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Stochastic Processes

M. Sami Fadali

Professor of Electrical Engineering

University of Nevada, Reno

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Outline

Stochastic (random) processes.

Autocorrelation.

Crosscorrelation.

Spectral density function.

Deterministic vs. Random Signals

Deterministic Signal: Exactly predictable.

e

Random Signal: Associated with a chanceoccurrence.

a) Continuous or discrete (time series).

b) May have a deterministic structure.

e

a Z (integer)

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4

Example: No deterministicstructure.

0 1 2 3 4 5 6 7 8 9 10

-0.1

-0.05

0

0.05

0.1

0.15

t

X(t)

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Random Processes

Map the elements of the sample space

to the set of continuous time functions .

For a fixed time point = random variable.

Example: Measurement of any physicalquantity (with additive noise) over time.

=ordinary time function =random variable

2.5

3

3.5

4

4.5

5

0

10

20

30

-4

-3

-2

-1

0

1

2

3

4

5

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Random Process

SampleFunction

t

Random Sequence

Map the elements of the sample space tothe set of discrete time functions .

For a fixed time point = random variable.

Example: Samples of any physical quantity(with additive noise) over time.

Discrete random process, time series.

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Example: Random Binary Signal

Random sequence of pulses s.t.1. Rectangular pulses of fixed durationT.

2. Amplitude +1 or 1, equally likely.

3. Statistically independent amplitudes.

4. Start time D for sequence uniformlydistributed in the interval [0,T].

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Random Binary Signal

D

t1

1

2 T

2

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Mathematical Description

= unit amplitude pulse of duration .

= binary r.v. in {1, 1}= amplitude of pulse.

= random start time, uniformlydistributed in .

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Moments of Random Process

Fix time to obtain a random variable.

Obtain moments as a function of time.

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Properties of Binary Signal

2

1 1

2 1

1

2 1

Second moment =variance (zero mean).

Special Case: Moments are constant not

functions of time.

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J oint Densities

Specify how fast changes withtime

Later: related to spectral content.

Higher order densities provide moreinformation (hard to compute).

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Statistically IndependentRandom Signals

jiji YYXXYYXX fff 2121

Any choice of and

Possibly

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Gaussian Random Process

All density functions (any order) normal.

Multivariate normal density: completelyspecified by the mean and covariancematrix.

/

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Autocorrelation

Autocorrelation is an ensemble averageusing thejointdensity function.

Recall: for fixed , random process = r.v.

Similarly define, autocovariance (same as

autocorrelation for zero mean).

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Stationary Random Process

Two definitions

Strictly stationary random process

Wide sense stationary randomprocess (WSS)

(Strictly) Stationary implies wide-sensestationary.

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Strictly Stationary

All pdfs describing the process are

unaffected by any time shift.Xi = X(ti), i = 1, 2, , k

Xi= X(ti+), i = 1, 2, , k

Both governed by the same pdfs

Wide Sense Stationary: Constant mean,shift-invariant autocorrelation.

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Wide Sense StationaryRandom Signal

Stationarity of the mean (constant).

Stationary of the autocorrelation.

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Nonstationary Signal

Y(t )=R+cos(t), R~N(0,9),mY(t)=cos(t)

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Example: WSS only

Equally probable outcomes

Joint pdf:Assume four possible joint outcomesonly with two sines or two cosines of the samesign

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Not Strict Sense Stationary

Take two time points (say and )

Values at

Values at

First-order distributions are different(even thought their mean is thesame).

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Single realization is enough.

Time average = ensemble average. Ergodicity, like stationarity, is an

idealization.

Can be approximately true inpractice.

Ergodic signals tend to lookrandom.

Ergodic Random Processes

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Stationarity Necessary

Explanation

Single realization has a singleaverage for any property: allmoments, autocorrelation etc.

Can only obtain expected value ifit is constant.

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Example: Stationary not Ergodic

Random constant.

Amplitude N

Sample realization with amplitude

Mean of

Stationarity is not sufficient.

