Joint Power and Channel Allocation for D2D Communication in Cellular Networks

Abstract

Device-to-device (D2D) communication is a technology that allows devices communicating with other devices directly instead of going through the base station (BS), and it can reduce the burden of base stations and increase the system capacity of cellular networks. In this paper, we raise and analyze a system of D2D communication in cellular networks, which includes a base station in the center and several cellular user equipments (CUEs) coexisting with D2D user equipments (DUEs). Considering the high rate for CUE and low outage probability for DUE, we propose a scheme of resource allocation to improve the ergodic sum rate of CUEs under the constraint to guarantee the outage preference of DUEs. The problem of resource allocation is a non-convex problem, which is usually mathematically intractable. We divided the optimal problem into two sub-questions, power allocation and channel allocation. We firstly maximize the ergodic rate of CUEs under the constraint of reliability of the DUEs to find the optimal power distribution, then a maximum weight bipartite matching is used to find the optimal channel allocation by the Hungarian method. Simulation results demonstrate that the scheme of resource allocation can achieve the desired performance of the system.

Appendix

The ergodic rate $R_{i,j} (p_i^c ,p_j^d )$ can be calculated as

$$\begin{aligned} R_{i,j} (p_i^c ,p_j^d ) & = {\mathbb{E}}\left[ {\log_2 \left( {1 + \gamma_i^c } \right)} \right] \\ & = {\mathbb{E}}\left[ {\log_2 \left( {1 + \frac{{p_i^c g_{i,B} L_{i,B} }}{{p_j^d g_{j,B} L_{j,B} + \sigma^2 }}} \right)} \right] \\ \end{aligned}$$

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We define $a = \frac{{p_i^c L_{i,B} }}{\sigma^2 }$, $b = \frac{{p_j^d L_{j,B} }}{\sigma^2 }$, $X = g_{i,B}$, $Y = g_{j,B}$, $Z = \frac{aX}{{bY + 1}}$. Assuming $g_{i,B} \ E(1)$, $g_{j,B} \ E(1)$, the cumulative distribution function (CDF) of Z can be written as

$$\begin{aligned} F_Z \left( z \right) & = \Pr \left\{ {\frac{{ag_{i,B} }}{{bg_{j,B} + 1}} \le z} \right\} \\ & = \int\limits_0^\infty {{\text{d}}y\int\limits_0^{\frac{z(1 + by)}{a}} {e^{ - (x + y)} } {\text{d}}x} \\ \end{aligned}$$

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Then, the close-formed expression of the ergodic rate can be given by

$$\begin{aligned} R_{i,j} \left( {p_i^c ,p_j^d } \right) & = E\left[ {\log_2 \left( {1 + Z} \right)} \right] \\ & = \frac{1}{\ln 2}\int\limits_0^\infty {\ln \left( {1 + z} \right)f_Z \left( z \right)} {\text{d}}z \\ & = \frac{a}{(a - b)\ln 2}\left[ {\int\limits_0^\infty {\frac{{e^{ - \tfrac{z}{a}} }}{z + 1}{\text{d}}z - } \int\limits_0^\infty {\frac{{e^{ - \tfrac{z}{a}} }}{{z + \tfrac{a}{b}}}{\text{d}}z} } \right] \\ & = \frac{a}{{\left( {a - b} \right)\ln 2}}\left[ {e^{\tfrac{1}{a}} E_1 \left( \frac{1}{a} \right) - e^{\tfrac{1}{b}} E_1 \left( \frac{1}{b} \right)} \right] \\ \end{aligned}$$

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