Resource Allocation in Spectrum-Sharing Cognitive Heterogeneous Networks

Abstract

Cognitive radio-enabled heterogeneous networks are an emerging technology to address the exponential increase of mobile traffic demand in the next-generation mobile communications. Recently, many technological issues such as resource allocation and interference mitigation pertaining to cognitive heterogeneous networks have been studied, but most studies focus on maximizing spectral efficiency. This chapter introduces the resource allocation problem in cognitive heterogeneous networks, where the cross-tier interference mitigation, imperfect spectrum sensing, and energy efficiency are considered. The optimization of power allocation is formulated as a non-convex optimization problem, which is then transformed to a convex optimization problem. An iterative power control algorithm is developed by considering imperfect spectrum sensing, cross-tier interference mitigation, and energy efficiency.

Appendix

The proof of Theorem 1.

Proof.

(1) Suppose that $\eta {_{13,n}}^{{\ast}}$ is the optimal solution of (22), the inequality can be obtained

$$\displaystyle{ \begin{array}{l} \eta _{13,n}^{{\ast}} = \dfrac{P(\mathcal{H}_{n}^{v})(1 - q_{n}^{f}(\varepsilon _{n},\hat{\tau }))R_{1,n}(\hat{\tau },\tilde{P}_{s,n}^{v}) + P(\mathcal{H}_{n}^{o})q_{n}^{m}(\varepsilon _{n},\hat{\tau })R_{3,n}(\hat{\tau },\tilde{P}_{s,n}^{v})} {\tilde{P}_{s,n}^{v} + P_{c}} \\ \geq \dfrac{P(\mathcal{H}_{n}^{v})(1 - q_{n}^{f}(\varepsilon _{n},\hat{\tau }))R_{1,n}(\hat{\tau },P_{s,n}^{v}) + P(\mathcal{H}_{n}^{o})q_{n}^{m}(\varepsilon _{n},\hat{\tau })R_{3,n}(\hat{\tau },P_{s,n}^{v})} {P_{s,n}^{v} + P_{c}}\end{array} }$$

(79)

$$\displaystyle{ \begin{array}{*{20}{l}} \mathop{\max }\limits _{P_{s,n}^{v}}\left \{P(\mathcal{H}_{n}^{v})(1 -q_{n}^{f}(\varepsilon _{n},\hat{\tau }))R_{1,n}(\hat{\tau },P_{s,n}^{v}) + P(\mathcal{H}_{n}^{o})q_{n}^{m}(\varepsilon _{n},\hat{\tau })R_{3,n}(\hat{\tau },P_{s,n}^{v}) -\eta _{13,n}^{{\ast}}(P_{s,n}^{v} + P_{c})\right \} \\ = P(\mathcal{H}_{n}^{v})(1 -q_{n}^{f}(\varepsilon _{n},\hat{\tau }))R_{1,n}(\hat{\tau },P_{s,n}^{v}) + P(\mathcal{H}_{n}^{o})q_{n}^{m}(\varepsilon _{n},\hat{\tau })R_{3,n}(\hat{\tau },P_{s,n}^{v}) -\eta _{13,n}^{{\ast}}(P_{s,n}^{v} + P_{c}) = 0.\end{array} }$$

(80)

Hence, we have (81)

$$\displaystyle{ \left \{\begin{array}{*{20}{l}} P(\mathcal{H}_{n}^{v})(1 -q_{n}^{f}(\varepsilon _{n},\hat{\tau }))R_{1,n}(\hat{\tau },\tilde{P}_{s,n}^{v}) + P(\mathcal{H}_{n}^{o})q_{n}^{m}(\varepsilon _{n},\hat{\tau })R_{3,n}(\hat{\tau },\tilde{P}_{s,n}^{v}) -\eta _{13,n}^{{\ast}}\left (\tilde{P}_{s,n}^{v} + P_{c}\right ) = 0 \\ P(\mathcal{H}_{n}^{v})(1 -q_{n}^{f}(\varepsilon _{n},\hat{\tau }))R_{1,n}(\hat{\tau },P_{s,n}^{v}) + P(\mathcal{H}_{n}^{o})q_{n}^{m}(\varepsilon _{n},\hat{\tau })R_{3,n}(\hat{\tau },P_{s,n}^{v}) -\eta _{13,n}^{{\ast}}\left (P_{s,n}^{v} + P_{c}\right ) \leq 0. \end{array}\right. }$$

(81)

