This document discusses Wiener filters and linear optimum filtering. It introduces Wiener filters, the mean square error (MSE) criterion, and derives the Wiener-Hopf equations which describe the optimum filter coefficients that minimize the MSE. It also discusses properties of the error performance surface including its canonical quadratic form and the minimum MSE value. Applications to channel equalization and linearly constrained minimum variance filtering are also covered.
In communication system, intersymbol interference (ISI) is a form of distortion of a signal in which one symbol interferes with subsequent symbols. This is an unwanted phenomenon as the previous symbols have similar effect as noise, thus making the communication less reliable.
In communication system, the Nyquist ISI criterion describes the conditions which when satisfied by a communication channel (including responses of transmit and receive filters), result in no intersymbol interference(ISI). It provides a method for constructing band-limited functions to overcome the effects of intersymbol interference.
the presentation consists of a brief description about ADAPTIVE LINEAR EQUALIZER , its classification and the associated attributes of ZERO FORCING EQUALIZER and MMSE EQUALIZER
In communication system, intersymbol interference (ISI) is a form of distortion of a signal in which one symbol interferes with subsequent symbols. This is an unwanted phenomenon as the previous symbols have similar effect as noise, thus making the communication less reliable.
In communication system, the Nyquist ISI criterion describes the conditions which when satisfied by a communication channel (including responses of transmit and receive filters), result in no intersymbol interference(ISI). It provides a method for constructing band-limited functions to overcome the effects of intersymbol interference.
the presentation consists of a brief description about ADAPTIVE LINEAR EQUALIZER , its classification and the associated attributes of ZERO FORCING EQUALIZER and MMSE EQUALIZER
Salient Features:
The magnitude response is nearly constant(equal to 1) at lower frequencies
There are no ripples in passband and stop band
The maximum gain occurs at Ω=0 and it is H(Ω)=1
The magnitude response is monotonically decreasing
As the order of the filter ‘N’ increases, the response of the filter is more close to the ideal 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.
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.
Salient Features:
The magnitude response is nearly constant(equal to 1) at lower frequencies
There are no ripples in passband and stop band
The maximum gain occurs at Ω=0 and it is H(Ω)=1
The magnitude response is monotonically decreasing
As the order of the filter ‘N’ increases, the response of the filter is more close to the ideal 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.
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.
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It is sometimes desirable to have circuits capable of selectively filtering one frequency or range of frequencies out of a mix of different frequencies in a circuit. A circuit designed to perform this frequency selection is called a filter circuit, or simply a filter. A common need for filter circuits is in high-performance stereo systems, where certain ranges of audio frequencies need to be amplified or suppressed for best sound quality and power efficiency. You may be familiar with equalizers, which allow the amplitudes of several frequency ranges to be adjusted to suit the listener's taste and acoustic properties of the listening area. You may also be familiar with crossover networks, which block certain ranges of frequencies from reaching speakers. A tweeter (high-frequency speaker) is inefficient at reproducing low-frequency signals such as drum beats, so a crossover circuit is connected between the tweeter and the stereo's output terminals to block low-frequency signals, only passing high-frequency signals to the speaker's connection terminals. This gives better audio system efficiency and thus better performance. Both equalizers and crossover networks are examples of filters, designed to accomplish filtering of certain frequencies.
I am Lawrence B. I am a Signal Processing Assignment Expert at matlabassignmentexperts.com. I hold a Masters's in Matlab from, Durham University, UK. I have been helping students with their assignments for the past 5 years. I solve assignments related to Signal Processing.
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Echo Cancellation Algorithms using Adaptive Filters: A Comparative Studyidescitation
An adaptive filter is a filter that self-adjusts its transfer function according to an
optimization algorithm driven by an error signal. Adaptive filter finds its essence in
applications such as echo cancellation, noise cancellation, system identification and many
others. This paper briefly discusses LMS, NLMS and RLS adaptive filter algorithms for
echo cancellation. For the analysis, an acoustic echo canceller is built using LMS, NLMS
and RLS algorithms and the echo cancelled samples are studied using Spectrogram. The
analysis is further extended with its cross-correlation and ERLE (Echo Return Loss
Enhancement) results. Finally, this paper concludes with a better adaptive filter algorithm
for Echo cancellation. The implementation and analysis is done using MATLAB®,
SIMULINK® and SPECTROGRAM V5.0®.
