Joint optimization of detection threshold and resource allocation in infrastructurebased multiband cognitive radio networks
 Cong Shi^{1},
 Ying Wang^{1}Email author,
 Tan Wang^{1},
 Ping Zhang^{1},
 Jorge MartinezBauset^{2} and
 Frank Y Li^{3}
https://doi.org/10.1186/168714992012334
© Shi et al.; licensee Springer. 2012
Received: 1 March 2012
Accepted: 17 August 2012
Published: 2 November 2012
Abstract
Consider an infrastructurebased multiband cognitive radio network (CRN) where secondary users (SUs) opportunistically access a set of subcarriers when sensed as idle. The carrier sensing threshold which affects the access opportunities of SUs is conventionally regarded as static and treated independently from the resource allocation in the model. In this article, we study jointly the optimization of detection threshold and resource allocation with the goal of maximizing the total downlink capacity of SUs in such CRNs. The optimization problem is formulated considering three sets of variables, i.e., detection threshold, subcarrier assignment and power allocation, with constraints on the PUs’ rate loss and the power budget of the CR base station. Two schemes, referred to as offline and online algorithms respectively, are proposed to solve the optimization problem. While the offline algorithm finds the global optimal solution with high complexity, the online algorithm provides a closetooptimal solution with much lower complexity and realtime capability. The performance of the proposed schemes is evaluated by extensive simulations and compared with the conventional static threshold selection algorithm specified in the IEEE 802.22 standard.
Keywords
1 Introduction
The rapid development of new wireless devices and services has lead to a growing demand for radio spectrum, making the problem of spectrum shortage more serious. Indeed, the problem of spectrum scarcity is the result of, or is exacerbated by, the traditional static spectrum allocation policies, which assign spectrum bands to license holders on a longterm basis over large geographical regions [1–3]. Consequently, the concept of cognitive radio (CR) has emerged as a promising technology to realize dynamic spectrum access and solve the problem of spectrum scarcity.
In a CR network (CRN), secondary (unlicensed) users (SUs) may coexist with primary (licensed) users (PUs) of a primary radio network (PRN) in two ways: spectrum underlay which means that SUs may operate under the noise floor of PUs, or spectrum overlay which allows SUs operate only when the spectrum allocated to PUs is sensed as idle [4]. The PRN that we are interested in uses a set of licensed nonoverlapping orthogonal frequency subcarriers for communication between primary base stations (PBS) and PUs. We envisage that the PRN is underutilized and more revenue can be obtained by deploying CRNs, which opportunistically utilize the temporarily underutilized frequency subcarriers available at the PRN and provide service to SUs. In this article, we focus on spectrum overlay and consider an infrastructurebased CRN, in which there exists a central entity or CR base station (CRBS) that controls and coordinates the spectrum allocation and access of SUs [1, 5].
Resource allocation in CRN is an important issue for improving the SUs’ performance and has been widely investigated (see for example[6–8] and the references therein). However, these studies focus solely on capacity optimization for SUs based on the assumption of perfect spectrum sensing of the PUs’ activities. How the choice of spectrum sensing techniques and sensing parameters, such as detection threshold and sensing time, may affect the performance of SUs remains as an open question.
Spectrum sensing is of significant importance for CR systems. Among various spectrum sensing techniques, energy detection is the most common approach due to its simplicity and low latency (in the order of several tens of microseconds [9]). A decision on whether a specific spectrum band is occupied by a PU or not can be taken by comparing the received energy with a predetermined detection threshold. The detection thresholds for some primary systems have been specified by the IEEE 802.22 standard [10]. However, these fixed and identical detection thresholds are not adequate for multiband CRNs, since different subcarriers may suffer from different fading. Thus, the precise operation of the energy detection technique may require distinct detection thresholds on each subcarrier in order to accurately detect PUs’ activities.
Moreover, due to imperfect spectrum sensing, the selection of detection threshold may affect two parameters associated with energy detection: misseddetection probability (MDP) and false alarm probability (FAP) [11, 12]. More specifically, as the detection threshold decreases, the MDP decreases while the FAP increases. From the perspective of PUs, the lower the MDP is, the better the PUs are protected. However, from the SUs’ perspective, the higher the FAP, the lower the achievable capacity, because fewer subcarriers are detected as idle and can be utilized by SUs. On the other hand, the capacity of SUs also depends closely on the resource allocation scheme deployed, typically in the form of a subcarrier assignment and power allocation (SAPA) scheme. Therefore, in order to maximize the capacity of SUs while adequately protecting PUs’ activities, it is crucial to jointly consider the selection of the detection threshold and the SAPA scheme.
