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

24.15 Discrete convolution and correlation 803
f [m]
g[m]
10
8
6
4
2
–2
–4
1 2
m
10
8
6
4
2
–2
–4
Figure24.24
Thesequences f[m] = 3,9,2, −1 andg[m] = −4,8,5.
1 2
m
multipliedtogetherandadded.Ifrequired,atablecanbeconstructedwhichsummarizes
all the necessary information as shown below.
m −2 −1 0 1 2 3 4 5
f[m] -- -- 3 9 2 −1 -- --
n=0 g[−m] 5 8 −4 -- -- -- -- --
n=1 g[1−m] -- 5 8 −4 -- -- -- --
n=2 g[2−m] -- -- 5 8 −4 -- -- --
n=3 g[3−m] -- -- -- 5 8 −4 -- --
n=4 g[4−m] -- -- -- -- 5 8 −4 --
n=5 g[5−m] -- -- -- -- -- 5 8 −4
EXERCISES24.15.1
1 Given f[n] = 1,2,2,1 andg[n] = 2,1,1,2, findthe
linearconvolution f ∗g.
2 (a) Thelinearconvolution ofthe sequences
3,9,2, −1 and −4,8,5 was obtainedin
Example24.27. Show thatthisconvolution is
equivalent to multiplying the two polynomials
3+9x+2x 2 −x 3 and−4+8x+5x 2 .
(b) Byusingpolynomial multiplicationfind the
linearconvolution ofthe sequences f[n] = 9, −8
andg[n] = 1,2, −4.
(c) Use the polynomial methodto findthe linear
convolution of f[n] = 9, −1,3 and
g[n] = 7,2, −4.
3 Find the linearconvolution of f[n] = 1, −1,1,3 and
g[n] = 7,2,0,1.
4 Prove from the definition thatlinearconvolution is
commutative, that is f ∗g=g ∗ f.
Solutions
1 2,5,7,8,7,5,2
3 7, −5,5,24,5,1,3
2 (b) 9,10, −52,32 (c) 63,11, −17,10, −12
24.15.2 Circularconvolution
InthissectionweconsiderperiodicsequenceshavingperiodN.Wecanselectoneperiod
forexaminationbylookingattheterms f[0], f[1], f[2],..., f[N−1],say.Forexample,
the sequence
f[n]=...−7,11,2,−7,11,2,−7,11,2...