Outage probability and ergodic channel capacity of underlay device-to-device communications over $\kappa -\mu$ shadowed fading channels

Abstract

Device-to-device (D2D) communication is observed as an emerging technique to offload the traffic from base station for improving the network performance. Outage probability (OP) and ergodic channel capacity (ECC) are the key performance metrics for wireless communication system. This paper mainly focuses on deriving the expressions of OP and the ECC for underlay D2D communication operating over $\kappa -\mu$ shadowed fading channels with arbitrary fading and shadowing parameters by utilizing stochastic geometry. The uplink radio resource of existing cellular network has been reused by D2D users. The $\kappa -\mu$ shadowed fading is a composite channel model of multipath fading and shadowing which contains many classical fading models as special cases. The signal of interest and interfering signals both follow the distribution of $\kappa -\mu$ fading with shadowing. The analytical expressions of OP and ECC can be expressed in terms of Appell’s function and Gauss’s hypergeometric function, which makes numerical evaluation easy. Finally, the results obtained from the analytical analysis are validated through Monte-Carlo simulations that show the good agreement.

Appendix

Interchanging integration with summation over n is required for solving (22) by applying Tonelli’s theorem for sums and integrals [30, Corollary 1.1.46].

$$\begin{aligned} C&= {\text{K}} \sum _{n=0}^{\infty }\frac{(\beta +\mu )_n(m)_n}{n!(\mu )_n} \left( \frac{(\Omega _2-\Omega _1) \lambda Tr^{\alpha }}{\Omega _2(\Omega _1+\lambda Tr^{\alpha })}\right) ^n \\&\quad\times \int _{x = 0}^{\infty } \sum _{i = 0}^{\infty }\frac{(1-\mu -n)_i(\beta )_i}{i!(\beta +1)_i}\left( \frac{\Omega _1}{\Omega _1+\lambda (e^x-1)r^{\alpha }}\right) ^{\beta +i} dx. \end{aligned}$$

(31)

In (31), it is required to interchange the integrals with summation over i and take the integration inside summation. For this Lebesgue’s dominated convergence theorem (DCT) should be applied [31, Theorem 10.28].

According to DCT theorem, Let $g_i : M \rightarrow {\mathbb{R}}$ be a sequence of measurable functions, and if $d_p : M \rightarrow {\mathbb{R}}$ is another sequence of non negative measurable function such that $|g_i(t)|\le d_p(t)$$\forall p$ and almost all t and $\sum _{p=0}^{\infty }d_p(t)$ converges and $\sum _{p=0}^{\infty }\int d_p(t)<\infty$, then $\int _{M}\sum _{p=0}^{\infty } d_p(t)d = \sum _{p=0}^{\infty }\int _{M} d_p(t)$.

consider $g_i(x)=\frac{(1-m-\mu )_i(\beta )_i}{i!(\beta +1)_i}\left( \frac{1}{1+\Omega _1\lambda (e^x-1)r^\alpha }\right) ^{\beta +i}$ therefore, $\sum _{i=0}^{\infty }$ can be re-written as

$$\begin{aligned} \sum _{i=0}^{\infty }g_i(x)&= \left( \frac{1}{1+\Omega _1\lambda (e^x-1)r^{\alpha }}\right) ^{\beta } \\&\quad {}_2F_1\left( 1-\mu -n,\beta ;\beta +1;\frac{\Omega _1}{\Omega _1+\lambda (e^x-1)r^{\alpha }}\right) . \\ \end{aligned}$$

(32)

Since Gauss hypergeometric function converges ${}_2F_1(a,b,c,x)$ for $|x|<1$, $\sum _{i=0}^{\infty }g_i(x)$ converges. Now it is required to show $\sum _{i=0}^{\infty }\int g_i(x)< \infty$, where

$$\begin{aligned} \int g_i(x)&= \frac{(1-m-\mu )_i(\beta )_i}{i!(\beta +1)_i(\beta +i)} \\&\quad {}_2F_1\left( 1,\beta +i;\beta +i+1,\frac{\Omega _1}{\Omega _1+\lambda e^xr^{\alpha }-\lambda r^{\alpha }}\right) . \\ \end{aligned}$$

(33)

The function ${}_2F_1(\cdot )$ in (32) will be bounded only if $|\frac{1}{1+\Omega _1\lambda e^xr^{\alpha }-\Omega _1\lambda r^{\alpha }}|<1$ therefore,

$$\begin{aligned}&{}_2F_1(1,\beta +i;\beta +i+1,y) = \sum _{q = 0}^{\infty }\frac{(1)_q(\beta +i)y^q}{(\beta +i+1)_qq!} \\&\quad = \sum _{q = 0}^{\infty }\frac{(\beta +i)y^q}{(\beta +i+q)}\le \sum _{q = 0}^{\infty }y^q, \end{aligned}$$

(34)

where $y = \frac{1}{1+\Omega _1\lambda e^xr^{\alpha }-\Omega _1\lambda r^{\alpha }}$ Similarly for the same condition $|y|\le 1$, $\int g_i(x)$, is also bounded. The term $\frac{(\beta )_i}{(\beta +1)_i}\le 1$ and $\frac{\Gamma (1-m-\mu +i)}{(\beta +i)\Gamma (1-m-\mu )}<\infty$ Now it is required to show that $\sum _{n=0}^{\infty }\int f_n(t)dt<\infty$ and $\sum _{n=0}^{\infty }f_p(x)<\infty$ where $\sum _{p = 0}^{\infty }r_p(x) < \infty$. where $r_p(x) = \frac{\Gamma (1-m-\mu +i)}{\Gamma (p+1)}$ Using Gamma fraction relation given in [32, eq. 1]

$$\begin{aligned} r_p(x) = \frac{1}{p^{1+\mu +m}}\left[ 1+\frac{(-\mu -m)(1-\mu -m)}{2p}+O(|p|^{-2})\right] \\ \end{aligned}$$

(35)

Therefore, it is clear that the $\sum _{n=0}^{\infty }\int f_n(t)dt<\infty$. Since, both the required condition of DCT are fulfilled, hence, the integral and sums over n in (22) can be easily interchanged.