Page 325 - Schaum's Outline of Theory and Problems of Signals and Systems

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

(a) The fundamental period of x[n] is No = 8, and Ro = 277/N0 = n/4. Rather than using
Eq. (6.8) to evaluate the Fourier coefficients c,, we use Euler's formula and get

Thus, the Fourier coefficients for x[n] are cl = f, c-, =c-,+,=c7 = $, and all other
c, = 0. Hence, the discrete Fourier series of x(n1 is

(b) From Prob. 1.16(i) the fundamental period of x[n] is No = 24, and R, = 277/N0 = 77/12.
Again by Euler's formula we have

I
I
Thus, c3 = -j(4),c4 = $,c-, = c-~+~~ ?,cP3 =c-~+~~ =I(?), and all other
=cZO
=c2]
=
c, = 0. Hence, the discrete Fourier series of x[n] is
(c) From Prob. l.l6( j) the fundamental period of x[n] is No = 8, and Ro = 277/No = n/4.
Again by Euler's formula we have

1
I
Thus, c0 = f, c1 = a, c- = c- +n = c7 = a, and all other c, = 0. Hence, the discrete
Fourier series of x[nl is

6.7. Let x[n] be a real periodic sequence with fundamental period N, and Fourier
coefficients ck =ak + jb,, where a, and b, are both real.

(a) Show that a-,=ak and b-,= -bk.
(b) Show that c,,,,, is real if No is even.
(c) Show that x[n] can also be expressed as a discrete trigonometric Fourier series of
the form

(Nu- 1)/2 27r
x[n] =co+2 (a,coskfl,n -b,sinkfl,n) fl, = - (6.123)
k= 1 No
if No is odd or
(No - 2)/2
x[n] = c, + (- 1)'ch/, + 2 (a, cos kR,n - bk sin kflon) (6.124)
k=l
if N,, is even.