1
ELE 360
Probability & Stochastic Processes
Random Processes
Dr. M. Hassan
2
Introduction
•
A random process is a family of functions
defined on a given
probability space, indexed by the parameter where varies over the index set .
•
A general random process can be described as:
–
–
An ensemble of sample functions
Collection of time functions (signals) that
corresponds to various outcomes of random
experiments
Collection of random variables observed at
different times
Example:
– A box has infinitely many resistors
of same resistance
– Let be the event that the resistor has
been picked up from the box
– Let
represents the voltage of the
thermal noise measured on this resistor
The probability density function describes the general distribution of the magnitude of the random
process, but it gives no information on the time or frequency content of the process
Dr. M. Hassan
3
Definition
• A random process
is a mapping function that assigns to every outcome, , a
function of time or a waveform denoted by
Sample space
Parameter space
• A general random process can be described as:
–
–
A collection of time functions (signals) that corresponds to various outcomes of random
experiments
Collection of random variables observed at different times
−
For a particular sample point
sample function
− For a particular time
, we have the value
the value of which depends on the outcome
−
For a particular sample point
is a constant (real number)
, is a single function of time 
, which is a random variable
and a particular time
are called states
– The set of possible values that
can take is called the state space
– The values taken by a RP
– A RP can be observed at index set
Dr. M. Hassan
called the parameter set
What is a Stochastic Process? (cont’d)
A random process
is a mapping function that assigns to every
outcome, , a function of time or a waveform
– The values that
takes are called states. The set of possible values that
can take is called the state space of the random process
– If the state space is discrete  the process is a discrete-state process
– A discrete-state process is always called a chain
– If the state space is continuous  the process is continuous-state process
– The index set
is called the parameter set
– If the index set
is discrete  the process is discrete-parameter or discrete-time
process
– A discrete-parameter process is called a random sequence
– A random sequence is denoted by
– If the index set
time process
Dr. M. Hassan
is continuous  the process is continuous-parameter or continuous-
4
Discrete Random Processes- An Example
5
Let
be a uniform random variable defined on the unit interval
. Define the
random process
. Draw few examples of the sample path. Then,
obtain the first order PDF.
Sample functions:
Note that for each
is a RV.
The sequence of PDFs of
is called the
first-order pdf of the process
Noting that
is given by:
Dr. M. Hassan
is a differentiable function of the continuous RV
, the first order PDF
Continuous Random Processes- An Example
Let
be a uniform random variable defined on the interval
. Define the
random process
. Draw few examples of the sample path.
Then, obtain the first order PDF.
Sample functions:
The first order PDF is given by:
Dr. M. Hassan
6
The Mean, Autocorrelation, and Autocovariance
Partial information about a random process can be obtained by inspecting its moments.
For a continuous-time continuous-state RP
, we have:
The mean function
The variance function
The autocorrelation function
Note that
The autocovariance function
Note that
The autocorrelation coefficient function
Dr. M. Hassan
where
7
8
Characterizing A Stochastic Process
•
In the previous examples the random process were specified by describing the set of
sample functions (sequences, paths) and explicit probability measures over the set of
events (subsets of sample functions)
•
This way of characterizing a random process has very limited applicability
 It is only suited for very simple processes
•
A general way of characterizing a random process is to specify all of its
CDFs (PDFs, PMFs), i.e., the joint CDF (PDF, PMF) of the samples
for all orders
•
and for every set of
points
When we sample the random process at time
define a RV denoted by
–
-th order
we obtain one RV i.e. we can
is typically characterized by its distribution function:
– We can also define a joint (vector) random variable, characterized by a joint
CDF that’s given by:
Vector of RVs
Dr. M. Hassan
Stochastic Process Characterization
•
When we sample the random process at time
a RV denoted by
–
we obtain a RV i.e., we can define
is typically characterized by its distribution function:
– We can also define a joint (vector) random variable, characterized by a joint
CDF that’s given by:
Classification in more details
(1st order distribution)
•
Similarly, we can define
order joint distribution:
(2nd order joint distribution)
where
and
•
order distribution
It’s typically difficult to compute the
Dr. M. Hassan
9
Stochastic Process Characterization (cont’d)
10
Density & joint density functions
We can obtain the density and joint density functions from the derivatives of the relevant
CDF
(1st order distribution)
(1st order PDF)
(2nd order joint distribution)
(2nd order joint PDF)
•
Similarly, the
order joint density is obtained by taking the
joint CDF given by:
where
Dr. M. Hassan
and
derivative of the
Stationary Random Processes
11
Strict stationarity
•
A strict stationary process is a random process the statistical properties of which do
not change over time/space. This means that the joint distribution function of the RVs
obtained by observing the RP at times
satisfy:
Wide-sense stationarity
A wide-sense stationary is mean-stationary and autocorrelation-stationary. i.e. it is
2nd order stationary. This means that
i.e. the mean is time invariant
The autocorrelation between
depends only on the time difference
Remarks
A strict stationary random process is WSS but a WSS process is not necessarily a strict
stationary random process
Dr. M. Hassan
Independent Random Processes
Can the
•
order distribution computations be simplified?
