Robust LMS-based Compressive Sensing Reconstruction Algorithm for Noisy Wireless Sensor Networks

Transcription

1 he Proceedngs of the nd Internatonal Conference on Intellgent Green Buldng and Smart Grd (IGBSG) Robust LMS-based Compressve Sensng Reconstructon Algorthm for Nosy Wreless Sensor Networks Yu-Mn Ln, Hung-Ch Kuo, and An-Yeu (Andy) Wu Graduate Insttute of Electroncs Engneerng, Natonal awan Unversty, ape, 06, awan, R.O.C. {yumn, Abstract Wreless sensor networks (WSNs) show mmense promse n many applcatons, such as envronmental montorng and remotely meterng. Compressve sensng (CS) s a novel sgnal processng that has been envsoned as a useful regme to address the energy and scalng constrants n WSNs. CS s able to move the burden of sensory nodes to central cloud/server. However, prevalng CS reconstructon algorthms are vulnerable to nose. In ths paper, we explot the natural nose-tolerance property of least mean square (LMS) adaptve flter and propose a greedy-lms algorthm for CS reconstructon. When SNR s 48dB, greedy-lms algorthm acheves 6% and 47% hgher successful rate than BPDN and OMP, respectvely. In addton, the computatonal complexty of greedy-lms s compettve wth OMP. Keywords CS-based wreless sensor networks; sparse sgnal reconstructon; least mean square (LMS); Fg.. Convectonal wreless sensor networks. I. INRODUCION Wreless sensor networks (WSNs) [] have a vast feld of applcatons, ncludng envronmental montorng, ntellgent buldng, and remotely meterng. Moreover, WSNs play an mportant role n Internet of hngs (Io) as the foundaton nfrastructure. he llustraton of WSNs s shown n Fg.. he WSNs are bult of a large number of sensor nodes and rare gateway sensor nodes. he sensor nodes are nexpensve, wth lmted resource for communcaton. Each sensor node s connected to several nodes. hese sensor nodes wll sequentally aggregate measured data to a gateway sensor node. he gateway sensor node, whch s more expansve, wll transmt the measured data to central cloud/server. Fg.. CS-based wreless sensor networks. [3] he conventonal WSNs encounter the ssues of data transmsson n energy harvestng and asymmetry. he lfetme of sensor nodes are lmted by avalable batteres. Hence, we would lke to acqure the nformaton of targeted envronment wth as less measurements as possble. In addton, the asymmetrc data transmsson leads to nconvenent to replace batteres perodcally. Snce the sensor nodes are deployed to montorng physcal phenomenon n space, such as temperature or humdty, the measured data are spatally correlated and compressble under proper bass. Compressve sensng (CS) [] had been envsoned to decrease requre measurements n WSNs. CS theory states that the envronmental sgnals can be recovered from far fewer measurements than conventonal WSNs f the sgnals are sparse n some doman. he llustraton of CS-based wreless sensor networks [3]-[4] s shown n Fg.. Wth the regme of CS, global communcaton cost can be reduced. Furthermore, the energy consumpton of sensor nodes becomes more symmetrcal that each sensor node transmts same amount of measurements. he CS-based wreless sensor networks [3]-[4] are emergng feld that has drawn a great deal of attentons. In the cloud/server, envronmental sgnals are recovered wth the receved fewer measurements. Snce the sensory node s nexpensve, the transmtted measurements are prone to contan nose. How to reconstruct the envronmental sgnals wth the nosy measurements s a crucal problem. Extng CS-based WSNs lteratures adopted prevalng bass pursut denosng (BPDN) algorthm [5] or orthogonal matchng pursut (OMP) hs work was supported n part by the Mnstry of Scence and echnology of awan under Grant MOS 03-0-E , and n part by NOVAEK Fellowshp /6/$ IEEE.

