Suboptimum Linear Receivers for CDMA Systems

For this problem, consider the same system description as in the previous problem. Thus, two users attempt to transmit simultaneously over an additive white Gaussian noise channel using the (not quite orthogonal) signal sets sown in problem 1. Here, we are interested in finding a linear receiver for demodulating the first user's signal which is capable of rejecting the interference from the second user.

In general, a linear receiver can be represented by the following block diagram:

$R_t$ is given by equation (1) in problem 1.

1. Assume that $g(t) = s_0^{(1)}(t) + a s_0^{(2)}(t)$ is chosen. How would you choose the constant $a$ such that the inner product $\langle~R_t,g(t)~\rangle$ is independent of the amplitude $A_2$ of the interfering signal.
2. Assume now that $a=-\langle~s_0^{(1)}(t),s_0^{(2)}(t)~\rangle$. Find the distribution of the random variable $R$ for both cases, $s_0^{(1)}(t)$ and $s_1^{(1)}(t)$ is transmitted.
3. How would you choose the threshold $K$ in the receiver above?
4. Find the probability of error for this receiver. Which value does the probability of error approach if the amplitude ${A_2}$ of the interfering user approaches $\infty$?
5. Indicate the decision boundary formed by this receiver in a suitably chosen and accurately labeled signal space.