5. Random Processes

7/30/2019 5. Random Processes

1/37

Random Processes

Random Process


2/37

Definition of Random Process

Consider we want to determine the probability oftemperature at 12 pm in certain city

The RV is X = temperature at 12 pm in city A

We have to record the data for many days to get pX(x)

However, the temperature is a function of time The temperature varies between time to time and days to days

The RV = temperature in city A is function of time

An RV that is a function of time (or any other variable) is

called a random process or stochastic process


3/37


The collection of all

possible waveform is known

as ensemble of the random

process x(t) A waveform in the

collection is called a sample

function

Sample-function amplitudesat some instant are the

values taken by the RV in

various trials


4/37



5/37


A discrete-time stochastic process is one when the indexset is a countable set

When the index set is continuous, it is called continuous-

time stochastic process

Example:


6/37


The number of waveforms in an ensemble may be finiteor infinite

Example of infinite waveform: temperature

Example of finite waveform: output of binary signal

generator Consider that we will examine the output over the period 0 to

2T

We will have only 22 = 4 waveforms


7/37


Note that the randomness occurs to the selection ofwaveforms

The waveforms (in the ensemble) themselves are

deterministic

Example: In the experiment of tossing a coin four times,there are 16 possible outcomes. The randomness is which

of the 16 outcomes will occur in a given trial


8/37

Specification of A Random Process

How to specify random process: Analytical

Experimental

Analytical: mathematical expression

Experimental: We need to find some quantitatibe measure

Random process can be seen as collection of infinite,

independent number of RV

We need pdf


9/37

Autocorrelation

The autocorrelation function is defined as 1 2 1 2 1 2,XR t t x t x t x x


10/37

Example

A random process X(t) is defined by

where Y is a discrete random variable with P(Y = 0) =

and P(Y = /2) = . Find X(1) and RXX(0,1)

Solution:

X(1) is a random variables with values and

With probability each

2cos 2X t t Y

2cos 2 2cos 2 2

11 22

P X 11 0 2P X


11/37

Example (Contd)

Thus,

The autocorrelation will be searched for t = 0 and t = 1

The values of P[X(0) = 0, X(1) = 0] =

P[X(0) = 2, X(1) = 2] =

Thus, the autocorrelation is

1 1

1 1 2 0 12 2

X E X

0,1 0 1

1 12 2 0 0 2

2 2

XXR E X X


12/37

Classification of Random Process

Stationary and Nonstationary random process A random process whose statistical characteristics do not

change with time

Random process X(t) and X(t+) have the same statistical

properties

In other words, time translation of a sample function results in

another sample function of the random process having the

same probability

Example of stationary random process: noise

If the statistical characteristics depend on time, it is called

nonstationary random process

Example of nonstationary random process: temperature


13/37


Wide-sense stationary process (WSS) The mean and autocorrelation function do not vary with a

time shift

All stationary processes are WSS, but not vice versa

constantE X t K

1 2 2 1, ,X XR t t R t t


14/37


Ergodic Ensemble averages are equal to time averages of any sample

function

x t x t

X XR


15/37



16/37

Example

Show that , where and areconstants and is a random variable that is uniformly

distributed over (0, 2), is ergodic

Solution:The ensemble average is

The time average is

0cosx t A t

A0

2

0

0

1cos 0

2x x f d A t d

0

0

0 0

1cos 0

T

x t A tT


17/37

Power Spectral Density of A Random

Process

PSD depicts the power spread in frequency domain It is easy to get the frequency domain for deterministic

signal by using Fourier transform

How to compute the PSD for random signal?

Random signals are power signals

Several questions of determining the PSD for random

process arise:

We may not be able to describe the sample function

analytically

Every sample function may be different from another one

PSD is defined for stationary (or WSS)


18/37


Process

We have to define the PSD of random process as theensemble average of the PSDs of all sample functions

Suppose that represents a sample function of a

random process . The truncated version of this sample

function is obtained by multiplying the signals withrectangular function

, ix t

x t

rect t T

, , 0.5

,0,

i

T i

x t t Tx t

elsewhere


19/37


Process

The Fourier transform is

The normalized energy in time interval (-T/2, T/2)

