## Digital Communications

###### Random Processes in Linear Systems Review Part 2

Output Autocorrelation Function Derivation (1/4)

1/19/15

Running Time: 6:57

Consider the continuous-time linear system with input random process X(t) and output random process Y(t).

We derive several different expressions for the autocorrelation function of the output random process depending on whether the input random process is wide-sense stationary, the system is time-invariant, and when both of these assumption are true simultaneously.

Output Autocorrelation Function Derivation (2/4)

1/19/15

Running Time: 7:20

In the first video of this series, a double-integral expression was derived that related the correlation function of the output random process to the autocorrelation function of the input random process and the (possibly time-varying) impulse response of the linear system.

This video simplifies the double-integral expression under two different assumptions: 1) The input random process is Wide Sense Stationary (WSS), and 2) The linear system is time-invariant (TI).

Output Autocorrelation Function Derivation (3/4)

1/19/15

Running Time: 8:40

The previous video in this series simplified the output autocorrelation function expression under two different assumptions: 1) The input random process is Wide Sense Stationary (WSS), and 2) The linear system is time-invariant (TI).

These assumptions were made one at a time.  This video assumes both these assumptions are true simultaneously and begins the derivation of the autocorrelation function output for WSS random process inputs to linear time-invariant (TI) systems.

Output Autocorrelation Function Derivation (4/4)

1/19/15

Running Time: 5:16

This video concludes the derivation of a compact expression for the output random process outcorrelation function.  We find that for wide-sense stationary (WSS) inputs to a linear time-invariant (TI) system, the final expression can be compactly written as a convolution operation.

Linear System Output Random Process Power

1/22/15

Running Time: 4:15

The 4-video series "Output Autocorrelation Function Derivation" developed a compact expression for computing the autocorrelation function of the random process as output/filtered by a linear system.

This previously developed equation allows the autocorrelation function to be computed for all time tau.

Often, we are just interested in computing the power of the output random process.  For the case of time-invariant systems with WSS random process input, this computation is much simpler.

This video shows that computing the output power for this special case requires the computation of just one simple integral.

Equivalent Input/Output Equations For Discrete Time Random Processes

1/24/15

Running Time: 7:43

The previous videos in this playlist derived different expressions relating the mean function and autocorrelation of a continuous-time random process to the mean function and autocorrelation function of a random process after being filtered by a linear system.

Similar equations can be derived for discrete-time linear systems.  This video summarizes these different input/output relationships for discrete-time linear systems.

Gaussian Random Process Input/Output Relationship

1/23/15

Running Time: 13:56

It's "well known" that when a Gaussian random process is input to a linear system, the output is also a Gaussian random process.

This video formally proves this result for a stable discrete-time linear system with a 2nd order Gaussian random process input.  The key steps in this proof are:

1) Since the system is stable and the input is a 2nd order process, the output random process Y(k) is well-defined, and for any arbitrary time in the limit is equal in a mean-square sense to a finite sum of Gaussian random variables.  We denote this finite summation of Gaussian random variables as YN(k).

2) Since the output random process is MS convergent to YN(k), it is also convergent in distribution.  This allows us to solve for the distribution of Y(k) by taking the limit of the distribution of YN(k).  By taking this limit we find that Y(k) has a Gaussian distribution as desired.