Jing Topics: Initial value theorem, Fi
troduction
orm is used in the study of various areas of
ce and engineering like communication, control
wy, system designing and digital signal processing
sforms discrete time domain to discrete
domain. It plays an important role in
analysis as Laplace Transform and Fourier
m in continuous systems.
perties of Laplace transform, and Z transform
pptained for
lar. The Z transforms can be
ces of discrete integer valued arguments,
Laplace transform is obtained for arguments
jous functions.
crete systems lead to difference equations
plution and analysis can be carried out with
[of transforms. In this chapter you will earn
d the Z transform of standard functions:
rent properties with examples are als?
You will also learn how to find inverse 2
by partial fraction method and convolution:
The Z Transform of @ $208
O= Fe) = Sp ayet
kt :
sabia), °C ablict sin(ak+p), (sinh akh{ck cosh ak).
f Z Transform : on
of Z Transform : Change of Scale, Shifting Property, Multiplication, and Division by k, Convolution
Z transform : Partial Fraction Method, Convolution Method.
al value theorem,
Inverse of Z Transform by Binomial Expansion
whenever the infinite series converges. Where 2 '5
complex number, Z is an operator of 2 Transform and F(2)
js the Z-Transform of a sequence f(K).
} where T denote the
Ex. 3.1.1 : Ith)
4 corresponds to k-0. Find Z (1(8)]
Soln.:
‘We Know, by definitior
£00,
mn of Z transform of a sequence
zirqg) = © f0d2™
since tothe left off there are three elements therefore
lower limit is -3
Since tothe right oft there are two elements therefore
upper limit is 2
2
zig = LF Gor*
. Z{FOO = FC3) af (-2) 2? +f (-1)2' +f (0) 2
+f()z+f(Q)27
Z{F(0Q)) = 22°+322+ 62! + 42° Tete?
a
2, BECO) = 22432? + 624 4 a?
ee
(3-2,4,-8,0,1,2,3,4) where T denote the
Bx. 9.1.2: Ii(h)= (-9.-2
‘+ elament corresponds to k=0. Find z/1(K)] -
‘Soln.:
We Know, by definition of Z transform of a sequence
£09, Z{FOO) = Endo a.
+A We know su! —
Since to the left of ¢ there are five elements therefore
lower limit is -S, Since to the right of 7 there are three
‘elements therefore upper limit is 3
3
= ZI) = Dragz*
3
DEQ] = £5) 2548 (4) 2* + (3) 23+ £ (-2) 2?
+ £1) 21+ £(0) 2% £ (1) 24 £2) 2
+f£@)z9
= ZIFQQ) = (3) 25+(-2) 24 + (4) 23+ (8) 27+ (0) 24
+ (1) 2+ @ 244 )z7+ (4) 29
Sac aia
Z (6G) = £(-2)2?+£(-1)2"+ £(0) 29
+80) 244 £(2)2?
et yi
+ 21609) = Beg hth
» Z{f (9) = Bet Ses 19g eoh
Ex. 3.14211 (k)=7%20.FindzZ [1].
Soin. :
We Know,
ZI) = Z sya
HO) = OMe Fayed (aye
: ee
200) = 142,20 2
78 +o
* Zitoo= 1+ @ @ * @
1m of infinite Geometric series
a= avereor?+ os oa
ete
“200 = 77
hk <0
24
ex. 3.15: ={3c Sg ten tind ZY (
Soin. :
‘We Know,
Zif ao) = E fu) z*
Here y(k)) = (..2°, 2-2, 2-1, 9°, 91, oe
ZAF()] = mnf(-3) 23 + f(-2)2? + FEA)
SCE" + $2)? + £3) 23
+ MFO] = (27) 29+ (2) 27 + 2 Het ay
+ (9) + (97) 27 + (97) 29
Benge 9 9
2 ayton = 2B od ore8
#0) - £8). sn6
8-0.
We know sum ofinfinite Geometric series _
S.= atartarésary
z
HALA] ces aSon ek
2709) = 52,2
ae
‘ = He-9)+2(2-
zi) = ayes
— ion, 2, 8
* 21f09) = a
~ 2109) =
=72
@-2 =a
“ = Ta
* ZEf09) = eae ZTransform
GYFRL),
a Y Y
Ejooe*
°
£(0) 2°+ F(1) 27+ £2) 27 +£(3) 2%,
2 93
8
Soest sta
Be 3
£000
sum of infinite Geometric series _
ROC, [zl
1! *
3.3_Z-Transform of Standard Functions —
Ex. 3.3.1 : Find Zia‘), k2 0.
Soin. :
We know
2709] = Zf0)2*
~ Ta] = Es09 x
Tak) = f0)2” oe
re
+5+
© sh
. Dat] = 145
aa) 6-0
‘We know sum of infinite Geometric series
a
peltl<4
go = atarsart+ar+iss.=
Ex. 3.3.2 : Find Zja*], k> 0.
