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.
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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
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:
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
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
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
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)
The converse is not true in general
Time Average and Ergodicity
An attribute (سمة )of stochastic systems; generally, a system that tends
in probability to a limitingform that is independent of the initial
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
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
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),
and the autocorrelation
are actually random variables
By taking the expected value for
Autocorrelation Function and Its Properties
The autocorrelation function of a random process X(t) is the correlation
of two random variables
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 ?
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
Question: How can we describe then the behavior of the input random process and
the output random process through a linear time-invariant system ?
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
Sxx(f) ( or Sxx(w)) be the Fourier Transform ofRxx(t)
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
The Mean of the output,
the mean is constant and the
autocorrelationof the output
is a function of
The Autocorrelation of the output,
Y(t) is WSS