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

Size: px
Start display at page:

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

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 n-1, then In other way If t 1 < t 2 <.. < t n, then Gauss-Markov Process A Gauss-Markov 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 n-1 o If the sequence {W n } is Gaussian, then the resulting process X(t) is Gauss-Markov 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 auto-correlation 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 un-correlated. 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 in-phase and quadratic components of X(t) and are lowpass process Theorem: o If X(t) is a zero-mean stationary random process, the processes are also zero-mean, 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 cross-correlation function of and is expressed as o The auto-correlation function of the bandpass process X(t) is expressed in terms of the autocorrelation function and the cross-correlation function as

11

Chapter 8 - Power Density Spectrum

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 information

Digital Transmission (Line Coding)

Digital 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 information

3.5.1 CORRELATION MODELS FOR FREQUENCY SELECTIVE FADING

3.5.1 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 information

5 Signal Design for Bandlimited Channels

5 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 information

NRZ Bandwidth - HF Cutoff vs. SNR

NRZ 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 information

EE 179 April 21, 2014 Digital and Analog Communication Systems Handout #16 Homework #2 Solutions

EE 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.3-3). For the signal g(t) shown below, sketch: a. g(t 4); b. g(t/.5); c. g(t

More information

MODULATION Systems (part 1)

MODULATION Systems (part 1) Technologies and Services on Digital Broadcasting (8) MODULATION Systems (part ) "Technologies and Services of Digital Broadcasting" (in Japanese, ISBN4-339-62-2) is published by CORONA publishing co.,

More information

The continuous and discrete Fourier transforms

The 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 information

2 The wireless channel

2 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 information

min ǫ = E{e 2 [n]}. (11.2)

min 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 mean-square-error (MMSE) estimation of a WSS random process of interest,

More information

Digital Baseband Modulation

Digital 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 information

RANDOM VIBRATION AN OVERVIEW by Barry Controls, Hopkinton, MA

RANDOM 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 information

Department of Electrical and Computer Engineering Ben-Gurion University of the Negev. LAB 1 - Introduction to USRP

Department of Electrical and Computer Engineering Ben-Gurion University of the Negev. LAB 1 - Introduction to USRP Department of Electrical and Computer Engineering Ben-Gurion University of the Negev LAB 1 - Introduction to USRP - 1-1 Introduction In this lab you will use software reconfigurable RF hardware from National

More information

Lecture 1-6: Noise and Filters

Lecture 1-6: Noise and Filters Lecture 1-6: 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 information

Signal Detection C H A P T E R 14 14.1 SIGNAL DETECTION AS HYPOTHESIS TESTING

Signal 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 information

Chapter 4 - Lecture 1 Probability Density Functions and Cumul. Distribution Functions

Chapter 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 information

Sampling Theorem Notes. Recall: That a time sampled signal is like taking a snap shot or picture of signal periodically.

Sampling 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 information

Lecture 8: Signal Detection and Noise Assumption

Lecture 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 information

Univariate and Multivariate Methods PEARSON. Addison Wesley

Univariate 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 information

RF SYSTEM DESIGN OF TRANSCEIVERS FOR WIRELESS COMMUNICATIONS

RF 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 information

TCOM 370 NOTES 99-4 BANDWIDTH, FREQUENCY RESPONSE, AND CAPACITY OF COMMUNICATION LINKS

TCOM 370 NOTES 99-4 BANDWIDTH, FREQUENCY RESPONSE, AND CAPACITY OF COMMUNICATION LINKS TCOM 370 NOTES 99-4 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 information

EE2/ISE2 Communications II

EE2/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 information

Power Spectral Density

Power 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 information

T = 10-' s. p(t)= ( (t-nt), T= 3. n=-oo. Figure P16.2

T = 10-' s. p(t)= ( (t-nt), T= 3. n=-oo. Figure P16.2 16 Sampling Recommended Problems P16.1 The sequence x[n] = (-1)' is obtained by sampling the continuous-time sinusoidal signal x(t) = cos oot at 1-ms intervals, i.e., cos(oont) = (-1)", Determine three

