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156 Chapter 5 THE DISCRETE FOURIER TRANSFORM
5.3.1 MATLAB IMPLEMENTATION
It is clear from the discussions at the top of this section that the DFS is
practically equivalent to the DFT when 0 ≤ n ≤ N − 1. Therefore the
implementation of the DFT can be done in a similar fashion. If x(n) and
X(k) are arranged as column vectors x and X, respectively, then from
(5.24) and (5.25) we have
X = W N x
x = 1 N W∗ N X (5.26)
where W N is the matrix defined in (5.7) and will now be called a DFT
matrix. Hence the earlier dfs and idfs MATLAB functions can be renamed
as the dft and idft functions to implement the discrete Fourier
transform computations.
function [Xk] = dft(xn,N)
% Computes Discrete Fourier Transform
% -----------------------------------
% [Xk] = dft(xn,N)
% Xk = DFT coeff. array over 0