OPTIMIZED COOPERATIVE SPECTRUM-SENSING IN CLUSTERED COGNITIVE RADIO NETWORKS

Transcription

1 International Journal of Wireless & Mobile Networks (IJWMN) Vol. 5, No. 4, August 3 OPTIMIZE COOPERATIVE SPECTRUM-SENSING IN CLUSTERE COGNITIVE RAIO NETWORKS ABSTRACT Birsen Sirkei-Mergen and Wafa-Iqbal Eletrial Engineering, San Jose State University, San Jose, CA birsen.sirkei@sjsu.edu, wafa.iqbal@gmail.om In ognitive radio networks, the ertinent task of setrum sensing at the Seondary Users (SUs) an be ahieved when the SUs ooerate in order to make a final deision about the resene of a ommuniating Primary User (PU). In this aer, we study a two-ho relaying system in whih SUs are groued into lusters. The SUs transmit a simle ower funtion (arameterized by ) of their observationto a Fusion Centre (FC) using orthogonal hannels. The FC ombines the reetions from ooerating nodes linearly. The goal of this work is to maximize the robability of detetion over the arameters (number of lusters), (ower funtion exonent), and w(linear ombining oeffiients) for a given false alarm robability. Overall, this work quantifies the advantages of otimal ooeration in rimary detetion in ognitive radio networks. KEYWORS Wireless Networks, Relaying, Setrum Sensing, Cognitive Radios, Cooeration, Fusion Centre.. INTROUCTION Wireless ommuniation is rogressing at an aelerated seed. Inreasing variety of aliations and features of wireless devies is leading to demands for higher and higher data rates. owever, the bandwidth liensed to radio ommuniation is limited. The infamous question is ow do we get better data rates under limited bandwidth requirements to meet the demand?.effiient setrum utilization is the key to answer this question. In, Federal Communiations Commission (FCC), the US government ageny that regulates the use of frequeny bands of the eletromagneti setrum, indiated that the liensed frequeny bands are unused 9% of the time[]. In 8, FCC ruled that unused ortions of the RF setrum will be made available for ubli use under ertain onditions. In the light of this rule, setrum effiieny an be imroved if radio devies are equied with tehnologies that take advantage of the liensed setrum when it is unused. An emerging advaned solution for effiient setrum utilization is the so-alled ognitive radios. A ognitive radio (CR) is a transeiver tehnology in whih frequeny setrum is ontinuously sensed for unouied saes. In a CR system, the rimary user (PU) is the one who has liensed rivilege to transmit in a artiular frequeny band and other users known as seondary users (SU) are the unliensed users who desire to share the setrum. The available unused frequeny bands are alled setrum holes. SUs sense the setrum for setrum holes ontinuously. A CR is aable of not only sensing the setrum, but also, monitoring, deteting and adating its ommuniation hannel aess. For examle, a CRan intelligently adjust its transmission arameters aording to the availability in the frequeny bands[],[3]. CRtehnology has gained a OI :.5/ijwmn.3.54

