Major Coefficients Recovery: a Compressed Data Gathering Scheme for Wireless Sensor Network

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1 This full text paper was peer reviewed at the directio of IEEE Commuicatios Society subject matter experts for publicatio i the IEEE Globecom proceedigs. Major Coefficiets Recovery: a Compressed Data Gatherig Scheme for Wireless Sesor Network Liwe Xu, Yuexua Wag, Yogcai Wag Istitute for Iterdiscipliary Iformatio Scieces Tsighua Uiversity, Beijig, 8 kyoi.c@gmail.com, {wagyuexua,wagyc}@mail.tsighua.edu.c Abstract For large-scale sesor etworks deployed for data gatherig, eergy efficiecy is critical. Elimiatig the data correlatio is a promisig techique for eergy efficiecy. Compressive Data Gatherig (CDG) [8], which employs distributed codig to compress data correlatio, is a importat approach i this area. However, the CDG scheme uses a uiform patter i data trasmissio, where all odes trasmit the same amout of data regardless of their hop distaces to the sik, makig it iefficiet i savig trasmissio costs i -D etworks. I this paper, the Major Coefficiet Recovery (MCR) scheme is proposed, where the Discrete Cosie Trasformatio (DCT) is applied i a distributed fashio to the origial sesed data. A o-uiform data trasmissio patter is proposed by exploitig the eergy cocetratio property of DCT ad QR decompositio techiques so that sesors with larger hop-cout ca trasmit fewer messages for etwork eergy efficiecy. The sik ode recovers oly the major coefficiets of the DCT to recostruct the origial data accurately. MCR reduces the trasmissio overhead to O(k k ), a improvemet by O(log ) over CDG i both -D ad -D cases. The recovery performace of MCR is verified by extesive simulatios. I. INTRODUCTION Oe of the major objectives of wireless sesor etworks (WSN) is to collect iformatio of physical pheomeo withi a large-scale area. Eergy efficiecy ad accurate observatio of physical iformatio are two goals of such data gatherig etworks. However, it is difficult to acheive both these goals at the same time, because accurate observatio geerally requires massive data gatherig, posig great challeges to eergy efficiecy. Cosiderig the fact that may physical measuremets are strogly correlated to the oes earby [], elimiatig data correlatio is a promisig techique for eergy efficiet data collectio. Such approaches [8][] solve the wireless etwork data gatherig problem by a distributed codig framework. y = Φd = ΦΨ x () where d R is a vector of odes origial readigs that ca be compressed. Ψ R is a selected basis, which maps d to a sparse represetatio, i.e., i d = Ψx. After compressio, x R is usually sparse. Φ is the trasmissio patter matrix i data gatherig etworks, which determies how sesors trasmit their data to the sik. y is the observatio at the sik. The objective is to accurately recover x from y. For eergy efficiecy ad accurate observatio purposes, the problem i the above framework are: ) How to select Ψ appropriately, so that the origial observatios ca be compressively coded, ad ) How to optimize the desig of Φ, so that the trasmissios i a etwork ca be reduced to prolog the etwork s lifetime. Some work has bee doe to address the above problems. Sice sesors work distributively, it is ecessary for sesors to carry out compressive codig ad trasmissio i a distributed fashio. Distributed Source Codig (DSC) was proposed i [], where multiple correlated sesors compress their data distributively ad sed the compressed outputs to a cetral poit for joit decodig. Based o Slepia-Wolf codig theory, [] showed that distributed ecodig ca achieve the same efficiecy as joit ecodig ca. This results esure that the framework () ca be carried out by distributed sesors. Furthermore, for the selectio of Ψ, Discrete Fourier Trasform (DFT), Discrete Cosie Trasform (DCT) ad Wavelet Trasform[] is geerally proposed as a trasformatio basis. For the optimizatio of Φ, Yoo et al. [] proposed a clustered aggregatio techique that first groups sesor odes accordig to their measuremets ad trasmits similar measuremets per group oly oce i the process of data gatherig. [][5] proposed a scheme to ecode relayed data for data gatherig. Most related to our work, a compressive data gatherig (CDG) scheme was