Page 50 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 11 SIGNALS AND SYSTEMS

Now, using Eq. (1.20) for 4(0), we obtain

for any 4(t). Then, by the equivalence property (1.991, we obtain

1
6(at) = -6(t)
la1
(6) Setting a = - 1 in the above equation, we obtain
1
6( -t) = -6(t) = S(t)
I- 11

which shows that S(t) is an even function.

1.26. (a) Verify Eq. (1.26):

x(t)8(t - to) =x(t,)S(l - 1,)
if x(t) is continuous at t = to.

(b) Verify Eq. (1.25):
x(r)S(t) =x(O)S(t)

if x(t) is continuous at t = 0.
(a) If x(t) is continuous at t =to, then by definition (1.22) we have

for all +(t) which are continuous at t = to. Hence, by the equivalence property (1.99) we
conclude that
x(t)6(t - to) =x(to)6(t - t,,)

(6) Setting to = 0 in the above expression, we obtain
x(t)b(t) =x(O)S(t)

1.27. Show that

(a) t8(t) = 0
(b) sin t8(r) = 0
(c) COS tS(t - r) = -S(t - T)