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

814 Chapter 24 The Fourier transform
(a)
(b)
(c)
(d)
(e)
g[m ]
g[m – 1]
g[m – 2]
g[m + 1]
g[m + 2]
10
6 8
4
2
– 4
10
6 8
4
2
– 4
10
6 8
4
2
– 4
10
6 8
4
2
– 4
10
6 8
4
2
– 4
m
m
m
m
m
Figure24.32
The effect oftranslatingg[m].
n = 0
sum of products:
(7)(1) + (2)(9) + (–3)(–1) = 7 + 18 + 3 = 28
n = 1
sum of products:
(2)(1) + (–3)(9) = 2 – 27 = –25
n = 2
sum of products:
(–3)(1) = –3
n=–1
sum of products:
(7)(9) + (2)( –1) = 63 – 2 = 61
n= –2
sum of products:
(7)( –1) = –7
The graph ofg[m] can be translatednunits to the left or right by changing the
argument ofgfromg[m] tog[m −n]. Ifnis positive the graph moves to the right.
Study the figure to observe this. Correlation is the sum of products of f[m] and
g[m − n]. For each value of n, the graph of f[m] is superimposed. We are only
interested in values ofnfor which the graphs overlap -- otherwise each product is
zero.Foreachvalueofnthesuperimposedgraphsmakeiteasytoseewhichvalues
must be multiplied together and added. Placing these results in order confirms the
resultobtained inpart(a), thatisthe correlation is −7,61,28,−25,−3.
Whenasequenceiscross-correlatedwithitselftheprocessisknownasautocorrelation.
Autocorrelationisusedtosearchforpossibleperiodicitiesinsignals,becauseifasignal
isperiodicwithperiodN itsautocorrelationsequencewillshowpeaksatintervalsofN.
EXERCISES24.15.4
1 Find the linearcross-correlationofthe sequences
f[n] = 4,5,9 andg[n] = 3,1,1.
2 Show thatthe linearcross-correlationof f[n] andg[n]
can be written in the alternative form
f⋆g= ∑ ∞
m=−∞ f[m +n]g[m].