The document discusses digital filter structures. It covers IIR and FIR filter structures. For IIR filters, it describes direct form I and II structures as well as cascade form using biquad sections. Cascade form implements the IIR filter as a product of second-order filter sections in a direct form structure. FIR filters can be implemented using direct form or cascade of direct form filter sections. The choice of structure depends on factors like complexity, memory requirements, and quantization effects.
In many applications one observes rapid change of the solution in the boundary region. Accurate and numerically efficient resolution of the solution close to the moving boundaries is considered to be an important problem. We develop an approach to the optimization of the discretization grids for finite-difference scheme. Using the suggested approach we are able to achieve the exponential convergence of the boundary Neumann- to-Dirichlet maps. It increases the convergence order without increasing the stencil size of the finite-difference scheme and preserves stability.
In many applications one observes rapid change of the solution in the boundary region. Accurate and numerically efficient resolution of the solution close to the moving boundaries is considered to be an important problem. We develop an approach to the optimization of the discretization grids for finite-difference scheme. Using the suggested approach we are able to achieve the exponential convergence of the boundary Neumann- to-Dirichlet maps. It increases the convergence order without increasing the stencil size of the finite-difference scheme and preserves stability.
A short course I taught in 2002 at the University of Hawaii's Kauai Community College. This course was offered to professionals working on radar tracking systems for air, missile, surface, and subsurface vehicle tracking. The material is a decade old, so it does not cover the latest technology. However, it is an excellent primer for those just starting on the subject.
Convex Optimization Modelling with CVXOPTandrewmart11
An introduction to convex optimization modelling using cvxopt in an IPython environment. The facility location problem is used as an example to demonstrate modelling in cvxopt.
Kernel based models for geo- and environmental sciences- Alexei Pozdnoukhov –...Beniamino Murgante
Kernel based models for geo- and environmental sciences- Alexei Pozdnoukhov – National Centre for Geocomputation, National University of Ireland , Maynooth (Ireland)
Intelligent Analysis of Environmental Data (S4 ENVISA Workshop 2009)
EXPERT SYSTEMS AND SOLUTIONS
Project Center For Research in Power Electronics and Power Systems
IEEE 2010 , IEEE 2011 BASED PROJECTS FOR FINAL YEAR STUDENTS OF B.E
Email: expertsyssol@gmail.com,
Cell: +919952749533, +918608603634
www.researchprojects.info
OMR, CHENNAI
IEEE based Projects For
Final year students of B.E in
EEE, ECE, EIE,CSE
M.E (Power Systems)
M.E (Applied Electronics)
M.E (Power Electronics)
Ph.D Electrical and Electronics.
Training
Students can assemble their hardware in our Research labs. Experts will be guiding the projects.
EXPERT GUIDANCE IN POWER SYSTEMS POWER ELECTRONICS
We provide guidance and codes for the for the following power systems areas.
1. Deregulated Systems,
2. Wind power Generation and Grid connection
3. Unit commitment
4. Economic Dispatch using AI methods
5. Voltage stability
6. FLC Control
7. Transformer Fault Identifications
8. SCADA - Power system Automation
we provide guidance and codes for the for the following power Electronics areas.
1. Three phase inverter and converters
2. Buck Boost Converter
3. Matrix Converter
4. Inverter and converter topologies
5. Fuzzy based control of Electric Drives.
6. Optimal design of Electrical Machines
7. BLDC and SR motor Drives
A short course I taught in 2002 at the University of Hawaii's Kauai Community College. This course was offered to professionals working on radar tracking systems for air, missile, surface, and subsurface vehicle tracking. The material is a decade old, so it does not cover the latest technology. However, it is an excellent primer for those just starting on the subject.
Convex Optimization Modelling with CVXOPTandrewmart11
An introduction to convex optimization modelling using cvxopt in an IPython environment. The facility location problem is used as an example to demonstrate modelling in cvxopt.
