Joint Power Control and Subchannel Allocation for D2D Communications Underlaying Cellular Networks: A Coalitional Game Perspective

Abstract

A coalition based joint subchannel and power allocation approach is studied to improve the performance of device-to-device (D2D) communication underlaying cellular networks with uplink spectrum sharing. To exploit the spectrum reuse gain, we formulate the problem as a coalition formation game. Furthermore, a distributed coalition formation algorithm is devised to assist D2D pairs in joining or leaving a coalition. During the coalition formation process, we introduce an iterative power control method. By using this method, D2D pairs can evaluate their current coalition with D2D sum rate maximization and cellular user equipment protection. Numerical results are provided to corroborate the proposed studies.

A  The Proof of Proposition 1

Proof

Let $\varvec{p}\left( 1\right)$ and $\varvec{p}\left( 2\right)$ be two different power allocation vectors. The solution to (8) can be shown as

$$\begin{aligned} {p_i}\left( m \right) = \left[ {\frac{1}{\lambda } - \frac{{\sum \nolimits _{j \in {D_k}\backslash \left\{ i \right\} } {{p_j}\left( {m} \right) h_{j,i}^k} + {N_0} + {p_k}g_i^k}}{{h_{i,i}^k}}} \right] _0^{{p_{\max }}} \end{aligned}$$

(13)

where the operator $\left[ x\right] _0^{p_{\max }}$ denotes the value of x is within $\left[ 0, p_{\max }\right]$, and the $m = 1,2$.

For a fixed price factor $\lambda$, the difference for (13) with different power vector $\varvec{p}{\left( 1\right) }$ and $\varvec{p}{\left( 2\right) }$ is derived as

(14)

From (14), we prove that (13) is a non-expansive operator; therefore, we conclude that the iterative procedure will converge to the unique fixed point (Miao et al. 2011, Theorem 3) when $\sum \limits _{j \in {D_k},j \ne i} {\frac{{h_{j,i}^k}}{{h_{i,i}^k\ln 2}}} < 1, \forall i \in D_k$.