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 Open Access
Optimal spectrum access and power control of secondary users in cognitive radio networks
 Yang Yang^{1}View ORCID ID profile,
 Linglong Dai^{1}Email author,
 Jianjun Li^{2},
 Shahid Mumtaz^{3} and
 Jonathan Rodriguez^{3}
https://doi.org/10.1186/s1363801708765
© The Author(s) 2017
 Received: 30 October 2016
 Accepted: 5 May 2017
 Published: 30 May 2017
Abstract
In future 5G communication system, radio resources can be effectively reused by cognitive radio networks (CRNs), where a lot of secondary users (SUs) are able to access the spectrum of primary users (PUs). In this paper, we analyze the optimal spectrum access and power control of SUs on multiple bands with the target of maximizing the average sum rate (ASR) of SUs. Specifically, based on the stochastic geometry, the random distributions of PUs and SUs are modeled by Poisson point processes (PPPs), based on which we derive out the closedform outage probabilities and obtain the ASR of SUs. Then, we formulate the maximization problem of ASR on multiple bands under the constraints of outage probabilities. With the help of convex optimization, the optimal density of SUs is obtained in closedform when the power of SUs is fixed. The convexity of ASR is also verified, and we evaluate the optimal power of SUs when the density of SUs is fixed. Based on these two obtained results, a spectrum access and power control algorithm is further proposed to maximize the ASR of SUs on multiple bands. Simulation results demonstrate that the proposed algorithm can achieve a higher maximum ASR of SUs over the average power allocation algorithm, and the density and power boundary of SUs are constrained by PUs as well as the interference in the networks.
Keywords
 Cognitive radio networks
 Spectrum access
 Power control
 Poisson point process
 Convex optimization
1 Introduction
The rapid growth of mobile traffic and the explosion of mobile users makes spectrum shortage more and more serious [1]. Cognitive radio networks (CRNs) are one of the most promising solution to improve spectrum efficiency, which can effectively alleviate the traffic demands by reusing the spectrum of primary users (PUs). In CRNs, secondary users (SUs) are able to access the spectrum of PUs and transmit signals without causing serious interference to PUs [2]. CRNs also brings a lot of benefits such as improving data rate, reducing power consumption, and enhancing spectral efficiency, which make it become an important part for future 5G wireless communications [3].
In CRNs, the random distribution of a large number of SUs significantly aggravate the interference caused by spectrum reuse. It is very difficult to model the dedicated interference for every SU in random geographical distribution, especially when the number of SUs is very large. Many research works have been proposed to model and analyze the CRNs with Poisson point process (PPP), which different network performances such as signal to interference plus noise ratio (SINR) distribution [4], coverage probability [5], and average spectral efficiency [6]. However, those performance analyses highly depend on the CRNs which PUs and SUs have already accessed the networks. Moreover, if the number of SUs is large, the optimization of spectrum access for performance enhancement in CRNs is challenging [7].
On the other hand, SUs accessing the spectrum of PUs makes the power control in CRNs difficult to operate, especially when multiple frequency spectrum bands are considered. This is due to the factor that different number of users may access different bands, and each user uses different powers to transmit signal on different bands. Moreover, each terminal in CRNs has its own power limitation, and the reliability of the PUs’ transmission should also be guaranteed. Therefore, power control is necessary to improve spectral efficiency [8] and network throughput [9] and reduce the interference in CRNs [10]. However, combining the spectrum access and power control of SUs to enhance the network performance, particularly on multiple bands with power limitation and random distribution in CRNs, is still needing further investigation.
1.1 Related works
Spectrum access is very important for SUs in CRNs, since the communication of PUs in CRNs must be protected at first [11]. Some early research works focused on the spectrum access in the uplink [12] and downlink [13] of PUs, where both the control and the scheduling were managed by the base stations (BSs) in CRNs. Based on these works, a cooperative spectrum sense of other SUs was introduced to reduce the false detection caused by shadowing fading [14]. Furthermore, it was proposed that both PUs and SUs can operate spectrum access in CRNs to improve the transmission rate, where the network connectivity and flexibility can be also enhanced [15].
