6908 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 66, NO. 8, AUGUST 2017

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1 6908 IEEE TRASACTIOS O VEHICUAR TECHOOGY, VO. 66, O. 8, AUGUST 017 Preecision or Wieban Spectrum Sensing With Sub-yquist Sampling Tianyi Xiong, Member, IEEE, Hongbin i, Senior Member, IEEE, PeihanQi,Zani, Senior Member, IEEE, an Shilian Zheng, Member, IEEE Abstract Built on compresse sensing theories, sub-yquist spectrum sensing SSS) has emerge as a promising solution to the wieban spectrum sensing problem. However, most o the existing SSS methos o not istinguish i primary users PUs) are present or absent in the concerne spectrum ban an irectly pursue support recovery o the PUs. This may lea to a high alse alarm rate an a waste o computational cost. To aress the issue, we propose a preecision algorithm, reerre to as the pairwise channel energy ratio PCER) etector, to etermine the presence or absence o PUs prior to signal support recovery. The propose etector is base on the popular moulate wieban converter MWC) ramework or SSS, which has several avantages over other SSS approaches. The PCER test statistic is constructe rom compresse samples obtaine by the MWC. The ecision threshol an the etection probability are erive in close orm ollowing the eyman Pearson criterion. umerical results are presente to veriy the theoretical calculation. The propose PCER etection metho is shown to be able to etect the existence o PUs in a wie range o signal-to-noise ratio, while being robust to noise uncertainty an oes not nee the prior knowlege o the PU signals. Aitionally, our results show that the use o the PCER etector leas to a signiicant improvement o the correct support recovery rate o the PU signals. Inex Terms Cognitive raio CR), compresse sampling, correct support recovery CSR), moulate wieban converter MWC), noise uncertainty, sub-yquist spectrum sensing SSS), wieban spectrum sensing. I. ITRODUCTIO THE exponential growth o ubiquitous wireless evices an services uring past ecaes has impose increasing Manuscript receive June 13, 016; revise October 3, 016 an December 15, 016; accepte January 6, 017. Date o publication January 3, 017; ate o current version August 11, 017. This work was supporte by the ational atural Science Founation o China uner Grant an Grant The work o H. i was supporte in part by the ational Science Founation uner Grant ECCS an Grant ECCS The review o this paper was coorinate by Pro. S.-H. eung. Corresponing author: Peihan Qi.) T. Xiong, P. Qi, an Z. i are with the State Key aboratory o Integrate Service etworks, Xiian University, Xi an , China xiongtianyi1989@163.com; phqi@xiian.eu.cn; zanli@xiian.eu.cn). H. i is with the Department o Electrical an Computer Engineering, Stevens Institute o Technology, Hoboken, J USA Hongbin.i@stevens.eu). S. Zheng is with the Science an Technology on Communication Inormation Security Control aboratory, Jiaxing , China lianshizheng@16.com). Color versions o one or more o the igures in this paper are available online at Digital Object Ientiier /TVT stress on the limite spectrum resources. The current spectrum policy, primarily base on allocating a ixe-requency ban to iniviual wireless services, has been shown to be highly ineicient, causing congestion in some bans while unerutilization in others 1 3. The cognitive raio CR) 4 6, which allows seconary users SUs) to opportunistically access unuse raio spectrum, calle white space, has been wiely consiere as a promising solution to the above problem. A key unction in the CR is spectrum sensing that involves monitoring the spectrum usage an etecting the presence/absence o primary users PUs) 7 9. Recently, wieban spectrum sensing has receive signiicant interest 10 13, where SUs perorm spectrum sensing in a wie range o requency ban to search or more access opportunities. However, the implementation o wieban spectrum sensing at the SU has some iiculties, e.g., the nee or a complex ront en, high sampling rate, an ast igital signal processing 14. Meanwhile, compresse sensing has been establishe as a new iel that oers powerul tools or recovering sparse signals rom a ew nonaaptive linear measurements, signiicantly reucing the sampling rate compare with the conventional yquist sampling ramework This has motivate the interest o using compresse sensing techniques to solve the wieban spectrum sensing problem. Sub-yquist spectrum sensing SSS) base on compresse sampling has been consiere in a multitue o stuies. In 19, a wieban spectrum sensing approach base on yquist sampling is introuce by using sparsity recovery techniques originate rom compresse sensing. The authors o 0 propose an autonomous compresse spectrum sensing algorithm base on a measurement proceure an the valiation approach. Qin et al. 1 combine compressive spectrum sensing with geolocation atabase to in spectrum holes in the CR. For compresse sampling o wieban signals, an analog-to-inormation converter AIC) that employs a wieban pseuoranom emoulator an a low-rate sampler was introuce in 5. Base on the AIC, several SSS methos consisting o two stages were propose in 6 9. In the irst stage, the original wieban signal or its power spectrum is reconstructe rom compresse samples; in the secon stage, wieban spectrum sensing is perorme to etermine the locations o the occupie requency bans. However, it has been oun that the AIC ramework can be aecte by some input signal moel mismatch problems 10, 30. Multicoset sampling is another compresse sampling ramework 10, 31 33, which uses multiple analog-to-igital converter ADCs) in a time inter IEEE. 