Here are the key steps to estimate the Doppler frequency from radar returns to determine vehicle speed:
1. Demodulate the received radar signal to baseband to extract the complex envelope. This will produce a signal of the form s[n] = (A/2)exp(j2πFDnΔ + φ), where FD is the Doppler frequency related to vehicle speed.
2. Sample the complex envelope at a rate Fs higher than twice the maximum expected Doppler frequency to avoid aliasing.
3. Estimate the Doppler frequency FD from the sampled complex envelope using techniques like periodogram or autoregressive spectral estimation. The Doppler estimate allows calculating the vehicle speed as v = cFD/2F0.
4.
Basics of probability in statistical simulation and stochastic programmingSSA KPI
AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 2.
More info at http://summerschool.ssa.org.ua
The following resources come from the 2009/10 BEng (Hons) in Digital Communications & Electronics (course number 2ELE0064) from the University of Hertfordshire. All the mini projects are designed as level two modules of the undergraduate programmes.
The objective of this module is to have built communication links using existing AM modulation, PSK modulation and demodulation blocks, constructed AM modulators and constructed PSK modulators using operational function blocks based on their mathematical expressions, and conducted simulations of the links and modulators, all in Simulink®.
Probability and random processes project based learning template.pdfVedant Srivastava
To understand the concept of Monte –Carlo Method and its various applications and it rely on repeated and random sampling to obtain numerical result.
Developing the computational algorithms to solve the problem related to random sampling.
Objective also contains simulation of specific problem in Matlab Software.
Basics of probability in statistical simulation and stochastic programmingSSA KPI
AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 2.
More info at http://summerschool.ssa.org.ua
The following resources come from the 2009/10 BEng (Hons) in Digital Communications & Electronics (course number 2ELE0064) from the University of Hertfordshire. All the mini projects are designed as level two modules of the undergraduate programmes.
The objective of this module is to have built communication links using existing AM modulation, PSK modulation and demodulation blocks, constructed AM modulators and constructed PSK modulators using operational function blocks based on their mathematical expressions, and conducted simulations of the links and modulators, all in Simulink®.
Probability and random processes project based learning template.pdfVedant Srivastava
To understand the concept of Monte –Carlo Method and its various applications and it rely on repeated and random sampling to obtain numerical result.
Developing the computational algorithms to solve the problem related to random sampling.
Objective also contains simulation of specific problem in Matlab Software.
Earned Value Probabilistic Forecasting Using Monte Carlo SimulationRicardo Viana Vargas
The aim of this article is to present a proposal of interconnection between models and probabilistic simulations of project as possible ways to determine EAC (Final cost) through Earned Value Analysis. The article proves that the use of the 3 main models of projection (constant index, CPI and SCI) as the basis of a triangular probabilistic distribution that, through Monte Carlo simulation will permit associate and determine the probability according to the accomplishment of budgets and costs of the project.
#NoEstimates project planning using Monte Carlo simulationDimitar Bakardzhiev
Here is the text behind the slides http://www.infoq.com/articles/noestimates-monte-carlo
Here is a video I prepared in order to help people understand how to plan a release using the Monte Carlo simulation in MS Excel http://youtu.be/r38a25ak4co
And here is an Excel file to show how Monte Carlo is done http://modernmanagement.bg/data/NoEstimate_Project_Planning_MonteCarlo.xlsx
Here are the SIPs for the baseline project http://modernmanagement.bg/data/SIPs_MonteCarlo_FVR.xlsx
Here is the planing simulation in Excel http://modernmanagement.bg/data/High_Level_Project_Planning.xlsx
The video ( after the 3:00 minute) http://youtu.be/GE9vrJ741WY on how to use the Excel files
In a communications system, the channel is affected by an additive white Gaussian noise (AWGN)
and a fading due to a distance between a transmitter and a receiver. Especially, there are many kinds of
channel fadings. Depending on the moving speeds of transmitters or receivers, a fading type can be a slow
fading or a fast fading (i.e., the product of 0.1 and coherence time than smaller or larger than the symbol
period of signal are corresponding to fast and slow fadings). Moreover, a channel can be referred as a
selective fading or a flat fading corresponding to the product of 0.1 and coherence bandwidth than smaller
or larger than the bandwidth of signal. These above effects can suffer received signals at a destination.
Hence the performance of received signals in term of bit-error-rate (BER) is much degraded.
In order to overcome these issues, communications systems would be carefully designed. In detail,
application systems operating over the AWGN channels would use coding schemes to combat an additive
white noise. However, if environment is affected by fading, coding techniques only solve a fast fading.
