Probability and Random Variables. Generation of random variables (r.v.)


 Roxanne James
 1 years ago
 Views:
Transcription
1 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 filtered random process P.S.D and Auto Correlation Generation of random variables (r.v.) o Simulate effects of noise signals o Random phenomenon in the real world o To study the effects through simulation of communication Systems Uniform r.v. A number between 0 and 1 with equal probability 0 <= A <=1 o A random variable ( r.v.) has the range 0 to 1 o The uniform probability density function (pdf) for the r.v. A is denoted by f(a) pdf : Probability density function f(a) The average value or mean value of a r.v. is m A m A =½ pdf can be defined for any r.v. while CDF or PDF can only be defined for continuous r.v.
2 Probability Distribution Function (PDF) or Cumulative distribution function (CDF) F(A) of a uniformly distributed r.v. A The integral of the probability density function (pdf) is called the Probability Distribution Function (PDF) or Cumulative Distribution Function (CDF) of the r.v. A is the area under f(a) denoted by F(A) or F X (x) ( i.e. P(X<=x) ) For any R.V. this area must always be unity ; maximum value of the distribution function. For the uniform random variable A the range of F(A) is 0<= F(A) <= 1 for 0 <= A <=1 Generating uniform distributed noise in interval (b,b+1) can be done by using output A of the random number generator and shifting it by an amount B=A+b B is new r.v. has mean m B = b+½ If b=½ then the r.v. B is uniformly distributed in interval (½, ½)
3 Generating the r.v. with other PDF Zero Mean uniformly distributed r.v. in the range (0,1) can be used to generate the r.v with other PDF. Suppose C with PDF F(C) Since the range of F(C) is the interval (0,1) o Begin with generating a uniformly distributed r.v. A in the range (0,1) and o Set F(C) =A hence C=F 1 (A). o This is the inverse mapping from A to C
4 Gaussian Process A random process X(t) is a Gaussian process if for all n and all (t 1, t 2, t 3,. t n ) the r.v.s have Joint Gaussian density function, f(x) as x vector of n r.vs x = (x 1, x 2,..., x n ) t, m mean, m=e(x) C n x n covariance matrix t transpose C 1 Properties inverse of covariance matrix C o At any time instant t 0 the r.v. X(t 0 ) is Gaussian o At any two points t 1, t 2 the r.vs. (X(t 1 ), X(t 2 )) are distributed according to a two dimensional Gaussian r.v. 1. For a Gaussian processes, the knowledge of the Mean m and the Covariance C provides a complete statistical description of the process. 2. If the Gaussian process X(t) is passed through a LTI system, the output of the system is also a Gaussian process. The effect of the system on X(t) is simply reflected by a change in the mean value and the covariance of X(t)
5 Markov Process o A Markov process X(t) is a random process whose past has no influence on the future if its present is specified. i.e. If t n > t n1, then In other way If t 1 < t 2 <.. < t n, then GaussMarkov Process A GaussMarkov process X(t) is a Markov process whose probability density function is Gaussian o The simplest method for generating a Markov process is by means of a simple recursive formula W n where is a sequence of zero mean i.i.d (white) r.v.s and is a parameter that determines the degree of correlation between X n and X n1 o If the sequence {W n } is Gaussian, then the resulting process X(t) is GaussMarkov FIGURE Gauss Markov Sequence (left) Auto  Correlation of the Gauss Markov process (right)
6 Power Spectrum of Random Processes and White processes A stationary process X(t) is characterize in the frequency domain by its power spectrum which is the F.T. of the autocorrelation function of the random process. That is The autocorrelation function of a stationary random process X(t) is obtained from the power spectrum by means of the IFT White Process A random process X(t) is called the white process i.e. if it has a flat power spectrum is constant for all f o Information sources are modeled as the output of LTI systems driven by a white process. o If then or all f the total power is infinite. No real Physical process can have infinite power and therefore, a white process may not be meaningful physical process