Kernel based models for geo- and environmental sciences- Alexei Pozdnoukhov –...Beniamino Murgante
Kernel based models for geo- and environmental sciences- Alexei Pozdnoukhov – National Centre for Geocomputation, National University of Ireland , Maynooth (Ireland)
Intelligent Analysis of Environmental Data (S4 ENVISA Workshop 2009)
EXPERT SYSTEMS AND SOLUTIONS
Project Center For Research in Power Electronics and Power Systems
IEEE 2010 , IEEE 2011 BASED PROJECTS FOR FINAL YEAR STUDENTS OF B.E
Email: expertsyssol@gmail.com,
Cell: +919952749533, +918608603634
www.researchprojects.info
OMR, CHENNAI
IEEE based Projects For
Final year students of B.E in
EEE, ECE, EIE,CSE
M.E (Power Systems)
M.E (Applied Electronics)
M.E (Power Electronics)
Ph.D Electrical and Electronics.
Training
Students can assemble their hardware in our Research labs. Experts will be guiding the projects.
EXPERT GUIDANCE IN POWER SYSTEMS POWER ELECTRONICS
We provide guidance and codes for the for the following power systems areas.
1. Deregulated Systems,
2. Wind power Generation and Grid connection
3. Unit commitment
4. Economic Dispatch using AI methods
5. Voltage stability
6. FLC Control
7. Transformer Fault Identifications
8. SCADA - Power system Automation
we provide guidance and codes for the for the following power Electronics areas.
1. Three phase inverter and converters
2. Buck Boost Converter
3. Matrix Converter
4. Inverter and converter topologies
5. Fuzzy based control of Electric Drives.
6. Optimal design of Electrical Machines
7. BLDC and SR motor Drives
EXPERT SYSTEMS AND SOLUTIONS
Center for Research Projects
Email: expertsyssol@gmail.com,
Cell: 9952749533,
Neyveli Tamil Nadu
IEEE and Non IEEE based Projects For
Final year students of B.E in EEE, ECE, EIE, Mech, Automobile, Mechatronics ,
M.E (Power Systems)
M.E (Applied Electronics)
M.E (Power Electronics)
M.E (VLSI Design)
M.E ( Control Systems )
M.E ( Process Control )
Ph.D Electrical and Electronics.
Power Systems
Power Electronics
Students can assemble their hardware in our Research labs. Experts will be guiding the projects.
We provide you with
• Confirmation letter
• Project Estimation
• Completion Certificate
• Abstracts Block Diagram, Circuit Diagram of the project
• Supporting documents- notes
• Datasheets, Reference Papers
• Software tools and Real time support.
• Online Classes for Master degree and PhD Projects
• Journal Publication for researchers in Power systems and Power electronics
Short term and Online courses
• C and C++, Embedded C,
• MicroController 8051
• PIC 16F877
• ARM 7 and ARM 9 Processors
• MATLAB Fundamental
• MATLAB Simpower Systems
• MATLAB Control systems
• MATLAB Fuzzy Toolbox
• MATLAB GA Toolbox
• MATLAB ANN Toolbox
• Engineering MATHS MI, MII and MIII
POWER SYSTEMS PROJECTS
Deregulated Systems, Wind power Generation and Grid connection, Profit based Unit commitment, Economic dispatch using AI methods for Voltage stability FLC Control Transformer Fault Identifications, SCADA in Power system Automation, FireFly Algorithm, PSO and QPSO , Genetic Algorithm, Fuzzy logic, Artifical Neural networks.
‘Mix tapes’ were a thing of love, a physical object which people would share with significant others and friends around them. They were naturally a social object and highly representative of a person’s identity. The knowledge of effort involved by the giver in selecting the songs and having to sit through each one was also part of the symbolism for the receiver. Objects can be generated and shaped from and by the media you “teach” them or existing objects you play (embed) media into. Thus the modern mix tape could become a linked series of small objects like lucky charms which are physically shareable in a form representing the tracks they contain. This is based on the idea that physical items often mean more to us as physical beings and adds a level of exclusivity and personalisation to the sharing process. Considering trans-platform engagement and the ability to engage users and viewers in co-creating media it is suggested that this may presented as a new modality for user co-creation and curation.
Become our Bella Ragazza Hats Model for Collection 2014BellaRagazzaHats
Buy a hat, get one for free and enter to become a fashion model for Bella Ragazza Hats Collection 2014 with all trip expenses paid to Galapagos Islands in april 2013.
Fine hats for women hand woven. Bella Ragazza Hats 2013 collection; high quality hats handmade; original pieces exclusively designed for girls who want to impose their style and natural beauty, you can wear it for any occasion.