It implies that, coding schemes degrade received signals when they go through slow fading channels. In
this case, an interleaving technique would be added to a communications system. In order to overcome
the fading channels, besides, using an interleaver as above, we can exploit the diversity of multi-path. It
implies that the effects of fading can be combated by transmitting the original signals over multiple paths
(experiencing independent fading) and then combining all received signals at the receiver. There are many
kinds of diversities to mitigate this issue, such as diversity in time, frequency, and space. Correspondingly,
a lot of state-of-art methods are given, viz. diversity receiving and transmitting, OFDM, space-time block
codes, MIMO, Cooperation and etc.
In summary, the main scope of this report is modeling a communications system. First, I create a
basic communications system, where it includes the modulation/demodulation using a QPSK modulation,
a channel type is an AWGN channel. Secondly, a coder/decoder scheme is added to a transmitter/receiver to
improve received signals. Thirdly, the fading channel is considered when a receiver/transmitter is moving.
It means that the slow fading is mentioned. The performance is shown to prove that the received signal
2
is degraded whether a coding scheme is used or not. Finally, an interleaver/deinterleaver is used to solve
this problem.
Besides, the performance in terms of BER is used to verify a validity of these above techniques in a
communications system.
MATLAB and Simulink for Communications System Design (Design Conference 2013)Analog Devices, Inc.
This session will show how Model-Based Design with MATLAB® and Simulink® can be used to model, simulate, and implement communications systems. Attendees will learn how multidomain modeling with continuous verification and automatic code generation can dramatically reduce system design time. A QPSK receiver model will be used as an example to highlight the design flow.
Earned Value Probabilistic Forecasting Using Monte Carlo SimulationRicardo Viana Vargas
The aim of this article is to present a proposal of interconnection between models and probabilistic simulations of project as possible ways to determine EAC (Final cost) through Earned Value Analysis. The article proves that the use of the 3 main models of projection (constant index, CPI and SCI) as the basis of a triangular probabilistic distribution that, through Monte Carlo simulation will permit associate and determine the probability according to the accomplishment of budgets and costs of the project.
#NoEstimates project planning using Monte Carlo simulationDimitar Bakardzhiev
Here is the text behind the slides http://www.infoq.com/articles/noestimates-monte-carlo
Here is a video I prepared in order to help people understand how to plan a release using the Monte Carlo simulation in MS Excel http://youtu.be/r38a25ak4co
And here is an Excel file to show how Monte Carlo is done http://modernmanagement.bg/data/NoEstimate_Project_Planning_MonteCarlo.xlsx
Here are the SIPs for the baseline project http://modernmanagement.bg/data/SIPs_MonteCarlo_FVR.xlsx
Here is the planing simulation in Excel http://modernmanagement.bg/data/High_Level_Project_Planning.xlsx
The video ( after the 3:00 minute) http://youtu.be/GE9vrJ741WY on how to use the Excel files
In a communications system, the channel is affected by an additive white Gaussian noise (AWGN)
and a fading due to a distance between a transmitter and a receiver. Especially, there are many kinds of
channel fadings. Depending on the moving speeds of transmitters or receivers, a fading type can be a slow
fading or a fast fading (i.e., the product of 0.1 and coherence time than smaller or larger than the symbol
period of signal are corresponding to fast and slow fadings). Moreover, a channel can be referred as a
selective fading or a flat fading corresponding to the product of 0.1 and coherence bandwidth than smaller
or larger than the bandwidth of signal. These above effects can suffer received signals at a destination.
Hence the performance of received signals in term of bit-error-rate (BER) is much degraded.
In order to overcome these issues, communications systems would be carefully designed. In detail,
application systems operating over the AWGN channels would use coding schemes to combat an additive
white noise. However, if environment is affected by fading, coding techniques only solve a fast fading.
It implies that, coding schemes degrade received signals when they go through slow fading channels. In
this case, an interleaving technique would be added to a communications system. In order to overcome
the fading channels, besides, using an interleaver as above, we can exploit the diversity of multi-path. It
implies that the effects of fading can be combated by transmitting the original signals over multiple paths
(experiencing independent fading) and then combining all received signals at the receiver. There are many
kinds of diversities to mitigate this issue, such as diversity in time, frequency, and space. Correspondingly,
a lot of state-of-art methods are given, viz. diversity receiving and transmitting, OFDM, space-time block
codes, MIMO, Cooperation and etc.
In summary, the main scope of this report is modeling a communications system. First, I create a
basic communications system, where it includes the modulation/demodulation using a QPSK modulation,
a channel type is an AWGN channel. Secondly, a coder/decoder scheme is added to a transmitter/receiver to
improve received signals. Thirdly, the fading channel is considered when a receiver/transmitter is moving.
It means that the slow fading is mentioned. The performance is shown to prove that the received signal
2
is degraded whether a coding scheme is used or not. Finally, an interleaver/deinterleaver is used to solve
this problem.
