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

690 Chapter 22 Difference equations and the z transform
x[n]
S
y[n]
ay[n – 1]
3 T
a
y[n – 1]
Figure22.7
Discrete-time filter.
Engineeringapplication22.3
Developmentofadifferenceequationtocalculatethe
accelerationofanobject
Engineersoftenneedtocalculatetheaccelerationofanobjectwhenallthatisknown
isthepositionofthatobject.Thiscanbeachievedbydevelopingadifferenceequation
that can be solved at regular time intervals by a computer. It is first necessary to
use a device that converts the position signal into a form suitable for analysis by a
computer. Usually the position signal will be continuous and therefore in analogue
form.Itisthereforenecessarytouseananalogue-to-digitalconvertertotransform
the analogue signal into a digital signal. Consider the following problem. Develop
a difference equation to convert a digital position signal into a digital acceleration
signal given that this position signal varies with time. Derive an associated block
diagramforthisdifference equation.
Lets =position, v =speed anda =acceleration:
v = ds a = dv = d2 s
dt dt dt 2
Therefore, in order to obtain an acceleration signal the position signal must be differentiatedtwice.ForasmalltimeintervalT,thederivative,y(t),ofasignalx(t)can
be approximated by
y(t) ≈ x(t)−x(t−T)
T
This follows directly from the definition of differentiation. If the signalx(t) is sampled
to givex[n] then the process of differentiation is represented by the difference
equation
x[n] −x[n −1]
y[n] =
T
Figure 22.8 shows a block diagram for the differentiator. It is important to note that
this difference equation is only an approximation to the process of differentiation.
This could be implemented using special-purpose hardware or by software on a microprocessor.
Itfollows thatthe speed of the object isgiven by
s[n] −s[n −1]
v[n] =
T
The problem of finding the acceleration, a[n], can now be solved by coupling two
differentiators together as shown in Figure 22.9. An alternative approach to this