Project titles for eee, ece, eie - M.E, B.E, Ph.D, EEE, ECE, EIE Projects Senthil Kumar
EXPERT SYSTEMS AND SOLUTIONS
Center for Research Projects
Email: expertsyssol@gmail.com,
Cell: 9952749533,
Neyveli, Tamil Nadu
IEEE and Non IEEE based Projects For
Final year students of B.E in EEE, ECE, EIE, Mech, Automobile, Mechatronics,
M.E (Power Systems)
M.E (Applied Electronics)
M.E (Power Electronics)
M.E (VLSI Design)
M.E (Control Systems)
M.E (Process Control)
Ph.D Electrical and Electronics.
Power Systems
Power Electronics
Students can assemble their hardware in our Research labs. Experts will be guiding the projects.
Engineering Research, Consultancy, Training and Design
Project Center For Research in Power Electronics and Power Systems
IEEE 2011 , IEEE 2012 BASED PROJECTS FOR FINAL YEAR STUDENTS OF
B.E - Electrical and Electronics Engineering,
B.E - Electronics and Communication Engineering,
B.E - Electronics and Instrumentation Engineering,
B.E - Bio medical Engineering,
B.E - Tele-communication Engineering,
B.E - Computer Science and Information Technology,
M.E - Power Systems Engineering,
M.E - Applied Electronics,
M.E - Power Electronics And Drive,
M.E - Mechatronics ,
MCA.
Project Cost for the above courses will be Rs 16,000 (INR) to Rs 20,000 (INR)
( including Consultation Fees)
Ph.D - Electrical Engineering - Power Systems Engineering - Power Electronics
- Project Cost Rs 3,50,000 to Rs 4,20,000
Ph.D - Electronics and Instrumentation Engineering, Control systems
- Project Cost Rs 3,50,000 to Rs 4,20,000
You’ll never believe the crazy tricks top mobile app developers have learned in order to make their mobile apps run faster and leaner. Embarrassing performance mistakes are much more common that than you might believe…but are insanely easy to fix. In our talk, we’ll walk through five simple steps that will help ensure your mobile application is the fastest, most efficient application on mobile devices today. You’ll walk out of our session with the tools to quickly identify the issues and the knowledge to solve them. It’ll be the most useful talk you’ll hear all day!
A Novel Methodology for Designing Linear Phase IIR FiltersIDES Editor
This paper presents a novel technique for
designing an Infinite Impulse Response (IIR) Filter with
Linear Phase Response. The design of IIR filter is always a
challenging task due to the reason that a Linear Phase
Response is not realizable in this kind. The conventional
techniques involve large number of samples and higher
order filter for better approximation resulting in complex
hardware for implementing the same. In addition, an
extensive computational resource for obtaining the inverse
of huge matrices is required. However, we propose a
technique, which uses the frequency domain sampling along
with the linear programming concept to achieve a filter
design, which gives a best approximation for the linear
phase response. The proposed method can give the closest
response with less number of samples (only 10) and is
computationally simple. We have presented the filter design
along with its formulation and solving methodology.
Numerical results are used to substantiate the efficiency of
the proposed method.
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.
A tale of scale & speed: How the US Navy is enabling software delivery from l...sonjaschweigert1
Rapid and secure feature delivery is a goal across every application team and every branch of the DoD. The Navy’s DevSecOps platform, Party Barge, has achieved:
- Reduction in onboarding time from 5 weeks to 1 day
- Improved developer experience and productivity through actionable findings and reduction of false positives
- Maintenance of superior security standards and inherent policy enforcement with Authorization to Operate (ATO)
Development teams can ship efficiently and ensure applications are cyber ready for Navy Authorizing Officials (AOs). In this webinar, Sigma Defense and Anchore will give attendees a look behind the scenes and demo secure pipeline automation and security artifacts that speed up application ATO and time to production.
We will cover:
- How to remove silos in DevSecOps
- How to build efficient development pipeline roles and component templates
- How to deliver security artifacts that matter for ATO’s (SBOMs, vulnerability reports, and policy evidence)
- How to streamline operations with automated policy checks on container images
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
Clients don’t know what they don’t know. What web solutions are right for them? How does WordPress come into the picture? How do you make sure you understand scope and timeline? What do you do if sometime changes?
