Optimal Cooperative Spectrum Sensing in Cognitive Sensor Networks

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1 Optimal Cooperative Spectrum Sensing in Cognitive Sensor Networks Hai Ngoc Pham, an Zhang, Paal E. Engelsta,,3, Tor Skeie,, Frank Eliassen, Department of Informatics, University of Oslo, Norway Simula Research Laboratory, Oslo, Norway 3 Telenor R&I, Oslo, Norway {hainp, paalee, tskeie, frank}@ifi.uio.no {yanzhang, paalee, tskeie, frank}@simula.no paal.engelsta@telenor.com ABSTRACT This paper aresses the problem of optimal cooperative spectrum sensing in a cognitive-enable sensor network where cognitive sensors can cooperate in the sensing of the spectrum. Such sensor networks are assume to be power resource constraine. With a given threshol for the accuracy of the spectrum etection, we fin the optimal number of cognitive sensors participating in the cooperative spectrum sensing an the optimal sensing interval that imize the total energy consumption of the cooperative sensing. First, the mathematical lower boun an upper boun for the number of cooperative cognitive sensors are foun. Then the optimization problem to imize the total energy consume by a group of sensors is presente. Finally, an efficient approximate solution to the optimization problem is propose. Numerical calculations valiate the accuracy an the performance of the propose scheme. The impact of the noise uncertainty, the choice of the energy etection threshol, an the spectrum banwith on the etection accuracy an the imum total energy consumption is also stuie. Categories an Subject Descriptors C.. [Computer-Communication Networks]: Network Architecture an Design Wireless communication General Terms Design, Performance, Reliability Keywors Spectrum Sensing, Optimization, Energy Consumption. INTRODUCTION The Feeral Communications Commission (FCC in the US estimates that only 5% - 85% of the assigne spectrum Permission to make igital or har copies of all or part of this work for personal or classroom use is grante without fee provie that copies are not mae or istribute for profit or commercial avantage an that copies bear this notice an the full citation on the first page. To copy otherwise, to republish, to post on servers or to reistribute to lists, requires prior specific permission an/or a fee. IWCMC 9, June -4, 9, Leipzig, Germany Copyright 9 ACM /9/6...$5.. is utilize, epening on geographical an temporal variations []. The main reason is that the policy of fixe or static assignment of the spectrum leas to spectrum unerutilization. Therefore, cognitive raio (CR has been envisione by J. Mitola as the emerging technology to accommoate ynamic spectrum access [9] []. In a cognitive raio network, the unlicense (seconary evices can utilize the license spectrum when it is unuse by any license (primary evices. However, the occupie spectrum will nee to be vacate instantly when a primary evice starts using it in orer to avoi interfering with the transmission of the primary evice. Thus, spectrum sensing, which enables a CR evice to etect an aapt to primary usage of the spectrum ban, plays a critical role in ynamic spectrum utilization. The use of cognitive-enable sensors is an emerging technology for spectrum sensing. These sensors sense the spectrum ban continuously an report the etection results to seconary evices that will make use of the spectrum. However, one single sensor might perform poorly when the communication channel experiences faing an shaowing. To overcome this problem, cooperative spectrum sensing by a group of collaborating sensors has been propose to exploit multi-user iversity in the sensing process [4, 8, ]. Spectrum sensing consumes energy for the receiver an base-ban circuitry an epletes the battery life-time of a cognitive sensor. On the one han, one wants to gain a high sensing (or etection reliability by using many collaborating cognitive sensors an a long sensing interval. On the other han, one wants to save as much energy as possible by using fewer sensors an a shorter sensing interval. This traeoff is explore in this paper. To the best of our knowlege, this key problem has not been stuie before. It is assume that each cognitive sensor performs spectrum sensing with the well-known energy