Page 37 - Schaum's Outline of Theory and Problems of Signals and Systems

SIGNALS AND SYSTEMS [CHAP. 1

1.8. Show that
(a) If x(t) and x[n] are even, then

(1.75a)

(I. 75b)

(b) If x(t) and x[n] are odd, then
x(0) = 0 and x[O] =O

k
dr
/a ~(r) = 0 and x x[n] =O
-a n= -k
(a) We can write

Letting t = -A in the first integral on the right-hand side, we get

Since x(t) is even, that is, x( -A) = x(A), we have

Hence,

Similarly,

Letting n = -m in the first term on the right-hand side, we get

Since x[n] is even, that is, x[ -m] =x[m], we have

Hence,

(6) Since x(t) and x[n] are odd, that is, x( - t) = -x(t) and x[ -n] = -x[n], we have
X( -0) = -x(O) and x[-01 = -x[O]