While there have been considerable works on spectrum access scheduled by some central operator such as BSs in CRNs, very little attention has been paid to the spectrum access under random conditions. The authors in [16] adopted the random spectrum access of SUs in CRNs, where the network latency can be reduced by some extra MAC layer information. Besides, a hybrid spectrum access method was investigated in [17], where both opportunistic spectrum access and exclusive spectrum access were used to optimize the network performance. In addition, some spectrum access schemes were also extended to multiple bands [18], relay scenarios [19], and so on. However, most of the existing works considered the spectrum access in only one or several cells, while the spectrum access of SUs in a large area, especially for the condition with large random geographical user distribution, which still requires further investigation.
1.2 Contributions

With the assumption of random distribution of both PUs and SUs in CRNs, we model the system as PPPs and use Laplace transformation to derive the outage probabilities of both PUs and SUs on one single band. These results show the requirements to the transmission of PUs and SUs, which are also the basic constraints for optimizing the spectrum access of SUs. Note that our derived results can be further optimized on multiple bands, which is more generalized and can include many previously published works [4, 5, 15, 20, 21] as special cases only on one single bands.

The power control is optimized for maximizing ASR of SUs in CRNs. Our analysis not only considers the constraints of outage probabilities, power, and density on multiple bands, but also combines the optimal spectrum access of SUs. We further prove the convexity of ASR in the power definition domain of a secondary network. Then, we get the optimal density of SUs and optimal power of SUs on each bands. Based on these results, a spectrum access and power control algorithm is proposed, which considers the joint interaction effect between density and power of SUs. Simulation results demonstrate that the proposed algorithm can achieve a higher maximum ASR of SUs over the average power allocation algorithm. In addition, the density and power boundaries are constrained by PUs as well as the interference in the networks, and the ASR of SUs can be impacted by different powers of PUs. This new method is different from the previous works only consider power control on multiple bands [16, 17, 22].
1.3 Outline
The rest of the paper is organized as follows. Section 2 describes the scenario and network model. Section 3 presents the outage probabilities and the definition of ASR of SUs on multiple bands. In Section 4, we derived the optimal density and power of SUs which access the spectrum of PUs. Moreover, a spectrum access and power control algorithm is proposed to get the maximum ASR of SUs. Simulation results are shown in Section 5. Last, the conclusions are summarized in Section 6.
2 Scenario description and network model
In this section, we first describe the scenario of the cognitive radio networks, and then we model the networks with PPPs. Last, the propagation model is explained in detail.
2.1 Scenario description
2.2 Network models
Based on stochastic geometry theory, following assumptions are made:
Assumption 1
The transmitters of secondary network form a PPP on the two dimensional plane ℜ, which is denoted as Π _{0} with the density λ _{0,i } on band i,i=1,2,…,N. The transmission powers of the SUs are denoted as P _{0,i },(i=1,2,…,N) on each band.
Assumption 2
The transmitters of PUs form a series of stationary PPPs on each band, which are denoted as \(\Pi _{1}^{i}\) with the density λ _{1,i },(i=1,2,…,N) on ℜ. The transmission powers of the PUs are denoted as P _{1,i },(i=1,2,…,N) on each band.
Assumption 3
According to Palm theory [23], a typical receiver of S _{ j },j∈{0,1} is located at the origin, which does not influence the statistics of the PPP.
2.3 Propagation models
where P _{ tx } and P _{ rx } represent the transmitter and receiver power, respectively, α is the path loss exponent, D is the distance between the transmitter and the receiver. δ stands for Rayleigh fading coefficient, which has an independent exponential distribution with unit mean for every communication link in the CRNs.
When SUs access the spectrum of PUs, the receiver suffers from the interference generated by transmitters in both primary and secondary networks. The distribution of the interfering users in a single network S _{ j },j∈{0,1} on band i can be modeled by mark PPP, which is denoted as Π _{ j }={(X _{ jk },δ _{ jk })}, where δ _{ jk } and X _{ jk } are Rayleigh fading and the distance from the origin to the node k of network S _{ j }.