In this article, we study the joint optimization of detection threshold and SAPA scheme with the goal of maximizing the total downlink capacity of SUs in such an infrastructurebased CRN. The optimization problem is formulated considering three sets of variables, i.e., detection threshold, SAPA, with constraints on PUs’ rate loss and the power budget of the CRBS. We propose two algorithms, referred to as offline and online respectively to solve this problem. Extensive simulation results demonstrate that our proposed algorithms outperform the conventional uniform detection threshold (UDT) selection algorithms significantly. In brief, the contributions of this article are twofold.
 (1)
We formulate the joint optimization problem with three sets of variables, including detection threshold, SAPA, with the objective of maximizing the total capacity of SUs, subject to the constraints on the PUs’ rate loss and the power budget of the CRBS. Different from the conventional way of using interference power as the constraint to protect PUs, our scheme instead bounds the PUs’ rate loss caused by SUs’ activities due to a nonzero MDP.
 (2)
We propose two algorithms, one offline and another online, to solve the optimization problem. The offline algorithm finds the global optimal solution with high computation complexity, but it is difficult to implement in practice. The online algorithm iteratively optimizes the detection threshold and SAPA with suboptimal performance, with much lower computation complexity.
The remainder of this article is organized as follows. In Section 2., we review some studies that are related to this article. The system model is described in Section 3. Section 4. formulates the joint optimization problem. Then the offline and online algorithms are proposed in Sections 5. and 6., respectively. The computation complexity is analyzed in Section 7., followed by the simulation results and discussions in Section 8. Finally the article is concluded in Section 9.
2 Related work
Previous studies on joint optimization of detection threshold and resource allocation mainly concentrate on evaluating the impact of sensing time or detection threshold on SUs’ performance, without considering the design of the SAPA scheme. In [13], a sensingthroughput tradeoff is achieved by searching the optimal sensing time that maximizes SUs’ throughput. In [14], an optimization problem is formulated to maximize the ergodic capacity of SUs over transmission power and sensing time, by assuming a constant detection threshold.
To obtain optimized detection threshold, most of the related studies focus on the performance of the energy detector. For instance, they aim at minimizing the MDP and the FAP, but the performance of SUs is barely considered. In [15], the tradeoff between transmission power and detection threshold is studied. However, the focus of their work is on how to reduce interference caused to PUs rather than on the optimization of SUs’ capacity. In [16], an adaptive detection threshold algorithm is proposed to minimize the impairments caused by wireless channel and nonstationary noise. The detection performance is evaluated in multichannel CRNs that perform opportunistic access.
So far, little work has been done on joint optimization of detection threshold and resource allocation in multiband CRNs. In [17], a joint optimal power allocation (OPA) and detection threshold scheme is proposed to maximize SUs’ capacity in spectrum sharing CRNs. In [18], the sensing threshold is determined to optimize two different sensing objectives: weighted network total capacity and the more traditional Bayesian cost, by exploiting location information. However, these two schemes are not designed for multiband CRNs. In [19], a joint crosslayer scheduling and sensing framework is designed to optimize average weighted SUs’ capacity, by adapting SAPA across SUs (under a constraint on average interference to PUs). However, this framework assumes fixed sensing parameters such as the MDP and the FAP when optimizing SUs’ capacity.
3 System model
We consider the downlink transmission of an infrastructurebased CRN, where a CRBS coexists in the vicinity of a PBS and there are a number of SUs and PUs covered by both of them. The CRBS is regarded as the control center for the secondary system. The available band of the primary system is divided into N orthogonal subcarriers, for which we assume frequency flat fading. There are K SUs which utilize the subcarriers in an overlay way, e.g., either when PUs are absent and no false alarm is generated, or when PUs are present but they are not detected.
The CRBS downlink is partitioned into time slots and each slot consists of a sensing period and a data transmission period [13]. The uninterrupted activity periods of the PUs are much longer than a single SU time slot. During the sensing period, each SU performs spectrum sensing and reports a set of preprocessed energy measurements to the CRBS. The reporting to the CRBS may be done, for example, via a control channel. Based on the sensing information received from the involved SUs, the CRBS can make a decision on whether a subcarrier is occupied by a PU or not. Note that the sensing results are not always reliable due to channel fading and imprecise energy detection.
Based on the instantaneous channel power gains and sensing results, the CRBS makes a decision on the channel occupancy status of each subcarrier and SAPA. The decisions are broadcast to the SUs through the downlink control channel. Note that in this article, we focus on the design of detection threshold and SAPA, and the sensing time is set as fixed through all slots. Thus, the data transmission period is considered only when formulating the optimization problem for the sake of simplicity.