etection scheme [3]. The performance of this scheme is represente by two essential parameters, i.e. the probability of etection P an the false alarm probability P f. A high etection probability means a high accuracy of etecting the activity of primary users. Furthermore, a low false alarm probability translates into a high usage of available spectrum by the seconary evice, ue to a low chance that the spectrum is mistakenly believe to be occupie when it is actually available. The previously mentione traeoff means to keep a high P an an acceptable level of P f an at the same time preserve the power resources of the cognitive sensors as much as possible. The rest of the paper is organize as follows. First, the

2 relate work is presente in Section. Next, the bouns for the number of cognitive sensors in cooperative spectrum sensing is formulate in Section 3. Then, Section 4 presents the energy imization problem an an approximation approach to efficiently solve the optimization uner the given accuracy of primary user etection. In Sections 5 numerical results are use to explore the optimization an valiate the performance an accuracy of the propose scheme. Finally, conclusions an future work are rawn in Section 6.. RELATED WORK There has been a high research focus on improving the accuracy of spectrum sensing as well as fining optimal spectrum sensing strategy, an a lot of literature on this issue is available. In [6], Lee an Akyiliz propose the interesting iea of optimizing the sensing parameters in orer to maximize the sensing efficiency subject to interference avoiance constraints in a single spectrum ban. They propose spectrum selection an scheuling methos where the bespectrum bans for sensing are selecte to maximize the sensing capacity. An aaptive an cooperative spectrum sensing metho where the sensing parameters are optimize aaptively to the number of cooperating users is also consiere. Furthermore, in [7], Liang et al. stuy the problem of esigning the sensing uration to maximize the achievable throughput for the seconary network uner the constraint that the primary users are sufficiently protecte. They formulate the sensing-throughput traeoff problem an use the energy etection sensing scheme in orer to prove that the formulate problem has one optimal sensing time, which yiels the highest throughput for the seconary network. In [], the authors consier sensor networks that attempt to reclaim some of the available spectrum for their own communications by using spectrum sensing to etect the absence of the primary user. Different nearby sensor networks cooperate to reuce uncertainty cause by the presence/absence of possible interference from other users. Peh an Liang show in [] that cooperation among all seconary users in the network oes not necessary achieve the optimum performance. Instea, optimum is achieve with a cooperation that involves only a certain number of seconary users, i.e. those users sensing the highesignal to noise ratio of the primary transmission. 3. LOWER BOUND AND UPPER BOUND FOR THE NUMBER OF SENSORS 3. Energy Detector for Spectrum Sensing Each cognitive sensor i is assume to erive the singlenoe etection probability P i an single-noe false alarm probability P fi using the energy etection scheme for spectrum sensing [3, 4]. P i an P fi can then be evaluate in terms of the Q-function [5], following the approach in [6]: «λ tsw(γ i + σn P i = Q ( 4tsW(γ i + σn P fi = Q «λ tswσn 4tsWσn 4 where is the sensing interval, which is assume to be the same for every sensor, W is the spectrum banwith, an λ is the energy etection threshol. σ n,i an σ n,i are the ( variance of the noise an of the receive signal aensor i, respectively. Here, the Signal-to-Noise-Ratio (SNR at the sensor i is γ i = σs,i/σ n,i. Without loss of generality, it is assume that the variance of the noise is the same at every sensor, an it can therefore simply be enote as σ n. Observe that P i is monotonically increasing with regar to the sensing interval an the SNR γ i. This means that if all sensors have the same sensing interval, the sensors that experience the lowest SNR will yiel the lowest etection probabilities or the least accurate etection. Thus, one might reuce the total energy consumption by excluing