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Wireless Sensor Network, 2011, 3, 24-37 doi:10.4236/wsn.2011.31004 Published Online January 2011 (http://www.SciRP.org/journal/wsn) Copyright © 2011 SciRes. WSN Collaborative Spectrum Sensing for Cognitive Radio: Diversity Combining Approach Oscar Filio-Rodriguez1, V. Kontorovich2, Serguei Primak1, F. Ramos-Alarcon2 1Department of Electrical and Compu ting Engineering, University of Western Ontario, London, Canada 2Electrical Engineering Department, Research and Advanced Studies Center, Mexico City, Mexico. E-mail: valeri@cinvestav.mx Received August 5, 2010; revised September 2, 2010; accepted January 15, 2011 Abstract In this paper it is shown that cyclostationary spectrum sensing for Cognitive Radio networks, applying mul- tiple cyclic frequencies for single user detection can be interpreted (with some assumptions) in terms of op- timal incoherent diversity addition for “virtual diversity branches” or SIMO radar. This approach allows proposing, by analogy to diversity combining, suboptimal algorithms which can provide near optimal cha- racteristics for the Neyman-Pearson Test (NPT) for single user detection. The analysis is based on the Gene- ralized Gaussian (Klovsky-Middleton) Channel Model, which allows obtaining the NPT noise immunity characteristics: probability of misdetection error (PM) and probability of false alarm (Pfa) or Receiver Opera- tional Characteristics (ROC) in the most general way. Some quasi-optimum algorithms such as energetic re- ceiver and selection addition algorithm are analyzed and their comparison with the noise immunity proper- ties (ROC) of the optimum approach is provided as well. Finally, the diversity combining approach is ap- plied for the collaborative spectrum sensing and censoring. It is shown how the diversity addition principles are applied for distributed detection algorithms, called hereafter as SIMO radar or distributed SIMO radar, implementing Majority Addition (MA) approach and Weighted Majority Addition (WMA) principle. Keywords: Spectrum Sensing, Cognitive Radio, Diversity Combining, Collaborative Sensing, Majority Diversity Addition, Sequential Analysis 1. Introduction Spectrum sensing is one of the most important elements for the functioning of Cognitive Radio (CR) networks. As it is well known [1], CR networks are made up of primary users (PU) which have “legal” use of certain frequency bands and secondary or cognitive users (CU), located in different space–distributed cells, which share the same frequencies as the PU in a part-time fashion. The CU produce undesired interferences to the PU and so they are allowed to share the same spectrum with the PU if and only if the Quality of Service (QoS) degradation provoked to the PU does not reach a pre-established level. The cognitive users have to make first a spectrum sensing in order to determine whether the primary users are “on” or “off”1 and then “adapt” their transmission rate and transmission power in order to avoid producing harmful interferences to the PU or take advantage of the “spectrum holes” free of PU [1,2], etc. The common approaches are based on the Interference Temperature (IT) and power spectrum estimations, ener- gy detection and cyclostationary feature detection (see [1-4] and the references therein). The last one was first proposed in [5] and generalized for multiple cyclic fre- quencies at [4]. It is worth mentioning here that the cyclostationarity, as phenomenon, is not a recent development at all (see for example [6]) but effective tests for indication of second order cyclostationarity using a Neyman-Pearson type test was proposed not long ago. Its natural generalization for multiple cyclic frequencies was recently proposed for PU identification in CR networks [4]. As it was already mentioned in [4,5] the cyclostatio- narity is present, practically, in many communication signals: multiple cyclic frequencies may be related to symbol rate, guard periods (as in the case of OFDM sys- tems), etc. 1Actually it is not a necessary condition for CU to have access to the frequencies allocated to PU [2]. O. FILIO-RODRIGUEZ ET AL. Copyright © 2011 SciRes. WSN 25 In the following it will be shown that single-user detec- tion algorithms (in the form of expected value estimation of the cyclic autocorrelation) can be interpreted as an specific form of the mixed frequency-delay incoherent “diversity combining” block with the number of virtual “branches” equal to the product numbers of cyclic fre- quencies and time delays; it can be also called as a SIMO radar. This is the main difference between this paper and the material presented in [4,5]. Based on practical reasons, it is possible to assume that these “branches” suffer from fading which in the general case can be modeled with the help of generalized Gaussian statistics, or Klovsky-Middleton model, (see for example [7]). Moreover, in the following, depending on the frequen- cy and delay diversity parameters, fading in these branches or antennas is assumed as non-homogeneous, homogeneous and totally correlated (the latter was consi- dered at [4] for flat Rayleigh fading) or statistically inde- pendent. Certainly those last two marginal cases are as- sumed in the following only in order to obtain tractable analytical results; generalized analysis for correlated branches, based on the statistical description of Gaussian quadratic forms (see [8,9], etc.) will be considered else- where later on, but one important special case of corre- lated branches is considered in the following as well. It is worth mentioning that the concept of “diversity approach” for multiple cyclic detection is useful not only for effective development of quasi-optimal approaches, but also allows to consider the necessary “trade-off” be- tween the number of delays and cyclic frequencies for the detection procedure and the statistical dependency of the corresponding “diversity branches” in order to fulfill the noise immunity or “Receiver Operating Characteristics” (ROC) requirements. Moreover, in the following, it will be shown that the “diversity” concept for spectrum sensing is rather con- structive for the analysis of collaborative sensing as well (see [9], etc.). In [9] the collaboration is tackled in a ra- ther different way, than in the following. For the latter the set of secondary users (SU) being collaborating between themselves or operating through a Fusion Center (FC) can be interpreted as virtual branches (antennas) of the distri- buted detection system, which can apply NPT detection technique or Sequential Analysis methods. This distributed system is nothing else as a distributed SIMO radar, where virtual receiving branches are af- fected by statistically independent flat fading (the above mentioned incoherent combining algorithm at SU is also working as an optimum SIMO radar, but not in a distri- buted fashion). See also some examples at [10,11], etc. for calculation of its noise immunity properties. Regularly proposed counting rules [2,12-14] for oper- ating at FC can be also interpreted as a special case of quasi-optimum incoherent diversity addition (see MA algorithm in the following) and can be modified in order to approach its ROC properties to the optimum SIMO radar case (see the WMA algorithm in the following). In this paper only the novel theoretical material, which is the kernel of a deeper and original insight into the Spectrum Sensing problem, is presented. Some simula- tions related to this problem have been included in [15]; however, comprehensive and thorough simulations are reserved for another work of the authors (a book chapter already in process for publication). The paper is organized as follows. Section 2 briefly presents some fundamental results concerned to the Ge- neralized Gaussian (GG) channel modeling. Section 3 is totally dedicated to single user multiple cyclic frequency detection and its relation to incoherent optimum diversity combining. In section 4 the noise immunity of the NPT for the GG channel is analyzed. In Section 5 some subop- timal algorithms for multiple frequency cyclostationary sensing are considered. Here some discussion of the re- sults is presented as well. Section 6 is totally dedicated to collaborative sensing issues. Conclusions will be pre- sented at Section 7. 2. Generalized Gaussian (Klovsky-Middleton) Channel Model Basically most of the existing fading channel models are based on the concept of the module and phase of the ran- dom vector with Gaussian Probability Density Functions (PDF) for orthogonal statistically independent quadrature components “x” and “y”2, i.e., [7,8]: 2 2 22 1 ,exp 222 y x xy xy ym xm Wxy (1) where 22 ,yx and mx, my are variances and expectations of the “x” and “y” quadrature components respectively. Then, defining the module 22 yx and the phase of the random vector x y arctan , one can get: 2 0 2 2 2 2 2 sin 2 cos exp 2d m m W y y x x yx (2) From (2) it is possible to obtain various representations for W(), which actually depend on four parameters: , x y mm and 22 ,yx , [7,8]. For this reason in the following the term “four parametric distribution” is used, and the rest of this section corresponds to [8]. 2Those issues were tackled comprehensively in the 60th-70th of the las t century by many authors. Here we would like to distinguish D. Klovsk y [ 8 ], D. Middleton [ 16 ], P. Beckman [ 17 ], etc. Details can be found at [ 7 ] . O. FILIO-RODRIGUEZ ET AL. Copyright © 2011 SciRes. WSN 26 Hereafter the following two equivalent forms for the four-parametric distribution W( )3 will be used: 2 2 2 22 22 exp x x y y yx m m W 0 222 2 211 2!!2 ky y k k yx k y y k k km I mk H (3a) 0 2 2 2 ! )( kk II k I k k k mm k R W 22 2 0 2 222 2 exp III III mmI mm (3b) where I0(z) is the modified Bessel function of order zero [18], 2 yx I mm m , 2 yx II mm m , 2 22 2yx , 22 22 yx xy R , 2 22 11 2y yx x m 22 x y . Other forms (see for example [7]) follow from the way the integrand in (2) is calculated, but they are not applied in the following. Beckman, Hoyt, Rice, Rayleigh and truncated Gaus- sian distributions follow from (3) directly. Let us introduce the new parameters: 22 22 2 yx yx mm q , 2 2 2 y x , 222 0yx mm , 222 0 2yx , x y m m arctan 0 Beckman distribution follows from (3) when my = 0, 0 = x m, while Hoyt PDF appears, when 22 yx 0 = mx = my = 0. Rayleigh PDF follows when 0 = 0 and mx = my = 0; truncated Gaussian when additionally to the latter 0 2 x . That is why (3) is named as Generalized Gaussian model. Next it is easy to find the parameter “m” for equivalent Nakagami distribution [19]: 22 22 42 2222 00 11 212 1cossin q q m (4) It is worth mentioning here that Nakagami distribution is only an approximation for the four-parameter case, but mainly it adequately represents the “dynamics” of the variation of the four-parameter PDF functional form. 3. Single User Multiple Cyclic Frequency Detection The cyclostationary (CL) properties of the communica- tion signals have been already widely investigated and applied (see [4,5,20] etc.). For the case of PU the signal shapes are known a- pri- ori, and so their cyclic frequencies of interest are known as well. Following here the material of [4], let us intro- duce the set P n A1 for cyclic frequencies of interest and let P nn NN 1 be the numbers of integers for time delays for the autocovariance function calculus for each cyclic frequency from A (here P denotes the number of cyclic frequencies). Thus, the estimation of the autocovariance function is [4]: M l xx ljlxlx M R 1 *2exp* 1 , ˆ , (5) where the time delay is an integer and is fixed, the cyc- lic frequency is fixed as well, M is the number of ob- servations at (5) and x(l) is an input complex sample, with x*(l) being its complex conjugate. Representing the complex exponent in (5) in a trigo- nometric form and assuming that x(l) is a sample of the ergodic stochastic process, one can easily see that when M1 or the time of analysis T is much more than