proposed by [8] for collectig data i tree structure etworks. Sesors observatios are projected usig radom coefficiets i Φ allowig each sesor to trasmit the same umber of messages, regardless of their hop distaces to the sik. Such a uiform trasmissio patter balaces the loads of the sesors, but this results i a trasmissio overhead of O(k log ): the same order as direct data collectio takes without compressive sesig (o-cs) i -D etworks. This is because whe compared with o-cs, CDG assigs more work/heavier load for odes further away from the sik ad assigs less work/ligher load for odes closer to the sik. I - D etworks, sesors far away from the sik are more umerous tha the sesors close to the sik, makig CDG iefficiet. [9] proposed hybrid compressive sesig (Hybrid-CS) to address such iefficiecy, i which, a o-cs scheme is applied i the earlier stages of data collectio (startig from the leaf odes), ad CDG is applied to odes whose icomig traffic becomes greater tha or equal to m, the umber of rouds //$. IEEE

2 This full text paper was peer reviewed at the directio of IEEE Commuicatios Society subject matter experts for publicatio i the IEEE Globecom proceedigs. d d m d d d d d md md S S S S i,id i d,i i m id, Sik ode Fig. : Illustratio of the relatio betwee the trasmissio patter ad distributed source codig of relay. But such a hybridized method eeds to optimize the switchig threshold ad is ot a uified framework for CSbased gatherig. Furthermore, CDG ad Hybrid-CS do ot fully exploit the features of the sparse sigal, which leaves space for further reductio of the umber of trasmissios. I this paper, we exploit the eergy cocetratio property of DCT-IV, ad propose Major Coefficiet Recovery (MCR) scheme to further reduce the umber of trasmissios at each sesor. I additio, we propose a QR decompositio-based algorithm to costruct a o-uiform trasmissio patter matrix for MCR, so that sesors far away from the sik ca trasmit fewer messages. MCR reduces the trasmissio overhead i -D etwork to O(k k ), which is a improvemet by O(log ) over CDG. To improve recovery accuracy, Exteded MCR is also proposed. The recovery performaces of MCR are verified by extesive simulatios. The structure of this paper is orgaized as follows: Sectio II itroduces the impact of trasmissio patter matrices ad desig priciples. Sectio III itroduces the properties of DCT- IV, the proposed MCR scheme ad the methods used i Exteded MCR. Sectio IV presets some simulatio experimet results. Sectio V cocludes thepaper with some discussio of future work. II. TRANSMISSION PATTERN AND OVERHEAD A. The relatio betwee the trasmissio patter matrix ad trasmissio overhead We defie trasmissio overhead as the umber of trasmissios. A trasmissio patter matrix determies how readigs are coded ad how they are trasmitted i the etwork. I DSC, the origial readigs are ecoded at each ode, ad the liearly combied with the data at each hop of the relay. Figure illustrates the trasmissio patter of a typical DSC i -D etworks. s has weight ϕ ad trasmits the product ϕ d to ode s, where d is the origial measuremet of s. The s trasmits the sum ϕ d +ϕ d to s. At last, the sik ode recieves i= ϕ id i from s, i.e., a liear combiatio of the measuremets of each ode. Repeatig this for m rouds the sik ode recieves a vector of combied measuremets y show i (). y = Φ m d = Φ m Ψ x () The trasmissio patter matrix determies the trasmissio overhead of a data gatherig etwork. I oe-dimesioal cases: Each row of the trasmissio patter matrix correlates to a sigle roud of trasmissio. The umber of tras- 5 Sik ode DSC with m=. The odes are labeled by the trasmissio orders. 