In a variation on Example 9.10, we use the observation vector Y = Y(n) = [Y1 ··· Yn] to estimate X = X1. The 21-dimensional vector X has correlation matrix RX with i, jth element
Find the LMSE estimate 
L (Y(n)) = 
(n)Y(n). Graph the mean square error e∗L (n) as a function of the number of observations n, and interpret your results for the cases
Receiver to make decisions on what bits were transmitted Based on the LMSE estimate i, the bit decision rule for useri is 
i = sgn (i) Following the approach in Problem 8.4.6, construct a simulation to estimate the BER for a system with processing gain n = 32, with each user operating at 6dB SNR. Graph your results as a function of the number of users k for k = 1, 2, 4, 8, 16, 32. Make sure to average your results over the choice of code vectors Si.
Example 9.10
The correlation matrix RX of a 21-dimensional random vector X has i, jth element
W is a random vector, independent of X, with expected value E[W] = 0 and diagonal correlation matrix RW = (0.1) I. Use the first n elements of Y = X+W to form a linear estimate of X21 and plot the mean square error of the optimum linear estimate as a function of n for
Problem 8.4.6
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 Sij are iid random variables such that PSij (1) = PSij (-1) = 1/2. Note that the factor 1/ √n is used so that each code vector Si has length 1: ||Si||2 = S'iSi
= 1.
- Each user transmits at 6dB SNR. For convenience, assume Pi= 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 {Si}, 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 ≠ Xi. 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,..., Sk} 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 ··· Xn] has iid components with PXi(1) = PXi(-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?