The document defines z-transforms and provides properties of z-transforms. It defines the two-sided and one-sided z-transforms as infinite sums involving the input signal x[n] and complex variable z. Key properties include: convolution in the time domain is equivalent to multiplication in the z-domain; time shifts correspond to multiplication by powers of z; the region of convergence is determined by poles of the z-transform.
A PowerPoint presentation on the Derivative as a Function. Includes example problems on finding the derivative using the definition, Power Rule, examining graphs of f(x) and f'(x), and local linearity.
A PowerPoint presentation on the Derivative as a Function. Includes example problems on finding the derivative using the definition, Power Rule, examining graphs of f(x) and f'(x), and local linearity.
Pseudo and Quasi Random Number GenerationAshwin Rao
Talk given at Morgan Stanley on efficient Monte Carlo simulation using Pseudo random numbers and low-discrepancy sequences (i.e., Quasi random numbers)
Limits of Wiener filtering. Non-linear shrinkage functions. Limits of Fourier representation. Continuous and discrete wavelet transforms. Sparsity and shrinkage in wavelet domain. Undecimated wavelet transforms, a trous algorithm. Regularization. Sparse regression, combinatorial optimization and matching pursuit. LASSO, non-smooth optimization, and proximal minimization. Link with implcit Euler scheme. ISTA algorithm. Syntehsis versus analysis regularized models. Applications to image deconvolution.
In this presentation we will learn Del operator, Gradient of scalar function , Directional Derivative, Divergence of vector function, Curl of a vector function and after that solved some example related to above.
Gradient in math
Directional derivative in math
Divergence in math
Curl in math
Gradient , Directional Derivative , Divergence , Curl in mathematics
Gradient , Directional Derivative , Divergence , Curl in math
Gradient , Directional Derivative , Divergence , Curl
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
What are the key things to consider when you're planning your content? From understanding your audience to measurement and optimisation, see how you can make your content work harder for you.
Pseudo and Quasi Random Number GenerationAshwin Rao
Talk given at Morgan Stanley on efficient Monte Carlo simulation using Pseudo random numbers and low-discrepancy sequences (i.e., Quasi random numbers)
Limits of Wiener filtering. Non-linear shrinkage functions. Limits of Fourier representation. Continuous and discrete wavelet transforms. Sparsity and shrinkage in wavelet domain. Undecimated wavelet transforms, a trous algorithm. Regularization. Sparse regression, combinatorial optimization and matching pursuit. LASSO, non-smooth optimization, and proximal minimization. Link with implcit Euler scheme. ISTA algorithm. Syntehsis versus analysis regularized models. Applications to image deconvolution.
In this presentation we will learn Del operator, Gradient of scalar function , Directional Derivative, Divergence of vector function, Curl of a vector function and after that solved some example related to above.
Gradient in math
Directional derivative in math
Divergence in math
Curl in math
Gradient , Directional Derivative , Divergence , Curl in mathematics
Gradient , Directional Derivative , Divergence , Curl in math
Gradient , Directional Derivative , Divergence , Curl
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
What are the key things to consider when you're planning your content? From understanding your audience to measurement and optimisation, see how you can make your content work harder for you.
Social media analysis of the Interactive Minds Summit in Brisbane 1st of August 2013. This is a sample report prepared as part of BuzzNumbers' sponsorship of the event.
Part three of FaBEducation.com's Financial Statements presentation on their active learning programme in business finance for non-financial managers and entrepreneurs
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
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf91mobiles
91mobiles recently conducted a Smart TV Buyer Insights Survey in which we asked over 3,000 respondents about the TV they own, aspects they look at on a new TV, and their TV buying preferences.
DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
Builder.ai Founder Sachin Dev Duggal's Strategic Approach to Create an Innova...Ramesh Iyer
In today's fast-changing business world, Companies that adapt and embrace new ideas often need help to keep up with the competition. However, fostering a culture of innovation takes much work. It takes vision, leadership and willingness to take risks in the right proportion. Sachin Dev Duggal, co-founder of Builder.ai, has perfected the art of this balance, creating a company culture where creativity and growth are nurtured at each stage.