3 Average sum rate of secondary network on multibands
3.1 The outage probability on one single band
where \(I_{n,i,0} = \sum \limits _{\left ({{X_{0k}},{\delta _{0k}}} \right) \in {\Pi _{0}}} {\left ({\frac {{{P_{0,i}}}}{{{P_{n,i}}}}} \right){\delta _{0k}}{{\left  {{X_{0k}}} \right }^{ \alpha }}}, I_{n,i,1} = \sum \limits _{\left ({{X_{1k}},{\delta _{1k}}} \right) \in {\Pi _{1}}} {\left ({\frac {{{P_{1,i}}}}{{{P_{n,i}}}}} \right){\delta _{1k}}{{\left  {{X_{1k}}} \right }^{ \alpha }}}\). Set T _{ n,i } as the threshold of SIR on the ith band, the following lemma shows the outage probability of a typical receiver:
Lemma 1
where Pr(∙) represent the probability, \(\varsigma _{n,i} = \left [ \pi \Gamma \left (1 + \frac {2}{\alpha } \right) \Gamma \left ({1  \frac {2}{\alpha }} \right) \right ]T_{n,i}^{\frac {2}{\alpha }}R_{n0,i}^{2}\).
Proof
Denote \(\varsigma _{n,i} = \pi \Gamma \left ({1 + \frac {2}{\alpha }} \right)\Gamma \left ({1  \frac {2}{\alpha }} \right)T_{n,i}^{\frac {2}{\alpha }}R_{n0,i}^{2}\), Eq. (4) is obtained. □
where λ _{ n,i } is the node density of S _{ n } on the ith band.
3.2 Average sum rate of the secondary network on multiple bands
where \(\omega _{i} = \frac {{{W_{i}}}}{{\sum \limits _{m=1}^{N} {{W_{m}}}}}, W_{i}\) is the bandwidth of the ith band which is reused by SUs, P _{0,i } is the power of SUs, and λ _{0,i } is the density of SUs on the ith band.
where θ _{0} and θ _{1} are the outage probability thresholds of SUs and PUs, respectively. λ _{max,i } and P _{max,i } are the maximum density and power of SUs on each band, respectively.
4 Maximizing the ASR of the secondary network on multiple bands
In this section, the ASR of the secondary network on multiple bands is maximized under the density and power constraints. Then, we get the optimal density and power of SUs in closedform. Last, a spectrum access and power control algorithm is proposed to get the maximum ASR of the secondary network.
4.1 Maximizing ASR of the secondary network on multibands with density constraints
Make \(\lambda _{0,i,{{\sup }_{1}}} \,=\, \frac {{  1}}{{{\varsigma _{0,i}}}}\ln \left ({1 \,\, {\theta _{0}}} \right) \,\, {\left (\! {\frac {{{P_{1,i}}}}{{{P_{0,i}}}}} \!\right)^{\frac {2}{\alpha }}}{\!\lambda _{1,i}}\) and \(\lambda _{0,i,{{\sup }_{2}}}={\left ({\frac {{{P_{1,i}}}}{{{P_{0,i}}}}} \right)^{\frac {2}{\alpha }}}\left ({\frac {{  1}}{{{\varsigma _{1,i}}}}\ln \left ({1  {\theta _{1}}} \right)  {\lambda _{1,i}}} \right)\), from constraint (13), the upper density limit of SUs on a single band is \(\phantom {\dot {i}\!}{\lambda _{0,i,\sup }} = \min \left \{ {{\lambda _{0,i,{{\sup }_{1}}}},{\lambda _{0,i,{{\sup }_{2}}}},{\lambda _{\max,i}}} \right \}, \left ({i = 1,2,\ldots,N} \right)\).
