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

22.7 Definition of the z transform 701
Table22.2
Theztransformsofsome common functions.
f[k] F(z) f[k] F(z)
{ 1 k=0
δ[k] =
0 k≠0
{ 1 k0
u[k] =
0 k<0
k
e −ak
a k
ka k
k 2 a k
1 k 2 z(z +1)
(z −1) 3
z
z −1
z
(z −1) 2 sinak
z
z−e −a
z
z −a
k 3 z(z 2 +4z +1)
(z −1) 4
cosak
e −ak sinbk
az
(z −a) 2 e −ak cosbk
az(z +a)
(z −a) 3
zsina
z 2 −2zcosa+1
z(z −cosa)
z 2 −2zcosa+1
ze −a sinb
z 2 −2ze −a cosb +e −2a
z 2 −ze −a cosb
z 2 −2ze −a cosb +e −2a
EXERCISES22.7
1 Using the definition oftheztransform, find
closed-form expressions fortheztransformsofthe
followingsequences f[k] where
(a) f[0]=0,f[1]=0,f[k]=1fork 2
{ 0 k=0,1,...,5
(b) f[k] =
4 k>5
(c) f[k]=3k,k 0
(d) f[k]=e −k ,k=0,1,2,...
(e) f[0]=1,f[1]=2,f[2]=3,f[k]=0,k 3
(f) f[0]=3,f[k]=0,k ≠0
{ 2 k0
(g) f[k] =
0 k<0
2 UseTable 22.2 to find theztransformsof
(a) cos3k (b) e k (c) e −2k cosk
(d) e 4k sin2k (e) 4 k (f) (−3)
( ) ( )
k
kπ kπ
(g) sin (h) cos
2 2
3 Find, fromTable 22.2, the sequences whichhave the
followingztransforms:
z 2z 3z
(a) (b) (c)
z +4 2z−1 3z+1
z z
(d)
z−e 3 (e)
z 2 +1
4 Byconsidering the Taylor seriesexpansion ofe 1/z ,
find theztransformofthe sequence
f[k]= 1 k!
k 0
Solutions
1 (a)
(c)
1 4
(b)
z(z −1) z 5 (z−1)
3z ez
(z −1) 2 (d)
ez−1
(e)
2 (a)
z 2 +2z+3
z 2 (f) 3 (g)
z(z −cos3)
z 2 −2zcos3+1
(b)
z
z −e
2z
z −1