one, the estimations * ˆxx R are nothing else but estimations of the complex Fourier coefficients for fixed and (see also [6]). If one forms a complex vector of (5) for different and , the Generalized Maximum Likelihood Ratio (GMLR) for its estimation (assuming asymptotic Gaus- sianity of the observation) is well known (see for exam- ple [4,8,21]): 0* 1 *ˆ ˆ ˆT xxxx rr, (6) where * ˆxx r is a complex vector of estimations of the Fourier coefficients (F-Coefficients); ˆ is a 2N × 2N covariance matrix of * ˆxx r (in the non-asymptotic case generally those coefficients are correlated), P nn NN 1 (7) Let us define an estimation of each “j” complex F- coeffi- cient as jjj VVV ~ ˆ , (8) where jj VV ~ , are real and imaginary parts of j V ˆ; here it is assumed that in the estimation process necessarily takes place n(t) – the additive white Gaussian noise (AWGN) with intensity N0 equal for all j. 3Taking into account, that in this paper the incoherent diversity combining will be applied, the PDF of the phase is not presented hereafter. O. FILIO-RODRIGUEZ ET AL. Copyright © 2011 SciRes. WSN 27 It is well known that “true” F-Coefficients are not cor- related, but their estimations for finite “M” and corrupted by the noise are not correlated only asymptotically, when M or T (or both) are much more than one. With this as- sumption, applied systematically in the following, (6) can be significantly simplified taking into account the total Gaussianity of the terms in (8) [8], (see also [4]): Q iii NP VV 1 1 2 11 21 ,ˆ ),( ˆ , NPQ , where 22 1 1 2 1 ,, 2 1 Q ihh diag . Finally the left side in (6) can be represented in the way: Q ii iih VV 1 2 22 2 ~ , (8a) where 2 0 2 ii hEN, 22 2~ ˆiii VVV , Ei = PiT and Pi is the average power of each F-coefficient (fading is not considered here). One can see that the algorithms (8), (8a) are nothing else but an optimum incoherent quadratic diversity com- bining of Q total virtual “branches”, where 2 2 1 i h are weighting coefficients for each branch, generally related to inhomogeneous conditions for combining. Note that quadratic combining to obtain the NPT can be presented in the way: 0 1 2 2 2 ˆ Q ii i h V, (9) where 0 is a detection threshold. Formula (9) is not only a formal analogy with diversity addition or SIMO radar test: it is an essential reflection of the analogy between the autocovariance estimation and diversity combining of statistically independent data (see also [22]). So, in absence of fading, all branches are asymptoti- cally statistically independent. In presence of fading 2 ˆi V can be statistically inde- pendent as well, but also might be totally correlated in scenarios of flat fading both in frequency and time do- mains. Both cases will be considered while noise immun- ity of this single user multiple cyclic frequency algorithm will be analyzed (see next section). 4. Noise Immunity of the Algorithm (9) in Generalized Gaussian Channels It is well known [23] that the Neyman-Pearson Test (NPT) in terms of hypothesis testing, can be formulated as fol- lowing: Q ii i Q ii i tn h H tn h V H 1 2 2 1 1 2 2 0 )( 2 : )( 2 ˆ : , (10) here 2 i are “true” F-coefficients, n(t) – white Gaussian noise with intensity N0. For simplicity, in the following let us suppose that all 2 2 1 i h are the same and inhomogeneous features of the virtual branches will be addressed to different 2 i 222 0ii yx . It means that in (10) one has to consider only the routine form for quadratic combining4: Q i i Vz 1 2 ˆ (11) As it is well known, the NPT is characterized by Pfa and PM which are respectively the probability of false alarm and the probability of misdetection error [23]. In absence of fading, the “z” is formed by squares of the normally distributed components and its PDF for dif- ferent hypothesis can be defined in the way [8,23]: PDF square-chi central-non ˆ ,: PDF square-chi central )(: 1 22 21 2 20 Q iiQ Q VzH zH (11a) where Q ii V 1 2 ˆ is the expectation of the sum of 2 ˆi V and is a parameter of the noncentral chi-square distribu- tion [11]. It is worth to notice that in presence of fading, the functional forms for these distributions will differ de- pending on the scenarios for GG channel model and will be considered in the following. 4.1. Statistically Independent Virtual Branches with Flat Generalized Gaussian Fading in Each Branch Let us assume that each 2 ˆi V, see (8, 8a), is: 222 ˆˆ ˆiii yxV , where, iiiVxx ˆ, iii Vyy ~ ˆ and , ii x y are quadrature Gaussian components of the GG fading model. Here we have to notice that for both hypothesis each quadrature components in 2 ˆi V now are Gaussian as well, as before, but their means are not equal and their variances are arbitrary. Now, if Q iii VVz 1 22 ~, (11b) the routine procedure for calculus of the noise immunity can be applied [7,8,23], etc. 4In the following the module sign will be omitted. O. FILIO-RODRIGUEZ ET AL. Copyright © 2011 SciRes. WSN 28 Particularly for hypothesis H0: 2 2 2 2 2 2 2 0 2 2 2 0 21 2 ~ , 21 2 21 2 2 1 2 ~ 21 2 2 1 2 i i i i i i i i i i i i y y i x x i y y y i yi x x x i xi h h VD h h VD h h h N E mVM h h h N E mVM (12) For hypothesis H1: 22 2 2 0 2 2 0 2 ~ ,2 21 2 2~ 21 2 2 ii i i i i i i yixi y y i yi x x i xi hVDhVD h h N E mVM h h N E mVM (13) where M{} and D{} are symbols of mean and variance respectively; 2 0 22 PN E hi xx ii ; 2 0 22 PN E hi yy ii . Here it is taken into account that for the frequency diversity case there are P out of Q virtual branches and the transmitted power has to be divided between them [24]. From (12) and (13) it follows that for both hypothesis the PDF W(z) is always a non-central chi-square distribu- tion. For analytical evaluation the special cases of “high re- liability detection” 1,22 ii yxhh , are considered. Then for these conditions, from (12) it follows that for H0, }{ i VM and } ~ {i VM are close to zero, while the variances are close to one, then 0 )( H zW is tending to 2 2Q central distribution. For this case, Pfa is well known [23]: 1 0 0 00 ! exp, !1 1 ~ Q q q fa q Q Q P (14) Fixing the level of Pfa one can find 0 and once more, applying the conditions 1,22 ii yx hh from (13) it fol- lows that for the hypothesis H1, the variances }{ i VD and } ~ {i VD are going to be extremely large. In this case PM is: 22 0 2 1 11 ~!2 QQii Miii q PQh ii i i ii q 0 22 0 2 2 22 sincos 2 1 exp (15) where 2222 2 0 22 iiii yxyx i imm PN E h ; the remaining parameters were introduced at (3). For one sided- Gaus- sian distribution (15) becomes: 0 2 2 1 2! Q Q i M Q PQ hQ (16) Now let us repeat the same analysis as before, but for Nakagami fading channels (see Section 5). Assuming non-correlated homogeneous conditions for the fading in all “virtual branches” one can get: Qm Q Mhm m Q P 2 0 2 2 ! (17) where can be found from (14) and “m” from (4). 