Primary Network Interference Compensation-Based Dynamic Spectrum Leasing and Secondary Network Power Control

Abstract

The underlay paradigm is an important spectrum sharing model for cognitive-based secondary networks, in which the interference temperature limit (ITL) quantifies the amount of resources that can be shared by a secondary network and power control is known to play a crucial role. Many recent studies have discussed ITL-based underlay spectrum sharing and power control, but issues such as ITL decision-making and the value range for the ITL have not been analyzed. We also know that the presence of secondary network interference may increase the transmission power of primary users (PUs) who have hard traffic quality of service requirements. This additional resource sharing cost (increased transmit power) may discourage the PUs from sharing resources with secondary networks; the motivation in favor of resource sharing has not yet been completely explored. To address these unresolved problems, underlay-based spectrum sharing is reconsidered in this paper, using the scenario in which a cell primary network coexists with a cell secondary network that shares the spectrum resource with the primary network. First, under a fixed interference cap (IC) design, a simple primary network power control scheme is introduced. We then discuss IC design issues from the primary network point of view. Finally, a primary network interference compensation-based dynamic spectrum leasing scheme is proposed, in which the primary network suboptimal price search and SU’s power iteration algorithm are also provided. We use simulation to demonstrate the performance of our proposed schemes.

Appendices

Appendices

1.1 Appendix 1: Proof of (20)

Using the results obtained in Sect. 3, we first simplify the SUs’ SINR defined in (2), and then we derive (20).

Let us define $I_p^{(s)} ={\sum _{i=1}^{K^{(p)}} {p_i^{(p)} g_{i,s}^{(p)}}}\big /{G_p^{(s)}}$ as the total primary network interference experienced by all SUs at the SUBS.¹⁰ From (8), we know that $p_i^{(p)} =\frac{G_p^{(p)} \gamma _{0,tar}^{(p)}}{G_p^{(p)} -(K^{(p)}-1)\gamma _{0,tar}^{(p)}}\left( {\eta _0^{(p)} +I_s^{(p)}}\right) \frac{1}{g_{i,p}^{(p)}}$. Because we have $\eta _i^{(s)} =\eta _0^{(s)},\forall i=1,\ldots ,K^{(s)}$, that is, all SUs’ signals at the SUBS are corrupted by the same noise power, then after some manipulations about the SINR, we ultimately derive the SINR for SU $i$ as

$$\begin{aligned} \gamma _i^{(s)} \left( {p_i^{(s)},\mathbf{p}_{-i}^{(s)}}\right) =\frac{p_i^{(s)} h_{i,s}^{(s)}}{\frac{a}{G_s^{(p)}}p_i^{(s)} h_{i,p}^{(s)} +\sum _{j\ne i}^{K^{(s)}} {p_j^{(s)} c_j} +b},\quad \forall i=1,\ldots ,K^{(s)} \end{aligned}$$

(23)

where $a=\left( {\frac{1}{G_p^{(s)}}\frac{G_p^{(p)} \gamma _{0,tar}^{(p)}}{G_p^{(p)} -\left( {K^{(p)}-1}\right) \gamma _{0,tar}^{(p)}}\sum _{l=1}^{K^{(p)}} {\frac{g_{l,s}^{(p)}}{g_{l,p}^{(p)}}}}\right) , b=a\eta _0^{(p)} +\eta _0^{(s)}, c_j ={ah_{j,p}^{(s)}}/{G_s^{(p)}}+{h_{j,s}^{(s)}}/{G_s^{(s)}}$. Based on this SINR formula, two properties can be obtained for (23).

Property 1

$(i)$ We have ${\partial \gamma _i^{(s)} \left( {p_i^{(s)},\mathbf{p}_{-i}^{(s)}}\right) }\big /{\partial p_i^{(s)}}>0$, that is, the SINR $\gamma _i^{(s)} \left( {p_i^{(s)}, \mathbf{p}_{-i}^{(s)}}\right)$ of SU $i$ is a monotonically increasing function of its transmission power $p_i^{(s)}$ for any given $\mathbf{p}_{-i}^{(s)}$; (ii) For SU $i$, we have $\gamma _i^{(s)} \in \left[ {0,\gamma _{i,\max }^{(s)}}\right)$, where $\gamma _{i,\max }^{(s)} ={G_s^{(p)} h_{i,s}^{(s)}}\big /{ah_{i,p}^{(s)}}$.

