Energy Minimization Approach for Optimal Cooperative Spectrum Sensing in Sensor-Aided Cognitive Radio Networks

Transcription

1 Energy Minimization Approach or Optimal Cooperative Spectrum Sensing in Sensor-Aie Cognitive Raio Networks Hai Ngoc Pham, Yan Zhang, Paal E. Engelsta, Tor Skeie an Frank Eliassen Department o Inormatics, University o Oslo, Norway {hainp, paalee, tskeie, rank}@ii.uio.no Simula Research Laboratory, Oslo, Norway {hainp,yanzhang, paalee, tskeie, rank}@simula.no Telenor R&I, Oslo, Norway Paal.Engelsta@telenor.com Abstract In a sensor-aie cognitive raio network, collaborating battery-powere sensors are eploye to ai the network in cooperative spectrum sensing. These sensors consume energy or spectrum sensing an thereore eplete their lie-time, thus we stuy the key issue in imizing the sensing energy consume by such group o collaborating sensors. The IEEE P80. stanar speciies spectrum sensing accuracy by the etection an alse alarm probabilities, hence we aress the energy imization problem uner this etection accuracy constraint. Firstly, we erive the bouns or the number o sensors to simultaneously guarantee the threshols or high etection probability an low alse alarm probability. With these bouns, we then ormulate the optimization problem to in the optimal sensing interval an the optimal number o sensor that imize the energy consumption. Thirly, the approximate analytical solutions are erive to solve the optimization accurately an eiciently in polynomial time. Finally, numerical results show that the imize energy is signiicantly lower than the energy consume by a group o ranomly selecte sensors. The mean absolute error o the approximate optimal sensing interval compare with the exact value is less than 4% an 8% uner goo an ba SNR conitions, respectively. The approximate optimal number o sensors is shown to be very close to the exact number. I. INTRODUCTION Cognitive Raio (CR envisione by J. Mitola in [1] has emerge as the innovative ynamic spectrum access technology [] to improve the current utilization o assigne spectrum. It is reporte by The Feeral Communications Commission (FCC in [3] that the spectrum is only 15% - 85% utilize epening on geographical an temporal variations. In a cognitive raio network, the unlicense (seconary evices can utilize the license spectrum when it is unuse by the license (primary evices. However, the occupie spectrum will nee to be vacate instantly when a primary evice starts using it in orer to avoi interering with the primary transmission. Thus, spectrum sensing is speciie as a manatory eature within the IEEE P80. stanar [4] to enable a CR evice to etect an aapt to the primary usage o a spectrum ban. The sensing perormance metric is summarize in IEEE P80. in terms o sensing receiver sensitivity, channel etection time (sensing interval, etection probability, an alse alarm probability. Hence, improving the sensing perormance has emerge as one o the most important issues in spectrum sensing recently. Collaborative spectrum sensing by multiple collaborating sensing evices is stuie in [5], [6], [7] to increase the etection probability. The cooperative spectrum sensing is also consiere in [8] to imize the total error rate given the number o sensing noes an their Signal-to-Noise-Ratio (SNR. In [9], the problem o maximizing the ratio o the transmission uration over the entire sensing cycle is stuie. However, or practical purpose o using energy-constraine sensor network or spectrum sensing in cognitive raio networks [10], it is critical to use ewer sensing sensors perorg in a shorter sensing interval in orer to preserve as much energy as possible while still satisying the requirement or spectrum etection accuracy. The present paper investigates this issue in terms o ining the optimal sensing interval an the optimal number o sensors in orer to imize the total energy consumption or cooperative spectrum sensing. This paper stuies cooperative spectrum sensing by a powerconstraine sensor network in sensor-aie cognitive raio networks [10]. These sensors can sense the spectrum ban continuously an reports the etection results to a usion center as emonstrate in Fig. 1. In the consiere cooperative Fig. 1. Cooperative spectrum sensing moel spectrum sensing scheme, the usion center invites a speciic number o sensors in the network to participate in a sensing Digital Object Ientiier: /ICST.WICON