4.2. Totally Dependent Virtual Branches (Flat Fading) in GG Channel In this case the fading processes at all the virtual branches are totally correlated. Obviously it means that the result- ing SNR after combining is: Q ii h P h 1 2 2 21 ; thus the problem can be transferred to the quadrature addition algorithm for one equivalent branch, i.e. without diversity but, with the GG model of flat fading: 0 22 ~ oo VV (18) where Q ii VV 1 22 0, Q ii VV 1 22 0 ~~ . Here formulas (12) and (13) are valid but for conditions of single channel, i.e., without index “i”. Then Qeqv 1 and from (14) 0 exp fa P, fa P 1 ln 0 and 2 22 2222 00 2 1 ln 11 1 2expcos sin 2 fa M i i q P Pq h (19) Dependence between Pfa and PM is usually called as “Receiver Operational Characteristic-ROC” and they are presented at Figures 1-3, where the continuous lines cor- respond to case a and the dotted lines to case b. Comparison of (19) and (15) deserves some comments. 1) When in both scenarios 2 i h are equal and Pfa is fixed, then PM from (19) is much more than PM from (15). The latter can be explained by the diversity effect at (19), see also [8,25]. 2) Then it is reasonable to choose a small set of delays and multiple frequencies (Q 5 [10,25]) in order to pro- vide (if the channel conditions allow it) statistically inde- O. FILIO-RODRIGUEZ ET AL. Copyright © 2011 SciRes. WSN 29 Figure 1. ROC, continuous (15), dotted (19). Figure 2. ROC (15) and (19) for another set of parameters. Figure 3. ROC (15) and (19) for a different set of parameters. pendent fading in those virtual branches, i.e. it is rea- sonable to “sacrifice” the numbers of P and N by big- ger intervals between and so as to artifi- cially create independent fading in the frequency and delay domains, which certainly leads to better noise immunity after “diversity combining”. So, appropriate choice of cyclostationary features Q = NP of the de- sired signals of PU can significantly improve their ROC properties. 4.3. One Special Case of Covariance Matrix for Correlated Branches at Quadratic Incohe- rent Addition Algorithm Let us consider in the following one special case of the covariance matrix for quadrature components Q l xx 1 and Q l yy1 : assume that across the branches all xl or yl Gaussian components are correlated with coefficients Rx or Ry and there is no cross-correlation at all between xl and yl Gaussian components. One can see, that this as- sumption restricts (in general) the type of the covariance matrix of the GG channel model but might be useful for the first step examination of the influence of the cova- riance between virtual (but not only virtual!) branches at the noise immunity characteristics of the SU: consider, for example, SU which applies multi-antenna receiving system, etc. It is well known that for each pair of x or y Gaussian variables, by the well known angle rotation linear trans- form it is possible to obtain a new set of statistically in- dependent Gaussian variables: rotating of the coordinate system (linear transform) by the angle 2 2 2 1 21 2arctan R, where R is a correlation co- efficient; 2 1 , 2 2 -are variances of two correlated Gaus- sian quadrature components, while new Gaussian va- riables are statistically independent [17,11]. In order to provide tractable analytical results, in the following only the case Q = 2 for the algorithm (11) will be considered. Then noise immunity analysis can be done in the same way as it was done at IIIa, but the means and variances for hypothesis H0 and H1 have to be calculated by the formulas: 22 2 12 2 ,22 222 12 12 22 12 21 4 111 III R R where 2 1 , 2 2 correspond to the variances of the qua- drature components, calculated for different hypothesis H0 and H1 (see (12), (13)); 2 ,III are new variances of the quadrature components after angle rotation (for each two branches). Moreover, assuming in the following for simplicity Rx = Ry = R, in the same way as before one gets: O. FILIO-RODRIGUEZ ET AL. Copyright © 2011 SciRes. WSN 30 cossin sincos 21 21 mmm mmm II I where m1, m2 are expectations of the initial quadrature components (see also (12),(13) ); mI, mII are new means after angle rotation. Now all the set of these parameters can be considered as new parameters of the GG model with the statistically in- dependent branches. So the noise immunity (ROC) can be calculated in the same way as in IIIa (see formulas (14), (15)). This calculation, (in general) is rather cumbersome, because the new parameters of the GG model come from rather complex expressions (see above). Therefore assuming that Pfa and PM. are much less than one as it was done earlier, it is possible to apply the asymptotic calculus (see [10,11]). Particularly, for the hypothesis H0 (see IIIa) all means will be close to zero and all variances will be equal to (1-R2). So Pfa can be calculated by (14) but with the new threshold ′ = 0/(1-R2), depending on R, but the method of calculus is the same as in IIIa. Then for the asymptotic case PM << 1, one can get: 2 22 2 22 2 222 222 2 ~exp 221 2121 11 21 1 21 y xx xx y yxx xxx y I III M IIII II II III II m mm P mm 22 22 222 222 22 222 11 211 21 1 1. 