On The Fundamental Aspects of DemodulationCSCJournals
When the instantaneous amplitude, phase and frequency of a carrier wave are modulated with the information signal for transmission, it is known that the receiver works on the basis of the received signal and a knowledge of the carrier frequency. The question is: If the receiver does not have the a priori information about the carrier frequency, is it possible to carry out the demodulation process? This tutorial lecture answers this question by looking into the very fundamental process by which the modulated wave is generated. It critically looks into the energy separation algorithm for signal analysis and suggests modification for distortionless demodulation of an FM signal, and recovery of sub-carrier signals
Images may contain different types of noises. Removing noise from image is often the first step in image processing, and remains a challenging problem in spite of sophistication of recent research. This ppt presents an efficient image denoising scheme and their reconstruction based on Discrete Wavelet Transform (DWT) and Inverse Discrete Wavelet Transform (IDWT).
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.
Courier management system project report.pdfKamal Acharya
It is now-a-days very important for the people to send or receive articles like imported furniture, electronic items, gifts, business goods and the like. People depend vastly on different transport systems which mostly use the manual way of receiving and delivering the articles. There is no way to track the articles till they are received and there is no way to let the customer know what happened in transit, once he booked some articles. In such a situation, we need a system which completely computerizes the cargo activities including time to time tracking of the articles sent. This need is fulfilled by Courier Management System software which is online software for the cargo management people that enables them to receive the goods from a source and send them to a required destination and track their status from time to time.
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.
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.
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.
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
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/
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptx
Wiener filters
1. Week 3 ELE 774 - Adaptive Signal Processing 1
WIENER FILTERS
2. ELE 774 - Adaptive Signal Processing2Week 3
Complex-valued stationary (at least w.s.s.) stochastic processes.
Linear discrete-time filter, w0, w1, w2, ... (IIR or FIR (inherently stable))
y(n) is the estimate of the desired response d(n)
e(n) is the estimation error, i.e., difference bw. the filter output and the
desired response
Linear Optimum Filtering: Statement
3. ELE 774 - Adaptive Signal Processing3Week 3
Linear Optimum Filtering: Statement
Problem statement:
Given
Filter input, u(n),
Desired response, d(n),
Find the optimum filter coefficients, w(n)
To make the estimation error “as small as possible”
How?
An optimization problem.
4. ELE 774 - Adaptive Signal Processing4Week 3
Linear Optimum Filtering: Statement
Optimization (minimization) criterion:
1. Expectation of the absolute value,
2. Expectation (mean) square value,
3. Expectation of higher powers of the absolute value
of the estimation error.
Minimization of the Mean Square value of the Error (MSE) is
mathematically tractable.
Problem becomes:
Design a linear discrete-time filter whose output y(n) provides an
estimate of a desired response d(n), given a set of input samples
u(0), u(1), u(2) ..., such that the mean-square value of the
estimation error e(n), defined as the difference between the
desired response d(n) and the actual response, is minimized.
5. ELE 774 - Adaptive Signal Processing5Week 3
Principle of Orthogonality
Filter output is the convolution of the filter IR and the input
6. ELE 774 - Adaptive Signal Processing6Week 3
Principle of Orthogonality
Error:
MSE (Mean-Square Error) criterion:
Square → Quadratic Func. → Convex Func.
Minimum is attained when
(Gradient w.r.t. optimization variable
w is zero.)
7. ELE 774 - Adaptive Signal Processing7Week 3
Derivative in complex variables
Let
then derivation w.r.t. wk is
Hence
or
!!! J: real, why? !!!
8. ELE 774 - Adaptive Signal Processing8Week 3
Principle of Orthogonality
Partial derivative of J is
Using and
Hence
9. ELE 774 - Adaptive Signal Processing9Week 3
Principle of Orthogonality
Since , or
The necessary and sufficient condition for the cost function J to
attain its minimum value is, for the corresponding value of the
estimation error eo(n) to be orthogonal to each input sample that
enters into the estimation of the desired response at time n.
Error at the minimum is uncorrelated with the filter input!
A good basis for testing whether the linear filter is operating in its
optimum condition.
10. ELE 774 - Adaptive Signal Processing10Week 3
Principle of Orthogonality
Corollary:
If the filter is operating in optimum conditions (in the MSE sense)
When the filter operates in its optimum condition, the estimate of the
desired response defined by the filter output yo(n) and the
corresponding estimation error eo(n) are orthogonal to each other.