2 group, say S. Then, the invite sensors inepenently start sensing the spectrum an report their observations back to the usion center who perorms the OR-rule usion mechanism [11] to make a ecision on the availability o the monitore spectrum. In this scenario, each sensor uses the energy etection scheme [1], [5] or spectrum sensing, whose perormance is evaluate by the etection an alse alarm probabilities. A high etection probability means a high accuracy o etecting the activity o a primary user. A low alse alarm probability inicates a high usage o available spectrum by the seconary users, ue to a low chance that the spectrum is mistakenly believe to be occupie when it is actually available. There is a traeo in keeping high etection probability an low alse alarm probability at the same time in the OR-rule usion mechanism. The more sensors the higher etection an alse alarm probabilities an vise versa. Hence, this paper irst ins the lower boun an upper boun or the number o sensors uner a given requirement or the spectrum sensing accuracy. Spectrum sensing consumes energy an thereore epletes the lie-time o the power-constraine sensors. Hence, energy imization is critical to prolong the lie-time o the sensor network. This paper ormulates an optimization problem to imize the total energy consumption. It is esirable to gain a high etection accuracy by using a cooperative group o many sensors perorg a long channel sensing interval, which in turn consumes more energy. On the other han, it is also highly esirable to save as much energy as possible by using ewer sensors an sensing or a shorter time. This traeo is aresse in the propose optimization. Finally, this paper proposes an eicient approximation approach to analytically an accurately solve the optimization in polynomial time, since the optimization is shown to be extremely iicult to solve irectly. The approximate analytical solutions or the optimal sensing interval an the optimal number o sensors are erive accurately. We in that uner goo SNR conitions, the mean absolute error o the approximate optimal sensing interval is less than 4% compare to the exact optimal one. In the worst SNR conitions, this error is aroun 8%. The rest o the paper is organize as ollows. The relate work is presente in Section II. Then Section III presents the system moel. Next, the energy imization problem is aresse in Section IV. Then, Section V proposes an approximation approach to analytically solve the optimization problem in polynomial time, which is prove in Appenices A, B & C. Numerical results are presente in Section VI to explore the optimization an valiate the accuracy o the approximate optimal solutions. Finally, conclusions an uture irection are state in Section VII. II. RELATED WORK There have been some recentuies on improving the perormance in cooperative spectrum sensing. Peh an Liang show in [13] that an optimum perormance can be achieve by the cooperation o only a certain number o seconary users, i.e. users who sense the highest SNR o the primary users transmission. They stuy the optimization o the etection probability an alse alarm probability separately with regar to (w.r.t the number o cooperation users. The traeo between keeping high etection probability an low alse alarm probability at the same time w.r.t the optimal number o cooperation users is not aresse yet. In [8], Zhang et al. ocus on ining an optimal usion rule to imize the summation o alse alarm an miss-etection probabilities by assug that the number o cognitive raios an their SNR are known. However, the SNR receive by all cognitive raios changes over the time ue to the changing communication environment. The present paper shows that in orer to in the optimal number o sensors uner the constraint o etection accuracy, knowing in avance the number o all sensors in the network is not require. In [9], Lee an Akyiliz stuy the problem o maximizing the ratio o the transmission uration over the entire sensing cycle. They report that the optimal sensing parameters will nee to be aapte to the number o cooperative