# DFT_Definition and Properties.

Cuthbert Nyack
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.
The Spectrum is discrete with N values, usually complex. The sequence x(n) is usually real but can be complex.
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 N is even then

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

Given 2 sequences x1(n) and x2(n) with spectra X1(m) and X2(m), then the DFT Spectrum of the convolution of the 2 sequences is the product of the individual spectra.