Problem on synchronous cdma transmission system

Question

Solve the following problem:

Consider the two-user, synchronous CDMA transmission system described in Problem 1. P(b1 = 1) = P(b₂ = 1) = ½ and P(b₁, b₂) = P(b₁)P(b₂). The jointly optimum detector makes decisions based on the maximum a posteriori probability (MAP) criterion. That is, the detector computes

max P[b₁,b₂|r(t), 0 ≤ t ≤ T]
b₁,b₂

a. For the equally likely information bits (b₁, b₂) show that the MAP criterion is equivalent to the maximum-likelihood (ML) criterion

max p[r(t), 0 ≤ t ≤ T | b₁,b₂]
b₁,b₂

b. Show that the ML criterion in (a) leads to the jointly optimum detector that makes decisions on b₁ and b₂ according to the following rule:

max (√ε₁b₁r₁ + √ε₂b₂r₂ - √ε₁ε₂pb₁b₂)
b₁,b₂

Problem 1:  Consider a two-user, synchronous CDMA transmission system, where the received signal is

r(t) = √ε₁b₁g₁(t) + √ε₂b₂g₂(t) + n(t) ,     0 ≤ t ≤ T

and (b₁, b₂) = (±1, ±1). The noise process n(t) is zero-mean Gaussian and white, with spectral density N₀/2. The demodulator for r(t) is shown in Figure .

a. Show that the correlator outputs r₁ and r₂ at t = T may be expressed as

r₁ = √ε₁b₁ + √ε₂pb₂ + n₁

r₂ = √ε₁b₁p + √ε₂b₂ + n₂

b. Determine the variances of n₁ and n₂ and the covariance of n₁ and n₂.

c. Determine the joint PDF p(r₁,r₂|b₁, b₂).