3.4. The Discrete Fourier Transform

Suppose that the function h(t) has been measured only at the discrete times t_k (either at the given times, or averaged over the corresponding time interval) given by

t_k =k k= 0,1,2,...,N-1

where N is the number of samples. (We will consider only cases in which N is even and, indeed, a power of 2.) The Discrete Fourier Transform (DFT) is then defined as

where the frequencies f_n are given by

This definition is based on the idea that the sampled function h_k is replicated indefinitely, in the sense that k₀ = k_N = k_2N = ..., k₀ = k_-N = k_-2N = ... and so forth for all the other elements. There is clearly a close relationship between the DFT and the continuous Fourier integral.