7 However thermal noise can be modeled for all practical purposes as white noise process with the power spectrum equating The value kt is denoted by N0 PSD of thermal noise is PSD of thermal noise is referred to two sided power spectral density For a white random process X(t) with power spectrum the auto correlation function is where is the unit impulse for all we have o If we sample a white process at two points t 1, t 2, r.v.s will be uncorrelated. the resulting o If in addition to being white, the random process is also Gaussian, the sampled r.v.s will be statistically independent Gaussian r.v.s
8 Linear Filtering of Random Processes o Suppose that a stationary random process X(t) is passed through a LTI filter that is characterized in time domain by Its I.R. and in the frequency domain by its F.R. o It follows that the output of the linear filter is the random process o The mean value of Y(t) is where H(0) is the F.R. H(f) of the filter evaluated at f = 0 o The autocorrelation function of Y(t)
9 o In the frequency domain, the power spectrum of the output Y(t) is related to the power spectrum of the input process X(t) and the frequency response of the linear filter by the expression o This is easily know by taking the F.T. of Low pass and Band Pass Processes o Random signals can also be characterized as low pass random processes. Definition o A random process is called lowpass if its power spectrum is larger in the vicinity of f = 0 and small ( approaching 0) at high frequencies. In other words a low pass random process has most of its power contracted at low frequencies. Definition o A lowpass random process X(t) is bandlimited if the power spectrum for the parameter B is called Bandwidth of the random process Definition o A random process is called bandpass if its power spectrum is large in the band of frequencies centered in the neighborhood of a central frequency and relatively small outside this band of frequencies. o A random process is called narrowband if its bandwidth the vicinity of f=0 and small ( approaching 0) at high frequencies. o In other words a low pass random process has most of its power contracted at low frequencies. in
10 Properties o Bandpass process are suitable for representing modulated signals. o The information bearing signal is usually a lowpass random process that modulates over a bandpass (narrowband) communication channel. o Modulated signal is a bandpass random process. o A bandpass random process X(t) can be represented as where and are called inphase and quadratic components of X(t) and are lowpass process Theorem: o If X(t) is a zeromean stationary random process, the processes are also zeromean, jointly stationary processes. o The autocorrelation functions of and are identical and may be expressed as and where is the autocorrelation function of the bandpass process X(t) and is the Hilbert transform of and is defined as o The crosscorrelation function of and is expressed as o The autocorrelation function of the bandpass process X(t) is expressed in terms of the autocorrelation function and the crosscorrelation function as
11
Chapter 8  Power Density Spectrum
EE385 Class Notes 8/8/03 John Stensby Chapter 8  Power Density Spectrum Let X(t) be a WSS random process. X(t) has an average power, given in watts, of E[X(t) ], a constant. his total average power is
More informationPurpose of Time Series Analysis. Autocovariance Function. Autocorrelation Function. Part 3: Time Series I
Part 3: Time Series I Purpose of Time Series Analysis (Figure from Panofsky and Brier 1968) Autocorrelation Function Harmonic Analysis Spectrum Analysis Data Window Significance Tests Some major purposes
More informationINTRODUCTION TO SIGNAL PROCESSING
INTRODUCTION TO SIGNAL PROCESSING Iasonas Kokkinos Ecole Centrale Paris Lecture 7 Introduction to Random Signals Sources of randomness Inherent in the signal generation Noise due to imaging Prostate MRI
More informationOutline. Random Variables. Examples. Random Variable
Outline Random Variables M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno Random variables. CDF and pdf. Joint random variables. Correlated, independent, orthogonal. Correlation,
More informationWhat is a Filter? Output Signal. Input Signal Amplitude. Frequency. Low Pass Filter
What is a Filter? Input Signal Amplitude Output Signal Frequency Time Sequence Low Pass Filter Time Sequence What is a Filter Input Signal Amplitude Output Signal Frequency Signal Noise Signal Noise Frequency
More informationELECE8104 Stochastics models and estimation, Lecture 3b: Linear Estimation in Static Systems
