ELEG 3143 Probability & Stochastic Process Ch. 6 Stochastic Process

Size: px
Start display at page:

Download "ELEG 3143 Probability & Stochastic Process Ch. 6 Stochastic Process"

Transcription

1 Department of Electrical Engineering University of Arkansas ELEG 3143 Probability & Stochastic Process Ch. 6 Stochastic Process Dr. Jingxian Wu wuj@uark.edu

2 OUTLINE 2 Definition of stochastic process (random process) Description of continuous-time random process Description of discrete-time random process Stationary and non-stationary Two random processes Ergodic and non-ergodic random process Power spectral density

3 DEFINITION 3 Stochastic process, or, random process A random variable changes with respect to time Example: the temperature in the room At any given moment, the temperature is random: random variable The temperature (the value of the random variable) changes with respect to time.

4 DEFINITION 4 Stochastic process (random process) Recall: Random variable is a mapping from random events to real number: (), where is a random event Stochastic process is a random variable changes with time Denoted as: ( t, ) It is a function of random event, and time t. Recall: random variable is a function of random event

5 Ensemble domain DEFINITION 5 Stochastic process (random process) Fix time: (, ) is a random variable t k ( t, ) pdf, CDF, mean, variance, moments, etc. fix event: t, ) is a deterministic time function ( k Defined as a realization of a random process Or: a sample function of a random process time domain

6 DEFINITION 6 Example Let be a Gaussian random variable with 0 mean and unit variance, then ( t, ) cos(2 ft) is a random process. If we fix t t 0, we get a Gaussian random variable The pdf of ( t 0, ) is: ( t 0, ) cos(2 ft0) If we fix 0, we get sample function It is a deterministic function of time ( 0 t, 0) cos(2 ft)

7 DEFINITION 7 Stochastic process: A random variable changes with respect to time ( t, ) Sample function: A deterministic time function t, ) associated with outcome ( k Ensemble: The set of all possible time functions of a stochastic process.

8 DEFINITION 8 Example Let be a random variable uniformly distributed between [, ], then ( t, ) Acos(2 ft ) is a random process, where A is a constant. At t t 0 Find the mean and variance of ( t 0, )

9 DEFINITION 9 Example Let be a random variable uniformly distributed between [, ], and 2 a Gaussian random variable with 0 mean and variance. Then cos( 2ft ) is a random process. and are independent. Find the mean and variance of the random process at t t 0

10 DEFINITION 10 Continuous-time random process The time is continuous E.g. ( t, ) cos(2 ft) Discrete-time random process The time is discrete E.g. ( n, fn) ) cos(2 Continuous random process At any time, ( t 0, ) is a continuous random variable E.g. ( t, ) cos(2 ft), is Gaussian distributed Discrete random process t R n 0,1,2, At any time, ( t 0, ) is a discrete random variable E.g. ( t, ) cos(2 ft), is a Bernoulli RV

11 DEFINITION 11 Classifications Continuous-time v.s. discrete-time Continuous v.s. discrete Stationary v.s. non-stationary Wide sense stationary (WSS) v.s. non-wss Ergodic v.s. non-ergodic Simple notation Random variable: (), usually denoted as Random process: ( t, ), usually denoted as (t)

12 OUTLINE 12 Definition of stochastic process (random process) Description of continuous-time random process Description of discrete-time random process Stationary and non-stationary Two random processes Ergodic and non-ergodic random process Power spectral density

13 DESCRIPTION 13 How do we describe a random process? A random variable is fully characterized by its pdf or CDF. How do we statistically describe random process? E.g. We need to describe the random process in both the ensemble domain and the time domain Descriptions of the random process Joint distribution of time samples: joint pdf, joint CDF Moments: mean, variance, correlation function, covariance function

14 DESCRIPTION 14 Joint distribution of time samples Consider a random process (t) Let joint CDF: Joint pdf: A random process is fully specified by the collections of all the joint CDFs (or joint pdfs) for any n and any choice of sampling instants. For a continuous-time random process, there will be infinite such joint CDFs.

15 DESCRIPTION 15 Moments of time samples Provide a partial description of the random process For most practical applications it is sufficient to have a partial description. Mean function m The mean function is a deterministic function of time. Variance function ( ( t) t) E[ ( t)] uf ( u) du 2 2 [ ( t) m ( t)] x m ( t) f ( x dx 2 ( ( t) t) E )

16 DESCRIPTION 16 Example For a random process ( t) ACos(2 ft), where A is a random variable with mean and variance. Find the mean function and variance function of (t). m 2 A A

17 DESCRIPTION 17 Autocorrelation function (ACF) The autocorrelation function of a random process (t) is defined as the R It describes the correlation of the random process in the time domain How are two events happened at different times related to each other. E.g. if there is a strong correlation between the temperature today and the temperature tomorrow, then we can predict tomorrow s temperature by using today s observation. E.g. the text in a book can be considered as a discrete random process» Stochasxxx» We can easily guess the contents in xxx by using the time correlation. 2 is the second moment of ( t1, t2) E ( t1) ( t2) R t, t ) E ( ) t ) ( 1 1 t1 ( 1

18 DESCRIPTION Autocovariance function The autocovariance function of a random process (t) is defined as the 18 ) ( ) ( )][ ( ) ( ), ( t m t t m t E t t C ) ( ) ( )] ( ) ( [ t m t m t t E 2 ) ( ), ( t t t C

19 DESCRIPTION 19 Example Consider a random process ( t) Acos(2 ft), where A is a random variable with mean and variance 2. Find the autocorrelation A function and autocovariance function. m A

20 DESCRIPTION 20 Example Consider a random process ( t) Acos(2 ft ), where A is a random 2 variable with mean and variance A. is uniformly distributed in [, ]. A and are independent. Find the autocorrelation function and autocovariance function. m A

21 DESCRIPTION 21 Example

22 OUTLINE 22 Definition of stochastic process (random process) Description of continuous-time random process Description of discrete-time random process Stationary and non-stationary Two random processes Ergodic and non-ergodic random process Power spectral density

