Pushing towards the Limit of Sampling Rate: Adaptive Chasing Sampling

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1 Pushing owards he Limi of Sampling Rae: Adapive Chasing Sampling Ying Li, Kun Xie, Xin Wang Dep of Elecrical and Compuer Engineering, Sony Brook Universiy, USA College of Compuer Science and Elecronics Engineering, Hunan Universiy, Changsha, China Absrac Measuremen samples are ofen aken in various monioring applicaions. To reduce he sensing cos, i is desirable o achieve beer sensing qualiy while using fewer samples. Compressive Sensing (CS) echnique finds is role when he signal o be sampled mees cerain sparsiy requiremens. In his paper we invesigae he possibiliy and basic echniques ha could furher reduce he number of samples involved in convenional CS heory by exploiing learning-based non-uniform adapive sampling. Based on a ypical signal sensing applicaion, we illusrae and evaluae he performance of wo of our algorihms, Individual Chasing and Cenroid Chasing, for signals of differen disribuion feaures. Our proposed learning-based adapive sampling schemes complemen exising effors in CS fields and do no depend on any specific signal reconsrucion echnique. Compared o convenional sparse sampling mehods, he simulaion resuls demonsrae ha our algorihms allow 46% less number of samples for accurae signal reconsrucion and achieve up o 57% smaller signal reconsrucion error under he same noise condiion. I. INTRODUCTION Efficien informaion collecion is criical for many applicaions, such as medical imaging, radar deecion and specrum sensing. Pracical signals are generally coninuous and can be sampled ino digial form for more efficien sorage, processing and communicaions. Obviously, a higher number of samples would lead o larger resource consumpion and higher processing complexiy. The fundamenal challenge is o achieve he desired degree of signal fideliy in an ofen noisy environmen wih he minimum number of samples. Compressive Sensing (CS) [2] [4], [12] has araced a lo of recen aenion, wih is capabiliy of reconsrucing sparse signals wih he number of samples much lower han ha of he Nyquis rae. The fundamenal works of CS include he inroducion of he l 1 -minimizaion o reconsruc he signal, and more recenly he greedy recovering of signal componens gradually [28] [13] [27] [32]. Aemps have been made o direcly apply CS in differen applicaion areas, including he reducion of raffic volume during signal acquisiion [24] [1], and finding arge locaions and numbers [1] [32] [17]. Given he limiaion of sensor resources and baery energy, i is ofen desirable o minimize he number of sensors involved in a single sensing ask. This will in urn conribue o a lower average duy cycle of all he sensors and he exension of he reliable working life of he sensor nework. In applicaions ha require he paricipaion of surrounding devices, for example he fas growing research field of crowd sensing, cerain rewards can be given o he conribuors o moivae heir involvemen and cooperaion in sensing. In hese cases, minimizing he number of involved sensors helps reduce he exra sensing cos besides power consumpion. Convenional compressive sensing schemes ake samples randomly and uniformly in a sampling space. This ineviably makes i difficul o deermine he minimum number of samples o ake. If he samples are no sufficien, he signal canno be accuraely recovered. Thus o be safe, usually more han enough samples are aken o guaranee he accurae reconsrucion a higher cos. Insead of aking samples all a once, we would like o find ou if he samples can be aken sequenially, wih subsequen number of samples deermined based on he esimaion resuls from previous samples. More specifically, we would like o find ways of adaping he sensing marix ha defines he sensing behaviors based on he informaion learn from previous observaions o guide subsequen measuremens o "focus" on he areas deeced wih possible exisence of signals, so ha he overall number of samples can be grealy reduced. Alhough i may incur some delay in collecing samples over muliple rounds, his mehod may help significanly reduce he sensing resources when here is a need o coninuously monior some arges or signals, while he chance of appearance of he arges or signals is very small. Fundamenally differen from some exising CS algorihms [3], [5], [14], [28] whose aims are o improve he qualiy of reconsruced signals wih a given number of samples, our proposed algorihms aims o reduce he number of samples needed for he whole sensing process hrough he adapaion of sampling disribuion, aking advanage of he learning process o achieve similar or even beer signal recovery accuracy. Differen from convenional schemes ha search for signals in random locaions which may miss he signals, sensors seleced in our scheme firs deec he poenial exisence of signals alhough heir received values may be low, and he knowledge is hen applied o guide efficien signal sensing laer. To illusrae he principle of our proposed learning-based adapive CS schemes, we consruc a sensing marix according o a pracical applicaion, which allows us o map he change of he sensing marix in heory o he choice of sensor measuremen locaions in pracice. Specifically, we consider he deecion and esimaion of he srenghs and locaions of sparsely disribued signal sources using sensors embedded in commonly accessible wireless devices such as cell phones, which have been considered for use in crowd sensing. Deecion of he signal srengh on differen locaions finds is use in many

