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

702 Chapter 22 Difference equations and the z transform
(c)
(d)
(e)
z 2 −ze −2 cos1
z 2 −2ze −2 cos1 +e −4
ze 4 sin2
z 2 −2ze 4 cos2 +e 8
z z
(f)
z −4 z +3
(g)
z
z 2 +1
(h)
z 2
z 2 +1
3 (a) (−4) k (b) (1/2) k (c) (−1/3) k
(d) e 3k
4 e 1/z
(e) sin(πk/2)
22.8 SAMPLINGACONTINUOUSSIGNAL
WehavealreadyintroducedsamplinginSection22.4.Wenowreturntothetopic.Most
of the signals that are encountered in the physical world are continuous in time. This
meansthattheyhaveasignallevelforeveryvalueoftimeoveraparticulartimeinterval
of interest. An example is the measured value of the temperature of an oven obtained
usinganelectronicthermometer.Thistypeofsignalcanbemodelledusingacontinuous
mathematical function in which for each value oft there is a continuous signal level,
f (t). Several engineering systems contain signals whose values are important only at
particularpointsintime.Thesepointsareusuallyequallyspacedandseparatedbyatime
interval, T. Such signals are referred to as discrete time, or more compactly, discrete
signals. They are modelled by a mathematical function that is only defined at certain
points in time. An example of a discrete system is a digital computer. It carries out
calculationsatfixed intervals governed by anelectronic clock.
Supposewehaveacontinuoussignal f (t),definedfort 0,whichwesample,that
is measure, at intervals of time, T. We obtain a sequence of sampled values of f (t),
thatis f[0], f[1], f[2],..., f[k],....Returningtotheexampleoftheoventemperature
signal, a discrete signal with a time interval of 5 seconds can be obtained by noting
the value of the electronic thermometer display every 5 seconds. Some textbooks use
the notation f[kT] as a reminder that the sequence has been obtained by sampling at
an interval T. We will not use this notation as it can become clumsy. However, it is
important to note that changing the value ofT changes theztransform as we shall see
in Example 22.13. It can be shown that sampling a continuous signal does not lose the
essence of the signal provided the sampling rate is sufficiently high, and it is in fact
possible to recreate the original continuous signal from the discrete signal, if required.
Itisoftenconvenienttorepresentadiscretesignalasaseriesofweightedimpulses.The
strengthofeachimpulseisthelevelofthesignalatthecorrespondingpointintime.We
write
f ∗ (t) =
∞∑
f[k]δ(t −kT) (22.8)
k=0
the * indicating that f (t) has been sampled. This representation is discussed in
Appendix I. This is a useful mathematical way of representing a discrete signal as the
propertiesoftheimpulsefunctionlendthemselvestoavaluethatonlyexistsforashort
interval of time. In practice, no sampling method has zero sampling time but provided
the sampling time is much smaller than the sampling interval, then this is a valid mathematical
model ofadiscrete signal.