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2 XIOG et al.: PREDECISIO FOR WIDEBAD SPECTRUM SESIG WITH SUB-YQUIST SAMPIG 6909 leaving ashion. However, maintaining accurate time shits an channel synchronization between sampling channels are iicult to implement, an the possible ront ens are ar below the wieban regime. A practical compresse sampling ramework base on a moulate wieban converter MWC) is propose to cope with the input moel mismatch problem 30, 34. The MWC ramework escribes the input signals as a sparse union o shitinvariant subspaces an processes the input signal in multiple channels simultaneously. In each channel, the input signal is multiplie by a perioic ±1 sequence, iltere by a low-pass ilter PF), an then sample by a low-rate ADC. By means o rame construction an solving a multiple measurement vector problem, the signal support can be estimate irectly without a ull signal recovery. It has been shown that the MWC-base SSS has the avantages o being able to aaptive to ierent types o signals, ast signal support recovery SR), an low computational loa 35. Most o the existing SSS methos are evelope uner the assumption that the PU signals are present in the concerne requency ban. However, there are cases where the requency ban may be completely vacant an only backgroun noise exists, e.g., in satellite an millimeter wave communications, 3, 36. In such cases, a irect application o the aorementione SSS methos is inappropriate, ue to the absence o a sparse signal, an woul lea to several problems: 1) high alse alarm rate SSS methos are likely to provie incorrect spectrum sensing results, causing a high alse alarm probability an making the vacant spectrum bans to be unerutilize; an ) a waste o computational cost an energy. In this paper, we propose a preecision algorithm, reerre to as the pairwise channel energy ratio PCER) etector, which is integrate with the MWC ramework or SSS. The PCER algorithm is to etermine the presence/absence o PU signals prior to signal SR. Only i the PU signals are etecte in the concerne requency ban, the unction o signal SR will be activate to estimate the location o the occupie bans; otherwise, signal SR is bypasse. It is necessary to point out that the propose preecision algorithm can also be extene to other compresse sampling ramework besies the MWC. We moel this preecision problem as a binary hypothesis testing an construct a test statistic with compresse samples to etect the PU signals. By exploiting the statistical properties o the test statistic, we erive the ecision threshol an the probability o etection in close orm, ollowing the eyman Pearson P) criterion. Theoretical erivation an numerical simulation show that our PCER etector is robust to noise uncertainty 37. Furthermore, it oes not require any prior knowlege o the PU signal. Some spectrum sensing techniques have been propose to overcome noise uncertainty 6, The algorithms propose in 6 an 38 employ the covariance matrix o the signal samples, an eigenvalue ecomposition is require. The cyclostationarity-base spectrum sensing in 39 nees to know the cyclic requency o the signal as a prior knowlege; otherwise, exhaustive search has to be carrie out. In recent work o Sun et al. 40, particle iltering technology is use to esign a spectrum sensing technique to overcome noise uncertainty. Fig. 1. Conventional MWC-base SSS. However, the above-mentione spectrum sensing algorithms are esigne in the conventional yquist sampling ramework an are not suitable or the MWC-base SSS ramework in this paper. With the PECR preecision algorithm, the presence/absence o PU signals is etermine beore signal SR, which avois unnecessary computational overhea an high alse alarm probability cause by incorrect SR. As a result, the unctions o compresse sampling, signal etection, an signal recovery are integrate, making the SSS more useul or wieban spectrum sensing. The rest o this paper is organize as ollows. Section II briely reviews the MWC-base SSS ramework an introuces the preecision problem concerne in this paper. In Section III, the PCER algorithm is summarize, an close-orm expressions o the ecision threshol an probability o etection are erive. Section IV provies numerical results an iscussions on the perormance o the propose algorithm. The whole work is conclue in Section V. otation: E an D represent the mathematical expectation an variance operator, respectively, Cov, enotes the covariance between two ranom variables, an R, I represent taking the real part an imaginary part, respectively. II. MWC-BASED SSS AD THE PROBEM To motivate the propose algorithm an acilitate its presentation, we irst briely review the MWC-base SSS ramework an then introuce the preecision problem concerne in this paper. A. MWC-Base SSS Fig. 1 shows the conventional MWC-base SSS 35, which consists o the MWC sampling moule an the SR moule. The ormer converts the input analog signal x t) into compresse samples, an the latter estimates the support o the PU signals. At the SU, the receive signal xt), which is banlimite in the range o YQ /, YQ /, isetok channels o the MWC moule simultaneously. The signal in the ith channel is multiplie by a T p -perioic sequence p i t), which contains M chips alternating between levels ±1. Speciically p i t) θ im m T p M t m + 1) T p M, 0 m M 1 1) where θ im {, +1} an p i t + T p )p i t). Aterwar, the mixe signal o each channel is iltere by an PF h t) with pass ban s /, s /, where s is the sampling rate o the subsequent low-rate ADCs. Thus, the output o the PF is

