Research Article Cooperative Spectrum Sensing and ...downloads.hindawi.com/journals/js/2013/606413.pdfResearch Article Cooperative Spectrum Sensing and Localization in Cognitive Radio

Hindawi Publishing CorporationJournal of SensorsVolume 2013 Article ID 606413 9 pageshttpdxdoiorg1011552013606413

Research ArticleCooperative Spectrum Sensing and Localization in CognitiveRadio Systems Using Compressed Sensing

Wael Guibegravene and Dirk Slock

Mobile Communications Department EURECOM SophiaTech Campus 450 Route des Chappes 06410 Biot France

Correspondence should be addressed to Wael Guibene guibeneeurecomfr

Received 18 April 2013 Revised 20 June 2013 Accepted 26 July 2013

Academic Editor Luca Reggiani

Copyright copy 2013 W Guibene and D Slock This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We propose to fuse two main enabling features in cognitive radio systems (CRS) spectrum sensing and location awareness in asingle compressed sensing based formalism In this way we exploit sparse characteristics of primary units to be detected both interms of spectrum used and location occupiedThe compressed sensing approach also allows to overcome hardware limitations interms of the incapacity to acquiremeasurements and signals at theNyquist ratewhen the spectrum to be scanned is large Simulationresults for realistic network topologies and different compressed sensing reconstruction algorithms testify to the performance andthe feasibility of the proposed technique to enable in a single formalism the two main features of cognitive sensor networks

1 Introduction

Cognitive radio (CR) is a smart wireless communication con-cept that is able to promote the efficiency of the spectrumusage by exploiting the free frequency bands in the spectrumnamely spectrum holes [1 2]

Detection of spectrum holes namely spectrum sensingis the first step towards implementing a cognitive radio sys-tem

The major problem for spectrum sensing arises in wide-band radio when the radio is not able to acquire signals at theNyquist sampling rate due to the current limitations in Ana-log-to-Digital Converter (ADC) technology [3] Compressivesensing makes it possible to reconstruct a sparse signal bytaking less samples than Nyquist sampling and thus wide-band spectrum sensing is doable by compressed sensing (CS)A sparse signal or a compressible signal is a signal that isessentially dependent on a number of degrees of freedomwhich is smaller than the dimension of the signal sampledat Nyquist rate In general signals of practical interest maybe only nearly sparse [3] And typically signals in open net-works are sparse in the frequency domain since dependingon location and at some times the percentage of spectrumoccupancy is low due to the idle radios [1 4]

In CS a signal with a sparse representation in some basiscan be recovered from a small set of nonadaptive linear mea-surements [5] A sensing matrix takes few measurements ofthe signal and the original signal can be reconstructed fromthe incomplete and contaminated observations accuratelyand sometimes exactly by solving a simple convex optimiza-tion problem [3 6] In [7 8] conditions on this sensingmatrix are introduced which are sufficient in order to recoverthe original signal stably And remarkably a random matrixfulfills the conditions with high probability and performs aneffective sensing [5 9]

Another step towards the feasibility and a real implemen-tation of cognitive radio systems is the problem of locationawareness [10 11]This problem arises when we do consider arealistic scenario in hybrid overlayunderlay systems [12 13]when these spectrum opportunities permit cognitive radiosto transmit below the primary users tolerance threshold Inthis case the cognitive radio (CR) has to estimate robustlythe primary users locations in the network in order to adjustits transmission power function of the estimated locationin the network The knowledge of position information inCR system (CRS) is also an enabler of location-based beam-forming as shown in [14] and also as shown in the ICT-WHERE2 project a whole framework of location-aided

2 Journal of Sensors

PHYMAC layer design for advanced cognitive radios [15]with novel concepts of spectrum sensing techniques basedon location information [16] to multicell multiuser MIMOsystems with location-based CSIT [17]

In our approach (part of the work presented in this paperwas accepted and presented in WIMOB 2012 [10] 8th IEEEInternational Conference on Wireless and Mobile Comput-ing Networking and Communications and CAMAD 2012[11] IEEE 17th International Workshop on Computer AidedModeling and Design of Communication Links and Net-works) we propose to analyze all these arisen problems Dur-ing the problem formulation and when analyzing moredeeply the equations related to each question apart we willmake the link between the formulation of spectrum sensinglocation awareness and the hardware limitation by describ-ing those problems in a unique compressed sensing formal-ism

The paper is organized as follows In Section 2 we start bygiving a first overview on the spectrum sensing techniquesSection 3 is dedicated to introducing the compressed sensingframework we are targeting during the problem formulationSection 4 is exposing the remonstration techniques we areusing to overcome the sparsity problem In Section 5 weintroduce the systemmodel of the consideredCR system andin Section 6 we make the link between spectrum sensing andlocalization in a common CS formalism Finally in Section 7we give the simulations results for realistic CR configurationand in Section 8 we conclude on the effectiveness of the pro-posed CS approach for sensinglocalization in CR systems

2 Related Work

As previously stated cognitive radio (CR) is presented [18]as a promising technology in order to handle this shortageand misuse of spectral resources According to FCC a radiois considered cognitive when it has the following capabilities

(i) Frequency agility the ability of a radio to change itsoperating frequency to optimize use under certainconditions

(ii) Dynamic frequency selection the ability to sense sig-nals from other nearby transmitters in an effort tochoose an optimum operating environment

(iii) Location awareness the ability for a device to deter-mine its location and the location of other transmit-ters that would help selecting the appropriate oper-ating parameters such as the power and frequencyallowed at the given location

(iv) Negotiated use incorporate a mechanism that wouldenable sharing of spectrum under the terms of a pre-arranged agreement between a licensed (primary)and nonlicensed (secondary) users

(v) Adaptive modulation the ability to modify and adapttransmission characteristics andwaveforms to exploitopportunities to use spectrum

(vi) Transmit power control allows transmission at fullpower limits when necessary

The presented work fits in the context of spectrum sens-inglocalization framework of cognitive radio systems (CRS)and more precisely cooperative transmitter detectionlocali-zation In this context many statistical approaches for spec-trum sensing have been developedThemost performing oneis the cyclostationary features detection technique [19 20]The main advantage of the cyclostationarity detection is thatit can distinguish between noise signal and PU transmitteddata Indeed noise has no spectral correlation whereas themodulated signals are usually cyclostationary with nonnullspectral correlation due to the embedded redundancy in thetransmitted signal The cyclostationary features detector isthus able to distinguish between noise and PU

The reference sensingmethod is the energy detector (ED)[19] as it is the easiest to implement Although the ED can beimplemented without any need for a priori knowledge of thePU signal some difficulties still remain for implementationFirst of all the only primary user (PU) signal that can bedetected is the one having an energy above some given thresh-old So the threshold selection in itself can be problematicas the threshold highly depends on the changing noise leveland the interference level Another challenging issue is thatthe ED approach cannot distinguish the PU from the othersecondary user (SU) sharing the same channel Cyclosta-tionary features detector (CFD) is more robust to noiseuncertainty than an ED Furthermore it can work with lowerSNR than ED

