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?