3.1 Energy detection based spectrum sensing
The simultaneous spectrum sensing over multiple subcarriers is performed by using a set of multiband joint detectors, as proposed in [21]. At each SU, N signal samples corresponding to N subcarriers are obtained using N independent energy detectors.
where Q(.) is the complementary distribution of a standard Gaussian random variable and ${\gamma}_{n}={\widehat{\gamma}}_{n}\sum _{k=1}^{K}{h}_{k,n}^{\text{ps}}{}^{2}/K$. ${\widehat{\gamma}}_{n}={p}^{\text{pu}}/{\delta}^{2}$ denotes the average signal to noise ratio (SNR) of PUs on subcarrier n, where p^{pu} is the transmission power of PU on subcarrier n.
As mentioned earlier, the SUs will utilize subcarrier n under two possible circumstances: when the subcarrier is idle and no false alarm is generated, or when the subcarrier is occupied but the occupancy is not detected. Similar to what is widely used in the literature [22, 23], we assume that the subcarriers are occupied by the PUs according to an ON/OFF model. Let us denote by Π_{0}the prior probability that PUs are absent on a specific subcarrier and by Π_{1}=1−Π_{0} the prior probability that they are active. Then, the two circumstances described above happen with probabilities $(1{P}_{n}^{\mathrm{\text{fa}}}){\Pi}_{0}$ and ${P}_{n}^{\mathrm{\text{md}}}{\Pi}_{1}$, respectively.
4 Optimization problem formulation
where δ^{2}denotes the AWGN noise variance. Throughout the context we use (6) as a normalized capacity expression to represent both the theoretical Shannon capacity and the achieved data rate, i.e., protocol overhead is ignored. Similarly, we use ${R}_{k,n}^{1}=C\left({p}_{k,n}{h}_{k,n}^{\mathrm{\text{cs}}}{}^{2},{p}^{\mathrm{\text{pu}}}{h}_{k,n}^{\mathrm{\text{ps}}}{}^{2}\right)$ to represent the data rate of SU k on subcarrier n when PUs are present, where ${p}^{\mathrm{\text{pu}}}{h}_{k,n}^{\mathrm{\text{ps}}}{}^{2}$ is the interference power of the PBS measured at SU k.
In order to formulate the joint detection threshold and resource allocation problem, the following system constraints must be taken into consideration:

CRBS power budget constraint: Let P_{ T }denote the maximum transmission power of the CRBS, the total power allocated on all subcarriers must be less than or equal to P_{ T }. That is:$\sum _{k=1}^{K}\sum _{n=1}^{N}{\rho}_{k,n}{p}_{k,n}\le {P}_{T}.$(8)

PUs’ rate loss constraint: Similar to [20], in order to protect the transmission of the PUs, we impose an upper bound on PUs’ rate loss,^{a} which is caused by unexpected SU transmissions. Let ${R}_{\mathrm{\text{pu}},n}^{\text{max}}=C\left({p}^{\mathrm{\text{pu}}}{h}_{n}^{\mathrm{\text{pu}}}{}^{2},0\right)$ be the maximum data rate that the PUs can achieve on subcarrier n. Denote by ${R}_{\mathrm{\text{pu}},n}^{\mathrm{\text{md}}}=C\left({p}^{\mathrm{\text{pu}}}{h}_{n}^{\mathrm{\text{pu}}}{}^{2},{p}_{k,n}{h}_{k,n}^{\mathrm{\text{cp}}}{}^{2}\right)$ the achievable data rate of the PUs on subcarrier n when SU k is also transmitting. Let ΔR be the maximum rate loss that the PUs can tolerate on each subcarrier. Then, the SU transmission on subcarrier n is allowed only when the following constraint is satisfied:${P}_{n}^{\mathrm{\text{md}}}\left({R}_{\mathrm{\text{pu}},n}^{\text{max}}{R}_{\mathrm{\text{pu}},n}^{\mathrm{\text{md}}}\right)\le \mathrm{\Delta R},\phantom{\rule{1em}{0ex}}n=1,2,\dots ,\mathrm{N.}$(9)
where constraints C4 and C5 take into account the fact that each subcarrier can only be assigned to one SU.
However, (P1) is a mixed integer programming problem including three sets of variables (λ,P,ρ) to be optimized, and it is computationally complex to solve. Moreover, Constraint C2 is nonconvex, making (P1) nonconvex. In the following, we propose two algorithms named as the offline and online algorithms to solve (P1). In the offline algorithm, an exhaustive search based on a large number of detection thresholds is done to optimize the SAPA. Then, the optimal solution is found among the candidate detection thresholds. In the online algorithm, a suboptimal solution is obtained iteratively. We show later that this algorithm converges after a few iterations.
5 The offline algorithm to solve (P1)
The main idea of the offline algorithm is to solve (P1) by fixing one of the three variables, λ, with a set of known values. This is based on the observation that if the detection threshold vector λ is given, then ${P}_{n}^{\mathrm{\text{fa}}}$ and ${P}_{n}^{\mathrm{\text{md}}}$ can be easily calculated by (4) and (5), respectively. Thus (P1) is shrunk to a SAPA optimization problem with only two variables, P and ρ, for any given λ. We can then obtain the global optimal solution by selecting the best λ value.
