Spread Spectrum Multiple Access

In a spread spectrum communication system users employ signals which occupy a significantly larger bandwidth than the symbol rate. Such a signalling scheme provides some advantages which are primarily of interest in secure communication systems, e.g., low probability of intercept or robustness to jamming. In this problem we explore the inherent multiple access capability of spread spectrum signalling, i.e., the ability to support simultaneous transmissions in the same frequency band.

In the sequel, assume that the communication channel is an additive white Gaussian noise channel with spectral height $\frac{N_0}{2}$.

1. One user employs the following signal set to transmit equally likely binary symbols
   
   
   

   Draw a block diagram of the receiver which minimizes the probability of a bit error for this signal set.

2. Compute the probability of error achieved by your receiver.
3. Now, a second users transmits one of the following signals with equal probability
   
   
   

   Both signals are transmitted simultaneously, such that the received signal is given by
   

   $$R_t = A_1 s_i^{(1)}(t) + A_2 s_j^{(2)}(t) + N_t,$$ (1)

   
   
   where $N_t$ is the noise process and $i,j \in \{0,1\}$ indicate which symbol each of the users is transmitting. We are interested in receiving the first user's signal in the presence of the second (interfering) user.

   Find the probability of error of your receiver from part (a) for distinguishing between $s_0^{(1)}(t)$ and $s_1^{(1)}(t)$ if the received signal is given by (1). Which value does the probability of error approach if the amplitude ${A_2}$ of the interfering user approaches $\infty$?

4. Find the minimum probability of error receiver for distinguishing between $s_0^{(1)}(t)$ and $s_1^{(1)}(t)$ in the presence of the interfering signal $s_j^{(2)}(t)$, i.e., if the received signal is given by (1). Note: You do not need to find the probability of error for this receiver.
5. Indicate the locations of the relevant signals and the decision regions for your receiver from part (d) in a suitably chosen and accurately labeled signal space. Indicate also the decision boundary formed by the receiver from part (a).