2 International Journal of Wireless & Mobile Networks (IJWMN) Vol. 5, No. 4, August 3 lot of attention in the last deade. Currently, ommuniation standards are adatingthis tehnology [4]. Cooerative setrum sensing is a sheme in whih SUs ooerate with eah other in a distributed or entralized manner, in order to make the deision about setrum availability. This ould be done via a Fusion Centre (FC). The SUs sense the hannel for the resene of PUs and relay a funtion of their observations to the FC for a olletive deision. The hoie of this relaying is ritial in order to otimize the overall erformane at the FC.In the next subsetion, we summarize the reent relevant work on ooerative setrum sensing... Cooerative Setrum Sensing In ooerative setrum sensing, the SUs ollaborate with eah other in sensing the setrum [5]. If otimized, ooeration redues the ower requirements at the SUsand imroves the sensing erformane even if it may introdue overhead for ertain ases. In the ase when SUs ooerate through a FC, every SU transmits its reeived signal to the FC that makes a deision about the resene of a PU based on the olletive information from all the SUs.This is also alled relayassisted ooerative setrum sensing [6]. The transmissions of the SUs to the fusion enter ould be on orthogonal hannels [6], [7]. In this ase, eah SUforwards a funtion of their observation to the fusion enter through and individualorthogonal hannel similar to the well-known time-division multile-aess (TMA), or frequeny-division multile-aess (FMA). On the other hand, transmissions of the SUs to the fusion enter ould be non-orthogonal, that is ooerating SUs transmit a funtion of their observation byusing the same hannel. In the non-orthogonal hannel model, it is assumed that SUs are synhronized so that the reeived signal in the fusion enter is the oherent sumof transmitted signals by SUs[8], [9]. For orthogonal hannels, the fusion enter anuse various ombining tehniques of the reeived vetor to obtain the final deision. It is shown in [][] that the robability of error for oherent orthogonal hannel system will not imrove with the inreasing number of SUs. On the otherside, the erformane of the non-orthogonal hannel systemimroves with the inreasing number of SUs due to the array gain [9], []. It iswell-known that in order to have an energy-effiient and reliable setrum sensing, it is imortant for SUs to ooerate with eah other when sensing for the PUs. owever, one has to arefully weigh the trade-offs between the ahievable Cooerative Gain and the inurred Cooerative Overhead[]. In the ase of orthogonal aess between SUs and a FC, eah radio is dediated anorthogonal hannel, and the requirement for bandwidth sales by the number of SUs. Then, the reetions from SUs at the FC are ombined. In general, the linear ombining tehniquesare attrative, beause, they are simle omared to non-linear tehniques, and when the weighting oeffiients are otimized, the imrovement in the robability of detetion at the fusion enter is signifiant. Onthe other hand, in the ase when non-orthogonal aess is utilized from SUs to the FC, bandwidth requirements are negligible. Furthermore, the additive noise in the non-orthogonal hannelis negligible, eseially for large networks, omared to orthogonal hannels sine it is indeendent of the number of SUs. The gains due to otimized weighting oeffiients in orthogonal hannels and the indeendene of noises from the number of SUs in non-orthogonal hannels osea trade-off. In order to otimizethis trade-off, one sheme roosed in [3]by the first author: grou-orthogonal multile aess hannel (MAC) aroah for setrum sensing. In grou- orthogonal MAC, SUs utilize the available orthogonal hannels in lusters, and eah SU transmit to the FC the energy of its reetion from the PUs. In [3], authors exloit the benefits of both orthogonal and non-orthogonal transmissions byfinding the otimal

3 International Journal of Wireless & Mobile Networks (IJWMN) Vol. 5, No. 4, August 3 number of users that should be in an orthogonal grou and the otimal linear weighting oeffiients at the FC. In this aer, we study otimal relaying funtion at SUs under different hannel aess shemes from the SUs to the FC. The onsidered ases are orthogonal, non-orthogonal and grouorthogonal multile-aess hannels (MACs). In the grou-orthogonal ase, SUs are lustered into grous that transmit on orthogonal hannels. In fat, orthogonal MAC and nonorthogonal MAC are seial ases of grou-orthogonal MAC when =number of SUs, and =, resetively. The exressionsfor the robability of detetion as a funtion of robability of false alarm under different grou sizes and maings arederived and analysed. This aer otimizes the erformane over a set of relay funtions and hannel aess shemes. This will hel the SUs to make intelligent deisions when seletinghow and what to send to FC,for given a robability of false alarm in deteting the setrum availability. The rest of the aer is organized as follows. In Setion, we give the roblem formulation. In Setion 3, we derive theotimal number of grous and weighting oeffiientunder ertain assumtions. Simulationresults are given in Setion 4. Finally, Setion 5 onludesthe aer.. SYSTEM MOEL We onsider a ognitive radio network that is omosed of apu, multile SUs and a FC-whih ould also be one of the SUs (see Fig. ). Although the setrum band under onsideration is liensed to the PUs, they may or may not be transmitting during the onsidered time-slot. ene, SUs need to deide whether thepuis idle (null hyothesis) or it is using the hannel (alternative hyothesis) in order to utilize the band effiiently. In the onsidered set-u, the deisions are made ooeratively- that is eah user makes deisions based on reetions from multile SUs whih also serve as relays. When ating as relays, eah SU makes an observation, and transmits a signal based on solely its observation to the FC. We assume the FC ombines the reeived signals linearly and makes final deision about the existene of the rimary based on the ombined signal. Linear ombining at the FC is an attrative method rimarily due to its simliity. The two hyotheses: (no rimary user exists) and (at least one rimary user exist) form a binary hyothesis test given as below: ( ) i ( ) ( ) ( ) ( ) : x k = v k i =,... M, k =,... N i : x k = h s k + v k i =,... M, k =,... N i i i wherex i (k) is the observed signal by the ith seondary user over N timeslots, s(k) is the transmitted signal by the PU in the kth timeslot and v i (k)is the additive noise at the ith user in the kth timeslot. The noisev i (k) is assumed be white Gaussian noise with zero mean and variane and also v i (k)sare assumed to be indeendent and identially distributed (i.i.d.) over time index k and user index i. The hannel gains from the PU to the SUsare assumed to stay onstant over the observation interval (slow fading senario). In the network, eah SU observes the hannel for N timeslots and then forwards a ower funtion of the observed signal to the FC: u i = f(x i ) = i x i where i = P i / E{ xi } is the saling fator so that average transmission ower is boundedbythe ower onstraint P i. A ommon relay oeration is to send the energy of the observedsignal [3], whih is equivalent to the ase when 3

