The z-transform is an important tool for analyzing digital systems and digital signal processing. It allows the design of digital filters and frequency analysis of digital signals. The z-transform represents a discrete-time signal as a function of a complex variable z. It is defined as the summation of the signal multiplied by z to the power of n from negative infinity to positive infinity. The values of z that satisfy the convergence of this summation form the region of convergence. The z-transform has properties of linearity, shift, and convolution that allow transforming difference equations into algebraic equations that can be solved. The inverse z-transform is used to obtain the original discrete-time signal from its z-transform.
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
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
Z Transform And Inverse Z Transform - Signal And SystemsMr. RahüL YøGi
The z-transform is the most general concept for the transformation of discrete-time series.
The Laplace transform is the more general concept for the transformation of continuous time processes.
For example, the Laplace transform allows you to transform a differential equation, and its corresponding initial and boundary value problems, into a space in which the equation can be solved by ordinary algebra.
The switching of spaces to transform calculus problems into algebraic operations on transforms is called operational calculus. The Laplace and z transforms are the most important methods for this purpose.
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.
The presentation covers sampling theorem, ideal sampling, flat top sampling, natural sampling, reconstruction of signals from samples, aliasing effect, zero order hold, upsampling, downsampling, and discrete time processing of continuous time signals.
Why Fourier Transform
General Properties & Symmetry relations
Formula and steps
magnitude and phase spectra
Convergence Condition
mean-square convergence
Gibbs phenomenon
Direct Delta
Energy Density Spectrum
Z Transform And Inverse Z Transform - Signal And SystemsMr. RahüL YøGi
The z-transform is the most general concept for the transformation of discrete-time series.
The Laplace transform is the more general concept for the transformation of continuous time processes.
For example, the Laplace transform allows you to transform a differential equation, and its corresponding initial and boundary value problems, into a space in which the equation can be solved by ordinary algebra.
The switching of spaces to transform calculus problems into algebraic operations on transforms is called operational calculus. The Laplace and z transforms are the most important methods for this purpose.
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.
The presentation covers sampling theorem, ideal sampling, flat top sampling, natural sampling, reconstruction of signals from samples, aliasing effect, zero order hold, upsampling, downsampling, and discrete time processing of continuous time signals.
Why Fourier Transform
General Properties & Symmetry relations
Formula and steps
magnitude and phase spectra
Convergence Condition
mean-square convergence
Gibbs phenomenon
Direct Delta
Energy Density Spectrum
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.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
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.
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.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
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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.
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/
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
1. Z - Transform
1
CEN352, Dr. Ghulam Muhammad
King Saud University
The z-transform is a very important tool in describing and analyzing digital
systems.
It offers the techniques for digital filter design and frequency analysis of
digital signals.
n
n
z
n
x
z
X ]
[
)
(
Definition of z-transform:
For causal sequence, x(n) = 0, n < 0:
Where z is a complex
variable
All the values of z that make the summation to exist form a region of convergence.
2. CEN352, Dr. Ghulam Muhammad
King Saud University
2
Example 1
Given the sequence, find the z transform of x(n).
Problem:
Solution:
We know,
Therefore,
1
1
1
1
1
1
)
( 1
z
z
z
z
z
X
When, 1
|
|
1
|
| 1
z
z
Region of convergence
3. CEN352, Dr. Ghulam Muhammad
King Saud University
3
Example 2
Given the sequence, find the z transform of x(n).
Problem:
Solution:
Therefore,
a
z
z
z
a
az
z
X
1
1
1
1
)
( 1
When, a
z
az
|
|
1
|
| 1
Region of convergence
5. CEN352, Dr. Ghulam Muhammad
King Saud University
5
Example 3
Find z-transform of the following sequences.
Problem:
a. b.
Solution:
a. From line 9 of the Table:
b. From line 14 of the Table:
6. CEN352, Dr. Ghulam Muhammad
King Saud University
6
Z- Transform Properties (1)
Linearity:
a and b are arbitrary constants.
Example 4
Find z- transform of
Line 3
Line 6
Using z- transform
table:
Therefore, we get
Problem:
Solution:
7. CEN352, Dr. Ghulam Muhammad
King Saud University
7
Z- Transform Properties (2)
Shift Theorem:
Verification:
Since x(n) is assumed to be causal:
Then we achieve,
n = m
8. CEN352, Dr. Ghulam Muhammad
King Saud University
8
Example 5
Find z- transform of
Problem:
Solution:
Using shift theorem,
Using z- transform table, line 6:
9. CEN352, Dr. Ghulam Muhammad
King Saud University
9
Z- Transform Properties (3)
Convolution
In time domain,
In z- transform domain,
Verification:
Eq. (1)
Using z- transform in Eq. (1)
10. CEN352, Dr. Ghulam Muhammad
King Saud University
10
Example 6
Problem:
Solution:
Given the sequences,
Find the z-transform of their
convolution.