11. ELE 774 - Adaptive Signal Processing11Week 3
Minimum Mean-Square Error
Let the estimate of the desired response that is optimized in the
MSE sense, depending on the inputs which span the space
i.e. ( ) be
Then the error in optimal conditions is
or
Also let the minimum MSE be (≠0)
HW: try to derive this
relation from the corollary.
12. ELE 774 - Adaptive Signal Processing12Week 3
Minimum Mean-Square Error
Normalized MSE: Let
Meaning
If ε is zero, the optimum filter operates perfectly, in the sense that
there is complete agreement bw. d(n) and . (Optimum case)
If ε is unity, there is no agreement whatsoever bw. d(n) and
(Worst case)
13. ELE 774 - Adaptive Signal Processing13Week 3
Wiener-Hopf Equations
We have (principle of orthogonality)
Rearranging
where
Wiener-Hopf
Equations
(set of
infinite eqn.s)
14. ELE 774 - Adaptive Signal Processing14Week 3
Wiener-Hopf Equations
Solution – Linear Transversal (FIR) Filter case
M simultaneous equations
15. ELE 774 - Adaptive Signal Processing15Week 3
Wiener-Hopf Equations (Matrix Form)
Let
Then
and
16. ELE 774 - Adaptive Signal Processing16Week 3
Wiener-Hopf Equations (Matrix Form)
Then the Wiener-Hopf equations can be written as
where
is composed of the optimum (FIR) filter coefficients.
The solution is found to be
Note that R is almost always positive-definite.
17. ELE 774 - Adaptive Signal Processing17Week 3
Substitute →
Rewriting
Error-Performance Surface
18. ELE 774 - Adaptive Signal Processing18Week 3
Error-Performance Surface
Quadratic function of the filter coefficients → convex function, then
or
Wiener-Hopf
Equations
19. ELE 774 - Adaptive Signal Processing19Week 3
Minimum value of Mean-Square Error
We calculated that
The estimate of the desired response is
Hence its variance is
Then
At wo.
(Jmin is independent of w)
20. ELE 774 - Adaptive Signal Processing20Week 3
Canonical Form of the Error-Performance Surface
Rewrite the cost function in matrix form
Next, express J(w) as a perfect square in w
Then, by substituting
In other words,
21. ELE 774 - Adaptive Signal Processing21Week 3
Canonical Form of the Error-Performance Surface
Observations:
J(w) is quadratic in w,
Minimum is attained at w=wo,
Jmin is bounded below, and is always a positive quantity,
Jmin>0 →
22. ELE 774 - Adaptive Signal Processing22Week 3
Canonical Form of the Error-Performance Surface
Transformations may significantly simplify the analysis,
Use Eigendecomposition for R
Then
Let
Substituting back into J
The transformed vector v is called as the principal axes of the
surface.
a vector
Canonical form
23. ELE 774 - Adaptive Signal Processing23Week 3
Canonical Form of the Error-Performance Surface
w1
w2
wo
J(wo)=Jmin
J(w)=c curve
v1
(λ1)
v2
(λ2)
Jmin
J(v)=c curve
Q
Transformation
24. ELE 774 - Adaptive Signal Processing24Week 3
Multiple Linear Regressor Model
Wiener Filter tries to match the filter coefficients to the model of the
desired response, d(n).
Desired response can be generated by
1. a linear model, a
2. with noisy observable data, d(n)
3. noise is additive and white.
Model order is m, i.e.
What should the length of the Wiener filter be to achive min. MSE?
25. ELE 774 - Adaptive Signal Processing25Week 3
Multiple Linear Regressor Model
The variance of the desired response is
But we know that
where wo is the filter optimized w.r.t. MSE (Wiener filter) of length M.
1. Underfitted model: M<m
Performance improves quadratically with increasing M.
Worst case: M=0,
2. Critically fitted model: M=m
wo=a, R=Rm,
26. ELE 774 - Adaptive Signal Processing26Week 3
Multiple Linear Regressor Model
3. Overfitted model: M>m
Filter longer than the model does not improve performance.
