Page 62 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 11 SIGNALS AND SYSTEMS

Setting t = 0, we obtain
to) =y(0)eS1"
Since to is arbitrary, by changing to to t,' we can rewrite Eq. (1.118) as

y(t) = y(0)eS1 = hes'
or T(eS'} = AeS'
where A = ~(0).

1.45. Let T represent a discrete-time LTI system. Then show that

T{zn) = hzn
where z is a complex variable and A is a complex constant.
Let y[n] be the output of the system with input x[n] = zn. Then
Tbnj =ybl
Since the system is time-invariant, we have

T{zn+"~j y[n + no]
=
for arbitrary integer no. Since the system is linear, we have

Hence, y[n +no ] = znOy[n]
Setting n = 0, we obtain

Since no is arbitrary, by changing no to n, we can rewrite Eq. (1.120) as
y[n 3 = y[O]zn = Azn
or T(zn) = Azn
where A = y[O].
In mathematical language, a function x(.) satisfying the equation

is called an eigenfunction (or characteristic function) of the operator T, and the constant A is
called the eigenvalue (or characteristic value) corresponding to the eigenfunction x(.). Thus
Eqs. (1.117) and (1.119) indicate that the complex exponential functions are eigenfunctions of
any LTI system.

Supplementary Problems

1.46. Express the signals shown in Fig. 1-41 in terms of unit step functions.
t
Ans. (a) x(t) = -[u(t) - u(t - 2)1
2
-
(6) ~(t) u(t + 1) + 2u(t) - u(t - 1) - u(t - 2) - ~(t 3)
=