A new derivation of the Nakagami-m distribution as a composite of the Rayleigh distribution

Mobile communications systems are affected by what is known as fading, which is a well-known problem largely studied for decades. The direct consequence of fading is the complete loss of signal (or a large decrease of the received power). Rayleigh fading is a reasonable model for wireless channels although, Nakagami-m distribution seems better suited to fitting experimental data. In this paper we obtain the Nakagami-m distribution as a composite (mixture) of the Rayleigh distribution, a result which as far as we know it has not been shown in the literature. This representation of the Nakagami-m distribution facilitates computations of the average BER (Bit Error Rate) for DPSK (Differential Phase Shift Keying) and MSK (Minimum-Shift Keying) modulations for this distribution and higher moments of them, which is of great applicability to modeling wireless fading channels. Furthermore, a simple, not depending on any special function, apart of the Gamma function, bivariate version of the Nakagami-m distribution is also proposed as a special case of the multivariate version which is also presented. The proposed composite distribution is simulated through the standard procedure of summation of phasors, and results for the new closed-form measures for the MSK modulation are also shown. From that it is clear that the alternative formulation of the Nakagami-m distribution allows for easier modeling of fading fading-shadowing wireless channels through the new explicit second order statistics metrics. is well suited for modelling fading-shadowing wireless channels.