Stochastics models and estimation, Lecture 3b: Linear Estimation in Static Systems Minimum Mean Square Error (MMSE) MMSE estimation of Gaussian random vectors Linear MMSE estimator for arbitrarily distributed
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 information3.5.1 CORRELATION MODELS FOR FREQUENCY SELECTIVE FADING
Environment Spread Flat Rural.5 µs Urban 5 µs Hilly 2 µs Mall.3 µs Indoors.1 µs able 3.1: ypical delay spreads for various environments. If W > 1 τ ds, then the fading is said to be frequency selective,
More informationmin ǫ = E{e 2 [n]}. (11.2)
C H A P T E R 11 Wiener Filtering INTRODUCTION In this chapter we will consider the use of LTI systems in order to perform minimum meansquareerror (MMSE) estimation of a WSS random process of interest,
More informationNRZ Bandwidth  HF Cutoff vs. SNR
Application Note: HFAN09.0. 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 informationLecture 3: Quantization Effects
Lecture 3: Quantization Effects Reading: 6.76.8. We have so far discussed the design of discretetime filters, not digital filters. To understand the characteristics of digital filters, we need first
More informationLecture Notes for ECE 361. Fall 1995
Introduction to Digital Communication Systems Lecture Notes for ECE 361 Fall 1995 Dilip V. Sarwate Department of Electrical and Computer Engineering University of Illinois at UrbanaChampaign Urbana, Illinois
More informationDigital Transmission (Line Coding)
Digital Transmission (Line Coding) Pulse Transmission Source Multiplexer Line Coder Line Coding: Output of the multiplexer (TDM) is coded into electrical pulses or waveforms for the purpose of transmission
More informationEE 179 April 21, 2014 Digital and Analog Communication Systems Handout #16 Homework #2 Solutions
EE 79 April, 04 Digital and Analog Communication Systems Handout #6 Homework # Solutions. Operations on signals (Lathi& Ding.33). For the signal g(t) shown below, sketch: a. g(t 4); b. g(t/.5); c. g(t
More informationEE4512 Analog and Digital Communications EE4513 Analog and Digital Communications Laboratory Dr. Dennis Silage
EE4512 Analog and Digital Communications EE4513 Analog and Digital Communications Laboratory Dr. Dennis Silage silage@temple.edu Course syllabus Course textbooks Course grades Course objectives EE4512
More informationMODULATION Systems (part 1)
Technologies and Services on Digital Broadcasting (8) MODULATION Systems (part ) "Technologies and Services of Digital Broadcasting" (in Japanese, ISBN4339622) is published by CORONA publishing co.,
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 informationLecture 16: Noise and Filters
Lecture 16: Noise and Filters Overview 1. Periodic and Aperiodic Signals Review: by periodic signals, we mean signals that have a waveform shape that repeats. The time taken for the waveform to repeat
More informationDepartment of Electrical and Computer Engineering BenGurion University of the Negev. LAB 1  Introduction to USRP
Department of Electrical and Computer Engineering BenGurion University of the Negev LAB 1  Introduction to USRP  11 Introduction In this lab you will use software reconfigurable RF hardware from National
More informationPower Spectral Density
C H A P E R 0 Power Spectral Density INRODUCION Understanding how the strength of a signal is distributed in the frequency domain, relative to the strengths of other ambient signals, is central to the
More informationSHOCK AND VIBRATION RESPONSE SPECTRA COURSE Unit 18. Filtering
SHOCK AND VIBRATION RESPONSE SPECTRA COURSE Unit 18. Filtering By Tom Irvine Email: tomirvine@aol.com Introduction Filtering is a tool for resolving signals. Filtering can be performed on either analog
More informationChapter 4  Lecture 1 Probability Density Functions and Cumul. Distribution Functions
Chapter 4  Lecture 1 Probability Density Functions and Cumulative Distribution Functions October 21st, 2009 Review Probability distribution function Useful results Relationship between the pdf and the
More informationThe continuous and discrete Fourier transforms
FYSA21 Mathematical Tools in Science The continuous and discrete Fourier transforms Lennart Lindegren Lund Observatory (Department of Astronomy, Lund University) 1 The continuous Fourier transform 1.1
More information2 The wireless channel
CHAPTER The wireless channel A good understanding of the wireless channel, its key physical parameters and the modeling issues, lays the foundation for the rest of the book. This is the goal of this chapter.