21 1 yy xx xxx yyy yy yyy II II II IIII IIIII II II IIII II m m m This formula is for the GG channel model and is rather general in the sense that it does not provide a “transparent picture” about the dependence of PM, for example, on R etc; it requires implicitly numerical calculus. Let us consider a special case: 222 yxx IIII 22 y II ; next, introducing 222 0yxI II mm and 222 0yxII IIII mm one can get: 4 2 0 4 2 0 2 2 0 2 2 0 4 2 22 3 22 exp 2 ~ IIIIII M P. For this special case it can be found that: 12 12 12 12 222 00 2 22 00 22 2 00 222 00 21 4 11(1 ) hh R hh hh Rhh when 2 2 0 2 I and 2 2 0 2 II are much less than one (strong fading), then: 12 22 4222 00 33 1 ~2 21 M Phh R 12 22 4222 00 33 1 ~2 21 M Phh R The last formula shows that, losses related to correla- tion between diversity branches depend mainly on 2 1 1R; this result was in some sense predictable (see [7, 8] for example). When the fading follows the truncated Gaussian PDF then: 22 2 22 22 13 ~22 2121 11 yy yy yy III M III III P when PM 1, then: 12 2 22 2 00 5 ~281 M PhhR One can see that losses once more depend on 2 1 1R as well. The same character of losses can be found for signifi- cantly Rician character of the GG model; so, it can be considered as a rather “universal” dependence of losses on the correlation coefficient value. Of course changes of the threshold, which depends on “R”, influence the character of the dependence of ROC on the correlation properties of the GG model in a nonlinear way, but this will be discussed elsewhere in the future. Concluding the material of this section, it is worth to mention, that from the theory of diversity combining it is well known [7,8] that correlation between branches has influence, mainly, on the noise immunity characteristics (ROC, in our case) when resulting SNR is rather high, i.e. PM is much less than one. 5. Suboptimal Algorithms and Their Noise Immunity The first suboptimal algorithm considered hereafter will be an energetic receiver where the desired signal is repre- sented in the way: B iiittx 1 )()( ; here B it1 )}({ -are orthonormal functions. According to [22,23] the corresponding algorithm (NPT) can be represented in the way: O. FILIO-RODRIGUEZ ET AL. Copyright © 2011 SciRes. WSN 31 2 0 10 : T B i j H zt tdtnt (20) 2 1 10 : T B i j H zt tdt , i.e. = * ˆ xx rnt, where B is the number of orthonormal functions B it1 )}({ applied for the expansion of the desired signal x(t); the received signal is ztxt nt, T is the time of analysis. Now, for the representation of x t and nt let us apply the F-basis in the same way as it was done at [10], (see also [22] and references therein). Then: 00 0 00 0 cos sin cos sin B kk k B kk k x taktbkt ntk tk t (21) where 02T , B = 2FT, 1 12 F kk F – fre- quency bandwidth, k2, k1 are upper and lower indexes taken into account here for the F-series expansion. Then: B kkkkk B kkk baH tnH 0 0 22 1 0 22 0 2 1 : 2 1 : (22) As all k a, k b, k and k are Gaussian distributed coefficients, the left side in (22) has central or non-central 2 2B distributions respectively. Defining those left sides in (22) as 1 and 2 (see [10,22] and Section 4 one has to apply: /2 1 /2 1 11 1/2 /2 1 22 22 2 1exp 2 2(/2) 1exp 22 B B B B WDD DB WI DDD (23) where xk B kkPba 2 2 0 2 is the average power of x(t) and the parameter 2 0TN D. Then the threshold 0 can be easily found from (14) where QB and Pfa are fixed. The detection probability PM is: 2 exp 2 1 ,, 2 exp 2 12 2 0 2h Bh D F h PM (24) where 1,,0 2 0 Bh D F is the Cumulative Distri- bution Function (CDF) of the non-central 2 2B PDF. The upper bound of PM for the GG channel model with flat fading is known from [8]: 22 0 2 1 11cos 1exp 21 1 M q qh PCqh 22 0 11sin 1 q h (25) where 22 2 0 2E hPN , and 2 12 21 11 h Cq An exact tractable analytical expression of PM for the GG model is not available. In absence of fading it is possible to obtain an analyti- cal result in the following way. First, representing the Bessel function as in [18] in the way: 21 22 2 21 0!2 Bk Bk hy IhykB k , (26) then the PM from (24) is: 1 2 042 2 1 0 2, 12 exp 22 !22 Bk Mk k Bk h hD PkB k (27) where B = 2F, (,x) – is the lower incomplete gamma function. Analysis of (27) shows that influence of B can be sig- nificant and it can be shown that for fixed Pfa or D 0 , while B grows, PM also grows. To the best of our know- ledge, influence of B and not only of 2 h on the noise immunity of the energetic (autocovariance) receiver was first stressed in [22]. Then for the multiple cyclic frequency case, when the number of frequencies P is rather large while T is fixed, F is large as well and PM grows. Therefore the energetic detector is not definitely a good candidate for spectrum sensing for this scenario, as its PM is much worse than for the optimum detector (see the previous section). In some sense this comment coincides with the simulations [4], besides that there the energetic O. FILIO-RODRIGUEZ ET AL. Copyright © 2011 SciRes. WSN 32 detection was not implemented in the same way as men- tioned above. Another option for suboptimal detection is to take ad- vantage of the analogy between multiple cyclic frequency detection and quadratic diversity combining and apply a suboptimal variant of incoherent diversity addition (see [7, 8]). Hereafter a selection (switching) combining method was chosen, assuming that fading has a Nakagami PDF (see [4]). See formula (4) to adjust parameters of Naka- gami PDF and four–parameter distribution. There are several different approaches for switching combining but in the following we will analyze only the algorithm of selection of the “virtual branch” with i max , 0 ,1Qi . Let us assume here, for simplicity, the homogeneous fading conditions then, the distribution of the maximum value of the identically distributed random values is [11]: 1 0 () Q WQW Wydy (28) If W() is [19]: 21 2 22 2exp mm m my y Wm m (29) Then it is necessary to average PM for one virtual branch without fading through (28) with the help of (29) while Pfa is: exp fa P, (30) where is a threshold. The PM of the channel without fading is: ,hQPM (31) Following the above mentioned procedure for the case Q = 2 and h > 1 one can get from (28)-(31) an approx- imate formula: 1 02 2 2 2 2 2 2)!1(! 