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

708 Chapter 22 Difference equations and the z transform
Now we consider the Laplace transformof the sampled signal f ∗ (t):
F ∗ (s)=L{f ∗ (t)} =
=
∞∑
f[k]e −skT
k=0
∞∑
(A 0
e a 0 kT +A 1
e a 1 kT + ··· +A n
e a n kT )e −skT
k=0
∑ ∞
∑ ∞
=A 0
e −kT(s−a 0 ) +A 1
e −kT(s−a 1 ) +···
k=0
+A n
∞
∑
k=0
e −kT(s−a n )
Now each of the summations can be converted into a closed form. For example,
∞∑
e −kT(s−a 0 ) =1+e −T(s−a 0 ) +e −2T(s−a 0 ) +e −3T(s−a 0 ) +···
k=0
Therefore,
F ∗ (s) =
1
=
1−e −T(s−a 0 )
k=0
A 0
1−e + A 1
−T(s−a 0 ) 1−e +···+ A n
−T(s−a 1 ) 1−e −T(s−a n )
It is possible to show that for each simple pole in F(s) there is now an infinite set of
poles. Consider the pole ats =a 0
. This contributes the term
A 0
1−e −T(s−a 0 )
toF ∗ (s). This term has poles whenever 1 − e −T(s−a 0 ) = 0, that is e −T(s−a 0 ) = 1. This
corresponds toT(s −a 0
) = 2πmj,m ∈ Z. Therefore,
T(s−a 0
)=2πmj
s−a 0
= 2πm
T j
s=a 0
+ 2πm
T j
m∈Z
The effect of sampling is to introduce an infinite set of poles. Each one is equal to the
pole of the original continuous signal but displaced by an imaginary component. This
is illustrated in Figure 22.19 for a real polea 0
. However, the proof is equally valid for
a complex conjugate pair of poles but the diagram is more cluttered and has, therefore,
not been shown.
Clearly discrete systems are not amenable tosplane design techniques. Fortunately,
thezplane can be used for analysing discrete systems in the same way that thesplane
can be used when analysing continuous systems. It is possible to map points from thes
plane to thezplane using the relationz = e sT which gives rise to the definition of thez
transform as described previously.