More informationEE2/ISE2 Communications II
EE2/ISE2 Communications II Part I Communications Principles Dr. Darren Ward Chapter 1 Introduction 1.1 Background Communication involves the transfer of information from one point to another. In general,
More informationSYSM 6304: Risk and Decision Analysis Lecture 3 Monte Carlo Simulation
SYSM 6304: Risk and Decision Analysis Lecture 3 Monte Carlo Simulation M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu September 19, 2015 Outline
More informationST 371 (VIII): Theory of Joint Distributions
ST 371 (VIII): Theory of Joint Distributions So far we have focused on probability distributions for single random variables. However, we are often interested in probability statements concerning two or
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 informationDigital Baseband Modulation
Digital Baseband Modulation Later Outline Baseband & Bandpass Waveforms Baseband & Bandpass Waveforms, Modulation A Communication System Dig. Baseband Modulators (Line Coders) Sequence of bits are modulated
More informationSampling Theorem Notes. Recall: That a time sampled signal is like taking a snap shot or picture of signal periodically.
Sampling Theorem We will show that a band limited signal can be reconstructed exactly from its discrete time samples. Recall: That a time sampled signal is like taking a snap shot or picture of signal
More informationUnivariate and Multivariate Methods PEARSON. Addison Wesley
Time Series Analysis Univariate and Multivariate Methods SECOND EDITION William W. S. Wei Department of Statistics The Fox School of Business and Management Temple University PEARSON Addison Wesley Boston
More informationTCOM 370 NOTES 994 BANDWIDTH, FREQUENCY RESPONSE, AND CAPACITY OF COMMUNICATION LINKS
TCOM 370 NOTES 994 BANDWIDTH, FREQUENCY RESPONSE, AND CAPACITY OF COMMUNICATION LINKS 1. Bandwidth: The bandwidth of a communication link, or in general any system, was loosely defined as the width of
More informationRF SYSTEM DESIGN OF TRANSCEIVERS FOR WIRELESS COMMUNICATIONS
RF SYSTEM DESIGN OF TRANSCEIVERS FOR WIRELESS COMMUNICATIONS Qizheng Gu Nokia Mobile Phones, Inc. 4y Springer Contents Preface xiii Chapter 1. Introduction 1 1.1. Wireless Systems 1 1.1.1. Mobile Communications
More information4.3 Moving Average Process MA(q)
66 CHAPTER 4. STATIONARY TS MODELS 4.3 Moving Average Process MA(q) Definition 4.5. {X t } is a movingaverage process of order q if X t = Z t + θ 1 Z t 1 +... + θ q Z t q, (4.9) where and θ 1,...,θ q
More informationMaximum Entropy. Information Theory 2013 Lecture 9 Chapter 12. Tohid Ardeshiri. May 22, 2013
Maximum Entropy Information Theory 2013 Lecture 9 Chapter 12 Tohid Ardeshiri May 22, 2013 Why Maximum Entropy distribution? max f (x) h(f ) subject to E r(x) = α Temperature of a gas corresponds to the
More informationElectromagnetic Spectrum
Introduction 1 Electromagnetic Spectrum The electromagnetic spectrum is the distribution of electromagnetic radiation according to energy, frequency, or wavelength. The electromagnetic radiation can be
More informationT = 10' s. p(t)= ( (tnt), T= 3. n=oo. Figure P16.2
16 Sampling Recommended Problems P16.1 The sequence x[n] = (1)' is obtained by sampling the continuoustime sinusoidal signal x(t) = cos oot at 1ms intervals, i.e., cos(oont) = (1)", Determine three
More informationINTRODUCTION TO PROBABILITY AND RANDOM PROCESSES