sensing users, which varies over time. Liang et al. also stuy the sensing uration problem in [14] as a sensing-throughput traeo to imize the alse alarm probability given the etection probability threshol. The present paper, on the other han, stuies the traeo in eriving the optimal sensing uration an the optimal number o sensors while preserving as much energy as possible uner a given etection accuracy constraint. Optimal cooperative spectrum sensing by imizing the energy consumption is also stuie in [15]. However, the traeo in keeping a high etection probability an a low alse alarm probability simultaneously in their optimization is notuie. In aition, the approach in [15] yiels a airly high error in the approximate results. The present paper iers rom the previous work in terms o comprehensively stuying an ormulating the energy imization problem or cooperative spectrum sensing while satisying a given threshol or etection accuracy. The traeo between the optimal number o sensors an the optimal sensing interval as well as the traeo in keeping a high etection probability an a low alse alarm probability simultaneously are consiere an ormulate in the propose optimization problem. III. SYSTEM MODEL AND PROBLEM FORMULATION Table I lists the main notations use in this paper. The system moel or cooperative spectrum sensing is illustrate in Fig. 1. The sensor network N is eploye to etect the activity o a primary system on a given spectrum ban. Each sensor i in N receives the primary signal with an instant SNR γ i an this signal-to-noise-ratio varies rom sensor to sensor epening on the surrouning wireless communication environment. The etails o the stuie cooperative spectrum sensing scheme is presente as ollows. Digital Object Ientiier: /ICST.WICON

3 TABLE I Symbol Deinition N The set all sensors in the network S The group o sensors or cooperative spectrum sensing γ i Signal-to-noise ratio (SNR aensor i (B γ The imum SNR among the sensors (B σ n The groun noise (B N(μ i,σi Chi-square istribution with mean μ i an variance σi λ Energy threshol use by the energy etector (B W The spectrum banwith (Hz ˆP i Single-noe etection probability o sensor i ˆP i Single-noe alse alarm probability o sensor i ˆP The imum single-noe etection probability ˆP max The maximum single-noe alse alarm probability Q Cooperative etection probability o the sensing group S Q Cooperative alse alarm probability o the sensing group Q Threshol or cooperative etection probability Q Threshol or cooperative alse alarm probability The spectrum sensing interval (sec t s The optimal spectrum sensing interval (sec n The number o sensors inclue in S n The optimal number o sensors inclue in S Q(z The Gaussian Q-unction o a ranom variable z [16] A. Maximum A Posteriori (MAP Energy Detection or Spectrum Sensing In this paper, we ollow the approach o MAP energy etection scheme escribing in [9] as ollows. By aopting the energy etection scheme [1], [5] or the spectrum sensing, each sensor i etects the presence o the primary user by the single-noe etection an alse alarm probabilities P i an P i, respectively. This sensor receives the primary signal r i (t in the ollowing orm [1]: { r i (t = n i (t hyphothesis H 0 s i (t+n i (t hyphothesis H 1 (1 where, H 0 an H 1 are the hypotheses corresponing to no signal transmitte an signal transmitte, respectively. s i (t is the receive signal waveorm an n i (t is a zero-mean aitive white Gaussian noise (AWGN. Hence P i an P i are erive as ollows [1]: P i = P r [Y i >λ H 1 ] ( P i = P r [Y i >λ H 0 ] where λ is the energy etection threshol or every sensor. The test or ecision statistic Y i N(μ i,σi is the Chisquare istribution an can be approximate as a Gaussian istribution as [1], [9, Re. 13]: { ( N uσ ni, uσ 4 ni, H0 Y i N ( u(σni + σsi, u(σni + σsi, H 1 where u = W is the number o samples. is assume to be the same or every sensor. σni an σ si are the variance o the noise n i (t an the receive signal s i (t, respectively. The SNR is erive as: γ i = σsi /σ ni. Without loss o generality, the variance o the noise is assume to be the same at every sensor an is simply enote by σ n. Thus, the tail probability o the Gaussian istribution, P r [Y > λ], can be erive in terms o the