Candidate Spectral Estimation for Cognitive Radio

Transcription

1 Proeedings of the th WE International Conferene on COMMUNICTION, gios Nikolaos, Crete Island, Greee, July 6-8, 7 9 Candidate petral Estimation for Cognitive Radio MIGUEL ROJ,3, MIGUEL LGUN,,N I PÉREZ, Centre Tenològi de Teleomuniaions de Catalunya TC Universitat Politènia de Catalunya 3 Universidad Veraruzana v Canal Olímpi /N Castelldefels Barelona, PIN malagunas@ttes bstrat: Introduing a different view point of traditional filter bank approah for spetral estimation it is derived a spetral estimation method able to detet a given spetral shape forming part or ontributing to a given data reord The method provides an estimated power level of the spetral shape ontribution to the data reord, and peaks lose to the frequeny where the spetral shape is loated Basially, the filter-bank, instead of being tuned to a single arrier, is tuned to the spetral shape The spetral shape used to tune the filter-bank is refereed as the andidate spetrum The major motivation for this proedure was the proper spetrum labelling of liensed users and interferers in ognitive radio appliations Key-Words: petral estimation, ognitive radio, filter bank, spetrum labeling Introdution Filter bank signal analysis has been the major tool for spetral estimation and the dominant tehnology for spetrum analyzers In general, filter bank represents in the past the way to implement Fourier analyzers from aoustis to optis The periodogram, or any DFT based proedure, move filter banks to domains of appliations where /D onverters annot afford, and being DFT-based tehnology dominant in those ases where the number of filters was high and omplexity ditates the appliation Nevertheless, most ommuniations systems as well as spetrum analyzers remain in the filter-bank philosophy For instrumentation issues the so-alled poly-phase networks provide the manner to upgrade a Fourier analysis to a filter-bank and they are urrently used for trans-multiplexers and ommuniation systems mathing strong mask onstrains Furthermore, it seems that filter-bank analysis, as an upgrade of OFDM, is well positioned to replae ompletely DFT tehnology in modern ommuniations systems The upgrade to the DFT approah in spetral estimation (ie the Periodogram) was reported by Capon where, instead of designing a low pass prototype (data window) that was modulated to the steering frequeny of the analyzer, propose to design a dediated filter whih depends on the steering frequeny and the data reord under analysis ine this is just an upgrade of the Fourier analysis, the Capon s method an be easily extended to multidimensional and ross-spetral estimation yet preserving the superior performane with respet any version based on Fourier analysis Filter bank analysis always estimate spetral density trough estimating power in a given frequeny band, in onsequene, onstant bandwidth analysis sales the power level independently of the frequeny; on the other hand, dediated filter bank tehniques may math the analysis bandwidth to better estimate spetral densities Motivated by the interest of, let us say, dediated spetral estimation for ognitive radio appliations, the so-alled traditional spetral estimation represented by the dediated filter bank approah of Capon is revisited in order to find fundamentals that allow to move the sanning by pure frequenies to sanning or tuning the filter to a speifi spetral shape or orrelation signature This spetral shape is refereed herein as the andidate spetrum, whih is assumed to be known in shape and bandwidth but remaining the power level and the frequeny loation unknown In finding the autoorrelation signature or andidate in the autoorrelation matrix of data it is shown that the notion of geodesi distane among them beomes relevant, onluding that the detetion and power level estimation depends diretly on the geodesi distane of the andidate and the and data orrelations matrix The motivation of this work was the interest that for ognitive radio air-interfaes has the ability to detet the presene or absene of a liensed user regardless other interferers, non-liensed users and noise The reported method shows good performane for moderate data lengths and low filter order in NR ranges above those demanded by the appliation

