082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

792 Chapter 24 The Fourier transform
24.12 MATRIXREPRESENTATIONOFTHED.F.T.
Whenitisnecessarytodevelopcomputercodeforperformingad.f.t.,anunderstanding
of the following matrix representation isuseful.
We have seen thatthe d.f.t. ofasequence f[n] isgiven by
F[k] =
∑N−1
n=0
f[n]e −2jnkπ/N
Considertheterme −2jnkπ/N whichcanbewrittenusingthelawsofindicesas (e −2jπ/N ) nk .
DefineW = e −2jπ/N so that
F[k] =
∑N−1
n=0
f[n]W nk
Note thatW does not depend uponnor k. For a fixed value of N we can calculateW
which isthen a constant. Writingout thissum explicitly we find
F[k] = f[0]W 0 + f[1]W k + f[2]W 2k + f[3]W 3k +···
+f[N−1]W (N−1)k
Writingthisout foreachkwefind
fork=0,1,2,...,N−1
F[0] = f[0]W 0 + f[1]W 0 + f[2]W 0 + f[3]W 0 +···+f[N−1]W 0
F[1] = f[0]W 0 + f[1]W 1 + f[2]W 2 + f[3]W 3 +···+f[N−1]W N−1
F[2] = f[0]W 0 + f[1]W 2 + f[2]W 4 + f[3]W 6 +···+f[N−1]W 2(N−1)
. = . .
F[N −1] = f[0]W 0 + f[1]W N−1 + f[2]W 2(N−1)
+ f[3]W 3(N−1) +···+f[N−1]W (N−1)(N−1)
These equations can be written inmatrix form asfollows:
⎛ ⎞ ⎛
⎞ ⎛
F[0] W 0 W 0 W 0 ... W 0
F[1]
W 0 W 1 W 2 ... W N−1
F[2]
=
W 0 W 2 W 4 ... W 2(N−1)
⎜
⎝
⎟ ⎜
. ⎠ ⎝
.
.
.
.
⎟ ⎜
. ⎠ ⎝
F[N −1] W 0 W N−1 W 2(N−1) ...W (N−1)(N−1)
whereW = e −2jπ/N .
f[0]
f[1]
f[2]
.
f[N−1]
⎞
⎟
⎠
Example24.23 (a) Find the matrix representing a three-point d.f.t.
(b) Usethe matrix tofind the d.f.t. of the sequence f[n] = 4,−7,11.
Solution (a) HereN = 3 and soW = e −2πj/3 . The required matrix is
⎛
⎞
1 1 1
⎝1 e −2πj/3 e −4πj/3 ⎠
1 e −4πj/3 e −8πj/3