these sensors from the group of sensing noes, an still keep the total etection probability above the require threshol. In aition, the primary activity has also an effect on the spectrum sensing performance. The traffic pattern of the primary user on a channel can be moele as a two state inepenent an ientically istribute (i.i. ON-OFF ranom process, whose ON an OFF istributions are exponentially istribute with means equal to T on an T off, respectively. Hence, the probabilities of the ON an OFF perios T are P on = on T on+t off an P off = T off T on+t off, respectively [6, Ref. 5]. Therefor sensor i can etect a spectrum by the single-noe etection an false alarm probabilities as [6]: i = P onp i (3 fi = P off P fi (4 3. A Cooperative Spectrum Sensing Scheme In this paper, it is assume that the cooperative spectrum sensing is coorinate by the common receiver. The common receiver invites a group of sensors to participate in spectrum sensing. After having receive the invitation, all the cognitive sensors of the invite group, say G, will inepenently starensing the spectrum an then return their observations back to the common receiver. It is further assume that the common receiver is employing theor-rule for ecision fusion an that the communication channels with the invite cognitive sensors are perfect. Hence, the cooperative etection probability Q an the cooperative false alarm probability Q f are given by [4]: Q = ( i (5 Q f = ( fi (6 where is the number of the invite sensors in G an i, fi are the sing-noe etection an false alarm probabilities estimate from sensor i accoring to (3 an (4, respectively. We observe from (5 an (6 that when the number of cooperative sensors for spectrum sensing, n (or, increases, then the cooperative etection probability Q of the group increases an as a result the accuracy of primary user being etecte also increases. However, the higher, the higher cooperative false alarm probability Q f which in turn leas to a higher chance that a spectrum opportunity will be misse. In aition, the more sensors participate in the cooperative group for spectrum sensing, the more energy is consume. This shoul be strictly avoie, since sensors are assume to be battery-powere an have limite energy re-

3 source. That is why fining an optimal size of such cooperative spectrum sensing group is an important issue. In aition, an energy-efficient problem of how to efficiently select the actual sensors that experience the highest SNR an that are well separate to avoi correlation shaowing is also critical. This problem is consiere as the future irection of the present paper. 3.3 Boun for the Size of the Cooperative Sensing Group Given a threshol Q for the cooperative etection probability, the conition Q Q is necessary to be confient that a primary user is etecte. Hence, (5 yiels: Q ( i (7 Let enote the imum of the single-noe etection probability among the group of sensor such that: { i, i = [... n]} «λ tsw(γ + σn =P on.q 4tsW(γ + σn where, the imum SNR is: γ {γ i, Hence, if is boune by: Q ( log( Q log( then the inequality in (7 will be satisfie as: Q ( ( i (8 i = [... n]}. Similarly, given a cooperative false alarm probability threshol Q f, the conition Q f Q f is neee to ensure that a spectrum opportunity is not misse. Likewise, (6 yiels: (9 Q f ( fi ( max f Here we enote as the maximum single-noe false alarm probability among the group such that: max f =max{ fi, i = [... n]} ( Then, ( is guarantee when: ( fi ( max f Q f which leas to the following upper boun: Q f ( max f log( Q f log( max f ( Combining (9 an (, the boun for is erive as: & log( Q $ log( Q % f (3 max log( log( f where. an. enote the ceiling an flooring functions for the rouning of a real number to an integer, respectively. It is observe that the higher the single-noe etection probability the fewer sensors are require to guarantee a given threshol. More importantly, the lower boun shows that the higher the imum SNR conition among the sensors, the fewer sensors nee to be inclue in the spectrum sensing. This gives a hint that one shoul only select as part of the sensing group only the sensors that experience a sufficiently high SNR. On the other han, the upper boun inicates an invaluable physical meaning for the requirement of the false alarm probability of the cooperative group. It means the require threshol Q f cannot be as small as possible, since a low Q f requires a small number of sensors, which may break the etection accuracy by breaking (3. 