APPENDIX H INTRODUCTION TO PROBABILITY AND RANDOM PROCESSES This appendix is not intended to be a definitive dissertation on the subject of random processes. The major concepts, definitions, and results
More informationM5MS09. Graphical Modelling
Course: MMS09 Setter: Walden Checker: Ginzberg Editor: Calderhead External: Wood Date: April, 0 MSc EXAMINATIONS (STATISTICS) MayJune 0 MMS09 Graphical Modelling Setter s signature Checker s signature
More informationABHELSINKI UNIVERSITY OF TECHNOLOGY
Fading Models S72.333 Physical Layer Methods in Wireless Communication Systems Fabio Belloni Helsinki University of Technology Signal Processing Laboratory fbelloni@wooster.hut.fi 23 November 2004 Belloni,F.;
More information568 Subject Index. exponential distribution, 106, 108, 539 extension field, 545, 551 eye diagram, 193
SUBJECT INDEX additive system, 182 additive white Gaussian noise, 99, 103 aliasing, 52, 54 amplifier AM/AM characteristic, 332, 342 AM/PM characteristic, 332, 342 nonlinear, 344 twobox model, 344 amplifiers,
More informationL10: Probability, statistics, and estimation theory
L10: Probability, statistics, and estimation theory Review of probability theory Bayes theorem Statistics and the Normal distribution Least Squares Error estimation Maximum Likelihood estimation Bayesian
More informationRoundoff Noise in IIR Digital Filters
Chapter 16 Roundoff Noise in IIR Digital Filters It will not be possible in this brief chapter to discuss all forms of IIR (infinite impulse response) digital filters and how quantization takes place in
More informationFrequency Response and Continuoustime Fourier Transform
Frequency Response and Continuoustime Fourier Transform Goals Signals and Systems in the FDpart II I. (Finiteenergy) signals in the Frequency Domain  The Fourier Transform of a signal  Classification
More informationEvaluation of measurement uncertainty for timedependent quantities
EPJ Web of Conferences 77, 00003 ( 2014) DOI: 10.1051/ epjconf/ 20147700003 C Owned by the authors, published by EDP Sciences, 2014 Evaluation of measurement uncertainty for timedependent quantities Sascha
More informationSTAT/MTHE 353: Probability II. STAT/MTHE 353: Multiple Random Variables. Review. Administrative details. Instructor: TamasLinder
STAT/MTHE 353: Probability II STAT/MTHE 353: Multiple Random Variables Administrative details Instructor: TamasLinder Email: linder@mast.queensu.ca T. Linder ueen s University Winter 2012 O ce: Je ery
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 informationLOWPASS/BANDPASS SIGNAL RECONSTRUCTION AND DIGITAL FILTERING FROM NONUNIFORM SAMPLES. David Bonacci and Bernard Lacaze
LOWPASS/BANDPASS SIGNAL RECONSTRUCTION AND DIGITAL FILTERING FROM NONUNIFORM SAMPLES David Bonacci and Bernard Lacaze TESA Laboratory  7 Boulevard de la Gare  315 Toulouse  France ABSTRACT This paper
More informationProbability and Statistics
CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b  0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute  Systems and Modeling GIGA  Bioinformatics ULg kristel.vansteen@ulg.ac.be
More informationChapter 3 RANDOM VARIATE GENERATION
Chapter 3 RANDOM VARIATE GENERATION In order to do a Monte Carlo simulation either by hand or by computer, techniques must be developed for generating values of random variables having known distributions.