In the rest of this section, we propose two algorithms, one optimal and another suboptimal, to solve the SAPA optimization problem, (P2).
5.1 The optimal algorithm to solve the SAPA problem
According to [24], the duality gap of the SAPA problem in multiband networks is nearly zero when the number of subcarriers is sufficiently large. Thus, the dual decomposition method can be used to solve (P2) when λ is given. In order to make (P2) tractable, we first transform constraint C2 of (P2) into a convex form. Redefine $\mathrm{\Delta R}=\eta {R}_{\mathrm{\text{pu}},n}^{\text{max}}$, where 0≤η≤1. η can be interpreted as the fraction of PUs’ rate loss. Using the following proposition, we can transform the PUs’ rate loss constraint into a maximum power allocation constraint.
Proposition 1
Hence, constraint C2 of (P2) is satisfied if $\sum _{k=1}^{K}{\rho}_{k,n}{p}_{k,n}<{P}_{n}^{\text{max}},\forall n$. If $\eta \ge {P}_{n}^{\text{md}}({\lambda}_{n})$, the PUs’ rate loss constraint is equivalent to $\sum _{k=1}^{K}{\rho}_{k,n}{p}_{k,n}\ge 0,\forall n$.
Proof
Please refer to Appendix 1.
where μ and β=[β_{1},…,β_{ N }] are the Lagrange multipliers corresponding to constraints C1 and C2 in (P2). Note that constraints C3, C4 and C5 are not taken into account in the Lagrangian function. However, these constraints will be satisfied in the dual domain when solving the problem, as shown later. □
Since each subcarrier can be assigned to only one SU, (16) actually indicates a rule for allocating subcarriers, which is to search the SU that maximizes (16) for a specific subcarrier. To solve (P2), a two step iterative process is needed. First, each subcarrier is allocated to the corresponding SU according to (16). Second, the Lagrangian multipliers are updated. The iterative process continues until convergence is achieved.
Proposition 2
otherwise ${\widehat{p}}_{k,n}=0$.
Proof
Please refer to Appendix 2. □
Based on the discussions above, the OPA ${\widehat{p}}_{k,n}$, now denoted as ${\widehat{p}}_{k,n}(\mu ,\mathit{\beta})$, can be obtained using Proposition 2 for given (μ,β). Then, substituting it into (16), ${\hat{\mathcal{P}}}_{n}(\mu ,\mathit{\beta})$ can be determined.
In order to minimize the Lagrangian dual function g(μ β), both subgradient and ellipsoid methods [25] can be used to update the Lagrangian multipliers (μ β). Without loss of generality, the subgradient method is adopted in this article.
Lemma 1
Proof
Please refer to Appendix 3. □
where ${\xi}_{\mu}^{j}$ and ${\xi}_{{\beta}_{n}}^{j}$ are the appropriate positive stepsize sequences, and j is the iteration index.
The initial value of (μ,β) could therefore be selected from these areas shown above.
After obtaining the optimal Lagrangian multipliers, denoted as (μ^{∗},β^{∗}), the OPA can be obtained as ${p}_{k,n}^{\ast}={\widehat{p}}_{k,n}({\mu}^{\ast},{\mathit{\beta}}^{\ast})$. Then, using (16), the optimal subcarrier assignment is determined. The pseudocode for the optimal SAPA algorithm is described in Algorithm 1.
