Compressed Edge Spectrum Sensing for Wideband Cognitive Radios

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1 Compreed Edge Spectrum Sening for Wideband Cognitive Radio Edgar Beck, Carten Bockelmann and Armin Dekory Department of Communication Engineering Univerity of Bremen, Bremen, Germany {beck, bockelmann, Abtract Free licened pectral band have become rare due to the increaing number of wirele uer and their demand for high data rate Likewie, the tatic allocation of thee band reult in an under-utilization of the pectrum Cognitive Radio (CR) ha emerged a a promiing olution to the dilemma by allowing opportunitic uer to tranmit in the abence of licened uer Spectrum ening i therefore the key component of CR and coexitence management in general In order to detect a much tranmiion opportunitie a poible, a large bandwidth ha to be monitored which according to Shannon-Nyquit neceitate high ampling rate For fat and accurate pectrum etimation, we propoe a novel approach called Compreed Edge Spectrum Sening (CESS) which exploit the parity of power pectrum edge and allow for ampling down to 6% of Nyquit without loe in the detection accuracy of occupied and unoccupied pectrum region I INTRODUCTION The number of device with a need for higher data rate i growing due to the tranition from voice-only to multimediatype application [1] The concept of Cognitive Radio (CR) [2] ha evolved a a reult of the ever-increaing demand for new pectral band and the under-utilization of thoe already tatically allocated Herein, econdary uer are allowed to ue vacant band in the abence of licened primary uer Similar, Coexitence Management (CM) [3] for Indutry 4 application hould enable coordination of everal wirele connected ytem in crowded indutrial environment Therefore, a crucial apect of CR and CM i pectrum ening to detect thee opportunitie for tranmiion, the o-called white pace In order to find a much white pace a poible and to enhance throughput, a wideband mut be ened with the reult of high Nyquit rate Otherwie, a bank of tunable narrowband bandpa filter, to earch one narrowband at a time, would become neceary, which mean a coniderable implementation challenge and reult in a high power conumption not favored in wirele device Fortunately, everal meaurement have hown the pectrum to be under-utilized [1] o that it can be aumed pare Therefore, Compreed Sening (CS) i often mentioned a a way of reducing the load [4], [5], [6] However, two problem reult On the one hand, CS algorithm add proceing complexity and delay Since the time for detecting udden interference to a primary uer ha to be a low a poible, algorithm have to be choen that can be executed very fat On the other hand, the Power Spectral Denity (PSD) may not alway be pare but crowded, eg, in a PSD white pace Fig 1 Spare pectrum with white pace buy environment, which make application of CS difficult and require repreentation in another bai Becaue many pectra in practical application like thoe of OFDM tranmiion exhibit harp boundarie and can be approximated a piecewie flat, the derivative ha a few non-zero element which depict edge Thu, it eem reaonable to ue the edge pectrum a the parity bai which we exploit in thi paper In the following, the underlying theory i decribed and extended to build a whole proceing chain which i able of eeking and claifying occupied and free frequency band II SYSTEM MODEL AND PROBLEM STATEMENT Uually, the Nyquit rate equaling the bandwidth of interet i choen and N ample x C N 1 of the continuou ignal x(t) are obtained in order to perform pectrum ening The ignal conit of the everal tranmiion x = i x i with a piecewie flat PSD of height σ 2 i a illutrated in Fig 1 The tranmiion have to be detected and are additionally uppoed to be zero-mean wide-ene tationary Thee aumption are valid for many ignal appearing in real world etting, eg, OFDM tranmiion [4], ince well-defined pectrum mak are employed To introduce CS, we conider a ubampling approach It i repreented by the ubampling matrix V C M N and generate compreed meaurement y C M 1 Relating the meaurement to the amplitude pectrum c through y = Vx = VF 1 c = Ac (1) with F C N N being the DFT matrix, we formulate the CS recontruction problem ĉ = arg min c c t y = Ac (2) f