Denoting the density of SUs as λ _{0}, we get \(\sum \limits _{i = 1}^{N} {{\lambda _{0,i}}} = {\lambda _{0}}\) and \({\lambda _{0}}\le \sum \limits _{i=1}^{N} {{\lambda _{0,i,\sup }}}\). While \({\lambda _{0}} > \sum \limits _{i = 1}^{N} {{\lambda _{0,i,\sup }}}\), the network should control the spectrum access of SUs to satisfy 0≤λ _{0,i }≤λ _{0,i,sup},(i=1,2,…,N). Then, we get the following two conditions:
Then, we get the optimal density of SUs in the following theorem:
Theorem 1
where \(\rho = \frac {v}{{{A_{i}}}}, {A_{i}} = {\omega _{i}}{e^{ {\varsigma _{0,i}}{{\left ({\frac {{{P_{1,i}}}}{{{P_{0,i}}}}} \right)}^{\frac {2}{\alpha }}}{\lambda _{1,i}}}}, \left ({i = 1,2,\ldots,N} \right)\). \(\psi _{u}=\left ({1  {\varsigma _{0,i}}{\lambda _{0,i,\sup }}} \right){e^{ {\varsigma _{0,i}}{\lambda _{0,i,\sup }}}}\). v is a Lagrange multiplier coefficient which satisfies \(\sum \limits _{i = 0}^{N} {\lambda _{0,i,opt2}^{*}} = {\lambda _{0}}\).
Proof
See Appendix 1. □
4.2 Maximizing the ASR of the secondary network on multibands with power constraints
Denote \({P_{0,i,{{\inf }_{1}}}} \,=\, {P_{1,i}}{\left [ {\frac {{  \ln \left ({1  {\theta _{0}}} \right)}}{{{\lambda _{1,i}}{\varsigma _{0,i}}}}  \frac {{{\lambda _{0,i}}}}{{{\lambda _{1,i}}}}} \right ]^{\frac {{  \alpha }}{2}}}\) and \({P_{0,i,{{\sup }_{1}}}} = {P_{1,i}}{\left [ {\frac {{  \ln \left ({1  {\theta _{1}}} \right)}}{{{\lambda _{0,i}}{\varsigma _{1,i}}}}  \frac {{{\lambda _{1,i}}}}{{{\lambda _{0,i}}}}} \right ]^{\frac {\alpha }{2}}}\), from constraint (14), the lower and upper power limit of SUs in a single band are \(\phantom {\dot {i}\!}{P_{0,i,\inf }} = \max \left \{ {0,{P_{0,i,{{\inf }_{1}}}}} \right \}\) and \(\phantom {\dot {i}\!}{P_{0,i,\sup }} = \min \left \{ {{P_{\max,i}},{P_{0,i,{{\sup }_{1}}}}} \right \}, i=1,2,\ldots,N\), respectively.
Define P _{0} as the power for one SUs on all the bands, so we have \(P_{0}\le \sum \limits _{i=1}^{N} {{P_{0,i,\sup }}}\); otherwise, we can control the spectrum access of SUs on each band to make P _{0,i,inf}≤P _{0,i }≤P _{0,i,sup},(i=1,2,…,N) established when \(P_{0} > \sum \limits _{i = 1}^{N} {{P_{0,i,\sup }}}\). Similarly, we get following two aspects:
Remark 1
When more SUs access the spectrum, the interference is becoming more and more serious. Once the density of SUs on the ith band is large enough to make inequality (27) established, the communication of the PUs cannot be ensured if SUs transmit signal on that band. Then power control must be operated, so P _{0,i } should be zero under this condition.
In cognitive radio networks, the outage probability threshold of PUs θ _{1} is usually defined as a very small value because the priority of PUs should be provided. Then, denoting \(B_{i} = {\omega _{i}}{\lambda _{0,i}}{e^{ {\varsigma _{0,i}}{\lambda _{0,i}}}}, D_{i} = {\varsigma _{0,i}}P_{1,i}^{\frac {2}{\alpha }}{\lambda _{1,i}}, (i = 1,2,\ldots,N)\), we get the following lemma and theorem:
Lemma 2
When outage probability threshold is \({\theta _{1}} \in \left ({0,1  {e^{ {\lambda _{1,i}}{\varsigma _{1,i}}}}} \right)\), the negative ASR of secondary network \(f\left ({{\lambda _{0,i}},{P_{0,i}}} \right) =  \sum \limits _{i = 1}^{N} {{\omega _{i}}{\lambda _{0,i}}{e^{ {\varsigma _{0,i}}\left [ {{\lambda _{0,i}} + {{\left ({\frac {{{P_{1,i}}}}{{{P_{0,i}}}}} \right)}^{\frac {2}{\alpha }}}{\lambda _{1,i}}} \right ]}}}\) is convex in the power definition domain of SUs [P _{0,i,inf},P _{0,i,sup}].
