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

806 Chapter 24 The Fourier transform
n=0 n=1 n=2
7
2
–4
9
9
9
–4
– 1 3
2
7
– 1 3
–4
2
– 1 3
7
Sums of products:
(9)(7) + ( –1)(– 4) + (3)(2) = 73
(9)(2) + ( –1)(7) + (3)( – 4) = –1
(9)( – 4) + ( –1)(2) + (3)(7) = –17
Figure24.28
The sequences f[m]drawn around the inner circle,andg[n −m] drawn around the outer circle,
forn=0,1,2.
Finally, letn = 2.
h[2] =
2∑
f[m]g[2 −m]
m=0
= f[0]g[2] + f[1]g[1] + f[2]g[0]
= (9)(−4) + (−1)(2) + (3)(7)
= −17
So the circular convolution f ○∗ g = 73,−1,−17.
If required, a table can be constructed which summarizes all the necessary informationaswasshowninExample24.26.Thesequencesmustbeextendedtoshow
their periodicity, and this time we are only interested in generating the convolution
sequence over one period, namely thatsection of the table form = 0,1,2.
m −2 −1 0 1 2 3 4 5
f[m] −1 3 9 −1 3 9 −1 3
n=0 g[−m] −4 2 7 −4 2 7 −4 2
n=1 g[1−m] 7 −4 2 7 −4 2 7 −4
n=2 g[2−m] 2 7 −4 2 7 −4 2 7
(b) A graphical representation can be developed by listing the fixed sequence f[m],
for m = 0,1,2, anticlockwise around an inner circle as shown in Figure 24.28.
We list g[−m] around an outer circle but do so clockwise to take account of the
folding.Byrotatingtheoutercircleanticlockwiseweobtaing[1−m]andg[2−m].
Bymultiplyingneighbouringtermsandaddingweobtaintherequiredconvolution.
The resultis73,−1,−17 as obtained inpart(a).
In the following section you will see how circular convolution can be performed using
the d.f.t.