Proof

$(i)$ can be directly obtained from (23) by derivative. With property $(i)$, we have that $\gamma _i^{(s)} \left( {p_i^{(s)},\mathbf{p}_{-i}^{(s)}}\right)$ is maximized at $p_i^{(s)} \rightarrow \infty$; therefore, we have $\mathop {\lim }\limits _{p_i^{(s)} \rightarrow \infty } \gamma _i^{(s)} \left( {p_i^{(s)},\mathbf{p}_{-i}^{(s)}}\right) =\gamma _{i,\max }^{(s)}$, that is, property (ii).

We can express the first-order partial derivative of $u_i^{(s)} \left( {p_i^{(s)},\mathbf{p}_{-i}^{(s)}}\right)$ with respect to $p_i^{(s)}$ as

$$\begin{aligned} \frac{\partial u_i^{(s)} \left( {p_i^{(s)},\mathbf{p}_{-i}^{(s)}}\right) }{\partial p_i^{(s)}}=\frac{\partial \gamma _i^{(s)} \left( {p_i^{(s)},\mathbf{p}_{-i}^{(s)}}\right) }{\partial p_i^{(s)}}\left( {\lambda _i -2\omega _i^{(s)} \left( {\gamma _{i,tar}^{(s)} -\gamma _i^{(s)} \left( {p_i^{(s)},\mathbf{p}_{-i}^{(s)}}\right) }\right) }\right) \end{aligned}$$

From Property 1, we know that ${\partial \gamma _i^{(s)} \left( {p_i^{(s)},\mathbf{p}_{-i}^{(s)}}\right) }\big /{\partial p_i^{(s)}}>0$ is always true, and we assume that $\lambda _{0}>0$.¹¹ Therefore, the best response of the SUs should consider two cases:

Case I: $\lambda _0 \ge 2\omega _i^{(s)} \gamma _{i,tar}^{(s)}$. It is clear that the term $\lambda _0 -2\omega _i^{(s)} \left( {\gamma _{i,tar}^{(s)} -\gamma _i^{(s)} \left( {p_i^{(s)}, \mathbf{p}_{-i}^{(s)}}\right) }\right)$ is positive for any feasible value of $\gamma _i^{(s)} \left( {p_i^{(s)}, \mathbf{p}_{-i}^{(s)}}\right)$, which also means that ${\partial \gamma _i^{(s)} \left( {p_i^{(s)}, \mathbf{p}_{-i}^{(s)}}\right) }\big /{\partial p_i^{(s)}}$ is positive and SU $i$’s utility function is monotonically increasing in $p_i^{(s)}$. Because the SUs aim to minimize their utility functions, the best response of SU $i$ is to transmit at zero power.

Case II: $0<\lambda _0 <2\omega _i^{(s)} \gamma _{i,tar}^{(s)}$. In this case, there exists a unique $p_i^{(*s)} \in P_i^{(s)}$ such that ${\partial \gamma _i^{(s)} \left( {p_i^{(s)},\mathbf{p}_{-i}^{(s)}}\right) }\big /{\partial p_i^{(s)}}=0$ and $\gamma _i^{(s)} \left( {p_i^{(s)},\mathbf{p}_{-i}^{(s)}}\right) =\gamma _{i,tar}^{(s)} -{\lambda _0}\big /{2\omega _i^2}$. With the secondary user SINR definition given in (23), we then have

$$\begin{aligned} p_i^{(*s)} =\left[ {\frac{f_i^{(s)} \left( {\gamma _{i,tar}^{(s)} ,\lambda _0}\right) \left( {\sum _{j\ne i}^{K^{(s)}} {p_j^{(s)} c_j} +b}\right) }{h_{i,s}^{(s)} \left( {1-{f_i^{(s)} \left( {\gamma _{i,tar}^{(s)} ,\lambda _0}\right) }\big /{\gamma _{i,\max }^{(s)}}}\right) }} \right] ^{+}. \end{aligned}$$