Elevating Tactical DDD Patterns Through Object CalisthenicsDorra BARTAGUIZ
After immersing yourself in the blue book and its red counterpart, attending DDD-focused conferences, and applying tactical patterns, you're left with a crucial question: How do I ensure my design is effective? Tactical patterns within Domain-Driven Design (DDD) serve as guiding principles for creating clear and manageable domain models. However, achieving success with these patterns requires additional guidance. Interestingly, we've observed that a set of constraints initially designed for training purposes remarkably aligns with effective pattern implementation, offering a more ‘mechanical’ approach. Let's explore together how Object Calisthenics can elevate the design of your tactical DDD patterns, offering concrete help for those venturing into DDD for the first time!
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024Albert Hoitingh
In this session I delve into the encryption technology used in Microsoft 365 and Microsoft Purview. Including the concepts of Customer Key and Double Key Encryption.
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
Key Trends Shaping the Future of Infrastructure.pdfCheryl Hung
Keynote at DIGIT West Expo, Glasgow on 29 May 2024.
Cheryl Hung, ochery.com
Sr Director, Infrastructure Ecosystem, Arm.
The key trends across hardware, cloud and open-source; exploring how these areas are likely to mature and develop over the short and long-term, and then considering how organisations can position themselves to adapt and thrive.
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
Guy Korland, CEO and Co-founder of FalkorDB, will review two articles on the integration of language models with knowledge graphs.
1. Unifying Large Language Models and Knowledge Graphs: A Roadmap.
https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
1. z-Transforms
1
Definitions
• Given an input signal x[n], the two-sided z-transform is given by
∞
X(z) =
x[n]z −n .
(1)
k=−∞
• Given an input signal x[n], the one-sided z-transform is given by
∞
X(z) =
x[n]z −n .
(2)
k=0
• By default, the term z-transform means the two-sided z-transform, i.e.,
if we simply say z-transform without explicitly specifying if it is a onesided z-transform or a two-sided z-transform, then we mean a two-sided
z-transform.
• A series of the form (1) is called a Laurent series.
• Notation: The two-sided z-transform of a signal x[n] is denoted by
z{x[n]}.
2
Basic Properties of the z-transform
• If x[n] is a causal signal, then the two-sided z-transform of x[n] is equal
to the one-sided z-transform of x[n], i.e., if x[n] is a causal signal, then
the z-transform of x[n] contains only negative powers of z.
• If x[n] is of finite length and is zero for n > N2 and n < −N1 , then the
limits of summation in (1) reduce to −N1 and N2 , i.e., the two-sided
z-transform of such a signal x[n] is
N2
X(z) =
x[n]z −n .
k=−N1
1
(3)
2. • Linearity: z{αx1 [n] + βx2 [n]} = αz{x1 [n]} + βz{x2 [n]}
• z{x[n − n0 ]} = z −n0 z{x[n]} ... time shift by n0 is equivalent to multiplication of the z-transform by z −n0 .
3
Examples
• z{δ[n]} = 1
• z{δ[n − n0 ]} = z −n0 for any constant n0 .
• z{αn u[n]} =
1
1−αz −1
for any constant α.
• z{−αn u[−n − 1]} =
4
1
1−αz −1
for any constant α.
More Properties of the z-transform
• z{x1 [n] ∗ x2 [n]} = z{x1 [n]}z{x2 [n]} ... convolution in the time domain is equivalent to multiplication in the z domain, i.e., convolution
is mathematically equivalent to polynomial multiplication.