2 Proeedings of the th WE International Conferene on COMMUNICTION, gios Nikolaos, Crete Island, Greee, July 6-8, 7 9 Next setion revisit the filter bank method for spetral estimation and explores the possibility of hanging the traditional single frequeny sanning by a shape sanning obtaining also power level and density in the data signal The spetral shape and its orrelation signatures are named as andidate spetrum etion 3 is dediated to onnet frequeny estimators and parametri methods for spetral estimation with the so-alled andidate spetral estimation etion 4 attempts to formalise, in terms of distane between orrelation matrixes, the formulation of the andidate spetral estimation Next, etion 5 desribes briefly the ognitive radio motivation and reports the proedure: Finally, some simulation and performane evaluation is inluded on the simulations setion Revisiting the filter bank approah The filter bank approah for spetral estimation is based on a dediated filter design, whih steered to a given frequeny, aims to estimate the spetral density of the input signal at this frequeny The estimate is given by the output power of the filter divided by the bandwidth of the analysis filter Independently of the frequeny, inluding optial ommuniations, spetrum analyzers use this priniple urrently and, being dependent on tehnology available, ost and omplexity the filter bank ould be data independent or data dependent From the seond lass the most popular approah is due to Capon, whih set a variational design of the analysis filter setting an objetive that was to minimize the filter output power onstrained to a ero db response at the steered frequeny The formulation of this design, also know as minimum variane, is shown in () MIN = () Where ontains the FIR Q filter oeffiients, R is the orrelation matrix of the input signal, [ exp( jw) exp( j( Q ) w] T steering frequeny vetor and w ( f ) = is the π is the frequeny where the estimate of the spetral density of the input signal is going to be produed The output power is given by (), the noise bandwidth of the analysis filter is given by (3), and the spetral estimate is (4) Px ( w) = () ( ) BN ( w) = (3) P R x ( w) x ( w) = = (4) B ( w) R N nd it is easy to prove (5) that evidenes how resolution sales among normalized estimate, power estimate and periodogram estimate R R Q R Q (5) The filter-bank framework an be used also for n- dimensional spetral estimation as well as for rossspetral densities s onerns with the design equations, it is lear that the response of ero db t the steering frequeny refers to the magnitude only and not neessarily over the filter phase at the steering frequeny When setting the magnitude onstrain only () is reformulated as (6) [ ] = MIN (6) The solution to this problem is formulated in (7), where λ is the Lagrange multiplier nd the solution for the multiplier and the null-eigenvetor is shown in (8) ( R ) = λ (7) R = λ = (8) 5 ( R ) R Regardless this solution may appears different, it produes exatly the same estimates than the traditional formulation In addition it provides additional viewpoints on the filter-bank framework First note that in (7) there is a spetral substration from the orrelation matrix of the original data and the orrelation matrix of, let us say, the andidate spetrum In other words, the formulation find out how muh andidate spetrum is ontained in the original data The andidate spetrum is the one that orresponds to a single line at the steered frequeny Clearly the Lagrange multiplier is basially how

3 Proeedings of the th WE International Conferene on COMMUNICTION, gios Nikolaos, Crete Island, Greee, July 6-8, 7 9 muh power seems to be ontained in the original data from the andidate spetrum Furthermore, the power an be reformulated as the maximum value of the andidate spetrum that an be subtrated from the original data orrelation The solution is that λ is set in order that the minimum eigenvalue of the resulting matrix is ero In summary, looking for a andidate spetrum, ie orrelation matrix, the maximum eigenvalue of (9) provides the power estimate of the andidate spetrum in the data spetrum R DT e = λ R e (9) CNDIDTE Next setion will use the same framework to enompass well-known spetral estimation proedures for line spetrum and parametri models 3 MUIC and Parametri petral Estimation ssuming that the problem is to find out if a given spetral andidate (w) (unity area) with autoorrelation matrix given by R in data ontaminated by other soures with different spetral shape, applying the framework of the previous setion the solution will be to solve for the maximum eigenvalue of () R e = λ R e () One the maximum eigenvalue is found, it is the power estimate of the andidate spetrum t the same time, denoting the eigenvetor with the filter notation, ie hanging e by, () an written as () ( R R ) = λ () Clearly, assuming exat spetral subtration () implies that the eigenvetor is orthogonal to the rest of omponents of the original data It is lear that when the andidate spetrum is oloured noise and the rest of the omponents are spetral lines, the inverse of the response of the filter peaks at the frequeny loation of the spetral lines Using the multipliity of the maximum eigenvetor we arrive to Musi To extent the same framework to parametri models, starting with the ase of a pure R proess, being the vetor ontaining the denominator oeffiients of the model, R the autoorrelation matrix of the data and σ the power of the input white noise to the R model, and b() the oeffiient of the numerator, equation () holds * = σ b() * b () () t the same time, [ ( ) ] b() h = (3) Where h() is the impulse response of the model In summary () hanges to (4), where vetor is equal to [ ] = σ h() ( ) (4) Clearly, matrix ( ) plays the role of the andidate orrelation matrix and the R problem redues to solve (5), and the solution is provided in (6) whih, of ourse, oinides with the orret one ( R ) = γ (5) R γ = = (6) R ( R ) 5 For RM models, just hanging equation (3) by the relationship between the impulse response and the model oeffiients we have: h() h() h( P) h() h( P ) = = B Where vetor B ontains the P+ numerator oeffiients The solution is given by (7) whih is just a generalization of the R formula ( R ( ) = γ (7) This proedure to estimate RM spetrums is useful in those ases where the impulse response of the model is available for a few samples, lose to the

