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

24.15 Discrete convolution and correlation 807
EXERCISES24.15.2
1 Calculate the circular convolution of f[n] = 1, −1,1,3 andg[n] = 7,2,0,1.
Solutions
1 12, −4,8,24
24.15.3 The(circular)convolutiontheorem
Suppose f[n]andg[n]areperiodicsequencesofperiodN.Supposefurtherthatthed.f.t.s
of f[n] andg[n] forn = 0,1,2,...,N −1 are calculated and are denoted byF[k] and
G[k]. The convolution theorem states that the d.f.t. of the circular convolution of f[n]
andg[n] isequal tothe product of the d.f.t.sof f[n]andg[n].
The convolution theorem:
D{f ○∗ g} =F[k]G[k]
This is important because it provides a technique for finding a circular convolution. It
follows fromthe theorem that
f ○∗ g = D −1 {F[k]G[k]}
So,tofind the circular convolution of f[n] andg[n] weproceed as follows:
(1) Find the corresponding d.f.t.s,F[k] andG[k].
(2) Multiply these together toobtainF[k]G[k].
(3) Find the inverse d.f.t. togive f ○∗ g.
Whilst this procedure may seem complicated, it is nevertheless an efficient way of calculating
a convolution.
Example24.28 Usethe convolution theorem tofind f ○∗ gwhen f[n] = 5,4 andg[n] = −1,3.
Solution Firstwefind the corresponding d.f.t.s,F[k] andG[k]. Using
F[k] =
∑N−1
n=0
withN = 2 gives
F[0] =
f[n]e −2jnkπ/N
1∑
f[n] =9, F[1] =
n=0
1∑
f[n]e −jnπ = 5 +4e −jπ = 1
n=0