082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

682 Chapter 22 Difference equations and the z transform
systemstothatplayedbytheLaplacetransformforcontinuousones.Inthischapteryou
will be introduced to theztransform and one of its applications -- the solution of linear
constant coefficient difference equations. The z transform is of increasing importance
as more and more engineering systems now contain a microprocessor or computer and
sohaveoneormorediscrete-timecomponents.Forexample,mostindustrialcontrollers
nowhaveanembeddedmicroprocessor,andoverallcontrolofafactoryisoftenbymeans
ofasupervisoryprocesscontrolcomputer.Manyfactoriesalsohaveaproductioncontrol
computer toschedule production.
22.2 BASICDEFINITIONS
Before classifying difference equations wewill first derive an example of one.
Suppose a microprocessor system is being used to capture and analyse images. The
number of instruction cycles,i, of the microprocessor needed to process an image depends
on the number of pixels, n, that the image is broken down into. Clearly n is a
non-negative integer, that is n ∈ N. Since i depends upon n we write i = i[n]. The
squarebracketsnotationreflectsthefactthatnisadiscretevariable(seeSection6.2).If
there arenitemsof data, the number of instruction cycles used isi[n].
Suppose that if there aren + 1 items of data, the number of instruction cycles used
increases by 10n +1.Then,
i[n+1]=i[n]+10n+1
Thisisanexampleofadifferenceequation.Thedependentvariableisi;theindependent
variable isn.
Puttingn = 0 inthe difference equation gives
i[1] =i[0] +10(0) +1 =1
Similarly,i[2] = 12,i[3] = 33,i[4] = 64andsoon.Weseethatthedifferenceequation
gives risetoasequence of values.
There are strong similarities between difference and differential equations. The important
point to note is that with difference equations, the independent variable is discrete,
not continuous. In the above example,nis the number of pixels; it can have only
integer values. This discrete property of the independent variable is an essential and
distinguishing feature of difference equations. Much of the terminology of differential
equations isapplied, withidentical meaning, todifference equations.
22.2.1 Dependentandindependentvariables
Consider a simple difference equation:
x[n +1] −x[n] = 10
The dependent variable isx; the independent variable isn. Inthe difference equation
y[k+1]−y[k]=3k+5
the dependent variable isyand the independent variable isk.