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

762 Chapter 24 The Fourier transform
Both of these properties follow directly from the definition and linearity properties of
integrals, and mean that F isalinear operator.
Example24.3 Find F{u(t)e −t +u(t)e −2t }.
Solution We sawinExample 24.1 that
F{u(t)e −t } = 1
1+jω
Furthermore,
F{u(t)e −2t } =
Therefore,
=
∫ ∞
−∞
∫ ∞
0
u(t)e −2t e −jωt dt
e −(2+jω)t dt
[ e
−(2+jω)t ∞
=
−(2+jω)]
0
= 1
2+jω
F{u(t)e −t +u(t)e −2t } = 1
1+jω + 1
2+jω
by linearity
= 2+jω+1+jω
(1+jω)(2+jω)
=
3+2jω
2−ω 2 +3jω
24.3.2 Firstshifttheorem
IfF(ω)isthe Fourier transform of f (t),then
F{e jat f (t)} =F(ω −a)
whereaisaconstant
Example24.4 (a) Show thatthe Fourier transform of
{ 3 −2t2
f(t)=
0 otherwise
isgivenbyF(ω) = 6sin2ω .
ω
(b) Usethe first shift theorem tofind the Fourier transform of e −jt f (t).
(c) VerifythefirstshifttheorembyobtainingtheFouriertransformofe −jt f (t)directly.