23 DESCRIPTION 23 Discrete-time random process (random sequence) A discrete-time random process E.g. Joint PMF Joint CDF

24 DESCRIPTION 24 Bernoulli process

25 DESCRIPTION 25 Moments of time samples Provide a partial description of the random process For most practical applications it is sufficient to have a partial description. Mean function The mean function is a deterministic function of time. Variance function It is a deterministic function of time

26 DESCRIPTION 26 Example For a Bernoulli (p) process, find the mean function and variance function.

27 DESCRIPTION 27 Autocorrelation function of a random sequence Autocovariance function of a random sequence

28 DESCRIPTION 28 Example

29 OUTLINE 29 Definition of stochastic process (random process) Description of continuous-time random process Description of discrete-time random process Stationary and non-stationary Two random processes Ergodic and non-ergodic random process Power spectral density

30 STATIONARY RANDOM PROCESS 30 Stationary random process A random process, (t), is stationary if the joint distribution of any set of samples does not depend on the time origin. For any value of and, and for any choice of

31 STATIONARY RANDOM PROCESS 31 1 st order distribution If (t) is a stationary random process, then the first order CDF or pdf must be independent of time Mean: F ( x) F ( t )( ( t) x f ( x) f ( t )( ( t) x ), ), t, t, The samples at different time instant have the same distribution. E E (t) ( t ) For a stationary random process, the mean is independent of time

32 STATIONARY RANDOM PROCESS 32 2 nd order distribution If (t) is a stationary random process, then the 2 nd order CDF and pdf are: F f ( x1, x2) F (0) ( )( x1, 2), ( t) ( t ) x ( x1, x2) f (0) ( )( x1, 2), ( t) ( t ) x t, t, The 2 nd order distribution only depends on the time difference between the two samples Autocorrelation function t, t 1 ) R ( 2 Autocovariance function: For a stationary random process, the autocorrelation function and autocovariance function only depends on the time difference: t2 t1

33 STATIONARY RANDOM PROCESS 33 Stationary random process In order to determine whether a random process is stationary, we need to find out the joint distribution of any group of samples Stationary random process the joint distribution of any group of time samples is independent of the starting time This is a very strict requirement, and sometimes it is difficult to determine whether a random process is stationary.

34 STATIONARY RANDOM PROCESS 34 Wide-Sense Stationary (WSS) Random Process A random process, (t), is wide-sense stationary (WSS), if the following two conditions are satisfied The first moment is independent of time E ( t) E[ ( t )] m The autocorrelation function depends only on the time difference E ( t) ( t ) R ( ) Only consider the 1 st order and 2 nd order distributions

35 STATIONARY RANDOM PROCESS 35 Stationary v.s. Wide-Sense Stationary If (t) is stationary (t) is WSS It is not true the other way around Stationary is a much stricter condition. It requires the joint distribution of any combination of samples to be independent of the absolute starting time WSS only considers the first moment (mean is a constant) and second order moment (autocorrelation function depends only on the time difference) E E ( t) E[ ( t )] m ( t) ( t ) R ( )

36 STATIONARY RANDOM PROCESS 36 Example A random process is described by ( t) A Bcos(2 ft ) where A is a random variable uniformly distributed between [-3, 3], B is an RV with zero mean and variance 4, and is a random variable uniformly distributed in [ / 2,3 / 2]. A, B, and are independent. Find the mean and autocorrelation function. Is (t) WSS?

37 STATIONARY RANDOM PROCESS 37 Autocorrelation function of a WSS random process R ( ) E ( t) ( t ) R (0) 2 ( t) is the average power of the signal (t) E ) R ( ) E ( t) ( t ) E ( t ) ( t) R ( is an even function R ( ) R (0) Cauchy-Schwartz inequality If R ( ) is non-periodic, then lim R ( ) m 2

38 STATIONARY RANDOM PROCESS 38 Example Consider a random process having an autocorrelation function Find the mean and variance of (t)

39 STATIONARY RANDOM PROCESS 39 Example A random process Z(t) is Z ( t ) ( t ) ( t t 0) Where (t) is a WSS random process with autocorrelation function R Z 2 ( ) exp( ) Find the autocorrelation function of Z(t). Is Z(t) WSS?

40 STATIONARY RANDOM PROCESS 40 Example A random process (t) = At + B, where A is a Gaussian random variable with 0 mean and variance 16, and B is uniformly distributed between 0 and 6. A and B are independent. Find the mean and auto-correlation function. Is it WSS?

41 OUTLINE 41 Definition of stochastic process (random process) Description of continuous-time random process Description of discrete-time random process Stationary and non-stationary Two random processes Ergodic and non-ergodic random process Power spectral density

42 TWO RANDOM PROCESSES 42 Two random processes (t) and Y(t) Cross-correlation function The cross-correlation function between two random processes (t) and Y(t) is defined as t, t ) E[ ( t ) Y ( )] R Y ( t2 Two random processes are said to be uncorrelated if for all and E[ ( t1) Y( t2)] E[ ( t1)] E[ Y( t2)] Two random processes are said to be orthogonal if for all and R Y ( t, t2) Cross-covariance function 1 t1 2 t1 2 The cross-covariance function between two random processes (t) and Y(t) is defined as C Y ( t1, t2) E[ ( t1) Y( t2)] m ( t1) m ( t2) 0 t t

43 TWO RANDOM PROCESSES 43 Example Consider two random processes ( t) cos(2 ft ) and Y( t) sin(2 ft ). Are they uncorrelated?

44 TWO RANDOM PROCESSES 44 Example Suppose signal Y(t) consists a desired signal (t) plus noise N(t) as Y(t) = (t) + N(t) The autocorrelation functions of (t) and N(t) are: R ( t, t ) and R ( t, t 1 2 NN 1 2), respectively. The mean function of (t) and N(t) are: m (t) and m N (t), respectively. Find the cross-correlation between (t) and Y(t).