3 6910 IEEE TRASACTIOS O VEHICUAR TECHOOGY, VO. 66, O. 8, AUGUST 017 expresse as y i t) x τ) p i τ) h t τ) τ ) where i 1,,...,K. Discretizing y i t) with a sampling interval T s 1/ s yiels y i n) y i t) tnts, n 0, 1,... 3) The iscrete-time Fourier transorm DTFT) o y i n) is given by 30 ) Y i e jπt s y i n)e jπnt s T p n c il X l p ), F 4) j π where c il 1 T T p 0 p i t)e lt p t are the Fourier series coeicients o p i t), p 1/T p, X ) is the Fourier transorm o x t), an 0 YQ + s ) / p ) is chosen as the smallest integer such that the summation in 4) contains all the nonzero subbans o X), which are aliase into F s /, s /. Equation 4) is key to the SR o the input signal, since it relates x t) to the DTFT o observations y i n), which are obtaine rom sub-yquist sampling. Deine Y ) Y ) 1 ),...,Y i ),...,Y K ) T, with Y i ) Y i e jπt s an Z ) Z1 ),...,Z l ),...,Z ) T, with Z l ) X +l 0 1) p ), where Consequently, 4) can be rewritten in a more compact orm as Y ) AZ ) 5) where F, an A C K with elements A il c i, l+0 +1, 1 i K an 1 l. It is necessary to point out that i xt), in the absence o noise, is a sparse multiban signal, i.e., there are a ew nonzeros bans in the requency omain, Z ) has the same sparsity support as xt). This is because each element o Z ) correspons to one subban o X) that is shite into s /, s /. Without loss o generality, we set p s in this paper, an equals M uner this setting 34. To implement SR with compresse measurements y i n), consier the autocorrelation matrix o Y ) +s / Q Y )Y H ) y n) y T n) 6) s / n where y n) y 1 n),...,y i n),...,y K n) T is the compresse samples output rom all the channels o the MWC at time instance nt s. Decompose the autocorrelation matrix as Q VV H, where V is the Hermitian square root o Q. Itis shown that the ollowing MMV sparse recovery problem 33 V AU 7) has a unique sparsest solution Ū with the ewest nonzero rows. I xt) is a sparse multiban signal, the support o vector Z ) can be obtaine as S supp Ū ). As a result, the number an location o occupie bans by PUs can be estimate. Fig.. Propose MWC-base SSS ramework. However, i the requency ban is vacant an xt) contains only white noise, the support o Ū solve rom 7) is totally irrelevant note that the support o Z ) oes not even exist in this case). This implies that the SR in the conventional MWC-base SSS will lea to incorrect spectrum sensing results, causing a high alse alarm rate an unnecessary computational expenses. Clearly, there is no nee to perorm SR i the requency ban is empty. B. Problem To aress the above issue, we propose to insert a preecision moule, reerre to as the PCER in this paper, between the MWC moule an the SR moule, as sketche in Fig., where enotes the hypothesis that there is only white noise in the input signal an H 1 represents that there exist PU signals. I the etection result o the PCER moule is, inicating that the whole requency ban is available an the SR is unnecessary an the SR moule is bypasse. On the other han, i the PCER moule ecies hypothesis H 1, SR is conucte subsequently, an the location o the occupie bans is estimate. By aing the PCER moule, the propose ramework integrates the unctions o ata acquisition, etection, an estimation, making the MWC-base SSS more useul or wieban spectrum sensing. As shown in Fig., the PCER moule has to solve the ollowing binary hypothesis etection problem base on compresse samples: y i n) { wi n), s i n)+w i n), H 1 8) where i 1,,...,K, w i n) is the noise sample output rom the ith channel o the MWC prouce by a zero-mean Gaussian white noise wt) at the input with power spectrum ensity σ w, an s i n) is the signal sample prouce by st) at the input which is sparse an compose o one or more PU signals. The etection perormance is evaluate by the probability o etection P an the probability o alse alarm P, which are eine as P Pr{ecision H 1 H 1 } P Pr{ecision H 1 } 9a) 9b) where P is the probability o etecting the PU sparse signal in the concerne requency ban when it is present, an P is the probability that the etector incorrectly ecies that the consiere requency ban is occupie, whereas there is only white noise.

4 XIOG et al.: PREDECISIO FOR WIDEBAD SPECTRUM SESIG WITH SUB-YQUIST SAMPIG 6911 perormance. As a result, a ecision matrix D is constructe with elements given by { 0, i γ <r <γ U Dj, h) 1, otherwise 1) Fig. 3. PCER Moule. III. PROPOSED PCER DETECTOR AD PERFORMACE AAYSIS In this section, we irst present the propose PCER preecision algorithm. Aterwar, we stuy the statistical characteristics o the PCER etector uner both hypothesis an H 1. Finally, close-orm expressions o the ecision threshol an the etection probability o the PCER etector are erive uner the P criterion. A. PCER Detector For the binary hypothesis etection problem 8), we construct a test statistic with the compresse measurements yn) an compare it with a ecision threshol to provie a ecision. The problem can be solve by the energy etector ED), which is a wiely use technique because o simplicity an low computational cost, on a channel-by-channel ashion. evertheless, the ecision threshol o the ED epens on the backgroun noise power, which makes its perormance sensitive to any uncertainty o the noise power. There are some work investigating the ED uner noise uncertainty 41 43, which requires estimating the noise power. Inaccurate estimation o the noise power will also aect the perormance o the ED, which makes such methos not suitable or the MWC-base SSS ramework. In the ollowing, we propose a novel test statistic, which oes not require the prior knowlege o the noise power an is shown to be robust to the noise uncertainty while has a similar complexity as the ED. Formally, as