2. Agenda
๏ฑ Difference Equation vs Differential Equati
๏ฑ Z-Transform
๏ง Examples
๏ง Properties
๏ฑ Inverse Z-Transform
๏ง Long Division
๏ง Partial Fraction
๏ฑ Solution of Difference Equation
๏ฑ Mapping between s-plane to z-plane
2
3. Difference Equation vs Differential Equation
๏ฑ A difference equation expresses the change in some variable as a result
of a finite change in the other variable.
๏ฑ A differential equation expresses the change in some variable as a result
of an infinitesimal change in the other variable.
3
4. Differential Equation
๏ฑ Following figure shows a mass-spring-damper-system. Where y is
position, F is applied force D is damping constant and K is spring
constant.
๏ฑ Rearranging above equation in following form
4
๐น ๐ก = ๐ ๐ฆ ๐ก + ๐ท ๐ฆ ๐ก + ๐พ๐ฆ(๐ก)
๐ฆ ๐ก =
1
๐
๐น ๐ก โ
๐ท
๐
๐ฆ ๐ก
๐พ
๐
๐ฆ(๐ก)
7. Difference Equations
๏ฑ Difference equations arise in problems where the time is assumed to
have a discrete set of possible values.
๏ฑ Where coefficients ๐ ๐โ1, ๐ ๐โ2,โฆ and ๐ ๐, ๐ ๐โ1,โฆ are constant.
๏ฑ ๐ข(๐) is forcing function
7
๐ฆ ๐ + ๐ + ๐ ๐โ1 ๐ฆ ๐ + ๐ โ 1 + โฏ + ๐1 ๐ฆ ๐ + 1 + ๐0 ๐ฆ ๐
= ๐ ๐ ๐ข ๐ + ๐ + ๐ ๐โ1 ๐ข ๐ + ๐ โ 1 + โฏ + ๐1 ๐ข ๐ + 1 + ๐0 ๐ข ๐
8. Difference Equations
๏ฑ Example 1: For the given difference equation, determine the (a)
order of the equation. Is the equation (b) linear, (c) time
invariant, or (d) homogeneous?
๐ ๐ + ๐ + ๐. ๐๐ ๐ + ๐ + ๐. ๐๐๐ ๐ = ๐ ๐
๏ฑ Solution:
a) The equation is second order.
b) All terms enter the equation linearly
c) All the terms if the equation have constant coefficients.
Therefore the equation is therefore LTI.
d) A forcing function appears in the equation, so it is
nonhomogeneous.
8
9. Z-Transform
๏ฑ Difference equations can be solved using z-transforms which provide a
convenient approach for solving LTI equations.
๏ฑ The z-transform is an important tool in the analysis and design of
discrete-time systems.
๏ฑ It simplifies the solution of discrete-time problems by converting LTI
difference equations to algebraic equations and convolution to
multiplication.
๏ฑ Thus, it plays a role similar to that served by Laplace transforms in
continuous-time problems.
9
10. Z-transform Definition
๏ฑ Given the causal sequence {x(ktT)}, its z-transform is defined as
๏ฑ The variable zโ1 in the above equation can be regarded as a time delay
operator.
