Throughput Optimization via Cooperative Spectrum Sensing with Novel Frame Structure

Abstract

In cognitive radio (CR) networks, cooperation can greatly improve the performance of spectrum sensing. In this paper, we propose a novel cooperative spectrum sensing (CSS) frame structure in which CR users conduct spectrum sensing and data transmission concurrently over two different parts of the primary user (PU) spectrum band. Energy detection sensing scheme is used to prove that there exists an optimal sensing bandwidth which yields the highest throughput for the CR network. Thus, we focus on the optimal sensing settings of the proposed sensing scheme in order to maximize the throughput of the CR network under the conditions of sufficient protection to PUs and required bandwidth for potential CR user data transmission. Some algorithms are also derived to jointly optimize the sensing bandwidth and the final decision threshold. Our simulation results show that optimizing the sensing bandwidth and the final decision threshold together will further increase the throughput of the CR network as compared to that which only optimizes the sensing bandwidth or the final decision threshold.

1. Introduction

Cognitive radio is a promising technology to improve the efficiency of spectrum usage [1]. The CR users should obey the two following etiquettes [2]: (1) they are allowed to use some unoccupied spectrum bands; (2) they must vacate the spectrum bands quickly whenever the PUs return. In order to avoid interference with PUs, a CR user needs to efficiently and effectively detect the presence of the PUs. CR users can use one of several common detection methods [3], such as matched filter, feature detection, and energy detection. Energy detection is the most popular method addressed in the literature [4–8]. Measuring only the received signal power, energy detection has much lower complexity than the other two schemes. Therefore, we consider energy detection for spectrum sensing in this paper.

The detection quality of spectrum sensing easily suffers from the fading and shadowing environment, which can cause hidden terminal problem. To combat these impacts, cooperative spectrum sensing has been introduced to obtain the space diversity in multiuser CR networks [9–12]. Game theory is suitable for analyzing conflict and cooperation among rational decision makers [13–16]. In [17], the authors focused on dynamical effects of coevolutionary rules on the evolution of cooperation. Game theory has been widely applied to study distributed optimization problems, such as power control, dynamic spectrum access and management [18, 19], and so on. In [20], the authors proposed an evolutionary game framework to answer the question of “how to collaborate” in multiuser decentralized cooperative spectrum sensing. The cooperative spectrum sensing involves sensing, reporting, and decision making steps [21]. In the sensing step, every CR user performs spectrum sensing independently using energy detection method and makes a local decision. In the reporting step, all the local sensing observations are reported to a fusion center (FC). The decision step is made to indicate the absence or the presence of the PU.

In the frame structure of periodic spectrum sensing (PSS), the CR user senses the status of the radio spectrum in the sensing slot and transmits data using the remaining frame duration [22]. Since the CR user must interrupt data transmission during the sensing slot, the CR user transmission delay will be long. Thus, in the case of delay sensitive applications, the QoS (quality of service) will not be guaranteed. In our previous work [23], we propose a novel cooperative spectrum sensing frame structure in which CR users conduct spectrum sensing and data transmission concurrently over two different parts of the primary user spectrum band. In this way, the CR users do not need to interrupt data transmission in the sensing stage, and the QoS can be guaranteed. The optimal multiminislot sensing scheme and the optimal fusion scheme that minimize the CR user transmission delay were analyzed in [23]. However, the performance of throughput with the novel CSS frame structure was not investigated. In this paper, we focus on analyzing the throughput of the CR network.

In the novel frame structure designed for CSS, one part of the PU transmission bandwidth is assigned exclusively to spectrum sensing and sensing results reporting, and the other part is assigned exclusively to potential CR user data transmission. According to this frame structure, under the condition of sufficient protection to PUs, an increase in the sensing bandwidth results in a lower false alarm probability, which leads to higher throughput of the CR network. However, the increase of the sensing bandwidth results in a decrease of the bandwidth assigned to potential CR user data transmission and hence the throughput of the CR network. Therefore, there could exist the optimal sensing bandwidth that maximizes the throughput of the CR network.

In this paper, we study the tradeoff problem for cooperative spectrum sensing with novel frame structure. We are interested in the problem of designing the sensing bandwidth to maximize the throughput of the CR network under the conditions of sufficient protection to PUs and required bandwidth for potential CR user data transmission. When energy detection is utilized for spectrum sensing, we prove that there indeed exists one optimal sensing bandwidth which yields the highest throughput for the CR network.

In the cooperative spectrum sensing with novel frame structure, we employ the counting rule as the fusion rule at the FC since it requires the least communication overhead and is easy to implement. Since CR is originally designed to improve the spectrum efficiency, maximizing the CR users’ throughput is one of the most practical interests. Our object is to find the optimal sensing settings to maximize the throughput of the CR network under the conditions of sufficient protection to PUs and required bandwidth for potential CR user data transmission. To achieve that, we propose an efficient search algorithm that jointly optimizes the sensing bandwidth and the final decision threshold. Computer simulations show that optimizing the sensing bandwidth and the final decision threshold together will further increase the throughput of the CR network as compared to that which only optimizes the sensing bandwidth or the final decision threshold.

The rest of this paper is organized as follows. The cooperative spectrum sensing with novel frame structure is analyzed in Section 2. The optimization problem of maximizing the throughput with sensing bandwidth and final decision threshold as the optimization variables is formulated in Section 3. Optimal sensing settings of throughput are analyzed in detail in Section 4. Simulation results are presented in Section 5, followed by concluding remarks in Section 6.

2. Cooperative Spectrum Sensing with Novel Frame Structure

We consider a CR network, with a number of 𝐾 CR users and a fusion center. The size of the secondary network is small compared with the distance between the primary network and the secondary network to ensure the QoS of primary link. Then the path loss of each CR user is almost identical and the primary signals received at the CR users are considered to be independent and identically distributed (i.i.d.) [24]. In cooperative spectrum sensing, local CR users individually sense the channels and then send information to the fusion center, and the fusion center will make the final decision.