2 applicaions, including he emerging cogniive radio field where cell phones can be used collaboraively o form he specrum sensing maps on he aciviies of primary users, localizing he sound sources for deecing evens such as gunsho or rios, and deecion of radiaion sources. Our adapive sampling scheme can be applied o broad caegories of energy signal sensing. We propose wo learning-based adapive sensing algorihms for differen applicaion scenarios o significanly improve he sensing efficiency and qualiy. Our algorihms do no depend on he underlying CS reconsrucion mehods, hus can be applied in many CS-suiable applicaions. The feaures of our work can be summarized as follows: Compared o convenional schemes which employ compleely random sampling, our adapive algorihms wisely choose where o sample based on he knowledge from previous sampling process, hus requiring much fewer samples o esimae he signals a desired accuracy level. Compared o a few exising adapive sensing schemes [6], [3] in he lieraure: The lieraure sudies ake samples uniformly alhough he sampling rae is adaped in he emporal or spaial domain o improve he sensing qualiy, while our schemes adap he sample disribuion based on he learning from previous esimaion resuls o grealy reduce he oal number of samples needed. Our unique sensing marix srucure allows convenien reuse of samples previously aken along wih laer samples for higher sensing accuracy while avoiding asking same sensors muliple imes o conserve sensing resources and reduce he overhead in ransmiing sensing signals in he nework. As we will heoreically show laer, if a sensor in our schemes is asked once, i only akes measuremen in one round in he whole adapive sensing process, hus he number of sensors used equals he number of samples in our work, and we refer o hem iner-changeably in he paper. Therefore, minimizing he number of samples is equivalen o reducing he number of involved sensors, which helps o conserve he sensing cos involved for boh sampling and communicaions. In addiion, our schemes achieve much lower reconsrucion error compared o convenional CS schemes under he same level of noise for he same number of sensors used. The res of he paper is organized as follows. In Secion II, some relaed works are discussed. Secion III gives he fundamenals of CS. Our sysem model and formulaion are described in Secion IV. Secion V inroduces he wo algorihms of adapive sensing. We evaluae he performance in Secion VI, and conclude he work in Secion VII. II. RELATED WORK The research problems on CS can be roughly divided ino wo caegories. One is o sudy and improve he reconsrucion echnique iself in order o reduce he compuaional complexiy or increase he recovery accuracy agains noises. Anoher is he applicaion of CS o solving pracical problems. For signal reconsrucion, he l 1 -minimizaion algorihm [3], [5], [14], also called basis pursui, ransforms he original reconsrucion problem ino a linear programming problem o solve wih convex opimizaion; Anoher family of mehods is greedy-based, including Orhogonal Maching Pursui (OMP) [28], Sagewise OMP (SOMP) [13] and Regularized OMP (ROMP) [27], which ackles he reconsrucion problem by gradually recovering he componens of he signal in each ieraion. The l 1 -minimizaion is generally believed o give beer reconsrucion performance. While he greedy-based mehods have he advanages of simpler implemenaion and ligher compuaional overhead, hey require more measuremens for he same reconsrucion qualiy. Oher reconsrucion algorihms include ieraive hresholding mehods [9], [2] and various Bayesian mehods [22], [31]. CS has been direcly applied in various nework and sensing applicaions. In he nework area, some schemes have been proposed o reduce he oal number of nework messages by aggregaing and forwarding he linearly weighed summaion of he messages on he pah owards he desinaion [24] or a neighbor [1], aking advanage of he capabiliy of CS o recover he original messages. The Greedy Maching Pursui (GMP) algorihm for arge couning in [32] assumes ha he nonzero values can only ake several possible inegers and exhausively ry ou each posiion of he vecor. This mehod is oo compuaional cosly for signal vecors of large dimensions, and nearly impossible for vecors wih arbirary values. [17] sraighforwardly adops CS ino he Access Poin received signal srengh based localizaion problem. CS also finds is use in he field of Cogniive Radio recenly. The work in [23] explois CS o esimae he occupied specrum channels and he locaions of Primary Radios. Ieraive CS algorihms have also been proposed in he lieraure. In he field of image/video processing where he compuing overhead is no a consrain, muliple rounds of CS reconsrucions are sequenced o achieve beer qualiy of recovered signal accuracy [29] [15] a he cos of higher number of samples and compuaion overhead. There are also aemps o adapively reconsruc componens of he signals for an overall beer performance [19], [21], [25]. Despie he increase of olerance o noise, hese adapive CS algorihms canno be applied o effecively reduce he number of samples. In he exising CS heory, he necessary number of samples o ake is given and derived for basic l 1 minimizaion based on random sampling. In his work, we invesigae he possibiliy and ways of reducing he number of samples needed as well as he compuaional complexiy while mainaining he sensing accuracy. Raher han aking random samples as in convenional CS schemes, o reduce he number of samples and improve he recovery qualiy, we propose o adap he sensing marix aking advanage of he ieraive learning process. III. FUNDAMENTALS OF COMPRESSIVE SENSING Convenional informaion heory mandaes a sampling rae o be a leas wice he bandwidh of he signal being sampled. Recen research shows ha a sparse signal can be reconsruced hrough Compressive Sensing wih a high probabiliy a much lower sampling rae. Moreover, mos signals ha are no sparse enough can also be projeced o oher domains o achieve he desired sparsiy.

