The problem of finding the optimal labeling (bit-to-symbol mapping) of multilevel coherent phase shift keying (PSK), pulse amplitude modulation (PAM), and quadrature amplitude modulation (QAM) constellations with respect to minimizing the bit-error probability (BEP) over a Gaussian channel is addressed. We show that using the binary reflected Gray code (BRGC) to label the signal constellation results in the lowest possible BEP for high enough signal energy-to-noise ratios and analyze what is "high enough" in this sense. It turns out that the BRGC is optimal for PSK and PAM systems whenever the target BEP is at most a few percent, which covers most systems of practical interest. New and simple closed-form expressions are presented for the BEP of PSK, PAM, and QAM using the BRGC

This paper is concerned with the problem of selecting a binary labeling for the signal constellation in an M-PSK communication system. A good starting point is labelings having the Gray property, but this is not altogether enough, since the number of distinct Gray labelings that result in different bit error probability grows rapidly with increasing constellation size. By introducing a recursive Gray labeling construction method called expansion, the paper answers the natural question of what labeling, among all possible constellation labelings (not only Gray), that will give the lowest possible average probability of bit errors. Under certain assumptions on the channel, the answer is that the labeling originally proposed by Gray, the binary reflected Gray code, is the optimal labeling for M-PSK systems, which has, surprisingly, never been proved before.

This paper concerns the problem of selecting a binary labeling for the signal constellation in M-PSK, M-PAM, and M-QAM communication systems. Gray labelings are discussed and the original work by Frank Gray is analyzed. As is noted, the number of distinct Gray labelings that result in different bit-error probability grows rapidly with increasing constellation size. By introducing a recursive Gray labeling construction method called expansion, the paper answers the natural question of what labeling, among all possible constellation labelings, will give the lowest possible average probability of bit errors for the considered constellations. Under certain assumptions on the channel, the answer is that the labeling proposed by Gray, the binary reflected Gray code, is the optimal labeling for all three constellations, which has, surprisingly, never been proved before.

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/spl ges/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.

To evaluate the performance of a convolutional code it is common to use a union bound on either the bit-error rate or the first-event error rate. Since these bounds are given as infinite sums they must be truncated to be evaluated. However, as the tail of the infinite sum is discarded, the upper bound turns into an approximation. Two methods to evaluate the union bound while taking the summation tail into account and thus maintaining the upper bound are described. The first method is based on a state matrix description of the convolutional encoder, while the second method is based on the transfer function. Results are given that compare the upper bounds and the truncated bound (approximation) to computer simulations. The standard rule-of-thumb that the union bound is tight as long as the error rates are below 10/sup -3/ is readily seen to be valid.