8 Kinematic Models for Target Tracking
|
|
- Annabel Snow
- 7 years ago
- Views:
Transcription
1 Technion Israel Institute of Technology, Department of Electrical Engineering Estimation and Identification in Dynamical Systems (04885) Lecture Notes, Fall 009, Prof. N. Shimkin 8 Kinematic Models for Target Tracking Target tracking is a major application area of Kalman Filtering. Typical applications include aircraft tracking using noisy remote sensors (radar, vision, IR, etc.), and tracking moving objects in video sequences. We consider here very briefly some basic models for target tracking, in order to illustrate: Physical modeling and discretization. The basic structure of a tracking filter. For simplicity we consider tracking in one cartesian coordinate only. 8.1 Modelling Target Motion A. Second-order models: random acceleration These are the simplest useful models. Here the velocity is constant except for a noise term. Let p(t) denote the object position, V (t) = ṗ(t) its velocity, and a(t) = p(t) its acceleration. The basic second-order model in continuous time is described by p(t) = w(t), where w(t) is white noise: R w (t) = σw δ(t). The corresponding state equations are: [ ] [ ] [ ] p x = ; ẋ(t) = x(t) + w(t). ṗ
2 As the measurements are typically obtained in discrete times, we need to consider discretized versions of this model. B. Discretization Consider the state-space equation: ẋ(t) = Ax(t) + Bu(t) + Gw(t) and sampling times t k = kt. As we know, x(t) = e A(t t k) x(t k ) + t ( ) e A(t tk) Bu(t ) + Gw(t ) dt. t k We now assume that the input u k changes slowly relative to the sampling period, so that u(t) u(t k ) on [t k, t k+1 ). This gives x(t k+1 ) = F x(t k ) + Bu(t k ) + w k, where F = e AT B = w k = T 0 tk+1 e A(T t ) B dt t k e A(t k+1 t ) Gw(t ) dt. If w(t) is a white noise process with R w (t) = Q w (t) δ(t), then {w k } is a white noise sequence with Q k = cov(w k ) = tk+1 If w(t) is also Gaussian, then w k N(0, Q k ). t k e A(t k+1 t ) GQ w (t )G T e AT (t k+1 t ) dt.
3 For our nd order model we get F = 1 T, Q = 1 T 3 1, T 3 σ T w., T We note that the order of magnitude of possible velocity change over [t k, t k+1 ] is V Q = T σw. This should guide the choice of σw. Typical measurements include position, velocity, or both. The measurement equation is of the usual form z(t k ) = Hx(t k ) + v k, where v k is the measurement noise (which depends on the sensor). C. Simplified discretization A slightly different model can be obtained from p(t) = w(t) by making the simplifying assumption that the noise is constant between sampling instants, that is w(t) w(t k ) for t [t k, t k+1 ). This gives the equations V (t k+1 ) = V (t k ) + T w(t k ) p(t k+1 ) = p(t k ) + T V (t k ) + T w k so that x k = 1 T x k + w k, 0 1 where w k = T / w(t k ), Qk = 1 T 4 1, T 3 4 σ 1 T T 3, T w. After scaling σ w by T, the model is similar to the previous one except for the 1 4 coefficient in Q 11. 3
4 D. Direct modeling in discrete time A simplified model can sometimes be directly constructed in discrete time. Let p k and V k denote the target position and velocity at time t k. Let x k = (p k, V k ) T, and let T = t k+1 t k The approximate state equations are: p k+1 = p k + T V k V k+1 = V k + w k where (w k ) is a white noise sequence. These equations reflect the following assumptions: The velocity V k is constant on [t k, t k+1 ]. The velocity increments (V k+1 V k ) are white. The last assumption can be interpreted as a white-noise acceleration. The variance σ w should reflect the possible change in velocity over period T. The resulting state model is: [ ] [ ] 1 T 0 x k+1 = x(t) + w k The model is similar to the previous ones except for zero noise in the position component. The previous models should be preferred unless T is very small. 4
