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Abstract—This paper proposes two methods for maximizing the lifetime of the cognitive sensor networks by focusing on node selection criteria for spectrum ...
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Lifetime Maximization in Cognitive Sensor Networks Based on the Node Selection Maryam Najimi, Ataollah Ebrahimzadeh, Seyed Mehdi Hosseini Andargoli, and Afshin Fallahi

Abstract— This paper proposes two methods for maximizing the lifetime of the cognitive sensor networks by focusing on node selection criteria for spectrum sensing in such networks. We maximize the network lifetime with constraints on the detection performance by determining the sensors, which sense the spectrum. This is a NP-complete problem and cannot be solved by standard methods. Therefore, we relax it to a more tractable form. Based on convex optimization framework, the optimal conditions are obtained and the sensors, which sense the spectrum, are determined. Simulation results show that our proposed algorithm is very efficient in terms of network lifetime maximization. Index Terms— Lifetime, wireless cognitive sensor networks, cooperative spectrum sensing, probabilities of detection and false alarm.

I. I NTRODUCTION

C

OGNITIVE radio (CR) is a radio which can change its transmitter parameters by interaction with its operating environment. This novel technology allows secondary (unlicensed) users to share the spectrum with primary (licensed) users without any harmful impact on primary users’ communications. Spectrum sensing has an important role in cognitive radio networks. With spectrum sensing the status of the activity of the corresponding channel is determined. However, attenuation impairments such as shadowing and fading can lead to unreliable detection of spectrum holes by the secondary user which degrades the primary network’s performance. It is believed that deploying multiple detectors and utilizing this distributed sensing information in a collaborative manner can result in a more accurate detection of primary signals. In cooperative spectrum sensing, different users share their results and decide on the status of the channel [1], [2]. A secondary user may declare a busy channel, called probability of detection, when it is already occupied by a primary user, or confirm the presence of a primary transmission when the spectrum is actually not in use. This is referred to as false alarm. Cognitive radio technology may also be used in wireless sensor networks (WSNs) [3]. Wireless cognitive sensor networks consist of tiny smart nodes with sensing, processing and communication

Manuscript received March 19, 2013; revised August 6, 2013; accepted March 3, 2014. Date of publication March 11, 2014; date of current version May 29, 2014. The associate editor coordinating the review of this paper and approving it for publication was Dr. Nitaigour P. Mahalik. M. Najimi, A. Ebrahimzadeh, and S. M. H. Andargoli are with the Faculty of Electrical and Computer Engineering, Babol University of Technology, Babol 47148-71167, Iran (e-mail: [email protected]; [email protected]; [email protected]). A. Fallahi is with Rightel, Tehran 14174, Iran (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JSEN.2014.2311154

capabilities. The sensor nodes are deployed densely enough to cooperate and obtain information about a specific target, area or environment, depending on the application of interest [4], [5]. Most wireless sensors are battery-backed and are small. These physical constraints of sensors and the prohibitive costs to replace the depleted sensors in the sensor network make energy a crucial consideration to design a long lifetime sensor network [6]. Thus, energy efficiency is a major concern, since it allows us to maximize the lifetime of the network [7]. There are several definitions for the network lifetime. Lifetime can be defined as the time until the first sensor runs out of energy [7]. This definition is too pessimistic because when one node fails, the rest of the nodes still can provide appropriate functionality. Others have used definitions that include fractions of surviving sensors [8]. Sensors should consume the energy in the optimal manner to maximize the network lifetime. Therefore, energy consumption and residual energy balance are both critical in WSN design. [9] presents an analytical model to estimate and evaluate the network lifetime. In [10], a realistic power consumption model for WSN devices is proposed to derive the conditions for minimum power consumption in data transmissions. In [11] fuzzy logic systems are used to analyze the lifetime problem. They show that a type-2 fuzzy membership function is the best model for a single node lifetime in wireless sensor networks. In this paper, the lifetime of a sensor network is defined as the time in which a certain percentage of the network nodes run out of power. We use the max-min approach to exploit the state information of the network to maximize the minimum residual energy across the network [12]. It means that the nodes with less remaining energy do not participate in spectrum sensing, while the sensors with more energy, sense the spectrum. In this way, the remaining energy of all sensors are kept almost the same, therefore, the network lifetime is improved significantly. We formulate the first problem and solve it mathematically using convex optimization methods and obtain Karush–Kuhn–Tucker (KKT) conditions for optimal solutions. Then, we employ an iterative algorithm to find the nodes which sense the spectrum. Each iteration involves the sensors selection search and the solution of a max-flow problem to check the feasibility of the network lifetime. In another aspect, we focus on the maximization of the weighted residual energy to improve the network lifetime. In our network, the nodes have different signal-to-noise ratio (SNR) and the assumption is that SNR and distance between each node and FC are known. The rest of the paper is organized as follows. In Section II, we describe the system model and obtain the global probability