More recently a detector based on the signal spacedimension based on the estimation of the number of thecovariance matrix independent eigenvalues has been devel-oped [21] It was presented that one can conclude on thenature of this signal based on the number of the independenteigenvectors of the observed signal covariance matrix TheAkaike information criterion (AIC) was chosen in order tosense the signal presence over the spectrum bandwidth Byanalyzing the number of significant eigenvalues minimizingthe AIC criterion one is able to conclude on the nature of thesensed subband Specifically it is shown that the number ofsignificant eigenvalues is related to the presence or not of datain the signal

Still in these state-of-the-art techniques a key assumptionof perfect signals acquisition is always made But real worldsignals and systems are intrinsically sparse as sparsity maycome from different factors A first one which is consideredin this paper is the inability of the ADC to acquire signals ata Nyquist rate resulting in inaccurate and incomplete data Asecond source of sparsity may be that spectrum use in openwireless networks is sparse that is to say few resources areused and in most of geographic areas andor period of timeresources are left idle

3 Compressed Sensing Framework

In this section we are considering sparse signalsvectorsreconstruction

A given 119889-dimensional vector is assumed to be 119904-sparse ifit has 119904 or fewer nonzero coordinates that is to say

119909 isin R119889 1199090 ≜

1003816100381610038161003816supp (119909)1003816100381610038161003816 le 119904 ≪ 119889 (1)

Journal of Sensors 3

where we denote by sdot 0 the quasinorm and for 1 le 119901 lt infin sdot 119901 ≜ (sum

119889

119894=1|119909119894|119901)1119901 is the usual 119901-norm In real world we

will not encounter perfect sparse signals but signals whosecoordinates satisfy a power law decay that is to say 119909 satisfythe following equation

1003816100381610038161003816119909lowast

119894

1003816100381610038161003816 le 119877119894(minus1119902)

(2)

where 119909lowast is a nonincreasingly rearranged version of 119909 119877 issome positive constant and 119902 is satisfying 0 lt 119902 lt 1 Sparsevectors recovery algorithms tend to reconstruct sparse vec-tors from a small set of measurements Each of these mea-surements can be viewed as an inner product between a givenvector say 120601119894 isin R119889 and the vector 119909 isin R119889 Collecting the119898 measurement in a single matrix we thus build an 119898 times 119889

measurement matrix sayΦ = [1206011 sdot sdot sdot 120601119898]Theoretically speaking recovering 119909 from its measure-

ment 119906 = Φ119909 isin R119898 is equivalent to solving the 1198970-mini-mization problem as follows

min119911isinR119889

1199110 subject to Φ119911 = 119906 (3)

If 119909 is 119904-sparse and Φ is one to one (a matrix is told to beone to one if it is representing an injective transformationfrom one space to another) on all 2119904-sparse vectors thenthe solution of (3) must be the signal 119909 Indeed say 119911 is asolution and given the fact that 119909 is an obvious solution then119911 minus 119909 must be a kernel of Φ But 119911 minus 119909 is a 2119904-sparse vectorand given the assumption that Φ is one to one then 119911 = 119909Thus theoretically speaking the 1198970-minimization is a perfectsolution to the reconstruction problem Unfortunately it isshown in the literature [22] that the 1198970-minimization is anNP-hard problem and numerically unfeasible

This problem can be overcome by means of compressedsensing (CS) A first approach is to use basis pursuit (BP)algorithm in order to relax the 1198970-minimization to an 1198971-minimization formalism BP requires stronger hypothesis onΦ so it does not have only to verify injection on sparse vectorsproperty but it has been shown that the relationship between119898 119889 and 119904 is given by 119898 = 119904 log 119874(1)119889 1198971-minimizationoften relies on linear programming and since there is nolinear bound for such techniques BP approaches are quiteslowly convergent techniques The second approach is touse greedy algorithms such as orthogonal matching pursuit(OMP) [23] stagewise orthogonalmatching pursuit (StOMP)[24] or iterative thresholding [25 26] Those approaches arebased on the iterative computation of the signalrsquos supportAs for BP 119898 119889 and 119904 are linked parameters such as 119898 =

119874(119904 log 119889) Once the support 119878 of the signal is computed 119909 isreconstructed from themeasurement vector 119906 as 119909 = (Φ119878)

dagger119906

where Φ119878 is the restriction of Φ to the columns indexed by119878 and dagger is the pseudoinverse operator (recalling that for agiven matrix 119860 119860dagger = (119860

lowast119860)minus1119860lowast) The main advantage of

greedy approaches is their convergence time as they are fasterthan BP but they lose in stability compared to BP Anotherclass of CS algorithms recently emerged in order to shortenthe gap between greedy algorithms and BP From these algo-rithms we may cite regularized orthogonal matching pursuit

(ROMP) [27] and compressive sampling matching pursuit(CoSaMP) [28] These two algorithms provide a similarguarantee of stability as BP with the same iterative propertyof fast convergence as greedy algorithms

4 Reconstruction Algorithms

In our work we will consider one algorithm per class to bestudied for spectrum sensing and localization purposes Wewill thus introduce first of all BP then OMP and finallyCoSaMP that will be used afterwards for the target applica-tions

41 Basis Pursuit Since the problem as formulated in (3) isan NP-hard problem and numerically unfeasible let us intro-duce a first approach to solve this problem Onemay considerfirst a mean square approach to solve the problem as follows


1199112 subject to Φ119911 = 119906 (4)

Since the minimizer say 119909lowast must satisfy Φ119909lowast = 119906 = Φ119909 119909lowastmust be in the subspace 119870 = 119909 + kerΦ Actually 119909lowast asdefined in (4) is the exact contact point between the smallestEuclidian ball centered at the origin and the subspace 119870 Asshown in Figure 1 in the mean square approach there is noneed to have119909lowast coincidingwith the actual signal119909This is dueto the fact that Euclidian geometry ball is not a good detectorof sparsity

In this case in order to solve (3) we may opt for an1198971 approach In this case 119909lowast would meet the contact pointbetween the 1198971 ball (also named octahedron) centered at theorigin and the subspace 119870

A first idea could be relaxing the problem into an 1198971minimization


1199111 subject to Φ119911 = 119906 (5)

Authors in [7] have proved that for measurement matricessatisfying a certain quantitative property called restrictedisometry property 1198970 and 1198971 become equivalent

42 OrthogonalMatching Pursuit OMP is based on subgaus-sian measurement matrices to reconstruct sparse signals IfΦ is verifying such condition (subgaussian property) thenΦlowastΦ is close to identity In this case a nonzero coordinate

of 119909 would maximize the observation 119910 = ΦlowastΦ119909 and that

is how we iteratively reconstruct the support of 119909 OMP isshown to be fast but not as stable as BP

Algorithm 1 gives the pseudocode for OMP implementa-tion

Once the support 119868 of the signal 119909 is found the estimate119909 can be reconstructed as 119909 = Φ

dagger

119868119906 The algorithm simplicity

allows a fast reconstruction as it iterates 119904 times and over eachiteration it selects one among 119889 elements in 119874(119889) time andmultiplies byΦlowast in an119874(119898119889) time period and finally solves aleast squares problem in119874(1199042119889) So the cost of such techniqueis 119874(119904119898119889) operations


x

xlowast

(a)

x = xlowast

(b)

