Revisiting the Basics - Digital Signal Processing: A Practitioner's Approach - page 27

2.2.1 Continuous to Discrete
Let us take a numerical method [5] known as the Tustin approximation for sampling
times of 0.001 s and 0.002 s and apply it to the continuous system characterised by
(2.11). We get the difference equations as

y k ¼ 1 : 5519 y k 1 0 : 8918 y k 2 þ 0 : 08497 u k þ 0 : 1699 u k 1 þ 0 : 08497 u k 2 ; ð 2 : 12 Þ

y k ¼ 0 : 7961 y k 1 0 : 8347 y k 2 þ 0 : 2596 u k þ 0 : 5193 u k 1 þ 0 : 2596 u k 2 : ð 2 : 13 Þ

Note that we have two different difference equations representing the same physical
system; this is due to a change in sampling rate. Equation (2.13) has two
components: an autoregressive (feedback) component and a moving-average
(feedforward) component. By using the notation y k 1 ¼ z 1 y k ;

we define vectors

y z and u z :

y z ¼½ 1 z 1 z 2 and u z ¼½ 1 z 1 z 2 y k u k :

And we define the coefficient vectors as

a t and b t ¼½ 1 0 : 7961 0 : 8347 ¼½ 0 : 2596 0 : 5193 0 : 2596 :

Using this notation, we write (2.13) in vector form as a t y z ½ y k ¼b t u z ½ u k . In this
textbook the delay operator z 1 and the complex variable z 1 have been used
interchangeably. We continue to do this even though mathematicians may object.
We believe that algebraically there is no loss of generality.

2.2.2 Nomenclature
For historical reasons, specialists from different backgrounds have different names
for (2.14) and (2.15). However, they are identical in a strict mathematical sense.
Statisticians use the term Autoregressive (AR) [6]; control engineers call it an all-
pole system; and communication engineers call it an infinite impulse response (IIR)
filters. The statisticians' moving-average (MA) process is called a feedforward
system by control engineers and a finite impulse response (FIR) filter by commu-
nication engineers so we have chosen names to suit the context of this textbook;
readers may choose other names to suit their own contexts.

2.2.3 Difference Equations
We can write (2.12) or (2.13) in the form

a t y z ð y k Þ¼b t u z ð u k Þ ,

ð 2 : 14 Þ