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where the first part is path loss component based on free- space path loss (FPL) model which involves d p j as the distance of every node from the primary user, f c as the working frequency denoted by 2.4 G H z and C is the speed of light. The second part is a real Gaussian random variable with zero mean and standard deviation 3 according to the large scale log-normal shadowing [13]. The probability of detection and probability of false alarm represent the detection performance. The higher probability of detection (Pd ) protects the primary user from the interference with the secondary user transmission and lower probability of false alarm (P f ) provides the opportunity of using idle channel for secondary users. Using an energy detector, the decision rule is defined as follows  1 δ f s 2 D j = 0 i f H0 H1 Ej = X ≷ : (5) D j = 1 i f H1 δ f s k=1 j k H0 Fig. 1.

Cooperative spectrum sensing configuration.

of detection and the global probability of false alarm for our problem. Our problem is defined in Section III. In SubSection A, we analyze how to solve the first problem using the analytical algorithms. In Sub-Section B and C, one solution for our problem is stated and our proposed algorithm is introduced. An alternative solution for the problem is presented in Section IV. Numerical and simulation results are given in Section V and conclusions are drawn in Section VI. II. S YSTEM M ODEL In this work, we consider a network with N cognitive sensors and a fusion center (FC) as it is shown in Fig. 1. T and δ are the frame duration and spectrum sensing time, respectively. fs is the sampling frequency of the received signal from the primary user. δ f s is considered as the number of samples. Each observation sample, X j [k], k=1,2,… δ f s has two hypotheses about the channel status. Hypothesis H0, means the primary user is inactive and hypothesis H1, means the channel is busy. H1: X j [k] = h j [k]s[k] + u j [k] H0: X j [k] = u j [k]

(1) (2)

where s[k] is the primary user signal which is assumed to be deterministic; h j is the channel gain between each node and primary user. The noise, u j [k] is a Gaussian, independent and identically distributed (i.i.d) random process with zero mean and variance σu2 . Each node has a special received signal-tonoise ratio (SNR) from the primary user, which is denoted as γ j for the j th sensor and is assumed to be well known. The channel model between each node and primary user has a model as follows −L j (3) h j = 10 20 · gj where g j is a complex Gaussian random process with zero mean and unit variance accounting for Raleigh fading and L j has two components   d p j 4π f c + nj L j = 20 log (4) C

 is the detection threshold for all cognitive sensors and D j denotes the decision about the channel. D j = 0 means that channel is idle while D j = 1 indicates that primary user is active. It shows each sensor sends one decision bit to the FC. E j is a random variable whose probability density function (PDF) is a chi-square distribution with 2δ f s degrees of freedom under H 0 . Under H 1 , its distribution is noncentral chi-square with 2δ f s degrees of freedom and a non-centrality parameter 2γ j · γ j is the received signal-to-noise ratio (SNR) of the primary user measured at the j th sensor. Using central limit theorem, for a large 2δ f s , E j has a Gaussian distribution, therefore, the local probability of detection and local probability of false alarm for each sensor is obtained with the following formulas [14]      δ fs ) (6) Pd j = P E j > |H1 = Q( 2 − γ j − 1 σu 2γ j + 1 and



P f j = P E j > |H0





  = Q( 2 − 1 δ fs ) σu

(7)

where Q is the complementary distribution function of the standard Gaussian. In order to improve the detection performance of cognitive sensor networks in deep fading and shadowing conditions, cooperative spectrum sensing is proposed [15], [16]. In cooperative spectrum sensing, several cognitive sensors decide about the channel status and send their results to the FC. Then FC makes the final decision using a fusion rule [17]. OR rule is used in this paper due to its simplicity. In this rule, if at least one sensor reports that the channel is busy, then the final decision is the activity of the primary user. So, the global probability of detection and global probability of false alarm in cooperative spectrum sensing are defined, respectively as

