For the CDMA system in Problem 11.3.8, we wish to use MATLAB to evaluate the bit error rate (BER) performance of the decorrelater introduced Problem 11.3.9. In particular, we want to estimate Pe, the probability that for a set of randomly chosen code vectors, that a randomly chosen user's bit is decoded incorrectly at the receiver.

For a k user system with a fixed set of code vectors S1,...,Sk, let S denote the matrix with Si as its ith column. Assuming that the matrix inverse (SʹS)-1 exists, write an expression for Pe,i(S), the probability of error for the transmitted bit of user i, in terms of S and the Q(∙) function. For the same fixed set of code vectors S, write an expression for Pe, the probability of error for the bit of a randomly chosen user.

(b) In the event that (SʹS)-1 does not exist, we assume the decorrelator flips a coin to guess the transmitted bit of each user. What are Pe,i and Pe in this case?

(c) For a CDMA system with processing gain n = 32 and k users, each with SNR 6 dB, write a MATLAB program that averages over randomly chosen matrices S to estimate Pe for the decorrelator. Note that unlike the case for Problem 11.4.6, simulating the transmission of bits is not necessary. Graph your estimate e as a function of k.