3 Le vecor x R N be a signal no sparse enough. Given an N N orhogonal basis Ψ = [Ψ 1, Ψ 2,..., Ψ N ] wih each Ψ i being a column vecor, we have: x = Ψs = N s i Ψ i (1) where s is he coefficiens of x in he ransformed domain Ψ. s is said o be k-sparse if i has a mos k nonzero enries and k N. The samples are hen i=1 y = Φx = ΦΨs = As (2) where Φ is an M N measuremen marix o be defined laer wih k M N, he combined M N marix A is called he sensing marix, and y is he sample vecor of M 1. Under he condiion of l minimizaion: min s l, insead of acquiring N samples of s, only M = 2k of noisefree measuremens are needed o reconsruc s [8]. However, his problem is NP-hard. If A mees he Resriced Isomery Propery (RIP) condiion, i is much easier o solve he l 1 - minimizaion problem below min s l1 (3) s.. Ax y l2 ε (4) where he parameer ε is he bound of he error, as long as M c k log(n/k) [2]. Noneheless, he lower bound requiremen of M = 2k measuremens is rarely achieved using convenional CS mehods. The major conribuion of his paper is o provide some insighs on he possibiliy and poenial sraegies ha can be exploied o push he sampling rae owards he limi. IV. PROBLEM FORMULATION To invesigae he possibiliy and mehods of improving he sampling efficiency and accuracy, in his paper, we insaniae a specific problem of using cogniive cell phones which can swich o oher specrum bands o deec he srengh and locaions of primary signal sources for he poenial cogniive use of he unoccupied specrum of he measured bands. This seing helps o picure he mapping of heoreical changes on he sensing marix ino he adjusmen of sensor posiions/sampling poins in pracice. We consider a general scenario where some cell phone users are willing o paricipae in esimaing he srengh and locaions of aggregae primary radio signals (e.g., TV or radar signals) which are generally locaed sparsely in he sensing field. Since we uilize cell phones o deec he signal srengh, we consider cell phones as specrum "sensors" in his paper. To provide a locaion reference and faciliae he scalable monioring, he sensing domain is pariioned ino N grids wih he size deermined based on he resoluion requiremen of a specific applicaion, and each grid could have no or several signal sources inside i. Similarly, a grid could have no or several cell phones, each can updae he base saion is grid locaion upon grid crossing by piggying back he informaion wih oher uplink messages. While some lieraure work [32] assume ha signal sources have idenical ransmiing power o coun he number of arges, we do no have his resricion or o differeniae individual signals wihin a grid. Insead, we consider signals wihin a grid as an aggregae signal source locaed a he grid cener. Generally, wihin he shor sensing period, we don expec a significan change on he srengh of he signals o be measured and a large number of signal sources or sensors o move across grids. Figure 1 shows an example sysem of 16 grids in a monioring region. Signal sources wih differen level of energy are indicaed by he do of differen sizes. For grids ha have several phones inside, we can always pick only one cell phone o do he ask in order o preserve he sensing resources. Thus i is safe o assume each grid has a mos one sensor for a specific sensing ask a a given ime. Cell phones asked for sensing he specrum will send heir samples o he base saion o be fused Legend Sensors wih measuremen Sensors no been acivaed Signal Sources wih differen power Φ = s = Fig. 1. A demo of he sysem wih he corresponding signal vecor s and measuremen marix Φ. To monior he energy of signals, radiionally, a large number of sensors are placed across he whole monioring domain and kep acive o mainain he coverage [18] [16]. In realiy, for a given sensing resoluion, i.e. grid size in his seup, only a small number of grids will conain signal sources a a given ime. This spaial domain sparsiy makes i possible o apply he compressive sensing echnique o reconsruc he locaions and srengh of he signal sources. Insead of asking an adequae number of sensors [32] randomly based on he CS recovery requiremen, o increase he recovery accuracy while reducing he oal number of sensors involved, we would like o sar wih a small number of sensors a random locaions. We hen adapively ask addiional sensors by learning from pas measuremen daa. Nex we will show mahemaically how our marix srucure helps o faciliae his process. The srengh or energy of many real-world signals decays over he disance, and could also be impaced by he environmen. Our proposed schemes do no rely on any specific channel models, herefore we denoe he channel facor beween he radio ransmier in grid i and receiver in grid j as C ij. The channel fading and environmen impac ofen add in random facors ino C ij, rendering he channel o be Gaussian.

