Laplace Transform
content:
PIERRE-SIMON LAPLACE
Existence of Laplace Transform
Laplace Transform of some basic functions
Piece Wise continuous function
Image Processing by using Laplace Transform
Real Life Application of Laplace Transform
Limitations of Laplace Transform
Conclusion
Laplace Transform
content:
PIERRE-SIMON LAPLACE
Existence of Laplace Transform
Laplace Transform of some basic functions
Piece Wise continuous function
Image Processing by using Laplace Transform
Real Life Application of Laplace Transform
Limitations of Laplace Transform
Conclusion
The z-Transform is often time more convenient to use
Definition:
Compare to DTFT definition:
z is a complex variable that can be represented as z=r ej
Substituting z=ej will reduce the z-transform to DTFT
Region of Convergence for a discrete time signal x[n] is defined as a continuous region in z plane where the Z-Transform converges.
The roots of the equation P(z) = 0 correspond to the ’zeros’ of X(z)
The roots of the equation Q(z) = 0 correspond to the ’poles’ of X(z)
The RoC of the Z-transform depends on the convergence of the polynomials P(z) and Q(z),
Uses to analysis of digital filters.
Used to simulate the continuous systems.
Analyze the linear discrete system.
Used to finding frequency response.
z-Transform is for the analysis and synthesis of discrete-time control systems.The z transform in discrete-time systems play a similar role as the Laplace transform in continuous-time systems
The z-Transform is often time more convenient to use
Definition:
Compare to DTFT definition:
z is a complex variable that can be represented as z=r ej
Substituting z=ej will reduce the z-transform to DTFT
Region of Convergence for a discrete time signal x[n] is defined as a continuous region in z plane where the Z-Transform converges.
The roots of the equation P(z) = 0 correspond to the ’zeros’ of X(z)
The roots of the equation Q(z) = 0 correspond to the ’poles’ of X(z)
The RoC of the Z-transform depends on the convergence of the polynomials P(z) and Q(z),
Uses to analysis of digital filters.
Used to simulate the continuous systems.
Analyze the linear discrete system.
Used to finding frequency response.
z-Transform is for the analysis and synthesis of discrete-time control systems.The z transform in discrete-time systems play a similar role as the Laplace transform in continuous-time systems
Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
When a customer search for a automobile, if the automobile is available, they will be taken to a page that shows the details of the automobile including automobile name, automobile ID, quantity, price etc. “Automobile Management System” is useful for maintaining automobiles, customers effectively and hence helps for establishing good relation between customer and automobile organization. It contains various customized modules for effectively maintaining automobiles and stock information accurately and safely.
When the automobile is sold to the customer, stock will be reduced automatically. When a new purchase is made, stock will be increased automatically. While selecting automobiles for sale, the proposed software will automatically check for total number of available stock of that particular item, if the total stock of that particular item is less than 5, software will notify the user to purchase the particular item.
Also when the user tries to sale items which are not in stock, the system will prompt the user that the stock is not enough. Customers of this system can search for a automobile; can purchase a automobile easily by selecting fast. On the other hand the stock of automobiles can be maintained perfectly by the automobile shop manager overcoming the drawbacks of existing system.
Forklift Classes Overview by Intella PartsIntella Parts
Discover the different forklift classes and their specific applications. Learn how to choose the right forklift for your needs to ensure safety, efficiency, and compliance in your operations.
For more technical information, visit our website https://intellaparts.com
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
Event Management System Vb Net Project Report.pdfKamal Acharya
In present era, the scopes of information technology growing with a very fast .We do not see any are untouched from this industry. The scope of information technology has become wider includes: Business and industry. Household Business, Communication, Education, Entertainment, Science, Medicine, Engineering, Distance Learning, Weather Forecasting. Carrier Searching and so on.
My project named “Event Management System” is software that store and maintained all events coordinated in college. It also helpful to print related reports. My project will help to record the events coordinated by faculties with their Name, Event subject, date & details in an efficient & effective ways.
In my system we have to make a system by which a user can record all events coordinated by a particular faculty. In our proposed system some more featured are added which differs it from the existing system such as security.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
Here's a great example: At a large natural gas-fired power plant, where they use waste heat to generate steam and energy, they were puzzled that their boiler wasn't producing as much steam as expected.
R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
Based on our results, Tetra Engineering installed covering plates to reduce the bypass flow. This improved the boiler's performance and increased electricity production.
