Random Processes

Video Lectures
 
Random Processes: Part 1
We provide a formal definition of a random process, and also define important related quantities such as the mean function and autocorrelation function.  Several examples of these quantities are provided, and then we begin to analyze several specific random processes.  The first is the Asynchronous Binary Signal (ABS) random process, and then we introduce the Poisson Process.
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Introduction

9/28/14

Running Time: 7:59

This video discusses the concept of a random process.  Similarities between random processes and random sequence (discussed early in the course) are made.  

 

This video also provides the formal/mathematical definition of a random process.  A random process is just an infinite collection of random variables organized by the continuous time-variable "t".

Definitions (Part 1)

9/28/14

Running Time: 9:42

This video provides several basic definitions related to random processes.  For a random process X(t) we define the following:

 

Statistically specified random process
Mean function of the random process
Autocorrelation function of the random process

We also review properties of the random process Autocorrelation function.  No formal proofs are provided for these properties as they were proven when discussing random sequences previously in the course.

Definitions (Part 2)

9/28/14

Running Time: 5:25

This video provides several basic definitions related to random processes.  For a random process X(t) we define the following:

 

Autocovariance function of the random process
Variance function of the random process
Power function of the random process

We also provide a useful equation that relates the autocorrelation function, autocovariance function, and mean function of the random process.

Mean and Autocorrelation Function Example

9/28/14

Running Time: 8:24

The previous videos provided definitions of the mean and autocorrelation function of a random process.

In this video we work with the random process X(t) = Asin(wc*t + theta) where both A and theta are random variables.  We compute the mean function and autocorrelation function of this random process.

Asynchronous Binary Signaling (Introduction)

9/28/14

Running Time: 5:44

This and the next 2 videos examine an Asynchronous Binary Signaling (ABS) random process.  This random process is a binary communication signal that randomly switches between amplitudes "-a" and "+a" at every symbol time "T".  It is an asynchrounous signal as it has a random displacement with respect to the time origin.

Asynchronous Binary Signaling (Mean Function)

9/28/14

Running Time: 2:51

We compute the mean function of the asynchronous binary signaling (ABS) random process defined in the previous video.  We use the basic definition of the mean function and exploit the fact that pulse amplitudes and waveform displacement are independent random variables to perform the computation.

Asynchronous Binary Signaling (Correlation Function)

9/28/14

Running Time: 14:03

We compute the autocorrelation function of the asynchronous binary signaling (ABS) random process defined in the previous video.  We use the basic definition of the autocorrelation function to compute the desired ensemble average.

Poisson Process (Introduction)

9/28/14

Running Time: 9:10

We introduce the Poisson random process.  This process, denoted N(t), can be thought of as a "counting process", indicating the number arrivals at time "t".  The interarrival time random sequence tau[n] and time-to-the-nth-arrival random sequence T[n] are used to define this random process.

Poisson Process Properties (Part 1)

9/28/14

Running Time: 11:12

We derive the probability mass function of the Poisson random process.  This derivation uses properties of the underyling random sequences tau[n] and T[n] (i.e. the fact that they are independent) to derive an expression for P(N(t) = n).  Not surprisingly, this expression is just the probability distribution of a Poisson random variable.  This makes it almost trivial to determine the mean function and variance function of the Poisson random process.

Poisson Process Properties (Part 2)

10/2/14

Running Time: 12:00

The Poisson random process has an "independent increments" property. For the Poisson process this means that the number of arrivals on non-overlapping time intervals are independent random variables.

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© 2020 by Adam Panagos

adam.panagos@gmail.com