Stochastic Processes

Stochastic Processes Elements of Stochastic Processes Lecture II

Overview • Reading Assignment • Chapter 9 of textbook • Further Resources • MIT Open Course Ware • S. Karlin and H. M. Taylor, A First Course in Stochastic Processes, 2nd ed., Academic Press, New York, 1975.

Outline • Basic Definitions • Stationary/Ergodic Processes • Stochastic Analysis of Systems • Power Spectrum

Basic Definitions • Suppose a set of random variables indexed by a parameter • Tracking these variables with respect to the parameter constructs a process that is called Stochastic Process. • i.e. The mapping of outcomes to the real (complex) numbers changes with respect to index.

Basic Definitions (cont’d) • In a random process, we will have a family of functions called an ensemble of functions

Basic Definitions (cont’d) • With fixed “beta”, we will have a “time” function called sample path. • Sometimes stochastic properties of a random process can be extracted just from a single sample path. (When?)

Basic Definitions (cont’d) • With fixed “t”, we will have a random variable. • With fixed “t” and “beta”, we will have a real (complex) number.

Basic Definitions (cont’d) • Example I • Brownian Motion • Motion of all particles (ensemble) • Motion of a specific particle (sample path) • Example II • Voltage of a generator with fixed frequency • Amplitude is a random variables

Basic Definitions (cont’d) • Equality • Ensembles should be equal for each “beta” and “t” • Equality (Mean Square Sense) • If the following equality holds • Sufficient in many applications

Basic Definitions (cont’d) • First-Order CDF of a random process • First-Order PDF of a random process

Basic Definitions (cont’d) • Second-Order CDF of a random process • Second-Order PDF of a random process

Basic Definitions (cont’d) • nth order can be defined. (How?) • Relation between first-order and second-order can be presented as • Relation between different orders can be obtained easily. (How?)

Basic Definitions (cont’d) • Mean of a random process • Autocorrelation of a random process • Fact: (Why?)

Basic Definitions (cont’d) • Autocovariance of a random process • Correlation Coefficient • Example

Basic Definitions (cont’d) • Example • Poisson Process • Mean • Autocorrelation • Autocovariance

Basic Definitions (cont’d) • Complex process • Definition • Specified in terms of the joint statistics of two real processes and • Vector Process • A family of some stochastic processes

Basic Definitions (cont’d) • Cross-Correlation • Orthogonal Processes • Cross-Covariance • Uncorrelated Processes

Basic Definitions (cont’d) • a-dependent processes • White Noise • Variance of Stochastic Process

Basic Definitions (cont’d) • Existence Theorem • For an arbitrary mean function • For an arbitrary covariance function • There exist a normal random process that its mean is and its covariance is

Stationary/Ergodic Processes • Strict Sense Stationary (SSS) • Statistical properties are invariant to shift of time origin • First order properties should be independent of “t” or • Second order properties should depends only on difference of times or • …

Stationary/Ergodic Processes (cont’d) • Wide Sense Stationary (WSS) • Mean is constant • Autocorrelation depends on the difference of times • First and Second order statistics are usually enough in applications.

Stationary/Ergodic Processes (cont’d) • Autocovariance of a WSS process • Correlation Coefficient

Stationary/Ergodic Processes (cont’d) • White Noise • If white noise is an stationary process, why do we call it “noise”? (maybe it is not stationary !?) • a-dependent Process • a is called “Correlation Time”

Stationary/Ergodic Processes (cont’d) • Example • SSS • Suppose a and b are normal random variables with zero mean. • WSS • Suppose “ ” has a uniform distribution in the interval

Stationary/Ergodic Processes (cont’d) • Example • Suppose for a WSS process • X(8) and X(5) are random variables

Stationary/Ergodic Processes (cont’d) • Ergodic Process • Equality of time properties and statistic properties. • First-Order Time average • Defined as • Mean Ergodic Process • Mean Ergodic Process in Mean Square Sense

Stationary/Ergodic Processes (cont’d) • Slutsky’s Theorem • A process X(t) is mean-ergodic iff • Sufficient Conditions • a) • b)

Stochastic Analysis of Systems • Linear Systems • Time-Invariant Systems • Linear Time-Invariant Systems • Where h(t) is called impulse response of the system

Stochastic Analysis of Systems (cont’d) • Memoryless Systems • Causal Systems • Only causal systems can be realized. (Why?)

Stochastic Analysis of Systems (cont’d) • Linear time-invariant systems • Mean • Autocorrelation

Stochastic Analysis of Systems (cont’d) • Example I • System: • Impulse response: • Output Mean: • Output Autocovariance:

Stochastic Analysis of Systems (cont’d) • Example II • System: • Impulse response: • Output Mean: • Output Autocovariance:

Power Spectrum • Definition • WSS process • Autocorrelation • Fourier Transform of autocorrelation

Power Spectrum (cont’d) • Inverse trnasform • For real processes

Power Spectrum (cont’d) • For a linear time invariant system • Fact (Why?)

Power Spectrum (cont’d) • Example I (Moving Average) • System • Impulse Response • Power Spectrum • Autocorrelation

Power Spectrum (cont’d) • Example II • System • Impulse Response • Power Spectrum

Next Lecture Markov Processes & Markov Chains

Stochastic Processes