Pulse Position Modulation

The following signal set is used to transmit equally likely messages over an AWGN channel with spectral height N₀/2:

1.

Draw a block diagram of the optimum receiver and compute the probability of error for that receiver.

2.

Determine a set of orthonormal basis functions $\{\phi_0(t),\phi_1(t)\}$ and represent the above signal set in terms of these basis functions. I.e., find the coefficients c_i,j of the expansion $s_i(t) = \sum_j c_{i,j} \phi_j(t)$.

3.

Sketch and accurately label the signal set in a suitably chosen signal space.

4.

Consider now the following receiver:

Is it possible that this receiver, with suitably chosen a, b, and $\gamma$, achieves the same probability of error as the optimum receiver? If your answer is ``yes'' determine f(R<sub>0</sub>,R<sub>1</sub>) and $\gamma$. If your answer is ``no'' explain why not.