Figure 1 The minimizers to the mean square (a) and 1198971 (b) approaches

RequireMeasurement matrixΦ measurement vector 119906 = Φ119909 sparsity level 119904(1) Initialize As a first step initialize 119868 = 0 and the residual 119903 = 119906 repeat 119904 times(2) Identify Select the largest coordinate 120582 of 119910 = Φ

lowast119903 in absolute value Break ties lexicographically

(3)Update Add the coordinate 120582 to the index set 119868 larr 119868⋃120582 and update the residual according to119909 = argmin

119911

10038171003817100381710038171003817119906 minusΦ|119868

119911100381710038171003817100381710038172 119903 = 119906 minusΦ119909

(4) return Index set 119868 isin 1 119889

Algorithm 1 Orthogonal matching pursuit (OMP) pseudocode

43 Compressive Sampling Matching Pursuit CompressiveSampling Matching Pursuit (CoSaMP) implementation isgiven byAlgorithm 2 As far as the CoSaMP algorithm is con-cerned the sampling operator Φ is supposed to satisfy therestricted isometry property (for more information and acomplete description of RIP please refer to [29]) and each 119904

coordinates of signal 119910 = ΦlowastΦ119909 also called proxy signal

are close in terms of Euclidian norm to the 119904 correspondingcoordinates of 119909

The algorithm operates according to the following steps

(1) Identification the algorithm takes the residual as aproxy and locates its highest coordinates

(2) Support merging at the iteration 119896 the set of recentlyidentified coordinates is merged with the set fromiteration 119896 minus 1

(3) Estimation based on the set of coordinates the algo-rithmperforms a least square to determine an approx-imation of the target signal

(4) Pruning in the estimated signal from least squaresthe algorithm retains only the highest coordinates

(5) Sample updating the samples are updates so that theyintegrate the residual part

44 Compressed Sensing for Spectrum Sensing and PrimaryUsers Localization In this paper wewill use the above frame-work in order to solve two major issues enabling CRspectrum sensing and terminals localization In order to do

so we will tend in our analysis to express the upcoming equa-tions as the following

119910 = Φ119909 (6)

where 119910 isin R119872 is the measured entity Φ isin R119872times119873 the meas-urement matrix and 119909 isin R119873 the 119870-sparse vector to bereconstructed

According to restricted isometry property definition Φwould verify the RIP if119872 ge 119874(119870 log(119873119870))

5 System Model

In the considered system model we will suppose that wedo dispose of 119873ch available channels in a wideband wirelessnetwork Over a large geographic area let119873119901 be the numberof deployed primary users using a different channel eachIn this large area we disperse 119873119888 cognitive radios that willcooperate to locate PUs in the network and detect their chan-nels usage and states The measures made by these cognitiveterminals will then be sent to the fusion center In order toenable SUs transmissions the secondary network has to beaware of the availability and the state of each channel ThusSUs have to estimate which channels are occupied and toidentify the PUs transmission powers and locations

For our system we adopt the path loss model given by

119871 (119891 119889) = 1198750 + 20lg (119891) + 10119899 lg (119889) [dB] (7)

where 1198750 is a constant related to antennas gain 119891 is the chan-nel frequency 119899 is the path loss exponent 119889 is the distance


RequireMeasurement matrixΦ measurement vector 119906 = Φ119909 sparsity level 119904(1) Initialize Set 1198860 = 0 V = 119906 119896 = 0

Repeat the following steps while increasing 119896 until achieving halt criterion(2) Signal Proxy Set 119910 = Φ

lowastV Ω = supp1199102119904 and merge the support 119879 = Ω⋃ supp119886119896minus1

(3) Signal Estimation Solving a least squares problem set 119887|119879 = Φdagger

119879119906 and 119887|

119879119888= 0

(4)Prune Preparing the next iteration set 119886119896 = 119887119904

(5) Sample Update Update the samples by V = 119906 minusΦ119886119896

(6) return 119904-sparse reconstructed vector 119909 = 119886

Algorithm 2 Compressive sampling matching pursuit (CoSaMP) pseudocode

separating the transmitting and receiving nodes lg(sdot) =

log10(sdot)In our case we dispose of 119873ch channels thus 119891 would

be assumed the central frequency of each band that is 119891 isin

1198910 1198911 119891119873chminus1

Let us keep in mind that the path loss is related to theunknown channel and location of the PUThe received signalpower is a combination of the unknown transmit power withthe path loss expressed in (7)

Our task is to infer from the received signal at the cog-nitive terminals all this unknown but valuable informationabout the primary users

6 Compressed Sensing forCognitive Radio Applications

61 Spectrum Sensing For discrete signals the time domainsamples 119905 are used to construct the spectrum in frequencydomain using directly the DFT transform as follows

119891 = F119905 (8)

where F is the normalized DFT matrixAs stated previously the sparsity in this contextmay come

from the inability of the ADC to acquire signals at a Nyquistrate The time samples 119905 are thus acquired at a sub-Nyquistrate which may result in a sparse vector

We will thus directly apply the CS formalism with thedifferent introduced algorithms to reconstruct the originaltime domain transmitted signal and spectrum sensing will beachieved using energy detection

62 Location Estimation Based on Compressed Sensing Oncespectrum reconstructed and spectrum sensing achievedmore information can be derived while looking deeper intochannels occupied by primary users

Let us assume that in a certain wide area PUs are locatedat coordinates (119909119901119898 119910119901119899) where 119909119901119898 isin 0 Δ119909119901 (119872 minus

1)Δ119909119901 are119872 possible 119909-axis positions (abscissaelig) of the PUs(when we say 119909119901119898 isin 0 Δ119909119901 (119872 minus 1)Δ119909119901 that does notmean that there are 119872 PUs but it means that 119873119901 primaryusers abscissaelig (for ordinates as well) do actually have a finiteldquodictionaryrdquo) 119910119901119898 isin 0 Δ119910119901 (119873minus1)Δ119910119901 are119873 possible119910-axis positions (ordinates) of the PUs Δ119909119901 and Δ119910119901 arerespectively the resolutions over 119909-axis and 119910-axis Here we

do impose and suppose to the PU coordinates to be in discrete119872times119873 dictionary (which actually is always true) It is goodto remind at this level that the exact positions of the119873119901 PUs(119909119901119894 119910119901119894) 119894 isin [1 119873119901] are unknown to our problem

The 119873119888 CRs positions in the network are located atpositions (119886119894 119887119894) 119894 isin [1 119873119888] (on which we do notimpose being in a finite set even if they necessarily are)

For the 119896th CR sensing the 119894th channel the contributionof the PU located at the (119909119901119898 119910119901119899) position on the receivedPSD is

119877119896119894 (119898 119899) = 119875 (119898 119899 119894) times 10119871(119891119894 119889(119898119899119896))10

119889 (119898 119899 119896) = radic(119909119901119898 minus 119886119896)2+ (119910119901119899 minus 119887119896)

2

(9)

where 119875(119898 119899 119894) is the power transmitted by a PU using the119894th channel located at (119909119901119898 119910119901119899) 119891119894 is the center frequencyof the 119894th channel 119889(119898 119899 119896) represents the distance betweenthe 119896th CR and the the PU located at (119909119901119898 119910119901119899)