This is true under some simplifying assumptions:
– Stationary (strict)
Notice the independence
of time
(
– Independence
Dr. M. Hassan
order distribution)
12
13
The Autocorrelation Function
• Recall that a random process is said to be wide-sense stationary if:
Properties of the autocorrelation function
, for all 𝑡
•
• The autocorrelation function
•
has its maximum magnitude at
,
•
• The autocorrelation function
is an even function of
• PSD is the Fourier transform of the autocorrelation
Dr. M. Hassan
:
14
The Autocovariance Function
Let
be a strictly stationary RP. The autocovariance of
is given by:
which is a function of the (lag) time difference
Note that
can always be determined if
and
are known
Remarks
1.
and
only provide a partial description of the RP
2.
If
and
stationary (stationary process)
3.
The class of strictly stationary processes with finite second-order moments
is a subclass of the class of all stationary processes
4.
The first- and second-order moments may not exist
Dr. M. Hassan
, then
is wide-sense
15
Example: A Sinusoid with Random Amplitude
Consider the RP
where the amplitude
Find the mean, autocorrelation and the autocovariance functions of
Solution
The mean of
is given by:
The autocorrelation of
The autocovariance of
is given by:
is given by:
What do you conclude?
Dr. M. Hassan
is a RV.
Example -Sinusoidal Wave with Random Phase
Consider the RP
where
. Find the power spectral density and the power of
Solution
a uniform RV is over
PSD is the Fourier transform of the ACF
Recall that
The power
is evaluated from the autocorrelation function when
Note that the power
Dr. M. Hassan
can also be obtained by integrating
with respect to
16
Ergodic Random Processes
A stationary process
is said to be ergodic when the (ensemble)
statistical averages of which are equal to its time averages.
•
17
This clearly means that the statistics, or expectations, of a stationary random process
are NOT necessarily equal to its time averages and hence a stationary random process
is NOT always ergodic.
Time averaging:
In time averaging, the properties are obtained by averaging over a single record in time
as follows:
The mean value:
The correlation:
The mean square value:
The autocovariance:
The variance:
when
The autocorrelation coefficient:
It is the average of the square of the
deviation of
from the mean value
Dr. M. Hassan
18
Example
Show that the random process given by
stationary process where is a uniformly distributed RV over
its autocorrelation function. Is
an ergodic process?
is a wide-sense
. Then, determine
Remarks
This ensemble consists of sinusoids all of which
is having constant amplitude of
and constant
frequency
but a random phase of
For any sample function, the phase
has an
equally-likely probability to take any value in
the range (0, 2π)
Since
is uniformly distributed over the range
(0, 2π), we can determine the PDF of
and
then
Ensemble of the random process
Dr. M. Hassan
2π
θ
Example (cont’d)
19
Solution
Since
is a function of the RV
, we have:
where
The ensemble mean at any instant is zero (i.e., a constant)
The autocorrelation function is determined as follows:
Contains no RV
(i.e., a constant)
Function of a RV
i.e., the mean of which is
the constant itself!
Dr. M. Hassan
Example (cont’d)
Contains no RV
Function of a RV
Therefore,
where
which is a function of the lag
Clearly,
Dr. M. Hassan
is a wide-sense stationary process
20
Example (cont’d)
To check whether
is ergodic or not, we compute the time averages of
The mean value:
The variance:
The correlation:
Clearly, the time averages and statistical averages are equal.
is an ergodic process
Dr. M. Hassan
21
22
Example
Consider a random process 𝑋 𝑡
𝐴 where 𝐴 is a binary-valued random
variable taking on values of 1 with equal probabilities. Is 𝑋 𝑡 ergodic?