5 Sik ode o-cs data collectio.the odes are labeled by the trasmissio orders. Fig. : Compariso of DSC ad o-cs data collectio. The umbers beside the liks are the umber of trasmissios missios i this roud is equal to the umber of etries that come after the first o-zero etry of this row. For example, if ϕ i = ϕ i =... = ϕ ij =, the i the i-th roud, odes s,,s j do t eed to trasmit data. The relatio i two-dimesioal cases is more complicated. However, uder some specific settigs, we ca derive the relatio similar to the oe i -D cases. Cosider the tree topology etwork as show i Figure. Each ode is labeled with a umber, which implies the order of its relay. If it ca be esured that a ode closer to the sik will ot be labeled with a umber smaller tha the oes further from the sik, the relatio betwee the trasmissio patter matrix ad the trasmissio overhead i -D etworks still holds for -D tree structured etworks. It is crucial to desig a suitable trasmissio patter matrix so that the etwork is eergy efficiet. We ca summarize two priciples i desigig the trasmissio patter: Fewer rows. Fewer rows meas fewer rouds of relay, which greatly reduces the total umber of trasmissios. More leadig zeros for each row. Leadig zeros i a row meas fewer trasmissios i this roud of relay. B. Trasmissio overhead of CDG aalysis CDG scheme proposed i [8] performs distributed codig, ad employs compressive sesig techique to recostruct origial readigs. The trasmissio patter Φ i CDG is a m radom matrix with each etry obeyig a ormal distributio, satisfyig the requiremet of compressive sesig that Φ exhibits the restricted isometry property with high probability []. Ψ is chose to be a DCT basis. Assumig the sparsity of x is k, oly if m ck log, c is a positive costat, ca x be recostructed with high probability accordig to the theory of compressive sesig []. So CDG oly adopts the first desig priciple to optimize the trasmissio patter matrix. The trasmissio overhead of CDG ca be evaluated as follows: Theorem. The trasmissio overhead of CDG is at least of the order O(k log ) if the topology of the etwork is a chai of odes or a tree with odes labeled by the trasmissio order as described above. Proof: ThesizeofΦ is m which shows the umber of trasmissios is O(m). Sice m ck log, m O(k log ). Because the trasmissio overhead of o-cs data collectio is O( ) i -D etwork ad is O( log ) i -D etworks[7], CDG reduces the trasmissio overhead i -D

3 This full text paper was peer reviewed at the directio of IEEE Commuicatios Society subject matter experts for publicatio i the IEEE Globecom proceedigs Fig. : A Markov-sequece-sigal ad its DCT etworks efficietly, but has the same order of overhead as the o-cs collectio i the -D etworks. III. MAJOR COEFFICIENTS RECOVERY Whe we defie MCR as a extesio of the distributed codig framework (), the the first several major coefficiets of x ca be recovered: y = Φ k d = Φ k Ψ x = A k x () where k is the umber of major coefficiets that MCR tries to recover. Note that the size of trasmissio patter matrix is k, where k rouds of relay are carried out. Ψ i MCR is a basis of Discrete Cosie Trasform type-iv (DCT-IV), whose properties of decorrelatio ad cocetratio allow MCR to achieve a accurate recostructio by recoverig oly very few first coefficiets of x. A. Eergy Cocetratio Property of DCT-IV There are several differet kids of DCT defiitios, ad DCT-IV is cosidered i this paper for its orthogoality, eergy cocetratio ad simplicity of defiitio. The trasform matrix of a -D DCT-IV (of legth ) is defied as Ψ = {ψ ij }, i, j [ ( π ψ ij = cos i + )( j + )] From the symmetry ad orthogoality of Ψ, we kow Ψ = Ψ T = Ψ () DCT exhibits two importat properties: ) Decorrelatio which ca trasform the correlated sigal ito urelated coefficiets that are usually sparse. ) Eergy cocetratio. The DCT coefficiets are the combiatios of cosie values of differet frequecies. [][] illustrated that DCT coefficiets ted to obey a Laplacia distributio i most practical cases. As k, i.e., the frequecy, gets higher, the scale parameter b of a Laplacia distributio gets smaller, which meas the correspodig DCT coefficiet has a greater probability of takig o a very small value, eve zero. Figure is the DCT represetatio of a typical sigal. Nearly 95% of the total eergy is cocetrated withi the first % of coefficiets. B. Trasmissio Patter Matrix Desig i MCR To address the eergy cocetratio property of DCT, we observe that the k-lowest frequecy coefficiets i x domiate the expressio of Ψ x. Therefore, a Major Coefficiet Recovery scheme is proposed to recover d usig oly the first k, i.e., the major coefficiets, of x, which substatially reduces the trasmissio cost. The k matrix A ca be divided ito two submatrices of size k k ad k ( k) deoted by A k, A r i (5). y =(A k A r ) x (5) If A r is further made to be, (5) becomes y =(A k ) x () O the coditio that A k is full