The total received power over all the existing PUs thatis over the 119872 times 119873 possible positions of the PUs can beformulated as the following

119884119896119894 = sum

119898

sum

119899

119877119896119894 (119898 119899)

119884119896119894 = sum

119898

sum

119899

10119871(119891119894 119889(119898119899119896))10 times 119875 (119898 119899 119894)

119884119896119894 = 119871119879(119896 119894) 119875 (119894)

(10)

where 119875(119894) is the vector containing the transmission power ofthe over all119872times119873 grid over the 119894th channel 119871(119896 119894) is the pathloss vector computed according to (7) from all PU possiblepositions at the level of the 119896th CR on the 119894th channelConsider

119871 (119896 119894) = 10119871dB(119896119894)10

119871dB (119896 119894) = [119871 (119891119894 119889 (0 0 119896)) 119871 (119891119894 119889 (1 0 119896))

119871 (119891119894 119889 (119898 119899 119896)) 119871 (119891119894 119889 (119872119873 119896))]119879

(11)

Let us denote by 119884119896 = [1198841198961 119884119896119873ch]119879 the received signal

power vector at the level of the 119896th CR over the119873ch available


channels This according to (10) and adopting the previousnotation can be expressed as

119884119896 = Lk119875 (12)

where 119875 is the vector containing the transmission powerof the 119872 times 119873 grid of PU locations over the 119873ch availablechannels of the119873119862 deployed CRs as follows

119875119896 = [119875119879(1198941) 119875

119879(1198942) 119875

119879(119894119873119862

)]119879

(13)

The matrix Lk is the fading gain matrix grouping at the levelof the 119896th CR the loss path contributions of the 119872 times 119873 PUpositions The 119895th row of Lk is

Lk (119895) = [0 0 119871119879(119896 119895) 0 0] (14)

Combining all the equations describing the 119873119862 CR systemwe do obtain

119884 = L119875 (15)

where 119884 = [1198841119879 119884119873119862

119879]119879 and L = [L1 LNC

]The equation we ended with in (15) reminds us of (6)

of the CS formalism we introduced previously as 119875 is anunknown but sparse vector because over the119872 times119873 area wehave been considering only119873119875 PUs are deployed in this area

The two stages spectrum sensing and localization seemthen to be attached to the same CS framework we haveintroduced before

63 Joint Spectrum Sensing and Primary User LocalizationWe have shown till now that both problems sensing andlocalization can separately be solved using CS formalism Itis now easy for us to combine these two operations

The joint framework would consider this dual sens-inglocalization problem as a 3D image reconstruction fromsparse observation The 119909- axis and 119910-axis of this unknownimage are the PU location information the 119911-axis wouldrepresent the sensed channel occupancy information and thevalue of a given pixel at (119909 119910 119911) is the reconstructed transmitpower

7 Simulations and Results

For our analysis we suggest a realistic network simulationIn the considered CRS we deploy 5 PUs and 3 SUs The 3deployed CRs are attempting to communicate opportunisti-cally and thus will perform the sensing and localization task

A hexagonal cellular system functioning at 18 GHzwith asecondary cell of radius119877 = 1000mand a primary protectionarea of radius 119877119901 = 600m is considered Secondary trans-mitters may communicate with their respective receivers ofdistances 119889 lt 119877119901 from the BS We assume that the PUs andthe SUs are randomly distributed in a two-dimensional planeas shown in Figure 2The BS is placed at the center (0 0) Thedistance 119889(119898 119899 119896) from the 119896th SU to the PU (119898 119899) is givenby


2 (16)

d(m n 3)

d(m n 2)

d(m n 1)

RP

R

SU1

PU1

SU3

PU4 (m n)

PU2

PU3

PU5

SU2

Figure 2 Two-dimensional plane of the cognitive radio systemtopology with five primary users and three secondary users

where (119909119901119898 119910119901119899) are the coordinates of the PU and (119886119896 119887119896) thecoordinates of the 119896th CRThe channel gains are based on theCOST-231 Hata model [30] including log-normal shadowingwith standard deviation of 10 dB plus fast-fading assumed tobe iid circularly symmetric with distributionCN(0 1)Thebasic path loss for the COST-231 Hata model which is in dBin an urban area at a distance 119889 is

119871 (119891 119889) = 463 + 339log10(119891119888) minus 1382log10 (ℎ119887)

minus 119860119872 + (449 minus 655log10 (ℎ119887)) log10 (119889) + 119862119872(17)

where 119891119888 is the carrier frequency equal to 15 GHz and ℎ119887 isthe base antenna height equal to 50 meters The distance 119889 iscomputed using (16) 119862119872 is 0 dB for medium-sized cities andsuburbs and is 3 dB formetropolitan areas In the simulationswe use119862119872 = 0 dBThe119860119872 is defined for urban environmentas

119860119872 = 320(log10(1175ℎ119898))

2minus 497 (18)

where ℎ119898 is themobile antenna height equal to 10metersTheshadowing variations of the path loss can be calculated fromthe log-normal distribution as follows

119892 (119909 | 120590) =1

120590radic2120587exp(minus119909

2

21205902) (19)

where 120590 is the variability of the signal equal to 10 dB Theshadowing variation is computed using the Matlab function119903119886119899119889119899 Shadowing reflects the differences in the measuredreceived signal power with relation to the theoretical valuecalculated by path loss formulas Averaging over many


received signal power values for the same distance howeveryields the exact value given by path loss

Furthermore for channels distribution we suppose thatthe total number of available channels is in [1 2 20] chan-nels and each of the five PUs is communicating over a singledifferent channel

For simulations we generate a random sequence with 5of the 20 channels (a single channel per user is assumed)where the amplitude (PSD) is also randomly generated peruser This generated spectrum is then fixed once and for allin the simulations For spectrum sensing in order to modelthe ADC imperfections we generate a random119872times119873matrixas stated in [3 6] and the sparse signal is then the generatedsignal passed through the random matrix plus a white noiseat a given SNR After this generation of the sparse noisy sig-nal we process using the several reconstruction algorithmsand compute the spectrum reconstruction error For locationestimation distance is derived using the CS proposed for-malism with pathloss expression given in (17) Unlike spec-trum generation here noise is implicitly inherited from thereceived noisy signal

71 Simulation Results Figures 3 and 4 give an example ofspectrum reconstruction MSE (we recall the definition ofmean square error as MSE = ||119883 minus 119883||

2 where119883 is the orig-inal observation and 119883 is the reconstructed signal) at 50sparsity for the simulated algorithms and the impact ofsparsity level on spectrum reconstruction MSE

The very first remark that we can give here is that thesimulation results are in line with the theoretical aspectsrelated to BP OMP and CoSaMP As we explained at thevery beginning performance related to BP is expected toovercome OMP which is the case in Figure 3 and CoSaMPis shown both in theory and simulations to outperformOMPbut it is still not as efficient as BP

