Invertible bounds for M-QAM in Rayleigh fading

In this letter, we derive tight invertible bounds on the bit-error probability (BEP) for the coherent detection of M-ary quadrature amplitude modulation with Gray code bit mapping in Rayleigh fading channels. These bounds enable us to easily obtain tight lower and upper bounds on the bit-error outage (BEO), i.e., BEP-based outage probability, in a log-normal shadowing environment. As examples of applications, these bounds are used to investigate the BEO and mean spectral efficiency for slow adaptive modulation.