Applying z-transform on the two sequences,
From the table, line 2
Therefore we get,
11. CEN352, Dr. Ghulam Muhammad
King Saud University
11
Inverse z- Transform: Examples
Find inverse z-transform of
We get,
Using table,
Find inverse z-transform of
We get,
Using table,
Example 7
Example 8
12. CEN352, Dr. Ghulam Muhammad
King Saud University
12
Inverse z- Transform: Examples
Find inverse z-transform of
Since,
By coefficient matching,
Therefore,
Find inverse z-transform of
Example 9
Example 10
13. CEN352, Dr. Ghulam Muhammad
King Saud University
13
Inverse z-Transform: Using Partial Fraction
Find inverse z-transform of
First eliminate the negative power of z.
Example 11
Dividing both sides by z:
Finding the
constants:
Therefore, inverse z-transform is:
Problem:
Solution:
14. CEN352, Dr. Ghulam Muhammad
King Saud University
14
Inverse z-Transform: Using Partial Fraction
Example 12
Dividing both sides by z:
We first find B:
Next find A:
Problem:
Solution:
15. CEN352, Dr. Ghulam Muhammad
King Saud University
15
Example 12 – contd.
Using polar form
Now we have:
Therefore, the inverse z-transform is:
16. CEN352, Dr. Ghulam Muhammad
King Saud University
16
Inverse z-Transform: Using Partial Fraction
Example 13
Dividing both sides by z:
m = 2, p = 0.5
Problem:
Solution:
17. CEN352, Dr. Ghulam Muhammad
King Saud University
17
Example 13 – contd.
From Table:
Finally we get,
18. CEN352, Dr. Ghulam Muhammad
King Saud University
18
Partial Function Expansion Using MATLAB
Example 14
The denominator polynomial can be
found using MATLAB:
Therefore,
The solution is:
Problem:
Solution:
19. CEN352, Dr. Ghulam Muhammad
King Saud University
19
Partial Function Expansion Using MATLAB
Example 15
Problem:
Solution:
20. CEN352, Dr. Ghulam Muhammad
King Saud University
20
Partial Function Expansion Using MATLAB
Example 16
Problem:
Solution:
21. CEN352, Dr. Ghulam Muhammad
King Saud University
21
Difference Equation Using Z-Transform
The procedure to solve difference equation using z-transform:
1. Apply z-transform to the difference equation.
2. Substitute the initial conditions.
3. Solve for the difference equation in z-transform domain.
4. Find the solution in time domain by applying the inverse z-transform.
22. CEN352, Dr. Ghulam Muhammad
King Saud University
22
Example 17
Solve the difference equation when the initial condition is
Taking z-transform on both sides:
Substituting the initial condition and z-transform on right hand side using Table:
Arranging Y(z) on left hand side:
Problem:
Solution:
23. CEN352, Dr. Ghulam Muhammad
King Saud University
23
Example 17 – contd.
Solving for A and B:
Therefore,
Taking inverse z-transform, we get the solution:
24. CEN352, Dr. Ghulam Muhammad
King Saud University
24
Example 18
A DSP system is described by the following differential equation with zero initial condition:
a. Determine the impulse response y(n) due to the impulse sequence x(n) = (n).
b. Determine system response y(n) due to the unit step function excitation, where
u(n) = 1 for n 0.
Taking z-transform on both sides:
Problem:
Solution:
Applying on right side
a.
25. CEN352, Dr. Ghulam Muhammad
King Saud University
25
Example 18 – contd.
We multiply the numerator and denominator by z2
Hnece the impulse
response:
Solving for A and B:
Therefore,
26. CEN352, Dr. Ghulam Muhammad
King Saud University
26
Example 18 – contd.
The input is step unit function:
Corresponding z-transform:
b.
[Slide 24]
Do the
middle
steps by
yourself!
![Z - Transform
1
CEN352, Dr. Ghulam Muhammad
King Saud University
The z-transform is a very important tool in describing and analyzing digital
systems.
It offers the techniques for digital filter design and frequency analysis of
digital signals.
n
n
z
n
x
z
X ]
[
)
(
Definition of z-transform:
For causal sequence, x(n) = 0, n < 0:
Where z is a complex
variable
All the values of z that make the summation to exist form a region of convergence.](https://image.slidesharecdn.com/referenceforzandinverseztransform-210223142402/85/Reference-for-z-and-inverse-z-transform-1-320.jpg)