More information

Chapter 3 RANDOM VARIATE GENERATION

Chapter 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 information

Ergodic Capacity of Continuous-Time, Frequency-Selective Rayleigh Fading Channels with Correlated Scattering

Ergodic Capacity of Continuous-Time, Frequency-Selective Rayleigh Fading Channels with Correlated Scattering Ergodic Capacity of Continuous-Time, Frequency-Selective Rayleigh Fading Channels with Correlated Scattering IEEE Information Theory Winter School 2009, Loen, Norway Christian Scheunert, Martin Mittelbach,

More information

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional 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 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

More information

UNIVERSITY 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 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 2-sided sheet of

More information

Master s Theory Exam Spring 2006

Master 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 information

Solutions to Exam in Speech Signal Processing EN2300

Solutions 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 information

ANALYZER BASICS WHAT IS AN FFT SPECTRUM ANALYZER? 2-1

ANALYZER BASICS WHAT IS AN FFT SPECTRUM ANALYZER? 2-1 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 information

Short-time FFT, Multi-taper analysis & Filtering in SPM12

Short-time FFT, Multi-taper analysis & Filtering in SPM12 Short-time FFT, Multi-taper analysis & Filtering in SPM12 Computational Psychiatry Seminar, FS 2015 Daniel Renz, Translational Neuromodeling Unit, ETHZ & UZH 20.03.2015 Overview Refresher Short-time Fourier

More information

TTT4120 Digital Signal Processing Suggested Solution to Exam Fall 2008

TTT4120 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 input-output

More information

INTRODUCTION TO PROBABILITY AND RANDOM PROCESSES

INTRODUCTION 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 information

SIGNAL PROCESSING & SIMULATION NEWSLETTER

SIGNAL 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 information

Introduction to Probability

Introduction 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 information

Some probability and statistics

Some 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 information

Example/ an analog signal f ( t) ) is sample by f s = 5000 Hz draw the sampling signal spectrum. Calculate min. sampling frequency.

Example/ 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 information

From Fundamentals of Digital Communication Copyright by Upamanyu Madhow, 2003-2006

From Fundamentals of Digital Communication Copyright by Upamanyu Madhow, 2003-2006 Chapter Introduction to Modulation From Fundamentals of Digital Communication Copyright by Upamanyu Madhow, 003-006 Modulation refers to the representation of digital information in terms of analog waveforms

More information

Section 6.1 Joint Distribution Functions

Section 6.1 Joint Distribution Functions Section 6.1 Joint Distribution Functions We often care about more than one random variable at a time. DEFINITION: For any two random variables X and Y the joint cumulative probability distribution function

More information

1 Short Introduction to Time Series

1 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 information

Convolution, Correlation, & Fourier Transforms. James R. Graham 10/25/2005

Convolution, 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

Introduction to IQ-demodulation of RF-data

Introduction to IQ-demodulation of RF-data Introduction to IQ-demodulation of RF-data by Johan Kirkhorn, IFBT, NTNU September 15, 1999 Table of Contents 1 INTRODUCTION...3 1.1 Abstract...3 1.2 Definitions/Abbreviations/Nomenclature...3 1.3 Referenced

More information

Correlation and Convolution Class Notes for CMSC 426, Fall 2005 David Jacobs

Correlation 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 information

Time series analysis Matlab tutorial. Joachim Gross

Time 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 information

Review Jeopardy. Blue vs. Orange. Review Jeopardy

Review Jeopardy. Blue vs. Orange. Review Jeopardy Review Jeopardy Blue vs. Orange Review Jeopardy Jeopardy Round Lectures 0-3 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 information

Chapter 10 Introduction to Time Series Analysis

Chapter 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 information

Voice---is analog in character and moves in the form of waves. 3-important wave-characteristics:

Voice---is analog in character and moves in the form of waves. 3-important wave-characteristics: Voice Transmission --Basic Concepts-- Voice---is analog in character and moves in the form of waves. 3-important wave-characteristics: Amplitude Frequency Phase Voice Digitization in the POTS Traditional

More information

Systems of Linear Equations

Systems of Linear Equations Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and

More information

Appendix D Digital Modulation and GMSK

Appendix D Digital Modulation and GMSK D1 Appendix D Digital Modulation and GMSK A brief introduction to digital modulation schemes is given, showing the logical development of GMSK from simpler schemes. GMSK is of interest since it is used

More information

Continuity. DEFINITION 1: A function f is continuous at a number a if. lim

Continuity. 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 information

Symbol 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 information

CCNY. BME I5100: Biomedical Signal Processing. Linear Discrimination. Lucas C. Parra Biomedical Engineering Department City College of New York

CCNY. 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 information

CITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION

CITY 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 information

Jitter Measurements in Serial Data Signals

Jitter Measurements in Serial Data Signals Jitter Measurements in Serial Data Signals Michael Schnecker, Product Manager LeCroy Corporation Introduction The increasing speed of serial data transmission systems places greater importance on measuring

More information

EE 570: Location and Navigation

EE 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 information

LECTURE 4. Last time: Lecture outline

LECTURE 4. Last time: Lecture outline LECTURE 4 Last time: Types of convergence Weak Law of Large Numbers Strong Law of Large Numbers Asymptotic Equipartition Property Lecture outline Stochastic processes Markov chains Entropy rate Random

More information

International Journal of Computer Sciences and Engineering. Research Paper Volume-4, Issue-4 E-ISSN: 2347-2693

International Journal of Computer Sciences and Engineering. Research Paper Volume-4, Issue-4 E-ISSN: 2347-2693 International Journal of Computer Sciences and Engineering Open Access Research Paper Volume-4, Issue-4 E-ISSN: 2347-2693 PAPR Reduction Method for the Localized and Distributed DFTS-OFDM System Using

More information

Lecture 8 ELE 301: Signals and Systems

Lecture 8 ELE 301: Signals and Systems Lecture 8 ELE 3: Signals and Systems Prof. Paul Cuff Princeton University Fall 2-2 Cuff (Lecture 7) ELE 3: Signals and Systems Fall 2-2 / 37 Properties of the Fourier Transform Properties of the Fourier

More information

EECS 556 Image Processing W 09. Interpolation. Interpolation techniques B splines

EECS 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 information

Generating Random Numbers Variance Reduction Quasi-Monte Carlo. Simulation Methods. Leonid Kogan. MIT, Sloan. 15.450, Fall 2010

Generating Random Numbers Variance Reduction Quasi-Monte 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 Quasi-Monte

More information

TTT4110 Information and Signal Theory Solution to exam

TTT4110 Information and Signal Theory Solution to exam Norwegian University of Science and Technology Department of Electronics and Telecommunications TTT4 Information and Signal Theory Solution to exam Problem I (a The frequency response is found by taking

More information

Load Balancing and Switch Scheduling

Load Balancing and Switch Scheduling EE384Y Project Final Report Load Balancing and Switch Scheduling Xiangheng Liu Department of Electrical Engineering Stanford University, Stanford CA 94305 Email: liuxh@systems.stanford.edu Abstract Load

More information

THE 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. 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

Principles of Digital Communication

Principles of Digital Communication Principles of Digital Communication Robert G. Gallager January 5, 2008 ii Preface: introduction and objectives The digital communication industry is an enormous and rapidly growing industry, roughly comparable

More information

Implementation of Digital Signal Processing: Some Background on GFSK Modulation

Implementation 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 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.

