UploadFile_6417

144 Chapter 5 THE DISCRETE FOURIER TRANSFORM
would be to use a matrix-vector multiplication for each of the relations
in (5.5). We have used this approach earlier in implementing a numerical
approximation to the discrete-time Fourier transform. Let ˜x and ˜X denote
column vectors corresponding to the primary periods of sequences ˜x(n)
and ˜X(k), respectively. Then (5.5) is given by
˜X = W N ˜x
˜x = 1 N W∗ N ˜X
(5.6)
where the matrix W N is given by
W N
△
=
[
⎡ n −→ ⎤
1 1 ··· 1
]
1 W
WN
kn
N 1 ··· W (N−1)
N
0≤k,n≤N−1
= k
↓⎢
⎣.
.
. .. . .. ⎥
⎦
1 W (N−1)
N
··· W (N−1)2
N
(5.7)
The matrix W N is a square matrix and is called a DFS matrix. The
following MATLAB function dfs implements this procedure.
function [Xk] = dfs(xn,N)
% Computes Discrete Fourier Series Coefficients
% ---------------------------------------------
% [Xk] = dfs(xn,N)
% Xk = DFS coeff. array over 0