4 Proeedings of the th WE International Conferene on COMMUNICTION, gios Nikolaos, Crete Island, Greee, July 6-8, 7 93 origin sine the NR use to be higher and very low for samples above P t the same time, it uses the orrelation values lose to the origin with better results than estimating the denominator oeffiients from extended Yules-Walker equations 4 distane riterion Conerning the problem of finding a andidate autoorrelation in the data autoorrelation it an be formulated in terms of distane between autoorrelation matrixes Where (8) defines the line that onnets the data autoorrelation with the andidate autoorrelation, the distane between these two matrixes is given by (9), sub index F indiates Frobenius norm ~ R = R + t d ( R R) ( R, R ) = ( R R ) F (8) (9) When the data matrix is equal to the andidate plus a single line with autoorrelation given by γ, then is lear that maximizing γ is equivalent to maximise the distane between the original orrelation matrix and the resulting one from the subtration of the line ontribution Nevertheless, it is not adequate to define the distane between matrixes from (8), sine the line defined does not guarantee that any matrix in the line is positive definite Using geodesis instead of (8), the geodesi from the two matrixes is () and the geodesi distane is () / / [ ( )] / R exp t Ln R R R / R ( R, R ) Ln( λ ( q) ) ; R e = λ R e ~ R = () d = geo () gain, when the original data ontains the andidate plus a rank-one ontribution, only one of the eigenvalues of the generalized problem is different from one, reduing the geodesi distane to the log square of the maximum eigenvalue In general, finding for andidate orrelation γ R in the original data, the resulting square of the geodesi distane between the two matrixes will be () ( R γ R ) λ( q) d, = geo Ln () γ This distane has a minimum when γ is equal to the geometri mean of the eigenvalues and inreases monotonially when the parameter is above the geometri mean t the same time, γ is bounded by the maximum eigenvalue In onsequene the maximum distane is obtained when γ oinides with the maximum eigenvalue In summary, in order to perform spetral subtration of the andidate ontribution from the original data orrelation the maximum eigenvalue of the generalized problem maximizes the geodesi distane between the original matrix and the ontribution of the andidate matrix 5 Cognitive Radio appliation The advane of radio tehnology put in evidene that regulation rules beome obsolete in some respets Regulators did not envisage today s tehnology or, at least, not to ope with what an be done with advaned and intelligent radio interfaes One the protetion that regulation provides to a liensed user is the spetral mask The design of the mask, most of the ases, represents a great wall to prelude any non-liensed user to aess to the spetrum Ultrawideband ommuniations are able to provide reliable ommuniations, mainly in the short range without violating the regulation mask Nevertheless, the major drawbak of urrent regulation is that the radio spetrum is over-liensed but not over-used Liensed worldwide a given frequeny band may be free of use depending on the loation sine within this loation there is not interest for a ommuniations operator This is the ase in low populated areas or hazardous senarios among others From the regulation point of view seems to be lear that the spetrum have to be liensed taking into aount the loation It is non-sense to fore to pay for spetrum in loations where is not going to be used by the bidder Radio spetrum liensed in frequeny and square meters (geographially) will allow frequeny reuse in non-liensed sites where other ommuniation demands exists Nevertheless, the urrent system for liensing radio spetrum implies that many time-frequeny slots are free even in highly populated areas This fat represents an opportunity for modern tehnology, whih may use these free timefrequeny slots, being able to shutdown the ommuniation link when a liensed user enters to

