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

782 Chapter 24 The Fourier transform
Whent0
Whent 0 the graphs overlap fort λ < ∞. Hence
f⋆g=
=
∫ ∞
t
∫ ∞
t
f(λ)g(λ −t)dλ
e −λ e −2(λ−t) dλ
∫ ∞
= e 2t e −3λ dλ
= e 2t [ e
−3λ
−3
t
= e 2te−3t
3
= 1 3 e−t
] ∞
t
Finally the complete expression forthe correlation is
⎧
1
⎪⎨
(f ⋆g)(t)=
3 e2t whent < 0
⎪⎩ 1
3 e−t whent 0
(b) UsingTable 24.1 the Fourier transforms of f andgare
F(ω) = 1 G(ω) = 1
1+jω 2+jω
and hence
F(ω)G(−ω) =
1
(1+jω)(2−jω)
Furthermore, taking the Fourier transformof f ⋆g, obtained inpart(a), we have
∫ 0
∫ ∞
1
F{f⋆g}=
−∞ 3 e2t e −jωt 1
dt +
0 3 e−t e −jωt dt
[ e
t(2−jω) 0 [ e
t(−1−jω) ∞
=
+
3(2−jω)]
−∞
3(−1 −jω)]
0
=
1
3(2−jω) + 1
3(1+jω)
= 3+3jω+6−3jω
9(2−jω)(1+jω)
=
1
(2−jω)(1+jω)
We have shown that F{f ⋆g} = F(ω)G(−ω) and so the correlation theorem has
been verified.