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

24.15 Discrete convolution and correlation 813
Note the similarity between this definition and that of linear convolution defined in
Section 24.15.1. In the formula for cross-correlation the sequence g is not folded. If
the two sequences are finite and of length N, that is f[n] and g[n] are non-zero only
when 0 nN−1, there are 2N −1 terms in the cross-correlation sequence and the
formulacan be written as follows:
Linear cross-correlation of two finite sequences f[n] andg[n]:
and
c[n]=f⋆g=
c[n]=f⋆g=
∑N−1
m=n
N+n−1
∑
m=0
f[m]g[m−n]
f[m]g[m−n]
forn=0,1,2,...,N−1
forn=0,−1,−2,...,−(N−1)
Example24.31 Suppose f[n] = 7,2,−3 andg[n] = 1,9,−1. Assume both sequences f andgstart at
n=0.
(a) Find the linear cross-correlationc[n] = f ⋆gusing the formulae above.
(b) Develop a graphical interpretation of thisprocess.
Solution (a) Both f and g are finite sequences of length N = 3. Their cross-correlation is a
sequencec[n], forn = −2,−1,0,1,2, of length 5.
Usingthe formulae above withn = −2 gives
c[−2] =
0∑
f[m]g[m +2]
m=0
= f[0]g[2]
= (7)(−1)
= −7
Whenn = −1 we have
c[−1] =
1∑
f[m]g[m +1]
m=0
= f[0]g[1] + f[1]g[2]
= (7)(9) + (2)(−1)
= 61
The remaining terms are calculated in a similar fashion. You should calculate one
or two terms yourself toverifythat the fullsequence is
c[n] = −7,61,28,−25,−3
n = −2,−1,0,1,2
(b) The graphical interpretation is developed along the same lines as for linear convolution
in Example 24.26. Figure 24.32 shows the sequence f[m], form = 0,1,2,
denoted by the symbols ◦. Also shown isthe sequenceg[m] denoted by •.