. (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 Kolmogorov-Smirnov 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 information

1 if 1 x 0 1 if 0 x 1

1 if 1 x 0 1 if 0 x 1 Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

More information

Application Note Noise Frequently Asked Questions

Application Note Noise Frequently Asked Questions : What is? is a random signal inherent in all physical components. It directly limits the detection and processing of all information. The common form of noise is white Gaussian due to the many random

More information

Time Series Analysis in WinIDAMS

Time Series Analysis in WinIDAMS Time Series Analysis in WinIDAMS P.S. Nagpaul, New Delhi, India April 2005 1 Introduction A time series is a sequence of observations, which are ordered in time (or space). In electrical engineering literature,

More information

Transformations and Expectations of random variables

Transformations and Expectations of random variables Transformations and Epectations of random variables X F X (): a random variable X distributed with CDF F X. Any function Y = g(x) is also a random variable. If both X, and Y are continuous random variables,

More information

Vector and Matrix Norms

Vector and Matrix Norms Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty

More information

Course Curriculum for Master Degree in Electrical Engineering/Wireless Communications

Course Curriculum for Master Degree in Electrical Engineering/Wireless Communications Course Curriculum for Master Degree in Electrical Engineering/Wireless Communications The Master Degree in Electrical Engineering/Wireless Communications, is awarded by the Faculty of Graduate Studies

More information

Least Squares Estimation

Least Squares Estimation Least Squares Estimation SARA A VAN DE GEER Volume 2, pp 1041 1045 in Encyclopedia of Statistics in Behavioral Science ISBN-13: 978-0-470-86080-9 ISBN-10: 0-470-86080-4 Editors Brian S Everitt & David

More information

Math 431 An Introduction to Probability. Final Exam Solutions

Math 431 An Introduction to Probability. Final Exam Solutions Math 43 An Introduction to Probability Final Eam Solutions. A continuous random variable X has cdf a for 0, F () = for 0 <

More information

Time Series Analysis

Time Series Analysis Time Series Analysis Autoregressive, MA and ARMA processes Andrés M. Alonso Carolina García-Martos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 212 Alonso and García-Martos

More information

Sampling and Interpolation. Yao Wang Polytechnic University, Brooklyn, NY11201

Sampling and Interpolation. Yao Wang Polytechnic University, Brooklyn, NY11201 Sampling and Interpolation Yao Wang Polytechnic University, Brooklyn, NY1121 http://eeweb.poly.edu/~yao Outline Basics of sampling and quantization A/D and D/A converters Sampling Nyquist sampling theorem

More information

Capacity Limits of MIMO Channels

Capacity Limits of MIMO Channels Tutorial and 4G Systems Capacity Limits of MIMO Channels Markku Juntti Contents 1. Introduction. Review of information theory 3. Fixed MIMO channels 4. Fading MIMO channels 5. Summary and Conclusions References

More information

Lecture 5: Variants of the LMS algorithm

Lecture 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 information

Experiment 3: Double Sideband Modulation (DSB)

Experiment 3: Double Sideband Modulation (DSB) Experiment 3: Double Sideband Modulation (DSB) This experiment examines the characteristics of the double-sideband (DSB) linear modulation process. The demodulation is performed coherently and its strict

More information

Advanced Signal Processing and Digital Noise Reduction

Advanced Signal Processing and Digital Noise Reduction Advanced Signal Processing and Digital Noise Reduction Saeed V. Vaseghi Queen's University of Belfast UK WILEY HTEUBNER A Partnership between John Wiley & Sons and B. G. Teubner Publishers Chichester New

More information

Scott C. Douglas, et. Al. Convergence Issues in the LMS Adaptive Filter. 2000 CRC Press LLC. <http://www.engnetbase.com>.

