Revisiting the Basics - Digital Signal Processing: A Practitioner's Approach - page 28

which can be written as

y k X

p

Þ ¼ X

q

ð a i þ 1 y k i ð b i u k i þ 1 Þ: ð 2 : 15 Þ

i ¼ 1 i ¼ 1

In (2.15) the coefficient a 1 is always one. Equations (2.14) and (2.15) are the
generic linear difference equations of a pth order AR process (left-hand side) and a
qth order MA process (right-hand side). There are many technical names for (2.14)
and (2.15).

2.2.4 Transfer Function
Recognising (2.14) as the dot product of the vectors, it can be written as

A ð z Þ¼a t y z and B ð z Þ¼b t u z ; ð 2 : 16 Þ

where B ð z Þ is called the numerator polynomial and A ð z Þ is called the denominator
polynomial. We can recast the equations as y k ¼ B ð z Þ= A ð z ½ u k . The roots of B ð z Þ

and A ð z Þ have greater significance in the theory of digital filters, and for under-
standing physical systems and their behaviour. The impulse response of the system
(2.13) is depicted in Figure 2.3; it is real. It is not necessary that the response be a
single sequence; it could be an ordered pair of sequences, making it complex. We
can write B ð z Þ= A ð z Þ as

2596 ð 1 þ 2z 1 þ z 2 B ð z Þ

A ð z Þ ¼

0 : Þ

ð 1 0 : 7961z 1 þ 0 : 8347z 2 Þ

2

2596 ð 1 z 1 0 : Þ

¼

Þ :

ð 2 :

17 Þ

ð 1 z 1 0 : 9136e j1 : 12 Þð 1 z 1 0 : 9136e j1 : 12

The same information is shown in Figure 2.5 as a pole-zero plot. We can
decompose it into partial fractions as

B ð z Þ
A ð z Þ ¼ 0

6080z
ð 1 z 1 0

j0 : j0 : 6080z

2

2596 ð 1 z 1 : Þ

Þ

;

: 9136e j1 : 12 ð 1 z 1 0 : 9136e j1 : 12 Þ

ð 2 : 18 Þ

as stated in (2.9). However, it is a recommended practice to leave it as a second-
order function if the roots are complex. Second-order systems have great signifi-
cance and the standard representation is given below, where r ¼ 0 : 9136 is the radial

pole position at an angle of ¼ 1 : 12 radians in the z-plane.

2596 ð 1 þ 2z 1 þ z 2

B ð z Þ
A ð z Þ ¼

0 : Þ

:

ð 2 : 19 Þ

2 z 2

½ 1 2 ð 0 : 9136 Þ cos ð 1 : 12 Þ z 1 þð 0 : 9136 Þ