Figure 4 is assessing the impact of sparsity level onspectrum reconstructionMSEHerewe clearly see that resultsare coherent with the definition we gave of sparsity sayingthat ldquoa 119889-dimensional vector is assumed to be 119904-sparse if it has119904 or fewer nonzero coordinatesrdquo According to this definitionthe more sparse the vector is the larger its support is so theless the reconstruction MSE would be which is perfectly inline with the results reported in Figure 4

Figures 5 and 6 give an example of PU location estimationat 50 sparsity for the simulated algorithms and the impactof sparsity level on PU location estimation error at 0 dB

Figure 5 gives an overview of PU location estimationerror at 50 sparsity level for BP OMP and CoSaMP Herewe validate again the previous results as the tendency of BPoutperformingCoSaMPandOMP is confirmedThe same forFigure 6 where we clearly see themore signal sparse is (in thesense the more we have nonzero entries) the more robust thePU localization is

Assuming a 15 meters limit of ldquogood localizationrdquo boundFigure 5 shows that the idea of cooperative localization couldbe exploited up to minus3 dB minus2 dB and 1 dB for BP CoSaMPand OMP respectively These performances are obtained fortotally autonomous GPS free techniques and with absolutelyno need for extra overhead data exchange in the network

SNR (dB)Sp

ectr

um re

cons

truc

tion

MSE

BP at 50OMP at 50CoSaMP at 50

minus20 minus18 minus16 minus14 minus12 minus10 minus8 minus610

minus3

10minus2

10minus1

100

101

Figure 3 Example of spectrum reconstructionMSE at 50 sparsitylevel for BP OMP and CoSaMP

20 30 40 50 60 70 80Sparsity level ()

Spec

trum

reco

nstr

uctio

n M

SE

BPOMPCoSaMP

10minus2

10minus1

100

101

Figure 4 Impact of sparsity on spectrum reconstruction MSE at0 dB


0 5 10 150

50

100

150

200

250

300

350

400

SNR (dB)

PU lo

catio

n er

ror (

m)


minus20 minus15 minus10 minus5

Figure 5 Error on PU location estimation at 50 sparsity level forBP OMP and CoSaMP

8 Conclusion

This work presents a first look towards a combined spectrumsensing and localization task These two tasks are fundamen-tal in order to enable cognition in wireless networks Withthe combination of the two tasks we also considered a real-istic data acquisition constraint which is sparsity due to theADC technology limits Simulation results of the proposedtechnique show promising and interesting results for com-pressed sensing techniques applied to this formalism

The formalism that we derived here is the starting pointof a whole location-aided cognitive radio framework that isbeing developed within the FP7 WHERE2 project In thisframework once the radio map [31] is built thanks to thesensing and localization tasks the CRS are thus able to com-municate in the available bands in the available directionsand thus the proposed framework is a key enabler for spacedivisionmultiple access (SDMA) systems and spatial versionsof CR paradigms (underlay overlay and interweave) Thisformalism can also be exploited for D2D (device to device)communication as the CRs can communicate directly witheach other while a fusion center can play the role of thecontrol entity in the network

Acknowledgments

EURECOMrsquos research is partially supported by its indus-trial members ORANGE BMW Group Swisscom SFRST Microelectronics Symantec SAP Monaco Telecom andiABG and also by the EU FP7 projects WHERE2 and

20 30 40 50 60 70 805

10

15

20

25

30

Sparsity level ()

PU lo

catio

n er

ror (

m)

BPOMPCoSaMP

Figure 6 Impact of sparsity on PU location estimation at 0 dB

NEWCOM The authors would like to thank the reviewersfor their constructive comments

References

[1] S Haykin ldquoCognitive radio brain-empowered wireless com-municationsrdquo IEEE Journal on Selected Areas in Communica-tions vol 23 no 2 pp 201ndash220 2005

[2] J Mitola III and G Q Maguire Jr ldquoCognitive radio makingsoftware radios more personalrdquo IEEE Personal Communica-tions vol 6 no 4 pp 13ndash18 1999

[3] E J Candes ldquoCompressive samplingrdquo in Proceedings of theInternational Congress of Mathematicians Invited LectureMadrid Spain August 2006

[4] Z Tian andG BGiannakis ldquoCompressed sensing forwidebandcognitive radiosrdquo in Proceedings of the IEEE International Con-ference on Acoustics Speech and Signal Processing (ICASSP rsquo07)pp IV1357ndashIV1360 April 2007

[5] M A Davenport M B Wakin and R G Baraniuk ldquoDetectionand estimation with compressive measurementsrdquo Tech RepDepartment of ECE Rice University 2006

[6] E J Candes J K Romberg and T Tao ldquoStable signal recoveryfrom incomplete and inaccurate measurementsrdquo Communica-tions on Pure and Applied Mathematics vol 59 no 8 pp 1207ndash1223 2006

[7] E J Candes and T Tao ldquoDecoding by linear programmingrdquoIEEE Transactions on Information Theory vol 51 no 12 pp4203ndash4215 2005

[8] E J Candes andT Tao ldquoNear-optimal signal recovery from ran-dom projections universal encoding strategiesrdquo IEEE Transac-tions on InformationTheory vol 52 no 12 pp 5406ndash5425 2006

[9] E J Candes and M B Wakin ldquoAn introduction to compressivesampling a sensingsampling paradigm that goes against the


common knowledge in data acquisitionrdquo IEEE Signal ProcessingMagazine vol 25 no 2 pp 21ndash30 2008

[10] W Guibene and D Slock ldquoA combined spectrum sensing andterminals localization technique for cognitive radio networksrdquoin Proceedings of the IEEE 8th International Conference onWire-less and Mobile Computing Networking and Commrsquos (WiMobrsquo12) 2012

[11] W Guibene and D Slock ldquoA compressive sampling approachfor spectrum sensing and terminals localization in cognitiveradio networksrdquo in Proceedings of the IEEE 17th InternationalWorkshop on Computer Aided Modeling and Design of Commu-nication Links and Networks (CAMAD rsquo12) Barcelona SpainOctober 2012

[12] M G Khoshkholgh K Navaie and H Yanikomeroglu ldquoAccessstrategies for spectrum sharing in fading environment overlayunderlay and mixedrdquo IEEE Transactions on Mobile Computingvol 9 no 12 pp 1780ndash1793 2010

[13] G Ding Q Wu F Song X Li and J Wang ldquoJoint explorationand exploitation of spatial-temporal spectrum hole for cogni-tive vehicle radiosrdquo in Proceedings of the IEEE InternationalConference on Signal Processing Communications and Comput-ing (ICSPCC rsquo11) September 2011

[14] S Mumtaz J Bastos J Rodriguez and C Verikoukis ldquoAdaptivebeamforming for OFDMA using positioning informationrdquo inProceedings of the Wireless Advanced (WiAd rsquo11) pp 7ndash14 June2011

[15] WHERE2 Consortium ldquoLocation-aided PHYMAC LayerDesign for Advanced Cognitive Radiosrdquo Deliverable D33 2012httpwwwkn-sdlrdewhere2indexphp

[16] P Cheraghi Y Ma and R Tafazolli ldquoFrequency-domain differ-ential energy detection based on extreme statistics For OFDMsource sensingrdquo in Proceedings of the IEEE 73rd VehicularTechnology Conference (VTC rsquo11-Spring) May 2011