10
X ๐ง = ๐ฅ 0 + ๐ฅ ๐ ๐งโ1
+ ๐ฅ 2๐ ๐งโ2
+ โฏ + ๐ฅ ๐๐ ๐งโ๐
๐ ๐ง =
๐=0
โ
๐ฅ ๐๐ ๐งโ๐
11. Example
๏ฑ Obtain the Z-transform of the sequence
๏ฑ Solution:
11
x ๐๐ = {1, 1, 3, 2, 0, 4, 0, 0, 0, โฆ }
X ๐ง = 1 + ๐งโ1
+3 ๐งโ2
+ 2 ๐งโ3
+ 4๐งโ5
12. Laplace Transform and Z-Transform
๏ฑ Given the sampled impulse train of a signal
๐ฅโ ๐ก = ๐ฅ 0 ๐ฟ ๐ก + ๐ฅ ๐ ๐ฟ ๐ก โ ๐ + โฏ + ๐ฅ ๐๐ ๐ฟ(๐ก โ ๐๐) + โฏ
๐ฅโ
๐ก =
๐=0
โ
๐ฅ ๐๐ ๐ฟ ๐ก โ ๐๐
12
13. Laplace Transform and Z-Transform
๐โ
๐ = ๐ฅ 0 + ๐ฅ ๐ ๐โ๐ ๐
+ ๐ฅ 2๐ ๐โ๐ 2๐
+ โฏ + ๐ฅ ๐๐ ๐โ๐ ๐๐
+ โฏ
๐โ ๐ =
๐=0
โ
๐ฅ ๐๐ ๐โ๐ ๐๐ =
๐=0
โ
๐ฅ ๐๐ (๐ ๐ ๐)โ๐ โ (1)
๏ฑ The z-transform is
๐ ๐ง =
๐=0
โ
๐ฅ ๐๐ ๐งโ๐
โ (2)
๏ฑ Comparing (1) and (2) yields
๐ง = ๐ ๐ ๐ where ๐ is the sample period
13
14. A Note
๏ฑ In general, given a transfer function in s-domain you cannot just replace
๐ by s =
๐๐๐ง
๐
(from ๐ง = ๐ ๐ ๐
) to get its z-domain transfer function.
๏ฑ The reason is that ๐ง = ๐ ๐ ๐ is true with the sampled signal.
14
16. z-transform of the Unit impulse
๏ฑ Let x(kT) = ฮด(kT)
๏ฑ Then
16
๏ฎ
๏ญ
๏ฌ ๏ฝ
๏ฝ
otherwise0
0kfor1
(kT)๏ค
๏จ ๏ฉ ๏ ๏ 1(kT)
00
๏ฝ๏ฝ๏ฝ ๏ฅ๏ฅ
๏ฅ
๏ฝ
๏ญ
๏ฅ
๏ฝ
๏ญ
k
k
k
k
zzkTxzX ๏ค
17. z-transform of the a Shifted Unit impulse
๏ฑ Let x(kT) = ฮด(kT-qT)
๏ฑ Then
17
๏ฎ
๏ญ
๏ฌ ๏ฝ
๏ฝ
otherwise0
qkfor1
qT)-(kT๏ค
๏จ ๏ฉ q
k
k
k
k
zzzkTxzX ๏ญ
๏ฅ
๏ฝ
๏ญ
๏ฅ
๏ฝ
๏ญ
๏ฝ๏ฝ๏ฝ ๏ฅ๏ฅ 00
qT)-(kT)( ๏ค
18. z-transform of a Unit-Step Function
๏ฑ Let x(kT) = u(kT)
๏ฑ Then
18
๏ฎ
๏ญ
๏ฌ ๏ณ
๏ฝ
0kfor0
0kfor1
(kT)
๏ฐ
u
๏จ ๏ฉ
11
1
(kT))( 1
000 ๏ญ
๏ฝ
๏ญ
๏ฝ๏ฝ๏ฝ๏ฝ ๏ญ
๏ฅ
๏ฝ
๏ญ
๏ฅ
๏ฝ
๏ญ
๏ฅ
๏ฝ
๏ญ
๏ฅ๏ฅ๏ฅ z
z
z
zzuzkTxzX
k
k
k
k
k
k
19. z-transform of Sample Exponential
๏ฑ Let x(kT) = ak u(kT)
๏ฑ Then
19
๏ฎ
๏ญ
๏ฌ ๏ณ
๏ฝ
0kfor0
0kfor
(kT)
๏ฐ
k
k a
ua
๏จ ๏ฉ ๏ฅ๏ฅ๏ฅ
๏ฅ
๏ฝ
๏ญ
๏ฅ
๏ฝ
๏ญ
๏ฅ
๏ฝ
๏ญ
๏ฝ๏ฝ๏ฝ
000
(kT))(
k
kk
k
kk