(8) Pd = 1 − Nj=1 (1 − Pd j )

N (9) P f = 1 − j =1 (1 − P f j ) But, it is not necessary for all sensors to participate in cooperative spectrum sensing because it leads to more energy consumption without improving the detection performance

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significantly [14]. Therefore, we can modify (8) and (9) as follows

Pd = 1 − Nj=1 (1 − ρ j Pd j ) (10)

N P f = 1 − j =1 (1 − ρ j P f j ) (11) where ρ j ∈ {0, 1} is the assignment index in which ‘1’ denotes sensing and ‘0’ for not sensing the spectrum by the sensor node. In fact, ρ j shows the priority of nodes for spectrum sensing. III. P ROBLEM D EFINITION To maximize the lifetime, we study the energy consumption of sensors in cooperative spectrum sensing. The total energy consumption has two main parts. First, energy consumption for sensing the channel and signal processing which is assumed to be the same for all sensors. This is denoted by Cs . The second part is the energy consumption for transmitting one reliable bit to FC. This type of energy depends on the distance between each node and FC; it is denoted by Ct j . Therefore, the total energy consumption is stated as follows [18]  (12) C T = Nj=1 ρ j (Cs + Ctj ) Moreover, Ct j has two parts as follows   Ctj d j = Ct −elec + eamp d 2j

u

where  N M is the maximum number of sensing nodes and n = j =1 ρ j is the number of sensing nodes. Therefore, (15-3) can be replaced with the constraint on the number of sensing nodes. Then, (15) can be modified as follows s.t.

Pd ≥ β j =1 ρ j ≤ M

(13)

ρj

(14)

ρ j ∈ {0, 1}

where Eremain j , E_i nt j and E_r j are the remaining energy after sending the result to FC, the initial battery energy, and the residual energy for each node j before selection, respectively. β and α are design parameters. It is desirable to have high probability of detection and small probability of false alarm. In (14), we intend to maximize the network lifetime with constraint on detection performance. It is clear that problem1 is equivalent to problem 2. Problem 2: Maxi mi ze T h

s.t.

ρj

  E r j − Cs + Ctj ≥ T h ∀ jN ρj

(15-1)

Pd ≥ β

(15-2)

Pf ≤ α ρ j ∈ {0, 1}

(15-3) (15-4)

(17-1) (17-2) (17-3) (17-4)

A. Solution Theory In (17), ρ j is a discrete parameter and solving this problem is very difficult. In order to reduce the complexity of (17), at first, ρ j is considered as a continuous parameter and then after solving the problem, ρ j is mapped to ‘0’ or ‘1’. This is similar to the time-sharing parameter in [20] and represents the priority of the sensor node j for sensing. Then, ρ j s for the nodes with the higher priority are denoted by ‘1’ and for other nodes are denoted by ‘0’. Therefore, we can replace (17-1) with another constraint to simplify our problem. Hence, the new problem is stated as

Pf ≤ α



maxi mi zeρ j T h   E_r j − Cs + Ctj ≥ Th ∀j ρj N

Ct −elec is the transmitter electronics energy, eamp is the required amplification and d j is the distance between the j th node and FC. We define the network lifetime to be the time until 25 percent of the sensors run out of energy [19]. Our goal is to maximize the network lifetime by maximizing the minimum remaining energy. This assumption helps the energy level of all the sensors to be balanced which leads to the network lifetime maximization. In this way, the nodes with less energy have more opportunity to remain in the network. This is formulated as follows Problem 1: max mi n {E remain j = (E r j − (Cs +Ctj ))} s.t. P d ≥ β

where T h is the minimum remaining energy of the sensing nodes. In fact, the purpose in (14) is maximizing Eremain j for all nodes. This means that above parameter must be more than a threshold, (i.e., T h) for all sensors to lead the remaining energy for all sensors stay in the same range. Therefore, maximizing Eremain j is equivalent to the maximizing T h in (15). We note that all sensors have the same P f j . On the other hand, P f is not dependent on γ j . Therefore, using (11), number of sensing nodes must satisfy the following condition ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ln (1 − α)   √  ⎦ = M n≤⎣  (16) ln 1 − Q σ2 − 1 δ fs

st. ρ j



maxi mi zeρ j T h   E r j − Cs + Ctj − ρ j T h ≥ 0 ∀ j (18-1) 

