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

24.15 Discrete convolution and correlation 815
3 There are variants ofthe definition ofcorrelation.
Showthat if f ⋆gisredefined to be
∑ ∞m=−∞
f[m −n]g[m] then the corresponding
correlation ofthe sequences in Question1is
27,24,26,9,4.
4 Find the linear autocorrelation ofthe sequence
f[n]=3,2,1.
Solutions
1 4,9,26,24,27 4 3,8,14,8,3
24.15.5 Circularcross-correlation
The circular cross-correlation of two periodic sequences of period N is defined in a
similarmanner totheir circular convolution. Itisasequencec[n] oflengthN.
The circular cross-correlation of two periodic sequences, f[n] and g[n], each of
periodN, isdefined as
c[n]=f○⋆ g =
∑N−1
m=0
f[m]g[m−n]
forn=0,1,2,...,N−1
Whenasequenceiscross-correlatedwithitselftheprocessisknownasautocorrelation.
Example24.32 (a) Find the circular autocorrelation of the sequence f[n] = 3,2,1 using the formula.
(b) Develop a graphical method forperforming thiscalculation.
Solution (a) HereN = 3.From the definition
2∑
c[n]=f○⋆ f = f[m]f[m−n]
m=0
forn=0,1,2
2∑
c[0] = f[m]f[m]
m=0
= (3)(3) + (2)(2) + (1)(1)
= 14
2∑
c[1] = f[m]f[m−1]
m=0
= f[0]f[−1]+f[1]f[0]+f[2]f[1]
= (3)(1) + (2)(3) + (1)(2)
= 11