4. ENERG CONSUMPTION MINIMIZA- TION The lower boun in (3 gives the imum number of sensors collaborating in spectrum sensing in orer to yiel a given accuracy of the primary user etection. Clearly, the fewer the sensors use for cooperative spectrum sensing, the less the total sensing energy consume. Hence, for energy efficiency, the lower boun of in (3 is selecte as the imum number of spectrum sensing sensors of group G: & log( Q (4 log( However, this formulation oes not mean that the size of G is optimal in terms of imizing the total sensing energy consumption for cooperative spectrum sensing. It can be seen from ( that the etection probability epens on the sensing interval. The higher the sensing interval, the more accurate the etection probability. Then, fewer sensors will nee to participate in the cooperative spectrum sensing (see (4, an consequently less energy will be spent on the spectrum sensing. On the other han, the higher the sensing interval, the more energy is consume for spectrum sensing. As a result, there is a traeoff in estimating or in the energy efficiency problem. Let δe ss enote the sensing energy consumption per unit of spectrum sensing interval, which is assume to be the same for every cognitive sensor in the network. Hence, for a sensing interval, each sensor i consumes the sensing energy = (δe ss.. The total sensing energy consume by the group G is imize as follow: E ss i Minimize: X E ss i.(δe ss. log( Minimize: Q log(.(δess. (5 To refine the above optimization problem, it is argue that the absolute function log( is monotonically increasing with regar to. Thus, (5 is equivalent to: Minimize: log( Q.δE ss. log( Maximize: log( Q.δE Maximize:. ss P Q on log( Q.δE. ss λ tsw(γ +σn 4tsW(γ +σn Without loss of generality, it is assume that δe ss is known an that Q is given. P on = is also known an T on T on+t off

4 inepenent of the sensing interval. Finally, we erive the following maximization problem with regar to the sensing interval for imizing the total sensing energy consume by the cooperative group G as: t s = arg max arg max Q λ tsw(γ +σ n 4tsW(γ +σ n Q(z = R π z where: z = λ tsw(γ + σn 4tsW(γ + σn e x x (6 The optimality of (6 can be foun by solving the roots t s of the first partial erivative as follows: Q(z (Q(z. z z t = s. Q(z = t s (Q(z. z. = Q(z (7 z It can be shown thaolving (6 analytically is extremely ifficult ue to the exponential characteristic of the Q-function. Therefore, an approximation approach to solve (6 is propose, using the well-known approximation for Q(z: 8 >< Q(z >: / e z / e z if z (8 if z < (9 When z, the approximate optimal sensing interval can be foun by substituting (8 into (7 as follows: ( z e. z. = z z z e ( z.. = ( Then z. is foun as: λ tsw(γ z +σ n 4tsW(γ. = +σn. = λ + tsw(γ + σ n 4 W.(γ + σ n Substituting ( into ( yiels: z. λ + tsw(γ + σn 4 = W.(γ + σn x + (γ + σn = λ + 4(γ + σn 4 where x is given by: x = W(γ + σ n ( The approximate optimal sensing interval (i.e. the approximate root of (7 can now be foun as: p t λ + 4(γ s = + σn 4 (γ + σn ( W(γ + σn On the other han, when z is negative the approximate optimal sensing interval is foun using (9 instea of (8 as the approximation. However, solving the root of (7 analytically by this approximation is still extremely ifficult. In this case, we solve the propose optimization problem numerically to valiate the accuracy of that approximation. The benefit here is that the complexity of fining the optimality will be significantly reuce compare to the complexity of solving (6 irectly. Proviing accurate analytical solutions for this is consiere as the future work. With the solve optimal sensing interval t s above, the optimal number of cooperative cognitive sensors can be foun as: & log( Q n = where: log( = P on.q «λ t s W(γ + σn 4t s W(γ + σn (3 5. NUMERICAL RESULTS In this section, numerical calculations are presente to valiate the approximate optimal sensing interval an the approximate optimal number of collaborating sensors that imize the total energy consumption of the