2 to find c with under-determined A auming parity in c Here, c count the number of non-zero element in c A Power Spectrum Sening To recontruct the PSD intead of the amplitude pectrum, we make ue of the autocorrelation matrix [5]: R y = E [ yy H] = E [ Ac(Ac) H] = AR c A H (3) By definition [5], = E [ c 2] hold o that we can find the entrie of on the diagonal of R c Owing to the wideene tationarity of all ignal, R c i a diagonal matrix After vectorization and exploitation of the diagonal tructure, the following relation can be obtained according to [5]: vec (R y ) = (A A) vec (R c ) = (A A) = Φ (4) Here, A denote the complex conjugate of A, the Kronecker product and the column-wie Kronecker product, alo known a Khatri-Rao product The new PSD ening matrix Φ ha the dimenion M 2 N and (4) may thu exhibit a unique olution for M 2 N However, the minimum number of meaurement required to olve (4) uniquely i derived in [5] and equal M > N/2 due to the pecific problem tructure In fact, thi allow for ubampling without auming parity and olving the overdetermined equation ytem where r y = vec (R y ) with the Leat Square (LS) method: ŝ = arg min r y Φ 2 2 ŝ = Φ r y (5) indicate the Moore-Penroe peudoinvere and enable olution alo for underdetermined equation ytem which can be treated a l 2 -approximation of the l -norm in the equivalent CS recontruction problem Thi problem can be tated a ŝ = arg min t r y = Φ (6) and allow for further ubampling ince it can be olved uniquely for M > a derived in [5] In other word, a minimum average ampling rate equaling the actual occupied bandwidth become neceary lowering the requirement in a practical implementation The compreion ratio read κ = M/N > /N Thi mean a reduction of 5% compared to (2) and i due to the aumption made about tationarity For the previou bound to be valid, A ha to be full park B Practical Conideration In a practical implementation, olving (6) require Q meaurement y i to approximate the expected value of the autocorrelation by the mean where high value of Q lead to a high noie or error reduction: ˆR y = 1 Q y i yi H (7) Q i=1 Thee meaurement are obtained in a time window of length t auming wide-ene tationarity a illutrated in Fig 2 It T = K t t t Fig 2 Occupation of a pectrum in time t and frequency f The pectrum i aumed tationary in one time window t equaling the time reolution how the varying occupation of pectrum which remain contant in one time window If T S denote the Nyquit ampling period and N the frequency reolution, the whole ampling time for (7) amount to t = QNT S Therefore, a trade-off between fat detection, reolution and noie uppreion ha to be conidered Becaue noie i preent in practical etting, (6) ha to be modified to include a bounded error ɛ of the l 2 -norm: ŝ = arg min t r y Φ 2 ɛ (8) III COMPRESSED EDGE SPECTRUM SENSING A One-dimenional Edge Spectrum Sening Unfortunately, there i no guarantee for the pectrum to be pare ince it can be crowded or full in wort cae One olution to overcome thi problem i to conider the edge z of the piecewie contant pectrum due to the fact that there are coniderably fewer edge (exactly J) than occupied entrie of the PSD and entrie in general (J = z N) The definition of a new CS problem where the power pectrum in (4) i replaced by the edge pectrum immediately ugget itelf Both are related to each other through a numerical derivation = Γ 1 z, (9) with Γ C N N being the difference matrix 1 Γ = 1 1 (1) 1 1 Thi lead to the CS problem ẑ = arg min z z t r y = ( ΦΓ 1) z (11) defining a new approach that combine the idea of [4] and [5] In the following, we deignate thi approach one-dimenional Compreed Edge Spectrum Sening () It allow unique recontruction of the edge pectrum for compreion ratio up to κ = M/N > J/N which i a reult of the fact that the new ening matrix ΦΓ 1 ha the ame park a the old one Jut the mapping to the null pace change, but not f