‘Soln. :
We know,
zyoo] = Ese
2. Ufa*) = Saket
°
Bfa*) = f(0)2°+ f(z? + f(2) 27+ {B) 2° +
s gee a
ate) = 143+ tat
if
fa") = ad ‘wp
We know sum of infinite Geometric series
tera sarvars ar suis. 72y. rl WH Engineering Mathematics-IV 3-6
Ex. 3.3.3 : Find z[al'!] (1-xy? = 142x43x7 + 4x3 +.
3.33: a
‘Soin. : We know, Zfka) = 44-9)
2001 = E509 2* Wun ®
i .
+ Zak) = Daksky Dalek ;
ee : ve 2 kat] = 2
Here f(K) = (..a5,a7, a, a°, al, a2, ..)
= Mal] =. €-3)23 + f(-2) 22+ sC-1) 2! + f(0) eye as
+ f(1) 21 + f(2) 2? + f@3)z%...
22a] =. (a8) 23+ (a2) 22+ (24+ (1) + Ex. 3.3.5 : Find Z[k" a"), k>0
(atx + (2) 27+ (a4) 2. coe
3
STAM) = Peat earer ete Wess:
Mab] =~ (a2)*+ (a2)? (az)! ¥1+(2) ayool = Esey2
-@-0):
beset ee cic omciic 5 series
“fay = Dvater
°
LU ah = 2 Sie-1 ak Oe
oe 2h" a 22h aka
a + 4a jaz} here f(g <1
By multiplication by K
Z(kF)]
a= 25)
Senate -» (ena
2K = 25 ange) = 23 F
da
2 mel = (G35)
i. es)
ot) (241
, ea)
z(erl)
G1?
: ge =
222+ 1-22" +2:
erm fe @-0*
ze
=. =a
’
Z-Transform
Ex, 3.4.21 : Find Zick+1)a4), k> 0
Soin. :
Ulket)a‘} = Aka + Z[a")
We know Za) = 34
Be property of Multiplication by K
aah = -28
aka") = ee
sDfkal] =
eaites na = ae
cages at) = SEE
oe
cmon = oe
3.4.5. Division by k
17400) = Fe ten | 22] = FE ee
us
We know Z[1] = 77
By property of division by K
1e2[f()] = Fle) then 4 F lee ce AL] = -zlog(i-2)
mie
3.4.6 Convolution Theorem
If (h(16)} the convolution of two
* zt] = -tog (4)
2 LE] = aoga-24
Ex. 3.4.23: Find A] ace (id) then pay et *2(g0)
‘Soin:
Where
Welnow Za] = 345 hoo = (£00) * (@C10)
ee ng = s(m) gtk-m)
1 k=0
Wz = Fe then LA)] -- Faw )
| Ex. 3.4.25: WK) = UlR) & aK) = 2° Ulk) Find 2
Soln. :
zo] = ZIU0)] =FF&
By change of scale property,
&
2
209] = Z [2 ugg] = —
2
By convolution theorem,
= FO) *809] = 2700} * ze}
ee aad
FAW) = 4 Ul) and g(k) =
ZIK{k) * g(k),
Soin. :
“214 uag) =
Soin,
; SZBEU(Q) =
We know from revio a0
low from py ous BE og )
By left shitting Property q
exon 'Y convolution theorem,
= F(z) then
ney ANS f e j
Riis is U0) * g(a) = 2EF0) *2{(g 0}
peo
209 * gag) = 2Eaky J
a W)*Z 3k yy
$ ir] = 2F(z) BY UG)
“AGS
)- ae Z-Transform
‘transforms of standard sequences
kenge gk aye fa
"Ca O 0 Ee aapireltt
a <6 c 2 (2 cosm )
+) "+ cosnk| = : 2) zcstna_ iE “|<
[@ oe Fed ~ 2zcosn+ C¥sinak, k>0 peeaerceay oF 1
io 2] GS ¢ 2(z+1)
3)'+ cos:
2) 22-W)\g +2241 2{2-ccosa
C*cosak, k= 0 in
1 =) sg “sinhak, k> 0 zesinha— |e| <1
[Qos] = (GH) Ga esa Ftaccoshare|
z-accosha___ |e
; [ze azecosha Jel <
ms of standard sequences CKcoshak, k>0 =a 1
Z transforms of standard seque o
F@),
,k20
| 1,k21
7 1,k20
ak21
ai kz
! (ah k22
as
‘impulse function Ga) k20
ay = {¥k=0
= lokeo oa
3.5 _ Inverse Z Transform
Inverse Z transform is a process of transforming the
discrete frequency domain to discrete time domain. The
Jnverse 2 transform of a function F(z) is denoted by
Z*(F@) = 109 Ca) ko
Ca) ke0
Soin. : Consider
z
F(@) =
A(z-2) + B(z-1)
2
Ba)
ZF)
2a
ZF) = 10) 4221
~ ZF) -1+2%ke1
DUF@) = -1+2kx>0
Ex. 3.8.2 : Find inverse Z transform of Fz) =
Soin. :
Consider
AG@-6) +B — 4)
4
4 = AC2)
Az -2
Putz
ce
S253
“Fel = 223.