shown in Fig. 3, the propose PCER preecision algorithm is summarize as ollows. Step 1:The-point energy Ti t o the output o the ith channel o the MWC moule is compute as Ti t y i n) 10) n0 where i 1,,...,K enotes the channel inex, an the superscript t represents the time omain. Step : Consiering the impact o noise uncertainty on the perormance o ED, branch test statistics r are constructe as PCERs r Tj t /Th t 11) where j, 3,...,K an h 1,,...,j 1. ote that the test statistics employ the compresse samples rom all the K channels o the MWC. Step 3: Since energy ratio is employe as test statistic, it is reasonable to compare r with a lower threshol γ an an upper threshol γ U to be etermine) to achieve better where j, 3,...,K, h 1,,...,j 1, an Dj, h) is the element o D in the jth row an the ith column. For other values o j an h, Dj, h) is regare as irrelevant an set to 0. ote that there are K K)/ relevant elements in the ecision matrix D. Step 4: Given the branch ecision results i.e., the relevant elements o D), there are several popular usion rules, e.g., the OR, AD, an K-out-o- rule 44 46, which can be use to obtain the inal ecision result. For simplicity, we use the OR rule in this paper, an the other rules can be easily accommoate. Thereore, the inal ecision result is obtaine as K j j h1 Dj, h) 0, ecie K j j h1 Dj, h) 1, ecie H 1. 13) For a single pair o vali j an h, we reerre to an as the branch probability o alse alarm an the branch probability o etection, respectively, which are eine as Pr { r γ r γ H U } 0 14a) Pr { r γ r γ H U } 1 14b) where enotes the logic or operator. ote that 13) uses all branch ratios in 11) to etermine the inal ecision, which is calle all-branch usion rule in the ollowing. It is clear to see that these ratios are statistically correlate with each other, since some o the ratios share the same nominator an enominator, which can cause some eviation i we irectly applying the OR usion equation that requires inepenent usion branches 46. However, we oun that, via numerical simulation, there is relatively small correlation among the branch ratios r j,j,j, 3,...,K, or which the OR usion equation can be applie with only slight mismatch between theoretical an simulation results. Also, we eine r 1,0 as T1 t t /TK, which also has small correlation with the above branch ratios. The corresponing γ1,0, γ1,0 U, an D1, 0) are also eine in the same way as beore. Thereore, similar to 13), we have K j1 Dj, j 1) 0, ecie 15) K j1 Dj, j 1) 1, ecie H 1 an the overall probability o alse alarm an etection are given by P 1 P 1 K j1 K j1 1 P j,j ) 16a) 1 P j,j ). 16b)

5 691 IEEE TRASACTIOS O VEHICUAR TECHOOGY, VO. 66, O. 8, AUGUST 017 We shoul point out that the ecision rule 15) only uses partial branch ratios eine in 11), which is calle partialbranch usion rule in the ollowing, while the ecision rule 13) uses all branch ratios that coul achieve better perormance at higher complexity. Thus, by proviing both etectors, a traeo between perormance an complexity is possible. To in the overall P an P in 16), we shoul irst stuy the statistical property o r uner both hypotheses, which is examine in the ollowing subsections, an then use 14) to in the branch probability an. It is important to point out that the above algorithm is not limite to the MWCbase sampling ramework an can easily be extene to other compresse sampling rameworks. B. False Alarm Probability Analysis Uner, the receive signal xt) contains only white noise. Following the P criterion, we set a target branch probability o alse alarm an then in the branch threshol γ an γ U, j 1,,...,K, h j 1, which is given by the ollowing theorem. Theorem 1: For a given target branch probability o alse alarm, the corresponing branch threshol o the etector 11) is given by 17a) an 17b) at the bottom o this page, where Φ x) is the inverse unction o the Gaussian cumulative istribution unction CDF) Φx) 1 x π e η η, is the number o compresse measurements use to compute the channel energy an where Δ ρ 0 + ρ 1 + ρ + ρ 3) / 18) ρ 0 Rc jl )Ic hl ) Rc hl )Ic jl ) ρ 1 Ic jl )Rc hl ) Rc jl )Ic hl ) ρ Rc jl )Rc hl )+Ic hl )Ic jl ) 19a) 19b) 19b) ρ 3 Rc jl )Rc hl )+Ic hl )Ic jl ). 19c) Proo: To prove Theorem 1, we start rom 14a), which requires the statistical property o r eine in 11). ote that the branch test statistic r is the ratio o the signal energy rom two ierent channels o the MWC moule. We irst nee to etermine the statistical property o the numerator an enominator, i.e., the channel energy Tj t an Th t, respectively. Consiering the Parseval theorem that the energy in the time omain can be equivalently compute in the requency omain, we use the channel energy Tj an Th t instea o Tj an Th t, respectively, where the superscript represents the requency omain. It is more convenient to carry out the analysis an computation in the requency omain. Speciically, by the Parseval theorem, T i 1 Y i k) n0 y i n) T t i 0) where Y i k),i 1,,...,K, k 0, 1,..., 1, are the iscrete Fourier transorm DFT) o y i n). Since the DFT o y i n) is obtaine by uniormly sampling the DTFT o y i n), which is given by 4), Y i k) uner is Y i k) c il X k l) c il W k l) 1) where W k l) is the DFT o the aitive Gaussian white noise wt) at the input o the SU. emma 1 in Appenix A inicates that Ti is a sum o inepenent ranom variables Y i k), k 0, 1,,..., 1, with ientical mean an variance uner. Thus, or large an ollowing the central limit theorem, Ti is approximately a Gaussian variable with mean an variance given by E Ti 1 E Y i k) σw D T i 1 D Y i k) σw 4. a) b) The covariance o two ierent channel energies is Cov Tj,Th 1 Cov Y j k), 1 Y h k) 1 σ4 w Cov Y j k), Y h k) ) ρ 0 + ρ 1 + ρ + ρ 3) 3) γ γ U Φ, Φ 1 ) Φ,U, ) +Φ ) ) 1 ) Φ Φ, ) ) ), + 1 ) 17a) ) ) ) 1,U 1 Φ Φ 1,U ) ) ) 1,U + 1 ) 17b)

