The κ-μ Shadowed Fading Model with Integer Fading Parameters

Abstract

We show that the popular and general κ-μ shad- owed fading model with integer fading parameters μ and m can be represented as a mixture of squared Nakagami (or Gamma) distributions. Thus, its PDF and CDF can be expressed in closed-form in terms of a finite number of elementary functions (powers and exponentials). The main implications arising from such connection are then discussed, which can be summarized as: (1) the performance evaluation of communication systems operating in κ-μ shadowed fading becomes as simple as if a Nakagami fading channel was assumed; (2) the κ-μ shadowed distribution can be used to approximate the κ-μ distribution us- ing a closed-form representation in terms of elementary functions, by choosing a sufficiently large value of m; and (3) restricting the parameters μ and m to take integer values has limited impact in practice when fitting the κ-μ shadowed fading model to field measurements. As an application example, the average channel capacity of communication systems operating under κ-μ shadowed fading is obtained in closed-form.

Figures (11)

• Figure 1: Convergence in distribution between the $\kappa$-$\mu$ shadowed distribution and the $\kappa$-$\mu$ distribution as $m\rightarrow\infty$. Parameter values $\kappa=5$, $\mu=3$ and $\bar{\gamma}=1$.
• Figure 2: Gamma approximation to the Rician distribution with parameter $K$ using the Rician shadowed distribution with parameter $K$ and integer $m$. Parameter values $K=3$ and $\bar{\gamma}=1$. Nakagami approximation Nakagami1960 uses $\hat{m} = (1+K)^2/(1+2K)=2.28$.
• Figure 3: Gamma approximation to the Rician distribution with parameter $K$ using the Rician shadowed distribution with parameter $K$ and integer $m$. Parameter values $K=10$ and $\bar{\gamma}=1$. Nakagami approximation Nakagami1960 uses $\hat{m} = (1+K)^2/(1+2K)=5.76$.
• Figure 4: Evolution of the $\kappa$-$\mu$ shadowed fading PDF for different values of $\mu$. Solid lines correspond to real-valued $\mu\in\{1.5,2.5,3.5,4.5\}$, whereas dashed lines correspond to the largest previous and the smallest following integer values $\mu\in\{1:5\}$. Parameter values $\kappa=6$, $m=2$ and $\bar{\gamma}=1$.
• Figure 5: Evolution of the $\kappa$-$\mu$ shadowed fading PDF for different values of $m$. Solid lines correspond to real-valued $m\in\{1.5,2.5,3.5,4.5,5.5\}$, whereas dashed lines correspond to the largest previous and the smallest following integer values $m\in\{1:6\}$. Parameter values $\kappa=6$, $\mu=2$ and $\bar{\gamma}=1$.

Theorems & Definitions (13)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• proof
• Corollary 1
• proof
• Lemma 1
• proof
• Lemma 2