Proof
See Appendix 2. □
Then, following theorem shows the optimal power of SUs in each band:
Theorem 2
where [h _{0,i,min},h _{0,i,max}] is the range of the function \(h\left ({{P_{0,i}}} \right) = \frac {{2{B_{i}}{D_{i}}}}{\alpha }{e^{ {D_{i}}P_{0,i}^{\frac {{  2}}{\alpha }}}}P_{0,i}^{ \left ({1 + \frac {2}{\alpha }} \right)}\), and \(P_{0,i,solution}^{*}\) is the solution of u−h(P _{0,i })=0. While u is a Lagrange multiplier coefficient which is determined by the condition \(\sum \limits _{i = 1}^{N} {{P_{0,i}}}={P_{0}}\).
Proof
See Appendix 3. □
4.3 Spectrum access and power control algorithm for maximizing ASR of the secondary network
Based on the previous analysis, the ASR of the secondary network is convex when the density or the power of SUs is fixed. Then, we propose a spectrum access and power control algorithm for maximizing ASR of the secondary network. The detail of the algorithm is described in Algorithm 1.
The algorithm aims to use the optimal results in Theorems 1 and 2 to get the optimal spectrum of SUs and the optimal power of SUs, respectively. Several interpretations are shown as follows:
From step 4 to step 8, the algorithm calculates the optimal density of SUs on each band (i=1,2,…,N), i.e., we can get the optimal number of SUs which access the spectrum of PUs. Step 9 updates the upper density bound of SUs on each band. Then, step 12 to step 16 calculate the optimal power of SUs on each band (i=1,2,…,N), which the calculation is according to the Theorem 2. Step 17 updates the lower and upper bound of SUs on each band.
According to the two ifelseend parts, each time we get the optimal density of SUs, the constraints of the power are changed. Similarly, when we calculate the optimal power of SUs, the value will change the constraints of the density. So, we make an iteration between the optimization between the density and power of SUs, which is controlled by the flag bit in step 21. We also calculate the ASR gap between each two iterations, which is shown as Δ C in step 20. The final optimal density and power of SUs will be reached when Δ C<ε, where ε is a predefined threshold of Δ C. In the end, we get the optimal density and power of SUs in CRNs.
5 Simulation results and discussions
In this section, the outage probability and ASR of secondary network on one single band are analyzed. Then, the density and power boundary which confine the optimization are discussed. Moreover, we present the maximum ASRs of the secondary network on multiple bands. Finally, the maximum ASR on five bands with the proposed algorithm is compared with average power allocation method to make the results more insightful.
5.1 Simulation analysis of outage probability, density and power boundaries of SUs on the single band
Simulation parameters on one single band
Parameter  Physical mean  Value 

λ _{1,i }  Density of PUs  0.0001user/m ^{2} 
λ _{0,i }  Density of SUs  0.0001user/m ^{2} 
P _{1,i }  Power of PUs  25 dBm 
P _{0,i }  Power of SUs  15 dBm 
α  Path loss coefficient  4 
T _{0,i } / T _{1,i }  SIR threshold of SUs / PUs  0 dB 
R _{10,i }  Average link distance in the primary network  50 m 
R _{00,i }  Average link distance in the secondary network  15 m 
θ _{0} / θ _{1}  Outage probability threshold of SUs / PUs  0.1 
5.2 Simulation analysis of ASR and maximum ASR of the secondary network
Key parameters of the simulation on five bands
Parameters as [band1 band2 band3 band4 band5]  

Parameter  Case 1  Case 2  Case 3 
Power of PUs (dBm)  [10 10 10 20 20]  [20 25 15 10 20]  [20 25 15 10 20] 
Density of PUs (10^{−5}·user/m ^{2})  [1 1 1 1 3]  [1 1 1 1 3]  [10 30 50 20 30] 
6 Conclusions
In this paper, we have studied the optimal spectrum access and power control of SUs on multiple bands. The outage probabilities and ASR of SUs have been obtained based on the CRNs which are modeled by PPPs. Then, we have obtained the constraints from both PUs and SUs. The convexity of the target ASR has also been verified. So, we have derived out the optimal densities and powers of SUs in closedform. Last, a spectrum access and power control algorithm of SUs has been proposed to get the maximum ASR of SUs. From the simulation, we can see that the outage probability, ASR, density, and power of SUs on each band are all constrained by the primary network and the interference in CRNs. The optimal values on multiple bands are reduced when the sum constraints are added. Simulation results have also verified the superiority of the proposed algorithm over the average power allocation algorithm.