45 OUTLINE 45 Definition of stochastic process (random process) Description of continuous-time random process Description of discrete-time random process Stationary and non-stationary Two random processes Ergodic and non-ergodic random process Power spectral density

46 Ensemble domain ERGODIC RANDOM PROCESS 46 Time average of a signal x(t) x( t) T 1 T T 0 x( t) dt Time average of a sample function of the random process x( t) ( t, 0) 1 T ( t, 0 ) T ( t, 0) dt T 0 Ensemble average (mean) of a random process ( 0 t, ) E t, ) ( xf ( t, )( x ) dx time domain

47 ERGODIC RANDOM PROCESS 47 Ergodic random process A stationary random process is also an ergodic random process if the n-th order ensemble average is the same as the n-th order time average. n n ( t) E ( t) x n ( t) 1 lim T T T 0 x n ( t) dt E n t) n ( x f ( t) ( x) dx If a random process is ergodic, we can find the moments by performing time average over a single sample function. Ergodic is only defined for stationary random process If a random process is ergodic, then it must be stationary Not true the other way around

48 ERGODIC RANDOM PROCESS 48 Example Consider a random process ( t) A where A is a Gaussian RV with 0 mean and variance 4. Is it ergodic?

49 ERGODIC RANDOM PROCESS 49 Mean ergodic A WSS random process is mean ergodic if the ensemble average is the same as the time average. ( t) E ( t) Mean ergodic v.s. ergodic Ergodic: Mean ergodic: n n ( t) E ( t) ( t) E ( t) for stationary process for WSS process If a random process is ergodic, then it must be mean ergodic Not true the other way around If a process is mean ergodic, it must be WSS Mean ergodic is defined for WSS process only.

50 ERGODIC RANDOM PROCESS 50 Example Consider a random process ( t) Acos(2 ft ) where A is a non-zero constant, and is uniformly distributed between 0 and. Is it mean ergodic?

51 OUTLINE 51 Definition of stochastic process (random process) Description of continuous-time random process Description of discrete-time random process Stationary and non-stationary Two random processes Ergodic and non-ergodic random process Power spectral density

52 POWER SPECTRAL DENSITY 52 Power spectral density (PSD) The distribution of the power in the frequency domain. For a WSS random process, the PSD is the Fourier transform of the autocorrelation function S ( f ) F[ ( )] R The density of power in the frequency domain. The power consumption between the frequency range P f 2 f 1 S ( f ) df [ f 1, f2]:

53 POWER SPECTRAL DENSITY 53 White noise Autocorrelation function R x any two samples in the time domain are uncorrelated. Power spectral density N0 ( ) ( ) 2 N 2 0 N0 S x ( f ) ( ) d 2 ( ) R N0/2 S f N0/2 f

Chapter 6. Random Processes

Chapter 6. Random Processes Chapter 6 Random Processes Random Process A random process is a time-varying function that assigns the outcome of a random experiment to each time instant: X(t). For a fixed (sample path): a random process

More information

Stochastic Processes. M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno

Stochastic Processes. M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno Stochastic Processes M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno 1 Outline Stochastic (random) processes. Autocorrelation. Crosscorrelation. Spectral density function.

More information

Stochastic Processes

Stochastic Processes Elements of Lecture II Hamid R. Rabiee with thanks to Ali Jalali Overview Reading Assignment Chapter 9 of textbook Further Resources MIT Open Course Ware S. Karlin and H. M. Taylor, A First Course in Stochastic

More information

Stochastic Processes. A stochastic process is a function of two variables:

Stochastic Processes. A stochastic process is a function of two variables: Stochastic Processes Stochastic: from Greek stochastikos, proceeding by guesswork, literally, skillful in aiming. A stochastic process is simply a collection of random variables labelled by some parameter:

More information

ENSC327 Communications Systems 19: Random Processes. Jie Liang School of Engineering Science Simon Fraser University

ENSC327 Communications Systems 19: Random Processes. Jie Liang School of Engineering Science Simon Fraser University ENSC327 Communications Systems 19: Random Processes Jie Liang School of Engineering Science Simon Fraser University 1 Outline Random processes Stationary random processes Autocorrelation of random processes

More information

ELEG 3143 Probability & Stochastic Process Ch. 4 Multiple Random Variables

ELEG 3143 Probability & Stochastic Process Ch. 4 Multiple Random Variables Department o Electrical Engineering University o Arkansas ELEG 3143 Probability & Stochastic Process Ch. 4 Multiple Random Variables Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Two discrete random variables

More information

for valid PSD. PART B (Answer all five units, 5 X 10 = 50 Marks) UNIT I

for valid PSD. PART B (Answer all five units, 5 X 10 = 50 Marks) UNIT I Code: 15A04304 R15 B.Tech II Year I Semester (R15) Regular Examinations November/December 016 PROBABILITY THEY & STOCHASTIC PROCESSES (Electronics and Communication Engineering) Time: 3 hours Max. Marks:

More information

Parametric Signal Modeling and Linear Prediction Theory 1. Discrete-time Stochastic Processes

Parametric Signal Modeling and Linear Prediction Theory 1. Discrete-time Stochastic Processes Parametric Signal Modeling and Linear Prediction Theory 1. Discrete-time Stochastic Processes Electrical & Computer Engineering North Carolina State University Acknowledgment: ECE792-41 slides were adapted

More information

ELEG 3143 Probability & Stochastic Process Ch. 2 Discrete Random Variables

ELEG 3143 Probability & Stochastic Process Ch. 2 Discrete Random Variables Department of Electrical Engineering University of Arkansas ELEG 3143 Probability & Stochastic Process Ch. 2 Discrete Random Variables Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Random Variable Discrete Random

More information

Lecture Notes 7 Stationary Random Processes. Strict-Sense and Wide-Sense Stationarity. Autocorrelation Function of a Stationary Process

Lecture Notes 7 Stationary Random Processes. Strict-Sense and Wide-Sense Stationarity. Autocorrelation Function of a Stationary Process Lecture Notes 7 Stationary Random Processes Strict-Sense and Wide-Sense Stationarity Autocorrelation Function of a Stationary Process Power Spectral Density Continuity and Integration of Random Processes

More information

16.584: Random (Stochastic) Processes

16.584: Random (Stochastic) Processes 1 16.584: Random (Stochastic) Processes X(t): X : RV : Continuous function of the independent variable t (time, space etc.) Random process : Collection of X(t, ζ) : Indexed on another independent variable