4 International Journal of Wireless & Mobile Networks (IJWMN) Vol. 5, No. 4, August 3 is hosen to be, and i = in our senario. Our goal is to otimize over the ower funtion exonent so that the erformane is imroved. Figure. Grou-orthogonal MAC with M = 3 seondary users and lusters In the network, SUsare groued into lusters (see Fig. ). Eah luster is dediated to on an orthogonal hannel, and users in the same luster transmit on the same orthogonal hannel. We all this set-u grou-orthogonal MAC (Multile Aess Channel). The lusters are assumed to be re-determined. For examle, one ould form lusters based on geograhial roximity or signal quality. owever, the question of how the lusters are formed is out of soe of this aer. Let S j denote a set of users in the jth grou where j=. For simliity in the analysis, we also assume that lusters have equal number of nodes. Then, the ombined signal in the jth orthogonal hannel an be written as: y = g u + n j =,... j m m j m S j where g m is the hannel gain from SU to the FC and n j is the noise added at eah hannel and is assumed to be i.i.d. white Gaussian noise with zero mean and variane σ. When information 4

5 International Journal of Wireless & Mobile Networks (IJWMN) Vol. 5, No. 4, August 3 from eah grou reahes the FC, it is linearly ombined after being weighted. The weighting vetor is defined as w = [w, w,, w ]. After ombining, the signal observed at the FC is denoted by: y = w j y j = w j g mum + n jw j j= j= m S j j= At the FC, the global test statisti y is omared with γ to make a deision about PUs, that is if y, deide that has ourred, if y <, deide that has ourred. 3. OPTIMIZE COOPERATIVE SPECTRUM-SENSING In this setion, first we desribe the erformane metris that are used, and then we desribe the otimization roblem. Solution for the otimization roblem is also rovided. 3.. Performane Metris We use the two imortant metris: robability of detetion P d = P( ) and robability of false alarm P f = P( ). Our goal is to maximize the robability of detetion, P d, for a given robability of false alarm,p f. The otimization is over the set of arameter: as the number of orthogonal hannels, therelaying funtion i x i, and weighting oeffiients w j s. In the following, we will use entral limit theory [4] to derive analytial exressions for P d and P f. We an argue that for large N (the observation time interval), x i an be assumed to be asymtotially normally distributed as well as y j s and y. For a normally distributed random variabley, the robability of detetion and false alarm an be exressed as follows for a given detetion threshold γ at the FC: E y Pd = P[ ] = P[ y ] = Q Var y () E y Pf = P[ ] = P[ y ] = Q Var y u / whe re Q() x = / e dudenotes the Q-funti on. x () In order to find detetion and false alarm robabilities based on the above mentioned formulas it is required to find the onditional mean and varianes under both hyotheses for a given arameter set. Note that the P d is atually a funtion of not only P f, but also a funtion network arameterssuh as the number of luster (), the relay funtion exonent (), the FC ombining oeffiients ( w), hannel oeffiients between rimary and ognitive radios ( h i s), the hannel oeffiients between the ognitive radios and the FC (g i s), transmission ower of the radios (P i s), and the noise owers and δ. Our goal is to maximize the robability of detetion over the 5

