Page 507 - Advanced engineering mathematics

14.3 The Fourier Transform 487

14.3.4 Low-Pass and Bandpass Filters
If f is a signal with finite energy, then the spectrum of f is given by its Fourier transform. If
ω 0 is a positive number and f is not band-limited, we can replace f with a band-limited signal
having bandwidth not exceeding ω 0 by applying a low-pass filter which cuts f (ω) off at
ˆ
f ω 0
frequencies outside the range [−ω 0 ,ω 0 ]. That is, let

ˆ
f (ω) for −ω 0 ≤ ω ≤ ω 0 ,
ˆ
f ω 0 (ω) =
0 for |ω| >ω 0 .
ˆ
This defined the transform f ω 0 , from which we recover f ω 0 by the inverse Fourier transform
1 ∞ 1 ω 0
(t) = ˆ ω)e iωt dω = ˆ (ω)e iωt dω.
f ω 0 f ω 0 f ω 0
2π 2π
−∞ −ω 0
Applying a low-pass filter is actually a windowing process. Define the characteristic function χ I
of an interval I by

1 for t in I,
χ I (t) =
0 for t not in I.
Then

ˆ ˆ
f ω 0 (ω) = χ [−ω 0 ,ω 0 ] (ω) f (ω) (14.15)
ˆ
so we have windowed f (ω) with the characteristic function χ [−ω 0 ,ω 0 ] . More succinctly,
ˆ = χ [−ω 0 ,ω 0 ] f .
ˆ
f ω 0
In this context, the window function χ [−ω 0 ,ω 0 ] is called the transfer function. The inverse Fourier
transform of the transfer function is
1 ω 0 sin(ω 0 t)
−1 iωt
F [χ [−ω 0 ,ω 0 ] ](t) = e dω = ,
2π −ω 0 πt
whose graph is given in Figure 14.5 for ω 0 = π.

1

0.8

0.6

0.4

0.2

0
–6 –4 –2 0 2 4 6
–0.2
t
FIGURE 14.5 sin(πt)/t

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October 14, 2010 16:43 THM/NEIL Page-487 27410_14_ch14_p465-504