All these questions and more will be explored as we talk about matching clients’ needs with what your agency offers without pulling teeth or pulling your hair out. Practical tips, and strategies for successful relationship building that leads to closing the deal.
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdfPaige Cruz
Monitoring and observability aren’t traditionally found in software curriculums and many of us cobble this knowledge together from whatever vendor or ecosystem we were first introduced to and whatever is a part of your current company’s observability stack.
While the dev and ops silo continues to crumble….many organizations still relegate monitoring & observability as the purview of ops, infra and SRE teams. This is a mistake - achieving a highly observable system requires collaboration up and down the stack.
I, a former op, would like to extend an invitation to all application developers to join the observability party will share these foundational concepts to build on:
Essentials of Automations: The Art of Triggers and Actions in FMESafe Software
In this second installment of our Essentials of Automations webinar series, we’ll explore the landscape of triggers and actions, guiding you through the nuances of authoring and adapting workspaces for seamless automations. Gain an understanding of the full spectrum of triggers and actions available in FME, empowering you to enhance your workspaces for efficient automation.
We’ll kick things off by showcasing the most commonly used event-based triggers, introducing you to various automation workflows like manual triggers, schedules, directory watchers, and more. Plus, see how these elements play out in real scenarios.
Whether you’re tweaking your current setup or building from the ground up, this session will arm you with the tools and insights needed to transform your FME usage into a powerhouse of productivity. Join us to discover effective strategies that simplify complex processes, enhancing your productivity and transforming your data management practices with FME. Let’s turn complexity into clarity and make your workspaces work wonders!
GridMate - End to end testing is a critical piece to ensure quality and avoid...ThomasParaiso2
End to end testing is a critical piece to ensure quality and avoid regressions. In this session, we share our journey building an E2E testing pipeline for GridMate components (LWC and Aura) using Cypress, JSForce, FakerJS…
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
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/
In his public lecture, Christian Timmerer provides insights into the fascinating history of video streaming, starting from its humble beginnings before YouTube to the groundbreaking technologies that now dominate platforms like Netflix and ORF ON. Timmerer also presents provocative contributions of his own that have significantly influenced the industry. He concludes by looking at future challenges and invites the audience to join in a discussion.
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.
Why You Should Replace Windows 11 with Nitrux Linux 3.5.0 for enhanced perfor...SOFTTECHHUB
The choice of an operating system plays a pivotal role in shaping our computing experience. For decades, Microsoft's Windows has dominated the market, offering a familiar and widely adopted platform for personal and professional use. However, as technological advancements continue to push the boundaries of innovation, alternative operating systems have emerged, challenging the status quo and offering users a fresh perspective on computing.
One such alternative that has garnered significant attention and acclaim is Nitrux Linux 3.5.0, a sleek, powerful, and user-friendly Linux distribution that promises to redefine the way we interact with our devices. With its focus on performance, security, and customization, Nitrux Linux presents a compelling case for those seeking to break free from the constraints of proprietary software and embrace the freedom and flexibility of open-source computing.
Threats to mobile devices are more prevalent and increasing in scope and complexity. Users of mobile devices desire to take full advantage of the features
available on those devices, but many of the features provide convenience and capability but sacrifice security. This best practices guide outlines steps the users can take to better protect personal devices and information.
National Security Agency - NSA mobile device best practices
Digital fiiter
1. EXPERT SYSTEMS AND SOLUTIONS
Email: expertsyssol@gmail.com
expertsyssol@yahoo.com
Cell: 9952749533
www.researchprojects.info
PAIYANOOR, OMR, CHENNAI
Call For Research Projects Final
year students of B.E in EEE, ECE,
EI, M.E (Power Systems), M.E
(Applied Electronics), M.E (Power
Electronics)
Ph.D Electrical and Electronics.
Students can assemble their hardware in our
Research labs. Experts will be guiding the
projects.
3. Introduction
What is a digital filter
A filter is a system that is designed to remove some
component or modify some characteristic of a
signal
A digital filter is a discrete-time LTI system which
can process the discrete-time signal.
There are various structures for the implementation
of digital filters
The actual implementation of an LTI digital filter can
be either in software or hardware form, depending
on applications
4. Introduction
Basic elements of digital filter structures
Adder has two inputs and one output.
Multiplier (gain) has single-input, single-output.