k
k
zazuazkTxzX
๏จ ๏ฉ
az
z
az
azzX
k
k
๏ญ
๏ฝ
๏ญ
๏ฝ๏ฝ ๏ญ
๏ฅ
๏ฝ
๏ญ
๏ฅ 1
0
1
1
1
)(
20. z-transform of kaku[k]
๏ฑ Let x(kT) = k ak u(kT)
๏ฑ Then
20
๏ฎ
๏ญ
๏ฌ ๏ณ
๏ฝ
0kfor0
0kfor
(kT)
๏ฐ
k
k ka
uka
๏จ ๏ฉ ๏ฅ๏ฅ๏ฅ
๏ฅ
๏ฝ
๏ญ
๏ฅ
๏ฝ
๏ญ
๏ฅ
๏ฝ
๏ญ
๏ฝ๏ฝ๏ฝ
000
k(kT))(
k
kk
k
kk
k
k
zazukazkTxzX
๏จ ๏ฉ 221
0
1
)()1(
1
)(k
az
z
az
azzX
k
k
๏ญ
๏ฝ
๏ญ
๏ฝ๏ฝ ๏ญ
๏ฅ
๏ฝ
๏ญ
๏ฅ
21. z-transform of Sinusoids
๏ฑ Let x(kT) = (cos ฮฉkT) u(kT)
๏ฑ Then
๏ฑ Similarly it can be shown:
21
๏ฎ
๏ญ
๏ฌ ๏ณ๏
๏ฝ๏
0kfor0
0kfor)cos(
(n))cos(
๏ฐ
kT
un
2
)k(cos
kTjkTj
ee
T
๏๏ญ๏
๏ซ
๏ฝ๏๏
)}()({
2
1
u(k)})k({cos kTuekTueZT kTjkTj ๏๏ญ๏
๏ซ๏ฝ๏๏
๏บ๏ป
๏น
๏ช๏ซ
๏ฉ
๏ญ
๏ซ
๏ญ
๏ฝ๏๏ ๏๏ญ๏ TjTj
ez
z
ez
z
kTkT
2
1
)}u()({cos
1)(cos2
)(cos
)(cos21
)(cos1
)}u()({cos 2
2
21
1
๏ซ๏๏ญ
๏๏ญ
๏ฝ
๏ซ๏๏ญ
๏๏ญ
๏ฝ๏๏ ๏ญ๏ญ
๏ญ
Tzz
Tzz
zTz
Tz
kTkT
1)(cos2
)(s
)(cos21
)(s
)}u()({sin 221
1
๏ซ๏๏ญ
๏
๏ฝ
๏ซ๏๏ญ
๏
๏ฝ๏๏ ๏ญ๏ญ
๏ญ
Tzz
Tinz
zTz
Tinz
kTkT
28. Z-Transform Properties: Complex Differentiation
28
๏จ ๏ฉ
dz
zdX
zkTxk Z
๏ญ๏พ๏ฎ๏ฌ)(
๏จ ๏ฉzX
dz
d
zkTxk
m
Zm
๏ท
๏ธ
๏ถ
๏ง
๏จ
๏ฆ
๏ญ๏พ๏ฎ๏ฌ)(
29. Z-Transform Properties: Convolution
๏ฑ Convolution in time domain is multiplication in z-domain
๏ฑ Example: Letโs calculate the convolution of
๏ฑ Multiplications of z-transforms is
29
๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉzXzXkTxkTx Z
2121 ๏พ๏ฎ๏ฌ๏ช
๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉkTukTkTuakTx kT
๏ฝ๏ฝ 21 xand
๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ
๏จ ๏ฉ๏จ ๏ฉ1121
11
1
๏ญ๏ญ
๏ญ๏ญ
๏ฝ๏ฝ
zaz
zXzXzY
๏จ ๏ฉ ๏จ ๏ฉzXkTxzXkTx ZZ
2211 )(&)( ๏พ๏ฎ๏ฌ๏พ๏ฎ๏ฌ
30. Z-Transform Properties: Initial and Final Value Theorems
๏ฑ Initial Value Theorem:
๏ง The value of x(n) as k โ 0 is given by:
๏ฑ Final Value Theorem:
๏ง The value of x(n) as k โ ๏ฅ is given by:
30