2

2

2

2

, ,

,

j ft

T i T i

T

j ft

T i

T

X f x t e dt

x t e dt

22

T T T

E x t dt X f df


20/37


Process

The mean normalized energy is obtained by taking theensemble average

The normalized average power is the energy expendedper unit time

2

22 2

2

T

T T T

T

E x t dt x t dt X f df

2

2 2

2

2 2

1 1lim lim

1lim

T

TT T

T

TT

P x t dt x t dtT T

X f df x tT


21/37


Process

The definition of PSD is

So that

xP S f df

2

limT

xT

X fS f

T


22/37


Process

Wiener-Khintchine TheoremWhen is a wide-sense stationary process, the PSD can be obtained from the Fourier transform of the autocorrelation function

x t

x xR S f

2j ftx x xS f F R f R e d

1 2j ftx x xR F S f S f e df


23/37


Process

Properties of PSDPSD is always real

Let , we have

Therefore, the PSD is even function when is real

xR x t x t xR x t x t

t

x xR x x R

x t


24/37


Process

Properties of PSD (Contd)When is wide-sense stationary x t

2 0x xS f df P x R


25/37

White Noise Processes

A random process is said to be a white noise processif the PSD is constant over all frequencies

where is a positive constant

The autocorrelation function for the white noise process

is obtained by taking the inverse Fourier transform

x t

02

x

NS f

0N

02

x NR


26/37

Example

Determine the autocorrelation and the power of a low-pass random process with white noise PSD

Solution:

From the figure we have

2

X

NS

2 4

x

NS f rect

B

sinc 2xR NB B


27/37

Example (Contd)

We calculate the power

We can also calculate the power by using integral

2 0x xP x R NB

0

0

2

22

x x

B

P S df

Ndf

NB


28/37

The Gaussian Random Process

A random process is said to be Gaussian if therandom variables have an N-

dimensional Gaussian PDF for any N and any

The N-dimensional Gaussian PDF can be written

compactly by using matrix notation Let x be the column vector denoting the N random

variables

x t

1 1 2 2, , , N Nx x t x x t x x t

1 2, , , Nt t t

1 1

2 2

N N

x x t

x x t

x x t

x


29/37


The N-dimensional Gaussian PDF is

where the mean vector is

11 2

2 1 2

1

2

T

Nf e

Det

x m C x m

Xx

C

1 1

22

NN

x m

mx

mx

m x


30/37


The covariance matrix C is defined by

where

For WSS random process

11 12 1

21 22 2

1 2

N

N

N N NN

c c c

c c c

c c c

C

ij i i j j i i j jc x m x m x t m x t m

i i j jm x t m x t m


31/37


The elements of the covariance matrix is then

If happen to be uncorrelated for the

covariance matrix becomes

where

2ij x j ic R t t m

ix i j i jx x x x i j

2

2

2

0 0

0 0

0 0

C

2 2 2 20xx m R m


32/37


Some properties of Gaussian processes:

depends only on C and on m, which is another way of

saying that the N-dimensional Gaussian PDF is completely

specified by the first- and second-order moments

Since the are jointly Gaussian, the are

individually Gaussian

When C is a diagonal matrix, the random variables are

uncorrelated

A linear transformation of a set of Gaussian random variables

produces another set of Gaussian random variables A WSS Gaussian process is also strict-sense stationary

Xf x

i ix x t i ix x t


33/37


Theorem:

If the input to a linear system is a Gaussian random

process, the system output is also a Gaussian process


34/37

Bandpass Random Process

Bandpass waveform can be represented by

or

and

where

Re cj tv t g t e

cos sinc cv t x t t y t t

cos cv t R t t t

j g tg t g t e x t jy t


35/37


The spectrum of bandpass waveform is

In communication systems, the random processes may be

random signals, noise, or signals corrupted by noise If is Gaussian process, , , and are Gaussian

processes since they are linear functions of

But, and are not Gaussian because they are

nonlinear functions of

*1

2c cV f G f f G f f

v t g t x t y t v t

R t g t

v t


36/37


Theorem:

If and are jointly WSS processes, the real bandpass

process will be WSS if

and only if

x t y t

Re cos sincj t c cv t g t e x t t y t t

0

xy yx

x y

x t y t

R R

R R


37/37


Theorem:

If and are jointly WSS processes, the real bandpass

process

will be WSS when is an independent random variable

uniformly distributed over (0,2

)

x t y t

Re cos sinc cj t c c c cv t g t e x t t y t t c

Documents

5. Random Processes