[17] W Guibene and D Slock ldquoDegrees of freedom of downlinksingle- and multi-cell multi-user MIMO systems with locationbased CSITrdquo in Proceedings of the IEEE 77th Vehicular Technol-ogy Conference (VTC rsquo13-Spring) Dresden Germany 2013

[18] I F Akyildiz W-Y Lee M C Vuran and S Mohanty ldquoNeXtgenerationdynamic spectrum accesscognitive radio wirelessnetworks a surveyrdquo Computer Networks vol 50 no 13 pp2127ndash2159 2006

[19] T Yucek and H Arslan ldquoA survey of spectrum sensing algo-rithms for cognitive radio applicationsrdquo IEEE CommunicationsSurveys and Tutorials vol 11 no 1 pp 116ndash130 2009

[20] S Xu Z Zhao and J Shang ldquoSpectrum sensing based oncyclostationarityrdquo inProceedings of the IEEEWorkshop on PowerElectronics and Intelligent Transportation System (PEITS rsquo08)pp 171ndash174 August 2008

[21] B Zayen Spectrum sensing and resource allocation strategies forcognitive radio [PhD thesis] Telecom PariTech 2010

[22] S Muthukrishnan Data Streams Algorithms and ApplicationsNow Publishers 2005

[23] J A Tropp and A C Gilbert ldquoSignal recovery from randommeasurements via orthogonal matching pursuitrdquo IEEE Trans-actions on Information Theory vol 53 no 12 pp 4655ndash46662007

[24] D L Donoho Y Tsaig I Drori and J-L Starck ldquoSparse solu-tion of underdetermined systems of linear equations by stage-wise orthogonal matching pursuitrdquo IEEE Transactions on Infor-mation Theory vol 58 no 2 pp 1094ndash1121 2012

[25] M Fornasier and H Rauhut ldquoIterative thresholding algo-rithmsrdquoApplied and Computational Harmonic Analysis vol 25no 2 pp 187ndash208 2008

[26] T Blumensath and M E Davies ldquoIterative hard thresholdingfor compressed sensingrdquo Applied and Computational HarmonicAnalysis vol 27 no 3 pp 265ndash274 2009

[27] D Needell and R Vershynin ldquoUniform uncertainty principleand signal recovery via regularized orthogonal matching pur-suitrdquo Foundations of Computational Mathematics vol 9 no 3pp 317ndash334 2009

[28] D Needell and J A Tropp ldquoCoSaMP iterative signal recoveryfrom incomplete and inaccurate samplesrdquo Applied and Compu-tational Harmonic Analysis vol 26 no 3 pp 301ndash321 2009

[29] E J Candes ldquoThe restricted isometry property and its implica-tions for compressed sensingrdquo Comptes Rendus Mathematiquevol 346 no 9-10 pp 589ndash592 2008

[30] COST 231 ldquoUrban transmission loss models for mobile radioin the 900- and 1800MHz bandsrdquo TD(90)119 Revision 2September 1991

[31] Y Zhao Enabling cognitive radios through radio environmentmaps [PhD thesis] Virginia Polytechnic Institute and StateUniversity 2007

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



PHYMAC layer design for advanced cognitive radios [15]with novel concepts of spectrum sensing techniques basedon location information [16] to multicell multiuser MIMOsystems with location-based CSIT [17]

In our approach (part of the work presented in this paperwas accepted and presented in WIMOB 2012 [10] 8th IEEEInternational Conference on Wireless and Mobile Comput-ing Networking and Communications and CAMAD 2012[11] IEEE 17th International Workshop on Computer AidedModeling and Design of Communication Links and Net-works) we propose to analyze all these arisen problems Dur-ing the problem formulation and when analyzing moredeeply the equations related to each question apart we willmake the link between the formulation of spectrum sensinglocation awareness and the hardware limitation by describ-ing those problems in a unique compressed sensing formal-ism

The paper is organized as follows In Section 2 we start bygiving a first overview on the spectrum sensing techniquesSection 3 is dedicated to introducing the compressed sensingframework we are targeting during the problem formulationSection 4 is exposing the remonstration techniques we areusing to overcome the sparsity problem In Section 5 weintroduce the systemmodel of the consideredCR system andin Section 6 we make the link between spectrum sensing andlocalization in a common CS formalism Finally in Section 7we give the simulations results for realistic CR configurationand in Section 8 we conclude on the effectiveness of the pro-posed CS approach for sensinglocalization in CR systems

2 Related Work

As previously stated cognitive radio (CR) is presented [18]as a promising technology in order to handle this shortageand misuse of spectral resources According to FCC a radiois considered cognitive when it has the following capabilities

(i) Frequency agility the ability of a radio to change itsoperating frequency to optimize use under certainconditions

(ii) Dynamic frequency selection the ability to sense sig-nals from other nearby transmitters in an effort tochoose an optimum operating environment

(iii) Location awareness the ability for a device to deter-mine its location and the location of other transmit-ters that would help selecting the appropriate oper-ating parameters such as the power and frequencyallowed at the given location

(iv) Negotiated use incorporate a mechanism that wouldenable sharing of spectrum under the terms of a pre-arranged agreement between a licensed (primary)and nonlicensed (secondary) users

(v) Adaptive modulation the ability to modify and adapttransmission characteristics andwaveforms to exploitopportunities to use spectrum

(vi) Transmit power control allows transmission at fullpower limits when necessary

The presented work fits in the context of spectrum sens-inglocalization framework of cognitive radio systems (CRS)and more precisely cooperative transmitter detectionlocali-zation In this context many statistical approaches for spec-trum sensing have been developedThemost performing oneis the cyclostationary features detection technique [19 20]The main advantage of the cyclostationarity detection is thatit can distinguish between noise signal and PU transmitteddata Indeed noise has no spectral correlation whereas themodulated signals are usually cyclostationary with nonnullspectral correlation due to the embedded redundancy in thetransmitted signal The cyclostationary features detector isthus able to distinguish between noise and PU

The reference sensingmethod is the energy detector (ED)[19] as it is the easiest to implement Although the ED can beimplemented without any need for a priori knowledge of thePU signal some difficulties still remain for implementationFirst of all the only primary user (PU) signal that can bedetected is the one having an energy above some given thresh-old So the threshold selection in itself can be problematicas the threshold highly depends on the changing noise leveland the interference level Another challenging issue is thatthe ED approach cannot distinguish the PU from the othersecondary user (SU) sharing the same channel Cyclosta-tionary features detector (CFD) is more robust to noiseuncertainty than an ED Furthermore it can work with lowerSNR than ED

More recently a detector based on the signal spacedimension based on the estimation of the number of thecovariance matrix independent eigenvalues has been devel-oped [21] It was presented that one can conclude on thenature of this signal based on the number of the independenteigenvectors of the observed signal covariance matrix TheAkaike information criterion (AIC) was chosen in order tosense the signal presence over the spectrum bandwidth Byanalyzing the number of significant eigenvalues minimizingthe AIC criterion one is able to conclude on the nature of thesensed subband Specifically it is shown that the number ofsignificant eigenvalues is related to the presence or not of datain the signal