More information

P 1.5 X 4.5 / X 2 and (iii) The smallest value of n for

P 1.5 X 4.5 / X 2 and (iii) The smallest value of n for DHANALAKSHMI COLLEGE OF ENEINEERING, CHENNAI DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING MA645 PROBABILITY AND RANDOM PROCESS UNIT I : RANDOM VARIABLES PART B (6 MARKS). A random variable X

More information

EAS 305 Random Processes Viewgraph 1 of 10. Random Processes

EAS 305 Random Processes Viewgraph 1 of 10. Random Processes EAS 305 Random Processes Viewgraph 1 of 10 Definitions: Random Processes A random process is a family of random variables indexed by a parameter t T, where T is called the index set λ i Experiment outcome

More information

Fundamentals of Digital Commun. Ch. 4: Random Variables and Random Processes

Fundamentals of Digital Commun. Ch. 4: Random Variables and Random Processes Fundamentals of Digital Commun. Ch. 4: Random Variables and Random Processes Klaus Witrisal witrisal@tugraz.at Signal Processing and Speech Communication Laboratory www.spsc.tugraz.at Graz University of

More information

Random Processes Why we Care

Random Processes Why we Care Random Processes Why we Care I Random processes describe signals that change randomly over time. I Compare: deterministic signals can be described by a mathematical expression that describes the signal

More information

Definition of a Stochastic Process

Definition of a Stochastic Process Definition of a Stochastic Process Balu Santhanam Dept. of E.C.E., University of New Mexico Fax: 505 277 8298 bsanthan@unm.edu August 26, 2018 Balu Santhanam (UNM) August 26, 2018 1 / 20 Overview 1 Stochastic

More information

Probability and Statistics for Final Year Engineering Students

Probability and Statistics for Final Year Engineering Students Probability and Statistics for Final Year Engineering Students By Yoni Nazarathy, Last Updated: May 24, 2011. Lecture 6p: Spectral Density, Passing Random Processes through LTI Systems, Filtering Terms

More information

Name of the Student: Problems on Discrete & Continuous R.Vs

Name of the Student: Problems on Discrete & Continuous R.Vs Engineering Mathematics 05 SUBJECT NAME : Probability & Random Process SUBJECT CODE : MA6 MATERIAL NAME : University Questions MATERIAL CODE : JM08AM004 REGULATION : R008 UPDATED ON : Nov-Dec 04 (Scan

More information

where r n = dn+1 x(t)

where r n = dn+1 x(t) Random Variables Overview Probability Random variables Transforms of pdfs Moments and cumulants Useful distributions Random vectors Linear transformations of random vectors The multivariate normal distribution

More information

Statistical signal processing

Statistical signal processing Statistical signal processing Short overview of the fundamentals Outline Random variables Random processes Stationarity Ergodicity Spectral analysis Random variable and processes Intuition: A random variable

More information

SRI VIDYA COLLEGE OF ENGINEERING AND TECHNOLOGY UNIT 3 RANDOM PROCESS TWO MARK QUESTIONS

SRI VIDYA COLLEGE OF ENGINEERING AND TECHNOLOGY UNIT 3 RANDOM PROCESS TWO MARK QUESTIONS UNIT 3 RANDOM PROCESS TWO MARK QUESTIONS 1. Define random process? The sample space composed of functions of time is called a random process. 2. Define Stationary process? If a random process is divided

More information

Deterministic. Deterministic data are those can be described by an explicit mathematical relationship

Deterministic. Deterministic data are those can be described by an explicit mathematical relationship Random data Deterministic Deterministic data are those can be described by an explicit mathematical relationship Deterministic x(t) =X cos r! k m t Non deterministic There is no way to predict an exact

More information

Signals and Spectra - Review

Signals and Spectra - Review Signals and Spectra - Review SIGNALS DETERMINISTIC No uncertainty w.r.t. the value of a signal at any time Modeled by mathematical epressions RANDOM some degree of uncertainty before the signal occurs

More information

Chapter 5 Random Variables and Processes

Chapter 5 Random Variables and Processes Chapter 5 Random Variables and Processes Wireless Information Transmission System Lab. Institute of Communications Engineering National Sun Yat-sen University Table of Contents 5.1 Introduction 5. Probability

More information

Name of the Student: Problems on Discrete & Continuous R.Vs

Name of the Student: Problems on Discrete & Continuous R.Vs Engineering Mathematics 08 SUBJECT NAME : Probability & Random Processes SUBJECT CODE : MA645 MATERIAL NAME : University Questions REGULATION : R03 UPDATED ON : November 07 (Upto N/D 07 Q.P) (Scan the

More information

Probability Space. J. McNames Portland State University ECE 538/638 Stochastic Signals Ver

Probability Space. J. McNames Portland State University ECE 538/638 Stochastic Signals Ver Stochastic Signals Overview Definitions Second order statistics Stationarity and ergodicity Random signal variability Power spectral density Linear systems with stationary inputs Random signal memory Correlation

More information

Stochastic Process II Dr.-Ing. Sudchai Boonto

Stochastic Process II Dr.-Ing. Sudchai Boonto Dr-Ing Sudchai Boonto Department of Control System and Instrumentation Engineering King Mongkuts Unniversity of Technology Thonburi Thailand Random process Consider a random experiment specified by the

More information

Fig 1: Stationary and Non Stationary Time Series

Fig 1: Stationary and Non Stationary Time Series Module 23 Independence and Stationarity Objective: To introduce the concepts of Statistical Independence, Stationarity and its types w.r.to random processes. This module also presents the concept of Ergodicity.

More information

PROBABILITY AND RANDOM PROCESSESS

PROBABILITY AND RANDOM PROCESSESS PROBABILITY AND RANDOM PROCESSESS SOLUTIONS TO UNIVERSITY QUESTION PAPER YEAR : JUNE 2014 CODE NO : 6074 /M PREPARED BY: D.B.V.RAVISANKAR ASSOCIATE PROFESSOR IT DEPARTMENT MVSR ENGINEERING COLLEGE, NADERGUL

More information

LECTURES 2-3 : Stochastic Processes, Autocorrelation function. Stationarity.