6 XIOG et al.: PREDECISIO FOR WIDEBAD SPECTRUM SESIG WITH SUB-YQUIST SAMPIG 6913 where ρ 0, ρ 1, ρ, an ρ 3 are eine in Theorem 1. The last equality o 3) ollows rom 39) o emma 1. Thereore, the correlation coeicient between Tj an Th can be calculate as Cov T j,t ρ h + ρ ρ + ) ρ3. D T j D T h 4) Accoring to the erivation above, it shows that r uner is the ratio o two statistically correlate Gaussian variables whose statistics are given by ) 4). Furthermore, to in the CDF o r, we employ the ollowing property. Property 1 see 47): The ratio R o two statistically correlate Gaussian ranom variable Z 1 an Z obeys the istribution with the ollowing CDF: F R r) PrR<r)Φ rμ μ 1 σ 1 rρσ 1 σ + r σ ) 5) where Φx) is eine in Theorem 1, μ 1, μ, σ1, an σ are the mean an variance o Z 1 an Z, respectively, an ρ is the correlation coeicient o Z 1 an Z. To apply Property 1 to the branch test statics eine in 11), we irst split the branch an upper part,u into two parts as lower part, accoring to 14), which satisy, Pr r γ H ) 0,U Pr r γ H U ) 0. Applying Property 1 to above equations yiels γ, Φ 1 γ ρ +γ ) γ U,U 1 Φ. 1 γ U ρ +γ U ) 6a) 6b) 7a) 7b) As a result, the branch threshol 17a) an 17b) can be obtaine by inverting 7a) an 7b), respectively. From the erivation above an the expressions o the branch threshol in 17), it is clear that the branch threshol γ an γ U o the propose preecision algorithm epens on the number o compresse measurements, the correlation coeicient o ierent channel energies, an the branch target alse alarm probability an the way to split the target.the correlation coeicient in 4) only epens on the measurement matrix A o the MWC. It is important to notice that the branch threshol γ an γu are not relate to σ w, i.e., the power o the backgroun Gaussian white noise, an no prior knowlege o the PU signal is neee. This is reason why our PCER etector 11) is robust to noise uncertainty. For the special case that the target branch probability o alse alarm is set to be ientical or ierent values o j an h, the overall probability o alse alarm P in 16a) reuces to P 1 1 ) K. 8) In turn, with the P criterion along with a targete overall P, the branch ecision threshol can be oun by inverting 8), splitting an inverting 7). C. Detection Probability Analysis Uner hypothesis H 1, the input signal x t) is a sum o the unknown eterministic PU signal st) an the Gaussian white noise wt). To apply 16b) to in the overall probability o etection, we irst stuy the branch probability o etection, which is given by Theorem as ollows. Theorem : Uner hypothesis H 1, the branch probability o etection, j 1,,...,K an h j 1, is expresse as, +,U 9) where, an P,U are given by 30a) an 30b), respectively, at the bottom o this page, where γ an γu have been given in Theorem 1 an H uner H 1 1 which is given by T h H 1 is the correlation coeicient o T j D T j Σ+Θ H 1 D T h H 1 an 31) where Σ an Θ are eine as Σ Δ σ4 w ρ 0 + ρ 1 + ρ + 3) ρ 3a) Θ Δ σw η r j ηhρ i 0 + ηj i ηhρ r 1 + ηj r ηhρ r + ηj i ηhρ i ) 3 3b) where ρ 0, ρ 1, ρ, an ρ 3 are given in Theorem 1, ηj r R 0 c jl Sk l), ηj i I 0 c jl Sk l), ηh r R 0 c hl Sk l), an ηh i I 0 c hl S k l), k 0, 1,..., 1. Proo: To prove Theorem, we irst procee to in the CDF o r uner H 1. Similar to the proo o Theorem 1, we, Φ,U 1 Φ D T j H 1 D T j H 1 γ E Th H 1 E T j H 1 γ H 1 D T j H 1 D T h H 1 γ U E Th H 1 E T j H 1 γ U H 1 D T j H 1 D T h H 1 +γ ) D T h H 1 +γ U ) D T h H 1 30a) 30b)

7 6914 IEEE TRASACTIOS O VEHICUAR TECHOOGY, VO. 66, O. 8, AUGUST 017 stuy the statistic property o the nominator an enominator o the etector r uner H 1. Uner H 1,theDFToy i n) in the ith channel, i 1,,...,K, n 0, 1,..., 1, can be written as Y i k) c il X k l) c il S k l)+ c il W k l) 33) where Sk l) an W k l) are sample rom the DTFT o st) an wt), respectively. emma given in Appenix B provies the statistics o Y i k), an Ti is shown to ollow a Gaussian istribution approximately by applying the central limit theorem when is large. Consequently, the expectation an variance o Ti can be erive rom emma as ollows, respectively: E Ti 1 H 1 c il S k l) + σw 34a) D Ti σ H 1 w c il S k l) + σw 4. 34b) The correlation coeicient between Tj is calculate as H 1 D T j where the covariance o Tj compute as Cov Tj,T h H 1 D T h H 1 Cov Tj,Th 1 Cov Y j k), 1 an Th, by einition, 35) an Th, accoring to 51), can be Y h k ) k 0 σ4 w ρ 0 + ρ 1 + ρ + ρ 3) + σw η r j ηhρ i 0 + ηj i ηhρ r 1 + ηj r ηhρ r + ηj i ηhρ i ) 3. 36) Upon obtaining the statistics o Ti, an the correlation coeicient o Tj an T, Property 1 is again applie to compute h the CDF o r uner H 1. Similarly, accoring to the einition o in 14b), we split into two parts as, an P,U, which satisy, Pr r γ H ) 1,U Pr r γ H U ) 1. 