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We study complex-valued Nakagami-m variates, establishing the moment determinance of the envelope, phase and joint envelope-phase Nakagami-m probability density functions (pdf's). Inspired by that result, we then show that Nakagami-m variates with arbitrary fading figure m can be accurately decomposed onto a mixture of Nakagami-m variates with integer or half-integer m, i.e, 2m ∈ ℕ+. The latter has the immediate theoretical implication that results currently known to hold for Nakagami-m channels under the constraint 2m ∈ ℕ+ can accurately be extended to arbitrary m via simple linear decomposition, with weights given by the random mixture probabilities, for which formulas are provided. The latter is illustrated with an example of the application of the random mixture decomposition to the simplification and generalization of bit error rate (BER) expressions for PSK modulation in the Nakagami-m relay channel. For completeness, implications on possible extensions and improvements of existing methods to generate Nakagami-m variates is also briefly discussed in the form of further examples.