Here you will so some calculations on the relation regarding Fourier series,, very similar to what is discussed in our printed lecture notes.
You will be asked to do some graphing. All that I require is that your graphs be .gif files, linked to from your Amaya .html file, using the Amaya Insert Image icon. I recommend that you use either gnuplot or jgraph to generate Encapsulated Postscript files, and then convert them to .gif using xv.
We will continue to assume the conditions in the example in the notes: We wish to send the bits 101010..., transmitting at the rate of 2 million bits per second. However, suppose we do this using 4B/5B coding.
A. State what actual repeating bit pattern (1s and 0s) will be put on the line. Assume that before the repeating pattern is placed on the line, the previous line state had been low (-1). Also, assume that we send 101010... as 010101..., which will make things more convenient. (Check: The period T should consist of 10 bits.)
B. Graph the truncated Fourier series up through the n-th harmonic, for n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (i.e. you will have 12 curves).
Use my program AnBn.C to find the coefficients an and bn, instead of evaluating the integrals yourself. However, you are required to set up the integrals for the case n = 1. These must be set up so that any calculus student could evaluate them, even if he/she has never heard of networks, x(t), etc.; the string "x(t)" must NOT appear in the integrands of your integrals. (Note: You are welcome to modify the output format of AnBn.C program, in order to make it generate gnuplot commands, which you can then use the mouse to copy into gnuplot, thus saving yourself some work.)
Make sure to superimpose the true x(t) on your graphs for comparison purposes. (I've recently added something at the end of my gnuplot Web page on how to do "cases" type functions, e.g. "z(t) is equal to 9 for t < 12.0 and equal to 7.2 for t > 12.0.".)
C. Suppose the bandwidth of our transmission medium is 0.4 MHz, and suppose each bit is sampled in the middle of its bit time, and count any positive line value as a 1 and any negative line value as a 0. Which bits, if any, will be received incorrectly? What if the bandwidth of the medium is 0.6 MHz?
Suppose we have messages of m = 5 bits each, and add a CRC field using a divisor of C = 1011. As you know, the CRC check will report an error if the division yields a nonzero remainder.
Assume each of the 8 bits has probability p of being in error, with bit errors being independent. Thus for example the error pattern E = 00110100 has probability p3(1-p)5.
Find the following probabilities, expressed in terms of p:
A. Find the probability that the CRC check reports an error.
B. Find the probability that the CRC check reports an error, given that there is an error in the message portion of the frame.
C. Find the probability that the message portion is not in error, given that CRC reports an error.