rak, () turs out to be a liear system of equatios whose solutio is exactly the first k coefficiets of x. The oly requiremet to esure is that matrix A cosists of a k k full-rak matrix ad a zero matrix. Because Ψ is fixed as log as the umber of odes is fixed, we ca oly maipulate the trasmissio patter for Φ i order to esure that A takes o the proper form. We propose a algorithm which ca guaratee this. Algorithm Trasmissio Patter Matrix Desig i MCR Iput:, the umber of odes i a lie Output: Φ, the trasmissio patter Step. Calculate Ψ =(ψ ij ), the DCT-IV basis of legth, where ψ ij = cos [ ( )( )] π i + j + Step. LetΨ u be a k submatrix cosistig of the first k rows of Ψ Step. Do QR decompositio to Ψ u, therefore Ψ u = Q k k R k, where Q is a orthogoal matrix ad R is a upper triagular matrix Step. LetΦ = R Claim. Φ geerated by Algorithm guaratees the successful recovery of the k major coefficiets of x. Proof: From the algorithm we kow, Ψ u = QR = QΦ Φ = Q Ψ u = ( Q ) ( Ψ u Ψ l where Ψ l is a ( k) submatrix cosistig of the last k rows of Ψ. Sowehave, Φ = ( Q ) Ψ ΦΨ = ( Q ) Sice Ψ is symmetric ad orthogoal as was preseted i (), we have ( Q ) = ΦΨ = A )

4 This full text paper was peer reviewed at the directio of IEEE Commuicatios Society subject matter experts for publicatio i the IEEE Globecom proceedigs. 8 TABLE I: MSE ad trasmissio overhead compariso EXP # MSE Trasmissio Overhead CDG MCR ExMCR CDG MCR/ExMCR Fig. : Recovery of d i MCR ad exteded MCR A is defied i (5), so we ca see that Φ guaratees that A cosists of a k k full-rak matrix ad a zero matrix. Put aother way, the sik is to solve the liear system of equatios (Q )x = y (7) C. Exteded MCR MCR recovers the major coefficiets i DCT while igorig the compoets of high-frequecy. Though the value of highfrequecy coefficiets are usually very small, their loss may lead to divergece of recovery i the latter part of d as is showifigure. Sice the former part of d ca be recovered accurately, we would be able to achieve a good global recovery accuracy if oly former part could be exteded. This ca be achieved by isertig virtual odes betwee the last ode (the ode ext to the sik) ad the sik ode. Accordigly, virtual measuremets ca be geerated ad subsequetly collected by the sik ode. I the recovery process, the sik ode ca the simply throw out the virtual parts ad keep the real measuremets, which will improve the recover accuracy of MCR. I practice, the simplest way to geerate these virtual measuremets is to repeat the last ode s readig, oe for each virtual ode we require. Figure gives a example of this method which shows the icrease i recovery accuracy. D. Trasmissio Overhead of MCR The trasmissio patter matrix of MCR is a k upper triagular matrix. ϕ ϕ ϕ k ϕ k ϕ ϕ ϕ k ϕ k ϕ ϕ k k ϕ k k ϕ k ϕ kk ϕ k Theorem. The umber of trasmissios is at least of the order O(k k ) if the topology of the etwork is a chai of odes or a tree with odes labeled the trasmissio order as described i Sectio II-A. Proof: Φ, the trasmissio patter of MCR, is a k upper triagular matrix. If i<j,wehaveϕ ij =.Soithejth roud of data trasmissio, the odes {,,..., j } do ot eed to trasmit data. Therefore, the umber of trasmissios is the umber of o-zero etries i Φ, i.e. k k(k ) O(k k ). Compared with CDG, MCR is much more eergy efficiet. I both -D etworks ad -D tree structured etworks, MCR s trasmissio overhead is of the order O(k k ),ao(log ) improvemet over CDG. IV. EXPERIMENTS AND ANALYSIS Our experimets are desiged to compare the recovery accuracy ad trasmissio overhead of MCR ad the exteded MCR (ExMCR for short) with CDG. The recovery accuracy is measured usig Mea Square Error (MSE). Accordig to the origial sigal types, the experimets are separated ito sets: polyomial sigal, polyomial sigal with white oise ad Markov Chais. Table I shows the results of these experimets Fig. 5: EXP #: Polyomial-sigal Fig. : EXP #: Polyomial-sigal with oise N(,.) I the first experimets, CDG, MCR ad ExMCR all recover out of DCT coefficiets. I the polyomial cases, CDG, MCR ad ExMCR provide very accurate recoveries while MCR ad ExMCR perform slightly better tha CDG. I

