Energy efficiency resource allocation for D2D communication network based on relay selection

Abstract

In order to solve the problem of spectrum resource shortage and energy consumption, we put forward a new model that combines with D2D communication and energy harvesting technology: energy harvesting-aided D2D communication network under the cognitive radio (EHA-CRD), where the D2D users harvest energy from the base station and the D2D source communicate with D2D destination by D2D relays. Our goals are to investigate the maximization energy efficiency (EE) of the network by joint time allocation and relay selection while taking into the constraints of the signal-to-noise ratio of D2D and the rates of the Cellular users. During this process, the energy collection time and communication time are randomly allocated. The maximization problem of EE can be divided into two sub-problems: (1) relay selection problem; (2) time optimization problem. For the first sub-problem, we propose a weighted sum maximum algorithm to select the best relay. For the last sub-problem, the EE maximization problem is non-convex problem with time. Thus, by using fractional programming theory, we transform it into a standard convex optimization problem, and we propose the optimization iterative algorithm to solve the convex optimization problem for obtaining the optimal solution. And, the simulation results show that the proposed relay selection algorithm and time optimization algorithm are significantly improved compared with the existing algorithms.

Appendices

Appendix 1

Theorem 1

The perspective function of concave function is still concave function.

Proof

We define $f\left( x \right) = \log \left( {1 + x} \right),x > 0.$□

Obviously, $f\left( x \right)$ is a concave function in its domain.

And the perspective function of $f\left( x \right)$ can be expressed as

$$g\left( {x,y} \right) = yf\left( {\frac{x}{y}} \right), dom g = \left\{ {\left( {x,y} \right)\left| {\frac{x}{y} \in dom f,y} \right\rangle 0} \right\}$$

Because the perspective operation is convex preserving, and $f\left( x \right)$ is concave function. So $g\left( {x,y} \right)$ is also concave function.

Appendix 2

Theorem 2

If the function of $f_{1} \left( x \right) {\text{and }}f_{2} \left( x \right)$ is concave function.

Thus, $f\left( x \right) = min\left\{ {f_{1} \left( x \right),f_{2} \left( x \right)} \right\}$ is concave function when $dom f = dom f_{1} \cap dom f_{2}$.

Proof: for any $0 \le\uptheta \le 1$. and $x,y \in dom f$, we have

$$\begin{aligned} & f\left( {\uptheta\,x + \left( {1 -\uptheta} \right)y} \right) \\ & = \,min\left\{ {f_{1} \left( {\uptheta\,x + \left( {1 -\uptheta} \right)y} \right),f_{2} \left( {\uptheta\,x + \left( {1 -\uptheta} \right)y} \right)} \right\} \\ & \ge \,min\left\{ {\uptheta\,f_{1} (x) + \left( {1 -\uptheta} \right)f_{1} (y),\uptheta\,f_{2} (x) + \left( {1 -\uptheta} \right)f_{2} (y)} \right\} \\ & \ge\uptheta\,min\left\{ {f_{1} (x),f_{2} \left( x \right)\} + \left( {1 -\uptheta} \right)min\{ f_{1} (y),f_{2} \left( y \right)} \right\} \\ & =\uptheta\,f\left( x \right) + \left( {1 -\uptheta} \right)f\left( y \right) \\ \end{aligned}$$

Appendix 3

Theorem 3

The nonnegative weighted sum of a concave function is still a concave function.

Proof

If $f\left( x \right)$ is a convex function and α $\ge 0$, thus, α $f\left( x \right)$ is also a convex function if $f_{1} \left( x \right)$ and $f_{2} \left( x \right)$ are convex function, thus, $f_{1} \left( x \right) + f_{2} \left( x \right)$ is also a convex function.□

By combining the operation of non-negative expansion and sum, it can be seen that the set of convex functions is itself a convex cone: the non-negative weighted sum of convex functions is still a convex function, that is:

$$f\left( x \right) = w_{1} f_{1} \left( x \right) + w_{2} f_{2} \left( x \right) + \cdots + w_{m} f_{m} \left( x \right)$$

f(x) is still a convex function.

Similarly, the nonnegative weighted sum of a concave function is still a concave function

Appendix 4

Theorem 4

$f\left( {x,y} \right) = {\text{xlog}}2\left( {1 + \frac{x}{x + y}} \right)$ belongs to convex function when $x \in \left( {0,1} \right),y \in \left( {0,1} \right)$

Proof

Taking the second derivative about $x,y$ in $f\left( {x,y} \right)$, we have$\nabla_{x}^{2} f\left( {x,y} \right) > 0;\nabla_{y}^{2} f\left( {x,y} \right) > 0;$ when $x \in \left( {0,1} \right),y \in \left( {0,1} \right)$Thus,$f\left( {x,y} \right)$ is a convex function when $x \in \left( {0,1} \right),y \in \left( {0,1} \right)$□