3 the propertie of the problem itelf One problem occurring now i that the invere Γ 1 i a ummation matrix umming up a certain amount of entrie of the repective row of Φ In every j-th column of ΦΓ 1, N j column φ i of Φ are ummed up: N i=j φ i Therefore, the coherence hould increae ignificantly which ugget that the CS problem (11) i in contrat to (6) not robut againt noie Thi indicate that the Retricted Iometry Property (RIP) i jut fulfilled barely B Two-dimenional Edge Spectrum Sening So far, only edge in the frequency domain have been exploited But the pectrum alo exhibit edge in the time domain a depicted in Fig 2 Thi tem from the fact that communication ytem are only active for a limited time to adhere to regulation or becaue of intermittent activity Hence, thi additional tructure can be ued to create an advanced pectrum ening algorithm baed on the Total Variation Norm (TVN) in two dimenion For thi purpoe, K equation from (11) obtained in time window of length t with ubampling matrice Φ i are tacked together into one equation ytem: r y1 Φ 1 1 r y2 r T = = Φ 2 2 = Φ T T r yk Φ K K (12) For maximum ampling efficiency, the ampling proce in a practical CS ytem hould be deigned to include randomne [7] Hence, the Φ i are different from each other o that additional information can be ued In order to recontruct the edge in the time and frequency domain, the 2D-derivative of T ha to be minimized which lead to the following TVN minimization problem: ŝ T = arg min T Γ 2D T 1 t r T = Φ T T (13) The 2D-difference matrix Γ 2D = [Γ f, Γ t ] T conit of the difference operation in the frequency domain Γ f = I K Γ 1, with Γ 1 { 1,, 1} N 1 N denoting 1 1 Γ 1 = (14) 1 1 and I K {, 1} K K the identity matrix, and the difference operation in the time domain Γ t { 1,, 1} (K 1)N KN : T 1 1,1 1 2,2 Γ t = 1 N+1,1 1 (K 1)N,(K 1)N (15) 1 N+2,2 1 KN,(K 1)N From here on, thi algorithm will be denoted a 2D CESS The motivation for applying it i imple: The neceary minimum number of meaurement hould intuitively decreae when conidering piecewie-contant pectra a depicted in Fig 2 ince more information i exploited In comparion to the onedimenional cae, the reult are expected to allow for greater compreion and performance, repectively But thi hould happen at the cot of a delay T = K t, the time for collecting the ample IV PROCESSING CHAIN Fig 3 how the whole propoed proceing chain to perform pectrum ening Firt, the ignal arrive through a mitigating channel H and are uperimpoed with AWGN After ubampling, the power pectrum i recontructed according to (5), (6), (11) and (13), repectively Now, the quetion ha to be aked where the band are located and whether they are occupied or not Therefore, we apply a imple feature detection It conit of the Wavelet Edge Detector (WED) and the Energy Detector (ED) [1], [4] Firt, the WED detect the band boundarie ˆf i by looking for the local maxima in the derivative z We notice that direct recontruction of edge with CESS hould make thi tep redundant ince boundarie are expected to be where the derivative i non-zero However, CS recontruction a well a pectrum hape are not perfect and noie make it impoible to find the true olution To uppre noie induced edge, we thu additionally apply a threholding operation For the ake of implicity, we do not apply a Gauian kernel on the PSD or ue it even in the recontruction tep, in contrat to [4] In order to chooe a reaonable threhold η WED, it relation to detection and fale alarm rate ha to be derived: To begin with, we model the amplitude pectrum a a multivariate complex gauian random vector c CN (, Σ) with covariance matrix Σ Here, Σ = diag {, Σ i, } + σn 2 I N i a block diagonal matrix where each band tranmiion i repreented by it power level Σ i = σi 2I B i and bandwidth B i with i B i = N Some band are empty (σi 2 = ) and conit only of the noie level σn 2 So the received Nyquitampled ignal x conit of additive correlated Gauian noie According to (3) and (7), the average of the quared abolute value of c i the approximated etimate of the power pectrum For Q = 1, one point i ditributed according to the chi-quare ditribution χ 2 2 with two degree of freedom and variance (σi 2 + σ2 n ) 2 If more realization are examined, the variance decreae by the number of frame Q Additionally, thi reult in convergence to a normal ditribution according to the central limit theorem After numerical derivation, the variance Var [Z] of one point in the edge pectrum z i the um of two neighboring variance in the PSD Hence, the rate,wed = 2 F N ( η WED / ) Var [Z] (16) for exceeding an abolute value η WED can be tated, with F N denoting the cumulative function of the tandard normal ditribution In order to generate a threhold atifying a fale alarm rate,wed, we invert the equation We chooe