27. ELE 774 - Adaptive Signal Processing27Week 3
Example
Let
the model length of the desired response d(n) be 3,
the autocorrelation matrix of the input u(n) be (for conseq. 3 samples)
The cross-correlation of the input and the (observable) desired
response be
The variance of the observable data (desired response) be
The variance of the additive white noise be
We do not know the values
28. ELE 774 - Adaptive Signal Processing28Week 3
Example
Question:
a) Find Jmin for a (Wiener) filter length of M=1,2,3,4
b) Draw the error-performance (cost) surface for M=2
c) Compute the canonical form of the error-performance surface.
Solution:
a) we know that and then
29. ELE 774 - Adaptive Signal Processing29Week 3
Example
Solution, b)
30. ELE 774 - Adaptive Signal Processing30Week 3
Example
Solution, c) we know that
where for M=2
Then
v1
(λ1)
v2
(λ2)
Jmin
31. ELE 774 - Adaptive Signal Processing31Week 3
Application – Channel Equalization
Transmitted signal passes through the dispersive channel and a
corrupted version (both channel & noise) of x(n) arrives at the receiver.
Problem: Design a receiver filter so that we can obtain a delayed
version of the transmitted signal at its output.
Criterion: 1. Zero Forcing (ZF)
2. Minimum Mean Square Error (MMSE)
Filter, wChannel, h + +
Delay, δ
x(n) y(n)
x(n-δ)
ε(n)z(n)
-
32. ELE 774 - Adaptive Signal Processing32Week 3
Application – Channel Equalization
MMSE cost function is:
Filter output
Filter input
Convolution
Convolution
33. ELE 774 - Adaptive Signal Processing33Week 3
Application – Channel Equalization
Combine last two equations
Compact form of the filter output
Desired signal is x(n-δ), or
Convolution
Toeplitz matrix performs convolution
34. ELE 774 - Adaptive Signal Processing34Week 3
Application – Channel Equalization
Rewrite the MMSE cost function
Expanding (data and noise are uncorrelated E{x(n)v(k)}=0 for all n,k)
Re-expressing the expectations
35. ELE 774 - Adaptive Signal Processing35Week 3
Application – Channel Equalization
Quadratic function → gradient is zero at minimum
The solution is found as
And Jmin is
Jmin depends on the design parameter δ
36. ELE 774 - Adaptive Signal Processing36Week 3
Application – Linearly Constrained
Minimum - Variance Filter
Problem:
1. We want to design an FIR filter which suppresses all frequency
components of the filter input except ωo, with a gain of g at ωo.
37. ELE 774 - Adaptive Signal Processing37Week 3
Application – Linearly Constrained
Minimum - Variance Filter
Problem:
2. We want to design a beamformer which can resolve an
incident wave coming from angle θo (with a scaling factor g),
while at the same time suppress all other waves coming from
other directions.
38. ELE 774 - Adaptive Signal Processing38Week 3
Application – Linearly Constrained
Minimum - Variance Filter
Although these problems are physically different, they are
mathematically equivalent.
They can be expressed as follows:
Suppress all components (freq. ω or dir. θ) of a signal while
setting the gain of a certain component constant (ωo or θo)
They can be formulated as a constrained optimization problem:
Cost function: variance of all components (to be minimized)
Constraint (equality): the gain of a single component has to be g.
Observe that there is no desired response!.
40. ELE 774 - Adaptive Signal Processing40Week 3
Application – Linearly Constrained
Minimum - Variance Filter
Cost function: output power → quadratic → convex
Constraint : linear
Method of Lagrange multipliers can be utilized to solve the problem.
Solution: Set the gradient of J to zero
Optimum beamformer weights are found from the set of equations
similar to Wiener-Hopf equations.
output power constraint
41. ELE 774 - Adaptive Signal Processing41Week 3
Application – Linearly Constrained
Minimum - Variance Filter
Rewrite the equations in matrix form:
Hence
How to find λ? Use the linear constraint:
to find
Therefore the solution becomes
For θo, wo is
the linearly Constrained Minimum-Variance (LCMV) beamformer
For ωo, wo is
the linearly Constrained Minimum-Variance (LCMV) filter
42. ELE 774 - Adaptive Signal Processing42Week 3
Minimum-Variance Distortionless Response
Beamformer/Filter
Distortionless → set g=1, then
We can show that (HW)
Jmin represents an estimate of the variance of the signal impinging on
the antenna array along the direction θ0.
Generalize the result to any direction θ (angular frequency ω):
minimum-variance distortionless response (MVDR) spectrum
An estimate of the power of the signal coming from direction θ
An estimate of the power of the signal coming from frequency ω