7 Appendix 1
7.1 Proof of Theorem 1
 (1)
\(x_{i}^{*} \ge 0, i = 1,2,\ldots,N\);
 (2)
k _{ i }≥0,i=1,2,…,N;
 (3)
l _{ i }≥0,i=1,2,…,N;
 (4)
\(x_{i}^{*}{\lambda _{0,i,\sup }}\le 0, i=1,2,\ldots,N\);
 (5)
\({k_{i}}x_{i}^{*} = 0, i=1,2,\ldots,N\);
 (6)
\({l_{i}}\left ({x_{i}^{*}{\lambda _{0,i,\sup }}}\right)=0, i=1,2,\ldots,N\);
 (7)
\({A_{i}}\left ({1{\varsigma _{0,i}}x_{i}^{*}}\right){e^{{\varsigma _{0,i}}x_{i}^{*}}}{k_{i}}+{l_{i}}+v=0, i=1,2,\ldots,N\);
 (8)
\(\sum \limits _{i=1}^{N}{x_{i}^{*}}=\lambda _{0}\).
Combine with (1) to (5), we can know:
If \(v\ge A_{i}, v{A_{i}}\left ({1{\varsigma _{0,i}}x_{i}^{*}} \right){e^{{\varsigma _{0,i}}x_{i}^{*}}}>0\), so \(x_{i}^{*}=0, l_{i}=0\).
If v<A _{ i }, the following results are obtained:
Bring \({e^{ {\varsigma _{0,i}}x_{i}^{*}}} \sim \left ({1 + {\varsigma _{0,i}}x_{i}^{*}} \right)\) into the equation, we get \(x_{i}^{*} = \frac {1}{{{\varsigma _{0,i}}}}\left ({1  \sqrt {\frac {v}{{{A_{i}}}}}} \right)\). Otherwise, we have \(x_{i}^{*} = {\lambda _{0,i,\sup }}\).
So from above, the results in Eq. (21) are obtained. □
8 Appendix 2
8.1 Proof of Lemma 2
The domain of P _{0,i } is P _{0,i,inf}≤P _{0,i }≤P _{0,i,sup}, and it is obvious that \( 2{D_{i}} + \left ({\alpha + 2} \right)P_{0,i}^{\frac {2}{\alpha }}\) is monotonously increasing with P _{0,i }, so if the lower limit of P _{0,i } makes the second partial derivative greater than zero, all the values of P _{0,i } in the domain make so. Because \({P_{0,i,\inf }} = \left \{ {0,{P_{1,i}}{{\left [ {  \frac {{\ln \left ({1  {\theta _{0}}} \right)}}{{{\lambda _{1,i}}{\varsigma _{0,i}}}}  \frac {{{\lambda _{0,i}}}}{{{\lambda _{1,i}}}}} \right ]}^{\frac {{  \alpha }}{2}}}} \right \}\), we have
(1) When P _{0,i,inf}=0, it is obvious that −f ^{″}(λ _{0,i },P _{0,i })≥0.
