ELEC 471

Prof. Rich Kozick

Prof. Rich Kozick

**Date Assigned:** Thursday, April 5, 2001

**Date Due: ** Thursday, April 12, 2001

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

- 1.
- 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. The random variables*S*and*N*are independent. The receiver observes the random variable*X*=*S*+*N*.

- (a)
- Suppose the receiver obtains the value
*X*=*x*. Find the signal estimate that minimizes the mean-squared error .Show all of the steps in your analysis, and display as a plot versus*x*. I suggest that you draw plots of*P*_{S}(*s*),*P*_{N}(*n*),*P*_{X,S}(*x*,*s*),*P*_{X}(*x*),*P*_{S|X}(*s*|*x*) as we did in class. - (b)
- How would you process the signal estimates
in order to recover the
*binary*values of*S*? In other words, what ``decision rule'' would you use to recover the bits from*X*based on ? - (c)
- For the decision rule that you developed in part b, what is the probability of a bit error for this system? Explain your reasoning.
- (d)
- 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!**

- 2.
- Please answer the following questions for the joint
PMF
*P*_{X,Y}(*x*,*y*) shown in the figure below.- (a)
- Find the marginal PMFs
*P*_{X}(*x*) and*P*_{Y}(*y*), and plot them. - (b)
- Find the mean and variance of
*X*and*Y*: . - (c)
- Find the correlation between
*X*and*Y*, defined as*r*_{X,Y}=*E*[*X Y*]. - (d)
- Find the covariance .
- (e)
- Find the correlation coefficient .
- (f)
- Are
*X*and*Y**independent*random variables? Recall that*X*and*Y*are independent if and only if*P*_{X,Y}(*x*,*y*) =*P*_{X}(*x*)*P*_{Y}(*y*). - (g)
- Find the conditional PMFs
*P*_{X|Y}(*x*|*y*) =*P*[*X*=*x*|*Y*=*y*]*P*_{Y|X}(*y*|*x*) =*P*[*Y*=*y*|*X*=*x*]*x*and*y*(on separate plots). - (h)
- Suppose you need to produce an estimate of
*X*that minimizes the mean squared error .You must do this with*no information*about*Y*. What value should you choose for , and why? (Your answer should be a number!) - (i)
- 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 ? Explain how to compute , and present a plot of versus*y*. - (j)
- Is in part (i) different from 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
that you computed in part (e)?
That is, does the value of lead you to expect
that information about
*Y*should be useful in predicting the value of*X*? - (k)
- Using your answer from part (h), compute
Using your answer from part (i), compute following quantities:

, , and

.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