Optimal binary communication with nonequal probabilities

Optimal signal energies are derived for optimal binary digital communication systems with arbitrary signal probabilities and correlation with both coherent and noncoherent detection. The resulting bit-error probability (BEP) is computed and compared with the BEP of the same systems with equal signal energies. One of the conclusions is that for the coherent system with nonnegative correlation, and for the noncoherent system with arbitrary correlation, the optimal signals are on-off keying (OOK), i.e., the signal with probability p≤0.5 has energy E/p, while the second signal has zero energy, where E is the average signal energy. The proposed system is also better than a system with source coding and equiprobable signals.