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

24.2 The Fourier transform -- definitions 759
The Fourier transform of f (t)is defined tobe
F{f(t)}=F(ω) =
∫ ∞
−∞
f (t)e −jωt dt
YouwillalsonotethesimilaritybetweenEquation(24.4)andthedefinitionoftheLaplace
transformof f (t):
L{f(t)} =
∫ ∞
0
f (t)e −st dt (24.5)
We see that, apart from the limits of integration, the substitution jω = s in Equation(24.4)resultsintheLaplacetransforminEquation(24.5).ThereisindeedanimportantrelationshipbetweenthetwotransformswhichweshalldiscussinSection24.7.We
notethatEquation(24.3)providesaformulafortheinverseFouriertransformofF(ω),
although the integral isfrequently difficulttoevaluate.
Example24.1 Find the Fourier transform of the function f (t) = u(t)e −t , whereu(t) is the unit step
function.
f(t)
1
Figure24.2
Graph ofu(t)e −t .
Solution The functionu(t)e −t is shown in Figure 24.2. Using Equation (24.4), its Fourier transformis
given by
t
thatis,
F(ω) =
=
=
∫ ∞
−∞
∫ ∞
0
∫ ∞
0
f (t)e −jωt dt
e −t e −jωt dt since f (t) = 0 fort < 0
e −(1+jω)t dt
[ e
−(1+jω)t ∞
=
−(1+jω)]
0
= 1
1+jω
F(ω) = 1
1+jω
sincee −(1+jω)t → 0ast → ∞
Example24.2 Use Equation (24.3) to find the Fourier integral representation of the function defined
by
{ 1 −1t1
f(t)=
0 |t|>1
Solution UsingEquation (24.3)wefind
f(t)= 1
2π
∫ ∞
−∞
F(ω)e jωt dω