2 he Proceedngs of the nd Internatonal Conference on Intellgent Green Buldng and Smart Grd (IGBSG) algorthm [5] to reconstruct spare sgnals. Because BPDN algorthm lays the foundaton of convex optmzaton and lnear programmng to be nose-tolerance, the computatonal complexty s extremely hgh. On the other hand, OMP s lowcomplexty but vulnerable to measurement nose. Hence, the prevalng CS reconstructon algorthms are tradeoff between nose-tolerance and low-complexty. In ths paper, we am to develop a CS reconstructon algorthm whch s not only robust to nose but also lowcomplexty. herefore, we propose a novel reconstructon algorthm named greedy-lms algorthm. he greedy-lms algorthm adopts the pursung process as OMP to dentfy nonzero terms of sparse sgnals and apples well-known least mean square (LMS) adaptve flter, whch has the natural property of nose-tolerance [6], to estmate the value of nonzero terms. When SNR s 48dB, the proposed greedy-lms algorthm acheves 6% and 47% hgher successful rate than BPDN and OMP, respectvely. In addton, the requred multplcatons of greedy-lms are compettve wth OMP. he rest of ths paper s organzed as follows. Secton II brefly ntroduces basc backgrounds of CS-based WSNs and prevalng CS reconstructon algorthms. Secton III presents the proposed greedy-lms algorthm. he analyss and smulaton results are dscussed n Secton IV. Fnally, we gve a concluson n Secton V. II. BACKGROUND A. Data aggraton of CS based wreless sensor networks CS based wreless sensor networks aggregate the data dfferent from conventonal wreless sensor networks. For conventonal WSNs as Fg., the gateway sensor nodes wll aggregate N- measurements form sensory nodes and transmt N measurements to the cloud or server, N s number of nodes n the network. o transmt such amount of data s a heavy burden; therefore, the CS based wreless sensor networks are proposed to reduce global communcaton cost. When the th sensor measured the nformaton x, an encodng process wll be appled to multply x by predefned Φ as d Φ x, () M x s a scalar, Φ s a column vector, and M d s called locally encoded data. hen, the transmtted measurements addng y of th sensory node can be calculated by d wth receved measurements such as y d yreceved. () A smple example s gven n Fg. 3. Base on the regme, the fnal transmtted measurement of gateway sensor node becomes yn Φx. (3) Suppose that the overall envronmental sgnal can be formulated as x, x = [ x, x, x 3, L,x N ] s a vector of length N. he receved data of the server can be formulated as y = Φx+ n, (4) M N, s the measurements vector, y y y Φ [ Φ, Φ, L, Φ N ] represents the M N samplng matrx, M and n s the measurement nose. he sgnals x s sad to be K-sparse f x can be well approxmated usng only K nonzero coeffcents under some lnear transform. Accordng to CS theory, x could be recovered from the measurement y when Φ s random Gaussan matrx [7], whch each entry s ndependently and dentcally dstrbuted (d) Gaussan varable, and M satsfed [8] M CK log( N/ K), (5) C s a constant dependng on each nstance. Fg. 3. A smple example for measurement accumulaton n CS-based WSNs. B. CS Reconstructon When CS reconstructon, the N N bass Ψ are nducted and the CS formulaton becomes y=φx+ n=φψs = Θs. (6) he model of compressve sensng n matrx formaton s shown n Fg. 4. he reconstructon process can be decomposed nto two procedures, ncludng estmaton of sparse soluton ŝ and reconstructon of x by ˆ ˆx=Ψs. (7) How to search the sparest vector s the man problem. he sparest soluton can be calculated by takng l 0 norm nto account as mn, s.t. Θ = y. (8) 0 However (8) s generally a non-polynomal hard (NP-hard) problem. o overcome the dffculty, l norm are appled to replace l 0 norm n bass pursut algorthm [5] by computng mn, s.t. Θ = y. (9) Afterwards, constrant n (8) s relaxed to recover x wth nosy measurement n bass pursut de-nosng (BPDN) algorthm [5]: mn, s.t. y- Θs ˆ s, (0) n.