It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
2. 3.1 The Z-Transform
• Counterpart of the Laplace transform for discrete-time signals
• Generalization of the Fourier Transform
Fourier Transform does not exist for all signals
• Definition:
• Compare to DTFT definition:
• z is a complex variable that can be represented as z=r ej
• Substituting z=ej will reduce the z-transform to DTFT
Chapter 3: The Z-Transform 1
n
n
z
n
x
z
X
n
j
n
j
e
n
x
e
X
3.
j
n
n
z
n
n
re
z
z
n
x
z
X
z
X
n
x
z
X
z
n
x
n
x
0
)
(
)
(
)
(
r
:
اندازه
:
فاز
تبدیل
z
طرفهیک
تبدیل
z
طرفهدو
3.1 The Z-Transform
4. The z-transform and the DTFT
• Convenient to describe on the complex z-plane
• If we plot z=ej for =0 to 2 we get the unit circle
Chapter 3: The Z-Transform 3
Re
Im
Unit Circle
r=1
0
2 0 2
j
e
X
5. Convergence of the z-Transform
• DTFT does not always converge
Example: x[n] = anu[n] for |a|>1 does not have a DTFT
• Complex variable z can be written as r ej so the z-
transform
convert to the DTFT of x[n] multiplied with exponential
sequence r –n
• For certain choices of r the sum
maybe made finite
Chapter 3: The Z-Transform 4
n
n
j
n
n
n
j
j
e
n
x
e
n
x
re
X
r
r
n
j
n
j
e
n
x
e
X
n
n
x r n
-
6. Region of Convergence (ROC)
• ROC: The set of values of z for which the z-transform converges
• The region of convergence is made of circles
Chapter 3: The Z-Transform 5
Re
Im
• Example: z-transform converges for
values of 0.5<r<2
ROC is shown on the left
In this example the ROC includes the unit circle,
so DTFT exists
7. • Example:
Doesn't converge for any r.
DTFT exists.
It has finite energy.
DTFT converges in a mean square sense.
• Example:
Doesn't converge for any r.
It doesn’t have even finite energy.
But we define a useful DTFT with
impulse function.
n
n
x o
cos
sin c n
x n
n
Region of Convergence (ROC)
8. Example 1: Right-Sided Exponential Sequence
• For Convergence we require
• Hence the ROC is defined as
• Inside the ROC series converges to
Chapter 3: The Z-Transform 7
0
n
n
1
n
n
n
n
az
z
n
u
a
z
X
n
u
a
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0
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1
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a 1
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• Region outside the circle of
radius a is the ROC
• Right-sided sequence ROCs
extend outside a circle
9. (
ارچپگدنباله
)
a
z
z
az
z
a
z
X
a
z
z
a
z
a
ROC
z
a
z
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1
1
1
:
1
1
1
n
u
a
n
x n
Example 2: Left-Sided Exponential Sequence
10. Example 3: Two-Sided Exponential Sequence
Chapter 3: The Z-Transform 9
1
-
n
-
u
2
1
-
n
u
3
1
n
x
n
n
1
1
1
0
1
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1
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ROC 1
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:
ROC 1
2
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1
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3
1
1
1
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X
1
1
Im
2
1
oo
12
1
x
x
3
1
11. Example 4: Finite Length Sequence
Chapter 3: The Z-Transform 10
otherwise
0
1
0 N
n
a
n
x
n
N=16
Pole-zero plot
N
n
u
n
u
a
n
x n
0
:
1
1
1
)
(
1
0
1
1
1
1
1
1
0
1
1
0
z
az
az
ROC
a
z
a
z
z
az
az
az
z
a
z
X
N
n
n
N
N
N
N
N
n
n
N
n
n
n
13. SEQUENCE TRANSFORM ROC
1
z
0
m
if
or
0
m
if
0
except
z
All
1
z
1
1
1
z
1
1
1
z
m
z
m
n
n
u
n
u
n
1
1 z
ALL
Some common Z-transform pairs
14.
1
:
cos
2
1
cos
1
cos
:
1
1
:
1
:
1
1
1
:
1
1
2
1
0
1
0
0
2
1
1
2
1
1
1
1
z
ROC
z
z
z
n
u
n
a
z
ROC
az
az
n
u
na
a
z
ROC
az
az
n
u
na
a
z
ROC
az
n
u
a
a
z
ROC
az
n
u
a
Z
Z
n
Z
n
Z
n
Z
n
Some common Z-transform pairs
15.
