Optimum Power Division in Coherent Communication Systems

This paper considers the optimum way to divide the available energy between the channel measurement and information transfer functions. Specifically considered is a binary, symmetric, phase-modulation system operating over a channel which imparts a random phase modulation to the signal and adds Gaussian noise. We specify an efficient demodulation scheme and optimize it for all possible energy divisions at the transmitter. The energy division is then varied to achieve the minimum probability of error. For the range of signal and channel parameters considered, it is shown that the probability of error is minimized by devoting all available energy to transmitting information and operating on the information sequence to measure the channel. At high E/No ratios there may be cases in which the probability of error is minimized by dividing the energy between pilot tone and modulation. We were unable to find a case that satisfied the restrictions of our model in which this was true. The exact quantitative results depend on the statistics of the channel variations. Given any set of statistics, the approach developed will enable one to find the optimum power division for that particular case. Even more important are the conclusions which one can draw about the general case from our specific results. First, we see that from the theoretical standpoint one should always use the modulation waveform as part of the synchronization system. In many operational systems this is not done.