Therefore, $\mathop{\max }\limits _{P_{s,n}^{v}}\left \{\begin{array}{*{20}{l}} P(\mathcal{H}_{n}^{v})(1 - q_{n}^{f}(\varepsilon _{n},\hat{\tau }))R_{1,n}(\hat{\tau },P_{s,n}^{v}) \\ \begin{array}{l} +\ P(\mathcal{H}_{n}^{o})q_{n}^{m}(\varepsilon _{n},\hat{\tau })R_{3,n}(\hat{\tau },P_{s,n}^{v}) \\ -\ \eta _{13,n}^{{\ast}}(P_{s,n}^{v} + P_{c}) \end{array} \end{array} \right \} = 0$ can be concluded. That is, eq. (80) is achieved.

(2) Suppose that $\widetilde{P}_{s,n}^{v}$ is a solution to the problem of (80). The definition of (80) implies that (82)

$$\displaystyle{ \begin{array}{l} \begin{array}{*{20}{l}} P(\mathcal{H}_{n}^{v})(1 - q_{n}^{f}(\varepsilon _{n},\hat{\tau }))R_{1,n}(\hat{\tau },P_{s,n}^{v}) + P(\mathcal{H}_{n}^{o})q_{n}^{m}(\varepsilon _{n},\hat{\tau })R_{3,n}(\hat{\tau },P_{s,n}^{v}) \\ -\ \eta _{13,n}^{{\ast}}\left (P_{s,n}^{v} + P_{c}\right )\leq P(\mathcal{H}_{n}^{v})(1 - q_{n}^{f}(\varepsilon _{n},\hat{\tau }))R_{1,n}(\hat{\tau },\tilde{P}_{s,n}^{v}) \\ +\ P(\mathcal{H}_{n}^{o})q_{n}^{m}(\varepsilon _{n},\hat{\tau })R_{3,n}(\hat{\tau },\tilde{P}_{s,n}^{v}) -\eta _{13,n}^{{\ast}}\left (\tilde{P}_{s,n}^{v} + P_{c}\right ) = 0\end{array}\\ or \\ \left \{\begin{array}{*{20}{l}} P(\mathcal{H}_{n}^{v})(1 - q_{n}^{f}(\varepsilon _{n},\hat{\tau }))R_{1,n}(\hat{\tau },P_{s,n}^{v}) + P(\mathcal{H}_{n}^{o})q_{n}^{m}(\varepsilon _{n},\hat{\tau })R_{3,n}(\hat{\tau },P_{s,n}^{v}) \\ -\ \eta _{13,n}^{{\ast}}\left (P_{s,n}^{v} + P_{c}\right ) \leq 0 \\ P(\mathcal{H}_{n}^{v})(1 - q_{n}^{f}(\varepsilon _{n},\hat{\tau }))R_{1,n}(\hat{\tau },\tilde{P}_{s,n}^{v}) + P(\mathcal{H}_{n}^{o})q_{n}^{m}(\varepsilon _{n},\hat{\tau })R_{3,n}(\hat{\tau },\tilde{P}_{s,n}^{v}) \\ -\ \eta _{13,n}^{{\ast}}\left (\tilde{P}_{s,n}^{v} + P_{c}\right ) = 0.\end{array} \right. \end{array} }$$

(82)

Therefore, we obtain

$$\displaystyle{ \frac{\left \{\begin{array}{l} P(\mathcal{H}_{n}^{v})(1 - q_{n}^{f}(\varepsilon _{n},\hat{\tau }))R_{1,n}(\hat{\tau },\tilde{P}_{s,n}^{v}) \\ + P(\mathcal{H}_{n}^{o})q_{n}^{m}(\varepsilon _{n},\hat{\tau })R_{3,n}(\hat{\tau },\tilde{P}_{s,n}^{v}) \end{array} \right \}} {\tilde{P}_{s,n}^{v} + P_{c}} =\eta _{ 13,n}^{{\ast}} }$$

(83)

and

$$\displaystyle{ \frac{\left \{\begin{array}{l} P(\mathcal{H}_{n}^{v})(1 - q_{n}^{f}(\varepsilon _{n},\hat{\tau }))R_{1,n}(\hat{\tau },P_{s,n}^{v}) \\ + P(\mathcal{H}_{n}^{o})q_{n}^{m}(\varepsilon _{n},\hat{\tau })R_{3,n}(\hat{\tau },P_{s,n}^{v}) \end{array} \right \}} {P_{s,n}^{v} + P_{c}} \leq \eta _{13,n}^{{\ast}}. }$$

(84)

□

Lemma 1.