Pd ≥ β j =1 ρ j ≤ M

N

(18-2) (18-3)

ρ j ≥ 0 ∀ j (18-4) (18-1) is equivalent to (17-1) and it shows that the nodes which their Eremain j becomes more than T h, should be candidate for spectrum sensing. In the above problem, the objective function, (18-1), (18-3) and (18-4) are convex with respect to ρ j while (18-2) is not. Although (18) is not a standard convex optimization problem, we can still exploit the convex optimization methods to find ρ j s. Further, we can also employ the exhaustive search method to solve the problem but this method has high complexity with the order of O(N!) [21]. Therefore, we search the algorithms with linear complexity to solve the problem and find the sensors which consume less

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energy and help to maximize the network lifetime. First, we use the Lagrangian function as follows

Therefore, we can decrease ρ j s so that Pd = β is satisfied. In this way, P f is decreased and T h is increased which is more desirable answer. Therefore, (23-1) is an optimal condition. In the next section, we propose a method for searching points which can satisfy the optimality conditions simultaneously.

N        L ρ j , λ, η, ζ j = T h + ζ j (ρ j Er j − Cs +Ct j − ρ j T h ) j =1

+ λ (Pd − β)−η(

N

j =1 ρ j

− M)

(19)

where ζ j , λ and η are the Lagrangian multipliers for (18-1), (18-2) and (18-3), respectively. Then, Karush–Kuhn–Tucker (KKT) conditions reveal that [21]     ∂L = ζ j Er j − Cs + Ctj − 2ρ j T h ∂ρ j  + λPd j (1 − ρk Pdk ) − η = 0 j ∈ {0, 1, 2, . . . N} (20) k = j

According to (20), the priority function for selecting the sensing nodes is determined as follows ρ j = Pri f un ( j )   1 = (ζ j (E r j − Cs + Ct −elec + eamp d 2j ) ζ j (2T h)

+λP d j kN= j (1 − ρk Pdk ) − η) (21) This states that the first M sensors with the highest priority function as defined in (21) are selected as spectrum sensing candidates. We assume that T h is the same for all sensors and it can be omitted from the priority function. We consider another assumption that the other sensors are not selected yet. Therefore, (21) can be simplified as   1  Pri f un ( j ) = ∗ E r j − Cs + Ct −elec + eamp d 2j 2 λ η + Pd − (22) 2ζ j j 2ζ j According to (22), the sensors with higher Pri f un have more priority for spectrum sensing. We note that the number of sensing nodes is equal or less than M complimentary slackness conditions imply that ⎧  λ = 0, Pd > β (23-1) ⎪ ⎪ λ − β) = 0 → (P ⎪ d ⎪ λ  = 0, P = β (23-2) ⎪ d ⎪    ⎪  ⎪ ζ j E r j − Cs + Ctj − ρ j Th = 0 → ⎪ ⎪  ⎪ ⎨ ζ j = 0, E r j − Cs + Ctj > ρ j T h (23-3)   E r j −(C s + C tj ) = Th (23-4) ζ j = 0, ⎪ ⎪ ρj ⎪ ⎪ ⎪ ⎪ ⎪   ⎪ ⎪   η = 0,  ρ j < M (23-5) ⎪ ⎪ ρj − M = 0 → ⎩η η = 0, ρ j = M (23-6) According to (23-5) and (23-6), we assume that the number of sensing nodes is less than or equals to M. (23-3) and (23-4) show that sensing nodes are selected so that the remaining energy of the selected nodes becomes more than T h or in the worst case, equals to T h. It means that, there is at least one node which satisfies (23-4) while the others satisfy (23-3). Based on (23-1) and (23-2), we can prove that the optimum λ is a positive and non zero parameter and then Pd = β. The reason is that Pd and P f are increasing functions of ρ j s while T h is a decreasing function of ρ j according to (17-1).