4 We absrac he disribuion of signal sources ino vecor presenaion. Le s = [s 1, s 2,..., s N ] T be an N 1 column vecor, where he i h enry s i is he aggregae signal srengh of grid i. s is k-sparse wih k N, which means a mos k grids ou of N acually have signal sources. Le Ψ be an N N ransformaion marix which embodies he signal energy decaying process over he radio channel: C 11 C 21 C N1 C 12 C 22 C N2 Ψ =..... (5). C 1N C 2N C NN Then x = Ψs is he received signal srengh vecor wih x(j) denoing he aggregae signal srengh received by he sensor a grid j from all signal sources. As illusraed in Figure 1, he sensors asked o ake samples can be "seleced" by an M N measuremen marix Φ. The m h row of he marix is a 1 N row vecor wih all elemens equal o zero excep Φ(m, j) = 1, where j is he index of he grid a which he sensor m is locaed. Each enry x(j) denoes he signal srengh received a he sensor in grid j, so he effec of lef-muliplying Φ wih he vecor x in Equaion(2) is o selec M ou of N rows of x, or equivalenly choose he M sensors a specific grid posiions o ake samples. Under his formulaion, a se of sensors can be flexibly and convenienly specified wih marix Φ o ake samples in each round of he adapive sensing process. We can ake advanage of he samples acquired in he previous ieraions of he adapive process and combine hem wih he new samples o form a more informaive sample vecor y, where y = ( y y ) = ΦΨs = ( Φ Φ ) Ψs (6) o recover he daa using he new Φ marix. In Equaion 6, y is he vecor of samples already colleced by sensors previously seleced by Φ, and y conains he new samples aken by sensors newly specified in Φ. A seleced sensor only needs o ake sample once during he whole sensing process, wih is sample saved and used wih laer samples o reconsruc he signal. Therefore, he oal number of samples aken in he whole sensing period equals he number of sensors used, hus we do no differeniae he number of samples from he number of sensors hereafer. The specrum map can be updaed periodically, and for each period, a new sequence of sensing can be iniiaed. The reuse of samples from previous rounds is enabled by our unique sensing marix srucure, which helps o grealy reduce he sensing resource consumpion and serves as he basis of our algorihms o be proposed. The sensing marix A = ΦΨ has been proven in [12], [32] o obey RIP condiion as long as marix Φ and Ψ are consruced as defined above. Therefore in each adapive ieraion, we can safely apply he l 1 -minimizaion mehod o perform he reconsrucion. Alhough he small number of sensors used in he beginning phase of he adapive process would resul in inaccurae reconsrucion, in he nex secion, we will show ha by learning he resul from each previous ieraion, he process evenually converges o an accurae reconsrucion, while he overall number of sensors used is significanly reduced. V. ADAPTATION OF SENSING MATRIX AND PROCESS FOR IMPROVED SENSING QUALITY In his work, we propose wo learning-based mehods in which a small se of sensors considered o be beer will be asked o ake samples in each round of sensing based on he reconsrucion resuls from he previous round, hen he new samples ogeher wih he exising samples will generae an improved reconsrucion, and so forh. In his secion, we firs presen our basic seing in Secion V-A, and hen inroduce in deails of our sensing algorihms in Secion V-B and V-C, respecively. Finally, we discuss our sraegy in avoiding he local opimal soluion in Secion V-D. A. Basic Seing and Design Consideraion In his paper, we use erms such as "ask", "urn on" or "add" when selecing a sensor o ake measuremen. The locaions of sensors in erms of which grid are denoed by he se L. For example, in Figure 1 sensors are insalled a grids 2, 4, 5, 6, ec., hen L will have he corresponding elemens. As an ieraive scheme, he algorihm needs an iniial value o sar wih. Specifically, a he beginning of our adapive sensing process, we need o decide he number of sensors o ake samples for he firs round, hen he adapive algorihm will handle he following ieraions auomaically. In order o deermine he opimal number of samples o ake, convenional CS echniques ofen assume he prior knowledge of he sparsiy value k or esimae k, and heir performances are highly impaced by he accuracy of k value aken. In conras, our schemes ake samples sequenially and deermine he number of addiional samples needed based on he previous esimaion resuls, and he iniial k is only used as a reference. As menioned in Secion III, 2k (k is he sparsiy) is believed o be he heoreical exreme of he number of necessary samples [8] for under-sampled signal recovery. Our algorihms hus sar he firs ieraion by acivaing M () = 2k sensors a locaions randomly chosen from he se L o ake samples. Our simulaion sudy on he impac of he saring number of sensors a he firs ieraion also reveals ha 2k is he opimal choice in erms of he final oal sensing resources needed. In our proposed adapive schemes, insead of disribuing he sensing resources evenly o randomly sample he signal vecor and repeaing he same sensing process over ime, in each following ieraion, we propose o jus ake a few new samples around he vecor enries deeced o have a higher chance of being nonzero from he reconsrucion resul of he previous ieraion. Then he new samples combined wih exising samples, as in Equaion (6), are applied o reconsruc he signal vecor. We use ŝ (i) o denoe he reconsruced signal afer he i h round of sensing. As we sar from sensing wih he number of sensors smaller han necessary for CS recovery, here exiss inaccuracy in geing each inermediae ŝ (i). Alhough neiher he posiions nor he values of he nonzero enries of ŝ (i) may be accurae, our preliminary sudies indicae ha he acual nonzero enries of he original vecor are close o he