1)!1( 1 2 1 m iim m h m m h m Mmi im m h P (32) for m–integers. One can compare this method of switching combining (with fixed 2 h and Q = 2) with the optimum approach (see (14), (17)). Please notice that in fading channel conditions when the number of virtual branches is growing, one encounters the so-called “hardening effect”, i.e. while Q is increasing, the increment of noise immunity might be low. Therefore, with Q = 2 there is a good option to com- pare the effectiveness of the selection combining method with the optimum one. Figure 4. Comparison of the ROC for the optimum (17), and quasioptimum (32), cases. In Figure 4 the ROC for this method is presented, where for comparison some of the “optimum” ROC’s, see (17), are presented. One can see that the energetic losses for PM = 10-4 are rather small and for m = 1 are negligible. In the same manner as above, the well known set of sub-optimum combining algorithms can be applied: other methods of switching combining, linear (weighted and non-weighted) addition, etc. Their application is rather straightforward and is not presented here. Some discussion regarding the obtained results One can ask: if both algorithms (11) and (12) rely on quadratic addition of the F-coefficients, then why their noise immunity is so different, particularly with the GG channel fading? What is going on? The answer is rather straightforward. At (11) the object of the quadratic addition are the F- coefficients, but from the autocovariance function of the output of the multiple cyclic frequency optimum detector, i.e. after optimum processing of the quadrature compo- nents of the input signals. It is also possible to provide statistically independent fading of the virtual branches for incoherent addition by properly choosing the cyclic fre- quencies and delays, etc. which drastically increase the noise immunity (through the diversity effect). In contrary, the energetic receiver, as it is in (20)-(22), does not apply specific properties of the cyclic frequen- cies and just extracts the total energy of the aggregate input signal. It is often hardly possible for this case to the F-coefficients of the input signal to exhibit statistical in- dependency in fading conditions. Moreover, for the energetic receiver (22) the noise immunity, even in the case of a constant channel (without fading), goes down while the bandwidth F grows (B = 2 FT, with T fixed) as the noise power grows. Therefore the energetic receiver for multiple cyclic frequency signals might be useless when FT1. O. FILIO-RODRIGUEZ ET AL. Copyright © 2011 SciRes. WSN 33 For a quasi-optimum alternative for optimum quadrat- ic combining it is possible to consider all the set of switching combining algorithms (see (30) for example), as well as a whole set of quasi-optimum algorithms of non-coherent diversity combining such as a set of linear combining methods with rather low energetic losses for the fixed Pfa and PM. 6. Collaborative Spectrum Sensing with Censoring Here as in previous section it is supposed that spectrum sensing is based on the cyclostationary properties of the signals of the primary users (PU) and the secondary users (SU) are spatially distributed within certain area. All the set of SU can sense the whole frequency band of interest, or each SU may sense just a partial band. Hereafter it will be assumed that all SU are sensing the same fre- quency band. In both cases of spectrum monitoring, SU have to share the sensing information between them or might be coordinated by a Fusion Center. It seems reasonable that, no matter what kind of ex- change is used, the local decision information has to be obtained by a minimum set of observations M in (5), while N and P are fixed5. In other words, the time of analysis in (5) has to be reduced as much as possible; meanwhile the amount of transmitted data has to be reduced as well. Then it is opportunistic to apply the sequential analy- sis of A. Wald [26] where ML test, in contrary to NPT, has to be compared with two thresholds related to re- quirements of Pfa and PM. Let us suppose that for the latter, highly reliable final results for the test are predefined, so Pfa and PM have to be rather low. This might be a rational way to make cen- soring for the local test as only reliable information has to be forwarded to the FC or other SU. One has to notice that each SU will obtain those relia- ble final results (whether PU exists or not) at different time instants. This information has to be sent to other fellow SU or to FC in binary way. Next let us consider several rather general but differ- ent scenarios of collaborative spectrum sensing. - Each n-th SU, Kn ,1, passes, after time “T”, the information of “zn” (not binary) to the system of qu- adratic addition at the FC. So, KQ jj K nn zZ zZ 1 1 or (33) Then Z might be analyzed by the NPT or by sequential analysis (see below) assuming hereafter that the channel SU FC is error free. So after final addition, the result of quadratic diversity addition of KQ virtual branches (or of K SU) is analyzed, assuming statistically independent fading along all summations (see (10) and (11) above). This scenario can be called as a distributed optimum in- coherent “SIMO passive radar” and its characteristics are equal to (14), (15) with the number of virtual branches KQ. - Each of the n-th SU make an individual decision re- garding to the presence of PU and then send the bi- nary decision to the FC by error free channels. As- suming that all those decisions are statistically inde- pendent, the final result at the FC can be obtained according to the majority rule (see for example [27]) with the majority not- weighted (or weighted) diver- sity addition method. This case can be also called “SIMO radar” but in contrary to the first one it is non-optimum. In the following the topics related to those