Stochastic Processes

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# Stochastic Processes - PowerPoint PPT Presentation

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

Elements of Stochastic Processes

Lecture II

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

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Markov Chains