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Random Process

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Random Process

The concept of random variable was defined previously as mapping from the

Sample Space S to the real line as shown below

The concept of random process can be extended to include time

and the outcome will be random functions of time as shown below

The functions

are one realizations

of many of the random process X(t)

A random process also represents a random variable when time is fixed

is a random variable

Stationary and Independence

The random process X(t) can be classified as follows:

First-order stationary

A random process is classified as first-order stationary if its first-order

probability density function remains equal regardless of any shift in time

to its time origin.

If we Xt1let represent a given value at time t1

then we define a first-order stationary as one that satisfies the

following equation:

The physical significance of this equation is that our density function,

is completely independent of t1

and thus any time shift t

For first-order stationary the mean is a constant, independent of

any time shift

Second-order stationary

A random process is classified as second-order stationary if its

second-order probability density function does not vary over any

time shift applied to both values.

In other words, for values Xt1 and Xt2 then we will have the following

be equal for an arbitrary time shiftt

From this equation we see that the absolute time does not affect

our functions, rather it only really depends on the time difference

between the two variables.

For a second-order stationary process, we need to look at the

autocorrelation function ( will be presented later) to see its most

important property.

Since we have already stated that a second-order stationary process

depends only on the time difference, then all of these types of processes

have the following property:

Wide-Sense Stationary (WSS)

A process that satisfies the following:

is aWide-Sense Stationary (WSS)

Second-order stationary

Wide-Sense Stationary

The converse is not true in general

Time Average and Ergodicity

Ergodicity

An attribute (سمة )of stochastic systems; generally, a system that tends

in probability to a limitingform that is independent of the initial

condititions

The time average of a quantity is defined as

Here A is used to denote time average in a manner analogous to

E for the statistical average.

The time average is taken over all time because, as applied to

random processes, sample functions of processes are presumed

to exist for all time.

Let x(t) be a sample of the random process X(t) were the lower case

letter imply a sample function (not random function).

X(t) the random process

x(t) a sample

of the random

process the

random process

Let x(t) be a sample of the random process X(t) were the lower

case letter imply a sample function.

We define the mean value

( a lowercase letter is used to imply a sample function)

and the time autocorrelation function

as follows:

For any one sample function ( i.e., x(t) ) of the random process X(t),

the last two integrals simply produce two numbers.

A number for the average

and a number for

for a specific value of t

Since the sample function x(t) is one out of other samples functions

of the random process X(t),

The average

and the autocorrelation

are actually random variables

By taking the expected value for

and

,we obtain

Correlation Function

Autocorrelation Function and Its Properties

The autocorrelation function of a random process X(t) is the correlation

of two random variables

and

by the process at times t1 and t2

Assuming a second-order stationary process

Linear System with Random Input

In application of random process, the Input-Output relation through a linear system

can be described as follows:

Here X(t) is a random process and h(t) (Deterministic Function) is the impulse

response of the linear system ( Filter or any other Linear System )

Linear System with Random Input

Now we can look at input output relation as follows:

The Time Domain

The output in the time domain is the convolution of the Input random process X(t)

and the impulse response h(t),

Question: Can you evaluate this convolution integral ?

Answer:

We can observe that we can not evaluate this convolution integral in general because

X(t) is random and there is no mathematical expression for X(t).

The Frequency Domain

The output in the Frequency Domain is the Product of the Input Fourier Transform

of the input random process X(t) , FX(f) and the Fourier Transform of the impulse

response h(t), H(f)

the Fourier Transform of the input random process X(t)

is a random process

the Fourier Transform of the deterministic impulse response

the Fourier Transform of the output random process Y(t)

is a random process

Question : Can you evaluate the Fourier Transform of the input random process X(t) , FX(f) ?

Answer: In general no , since the function X(t) in general is random and has no mathematical

expression.

Question: How can we describe then the behavior of the input random process and

the output random process through a linear time-invariant system ?

as

We defined previously the autocorrelation

The auto correlation tell us how the random process is varying

Is it a slow varying process or a high varying process.

Next we will define another function that will help us on looking at the behavior of the

random process

Let

Sxx(f) ( or Sxx(w)) be the Fourier Transform ofRxx(t)

OR

OR

Since

Average Power

Then SXX(f) is Power Spectral Density ( PSD ) of the Random Process X(t)

Properties of PSD

Now let us look at the input-output linear system shown below in the time domain

and Frequency domain assuming the Random Process X(t) is WSS

Time Domain

The Mean of the output,

Constant

the mean is constant and the

autocorrelationof the output

is a function of

The Autocorrelation of the output,

Y(t) is WSS