5 8. Steady-State Filter for nd Order Models: The α-β Filter Assuming a fixed sampling interval T, we have arrived at the stationary model: x k+1 = F x k + w k z k = Hx k + v k with x = ṗ, F = 1 T, Q = p 0 1 T 4 4 T 3 T 3 T σ w Assume a noisy position measurement: z k = p(t k ) + v k, so that H = [1, 0], R = σ v. The steady-state filter will be of the form ˆx k+1 k = F ˆx k k ˆx k k = ˆx k k 1 + K 1 z k = ˆxk k 1 + α z k β/t K where, as usual, z k = z k H ˆx k k 1. Note that this filter gives both position and velocity estimates. We wish to compute the Kalman gain K, namely the coefficients α and β, and the error covariance P. 5
6 Recall the Ricatti equation for P P : P = F [P P H T S 1 HP ] F T + Q with S = HPH T + R, and K = P H T S 1. Denote ( ) m P = 11 m 1. m 1 m After some algebra, we obtain 3 quadratic equations in (m 11, m 1, m ), which can be solved (excercise). The Kalman gain elements may be expressed as: where is the maneuvering index. α = K 1 = 1 ( λ 8λ + (λ + 4) ) λ 8 + 8λ β = T K = 1 ( λ + 4λ λ ) λ 4 + 8λ, λ = T σ w σ v Essentially, it is the ratio of the state noise to the measurement noise. The behavior of the optimal gain parameters as a function of λ is illustrated in the next figure. It can also be shown that (P + ) 11 = α σ v so that α also represents the improvement in position estimation variance as compared with that of a single measurement. 6
7 8.3 Higher-Order Models Our basic model so far was a(t) = w(t), which essentially corresponds to a constant velocity motion (with white noise perturbation). In some cases a constant acceleration model may be more appropriate. In this case we can increase the model order and consider ȧ(t) = w(t). This leads to a 3rd order system. The resulting steady-state filer is called the α-β-γ filter, and has the form: α ˆx k k = ˆx k k 1 + β/t z k. γ/t The coefficients α, β, γ again depend only on the maneuvering index λ = T σ w σ v β 1.0 α λ Figure 1: The gain coefficients as function of λ (α-β filter) 7
8 Another option to regularize the velocity change in the nd-order filter is to use filtered noise in place of white noise. This simplest such model is: ȧ(t) = b a(t) + w(t). Note that this may be viewed as the (second-order) noise acceleration model a(t) = w(t) with low-pass filtered noise w = w/(s + b). This model is useful for tracking maneuvering targets, where velocity changes cannot be too abrupt. 8
9 8.4 Target Tracking in General Target tracking in practice involves many additional issues with varying degrees of difficulty. Among those we mention: Correlated motion in several dimensions. Polar measurements (leading to nonlinear filters). Partial measurements (such as bearings-only measurements in sonar). Spurious measurements ( clutter ). Different target maneuvers (which requires adaptive or multiple models). Multi-target tracking. 9
A Multi-Model Filter for Mobile Terminal Location Tracking
A Multi-Model Filter for Mobile Terminal Location Tracking M. McGuire, K.N. Plataniotis The Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, 1 King s College
More informationProbability and Random Variables. Generation of random variables (r.v.)
Probability and Random Variables Method for generating random variables with a specified probability distribution function. Gaussian And Markov Processes Characterization of Stationary Random Process Linearly
More informationCourse 8. An Introduction to the Kalman Filter
Course 8 An Introduction to the Kalman Filter Speakers Greg Welch Gary Bishop Kalman Filters in 2 hours? Hah! No magic. Pretty simple to apply. Tolerant of abuse. Notes are a standalone reference. These
More informationUnderstanding and Applying Kalman Filtering
Understanding and Applying Kalman Filtering Lindsay Kleeman Department of Electrical and Computer Systems Engineering Monash University, Clayton 1 Introduction Objectives: 1. Provide a basic understanding
More informationLecture 8: Signal Detection and Noise Assumption
ECE 83 Fall Statistical Signal Processing instructor: R. Nowak, scribe: Feng Ju Lecture 8: Signal Detection and Noise Assumption Signal Detection : X = W H : X = S + W where W N(, σ I n n and S = [s, s,...,
More informationA Reliability Point and Kalman Filter-based Vehicle Tracking Technique