5.2 Algorithm 1 Optimal SAPA (OSAPA)
1. Subcarrier set: $\mathcal{N}=\{1,2,\dots ,N\}$, SU set: $\mathcal{K}=\{1,2,\dots ,K\}$
2. Calculate optimal Lagrangian multipliers
3. Initialization: (μ ^{(0)},β ^{(0)})
4.while stopping rule is not satisfied
5. 1) Compute ${\widehat{p}}_{k,n}({\mu}^{(0)},{\mathit{\beta}}^{(0)})$ using (18) and Proposition 2;
6. 2) Compute ${\mathcal{K}}_{n}({\mu}^{(0)},{\mathit{\beta}}^{(0)})$, $\forall n\in \mathcal{N}$ using (16);
7. 3) Compute $g({\mu}^{(0)},{\mathit{\beta}}^{(0)})$ using (15);
8. 4) Update (μ ^{(0)},β ^{(0)}) using (20);
9.end while
10.Subcarrier assignment and power allocation
11.while $(\mathcal{N}\ne \varnothing )$do
12. 1) Compute optimal ${p}_{k,n}^{\ast}={\widehat{p}}_{k,n}({\mu}^{\ast},{\mathit{\beta}}^{\ast})$ using (18) and Proposition 2, $\forall k\in \mathcal{K}$, $\forall n\in \mathcal{N}$;
13. 2)Find a pair of (n ^{∗},k ^{∗}) such that (16) is maximum;
14. 3)Assgin subcarrier n ^{∗}to SU k ^{∗}, and update ${\rho}_{{k}^{\ast},{n}^{\ast}}=1$;
15. 4) Set ${p}_{k,n}={p}_{k,n}^{\ast}$, k=k ^{∗}and p _{k,n}=0, ∀k≠k ^{∗};
16. 5) $\mathcal{N}=\mathcal{N}\left\{{n}^{\ast}\right\}$;
17.end while
5.3 Suboptimal SAPA algorithm
The optimal SAPA algorithm presented above jointly optimizes the SAPA, but it is computationally costly (its complexity is analyzed in Section 7.). Generalizing the proof given in [26] for multiband CRN, the maximum capacity in the downlink can be obtained when each subcarrier is assigned to the user that has the best channel gain. However, this subcarrier assignment is not optimal in the CRN, as we need to consider the constraint on PUs’ rate loss for each subcarrier. Based on this consideration, a lowcomplexity but yet effective suboptimal SAPA scheme is proposed here.
More specifically, the optimization problem (P2) is divided into two parts, which are suboptimal subcarrier assignment (SSA) and OPA. Once subcarriers have been assigned to users, the OPA can be obtained using (18) and (20). The details of the suboptimal SAPA are given in Algorithm 2.
5.4 Algorithm 2 Suboptimal SAPA (SSAPA)
1: Subcarrier set: $\mathcal{N}=\{1,2,\dots ,N\}$, SU set: $\mathcal{K}=\{1,2,\dots ,K\}$
2:Suboptimal subcarrier Assignment
3:for k=1 to K do
4: 1) Compute $\frac{{h}_{k,n}^{\text{cs}}{}^{2}}{{\delta}^{2}}$, search ${n}^{\ast}=\text{arg}\underset{n\in \mathcal{N}}{\text{max}}\frac{\underset{k,n}{\overset{\text{cs}}{h}}{}^{2}}{{\delta}^{2}}$;
5: 2) Set ${\rho}_{k,{n}^{\ast}}=1$;
6: 3) $\mathcal{N}=\mathcal{N}\left\{{n}^{\ast}\right\}$;
7:end for
8:WHILE$(\mathcal{N}\ne \varnothing )$do
10: 2) Assign subcarrier n^{∗}to SU k^{∗}, ${\rho}_{{k}^{\ast},{n}^{\ast}}=1$;
11: 3) $\mathcal{N}=\mathcal{N}\left\{{n}^{\ast}\right\}$;
12:end while
13:Optimal Power Allocation
14: Compute Lagrangian multipliers (μ^{∗},β^{∗}) similar to Algorithm 1;
15: Compute optimal $\left\{{p}_{k,n}^{\ast}\right\}$, using ${\rho}_{{k}^{\ast},{n}^{\ast}}$ and (18);
6 The online algorithm to solve (P1)
The computational cost of the offline solution presented in Section 5. is quite high as an exhaustive search over a large number of candidate detection thresholds λ must be done, which makes it unfeasible for online deployment. Note that the number of λ is also a function of granularity between any two consecutive λ values. In this section, we propose a new algorithm named as iterative optimization of detection threshold and throughput (IODTT) to solve (P1). Unlike the offline algorithm, the IODTT algorithm finds the suboptimal detection thresholds (SDTs) and SAPA iteratively, thus it can be used online. In Section 8., we show through simulations that the algorithm converges after a small number of iterations.
The core idea of the IODTT algorithm is to divide (P1) into two subproblems, and then solve the two subproblems iteratively. At the i th iteration, the SDTs, λ^{(i)}, are obtained by solving the first subproblem. Using λ^{(i)}, the second subproblem SAPA {P^{(i)},ρ^{(i)}} is solved using either the optimal SAPA algorithm or the suboptimal SAPA algorithm proposed in Section 5. The iteration process stops once convergence is achieved.
6.1 Suboptimal detection threshold determination
where ${R}_{n}^{0}({\mathit{P}}^{(0)},{\mathit{\rho}}^{(0)})=\sum _{k=1}^{K}{\rho}_{k,n}^{(0)}C\left({p}_{k,n}^{(0)}{h}_{k,n}^{\mathrm{\text{cs}}}{}^{2},0\right)$ and ${R}_{n}^{1}({\mathit{P}}^{(0)},{\mathit{\rho}}^{(0)})=\sum _{k=1}^{K}{\rho}_{k,n}^{(0)}C\left({p}_{k,n}^{(0)}{h}_{k,n}^{\mathrm{\text{cs}}}{}^{2},{p}^{\mathrm{\text{pu}}}{h}_{k,n}^{\mathrm{\text{ps}}}{}^{2}\right)$ are the total data rates on subcarrier n given (P^{(0)},ρ^{(0)}), when the PUs are absent and present, respectively.