Figure 1 illustrates the novel frame structure designed for cooperative spectrum sensing. In the frequency domain, considering the case that the CR users know the PU transmission bandwidth 𝑊, 𝑊 is divided into two parts, namely, 𝑊_𝑠 and 𝑊_𝑡 (𝑊_𝑡=𝑊−𝑊_𝑠). 𝑊_𝑠 is assigned exclusively to spectrum sensing and sensing results reporting, which means that the status of PU can be decided by sensing a portion of PU bandwidth, and we do not need to sense the whole PU bandwidth. The other part 𝑊_𝑡 is assigned exclusively to potential CR user data transmission. The CR users conduct spectrum sensing and data transmission concurrently over two different parts of the primary user spectrum band.

Figure 1 

Novel frame structure for cooperative spectrum sensing.

In the time domain, it is assumed that the frame duration is 𝑇 and the individual reporting duration is 𝑇_𝑟. Since each CR user continues the spectrum sensing after sending its sensing result to the fusion center, the sensing duration for each CR user is 𝑇_𝑠=𝑇−𝑇_𝑟.

The essence of spectrum sensing is a binary hypothesis-testing problem [25]: ℋ₀ℋ:thePUisabsent;₁:thePUispresent. (1)

The received signal 𝑟_𝑖(𝑘) at the 𝑖th CR user can be formulated as 𝑟_𝑖(𝑘)=𝜀_𝑖(𝑘),ℋ₀,ℎ_𝑝𝑠(𝑘)+𝜀_𝑖(𝑘),ℋ₁, (2) where 𝑟_𝑖(𝑘) is the received signal at the 𝑖th CR user, 𝜀_𝑖(𝑘) is the noise and 𝜀_𝑖(𝑘)∼𝒞𝒩(0,𝜎²), 𝑠(𝑘) is the signal of PU and 𝑠(𝑘)∼𝒞𝒩(0,𝜎²_𝑠), and ℎ_𝑝 is the channel gain between PU and the 𝑖th CR user. The power spectrum density (PSD) of the noise and the PSD of the primary signal are evenly distributed, with values of 𝑆_𝑛 and 𝑆_𝑝, respectively. The SNR (signal-to-noise ratio) of PU’s signal at the 𝑖th CR user is 𝛾_𝑖=|ℎ_𝑝|²𝜎²_𝑠/𝜎².

Since the primary signals received at the CR users are considered to be i.i.d., we can omit the subscript “𝑖.” Suppose the sampling frequency is 2𝑊_𝑠 and the decision statistic of energy detection at each CR user is given by ℛ=(1/𝜎²)∑^{2𝑇𝑠𝑊𝑠}_𝑘=1|𝑟(𝑘)|². It is shown in [26] that when 2𝑇_𝑠𝑊_𝑠 is large enough, according to central limit theorem, we have 𝒩ℛ∼2𝑇_𝑠𝑊_𝑠,2𝑇_𝑠𝑊_𝑠,ℋ₀,𝒩2𝑇_𝑠𝑊_𝑠1+𝛾,2𝑇_𝑠𝑊_𝑠1+𝛾²,ℋ₁. (3) The probability density function (PDF) of ℛ can then be written as 𝑓_ℛ(𝑟)=12𝜋𝑇_𝑠𝑊_𝑠𝑒^{−(𝑟−2𝑇𝑠𝑊𝑠)2/4𝑇𝑠𝑊𝑠},ℋ₀.𝑓_ℛ(𝑟)=121+𝛾𝜋𝑇_𝑠𝑊_𝑠𝑒^{−[𝑟−2𝑇𝑠𝑊𝑠(1+𝛾)]2/4𝑇𝑠𝑊𝑠(1+𝛾)2},ℋ₁. (4)

For a nonfading environment, the probability of false alarm and the probability of detection at each CR user can be computed by 𝑝_𝑓=𝑃ℛ>𝜆∣ℋ₀=^∞_𝜆𝑓_ℛ∣ℋ0(𝑟)𝜆𝑑𝑟=𝒬2𝑇_𝑠𝑊_𝑠−2𝑇_𝑠𝑊_𝑠,𝑝_𝑑=𝑃ℛ>𝜆∣ℋ₁=^∞_𝜆𝑓_ℛ∣ℋ1(𝑟)𝜆𝑑𝑟=𝒬1+𝛾2𝑇_𝑠𝑊_𝑠−2𝑇_𝑠𝑊_𝑠, (5) where 𝜆 denotes the threshold of the energy detection and 𝒬(⋅) is the 𝒬-function defined as 𝒬(𝑥)=(1/√2𝜋)∫^∞_𝑥𝑒^−𝑡2/2𝑑𝑡.

By (5), we have 𝑝_𝑓=𝒬1+𝛾⋅𝒬⁻¹𝑝_𝑑+𝛾2𝑇_𝑠𝑊_𝑠. (6)

In CSS, each CR user makes a “one bit” local decision (1 standing for the presence of PU and 0 standing for the absence of PU) on the primary user activity and then sends the individual decision to the fusion center over a reporting channel. Let Λ denote the number of CR users reporting presence of PU. In the FC, the final decision strategy Φ(⋅) is given by [27] ℋΦ=⁰ℋ,ifΛ<𝑛,₁,ifΛ≥𝑛, (7) where 𝑛 is an integer, and 𝑛=1,2,…, 𝐾 is the final decision threshold at FC. It is seen that OR rule corresponds to the case of 𝑛=1, AND rule corresponds to the case of 𝑛=𝐾, and Majority rule corresponds to the case of 𝑛=⌈𝐾/2⌉. The final false alarm probability and final detection probability can be calculated as [27] 𝑄_𝑓=^𝐾_𝑖=𝑛𝐾𝑖𝑝^𝑖_𝑓1−𝑝_𝑓^𝐾−𝑖𝑄_𝑑,=^𝐾_𝑖=𝑛𝐾𝑖𝑝^𝑖_𝑑1−𝑝_𝑑^𝐾−𝑖. (8)(9) Lets define ℱ_𝑛(𝑥)=∑^𝐾_𝑖=𝑛^𝐾_𝑖𝑥^𝑖(1−𝑥)^𝐾−𝑖 and 𝒢_𝑛(𝑥)=𝑑ℱ_𝑛(𝑥)/𝑑𝑥=𝐾^𝐾−1_𝑛−1𝑥^𝑛−1(1−𝑥)^𝐾−𝑛. Then, we have 𝑄_𝑓=ℱ_𝑛(𝑝_𝑓), 𝑄_𝑑=ℱ_𝑛(𝑝_𝑑), 𝑑𝑄_𝑓/𝑑𝑝_𝑓=𝒢_𝑛(𝑝_𝑓), and 𝑑𝑄_𝑑/𝑑𝑝_𝑑=𝒢_𝑛(𝑝_𝑑).

