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

22.4 Block diagram representation of difference equations 689
x[n] x[n – 1]
T
Figure22.2
Adelay block delaysasequence
byatime interval,T.
x[n] x[n – 2]
T
T
Figure22.3
Two delay blocks in series.
x[n]
x[n]
p[n]
x[n] + a
1
kx[n]
3
q[n]
p[n] + q[n] + r[n]
S
a
k
r[n]
Figure22.4
Adding aconstant
to a sequence.
Figure22.5
Scalingasequence
byaconstant.
Figure22.6
Adding sequences together
usingasummer.
Notethatx[n −1]isundefinedwhenn = 0andsothisisassignedavalueof0.Adelay
of two sampling intervals results inthe sequencex[n −2]as shown inFigure 22.3.
Another block diagram element represents the addition of a constant to a sequence
and is shown in Figure 22.4. A sequence can be scaled by a constant. This is shown in
Figure 22.5. Finally, sequences can be added together using a summer. This is shown
inFigure 22.6.
Engineeringapplication22.2
Discrete-timefilter
The term filter is used by engineers to describe an electronic device that selectively
processessignalsofdifferentfrequencies.So,forexample,alowpassfilterisafilter
that allows low frequency signals to pass with little attenuation of their amplitude.
However, high frequency signals tend to be severely attenuated. In contrast, a high
passfilterattenuateslowfrequencysignalswhileallowinghighfrequencysignalsto
pass with relative ease. A discrete-time filter or, as more commonly termed, a digital
filter is an electronic device that uses digital methods to selectively filter digital
signals.
A simple example of a discrete-time filter is one described by the difference
equation
y[n] −ay[n −1] =x[n]
wherex[n]istheinputsequence,y[n]istheoutputsequenceandaisaconstant.Ifa
is positive then the filter behaves as a low-pass filter which rejects high frequencies
but allows low frequencies to pass. If a is negative then the filter behaves as a
high-pass filter. A block diagram for the filter is shown in Figure 22.7. Note that
this is a recursive filter because calculation ofy[n] requires knowledge of previous
values of the output sequence. Note also that the block diagram contains a feedback
path. This isafeatureofrecursive difference equations.
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