Discrete Fourier Transform

Discrete Fourier Transform

In most situations, the function h(t) is sampled at constant time intervals. Moreover, the number of sampled points is a finite number.

Let us call the sampling interval T, and the number of sampled points N. The integration on an infinite domain takes the form of a finite sum. The Fourier transformation becomes:

with hk = h(k.T), p = 0..N/2

Note that Hi = H(fi) and in the equation defining Hp, values of p comprised in range N/2+1, N-1 concern negative frequencies. As h is real, its Discrete Fourier Transform is symmetrical around f0. Thus, Hj = HN-j. For that reason, spectral elements of index greater than N/2 are ignored.

Note

Hp is a complex number. The FFT curves plotted by Simcenter Amesim are the magnitude and the phase of Hp.

The Nyquist critical frequency Nyquist critical frequency (the highest observable frequency) is defined as

This means that the sampling interval (Simcenter Amesim print interval) is directly linked to the maximum observable frequency in this signal. Forgetting this rule may lead to aliasing problems.