# Consider the jointly optimum detector defined in Problem 16.8 for the two-user, synchronous CDMA…

Consider the jointly optimum detector defined in Problem 16.8 for the two-user, synchronous CDMA system. Show that the (symbol) error probability for this detector may be upper-bounded as

Problem 16.8

Consider the two-user, synchronous CDMA transmission system described in Problem 16.6. 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

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

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:

Problem 16.6

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

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 P16.6.

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

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).