37a) 37b) Consequently, with the CDF unction o r,, an P,U can be obtaine as 30a) an 30b), respectively. With the branch probability o etection, j 1,,...,K, h j 1, we can compute the overall probability o etection P by using them as 16b). IV. PERFORMACE SIMUATIO AD DISCUSSIO In previous section, we have obtaine the alse alarm probability P, the ecision threshol, an the etection probability P. Here, we present numerical results to valiate the correctness o the theoretical analysis an illustrate the eectiveness o the propose PCER preecision algorithm. A. Parameters Setting In the ollowing simulations, the system parameters are set as ollows. The equivalent yquist sampling rate YQ 600 MHz, an the number o sampling channels K 0. The perio o p i t), i 1,,...,K,isT p 75 ns, containing M 45 chips o ±1 levels in each perio. The sampling rate o a single channel is s p 1/T p, an the cuto requency o the PF in each channel is s /. The PU signal consists o three igital moulation signals i.e., the number o occupie bans is B 6), the symbol rate o each signal is 1 MBau, an the carrier requency o each signal is l p, where l is ranomly rawn rom the set {1,,..., 0 } with equal probability. The powers o the three signals are set to be ientical, an the signal-to-noise ratio SR) is eine as P/σw ) B, where P is the total power o the PU signals an σw is the power spectrum ensity o the back groun noise w t). To evaluate the robustness o the propose etection technique to the variation o the noise level, the noise uncertainty enote by ρ is also eine. In the presence o noise uncertainty, the value o noise power o each realization uniormly raws rom the range o σw /ρ, σw ρ, where ρ 1. An ρ 1 implies that the noise power is ixe as a constant an there is no noise uncertainty. Following the P criterion, the probability o alse alarm P is preset as Without loss o generality, the branch is split into two equal parts, which means,,u. All the simulation results are average over 10 5 Monte Carlo experiments. B. Simulation Results Fig. 4 presents the etection probability P versus SR o ierent spectrum sensing methos in the presence/absence o noise uncertainty. The probability o etection o ED is obtaine by combining the energy etection results o all the output channels o MWC with the OR usion rule. It shows that the theoretical an simulation results o ED uner the case o no noise uncertainty match well with each other. ote that the theoretical result o ED can be obtaine by using the statistical istribution o the energy Ti, which has been erive beore. I introucing the noise uncertainty, i.e., ρ>1, which is commonly seen in practical scenarios, ED suers a signiicant perormance loss in the etection probability. For our propose PCER etector, two sets o curves are provie corresponing to the all-branch usion rule 13) with simulate results an the partial-branch usion rule 15) with both theoretical an simulate results, respectively. It can be observe that the all-branch usion outperorms the partial-branch usion, since it uses more branch ratios to obtain the inal ecision at higher complexity. A small gap between the theoretical an simulation results o partial-branch usion is cause by irectly

8 XIOG et al.: PREDECISIO FOR WIDEBAD SPECTRUM SESIG WITH SUB-YQUIST SAMPIG 6915 Fig. 4. Probability o etection o ierent spectrum sensing methos versus SR. applying the OR usion rule an ignoring the small correlation between the usion branches. Overall, unlike ED, both all-branch usion rule an partial-branch usion rule or PCER etector are robust to noise uncertainty. The recently introuce eigenvalue-base etector 38 has been moiie to it in the wieban setup. The resulting etector is inclue in Fig. 4. Speciically, we irst use compresse samples to construct the autocorrelation matrix Q as eine in 6) an then employ the ratio o the maximal eigenvalue to the minimal eigenvalue o Q as the test statistic. The ecision threshol can be approximately compute using the result o 48. However, ierent rom the conventional eigenvalue etector, which is use in the yquist sampling ramework, the compresse samples rom each channel o MWC are statistically correlate, an the prewhitening technique propose in 38 is not applicable here. The consequence is that the equivalent number o signal sample has been reuce, which leas to a perormance loss. Various cases o the number o compresse measurements or partial-branch usion rule are consiere in Fig. 5. The value o is chosen as 00, 400, 800, an 1600, respectively. It is shown that the propose PCER etector can achieve goo perormance with only 00 compresse samples uner relatively low SR, an the etection perormance can be promisingly improve by increasing the number o compresse samples. When there exists noise uncertainty, the ecision threshol compute by 17) oes not change or a given such that the perormance uner noise uncertainty is consistent with the case o no noise uncertainty. The results in Fig. 5 urther veriy our theoretical analysis uner ierent o parameter setups. The alse alarm probability o the PCER etector or the partial-branch usion rule with various number o compresse measurements an ierent values o noise uncertainty ρ is illustrate in Fig. 6. Curves are obtaine using the theoretical threshol erive by 17). In Fig. 6, the term Trial inex represents the number o the inepenent experiments, an in each experiments, 10 5 Monte Carlo simulations are conucte. Fig. 5. Probability o etection o PCER o partial-branch usion rule versus SR with ierent an ρ. Fig. 6. False alarm probability o the PCER or the partial-branch usion rule. a) ρ 1. b) ρ 1.3. It is shown that the alse alarm probability o the propose PCER is slightly luctuating aroun the preset target value 0.01) uner ierent values o an ρ, implying the correctness o our erivation. Fig. 7 presents the receiver operating characteristic ROC) curves o the PCER etector. The probability o alse alarm P ranges rom 10 3 to 10, an SR 4 B. It is observe that when there exists noise uncertainty, the perormance o the PCER remains nearly unchange compare with that in the absence o noise uncertainty. From the results presente in Figs. 4 7, it can be conclue that the propose PCER etection metho, which oes not require the knowlege o noise power an the PU signals, provies goo perormance in a wie range o SR uner ierent parameter setups.

