Computation of the exact bit-error rate of coherent M-ary PSK with Gray code bit mapping

The problem of calculating the average bit-error probability (BEP) of coherent -ary phase-shift keying (PSK) over a Gaussian channel has been studied previously in the literature. A solution to the problem for systems using a binary reflected Gray code (BRGC) to map bits to symbols was first presented by Lee. In this letter, we show that the results obtained by Lee are incorrect for 16. We show that the reason for this is an invalid assumption that the bit-error rate (BER) is independent of the transmitted symbols, an assumption which has also propagated to textbooks. We give a new expression for the BER of -PSK systems using the BRGC and compare this with Lee’s results.