Proof: Let x[n] = x1 [n] ∗ x2 [n], i.e., x[n] =
∞
k=−∞
x1 [k]x2 [n − k]. Then,
∞
z{x[n]} =
x[n]z −n
n=−∞
∞
∞
x1 [k]x2 [n − k]z −n
=
n=−∞ k=−∞
∞
∞
x2 [n − k]z −n
x1 [k]
=
n=−∞
k=−∞
∞
=
∞
x1 [k]z
x2 [n − k]z −(n−k)
−k
n=−∞
∞
k=−∞
∞
x1 [k]z −k
=
x2 [n1 ]z −n1
n1 =−∞
k=−∞
2
(4)
3. where n1 = n − k. Hence,
∞
∞
x1 [k]z −k
z{x[n]} =
x2 [n1 ]z −n1
n1 =−∞
k=−∞
= z{x1 [n]}z{x2 [n]}.
(5)
• If z{x[n]} = X(z), then z{an x[n]} = X(a−1 z).
Proof:
∞
z{an x[n]} =
an x[n]z −n
n=−∞
∞
=
x[n](a−1 z)−n
n=−∞
−1
= X(a z).
(6)
• If z{x[n]} = X(z), then z{nx[n]} = −z dX(z) .
dz
Proof: Note that
∞
d
dX(z)
=
dz
dz
x[n]z −n
n=−∞
∞
(−n)x[n]z −n−1
=
n=−∞
∞
= −z
nx[n]z −n .
−1
(7)
n=−∞
Hence,
∞
z{nx[n]} =
nx[n]z −n = −z
n=−∞
dX(z)
.
dz
(8)
• If x[n] is a causal signal, then x[0] = limz→∞ X(z).
Proof: If x[n] is a causal signal, then x[n] = 0 for n < 0 so that X(z)
contains only negative powers of z, i.e., X(z) is of the form
X(z) = x[0] + x[1]z −1 + x[2]z −2 + x[3]z −3 + . . .
(9)
Taking z → ∞, only the first term in (9) remains so that
lim X(z) = x[0].
z→∞
3
(10)
4. • If z{x[n]} = X(z), then z{x[−n]} = X(z −1 ).
Proof:
∞
z{x[−n]} =
x[−n]z −n
n=−∞
−∞
x[n1 ]z n1
=
(11)
n1 =∞
where n1 = −n. Hence,
∞
z{x[−n]} =
x[n1 ](z −1 )−n1
n1 =−∞
−1
= X(z ).
(12)
• If z{x[n]} = X(z), then z{x∗ [n]} = X ∗ (z ∗ ).
Proof: Note that
∞
X ∗ (z ∗ ) =
x[n](z ∗ )−n
∗
n=−∞
∞
=
x∗ [n]z −n
(13)
n=−∞
since (z ∗ )∗ = z. Hence,
∞
x∗ [n]z −n = X ∗ (z ∗ ).
z{x∗ [n]} =
(14)
n=−∞
1
• If z{x[n]} = X(z), then z{Re(x[n])} = 2 [X(z) + X ∗ (z ∗ )].
Proof: From the property proved above, we know that z{x∗ [n]} =
X ∗ (z ∗ ). Noting that
1
(x[n] + x∗ [n]),
(15)
Re(x[n]) =
2
the result follows.
• If z{x[n]} = X(z), then z{Im(x[n])} =
1
[X(z)
2j
− X ∗ (z ∗ )].
Proof: We know that z{x∗ [n]} = X ∗ (z ∗ ). Noting that
1
(x[n] − x∗ [n]),
Im(x[n]) =
2j
the result follows.
4
(16)
5. 5
Region of Convergence
• The set of values of z for which the summation in the definition of the
z-transform (1) converges is called the Region of Convergence (ROC).
The ROC is a subset of the complex plane.
• The ROC of the z-transform of a finite length signal is the entire complex plane. Since the z-transform of a finite length signal has only a
finite number of terms in the summation, the z-transform converges for
all z.
• From the theory of complex analysis, it is known that the ROC of the
z-transform of any signal is a connected set and is bounded by circles
(with the centers of the circles being at the origin of the complex plane).