3. Optimization Problem Formulation

In our proposed novel cooperative spectrum sensing frame structure, the CR user transmits data over the bandwidth 𝑊_𝑡 only when the PU is sensed to be absent. Suppose that the power spectrum density of the CR user signal is evenly distributed, with the value of 𝑆_𝑠. Then, 𝜎²_𝑠=𝑆_𝑠𝑊_𝑡. There are two scenarios for the CR users to transmit data.

(1) The PU Is Correctly Detected to Be Absent. The probability of this scenario happening is 𝑝(ℋ₀)(1−𝑄_𝑓), where 𝑝(ℋ₀) denotes the prior probability of the absence of the PU. In this case, the achievable throughput of the CR network is 𝕋₀=𝑊_𝑡log₂1+𝛾_𝑠𝑝ℋ₀1−𝑄_𝑓, (10) where 𝛾_𝑠 is the SNR for the secondary link, and 𝛾_𝑠=ℎ_𝑠²𝑆_𝑠𝑊_𝑡𝑆_𝑛𝑊_𝑡, (11) where ℎ_𝑠 is the channel gain of the secondary link.

(2) The PU Is Falsely Detected to Be Absent. The probability of this scenario happening is 𝑝(ℋ₁)(1−𝑄_𝑑), where 𝑝(ℋ₁) denotes the prior probability of the presence of the PU. In this case, the primary signal is considered as an interference to the secondary receiver, and the achievable throughput of the CR network is 𝕋₁=𝑊_𝑡log₂1+𝛾_𝑆𝐼𝑝ℋ₁1−𝑄_𝑑, (12) where 𝛾_𝑆𝐼 is the signal-to-noise and interference ratio for the secondary link, and 𝛾_𝑆𝐼=ℎ_𝑠²𝑆_𝑠𝑊_𝑡𝑆_𝑝ℎ_𝑝²+𝑆_𝑛𝑊_𝑡=𝛾_𝑠,1+𝛾 (13) where ℎ_𝑝 is the channel gain between the PU and the CR user.

Then, the achievable throughput of the CR network can be computed by 𝕋=𝕋₀+𝕋₁=𝑊_𝑡log₂1+𝛾_𝑠𝑝ℋ₀1−𝑄_𝑓+𝑊_𝑡log₂𝛾1+^𝑠𝑝ℋ1+𝛾¹1−𝑄_𝑑. (14)

Considering the fact that the priority of a CR system is the protection of the QoS of the primary link, a high probability of detection is required to ensure that no harmful interference is caused by the CR network. Our object is to find the optimal sensing settings to maximize the CR users’ throughput under the condition of sufficient protection to PUs. To satisfy the required bandwidth for potential CR user data transmission, we set 𝑊^th_𝑡≤𝑊_𝑡<𝑊; namely, 0<𝑊_𝑠≤𝑊−𝑊^th_𝑡. Mathematically, the optimization problem can be stated as max𝕋s.t.𝑄_𝑑≥𝑄^th_𝑑,0<𝑊_𝑠≤𝑊−𝑊^th_𝑡,1≤𝑛≤𝐾, (15)(16) where 𝑄^th_𝑑 is the target detection probability with which the PUs are defined as being sufficiently protected. In the second scenario, the primary signal is considered as an interference with the secondary receiver, and we have log₂(1+𝛾_𝑠)>log₂(1+𝛾_𝑆𝐼). Suppose that the prior probability of the presence of the PU is small; say less than 0.3; thus it is economically advisable to explore the secondary usage for PU spectrum band. Since 𝒬(𝑥) is a decreasing function of 𝑥, according to (5), we have 𝑝_𝑓<𝑝_𝑑. Furthermore, since 𝒢_𝑛(𝑥)≥0 for 0≤𝑥≤1, ℱ_𝑛(𝑥) is an increasing function of 𝑥 for 0≤𝑥≤1; thus ℱ_𝑛(𝑝_𝑓)<ℱ_𝑛(𝑝_𝑑) and 𝑄_𝑓<𝑄_𝑑. Therefore, the optimization problem can be approximated by maximizing 𝕋₀ subject to (16).

Theorem 1. 𝕋₀ is a decreasing function of 𝑄_𝑑.

Proof. For a given 𝑊_𝑠 and 𝑛, we have 𝑑𝕋₀𝑑𝑄_𝑑=−𝑝ℋ₀𝑊−𝑊_𝑠log₂1+𝛾_𝑠𝑑𝑄_𝑓𝑑𝑄_𝑑, (17) where 𝑑𝑄_𝑓𝑑𝑄_𝑑=𝑑𝑄_𝑓/𝑑𝑝_𝑑𝑑𝑄_𝑑/𝑑𝑝_𝑑=𝒢_𝑛𝑝_𝑓𝒢_𝑛𝑝_𝑑⋅𝑑𝑝_𝑓𝑑𝑝_𝑑,𝑑𝑝_𝑓𝑑𝑝_𝑑=𝑑𝑝_𝑓/(𝑑𝜆)𝑑𝑝_𝑑/()=𝑑𝜆1+𝛾exp𝜆𝛾−𝜆1+𝛾²𝛾𝛾+24𝑇_𝑠𝑊_𝑠1+𝛾²>0. (18) Since 𝒢_𝑛(𝑝_𝑓)>0 and 𝒢_𝑛(𝑝_𝑑)>0, it is derived that (𝑑𝕋₀)/𝑑𝑄_𝑑<0. Thus, 𝕋₀ is a decreasing function of 𝑄_𝑑. Theorem 1 is proved.