Thus, we know −f ^{″}(λ _{0,i },P _{0,i })>0 when P _{0,i }=P _{0,i,inf}. While \(2{D_{i}} + \left ({\alpha + 2} \right)P_{0,i}^{\frac {2}{\alpha }}\) is monotonously increasing with P _{0,i }, so −f(λ _{0,i },P _{0,i }) is convex when P _{0,i }∈[P _{0,i,inf},P _{0,i,sup}]. □
9 Appendix 3
9.1 Proof of Theorem 2
 (1)
s _{ i }≥0,i=1,2,…,N
 (2)
t _{ i }≥0,i=1,2,…,N
 (3)
\(p_{i}^{*}  {P_{0,i,\inf }} \ge 0, i=1,2,\ldots,N\)
 (4)
\(p_{i}^{*}  {P_{0,i,\sup }} \le 0, i=1,2,\ldots,N\)
 (5)
\({s_{i}}\left ({p_{i}^{*}  {P_{0,i,\inf }}} \right) = 0, i=1,2,\ldots,N\)
 (6)
\({t_{i}}\left ({p_{i}^{*}  {P_{0,i,\sup }}} \right) = 0, i=1,2,\ldots,N\)
 (7)
\( \frac {{2{B_{i}}{D_{i}}}}{\alpha }{e^{ {D_{i}}{{p_{i}^{*}}^{\frac { 2}{\alpha }}}}}{{p_{i}^{*}}^{ \left ({1 + \frac {2}{\alpha }} \right)}}  {s_{i}} + {t_{i}} + u = 0\);
 (8)
\(\sum \limits _{i = 1}^{N} {p_{i}^{*}} = {P_{0}}\).
If P _{0,i,sup}≤P _{0,i,inf}, we get \(p_{i}^{*}=0\); otherwise, make \(h(p_{i}^{*})=\frac {2B_{i}D_{i}}{\alpha }e^{D_{i}{p_{i}^{*}}^{\frac {2}{\alpha }}}{p_{i}^{*}}^{(1+\frac {2}{\alpha })}\), then it is a continuous function with its definition domain a compact closed set. So, defining its range [h _{0,i,min},h _{0,i,max}], we have the following:
(1) When u≤h _{0,i,min}, we have \(p_{i}^{*} = {P_{0,i,\inf }}, t_{i}=0\).
(2) When h _{0,i,min} < u ≤ h _{0,i,max},t _{ i } = 0, and \(u \frac {2B_{i}D_{i}}{\alpha }e^{D_{i}{p_{i}^{*}}^{\frac {2}{\alpha }}}{p_{i}^{*}}^{(1+\frac {2}{\alpha })} \,=\, 0\). According to equivalent infinite \(e^{D_{i}{p_{i}^{*}}^{\frac {2}{\alpha }}} \!\sim \! \left (\! {1 \,\, {D_{i}}{{p_{i}^{*}}^{\frac {{  2}}{\alpha }}}} \right)\) and substitute it into the equation before, we have \(u\frac {2B_{i}D_{i}}{\alpha }\left ({\vphantom {{{D_{i}}{{p_{i}^{*}}^{\frac {{  2}}{\alpha }}}}}} {1 }{{D_{i}}{{p_{i}^{*}}^{\frac {{  2}}{\alpha }}}} \right){p_{i}^{*}}^{(1+\frac {2}{\alpha })}=0\). When all the parameters are fixed, the numerical solution of \(p_{i}^{*}\) can be obtained, and we define it as \(P_{0,i,solution}^{*}\).
(3) When u>h _{0,i,max}, we have \(p_{i}^{*} = {P_{0,i,\sup }}\) which satisfies \(u\frac {2B_{i}D_{i}}{\alpha }e^{D_{i}P_{0,i,sup}^{\frac {2}{\alpha }}}P_{0,i,sup}^{(1+\frac {2}{\alpha })}+t_{i}=0\).
Substitute the results above into (8), i.e., \(\sum \limits _{i = 1}^{N} {p_{i}^{*}}=P_{0}\). We can get the numerical solution of u, so we can obtain the specific value of \(p_{i}^{*}\) on each band. □
Declarations
Acknowledgements
This work was supported by the the National Key Basic Research Program of China (Grant No. 2013CB329203), the National Natural Science Foundation of China (Grant Nos. 61571270 and 61271266), the China Postdoctoral Science Foundation (Grant No. 2016M591177), and the British Telecom and Tsinghua SEM Advanced ICT LAB. The research leading to these results also received funding from the European Commission H2020 programme under grant agreement no. 671705 (SPEED5G project).
Competing interests
The authors declare that they have no competing interests.
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