cooperative spectrum sensing. Mean absolute error is calculate as the accuracy metric for the approximate results. In all the numerical calculations, we use the following settings: δe ss =.5 J, T on = sec, T off = sec, Q =.9, Qf =., an banwith W = MHz. The imum SNR threshol (γ varies from to B for comparison. First of all, the valiation of the accuracy an performance of the propose approach is presente in Fig. - Fig. 3 with the chosen parameters for the energy threshol an signal noise as λ = 5 B, σ n = 5 B, respectively. Fig. illustrates the imum total energy consumption for cooperative spectrum sensing. The soli curve illustrates the imum value of the energy consumption foun by solving the original optimization (6 numerically. The ash-otasterisk curve, on the other han, shows the corresponing imum value foun by our propose analytical approximate solution. The results inicate that our approximation gives a rather goo solution to the original optimization problem, as the error is aroun 8.6% on average. It is also observe that when the imum SNR increases, less energy is neee for the cooperative spectrum sensing while still satisfying the require spectrum sensing accuracy. Minimum Total Consume Energy (J x 5 Original Optimization Approximation Minimum SNR (B Figure : The imum total energy consumption. A similar tren can also be observe from Fig.. The figure reconfirms that the optimal sensing interval solve by our propose approximation (the ash-ot-asterisk curve follows closely to that of the original optimization (the soli

5 curve. Again it is seen that the higher the imum SNR Optimal Sensing Interval (sec x 4 Original Optimization Approximation Minimum SNR (B Figure : The optimal sensing interval. threshol, the shorter time the cooperative cognitive sensors will nee to complete the spectrum sensing while still satisfying the require accuracy of the sensing. However, the optimal sensing interval solve by our approach is a bit longer (about 3.6% on average than that of the original optimization. This inaccuracy occurs as a consequence of approximating the Q-function. Fig. 3 presents the optimal number of cognitive sensors foun by our approach (the ashe bar an that of the original optimization (the soli bar. The results show that 4 Original Optimization Approximation the error between our approach an the original one is between an sensors on average. When the SNR conition is better, the optimal number of sensors is higher, which is ue to the reason that the energy imization prouces shorter optimal sensing interval uner the goo SNR conition. However, it nees to satisfy the requirement for etection accuracy, hence the sufficient number of sensors will nee to be inclue uring the optimization. It is also observe here that a higher error is cause when the imum SNR is higher than B, which is ue to the inaccuracy of the propose approximation. Improving this high inaccuracy is consiere as the future work of the present paper. The performance metric for the spectrum etection accuracy is evaluate in terms of the summation (the maximum total etection error probability: err max of the missetection an false alarm probabilities as: err max = ( Q +Q max f = ( n + ( f max n. Fig. 4 presents this maximum total etection error probability with regar to the noise uncertainty. It can be seen the Minimum SNR (B Signal noise = B Signal noise = 8 B Signal noise = 5 B Signal noise = B Signal noise = 5 B Signal noise = B Signal noise = B Signal noise = 8 B Signal noise = 5 B Signal noise = B Signal noise = 5 B Signal noise = B Number of Sensor Minimum SNR (B Figure 3: The optimal number of sensor Minimum SNR (B Figure 4: Two graphs of the maximum total etection error probability when the ceiling function is use an not use for n in (3. λ = B slightly ifference between the upper an lower graphs in Fig. 4. It is cause by using the ceiling function in (3 to roun the optimal number of sensors up to an integer. Since the number of sensors must be an integer, so it can be also observe from all the graphs in this section that they are not as smooth as the graphs resulte by not using the ceiling function in (3. The results in Fig. 4 show that the signal noise uncertainty has strong effect on the etection accuracy of the