A Reliability Point and Kalman Filter-based Vehicle Tracing Technique Soo Siang Teoh and Thomas Bräunl Abstract This paper introduces a technique for tracing the movement of vehicles in consecutive video
More informationVector Spaces; the Space R n
Vector Spaces; the Space R n Vector Spaces A vector space (over the real numbers) is a set V of mathematical entities, called vectors, U, V, W, etc, in which an addition operation + is defined and in which
More informationEE 570: Location and Navigation
EE 570: Location and Navigation On-Line Bayesian Tracking Aly El-Osery 1 Stephen Bruder 2 1 Electrical Engineering Department, New Mexico Tech Socorro, New Mexico, USA 2 Electrical and Computer Engineering
More informationChapter 4 One Dimensional Kinematics
Chapter 4 One Dimensional Kinematics 41 Introduction 1 4 Position, Time Interval, Displacement 41 Position 4 Time Interval 43 Displacement 43 Velocity 3 431 Average Velocity 3 433 Instantaneous Velocity
More informationDynamic data processing
Dynamic data processing recursive least-squares P.J.G. Teunissen Series on Mathematical Geodesy and Positioning Dynamic data processing recursive least-squares Dynamic data processing recursive least-squares
More informationPhysics Notes Class 11 CHAPTER 3 MOTION IN A STRAIGHT LINE
1 P a g e Motion Physics Notes Class 11 CHAPTER 3 MOTION IN A STRAIGHT LINE If an object changes its position with respect to its surroundings with time, then it is called in motion. Rest If an object
More informationSignal Detection C H A P T E R 14 14.1 SIGNAL DETECTION AS HYPOTHESIS TESTING
C H A P T E R 4 Signal Detection 4. SIGNAL DETECTION AS HYPOTHESIS TESTING In Chapter 3 we considered hypothesis testing in the context of random variables. The detector resulting in the minimum probability
More informationROBOTICS 01PEEQW. Basilio Bona DAUIN Politecnico di Torino
ROBOTICS 01PEEQW Basilio Bona DAUIN Politecnico di Torino Probabilistic Fundamentals in Robotics Robot Motion Probabilistic models of mobile robots Robot motion Kinematics Velocity motion model Odometry
More informationIntroduction to Engineering System Dynamics
CHAPTER 0 Introduction to Engineering System Dynamics 0.1 INTRODUCTION The objective of an engineering analysis of a dynamic system is prediction of its behaviour or performance. Real dynamic systems are
More informationOptimal Design of α-β-(γ) Filters
Optimal Design of --(γ) Filters Dirk Tenne Tarunraj Singh, Center for Multisource Information Fusion State University of New York at Buffalo Buffalo, NY 426 Abstract Optimal sets of the smoothing parameter
More informationContent. Professur für Steuerung, Regelung und Systemdynamik. Lecture: Vehicle Dynamics Tutor: T. Wey Date: 01.01.08, 20:11:52
1 Content Overview 1. Basics on Signal Analysis 2. System Theory 3. Vehicle Dynamics Modeling 4. Active Chassis Control Systems 5. Signals & Systems 6. Statistical System Analysis 7. Filtering 8. Modeling,
More information1 Short Introduction to Time Series
ECONOMICS 7344, Spring 202 Bent E. Sørensen January 24, 202 Short Introduction to Time Series A time series is a collection of stochastic variables x,.., x t,.., x T indexed by an integer value t. The
More informationThe Performance of Option Trading Software Agents: Initial Results
The Performance of Option Trading Software Agents: Initial Results Omar Baqueiro, Wiebe van der Hoek, and Peter McBurney Department of Computer Science, University of Liverpool, Liverpool, UK {omar, wiebe,
More informationIntroduction to Mobile Robotics Bayes Filter Particle Filter and Monte Carlo Localization
Introduction to Mobile Robotics Bayes Filter Particle Filter and Monte Carlo Localization Wolfram Burgard, Maren Bennewitz, Diego Tipaldi, Luciano Spinello 1 Motivation Recall: Discrete filter Discretize
More informationω h (t) = Ae t/τ. (3) + 1 = 0 τ =.
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering 2.004 Dynamics and Control II Fall 2007 Lecture 2 Solving the Equation of Motion Goals for today Modeling of the 2.004 La s rotational
More informationTo define concepts such as distance, displacement, speed, velocity, and acceleration.