๏จ ๏ฉ )(limlim)0(
0
zXkTxx
zk ๏ฅ๏ฎ๏ฎ
๏ฝ๏ฝ
๏จ ๏ฉ )]()1[(limlim)(
1
zXzkTxx
zk
๏ญ๏ฝ๏ฝ๏ฅ
๏ฎ๏ฅ๏ฎ
32. 32
The Inverse Z-Transform
๏ฑ Formal inverse z-transform is based on a Cauchy integral
๏ฑ Less formal ways sufficient most of the time
1) Direct or Long Division Method
2) Partial fraction expansion and Look-up Table
3) Inversion Integral Method (Residue-theorem)
๏จ ๏ฉ ๏จ ๏ฉ๏ ๏ ๏จ ๏ฉ dzzzX
j
zXZkTx k
C
11
2
1 ๏ญ๏ญ
๏ฒ๏ฝ๏ฝ
๏ฐ
33. 33
Inverse Z-Transform: Power Series Expansion
๏ฑ Using Long Division to expand X(z) as a series
๏ฑ Write the inverse transform as the sequence
๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ๏ฅ
๏ฅ
๏ฝ
๏ญ๏ญ๏ญ
๏ฝ๏ซ๏ซ๏ซ๏ฝ
0
21
...20
k
k
zkTxzTxzTxxzX
๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ ...},2,,0{ TxTxxkTx ๏ฝ
35. Inverse Z-Transform: Power Series Expansion
๏ฑ Example: Find x(n) for n = 0, 1, 2, 3, 4, when X(z) is given by:
๏ฑ Solution:
๏ง First, rewrite X(z) as a ratio of polynomial in ๐งโ1
, as follows:
๏ง Dividing the numerator by the denominator, we have:
35
๏จ ๏ฉ
๏จ ๏ฉ๏จ ๏ฉ2.01
510
๏ญ๏ญ
๏ซ
๏ฝ
zz
z
zX
๏จ ๏ฉ 21
21
2.02.11
510
๏ญ๏ญ
๏ญ๏ญ
๏ซ๏ญ
๏ซ
๏ฝ
zz
zz
zX
4321
68.184.181710 ๏ญ๏ญ๏ญ๏ญ
๏ซ๏ซ๏ซ zzzz
21
2.02.11 ๏ญ๏ญ
๏ซ๏ญ zz 21
510 ๏ญ๏ญ
๏ซ zz
32
217 ๏ญ๏ญ
๏ญ zz
432
4.34.2017 ๏ญ๏ญ๏ญ
๏ซ๏ญ zzz
43
4.34.18 ๏ญ๏ญ
๏ญ zz
543
68.308.224.18 ๏ญ๏ญ๏ญ
๏ซ๏ญ zzz
54
68.368.18 ๏ญ๏ญ
๏ญ zz
654
736.3416.2268.18 ๏ญ๏ญ๏ญ
๏ซ๏ญ zzz
321
21210 ๏ญ๏ญ๏ญ
๏ซ๏ญ zzz Therefore,
โข x(0) = 0,
โข x(1) = 10,
โข x(2) = 17,
โข x(3) = 18.4
โข x(4) = 18.68
36. Inverse Z-Transform: Partial Fraction Expansion
๏ฑ Assume that a given z-transform can be expressed as
๏ฑ Apply partial fractional expansion
๏ฑ First term exist only if M>N
๏ง Br is obtained by long division
๏ฑ Second term represents all first order poles