Gaussian Q-unction [16] as: [ Y μ P r [Y > λ]=p r > λ μ ] ( λ μ = Q σ σ σ Q(z = 1 e x x (3 π Then, P i an P i can be easily erive as ollows: ( λ ts W (γ i +1σ n P i = Q (4 W (γ i +1σn ( λ ts Wσ n P i = Q (5 Wσn In aition, the traic pattern o the primary user can be moele as a two state inepenent an ientically istribute (i.i. ON-OFF ranom process [17], whose ON an OFF perios are exponentially istribute with the means in terms o time as T on an T o, respectively. Hence, sensor i etects the monitore spectrum availability by the ollowing singlenoe etection an alse alarm probabilities [9]: ˆP i =P on.p i =.P i T on + T o ˆP i =P o.p i = z T on (6 T o.p i T on + T o Since, P i is monotonically increasing w.r.t the sensing interval an the SNR, the sensors that experience the lowest SNR will yiel the lowest etection probabilities or the least accurate etection. Thus, by excluing these weak-snr sensors rom the spectrum sensing group, the total energy consumption might be reuce while the total etection probability is still kept high. B. A Cooperative Scheme or Spectrum Sensing As escribe earlier in Section I an Fig. 1, the usion center perorms the OR-rule [11] to erive the cooperative etection an alse alarm probabilities Q an Q rom aggregating the single-noe etection an alse alarm probabilities, which are estimate rom the test or ecision statistic ( provie by the sensors in the cooperative sensing group S, respectively. The ecision on the occupancy o the monitoring spectrum will then be conclue by comparing Q an Q with the given threshols or etection accuracy Q an Q, respectively. By perorg the OR-rule, Q an Q can be erive as ollows [5]: Q =1 (1 ˆP i Q =1 (1 ˆP i where n is the number o the sensors in S. The single-noe probabilities ˆP i an ˆP i erive by (6 are reporte to the usion center by each iniviual sensor i in the sensing group S. This scheme shows that when n increases, Q will increase an as a consequence the accuracy o the primary user being etecte also increases. However, the higher the value o n, (7 Digital Object Ientiier: /ICST.WICON

4 the higher the cooperative alse alarm probability Q which in turn causes a higher chance that a spectrum opportunity will be misse. In aition, the more sensors inclue in S, the more energy is consume or spectrum sensing, which is unesirable since the sensors have limite power resource. Hence, ining an optimal size o the group S is an important issue to be solve in this paper. Furthermore, energy-eicienelection o the appropriate sensors to be inclue in S is also an important problem. For example, how to eiciently coorinate an select the sensors that experience the highest SNR an that are well separate rom each other in orer to avoi correlation shaowing in the cooperative spectrum sensing is an essential question. This issue is raise as the uture work o this paper. IV. ENERGY MINIMIZING IN COOPERATIVE SPECTRUM SENSING A. Boun or the Number o Sensors Given the threshols Q an Q or cooperative etection an alse alarm probabilities, respectively, the conitions Q Q an Q Q are neee to satisy the etection accuracy an to be conient that a spectrum opportunity is not misse. Thus, the cooperative scheme (7 yiels: 1 (1 ˆP i Q 1 Q (1 ˆP i (8 1 (1 ˆP i Q 1 Q As enote in Table I, ˆP ˆP =P on.q ˆP max (1 ˆP i (9 an can be erive as: ( λ ts W (γ +1σn W (γ +1σn (10 ˆP max =max{ ˆP i, i =[1...n]} where the imum SNR: γ ={γ i }. Then: (1 ˆP n (1 ˆP i (1 max ˆP n (1 ˆP i Hence, the conitions (8 an (9 will be satisie i the ollowing inequalities are kept: { 1 Q (1 n 1 Q (1 ˆP ˆP max which require the bouns or n as ollows: log(1 Q log(1 n Q (11 max where. an. enote the ceiling an looring unctions or the rouning o a real number to an integer, respectively. The lower boun shows that the higher single-noe etection probability, the ewer sensors are neee to guarantee a n given threshol. More importantly, the higher the imum SNR among the sensors, the ewer sensors are require. Thus, the usion center shoul only invite the suiciently high SNR sensors. Furthermore, the upper boun inicates an invaluable physical meaning on the speciication o the threshol Q. The threshol Q cannot be as low as possible, since the low Q requires the small number o sensors, which