Page 494 - Advanced engineering mathematics

474 CHAPTER 14 The Fourier Integral and Transforms

As an illustration, we will compute the inverse of this Fourier transform. By equation
(14.11),
1 ∞
−1 ˆ ˆ iωt
F [ f ](t) = f (ω)e dω
2π
−∞
1 ∞ 1 − cos(ω)
iωt
= e dω
π ω 2
−∞
= π(t + 1)sgn(t + 1) + π(t − 1)sgn(t − 1) − 2sgn(t).
This integration was done using MAPLE, in which

⎧
⎪1 for t > 0
⎨
sgn(t) = −1 for t < 0
⎪
0 for t = 0.
⎩
−1 ˆ
By considering cases t <−1, −1<t <1 and t >1, it is routine to verify that indeed F [ f ](t)=
f (t) in this example.

In the context of the Fourier transform, the amplitude spectrum of a signal f (t) is the graph
ˆ
of | f (ω)|.

EXAMPLE 14.7

Let a and k be positive numbers and let

k for −a ≤ t ≤ a
f (t) =
0 for t < −a and for t > a.

4

3

2

1

0
–8 –4 0 4 8
x
FIGURE 14.1 Graph of | ˆ f (ω)| in Example
14.7, for k = 1,a = 2.

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October 14, 2010 16:43 THM/NEIL Page-474 27410_14_ch14_p465-504