๏ฑ Third term represents an order s pole
๏ง There will be a similar term for every high-order pole
๏ฑ Each term can be inverse transformed by inspection
36
๏จ ๏ฉ
๏ฅ
๏ฅ
๏ฝ
๏ญ
๏ฝ
๏ญ
๏ฝ N
0k
k
k
M
0k
k
k
za
zb
zX
๏จ ๏ฉ
๏จ ๏ฉ๏ฅ๏ฅ๏ฅ ๏ฝ
๏ญ
๏น๏ฝ
๏ญ
๏ญ
๏ฝ
๏ญ
๏ญ
๏ซ
๏ญ
๏ซ๏ฝ
s
1m
m1
i
m
N
ik,1k
1
k
k
NM
0r
r
r
zd1
C
zd1
A
zBzX
37. Inverse Z-Transform: Partial Fraction Expansion
๏ฑ Coefficients are given as
๏ฑ Easier to understand with examples
37
๏จ ๏ฉ
๏จ ๏ฉ๏ฅ๏ฅ๏ฅ ๏ฝ
๏ญ
๏น๏ฝ
๏ญ
๏ญ
๏ฝ
๏ญ
๏ญ
๏ซ
๏ญ
๏ซ๏ฝ
s
m
m
i
m
N
ikk k
k
NM
r
r
r
zd
C
zd
A
zBzX
1
1
,1
1
0 11
๏จ ๏ฉ ๏จ ๏ฉ kdzkk zXzdA ๏ฝ
๏ญ
๏ญ๏ฝ 1
1
๏จ ๏ฉ ๏จ ๏ฉ
๏จ ๏ฉ ๏จ ๏ฉ๏ ๏ 1
1
1
!
1
๏ญ
๏ฝ
๏ญ
๏ญ
๏ญ
๏ญ
๏พ
๏ฝ
๏ผ
๏ฎ
๏ญ
๏ฌ
๏ญ
๏ญ๏ญ
๏ฝ
idw
s
ims
ms
ms
i
m wXwd
dw
d
dms
C
38. Example:
๏ฑ Find x(kT) if X(z) is given by:
๏ฑ Solution: We first expand X(z)/z into partial fractions as follows :
๏ฑ Then we obtain
๏ฑ then
38
๏จ ๏ฉ
๏จ ๏ฉ๏จ ๏ฉ2.01
10
๏ญ๏ญ
๏ฝ
zz
z
zX
๏จ ๏ฉ
๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ2.0
5.12
1
5.12
2.01
10
๏ญ
๏ญ
๏ญ
๏ฝ
๏ญ๏ญ
๏ฝ
zzzzz
zX
๏จ ๏ฉ ๏ท
๏ธ
๏ถ
๏ง
๏จ
๏ฆ
๏ญ
๏ญ
๏ญ
๏ฝ
2.01
5.12
z
z
z
z
zX
๏จ ๏ฉ )()2.0(5.12)(5.12 kTukTukTx k
๏ญ๏ฝ
๏จ ๏ฉ )(11
kTZ ๏ค๏ฝ๏ญ
๏จ ๏ฉkTu
z
z
Z ๏ฝ๏ท
๏ธ
๏ถ
๏ง
๏จ
๏ฆ
๏ญ
๏ญ
1
1
๏จ ๏ฉkTua
az
z
Z n
๏ฝ๏ท
๏ธ
๏ถ
๏ง
๏จ
๏ฆ
๏ญ
๏ญ1
39. Another Solution
๏ฑ In this example, if X(z), rather than X(z)/z, is expanded into partial
fractions, then we obtain:
๏ฑ However, by use of the shifting theorem we find :
๏ฑ Then
39
๏จ ๏ฉ
๏จ ๏ฉ๏จ ๏ฉ 2.0
5.2
1
5.12
2.01
10
๏ญ
๏ญ
๏ญ
๏ฝ
๏ญ๏ญ
๏ฝ
zzzz
z
zX
๏จ ๏ฉ ๏ท
๏ธ
๏ถ
๏ง
๏จ
๏ฆ
๏ญ
๏ญ
๏ญ
๏ฝ ๏ญ๏ญ
2.0
5.2
1
5.12 11
z
z
z
z
z
zzX
๏จ ๏ฉ )(2.015.12)()2.0(
2.0
5.2
)(5.12)(
)()2.0(5.2)(5.12)( 1
TkTuTkTuTkTukTx
TkTuTkTukTx
kk
k
๏ญ๏ญ๏ฝ๏ญ๏ญ๏ญ๏ฝ
๏ญ๏ญ๏ญ๏ฝ ๏ญ
๏จ ๏ฉ
๏จ ๏ฉ
๏จ ๏ฉ
๏จ ๏ฉ
๏จ ๏ฉ
..