Besides, the performance in terms of BER is used to verify a validity of these above techniques in a
communications system.
MATLAB and Simulink for Communications System Design (Design Conference 2013)Analog Devices, Inc.
This session will show how Model-Based Design with MATLAB® and Simulink® can be used to model, simulate, and implement communications systems. Attendees will learn how multidomain modeling with continuous verification and automatic code generation can dramatically reduce system design time. A QPSK receiver model will be used as an example to highlight the design flow.
ATI Courses Professional Development Short Course Applied Measurement Engin...Jim Jenkins
How do you know your test measurements are valid? Since NIST traceability actually guarantees little about your test data, how do you know? Could you prove validity to your customer? What is the right measurements solution for your testing requirements? Is it really as simple as the vendors say? What is your real cost of invalid, ambiguous data causing retest or, worst of all, hardware redesign?
This course is for engineers, scientists, and managers who must use systems to understand experimental test measurements on a daily basis. Learn how to design, buy and operate effective automated measurement systems providing demonstrably valid test data, the first time.
Fundamental & underlying engineering principles governing the design and operation of effective automated systems are demonstrated experimentally.
Welcome to MatlabAssignmentExperts, where we understand the importance of tight budgets. We empathize with the financial challenges you may be facing, which is why we are thrilled to offer you an exclusive solution. Our commitment to your success is unwavering, and to ensure that you can afford our exceptional services, we are offering an incredible 20% discount. This limited-time offer allows you to save your hard-earned money while still enjoying top-notch assistance. Don't let budget constraints hinder your academic progress. Visit our website now, avail this remarkable discount, and experience the best in Matlab assignment help. Your savings are our priority!
Visit us at www.matlabassignmentexperts.com
Email: info@matlabassignmentexperts.com
WhatsApp: +1(315)557-6473
Slack (or Teams) Automation for Bonterra Impact Management (fka Social Soluti...Jeffrey Haguewood
Sidekick Solutions uses Bonterra Impact Management (fka Social Solutions Apricot) and automation solutions to integrate data for business workflows.
We believe integration and automation are essential to user experience and the promise of efficient work through technology. Automation is the critical ingredient to realizing that full vision. We develop integration products and services for Bonterra Case Management software to support the deployment of automations for a variety of use cases.
This video focuses on the notifications, alerts, and approval requests using Slack for Bonterra Impact Management. The solutions covered in this webinar can also be deployed for Microsoft Teams.
Interested in deploying notification automations for Bonterra Impact Management? Contact us at sales@sidekicksolutionsllc.com to discuss next steps.
Securing your Kubernetes cluster_ a step-by-step guide to success !KatiaHIMEUR1
Today, after several years of existence, an extremely active community and an ultra-dynamic ecosystem, Kubernetes has established itself as the de facto standard in container orchestration. Thanks to a wide range of managed services, it has never been so easy to set up a ready-to-use Kubernetes cluster.
However, this ease of use means that the subject of security in Kubernetes is often left for later, or even neglected. This exposes companies to significant risks.
In this talk, I'll show you step-by-step how to secure your Kubernetes cluster for greater peace of mind and reliability.
Accelerate your Kubernetes clusters with Varnish CachingThijs Feryn
A presentation about the usage and availability of Varnish on Kubernetes. This talk explores the capabilities of Varnish caching and shows how to use the Varnish Helm chart to deploy it to Kubernetes.
This presentation was delivered at K8SUG Singapore. See https://feryn.eu/presentations/accelerate-your-kubernetes-clusters-with-varnish-caching-k8sug-singapore-28-2024 for more details.
Software Delivery At the Speed of AI: Inflectra Invests In AI-Powered QualityInflectra
In this insightful webinar, Inflectra explores how artificial intelligence (AI) is transforming software development and testing. Discover how AI-powered tools are revolutionizing every stage of the software development lifecycle (SDLC), from design and prototyping to testing, deployment, and monitoring.
Learn about:
• The Future of Testing: How AI is shifting testing towards verification, analysis, and higher-level skills, while reducing repetitive tasks.
• Test Automation: How AI-powered test case generation, optimization, and self-healing tests are making testing more efficient and effective.
• Visual Testing: Explore the emerging capabilities of AI in visual testing and how it's set to revolutionize UI verification.
• Inflectra's AI Solutions: See demonstrations of Inflectra's cutting-edge AI tools like the ChatGPT plugin and Azure Open AI platform, designed to streamline your testing process.
Whether you're a developer, tester, or QA professional, this webinar will give you valuable insights into how AI is shaping the future of software delivery.