3 he Proceedngs of the nd Internatonal Conference on Intellgent Green Buldng and Smart Grd (IGBSG) Alternatve prevalng CS reconstructon algorthms are greedy-type algorthms, such as orthogonal matchng pursut (OMP) [5]. Instead of computng optmzaton problem wth l norm, OMP algorthm draws pursung process nto solvng of sparse soluton. he pursung process wll dentfy the locaton of a non-zero term by teratvely calculates the correlaton of the resdual along wth each column of Θ. hen, OMP selects the column whch has the largest correlaton to chosen matrx Θ. hen, least-square (LS) process of OMP wll estmate sparse solutons by projectng observatons onto the subspace spanned by columns of chosen matrx Θ as = (ΘW ΘW )- ΘW y. he proposed greedy-lms algorthm has the property of nose reslence and low-complexty. he detaled procedures are descrbed n the followng subsectons. () he above mentoned CS reconstructon algorthms are tradeoff between nose-tolerance and low-complexty. BPDN algorthm s able to reduce mpact of nose but computatonal complexty s extremely hgh. OMP s low-complexty but vulnerable to measurement nose. Fg. 5. he flowchart of the proposed greedy-lms algorthm. B. Intalzaton and Iteratve Process In proposed greedy-lms algorthm, the pursung process of exstng greedy algorthm s adopted to dentfy the poston of non-zeros terms and correspondng columns n sensng matrx Fg. 4. Matrx formaton of CS reconstructon. III. Θ. In ntalzaton, estmated sparse s 0 s set to be N zero vector, t s the estmated sparse sgnal after tth teraton. On the other hand, 0 s set to be empty set, PROPOSED ROBUS GREDDY-LMS ALGORIHM A. Overvew of the Proposed Greedy-LMS Algorthm hs work s to develop a CS reconstructon algorthm wth robustness and low computatonal cost. Snce the LMS adaptve flter has the natural property of robustness, we combne LMS to exstng teratve greedy algorthm. he proposed algorthm s named greedy-lms. t denotes support locaton set n tth teraton. At last, resdual vector s ntalzed as r0 = y, rt s the resdual of tth teraton. Fg. 6 llustrates the dea of pursung process. Correlaton s appled to select the chosen matrx Θ, and the correspondng locaton n sgnal s assumed to be non-zero terms. he N correlaton vector Corr can be calculated as he flowchart of proposed greedy-lms algorthm s shown n Fg 5. he detaled steps are lsted as follows, Step ) Intalze the resduals r, support locaton set, and estmated sparse sgnal s. Step ) Calculate correlaton between r and Θ to dentfy the most correlated column of Θ. he column wll be updated to selected column set. Step 3) Derve the estmated sparse sgnal s through least mean square (LMS) process. Step 4) Update the resdual and calculate the square of norm of the resdual, whch s denoted by rnorm-square. Step 5) Check f termnaton condtons are reached or not. he process wll go to Step 3 f all of the condtons are not satsfed, or else output the reconstructed envronmental sgnal x. Corr = Θ r t. () hen, the pursung process wll select the maxmum element n Corr as w t + º arg max{ Corr (w ) }, wî[ N ] (3) t s the maxmum element. he new t wll be added to the support locaton set as W t + = W t È {w t + }. (4) We denote the sze of support locaton set W t + as L. Afterward, the chosen matrx Θ s composed accordng to 3

4 he Proceedngs of the nd Internatonal Conference on Intellgent Green Buldng and Smart Grd (IGBSG) t + the support locaton set W as Θ t { v j t j }, (5) v j s the jth column of Θ. he chosen matrx Θ, whch s M L, wll be nputted to LMS process to calculate the value of non-zero terms, whch t s denoted as % s. After LMS process, the resdual vector r and square of -norm of updated resdual r norm-square wll calculated as t t t r y Θ ˆs, (6) M normsquare r r, (7) t ŝ s L vector of non-zero terms, and r s th element n r. he resdual wll approach to zero vector, whch ndcates r norm-square beng zero. If maxmum teraton number s reached or r norm-square s less than predefned early termnaton threshold HR, the reconstructed envronmental sgnal ˆx wll be computed and outputted. Otherwse, greedy-lms algorthm wll go back to calculaton of correlaton and procedure terates. y = Θs = Θ s%, (8) s% s L vector of non-zero terms. he measurements y are regarded as desred sgnal, s% can be regarded as flter coeffcent vector, and Θ s the flter nput matrx. herefore, the estmaton error of LMS adaptve flter wth correspondng desred sgnal y can be formulated as W l = yl -Θlsl e % %, (9) Θ % denotes the th row of the chosen matrx Θ, and y denotes th element of measurements y. hen, the gradent decent recurson of the flter coeffcent vector s formulated as m s step-sze. s % s % Θ%, (0) l+ = l + m el l Fg. 7. Illustraton of LMS process. Fg. 6. Illustraton of pursung process. C. LMS Process We ntroduce stochastc gradent approach for robust compressve sensng because stochastc gradent approach has the natural property of denosng [6]. Least mean square (LMS) algorthm s a well-known stochastc gradent approach for low complexty and smplcty. he core dea of LMS s to mnmze the mean square error between the desred sgnal and the output of adaptve flter. he framework of LMS process of proposed algorthm s llustrated n Fg. 7. If all non-zero terms are