More informationErgodic Capacity of ContinuousTime, FrequencySelective Rayleigh Fading Channels with Correlated Scattering
Ergodic Capacity of ContinuousTime, FrequencySelective Rayleigh Fading Channels with Correlated Scattering IEEE Information Theory Winter School 2009, Loen, Norway Christian Scheunert, Martin Mittelbach,
More informationTime Series Analysis
Time Series Analysis Time series and stochastic processes Andrés M. Alonso Carolina GarcíaMartos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 2012 Alonso and GarcíaMartos
More informationUNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical & Computer Engineering Department ECE 101  Fall 2010 Linear Systems Fundamentals
UNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical & Computer Engineering Department ECE 101  Fall 2010 Linear Systems Fundamentals FINAL EXAM WITH SOLUTIONS (YOURS!) You are allowed one 2sided sheet of
More informationMaster s Theory Exam Spring 2006
Spring 2006 This exam contains 7 questions. You should attempt them all. Each question is divided into parts to help lead you through the material. You should attempt to complete as much of each problem
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 information1 1. The ROC of the ztransform H(z) of the impulse response sequence h[n] is defined
A causal LTI digital filter is BIBO stable if and only if its impulse response h[ is absolutely summable, i.e., S h [ < n We now develop a stability condition in terms of the pole locations of the transfer
More informationTTT4120 Digital Signal Processing Suggested Solution to Exam Fall 2008
Norwegian University of Science and Technology Department of Electronics and Telecommunications TTT40 Digital Signal Processing Suggested Solution to Exam Fall 008 Problem (a) The input and the inputoutput
More informationANALYZER BASICS WHAT IS AN FFT SPECTRUM ANALYZER? 21
WHAT IS AN FFT SPECTRUM ANALYZER? ANALYZER BASICS The SR760 FFT Spectrum Analyzer takes a time varying input signal, like you would see on an oscilloscope trace, and computes its frequency spectrum. Fourier's
More informationModulation methods. S72. 333 Physical layer methods in wireless communication systems. Sylvain Ranvier / Radio Laboratory / TKK 16 November 2004
Modulation methods S72. 333 Physical layer methods in wireless communication systems Sylvain Ranvier / Radio Laboratory / TKK 16 November 2004 sylvain.ranvier@hut.fi SMARAD / Radio Laboratory 1 Line out
More informationNoise Terminology: An Overview of Noise Terminology and Applications Author: Bob Muro, Applications Engineer
Noise Terminology: An Overview of Noise Terminology and Applications Author: Bob Muro, Applications Engineer Today s Webinar Important noise characteristics Technologies effected by noise Noise applications
More informationStefanos D. Georgiadis Perttu O. Rantaaho Mika P. Tarvainen Pasi A. Karjalainen. University of Kuopio Department of Applied Physics Kuopio, FINLAND
5 Finnish Signal Processing Symposium (Finsig 5) Kuopio, Finland Stefanos D. Georgiadis Perttu O. Rantaaho Mika P. Tarvainen Pasi A. Karjalainen University of Kuopio Department of Applied Physics Kuopio,
More informationIntroduction to Probability
Introduction to Probability EE 179, Lecture 15, Handout #24 Probability theory gives a mathematical characterization for experiments with random outcomes. coin toss life of lightbulb binary data sequence
More informationLinear Dependence Tests
Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks
More informationFading multipath radio channels
Fading multipath radio channels Narrowband channel modelling Wideband channel modelling Wideband WSSUS channel (functions, variables & distributions) Lowpass equivalent (LPE) signal ( ) = Re ( ) s t RF
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More informationEstimation with Minimum Mean Square Error
C H A P T E R 8 Estimation with Minimum Mean Square Error INTRODUCTION A recurring theme in this text and in much of communication, control and signal processing is that of making systematic estimates,
More information4.6 Null Space, Column Space, Row Space
NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear
More informationNotes for STA 437/1005 Methods for Multivariate Data
Notes for STA 437/1005 Methods for Multivariate Data Radford M. Neal, 26 November 2010 Random Vectors Notation: Let X be a random vector with p elements, so that X = [X 1,..., X p ], where denotes transpose.
