ELEC 321

Review Session

Updated 2017-12-06

Problems

Problem 1

Suppose we have with where

We can start by finding the Fourier transform of :

Now factor out and find the :

We recognize that the signal in time domain is

Thus we know that the output has convolved with the input (product in frequency domain)

What is the expected value of ?

Since , then

Since the second term is a geometric series.

We see that as time increases, the mean of is changing, thus is not a stationary process.

Problem 2

Packets arrive at probability ; packets depart at probability . The buffer can hold up to packets. Let be the number of packets in the buffer at time .

Show that the system can be modelled by Markov Chain:

Since at any time we don’t care about the number of packets in the buffer at time (history) if we already have all the information . In particular,

The conditional stuff (after that symbol) in the probability is useless information as far as the buffer is concerned.

There are total of states in the Markov chain: state

For state 0: there are two possible states to go to:

For state : there are also only two possible states to go

For any state in between (state 1 to state ) there are three possible outcomes:

Therefore, the transition matrix is:

To find the stationary distribution, we use the fact that

Do the matrix multiplication and we obtain equations for variables: .

The equations are:

Then we substitute every equation in terms of , and for general , we find the pattern:

Then find by setting an initial condition.

Office Hour

The variance for two normal random variables added together is the sum of two variances. (Proof later)