5 nonzero posiions indicaed by ŝ (i). In our signal deecion example, he signal sources are possibly locaed in he region close o he grids corresponding o he nonzero posiions of ŝ (i). So by "moving" sensing focus (i.e., selecing sensors a he desired locaions) owards he esimaed locaions of he signal sources sep by sep, he algorihms will improve he reconsrucion resuls unil he posiions and he values of he nonzero enries no longer change. This way we can find he accurae posiions and energy levels of he signal sources. This also helps o increase he sensing efficiency and reduce he number of sensors needed. This is he fundamenal principle of our adapive algorihms. The learning process ieraes unil he recovered resul has reached a desired qualiy or canno be furher improved. Our resuls in Secion VI will show ha he oal number of sensors used in he overall adapive process is much less han ha needed for a single-ime convenional CS recovery for achieving he same level of accuracy. One hing o noe is ha ŝ (i) may conain many nonzero enries wih very small values, which are insignifican and edious o consider as he possible posiions of arges. I has o be modified before each nex ieraion. To be specific, firs all he negaive enries of ŝ (i 1) will be se o zero. Then for he posiive enries, all hose wih values below α% ha of he larges posiive enry value in ŝ (i 1) will be se o zeros. The impac of α on performance is inspeced in he simulaions. Based on he same adapive principle, we propose wo differen algorihms on choosing new sampling locaions in he nex round for he signal srengh deecion problem, depending on signal source locaion disribuions. B. Individual Chasing If here is no knowledge on he locaions or disribuion feaures of signal sources, i is reasonable o assume hey are randomly disribued in he monioring area. In his case, we will adap he posiions for sampling owards he esimaed locaion of each individual signal source given by he las ieraion. In he i h ieraion, according o he previous reconsrucion vecor ŝ (i 1), for each of is nonzero enry ŝ (i 1) (n), a sensor in se L whose locaion is closes o grid n is seleced o ake a sample if a measuremen is no already aken here before. I is bes o choose he sensor righ inside grid n for sampling. However if here is no sensor locaed in he grid n, anoher sensor in he sensor locaion se L wih he smalles euclidian disance o grid n will be seleced. Afer each non-zero posiion n is ensured o have one sampling in he corresponding grid, he l 1 -minimizaion process is invoked o ge he reconsrucion ŝ (i) based on he combined samples y and combined Φ in Equaion (6). The reconsrucion resul ŝ (i) is fed ino Algorihm 2 for erminaion condiion check o deermine wheher he algorihm should end or coninue wih more ieraions. Algorihm 1 oulines he deails of Individual Chasing in each ieraion. Figure 2-(a) shows he sensor locaions when he Individual Chasing algorihm erminaes. I can be observed ha sampling measuremens have been aken a each grid wih signal source ha also has a sensor inside. For he grids wih signal sources (a) Individual Chasing (b) Cenroid Chasing Fig. 2. Illusraion of Individual Chasing and Cenroid Chasing. bu wihou sensors deployed, samples are aken a he closes grids, i.e. samples have been aken by sensors in grid 5 and 15 for nearby signals in grid 9 and 14. The Individual Chasing scheme adaps well when signal sources are uniformly disribued in he monioring field. More imporanly, he simulaions in Secion VI demonsrae ha he Individual Chasing algorihm converges fas wih superior recovery accuracy regardless of how he signal source locaions are disribued. Algorihm 1 Individual Chasing 1: In he i h ieraion: 2: for each nonzero posiion n of ŝ (i 1) do 3: find a grid posiion p in L wih he smalles euclidian disance o grid n. 4: if no measuremen has previously been aken a grid p hen 5: ake sampling a grid p s sensor. 6: end if 7: end for 8: combine new samples wih exising ones for y. 9: do l 1 -minimizaion on y and Φ o ge ŝ (i). 1: call Algorihm 2 o check he erminaion condiion. 11: if algorihm does no erminae in his ieraion hen 12: i = i + 1, go back o Line 1 and sar he nex ieraion. 13: end if Algorihm 2 Terminaion Condiion Check 1: if he nonzero posiions of ŝ (i) are all he same as ŝ (i 1) hen 2: if he numeric difference of each nonzero value beween ŝ (i) and ŝ (i 1) is smaller han a percenage hreshold of ŝ (i 1) hen 3: reconsrucion process erminaes in his ieraion. 4: else 5: coninue wih he nex ieraion of reconsrucion. 6: end if 7: else 8: coninue wih he nex ieraion of reconsrucion. 9: end if 1: he choice of is given in SecionVI

6 C. Cenroid Chasing In some scenarios, he signal sources may locae closely and form clusers. The clusering paerns of signal sources may be exploied o guide he sampling locaions o faciliae he finding of signal sources wih even fewer sensor measuremens. For example, in an urban area, a region wih high noise ofen indicaes he exisence of some evens, such as sree concer, parade, or rio. In his case, many sound sources may say closely, and here is a need o deec he high noise areas o idenify he evens and proec agains poenial harm o he communiy. The microphones of cell phones can be exploied o collaboraively deec he srengh and locaion of hese sound sources. The Cenroid Chasing scheme iniializes similarly as Individual Chasing by randomly acivaing M () = 2k sensors from he se L o sar he firs ieraion. Algorihm 3 deails he acions for each consecuive ieraion. Le T (i 1) denoes he se of grid