563489578934

Sec. 2–2 Fourier Transform and Spectra 51
Thus, for either a 7 0 or a 6 0, we get
w(at) 4 1
|a| Waf a b
The other theorems in Table 2–1 are proved in a similar straightforward manner, except
for the integral theorem, which is more difficult to derive because the transform result
involves a Dirac delta function d( f ). This theorem may be proved by the use of the convolution
theorem, as illustrated by Prob. 2–38. The bandpass signal theorem will be discussed in
more detail in Chapter 4. It is the basis for the digital and analog modulation techniques covered
in Chapters 4 and 5. In Sec. 2–8, the relationship between the Fourier transform and the
discrete Fourier transform (DFT) will be studied.
As we will see in the examples that follow, these theorems can greatly simplify the calculations
required to solve Fourier transform problems. The reader should study Table 2–1
and be prepared to use it when needed. After the Fourier transform is evaluated, one should
check that the easy-to-verify properties of Fourier transforms are satisfied; otherwise, there is
a mistake. For example, if w(t) is real,
• W(-f) = W * (f), or |W( f )| is even and u( f ) is odd.
• W(f) is real when w(t) is even.
• W(f) is imaginary when w(t) is odd.
Example 2–4 SPECTRUM OF A DAMPED SINUSOID
Let the damped sinusoid be given by
The spectrum of this waveform is obtained by evaluating the FT. This is easily accomplished by
using the result of the previous example plus some of the Fourier theorems. From Eq. (2–34), and
using the scale-change theorem of Table 2–1, where a = 1/T, we find that
Using the real signal frequency translation theorem with u = -p/2, we get
W(f) = 1 2 e e-jp/2
w(t) = e e-t/T sin v 0 t,
0,
e e-t/T , t 7 0
0, t 6 0 f 4 T
1 + j(2pf T)
T
1 + j2pT(f - f 0 )
= T 2j e 1
1 + j2pT(f - f 0 )
(2–44)
where e ; jp/2 = cos(p/2) ; j sin(p/2) = ;j. This spectrum is complex (i.e., neither real nor
imaginary), because w(t) does not have even or odd symmetry about t = 0.
See Example2_04.m for a plot of (2–44). As expected, this shows that the peak of the magnitude
spectrum for the damped sinusoid occurs at f =;f 0 . Compare this with the peak of the
magnitude spectrum for the exponential decay (Example 2–3) that occurs at f = 0. That is, the
sin v0t factor caused the spectral peak to move from f = 0 to f =;f 0 .
-
t > 0, T > 0
t < 0
+ e jp/2 T
1 + j2pT(f + f 0 ) f
1
1 + j2pT(f + f 0 ) f