Page 495 - Advanced engineering mathematics

14.3 The Fourier Transform 475

By Example 14.5,
∞

ˆ −iωt
f (ω) = f (t)e dt
−∞
a k
iωt
= ke −iωt dt =− (e −iωt − e )
−a iω
2k
= sin(aω).
ω
The amplitude spectrum of f is the graph of

ˆ sin(aω) ,
| f (ω)|= 2k
ω
shown in Figure 14.1 for k = 1 and a = 2.
We will list some properties and computational rules for the Fourier transform.
Linearity

F[ f + g]= F[ f ]+ F[g]
and, for any number k,
F[kf ]= kF[ f ].

Time Shifting If t 0 is a real number then

F[ f (t − t 0 )](ω) = e −iωt 0 ˆ
f (ω).
The Fourier transform of a shifted function f (t −t 0 ) is the Fourier transform of f , multiplied
by e −iωt 0 . This is similar to the second shifting theorem for the Laplace transform.

Proof From the definition of the Fourier transform,
∞

F[ f (t − t 0 )](ω) = f (t − t 0 )e −iωt dt
−∞

∞
= e −iωt 0 f (t − t 0 )e −iω(t−t 0 ) dt.
−∞
Upon setting u = t − t 0 we have

∞
F[ f (t − t 0 )](ω) = e −iωt 0 f (u)e −iωu du = e −iωt 0 ˆ
f (ω),
−∞
completing the proof.
The inverse version of the time shifting theorem is
−1
F [e −iωt 0 ˆ (14.12)
f (ω)](t) = f (t − t 0 ).
EXAMPLE 14.8

We will compute
2iω
e
−1
F .
5 + iω

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October 14, 2010 16:43 THM/NEIL Page-475 27410_14_ch14_p465-504