5 This full text paper was peer reviewed at the directio of IEEE Commuicatios Society subject matter experts for publicatio i the IEEE Globecom proceedigs. 8 8 Fig. 7: EXP #: Markov-sequece-sigal CDG Recovery Fig. 8: EXP #: Polyomial-sigal recovered at differet accuracy settigs experimet #, the origial sigals are geerated by Markov Chais. The experimet is repeated 5 times ad the average values of MSE are preseted. We ca see MCR ad ExMCR also perform better i this experimet. I exprimet #, a sigal of samples is geerated by a polyomial fuctio, ad CDG tries to recover 5 out of coefficiets ad MCR/ExMCR tries to recover. As show i Figure 8, CDG recovery diverges from the origial sigal throughout, MCR begis to diverge i the latter part ad ExMCR recovers better tha both. The result of MSE also supports this observatio. We summarize the results of our experimets ad highlight two poits: More accurate. MCR ad ExMCR provide accurate recoveries as good as or better tha CDG i most cases at the same recovery settig. Less overhead. I CDG, 5 rouds of trasmissio are carried out i order to meet the requiremet of compressive sesig, while i MCR, there are oly rouds of trasmissio, i.e. equal to the umber of DCT coefficiets we choose to recover. The umber of trasmissios i CDG is 5, imcris955, i.e. 9.% of the umber of trasmissios i CDG. This overhead ratio is 9.78% i experimet #. property of DCT. We further propose Exteded MCR i order to avoid the divergece i data recovery that MCR suffers. Compared with CDG, the umber of trasmissios is greatly reduced i MCR. The trasmissio overhead is O(k k ) i both -D etworks ad -D tree structured etworks, a O(log ) improvemet over CDG. I the future, this work ca be exteded by distributig the DCT trasformatio ad explorig the ode umberig strategy i a greater umber of -D topologies. ACKNOWLEDGMENT This work is supported i part by the Natioal Basic Research Program of Chia Grat 7CB879, 7CB879, the Natioal Natural Sciece Foudatio of Chia Grat 77,, 5, ad the Hi- Tech research ad Developmet Program of Chia Grat AAZ. REFERENCES [] E. Cadès,T.Tao.Near-optimal sigal recovery from radom projectios ad uiversal ecodig strategies. IEEE Tras. Iform. Theory 5:89-59,. [] E. Cadès. Compressive samplig. I Proc. of ICM,. [] A. Ciacio, S. Pattem, A. Ortega, ad B. Krishamachari. Eergyefficiet data represetatio ad routig for wireless sesor etworks based o a distributed wavelet compressio algorithm. I Proc. of IPSN, pages 9-,. [] R. Cristescu, B. Beferull-Lozao, ad M. Vetterli. O etwork correlated data gatherig. I Proc. of IEEE Ifocom, volume, pages 57-58, Mar.. [5] R. Cristescu, B. Beferull-Lozao, M. Vetterli, ad R. Wattehofer. Network correlated data gatherig with explicit commuicatio: Npcompleteess ad algorithms. IEEE/ACM Tras. o Networkig, ():-5, Feb.. [] E. Y. Lam ad J. W. Goodma, A Mathematical Aalysis of the DCT Coefficiet Distributios for Images, IEEE Tras. Image Process., vol. 9, o., pp. -, Oct.. [7] X. Li, Y. Wag, ad Y. Wag, Complexity of Data Collectio, Aggregatio, ad Selectio for Wireless Sesor Networks, IEEE Tra. o Computers, vol. o., pp. 8-99, [8] C Luo, F Wu, J Su, Compressive Data Gatherig for Large-scale Wireless Sesor Networks, i Proc. ACM Mobicom 9, pp. 5-5, Sep. 9. [9] J. Luo, L. Xiag, ad C. Roseberg, Does Compressed Sesig Improve the Throughput of Wireless Sesor Networks?, ICC 9, pp.-, 9 [] R. C. Reiiger ad J. D. Gibso, Distributios of the Two-dimesioal DCT Coefficiets for Images, IEEE Tras. Commu., vol. COM-, o., pp , Ju. 98. [] D. Slepia ad J. K. Wolf. Noiseless ecodig of correlated iformatio sources. 9:7-8, Jul. 97. [] M. C. Vura, Ö. B. Aka, ad I. F. Akyildiz, Spatio-temporal Correlatio: Theory ad Applicatios for Wireless Sesor Networks, Computer Networks Joural (Elsevier), vol. 5, o., Jue. [] Z. Xiog, A. Liveris, ad S. Cheg, Distributed source codig for sesor etworks, IEEE Sigal Processig Mag., vol., pp. 8C9, Sept.. [] S. Yoo ad C. Shahabi. The Clustered Aggregatio (CAG) Techique Leveragig Spatial ad Temporal Correlatios i Wireless Sesor Networks. ACM Tras. o Sesor Networks, (), Mar. 7. V. CONCLUSION I this paper, we propose the Major Coefficiets Recovery, or MCR, scheme for data gatherig i wireless sesor etwork. MCR oly recovers the first k coefficiets that are cosidered as major because they are likely to be larger tha the rest with high probability accordig to the eergy cocetratio