Page 263 - Schaum's Outline of Theory and Problems of Signals and Systems

252 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS [CHAP. 5

5.25. Find the Fourier transform of the periodic impulse train [Fig. 5-22(a)]

From Eq. (5.115) in Prob. 5.8, the complex exponential Fourier series of 6,,,(t) is given by

Using Eq. (5.1461, we get

m
= W() C S(w - kw,) = w,G,,(w)
k= - m
(5.147)

Thus, the Fourier transform of a unit impulse train is also a similar impulse train [Fig. 5-22(b)].

(0) (6)
Fig. 5-22 Unit impulse train and its Fourier transform.

5.26. Show that
x(t) cos o,,t - iX(o - o,,) + iX(o + oo)
+
and x(t ) sin w,,t - -j [ :~(w - w,) - :~(o w,)]
Equation (5.148) is known as the modulation theorem.
From Euler's formula we have

cos wol = i(ejW + -jW 1
Then by the frequency-shifting property (5.51) and the linearity property (5.491, we obtain
I
~[x(t)cos wet] = .~[ix(t) ei"ut + 'x(t) 2 e-jU~lt
= ;x(w - wo) + fx(w + w,)
Hence,

X(~)COS ~()t $X(W - w,) + ;x(w + w")
c-'