More informationExercises with solutions (1)
Exercises with solutions (). Investigate the relationship between independence and correlation. (a) Two random variables X and Y are said to be correlated if and only if their covariance C XY is not equal
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 informationFrom Fundamentals of Digital Communication Copyright by Upamanyu Madhow, 20032006
Chapter Introduction to Modulation From Fundamentals of Digital Communication Copyright by Upamanyu Madhow, 003006 Modulation refers to the representation of digital information in terms of analog waveforms
More informationTopic 4: Multivariate random variables. Multiple random variables
Topic 4: Multivariate random variables Joint, marginal, and conditional pmf Joint, marginal, and conditional pdf and cdf Independence Expectation, covariance, correlation Conditional expectation Two jointly
More informationShorttime FFT, Multitaper analysis & Filtering in SPM12
Shorttime FFT, Multitaper analysis & Filtering in SPM12 Computational Psychiatry Seminar, FS 2015 Daniel Renz, Translational Neuromodeling Unit, ETHZ & UZH 20.03.2015 Overview Refresher Shorttime Fourier
More informationChapter 10 Introduction to Time Series Analysis
Chapter 1 Introduction to Time Series Analysis A time series is a collection of observations made sequentially in time. Examples are daily mortality counts, particulate air pollution measurements, and
More informationTime series analysis Matlab tutorial. Joachim Gross
Time series analysis Matlab tutorial Joachim Gross Outline Terminology Sampling theorem Plotting Baseline correction Detrending Smoothing Filtering Decimation Remarks Focus on practical aspects, exercises,
More informationReview Jeopardy. Blue vs. Orange. Review Jeopardy
Review Jeopardy Blue vs. Orange Review Jeopardy Jeopardy Round Lectures 03 Jeopardy Round $200 How could I measure how far apart (i.e. how different) two observations, y 1 and y 2, are from each other?
More informationFILTER CIRCUITS. A filter is a circuit whose transfer function, that is the ratio of its output to its input, depends upon frequency.
FILTER CIRCUITS Introduction Circuits with a response that depends upon the frequency of the input voltage are known as filters. Filter circuits can be used to perform a number of important functions in
More informationContinuity. DEFINITION 1: A function f is continuous at a number a if. lim
Continuity DEFINITION : A function f is continuous at a number a if f(x) = f(a) REMARK: It follows from the definition that f is continuous at a if and only if. f(a) is defined. 2. f(x) and +f(x) exist.
More informationEE 179, Lecture 18, Handout #31 Line Coding for Digital Communication
EE 179, Lecture 18, Handout #31 Line Coding for Digital Communication Goal is to transmit binary data (e.g., PCM encoded voice, MPEG encoded video, financial information) Transmission distance is large
More information3.6: General Hypothesis Tests
3.6: General Hypothesis Tests The χ 2 goodness of fit tests which we introduced in the previous section were an example of a hypothesis test. In this section we now consider hypothesis tests more generally.
More informationVoiceis analog in character and moves in the form of waves. 3important wavecharacteristics:
Voice Transmission Basic Concepts Voiceis analog in character and moves in the form of waves. 3important wavecharacteristics: Amplitude Frequency Phase Voice Digitization in the POTS Traditional
More informationChapter 5: Joint Probability Distributions. Chapter Learning Objectives. The Joint Probability Distribution for a Pair of Discrete Random
Chapter 5: Joint Probability Distributions 51 Two or More Random Variables 51.1 Joint Probability Distributions 51.2 Marginal Probability Distributions 51.3 Conditional Probability Distributions 51.4
More informationSome probability and statistics
Appendix A Some probability and statistics A Probabilities, random variables and their distribution We summarize a few of the basic concepts of random variables, usually denoted by capital letters, X,Y,
More informationSymbol interval T=1/(2B); symbol rate = 1/T=2B transmissions/sec (The transmitted baseband signal is assumed to be real here) Noise power = (N_0/2)(2B)=N_0B \Gamma is no smaller than 1 The encoded PAM
More informationCITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION
No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August
More informationEE 570: Location and Navigation
EE 570: Location and Navigation OnLine Bayesian Tracking Aly ElOsery 1 Stephen Bruder 2 1 Electrical Engineering Department, New Mexico Tech Socorro, New Mexico, USA 2 Electrical and Computer Engineering
More informationSIGNAL PROCESSING & SIMULATION NEWSLETTER
1 of 10 1/25/2008 3:38 AM SIGNAL PROCESSING & SIMULATION NEWSLETTER Note: This is not a particularly interesting topic for anyone other than those who ar e involved in simulation. So if you have difficulty
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 informationExample/ an analog signal f ( t) ) is sample by f s = 5000 Hz draw the sampling signal spectrum. Calculate min. sampling frequency.