4 Channel H + AWGN Subampling Power Spectrum Recontruction LS CS Feature Detection Wavelet Edge Detector 3 2 Energy Detector 1 Reult Boundarie ˆf i Occupation ˆb Fig 3 Propoed deign of the pectral etimator κ [%] Fig 4 Detection and fale alarm rate of in comparion to LS and CS recontruction for variou compreion κ TABLE I DEFAULT SIMULATION PARAMETER SET parameter N Q κ β σn 2,WED value % the part of the pectrum with highet ignal power σi 2 for calculating the variance Var [Z] = 2 ( σi 2 + ) 2 σ2 n /Q of the random variable ince we have to cover the wort cae Analogouly, the mied detection rate ( ) ( ) η WED µ η P MD,WED = F N WED µ F N (17) Var [Z] Var [Z] ( at one noie-ignal ) edge with variance Var [Z] = σn 4 /Q + σ 2 i + σn 2 2 /Q and height µ = σ 2 i σn 2 can be derived In the lat tep, the energy detector imilarly compare the average carrier amplitude between two boundarie to a threhold depending on the noie level In contrat to literature [1], [8], we chooe it to be the ame except for the noie level that wa not conidered at the WED tage and ha to be added: η ED = η WED + σn 2 Thi i reaonable ince the height of rectangular hape correpond to the height of the repective edge Becaue the pectrum i expected not to be fully occupied, the noie floor can be extracted from a band with the lowet energy which wa done in the imulation In the end, one receive the binary vector ˆb compried of the information regarding occupation in every ingle band A Tet Setup V SIMULATION RESULTS To evaluate CESS, we aumed allocation of the 245GHz- ISM-band in accordance with the 8211g/n-tandard In addition, we added a fifth band at 9 MHz to thoe at 1, 3, 5 and 7 MHz to enable full occupation of a pectrum with 1 MHz bandwidth Signal were modeled a lowpa-filtered AWGN a before, with a bandwidth of 2 MHz and carrier power σi 2 {4, 6, 8, 1, 12} We fixed occupation β = /N, and thu the number of ignal, to maintain the ame condition and modulated them to a randomly elected carrier For invetigation of 2D CESS, alo time behavior had to be modeled Hence, a Markov model with the tate empty and occupied wa defined for every ingle band with the ame mean and tarting occupation β Here, the probability for changing the tate after one time lot t from empty to occupied wa et to 1/3 and 1/2 vice vera For CS recontruction, we ued the OMP (Orthogonal Matching Puruit) algorithm, and for TVN minimization, the convex optimization toolbox CVX [9] with the olver SDPT3 In both cae, we choe the topping criterion to be the true reidual ɛ = R y ˆR y F calculated with the ideal pectrum hape in order to how the bet theoretical performance In practice, other criteria have to be ued Table I how the default imulation parameter reflecting practical etting An appropriate threhold η WED to realize a ufficient dynamic range for the tet ignal can be obtained by dynamically etimating Var ˆ [Z] = 2 max(ŝ) 2 /Q and chooing,wed = 1% To fulfill null pace criterion and RIP with high probability, the ubampling matrix V wa et to a Gauian random matrix with normalized column Finally, the number of Monte Carlo trial wa choen a 1 B In Fig 4 the detection rate and the fale alarm rate for different method are illutrated a a function of the compreion κ Above κ = 15%, LS, CS and all how perfect recontruction Below 15%, LS ha a higher detection rate than CS, but vatly more fale alarm In contrat, begin to deteriorate ignificantly at a very low compreion ratio of 6% In the choen tet etup the number of non-zero element i reduced from 4 pectral point to J = 5 edge Thu, the recontruction quality i adequate even at the minimal compreion ratio κ = 6/1 > J/N = 5/1 which offer unique recontruction Furthermore, thi lead to reduced complexity in term of OMP iteration Altogether, it