B. Subgradient Search Method In this work, we intend to improve the lifetime in cognitive sensor networks. The objective function in the Lagrange problem is a convex and differentiable function. Therefore, we can use subgradient method [21] to find the maximum of  the objective function. If the step size ik , i = 1, 2, ik > 0 follows a non-summable diminishing rule at the kth iteration as follows ∞ k (24) li m ik = 0, k=1 i = ∞ k→∞

then, subgradient method is guaranteed to converge to the optimal value [22], [23]. With subgradient method, the dual variable at the (k + 1)th iteration is updated by (25) λk+1 = λk − 1k (Pd − β)     k+1 k k = ζ j − 2 Er j − Cs + Ctj − T h ∀ j (26) ζj wi , where The step size that we use in our algorithm is ik = √ k wi 1. It follows the non-summable diminishing rule.

C. Proposed Algorithm We propose an iterative algorithm to find the optimum λ, ζ j s and the nodes which sense the spectrum. At each iteration,  first, the  nodes which have enough energy (i.e., (Er j − Cs + Ctj > 0)) are candidates for spectrum sensing (in this case, number of nodes with sufficient energy is denoted by count in our algorithm). Then, the priority function in (22) is calculated for these sensors and sorted in descending order. The sensors with higher priority are selected for spectrum sensing until the global Pd ≥ β is satisfied. Note that maximum number of sensing nodes is less than or equal to count. Then, λ and ζ j for each node are updated according to the computed Pd and T h, respectively. T h is selected as the minimum remaining energy of the sensing sensors. This iterative algorithm terminates when the convergence metric is satisfied. Pseudo code for Network Lifetime Improvement with Sensor Selection (NLISS) is shown in Algorithm 1. IV. L IFETIME M AXIMIZATION BASED ON THE W EIGHTED R EMAINING E NERGY In the previous section, we introduced an algorithm to improve the network lifetime. Another way is to consider a weight value for each sensor which is defined as follows E_ i nt j (27) E_ r j where E_ i nt j is the initial energy which is assumed to be the same for all sensors and E_ r j is the residual energy for each node before selecting as a sensing node. Therefore, the new problem is defined as follows Wj =

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Algorithm 1 NLISS Algorithm λmin = 0 λmax = χ (a large enough number) ζmax = υ (a large enough number for each sensor) ζmin = 0 λ = λmax ζ j = ζmax Iteration = α(a big number) ε1 = small parameter ε2 = small parameter       While (λk+1 − λk  > ε1 && ζ jk+1 − ζ jk  > ε2 ) number of sensing nodes(n) =0 Determine the nodes which their remaining energy is more than their energy consumption and their numbers are denoted by count Compute  Pri f un( j ) =  λ Pd −η 1 2 ∗ E − C + C + e d + 2ζj r s t −elec amp j j 2 j

While (select n sensor with higher priority β, break, end n =n+1 end compute C T for all sensing nodes compute remaining energy for the sensing nodes update λ and ζ j according to (25) and (26) End End

Problem 3: max ρ j

N

j =1

Wjρj



  E_r j − ρ j Cs + Ctj

st. Pd ≥ β N j =1 ρ j ≤ M ρ j ∈ {0, 1}

(28)

Solving this problem is similar to (17); therefore, the priority function in (22) is modified and we have pri f un ( j ) =

λPd j E rj  +   2 Cs + Ctj 2 Cs + Ctj W j

(29)

This shows that the sensors with less W j s, (i.e., more residual energy before selecting as a sensing node), less energy consumption and more local probability of detection have more priority for spectrum sensing. This algorithm is similar to NLISS. The only issue is that they have different priority functions for selection of the sensing nodes. Here, our proposed algorithm is named Network Lifetime Improvement with Sensor Selection and Weighting (NLISSW). V. N UMERICAL AND S IMULATION R ESULTS In our wireless network, number of sensors changes from five to fifty. The nodes and primary user are uniformly distributed in a square field with a length of 200 m while FC is located in the center of the square. E int = 0.2m J is assumed as the initial energy for each sensor and the alive nodes are considered as the nodes that their remaining energies are more than the energy required for their spectrum sensing. In all