Chapter 7 Kinematics of a particle Overview In kinematics we are concerned with describing a particle s motion without analysing what causes or changes that motion (forces). In this chapter we look at
More informationKalman Filter Applied to a Active Queue Management Problem
IOSR Journal of Electrical and Electronics Engineering (IOSR-JEEE) e-issn: 2278-1676,p-ISSN: 2320-3331, Volume 9, Issue 4 Ver. III (Jul Aug. 2014), PP 23-27 Jyoti Pandey 1 and Prof. Aashih Hiradhar 2 Department
More informationVehicle Tracking in Occlusion and Clutter
Vehicle Tracking in Occlusion and Clutter by KURTIS NORMAN MCBRIDE A thesis presented to the University of Waterloo in fulfilment of the thesis requirement for the degree of Master of Applied Science in
More informationIntroduction to Kalman Filtering
Introduction to Kalman Filtering A set of two lectures Maria Isabel Ribeiro Associate Professor Instituto Superior écnico / Instituto de Sistemas e Robótica June All rights reserved INRODUCION O KALMAN
More informationSample Problems. Practice Problems
Lecture Notes Quadratic Word Problems page 1 Sample Problems 1. The sum of two numbers is 31, their di erence is 41. Find these numbers.. The product of two numbers is 640. Their di erence is 1. Find these
More informationObject tracking & Motion detection in video sequences
Introduction Object tracking & Motion detection in video sequences Recomended link: http://cmp.felk.cvut.cz/~hlavac/teachpresen/17compvision3d/41imagemotion.pdf 1 2 DYNAMIC SCENE ANALYSIS The input to
More informationMechanics lecture 7 Moment of a force, torque, equilibrium of a body
G.1 EE1.el3 (EEE1023): Electronics III Mechanics lecture 7 Moment of a force, torque, equilibrium of a body Dr Philip Jackson http://www.ee.surrey.ac.uk/teaching/courses/ee1.el3/ G.2 Moments, torque and
More informationEnhancing the SNR of the Fiber Optic Rotation Sensor using the LMS Algorithm
1 Enhancing the SNR of the Fiber Optic Rotation Sensor using the LMS Algorithm Hani Mehrpouyan, Student Member, IEEE, Department of Electrical and Computer Engineering Queen s University, Kingston, Ontario,
More informationMechanics 1: Conservation of Energy and Momentum
Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation
More informationLecture 5: Variants of the LMS algorithm
1 Standard LMS Algorithm FIR filters: Lecture 5: Variants of the LMS algorithm y(n) = w 0 (n)u(n)+w 1 (n)u(n 1) +...+ w M 1 (n)u(n M +1) = M 1 k=0 w k (n)u(n k) =w(n) T u(n), Error between filter output
More informationLecture L5 - Other Coordinate Systems
S. Widnall, J. Peraire 16.07 Dynamics Fall 008 Version.0 Lecture L5 - Other Coordinate Systems In this lecture, we will look at some other common systems of coordinates. We will present polar coordinates
More informationApplication of the IMM-JPDA Filter to Multiple Target Tracking in Total Internal Reflection Fluorescence Microscopy Images
Application of the IMM-JPDA Filter to Multiple Target Tracking in Total Internal Reflection Fluorescence Microscopy Images Seyed Hamid Rezatofighi 1,2, Stephen Gould 1, Richard Hartley 1,3, Katarina Mele
More informationFric-3. force F k and the equation (4.2) may be used. The sense of F k is opposite
4. FRICTION 4.1 Laws of friction. We know from experience that when two bodies tend to slide on each other a resisting force appears at their surface of contact which opposes their relative motion. The
More informationSolutions to Exam in Speech Signal Processing EN2300
Solutions to Exam in Speech Signal Processing EN23 Date: Thursday, Dec 2, 8: 3: Place: Allowed: Grades: Language: Solutions: Q34, Q36 Beta Math Handbook (or corresponding), calculator with empty memory.
