Page 70 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 21 LINEAR TIME-INVARIANT SYSTEMS

LTI system is expressed as

Alternatively, applying the causality condition ( 2.16) to Eq. (2.61, we have

y(t) = lt x(r)h(t - T) dr (2.18)
- w
Equation (2.18) shows that the only values of the input x(t) used to evaluate the output
y( t) are those for r 5 t.
Based on the causality condition (2.161, any signal x(t) is called causal if

and is called anticausal if
x(t) = 0 t>O
Then, from Eqs. (2.17), (2. I8), and (2. Iga), when the input x(t) is causal, the output y(t )
of a causal continuous-time LTI system is given by

C. Stability:

The BIBO (bounded-input/bounded-output) stability of an LTI system (Sec. 1.5H) is
readily ascertained from its impulse response. It can be shown (Prob. 2.13) that a
continuous-time LTI system is BIBO stable if its impulse response is absolutely integrable,
that is,

2.4 EIGENFUNCTIONS OF CONTINUOUS-TIME LTI SYSTEMS
In Chap. 1 (Prob. 1.44) we saw that the eigenfunctions of continuous-time LTI systems
represented by T are the complex exponentials eS', with s a complex variable. That is,

where h is the eigenvalue of T associated with e"'. Setting x(t) = es' in Eq. (2.10), we have

where

Thus, the eigenvalue of a continuous-time LTI system associated with the eigenfunction es'
is given by H(s) which is a complex constant whose value is determined by the value of s
via Eq. (2.24). Note from Eq. (2.23) that y(0) = H(s) (see Prob. 1.44).
The above results underlie the definitions of the Laplace transform and Fourier
transform which will be discussed in Chaps. 3 and 5.