Processing of Signals - Digital Signal Processing: A Practitioner's Approach - page 10

Noise

y k

x k

1
A ( z )

∑

B ( z )

Equations (1.19) and (1.20)

where k is white noise and y k is the output of the system. Equation (1.19) can also

be represented as a transfer function (TF) [3, 4]:

B ð z Þ
A ð z Þ

H ð z Þ¼ ð 1 : 21 Þ

or

ð z Þ

B ð z Þ
A ð z Þ

X ð z Þ¼ ð 1 : 22 Þ

or in the delay operator notation 8

k ;

B ð z Þ
A ð z Þ

x k ¼ ð 1 : 23 Þ

where X ð z Þ and ð z Þ are the z-transforms of x k and k , respectively, and

A ð z Þ¼ 1 X

p

a i z i

; ð 1 :

24 Þ

i ¼ 1

B ð z Þ¼ X

q 1

b j þ 1 z j

; ð 1 : 25 Þ

j ¼ 0

are the z-transforms of f 1 ; a 1 ; a 2 ; ...; a p g and f b 1 ; b 2 ; ...; b q g , respectively.

We define the parameter vector as

T

p ¼½ a 1 ; a 2 ; ...; a p ; b 1 ; b 2 ; ...; b q : ð 1 : 26 Þ

The parameter vector completely characterises the signal x k . The system defined

by (1.19) is also known as an autoregressive moving average (ARMA) model [1, 6].
Note that the TF H ð z Þ of this model is a rational function of z 1 , with p poles and q
zeros in terms of z 1 . An ARMA model or system with p poles and q zeros is
conventionally written ARMA ð p

p

; q Þ .

8 Some authors prefer to use the delay operator and the complex variable z 1 interchangeably for
convenience. In the representation x k 1 ¼ z 1 x k here, we have to treat z 1 as a delay operator and not as a
complex variable. In a loose sense, we can exchange the complex variable and the delay operator without
much loss of generality.