Page 505 - Advanced engineering mathematics

14.3 The Fourier Transform 485

Engineers refer to the windowing process as time-frequency localization.The center of the
window function w is defined to be
∞ 2

t|w(t)| dt
−∞
t C = ∞ .
|w(t)| dt
2
−∞
The number
1/2
∞ 2 2

(t − t C ) |w(t)| dt
−∞
t R =
∞ 2

|w(t)| dt
−∞
is the radius of the window function. The width of the window function is 2t R , a number
referred to as the RMS duration of the window.
Similar terminology applies when we deal with the the Fourier transform of the window
function:
∞ 2

ω|ˆw(ω)| dω
−∞
center of ˆw = ω C =
∞ 2
|ˆw(ω)| dω
−∞
and
1/2

∞ 2 2
(ω − ω C ) |ˆw(ω)| dω
−∞
radius of ˆw = ω R = .
∞ 2

|ˆw(ω)| dω
−∞
The width of ˆw is 2ω R , a number referred to as the RMS bandwidth of the window function.
14.3.3 The Shannon Sampling Theorem
A signal f (t) is band-limited if its Fourier transform f (ω) has nonzero values only on
ˆ
some interval [−L, L].If f is band-limited, the smallest positive L for which this is true
is called the bandwidth of f . For such L we have
f (ω) = 0if |ω| > L.
ˆ
The total frequency content of such a signal lies in the band [−L, L].
We will show that a band-limited signal can be reconstructed from samples taken at
appropriately chosen times. Begin with the integral for the inverse Fourier transform:
1 ∞
ˆ
f (t) = f (ω)e iωt dω.
2π
−∞
Because f is assumed to have bandwidth L, we actually have
1 L iωt
ˆ
f (t) = f (ω)e dω. (14.13)
2π −L
ˆ
Now expand f (ω) in a complex Fourier series on [−L, L]:
∞

ˆ
f (ω) = c n e nπiω/L , (14.14)
n=−∞

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October 14, 2010 16:43 THM/NEIL Page-485 27410_14_ch14_p465-504