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14.3 The Fourier Transform 483

14.3.2 The Windowed Fourier Transform

∞ 2
Let f (t) be a signal (function). We assume that | f (t)| dt is finite. This integral is
−∞
defined to be the energy of the signal.

In analyzing a signal, we sometimes want to localize the frequency content with respect to
ˆ
time. We know that f (ω) carries information about the frequencies ω of the signal. However,
ˆ
f (ω) does not particularize this information to specific time intervals, since

∞
ˆ
f (ω) = f (t)e −iωt dt,
−∞
and this integration is over all time. From this we can compute the total amplitude spectrum
ˆ
| f (ω)|, but cannot look at small time intervals. If we think of f (t) as a piece of music, we have
to wait until the entire piece is done before computing this amplitude spectrum.
We can obtain a picture of the frequency content of f (t) within a given time interval
by windowing the signal before taking its transform. The idea is to use a window func-
tion w(t) that is nonzero only on a finite interval, often [0, T ] or [−T, T ]. Window f (t)
with w(t) by forming the product w(t) f (t), which can be nonzero only on the selected
interval. The windowed Fourier transform of f , with respect to the particular window
function w,is

∞

f ˆ win (ω) = w(t) f (t)e −iωt dt.
−∞
EXAMPLE 14.11
Let f (t) = 6e −|t| . Then

∞ 12

ˆ
e
f (ω) = 6e −|t| −iωt dt = .
1 + ω 2
−∞
We will window f with the window function

1 for −2 ≤ t ≤ 2,
w(t) =
0 for |t| > 2.
Figures 14.2, 14.3, and 14.4 show, respectively, f (t), the window function w(t), and w(t) f (t).
The effect of windowing on this signal is to cut the signal off for times |t| > 2. The windowed
Fourier transform is therefore an integral only over [−2,2] instead of the entire real line:

∞

e
f ˆ (ω) = 6w(t)e −|t| −iωt dt
win
−∞
2
e
= 6e −|t| −iωt dt
−2
12
−2
−2
2
= −2e cos (ω) + e −2 + e ω sin(2ω) + 1 .
1 + ω 2
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October 14, 2010 16:43 THM/NEIL Page-483 27410_14_ch14_p465-504