Lets assume "dt" is the time increment and "dx" is the spatial increment... "Nt" is the number of time samples and "Nx" is the number of spatial samples, then:
Nyq_fq = 1/(2*dt);
Nyq_kx = 1/(2*dx);
dfq = 1/(Nt*dt);
dkx = 1/(Nx*dx);
if mod(Nt,2) == 0 % Nt is even
     fq = -Nyq_fq : dfq : Nyq_fq-dfq;
else            % Nt is odd
     fq = [sort(-1*(dfq:dfq:Nyq_fq)) (0:dfq:Nyq_fq)];
end
if mod(Nx,2) == 0 % Nx is even
     kx = -Nyq_kx : dkx : Nyq_kx-dkx;
else            % Nx is odd
     kx = [sort(-1*(dkx:dkx:Nyq_kx)) (0:dkx:Nyq_kx)];
end
Does this help solve things?
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