might break the etection accuracy by violating (11. Hence, the traeo in keeping Q high an Q low simultaneously is aresse in ormulating the optimization problem in this paper. B. Optimal Sensing Interval & Optimal Number o Sensors to Minimize the Energy Consumption For energy eiciency, the lower boun or n in (11 is use as the imum number o sensors inclue in the sensing group S. However, it oes not mean that n is optimal in terms o imizing the total energy consume by group S or cooperative spectrum sensing. Equation (4 shows that the longer the sensing interval, the higher the etection accuracy, hence ewer sensors are neee an consequently less energy will be spent. On the other han, the higher, the more energy is consume or spectrum sensing. This paper aresses that important traeo in ormulating the energy imization problem as ollows. Let δe ss enote the sensing energy consumption per time unit uring the spectrum sensing interval. δe ss is assume to be the same or every sensor in the network. Hence, uring, each sensor i consumes a sensing energy ΔEi ss = δe ss. The imization o the total sensing energy consume by group S is then ormulate as: Minimize: n ΔE ss i n δe ss log(1 Minimize: Q δe ss (1 Equation (1 can be urther reine by the observation that the absolute unction is monotonically increasing w.r.t ˆP as: Minimize: log(1 Q.δE ss. log(1 1 ˆP Maximize: log(1 Q..δEss Maximize: P Q on log(1 Q.δE. ss ˆP t ( s λ tsw (γ +1σ n W (γ +1σn Without loss o generality, it is assume that δe ss, T on, an T o are known an inepenent o the sensing interval an that Q is given. Thus, the optimal sensing interval t s that imizes the total sensing energy consume by the cooperative spectrum sensing group S can be solve by the Digital Object Ientiier: /ICST.WICON

5 ollowing maximization problem: ( λ tsw (γ Q +1σ n t W (γ s =argmax +1σn (13 subject to: c 1 : 0 (13a c : n log(1 Q (13b max where: n = log(1 Q ( λ ˆP ts W (γ +1σ n = P on.q W (γ +1σ ( n ˆP max λ Wσn = P o.q Wσn 4 Obviously, solving (13 irectly an analytically is extremely iicult ue to the exponential characteristic o the Q-unction. Hence, Section V presents an approximation approach to eiciently solve this optimization problem. C. Discussion on the Optimization s Constraint As iscusse earlier in subsection IV-A, there is a traeo in satisying a high threshol Q or the cooperative etection probability an a low threshol Q or the cooperative alse alarm probability at the same time. The meaning o this traeo inicates in keeping the upper boun o n (w.r.t a given threshol Q satisying the constraint (13b o the optimization problem (13. The etaile iscussion on this issue is presente as ollows. Recall the constraint (13b as: n = log(1 Q log(1 Q max The ollowing transormations are then erive to reason about the speciic requirement o the threshol Q : log(1 Q max. log(1 Q e log(1 Q e log(1 ˆP max log(1 ˆP. log(1 Q log(1 ˆP max ˆP Q Q =1 (1 Q log(1 where t s is the optimal sensing interval o (13 an: ( ˆP λ t = P on.q sw (γ +1σn t sw (γ +1σn ( ˆP max λ t = P o.q swσn t swσn (14 The inequality (14 inicates that with a given threshol Q or the cooperative etection probability, the requirement threshol Q or the cooperative alse alarm probability must be lower-boune by Q in orer to hol the optimality o the propose optimization problem. V. ANALYTICAL SOLUTIONS FOR THE ENERGY MINIMIZATION PROBLEM This section presents an approximation approach to accurately solve the optimization (13 in polynomial time. Recall the ormulation o the Gaussian Q-unction as [16]: Q(z 1 π z e x x (15 where: z = λ W (γ +1σ n W (γ +1σ n The exponential characteristic o Q(z implies thaolving (13 analytically is extremely iicult. Hence, approximation approaches can be propose to make (13 solvable. For example, [15] consiers these approximations: 1 Q(z e z / 1 1 / e z i z 0 (16 i z<0 (17 an shows that when z is positive, t s can be oun as: t s = 1 W [ λ 4(γ +1 σn However, this approximation prouces aroun 0% error compare to the exact result o the original optimization, which is mainly ue to the high inaccuracy o the approximation (16 or Q(z >0. In the ollowing, more accurate an polynomial time analytical solutions are erive. A. Linearization when 0.5 z 0.5 It is observe rom the Q-unction that its curvature is close to linear when z varies rom 0.5 to 0.5. The ollowing linearization is propose or that variation o z to accurately an analytically solve the optimization (13. Recall the partial erivative (Slope o the Q-unction as: Slope (Q(z z = 1 π e z The curvature o the linearization can be approximate as Slope(z =0= 1 π, hence the linearization is erive: Q(0.5 z ] z π (18 Substituting (18 into (13 an ollowing the transormations in Appenix A, t s can be oun analytically as: [ ] t s = 1 3λ 3λ π 1+ W π(γ +1σn + π + (γ +1σn (19 Digital Object Ientiier: /ICST.WICON