Therefore, the optimal solution must occur when the following equation stands: 𝑄_𝑑=^𝐾_𝑖=𝑛𝐾𝑖𝑝^𝑖_𝑑1−𝑝_𝑑^𝐾−𝑖=𝑄^th_𝑑. (19) To get the optimal solution of (15), we need to calculate the root of Ψ(𝑝_𝑑)=𝑄_𝑑−𝑄^th_𝑑=0 numerically. Newton-Raphson algorithm can be employed to find the root of Ψ(𝑝_𝑑)=𝑄_𝑑−𝑄^th_𝑑=0 [28]. The process of the Newton-Raphson algorithm is stated as follows.(1)Choose tolerance 𝛿 and initial guess 𝑝_𝑑,1; let 𝑗=1.(2)If |Ψ(𝑝_𝑑,𝑗)|<𝛿, stop; otherwise, go to step (3).(3)Let 𝑝_𝑑,𝑗+1=𝑝_𝑑,𝑗−Ψ(𝑝_𝑑,𝑗)/𝒢_𝑛(𝑝_𝑑,𝑗); let 𝑗=𝑗+1; go to step (2).

The optimization problem is reduced to max𝕋₀s.t.𝑄_𝑑=𝑄^th_𝑑,0<𝑊_𝑠≤𝑊−𝑊^th_𝑡,1≤𝑛≤𝐾. (20)(21)

In the next section, we will find the optimal solutions of 𝑊_𝑠 and 𝑛 to maximize the throughput of the CR network.

4. Optimal Sensing Settings of Throughput

First we will prove that, for any target detection probability 𝑄^th_𝑑 and the final decision threshold 𝑛, there exists an optimal value of 𝑊_𝑠 that maximizes the throughput of the CR network for the CSS with novel frame structure. The derivative of 𝕋₀ with respect to 𝑊_𝑠 is 𝑑𝕋₀𝑑𝑊_𝑠=−𝑝ℋ₀1−𝑄_𝑓log₂1+𝛾_𝑠−𝑝ℋ₀𝑑𝑄_𝑓𝑑𝑊_𝑠𝑊−𝑊_𝑠log₂1+𝛾_𝑠, (22) where 𝑑𝑄_𝑓𝑑𝑊_𝑠=𝑑𝑄_𝑓𝑑𝑝_𝑓⋅𝑑𝑝_𝑓𝑑𝑊_𝑠=𝒢_𝑛𝑝_𝑓𝑑𝑝_𝑓𝑑𝑊_𝑠,𝑑𝑝_𝑓𝑑𝑊_𝑠𝛾=−2𝑇_𝑠𝜋𝑊_𝑠−1exp21+𝛾𝒬⁻¹𝑝_𝑑+𝛾2𝑇_𝑠𝑊_𝑠². (23) Obviously, lim_𝑊𝑠→0𝑑𝕋₀𝑑𝑊_𝑠=∞,lim_{𝑊𝑠→𝑊}𝑑𝕋₀𝑑𝑊_𝑠=−𝑝ℋ₀1−𝑄_𝑓log₂1+𝛾_𝑠<0. (24)

Equation (24) means that 𝕋₀ increases when 𝑊_𝑠 is small and decreases when 𝑊_𝑠 approaches 𝑊. Hence, there must be a maximum point of 𝕋₀ for 𝑊_𝑠∈(0,𝑊). Next we will prove that the maximum point of 𝕋₀ is unique in this range; that is, there is a unique 𝑊^∗_𝑠 where 𝑊^∗_𝑠∈(0,𝑊) such that 𝑑𝕋₀/𝑑𝑊_𝑠|_{𝑊𝑠=𝑊∗𝑠}=0. Setting 𝑑𝕋₀/𝑑𝑊_𝑠=0 and after some algebraic manipulations, it is derived that 𝒜(𝑊_𝑠)=ℬ(𝑊_𝑠), where 𝒜𝑊_𝑠2()!()!=−2ln𝑛−1𝐾−𝑛𝛾𝐾!𝜋𝑇_𝑠𝑊_𝑠𝑊−𝑊_𝑠×^𝑛−1_𝑖=0𝐾𝑖𝑝_𝑓^{𝑖−𝑛+1}1−𝑝_𝑓^𝑛−𝑖,ℬ𝑊_𝑠=1+𝛾𝒬⁻¹𝑝_𝑑+𝛾2𝑇_𝑠𝑊_𝑠². (25)

If functions 𝒜(𝑊_𝑠) and ℬ(𝑊_𝑠) intersect each other only once for 𝑊_𝑠∈(0,𝑊), we can conclude that the root of 𝑑𝕋₀/𝑑𝑊_𝑠=0 is unique. The derivative of 𝒜(𝑊_𝑠) with respect to 𝑊_𝑠 is 𝑑𝒜𝑊_𝑠𝑑𝑊_𝑠=−𝑊+𝑊_𝑠𝑊_𝑠𝑊−𝑊_𝑠+2∑^𝑛−1_𝑖=0^𝐾_𝑖𝑝_𝑓^{𝑖−𝑛+1}1−𝑝_𝑓^𝑛−𝑖×^𝑛−1_𝑖=0𝐾𝑖(𝑛−1−𝑖)𝑝_𝑓^𝑖−𝑛1−𝑝_𝑓^𝑛−𝑖+(𝑛−𝑖)𝑝_𝑓^{𝑖−𝑛+1}1−𝑝_𝑓^{𝑛−𝑖−1}𝑑𝑝_𝑓𝑑𝑊_𝑠. (26)

