Optimal M-amplitude shift keying/quadrature amplitude shift keying with non-equal symbol probabilities

In this study the authors derive a formula to compute the optimal signal constellation, bit error probabilities and decision thresholds for M-ASK (amplitude shift keying) with non-equal symbol probabilities. The resulting signal constellation is not equally spaced. The authors demonstrate the importance of assigning lower energy signals to symbols with higher probabilities. The authors show that the optimal system has a lower error probability than a conventional system, that is, a system with source coding, equiprobable symbols and equal space constellation. In several cases presented in this study the error probability of the optimal system is significantly lower than in the conventional system. The authors show that even with equal symbol probabilities the equal space constellation is only asymptotically optimal, not optimal for M ≥ 4. However in the several cases that the authors have studied the difference in the error probabilities between ASK with optimal constellation and equal space constellation is insignificant in practice. The authors further extend our results in M-ASK to M-QASK (quadrature ASK) and similar improvements can be obtained.