Now the optimization problem is translated into a linear problem, as shown in (26) and it can be solved using the interiorpoint method or other numerical search algorithms.
6.2 The IODTT algorithm
Given the SDTs, λ^{(i)}, obtained at the i th iteration of (P3), (P1) is equivalent to (P2). Then, it can be correspondingly solved using either the optimal or the suboptimal SAPA algorithms presented in Section 5. In other words, (P1) is split into a set of (P2)s and it can be iteratively solved using the results from (P3). However, the solution obtained may not always be globally optimum.
where i means the index of iteration. Since P and λ are upper bounded, convergence is guaranteed. An important property of the IODTT algorithm is its insensitivity to the initial value P^{(0)}and ρ^{(0)}, as shown by the simulation results presented in Section 8.
6.3 Algorithm 3 IODTT
1: Initialization: P^{(0)}, ρ^{(0)}and i=0;
2:while$\parallel {p}_{k,n}^{(i+1)}{p}_{k,n}^{(i)}\parallel >\varphi \phantom{\rule{1em}{0ex}}\mathtt{\text{and}}\phantom{\rule{1em}{0ex}}\parallel {\lambda}_{n}^{(i+1)}{\lambda}_{n}^{(i)}\parallel >\varphi $
3: 1) Compute λ^{ opt }by solving (P2), using P^{(i)}, ρ^{(i)};
4: 2) λ^{(i + 1)}=λ^{ opt };
5: 3) Compute P^{ opt }, ρ^{ opt }by optimal or suboptimal SAPA, using λ^{(i + 1)};
6: 4) P^{(i + 1)}=P^{ opt }, ρ^{(i + 1)}=ρ^{ opt }and i=i + 1;
7:end while
8: Output P^{(i + 1)}, ρ^{(i + 1)};
7 Complexity analysis
For the offline solution, the complexity depends on the size of the region where possible detection thresholds are sought, the calculation of the SAPA, and the granularity of the thresholds. For the optimal SAPA, the number of iterations required to obtain the εoptimal Lagrangian multipliers (μ^{∗}β^{∗}), i.e., g(μ β)−g(μ^{∗}β^{∗})<ε, is in the order of $O\left(\frac{1}{{\epsilon}^{2}}\right)$[25]. In each iteration, the computation of (16) requires K comparisons for each of N subcarriers. Thus the total complexity of the optimal SAPA is $O\left(\frac{\text{NK}}{{\epsilon}^{2}}\right)$. For the suboptimal SAPA algorithm, the complexity of the SSA is $O\left(\frac{4\text{NK}+K3{K}^{2}}{2}\right)$, thus the total complexity of the suboptimal SAPA is $O\left(\frac{4(N+1)K3{K}^{2}}{2}+\frac{1}{{\epsilon}^{2}}\right)$. The complexity of the suboptimal SAPA is much lower than that of the optimal SAPA when the number of the SUs is large and ε is small. Assuming that for each subcarrier the same detection threshold value is used, and there are m possible detection thresholds to search, then the total computation complexity of the offline solution is $O\left(\frac{\text{mNK}}{{\epsilon}^{2}}\right)$ when using the optimal SAPA and $O\left(m\left(\frac{4(N+1)K3{K}^{2}}{2}+\frac{1}{{\epsilon}^{2}}\right)\right)$ when using the suboptimal SAPA. If subcarriers have different detection thresholds, the computation complexity will be prohibitively high in both cases.
The computation complexity of the IODTT algorithm is related to the number of iterations and the calculation of the detection threshold and SAPA. Since (26) is a linear programming problem, the computation complexity required to determine the detection thresholds is in the order of O(N^{1.5}) [27], where N is the total number of subcarriers. Denote the number of iterations required for the convergence as r. The total computation complexity of the IODTT algorithm is $O\left(r\left(\frac{4(N+1)K3{K}^{2}}{2}+\frac{1}{{\epsilon}^{2}}+{N}^{1.5}\right)\right)$ when the suboptimal SAPA is employed. With our parameter configuration presented below in Section 8., the average number of iterations required for convergence is found to be 4<E r<6.