numbers ha correspond o he non-zero posiions of he reconsruced ŝ (i 1), which indicae he possible grid posiions where signal sources may reside esimaed in he i h round. In he i h ieraion, grid numbers in T (i 1) are grouped ino C clusers based on heir muual euclidian disances, wih T (i 1) denoing he se of grids belonging o he h ( = 1,..., C) cluser. The opimal clusering crieria is given laer. The minimum recangular region ha covers all he grids of he h cluser is called he cluser region of he h cluser, and is size R (i 1) is he number of grids i covers. For each cluser, based on is signal arge densiy defined as T (i 1) /R (i 1), closes sensors (hese sensors could be eiher wihin or M (i 1) ouside of he recangular cluser region) will be seleced for samplings a his ieraion, where M (i 1) = T (i 1) R (i 1) (1 T(i 1) ). (7) Equaion (7) reflecs he endency o pu less sensing resource in a cluser region when he arge sources are denser. The number of sampling posiions o be considered for adding a each ieraion is conrolled o be less han ha of he Individual Chasing mehod, which is T (i 1), by muliplying i wih he densiy which is a value smaller han 1. This feaure, however, may resul in more ieraions needed hus more sensors involved han he Individual Chasing algorihm when he signals are sparsely disribued and poorly clusered. The performances of wo algorihms are compared under differen scenarios in he simulaion. Again, he new measuremens obained a he curren ieraion will be combined wih exising samples for CS reconsrucion, which will be checked by erminaion condiion aferwards. Specifically, Line 2 of Algorihm 3 clusers he possible signal sources whose posiions are esimaed by he nonzero posiions of ŝ (i 1). This procedure sars a an iniial empirical number of clusers C = T (i 1) /2, where T (i 1) is he oal number of elemens in he se o be clusered, as given in [26]. The opimal number of clusers o form is found by varying he number of clusers C and picking he one under which he average disance beween each grid posiion o he Algorihm 3 Cenroid Chasing 1: In he i h ieraion: 2: group he grids in T (i 1) ino C clusers. 3: for each of he cluser T (i 1) do 4: find he M (i 1) sensors in L ha are closes in euclidian disance o he cener of he h cluser region. 5: if any of hese M (i 1) chosen sensor grids has no sampled before hen 6: acivae he sensors a hese grids. 7: end if 8: end for 9: combine new samples wih exising ones for y. 1: do l 1 -minimizaion on y and Φ o ge ŝ (i). 11: call Algorihm 2 o check he erminaion condiion. 12: if algorihm does no erminae in his ieraion hen 13: i = i + 1, go back o Line 1 and sar he nex ieraion. 14: end if cenroid of cluser i belongs o is minimal. Afer clusering, a smaller number of sensors can be asked in he signal concenraed area o achieve even higher sensing efficiency. Figure 2-(b) shows he sensing condiion a he end of he Cenroid Chasing algorihm. Grids wih signals are grouped ino 3 clusers. Wihin each cluser, a porion of he closes sensors are acivaed for sampling. Compared wih Individual Chasing scheme on he lef of he figure, fewer number of sensors are used given he signal sources have a nice clusering feaure. The Cenroid Chasing algorihm is specially suiable for applicaions wih knowledge of clusered disribuion, which, noneheless, does no preven is applicabiliy in general cases. The simulaions in secion VI-D show ha Cenroid Chasing algorihm ouperforms he Individual Chasing algorihm when he signal sources are denser in clusers, and works almos equally efficien in general scenarios where he signal sources are sparsely disribued. D. Local Opimum Avoidance - Random Exploraion Very unlikely bu possible, adapive algorihms could converge a he local opimum. In our problem, local opimum does no give accurae reconsrucion a he algorihm erminaion ime. In an ieraive scenario, one can only look a he inermediae resuls o decide wheher he ieraion should sop. The Individual Chasing and Cenroid Chasing algorihms will sop when consecuive ieraions presen no more change in he resuls. In order o avoid possible local opimum, we inroduce an exra sep called Random Exploraion. To be specific, for boh algorihms, when he Terminaion Condiion Alg.2 is saisfied a cerain ieraion, we do no erminae he program immediaely. Insead, we randomly pick up some sensors ha have no been acivaed before o ake samples, hen re-ener he adapive process and le i converge again. This random walk procedure is proved o work exremely well in our simulaions. VI. SIMULATION AND PERFORMANCE EVALUATION We evaluae he performance of our wo proposed learningbased compressive sensing schemes hrough exensive simula-

7 ions. Before showing our simulaion resuls, we firs inroduce our performance merics and simulaion se-up. A. Performance Merics Reconsrucion Error: defined as he Sum of Absolue Difference (SAD) beween he recovered and he original signal vecor: SAD = ŝ s l1 = N ŝ i s i (8) i=1 This measuremen meric evaluaes he accuracy of vecor reconsrucion. I reflecs no only he degree of error due o he posiion mismach of nonzero enries, bu also he difference in magniude for each unmached enry. Number of Sensors Needed (M): in our adapive algorihms, new sensors may be added for sampling in each new ieraion. The number of sensors needed for our algorihms are defined as he oal number of sensors used during he whole sensing process, which is he leas number of sensors needed in our performance sudies o give 1% reconsrucion accuracy. B. Simulaion Se-Up We simulae he general problem on forming he energy measuremen map and compare he performances of our proposed schemes wih peer works. dbm is