BER computation of generalized QAM constellations

Based on a Karnaugh map style Gray coding of constellations, we derive a recursive algorithm for the exact bit error rate computation of generalized hierarchical M-ary pulse amplitude modulations (PAM) over additive white Gaussian noise (AWGN) and fading channels. We also show how this algorithm can be extended to generalized hierarchical quadrature amplitude modulations constellations (square and rectangular). For the AWGN case, these expressions are in the form of a weighted sum of complementary error functions and are solely dependent on the constellation size M, the carrier-to-noise ratio, and a constellation parameter which controls the relative message importance.