6 B. Approximation when z>0.5 This approximation is erive similar to (16 as: Q(z >0.5 1 (z+0.5 e (0 Then, ollowing the erivations in Appenix B in solving the optimization (13, t s = 1 W u can be oun as the root o the ollowing polynomial o egree our: p 4.u 4 + p 3.u 3 + p.u + p 1.u + p 0 =0 (1 where: u = W, an A =(γ +1σn p 4 =4A 4 p 3 =15A 4 p =A (8A 4λ λa p 1 =17λ A p 0 =4λ 4 C. Approximation when z< 0.5 When z< 0.5, (17 cannot be use in solving (13. Thus, this subsection ocuses on ining an approximation that has the similar Slope as that o the original Q-unction. Since Q(z < 0 is monotonically increasing when z is ecreasing below zero, then the ollowing approximation is propose when z< 0.5: Q(z <0 z.slope = z e z ( π Hence, ollowing the erivations in Appenix C, the approximate optimal sensing interval can be oun as the root o the ollowing cubic unction: u 3 + a.u + b.u + c =0 (3 where: u = W a = (γ +1σn λ (γ +1σn b = λ +6λ(γ +1σn ((γ +1σn ( 3 λ c = (γ +1σn VI. NUMERICAL RESULTS This section presents numerical calculation to valiate the approximate optimal results solve by the propose approach compare with the exact optimal results o the optimization. The imum energy consumption is also valiate comparing with the ranom energy consume by the ranom group orme by a ranom number o sensors. In all comparisions, the exact optimal results are numerically estimate rom (13 in Matlab. Mean absolute error (MAE is use to valiate o the accuracy o the approximate optimal results. In all the calculations, the ollowing setting are use: δe ss =0.05 J; the ON-OFF perio o the primary user is moele as T on = 1 s, T o = s [9]. The monitore spectrum banwith is W =10kHz. The energy etection threshol an the groun noise are chosen as λ =4.5 B an σ n = 10 B, respectively. The etection accuracy threshols Q = 0.9, an Q = 0.1 are ollowe the IEEE P80. stanar. The perormance o the propose optimization an approximation approach is valiate through a wie range o the imum SNR rom 30 B to 50 B. Firstly, the imum energy consumption is valiate comparing with the energy consume by a ranom group orme by ranomly selecting the number o sensors rom the range [1, 60]. This range is similar to that o the approximate optimal number o sensor solve by the propose approximate solutions. Even that the sensing interval yiele by the ranom case (the thick-ashe curves in Fig. 3 is sometimes smaller than the optimal value, the optimization always prouces the imum energy consumption as shown in Fig.. The total Minimum Total Consume Energy (J Exact Minimum Energy Approximate Minimum Energy Ranom Energy Minimum SNR (B Fig.. Comparing the total energy consumption. energy consume by the optimal group o sensors is aroun % less than the total energy consume by the ranom group. Huge energy saving uner very high SNR conition is yiele since the propose optimization prouces much shorter optimal sensing interval. Fig. also valiates the accuracy o the approximate solutions compare with the exact results as shown in the ash-ot-asterisk curve an the soli curve, respectively. The MAE between these curves is aroun 8.8%, which is cause mainly uner very low SNR conitions. The results also show that when the imum SNR increases, less energy is consume, which conirms the observation iscusse earlier on the inluence o the SNR conition to the cooperative spectrum sensing problem. In particular, the high energy consumption uner very low SNR conition implies the weakness o the energy etector scheme at low SNR. The valiation o the accuracy o the propose solutions is also presente in Fig. 3 where the approximate optimal sensing interval (the ash-ot-asterisk curve is very close to the exact optimal results (the soli curve. Ihows here again that the higher the imum SNR, the shorter time the sensing group perorms spectrum sensing while still satisying Digital Object Ientiier: /ICST.WICON