To sum up, we have the SUs’ best response (20). $\square$

1.2 Appendix 2: Proof of Theorem 1

We first prove the existence of the NE. From the well-known result due to Debreu et al. [42], an NE exists in game $\Xi ^{(s)}=\left\{ {\Omega ,\left\{ {P_i^{(s)}}\right\} _{i\in \Omega } ,\left\{ {u_i^{(s)} \left( {p_i^{(s)},\mathbf{p}_{-i}^{(s)}}\right) }\right\} _{i\in \Omega }}\right\}$ if for all $i\in \Omega , P_i^{(s)}$ is a nonempty, convex, and compact subset of some Euclidean space $\mathfrak {R}$, and $u_i^{(s)} \left( {p_i^{(s)}, \mathbf{p}_{-i}^{(s)}}\right)$ is continuous in $\left( {p_i^{(s)}, \mathbf{p}_{-i}^{(s)}}\right)$ and quasi-convex in $p_i^{(s)}$. We can note that for game $\Xi ^{(s)}$, the strategy set for SU $i$is characterized by $P_i^{(s)} =\left\{ {p_i^{(s)} :0\le p_i^{(s)} <\infty }\right\}$, that is, it is not a compact set. However, we can take $p_{i,\max }^{(s)} \rightarrow \infty$ for SU $i$ as its available MTPC, which will not affect the analysis of the paper. Thus, the strategy set for each SU can be observed as compact. Following the proof of (20), we know that the utility function is continuous in $\left( {p_i^{(s)},\mathbf{p}_{-i}^{(s)}}\right)$ and quasi-convex in $p_i^{(s)}$. Therefore, the secondary user power control game $\Xi ^{(s)}$ has at least one NE.

To prove the uniqueness of the NE and also the convergence of the SUs’ power iteration, using the best response-based NE definition, we just need to demonstrate that each SU’s best response can converge to a unique point [4]. From (25), we know that for a given primary network price $\lambda _{0}$, not all SUs will be active to share the primary network resource. Let ${\mathcal {A}}(\lambda _0)=\left\{ {i|i\in \Omega ,f_i^{(s)} \left( {\gamma _{i,tar}^{(s)},\lambda _0}\right) >0}\right\}$ denote the active SU sets, and define

$$\begin{aligned} \delta _i^{(s)} =\frac{f_i^{(s)} \left( {\gamma _{i,tar}^{(s)} ,\lambda _0}\right) }{h_{i,s}^{(s)} \left( {1-{f_i^{(s)} \left( {\gamma _{i,tar}^{(s)} ,\lambda _0}\right) }\big /{\gamma _{i,\max }^{(s)}}}\right) },\quad \forall i\in {\mathcal {A}}\left( {\lambda _0}\right) \end{aligned}$$

SU $i$’s power iteration is then

$$\begin{aligned} p_i^{(s)} =\delta _i^{(s)} \left( {h_{i,s}^{(s)} ,\gamma _{i,\max }^{(s)},\gamma _{i,tar}^{(s)} ,\lambda _0}\right) \left( {\sum \limits _{j\ne i}^{K^{(s)}} {p_j^{(s)} c_j} +b}\right) \end{aligned}$$

We rewrite the above power iteration in a linear matrix below, so that if the game has a unique NE, then it is the unique component-wise non-negative solution to

$$\begin{aligned} \left( {\mathbf{I}-\mathbf{F}^{(s)}}\right) \mathbf{p}^{(s)}=\mathbf{v}^{(s)} \end{aligned}$$

where $\left( {\mathbf{F}^{(s)}}\right) _{i,j\in {\mathcal {A}}\left( {\lambda _0}\right) } =\delta _i^{(s)} c_j$ if $i \ne j$ and $\left( {\mathbf{F}^{(s)}}\right) _{i,j\in {\mathcal {A}}\left( {\lambda _0}\right) } =0$ if $i=j$, and $\mathbf{v}^{(s)}=b\cdot \left( {\delta _1^{(s)},\delta _2^{(s)} ,\ldots ,\delta _{K^{(s)}}^{(s)}}\right) ^{T}$. For $0<\lambda _0 <\max _{i\in {\mathcal {A}}\left( {\lambda _0}\right) } \left( {2\omega _i^{(s)} \gamma _{i,tar}^{(s)}}\right)$ ¹², $\mathbf{F}^{(s)}$ is a non-negative matrix (i.e., all entries are non-negative) and is also non-negative component-wise. As [4], let $\rho (\mathbf{F}^{(p)})$ be the spectral radius of matrix $\mathbf{F}^{(s)}$. If $\rho (\mathbf{F}^{(p)})<1$, then $\left( {\mathbf{I}-\mathbf{F}^{(s)}}\right) ^{-1}=\sum _{n=0}^\infty {\left( {\mathbf{F}^{(s)}}\right) }^{n}$ exists and is non-negative. In that case, there is a unique component-wise non-negative solution to $\left( {\mathbf{I}-\mathbf{F}^{(s)}}\right) \mathbf{p}^{(s)}=\mathbf{v}^{(s)}$ given by

$$\begin{aligned} \mathbf{p}^{(*s)}=\sum \limits _{n=0}^\infty {\left( {\mathbf{F}^{(s)}}\right) }^{n}\mathbf{v}^{(s)} \end{aligned}$$