5 Proeedings of the th WE International Conferene on COMMUNICTION, gios Nikolaos, Crete Island, Greee, July 6-8, 7 94 the pretended slot radio interfae with suh apabilities requires fast and reliable methods of spetrum labelling within the frequeny band where it aims to transmit Enompassed within the so-alled ognitive radio, spetrum labelling is one of the major subsystems omposing a ognitive radio transeiver The objetive of spetrum labelling is to detet any spetral oupany and, more important than this, to label spetral oupanies depending if they belong to a liensed user or to an opportunisti one In this setion, the onepts revisited previously will be used to propose a solution to this problem Let us assume that the liensed user when aess to the radio spetrum is sensed without knowing the power level neither the arrier frequeny The first assumption is general sine this level depends on the loation of the sensing equipment The seond orresponds to the ase where the ognitive radio searh among different liensed bands and it is agile to move among them The liensed user is assumed to use a known power spetral density, whih mostly depends on the baud rate and the symbol shape This andidate spetrum implies a spetral oupany defined by F(w) for a arrier frequeny ero κ ( w) F( w) = ( ) = F w dw π ( πb) w ( πb) elsewhere (3) From this andidate spetrum, the orresponding autoorrelation funtion is obtained by inverse DFT, and the Toeplitz matrix R is derived in aordane with the order of the spetral estimation proedure The greater the order is the best will be the performane of the proedure and, at the same time, it will inrease the number of samples required to detet the presene or absene of the liensed user In order to explore the arrier frequeny and the power level, the andidate orrelation is saled by a fator γ and modulated using the omponentprodut (the produt of the two matrixes is the diret produt of eah orresponding entries) of the ero frequeny orrelation andidate with a rank-one matrix formed with the steering vetor at the sensed arrier frequeny The andidate orrelation is shown in (4), where denotes the omponent-byomponent produt [( ) R ] γ (4) Note that the andidate spetrum redues to traditional spetral estimation when only the ero frequeny on (3) has a value different from ero (equal to one due to the power normalization The power level estimate is found as the maximum λ of the following problem: ( R [( ) R ]) = λ (5) s it is expeted this power level estimate shows the same reliability and low resolution that it experienes for deteting a single line spetrum In (5) the filter proves that we an measure at its output a power level equal to whih, thanks to the frequeny response of the filter is proportional to the output power when the andidate spetrum is the only ontribution to the input spetrum [( ) R ] ( ) = λ = λ ψ (6) Thus, the power level estimate an be viewed as the power output of the filter, normalized by the response to the andidate spetrum γ ˆ = = λ (7) ψ ( ) In order to ompute the noise bandwidth of the filter, we need to assume that the bakground noise of the input spetrum is white with power density No In this ase the power level estimate is formed by the ontribution of the andidate plus the white noise density multiplied by the equivalent white noise bandwidth γˆ = = γ + N = γ + No B ψ o ( ) ψ ( ) (8) ine the filter is an eigenvetor whih norm is one The noise bandwidth is ψ ( ) In summary, the spetral estimation proedure is summarized below: N ( )

6 Proeedings of the th WE International Conferene on COMMUNICTION, gios Nikolaos, Crete Island, Greee, July 6-8, 7 95 Defineandidateautoorrelationmatrix atunitpowerleveland basebandfrequeny Findthemaximumeigenvalueandtheeigenvetor asoiatedof λr e= R e Power level estimate γˆ = λ ( ) ( ) Densityestimate sˆ λ e R = ( R e ) (9) Needless to say that the response, mainly of the density estimate, depends on how lose in geodesi distane terms is the andidate spetrum with the atual omponent within the data matrix t the same time, those omponents, whih are far away from the andidate, will not be shown by the density estimates lso, it is worthwhile to remark that the proedure redues to lassial spetral estimation when the andidate spetrum is a single line (ie orrelation matrix of the andidate is all ones) 6 imulations In order to illustrate the performane of the proedure, a data reord of 8 samples orresponding to the ase of two unmodulated arriers in white noise is analyzed The NR of the arriers are and db respetively, the frequenies of the arriers are 667 and 4, and the filter is 8 The andidate spetrum is γ ones(8,8) Note that this ase orresponds with lassi spetral estimation The periodogram is also inluded in the plot for referene Figure ontains the periodogram, power level and power density obtained with the proedure s mentioned previously the periodogram is above the power level and the power level is above the density estimate Next, the unmodulated arrier at 667 is replaed by a wideband spetrum uniformly oupying the frequeny range of 4 up to 49 yet preserving the same signal to noise ratio of db Figure illustrates the performane of traditional periodogram, power level and density estimates s it an be viewed only the unmodulated arrier is deteted with an inreased floor on its left orresponding to the wideband spetrum petral Estimates in db Frequeny Fig Top-Down, Periodogram, Power level estimate, petral density estimate for a andidate spetrum equal to an unmodulated arrier 5 5 petral Estimates in db Frequeny Fig Top-Down, Periodogram, power and density estimates (traditional spetral estimates) and with a wideband (Bandwidth 5) entred at 667 Changing the andidate spetrum to the wideband spetrum with the same bandwidth the situation hanges and as an be viewed in Figure (3) the unmodulated arrier disappears meanwhile a lear peak appears in the entral frequeny of the andidate spetrum Note that the power level indiates 9 db (tual db) and the density peaks lose to the atual loation In any ase, the performane is remarkable