LECTURES 2-3 : Stochastic Processes, Autocorrelation function. Stationarity. LECTURES 2-3 : Stochastic Processes, Autocorrelation function. Stationarity. Important points of Lecture 1: A time series {X t } is a series of observations taken sequentially over time: x t is an observation

More information

Stochastic Processes. Monday, November 14, 11

Stochastic Processes. Monday, November 14, 11 Stochastic Processes 1 Definition and Classification X(, t): stochastic process: X : T! R (, t) X(, t) where is a sample space and T is time. {X(, t) is a family of r.v. defined on {, A, P and indexed

More information

Probability and Statistics

Probability and Statistics Probability and Statistics 1 Contents some stochastic processes Stationary Stochastic Processes 2 4. Some Stochastic Processes 4.1 Bernoulli process 4.2 Binomial process 4.3 Sine wave process 4.4 Random-telegraph

More information

Prof. Dr.-Ing. Armin Dekorsy Department of Communications Engineering. Stochastic Processes and Linear Algebra Recap Slides

Prof. Dr.-Ing. Armin Dekorsy Department of Communications Engineering. Stochastic Processes and Linear Algebra Recap Slides Prof. Dr.-Ing. Armin Dekorsy Department of Communications Engineering Stochastic Processes and Linear Algebra Recap Slides Stochastic processes and variables XX tt 0 = XX xx nn (tt) xx 2 (tt) XX tt XX

More information

Chapter 6: Random Processes 1

Chapter 6: Random Processes 1 Chapter 6: Random Processes 1 Yunghsiang S. Han Graduate Institute of Communication Engineering, National Taipei University Taiwan E-mail: yshan@mail.ntpu.edu.tw 1 Modified from the lecture notes by Prof.

More information

3. ESTIMATION OF SIGNALS USING A LEAST SQUARES TECHNIQUE

3. ESTIMATION OF SIGNALS USING A LEAST SQUARES TECHNIQUE 3. ESTIMATION OF SIGNALS USING A LEAST SQUARES TECHNIQUE 3.0 INTRODUCTION The purpose of this chapter is to introduce estimators shortly. More elaborated courses on System Identification, which are given

More information

Communication Theory II

Communication Theory II Communication Theory II Lecture 8: Stochastic Processes Ahmed Elnakib, PhD Assistant Professor, Mansoura University, Egypt March 5 th, 2015 1 o Stochastic processes What is a stochastic process? Types:

More information

Stochastic Processes: I. consider bowl of worms model for oscilloscope experiment:

Stochastic Processes: I. consider bowl of worms model for oscilloscope experiment: Stochastic Processes: I consider bowl of worms model for oscilloscope experiment: SAPAscope 2.0 / 0 1 RESET SAPA2e 22, 23 II 1 stochastic process is: Stochastic Processes: II informally: bowl + drawing

More information

STOCHASTIC PROBABILITY THEORY PROCESSES. Universities Press. Y Mallikarjuna Reddy EDITION

STOCHASTIC PROBABILITY THEORY PROCESSES. Universities Press. Y Mallikarjuna Reddy EDITION PROBABILITY THEORY STOCHASTIC PROCESSES FOURTH EDITION Y Mallikarjuna Reddy Department of Electronics and Communication Engineering Vasireddy Venkatadri Institute of Technology, Guntur, A.R < Universities

More information

7 The Waveform Channel

7 The Waveform Channel 7 The Waveform Channel The waveform transmitted by the digital demodulator will be corrupted by the channel before it reaches the digital demodulator in the receiver. One important part of the channel

More information

Massachusetts Institute of Technology

Massachusetts Institute of Technology Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.011: Introduction to Communication, Control and Signal Processing QUIZ, April 1, 010 QUESTION BOOKLET Your

More information

ECE 650 Lecture #10 (was Part 1 & 2) D. van Alphen. D. van Alphen 1

ECE 650 Lecture #10 (was Part 1 & 2) D. van Alphen. D. van Alphen 1 ECE 650 Lecture #10 (was Part 1 & 2) D. van Alphen D. van Alphen 1 Lecture 10 Overview Part 1 Review of Lecture 9 Continuing: Systems with Random Inputs More about Poisson RV s Intro. to Poisson Processes

More information

PROBABILITY THEORY. Prof. S. J. Soni. Assistant Professor Computer Engg. Department SPCE, Visnagar

PROBABILITY THEORY. Prof. S. J. Soni. Assistant Professor Computer Engg. Department SPCE, Visnagar PROBABILITY THEORY By Prof. S. J. Soni Assistant Professor Computer Engg. Department SPCE, Visnagar Introduction Signals whose values at any instant t are determined by their analytical or graphical description

More information

Gaussian, Markov and stationary processes

Gaussian, Markov and stationary processes Gaussian, Markov and stationary processes Gonzalo Mateos Dept. of ECE and Goergen Institute for Data Science University of Rochester gmateosb@ece.rochester.edu http://www.ece.rochester.edu/~gmateosb/ November

More information

Module 4. Signal Representation and Baseband Processing. Version 2 ECE IIT, Kharagpur

Module 4. Signal Representation and Baseband Processing. Version 2 ECE IIT, Kharagpur Module Signal Representation and Baseband Processing Version ECE II, Kharagpur Lesson 8 Response of Linear System to Random Processes Version ECE II, Kharagpur After reading this lesson, you will learn

More information

5.9 Power Spectral Density Gaussian Process 5.10 Noise 5.11 Narrowband Noise

5.9 Power Spectral Density Gaussian Process 5.10 Noise 5.11 Narrowband Noise Chapter 5 Random Variables and Processes Wireless Information Transmission System Lab. Institute of Communications Engineering g National Sun Yat-sen University Table of Contents 5.1 Introduction 5. Probability

More information

Introduction to Probability and Stochastic Processes I

Introduction to Probability and Stochastic Processes I Introduction to Probability and Stochastic Processes I Lecture 3 Henrik Vie Christensen vie@control.auc.dk Department of Control Engineering Institute of Electronic Systems Aalborg University Denmark Slides

More information

Lecture - 30 Stationary Processes

Lecture - 30 Stationary Processes Probability and Random Variables Prof. M. Chakraborty Department of Electronics and Electrical Communication Engineering Indian Institute of Technology, Kharagpur Lecture - 30 Stationary Processes So,

More information

Problems on Discrete & Continuous R.Vs

Problems on Discrete & Continuous R.Vs 013 SUBJECT NAME SUBJECT CODE MATERIAL NAME MATERIAL CODE : Probability & Random Process : MA 61 : University Questions : SKMA1004 Name of the Student: Branch: Unit I (Random Variables) Problems on Discrete

More information

E X A M. Probability Theory and Stochastic Processes Date: December 13, 2016 Duration: 4 hours. Number of pages incl.

