Homework 3
Due Thursday, October 30.

Directions for submission: Set up directories, ProblemA and ProblemB, each one containing the .tex and .ps files for that problem. (The .tex files must do "includes" of the .ps files.) Use tar to combine the directories, and then use gzip and uuencode as usual. In the latter case, set things up so that when the Reader uses uudecode, it will produce a file named Hwk3.tar.gz.

Your analyses in both problems must be written up in LaTeX.

You will be asked to do some graphing. Use either gnuplot or jgraph for this, generating Encapsulated Postscript files. See

http://heather.cs.ucdavis.edu/~matloff/gnuplot.html
http://heather.cs.ucdavis.edu/~matloff/jgraph.html

Problem A:

Look at Section 7 (``Effect of Bandwidth on Transmission Rates'') in the unit in the printed lecture notes on the physical layer. Suppose, as in the example there, we are sending at the rate of 2 MHz. Consider the bit pattern 1001100110011... The waveform would have the value 1 between -T/2 and -T/4, -1 between -T/4 and T/4, 1 between T/4 and T/2, and so on.

Graph the third, fifth and seventh Fourier approximations to the desired waveform. By this I mean that k, in 2*pi*k*f0*t has a maximal value of 3, 5 and 7, respectively. (The graph in the handout has k = 5.)

State the minimal bandwidths a transmission medium would need to transmit each of those three Fourier approximations.

Problem B:

In this problem you will investigate the error-catching abilities of CRC, and write up your findings in LaTeX. Here we will have m = 5 and c = 4. We will take our C to be 1011.

Assume each of the eight bits has probability p of being in error, with bit errors being independent. Thus for example the error pattern E = 00110100 has probability

        3    5
       p (1-p)

Find all eight-bit E patterns which that C will NOT detect. Note that this is easy, because they consist of all possible products C*Q, where Q ranges over all 31 nonzero five-bit strings. List all these strings, and the probability of occurrence for each (one probability for each string).

(Do the multiplication C*Q just like the division--mod 2, and bitwise. To check you are doing it right, look at the division example in our CRC handout; there C*Q+R should come out to the original dividend, M'.)

Plot the probability of an undetected error, that is P(E is nonzero and C divides E), as a function of p.