Generating a custom Ruby SDK for your web service or Rails API using Smithyg2nightmarescribd
Have you ever wanted a Ruby client API to communicate with your web service? Smithy is a protocol-agnostic language for defining services and SDKs. Smithy Ruby is an implementation of Smithy that generates a Ruby SDK using a Smithy model. In this talk, we will explore Smithy and Smithy Ruby to learn how to generate custom feature-rich SDKs that can communicate with any web service, such as a Rails JSON API.
JMeter webinar - integration with InfluxDB and GrafanaRTTS
Watch this recorded webinar about real-time monitoring of application performance. See how to integrate Apache JMeter, the open-source leader in performance testing, with InfluxDB, the open-source time-series database, and Grafana, the open-source analytics and visualization application.
In this webinar, we will review the benefits of leveraging InfluxDB and Grafana when executing load tests and demonstrate how these tools are used to visualize performance metrics.
Length: 30 minutes
Session Overview
-------------------------------------------
During this webinar, we will cover the following topics while demonstrating the integrations of JMeter, InfluxDB and Grafana:
- What out-of-the-box solutions are available for real-time monitoring JMeter tests?
- What are the benefits of integrating InfluxDB and Grafana into the load testing stack?
- Which features are provided by Grafana?
- Demonstration of InfluxDB and Grafana using a practice web application
To view the webinar recording, go to:
https://www.rttsweb.com/jmeter-integration-webinar
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.
Neuro-symbolic is not enough, we need neuro-*semantic*Frank van Harmelen
Neuro-symbolic (NeSy) AI is on the rise. However, simply machine learning on just any symbolic structure is not sufficient to really harvest the gains of NeSy. These will only be gained when the symbolic structures have an actual semantics. I give an operational definition of semantics as “predictable inference”.
All of this illustrated with link prediction over knowledge graphs, but the argument is general.
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
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
Are you looking to streamline your workflows and boost your projects’ efficiency? Do you find yourself searching for ways to add flexibility and control over your FME workflows? If so, you’re in the right place.
Join us for an insightful dive into the world of FME parameters, a critical element in optimizing workflow efficiency. This webinar marks the beginning of our three-part “Essentials of Automation” series. This first webinar is designed to equip you with the knowledge and skills to utilize parameters effectively: enhancing the flexibility, maintainability, and user control of your FME projects.
Here’s what you’ll gain:
- Essentials of FME Parameters: Understand the pivotal role of parameters, including Reader/Writer, Transformer, User, and FME Flow categories. Discover how they are the key to unlocking automation and optimization within your workflows.
- Practical Applications in FME Form: Delve into key user parameter types including choice, connections, and file URLs. Allow users to control how a workflow runs, making your workflows more reusable. Learn to import values and deliver the best user experience for your workflows while enhancing accuracy.
- Optimization Strategies in FME Flow: Explore the creation and strategic deployment of parameters in FME Flow, including the use of deployment and geometry parameters, to maximize workflow efficiency.
- Pro Tips for Success: Gain insights on parameterizing connections and leveraging new features like Conditional Visibility for clarity and simplicity.
We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
1. Professional Development Short Course On:
Practical Statistical Signal Processing — using MATLAB
Instructor:
Dr. Steven Kay
ATI Course Schedule: http://www.ATIcourses.com/schedule.htm
ATI's Practical Statistical Signal Processing: http://www.aticourses.com/practical_statistical_signal.htm
349 Berkshire Drive • Riva, Maryland 21140
888-501-2100 • 410-956-8805
Website: www.ATIcourses.com • Email: ATI@ATIcourses.com
2. Practical Statistical Signal Processing Using MATLAB
with Radar, Sonar, Communications, Speech & Imaging Applications
June 22-25, 2009
Middletown, Rhode Island
$1895 (8:30am - 4:00pm)
"Register 3 or More & Receive $10000 each
Off The Course Tuition."
Course Outline
1. MATLAB Basics. M-files, logical flow, graphing,
debugging, special characters, array manipulation,
Summary vectorizing computations, useful toolboxes.
This 4-day course covers signal processing 2. Computer Data Generation. Signals, Gaussian
systems for radar, sonar, communications, speech, noise, nonGaussian noise, colored and white noise,
imaging and other applications based on state-of- AR/ARMA time series, real vs. complex data, linear
the-art computer algorithms. These algorithms models, complex envelopes and demodulation.
include important tasks such as data simulation,
3. Parameter Estimation. Maximum likelihood,
parameter estimation, filtering, interpolation, best linear unbiased, linear and nonlinear least
detection, spectral analysis, beamforming, squares, recursive and sequential least squares,
classification, and tracking. Until now these minimum mean square error, maximum a posteriori,
algorithms could only be learned by reading the general linear model, performance evaluation via Taylor
latest technical journals. This course will take the series and computer simulation methods.
