Interpreting FFT Results

Just as the sampled time data represents the value of a signal at discrete points in time, the result of a (forward) Fast Fourier Transform represents the spectrum of the signal at discrete frequencies. These discrete frequencies are a function of the frequency index (m), the number of samples collected (N), and the sampling interval (ξ):

The frequencies for which the FFT of a sampled signal are defined are sometimes called frequency bins, which refers to the histogram-like nature of a discrete spectrum. The width of each frequency bin is 1/(N * ξ).

Due to the complex exponential in the definition of the DFT, the spectrum has a cyclic dependence on the frequency index m. That is:

for p = any integer.

The frequency spectrum computed by IDL's FFT function for a one-dimensional time sequence is stored in a vector with indices running from 0 to N–1, which is also a valid range for the frequency index m. However, the frequencies associated with frequency indices greater than N/2 are above the Nyquist frequency and are not physically meaningful for sampled signals. Many textbooks choose to define the range of the frequency index m to be from – (N/2 – 1) to N/2 so that it is (nearly) centered around zero. From the cyclic relation above with p = –1:

v(– (N/2 – 1)) = v(N/2 + 1 – N) = v(N/2 + 1)

v(– (N/2 – 2)) = v(N/2 + 2 – N) = v(N/2 + 2)

...

v(–2) = v(N – 2 – N) = v(N – 2)

v(–1) = v(N – 1 – N) = v(N – 1)

This index shift is easily accomplished in IDL with the SHIFT function. See Real and Imaginary Components for an example.