In a code division multiple access cdma communications

Question

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?