Double Shadowing the Rician Fading Model

Abstract

In this letter, we consider a Rician fading envelope which is impacted by dual shadowing processes. We conveniently refer to this as the double shadowed Rician fading model which can appear in two different formats, each underpinned by a different physical signal reception model. The first format assumes a Rician envelope where the dominant component is fluctuated by a Nakagami- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${m}$ </tex-math></inline-formula> random variable (RV) which is preceded (or succeeded) by a secondary round of shadowing brought about by an inverse Nakagami- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${m}$ </tex-math></inline-formula> RV. The second format considers that the dominant component and scattered waves of a Rician envelope are perturbed by two different shadowing processes. In particular, the dominant component experiences variations characterized by the product of a Nakagami- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${m}$ </tex-math></inline-formula> and an inverse Nakagami- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${m}$ </tex-math></inline-formula> RV, whereas the scattered waves are subject to fluctuations influenced by an inverse Nakagami- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${m}$ </tex-math></inline-formula> RV. Using the relationship between the shadowing properties of the two formats, we develop unified closed-form and analytical expressions for their probability density function, cumulative distribution function, moment-generating function and moments. All derived expressions are validated through Monte Carlo simulations and reduction to a number of special cases.