adoped as he measuremen uni of signal srengh in our simulaion. One grid could have muliple signal sources, and he overall signal srengh for a grid is he aggregae of hese signals. Being aware ha he numeric scale of he nonzero enries of signal vecor is no criical o he problem of compressive sensing recovery bu acually he sparsiy is, we assume a range of 3-5 dbm for he possible aggregaed signal srengh inside any single grid. For every simulaion run, signal sources are generaed a random locaions across he sensing region wih he aggregae signal srengh of a grid o be a random value seleced from he 3-5 range. The simulaion is carried ou wih MATLAB. Our schemes improve he sensing performance benefiing from he adapive chasing process, bu do no rely on any adoped specific channel model as previously poined ou. In he simulaion, we would like o use a specific radio channel model o es our schemes. We define he channel as he following. The srengh a locaion j for a signal source a locaion i is roughly approximaed as: P ij = P ig ij d β, where G ij = X ij + Y ij i (9) ij P i is he signal srengh a is source locaion i, which is essenially he aggregaed srengh of signal sources a grid i, i.e. s i The denominaor represens he pah loss due o he disance d ij beween locaions i and j. β is he decaying facor wih possible value in [2., 5.], depending on he environmen. i is he imaginary sign. G ij is a complex random Gaussian variable wih real and imaginary componens boh being independen and idenically disribued zero-mean Gaussian variables X ij, Y ij N (, σ), 2 which capures he Raleigh disribuion for muli-pah fas fading of he signal [7], [11]. The variance σ is se o.5 as in [32] for fair comparison G ha follows. ij corresponds o C ij in he channel model of d β ij Equaion (5). In he shor duraion of performing he sensing ask, sensors (essenially cell phones in our applicaion) will no have significan moving disance or move across grids. For each sensing ask, 4 sensors are randomly deployed in an area of N = 3 3 = 9 grids wih a mos one sensor inside one grid, as we can always selec one sensor o paricipae in he sensing ask when a grid has muliple sensors inside as discussed previously. The size of each square grid is se o 3 meer which can reasonably reflec he disance effec in signal propagaion. Alhough he grid size deermines he signal monioring resoluion, i does no have significan impac on he performance of our algorihms once he resoluion requiremen is given. There are k(k << N) grids wih signal sources a a given ime insan. k is he sparsiy value, which is varied in differen simulaion sudies. The erminaion condiion hreshold in Alg.2 is se o 5% which will generally guaranee he recovery resul o be accurae a he algorihm erminaion ime wih he recovery error in he order of 1 4 even for real valued signal vecors based on our preliminary sudies. To compare our wo schemes, he posiions of k grids wih signal sources can be eiher disribued in a clusered fashion, or uniformly disribued. To examine he reliabiliy performance of our schemes, Gaussian Whie noise N(, σ 2 ) is added o he observed sample vecor y in some of he simulaion runs, and SNR measure is exploied o quanify he noise srengh. Each presened resul is he average of many runs. Our proposed wo algorihms Individual Chasing and Cenroid Chasing, also referred o as IC and CC from now on, do no depend on any specific CS reconsrucion echnique. Thus we chose wo fundamenal and mos prevalen ypes of work for performance evaluaions-l 1 minimizaion based CS and greedy based CS. GMP [32] provides a greedy based reconsrucion algorihm for CS, and also explois he received signal srengh a differen grid posiions o help solve he arge localizaion and couning problem. l 1 -magic is a concise and dominan realizaion of l 1 minimizaion based CS scheme, which can be direcly applied o and is hus worh comparing wih our simulaion scenario of signal srengh vecor reconsrucion. C. Parameer Sudy For our adapive algorihms, he oal number of sensors used increases afer each ieraion. Naurally, he number of sensors aken in he firs ieraion would have impac on he final oal number of sensors used. In Figure3-(a), we sudy he opimal saring number of sensors for our Individual Chasing algorihm. For each sparsiy k, he opimal choice on he number of sensors o sar wih is clearly around 2k. Inuiively, if we sar wih oo few sensors, he inermediae recovered vecors will be very inaccurae and resul in much more ieraions hus more sensors involved in oal; however saring wih oo many sensors a he beginning would provide more sensing resources han needed which again causes more sensors in oal. Therefore he number of sensors o sar wih for boh Individual Chasing and Cenroid Chasing algorihms

8 will be se o wice he value of k in he following simulaions if no oherwise specified. requires 46% 25% fewer sensors han l 1 -magic for differen k. Number of samplings used evenually (M) (a) 15 sparsiy k=5 sparsiy k=4 1 sparsiy k=3 sparsiy k= Raio = (number of samplings aken iniially) / k Number of samplings used evenually (M) (b) Aver. power=1dbm Aver. power=2dbm Aver. power=5dbm log(α%) α is nonzero enry rimming hreshold Fig. 3. (a) The number of sensors ulimaely used vs. he number of sensors sared wih. (b) The number of sensors ulimaely used vs. he rimming hreshold α In Secion V, we menioned he rimming hreshold α, based on which a porion of insignifican nonzero enries of he recovered vecor are se o before he nex ieraion in order o shrink he arge range of nonzero enries o work on. In Figure 3-(b), we fix k a 5, bu vary he average signal source ransmiing power (i.e. magniude of each vecor