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

704 Chapter 22 Difference equations and the z transform
Solution (a) The sampled sequence is
f[k] = cos3kT = cos((3T)k)
The z transform of this sequence can be obtained directly from Table 22.2 from
which we have
Z{cosak} =
Writinga = 3T wefind
Z{cos3kT} =
z(z −cosa)
z 2 −2zcosa+1
z(z −cos3T)
z 2 −2zcos3T +1
(b) WhenT = 0.2,the first four termsare
1 cos0.6 cos1.2 cos1.8
From Equation (22.8), the sequence of sampled values can be represented as the
following series of weighted impulses:
f ∗ (t) =
=
∞∑
cos3kTδ(t −kT)
k=0
∞∑
cos0.6kδ(t −0.2k)
k=0
= δ(t) +cos0.6δ(t −0.2) +cos1.2δ(t −0.4) +cos1.8δ(t −0.6) + ···
22.9 THERELATIONSHIPBETWEENTHEzTRANSFORM
ANDTHELAPLACETRANSFORM
We have defined theztransform quite independently of any other transform. However,
there is a close relationship between the z transform and the Laplace transform, the z
being regarded as the discrete equivalent of the Laplace. This can be seen from the following
argument.
Ifthecontinuoussignal f (t)issampledatintervalsoftime,T,weobtainasequence
of sampled values f[k], k ∈ N. From Section 22.8 we note that this sequence can be
regarded asatrainofimpulses.
f ∗ (t) =
∞∑
f[k]δ(t −kT)
k=0
Taking the Laplace transform,wehave
L{f ∗ (t)} =
=
∫ ∞
0
∑ ∞
e −st f[k]δ(t −kT)dt
∞∑
f[k]
k=0
k=0
∫ ∞
0
e −st δ(t −kT)dt