Since 𝑛−1−𝑖≥0 and 𝑛−𝑖>0 for 𝑖=0,1,…,𝑛−1, and 𝑑𝑝_𝑓/𝑑𝑊_𝑠<0, we can obtain that 𝑑𝒜(𝑊_𝑠)/𝑑𝑊_𝑠<0 for 𝑊_𝑠∈(0,𝑊). Thus, 𝒜(𝑊_𝑠) is a decreasing function of 𝑊_𝑠 for 𝑊_𝑠∈(0,𝑊). The derivative of ℬ(𝑊_𝑠) with respect to 𝑊_𝑠 is 𝑑ℬ𝑊_𝑠𝑑𝑊_𝑠=𝛾2𝑇_𝑠𝑊_𝑠1+𝛾𝒬⁻¹𝑝_𝑑+𝛾2𝑇_𝑠𝑊_𝑠. (27)

Case 1. If 𝒬⁻¹(𝑝_𝑑)≥0, we have (1+𝛾)𝒬⁻¹(𝑝_𝑑)+𝛾2𝑇_𝑠𝑊_𝑠>0 for 𝑊_𝑠∈(0,𝑊). According to (27), we can obtain that 𝑑ℬ(𝑊_𝑠)/𝑑𝑊_𝑠>0. In this case, ℬ(𝑊_𝑠) is an increasing function of 𝑊_𝑠 for 𝑊_𝑠∈(0,𝑊). Thus, 𝒜(𝑊_𝑠) and ℬ(𝑊_𝑠) can intersect at most once. Therefore, there is only one intersection between 𝒜(𝑊_𝑠) and ℬ(𝑊_𝑠) for 𝑊_𝑠∈(0,𝑊). The root of 𝑑𝕋₀/𝑑𝑊_𝑠=0 is unique in this case.

Case 2. If 𝒬⁻¹(𝑝_𝑑)<0, when 0<𝑊_𝑠<[(1+𝛾)𝒬⁻¹(𝑝_𝑑)]²/2𝑇_𝑠𝛾², we have (1+𝛾)𝒬⁻¹(𝑝_𝑑)+𝛾2𝑇_𝑠𝑊_𝑠<0 and 𝑑ℬ(𝑊_𝑠)/𝑑𝑊_𝑠<0. Thus, ℬ(𝑊_𝑠) is a decreasing function of 𝑊_𝑠 for 𝑊_𝑠∈(0,[(1+𝛾)𝒬⁻¹(𝑝_𝑑)]²/2𝑇_𝑠𝛾²). When [(1+𝛾)𝒬⁻¹(𝑝_𝑑)]²/2𝑇_𝑠𝛾²≤𝑊_𝑠<𝑊, we have (1+𝛾)𝒬⁻¹(𝑝_𝑑)+𝛾2𝑇_𝑠𝑊_𝑠≥0 and 𝑑ℬ(𝑊_𝑠)/𝑑𝑊_𝑠≥0. Thus, ℬ(𝑊_𝑠) is an increasing function of 𝑊_𝑠 for 𝑊_𝑠∈[[(1+𝛾)𝒬⁻¹(𝑝_𝑑)]²/2𝑇_𝑠𝛾²,𝑊). In this case, we will prove that 𝒜(𝑊_𝑠) and ℬ(𝑊_𝑠) can intersect each other only once for 𝑊_𝑠∈(0,𝑊).

Theorem 2. 𝑑𝒜(𝑊_𝑠)/𝑑𝑊_𝑠<𝑑ℬ(𝑊_𝑠)/𝑑𝑊_𝑠 for 𝑊_𝑠∈(0,𝑊).