mystery out of these designs by introducing the 4. Filtering/Interpolation/Extrapolation. Wiener,
algorithms with a minimum of mathematics and linear Kalman approaches, time series methods.
illustrating the key ideas via numerous examples
using MATLAB. 5. Detection. Matched filters, generalized matched
filters, estimator-correlators, energy detectors,
Designed for engineers, scientists, and other detection of abrupt changes, min probability of error
professionals who wish to study the practice of receivers, communication receivers, nonGaussian
statistical signal processing without the headaches, approaches, likelihood and generalized likelihood
this course will make extensive use of hands-on detectors, receiver operating characteristics, CFAR
MATLAB implementations and demonstrations. receivers, performance evaluation by computer
Attendees will receive a suite of software source simulation.
code and are encouraged to bring their own laptops 6. Spectral Analysis. Periodogram, Blackman-
to follow along with the demonstrations. Tukey, autoregressive and other high resolution
Each participant will receive two books methods, eigenanalysis methods for sinusoids in noise.
Fundamentals of Statistical Signal Processing: Vol. I 7. Array Processing. Beamforming, narrowband
and Vol. 2 by instructor Dr. Kay. A complete set of vs. wideband considerations, space-time processing,
notes and a suite of MATLAB m-files will be interference suppression.
distributed in source format for direct use or 8. Signal Processing Systems. Image processing,
modification by the user. active sonar receiver, passive sonar receiver, adaptive
noise canceler, time difference of arrival localization,
channel identification and tracking, adaptive
Instructor beamforming, data analysis.
Dr. Steven Kay is a Professor of Electrical 9. Case Studies. Fault detection in bearings,
Engineering at the University of acoustic imaging, active sonar detection, passive sonar
Rhode Island and the President of detection, infrared surveillance, radar Doppler
estimation, speaker separation, stock market data
Signal Processing Systems, a analysis.
consulting firm to industry and the
government. He has over 25 years
of research and development What You Will Learn
experience in designing optimal • To translate system requirements into algorithms
statistical signal processing algorithms for radar, that work.
sonar, speech, image, communications, vibration, • To simulate and assess performance of key
and financial data analysis. Much of his work has algorithms.
been published in over 100 technical papers and • To tradeoff algorithm performance for
the three textbooks, Modern Spectral Estimation: computational complexity.
Theory and Application, Fundamentals of • The limitations to signal processing performance.
Statistical Signal Processing: Estimation Theory, • To recognize and avoid common pitfalls and traps
and Fundamentals of Statistical Signal in algorithmic development.
Processing: Detection Theory. Dr. Kay is a Fellow • To generalize and solve practical problems using
of the IEEE. the provided suite of MATLAB code.
Register online at www.ATIcourses.com or call ATI at 888.501.2100 or 410.956.8805 Vol. 97 – 41
3. www.ATIcourses.com
Boost Your Skills 349 Berkshire Drive
Riva, Maryland 21140
with On-Site Courses Telephone 1-888-501-2100 / (410) 965-8805
Tailored to Your Needs
Fax (410) 956-5785
Email: ATI@ATIcourses.com
The Applied Technology Institute specializes in training programs for technical professionals. Our courses keep you
current in the state-of-the-art technology that is essential to keep your company on the cutting edge in today’s highly
competitive marketplace. Since 1984, ATI has earned the trust of training departments nationwide, and has presented
on-site training at the major Navy, Air Force and NASA centers, and for a large number of contractors. Our training
increases effectiveness and productivity. Learn from the proven best.
For a Free On-Site Quote Visit Us At: http://www.ATIcourses.com/free_onsite_quote.asp
For Our Current Public Course Schedule Go To: http://www.ATIcourses.com/schedule.htm
4. References
1∗. S. Kay, Fundamentals of Statistical Signal
Processing: Estimation Theory, Prentice-Hall, 1993
2∗. S. Kay, Fundamentals of Statistical Signal
Processing: Detection Theory, Prentice-Hall, 1998
3. L. Scharf, Statistical Signal Processing, Addison-
Wesley,
Reading, MA, 1991 (more advanced treatment)
4. R.N. McDonough, A.D. Whalen, Detection of
Signals in Noise, Academic Press, New York, 1995
5. H.L. Van Trees, Detection, Estimation, and
Modulation Theory, Vol. I, J. Wiley, New York, 1968
(fairly involved but a classic)
