Principles of System Identification

5

Transform-Domain Models for Linear

TIme-Invariant Systems

This chapter presents models of the LTI system in a transform (frequency) domain. A review of

the frequency-domain descriptions of discrete-time, LTI systems is presented. The objectives

are to provide insights into FRFs and their characteristics, introduce and review transfer func-

tion (z-domain) representations. The chapter also connects time-domain models of Chapter 4

with their frequency- and z-domain counterparts.

In the previous chapter, specifically in §4.2.3, we studied the frequency response function (FRF)

as one of the elementary response models for an LTI system.

The purpose of this chapter is review certain useful facets of the FRF, again with relevance to

identification. Subsequently, we advance to the subject of complex-frequency (damped sine waves)

representations of LTI systems, also known as the transfer functions, through the use of what are

known as z-transforms. We shall in due course observe an interesting resemblance between the

transfer function and the transfer function operator representation presented earlier in §4.3.

5.1 FREQUENCY RESPONSE FUNCTION

To recall, the FRF is the discrete-time Fourier transform of the IR sequence

G(e

jω

) =

∞

X

k=0

g[k]e

−jωk

(5.1)

It characterizes the response of an LTI system to sinusoidal (oscillatory) inputs. A point to reckon is

that it is complex-valued unlike the step and impulse response sequences. One of the most important

uses of FRF, recall, is that it characterizes the filtering nature of an LTI system1.

Filters play tremendous roles in communications, signal processing, control, estimation, identifi-

cation and other allied fields. Understanding the characteristics of filters from the FRF is essential to

their proper use in these applications. Numerous texts discuss this topic in great detail. The purpose

of the following discussion is only to spotlight aspects relevant to identification.

5.1.1 CHARACTERISTICS OF FRF

The main characteristics of FRF are discussed below.

i. A.C. Gain (|G(e

jω

)|): This is an alternative term for AR(ω), which quantifies the amplitude

change in the output to an input of unit amplitude at a given frequency. The D.C. Gain G

p

defined earlier in the context of a step response is simply the A.C. Gain evaluated at ω = 0,

naturally since at steady-state the step is a zero frequency signal. The A.C. Gain is one of the

most useful characteristics of a filter since it governs the filtering nature of the system.

1Any LTI system is technically identical to a linear filter and vice versa. The difference is that a system is one which is

already in place, whereas a filter is usually designed.

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