Page 504 - Advanced engineering mathematics

484 CHAPTER 14 The Fourier Integral and Transforms

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5
0.8
4
0.6

3
0.4
2
0.2

1
0
–4 –2 0 2 4
0
t
–4 –2 0 2 4
t
FIGURE 14.3 Window function w(t) in
FIGURE 14.2 f (t) = 6e −|t| Example 14.11.

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5

4

3

2

1
0
–4 –2 0 2 4
t
FIGURE 14.4 Windowed function w(t) f (t)
in Example 14.11.

Sometimes we use a shifted window function.If w(t) is nonzero only on [−T, T ], then
the shifted function w(t − t 0 ) is the graph of w(t) shifted t 0 units to the right and is nonzero
only on [t 0 − T,t 0 + T ]. In this case, the shifted windowed Fourier transform is the transform of
w(t − t 0 ) f (t):

f ˆ (ω) = F[w(t − t 0 ) f (t)](ω)
win,t 0
t 0 +T
= w(t − t 0 ) f (t)e −iωt dt.
t 0 −T
This gives the frequency content of the signal in the time interval [t 0 − T,t 0 + T ].

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October 14, 2010 16:43 THM/NEIL Page-484 27410_14_ch14_p465-504