6. G.M. Jenkins, D.G. Watts, Spectral Analysis and its
Applications, Holden-Day, 1968
7. S. Kay, Modern Spectral Estimation: Theory and
Application, Prentice-Hall, 1988
8. M.B. Priestley, Spectral Analysis and Time Series,
Academic Press, 1981
9. R.A. Monzingo, T.W. Miller, Adaptive Arrays, J.
Wiley, 1980
10. D.H. Johnson, D.E. Dudgeon, Array Signal
Processing, Prentice-Hall, 1993
∗
Provided as part of course materials
6. MATLAB Basics
Version: 5.2 for Windows
Useful toolboxes: signal processing, statistics, symbolic
m files: script files
Fortran vs. MATLAB example:
Signal generation
Math: s[n ] = cos(2π f 0n ) n = 0,1, K, N − 1
Fortran: pi=3.14159
f0=0.25
N=25
do 10 I=1,N
10 s(I)=cos(2*pi*f0*(I-1))
MATLAB: f0=0.25;N=25;
s=cos(2*pi*f0*[0:N-1]’);
7. Notes: pi already defined, [0:N-1]’ is a column vector,
cosine of vector of samples produces a vector output,
MATLAB treats vectors and matrices as elements
8. Noise Generation
Simplest model for observation noise is white Gaussian
noise (WGN)
Definition: zero mean, all samples are uncorrelated,
power
spectral density (PSD) is flat, and first order
probability density function (PDF) is
Gaussian
exp ⎛ − 2 x 2 ⎞
1 1
PDF: p( x ) = ⎜ ⎟
⎝ 2σ
p(x )
2πσ 2
⎠
where σ 2 = variance x
MATLAB Example: σ2 = 1
4
2
] 0
n
[
x
-2
-4
0 20 40 60 80 100
n
wgn.m
0 25
0
1
f
o 20
t
u
o
s 15
e
m
o
c 10
t
u
o
f 5
o
r
e
b 0
m
u -3 -2 -1 0 1 2 3
n x
9. wgn.m
1
0.9
0.8
0.7
0.6
)
x
(
p 0.5
,
F
D
P 0.4
0.3
0.2
0.1
0
-3 -2 -1 0 1 2 3
x
Note: randn(‘state’,0) sets random number generator to
default seed and thus generates the same set of
random numbers each time the program is run.
MATLAB code:
% wgn.m
%
% This program generates and plots
the time series, histogram, and
10. % estimated PDF for real white
Gaussian noise.
randn('state',0)
x=randn(100,1);
subplot(2,1,1)
plot(x)
xlabel('n')
ylabel('x[n]')
grid
subplot(2,1,2)
hist(x)
xlabel('x')
ylabel('number of outcomes out of
100')
title('wgn.m')
figure
pdf(x,100,10,-3,3,1)
xlabel('x')
ylabel('PDF, p(x)')
title('wgn.m')
% pdf.m
%
function
pdf(x,N,nbins,xmin,xmax,ymax)
%
11. % This function subprogram computes
and plots the
% PDF of a set of data.
%
% Input parameters:
%
% x - Nx1 data array
% N - number of data points
% nbins - number of bins (<N/10)
% xmin,xmax,ymax - axis scaling
%
[y,xx]=hist(x(1:N),nbins);
delx=xx(2)-xx(1);
bar(xx,y/(N*delx))
grid
axis([xmin xmax 0 ymax]);
12. Complex White Gaussian Noise
Definition: x [n ] = w1[n ] + jw2 [n ]
where w1[n ] and w2 [n ] are independent of each other
and
each one is real WGN with variance of σ 2 / 2
Mean: E (x [n ]) = 0
Variance: var(x [n ]) = var(w1[n ]) + var(w 2 [n ]) = σ 2
MATLAB code:
% cwgn.m
%
% This program generates complex
white Gaussian noise and
% then estimates its mean and
variance.
%
N=100;
varw=1;
x=sqrt(varw/2)*randn(N,1)+j*sqrt(varw
/2)*randn(N,1);
muest=mean(x)
varest=cov(x)
13. NonGaussian Noise
Generation: transform WGN using a nonlinear
memoryless
transformation
Example: Laplacian noise
1 ⎛ 2 ⎞
p( x ) = exp ⎜ − 2 x ⎟
2σ 2 ⎝ σ ⎠
Use the transformation
x = F −1 (w )
where w is uniform random variable on the interval
[0,1]
and F is the cumulative distribution function of the
Laplacian
PDF.
F (x )
1
x
14. Example: σ 2 = 1
5
] 0
n
[
x
-5
0 200 400 600 800 1000
n
laplaciannoise.m
1
0.8
)
x 0.6
(
p
,
F 0.4
D
P
0.2
0
-5 0 5
x
15. MATLAB Code:
% laplaciannoise.m
%
% This program uses a memoryless
transformation of a uniform
% random variable to generate a set
of independent Laplacian
% noise samples.