More informationLinear regression methods for large n and streaming data
Linear regression methods for large n and streaming data Large n and small or moderate p is a fairly simple problem. The sufficient statistic for β in OLS (and ridge) is: The concept of sufficiency is
More informationFEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL
FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL STATIsTICs 4 IV. RANDOm VECTORs 1. JOINTLY DIsTRIBUTED RANDOm VARIABLEs If are two rom variables defined on the same sample space we define the joint
More informationA defensive investment strategy for portfolio alpha return and market risk reduction
Università LUISS Guido Carli Dottorato di Ricerca in Metodi Matematici per l Economia, l Azienda, la Finanza e le Assicurazioni XXII Ciclo Anno IV Tesi di Dottorato A defensive investment strategy for
More informationApplications to Data Smoothing and Image Processing I
Applications to Data Smoothing and Image Processing I MA 348 Kurt Bryan Signals and Images Let t denote time and consider a signal a(t) on some time interval, say t. We ll assume that the signal a(t) is
More informationLinear Threshold Units
Linear Threshold Units w x hx (... w n x n w We assume that each feature x j and each weight w j is a real number (we will relax this later) We will study three different algorithms for learning linear
More informationKINEMATICS OF PARTICLES RELATIVE MOTION WITH RESPECT TO TRANSLATING AXES
KINEMTICS OF PRTICLES RELTIVE MOTION WITH RESPECT TO TRNSLTING XES In the previous articles, we have described particle motion using coordinates with respect to fixed reference axes. The displacements,
More informationRobotics. Chapter 25. Chapter 25 1
Robotics Chapter 25 Chapter 25 1 Outline Robots, Effectors, and Sensors Localization and Mapping Motion Planning Motor Control Chapter 25 2 Mobile Robots Chapter 25 3 Manipulators P R R R R R Configuration
More informationA Movement Tracking Management Model with Kalman Filtering Global Optimization Techniques and Mahalanobis Distance
Loutraki, 21 26 October 2005 A Movement Tracking Management Model with ing Global Optimization Techniques and Raquel Ramos Pinho, João Manuel R. S. Tavares, Miguel Velhote Correia Laboratório de Óptica
More informationAn Introduction to the Kalman Filter
An Introduction to the Kalman Filter Greg Welch 1 and Gary Bishop 2 TR 95041 Department of Computer Science University of North Carolina at Chapel Hill Chapel Hill, NC 275993175 Updated: Monday, July 24,
More informationGeneral Theory of Differential Equations Sections 2.8, 3.1-3.2, 4.1
A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics General Theory of Differential Equations Sections 2.8, 3.1-3.2, 4.1 Dr. John Ehrke Department of Mathematics Fall 2012 Questions
More information1 Simulating Brownian motion (BM) and geometric Brownian motion (GBM)
Copyright c 2013 by Karl Sigman 1 Simulating Brownian motion (BM) and geometric Brownian motion (GBM) For an introduction to how one can construct BM, see the Appendix at the end of these notes A stochastic
More informationThe Heat Equation. Lectures INF2320 p. 1/88
The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)
More informationPerformance. 13. Climbing Flight
Performance 13. Climbing Flight In order to increase altitude, we must add energy to the aircraft. We can do this by increasing the thrust or power available. If we do that, one of three things can happen:
More informationBildverarbeitung und Mustererkennung Image Processing and Pattern Recognition
Bildverarbeitung und Mustererkennung Image Processing and Pattern Recognition 1. Image Pre-Processing - Pixel Brightness Transformation - Geometric Transformation - Image Denoising 1 1. Image Pre-Processing
More informationStochastic Gradient Method: Applications
Stochastic Gradient Method: Applications February 03, 2015 P. Carpentier Master MMMEF Cours MNOS 2014-2015 114 / 267 Lecture Outline 1 Two Elementary Exercices on the Stochastic Gradient Two-Stage Recourse
More informatione.g. arrival of a customer to a service station or breakdown of a component in some system.
Poisson process Events occur at random instants of time at an average rate of λ events per second. e.g. arrival of a customer to a service station or breakdown of a component in some system. Let N(t) be
More informationKristine L. Bell and Harry L. Van Trees. Center of Excellence in C 3 I George Mason University Fairfax, VA 22030-4444, USA kbell@gmu.edu, hlv@gmu.