Scott C. Douglas, et. Al. Convergence Issues in the LMS Adaptive Filter. 2000 CRC Press LLC. <http://www.engnetbase.com>. Scott C. Douglas, et. Al. Convergence Issues in the LMS Adaptive Filter. 2000 CRC Press LLC. . Convergence Issues in the LMS Adaptive Filter Scott C. Douglas University of Utah

More information

1 The Brownian bridge construction

1 The Brownian bridge construction The Brownian bridge construction The Brownian bridge construction is a way to build a Brownian motion path by successively adding finer scale detail. This construction leads to a relatively easy proof

More information

FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL

FEGYVERNEKI 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 information

Michael Hiebel. Fundamentals of Vector Network Analysis

Michael Hiebel. Fundamentals of Vector Network Analysis Michael Hiebel Fundamentals of Vector Network Analysis TABIH OF CONTENTS Table of contents 1 Introduction 12 1.1 What is a network analyzer? 12 1.2 Wave quantities and S-parameters 13 1.3 Why vector network

More information

5. Continuous Random Variables

5. Continuous Random Variables 5. Continuous Random Variables Continuous random variables can take any value in an interval. They are used to model physical characteristics such as time, length, position, etc. Examples (i) Let X be

More information

Trend and Seasonal Components

Trend and Seasonal Components Chapter 2 Trend and Seasonal Components If the plot of a TS reveals an increase of the seasonal and noise fluctuations with the level of the process then some transformation may be necessary before doing

More information

6.025J Medical Device Design Lecture 3: Analog-to-Digital Conversion Prof. Joel L. Dawson

6.025J Medical Device Design Lecture 3: Analog-to-Digital Conversion Prof. Joel L. Dawson Let s go back briefly to lecture 1, and look at where ADC s and DAC s fit into our overall picture. I m going in a little extra detail now since this is our eighth lecture on electronics and we are more

More information

Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

More information

BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBERT SPACE REVIEW BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

More information

Aggregate Loss Models

Aggregate Loss Models Aggregate Loss Models Chapter 9 Stat 477 - Loss Models Chapter 9 (Stat 477) Aggregate Loss Models Brian Hartman - BYU 1 / 22 Objectives Objectives Individual risk model Collective risk model Computing

More information

Chapter 1. Vector autoregressions. 1.1 VARs and the identi cation problem

Chapter 1. Vector autoregressions. 1.1 VARs and the identi cation problem Chapter Vector autoregressions We begin by taking a look at the data of macroeconomics. A way to summarize the dynamics of macroeconomic data is to make use of vector autoregressions. VAR models have become

More information

I. Pointwise convergence

I. Pointwise convergence MATH 40 - NOTES Sequences of functions Pointwise and Uniform Convergence Fall 2005 Previously, we have studied sequences of real numbers. Now we discuss the topic of sequences of real valued functions.

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete

More information

L9: Cepstral analysis

L9: Cepstral analysis L9: Cepstral analysis The cepstrum Homomorphic filtering The cepstrum and voicing/pitch detection Linear prediction cepstral coefficients Mel frequency cepstral coefficients This lecture is based on [Taylor,

More information

Parametric Statistical Modeling

Parametric Statistical Modeling Parametric Statistical Modeling ECE 275A Statistical Parameter Estimation Ken Kreutz-Delgado ECE Department, UC San Diego Ken Kreutz-Delgado (UC San Diego) ECE 275A SPE Version 1.1 Fall 2012 1 / 12 Why

More information

Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab

Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab Monte Carlo Simulation: IEOR E4703 Fall 2004 c 2004 by Martin Haugh Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab 1 Overview of Monte Carlo Simulation 1.1 Why use simulation?

More information

PSTAT 120B Probability and Statistics

PSTAT 120B Probability and Statistics - Week University of California, Santa Barbara April 10, 013 Discussion section for 10B Information about TA: Fang-I CHU Office: South Hall 5431 T Office hour: TBA email: chu@pstat.ucsb.edu Slides will

More information

Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model

Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written

More information

VCO Phase noise. Characterizing Phase Noise

VCO Phase noise. Characterizing Phase Noise VCO Phase noise Characterizing Phase Noise The term phase noise is widely used for describing short term random frequency fluctuations of a signal. Frequency stability is a measure of the degree to which

More information