TABLE OF CONTENTS In this book we consider two methods of analysis of linear time-invariant (LTI) systems: the time-domain method and the frequency-domain method. In this chapter we discuss the time-domain analysis of linear, time-invariant, continuous-time (LTIC) systems.
For the purpose of analysis, we shall consider linear differential systems. This is the class of LTIC systems introduced in Chapter 1, for which the input x( t) and the output y( t) are related by linear differential equations of the form
where all the coefficients a i and b i are constants. Using operational notation D to represent d/ dt, we can express this equation as
or
where the polynomials Q( D) and P( D) are
Theoretically the powers M and N in the foregoing equations can take on any value. However, practical considerations make M > N undesirable for two reasons. In Section 4.3-2, we shall show that an LTIC system specified by Eq. (2.1) acts as an ( M ? N)th-order differentiator.
A differentiator represents an unstable system because a bounded input like the step input results in an unbounded output, ? ( t). Second, the noise is enhanced by a differentiator. Noise is a wideband signal containing components of all frequencies from 0 to a very high frequency approaching ? [ ]. Hence, noise contains a significant amount of rapidly varying components. We know that the derivative...
