Page 69 - Schaum's Outline of Theory and Problems of Signals and Systems

5 8 LINEAR TIME-INVARIANT SYSTEMS [CHAP. 2

3. The product x(~)h(t - T) is integrated over all T to produce a single output value
At).
4. Steps 1 to 3 are repeated as I varies over - 03 to 03 to produce the entire output y( t ).
Examples of the above convolution integral operation are given in Probs. 2.4 to 2.6.

F. Step Response:
The step response s(t) of a continuous-time LTI system (represented by T) is defined to
be the response of the system when the input is 41); that is,

In many applications, the step response dt) is also a useful characterization of the system.
The step response s(t) can be easily determined by Eq. (2.10); that is,

Thus, the step response s(t) can be obtained by integrating the impulse response h(t).
Differentiating Eq. (2.12) with respect to t, we get

Thus, the impulse response h(t) can be determined by differentiating the step response
dl).

2.3 PROPERTIES OF CONTINUOUS-TIME LTI SYSTEMS

A. Systems with or without Memory:
Since the output y(t) of a memoryless system depends on only the present input x(t),
then, if the system is also linear and time-invariant, this relationship can only be of the
form

(2.14)
Y([) = Kx(t)
where K is a (gain) constant. Thus, the corresponding impulse response h(f) is simply

h(t) = K6(t) (2.15)

Therefore, if h(tJ # 0 for I,, # 0, the continuous-time LTI system has memory.

B. Causality:
As discussed in Sec. 1.5D, a causal system does not respond to an input event until that
event actually occurs. Therefore, for a causal continuous-time LTI system, we have

Applying the causality condition (2.16) to Eq. (2.101, the output of a causal continuous-time