A simple technique for evaluating the bit-error probability of coherent M-ary phase-shift keying (PSK) with any bit-mapping is proposed. Closed-form expressions are given in terms of error functions for half- and quadrant-plane probabilities in the decision space, avoiding the numerical evaluation of complicated integrals that occurs in the direct method. Bit error probability expressions for M-ary PSK with Gray, natural binary, and folded binary bit-mappings are derived.>

A simple method for determining the sequency ordering of any row in any Hadamard matrix directly from its binary representation is developed. This proposed method is proved to be much simpler than the well-known bit-reverse inverse Gray code method.>

Contrary to what has been believed, it is found that weights can be associated with different bit positions of the standard binary reflected Gray code. Therefore, two algorithms are devised for direct Gray-to-decimal and decimal-to-Gray conversions. These algorithms are much simpler than old algorithms which involve conversions to and from binary code and exclusive-OR operations.