E X A M. Probability Theory and Stochastic Processes Date: December 13, 2016 Duration: 4 hours. Number of pages incl. E X A M Course code: Course name: Number of pages incl. front page: 6 MA430-G Probability Theory and Stochastic Processes Date: December 13, 2016 Duration: 4 hours Resources allowed: Notes: Pocket calculator,

More information

2. (a) What is gaussian random variable? Develop an equation for guassian distribution

2. (a) What is gaussian random variable? Develop an equation for guassian distribution Code No: R059210401 Set No. 1 II B.Tech I Semester Supplementary Examinations, February 2007 PROBABILITY THEORY AND STOCHASTIC PROCESS ( Common to Electronics & Communication Engineering, Electronics &

More information

Communication Systems Lecture 21, 22. Dong In Kim School of Information & Comm. Eng. Sungkyunkwan University

Communication Systems Lecture 21, 22. Dong In Kim School of Information & Comm. Eng. Sungkyunkwan University Communication Systems Lecture 1, Dong In Kim School of Information & Comm. Eng. Sungkyunkwan University 1 Outline Linear Systems with WSS Inputs Noise White noise, Gaussian noise, White Gaussian noise

More information

Review of Probability

Review of Probability Review of robabilit robabilit Theor: Man techniques in speech processing require the manipulation of probabilities and statistics. The two principal application areas we will encounter are: Statistical

More information

The distribution inherited by Y is called the Cauchy distribution. Using that. d dy ln(1 + y2 ) = 1 arctan(y)

The distribution inherited by Y is called the Cauchy distribution. Using that. d dy ln(1 + y2 ) = 1 arctan(y) Stochastic Processes - MM3 - Solutions MM3 - Review Exercise Let X N (0, ), i.e. X is a standard Gaussian/normal random variable, and denote by f X the pdf of X. Consider also a continuous random variable

More information

Problem Sheet 1 Examples of Random Processes

Problem Sheet 1 Examples of Random Processes RANDOM'PROCESSES'AND'TIME'SERIES'ANALYSIS.'PART'II:'RANDOM'PROCESSES' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''Problem'Sheets' Problem Sheet 1 Examples of Random Processes 1. Give

More information

Module 9: Stationary Processes

Module 9: Stationary Processes Module 9: Stationary Processes Lecture 1 Stationary Processes 1 Introduction A stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space.

More information

Fundamentals of Noise

Fundamentals of Noise Fundamentals of Noise V.Vasudevan, Department of Electrical Engineering, Indian Institute of Technology Madras Noise in resistors Random voltage fluctuations across a resistor Mean square value in a frequency

More information

IV. Covariance Analysis

IV. Covariance Analysis IV. Covariance Analysis Autocovariance Remember that when a stochastic process has time values that are interdependent, then we can characterize that interdependency by computing the autocovariance function.

More information

ECE 450 Homework #3. 1. Given the joint density function f XY (x,y) = 0.5 1<x<2, 2<y< <x<4, 2<y<3 0 else

ECE 450 Homework #3. 1. Given the joint density function f XY (x,y) = 0.5 1<x<2, 2<y< <x<4, 2<y<3 0 else ECE 450 Homework #3 0. Consider the random variables X and Y, whose values are a function of the number showing when a single die is tossed, as show below: Exp. Outcome 1 3 4 5 6 X 3 3 4 4 Y 0 1 3 4 5

More information

ECE302 Spring 2006 Practice Final Exam Solution May 4, Name: Score: /100

ECE302 Spring 2006 Practice Final Exam Solution May 4, Name: Score: /100 ECE302 Spring 2006 Practice Final Exam Solution May 4, 2006 1 Name: Score: /100 You must show ALL of your work for full credit. This exam is open-book. Calculators may NOT be used. 1. As a function of

More information

Stat 248 Lab 2: Stationarity, More EDA, Basic TS Models

Stat 248 Lab 2: Stationarity, More EDA, Basic TS Models Stat 248 Lab 2: Stationarity, More EDA, Basic TS Models Tessa L. Childers-Day February 8, 2013 1 Introduction Today s section will deal with topics such as: the mean function, the auto- and cross-covariance

More information

STAT 248: EDA & Stationarity Handout 3

STAT 248: EDA & Stationarity Handout 3 STAT 248: EDA & Stationarity Handout 3 GSI: Gido van de Ven September 17th, 2010 1 Introduction Today s section we will deal with the following topics: the mean function, the auto- and crosscovariance

More information

A SIGNAL THEORETIC INTRODUCTION TO RANDOM PROCESSES

A SIGNAL THEORETIC INTRODUCTION TO RANDOM PROCESSES A SIGNAL THEORETIC INTRODUCTION TO RANDOM PROCESSES ROY M. HOWARD Department of Electrical Engineering & Computing Curtin University of Technology Perth, Australia WILEY CONTENTS Preface xiii 1 A Signal

More information

ECE 636: Systems identification

ECE 636: Systems identification ECE 636: Systems identification Lectures 3 4 Random variables/signals (continued) Random/stochastic vectors Random signals and linear systems Random signals in the frequency domain υ ε x S z + y Experimental

More information

ECE353: Probability and Random Processes. Lecture 18 - Stochastic Processes

ECE353: Probability and Random Processes. Lecture 18 - Stochastic Processes ECE353: Probability and Random Processes Lecture 18 - Stochastic Processes Xiao Fu School of Electrical Engineering and Computer Science Oregon State University E-mail: xiao.fu@oregonstate.edu From RV