which represents the unique NE of the secondary network power game. On the other hand, if $\rho (\mathbf{F}^{(p)})>1$, then $\sum _{n=0}^\infty {\left( {\mathbf{F}^{(s)}}\right) }^{n}\rightarrow \infty$ and the game $\Xi ^{(s)}$ has no NE. Therefore, to prove the uniqueness of the NE and also the convergence of the algorithm is equivalent to proving $\rho (\mathbf{F}^{(p)})<1$ under the condition $\lambda _{0} \ge \lambda _{0,ne}$. Because each eigenvalue of a square matrix depends continuously upon its entries (as [43]), that is, $\rho (\mathbf{F}^{(p)})$ is continuous and non-increasing in primary network price $\lambda _{0}$. Then, following the proof of Theorem 1 in [4], we know that there exists a threshold $\lambda _{0,ne;}$ if $\lambda _{0} \ge \lambda _{0,ne}$, the algorithm converges to the unique NE. To sum up, we have Theorem 1. $\square$

1.3 Appendix 3: Proof of Theorem 2

First, from Theorem 1, we know that if $\lambda _{0,opt}<\lambda _{0,ne}$, then the secondary user power iteration algorithm will not converge to a fixed power level and $\lim _{n\rightarrow \infty } p_i^{(s)} (n)=p_{i,\max }^{(s)} \rightarrow \infty , \forall i\in {\mathcal {A}}\left( {\lambda _{0,opt}}\right)$. Thus, $\lim _{n\rightarrow \infty } \rho \omega ^{(p)}\left( {\sum _{i\in {\mathcal {A}}\left( {\lambda _{0,opt}}\right) } {p_i^{(s)} (n)} h_{i,p}^{(s)}}\right) \rightarrow \infty$. However, we know that the term $-\lambda _0^2 \sum _{i\in {\mathcal {A}}\left( {\lambda _{0,opt}}\right) } {1/{2\omega _i^2}} +\lambda _0 \sum _{i\in {\mathcal {A}}\left( {\lambda _{0,opt}}\right) } {\gamma _{i,tar}^{(s)}}$ is bounded above, then we will have $u^{(p)}(\lambda _{0})<0$, which violates the law of primary network resource sharing, thus, $\lambda _{0,opt}\ge \lambda _{0,ne}$.

For the primary network, its amount of resources shared is limited by constraint (18). From (20), we know that the secondary transmission power is a continuously and non-increasing function of primary network price; thus, there is always a price $\lambda _{0,th}(\le \lambda _{0,max})$ such that the secondary network interference is less than $I_{th}$ if $\lambda _{0}\ge \lambda _{0,th}$.

With the proof above, the constrained optimization problem defined in (22) can be transformed to an unconstrained optimization problem as $\max _{\max \left\{ {\lambda _{0,th} ,\lambda _{0,ne}}\right\} \le \lambda _0 \le \lambda _{0,\max }} u^{(p)}\left( {\lambda _0}\right) =\sum _{i\in {\mathcal {A}}\left( {\lambda _0}\right) } {\left( {-\frac{\lambda _0^2}{2\omega _i^2}+\gamma _{i,tar}^{(s)} \lambda _0}\right) } -\rho \omega ^{(p)}\left( {\sum _{i\in {\mathcal {A}}\left( {\lambda _0}\right) } {p_i^{(s)} \left( {\lambda _0}\right) } h_{i,p}^{(s)}}\right)$ in which, though the objective function $u^{(p)}(\lambda _{0})$ is non-concave, we have the knowledge that the primary network utility function is monotonically increasing in the interval $\left[ {0,\omega _{[1]}^{(s)} \gamma _{[1],tar}^{(s)}} \right]$, and how it will be if $\lambda _0 >\omega _{[1]}^{(s)} \gamma _{[1],tar}^{(s)}$ is unclear. Thus, $\lambda _{0,opt}$ is not less than $\omega _{[1]}^{(s)} \gamma _{[1],tar}^{(s)}$. To sum up, we have the conclusion. $\square$

1.4 Appendix 4: Proof of Proposition 3

From the proof of Theorem 1, we know that if $\lambda _{0} \ge \lambda _{0,th}$, then the SUs’ power control algorithm can always converge to a unique solution. Because the primary network suboptimal price search is from the maximum price, i.e., $\lambda _{0}=\lambda _{0,max}$, to price zero and based on the suboptimal price search stop condition, we know that this procedure will surely converge and can also guarantee the convergence of the SUs’ power iteration to the unique solution, that is, the Proposition 3. $\square$