The $\mu$ Process Type I

Abstract

A statistical process was proposed elsewhere and used to model the Nakagami- $\boldsymbol {m}$ , $\boldsymbol {\eta }$ - $\boldsymbol {\mu }$ , and $\boldsymbol {\alpha }$ - $\boldsymbol {\mu }$ complex signals. Since then, very little has been built upon its statistics. The aim of this letter is to deepen the understanding of this process, which includes the Gaussian one as a particular case. In particular, the following are found in an exact manner: (i) moments and the cumulative distribution function of the marginal statistics; (ii) bivariate probability density function (PDF); (iii) joint PDF of the variate and its time derivative; (iv) level crossing rate and average fade duration; (v) bit error probability in a power line communications; (vi) correlation coefficient; (vii) autocorrelation function; and (viii) power spectrum density. The items in (i) through (v) are all given in closed-form formulas. It is noteworthy that to arrive at one of our main results we solve a very general statistical problem with a broader application, namely, given the joint PDF of the modulus of two variates, find the joint PDF of these variates.