In the CDMA multiuser communications system introduced in Problem 8.3.9, each user i transmits an independent data bit 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
(a) Based on the observation Y, find the LMSE estimate Xˆi(Y) = 
iY of Xi.
(b) Let
Denote the vector of LMSE bits estimates for users 1,..., k. Show that
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?