8 Numerical evaluation
To evaluate the performance of the proposed algorithms, we perform extensive simulations using a custombuilt MATLAB simulator. For simplicity, we consider a multiband CRN with K=2 SUs and 2 PUs, and the variance of the AWGN is normalized to 1. We assume that the channel gains, e.g., ${h}_{n}^{\mathrm{\text{pu}}}$, ${h}_{k,n}^{\mathrm{\text{cs}}}$, ${h}_{k,n}^{\mathrm{\text{ps}}}$ and ${h}_{k,n}^{\mathrm{\text{cp}}}$, are independent, identically distributed Rayleigh random variables with average channel power gains equaling to 0.4, 0.4, 0.1 and 0.1 respectively. The number of subcarriers is N=8 and the number of samples is L=100. The PUs activity factor is assumed to be small, as Π_{1}=0.3. Note that the proposed algorithms also apply to more realistic scenarios with larger numbers of SUs/PUs and subcarriers. The reason that we configure the system with these comparatively small values is to run our simulations faster according to the complexity analysis in Section 7. Meanwhile, these values are also very representative when evaluating the proposed algorithms according to [21].
To run the offline and online algorithms, the initial value of detection threshold should be properly selected. For the offline algorithm, in order to determine the initial detection threshold vector λ, we set a set of candidates for FAP or MDP, e.g. FAP∈[10^{−6},10^{−5},…,10^{0}. Once the values of FAP or MDP are determined, the initial detection threshold is determined. This method for selecting the initial detection threshold has been widely used in [13, 15, 28]. For the online algorithm, since we iteratively find the best values of λ and (P ρ), the subcarrier assignment ρ and power allocation P are determined by random subcarrier assignment and equal power allocation. The initial λ value can be obtained according to (26).
Algorithm setup and complexity
Algorithm  Offline  Online/iterative  OSAPA  SSAPA  SDT  ADT  UDT  Complexity 

OFLO  ✓  ✓  $O\left(m(\frac{\text{NK}}{{\epsilon}^{2}})\right)$  
OFLS  ✓  ✓  $O\left(m(\frac{4\text{NK}+K3{K}^{2}}{2}+\frac{1}{{\epsilon}^{2}})\right)$  
ONLO  ✓  ✓  ✓  $O\left(r(\frac{\text{NK}}{{\epsilon}^{2}}+{N}^{1.5})\right)$  
ONLS  ✓  ✓  ✓  $O\left(r(\frac{4\text{NK}+K3{K}^{2}}{2}+\frac{1}{{\epsilon}^{2}}+{N}^{1.5})\right)$  
ADTO  ✓  ✓  ✓  $O\left(r(\frac{\text{NK}}{{\epsilon}^{2}}+{N}^{1.5})\right)$  
ADTS  ✓  ✓  ✓  $O\left(r(\frac{4\text{NK}+K3{K}^{2}}{2}+\frac{1}{{\epsilon}^{2}}+{N}^{1.5})\right)$  
UDTO  ✓  ✓  $O\left(\frac{\text{NK}}{{\epsilon}^{2}}\right)$  
UDTO  ✓  ✓  $O\left(\frac{4\text{NK}+K3{K}^{2}}{2}+\frac{1}{{\epsilon}^{2}}\right)$ 
Figures 2 and 4 further depict the impact of the power budget, P_{ T } of the CRBS, on the optimal detection thresholds, λ^{∗}, for the optimal and the suboptimal SAPA algorithms respectively. Two operating regions for λ are identified as λ^{∗}∈[0.9,1.04] and λ^{∗}∈[1.05,1.1].
In the first region, an increase in P_{ T }leads to a significant increase in $\overline{{R}_{\mathrm{\text{tot}}}}$. Clearly, a small value of λ leads to a high FAP and a low MDP, and consequently the number of subcarriers available for SU transmission is small. For a constant λ, if we increase P_{ T }, $\overline{{R}_{\mathrm{\text{tot}}}}$ increases. On the other hand, for a constant $\overline{{R}_{\mathrm{\text{tot}}}}$, λ increases significantly as P_{ T } decreases. This is because as P_{ T }decreases, the PUs will be less affected by the SUs, and then a higher MDP or a lower FAP can provide more transmission opportunities for the SUs. The higher the P_{ T }, the more sensitive the $\overline{{R}_{\mathrm{\text{tot}}}}$ with respect to λ. To achieve the peak $\overline{{R}_{\mathrm{\text{tot}}}}$ at a reduced power level, it is required to increase λ. In the second region with λ^{∗}∈[1.05,1.1], $\overline{{R}_{\mathrm{\text{tot}}}}$ is insensitive to P_{ T }, i.e., for a constant λ, different values of P_{ T } achieve the same $\overline{{R}_{\mathrm{\text{tot}}}}$. This result is expected, because as λ increases, the MDP increases, and then the constraint on power budget becomes insignificant.