0
:
1
1
0
1
0
:
cos
2
1
sin
sin
1
2
2
1
0
1
0
0
z
ROC
az
z
a
otherwise
N
n
a
r
z
ROC
z
r
z
r
z
r
n
u
n
r
N
N
Z
n
Z
n
r
z
ROC
z
r
z
r
z
r
n
u
n
r
z
ROC
z
z
z
n
u
n
Z
n
Z
:
cos
2
1
cos
1
cos
1
:
cos
2
1
sin
sin
2
2
1
0
1
0
0
2
1
0
1
0
0
Some common Z-transform pairs
16. 3.2 Properties of The ROC of Z-Transform
• The ROC is a ring or disk centered at the origin
• DTFT exists if and only if the ROC includes the unit circle
• The ROC cannot contain any poles
• The ROC for finite-length sequence is the entire z-plane
except possibly z=0 and z=
• The ROC for a right-handed sequence extends outward from the
outermost pole possibly including z=
• The ROC for a left-handed sequence extends inward from the
innermost pole possibly including z=0
• The ROC of a two-sided sequence is a ring bounded by poles
• The ROC must be a connected region
• A z-transform does not uniquely determine a sequence without
specifying the ROC
Chapter 3: The Z-Transform 15
17. Stability, Causality, and the ROC
• Consider a system with impulse response h[n]
• The z-transform H(z) and the pole-zero plot shown below
• Without any other information h[n] is not uniquely determined
|z|>2 or |z|<½ or ½<|z|<2
• If system stable ROC must include unit-circle: ½<|z|<2
• If system is causal must be right sided: |z|>2
Chapter 3: The Z-Transform 16
18. 3.4 Z-Transform Properties: Linearity
• Notation
• Linearity
– Note that the ROC of combined sequence may be larger than either ROC
– This would happen if some pole/zero cancellation occurs
– Example:
•Both sequences are right-sided
•Both sequences have a pole z=a
•Both have a ROC defined as |z|>|a|
•In the combined sequence the pole at z=a cancels with a zero at z=a
•The combined ROC is the entire z plane except z=0
Chapter 3: The Z-Transform 17
x
Z
R
ROC
z
X
n
x
2
1 x
x
2
1
Z
2
1 R
R
ROC
z
bX
z
aX
n
bx
n
ax
N
-
n
u
a
-
n
u
a
n
x n
n
19. Z-Transform Properties: Time Shifting
• Here no is an integer
– If positive the sequence is shifted right
– If negative the sequence is shifted left
• The ROC can change
– The new term may add or remove poles at z=0 or z=
• Example
Chapter 3: The Z-Transform 18
x
n
Z
o R
ROC
z
X
z
n
n
x o
4
1
z
z
4
1
1
1
z
z
X
1
1
1
-
n
u
4
1
n
x
1
-
n
20. Z-Transform Properties: Multiplication by
Exponential
• ROC is scaled by |zo|
• All pole/zero locations are scaled
• If zo is a positive real number: z-plane shrinks or expands
• If zo is a complex number with unit magnitude it rotates
• Example: We know the z-transform pair
• Let’s find the z-transform of
Chapter 3: The Z-Transform 19
x
o
o
Z
n
o R
z
ROC
z
/
z
X
n
x
z
1
z
:
ROC
z
-
1
1
n
u 1
-
Z
n
u
re
2
1
n
u
re
2
1
n
u
n
cos
r
n
x
n
j
n
j
o
n o
o
r
z
z
re
1
2
/
1
z
re
1
2
/
1
z
X 1
j
1
j o
o
21. Z-Transform Properties: Differentiation
• Example: We want the inverse z-transform of
• Let’s differentiate to obtain rational expression
• Making use of z-transform properties and ROC
Chapter 3: The Z-Transform 20
x
Z
R
ROC
dz
z
dX
z
n
nx
a
z
az
1
log
z
X 1
1
1
1
2
az
1
1
az
dz
z
dX
z
az
1
az
dz
z
dX
1
n
u
a
a
n
nx
1
n
1
n
u
n
a
1
n
x
n
1
n
22. Z-Transform Properties: Conjugation
Chapter 3: The Z-Transform 21
x
*
*
Z
*
R
ROC
z
X
n
x
n
n
n n
n n
n n
n n
X z x n z
X z x n z x n z
X z x n z x n z Z x n
23. Z-Transform Properties: Time Reversal