comparisons the exact optimal results are numerically obtained in MATLAB. Simulation results are shown for α = 0.1 and β = 0.9 which are the design parameters. Every simulation result in this section is averaged over 25000 realizations. We use the 2.4 GHz IEEE 802.15.4/ZigBee as the communication technology for our cognitive sensors which have to cooperate with each other to sense the spectrum. We model the wireless channel between each sensor and FC using a freespace path loss model. Considering the fact that the typical circuit power consumption of ZigBee is approximately 40 mW, the energy consumed for listening is approximately 40 nJ. The processing energy related to the signal processing part in the transmit mode for a data rate of 250 kb/s, a voltage of 2.1 V, and current of 17.4 mA is approximately 150 nJ/bit. Since we use one bit per decision, the sensing energy of each cognitive sensor is Cs = 190nJ. Assuming a data rate of 250 kb/s and a transmit power of 20 mW, Ct −elec = 80 nJ. The eamp to satisfy a receiver sensitivity of -90 dBm is 40.4pJ/m2 [24]. We also select decision threshold () as a multiple of the noise power. We compare our proposed algorithm with the following algorithms • MEESS algorithm: This algorithm in [24] is shown to be very efficient in saving energy for cooperative spectrum sensing. In this algorithm, the sensing nodes are selected to minimize the energy consumption and satisfy the detection performance constraints. But in this algorithm, the network lifetime is not considered. • Minimum Energy algorithm (MEA): In this algorithm, the nodes which are closer to FC and their remaining energies are more than their energy consumption, are selected for spectrum sensing until Pd ≥ β constraint is satisfied. • Maximum probability of detection algorithm (MPDA): In this algorithm, the nodes which their residual energies are more than their energy consumption and have the maximum Pd j , are selected for spectrum sensing untill Pd ≥ β constraint is satisfied. • Random Sensor Selection Algorithm (RSSA): In this algorithm, the nodes that their residual energies are more than their energy consumption are selected randomly for spectrum sensing so that the number of selecting nodes is less than M. This algorithm has the minimum complexity to find the solution for our problem. Fig. 2 shows the network lifetime versus different number of nodes. The network lifetime is defined as the number of iterations in which more than 25 percent of sensors are kept alive multiplied by the duration between the sensors selections. It is clear that when NLISS algorithm is used rather than MEESS, the network lifetime is improved effectively. NLISSW can also improve the lifetime of the network. The reason is that NLISS and NLISSW select the sensing nodes according to their remaining energies and their probabilities of detection while in MEESS, in the selection of the nodes, the remaining energy is not considered. MPDA, MEA and RSSA have less network lifetime. In MEA and MPDA algorithms, only the lowest energy consumption or the maximum probability of detection is important, respectively. In RSSA, the nodes selection is completely random.

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Fig. 4. Fig. 2.

Successful percent of finding the answer for different iterations.

Lifetime for different numbers of total nodes.

Fig. 3. Average successful percent of finding the answer for different number of nodes.

Fig. 5. Average minimum remaining energy for different numbers of cognitive sensors.

Lifetime is very important in wireless cognitive sensor networks; however, the other issue is the high probability of detection to determine the activity of the primary user. It means that, the proper algorithm not only increases the lifetime, finds the answer if the problem has any solution. One of the metrics in finding the proper solution is the rate of satisfying the constraint of the detection performance successfully. In Fig. 3, we show the average successful percentage of finding the answer as the number of cognitive sensors in the network increases. This metric shows the ability of the algorithms in satisfying the total probability of detection constraint. It is clear that NLISS and NLISSW have the maximum success ratio in finding the answer. The success ratio for MPDA and MEESS are close to each other. MEA and RSSA have the minimum percentage of finding the answer. In MEA, selection of the nodes is due to their distance from FC, which might have small probabilities of detection. This metric is averaged over the maximum iteration which is the lifetime of the best algorithm.

Fig. 4 shows the successful percent of finding the answer for NLISS and MPDA, with respect to the number of iterations. The number of total nodes is considered 30 and 40. We know that MPDA selects the sensors with maximum probability of detection for spectrum sensing (i.e., if the problem has an answer, MPDA can find it). In this algorithm, when the number of iterations increases, it is possible that the nodes with less probability of detection, remain in the network, or most of the sensors run out of energy. Therefore, they cannot satisfy the detection performance while NLISS selects the nodes according to their remaining energy and probability of detection. It means that, NLISS increases the network lifetime and more nodes are alive. Hence, the remaining nodes can maintain the probability of detection constraint for more iterations. In Fig. 5, the average minimum remaining energy of the nodes versus different number of sensors is shown. For this simulation, the minimum remaining energy is averaged over the iterations which satisfy Pd ≥ β constraint. NLISS and NLISSW algorithms have the highest minimum remaining

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Fig. 6.

Minimum remaining energy for different iterations. Fig. 8.