..
48.124
4.123
122
101
00
๏ฝ
๏ฝ
๏ฝ
๏ฝ
๏ฝ
x
x
x
x
x
๏จ ๏ฉ๏ ๏ ๏จ ๏ฉ๏ ๏TnkTxzXzZ o
no
๏ญ๏ฝ๏ญ๏ญ1
40. Example:
๏ฑ Find x(kT) if X(z) is given by:
๏ฑ Solution: Expand X(z)/z into partial fraction as follows:
๏ฑ Then:
๏ฑ By referring to tables we find that x(kT) is given by:
๏ฑ and therefore,
๏ฑ x(0) = 0 -1 +3 = 2
๏ฑ x(1) = 18 โ 2 + 3= 19
40
๏จ ๏ฉ
๏จ ๏ฉ ๏จ ๏ฉ12
2
2
3
๏ญ๏ญ
๏ซ
๏ฝ
zz
zz
zX
๏จ ๏ฉ
๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ 1
3
2
1
2
9
12
12
22
2
๏ญ
๏ซ
๏ญ
๏ญ
๏ญ
๏ฝ
๏ญ๏ญ
๏ซ
๏ฝ
zzzzz
z
z
zX
๏จ ๏ฉ
๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ1
3
22
9
2
๏ญ
๏ซ
๏ญ
๏ญ
๏ญ
๏ฝ
z
z
z
z
z
z
zX
๏จ ๏ฉ )(3)()2()()2(5.4 kTukTukTrkTx kk
๏ซ๏ญ๏ฝ
๏จ ๏ฉkTuk
k
kk
kTr ๏ฝ
๏ฎ
๏ญ
๏ฌ ๏ณ
๏ฝ
00
0
)(
๏ฐ
๏จ ๏ฉ
๏จ ๏ฉkTr
z
z
Z ๏ฝ๏บ
๏ป
๏น
๏ช
๏ซ
๏ฉ
๏ญ
๏ญ
2
1
1
๏จ ๏ฉ
๏จ ๏ฉkTra
az
az
Z k
๏ฝ๏บ
๏ป
๏น
๏ช
๏ซ
๏ฉ
๏ญ
๏ญ
2
1
44. Mapping between s-plane to z-plane
๏ฑ Where ๐ = ๐ + ๐๐ for real number ๐ and real number ๐.
๏ฑ Then ๐ง in polar coordinates is given by
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๐ง = ๐(๐+๐๐)๐
๐ง = ๐ ๐๐
๐ ๐๐๐
โ ๐ง = ๐๐๐ง = ๐ ๐๐
๐ง = ๐ ๐ ๐
45. Mapping between s-plane to z-plane
๏ฑ We will discuss following cases to map given points on s-plane to z-
plane.
๏ง Case-1: Real pole in s-plane (๐ = ๐) [on the left hand-side]
๏ง Case-2: Imaginary Pole in s-plane (๐ = ๐๐)
๏ง Case-3: Complex Poles (๐ = ๐ + ๐๐)
45๐ โ ๐๐๐๐๐ ๐ง โ ๐๐๐๐๐
46. Mapping between s-plane to z-plane
๏ฑ Case-1: Real pole in s-plane (๐ = ๐)
๏ฑ When ๐ = 0
๏ฑ When ๐ = โโ
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โ ๐ง = ๐๐๐ง = ๐ ๐ ๐
๐ง = ๐0๐ = 1 โ ๐ง = 0๐ = 0
๐ = 0
๐ โ ๐๐๐๐๐ ๐ง โ ๐๐๐๐๐
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๐ง = ๐โโ๐ = 1 โ ๐ง = 0๐ = 0