Proof. For 𝑊_𝑠∈[[(1+𝛾)𝒬⁻¹(𝑝_𝑑)]²/2𝑇_𝑠𝛾²,𝑊), we have 𝑑𝒜(𝑊_𝑠)/𝑑𝑊_𝑠<0 and 𝑑ℬ(𝑊_𝑠)/𝑑𝑊_𝑠≥0. Thus, 𝑑𝒜(𝑊_𝑠)/𝑑𝑊_𝑠<𝑑ℬ(𝑊_𝑠)/𝑑𝑊_𝑠.
For 𝑊_𝑠∈(0,[(1+𝛾)𝒬⁻¹(𝑝_𝑑)]²/2𝑇_𝑠𝛾²), we have 𝑑𝒜(𝑊_𝑠)/𝑑𝑊_𝑠<0 and 𝑑ℬ(𝑊_𝑠)/𝑑𝑊_𝑠<0. We first prove that, when 𝑛=1, 𝑑𝒜(𝑊_𝑠)/𝑑𝑊_𝑠<𝑑ℬ(𝑊_𝑠)/𝑑𝑊_𝑠. According to (26), when 𝑛=1, we have 𝑑𝒜𝑊_𝑠𝑑𝑊_𝑠𝑛=1=−𝑊+𝑊_𝑠𝑊_𝑠𝑊−𝑊_𝑠+21−𝑝_𝑓⋅𝑑𝑝_𝑓𝑑𝑊_𝑠. (28) According to (27) and (28), 𝑑𝒜(𝑊_𝑠)/𝑑𝑊_𝑠|_𝑛=1<𝑑ℬ(𝑊_𝑠)/𝑑𝑊_𝑠 is given as −𝑊+𝑊_𝑠𝑊_𝑠𝑊−𝑊_𝑠−𝛾1−𝒬1+𝛾⋅𝒬⁻¹𝑝_𝑑+𝛾2𝑇_𝑠𝑊_𝑠𝑇_𝑠𝜋𝑊_𝑠−1×exp21+𝛾𝒬⁻¹𝑝_𝑑+𝛾2𝑇_𝑠𝑊_𝑠²<𝛾2𝑇_𝑠𝑊_𝑠1+𝛾𝒬⁻¹𝑝_𝑑+𝛾2𝑇_𝑠𝑊_𝑠. (29)
For 𝑊_𝑠∈(0,[(1+𝛾)𝒬⁻¹(𝑝_𝑑)]²/2𝑇_𝑠𝛾²), (1+𝛾)𝒬⁻¹(𝑝_𝑑)+𝛾2𝑇_𝑠𝑊_𝑠<0. Let 𝑥=−[(1+𝛾)𝒬⁻¹(𝑝_𝑑)+𝛾2𝑇_𝑠𝑊_𝑠]>0. Substituting 𝑥 into (29), it is derived that 𝑊+𝑊_𝑠𝛾𝑊−𝑊_𝑠2𝑇_𝑠𝑊_𝑠+1√(𝑥)𝑒2𝜋𝒬^−𝑥2/2>𝑥. (30)
Since (1/√2𝜋𝒬(𝑥))𝑒^−𝑥2/2>𝑥 for 𝑥≥0 [29], the inequality at (30) is verified. Thus, 𝑑𝒜(𝑊_𝑠)/𝑑𝑊_𝑠|_𝑛=1<𝑑ℬ(𝑊_𝑠)/𝑑𝑊_𝑠 has been proved. Next we will prove that 𝑑𝒜(𝑊_𝑠)/𝑑𝑊_𝑠<𝑑ℬ(𝑊_𝑠)/𝑑𝑊_𝑠 for 𝑛=2,3,…,𝐾. Since 𝑛−𝑖≥1 for 𝑖=0,1,…,𝑛−1, we have 2∑^𝑛−1_𝑖=0^𝐾_𝑖𝑝_𝑓^{𝑖−𝑛+1}1−𝑝_𝑓^𝑛−𝑖×^𝑛−1_𝑖=0𝐾𝑖(𝑛−1−𝑖)𝑝_𝑓^𝑖−𝑛1−𝑝_𝑓^𝑛−𝑖+(𝑛−𝑖)𝑝_𝑓^{𝑖−𝑛+1}1−𝑝_𝑓^{𝑛−𝑖−1}∑>2^{𝑛−1𝑖=0𝐾𝑖}(𝑛−𝑖)𝑝_𝑓^{𝑖−𝑛+1}1−𝑝_𝑓^{𝑛−𝑖−1}∑^𝑛−1_𝑖=0^𝐾_𝑖𝑝_𝑓^{𝑖−𝑛+1}1−𝑝_𝑓^{𝑛−𝑖−1}⋅11−𝑝_𝑓>21−𝑝_𝑓. (31)
According to (26), (28), and (31), we obtain 𝑑𝒜(𝑊_𝑠)/𝑑𝑊_𝑠|_{𝑛=2,3,…,𝐾}<𝑑𝒜(𝑊_𝑠)/𝑑𝑊_𝑠|_𝑛=1. Therefore, 𝑑𝒜(𝑊_𝑠)/𝑑𝑊_𝑠<𝑑ℬ(𝑊_𝑠)/𝑑𝑊_𝑠 for 𝑊_𝑠∈(0,𝑊). Theorem 2 is proved.

In Case 2, there are two possible scenarios for the intersection between 𝒜(𝑊_𝑠) and ℬ(𝑊_𝑠).(i)𝒜(𝑊_𝑠) and ℬ(𝑊_𝑠) intersect each other in the region of 𝑊_𝑠∈(0,[(1+𝛾)𝒬⁻¹(𝑝_𝑑)]²/2𝑇_𝑠𝛾²). Since 𝑑𝒜(𝑊_𝑠)/𝑑𝑊_𝑠<𝑑ℬ(𝑊_𝑠)/𝑑𝑊_𝑠, it is impossible for them to intersect more than once because 𝒜(𝑊_𝑠) decreases at a faster rate than ℬ(𝑊_𝑠) for 𝑊_𝑠∈(0,[(1+𝛾)𝒬⁻¹(𝑝_𝑑)]²/2𝑇_𝑠𝛾²). For 𝑊_𝑠∈[[(1+𝛾)𝒬⁻¹(𝑝_𝑑)]²/2𝑇_𝑠𝛾²,𝑊), ℬ(𝑊_𝑠) is always larger than 𝒜(𝑊_𝑠) since ℬ(𝑊_𝑠) is an increasing function of 𝑊_𝑠 while 𝒜(𝑊_𝑠) is a decreasing function of 𝑊_𝑠. It is impossible for them to intersect each other for 𝑊_𝑠∈[[(1+𝛾)𝒬⁻¹(𝑝_𝑑)]²/2𝑇_𝑠𝛾²,𝑊). Therefore, in this scenario, there is only one intersection between 𝒜(𝑊_𝑠) and ℬ(𝑊_𝑠), and it occurs in the region of 𝑊_𝑠∈(0,[(1+𝛾)𝒬⁻¹(𝑝_𝑑)]²/2𝑇_𝑠𝛾²).(ii)𝒜(𝑊_𝑠) and ℬ(𝑊_𝑠) do not intersect each other in the region of 𝑊_𝑠∈(0,[(1+𝛾)𝒬⁻¹(𝑝_𝑑)]²/2𝑇_𝑠𝛾²). In this scenario, they must intersect in the region of 𝑊_𝑠∈[[(1+𝛾)𝒬⁻¹(𝑝_𝑑)]²/2𝑇_𝑠𝛾²,𝑊) since there must be at least one intersection in the entire range of 0<𝑊_𝑠<𝑊. For 𝑊_𝑠∈[[(1+𝛾)𝒬⁻¹(𝑝_𝑑)]²/2𝑇_𝑠𝛾²,𝑊), 𝒜(𝑊_𝑠) is a decreasing function of 𝑊_𝑠 and ℬ(𝑊_𝑠) is an increasing function of 𝑊_𝑠. Thus, they can intersect each other at most once. Therefore, in this scenario, there is only one intersection between 𝒜(𝑊_𝑠) and ℬ(𝑊_𝑠), and it occurs in the region of 𝑊_𝑠∈[[(1+𝛾)𝒬⁻¹(𝑝_𝑑)]²/2𝑇_𝑠𝛾²,𝑊).