Delay element delays the signal passing through it by
one sample. It is implemented by using a shift register.
a
z-1
a z-1
5. x (n) b0 y(n)
1 a2 2
z-1
z-1
a1
a1
5 3
z-1
x (n) y(n)
0 b a2 z-1
4
y( n) = b0 x ( n) + a1 y( n − 1) + a 2 y( n − 2)
w 2 ( n) = y( n)
w 3 ( n) = w 2 ( n − 1) = y( n − 1)
w 4 ( n) = w 3 ( n − 1) = y( n − 2)
w 5 ( n) = a1 w 3 ( n) + a 2 w4 ( n) = a1 y( n − 1) + a 2 y( n − 2)
w1 ( n) = b0 x( n) + w5 ( n) = b0 x ( n) + a1 y( n − 1) + a 2 y( n − 2)
6. Introduction
The major factors that influence the choice of a
specific structure
Computational complexity
refers to the number of arithmetic operations (multiplications,
divisions, and additions) required to compute an output value y(n)
for the system.
Memory requirements
refers to the number of memory locations required to store the
system parameters, past inputs, past outputs, and any
intermediate computed values.
Finite-word-length effects in the computations
refers to the quantization effects that are inherent in any digital
implementation of the system, either in hardware or in software.
7. IIR Filter Structures
The characteristics of the IIR filter
IIR filters have Infinite-duration Impulse Responses
The system function H(z) has poles in 0 <| z |< ∞
An IIR filter is a recursive system
M
Y (z) ∑ bk z − k
b0 + b1 z −1 + + bM z − M
H (z) = = k =0
=
X (z) N
1 − (a1 z −1 + + a N z − N )
1 − ∑ ak z − k
k =1
N M
y( n) = ∑ a k y( n − k ) + ∑ bk x( n − k )
k =1 k =0
The order of such an IIR filter is called N if aN≠0
8. IIR Filter Structures
Direct form
In this form the difference equation is implemented
directly as given. There are two parts to this filter,
namely the moving average part and the recursive
part (or the numerator and denominator parts).
Therefore this implementation leads to two versions:
direct form I and direct form II structures
M
M N ∑ bk z − k
y( n) = ∑ bk x ( n − k ) + ∑ a k y( n − k ) H ( z ) = k =0
N
k =0 k =1
1 − ∑ ak z − k
k =1
9. y(n)
b 0
z-1 a1 x (n)
y( n − 1)
b1 z-1
z-1 a2 x ( n − 1)
y ( n − 2) z-1
b 2
x ( n − 2)
aN-1
y( n − N + 1) bM-1
x ( n − M + 1)
z-1 aN
b
z-1
y( n − N ) M
y2 ( n) y1 ( n) x( n − M )
k =1 k =0
y( n) = ∑ bk x ( n − k ) + ∑ a k y( n − k )
N M
Direct form I
10. Direct form II
For an LTI cascade system, we can change the order
of the systems without changing the overall system
response
x (n) y(n)
bM
z-1
bM-1 a1 z-1
a2 z-1
b2 z
-1
b1 z
-1
b0
aN-1 z-1
aN
11. IIR Filter Structures
Cascade form
In this form the system function H(z) is written as a
product of second-order sections with real coefficients
M M1 M2
∑ bk z − k ∏ (1 − pk z −1 )∏ (1 − qk z −1 )(1 − qk z −1 )
∗
H (z) = k =0
N
=A k =1
N1
k =1
N2
1 − ∑ ak z −k ∏ (1 − c k z −1 )∏ (1 − d k z −1 )(1 − d k z −1 )
∗
k =1 k =1 k =1
M1 M2
M = M1 + 2M 2
∏ (1 − p z )∏ (1 + b
k
−1
1k
−1
z + b2 k z ) −2
N = N1 + 2N 2
H (z) = A k =1
N1
k =1
N2
∏ (1 − c k z −1 )∏ (1 − a1k z −1 − a 2 k z − 2 )
k =1 k =1
12. IIR Filter Structures
1 + b1k z −1 + b2 k z −2
H ( z ) = A∏ −1 −2
= A∏ H k ( z )
k 1 − a1 k z − a2k z k
Each second-order section (called biquads) is implemented in
a direct form ,and the entire system function is implemented
as a cascade of biquads.
If N=M, there are totally N + 1 biquads.
