Consider the opportunistic orthogonal signaling scheme described in Section 10.3.3. Each of the M messages corresponds to K (real) orthogonal signals. The encoder transmits the signal that has the largest correlation (among the K possible choices corresponding to the message to be conveyed) with the interference (real white Gaussian process with power spectral density Ns /2). The decoder decides the most likely transmit signal (among the MK possible choices) and then decides on the message corresponding to the most likely transmit signal. Fix the number of messages, M, and the number of signals for each message, K. Suppose that message 1 is to be conveyed.

1. Derive a good upper bound on the error probability of opportunistic orthogonal signaling. What is the appropriate choice of the threshold, y, as a function of M, K and the power spectral densities Ns/2, N0 /2?

2. By an appropriate choice of K as a function of M, Ns, N0 show that the upper bound you have derived converges to zero as M goes to infinity as long as cb/N0 is larger than -1.59 dB.

3. Can you explain why opportunistic orthogonal signaling achieves the capacity of the infinite bandwidth AWGN channel with no interference by interpreting the correct choice of K?

4. We have worked with the assumption that the interference s(t) is white Gaussian. Suppose s(t) is still white but not Gaussian. Can you think of a simple way to modify the opportunistic orthogonal signaling scheme presented in the text so that we still achieve the same minimal cb/N0 of -1.59 dB?