From the analysis above, one can conclude that there is a unique 𝑊^∗_𝑠 where 𝑊^∗_𝑠∈(0,𝑊) such that 𝑑𝕋₀/𝑑𝑊_𝑠|_{𝑊𝑠=𝑊∗𝑠}=0. Bisection method can be used to find the root of 𝑑𝕋₀/𝑑𝑊_𝑠=0 [28]. The process of bisection method is stated as follows.(1)Choose lower guess 𝑊^𝑙_𝑠 and upper guess 𝑊^𝑢_𝑠 for the root such that the function changes sign over the interval. This can be checked by ensuring that 𝑑𝕋₀/𝑑𝑊_𝑠|_{𝑊𝑠=𝑊𝑙𝑠}⋅𝑑𝕋₀/𝑑𝑊_𝑠|_{𝑊𝑠=𝑊𝑢𝑠}<0.(2)An estimate of the root 𝑊^𝑟_𝑠 is determined by 𝑊^𝑟_𝑠=(𝑊^𝑙_𝑠+𝑊^𝑢_𝑠)/2.(3)Make the following evaluations to determine in which subinterval the root lies.(a)If 𝑑𝕋₀/𝑑𝑊_𝑠|_{𝑊𝑠=𝑊𝑙𝑠}⋅𝑑𝕋₀/𝑑𝑊_𝑠|_{𝑊𝑠=𝑊𝑟𝑠}<0, the root lies in the lower subinterval. Therefore, set 𝑊^𝑢_𝑠=𝑊^𝑟_𝑠 and return to step (2).(b)If 𝑑𝕋₀/𝑑𝑊_𝑠|_{𝑊𝑠=𝑊𝑙𝑠}⋅𝑑𝕋₀/𝑑𝑊_𝑠|_{𝑊𝑠=𝑊𝑟𝑠}>0, the root lies in the upper subinterval. Therefore, set 𝑊^𝑙_𝑠=𝑊^𝑟_𝑠 and return to step (2).(c)If 𝑑𝕋₀/𝑑𝑊_𝑠|_{𝑊𝑠=𝑊𝑙𝑠}⋅𝑑𝕋₀/𝑑𝑊_𝑠|_{𝑊𝑠=𝑊𝑟𝑠}=0, the root equals 𝑊^𝑟_𝑠; terminate the computation.

Note that the second constraint of the optimization problem is 0<𝑊_𝑠≤𝑊−𝑊^th_𝑡. If 𝑊^∗_𝑠≥𝑊−𝑊^th_𝑡, 𝕋₀ is monotonically increasing in the range of 0<𝑊_𝑠≤𝑊−𝑊^th_𝑡. In this case, choose 𝑊_𝑠=𝑊−𝑊^th_𝑡; the optimization is achieved. If 0<𝑊^∗_𝑠<𝑊−𝑊^th_𝑡, 𝕋₀ is monotonically increasing in the range of 0<𝑊_𝑠≤𝑊^∗_𝑠 and is monotonically decreasing in the range of 𝑊^∗_𝑠<𝑊_𝑠<𝑊−𝑊^th_𝑡. In this case, choose 𝑊_𝑠=𝑊^∗_𝑠; the optimization is achieved.

For the third constraint of the optimization problem, 1≤𝑛≤𝐾, no closed-form solution for the optimal 𝑛^∗ is available. However, since 𝑛 is an integer, it is not computationally expensive to search through 𝑛 from 1 to 𝐾 to obtain the optimal 𝑛^∗ that maximizes (20). Therefore, the process to achieve the maximum throughput for the CSS with novel frame structure is listed as follows.(1)For each 𝑛(1≤𝑛≤𝐾), calculate the root 𝑝_{𝑑_𝑛} of Ψ(𝑝_{𝑑_𝑛})=0 by using the Newton-Raphson algorithm.(2)For each 𝑝_{𝑑_𝑛}, calculate the root 𝑊^∗_{𝑠_𝑛} of 𝑑𝕋₀/𝑑𝑊_𝑠=0 by using the bisection method. If 𝑊^∗_{𝑠_𝑛}≥𝑊−𝑊^th_𝑡 is satisfied, choose 𝑊^opt_{𝑠_𝑛}=𝑊−𝑊^th_𝑡; otherwise, choose 𝑊^opt_{𝑠_𝑛}=𝑊^∗_{𝑠_𝑛}.(3)For each 𝑊^opt_{𝑠_𝑛}, calculate the corresponding individual false alarm probability 𝑝_{𝑓_𝑛} by (6), the final false alarm probability 𝑄_{𝑓_𝑛} by (8), and the throughput 𝕋_{0_𝑛} by (10).(4)Compare 𝕋_{0_𝑛} and choose the maximum one.

5. Simulation Results

To get insight into the effectiveness of the proposed sensing methods and validate some related theorems, computer simulations have been conducted to evaluate the performance of throughput for various spectrum sensing schemes. In the simulations, we set the bandwidth of PU as 𝑊=20000 Hz; the frame duration is 𝑇=20 ms; the individual reporting duration is 𝑇_𝑟=1 ms; the target detection probability is 𝑄^th_𝑑=99%; the SNR for the secondary link is 𝛾_𝑠=20 dB; the prior probability of the absence of PU is 𝑝(ℋ₀)=0.8 and 𝑝(ℋ₁)=1−𝑝(ℋ₀)=0.2. To satisfy the required bandwidth for potential CR user data transmission, we set 𝑊^th_𝑡=𝑊/5.

