Total Throughput Capacity Maximization in Cooperative Cognitive Radio Relay Networks

Abstract

Wireless networks are burdened with the demand of providing higher data rates for the user devices connected to the wireless networks. This demand presents a huge challenge for the traditional radio resource constrained wireless communication networks. The cognitive radios and cooperative communication techniques have emerged to offer promising solutions to the burden of higher data rate demands. This paper addresses the problem of radio resource management in cooperative cognitive radio relay networks (CCRRN). In resource-constrained networks, a uniform bandwidth allocation scheme may not be efficient for the connected wireless multi-users. In this paper, a convex optimization analytical framework is used to investigate the problem of radio resource management in CCRRN. A joint optimal resource allocation (JORA) for power and bandwidth allocation strategy for the maximization of the total throughput capacity of all users in the networks is proposed. The formulated radio resource management problem is proved to be a standard convex optimization problem. The convex optimization problem is solved efficiently using the SNOPT solver package. The SNOPT solver is implemented in the MATLAB/TOMLAB software environment. The numerical results showed that the JORA scheme significantly outperformed the equal bandwidth (EQBW) scheme in terms of throughput capacity.

Appendix

Negative semi-definite functions are concave functions. Our goal is to prove that the objective function of the optimization problem is a negative semi-definite function. A function F(x, y) is negative semi-definite if and only if the eigenvalues ( $\lambda _l$) of the Hessian of F(x, y) are non-positive. That is $\lambda _l \le 0, l=1,2,\ldots L.$, where L is the dimension of the Hessian matrix of F(x, y). Note that at least one of the eigenvalues must be zero. Let the achievable rate in the first-transmission time slot be denoted as $F_1(b_1,p_1)$ and $F_1(b_1,p_1)\Rightarrow AR^i_{SR_k}$, where $b_1 \equiv B^i_S$, $g_1 \equiv G^i_{SR_k}$ and $p_1 \equiv P^i_S$, similarly let the achievable rate in the second transmission time slot be denoted as $F_2(b_2,p_2)$ and $F_2(b_2,p_2)\Rightarrow AR^i_{R_kD_i}$, where $b_2 \equiv B^i_{R_k}$, $p_2 \equiv P^i_R$ and. We need to prove that $F_1(b_1,p_1) = b_1 \log (1+ \frac{p_1 g_1}{N_0b_1})$ and $F_2(b_2,p_2) = b_2 \log (1+ \frac{p_2 g_2}{N_0b_2})$ are concave functions.

Firstly, we prove that $F_1(b_1, p_1)$ is negative semi-definite. We define the Hessian matrix $H_{F_1(b_1,p_1)}$ as;

$$\begin{aligned}{} & {} H_{F_1(b_1,p_1)} \Rightarrow \left[ \begin{array}{cc} \frac{ \partial ^2 F_1(b_1,p_1)}{\partial b^2_1} &{} \frac{ \partial ^2 F_1(b_1,p_1)}{\partial b_1 p_1} \\ \frac{ \partial ^2 F_1(b_1,p_1)}{\partial p_1 b_1} &{} \frac{ \partial ^2 F_1(b_1,p_1)}{\partial p^2_1} \end{array} \right] , \end{aligned}$$

(33)

$$\begin{aligned}{} & {} \frac{ \partial ^2 F_1(b_1,p_1)}{\partial b^2_1} = - \frac{g^2_1p^2_1}{(b_1(b_1 N_0 +p_1 g_1)^2)}, \end{aligned}$$

(34)

$$\begin{aligned}{} & {} \frac{ \partial ^2 F_1(b_1,p_1)}{\partial p^2_1} = - \frac{b_1 g^2_1}{(b_1 N_0 +p_1 g_1)^2}, \end{aligned}$$

(35)

$$\begin{aligned}{} & {} \frac{ \partial ^2 F_1(b_1,p_1)}{\partial b_1 \partial p_1} = \frac{ g^2_1 p_1}{(b_1 N_0 +p_1 g_1)^2}, \end{aligned}$$

(36)

$$\begin{aligned}{} & {} \frac{ \partial ^2 F_1(b_1,p_1)}{ \partial p_1 \partial b_1} = \frac{ g^2_1 p_1}{(b_1 N_0 +p_1 g_1)^2}, \end{aligned}$$

(37)

Therefore, we have the Hessian matrix $H_{F_1(b_1,p_1)}$ as;

$$\begin{aligned} H_{F_1(b_1,p_1)} =\left[ \begin{array}{cc} - \frac{g^2_1p^2_1}{(b_1(b_1 N_0 +p_1 g_1)^2)} &{} \frac{ g^2_1 p_1}{(b_1 N_0 +p_1 g_1)^2} \\ \frac{ g^2_1 p_1}{(b_1 N_0 +p_1 g_1)^2} &{} - \frac{b_1 g^2_1}{(b_1 N_0 +p_1 g_1)^2} \end{array} \right] . \end{aligned}$$

(38)

The eigenvalues $\lambda _1$ and $\lambda _2$ of $H_{F_1(b_1,p_1)}$ are the first principal minors of $H_{F_1(b_1,p_1)}$. Hence,

$$\begin{aligned}{} & {} \lambda _1 = - \frac{g^2_1p^2_1}{(b_1(b_1 N_0 +p_1 g_1)^2)} <0, \end{aligned}$$