POSERIOR CRAMÉR-RAO BOUND FOR RACKING ARGE BEARING Kristine L. Bell and Harry L. Van rees Center of Excellence in C 3 I George Mason University Fairfax, VA 22030-4444, USA bell@gmu.edu, hlv@gmu.edu ABSRAC
More informationMapping an Application to a Control Architecture: Specification of the Problem
Mapping an Application to a Control Architecture: Specification of the Problem Mieczyslaw M. Kokar 1, Kevin M. Passino 2, Kenneth Baclawski 1, and Jeffrey E. Smith 3 1 Northeastern University, Boston,
More informationIsaac Newton s (1642-1727) Laws of Motion
Big Picture 1 2.003J/1.053J Dynamics and Control I, Spring 2007 Professor Thomas Peacock 2/7/2007 Lecture 1 Newton s Laws, Cartesian and Polar Coordinates, Dynamics of a Single Particle Big Picture First
More informationDifferentiation of vectors
Chapter 4 Differentiation of vectors 4.1 Vector-valued functions In the previous chapters we have considered real functions of several (usually two) variables f : D R, where D is a subset of R n, where
More informationRANDOM VIBRATION AN OVERVIEW by Barry Controls, Hopkinton, MA
RANDOM VIBRATION AN OVERVIEW by Barry Controls, Hopkinton, MA ABSTRACT Random vibration is becoming increasingly recognized as the most realistic method of simulating the dynamic environment of military
More informationGround Rules. PC1221 Fundamentals of Physics I. Kinematics. Position. Lectures 3 and 4 Motion in One Dimension. Dr Tay Seng Chuan
Ground Rules PC11 Fundamentals of Physics I Lectures 3 and 4 Motion in One Dimension Dr Tay Seng Chuan 1 Switch off your handphone and pager Switch off your laptop computer and keep it No talking while
More informationIEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 7, JULY 2005 2475. G. George Yin, Fellow, IEEE, and Vikram Krishnamurthy, Fellow, IEEE
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 7, JULY 2005 2475 LMS Algorithms for Tracking Slow Markov Chains With Applications to Hidden Markov Estimation and Adaptive Multiuser Detection G.
More informationFinancial TIme Series Analysis: Part II
Department of Mathematics and Statistics, University of Vaasa, Finland January 29 February 13, 2015 Feb 14, 2015 1 Univariate linear stochastic models: further topics Unobserved component model Signal
More informationTexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA
2015 School of Information Technology and Electrical Engineering at the University of Queensland TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA Schedule Week Date
More informationLecture 4: Seasonal Time Series, Trend Analysis & Component Model Bus 41910, Time Series Analysis, Mr. R. Tsay
Lecture 4: Seasonal Time Series, Trend Analysis & Component Model Bus 41910, Time Series Analysis, Mr. R. Tsay Business cycle plays an important role in economics. In time series analysis, business cycle
More informationSensorless Control of a Brushless DC motor using an Extended Kalman estimator.
Sensorless Control of a Brushless DC motor using an Extended Kalman estimator. Paul Kettle, Aengus Murray & Finbarr Moynihan. Analog Devices, Motion Control Group Wilmington, MA 1887,USA. Paul.Kettle@analog.com
More informationBayesian Adaptive Trading with a Daily Cycle
Bayesian Adaptive Trading with a Daily Cycle Robert Almgren and Julian Lorenz July 28, 26 Abstract Standard models of algorithmic trading neglect the presence of a daily cycle. We construct a model in
More informationStatistical machine learning, high dimension and big data
Statistical machine learning, high dimension and big data S. Gaïffas 1 14 mars 2014 1 CMAP - Ecole Polytechnique Agenda for today Divide and Conquer principle for collaborative filtering Graphical modelling,
More informationStatistics Graduate Courses
Statistics Graduate Courses STAT 7002--Topics in Statistics-Biological/Physical/Mathematics (cr.arr.).organized study of selected topics. Subjects and earnable credit may vary from semester to semester.
More informationStatistics in Retail Finance. Chapter 6: Behavioural models
Statistics in Retail Finance 1 Overview > So far we have focussed mainly on application scorecards. In this chapter we shall look at behavioural models. We shall cover the following topics:- Behavioural
More informationAdaptive Equalization of binary encoded signals Using LMS Algorithm
SSRG International Journal of Electronics and Communication Engineering (SSRG-IJECE) volume issue7 Sep Adaptive Equalization of binary encoded signals Using LMS Algorithm Dr.K.Nagi Reddy Professor of ECE,NBKR
More informationBroadband Networks. Prof. Dr. Abhay Karandikar. Electrical Engineering Department. Indian Institute of Technology, Bombay. Lecture - 29.
