ECE 5345 Random Processes Stationary Random Processes copyright

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ECE 5345 Random Processes - Stationary Random Processes copyright Robert J. Marks II

ECE 5345 Random Processes - Stationary Random Processes copyright Robert J. Marks II

Random Processes Stationary Random Processes: The stochastic process does not change character with respect

Random Processes Stationary Random Processes: The stochastic process does not change character with respect to time. Examples Types n n Strict stationarity Stationary in the Wide Sense (Wide Sense Stationary) Cyclostationary - stationary in a periodic sense copyright Robert J. Marks II

Strict Stationarity X(t) is stationary in the strict sense if, for all k and

Strict Stationarity X(t) is stationary in the strict sense if, for all k and choices of (t 1, t 2…, tk ), In other words, all CDF’s are independent of the choice of time origin. copyright Robert J. Marks II

Strict Stationarity-example All iid random processes are strictly stationary. Using continuous notation… copyright Robert

Strict Stationarity-example All iid random processes are strictly stationary. Using continuous notation… copyright Robert J. Marks II

Strict Stationarity-example The telegraph signal is strict sense stationary when the origin is randomized

Strict Stationarity-example The telegraph signal is strict sense stationary when the origin is randomized with a 50 -50 coin flip (see the Flip Theorem). Using independent increment property: Substituting gives the same results! copyright Robert J. Marks II

Strict Stationarity: necessary conditions If X(t) is stationary in the strict sense, 1. the

Strict Stationarity: necessary conditions If X(t) is stationary in the strict sense, 1. the mean is a constant for all time 2. The autocorrelation is a function of the distance between the points only Note: In 2, autocovariance could be substituted with autocorrelation with the same result. copyright Robert J. Marks II

Wide Sense Stationarity RP’s X(t) is wide sense stationary if 1. The mean is

Wide Sense Stationarity RP’s X(t) is wide sense stationary if 1. The mean is a constant for all time 2. The autocovariance is a function of the distance between the points only Notes: n In 2, autocovariance could be substituted with autocorrelation with the same result. n All strictly stationary processes are wide sense stationary copyright Robert J. Marks II

Wide Sense Stationarity RP’s: Average Power Recall average power In general Thus If X(t)

Wide Sense Stationarity RP’s: Average Power Recall average power In general Thus If X(t) is wide sense stationary, this means If you stick your finger in a socket, this is what you feel copyright Robert J. Marks II

Wide Sense Stationarity RP’s: Average Power Example Let everything but be fixed in the

Wide Sense Stationarity RP’s: Average Power Example Let everything but be fixed in the stochastic process Then And Then X(t) is wide sense stationary, with And Recall rms voltage of a sinusoid copyright Robert J. Marks II

Wide Sense Stationarity RP’s: Autocorrelation Properties 1. 2. Proof quod erat demonstrandum copyright Robert

Wide Sense Stationarity RP’s: Autocorrelation Properties 1. 2. Proof quod erat demonstrandum copyright Robert J. Marks II

Wide Sense Stationarity RP’s: Autocorrelation Properties 3. Proof: Recall If X(t) is zero mean,

Wide Sense Stationarity RP’s: Autocorrelation Properties 3. Proof: Recall If X(t) is zero mean, or This is even true when X(t) is not zero mean. The desired result follows immediately. quod erat demonstrandum copyright Robert J. Marks II

Wide Sense Stationarity RP’s: Autocorrelation Properties 4. If for some > 0, then is

Wide Sense Stationarity RP’s: Autocorrelation Properties 4. If for some > 0, then is a periodic function. Example: Recall sinusoid with random phase. p. 361 copyright Robert J. Marks II

Cyclostationary RP’s: For a given time interval (period) T, the behavior of the process

Cyclostationary RP’s: For a given time interval (period) T, the behavior of the process on each period has the same character. Cyclostationary in the strict sense copyright Robert J. Marks II

Cyclostationary RP’s: Cyclostationary in the strict sense example. Let A be a random variable

Cyclostationary RP’s: Cyclostationary in the strict sense example. Let A be a random variable and define This RP is stationary in the strict sense with copyright Robert J. Marks II

Wide Sense Cyclostationary RP’s: copyright Robert J. Marks II

Wide Sense Cyclostationary RP’s: copyright Robert J. Marks II

ENGR 5345 Review of Probability Random Variables RobertENGR 5345 Review of Probability Random Variables Robert
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EE 5345 Multiple Random Variables Cdfs and pdfsEE 5345 Multiple Random Variables Cdfs and pdfs
EE 5345 Functions of a RV copyright RobertEE 5345 Functions of a RV copyright Robert
Stationary Stochastic Process A stochastic process is stationaryStationary Stochastic Process A stochastic process is stationary
Using Differentiation Stationary Points Christine Crisp Stationary PointsUsing Differentiation Stationary Points Christine Crisp Stationary Points
Stationary electric field in conducting medium The stationaryStationary electric field in conducting medium The stationary
Stationary Stochastic Process A stochastic process is stationaryStationary Stochastic Process A stochastic process is stationary
The Spectral Representation of Stationary Time Series StationaryThe Spectral Representation of Stationary Time Series Stationary
Identifying Stationary Points Differentiation 3 6 The stationaryIdentifying Stationary Points Differentiation 3 6 The stationary
15 6 Stationary wave x0 L x Stationary15 6 Stationary wave x0 L x Stationary
Random 1 Random 2 Random 3 Random 4Random 1 Random 2 Random 3 Random 4
Random Sampling using RANDOM Random Sampling using RANDOMRandom Sampling using RANDOM Random Sampling using RANDOM
Random Sampling using RANDOM Random Sampling using RANDOMRandom Sampling using RANDOM Random Sampling using RANDOM