issues will be thoroughly considered. Majority diversity addition and weighted majority ad- dition (WMA) in collaborative spectrum sensing If the majority principle is applied at FC, then the de- cision is made by analysis of the partial decisions at each SU (here SU acts as a “virtual” diversity branch) and the decision which takes place at the majority of the branches is favored. This method is called “majority di- versity addition”6. If partial solutions are binary and the number of virtual branches is odd, there cannot be any collision in the final decisions for such method. Let K = 2q-1 and “P0” denote the existence of PU after “q” tests on the branches. So if after “m-1” probes on the virtual branches one gets “q-1” results of existence of PU and “m-th” probe gives the same, then for the “q” test one gets the proba- bility of this event as 1 0111 1mq qq m PCPP (34) The probability of P(P0) is a sum of statistically inde- pendent probabilities of probes (34) through all “m” from m-q up to m = 2q-1, i.e.: 21 1 0111 1 qmq qq m mq PCPP , (35) where P1 can be Pfa or PM, so P(P0) is a final probability of false alarm or error detection (see for example [11,27] as well), depending which one of hypothesis is consi- dered. If one defines “n” in the way n = m – q, it yields: 1 1 0111 0 1 qn q qn n PCPP (36) 5Meanwhile one has to notice, that asymptotic conditions for M (T) are assumed to be valid here in order to preserve the uncorrelated conditions for F-coefficients in (5). 6Some modifications of the majority diversity addition can be found a t [13,28], etc. O. FILIO-RODRIGUEZ ET AL. Copyright © 2011 SciRes. WSN 34 At the same time, from the theory of diversity com- bining it is known that majority addition is equivalent to the optimal incoherent addition with the number of branches (here virtual ones) “q”, i.e., to incoherent (qua- dratic) addition with almost twice less branches. So, comparing the characteristics of majority addition with those of the optimum SIMO radar one can see sev- eral limitations of the former: - Optimum SIMO radar with incoherent addition ac- tually operates with almost twice more virtual bran- ches and therefore provides significantly better de- tection characteristics (ROC’s); - The majority addition operates successfully only with odd number of virtual branches, while optimum SI- MO radar operates with any number of branches. The price one has to pay for the advantages of the op- timum SIMO radar is a more complex data transmission scheme: in the majority addition, simply binary results are transmitted, and for SIMO radar the information of the “z” value for each SU has to be transmitted to FC through error free channels. Is it possible to improve the ROC properties of the majority addition in order to make them approach to those of the optimum SIMO radar? In order to approach the noise immunity properties of the majority addition to those of the optimum incoherent addi- tion, some modifications of the former were proposed. One of them, the so-called “weighted majority addi- tion” was proposed at [29,30]. The idea of this method is rather simple: introduce in the majority addition algo- rithm information of the channel gains for each partial solution, or in other words introduce “weights” in the procedure of the majority addition algorithm. In this way the channel gains for the diversity “branches” work as weighting coefficients in the process of majority selec- tion. It was shown that this suboptimal method provides results very close to those of the optimum incoherent addition [29,30] if the communication scenario allows taking advantages of channel gains. One can see that it is not the case for one of the scena- rios at FC: each result of detection at SU was obtained through the optimum quadratic addition by the SU itself, so the resulting fading at SU has a very low variance when Q is rather large (hardening effect) (see, for exam- ple [10,25]). Therefore, it is hardly possible to improve the results of majority addition by introducing weighting coefficients as all the weights might be practically equal. But it is known that if the channels are sufficiently hete- rogeneous, the hardening effect does not even appear or it appears very slowly, while Q at the SU. So, let us consider another extreme special case. Let us assume that the fading at the SU’s are so heterogeneous, that practi- cally all quadrature addition algorithms do not work as the diversity combining algorithm and each SU have Q 1 (single reception) with m-distributed fading7 and the fading is generally heterogeneous. With this assumption one can see that the problem is converted to the case of SIMO Radar: SU are sending to the FC binary information of partial decisions together with the information of their weights in order to provide to the FC with weighted addition (the channel SU FC is supposed to be error free). Let us formulate here an assumption: if the final deci- sions are taken at FC by applying the technique of weighted addition of partial decisions, then the SU’s have to transmit to the FC not only the information of partial decisions, but information of their reliability as well and all the system (PU, SU, FC) is working as a distributed quasi-optimum SIMO Radar. Returning back to the above mentioned scenario, one can see that the decision of PU existence in the majority of “branches” can be obtained by the algorithm: 10 11 q K j HjH jq j , (37) where 1 K j – are magnitudes of the channel gains; hypothesis H1 and H0 have the same sense as in (10). From Bayes theorem each of the summands in (37) have the following PDFs: 0 1 1 1 j j jfa jH fa jd jH d WP WP WP WP (38) and the error PM after addition, finally will take place if the sign of the inequality in (37) changes to the opposite one. Conditions for false alarm are defined in a similar way when PU really does not exist in observations at SU. Let us assume that [19]: 21 2 2 2 2exp j j m mj jjj jj m i ji m Wm m (39) where mj and the corresponding