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

24.15 Discrete convolution and correlation 805
f [0]= f [3]
–7
f [0]
–7
11 2
f [1]= f [4] f [2]= f [5]
(a)
11 2
f [–2] f [–1]
(b)
Figure24.26
Aperiodic sequence can be visualized
bylisting its termsaround acircle.
isaperiodicsequencewithperiodN = 3.Wecanselecttheterms f[0], f[1], f[2],thatis
terms −7,11,2, and use these tostudythe entire sequence.
Suchasequencecanberepresentedgraphicallybylistingitstermsaroundacircleas
shown in Figure 24.26. Doing this allows us to calculate further terms in the sequence
as we require them. Rotation anticlockwise represents increasingnas in (a). Rotation
clockwise represents decreasingnas in(b).
Suppose f[n] andg[n] are two periodic sequences having periodN. Their circular
convolution,orperiodicconvolution,isthesequenceh[n],whichwedenoteby f ○∗ g
and which isdefined as follows:
The circular convolution of two periodic sequences each of periodN isdefined as
h[n]=f○∗g=
∑N−1
m=0
f[m]g[n−m] forn=0,1,2,...,N−1
The sequence is periodic with period N and so we can state it for n = 0,1,2,...,
N−1.
Example24.27 (a) Calculate the circular convolution,h[n] = f ○∗ g, of the two periodic sequences
f[n] = 9,−1,3 andg[n] = 7,2,−4.
(b) Develop a graphical representation of thisprocess.
Solution (a) The sequenceg[n] isdepicted inFigure 24.27.
We use the formulagiven above. InthisExample,N = 3.First letn = 0.
g[0]
2∑
h[0] = f[m]g[0 −m]
7
2 – 4
g[1] g[2]
g[–2]
g[–1]
Figure24.27
Theperiodic sequence
g[n] = 7,2, −4.
m=0
= f[0]g[0] + f[1]g[−1] + f[2]g[−2]
= (9)(7) + (−1)(−4) + (3)(2)
= 73
Next letn = 1.
2∑
h[1] = f[m]g[1 −m]
m=0
= f[0]g[1] + f[1]g[0] + f[2]g[−1]
= (9)(2) + (−1)(7) + (3)(−4)
= −1