6 International Journal of Wireless & Mobile Networks (IJWMN) Vol. 5, No. 4, August 3 arameters: (i) the number of lusters (); (ii) the relaying funtion exonent ( ); and (iii) weighting oeffiients at the FC, w, when the rest of the arameters are given.this imlies that the hannel state information (CSI) is known for both links between PUs and SUs, and SUs and the FC. Note that the hannel fading oeffiients h i s and g i s are assumed to be slowly varying; hene the CSI assumtion is sensible. The following lemma rovides an exliit analytial exression for P d as a funtion of P f. Lemma : For given robability of false alarm P f, number of lusters, relaying funtion exonent, and weighting oeffiients w, if the hannel gains h i s are all equal ( h i = α, i), then the robability of detetion P d is given as follows for large N: where and ( ) E y E y Q Pf Var y + Pd (,,) w = Q (3) Var y g E[ y ] = A g w, E[ y ] = B g w, Var[ ] y = C w G w + w w, Var[ ] y = w G w + w w, = [ g P,..., g P ], G = diag( g P,...,). g P i i i i i i i i i S i S i S i S The oeffiients A, B, C, and deend on the relay funtion exonent and are given as follows: + + N + A = Γ / Γ + Γ N N B C + Γ N N / =, k / Φ Φ, k + Φ, k Φ, k k = + k = k < k Γ N + = Γ Γ / Γ + Γ N N = Γ Φ, k Γ Φ, k /, k, k, k k = Γ Φ + Γ Φ Φ k = k k < k () s k where Φ, k = F,,, () x denotes the gamma funtion, Γ and hyer-geometri funtion[5]. F denotes the onf luent Proof: Using Eqn. (), for any givenp f, threshold γ an be written as: ( ) = E y + Q Pf Var y (4) 6

7 International Journal of Wireless & Mobile Networks (IJWMN) Vol. 5, No. 4, August 3 By substituting Eqn. (4) in Eqn. (), we obtain Eqn. (3). The derivations for E[y ], E[y ], Var[y ], and Var[y ] are given in the Aendix 3.. It is imortant to note that Lemma rovides a formulation in whih the deendenes on the otimization arameters (,, w) are artially deouled. The variables w, g and G deend on the luster size, the onstants A, B, C, and deend on the relay funtion exonent, and the weighting oeffiient w shows u exliitly in the exression. This will hel solve the otimization roblem. 3.. Otimized Cooerative Transmission and Reetion We formulate the otimization roblem as follows. Given the hannel gains ( h i s, and g i s), the transmission owers of the SUs (P i s), and the noise owers and δ, the goal is to maximize the P d for a given limit on false alarm robabilityp f : max P(,,),, w d w We make the following assumtions in order to solve this roblem.. Uniform hannel gains: h i = α, and g i = >.. Uniform transmission owers: P i = Pfor all i. 3. The orthogonal grous have equal number of SUs (assuming M/ is an integer). 4. The weighting oeffiients,w i s,are nonnegative. Under these assumtions the following theorem rovides the otimal,and wfor a given value. Theorem :In the grou-orthogonalmac system, for a given relay ower funtion with exonent, if hannelgains are equal ( h i = α,g i = ), then the otimalweighting oeffiients that maximizes P d (Eqn. (3)) are uniform for a given, that is w i = /, i =.... And the otimal for given w and is given as (A) B C PM + Pf > Q () C P (A) B C PM + M M Pf < Q ( C ) P Q ()() f C P C PM otherwise (A) B M where X denotes the divisor of M that is losed to X. M Proof:See Aendix 3.. 7