• ROC is inverted
• Example:
• Time reversed version of
Chapter 3: The Z-Transform 22
x
Z
R
1
ROC
z
/
1
X
n
x
n
u
a
n
x n
n
u
an
1
1
1
-
1
-1
a
z
z
a
-
1
z
a
-
az
1
1
z
X
24. Z-Transform Properties: Convolution
• Convolution in time domain is multiplication in z-domain
• Example: Let’s calculate the convolution of
• Multiplications of z-transforms is
• ROC: if |a|<1 ROC is |z|>1 if |a|>1 ROC is |z|>|a|
• Partial fractional expansion of Y(z)
Chapter 3: The Z-Transform 23
2
x
1
x
2
1
Z
2
1 R
R
:
ROC
z
X
z
X
n
x
n
x
n
u
n
x
and
n
u
a
n
x 2
n
1
a
z
:
ROC
az
1
1
z
X 1
1
1
z
:
ROC
z
1
1
z
X 1
2
1
1
2
1
z
1
az
1
1
z
X
z
X
z
Y
1
z
:
ROC
assume
1
1
1
1
1
1
1
az
a
z
a
z
Y
n
u
a
n
u
a
1
1
n
y 1
n
26. 3.3 The Inverse Z-Transform
• Formal inverse z-transform is based on a Cauchy integral
• Less formal ways sufficient most of the time
– Inspection method
– Partial fraction expansion
– Power series expansion
• Inspection Method
Make use of known z-transform pairs such as
Example: The inverse z-transform of
Chapter 3: The Z-Transform 25
a
z
az
1
1
n
u
a 1
Z
n
n
u
2
1
n
x
2
1
z
z
2
1
1
1
z
X
n
1
27. Inverse Z-Transform by Partial Fraction
Expansion
• Assume that a given z-transform can be expressed as
• Apply partial fractional expansion
• First term exist only if M>N
– Br is obtained by long division
• Second term represents all first order poles
• Third term represents an order s pole
– There will be a similar term for every high-order pole
• Each term can be inverse transformed by inspection
Chapter 3: The Z-Transform 26
N
0
k
k
k
M
0
k
k
k
z
a
z
b
z
X
s
1
m
m
1
i
m
N
i
k
,
1
k
1
k
k
N
M
0
r
r
r
z
d
1
C
z
d
1
A
z
B
z
X
28. Inverse Z-Transform by Partial Fraction
Expansion
• Coefficients are given as
• Easier to understand with examples
Chapter 3: The Z-Transform 27
s
1
m
m
1
i
m
N
i
k
,
1
k
1
k
k
N
M
0
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s
i
m w
X
w
d
1
dw
d
d
!
m
s
1
C
29. Example 5: 2nd Order Z-Transform
Chapter 3: The Z-Transform 28
2
1
z
:
ROC
z
2
1
1
z
4
1
1
1
z
X
1
1
1
2
1
1
z
2
1
1
A
z
4
1
1
A
z
X
1
4
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2
1
1
1
z
X
z
4
1
1
A 1
4
1
z
1
1
2
2
1
4
1
1
1
z
X
z
2
1
1
A 1
2
1
z
1
2
30. Example 5 Continued
• ROC extends to infinity
– Indicates right sided sequence
Chapter 3: The Z-Transform 29
2
1
z
z
2
1
1
2
z
4
1
1
1
z
X
1
1
n
u
4
1
-
n
u
2
1
2
n
x
n
n
31. Example 6
• Long division to obtain Bo
Chapter 3: The Z-Transform 30
1
z
z
1
z
2
1
1
z
1
z
2
1
z
2
3
1
z
z
2
1
z
X
1
1
2
1
2
1
2
1
1
z
5
2
z
3
z
2
1
z
2
z
1
z
2
3
z
2
1
1
1
2
1
2
1
2
1
1
1
z
1
z
2
1
1
z
5
1
2
z
X
1
2
1
1
z
1
A
z
2
1
1
A
2
z
X
9
z
X
z
2
1
1
A
2
1
z
1
1
8
z
X
z
1
A
1
z
1
2
32. Example 5 Continued
• ROC extends to infinity
– Indicates right-sided sequence
Chapter 3: The Z-Transform 31
1
z
z
1
8
z
2
1
1
9
2
z
X 1
1
n
8u
-
n
u
2
1
9
n
2
n
x
n
33. Inverse Z-Transform by Power Series
Expansion
• The z-transform is power series
• In expanded form
• Z-transforms of this form can generally be inversed easily
• Especially useful for finite-length series
Chapter 3: The Z-Transform 32
n
n
z
n
x
z
X
2
1
1
2
2
1
0
1
2 z
x
z
x
x
z
x
z
x
z
X
34.
1
2
1
1
1
2
z
2
1
1
z
2
1
z
z
1
z
1
z
2
1
1
z
z
X
1
n
2
1
n
1
n
2
1
2
n
n
x
2
n
0
1
n
2
1
0
n
1
1
n
2
1
2
n
1
n
x
Example 6