Let $\boldsymbol{A}$ be an N × N symmetric matrix, $\boldsymbol{A}$ is negative semidefinite if and only if all the kth order principal minors of $\boldsymbol{A}$ are no larger than zero if k is odd, and not less than zero if k is even, where 1 ≤ k ≤ N.

The proof of Theorem 2.

Proof.

First, define the element $\tau _{k,i,n}\widehat{R}_{k,i,n}^{\mathrm{F}}$ in (59) as $f(\tau _{k,i,n},\widehat{p}_{k,i,n}) = \tau _{k,i,n}\widehat{R}_{k,i,n}^{\mathrm{F}}$. The objective function in (59) is the sum of $f(\tau _{k,i,n},\widehat{p}_{k,i,n})$ over all possible values of k, i, and n. Substituting $\widehat{R}_{k,i,n}^{\mathrm{F}} = \log _{2}\left (1 + \frac{\widehat{p}_{k,i,n}\hslash _{k,k,i,n}^{\mathrm{FF}}} {\tau _{k,i,n}I_{k,i,n}} \right )$ into $f(\tau _{k,i,n},\widehat{p}_{k,i,n})$, so we have

$$\displaystyle{ f(\tau _{k,i,n},\widehat{p}_{k,i,n}) = \tau _{k,i,n}\log _{2}\left (1 + \frac{\widehat{p}_{k,i,n}\hslash _{k,k,i,n}^{\mathrm{FF}}} {\tau _{k,i,n}I_{k,i,n}} \right ). }$$

(85)

Based on (85), one obtains

$$\displaystyle{ \frac{\partial ^{2}f} {\partial \tau _{k,i,n}^{2}} = -\frac{1} {\ln 2} \frac{(\widehat{p}_{k,i,n}\hslash _{k,k,i,n}^{\mathrm{FF}})^{2}} {\tau _{k,i,n}(\tau _{k,i,n}I_{k,i,n} + \widehat{p}_{k,i,n}\hslash _{k,k,i,n}^{\mathrm{FF}})^{2}}, }$$

(86)

$$\displaystyle{ \frac{\partial ^{2}f} {\partial \tau _{k,i,n}\partial \widehat{p}_{k,i,n}} = \frac{\partial ^{2}f} {\partial \widehat{p}_{k,i,n}\partial \tau _{k,i,n}} = \frac{1} {\ln 2} \frac{\widehat{p}_{k,i,n}(\hslash _{k,k,i,n}^{\mathrm{FF}})^{2}} {(\tau _{k,i,n}I_{k,i,n} + \widehat{p}_{k,i,n}\hslash _{k,k,i,n}^{\mathrm{FF}})^{2}}, }$$

(87)

$$\displaystyle{ \frac{\partial ^{2}f} {\partial \widehat{p}_{k,i,n}^{2}} = -\frac{1} {\ln 2} \frac{\tau _{k,i,n}(\hslash _{k,k,i,n}^{\mathrm{FF}})^{2}} {(\tau _{k,i,n}I_{k,i,n} + \widehat{p}_{k,i,n}\hslash _{k,k,i,n}^{\mathrm{FF}})^{2}}. }$$

(88)

Consequently, the Hessian matrix of $f(\tau _{k,i,n},\widehat{p}_{k,i,n})$ can be written as

$$\displaystyle{ \boldsymbol{H} = \left [\begin{array}{*{20}{c}} \frac{\partial ^{2}f} {\partial \tau _{k,i,n}^{2}} & \frac{\partial ^{2}f} {\partial \tau _{k,i,n}\partial \widehat{p}_{k,i,n}} \\ \frac{\partial ^{2}f} {\partial \widehat{p}_{k,i,n}\partial \tau _{k,i,n}} & \frac{\partial ^{2}f} {\partial \widehat{p}_{k,i,n}^{2}} \end{array} \right ]. }$$

(89)

Substituting (86), (87), (88) to (89), we can show that the first-order principal minors of $\boldsymbol{H}$ are negative, and the second-order principal minor of $\boldsymbol{H}$ is zero. Therefore, $\boldsymbol{H}$ is negative semidefinite according to Lemma 1, and $f(\tau _{k,i,n},\widehat{p}_{k,i,n})$ is concave. The objective function of (59) is concave because any positive linear combination of concave functions is concave [23, 33]. As the inequality constraints in (59) are convex, the feasible set of the objective function in (59) is convex, and the corresponding optimization problem is a convex problem. This completes the proof . ⊓ ⊔