2
If N>M, some of the biquads have numerator coefficients
that are zero, that is, either b2k = 0 or b1k = 0 or both b2k
= b1k = 0 for some k.
if N>M and N is odd, one of the biquads must have ak2 = 0,
so that this biquad become a first-order section.
13. biquad
−1 −2
1 + b1k z + b2 k z
a1k z-1 b1k H k (z) =
1 − a1k z −1 − a 2 k z − 2
a2k z-1 b2k
First-order section
a1k z-1 b1k a1k z-1 b1k
a2k z-1
a1k z-1 a1k z-1
a2k z-1
14. 8 − 4 z −1 + 11z −2 − 2 z −3
Example H (z) =
5 3 1
1 − z −1 + z − 2 − z − 3
4 4 8
( 2 − 0.3799 z −1 )(4 − 1.2402 z −1 + 5.2644 z −2 )
H ( z )=
(1 − 0.25 z −1 )(1 − z −1 + 0.5 z − 2 )
x (n) 4 2 y(n)
z-1 -1.2402
0.25
z-1 -0.3799
-0.5 z-1 5.2644
15. IIR Filter Structures
Parallel form
In this form the system function H(z) is written as a sum
of sections using partial fraction expansion. Each section
is implemented in a direct form. The entire system
function is implemented as a parallel of every section.
Suppose M=N
M
∑ bk z − k N1
Ak N2
b0 k + b1k z −1
H (z) = k =0
= G0 + ∑ −1
+∑ −1
N
k =1 1 − c k z k =1 1 − a1 k z − a2k z −2
1 − ∑ ak z −k
k =1
N = N 1 + 2N 2
16. IIR Filter Structures
N +1
2
b0 k + b1k z −1
H ( z ) = G0 + ∑
k =1 1 − a1k z −1 − a 2 k z −1
if N is odd, the system has one first-order
N −1
section and 2 second-order sections.
N
if N is even, the system has 2 second-order
sections.
17. Example
1 −1 2
10(1 − z )(1 − z −1 )(1 + 2 z −1 )
H (z) = 2 3
(1 − z −1 )(1 − z −1 )1 − ( + j ) z −1 1 − ( − j ) z −1
3 1 1 1 1 1
4 8
2 2
2 2
A1 A2 A3 A4
H (z) = + + +
3 −1 1 −1 1 1 −1 1 1 −1
(1 − z ) (1 − z ) 1 − ( + j ) z 1 − ( − j )z
4 8 2 2 2 2
A1 = 2.93, A2 = −17.68, A3 = 12.25 − j14.57, A4 = 12.25 + j14.57
− 14.75 − 12.90 z −1 24.50 + 26.82 z −1
H (z) = +
7 −1 3 − 2 −1 1 −2
1− z + z 1− z + z
8 32 2
18. − 14.75 − 12.90 z −1 24.50 + 26.82 z −1
H (z) = +
7 −1 3 − 2 −1 1 −2
1− z + z 1− z + z
8 32 2
-14.75
7/8 z-1 -12.9
x (n) y(n)
-3/32 z-1
24. 5
1 z-1 26.82
-1/2 z-1
19. IIR Filter Structures
Transposition theorem
If we reverse the directions of all branch transmittances
and interchange the input and output in the flow graph,
the system function remains unchanged.
The resulting structure is called a transposed structure
or a transposed form.
20. FIR Filter Structures
The characteristics of the FIR filter
FIR filters have Finite-duration Impulse Responses,
thus they can be realized by means of DFT
The system function H(z) has the ROC of | z |> 0 ,
thus it is a causal system
An FIR filter is a nonrecursive system
FIR filters can be designed to have a linear-phase
response
N −1 It has N-1 order poles at z = 0
H ( z ) = ∑ h( n) z −n
and N-1 zeros in | z |> 0
n=0
The order of such an FIR filter is N-1
21. FIR Filter Structures
Direct form
In this form the difference equation is implemented
directly as given. N −1
y( n) = ∑ h( m ) x( n − m )
m =0
z-1 z-1 z-1
x (n)
h(0) h(1) h(2) h(N-2) h(N-1)
y(n)
It requires N multiplications
22. FIR Filter Structures
Cascade form
In this form the system function H(z) is converted into
products of second-order sections with real coefficients
N It requires (3N/2) multiplications
N −1 2
H ( z ) = ∑ h( n) z −n
= ∏ (b0 k + b1k z + b2 k z )
−1 −2
n=0 k =1
x (n) b01 b02 b0x
y(n)
z-1 b11 z-1 b12 z-1 b1x
z-1 b21 z-1 b22 z-1 b2x
23. FIR Filter Structures
Linear-phase form
Linear phase:
The phase response is a linear function of frequency
Linear-phase condition
h( n) = h( N − 1 − n) Symmetric impulse response
h( n) = − h( N − 1 − n) Antisymmetric impulse response
When an FIR filter has a linear phase response, its
impulse response exhibits the above symmetry
conditions. In this form we exploit these symmetry
relations to reduce multiplications by about half.
