In this problem we evaluate the bit error rate ber

Question

In this problem, we evaluate the bit error rate (BER) performance of the CDMA communications system introduced in Problem 8.3.9. In our experiments, we will make the following additional assumptions.

• In practical systems, code vectors are generated pseudorandomly. We will assume the code vectors are random. For each transmitted data vector X, the code vector of user i will be

 Where the components S_ij are iid random variables such that P_Sij (1) = P_Sij (-1) = 1/2. Note that the factor 1/ √n is used so that each code vector Si has length 1: ||S_i||² = S'_iS_i

= 1.

• Each user transmits at 6dB SNR. For convenience, assume P_i= p = 4 and σ²= 1.

(a) Use Matlab to simulate a CDMA system with processing gain n = 16. For each experimental trial, generate a random set of code vectors {S_i}, data vector X, and noise vector N. Find the ML estimate x∗ and count the number of bit errors; i.e., the number of positions in which x∗ i ≠ X_i. Use the relative frequency of bit errors as an estimate of the probability of bit error. Consider k = 2, 4, 8, 16 users. For each value of k, perform enough trials so that bit errors are generated on 100 independent trials. Explain why your simulations take so long.

(b) For a simpler detector known as the matched filter, when Y = y, the detector decision for user i is 

Where sgn (x) = 1 if x > 0, sgn (x) = -1 if x

Problem 8.3.9

In a code division multiple access (CDMA) communications system, k users share a radio channel using a set of n-dimensional code vectors {S₁,..., S_k} to distinguish their signals. The dimensionality factor n is known as the processing gain. Each user i transmits independent data bits Xi such that the vector X = [X₁ ··· X_n] has iid components with PX_i(1) = PX_i(-1) = 1/2. The received signal is

Where N is a Gaussian (0, σ²I) noise vector From the observation Y, the receiver performs a multiple hypothesis test to decode the data bit vector X.

(a) Show that in terms of vectors,

(b) Given Y = y, show that the MAP and ML detectors for X are the same and are given by

Where Bn is the set of all n dimensional vectors with ±1 elements

(c) How many hypotheses does the ML detector need to evaluate?