Hence, there are three possibilities for the ROC as shown in Figure 1:
– Interior of a circle
– Exterior of a circle
– Annular region between two circles
−n −jωn
• If z = rejω , then X(z) = ∞
e
. It can be shown that
n=−∞ x[n]r
X(z) converges if and only if
∞
|x[n]r −n | < ∞.
(17)
n=−∞
Example: If x[n] = u[n], then (17) reduces to ∞ |r−n | < ∞ which
n=0
holds if |r −1 | < 1, i.e., if |r| > 1. Hence, the ROC of the z-transform
of u[n] is 1 < |z| < ∞, i.e., the exterior of a circle with radius 1.
Example: If x[n] = αn u[n], then (17) reduces to ∞ |αn r−n | < ∞
n=0
which holds if |αr −1 | < 1, i.e., if |r| > |α|. Hence, the ROC of the
z-transform of αn u[n] is |α| < |z| < ∞, i.e., the exterior of a circle with
radius |α|.
Example: If x[n] = −αn u[−n−1], then (17) reduces to −∞ |αn r−n | <
n=−1
∞ which holds if |α−1 r| < 1, i.e., if |r| < |α|. Hence, the ROC of the
z-transform of −αn u[−n−1] is 0 ≤ |z| < |α|, i.e., the interior of a circle
with radius |α|.
5
6. Figure 1: Possibilities for the region of convergence: (a) interior of a circle;
(b) exterior of a circle; (c) annular region between two circles.
• From the above examples, it can be generalized that the ROC of the
z-transform of a causal signal is the exterior of a circle and the ROC
of the z-transform of an anti-causal signal (i.e., a signal which is zero
for n > 0) is the interior of a circle.
• The above statement can be generalized to right-sided and left-sided
signals. A right-sided signal is a signal which is zero for all n smaller
than some constant n1 . A left-sided signal is a signal which is zero
for all n greater than some constant n1 . The ROC of the z-transform
of a right-sided signal is the exterior of a circle while the ROC of the
z-transform of a left-sided signal is the interior of a circle (possibly not
including the origin).
• Example: Consider the signal
αn for n ≥ 0
β n for n ≤ −1.
x[n] =
(18)
The signal x[n] in (18) is a two-sided signal. If |α| < |β|, then the ROC
6
7. of the z-transform of the signal x[n] is the annular region |α| < |z| < |β|.
If |α| ≥ |β|, then the z-transform of x[n] does not converge for any z.
• Special Case: If X(z) is a rational function (i.e., a ratio of polynomials), then the ROC is bounded by the poles of X(z), i.e., the values of
z for which X(z) is infinity.
Example: For the signal in (18), the z-transform is
∞
−∞
αn z −n +
X(z) =
n=0
∞
β n z −n
n=−1
∞
β −n z n
αn z −n +
=
n=1
n=0
1
1
+
−1
=
−1
1 − αz
1 − β −1 z
β −1 z
1
+
=
1 − αz −1 1 − β −1 z
1
1
=
−
.
−1
1 − αz
1 − βz −1
(19)
From (19), the poles of X(z) are z = α and z = β. As seen above, if
|α| < |β| the ROC of X(z) is the annular region |α| < |z| < |β|.
Example: If X(z) is a rational function, then the ROC of X(z) is
always bounded by poles. This means that the boundaries of the ROC
have poles on them. It does not mean that if X(z) has two poles, then
the ROC is the annular region between the poles. For example, consider
the signal x[n] = αn u[n] + β n u[n]. It can be shown that the ROC of
the z-transform of X(z) is max(|α|, |β|) < |z| < ∞, i.e., the exterior
of a circle with radius max(|α|, |β|). The boundary of this ROC is the
circle with radius max(|α|, |β|) which has a pole on it (the pole being
the larger of α and β). Note that the poles of X(z) are α and β.
7