An Alternative Approach in Derivation of Nakagami-m Distribution

Nakagami-m probability density function (pdf) is one of the frequently used distributions for describing fast received signal variations in radio channels, obtained as a result of multipath phenomenon. It is foremost derived by assuming the most general multipath channel model but applying mathematical approximations. Afterward, it is derived without approximations, but based on dedicated physical models with many constraints. Consequently, neither approach can be considered both, universally applicable and exact. Accordingly, in this paper, a novel approach in deriving Nakagami-m pdf is provided, being based on fewer constraints on propagation phenomena than others. Herein, it is shown that Nakagami-m pdf can be obtained as a distribution of a Euclidean distance of a point orthogonally projected from homogeneous distributed n-dimensional hypersphere on N-dimensional space, where received signal envelope is interpreted as mentioned Euclidean distance, with $n$ being a total number of orthogonal multipath components which can reach the receiver in idealized condition and $N$ being a number of these components which reach the receiver in reality (with N < n).