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 M-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 P.J. Lee (see ibid., vol.COM-34, p.488-91, 1986). We show that the results obtained by Lee are incorrect for M≥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 M-PSK systems using the BRGC and compare this with Lee´s results.