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

22.9 The relationship between the z transform and the Laplace transform 707
jv
5p —2
3
s plane
3p —2
p —2
3
3
– p —2
–3p —2
3
3
s
jy
1 3
z plane
–5p —2
3
Figure22.17
Thesampled signal has an infinite
number ofpoles.
–1 3
Figure22.18
There are two polesatz = ±j.
x
thatis,when σ = 0and ω = −(2n −1)π/2.Thusthereexistaninfinitenumberof
poles occurring when
s=−(2n−1) π 2 j
n∈Z
Some of these areillustratedinFigure 22.17.
(c) Theztransformof the sampled signalis
Z{f ∗ (t)}=1+ 0 z − 1 z 2 + 0 z 3 + 1 z 4 +···
=
1
1 − (−1/z 2 )
= z2
z 2 +1
which has justtwo poles atz = ±j asshown inFigure 22.18.
It is possible to show in general that the result of sampling a continuous signal is
to convert each simple pole of the Laplace transform into an infinite set of poles. Suppose
that the Laplace transform of the signal f (t) can be broken down by using partial
fractions into a series of n + 1 terms with simple poles a 0
,a 1
,...,a n
. For simplicity,
repeatingpoleswill notbeconsideredbut the proof forsuchacase issimilar. So,
F(s) = A 0
s−a 0
+ A 1
s−a 1
+ A 2
s−a 2
+···+
Inthe time domain thiscorresponds to
f(t) =A 0
e a 0 t +A 1
e a 1 t +···+A n
e a n t
A n
s−a n