5 D CESS D CESS D v 1D κ [%] Fig 5 Detection and fale alarm rate of 2D and for K = 2 time intance veru compreion κ , 1, Q Fig 6 Detection and fale alarm rate of 2D and for K = 2 time intance a a function of frame Q Fixed η WED = 15 and κ = 6% TABLE II NMSE OF 2D AND FOR K = 2 TIME INSTANCES AND κ κ [%] D CESS [db] [db] can be clearly een that outperform the other two approache in the tet etup C 2D CESS The performance of the pectral etimator in term of detection and fale alarm rate when uing 2D CESS and K = 2 can be deduced from Fig 5 In comparion to, lightly higher detection rate are achieved for relevant compreion up to 3% wherea fale alarm can be ignificantly reduced at leat by half, eg, from 75% to 1% at κ = 6% Another meaure for comparing the performance regarding recontruction accuracy i the normalized Mean Square Error (nmse) which i related to the ideal pectrum and given for ome value of κ in table II It can be clearly een that 2D CESS achieve an at leat 3 db maller nmse for relevant compreion than In ummary, it can be tated that in accordance with the firt gue 2D CESS offer performance benefit becaue it utilize the additional tructural information in the time domain However, 2D CESS introduce a delay due to the longer time window T = K t For a delay oriented comparion, we fixed the threhold to η WED = 15 and depicted Q in Fig 6 We can ee that at lower Q mainly fale alarm performance i better for 2D CESS A a reult, Q can be reduced, eg, by a factor of K = 2 from 2 to 1 for contant 5%, in comparion to Hence, both 2D and can have the ame ampling time at a comparable fale alarm rate But thi doe not hold for detection rate: A reduction by K 67 from 2 to 3 i poible here In concluion, 2D CESS introduce a delay and additionally ha a higher computational complexity VI CONCLUSION The main reult of called work can be ummarized a follow: can provide a lower compreion ratio than the CS and LS approach due to exploitation of the inherent ignal tructure Good performance can be achieved until the actual edge pectrum occupation of 5% i reached 2D CESS allow for an even better performance if time-domain edge are exploited Therefore, we aume CESS to be a proper candidate for future CR and CM application ACKNOWLEDGMENT Thi reearch ha been funded by the Federal Minitry for Economic Affair and Energy of Germany through the AiF in the project KoMe (project number 1835 BG/2) REFERENCES [1] E Axell, G Leu, E G Laron, and H V Poor, Spectrum Sening for Cognitive Radio [State-of-the-art and recent advance], IEEE Signal Proceing Magazine, vol 29, no 3, pp , May 212 [2] T Yücek and H Arlan, A Survey of Spectrum Sening Algorithm for Cognitive Radio Application, IEEE Communication Survey & Tutorial, vol 11, no 1, pp , Mar 29 [3] IEEE Standard for Information technology Telecommunication and information exchange between ytem Local and metropolitan area network Specific requirement Part 19: TV White Space Coexitence Method, IEEE Std , pp 1 326, Jun 214 [4] Z Tian and G B Giannaki, Compreed Sening for Wideband Cognitive Radio, in IEEE International Conference on Acoutic, Speech and Signal Proceing (ICASSP), Apr 27, pp [5] D Cohen and Y C Eldar, Sub-Nyquit Sampling for Power Spectrum Sening in Cognitive Radio: A Unified Approach, IEEE Tranaction on Signal Proceing, vol 62, no 15, pp , Aug 214 [6] D Cohen and Y C Eldar, Sub-Nyquit Cyclotationary Detection for Cognitive Radio, IEEE Tranaction on Signal Proceing, vol 65, no 11, pp , Jun 217 [7] Y C Eldar and G Kutyniok, Compreed Sening - Theory and Application Cambridge Univerity Pre, 212 [8] Y L Polo, Y Wang, A Pandharipande, and G Leu, Compreive Wideband Spectrum Sening, in IEEE International Conference on Acoutic, Speech and Signal Proceing (ICASSP), Apr 29, pp [9] M Grant and S Boyd, CVX: Matlab Software for Diciplined Convex Programming, verion 21, Dec 217