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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.

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stochastic processes

Stochastic Processes

Elements of Stochastic Processes

Lecture II

overview
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
Outline
  • Basic Definitions
  • Stationary/Ergodic Processes
  • Stochastic Analysis of Systems
  • Power Spectrum
basic definitions
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
Basic Definitions (cont’d)
  • In a random process, we will have a family of functions called an ensemble of functions
basic definitions cont d1
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 d2
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 d3
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 d4
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 d5
Basic Definitions (cont’d)
  • First-Order CDF of a random process
  • First-Order PDF of a random process
basic definitions cont d6
Basic Definitions (cont’d)
  • Second-Order CDF of a random process
  • Second-Order PDF of a random process
basic definitions cont d7
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 d8
Basic Definitions (cont’d)
  • Mean of a random process
  • Autocorrelation of a random process
  • Fact: (Why?)
basic definitions cont d9
Basic Definitions (cont’d)
  • Autocovariance of a random process
  • Correlation Coefficient
  • Example
basic definitions cont d10
Basic Definitions (cont’d)
  • Example
    • Poisson Process
    • Mean
    • Autocorrelation
    • Autocovariance
basic definitions cont d11
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 d12
Basic Definitions (cont’d)
  • Cross-Correlation
    • Orthogonal Processes
  • Cross-Covariance
    • Uncorrelated Processes
basic definitions cont d13
Basic Definitions (cont’d)
  • a-dependent processes
  • White Noise
  • Variance of Stochastic Process
basic definitions cont d14
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
outline1
Outline
  • Basic Definitions
  • Stationary/Ergodic Processes
  • Stochastic Analysis of Systems
  • Power Spectrum
stationary ergodic processes
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
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 d1
Stationary/Ergodic Processes (cont’d)
  • Autocovariance of a WSS process
  • Correlation Coefficient
stationary ergodic processes cont d2
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 d3
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 d4
Stationary/Ergodic Processes (cont’d)
  • Example
    • Suppose for a WSS process
    • X(8) and X(5) are random variables
stationary ergodic processes cont d5
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 d6
Stationary/Ergodic Processes (cont’d)
  • Slutsky’s Theorem
    • A process X(t) is mean-ergodic iff
    • Sufficient Conditions
      • a)
      • b)
outline2
Outline
  • Basic Definitions
  • Stationary/Ergodic Processes
  • Stochastic Analysis of Systems
  • Power Spectrum
stochastic analysis of systems
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
Stochastic Analysis of Systems (cont’d)
  • Memoryless Systems
  • Causal Systems
    • Only causal systems can be realized. (Why?)
stochastic analysis of systems cont d1
Stochastic Analysis of Systems (cont’d)
  • Linear time-invariant systems
    • Mean
    • Autocorrelation
stochastic analysis of systems cont d2
Stochastic Analysis of Systems (cont’d)
  • Example I
    • System:
    • Impulse response:
    • Output Mean:
    • Output Autocovariance:
stochastic analysis of systems cont d3
Stochastic Analysis of Systems (cont’d)
  • Example II
    • System:
    • Impulse response:
    • Output Mean:
    • Output Autocovariance:
outline3
Outline
  • Basic Definitions
  • Stationary/Ergodic Processes
  • Stochastic Analysis of Systems
  • Power Spectrum
power spectrum
Power Spectrum
  • Definition
    • WSS process
    • Autocorrelation
    • Fourier Transform of autocorrelation
power spectrum cont d
Power Spectrum (cont’d)
  • Inverse trnasform
  • For real processes
power spectrum cont d1
Power Spectrum (cont’d)
  • For a linear time invariant system
  • Fact (Why?)
power spectrum cont d2
Power Spectrum (cont’d)
  • Example I (Moving Average)
    • System
    • Impulse Response
    • Power Spectrum
    • Autocorrelation
power spectrum cont d3
Power Spectrum (cont’d)
  • Example II
    • System
    • Impulse Response
    • Power Spectrum
slide41
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