Random Processes Gaussian and Gauss-Markov processes Power spectrum of random processes and white processes

Random Process Random process X(t) Random Variable + Time X 1 (t) X 2 (t) X N (t) Sample function t The totality of all sample functions is called an ensemble For a specific time X(tk) is a random variable

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

Property 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 LTI 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 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.

Gauss-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 Zero mean i.i.d RV Degree of correlation between X n and X n-1

Stationary process Definition of Mean Definition of Autocorrelation Where X(t 1 ),X(t 2 ) are random variables obtained at t 1,t 2 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

Power spectrum of RP Power spectrum of X(t) Autocorrelation of X(t) Power spectrum and Autocorrelation function are Fourier transform pair

Def. of white (random) process A random process X(t) is called a white process if it has a flat power spectrum. If S x (f) is constant for all f It closely represent thermal noise f Sx(f) The area is infinite (Infinite power !)

In practice From quantum physics And we are interested in small range of frequency f Sn(f) Can be modeled as constant in this region Power spectrum (or Power spectral density)

Autocorrelation of white process Power spectrum Autocorrelation Sn(f) N 0 /2 f Rx(  )  Rx(  )=0 if  =t 1 -t 2  0 X(t 1 ) and X(t 2 ) are uncorrelated if t 1  t 2

White Gaussian Process The sampled random variables will be statistically independent Gaussian random variables Sn(f) N 0 /2 f Rx(  )   =0  0

Bandlimited random process Power spectrum Autocorrelation function

Homework Illustrative problem 2.4 and 2.5 Problems 2.8, 2.10