A closed-form expression for the exact BER of generalized PAM and QAM constellations

We derive an explicit closed-form expression for the exact bit-error rate (BER) computation of generalized hierarchical M-ary pulse amplitude modulations over additive white Gaussian noise (AWGN) and fading channels. This expression can also be used to obtain the exact BER of generalized hierarchical quadrature amplitude modulations (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.