%
rand('state',0)
varx=1;N=1000;
u=rand(N,1);
for i=1:N
if u(i)>0.5
x(i,1)=sqrt(varx)*(1/sqrt(2))*log(1/(
2*(1-u(i))));
else
x(i,1)=sqrt(varx)*(1/sqrt(2))*log(2*u
(i));
end
end
subplot(2,1,1)
plot(x)
xlabel('n')
ylabel('x[n]')
17. Solving Parameter Estimation Problems
Approach:
1. Translate problem into manageable estimation
problem
2. Evaluate best possible performance (bounds)
3. Choose optimal or suboptimal procedure
4. Evaluate actual performance
a. Analytically – exact or approximate
b. By computer simulation
18. Radar Doppler Estimation
(Step 1)
Problem: Given radar returns from automobile,
determine speed to within 0.5 mph
transmit
Physical basis: Doppler effect
receive – receive-
approaching moving away
19. Received frequency is
2v
F = F0 + F0
c
{
FD
where v = velocity, c= speed of light, F0 = sinusoidal
transmit
frequency
To measure the velocity use
c F − F0
v =
2 F0
and estimate the frequency to yield
c F − F0
ˆ
v =
ˆ
2 F0
20. Modeling and Best Possible Performance
(Step 2)
Preprocessing: first demodulate to baseband to produce
the
sampled complex envelope or
s [n ] = (A / 2) exp( j 2π FD n ∆ + ϕ )
%
⎛ F = 2v F ⎞
⎜ D ⎟
⎝ c 0⎠
Fs = 1/ ∆ > 2FD = 2 ⎛ max F0 ⎞
2v
Must sample at ⎜ ⎟
⎝ c ⎠
Example: v max =300 mph, F0 =10.5 Ghz (X-band),
c = 3x108 m/s
2v max
FD −max = F0 ≈ 9388 Hz
c
⇒ Fs > 18, 776 complex samples/sec
How many samples do we need?
Spec: error must be less than 0.5 mph for
21. (A / 2)2
SNR = 10 log10 > −10 dB
σ 2
Cramer-Rao Lower Bound for Frequency
• tells us the minimum possible variance for estimator
– very useful for feasibility studies
6
var( fD ) ≥
ˆ (*) (see [Kay 1988])
(2π )2ηN (N 2 − 1)
where fD = FD / Fs , N = number of complex samples,
η =linear SNR
cFs
Since FD = (2v / c )F0 ⇒ v = fD
2F0
and we can show that
2
⎛ cFs ⎞
var(v ) = ⎜
ˆ ˆ
⎟ var( fD )
⎝ 2F0 ⎠
For an error of 0.5 mph (0.22 m/s) set
22. 3 var(v ) = 0.22 ⇒ var( fD ) = 7.47x10 −8
ˆ ˆ
99.8%
ˆ
v
v − 0.5 v v + 0.5
and finally we have from (*) that
1/ 3
⎡ 6 ⎤
N >⎢ ≈ 272 samples
ˆ )⎥
⎢ (2π ) η var( fD ⎥
2
⎣ ⎦
23. Descriptions of MATLAB Programs
1. analogsim – simulates the action of an RC filter on a
pulse
2. arcov - estimates the AR power spectral density
using he covariance method for AR parameter
estimation for real data.
3. arexamples - gives examples of the time series and
corresponding power spectral density for various AR
models. It requires the function subprograms:
gendata.m and armapsd.m.
4. armapsd - computes a set of PSD values, given the
parameters of a complex or real AR or MA or ARMA
model.
5. arpsd - plots the AR power spectral density for some
simple cases. The external subprogram armapsd.m is
required.
6. arpsdexample - estimates the power spectral density
of two real sinusoids in white Gaussian noise using the
periodogram and AR spectral estimators.
It requires the functions subprograms: per.m and
arcov.m.
24. 7. arrivaltimeest - simulates the performance of an
arrival time estimator for a DC pulse. The estimator is
a running correlator which is the MLE for white
Gaussian noise.
8. avper - illustrates the effect of block averaging on
the periodogram for white Gaussian noise.
9. classicalbayesian - demonstrates the difference
between the classical approach and the Bayesian
approaches to parameter modeling.
10. cwgn - generates complex white Gaussian noise and
then estimates its mean and variance.
11. DClevelhist - generates Figures 1.4, 1.5 in
"Fundamentals of Statistical Signal Processing:
Detection Theory", S. Kay
12. DCleveltime - generates a data set of white
Gaussian noise only and also a DC level A in white
Gaussian noise
13. discretesinc – plots the graph in linear and dB
quantities of a discrete sinc pulse in frequency
25. 14. estperform - compares the frequency estimation
performance for a single complex sinusoid in complex
white Gaussian using the peak location of the
periodogram and an AR(1) estimator.