7 Proeedings of the th WE International Conferene on COMMUNICTION, gios Nikolaos, Crete Island, Greee, July 6-8, Fig 3 Density estimate (solid) and power level estimate (dashed) when using a wideband andidate of the same bandwidth Figure 4 shows the performane of the proedure, ompared with traditional estimates, when the power of the wideband signal inreases up to db 5 petral Estimates in db Finally, for a andidate spetrum equal to a BPK andidate of five samples per symbol using retangular NRZ signalling, the andidate orrelation is just a triangle of four samples per side and maximum one at the ero lag Note that this spetral andidate is valid also for QPK, MK and QM whenever the baud rate and the pulse shaping are the same Figure 5 depits the results for the ase that the BPK signal has a power of db above white noise power level The rest of the data for these graphis are the same as before db petral Estimates Frequeny Frequeny 4 Candidate estimates Candidate petrum Frequeny Frequeny Fig 4 Estimates orresponding with figures (upper) and 3 (lower) when the power of the wideband signal is db Fig 5 Traditional (upper) Candidate (lower) for a BPK signal entred at frequeny together with an unmodulated arrier at 4 Power levels above noise power of and db respetively 7 Conlusions generalization of the filter bank analysis for spetral estimation has been generalized, providing power level and density estimates of spetral omponents The shape and bandwidth of the spetral omponent, referred as the andidate spetrum, is introdued in the filter bank framework just in the same manner that single spetral lines are used to san the input data in order to estimate power

8 Proeedings of the th WE International Conferene on COMMUNICTION, gios Nikolaos, Crete Island, Greee, July 6-8, 7 97 and density The quality of the estimates reported depends diretly on the geodesi distane of the orrelation matrix of the spetral omponent present in data with the orresponding matrix of the andidate [6] aykin Cognitive Radio: Brain-Empowered Wireless Communiations IEEE Journal on eleted reas in Communiations, 3(5), pp -, Feb 5 The proedure was motivated to solve the problem faed in ognitive radio systems where different spetral labelling has to be provided to liensed users negleting, at the first stage other nonliensed users or interferers Finally, the framework reported provides a different view of major spetral estimation methods inluding frequeny detetors and parametri spetral estimation proedures knowledgements: My aknowledgement to the Mexian Government, for providing me the PROMEP fellowship for studying PhD program at Universitat Politènia de Catalunya Referenes: [] P toia, R Moses Introdution to spetral analysis Upper adle River, NJ Prentie all, 997 [] J Capon igh resolution frequenywavenumber spetrum analysis Pro IEEE, Vol 57, pp 48-48, ug 969 [3] M Lagunas, Gasull n improved Maximum Likelihood petral Estimate, IEEE Trans on oustis peeh and ignal Proessing, Vol P-3, no, pp 7-73, February 984 [4] M Lagunas et al Maximum Likelihood Filters in petral Estimation, ignal Proessing, EURIP, "peial Issue on Major Trends in petral nalysis" Volume, No, IN , pp 9-35, January 986 [5] P toia, Jakobsson, J Li «Mathed filter bank interpretation of some spetral estimators» ignal Proessing, Vol 66, pp 45-59, pr 998 [5] JMitola Cognitive Radio: n Integrated gent rhiteture for oftware Defined Radio Dissertation, Royal Institute of Tehnology