Exact expression for the BER of rectangular QAM with arbitrary constellation mapping

The exact closed-form expression for the bit-error rate (BER) of rectangular quadrature amplitude modulation (QAM) is given. The presented formula is independent of the bit mapping and it is thus particularly useful in the design and analysis of modulation schemes employing non-Gray mapping. Compared with the so-called expurgated bound and the union bound, our expression is shown to accurately predict the BER in the low signal-to-noise ratio range where the bounding techniques fail.