Fig. 7.

Energy consumption for different numbers of total nodes.

energy, because in these algorithms, the remaining energy of nodes is considered in sensor selection. It means that in these algorithms, there is equilibrium between the remaining energies of the sensors while the other algorithms have less minimum remaining energy due to not considering this parameter in the selection of the sensing nodes. It is clear that MEA has the minimum remaining energy, because in this algorithm, the number of sensing nodes is increased to maintain the constraint on the detection performance. Fig. 6 shows decreasing remaining energy as the time (number of iterations) is increased. Number of total nodes is 30 sensors. In fact, this indicates how the nodes run out of energy in different algorithms. In (15), our purpose was maximizing the remaining energy, which is obtained by NLISS in this figure. Therefore, maximizing the network lifetime in (14) is equal to the remaining energy maximization in (15). It is clear that by increasing the iteration, the difference between NLISS and NLISSW becomes more than the other algorithms.

Convergence of the threshold for different iterations.

In Fig. 7, the energy consumption in cooperative spectrum sensing versus different number of nodes is shown. MEESS has the minimum energy consumption, because, in this algorithm, the main concern is the energy consumption. RSSA has the maximum energy consumption because this algorithm selects the sensors randomly. In MEA the nodes are selected according to their distances from FC; therefore, when the probability of detection for each node is small, MEA needs more nodes for spectrum sensing to satisfy the detection constraints and energy consumption is increased. NLISSW and NLISS consume more energy in comparison with MEESS because these algorithms improve the network lifetime rather than energy consumption. Note that all algorithms are compared when they have an answer, meaning that Pd ≥ β constraint is satisfied. Fig. 8 shows the changes of the threshold (T h) versus the iterations that λ and ζ j s are updated. The iterations are considered between 850 and 1100. Number of total nodes is 50 sensors. We can see that in the 1000t h iteration, threshold converges to the fixed value. The value of the convergence iteration is increased due to the large number of the Lagrangian multipliers. VI. C ONCLUSION In this paper, we proposed two algorithms for improving the network lifetime. We formulated our problems and solved them analytically. We used KKT conditions to find the optimal sensing nodes which participate in cooperative spectrum sensing and maximize the network lifetime. Simulation results show that our proposed algorithms are very effective for increasing the lifetime of the network and satisfying the detection performance. Although the energy consumption was not minimized in our proposed algorithms, balancing the residual energy between all the sensors improved the network lifetime. Our algorithms are independent from the type of the fusion rule and it can be used for different fusion rules.

NAJIMI et al.: LIFETIME MAXIMIZATION IN COGNITIVE SENSOR NETWORKS

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Maryam Najimi received the B.Sc. degree in electronics from Sistan and Baloochestan University, Zahedan, Iran, in 2004, and the M.Sc. degree in telecommunication systems engineering from the K.N. Toosi University of Technology. She is currently pursuing the Ph.D. degree in communications at the Babol University of Technology. Her interests include spectrum sensing in wireless cognitive sensor networks.

Ataollah Ebrahimzadeh received the Ph.D. degree in electrical engineering from Ferdosi University, Mashhad, Iran, in 2007. He is currently an Associate Professor with the Faculty of Electrical and Computer Engineering, Babol University of Technology. His current scientific interests include general area of signal processing and artificial intelligence. He is the reviewer of international conferences and journals. He has authored more than 50 papers in international journals.

Seyed Mehdi Hosseini Andargoli received the B.Sc. degree in electronics engineering from Shahed University, Tehran, Iran, and the M.Sc. degree and Ph.D. degree in telecommunication systems engineering from the K.N. Toosi University of Technology, Tehran, Iran, in 2004, 2009, and 2011, respectively. He is currently an Assistant Professor with the Department of Electrical and Computer Engineering, Babol university of Technology, Babol, Iran. His research interests include resource allocation of cellular networks, cognitive radio networks, relay networks, sensor networks, optimization, and MIMO-OFDM systems.

Afshin Fallahi received the B.Sc. degree in electrical engineering from the University of Tehran, Iran, the M.Sc. degree in electrical engineering from Tarbiat Modarres University, Iran, and the Ph.D. degree in electrical engineering from the University of Manitoba, Canada, in 1996, 1999, and 2008, respectively. His main research interests include the area of modeling, analysis, and optimization for wireless networks.