8 International Journal of Wireless & Mobile Networks (IJWMN) Vol. 5, No. 4, August 3 The theorem states otimal linear ombining oeffiients at the FC should be uniform for orthogonal hannels and should sum to. This is very intuitive due to the assumtions on the equal hannel gains and noise owers, and also idential relay funtions at the relays. On the other hand, otimal has three different regions: (i) when the false alarm robability is high (P f >P L ), the otimal is equal to whih imlies that non-orthogonal transmission is otimal; (ii) when the false alarm robability is low (P f <P ), the otimal is equal to M whih imlies that eah SU should transmit on an orthogonal hannel and no lustering of SUs; and (iii) when the false alarm robability is between P L and P, otimal sheme is grou-orthogonal transmission. Note that for some seial senarios, the third region may merge with one of the other regions, that is the rounding oeration in the above equation may lead to = or =M. M Otimization over the relaying funtion exonent an be done by relaing the otimal values for and w obtained in Theorem in Eqn. 3, and by using an otimization toolbox for nonlinear integer rogramming. 4. SIMULATIONS In this setion, we rovide robability of detetion versus robability of false alarmurves for different values of and. For all the simulations, the number of users M=4, observation time N=, the hannel gains h i = α =, g i = β=, relay transmission owers P =, and noise owers σ =, and δ = 5. Below,P d denotes the robability of detetion and P f denotes the robability of false alarm. We assume the rimary signal s(k) = /N, for all k, for simliity. In Fig., we dislay the P d as a funtion of P f when the relay funtion has exonent = and =3. The urves for various hannel aess senarios between relays and FCare shown: orthogonal aess (=M =4), non-orthogonal aess (=), and grou-orthogonal aess (=). It is observed that for lower robability of false alarms, orthogonal MACgives the best robability of detetion, and for higher robability of false alarms, non-orthogonal MACgives the best robability of detetion. Using Theorem, we an obtain the boundaries of these two different regions:p f >.797 and P f <.8 for = and P f >.8 and P f <885 for =3, whih are onsistent with the simulations.in Fig. 3, we dislay the zoomed urves orresonding to the region.8<p f <.797for =. Aording to Theorem 3, in this region otimal ould be,, and 4 whih is what we observe in Fig. 3. Overall, the relay funtion with exonent =3 outerforms the relay funtion with =. Furthermore, we observe that therange of P f where grou-orthogonal MAC is otimal is getting smaller with the inrease in. 8

9 International Journal of Wireless & Mobile Networks (IJWMN) Vol. 5, No. 4, August 3 Figure Probability of detetion (P d ) vs. Probability of false alarm (P f ) with = and =3 Figure 3P d vs.p f for different ranges of P f : (.<P f <.9) Next, we analyse different relay funtions for a given hannel aess sheme in detail. In Fig. 4 and Fig.5, we lot the P d vs.p f urves for the non-orthogonal MAC (=), and orthogonal MAC(=M=4), resetively.it an be onluded that for a given, there does not exist a single relay funtion that erforms otimally for all P f values. In the limit where relay funtion 9

10 International Journal of Wireless & Mobile Networks (IJWMN) Vol. 5, No. 4, August 3 exonent is large, the urves redues to P d = P f line for any value. Similar behaviour is observed for other value. Figure 4P d vs. P f for non-orthogonal hannel ( = ) Figure 5Pd vs. Pf for orthogonal hannel ( = M) In Table, we dislay the otimal relay funtion exonent and otimal luster size for various P f values. It is imortant to note that otimal is always equal to, whih imlies that when otimal relay funtion (or equivalently ) is seleted for a given false alarm robability, then the

11 International Journal of Wireless & Mobile Networks (IJWMN) Vol. 5, No. 4, August 3 non-orthogonal sheme beomes otimal globally. In addition, as P f inreases, the otimal relay funtion exonent dereases. Table Otimal and Range of P f Otimal Otimal <P f <. >.<P f <.3.4<P f <.9.<P f <.6 9.7<P f < <P f <.6 7.7<P f <.5 6.5<P f < <P f < <P f < <P f <. P f = Overall, the otimal relay funtion for any of the hannel aess shemes is always the ower funtion with high exonents for lower robability of false alarms.owever, if the system is robust enough to handle higher robability of false alarms, small ower exonents suh as =, or 3is the otimal hoie of for the relay funtions.it an also be onluded that globally nonorthogonal sheme is otimal under the given assumtions. 5. CONCLUSIONS In this aer, we studied a ognitive radio network in whih SUs ooerate in order to make a deision about the rimary existene. The roosed sheme is distributed in the sense that the ooerating SUs transmit a ower funtion (arameterized with exonent ) of their loal observation, hene does not require any overhead due to ooeration. The SUs transmit to afc (whih ould also be one of the SUs) over orthogonal hannels, and FCombines these reetions linearly using weighting oeffiients w. We rovided analytial solutions and simulations for maximizing the robability of detetion at the fusion entre for a given false alarm robability over the arameters,, and w. It is interesting that non-orthogonal hannel aess beomes otimal globally when the best relaying funtion is utilized even though the orthogonal or grou-orthogonal aess shemes require more bandwidth. This behaviour is not observed in ooeration strategies whererelays simly send their energy to the fusion entre [3]. In summary, this work shows the imortane of otimization in ooerative ognitive radio networks in order to extrat the gains of ooeration for setrum sensing with negligible overhead. 6. APPENICES 6.. Proof of Lemma erivation of E y : For the first hyothesis we derive the exeted value as: N N N i () k E vi ( k) vi ( ) = = k = k = k = E x E k