2 TFG) =
zak)
ZR) =
oD RQ) = When region of convergence is given then to find 2:
transform of fnetions we use the following method.
mee
Ex. 3.55 : Find the inverse Z transform “
it Region of convergence (ROC) is. Iz
‘soln. :
“The given function
partial fraction as Since in the denominator of above inequalities we
‘can take 2 common from the first and second term of
Fe
Se Aited ote evens ertlo en 13
4
=F + t.2.
2 Fla)
= FQ)
We know (1-x)"?
2:
= 4eet
0 tee
Soin,
The given funcio
(46. 23H
2
rer
4 2
CHT
9" G3"
= exes.
+m0$G)--0).-)
#0) O'+-@--)
oo )
Se ea
Zao 23") rhyey
234 k21
= 4
15.8 : Find the inverse Z vanetorm of F(z) «223
‘Region of convergence (ROC) is
Paeg
) can be writen as by method of
= AGB
Pur = 0
29 = hep
3 = 2As5,
ites
Fe) = a ‘We know (1-3) = 149x242" + =
wry = 21+ @-O+-Os-)
AEG
are = (15-5 +8- 1. OEE
Som:
: i
oe 10 etre te Ans, :——Z8in7_
2-22008744 | Q.37 Fat]
‘Ans, :-;22£08; (ka) Ans. :
222008744 | 0.38 Fina A]
Ans. 22-2082) _ Meets eta)?
2zcos244 | 2.39 Find 2 [sin (k + 1) 6] ising
Rieke ys A ace Tt
BR 2ecoahae | TAZ lsn+3)9) s
Raps x conns, ‘Ans, ; 28108 (2200826-2c0s0+2)
are ai 2? -22.c080+1
(oath e Q.41 Find Zi(k+2)a¥], k>0 ans, ; 22 ==
as0-5 | 5 os
er By 1. 42. Find Z]ka*), k>0 nce —ahe
7WayEsy alles . “ea?
reo et ind Zike™}, k> 0 Ane 222
Ans. = 32) (@-3) ee
i Q. 44 Find ZK oP), k > Pz (zre)
analls 4 Lk20 0
2 —2zc0s1 +1 ok 0
it i e 45 ind]. K>1 Ans. :~ log (1-227!)
2—2zc081 +1 ae 1
ak (zsin1 + sin(2 0.46 Find 42) 21 Ans. :~5zlog (1-22-1)
2zcos3
Hae aging an Q.47 reg a Ans. =
= 2zc0s6 +1 ee
14
se Hensal) sain(§ 4) @.48 Find = I: Ans. Bay A)
© Bee | aaa Find Zh) ait sh) = (3) and ot = snc
tein an(§-8) te
emi eer
ae @..50 Find the inverse Z transform of Fiz) = Garey by
Ans. 279 z cos +25 ke : ioe
‘i ‘az sint 0.51 Find the inverse Z transform of F(z) = area by
faeces! +16 partial fraction method. (Ans. : 2(5- 45)
Tasedt €40"| 0.82 Find the inverse Z vanstorn of F2) = 32 — #2 — by
partial fraction method. (Ans. : (-1) "2°**S_ 7(-3)§)
Ze 4z
@.53 Find inverse Z transform of F(z) = Brees
partial fraction method. in,
@.54 Find the inverse Z transform ot F(z) =
‘by partial traction method.
(ans. : ee 11K")
= 107
0.58 Find the inverse 2 waster of Fa) = = a ear
(ans. :§(2 ka" + 32 9)
+ 8rd
partial fraction method. W Engineering Mathematics-IV
.56 Find the inverse Z transform of Fla) = tap eay i
Region of convergence (ROC) is (i) [z| > 6
@.61 Find the inverse Z transform of
convolution theorem.
Let F(a) = 72g and G2) =2—
2) e+8)
Region of convergence (ROC) is (i) jz] > 6
(Ans, : -52*%((-3)*- 1)
@.58 Find the inverse Z transform of Fa) = 25 it
Region of convergence (ROC) is (i) |z| >3
(Ans. : 3° (8k-5))
=e
Q.59 Find the eee
ed inverse Z transtom of F(z) = ES -
on of convergence (ROC) se] >5 S yatgew
(ans. : pa
Z*(F@) -G(2))
Z*TF@) Ge]
a otra = 7 {3%5] =s*=s09
2 EGG) = rit] =6 "= g(k)
+ TFG) GG] = f0W) * g(k) = 5** 6X
k
+ DF) -G@)) 2 sm) g(k-m)
5
Bart: G@]
a a FFE) -c~y = 6(6 6-@")
ber
+ DMF@)-c@y = of {=}}
6
+ Z*[FG])-G(z)) = giet_gke
Q. 62 Find the inverse Fe
TIF) -G(@)) = fg *g(e) = 14-2
: P
* TAF) cto) = E semgtem= E rmgen
aN m=0
Z*(FG@)-G(e] = 2s 21,22 44
+ ZAPG)-G(a)) = 24214 22 okk
2 TFQ)-G(a)) = Aes eee 3)
2
4-0"
7G) Gan = ke apy 4
2") Ga) = 2(2-2%
7" [F(2) Gea) = ght
“T*TF(2)- Gey] =