(39)

$$\begin{aligned}{} & {} \lambda _2 = - \frac{b_1 g^2_1}{(b_1 N_0 +p_1 g_1)^2}<0. \end{aligned}$$

(40)

The last eigenvalue $\lambda _3$ is the determinant of the Hessian matrix

$$\begin{aligned} \lambda _3 =\begin{array}{|cc|} - \frac{g^2_1p^2_1}{(b_1(b_1 N_0 +p_1 g_1)^2)} &{} \frac{ g^2_1 p_1}{(b_1 N_0 +p_1 g_1)^2} \\ \frac{ g^2_1 p_1}{(b_1 N_0 +p_1 g_1)^2} &{} - \frac{b_1 g^2_1}{(b_1 N_0 +p_1 g_1)^2} \end{array} =0. \end{aligned}$$

(41)

Therefore, with $\lambda _1 <0$, $\lambda _2 <0$ and $\lambda _3 = 0$, consequently $H_{F_1(b_1,p_1)}$ is negative semi-definite and hence, $AR^i_{SR_k}$ is a concave function in $B^i_S$ and $P^i_S$.

Secondly, we prove that $F_2 (b_2, p_2)$ is negative semi-definite. Defining the Hessian matrix $H_{F_2(b_2,p_2)}$ as;

$$\begin{aligned}{} & {} H_{F_2(b_2,p_2)} =\left[ \begin{array}{cc} \frac{ \partial ^2 F_2(b_2,p_2)}{\partial b^2_2} &{} \frac{ \partial ^2 F_2(b_2,p_2)}{\partial b_2 p_2} \\ \frac{ \partial ^2 F_2(b_2,p_2)}{\partial p_2 b_2} &{} \frac{ \partial ^2 F_2(b_2,p_2)}{\partial p^2_2} \end{array} \right] , \end{aligned}$$

(42)

$$\begin{aligned}{} & {} \frac{ \partial ^2 F_2(b_2,p_2)}{\partial b^2_2} = - \frac{g^2_2p^2_2}{(b_2(b_2 N_0 +p_2 g_2)^2)}, \end{aligned}$$

(43)

$$\begin{aligned}{} & {} \frac{ \partial ^2 F_2(b_2,p_2)}{\partial p^2_2} = - \frac{b_2 g^2_2}{(b_2 N_0 +p_2 g_2)^2}, \end{aligned}$$

(44)

$$\begin{aligned}{} & {} \frac{ \partial ^2 F_2(b_2,p_2)}{\partial b_2 \partial p_2} = \frac{ g^2_2 p_2}{(b_2 N_0 +p_2 g_2)^2}, \end{aligned}$$

(45)

$$\begin{aligned}{} & {} \frac{ \partial ^2 F_2(b_2,p_2)}{ \partial p_2 \partial b_2} = \frac{ g^2_2 p_2}{(b_2 N_0 +p_2 g_2)^2}, \end{aligned}$$

(46)

Thus, the Hessian matrix $H_{F_2(b_2,p_2)}$ is;

$$\begin{aligned} H_{F_2(b_2,p_2)} =\left[ \begin{array}{cc} - \frac{g^2_2p^2_2}{(b_2(b_2 N_0 +p_2 g_2)^2)} &{} \frac{ g^2_2 p_2}{(b_2 N_0 +p_2 g_2)^2} \\ \frac{ g^2_2 p_2}{(b_2 N_0 +p_2 g_2)^2} &{} - \frac{b_2 g^2_2}{(b_2 N_0 +p_2 g_2)^2} \end{array} \right] . \end{aligned}$$

(47)

The corresponding the eigenvalues $\lambda _1$ and $\lambda _2$ of $H_{F_2(b_2,p_2)}$ are the first principal minors of $H_{F_2(b_2,p_2)}$. Hence,

$$\begin{aligned} \lambda _1= & {} - \frac{g^2_2p^2_2}{(b_2(b_2 N_0 +p_2 g_2)^2)} <0, \end{aligned}$$

(48)

$$\begin{aligned} \lambda _2= & {} - \frac{b_2 g^2_2}{(b_2 N_0 +p_2 g_2)^2}<0, \end{aligned}$$

(49)

while the last eigenvalue $\lambda _3$ is the determinant of the Hessian matrix

$$\begin{aligned} \lambda _3 =\begin{array}{|cc|} - \frac{g^2_2p^2_2}{(b_2(b_2 N_0 +p_2 g_2)^2)} &{} \frac{ g^2_2 p_2}{(b_2 N_0 +p_2 g_2)^2} \\ \frac{ g^2_2 p_2}{(b_2 N_0 +p_2 g_2)^2} &{} - \frac{b_2 g^2_2}{(b_2 N_0 +p_2 g_2)^2} \end{array} =0. \end{aligned}$$

(50)

Therefore, with $\lambda _1 <0$, $\lambda _2 <0$ and $\lambda _3 = 0$, thus $H_{F_2(b_2,p_2)}$ is negative semi-definite and hence, $AR^i_{R_k D_i}$ is a concave function in $B^i_{R_k}$ and $P^i_R$.

Finally, since both $AR^i_{S R_k}$ and $AR^i_{R_k D_i}$ are concave functions, therefore, problem $\textbf{ P1}$ is a convex optimization problem. Consequently, problem $\textbf{P2}$ is a convex optimization problem.