Broadband Networks Prof. Dr. Abhay Karandikar Electrical Engineering Department Indian Institute of Technology, Bombay Lecture - 29 Voice over IP So, today we will discuss about voice over IP and internet
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationCE801: Intelligent Systems and Robotics Lecture 3: Actuators and Localisation. Prof. Dr. Hani Hagras
1 CE801: Intelligent Systems and Robotics Lecture 3: Actuators and Localisation Prof. Dr. Hani Hagras Robot Locomotion Robots might want to move in water, in the air, on land, in space.. 2 Most of the
More informationHow do we obtain the solution, if we are given F (t)? First we note that suppose someone did give us one solution of this equation
1 Green s functions The harmonic oscillator equation is This has the solution mẍ + kx = 0 (1) x = A sin(ωt) + B cos(ωt), ω = k m where A, B are arbitrary constants reflecting the fact that we have two
More informationCCNY. BME I5100: Biomedical Signal Processing. Linear Discrimination. Lucas C. Parra Biomedical Engineering Department City College of New York
BME I5100: Biomedical Signal Processing Linear Discrimination Lucas C. Parra Biomedical Engineering Department CCNY 1 Schedule Week 1: Introduction Linear, stationary, normal - the stuff biology is not
More informationPhysics 2048 Test 1 Solution (solutions to problems 2-5 are from student papers) Problem 1 (Short Answer: 20 points)
Physics 248 Test 1 Solution (solutions to problems 25 are from student papers) Problem 1 (Short Answer: 2 points) An object's motion is restricted to one dimension along the distance axis. Answer each
More informationSteady state approximation
Steady state approximation Supplementary notes for the course Chemistry for Physicists Course coordinator: Prof. Dr. Mathias Nest Teaching assistant: Dr. Raghunathan Ramakrishnan contact: rama@mytum.de,
More informationSolutions to old Exam 1 problems
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
More information8. Linear least-squares
8. Linear least-squares EE13 (Fall 211-12) definition examples and applications solution of a least-squares problem, normal equations 8-1 Definition overdetermined linear equations if b range(a), cannot
More information4F7 Adaptive Filters (and Spectrum Estimation) Least Mean Square (LMS) Algorithm Sumeetpal Singh Engineering Department Email : sss40@eng.cam.ac.
4F7 Adaptive Filters (and Spectrum Estimation) Least Mean Square (LMS) Algorithm Sumeetpal Singh Engineering Department Email : sss40@eng.cam.ac.uk 1 1 Outline The LMS algorithm Overview of LMS issues
More informationNonlinear Systems of Ordinary Differential Equations
Differential Equations Massoud Malek Nonlinear Systems of Ordinary Differential Equations Dynamical System. A dynamical system has a state determined by a collection of real numbers, or more generally
More information11. Time series and dynamic linear models
11. Time series and dynamic linear models Objective To introduce the Bayesian approach to the modeling and forecasting of time series. Recommended reading West, M. and Harrison, J. (1997). models, (2 nd
More information5 Signal Design for Bandlimited Channels
225 5 Signal Design for Bandlimited Channels So far, we have not imposed any bandwidth constraints on the transmitted passband signal, or equivalently, on the transmitted baseband signal s b (t) I[k]g
More information4F7 Adaptive Filters (and Spectrum Estimation) Kalman Filter. Sumeetpal Singh Email : sss40@eng.cam.ac.uk
4F7 Adaptive Filters (and Spectrum Estimation) Kalman Filter Sumeetpal Singh Email : sss40@eng.cam.ac.uk 1 1 Outline State space model Kalman filter Examples 2 2 Parameter Estimation We have repeated observations
More informationEngineering Feasibility Study: Vehicle Shock Absorption System
Engineering Feasibility Study: Vehicle Shock Absorption System Neil R. Kennedy AME40463 Senior Design February 28, 2008 1 Abstract The purpose of this study is to explore the possibilities for the springs
More informationSignal Detection. Outline. Detection Theory. Example Applications of Detection Theory
Outline Signal Detection M. Sami Fadali Professor of lectrical ngineering University of Nevada, Reno Hypothesis testing. Neyman-Pearson (NP) detector for a known signal in white Gaussian noise (WGN). Matched
More informationExperiment: Static and Kinetic Friction
PHY 201: General Physics I Lab page 1 of 6 OBJECTIVES Experiment: Static and Kinetic Friction Use a Force Sensor to measure the force of static friction. Determine the relationship between force of static