In Figures 3 and 5, the impact of the PUs’ rate loss fraction, η, on the optimal detection threshold, λ^{∗}, is evaluated. Similar to the analysis of Figures 2 and 4 above, two regions for λ are identified. In the first region, e.g., [0.90,0.95], $\overline{{R}_{\mathrm{\text{tot}}}}$ is insensitive to η for a constant λ. Clearly, for small values of λ, the MDP is small, meaning that all PU activities will be correctly detected. Then the PUs’ rate loss will be negligible. Consequently, $\overline{{R}_{\mathrm{\text{tot}}}}$ will not improve appreciably for different values of the PUs’ rate loss constraint η. In the second region, e.g., [0.96,1.1], as λ increases, the MDP increases and the FAP decreases, and η has higher impact on the perfomathrmance of $\overline{{R}_{\mathrm{\text{tot}}}}$. The reasons for this behavior are twofold. Firstly, for a constant λ, if we increase η, then $\overline{{R}_{\mathrm{\text{tot}}}}$ will increase in approximately the same proportion. This can be explained because that the higher the PUs’ rate loss we allow, the more power can be allocated to the SUs’ transmission. Secondly, for a constant $\overline{{R}_{\mathrm{\text{tot}}}}$, if we decrease η, then λ will decrease in approximately the same proportion. Again, this is because that by decreasing the PUs’ rate loss, the MDP must decrease in order to detect the PU activities more precisely, leading to a reduction of λ.
Figure 7 shows the variation of $\overline{{R}_{\text{tot}}}$ with η, while keeping P_{ T }=10 W. The performance relationship among the different algorithms exhibits the same trend as we may observe in Figure 6. However, it is worth mentioning that the performance difference between OFLO and ONLO becomes marginal in this case, compared with that in Figure 6. Again, our proposed algorithms outperform UDTO and UDTS. Note that UDTO and UDTS are insensitive to η as they select a low detection threshold to achieve a low MDP, which prevents the SUs from sharing the resources with the PUs. In addition, selecting a low detection threshold leads to a high FAP, which further exacerbates the reduction of available resources for the SUs.
9 Conclusions
In this article, we study the joint optimization of detection threshold and resource allocation in infrastructurebased multiband CRN. The optimization problem has been formulated to maximize the total downlink capacity of SUs, considering three sets of variables, i.e., detection threshold, SAPA, with constraints on the PUs’ rate loss and the power budget of the CRBS. Two schemes, referred to as offine and online algorithms respectively, are proposed to solve the optimization problem. The offline algorithm is able to achieve global optimization however with prohibitively high computation complexity. The online algorithm, on the other hand, is able to achieve closetooptimal performance with realtime operations. Lastly, we have shown through extensive simulations that by jointly optimizing the detection threshold together with the SAPA strategies, the downlink capacity of CRN can be improved significantly, outperforming the traditional static detection threshold based algorithms.
Appendix 1: proof of proposition 1
With $\mathrm{\Delta R}=\eta {R}_{\text{pu},n}^{\text{max}}$, the C2 of (P2) can be transformed into $({P}_{n}^{\text{md}}({\lambda}_{n})\eta ){R}_{\text{pu},n}^{\text{max}}\le {P}_{n}^{\text{md}}({\lambda}_{n})\underset{2}{\text{log}}\phantom{\rule{1em}{0ex}}\left(1+\frac{{p}^{\text{pu}}{h}_{n}^{\text{pu}}{}^{2}}{{p}_{k,n}{h}_{k,n}^{\text{cp}}{}^{2}+{\delta}^{2}}\right)$. If $\eta <{P}_{n}^{\text{md}}({\lambda}_{n})$, after straightforward mathematical manipulation, the C2 of (P2) is shown to be equivalent to $\sum _{k=1}^{K}{\rho}_{k,n}{p}_{k,n}<{P}_{n}^{\text{max}},\forall n$; If $\eta \ge {P}_{n}^{\text{md}}({\lambda}_{n}^{\text{opt}})$, we have $({P}_{n}^{\text{md}}({\lambda}_{n})\eta ){R}_{\text{pu},n}^{\text{max}}\le 0\le {P}_{n}^{\text{md}}({\lambda}_{n}){R}_{\text{pu},n}^{\text{md}}$. Thus C2 is equivalent to $\sum _{k=1}^{K}{\rho}_{k,n}{p}_{k,n}\ge 0,\forall n$.
Appendix 2: proof of proposition 2
Appendix 3: proof of lemma 1
which verifies the definition of subgradient and completes the proof.
Endnote
^{a}Rate loss in our context indicates the lost data rate or capacity of PUs due to the coexistence of SUs in the case of missed detection.
Declarations
Acknowledgements
This work is supported by the EU FP7 S2EuNet project (247083), the National Nature Science Foundation of China (NSF61121001), Program for New Century Excellent Talents in University (NCET) and the Spanish Ministry of Education and Science under project (TIN200806739C0402).
Authors’ Affiliations
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