Understanding the FFT documentation
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As I understand, for a signal x, the frequency spectrum X = fft(x) is two-sided - the first half of the array is frequencies 0 to FS/2 (half the sampling frequency, i.e: Nyquist limit), while the second half has negative frequencies from -FS/2 to the lowest non-zero frequency (-1/N for a signal with N points).
Meanwhile the documentation for FFT (found using "help fft" in MATLAB R2019a) says:
N
X(k) = sum x(n)*exp(-j*2*pi*(k-1)*(n-1)/N), 1 <= k <= N.
n=1
This aligns with my belief that X(1) contains DC offset (0 frequency), however I reckon this would mean that X(N) contains frequency N-1. It would also mean that X(N/2) contains frequency (N/2 - 1), which actually aligns with my statement at the top of the question.
Can you help me with my confusion around the equation from the documentation?
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Star Strider
2020-10-20
0 个投票
Your understanding essentially equates to my understanding of fft. It produces a ‘two-sided’ Fourier transform with the first value being the value of the d-c offset (mean of the time-domain signal). The mid-frequency is the Nyquist frequency. The second half is the flipped (not transposed) complex-conjugate of the first half. So the frequency is actually defined for the first half from 0 to the Nyquist frequency, and for the second half, from the negative Nyquist frequency to 0.
The fftshift function creates an accurate representation of this, so the frequency vector of the result goes from the -Nyquist frequency to the +Nyquist frequency.
I doubt if I clarified anything however, I likely just re-stated it.
8 个评论
Will Hardiman
2020-10-21
Star Strider
2020-10-21
It simply muktiplies the complex exponential (from the Euler identities) by the value of the time domain funciton.
Will Hardiman
2020-10-21
Star Strider
2020-10-21
It computes a two-sided Fourier transform, with the frequencieds defined as I previously described. It is not 2*Nyquist frequency, instead the frequency vector for the first half and the complex-conjugate of the first half of the resuult are flipped and concatenated to the end of the first half.
Will Hardiman
2020-10-23
编辑：Will Hardiman
2020-10-23
Star Strider
2020-10-23
Well, the upper limit of the frequency is in reality not ‘2*Nyquist’ (actually 0 to 
) , since the frerquency folds on itself as I described. The fft function cares not one whit about the actual frequency. It simply does the Fourier transform and lets your handle the frequency vector appropriately with respect to the time-domain signal.
) , since the frerquency folds on itself as I described. The fft function cares not one whit about the actual frequency. It simply does the Fourier transform and lets your handle the frequency vector appropriately with respect to the time-domain signal.
Will Hardiman
2020-10-24
Star Strider
2020-10-24
编辑：Star Strider
2020-10-24
As always, my pleasure!
It’s definitely different from the result the Symbolic Math Toolbox would produce for a Fourier transform:
syms t w
FTsp(w) = int(1*exp(1j*w*t), t, -1, 1);
[FTspn(w),FTspd(w)]= numden(FTsp);
FTsplh(w) = diff(FTspn)/diff(FTspd); % L’Hospital
FTsp(w) = piecewise(w<0,FTsp(w), w==0,FTsplh(w), w>0,FTsp(w));
figure
fplot(abs(FTsp), [-10*pi 10*pi])
xlabel('\omega')
ylabel('|F(\omega)|')
title('Fourier Transform Of A Pulse')
grid
EDIT — (24 Oct 2020 at 16:33)
Corrected typographical error.
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