9 6916 IEEE TRASACTIOS O VEHICUAR TECHOOGY, VO. 66, O. 8, AUGUST 017 propose SSS ramework over the conventional MWC-base SSS relies on the accurate etection o PU signals by the PCER moule, avoiing the incorrect SR when there is only white noise. In the aspect o complexity, it nees K multiplications, K 1) aitions, an KK 1)/ ivisions or the allbranch usion rule an only K ivisions or the partial-branch usion rule. However, support recovery requires correlation matrix construction, matrix ecomposition, an orthogonal matching pursuit process. Correlation matrix construction nees K multiplications an 1)K aitions. Matrix ecomposition an orthogonal matching pursuit have computational complexity OK 3 ) an OBK ), respectively. Thereore, the propose PCER moule can reuce the average computational cost while improving the CSR rate. Fig. 8. Fig. 7. ROC curves o the PCER. CSR rate with an without the PCER. To show the necessity an signiicance o the propose PCER etector, we procee to investigate the correct support recovery CSR) rate P csr with an without the PCER moule, as shown in the Fig. 8. The CSR rate is eine as the ratio o the number o CSR, i.e., {S S }, to the total number o simulations. Here, S an S represent the recovery support an the true support o the input signal, respectively. ote that the recovery support S shoul be a null set when there is only white noise in the requency ban. With the PCER moule, which is inserte between the compresse sampling moule an the SR moule, the unction o SR will be conucte only i the PCER moule etects the presence o the PU signals; otherwise, the SR moule is bypasse. Without the PCER moule, the SR moule is always active an output a support set even there is only white noise in the concerne requency ban. Both cases are experimente with the maximum number o occupie ban B max 4, 6. Once B max is given, the number o occupie ban, which is an even number incluing zero), will be equiprobably selecte in the range o 0,B max in a simulation. AsshowninFig.8,P csr with the PCER moule outperorms that without the PCER moule, where the perormance improvement epens on the probability that there are no PU signals in the given wieban. The perormance improvement o the V. COCUSIO Base on a MWC ramework or SSS, a preecision etector reerre to as the PCER is propose to etermine the presence/absence o PU signals prior to signal SR. The PCER etector utilizes the ratio o pairwise energy rom ierent channels o the MWC as the test statistic, whose CDFs uner both an H 1 have been erive. The ecision threshol an the probability o etection uner the P criterion are provie in close orm, which show that the ecision threshol is unrelate to the noise power. The theoretical analysis is shown to match well with simulation results uner various simulation setups. The propose PCER etector is observe to oer goo etection perormance without requiring any prior knowlege o the PU signals in a wie range o SR. Furthermore, unlike the conventional ED, the PCER etector is robust to the noise uncertainty. Our results inicate that the integration o the PCER etector in the MWC ramework can lea to a signiicant improvement o the CSR rate o the PU signals with low computational cost, making the MWC-base SSS ramework more useul or wieban spectrum sensing. APPEDIX A EMMA 1 emma 1: Uner, the mean an variance o Y i k) are E Y i k) σw 38a) D Y i k) σw 4 38b) respectively, where i 1,,...,K, k 0, 1,..., 1. The covariance o the power spectrum lines Y j u) an Y h v) or ierent values o j, h) an u, v) is 0, j h, u v Cov Y j u), Y h v) σ 4 3 w ρ i, j h, u v i 0 0, j h, u v 39) where j, h 1,,...,K, u, v 0, 1,..., 1, ρ 0, ρ 1, ρ, an ρ 3 are eine in Theorem 1.

10 XIOG et al.: PREDECISIO FOR WIDEBAD SPECTRUM SESIG WITH SUB-YQUIST SAMPIG 6917 Proo: Consiering the linear property o DFT operation, Xk l) uner is a complex Gaussian ranom variable with zero mean an variance σw. Y i k), the weighte sum o X k l) accoring to 1), is a complex Gaussian ranom variable with zero mean an variance 0 c il σw.in the MWC system, the power o the perioic ±1 sequences is unit; thereore, the mean an variance o Y i k) is compute as EY i k) 0 an DY i k) σw, an the mean an variance o RY i k)) an IY i k)), respectively, are given by E RY i k)) E IY i k)) 0 D RY i k)) D IY i k)) σ w /. 40a) 40b) Also note that the covariance o RXk)) an IXk)) is zero 49, an the covariance o RY i k)) an IY i k)) can be obtaine by 41), shown at the bottom o this page. Thus, Y i k) RY i k)) + IYi k)) is a sum o the squares o two inepenent an ientically istribute Gaussian ranom variable, an we have E Y i k)) σ w /) σ w 4a) D Y i k)) 4 σ w /) 4 σ 4 w 4b) which is consistent with 38). To prove 39), we note that Xk l) in 1) is equivalent to the DFT o the compresse samples obtaine rom xt)e πl s t, the requency-shit version o the original xt). Thereore, ierent spectrum lines Xu l) an Xv l), u v, shite rom the same requency ban are uncorrelate 49, an the spectrum lines Xu l) an Xv l ), l l, are also uncorrelate, since the samples use to compute DFT are obtaine rom ierent narrow ban white noise uner, which means Cov Y j u),y h v) 0, j h, u v. Meanwhile, Y j u) an Y h v) are continuous unctions o Y j u) an Y h v), respectively, implying Cov Y j u), Y h v) 0 43) where j h an u v. To compute the covariance o Y j u) an Y h v) or j h an u v, we irst etermine the ollowing our items: Cov RY j u)), RY h v)), Cov RY j u)), IY h v)), Cov IY j u)), RY h v)), Cov IY j u)), IY h v)), where Cov RY j u)), RY h v)) is compute by 44), shown at the bottom o the page. The other three items can be compute in the same ashion. Using the result o 40), the correlation coeicient o RY p k)) an RY q k)) can be obtaine as ρ ; thus, the correlation coeicient o RY p k)) an RY q k)) is ρ 50. Consequently Cov R Y j u)), R Y h v)) ρ D R Y j u)) D R Y h v)) σw 4 ρ. 45) Similarly, we have Cov R Y j u)), I Y h v)) σw 4 ρ 0 46a) Cov I Y j u)), R Y h v)) σw 4 ρ 1 46b) Cov I Y j u)), I Y h v)) σw 4 ρ 3. 46c) Cov RY i k)), IY i k)) E RY i k))iy i k)) E RY i k)) E IY i k)) ) E R c il ) R X k l)) Ic il ) I X k l))) ) R c il ) I X k l)) + I c il ) R X k l))) 0. Cov RY j u)), RY h v)) E RY j u))ry h v)) E RY j u)) E RY h v)) ) E R c jl ) R X u l)) Ic jl ) I X u l))) ) R c hl ) R X v l)) Ic hl ) I X v l))) R c jl ) R c hl ) σ w + I c jl ) I c hl ) σ w σ w ρ. 44) 41)