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Ergodicity in Mean

Time Average

Ergodicity in the mean:

, as

For zero mean , as

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Ergodicity in AutocorrelationTime Autocorrelation

e

Ergodicity in autocorrelation:

as

Need 4th moment.36

Example

Deterministic structure

N(0,2), constant

Sample realization

, fix

Compare time autocorrelation &autocorrelation.

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Time Autocorrelation

cos2

2coscos2

sinsin1

)()(1

)(

21

0

21

0

21

0

A

dttT

A

dtttAT

dttXtX

T

T

T

T

T

T

AA

T

XA

_|

_|

_|e

(finite integral)/T

TT

dtttdtt00

2sinsin2coscos2cos

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Autocorrelation

Not a function of the time shift only.

Not equal to time autocorrelation.

Not an ergodic process.

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Properties of Autocorrelation

Useful general properties usedthroughout the course.

Several apply to the stationarycase only.

Assume real scalar processes.

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Mean Square Value

From autocorrelation

For stationary

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Zero-mean

Zero-mean, ergodic process

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Autocorrelation for Vector

casestationary)()(

),(

)()(

)()(),(

*

2112

*12

*

1*

2

2

*

121

ttRttR

ttR

ttE

ttEttR

XXXX

xx

XX

xx

xx

=conjugate transpose

=transpose for real

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Crosscorrelation Function

Stationary: skew-symmetric

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Properties of Crosscorrelation

eduncorrelat,

orthogonal,0

)(

)0()0(2

1)(

)0()0()(

)()(

YXXY

YYXXXY

YYXXXY

YXXY

mmR

RRR

RRR

RR

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Example: Cross-correlation

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sin21

sin22sin.2

cossin.

)()()(

2,0~,cos)(

2,0,sin)(

AB

tEAB

ttEAB

tYtXER

UtBtY

UtAtX

XY

Zero at =0 not maximum

Zero mean asshown earlier

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Combination of Random Processes

Sum of two random processes.

Uncorrelated and at least one zero-mean.

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Time Delay Estimation

Transmit signal to object.

Receive signal.

Find cross-correlation of two signals.

Location of peak = time delay.

-8 -6 -4 -2 0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

dT

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Power Spectral DensityFunction (SDF)

Fourier transform of autocorrelation.

Wiener-Khinchine Relation

_

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Example

Same as Gauss-Markov process.

22

2

2

22

2

2

2)()(

2)()(

)(

s

RsS

RjS

eR

XXXX

XXXX

XX

_

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Autocorrelation and PSD

R() S(j)

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Mean-square Value of Output

For = 0

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Power in a Finite Band

Power spectral density gives

distribution of power overfrequencies.

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Example

Integration tables.

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Example: Mean Square

Later: Table 3.1 for s-domain integral.

212

2

tan1

2

2

)(2

1)(2

1

)(

j

jXX

XX

dssSj

djStXE

Cross Spectral Density Function

Skew-symmetry of cross-correlation: = .

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Properties of SDF

The autocorrelation is real, even.

Real:

Even:

Nonnegative

since integral=energy over BW.

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Spectral Density of Sum

+

+

Zero cross-correlation:

+

X& Yuncorrelated and X orY zero mean.

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Coherence Function

Used in applications (e.g. acoustics).

lationcrosscorrezero)(&)(,0

)()(,1)(

,1)()(

)()(

2

2

2

tYtX

tYtXj

jSjS

jSj

XY

YYXX

XY

XY

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Conclusion

Probabilistic description of randomsignals.

Autocorrelation and crosscorrelation.

Power spectral density function.

Experimental determination:necessary but difficult.

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References

R. G. Brown and P. Y. C. Hwang, Introduction toRandom Signals and Applied Kalman Filtering,

3ed, J. Wiley, NY, 1997.G. R. Cooper and C. D. MaGillem, ProbabilisticMethods of Signal and System Analysis, OxfordUniv. Press, NY, 1999

Binary signal example is from

K. S. Shanmugan & A. M. Breipohl, Detection,Estimation & Data Analysis, J. Wiley, NY, 1988,

pp. 132-134.