enry) a hree disinc levels-1, 2 and 5 dbm, and invesigae he number of sensors needed for accurae recovery using each differen rimming hreshold. Since he range of α picked spans over several orders of magniude, we apply base-1 logarihm on α% as he x-axis. I can be clearly observed ha for each seing of signal vecor magniude, here is an opimal range of α ha gives he minimum number of sensors needed. Ou of his range, he performance deerioraes and quickly reach a seady level. This is because only when he α% is appropriaely chosen ha no oo many componens of he vecor are wiped ou nor oo many insignifican ones are preserved will he reconsrucion process requires he leas number of sensors. Wih he aggregae signal srengh ranging beween 3-5 dbm, he average signal srengh in our simulaion is close o 2 dbm. According o he figure, 1% is adoped as he defaul rimming hreshold in he simulaions ha follow. D. Number of Sensors Needed Pursuing a minimum number of sensors needed for an accurae signal reconsrucion is he major challenge and research focus in he closely relaed research fields. We evaluae he minimum number of sensors needed for accurae signal vecor reconsrucion (zero reconsrucion error) under differen levels of signal sparsiy for each scheme in Figure 4. As expeced, he number of sensors needed increases as k grows for all he algorihms. Paricularly, Figure 4-(a) is under he scenario where he signal sources are randomly uniformly disribued across he nework grids. IC performs slighly beer han CC as expeced. The clusering funcion of CC is no effecive when he signal sources are uniformly disribued, which leads o more ieraions o converge and more sensors needed. Compared o GMP, IC requires 45% fewer sensors when k is small, and abou 23% fewer sensors when k ges bigger. IC Number of samplings needed (M) for accurae reconsrucion (a) IC CC L1 Magic GMP Number of grids ha have signal sources (sparsiy k) Number of samplings needed (M) for accurae reconsrucion (b) IC CC L1 Magic GMP Number of grids ha have signal sources (sparsiy k) Fig. 4. (a) scaered signal sources. (b) clusered signal sources. In Figure 4-(b), he signal sources are disribued in clusered fashion across grids, which benefis he clusering process of CC algorihm. Thus CC is observed o require fewer sensors han IC as he number of signal sources exceeds cerain value. While he performance difference beween IC and CC is small, CC requires 4% 33% fewer sensors han GMP, and 4% 3% fewer han l 1 -magic, which are big improvemens. In general, boh IC and CC work exremely well no maer he signal sources are concenraed or scaered, and far ouperform he oher wo schemes under he same k. Given he performance difference beween IC and CC is small and boh algorihms follow he same principle, we only sudy and compare IC wih he oher wo schemes and assume he signal sources are randomly and uniformly disribued in he simulaions ha follow. E. Reconsrucion Error 1) Convergency Sudy: We sudy he convergency of IC in Figure 5-(a). I is clear ha under all k, IC is able o converge wihin 3-6 ieraions o ge accurae reconsrucion wih error, and i exhibis a raher seady (approximaelinear) improvemen in reducing he reconsrucion error in each ieraion. I converges faser for larger k. This is due o he fac ha we iniialize 2k number of sensors for he opimal performance. Wih a larger k, here are more samples aken a he beginning, herefore i needs fewer ieraions o ge enough overall sensor samples for accurae recovery. Reconsrucion Error (SAD) (a) sparsiy k=5 sparsiy k=4 sparsiy k=3 sparsiy k= Ieraion number Reconsrucion Error (SAD) (b) IC L1 Magic GMP Signal o Noise Raio (SNR/dB) Fig. 5. (a) Convergency sudy of Individual Chasing algorihm. (b) The reconsrucion error comparison due o noise under he same k = 5 and M = 25.

9 2) Performance Under Noise: The adapive algorihms can always find he accurae reconsrucion given enough ieraions and proper handling wih local opimum avoidance. However in a noisy environmen where he sample y is conaminaed, he final reconsruced resul could be differen from he acual signal vecor. The reconsrucion errors due o sampling noise for differen algorihms are compared in Figure 5-(b). The reconsrucion error reduces as signal-o-noise raio increases for all hree schemes. Under he same sensing condiion of k = 5 and M = 25, a each SNR level, IC gives much more accurae resul han he oher wo. In he wors scenario wih he sronges noise a 15dB SNR in our es seing, he reconsrucion error for IC is approximaely (14 6)/14 = 57% smaller han ha of l 1 -magic. GMP is slighly more accurae han l 1 - magic under he same sensing seing, because i enumeraes all possible values for each possible nonzero posiion of he vecor a he cos of higher compuaional overhead. VII. CONCLUSION Raher han using convenional random sampling in compressive sensing, we observe and heoreically prove ha by learning from he sensing resuls and adjusing he srucure of he sensing marix adapively, he number of samples needed for high-qualiy signal recovery in compressive sensing can be significanly lower han ha required by he basic l 1 minimizaion soluion. We propose wo learning-based adapive algorihms, Individual Chasing and Cenroid Chasing, for differen signal source disribuion scenarios. Boh schemes adapively concenrae sensing resources o proper signal subspace owards beer acquisiion of signals, and do no depend on any specific CS reconsrucion mehods. The insaniaion of our algorihms o solve he general signal srengh sensing problem wih disance fading can be convenienly generalized o various similar applicaions. 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