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

24.8 Convolution and correlation 781
Example24.18 (a) Usingthe definition ofcorrelation, calculate the correlation of f (t) =u(t)e −t and
g(t) =u(t)e −2t , whereu(t) isthe unit stepfunction.
(b) Verifythe correlation theorem for these functions.
Solution (a) The correlation of f andgisgiven by
f⋆g=
∫ ∞
−∞
f(λ)g(λ −t)dλ
1
f(t)= u(t) e –t
g(t) = u(t) e – 2t
Figure24.16
Graphsof f (t) =u(t)e −t
andg(t) =u(t)e −2t .
t
Graphs of f (t)andg(t) areshown inFigure 24.16.
InFigure24.17wehavesuperimposedthegraphsof f (λ),showndashed,and
g(λ −t), fort being negative, and then positive. Where the graphs do not overlap,
the product f (λ)g(λ −t), and hence the correlation, iszero.
Whent<0
Whent < 0 the graphs overlap for 0 λ < ∞. Hence
f⋆g=
=
∫ ∞
0
∫ ∞
0
f(λ)g(λ −t)dλ
e −λ e −2(λ−t) dλ
∫ ∞
= e 2t e −3λ dλ
0
= e 2t [ e
−3λ
−3
] ∞
0
= 1 3 e2t
f( l)= u( l)e –
l
(a)
l
g( l – t) when t < 0
(b)
t
l
(c)
g( l – t )when t > 0
t
l
Figure24.17
Graphsof f(λ),andg(λ −t)for(b)t < 0,
(c)t>0.