1 2 3 4 Example/ an analog signal f ( t) = 1+ cos(4000πt ) is sample by f s = 5000 Hz draw the sampling signal spectrum. Calculate min. sampling frequency. Sol/ H(f) 7KHz 5KHz 3KHz 2KHz 0 2KHz 3KHz
More informationInternational Journal of Computer Sciences and Engineering. Research Paper Volume4, Issue4 EISSN: 23472693
International Journal of Computer Sciences and Engineering Open Access Research Paper Volume4, Issue4 EISSN: 23472693 PAPR Reduction Method for the Localized and Distributed DFTSOFDM System Using
More informationDigital Speech Processing Lectures 78. Time Domain Methods in Speech Processing
Digital Speech Processing Lectures 78 Time Domain Methods in Speech Processing 1 General Synthesis Model voiced sound amplitude Log Areas, Reflection Coefficients, Formants, Vocal Tract Polynomial, Articulatory
More informationWelcome to Stochastic Processes 1. Welcome to Aalborg University No. 1 of 31
Welcome to Stochastic Processes 1 Welcome to Aalborg University No. 1 of 31 Welcome to Aalborg University No. 2 of 31 Course Plan Part 1: Probability concepts, random variables and random processes Lecturer:
More informationGenerating Random Numbers Variance Reduction QuasiMonte Carlo. Simulation Methods. Leonid Kogan. MIT, Sloan. 15.450, Fall 2010
Simulation Methods Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Simulation Methods 15.450, Fall 2010 1 / 35 Outline 1 Generating Random Numbers 2 Variance Reduction 3 QuasiMonte
More informationLecture 8 ELE 301: Signals and Systems
Lecture 8 ELE 3: Signals and Systems Prof. Paul Cuff Princeton University Fall 22 Cuff (Lecture 7) ELE 3: Signals and Systems Fall 22 / 37 Properties of the Fourier Transform Properties of the Fourier
More informationEECS 556 Image Processing W 09. Interpolation. Interpolation techniques B splines
EECS 556 Image Processing W 09 Interpolation Interpolation techniques B splines What is image processing? Image processing is the application of 2D signal processing methods to images Image representation
More informationTHE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE. Alexander Barvinok
THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE Alexer Barvinok Papers are available at http://www.math.lsa.umich.edu/ barvinok/papers.html This is a joint work with J.A. Hartigan
More information. (3.3) n Note that supremum (3.2) must occur at one of the observed values x i or to the left of x i.
Chapter 3 KolmogorovSmirnov Tests There are many situations where experimenters need to know what is the distribution of the population of their interest. For example, if they want to use a parametric
More informationCorrelation and Convolution Class Notes for CMSC 426, Fall 2005 David Jacobs
Correlation and Convolution Class otes for CMSC 46, Fall 5 David Jacobs Introduction Correlation and Convolution are basic operations that we will perform to extract information from images. They are in
More informationImplementation of Digital Signal Processing: Some Background on GFSK Modulation
Implementation of Digital Signal Processing: Some Background on GFSK Modulation Sabih H. Gerez University of Twente, Department of Electrical Engineering s.h.gerez@utwente.nl Version 4 (February 7, 2013)
More informationConvolution, Correlation, & Fourier Transforms. James R. Graham 10/25/2005
Convolution, Correlation, & Fourier Transforms James R. Graham 10/25/2005 Introduction A large class of signal processing techniques fall under the category of Fourier transform methods These methods fall
More information