More information

Spectral Analysis of Random Processes

Spectral Analysis of Random Processes Spectral Analysis of Random Processes Spectral Analysis of Random Processes Generally, all properties of a random process should be defined by averaging over the ensemble of realizations. Generally, all

More information

Question Paper Code : AEC11T03

Question Paper Code : AEC11T03 Hall Ticket No Question Paper Code : AEC11T03 VARDHAMAN COLLEGE OF ENGINEERING (AUTONOMOUS) Affiliated to JNTUH, Hyderabad Four Year B Tech III Semester Tutorial Question Bank 2013-14 (Regulations: VCE-R11)

More information

EEM 409. Random Signals. Problem Set-2: (Power Spectral Density, LTI Systems with Random Inputs) Problem 1: Problem 2:

EEM 409. Random Signals. Problem Set-2: (Power Spectral Density, LTI Systems with Random Inputs) Problem 1: Problem 2: EEM 409 Random Signals Problem Set-2: (Power Spectral Density, LTI Systems with Random Inputs) Problem 1: Consider a random process of the form = + Problem 2: X(t) = b cos(2π t + ), where b is a constant,

More information

ECE Homework Set 3

ECE Homework Set 3 ECE 450 1 Homework Set 3 0. Consider the random variables X and Y, whose values are a function of the number showing when a single die is tossed, as show below: Exp. Outcome 1 3 4 5 6 X 3 3 4 4 Y 0 1 3

More information

Fourier Analysis Linear transformations and lters. 3. Fourier Analysis. Alex Sheremet. April 11, 2007

Fourier Analysis Linear transformations and lters. 3. Fourier Analysis. Alex Sheremet. April 11, 2007 Stochastic processes review 3. Data Analysis Techniques in Oceanography OCP668 April, 27 Stochastic processes review Denition Fixed ζ = ζ : Function X (t) = X (t, ζ). Fixed t = t: Random Variable X (ζ)

More information

Lecture 2: Repetition of probability theory and statistics

Lecture 2: Repetition of probability theory and statistics Algorithms for Uncertainty Quantification SS8, IN2345 Tobias Neckel Scientific Computing in Computer Science TUM Lecture 2: Repetition of probability theory and statistics Concept of Building Block: Prerequisites:

More information

ECE Lecture #10 Overview

ECE Lecture #10 Overview ECE 450 - Lecture #0 Overview Introduction to Random Vectors CDF, PDF Mean Vector, Covariance Matrix Jointly Gaussian RV s: vector form of pdf Introduction to Random (or Stochastic) Processes Definitions

More information

Fundamentals of Applied Probability and Random Processes

Fundamentals of Applied Probability and Random Processes Fundamentals of Applied Probability and Random Processes,nd 2 na Edition Oliver C. Ibe University of Massachusetts, LoweLL, Massachusetts ip^ W >!^ AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARIS

More information

Econ 424 Time Series Concepts

Econ 424 Time Series Concepts Econ 424 Time Series Concepts Eric Zivot January 20 2015 Time Series Processes Stochastic (Random) Process { 1 2 +1 } = { } = sequence of random variables indexed by time Observed time series of length

More information

Final. Fall 2016 (Dec 16, 2016) Please copy and write the following statement:

Final. Fall 2016 (Dec 16, 2016) Please copy and write the following statement: ECE 30: Probabilistic Methods in Electrical and Computer Engineering Fall 06 Instructor: Prof. Stanley H. Chan Final Fall 06 (Dec 6, 06) Name: PUID: Please copy and write the following statement: I certify

More information

Basics on 2-D 2 D Random Signal

Basics on 2-D 2 D Random Signal Basics on -D D Random Signal Spring 06 Instructor: K. J. Ray Liu ECE Department, Univ. of Maryland, College Park Overview Last Time: Fourier Analysis for -D signals Image enhancement via spatial filtering

More information

13. Power Spectrum. For a deterministic signal x(t), the spectrum is well defined: If represents its Fourier transform, i.e., if.

13. Power Spectrum. For a deterministic signal x(t), the spectrum is well defined: If represents its Fourier transform, i.e., if. For a deterministic signal x(t), the spectrum is well defined: If represents its Fourier transform, i.e., if jt X ( ) = xte ( ) dt, (3-) then X ( ) represents its energy spectrum. his follows from Parseval

More information

Recitation 2: Probability

Recitation 2: Probability Recitation 2: Probability Colin White, Kenny Marino January 23, 2018 Outline Facts about sets Definitions and facts about probability Random Variables and Joint Distributions Characteristics of distributions

More information

Northwestern University Department of Electrical Engineering and Computer Science

Northwestern University Department of Electrical Engineering and Computer Science Northwestern University Department of Electrical Engineering and Computer Science EECS 454: Modeling and Analysis of Communication Networks Spring 2008 Probability Review As discussed in Lecture 1, probability

More information

2. SPECTRAL ANALYSIS APPLIED TO STOCHASTIC PROCESSES

2. SPECTRAL ANALYSIS APPLIED TO STOCHASTIC PROCESSES 2. SPECTRAL ANALYSIS APPLIED TO STOCHASTIC PROCESSES 2.0 THEOREM OF WIENER- KHINTCHINE An important technique in the study of deterministic signals consists in using harmonic functions to gain the spectral

More information

Stochastic processes Lecture 1: Multiple Random Variables Ch. 5

Stochastic processes Lecture 1: Multiple Random Variables Ch. 5 Stochastic processes Lecture : Multiple Random Variables Ch. 5 Dr. Ir. Richard C. Hendriks 26/04/8 Delft University of Technology Challenge the future Organization Plenary Lectures Book: R.D. Yates and

More information

Chapter 2 Random Processes

Chapter 2 Random Processes Chapter 2 Random Processes 21 Introduction We saw in Section 111 on page 10 that many systems are best studied using the concept of random variables where the outcome of a random experiment was associated

More information

UCSD ECE153 Handout #40 Prof. Young-Han Kim Thursday, May 29, Homework Set #8 Due: Thursday, June 5, 2011