12 International Journal of Wireless & Mobile Networks (IJWMN) Vol. 5, No. 4, August 3 We know that ( ) i v k is a zero-mean Gaussian (, i ) and for suh a random variable the exeted value of the absolute funtion is given by [6]: E + X =.. Γ ene, + Γ N N E N vi ( k ) + = = Γ k = k = (5) erivation of Normalization fator for yothesis : All the above derived exressions are yet not normalized there we need to find exressions for the normalization fators for every hyothesis. The normalization fator hyothesis an be derived as follows: N N E xi (k) = E(k) vi k = k = It an be easily derived that the exeted value of square of sums is given by: N N E k = E k E k E X X + X X k k = k = k < k (6) Sine v i (k) is indeendent and identially distributed for eah k, substituting the values of the exetations in the above equation gives: N N + N ( N ) + E xi (k) = Γ + Γ k = (7) Sine y is a linear ombination of xi after being saled by hannel gain and weighting fator, so the exeted value of y an be written as: E y = N + Γ N + N ( N ) + Γ + Γ g w erivation of E y : For the seond yothesis, we derive the exeted value as N N E xi (k) E = hi s k vi k k = k = ( ) + ( ) We see that x i (k) is a non-zero mean Gaussian random variable N(h i s(k), σ i) and for a Gaussian random variable with mean µ x and variane σ x +.. x Γ x E X =. F,, x where F is the onfluent hyer geometri funtion.ene, (8)

13 International Journal of Wireless & Mobile Networks (IJWMN) Vol. 5, No. 4, August Γ N N E hi s ( k ) + vi ( k ) hi. s ( k ) = F,, k = k = Using the assumtions we have made about the hannel gain h i, we an simlify the exression as following: N N () E hi s ( k ) vi ( k ) + s k + F,, k = k = = Γ erivation of Normalization fator for yothesis : Using(6), the normalization fator hyothesis an be derived as follows: + Γ N () s k E xi (k) = F,, + k = k k < k + s()() k s k Γ F,, F,, Therefore, we an write the E y as: (9) E y = where N s() k F,, k= + () Γ N s k F,,, k, k + k + Φ Φ = k< k Γ g w () s() k Φ, k = F,,,. () erivation ofvar y : For the first yothesis, we derive the variane value as Var x E v k E v ( ) N N N i (k) = Var vi ( k ) i ( ) i ( k ) = k = k = k = N + + = Γ N Γ () Using this exression and ower normalization fator(7), we an find the Var y as: 3

14 International Journal of Wireless & Mobile Networks (IJWMN) Vol. 5, No. 4, August 3 N + + Γ N Γ Var y = N N ( N ) w Gw Γ + Γ w w (3) erivation of Var y : For the seond yothesis, we derive the variane value as ( ) N N Var xi (k) E hi s ( k ) vi ( k ) E hi s ( k ) vi ( k ) = + + k = k = N + () s k () + s k = Γ F,, Γ F,, k= (4) Using the ower normalization fator in Eqns.(9), (4), and (), we obtain N N + + Γ Φ, k Γ Φ, k k = k= Var y = w Gw + w w + + Γ Φ, k + Γ Φ, k Φ, k k k < k 6.. Proof of Theorem (5) Under the given assumtions P d simlifies as PM PM ()() A B wi + Q Pf C + w i i= i= Pd = Q PM + wi i= This formulation of P d shows that P d is a funtion of Σw i and Σw i. owever, note that P d is indeendent of Σw i. This an be shown easily by relaing γw instead w. Then, P d beomes indeendent of γ, hene we an laim that otimal w is suh that Σw i =.Furthermore, using the above equation, we an see that the P d is maximized when is Σw i minimized assuming Σw i =. This is ahieved when w = (/)[ ]. Assuming w= (/)[ ], we an take the derivative of P d wrt. and find the otimal when є {,,,M}. Note that is an integer, and one has to ay attention to the boundary of the set {,,,M} while finding the that maximized P d. This oeration will give us the otimal solution sine should be an integer. 4

