B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

74 ANALYSIS AND TRANSMISSION OF SIGNALS
If an impulse at J = 0 is a spectrum of a de signal, what does an impulse at J = J o
represent? We shall answer this question in the next example.
Exam ple 3 .5 Find the inverse Fourier transform of 8(f - Jo).
We the sampling property of the impulse function to obtain
.F - 1 [8(f - Jo)] = 1_: 8(f -J o )J 2n ft dJ = J 2n f0t
Therefore,
J 2n fo t o(f _ Jo)
(3.22a)
This result shows that the spectrum of an everlasting exponential ei 2 nfo t is a single impulse
atJ = Jo. We reach the same conclusion by qualitative reasoning. To represent the everlasting
exponential ei 2 nfo t , we need a single everlasting exponential ei 2n ft
with cv = 2nJ 0 .
Therefore, the spectrum consists of a single component at frequency J = J o .
From Eq. (3.22a) it follows that
e-j2n fo
t
o(f + Jo)
(3.22b)
Exam ple 3 .6 Find the Fourier transforms of the everlasting sinusoid cos 2nJo.
Recall the Euler formula
cos 2nJot = ! c
J 2 nfot
+ e-j 2n fo t )
Adding Eqs. (3.22a) and (3.22b), and using the preceding formula, we obtain
2
1
cos 2nJot [8(f +Jo) + 8(f - Jo)] (3.23)
2
Figure 3.12
Cosine signal
and its Fourier
spectrum.
The spectrum of cos 2nJ o t consists of two impulses at Jo and -J o in the J-domain, or,
two impulses at ±cv o = ±2nJ o in the cu-domain as shown in Fig. 3.12. The result also
follows from qualitative reasoning. An everlasting sinusoid cos w o t can be synthesized by
two everlasting exponentials, J w ot
and e-j w ot . Therefore, the Fourier spectrum consists
of only two components of frequencies cv o and -w o .
-
G(f)
0.5 0.5
-f o
0
fo f -
(a)
(b)