Figure 2 compares the throughput of the CR network when the novel CSS frame structure is employed to the case when the previous frame structure is employed. For the previous frame structure, we can refer to Figure 1 in [25]; for the novel CSS scheme, the number of CR users is 𝐾=9; the SNR of the PU’s signal at the CR user is 𝛾=−10 dB. It is seen that using the novel CSS frame structure can achieve a much higher throughput than that using the previous frame structure, especially when the final decision threshold 𝑛 is optimized. For the previous frame structure, the maximum achievable throughput is approximately 1.55×10⁴ bits/s. However, when our proposed novel CSS frame structure is employed and 𝑛 is optimized, the maximum achievable throughput is approximately 6.64×10⁴ bits/s. From this figure, it can be seen that, for a given sensing bandwidth, the optimal sensing settings improve the throughput of the CR network. When 𝑊_𝑠≥1.6 kHz, Majority rule (𝑛=5) is suboptimal. However, OR rule (𝑛=1) outperforms Majority rule when 0<𝑊_𝑠<1.6 kHz. It is also observed that there exists an optimal sensing bandwidth which yields the highest throughput for the secondary network. The optimal sensing bandwidth varies with different values of 𝑛.

Figure 2 

The throughput of the CR network versus the sensing bandwidth 𝑊𝑠 for various sensing schemes.

In Figure 3, it is also observed that the throughput of the novel CSS scheme is much higher than that of the previous scheme. For the novel CSS scheme, optimal 𝑛 values are used for each fixed 𝑊_𝑠. The optimal sensing settings can achieve a higher throughput than that using fixed sensing bandwidth. In addition, one can clearly see that the disadvantage with a fixed sensing bandwidth is that, at high SNR levels where the PU can be easily detected, the throughput of the CR network is bounded by the percentage of the total PU transmission bandwidth that is assigned to spectrum sensing. For the novel CSS scheme, by comparing the curves of 𝑊_𝑠=𝑊/5 and 𝑊_𝑠=𝑊/4, we can see that, in low SNR region, −20∼−9 dB, the throughput of the former scheme (𝑊_𝑠=𝑊/5) is lower than that of the latter scheme (𝑊_𝑠=𝑊/4). However, when SNR is larger than −9 dB, the two curves approach constants and the throughput of the former scheme is larger than that of the latter scheme. This indicates that no fixed 𝑊_𝑠 is optimal for all SNR values. Thus, there is a need to optimize the value of 𝑊_𝑠 to enhance the throughput of the CR network.

Figure 3 

The throughput of the CR network versus SNR for various values of 𝑊𝑠; optimal 𝑛 values are used for each fixed 𝑊𝑠 in the novel CSS scheme.

Figure 4 illustrates the maximum throughput of the CR network versus the average SNR for the novel CSS scheme with various counting rules. Optimal sensing bandwidth is used in each of the counting rules. One can see clearly that the optimal sensing settings can achieve a higher throughput than that using uniform thresholds. It is seen that, when −15 dB≤SNR≤0 dB, Majority rule (𝑛=5) is suboptimal and AND rule (𝑛=9) performs the worst. In particular, at −10 dB SNR, the optimal sensing settings can achieve almost 1.5 times throughput than that when AND rule is used. However, when SNR is lower than −15 dB, AND rule outperforms Majority rule and OR rule (𝑛=1). And it achieves the same throughput as the optimal setting when SNR is extremely weak. Therefore, there is no single decision threshold 𝑛 that is optimal for all cases. To maximize the throughput of the CR network, different SNR levels require different optimal 𝑛 values.

Figure 4 

The maximum throughput versus SNR for various counting rules; optimal sensing bandwidth is used in each of the counting rules.

Figure 5 is simulated to show the maximum throughput of the CR network versus number of CR users 𝐾 for various sensing schemes with different values of 𝑊_𝑠, in which optimal 𝑛 values are used for each fixed 𝑊_𝑠. The SNR of PU’s signal at the CR user is 𝛾=−8 dB. It has been shown that the throughput with optimal sensing settings is larger than that using fixed sensing bandwidth and grows as the number of CR users increases. For the sensing scheme with fixed sensing bandwidth, the throughput of the CR network approaches constants when the number of CR users is extremely large and is bounded by the percentage of the total PU transmission bandwidth that is assigned to spectrum sensing. By comparing the curves of the sensing schemes with different values of 𝑊_𝑠, we can also find that no fixed 𝑊_𝑠 is optimal for all cases. To enhance the throughput of the CR network, the value of 𝑊_𝑠 needs to be optimized.

Figure 5 

The maximum throughput versus the number of CR users for various values of 𝑊𝑠; optimal 𝑛 values are used for each fixed 𝑊𝑠.

In Figure 6, the maximum throughput of the CR network is presented versus the number of CR users 𝐾 for the novel CSS scheme with various counting rules. Optimal sensing bandwidth is used in each of the counting rules. The SNR of PU’s signal at the CR user is 𝛾=−8 dB. It is observed that the optimal sensing settings can achieve a higher throughput than that using uniform thresholds. Majority rule (𝑛=5) is suboptimal and AND rule (𝑛=9) always performs the worst. It is also seen that the throughput of the CR network increases as the number of CR users increases. However, the complexity of our proposed algorithm grows approximately linearly with the number of cooperating CR users.

Figure 6 

The maximum throughput versus the number of CR users for various counting rules; optimal sensing bandwidth is used in each of the counting rules.

6. Conclusions

In this paper, we propose a novel frame structure for cooperative spectrum sensing. It has been proved that there exists an optimal sensing bandwidth which yields the highest throughput for the CR network. The optimal sensing settings to maximize the throughput of the CR network under the conditions of sufficient protection to PUs and required bandwidth for potential CR user data transmission have been proposed. The proposed optimal sensing settings are analyzed and calculated in detail by using some simple but reliable methods. Computer simulations have shown that significant improvement in the throughput of the CR network has been achieved when the sensing bandwidth and the final decision threshold are jointly optimized.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Fundament Research of China (973 no. 2009CB3020400) and the Jiangsu Province Natural Science Foundation under Grant BK2011002.