15 International Journal of Wireless & Mobile Networks (IJWMN) Vol. 5, No. 4, August 3 REFERENCES [] Federal Communiations Commission, Setrum Poliy Task Fore, Nov.. [] S.aykin, Cognitive Radio: Brain-emowered wireless ommuniations, IEEE transations on Signal Proessing, vol. 57, no. 9, , Set 9. [3] J. Mitola, Cognitive Radio: An integrated agent arhiteture for software defined radio, Stokholm, Sweden,. [4] C. Stevenson, G. Chouinard, L. Zhongding,. Wendong, S. J. Shellhammer and W. Caldwell, IEEE 8.: The first ognitive radio wireless regional area network standard, IEEE Communiations Magazine, vol. 47, no.,. 3-38, January 9. [5] S. Mishra, A. Sahai and R. Broderson, Cooerative sensing among ognitive radios, in Pro. IEEE International Conferene on Communiations (ICC), Istanbul, Turkey, 6. [6] J. Shen, S. Liu, L. Zeng, J. G. G. Xie and Y. Liu, Otimisation of ooerative setrum sensing in ognitive radio network, IET Commun, vol. 3,. 7 78, Jun. 9. [7] Z. Quan, S. Cui and A. Sayed, Otimal linear ooeration for setrum sensing in ognitive radio networks, IEEE Journal of Seleted Tois in Signal Proessing, vol., no.,. 8-4, Feb 8. [8] J. Xiao, S. Cui, Z.-Q. Luo and A. Goldsmith, Linear oherent deentralized estimation, IEEE Transations on Signal Proessing, vol. 56, no., , Feb 8. [9] C. Berger, M. Guerriero, S. Zhou and P. Willett, PAC vs. MAC for deentralized detetion using nonoherent modulation, IEEE Journal on Seleted Areas in Communiations, vol. 3, no.,. -, Feb 5. [] M. Gastar, M. Vetterli and P. ragotti, Sensing reality and ommuniating bits: A dangerous liason, IEEE Signal Proessing Magazine, vol. 4, no. 3,. 7-83, Aril 6. [] M. Gatsar and M. Vetterli, On the aaity of wireless networks: The relay ase, in Pro. IEEE INFOCOM,. [] I. Akyildiz, B. Lo and R. Balakrishnan, Cooerative setrum sensing in Cognitive Radio Networks: A Survey, Physial Communiations (Elsevier) Journal, vol. 4, no.,. 4-6, Marh. [3] B. Sirkei-Mergen and X. Liu, Grou-Orthogonal MAC for Cooerative setrum sensing in Cognitive Radios, in Pro. of MILCOM, San Jose, CA,. [4]. Stark and J. Woods, Probability, Random Proesses and Estimation Theory for Engineers, ed., Uer Saddle River, NJ: Prentie-all In., 994. [5] Wolfram Researh, [Online]. Available: htt://funtions.wolfram.om/yergeometrifuntions/yergeometrif/3//. [6] Wikiedia, [Online]. Available: htt://en.wikiedia.org/wiki/normal_distribution. [7] Wolfram Researh, [Online]. Available: htt://funtions.wolfram.om/yergeometrifuntions/yergeometrif/3//5/9/. [8] R. Morelos-Zaragoza, Faulty Publiations, 7. [Online]. Available: htt://sholarworks.sjsu.edu/ee_ub/8. [9] W. Iqbal, Otimal Relay Maing for Cooerative Setrum Sensing in Cognitive Radios, Master's Projet Reort, San Jose State University, San Jose, CA, May 3. 5