cooperative spectrum sensing using the energy etector, especially when the imum SNR conition is ba. The main reason is ue to the fact that the energy etection scheme is not robust in the low SNR conitions an is sensitive to the signal noise [3]. It also confirms that our approach prouces goo etection accuracy when the signal noise uncertainty is not very high an the SNR conition is not very low. Tackling the etection inaccuracy uner very low imum SNR conitions an strong noise uncertainty is the future work of the present paper. Secon of all, the impact of the noise uncertainty σ n, the choice of the energy threshol λ, an the spectrum banwith W on the etection accuracy an the imum total

6 Noise = B SNR = B SNR = B SNR = B SNR = 3 B SNR = 4 B SNR = 5 B Noise = B SNR = B SNR = B SNR = B SNR = 3 B SNR = 4 B SNR = 5 B lamba (B lamba (B Noise = 5 B SNR = B SNR = B SNR = B SNR = 3 B SNR = 4 B SNR = 5 B Noise = 5 B SNR = B SNR = B SNR = B SNR = 3 B SNR = 4 B SNR = 5 B lamba (B lamba (B Figure 5: The maximum total etection error probability vs. energy threshol an noise uncertainty. energy consumption for the cooperative spectrum sensing scheme, especially uner the low imum SNR conitions, is stuie in Fig. 5 - Fig. 7. The results in Fig. 5 show how the energy threshol an the noise uncertainty affect the maximum total etection error probability err max uner ifferent imum SNR conitions. It can be observe that uner noise uncertainty, the energy etector can only perform well uner the low imum SNR conitions if the energy threshol is set high enough. For example, when σ n = 5 B, it is better to choose λ = 5 B, as we use in the calculations for Fig. - Fig. 3, in orer to keep err max. when the imum SNR γ 3 B. In aition, the four graphs in Fig. 5 illustrate that when the noise uncertainties are B, 5 B, B, an 5 B, the corresponing energy threshols shoul be selecte as about 5 B, 5 B, 5 B, an 35 B, respectively in orer to keep the energy etector etecting the spectrum accurately uner the low imum SNR conitions. This can be implie generally that when the noise uncertainty increases by 5 B, the energy threshol shoul have to be increase by aroun B, which requires much more energy consumption in orer to prouce high etection accuracy uner the low imum SNR conitions. It inicates again that the cooperative spectrum sensing using energy etector is highly sensitive to the noise uncertainty an the choice of the energy threshol. Thus, the important issue of fining an analytical form of the optimal energy threshol uner a specific range of the noise uncertainty while satisfying a given spectrum etection accuracy is consiere as the future work of the present paper. Furthermore, we observe how the choice of the energy threshol λ for the energy etector affects the imum total energy consumption when the noise uncertainty is fixe as σ n = 5 B in Fig. 6. Obviously, the higher energy threshol, the more sensing energy will nee to be consume by the group of collaborating sensors. However, uner the noise σ n = 5 B, the results in Fig. 5 show that the energy threshol shoul be selecte as λ = 5 B in orer to yiel err max. uner the low imum SNR conitions. Hence there is a traeoff in selecting the energy threshol to keep the imum total energy consumption an the maximum total etection error probability as low as possible at the same time, especially when the imum SNR conition is ba. On the other han, uner the goo imum SNR conitions, for example when γ 5 B, a wie range of λ coul be selecte to yiel err max., hence the lower the selecte λ, the lower the imum total energy consumption. It again inicates that the cooperative Minimum Total Consume Energy (J x 4 lamba = 5 B lamba = 5 B lamba = 5 B Minimum SNR (B Figure 6: Minimum total energy consumption vs. energy threshol λ for the energy etector. spectrum sensing scheme using energy etector is highly sensitive to the energy threshol uner the low imum SNR conitions. Thus, making this scheme be less sensitive to the noise uncertainty an/or the energy threshol in a wie range of the imum SNR conitions is seen to be an important issue. We consier this observation as the future

