When the FFT is performed on a function of time u in Matlab, a complex spectrum uf is returned. To plot the spectral amplitude abs(uf) against its frequency content, a frequency grid can be made to accommodate uf. I can associate a wavelength grid with the frequency grid, and plot uf against that also. The spacing between each element in the frequency array is constant, but since wavelength ~ 1/frequency, the spacing between each point in the wavelength array varies over the array index. I am curious if there is a way to take the FFT of a function of time to yield a spectrum in wavelength that has constant spacing. Here is my code in Matlab:
clc;
close all;
clear all;
lam = 800e-9; % Wavelength (m)
c = 3e8; % Light speed (m/s)
nt = 8192; % Temporal grid resolution
T = 400*1e-15; % Temporal grid size (s)
dt = T/nt; % Temporal pixel spacing
df = 1/(nt*dt); % Frequency pixel spacing
ff = [(0:nt/2-1) (-nt/2:-1)]*df; % Frequency grid
ff = fftshift(ff);
wav = c./ff; % Wavelength array (spacing is not constant between each element)
for k = 1:nt
tt(k) = (-nt/2+k-1)*dt; % Time array
u(k) = cos(2*pi*c/lam*tt(k)); % Function of time
end
%Now I can take FFT:
uf = fftshift(fft(u)); % The spectrum of my function. The FFT has yielded a spectrum associated with a frequency array of linearly spaced elements (ff).
Both plots of spectral amplitude vs. wavelength and vs. frequency yield good results.
figure(1)
plot(ff,abs(uf))
title('Spectral amplitude vs frequency')
xlabel('Frequency (Hz)')
ylabel('Spectral amplitude')
figure(2)
plot(wav,abs(uf))
title('Spectral amplitude vs wavelength')
xlabel('Wavelength (m)')
ylabel('Spectral amplitude');
But my wavelength array does not have constant spacing:
figure(3)
plot(ff)
title('Frequency array')
ylabel('Frequency (Hz)')
xlabel('Index')
figure(4)
plot(wav)
xlim([(nt/2 +1) (nt/2 + 100)])
title('Wavelength array')
ylabel('Wavelength (m)')
xlabel('Index')
wav = c./ff). You can interpolate if needed. – m7913d