Still in these state-of-the-art techniques a key assumptionof perfect signals acquisition is always made But real worldsignals and systems are intrinsically sparse as sparsity maycome from different factors A first one which is consideredin this paper is the inability of the ADC to acquire signals ata Nyquist rate resulting in inaccurate and incomplete data Asecond source of sparsity may be that spectrum use in openwireless networks is sparse that is to say few resources areused and in most of geographic areas andor period of timeresources are left idle

3 Compressed Sensing Framework

In this section we are considering sparse signalsvectorsreconstruction

A given 119889-dimensional vector is assumed to be 119904-sparse ifit has 119904 or fewer nonzero coordinates that is to say

119909 isin R119889 1199090 ≜

1003816100381610038161003816supp (119909)1003816100381610038161003816 le 119904 ≪ 119889 (1)



119889



1003816100381610038161003816119909lowast

119894

1003816100381610038161003816 le 119877119894(minus1119902)

(2)





1199110 subject to Φ119911 = 119906 (3)




dagger119906









1199112 subject to Φ119911 = 119906 (4)





1199111 subject to Φ119911 = 119906 (5)







dagger




x

xlowast

(a)

x = xlowast

(b)





119911

10038171003817100381710038171003817119906 minusΦ|119868

119911100381710038171003817100381710038172 119903 = 119906 minusΦ119909














119910 = Φ119909 (6)



5 System Model



119871 (119891 119889) = 1198750 + 20lg (119891) + 10119899 lg (119889) [dB] (7)







119879119906 and 119887|

119879119888= 0








1198910 1198911 119891119873chminus1





119891 = F119905 (8)










119877119896119894 (119898 119899) = 119875 (119898 119899 119894) times 10119871(119891119894 119889(119898119899119896))10


2

(9)



119884119896119894 = sum

119898

sum

119899

119877119896119894 (119898 119899)

119884119896119894 = sum

119898

sum

119899

10119871(119891119894 119889(119898119899119896))10 times 119875 (119898 119899 119894)

119884119896119894 = 119871119879(119896 119894) 119875 (119894)

(10)


119871 (119896 119894) = 10119871dB(119896119894)10

119871dB (119896 119894) = [119871 (119891119894 119889 (0 0 119896)) 119871 (119891119894 119889 (1 0 119896))

119871 (119891119894 119889 (119898 119899 119896)) 119871 (119891119894 119889 (119872119873 119896))]119879

(11)





119884119896 = Lk119875 (12)


119875119896 = [119875119879(1198941) 119875

119879(1198942) 119875

119879(119894119873119862

)]119879

(13)


Lk (119895) = [0 0 119871119879(119896 119895) 0 0] (14)


119884 = L119875 (15)

where 119884 = [1198841119879 119884119873119862

119879]119879 and L = [L1 LNC










2 (16)

d(m n 3)

d(m n 2)

d(m n 1)

RP

R

SU1

PU1

SU3

PU4 (m n)

PU2

PU3

PU5

SU2



119871 (119891 119889) = 463 + 339log10(119891119888) minus 1382log10 (ℎ119887)



119860119872 = 320(log10(1175ℎ119898))

2minus 497 (18)


119892 (119909 | 120590) =1

120590radic2120587exp(minus119909

2

21205902) (19)













SNR (dB)Sp

ectr

um re

cons

truc

tion

MSE



minus3

10minus2

10minus1

100

101


20 30 40 50 60 70 80Sparsity level ()

Spec

trum

reco

nstr

uctio

n M

SE

BPOMPCoSaMP

10minus2

10minus1

100

101



0 5 10 150

50

100

150

200

250

300

350

400

SNR (dB)

PU lo

catio

n er

ror (

m)




8 Conclusion



Acknowledgments


20 30 40 50 60 70 805

10

15

20

25

30

Sparsity level ()

PU lo

catio

n er

ror (

m)

BPOMPCoSaMP



References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119889



1003816100381610038161003816119909lowast

119894

1003816100381610038161003816 le 119877119894(minus1119902)

(2)





1199110 subject to Φ119911 = 119906 (3)




dagger119906









1199112 subject to Φ119911 = 119906 (4)





1199111 subject to Φ119911 = 119906 (5)







dagger




x

xlowast

(a)

x = xlowast

(b)





119911

10038171003817100381710038171003817119906 minusΦ|119868

119911100381710038171003817100381710038172 119903 = 119906 minusΦ119909














119910 = Φ119909 (6)



5 System Model



119871 (119891 119889) = 1198750 + 20lg (119891) + 10119899 lg (119889) [dB] (7)







119879119906 and 119887|

119879119888= 0








1198910 1198911 119891119873chminus1





119891 = F119905 (8)










119877119896119894 (119898 119899) = 119875 (119898 119899 119894) times 10119871(119891119894 119889(119898119899119896))10


2

(9)



119884119896119894 = sum

119898

sum

119899

119877119896119894 (119898 119899)

119884119896119894 = sum

119898

sum

119899

10119871(119891119894 119889(119898119899119896))10 times 119875 (119898 119899 119894)

119884119896119894 = 119871119879(119896 119894) 119875 (119894)

(10)


119871 (119896 119894) = 10119871dB(119896119894)10

119871dB (119896 119894) = [119871 (119891119894 119889 (0 0 119896)) 119871 (119891119894 119889 (1 0 119896))

119871 (119891119894 119889 (119898 119899 119896)) 119871 (119891119894 119889 (119872119873 119896))]119879

(11)





119884119896 = Lk119875 (12)


119875119896 = [119875119879(1198941) 119875

119879(1198942) 119875

119879(119894119873119862

)]119879

(13)


Lk (119895) = [0 0 119871119879(119896 119895) 0 0] (14)


119884 = L119875 (15)

where 119884 = [1198841119879 119884119873119862

119879]119879 and L = [L1 LNC










2 (16)

d(m n 3)

d(m n 2)

d(m n 1)

RP

R

SU1

PU1

SU3

PU4 (m n)

PU2

PU3

PU5

SU2



119871 (119891 119889) = 463 + 339log10(119891119888) minus 1382log10 (ℎ119887)



119860119872 = 320(log10(1175ℎ119898))

2minus 497 (18)


119892 (119909 | 120590) =1

120590radic2120587exp(minus119909

2

21205902) (19)













SNR (dB)Sp

ectr

um re

cons

truc

tion

MSE



minus3

10minus2

10minus1

100

101


20 30 40 50 60 70 80Sparsity level ()

Spec

trum

reco

nstr

uctio

n M

SE

BPOMPCoSaMP

10minus2

10minus1

100

101



0 5 10 150

50

100

150

200

250

300

350

400

SNR (dB)

PU lo

catio

n er

ror (

m)




8 Conclusion



Acknowledgments


20 30 40 50 60 70 805

10

15

20

25

30

Sparsity level ()

PU lo

catio

n er

ror (

m)

BPOMPCoSaMP



References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










x

xlowast

(a)

x = xlowast

(b)





119911

10038171003817100381710038171003817119906 minusΦ|119868

119911100381710038171003817100381710038172 119903 = 119906 minusΦ119909














119910 = Φ119909 (6)



5 System Model



119871 (119891 119889) = 1198750 + 20lg (119891) + 10119899 lg (119889) [dB] (7)







