ELEC 471
Prof. Rich Kozick

Homework 9

Date Assigned: Thursday, April 5, 2001
Date Due: Thursday, April 12, 2001

Reading: Please read Chapter 4, Sections 4.1 - 4.5.

In this problem, I would like you to perform an analysis of a digital communication system that is similar to the case that we studied in class on April 5. The only difference is that the signal S and noise N have PMFs as follows.

P_S(s) = \left\{ \begin{array}
0.25, & s = -1 \ 0.75, ...
 ... 0 \ 0.25, & n = +1 \ 0, & {\rm otherwise}\end{array} \right.\end{displaymath}

The random variables S and N are independent. The receiver observes the random variable X = S + N.
Suppose the receiver obtains the value X = x. Find the signal estimate $\hat{s}(x)$ that minimizes the mean-squared error $E [ ( S - \hat{s}(x))^2]$.Show all of the steps in your analysis, and display $\hat{s}(x)$ as a plot versus x. I suggest that you draw plots of PS(s), PN(n), PX,S(x,s), PX(x), PS|X(s|x) as we did in class.
How would you process the signal estimates $\hat{s}(x)$ in order to recover the binary values $\{ -1, +1 \}$ of S? In other words, what ``decision rule'' would you use to recover the bits from X based on $\hat{s}(x)$?

For the decision rule that you developed in part b, what is the probability of a bit error for this system? Explain your reasoning.

Write a MATLAB program to simulate this system, including the decision rule developed in part b. Compare the bit error rate (BER) in your simulation with the analytical probability of a bit error computed in part c.

Be sure to compare the simulated and analytical BER!

Please answer the following questions for the joint PMF PX,Y(x,y) shown in the figure below.

Find the marginal PMFs PX(x) and PY(y), and plot them.
Find the mean and variance of X and Y: $\mu_X, \mu_Y, \sigma_X^2, \sigma_Y^2$.
Find the correlation between X and Y, defined as rX,Y = E [ X Y].
Find the covariance ${\rm Cov} [ X, Y ]$.
Find the correlation coefficient $\rho_{X,Y}$.
Are X and Y independent random variables? Recall that X and Y are independent if and only if PX,Y(x,y) = PX(x) PY(y).
Find the conditional PMFs

PX|Y(x|y) = P [ X = x | Y = y]

PY|X(y|x) = P [ Y = y | X = x]

and plot each of these versus x and y (on separate plots).
Suppose you need to produce an estimate $\hat{x}$ of X that minimizes the mean squared error $E [ (X - \hat{x})^2 ]$.You must do this with no information about Y. What value should you choose for $\hat{x}$, and why? (Your answer should be a number!)
Suppose that you observe that the random variable Y takes on the value y (i.e., Y = y). We would like to incorporate the knowledge that Y=y to improve our estimate of X. For each possible value of y, what is your estimate of X, denoted by $\hat{x}(y)$? Explain how to compute $\hat{x}(y)$, and present a plot of $\hat{x}(y)$ versus y.
Is $\hat{x}(y)$ in part (i) different from $\hat{x}$ in part (h)? Is this reasonable based on the probability values in the joint PMF? Is this reasonable based on the value of correlation coefficient $\rho_{X,Y}$ that you computed in part (e)? That is, does the value of $\rho_{X,Y}$ lead you to expect that information about Y should be useful in predicting the value of X?
Using your answer from part (h), compute $E [ (X - \hat{x})^2 ]$

Using your answer from part (i), compute following quantities:
$E[(X-\hat{x}(-1))^2 \vert Y=-1]$, $E[(X-\hat{x}(0))^2 \vert Y=0]$, and
$E[(X-\hat{x}(1))^2 \vert Y=1]$.

Do these results show that we get better estimates for X when the value of Y is known? Please explain.

Presentations: The following students are asked to present their solution to these problems. The presentation will be done as a group.

Item 1
Shawn Mattews and Matt Murdock

Item 2
Kevin Murphy and Dan Oh