dentfed after teratons, the equaton (6) becomes IV. PERFORMANCE ANALYSIS A. CS Reconstructon Performance We conduct a smulaton of CS reconstructon for CS wreless sensor networks, whch measurements are nosy. he proposed greedy-lms algorthm, BPDN algorthm, and OMP algorthm are smulated. We fx the N, M, K, and modfy the sgnal-to-nose rato (SNR) to observe the recovery performance. he smulaton setup s summarzed n able II, whch N, M, K are set accordng to [4]. he bass Ψ s set to be dscrete cosne transform (DC) matrx as [4]. Maxmum number of teratons of greed-lms s set to be M as OMP algorthm. he quantty of smulaton data s 0 4 for each pont of SNR. We employ the normalzed root mean square error (NRMSE) to evaluate the reconstructon performance as 4

5 he Proceedngs of the nd Internatonal Conference on Intellgent Green Buldng and Smart Grd (IGBSG) N Â( xˆ -x) N = () NRMSE =, max( x) - mn( x) x and xˆ are th element n x and ˆx. We regard a recovery as successful reconstructon f the NRMSE of the tral s less than 0-3. he successful rate (SR), -3 # of rals that NRMSE < 0 SR, (3) # of rals s defned as a performance ndex. A performance smulaton s gven n Fg. 8. he proposed greedy-lms algorthm outperforms exstng BPDN, and OMP algorthm. When SNR s 48dB, the SR of greedy-lms, BPDN, and OMP s 00%, 84%, and 53%, respectvely. herefore, the proposed greedy- LMS algorthm acheves 6% and 47% hgher successful rate than BPDN and OMP, respectvely. C. Analyss of Computatonal Complexty Snce OMP has been regarded as representatve lowcomplexty CS reconstructon algorthm, we would lke to analyze the computatonal complexty of OMP and greedy- LMS algorthm n terms of N, M, K, and L. able II presents the computatonal complexty of multplcatons of sngle teraton n both OMP and greedy- LMS algorthms. he dfference of OMP and SGP les on the LS process and LMS process. In teraton, the OMP requres a matrx nverson of L by L matrx, L s number of columns of ( Θ Θ ). When sze of chosen matrx s large, ABLE I. SIMULAION PARAMEERS Input length (N) 04 Measurement length (M) 56 Sparsty (K) 3 Bass ( Ψ ) DC HR 0-5 Maxmum number of teratons (ter max) 56 Sgnal-to-nose rato (SNR) 6::48 rals 000 Fg. 8. Successful rate of proposed greedy-lms, BPDN, OMP algorthm. the multplcatons requred by OMP ncreases n O(L ). On the other hand, the LMS process needs to calculate M (L+) multplcatons, whch s O(L). he requred multplcatons of greedy-lms are compettve wth OMP. Moreover, the multplcatons of greedy-lms are far less than OMP when L enlarges. ABLE II. COMPUAIONAL COMPLEXIY OF MULIPLICAIONS OF SINGLE IERAION OMP a V. CONCLUSION We are the frst to propose a novel greedy-type CS reconstructon algorthm based on LMS adaptve flter. Wth the natural nose-tolerance property of least mean square (LMS), proposed greedy-lms algorthm s not only robust but wth low-complexty. REFERENCES Proposed Greedy-LMS Correlaton NM NM LS/LMS Process M(L ) 3 L ( M ) L Resdual Update ML ML Haltng Condton M M a. he requred multplcatons s estmated by [0], whch appled Cholesky decomposton for matrx nverson. Besdes, the multplcatons n [0] accumulate form L=~K. [] I. F. Akyldz, W. Su, Y. Sankarasubramanam, and E. Cayrc, A survey on sensor networks, IEEE Comm. Magazne, vol.40, no.8, pp. 0-4, Aug. 00. [] E. J. Candes, and M. B. Wakn, An Introducton to Compressve Samplng, IEEE Sgnal Process. Mag., vol. 5, no., pp. -30, Mar [3] J. Luo, L. Xang, and C. Rosenberg, Does Compressed Sensng Improve the hroughput of Wreless Sensor Networks?, n Proc. IEEE Int. Conf. Commun., May 00, pp. -6. [4] S. L, L. Xu, and X. Wang, Compressed Sensng Sgnal and Data Acquston n Wreless Sensor Networks and Internet of hngs, IEEE rans. Ind. Informat., vol.9, no.4, pp , Nov. 03. [5] M.F. Duarte and Y.C. Eldar, Structured Compressed Sensng: From heory to Applcatons, IEEE rans. Sgnal Process., vol.59, no.9, pp , Sept. 0. [6] S. Haykn, Adaptve Flter heory. 3rd ed. Englewood Clffs, NJ: PrentceHall, 996. [7] E.J. Candes and. ao, Near-Optmal Sgnal Recovery From Random Projectons: Unversal Encodng Strateges?," IEEE rans. Inf. heory, vol.5, no., pp , Dec. 006 [8] E. Candes and. ao, Decodng by lnear programmng, IEEE rans. on Informaton heory, vol. 5, no., pp , Dec [9] R. H. Kwong and E. W. Johnston, A varable step sze LMS algorthm. IEEE rans. Sgnal Process., vol. 40, no. 7, pp , Jul. 99. [0] H. Rabah, A. Amra, B. K. Mohanty, S. Almaadeed, and P. K. Meher, FPGA Implementaton of Orthogonal Matchng Pursut for Compressve Sensng Reconstructon, IEEE ran. on Very Large Scale Integr. Syst., vol.3, no.0, pp. 09-0, Oct