Principles of System Identification

4

Models for Discrete-Time LTI Systems

This chapter presents a self-contained review of the models for discrete-time, linear time-

invariant systems. The reader is recommended to pay particular attention to the identification

viewpoints of each model structure. The material in this chapter is essential for understanding

the remainder of this textbook.

In Chapter 3, we learnt the definition of a linear time-invariant (LTI) system. To recap, an LTI

system satisfies two important properties:

i. Superposition: A linear combination of inputs produce the same linear combination of the re-

spective outputs.

ii. Time-invariance: Input delayed (or shifted) by a time T produces the same output as the original

input, but shifted by the same amount of time.

These two properties are central to the derivation of the different model forms that one can write

for an LTI system. Fundamental to all these forms is the convolution equation model.

4.1 CONVOLUTION MODEL

The discrete-time convolution equation is given by

y[k] =

∞

X

n=−∞

g[n]u[k − n] =

∞

X

n=−∞

g[k − n]u[n] (4.1)

Interpretation: The output of an LTI system at any instant is a infinite sum of weighted past,

present and future inputs.

The weights {g[n]} are known as the impulse response (IR) coefficients of the system

We first discuss how the convolution equation results as a fundamental description. Any arbi-

trary discrete-time signal u[k] can be expressed as a weighted combination of shifted discrete-time

impulses

u[k] =

∞

X

n=−∞

u[n]δ[k − n] (4.2)

where δ[k] is the Kronecker delta function,

δ[k] =

(

1, k = 0

0, k , 0

k

δ[k]

1

0 1 2 3 4-1-2-3-4-5 5

68

Models for Discrete-Time LTI Systems 69

G

δ[k] g[k]

FIGURE 4.1 Impulse response of an LTI system.

Denote the response of the LTI system to the impulse input δ[k] by g[k] as shown in Figure 4.1.

The LTI system is denoted by G. Each term of Equation (4.2) produces an output u[n]g[k − n] by

virtue of the linearity and time-invariance properties of an LTI system.

Invoking the linearity property, the output is a superposition of the individual outputs as illus-

trated in Figure 4.2. Equation (4.1) then follows.

u[k]

u[#2]δ[k%+%2]

u[#1]δ[k%+%1]

u[0]δ[k%#%0]

u[1]δ[k%#%1]

u[2]δ[k%#%2]

...

...

G

G

G

G

G

u[#2]g[k%+%2]

u[#1]g[k%+%1]

u[0]g[k%+%0]

u[1]g[k%#%1]

u[2]g[k%#%2]

y[k]

FIGURE 4.2 Schematic depicting the derivation of a convolution operation.

4.1.1 IMPULSE RESPONSE

The name impulse response model is used synonymously with the convolution equation model.

An important observation is that for an LTI system, merely knowing the impulse response g[k] is

sufficient to predict the response of the system G to an arbitrary input. Additionally, several key

properties of the LTI system (e.g., stability, causality) can be easily inferred by an analysis of its IR

coefficients. A few useful facts are discussed below in this context.

A few useful facts

a. Knowing the IR coefficients, one can compute the response of a system to an arbitrary input.

Example 4.1: Compute Response to an Arbitrary Input Using the IR

Compute the response of a system whose impulse response is

g[k] =











(0.5)

k−1

k ≥ 0

0 k ≤ 0

(4.3)

to a unit step input u[k] = 1, k ≥ 0.

Solution: Denoting the response by y

s

[k] and using Equation (4.1),

y

s

[k] =

∞

X

n=−∞

g[n]u[k − n] =

k

X

n=1

(0.5)

n−1

(4.4)

= 1 − 0.5

k

− k ≥ 1 (4.5)