UCSD ECE153 Handout #40 Prof. Young-Han Kim Thursday, May 29, Homework Set #8 Due: Thursday, June 5, 2011 UCSD ECE53 Handout #40 Prof. Young-Han Kim Thursday, May 9, 04 Homework Set #8 Due: Thursday, June 5, 0. Discrete-time Wiener process. Let Z n, n 0 be a discrete time white Gaussian noise (WGN) process,

More information

conditional cdf, conditional pdf, total probability theorem?

conditional cdf, conditional pdf, total probability theorem? 6 Multiple Random Variables 6.0 INTRODUCTION scalar vs. random variable cdf, pdf transformation of a random variable conditional cdf, conditional pdf, total probability theorem expectation of a random

More information

Some Time-Series Models

Some Time-Series Models Some Time-Series Models Outline 1. Stochastic processes and their properties 2. Stationary processes 3. Some properties of the autocorrelation function 4. Some useful models Purely random processes, random

More information

STA 6857 Autocorrelation and Cross-Correlation & Stationary Time Series ( 1.4, 1.5)

STA 6857 Autocorrelation and Cross-Correlation & Stationary Time Series ( 1.4, 1.5) STA 6857 Autocorrelation and Cross-Correlation & Stationary Time Series ( 1.4, 1.5) Outline 1 Announcements 2 Autocorrelation and Cross-Correlation 3 Stationary Time Series 4 Homework 1c Arthur Berg STA

More information

Name of the Student: Problems on Discrete & Continuous R.Vs

Name of the Student: Problems on Discrete & Continuous R.Vs SUBJECT NAME : Probability & Random Processes SUBJECT CODE : MA645 MATERIAL NAME : Additional Problems MATERIAL CODE : JM08AM004 REGULATION : R03 UPDATED ON : March 05 (Scan the above QR code for the direct

More information

Chapter 5,6 Multiple RandomVariables

Chapter 5,6 Multiple RandomVariables Chapter 5,6 Multiple RandomVariables ENCS66 - Probabilityand Stochastic Processes Concordia University Vector RandomVariables A vector r.v. is a function where is the sample space of a random experiment.

More information

TSKS01 Digital Communication Lecture 1

TSKS01 Digital Communication Lecture 1 TSKS01 Digital Communication Lecture 1 Introduction, Repetition, and Noise Modeling Emil Björnson Department of Electrical Engineering (ISY) Division of Communication Systems Emil Björnson Course Director

More information

System Identification

System Identification System Identification Arun K. Tangirala Department of Chemical Engineering IIT Madras July 27, 2013 Module 3 Lecture 1 Arun K. Tangirala System Identification July 27, 2013 1 Objectives of this Module

More information

Stochastic Processes (Master degree in Engineering) Franco Flandoli

Stochastic Processes (Master degree in Engineering) Franco Flandoli Stochastic Processes (Master degree in Engineering) Franco Flandoli Contents Preface v Chapter. Preliminaries of Probability. Transformation of densities. About covariance matrices 3 3. Gaussian vectors

More information

Stochastic Processes. Chapter Definitions

Stochastic Processes. Chapter Definitions Chapter 4 Stochastic Processes Clearly data assimilation schemes such as Optimal Interpolation are crucially dependent on the estimates of background and observation error statistics. Yet, we don t know

More information

Lecture 15. Theory of random processes Part III: Poisson random processes. Harrison H. Barrett University of Arizona

Lecture 15. Theory of random processes Part III: Poisson random processes. Harrison H. Barrett University of Arizona Lecture 15 Theory of random processes Part III: Poisson random processes Harrison H. Barrett University of Arizona 1 OUTLINE Poisson and independence Poisson and rarity; binomial selection Poisson point

More information

Econometría 2: Análisis de series de Tiempo

Econometría 2: Análisis de series de Tiempo Econometría 2: Análisis de series de Tiempo Karoll GOMEZ kgomezp@unal.edu.co http://karollgomez.wordpress.com Segundo semestre 2016 II. Basic definitions A time series is a set of observations X t, each

More information

Random Process Review

Random Process Review Rando Process Review Consider a rando process t, and take k saples. For siplicity, we will set k. However it should ean any nuber of saples. t () t x t, t, t We have a rando vector t, t, t. If we find

More information

3F1 Random Processes Examples Paper (for all 6 lectures)

3F1 Random Processes Examples Paper (for all 6 lectures) 3F Random Processes Examples Paper (for all 6 lectures). Three factories make the same electrical component. Factory A supplies half of the total number of components to the central depot, while factories

More information

Financial Econometrics and Quantitative Risk Managenent Return Properties

Financial Econometrics and Quantitative Risk Managenent Return Properties Financial Econometrics and Quantitative Risk Managenent Return Properties Eric Zivot Updated: April 1, 2013 Lecture Outline Course introduction Return definitions Empirical properties of returns Reading

More information

FINAL EXAM: 3:30-5:30pm

FINAL EXAM: 3:30-5:30pm ECE 30: Probabilistic Methods in Electrical and Computer Engineering Spring 016 Instructor: Prof. A. R. Reibman FINAL EXAM: 3:30-5:30pm Spring 016, MWF 1:30-1:0pm (May 6, 016) This is a closed book exam.

More information

This examination consists of 10 pages. Please check that you have a complete copy. Time: 2.5 hrs INSTRUCTIONS

This examination consists of 10 pages. Please check that you have a complete copy. Time: 2.5 hrs INSTRUCTIONS THE UNIVERSITY OF BRITISH COLUMBIA Department of Electrical and Computer Engineering EECE 564 Detection and Estimation of Signals in Noise Final Examination 08 December 2009 This examination consists of

More information

f (1 0.5)/n Z =

f (1 0.5)/n Z = Math 466/566 - Homework 4. We want to test a hypothesis involving a population proportion. The unknown population proportion is p. The null hypothesis is p = / and the alternative hypothesis is p > /.

More information

ELEG 3124 SYSTEMS AND SIGNALS Ch. 5 Fourier Transform

ELEG 3124 SYSTEMS AND SIGNALS Ch. 5 Fourier Transform Department of Electrical Engineering University of Arkansas ELEG 3124 SYSTEMS AND SIGNALS Ch. 5 Fourier Transform Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Introduction Fourier Transform Properties of Fourier

More information