Random ProcessesRandom Processes

Gaussian and Gauss-Markov processes

Power spectrum of random processes and white processes

Random ProcessRandom Process

Random process X(t) Random Variable + Time

X1(t)

X2(t)

XN(t)

Sample function

t

The totality of all sample functions is called

an ensemble

For a specific timeX(tk) is a random variable

Why Gaussian Process ?Why Gaussian Process ? Central limit theorem

The sum of a large number of independent and identically distributed(i.i.d) random variables getting closer to Gaussian distribution

Thermal noise can be closely modeled by Gaussian process

Mathematically tractable Only 2 variables are needed for pdf

Mean and Covariance (or Standard deviation)

Definition of Gaussian ProcessDefinition of Gaussian Process

A random process X(t) is a Gaussian process if for all n and for all , the random variables has a jointly Gaussian density function, which may expressed as

Where

1 2( , , , )nt t t

2{ ( ), ( ), , ( )}i nX t X t X t

1/ 2 1/ 2

1 1( ) exp[ ( ) ( )]2(2 ) [det( )]

Tnf x x m C x m

C

2[ ( ), ( ), , ( )]Ti nx X t X t X t ( )m E X

{ } (( )( ))ij i i j jC c E x m x m

: n random variables: mean value vector: nxn covariance matrix

Property of Gaussian ProcessProperty of Gaussian Process Property 1

For Gaussian process, knowledge of the mean(m) and covariance(C) provides a complete statistical description of process

Property 2 If a Gaussian process X(t) is passed through a LT

I system, the output of the system is also a Gaussian process. The effect of the system on X(t) is simply reflected by the change in mean(m) and covariance(C) of X(t)

Definition of Markov ProcessDefinition of Markov Process

Markov process X(t) is a random process whose past has no influence on the future if its present is specified. If , then

Or if

현재는 직전의 상태에만 영향을 받는다 .

1n nt t

1 1[ ( ) | ( ) ] [ ( ) | ( )]n n n n n nP X t x X t t t P X t x X t

2 1...nt t t

1 2 1 1[ ( ) | ( ), ( ),..., ( )] [ ( ) | ( )]n n n n n n nP X t x X t X t X t P X t x X t

Gauss-Markov ProcessGauss-Markov Process Definition

A Gauss-Markov process X(t) is a Markov process whose probability density function is Gaussian

Generating Gauss-Markov process

If {wn} is Gaussian, then X(t) is Gauss-Markov process Homework

Illustrative Problem 2.3

1( ) ( )n n nX t X t w Zero mean i.i.d RVDegree of correlation between Xn and Xn-1

2 2 2 21 1 1( ) ( )n n n nE X X E X

Stationary processStationary process Definition of Mean

Definition of Autocorrelation

Where X(t1),X(t2) are random variables obtained at t1,t2

Definition of stationary A random process is said to be Wide-sense stationary, if

its mean(m) and covariance(C) do not vary with a shift in the time origin

A process is (wide-sense) stationary if

( )( ( )) ( ) ( )kk X t X kE X t xp x dx m t

( ( ) constantk XE X t m

1 2 1 2( , ) [ ( ) ( )]XR t t E X t X t

1 2 1 2( , ) ( ) ( )X X XR t t R t t R

Power spectrum of RPPower spectrum of RP

Power spectrum of X(t)

Autocorrelation of X(t)

Power spectrum and Autocorrelation function are Fourier transform pair

2( ) [ ( )] ( ) j fx x xS f F R R e d

1 2( ) [ ( )] ( ) j fx x xR F S f S f e df

Def. of white (random) processDef. of white (random) process

A random process X(t) is called a white process if it has a flat power spectrum. If Sx(f) is constant for all f

It closely represent thermal noise

f

Sx(f)

The area is infinite(Infinite power !)

In practiceIn practice

From quantum physics And we are interested in small range of

frequency

f

Sn(f)

Can be modeled as constant in this region

0( ) constant2 2n

NkTS f

Power spectrum(or Power spectral density)

Boltzmann's constanttemperature in Kelvin

kT

Autocorrelation of white processAutocorrelation of white process

Power spectrum

Autocorrelation

0( )2nN

S f

2 20 0( ) ( ) ( )2 2

j f j fx x

N NR S f e df e df

Sn(f)N0/2

fN0/2

Rx()

Rx()=0 if =t1-t20X(t1) and X(t2) are uncorrelated if t1 t2

White Gaussian ProcessWhite Gaussian Process

The sampled random variables will be statistically independent Gaussian random variables

Sn(f)N0/2

f

N0/2Rx()

=0 0

Bandlimited random processBandlimited random process

Power spectrum

Autocorrelation function

0 ,( ) 2

0,n

Nf B

S ff B

2 20

0 0

( ) ( )2

sin(2 ) sinc(2 )2

Bj f j fx x B

NR S f e df e df

BN B N B BB

HomeworkHomework

Illustrative problem 2.4 and 2.5 Problems

2.8, 2.10