119879119906 and 119887|

119879119888= 0








1198910 1198911 119891119873chminus1





119891 = F119905 (8)










119877119896119894 (119898 119899) = 119875 (119898 119899 119894) times 10119871(119891119894 119889(119898119899119896))10


2

(9)



119884119896119894 = sum

119898

sum

119899

119877119896119894 (119898 119899)

119884119896119894 = sum

119898

sum

119899

10119871(119891119894 119889(119898119899119896))10 times 119875 (119898 119899 119894)

119884119896119894 = 119871119879(119896 119894) 119875 (119894)

(10)


119871 (119896 119894) = 10119871dB(119896119894)10

119871dB (119896 119894) = [119871 (119891119894 119889 (0 0 119896)) 119871 (119891119894 119889 (1 0 119896))

119871 (119891119894 119889 (119898 119899 119896)) 119871 (119891119894 119889 (119872119873 119896))]119879

(11)





119884119896 = Lk119875 (12)


119875119896 = [119875119879(1198941) 119875

119879(1198942) 119875

119879(119894119873119862

)]119879

(13)


Lk (119895) = [0 0 119871119879(119896 119895) 0 0] (14)


119884 = L119875 (15)

where 119884 = [1198841119879 119884119873119862

119879]119879 and L = [L1 LNC










2 (16)

d(m n 3)

d(m n 2)

d(m n 1)

RP

R

SU1

PU1

SU3

PU4 (m n)

PU2

PU3

PU5

SU2



119871 (119891 119889) = 463 + 339log10(119891119888) minus 1382log10 (ℎ119887)



119860119872 = 320(log10(1175ℎ119898))

2minus 497 (18)


119892 (119909 | 120590) =1

120590radic2120587exp(minus119909

2

21205902) (19)













SNR (dB)Sp

ectr

um re

cons

truc

tion

MSE



minus3

10minus2

10minus1

100

101


20 30 40 50 60 70 80Sparsity level ()

Spec

trum

reco

nstr

uctio

n M

SE

BPOMPCoSaMP

10minus2

10minus1

100

101



0 5 10 150

50

100

150

200

250

300

350

400

SNR (dB)

PU lo

catio

n er

ror (

m)




8 Conclusion



Acknowledgments


20 30 40 50 60 70 805

10

15

20

25

30

Sparsity level ()

PU lo

catio

n er

ror (

m)

BPOMPCoSaMP



References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














119879119906 and 119887|

119879119888= 0








1198910 1198911 119891119873chminus1





119891 = F119905 (8)










119877119896119894 (119898 119899) = 119875 (119898 119899 119894) times 10119871(119891119894 119889(119898119899119896))10


2

(9)



119884119896119894 = sum

119898

sum

119899

119877119896119894 (119898 119899)

119884119896119894 = sum

119898

sum

119899

10119871(119891119894 119889(119898119899119896))10 times 119875 (119898 119899 119894)

119884119896119894 = 119871119879(119896 119894) 119875 (119894)

(10)


119871 (119896 119894) = 10119871dB(119896119894)10

119871dB (119896 119894) = [119871 (119891119894 119889 (0 0 119896)) 119871 (119891119894 119889 (1 0 119896))

119871 (119891119894 119889 (119898 119899 119896)) 119871 (119891119894 119889 (119872119873 119896))]119879

(11)





119884119896 = Lk119875 (12)


119875119896 = [119875119879(1198941) 119875

119879(1198942) 119875

119879(119894119873119862

)]119879

(13)


Lk (119895) = [0 0 119871119879(119896 119895) 0 0] (14)


119884 = L119875 (15)

where 119884 = [1198841119879 119884119873119862

119879]119879 and L = [L1 LNC










2 (16)

d(m n 3)

d(m n 2)

d(m n 1)

RP

R

SU1

PU1

SU3

PU4 (m n)

PU2

PU3

PU5

SU2



119871 (119891 119889) = 463 + 339log10(119891119888) minus 1382log10 (ℎ119887)



119860119872 = 320(log10(1175ℎ119898))

2minus 497 (18)


119892 (119909 | 120590) =1

120590radic2120587exp(minus119909

2

21205902) (19)













SNR (dB)Sp

ectr

um re

cons

truc

tion

MSE



minus3

10minus2

10minus1

100

101


20 30 40 50 60 70 80Sparsity level ()

Spec

trum

reco

nstr

uctio

n M

SE

BPOMPCoSaMP

10minus2

10minus1

100

101



0 5 10 150

50

100

150

200

250

300

350

400

SNR (dB)

PU lo

catio

n er

ror (

m)




8 Conclusion



Acknowledgments


20 30 40 50 60 70 805

10

15

20

25

30

Sparsity level ()

PU lo

catio

n er

ror (

m)

BPOMPCoSaMP



References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119884119896 = Lk119875 (12)


119875119896 = [119875119879(1198941) 119875

119879(1198942) 119875

119879(119894119873119862

)]119879

(13)


Lk (119895) = [0 0 119871119879(119896 119895) 0 0] (14)


119884 = L119875 (15)

where 119884 = [1198841119879 119884119873119862

119879]119879 and L = [L1 LNC










2 (16)

d(m n 3)

d(m n 2)

d(m n 1)

RP

R

SU1

PU1

SU3

PU4 (m n)

PU2

PU3

PU5

SU2



119871 (119891 119889) = 463 + 339log10(119891119888) minus 1382log10 (ℎ119887)



119860119872 = 320(log10(1175ℎ119898))

2minus 497 (18)


119892 (119909 | 120590) =1

120590radic2120587exp(minus119909

2

21205902) (19)













SNR (dB)Sp

ectr

um re

cons

truc

tion

MSE



minus3

10minus2

10minus1

100

101


20 30 40 50 60 70 80Sparsity level ()

Spec

trum

reco

nstr

uctio

n M

SE

BPOMPCoSaMP

10minus2

10minus1

100

101



0 5 10 150

50

100

150

200

250

300

350

400

SNR (dB)

PU lo

catio

n er

ror (

m)




8 Conclusion



Acknowledgments


20 30 40 50 60 70 805

10

15

20

25

30

Sparsity level ()

PU lo

catio

n er

ror (

m)

BPOMPCoSaMP



References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




















SNR (dB)Sp

ectr

um re

cons

truc

tion

MSE



minus3

10minus2

10minus1

100

101


20 30 40 50 60 70 80Sparsity level ()

Spec

trum

reco

nstr

uctio

n M

SE

BPOMPCoSaMP

10minus2

10minus1

100

101



0 5 10 150

50

100

150

200

250

300

350

400

SNR (dB)

PU lo

catio

n er

ror (

m)




8 Conclusion



Acknowledgments


20 30 40 50 60 70 805

10

15

20

25

30

Sparsity level ()

PU lo

catio

n er

ror (

m)

BPOMPCoSaMP



References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 5 10 150

50

100

150

200

250

300

350

400

SNR (dB)

PU lo

catio

n er

ror (

m)




8 Conclusion



Acknowledgments


20 30 40 50 60 70 805

10

15

20

25

30

Sparsity level ()

PU lo

catio

n er

ror (

m)

BPOMPCoSaMP



References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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