7 Optimal Sensing Interval (sec Exact Optimal Result Approximate Optimal Result Ranom Result Minimum Coop. Detection Probability Q Exact Result Approximate Result Threshol Minimum SNR (B Minimum SNR (B Fig. 3. Comparing the sensing interval. Fig. 5. The imum cooperative etection probability Q. the require etection accuracy. In aition, in the range 30 B to 50 B or the imum SNR, the MAE prouce by the propose solutions (3, (19, an (1 are aroun 8.1%, 4.0%, an.13%, respectively. The highest error cause uner the lowest SNR conition. Fig. 4 presents the approximate optimal number o sensors (the ashe bar an the exact optimal result (the soli bar. Ihows that the ierence between the approximate an the Number o Sensor S Exact Optimal Result Approximate Optimal Result Minimum SNR (B Fig. 4. The optimal number o sensor. exact results is very small. It is also observe that the better the SNR conition the higher optimal number o sensors. The reason is that the propose energy imization prouces much shorter optimal sensing interval uner the goo SNR conition. However, it nees to guarantee the given etection accuracy, hence the suicient number o sensors will nee to be inclue uring the optimization as shown in Fig. 4. Finally, igures 5 an 6 show the valiations o the imum cooperative etection probability an the maximum alse alarm probability yiele by the optimization. The result in Fig. 5 inicates that the propose solutions prouce accurate etection by keeping the imum cooperative etection probability above a given threshol. Fig. 6, on the other han, restates the traeo in keeping very low cooperative alse alarm probability when the SNR conition is very ba. Iuggests Maximum Coop. False Alarm Probability Q Exact Result Approximate Result Minimum SNR (B Fig. 6. The maximum cooperative alse alarm probability. that the usion center shoul either only invite the high SNR sensors or cooperative spectrum sensing or accept higher tolerance or the cooperative alse alarm probability uner very low SNR situations. VII. CONCLUSION This paper has stuie the imization o the total energy consume by a group o power-constrainensors or cooperative spectrum sensing in sensor-aie cognitive raio networks. Firstly, the lower boun an upper boun or the number o sensors are oun uner the etection accuracy threshols. Then, with the erive bouns, the optimization problem to imize the total sensing energy consumption is ormulate. Next, the approximate analytical solutions are oun to solve the optimization accurately an eiciently in polynomial time. Finally, numerical calculations show that the imize energy is signiicantly lower than the energy consume by a group o ranomly selecte sensors. The approximate optimal number o sensors is shown to be very close to the exact number. Uner goo SNR conitions, the Digital Object Ientiier: /ICST.WICON

8 mean absolute error o the approximate optimal sensing interval is less than 4% compare to the exact value. In the worst SNR conitions, this error is aroun 8%. It has also observe that the energy etector scheme oes not perorm well at very low SNR conitions, which cause higher energy consumption or cooperative spectrum sensing. This paper has also observe an iscusse that energyeicient selection o the appropriate sensors to be inclue in the cooperative spectrum sensing group is an important problem. The question o how to eiciently coorinate an select the sensors that experience the highest SNR an that are well separate rom each other in orer to avoi correlation shaowing in cooperative spectrum sensing is aresse as the uture irection o this paper. by substituting (A. as: λ WA ts WA + λ + WA WA = 3λ WA ts WA = π (3λ WA =π WA π [ WA (3λ + πa] =6λπA + π A Finally, the optimal sensing interval is oun as: [ ] t s = 1 3λ 3λ π 1+ W π(γ +1σn + π + (γ +1σn (A.4 APPENDIX A PROOF OF THE OPTIMAL RESULT (19 Proo: The irst erivative o (13 can be erive as: ( Q(z =0 (Q(z z. z. Q(z t s (Q(z. z. = Q(z z where z = λ tsw (γ +1σ n. Then this is erive: W (γ +1σn ( λ tsw (γ z +1σ n W (γ = +1σn = (λ W (γ +1σn =0. W (γ +1σn ( ts W (γ +1σn (A.1 ( λ ts W (γ +1σ n. ( W (γ +1σ n ( ts W (γ +1σ n = W (γ +1σn. W (γ +1σn 4W (γ +1 σn 4 ( λ ts W (γ +1σn. tsw W (γ +1σn APPENDIX B PROOF OF THE OPTIMAL RESULT (1 Proo: Substituting (0 into (A.1 yiels: ( 1 (z+0.5 e z. z. = 1 (z+0.5 e (z + 1 z.. =1 (B.1 By enoting A (γ +1σn an substituting (A. into (B.1, t s can be oun as ollows: ( λ ts WA ts WA + 1 ( λ + ts WA WA λ + WA 4 WA =1 λ ( WA WA =1 WA(λ + WA=( WA +4 WA λ WA (λ + WA = [ ( WA +4 WA λ ] p 4 ( W 4 + p 3 ( W 3 + p ( W + p 1 W + p 0 =0 where: p 4 = 4A 4 p 3 = 15A 4 p = A (8A 4λ λa p 1 = 17λ A p 0 = 4λ 4 (B. 4W (γ +1 σ 4 n = λ +W (γ +1σ n 4 W.(γ +1σ n Using the approximation (18 yiels the ollowing: (A. APPENDIX C PROOF OF THE OPTIMAL RESULT (3 Proo: When z< 0.5, ( is use, then: ( 1 z z π z z. =. z. = 1 π z π (A.3 Denote A (γ +1σ n, then (A.3 is urther erive ( z π e z. z. = z e z z t [ s π 1 ] e z (1 z. z. = z e z π π [ 1 z ]. z. = z (C.1 Digital Object Ientiier: /ICST.WICON

9 Substituting (A. into (C.1 leas to the ollowing result: [11] P. K. Varshney, Distribute Detection an Data Fusion, ser. Signal [ ( λ W (γ +1σn ] [ λ +ts W (γ +1σn ] processing an ata usion, C. S. Burrus, E. New York: Springer, 1. W (γ +1σn W.(γ +1σn [1] F. Digham, M.-S. Alouini, an M. K. Simon, On the energy etection = λ W (γ +1σn o unknown signals on the energy etection o unknown signals over aing channels, IEEE ICC, vol. 5, pp , May 003. W (γ +1σn [13] E. Peh an Y.-C. Liang, Optimization or cooperative sensing in cognitive raio networks, IEEE WCNC, pp. 7 3, March 007. [14] Y.-C. Liang, Y. Zeng, E. C. Y. Peh, an A. T. Hoang, Sensingthroughput traeo or cognitive raio networks, IEEE Transactions on Wireless Communications, vol. 7, no. 4, pp , April 008. [15] H. N. Pham, Y. Zhang, P. E. Engelsta, T. Skeie, an F. Eliassen, Optimal cooperative spectrum sensing in cognitive sensor networks, In The 5th International Wireless Communications an Mobile Computing Conerence (IWCMC, June 009. [16] I. S. Grashteyn an I. M. Ryzhik, Table o Integrals, Series, an Denoting A (γ +1σn,then: [ ( λ t ] [ ] swa λ + ts WA ts WA 1. = λ WA WA ts WA λ (λw A + WA +ts W A WA. λ + WA = λ WA ( λ (λw A + WA +ts W A.(λ + WA = WA (λ WA t 3 sw 3 A 3 + t s(w A 3 λw A (λ WA+3λW A + λ 3 =0 ( ( A λ λ u 3 + u ( 3 +3λA λ u A A + =0 A u 3 + au + bu + c =0, where: u = W (C. Thus, t s is solve as a root o (C. as ollows: t s = 1 W [max(x 1,x,x 3 ] x 1 = s + t b 3a x = 1 b (s + t x 3 = 1 b (s + t { s =(r + q 3 + r 1 3 t =(r q 3 + r 1 3 3a + 3 3a 3, where: an (s tj (s tj {q = 3ac b 9a r = 9abc 7a b 3 54a 3 Proucts, 7th e., A. Jerey an D. Zwillinger, Es. Elsevier, Feb [17] C.-T. Chou, S. N. Shankar, H. Kim, an K. G. Shin, What an how much to gain by spectrum agility? IEEE Journal on Selecte Areas in Communications (JSAC, vol. 5, no. 3, pp , April 007. REFERENCES [1] J. Mitola, Cognitive raio or lexible mobile multimeia communications, Mobile Networks an Applications, vol. 6, Sept [] I. F. Akyiliz, W.-Y. Lee, M. C. Vuran, an S. Mohanty, Next generation/ynamic spectrum access/cognitive raio wireless networks: asurvey, Comput. Netw., vol. 50, no. 13, pp , 006. [3] F. C. Commission, Spectrum policy task orce report, Tech. Rep. ET Docket No , Nov. 00. [4] IEEE, Ieee p80./1.0 ratanar or wireless regional area networks part : Cognitive wireless ran meium access control (mac an physical layer (phy speciications: Policies an proceures or operation in the tv bans, 008. [5] A. Ghasemi an E. S. Sousa, Collaborative spectrum sensing or opportunistic access in aing environments, IEEE DySPAN, pp , Nov [6] S. M. Mishra, A. Sahai, an R. W. Broersen, Cooperative sensing among cognitive raios, IEEE ICC, vol. 4, pp , June 006. [7] A. Sahai, R. Tanra, an N. Hoven, Opportunistic spectrum use or sensor networks: The nee or local cooperation, IEEE IPSN, 006. [8] W. Zhang, R. K. Mallik, an K. B. Letaie, Cooperative spectrum sensing optimization in cognitive raio networks, IEEE ICC, pp , May 008. [9] W.-Y. Lee an I. F. Akyiliz, Optimal spectrum sensing ramework or cognitive raio networks, IEEE Transactions on Wireless Communications, vol. 7, no. 10, pp , Oct [10] Senora project, 008. [Online]. Available: Digital Object Ientiier: /ICST.WICON