Solution
The statistical mean of this process is:
𝐸𝐴
1
2
1
1
2
1
0
This is clearly explained by the fact that each sample function of 𝑋 𝑡 is a constant
function of time (i.e., the sample functions are DC waveforms of two different
magnitudes 1 or 1  the statistical mean of this random process (computed
above) is zero.
To compute the time-averaged mean of 𝑋 𝑡 , let us assume that the given sample
function is 𝑋 𝑡
𝐴 . Then the time-averaged mean for this sample function is
Since 𝐴 is only either 1 or 1 but never 0, the time-averaged mean and the
statistical mean are NOT the same.
 As a result, this is NOT an ergodic random process.
Dr. M. Hassan
Noise
• The term noise refers to any undesired superimposed signals, the presence
of which limits the receiver from the efficient recovery of the desired
baseband-transmitted message
– A major objective of communications’ engineers is to reduce
(minimize/eliminate if possible) the effect of noise on the communication
process. This is typically done through:
• The choice of the proper modulation scheme
• Filtering
• Optimum receivers
Gaussian noise
•
When noise results from a large number of statistically independent & random
contributions it is typically of a Gaussian distribution with zero mean
σ2 is the variance of noise voltage
For zero mean, the normalized power or the
mean square voltage is given by:
The PDF of a RV with zero mean
& standard deviation σ
Dr. M. Hassan
23
Additive White Gaussian Noise (AWGN)
• Noise is usually assumed to be zero-mean AWGN
• AWGN: additive white Gaussian noise
– Additive: noise is added (not multiplied) to the signal
– White: has constant PSD (equal power for all frequency)
– Gaussian: in every time-instant (sampling instant), the noise is a Gaussian
random variable
Properties of the zero-mean AWGN n(t)
Σ
PSD
Autocorrelation:
Signal model
White Noise
Dr. M. Hassan
PDF:
watts/Hz
24
25
Input-Output Relationships
In many situations we need to know how an input random
Input
process
is modified by a system to result in a random
output process
Impulse
Response
Output
Transmission of a RP through an LTI system
Consider the system shown in figure, the output RP
is given by:
convolution
which is typically evaluated using the convolution integral given by:
where
The mean of the output process
is the impulse response of the system
is given by:
mean of the input
process
When
is a stationary RP 
is finite and the system is stable
the system DC response
Dr. M. Hassan
Transmission of a RP Through an LTI System
The autocorrelation of the output process
When
is a stationary,
At
is given by:
is finite, and the system is stable, we have:
the autocorrelation is given by:
This means that when the input is stationary  the output is stationary
Dr. M. Hassan
26
27
Power Spectral Density (PSD)
Recall that the Fourier transform of a continuous time function
where
Thus, the impulse response
by
as follows:
The autocorrelation
is given by:
can be obtained using IFT
can be recovered from the frequency response denoted
is now obtained by:
Let
Complex conjugate of the frequency response of the filter
Dr. M. Hassan
Power Spectral Density (PSD)-(cont’d)
where
is the magnitude response
Recall that the Power Spectral Density (Fourier transform of
) is given by:
Example:
Let
be the magnitude response of an ideal narrowband filter
If
Dr. M. Hassan
and
is continuous
in
: filter bandwidth
28
29
Relation between the PSD of the I/O Random Processes
Consider the RP
with the PSD
. We are
interested in the PSD
of the output process
of an LTI system when
is the input to the system
The autocorrelation of the output process is given by:
The PSD is the FT of the autocorrelation function
Thus, we have:
Let
Dr. M. Hassan
30
Summary of Input-Output Relationships
In many situations, we need to know how an input random
process
is modified by a system to result in a random
output process
Input
Impulse
Response
Output
Transmission of a RP through an LTI system
Consider the system shown in figure, the output RP
is given by:
convolution which is typically evaluated using the
convolution integral given by:
where
is the impulse response of the system
When the input is stationary  the output is stationary
1. The mean of the output process
is given by:
the system DC response
1.
2. The 2nd moment of
is
1.
2.
3. The spectral density of
Dr. M. Hassan
is