# How to obtain the exact value of wavelength from a 2D FFT amplitude vs wavenumber plot like it is obtainable from 1D FFT amplitude vs wavenumber plot?

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S Ban on 7 May 2021
Answered: David Goodmanson on 9 May 2021
I have a two dimensional multi modal spatial signal generated from a MATLAB code using sinusoidal functions of different wave numbers, amplitudes and phases. What I want to know is that if I have the amplitude vs wave number plot of that signal, how can I extract the wavelength of the different spatial structures. For 1D signal and 1D FFT I know that it is possible to extract the wavelength from the amplitude vs wavenumber plot by simply taking the reciprocals of the wave numbers with non zero amplitudes. But how can it be done for 2D signal and 2D FFT? I am attaching my code below.
Ws=160; % Sampling wavenumber @ 160 Hz
L=10; % Length of domain = 10cm
N = L*Ws; % Length of signal
x = (0:N-1)*(1/Ws); % Space vector
y = [(0:N-1)*(1/Ws)]'; % Space vector
x = repmat(x,1600,1); % Space matrix
y = repmat(y,1,1600); % Space matrix
V = sin((4*pi*x)/L)+sin((4*pi*y)/L); % Function in the spatial domain
fx = linspace((-Ws/2),Ws/2,N); % computing wavenumber vector fx
fy = [linspace((-Ws/2),Ws/2,N)]'; % computing wavenumber vector fy
fx = repmat(fx,1600,1); % computing wavenumber matrix fx
fy = repmat(fy,1,1600); % computing wavenumber matrix fy
T = fft2(V); % 2D FFT
subplot(1,2,1)
pcolor(x,y,V) %%%% Contour plot in spatial domain
ax = gca;
ax.LineWidth = 2;
colormap jet
colorbar
pbaspect([1 1 1])
xlabel('X(cm)-->.')
ylabel('Y(cm)-->.')
title('V = sin((4*pi*x)/L)+sin((4*pi*y)/L)')
subplot(1,2,2)
pcolor(fx,fy,abs(fftshift(T))) % Contour plot of FFT amplitude vs wavenumber
lin = zeros(1,N);
hold on
plot(lin,fy(1:1600),'b') % Vertical line passing through (0,0) in fft amplitude plot
hold on
plot(fx(1:1,1:1600),lin,'k') % Horizontal line passing through (0,0) in fft amplitude plot
colormap jet
colorbar
pbaspect([1 1 1])
xlabel('Wx(cm^{-1})')
ylabel('Wy(cm^{-1})')
title('FFT amplitude')
I am attaching the plot which I got as well. Any help would be greatly appreciated. David Goodmanson on 9 May 2021
Hello SB,
I am speculating that the issue is the offset from the correct frequencies in the fequency domain plot. The culprit is the linspace statements, which define the frequency grid with slightly incorrect spacing.
The code below takes up where your linspace statements are, and defines the frequencies in a similar manner as you did for the spacial array. In fig(2) the frequency peaks are now centered exactly where they should be, i.e. (0, .2), (0, -.2), (-.2, 0) and (.2, 0). In case you don't like the look of the funny hexagonal spots, which are due to some manipulation by pcolor, fig(3) shows individual pixels of abs(fftshift(T)), which are in the correct locations.
The two repmat statements can be replaced by a single meshgrid statement as below. In that case fyy can be defined as a row vector because meshgrid will accept either row or column.
fxx = ((-N/2:N/2-1)/N)*Ws;
fyy = ((-N/2:N/2-1)/N)*Ws;
[fx fy] = meshgrid(fxx,fyy);
% fx = repmat(fxx,1600,1); % computing wavenumber matrix fx
% fy = repmat(fyy,1,1600); % computing wavenumber matrix fy
T = fft2(V); % 2D FFT
figure(2) % (leaving out the extra lines)
pcolor(fx,fy,abs(fftshift(T))) % Contour plot of FFT amplitude vs wavenumber
colormap jet
colorbar
pbaspect([1 1 1])
xlabel('Wx(cm^{-1})')
ylabel('Wy(cm^{-1})')
title('FFT amplitude')
xlim([-.4 .4])
ylim([-.4 .4])
figure(3)
imagesc(fxx,fyy,abs(fftshift(T)))
colormap spring
colorbar
xlim([-.4 .4])
ylim([-.4 .4])