The Fourier Series and Transform depends on knowing a function f(t) in the time domain. However in many applications, the time domain data consists of a set of samples of the variable being measured obtained at times t = nT where T is the time between samples.

For these cases the DFT or related FFT is more suitable. If the time data is contained in N samples x(n) and the frequency spectrum is X(m), then the DFT and its inverse is defined by the following equations.

if x(n) is real, then the following symmetric relations hold. In this way the n independent variables in x(n) matches to n independent parameters in X(m). If x(n) is imaginary, then the symmetry relations for the real and imaginary parts are reversed.

If the sequence x(n) is delayed by k sample times, then the change in X(m) is shown below.

Given 2 sequences x

COPYRIGHT © 2007 Cuthbert Nyack.