For the CDMA system in Problem 8.3.9, we wish to use Matlab to evaluate the bit error rate (BER) performance of the decorrelater introduced Problem 8.3.10. 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.
(a) For a k user system with a fixed set of code vectors {S}1k, let S denote the matrix with Si as its ith column. Assuming that the matrix inverse (S'S)-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 (S'S)-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 6dB, write a Matlab program that averages over randomly chosen matrices S to estimate Pe for the decorrelator. Note that unlike the case for Problem 8.4.6, simulating the transmission of bits is not necessary. Graph your estimate 
e as a function of k.
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?
Problem 8.3.10
For the CDMA communications system of Problem 8.3.9, a detection strategy known as decorrelation applies a transformation to Y to generate
Where 
= (S'S)-1S'N is still a Gaussian noise vector with expected value E[] = 0. Decorrelation separates the signals in that the ith component of 
is
Which is the same as a single user receiver output of the binary communication system of Example 8.6 For equally likely inputs Xi = 1 and Xi = -1, Example 8.6 showed that the optimal (minimum probability of bit error) decision rule based on the receiver output i is
Although this technique requires the code vectors S1,..., Sk to be linearly independent, the number of hypotheses that must be tested is greatly reduced in comparison to the optimal ML detector introduced in Problem 8.3.9. In the case of linearly independent code vectors, is the decorrelator optimal? That is, does it achieve the same BER as the optimal ML detector?
Example 8.6
With probability p, a digital communications system transmits a 0. It transmits a 1 with probability 1 - p. The received signal is either X = -v + N volts, if the transmitted bit is 0; or v + N volts, if the transmitted bit is 1. The voltage ±v is the information component of the received signal, and N, a Gaussian (0,σ) random variable, is the noise component. Given the received signal X, what is the minimum probability of error rule for deciding whether 0 or 1 was sent?
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?
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