What frequency range does fft compute?

So according to the documentation, the fft function for a time series of n samples computes the following:

The question is what are the values of k and how is it ordered from the output of fft? Looking at the example code, they truncate the fft output by about a half to get the "positive" frequencies. When I plot out the full fft output without truncating the vector, it appears that the negative frequencies are shown on the right half of the fft output, i.e., values of k would be [0:n/2-1,-n/2:-1] where n>2. Is this correct?

If so, why the offset in the negative frequencies, and what about frequencies for ? Are those frequencies ignored because they are not practical due to the sampling theorem?

Answer 1

[Star Strider]

The first element of the output of the fft function is the constant or DC offsets (mean values) of the original time domain vector. As a general rule, the second half of the original vector is the flipped complex conjugate of the first half of the vector. For a one-sided Fourier transform, the frequencies go from 0 Hz (DC) to the Nyquist frequency (half the sampling frequency). The sampling frequency (assuming a time vector with uniform increments, so a constant sampling frequency) is the inverse of the common sampling interval.

For a two-sided fft result (after using the fftshift function so that the centre frequency is 0 Hz) the frequency vector is:

Fv = linspace(-Fs/2, Fs/2-Fs/length(s), length(s)); % EVEN 'length(s)' (Asymmetric)

Fv = linspace(-Fs/2, Fs/2, length(s)); % ODD 'length(s)' (Symmetric)

This results in a two-sided Fourier transform with the ‘negative’ frequencies to the left of centre and the positive frequencies to the right of centre.

I generally zero-pad the fft argument using:

NFFT = 2^nextpow2(L);

FTs = fft(s, NFFT)/L;

where ‘L’ is the length of the associated time vector (since the signal can be either a vector or a column-oriented matrix where each column is a separate signal) and ‘s’ is the signal (vector or matrix). This improves the efficiency of the fft calculation, and also makes the subsequent analysis a bit easier.

.

[Paul]

Hi Star,

For an odd length sequence I believe that Fv is

Fv = linspace(-F2/2, Fs/2, length(s)) * (length(s) - 1)/length(s)