ELEC 471, Spring 2001
Prof. Rich Kozick

## Homework 4

Date Assigned: Thursday, February 8, 2001
Date Due: Tuesday, February 13 and Tuesday, February 20, 2001

** DUE DATE FOR DIGITAL COMMUNICATION PART CHANGED TO THURSDAY, FEBRUARY 22 **

1. Exam 1 will be on Thursday, February 15. The topics for the exam include Chapter 1 of the text and the digital communication application.

2. Reading: Please begin reading Chapter 2, Sections 2.1 and 2.2.

3. Please solve the following problems in the text, and submit your solutions on February 13:
Chapter 1: 1.8.2, 1.9.1, and 1.9.6.
We will have a short quiz on February 13.

4. Each student should individually implement MATLAB simulations of the digital communication system that we discussed in class. Please do the following, and submit your work on Tuesday, February 20. (You may want to complete this assignment by February 15 to help you study for Exam 1, but the due date is February 20.)

** DUE DATE FOR DIGITAL COMMUNICATION PART CHANGED TO THURSDAY, FEBRUARY 22 **

Recall that the observations at the receiver are modeled as

x = s + n

where the signal s and noise n are independent and take on the following values with the indicated probabilities:

```n       P[n]    s     P[s]
--      ----    --    ----

-2      0.1     -1    0.5
-1      0.2     +1    0.5
0      0.4
+1      0.2
+2      0.1
```

We discussed various "decision rules" in class whose purpose is to decide whether a -1 or +1 was transmitted based on the noisy observation x. We also calculated the probability of a bit error (also called bit error rate or BER) using mathematical analysis. Now we will use MATLAB to simulate the digital communication system and evaluate the BER for various situations, as described below.

1. Simulate the system with a decision rule that always chooses signal +1 when the received data is x = 0. Recall that the BER should be 0.2 for this case.

Sample MATLAB programs digcom1.m and digcom2.m are available to help you get started. These are the programs that we discussed in class on February 8.

2. Modify your previous simulation so that when x = 0 is received, your decision rule randomly chooses +1 with probability r and -1 with probability 1-r. Run for the cases r = 1, 0.5, and 0. Does changing r have any effect on the BER? Compare with the analytical BER.

3. Next modify your simulation program so that each bit is transmitted three times, with independent noise added on each transmission. Make your "overall" decision for each bit as the majority of the three individual decisions. (You can do the majority conveniently in MATLAB using the median function, as in the sample program digcom2.m.) What is the BER for this system, for the cases of r = 1, 0.5, and 0 in your decision algorithm? Does the BER vary with r?

Derive a formula for the BER as a function of r for this case in which each bit is transmitted three times. Compare your simulation results with the analytical BER.

Hint: When developing the analytical BER, think of the conditional probability of a bit error given that s = -1 was transmitted, and do the same given that s = +1 was transmitted. Then you should be able to obtain the correct analytical BER.

If you would like, repeat using five transmissions of each bit, and compare analysis with simulation. (This is not required.)

Please submit printouts of your MATLAB programs, BER results from running your programs (you will need to decide how many "trials" should be performed to get accurate results), and compare your simulation results with the analytical BER. You should explain clearly and completely how the analytical BER is calculated for each case. If you do things correctly, you should obtain excellent agreement between the simulated BER and the analytical BER!

Thank you.