Page 333 - Schaum's Outline of Theory and Problems of Signals and Systems

320 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS [CHAP. 6

6.19. Find the Fourier transform of the sinusoidal sequence
x[n] = cos Ron lflolS

From Euler's formula we have
cos Ron = ;(ein~" + e-' IJ n,
Thus, using Eq. (6.135) and the linearity property (6.42), we get
X(R) = T[S(~ 0,) + 6(R + a,)] IRI, Ia0I 5
-
which is illustrated in Fig. 6-15. Thus,

cos Ron - a[6(R - Ro) + 6(R+ a,)] Ial, IRol r T (6.137)

Fig. 6-15 A cosine sequence and its Fourier transform.

6.20. Verify the conjugation property (6.45), that is,
x*[n] -X*(-R)
From Eq. (6.27)
m
.F(x*[n]) = C x*[n] e-inn =
n = -m n= -m

Hence,
x*[n] -X*( -0)

6.21. Verify the time-scaling property (6.491, that is,

From Eq. (6.48)

x[n/m] =x[k] if n = km, k = integer
ifnzkm
Then, by Eq. (6.27)
m
.F(-qrn)b1) C ~(,)bl e-jnn
=
,,= -m