Page 67 - Schaum's Outline of Theory and Problems of Signals and Systems

Chapter 2

Linear Time-Invariant Systems

2.1 INTRODUCTION
Two most important attributes of systems are linearity and time-invariance. In this
chapter we develop the fundamental input-output relationship for systems having these
attributes. It will be shown that the input-output relationship for LTI systems is described
in terms of a convolution operation. The importance of the convolution operation in LTI
systems stems from the fact that knowledge of the response of an LTI system to the unit
impulse input allows us to find its output to any input signals. Specifying the input-output
relationships for LTI systems by differential and difference equations will also be dis-
cussed.

2.2 RESPONSE OF A CONTINUOUS-TIME LTI SYSTEM AND
THE CONVOLUTION INTEGRAL

A. Impulse Response:
The impulse response h(t) of a continuous-time LTI system (represented by T) is
defined to be the response of the system when the input is 6(t), that is,

B. Response to an Arbitrary Input:
From Eq. (1.27) the input x( t) can be expressed as

Since the system is linear, the response y( t of the system to an arbitrary input x( t ) can be
expressed as

Since the system is time-invariant, we have

Substituting Eq. (2.4) into Eq. (2.31, we obtain