24. If N is odd
N −1
H ( z ) = ∑ h( n) z −n
n=0
N −1
−1
N −1 N −1
2
N −1 −
= ∑ h(n)z
n=0
−n
+ h(
2
)z 2
+ ∑ h(n)z
N −1
−n
n= +1
2
N −1
−1
N −1
−1 let n = N − 1 − m
N −1
2
N −1 − 2
= ∑ h( n) z − n + h(
n=0 2
)z 2
+ ∑ h( N − 1 − m ) z −( N −1− m )
m =0
N −1
−1 let n ← m
2
N − 1 − N2−1
= ∑ h( n)[ z − n ± z −( N −1− n ) ] + h( )z
n=0 2
h( n) = ± h( N − 1 − n)
25. If N is odd
N −1
−1
N −1
2
N −1 −
H (z) = ∑ h(n)[ z
n=0
−n
±z − ( N −1− n )
] + h(
2
)z 2
x (n)
z-1 z-1 z-1
±1 ±1 ±1 ±1
z-1
z-1
z-1
h(0) h(1) h( 2) N −1 N −1
h( − 1) h( )
2 2
y(n)
1 for symmetric
-1 for antisymmetric
26. If N is even
N
−1
N −1 2 N −1
H ( z ) = ∑ h( n) z − n = ∑ h( n) z − n + ∑ h( n) z − n
n=0 n=0 N
n=
2
N
2
−1
N
2
−1 let n = N − 1 − m
= ∑ h( n) z − n + ∑ h( N − 1 − m ) z −( N −1− m )
n=0 m =0
N
−1 let n ← m
2
= ∑ h( n)[ z − n ± z −( N −1− n ) ]
n=0
h( n) = ± h( N − 1 − n)
27. N
If N is even 2
−1
H (z) = ∑ h( n)[ z − n ± z −( N −1− n ) ]
n=0
x (n) z-1
z-1 z-1
±1 ±1 ±1 ±1 ±1 z-1
z-1 z-1 z-1
h(0) h(1) h( 2) N N
h( − 2) h( − 1)
2 2
y(n)
1 for symmetric
-1 for antisymmetric
The linear-phase filter structure requires 50% fewer
multiplications than the direct form.
28. FIR Filter Structures
Frequency sampling form
This structure is based on the DFT of the impulse
response h(n) and leads to a parallel structure. It is also
suitable for a design technique based on the sampling
of frequency response H(z)
N −1 N −1
1 H (k ) 1
H ( z ) = (1 − z − N ) ∑ − k −1
= H c ( z )∑ H k ( z )
′
N k =0 1 − W N z N k =0
29. Hc (z) = 1 − z − N
− z−N
It has N equally spaced j
2π
k
zeros on the unit circle zk = e N
, k = 0,1, , N − 1
ωN
jω − jω N −j ωN
H c (e ) = 1 − e = 2 je 2
sin( )
2
ωN
H c (e jω ) = 2 sin( ) All-zero filter
2
H c ( e jω ) or comb filter
2
2π 4π
ω
0
N N
30. N −1 N −1
H (k )
∑ H k′ ( z ) = ∑ 1 − W −k z −1
k =0 k =0
resonant filter
N
It has N equally spaced poles on the unit circle
2π
j k
zk = e N
, k = 0,1, , N − 1
The pole locations are identical to the zero locations and that
2π
both occur at ω= k , which are the frequencies at which
N
the designed frequency response is specified. The gains of
the filter at these frequencies are simply the complex-valued
parameters H (k )
31. N −1 N −1
1 H (k ) 1
H ( z ) = (1 − z −N
)
N ∑ 1 − W −k z −1 = N H c ( z ) ∑ H k′ ( z )
k =0 k =0
N
H ( 0)
0
WN z-1
H (1)
x (n) y(n)
−
WN1 z-1
− z−N
H ( N − 1)
W N ( N −1) z-1
−
32. FIR Filter Structures
Problems
It requires a complex arithmetic implementation
It is possible to obtain an alternate realization in which only
real arithmetic is used. This realization is derived using the
−
symmetric properties of the DFT and the W N k factor.