11 6918 IEEE TRASACTIOS O VEHICUAR TECHOOGY, VO. 66, O. 8, AUGUST 017 With 45) an 46), the covariance o Y j u) an Y h v) is compute as Cov Y j u), Y h v) σw 4 ρ 0 + ρ 1 + ρ + 3) ρ 47) where j h an u v. For the case j h an u v, in the same process, it is easy to obtain Cov R Y j u)), R Y h v)) 0 48a) Cov R Y j u)), I Y h v)) 0 48b) Cov I Y j u)), R Y h v)) 0 48c) Cov I Y j u)), I Y h v)) 0 48) which leas to Cov Y j u), Y h v) 0 49) where j h an u v. With 43), 47), an 49), we can erive 39) in emma 1. APPEDIX B EMMA emma : Uner H 1, the mean an variance o Y i k) are E Y i k) c il S k l) + σw 50a) D Y i k) 0 σw c il S k l) + σw 4 50b) respectively, where i 1,,...,K, k 0, 1,..., 1. The covariance o spectrum lines Y j u) an Y h v) or ierent values o j, h) an u, v) is 0, j h, u v Cov Y j u), Y h v) Σ + Θ), j h, u v 0, j h, u v 51) where j, h 1,,...,K, u, v 0, 1,..., 1, an Σ an Θ are eine by 3a) an 3b), respectively. 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Tianyi Xiong S 15 M 16) was born in Sichuan Province, China, in He receive the B.S. egree in optical inormation science an technology rom South China University o Technology, Guangzhou, China, in 011. He is currently working towar the Ph.D. egree at the School o Telecommunications Engineering, Xiian University, Xi an, China. His research interests inclue wireless communication, spectrum sensing, an cognitive raio networks. Hongbin i M 99 SM 08) receive the B.S. an M.S. egrees rom the University o Electronic Science an Technology o China, Chengu, China, in 1991 an 1994, respectively, an the Ph.D. egree rom the University o Floria, Gainesville, F, USA, in 1999, all in electrical engineering. From July 1996 to May 1999, he was a Research Assistant with the Department o Electrical an Computer Engineering, University o Floria. Since July 1999, he has been with the Department o Electrical an Computer Engineering, Stevens Institute o Technology, Hoboken, J, USA, where he became a Proessor in 010. He was a Summer Visiting Faculty Member at the Air Force Research aboratory in the summers o 003, 004, an 009. His general research interests inclue statistical signal processing, wireless communications, an raars. Dr. i is a member o Tau Beta Pi an Phi Kappa Phi. He was the recipient o the IEEE Jack eubauer Memorial Awar in 013 rom the IEEE Vehicular Technology Society, the Outstaning Paper Awar rom the IEEE AFICO Conerence in 011, the Harvey. Davis Teaching Awar in 003, the Jess H. Davis Memorial Awar or Excellence in Research in 001 rom Stevens Institute o Technology, an the Sigma Xi Grauate Research Awar rom the University o Floria in He has been a member o the IEEE Signal Processing Society SPS) Signal Processing Theory an Methos Technical Committee TC) an the IEEE SPS Sensor Array an Multichannel TC, an Associate Eitor or Signal Processing Elsevier), the IEEE TRASACTIOS O SIGA PROCESSIG, IEEE SIGA PROCESSIG ETTERS, an the IEEE TRASACTIOS O WIREESS COMMUICATIOS, an a Guest Eitor or the IEEE JOURA OF SEECTED TOPICS I SIGA PROCESSIG an the EURASIP Journal on Applie Signal Processing. He has been involve in various conerence organization activities, incluing serving as a General co-chair or the 7th IEEE Sensor Array an Multichannel Signal Processing Workshop, Hoboken, J, USA, June 17 0, 01. Peihan Qi was born in Henan Province, China, in He receive the B.S. egree in telecommunications engineering rom Changan University, Xi an, China, in 006, an the M.S. egree in communication an inormation system an Ph.D. egree in military communication rom Xiian University, Xi an, in 011 an 014, respectively. Since January 015, he has been a Postoctoral Researcher with the School o Telecommunications Engineering, Xiian University. His research interests inclue compresse sensing, spectrum sensing in cognitive raio networks, an high-spee igital signal processing.

13 690 IEEE TRASACTIOS O VEHICUAR TECHOOGY, VO. 66, O. 8, AUGUST 017 Zan i SM 14) was born in Shaanxi Province, China, in She receive the B.S. egree in telecommunications engineering an M.S. an Ph.D. egrees in communication an inormation system rom Xiian University, Xi an, China, in 1988, 001, an 006, respectively. She is a Proessor with the School o Telecommunications Engineering, Xiian University. Her research interests inclue wireless communication system, cognitive raio networks, an igital signal processing. Shilian Zheng M 11) receive the B.S. egree in telecommunications engineering an the M.S. egree in signal an inormation processing rom Hangzhou Dianzi University, Hangzhou, China, in 005 an 008, respectively, an the Ph.D. egree in inormation an communications engineering rom Xiian University, Xi an, China, in 014. His current research interests inclue cognitive raio, compresse sensing, an big ata analytics.