parameters for four- parametric distribution are related by 1 2 j m. Then introducing the new variable 2 2 j jj j h m xm 7In relation to the fading model assumed here for simplicity, see formula (4) for the definition of the parameter m through the parameters of the Generalized Gaussian model. O. FILIO-RODRIGUEZ ET AL. Copyright © 2011 SciRes. WSN 35 one can get: 0 02 21 2 2 2 21 2 2 22 2 2exp 2exp 2 j j j j j m mj jj j jH j m i ji m mj jj j jH j mhi m ji j mx x Wx m m mx x Wx m h mm (40) where Pfa 1, PM < 1. From (40) it can be seen that 0 jH Wx and 1 jH Wx have the Nakagami PDF form. Getting back to (37) and introducing the variables [29,30]: 0 1 1 1 1 q qjH j Q Qqj H jq x x (41) one can formally calculate the error probability in (37). In the general case of the heterogeneous scenarios, according to [31] it is possible to find distributions of q and 1Qq in a Nakagami PDF form after rather cumbersome calculus. In the most tractable way, ac- cording to (77)-(89) in [19], it is possible to provide the error analysis for the following special case, when8: 2 2 1 1 n n mm here n is the number of Nakagami variables at (41). Then the sums in (41) will have equivalent Nakagami parameters 0mmn and 22 0n . Note, that for the general case [31], the calculus of 0m and 2 0 can be done mainly numerically. Then from (37) is possible to get (see [30]) for the conditional error probability (with q fixed): 0( ) 0 0 2 00 0 () 12 mK q mK q qmK Pq hKq mK qmKm (42) But the number of virtual branches “q” with errors, both for PM and Pfa, is a random variable with Bernoulli PDF, when the virtual branches have statistically inde- pendent fading [29,30]. Then: 1 1 () K error KK q PPqPqPK (43) where 11 1 K q qq KK Pq CPP ; P1 – is Pfa or PM from (14), (15), when Q = 1. Note that at (14), (15) the four parameters have to be previously adjusted by (4) with the value of 0m and 2 0 at 2 h . The formula (43) is “universal” in the sense that the final PM and Pfa can be calculated through it, because as it was mentioned above, the inequality of the (37) type can be applied for calculus of false alarm as well. The ROC’s for WMA is presented at Figures 5-7. For comparison purposes we have included the plots corres- ponding to sections IVa and IVb denoted with conti- nuous lines and with dotted lines the plots corresponding to (43). One can see that for PM = 10-4 energetic losses are less than 1.5 - 2 dB. Finally let us compare the “ideology” of the weighted majority addition with some of the approaches mentioned at [32], see also the references therein, (in [32] it is also assumed an error free channel between SUi FC). The “simple counting” approach [32] is nothing else than selecting for FC decision only “highly weighted” SU. For sure this addition is “less optimum” than the approach in [29] because some of the SU’s with small weights do not participate in the decision-taking process at the FC. Other two methods, namely the Partial Agreement Counting and the Collision Detection, assume the exis- tence of a feedback channel between SU and FC which can be used for comparing partial decisions at the SU and final decisions at the FC in order to select the “true” final decision. This option was not considered in the current analysis. Figure 5. ROC for WMA (15) and (19) –––, (43) ---. 8Formally the following analysis is valid for the general case [30] as well. O. FILIO-RODRIGUEZ ET AL. Copyright © 2011 SciRes. WSN 36 Figure 6. ROC for WMA with different parameters. Figure 7. ROC for WMA for different parameters. For sure, application of the feedback channel opens the possibility to improve the reliability of the final deci- sion at the FC and taking into account that weighted ma- jority addition is a practically optimum incoherent addi- tion, the final characteristics might be better than what it has been mentioned at [32]. 7. Conclusions In this paper we have shown that cyclostationary spec- trum sensing as well as collaborative spectrum sensing for Cognitive Radio networks can be interpreted as a special case of the concept of optimum or sub-optimum incoherent diversity combining approach (SIMO radar). It was shown, that as sub-optimum algorithms for this purpose it is possible to apply the whole “gamma” of well known algorithms such as all types of switching combining, as well as linear combining and counting rules (discrete addition), etc. The concrete detection algorithms (distributed or not) utilizing NPT or sequential tests leads to the so-called SI- MO radar algorithms and their ROC’s were analyzed here for GG channel fading models in the most general way. It is worth mentioning here that, application of the cyclostationary properties of the PU signals (through the estimation of the F-coefficients of the autocovariance function) is a convenient but obviously not the unique approach that allows construction of statistically inde- pendent virtual diversity branches for the spectrum sens- ing detection algorithms. For example, for the broad band GG communication channels, virtual branches can be constructed through channel orthogonalizations in the frequency and time domains (in the same way as it was done in [33]), or by choosing statistically independent fading sub-carriers of OFDMA signals (see [34]), etc. Other emerging problems, such as detailed analysis for correlated virtual branches of the sensing algorithms, adaptive methods of sensing for unknown parameters of GG channels, application of the ideas of the feedback algorithms for collaborative sensing, etc. will be pre- sented by the authors elsewhere. 8. Acknowledgements With the material presented above, the authors would like to acknowledge the outstanding contribution of Prof. D. D. Klovsky, who survived to the Holocaust and passed away almost ten years ago, to the theory of diver- sity combining in fading channels which as it was shown above is fully operating for the new challenges in com- munications. 9. References [1] S. 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