It has poles on the unit circle, which makes this filter
critically unstable
We can avoid this problem by sampling H(z) on a circle |z|
=r where the radius r is very close to one but is less than
one.
33. jIm[z]
− N N −1
1− z H (k )
∑ 1 − W − k z −1
H (k )
H (z) = unit circle
N k =0 N
r
Re[z]
N −1
1 − r N z−N H r (k )
H (z) =
N ∑ 1 − rW − k z −1
k =0
H r (k )
N
H r (k ) ≈ H (k )
− N N −1
1− r z N
H (k )
H (z) ≈
N ∑ 1 − rW − k z −1
k =0 N
34. N −1
1 − r N z−N H (k )
H (z) ≈
N ∑ 1 − rW − k z −1
k =0 N
−
By using the symmetric properties of the DFT and the W N k
factor, a pair of single-pole filters can be combined to form
a single two-pole filter with real-valued parameters.
2π
j k
For poles zk = e N
, k = 0,1, , N − 1
∗
z N −k = zk
2π 2π
−( N − k ) j ( N −k ) j k ∗ −k ∗
rW N = re N
= r (e N
) = r (W N ) = rW k
N
h(n) is a real-valued sequence, so
∗
H ( k ) = H (( N − k )) N RN ( k )
35. So the kth and (N-k)th resonant filters can be combined to form
a second-order section H k (z ) with real-valued coefficients
H (k ) H(N − k)
H k (z) = − k −1
+
1 − rW N z 1 − rW N ( N − k ) z −1
−
H (k ) ∗
H (k ) b0 k + b1k z −1
= + =
− k −1
1 − rW N z k −1
1 − rW N z −1 2π 2 −2
1 − z 2r cos( k ) + r z
N
k = 1,2, , N − 1 , N is odd
2 b0 k = 2 Re[ H ( k )]
N
k = 1,2, , − 1, N is even b1k = −2r Re[ H ( k )W N ]
k
2
36. N is even jIm[z] N is odd jIm[z]
|z|=r |z|=r
k=N k=0 k=N k=0
2 2
Re[z] Re[z]
If N is even, the filter has a pair of real-valued poles z = ±r
H ( 0) H(N )
H 0 (z) = H N (z) = 2
1 − rz −1 2 1 + rz −1
If N is odd, the filter has a single real-valued pole z=r
H ( 0)
H 0 (z) = −1
1 − rz
37. second-order section
b0k
2π z-1 b1k
2r cos( k)
N
− r2 z-1
first-order sections
H0 (z) H N (z)
2
H ( 0) H(N )
2
r z-1 r z-1
38. 1 −r z
N −N N −1
2
N is even H (z) = H 0 ( z ) + H N ( z ) + ∑ H k ( z )
N 2 k =1
−N ( N −1 )
N is odd 1 −r z N 2
H 0 ( z ) + ∑H k ( z )
H (z) =
N k =1
H0 (z)
1
H1 ( z)
N
x (n) y(n)
H k (z )
− r N z−N
H N (z)
2
39. FIR Filter Structures
Fast convolution form
x(n): N1-point sequence
h(n): N2-point sequence
L ≥ N1 + N2 - 1
x (n) X (k )
L-point
DFT
Y (k ) y(n)
L-point
IDFT
h(n)
L-point
DFT
H (k )
直接 II 型比直接 I 型节省存储单元(软件实现),或节省寄存器(硬件实现) 但不论是直接 I 型还是直接 II 型,其共同的缺点是系数对滤波器的性能控制作用不明显,这是因为它们与系统函数的零极点关系不明确因而调整困难。 另外,这两种结构极点对系数的变化过于敏感,从而使系统频率响应对系数的变化过于灵敏,也就是对有限精度(有限字长)运算过于灵敏,容易出现不稳定或产生较大误差。