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

816 Chapter 24 The Fourier transform
n=0 n=1 n=2
3
1
2
3
3
3
2
2 1
1
3
2 1
2
1
2 1
3
Sums of products:
(3)(3) + (2)(2) + (1)(1) = 14 (3)(1) + (2)(3) + (1)(2) = 11 (3)(2) + (2)(1) + (1)(3) = 11
Figure24.33
Graphical calculation ofthe autocorrelation of f[n] = 3,2,1.
c[2] =
2∑
f[m]f[m−2]
m=0
= (3)(2) + (2)(1) + (1)(3)
= 11
Hencec[n] = 14,11,11.
(b) The graphical method involves listing the sequence f[m], m = 0,1,2, around an
inner circle. Around an outer circle we listitagain. This method is identical to that
used in Example 24.27 for circular convolution, but because now there is no folding,
the sequence on the outer circle is not reversed. The calculation can be seen in
Figure 24.33.
24.15.6 (Circular)correlationtheorem
For real sequences f[n]andg[n] the correlation theorem states:
D{f ○⋆ g} =F[k]G[k]
whereG[k] denotes the complex conjugate ofG[k].
This provides a technique forcalculating a correlation usingthe d.f.t.
Example24.33 Findthe circularcorrelation f ○⋆ gwhen f[n] = 8,−9,3,2 andg[n] = 11,4,−1,−5.
Solution Either directly from the definition of the d.f.t., or by using a computer package, we can
show that
F[k] = 4,5 +11j,18,5 −11j G[k] = 9,12 −9j,11,12 +9j
Theconjugate ofG[k] isG[k] = 9,12 +9j,11,12 −9j. Then
F[k]G[k] = 36,−39 +177j,198,−39 −177j