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Sec. 2–2 Fourier Transform and Spectra 47
A weaker sufficient condition for the existence of the Fourier transform is
q
E = |w(t)| 2 dt 6q
L
-q
(2–32)
where E is the normalized energy [Goldberg, 1961]. This is the finite-energy condition that is
satisfied by all physically realizable waveforms. Thus, all physical waveforms encountered in
engineering practice are Fourier transformable.
It should also be noted that mathematicians sometimes use other definitions for the
Fourier transform rather than Eq. (2–26). However, in these cases, the equation for the corresponding
inverse transforms, equivalent to Eq. (2–30), would also be different, so that when
the transform and its inverse are used together, the original w(t) would be recovered. This is a
consequence of the Fourier integral theorem, which is
w(t) =
L
q
q
-q L-q
w(l)e j2pf(t-l) dl df
(2–33)
Equation (2–33) may be decomposed into Eqs. (2–26) and (2–30), as well as other definitions
for Fourier transform pairs. The Fourier integral theorem is strictly true only for wellbehaved
functions (i.e., physical waveforms). For example, if w(t) is an ideal square wave, then
at a discontinuous point of w (l 0 ), denoted by l 0 , w(t) will have a value that is the average of
the two values that are obtained for w(l)
on each side of the discontinuous point .
l 0
Example 2–3 SPECTRUM OF AN EXPONENTIAL PULSE
Let w(t) be a decaying exponential pulse that is switched on at t = 0. That is,
Directly integrating the FT integral, we get
or
In other words, the FT pair is
e e-t , t 7 0
(2–34)
0, t 6 0 f 4 1
1 + j2pf
The spectrum can also be expressed in terms of the quadrature functions by rationalizing the
denominator of Eq. (2–34); thus,
X(f) =
See Example1_03A.m for plots.
q
W(f) = e -t e -j2pft dt = `
L
0
w(t) = e e-t , t 7 0
0, t 6 0
W(f) =
1
1 + j2pf
1
1 + (2pf) 2 and Y(f) =
-e -(1+j2pf)t q
1 + j2pf `
0
-2pf
1 + (2pf) 2