More informationLecture L6 - Intrinsic Coordinates
S. Widnall, J. Peraire 16.07 Dynamics Fall 2009 Version 2.0 Lecture L6 - Intrinsic Coordinates In lecture L4, we introduced the position, velocity and acceleration vectors and referred them to a fixed
More informationLeast-Squares Intersection of Lines
Least-Squares Intersection of Lines Johannes Traa - UIUC 2013 This write-up derives the least-squares solution for the intersection of lines. In the general case, a set of lines will not intersect at a
More informationTime series Forecasting using Holt-Winters Exponential Smoothing
Time series Forecasting using Holt-Winters Exponential Smoothing Prajakta S. Kalekar(04329008) Kanwal Rekhi School of Information Technology Under the guidance of Prof. Bernard December 6, 2004 Abstract
More information1 of 7 9/5/2009 6:12 PM
1 of 7 9/5/2009 6:12 PM Chapter 2 Homework Due: 9:00am on Tuesday, September 8, 2009 Note: To understand how points are awarded, read your instructor's Grading Policy. [Return to Standard Assignment View]
More informationTMA4213/4215 Matematikk 4M/N Vår 2013
Norges teknisk naturvitenskapelige universitet Institutt for matematiske fag TMA43/45 Matematikk 4M/N Vår 3 Løsningsforslag Øving a) The Fourier series of the signal is f(x) =.4 cos ( 4 L x) +cos ( 5 L
More informationSince it is necessary to consider the ability of the lter to predict many data over a period of time a more meaningful metric is the expected value of
Chapter 11 Tutorial: The Kalman Filter Tony Lacey. 11.1 Introduction The Kalman lter ë1ë has long been regarded as the optimal solution to many tracing and data prediction tass, ë2ë. Its use in the analysis
More informationHigh Quality Image Magnification using Cross-Scale Self-Similarity
High Quality Image Magnification using Cross-Scale Self-Similarity André Gooßen 1, Arne Ehlers 1, Thomas Pralow 2, Rolf-Rainer Grigat 1 1 Vision Systems, Hamburg University of Technology, D-21079 Hamburg
More informationLecture 7: Finding Lyapunov Functions 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 7: Finding Lyapunov Functions 1
More informationMaster s thesis tutorial: part III
for the Autonomous Compliant Research group Tinne De Laet, Wilm Decré, Diederik Verscheure Katholieke Universiteit Leuven, Department of Mechanical Engineering, PMA Division 30 oktober 2006 Outline General
More informationPhysics Kinematics Model
Physics Kinematics Model I. Overview Active Physics introduces the concept of average velocity and average acceleration. This unit supplements Active Physics by addressing the concept of instantaneous
More informationThe purposes of this experiment are to test Faraday's Law qualitatively and to test Lenz's Law.
260 17-1 I. THEORY EXPERIMENT 17 QUALITATIVE STUDY OF INDUCED EMF Along the extended central axis of a bar magnet, the magnetic field vector B r, on the side nearer the North pole, points away from this
More informationChapter 5 Discrete Probability Distribution. Learning objectives
Chapter 5 Discrete Probability Distribution Slide 1 Learning objectives 1. Understand random variables and probability distributions. 1.1. Distinguish discrete and continuous random variables. 2. Able
More informationNRZ Bandwidth - HF Cutoff vs. SNR
Application Note: HFAN-09.0. Rev.2; 04/08 NRZ Bandwidth - HF Cutoff vs. SNR Functional Diagrams Pin Configurations appear at end of data sheet. Functional Diagrams continued at end of data sheet. UCSP
More informationEvaluating System Suitability CE, GC, LC and A/D ChemStation Revisions: A.03.0x- A.08.0x
CE, GC, LC and A/D ChemStation Revisions: A.03.0x- A.08.0x This document is believed to be accurate and up-to-date. However, Agilent Technologies, Inc. cannot assume responsibility for the use of this
More informationLecture 5 Rational functions and partial fraction expansion
S. Boyd EE102 Lecture 5 Rational functions and partial fraction expansion (review of) polynomials rational functions pole-zero plots partial fraction expansion repeated poles nonproper rational functions
More informationNumerical Solution of Differential
Chapter 13 Numerical Solution of Differential Equations We have considered numerical solution procedures for two kinds of equations: In chapter 10 the unknown was a real number; in chapter 6 the unknown
More informationIntroduction to Time Series Analysis. Lecture 1.
Introduction to Time Series Analysis. Lecture 1. Peter Bartlett 1. Organizational issues. 2. Objectives of time series analysis. Examples. 3. Overview of the course. 4. Time series models. 5. Time series
More information