15. Fig35new - computes Figure 3.5 (same as Figure
4.5) in "Fundamentals of Statistical Signal Processing:
Detection Theory", S. Kay. The function subprograms
Q.m and Qinv.m are required.
16. Fig39new - computes Figure 3.9 in "Fundamentals
of Statistical Signal Processing: Detection Theory", S.
Kay. The function subprograms Q.m and Qinv.m are
required.
17. Fig77new - computes Figure 7.7 in "Fundamentals
of Statistical Signal Processing: Detection Theory", S.
Kay.
18. gendata - generates a complex or real AR, MA, or
ARMA time series given the filter parameters and
excitation noise variance.
19. kalman - implementation of the vector state-scalar
observation linear Kalman filter. See (13.50)-(13.54) of
"Fundamentals of Statistical Signal Processing:
Estimation Theory" by S. Kay for more details.
26. 20. kalmanexample - uses the linear Kalman filter to
estimate the tap weights for a random TDL channel. It
generates Figures 13.16-13.18 in "Fundamentals of
Statistical Signal Processing: Estimation Theory", S.
Kay. It requires the function subprogram kalman.m.
21. laplaciannoise - uses a memoryless transformation
of a uniform random variable to generate a set of
independent Laplacian noise samples.
22. linearmodel - computes the optimal estimator of
the parameters of a real or complex linear model.
Alternatively, it is just the least squares estimator.
23. linearmodelexample - implements a line fit to a
noise corrupted line. The linear model or least squares
estimator is used. The function subprogram
linearmodel.m is required.
24. MAexample – plots out the PDF of an MA process
25. mlevar - computes the mean, variance, PDF of the
MLE for the power of a WGN process and compares it
to the CRLB.
26. montecarloroc - uses a Monte Carlo approach to
determine the detection performance of a Neyman-
Pearson detector for a DC level in WGN. The true
27. performance is shown in "Fundamentals of Statistical
Signal Processing: Detection Theory", S. Kay, in Figure
3.9 for d^2=1. The function subprogram roccurve.m is
required.
27. pcar - estimates the frequencies of real sinusoids
by using the principal component AR approach. Futher
details can be found in "Modern Spectral Estimation:
Theory and Application", by S. Kay.
28. pdf - computes and plots the PDF of a set of data.
29. per - computes the periodogram spectral estimator.
Futher details can be found in "Modern Spectral
Estimation: Theory and Application", by S. Kay.
30. perdetectexample - illustrates the detection
performance of a periodogram, which is an incoherent
matched filter.
31. perexamples - illustrates the capability of the
periodogram for resolving spectral lines.
32. plot1 – plots a sinusoid
33. psk - implements a matched filter receiver for the
detection of a PSK signal. The data are assumed real.
28. 34. pskexample - illustrates the optimal
detection/decoding of a PSK encoded digital sequence.
The bits are decoded and the probability of error is
computed and compared to the number of actual errors.
The external function subprogram psk.m is required.
35. Q - computes the right-tail probability
(complementary cumulative distribution function) for a
N(0,1) random variable.
36. Qinv - computes the inverse Q function or the
value which is exceeded by a N(0,1) random variable
with a probability of x.
37. repcorr - implements a replica correlator for either
real or complex data.
38. repcorrexample - illustrates the replica-correlator.
It requires the subprogram repcorr.m.
39. roccurve - determines the ROCs for a given set of
detector outputs under H0 and H1.
40. sampling – plots out an analog sinusoid and the
samples taken
41. seqls - implements a sequential least squares
estimator for a DC level
29. in WGN of constant variance.
42. shift - shifts the given sequence by a specified
number of samples. Zeros are shifted in either from
the left or right.
43. signdetexample - implements a sign detector for a
DC level in Gaussian-mixture noise. A comparison is
made to a replica correlator, which is just the sample
mean.
44. sinusoid - generates a sinusoid
45. stepdown - implements the step-down procedure to
find the coefficients and prediction error powers for all
the lower order predictors given the filter parameters
and white noise variance of a pth order AR model. See
(6.51) and (6.52). This program has been translated
from Fortran into Matlab. See "Modern Spectral
Estimation" by S. Kay for further details.
46. timedelaybfr - implements a time delay
beamformer for a line array of 3 sensors. The emitted
signal is sinusoidal and is assumed to be at
broadside or at 90 degrees (perpendicular to line array).
30. 47. wgn - generates and plots the time series,
histogram, and estimated PDF for real white Gaussian
noise.
48. wiener - implements a Wiener smoother for
extracting an AR(1) signal in white Gaussian noise and
also for predicting an AR(1) signal for no observation
noise present.
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