7 work of the present paper. Finally, the impact of the spectrum banwith on the imum total energy consumption for cooperative spectrum sensing is presente in Fig. 7. In this figure, the energy threshol an the signal noise are set as λ = 5 B, σ n = 5 B, respectively. The similar observation can also be seen here that the higher the imum SNR, the lower the total energy consumption. However, the larger the spectrum banwith to be sense, the lower the total energy consumption, which is cause by perforg shorter sensing interval in cooperative spectrum sensing. Minimum Total Consume Energy (J 3.5 x W = MHz W = MHz W = 3 MHz W = 6 MHz W = MHz Minimum SNR (B Figure 7: Minimum total energy consumption vs. banwith W. 6. CONCLUSION AND FUTURE WORK This paper explores the problem of optimal cooperative spectrum sensing in cognitive-enable sensor networks. The cognitive-enable sensors are assume to be battery powere an constraine by limite power resources. Thus, the key problem is how to imize the total energy consumption for cooperative spectrum sensing by a group of cognitive sensors while still satisfying the given threshol for the spectrum etection accuracy. First, an expression for the lower an upper boun for the number of cognitive sensors participating in the spectrum sensing is foun. The optimization problem for imizing the total energy consume by that group for cooperative spectrum sensing is also propose. Furthermore, an approximation approach to solve the propose energy consumption imization is presente. Finally the accuracy of the approximation is valiate using numerical calculation to compare the approximate optimization with the original propose optimization. The impact of the noise uncertainty, the choice of the energy threshol for the energy etector, an the spectrum banwith on the spectrum etection accuracy an the imum total energy consumption for the cooperative spectrum sensing scheme is also iscusse. In the future, improving the etection accuracy uner extremely low imum SNR conitions an high noise uncertainty as well as fining more accurate approximations for analytically solving the optimal results are necessary. Furthermore, fining an analytical form of the optimal energy threshol uner a specific range of the noise uncertainty while satisfying a given spectrum etection accuracy is aresse as an important issue for the future work. Finally, making the cooperative spectrum sensing scheme using energy etector be less sensitive to the noise uncertainty an/or the energy threshol in a wie range of the imum SNR conitions is the future work as well. 7. REFERENCES [] I. F. Akyiliz, W.-. Lee, M. C. Vuran, an S. Mohanty. Next generation/ynamic spectrum access/cognitive raio wireless networks: a survey. Comput. Netw., 5(3:7 59, 6. [] F. C. Commission. Spectrum policy task force report. Technical Report ET Docket No. -55, Nov.. [3] F. Digham, M.-S. Alouini, an M. K. Simon. On the energy etection of unknown signals on the energy etection of unknown signals over faing channels. IEEE ICC, 5: , May 3. [4] A. Ghasemi an E. S. Sousa. Collaborative spectrum sensing for opportunistic access in faing environments. IEEE DySPAN, pages 3 36, Nov. 5. [5] I. S. Grashteyn an I. M. Ryzhik. Table of Integrals, Series, an Proucts. Amsteram: Elsevier, 7th eition, Feb. 7. [6] W.-. Lee an I. F. Akyiliz. Optimal spectrum sensing framework for cognitive raio networks. IEEE Transactions on Wireless Communications, 7(: , Oct. 8. [7].-C. Liang,. Zeng, E. C.. Peh, an A. T. Hoang. Sensing-throughput traeoff for cognitive raio networks. IEEE Transactions on Wireless Communications, 7(4:36 337, April 8. [8] S. M. Mishra, A. Sahai, an R. W. Broersen. Cooperative sensing among cognitive raios. IEEE ICC, 4: , June 6. [9] J. Mitola. Cognitive raio for flexible mobile multimeia communications. Mobile Networks an Applications, 6, Sept.. [] E. Peh an.-c. Liang. Optimization for cooperative sensing in cognitive raio networks. IEEE WCNC, pages 7 3, March 7. [] A. Sahai, R. Tanra, an N. Hoven. Opportunistic spectrum use for sensor networks: the nee for local cooperation. IEEE ICC, 6. [] S. J. Shellhammer, S. S. N, R. Tanra, an J. Tomcik. Performance of power etector sensors of tv signals in ieee 8. wrans. In TAPAS Workshop, page 4, 6. [3] R. Tanra an A. Sahai. Snr walls for signal etection. IEEE Journal of Selecte Topics in Signal Processing, (:4 7, Feb. 8.