Research Article Capacity of Data Collection in Wireless Sensor Networks Based on Mutual Information and MMSE Estimation

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1 Hndaw Publshng Corporaton ISRN Sensor Networks, Artcle ID 38945, 9 pages Research Artcle Capacty of Data Collecton n Wreless Sensor Networks Based on Mutual Informaton and MMSE Estmaton Ajb Setyo Arfn and Tomoak Ohtsuk Graduate School of Scence and Technology, Keo Unversty, 3-4- Hyosh, Kohoku-ku, Yokohama 3-85, Japan Correspondence should be addressed to Tomoak Ohtsuk; ohtsuk@cs.keo.ac.jp Receved 4 December 03; Accepted 30 December 03; Publshed 7 February 04 Academc Edtors: T.-S. Chen, J., and Y. Yu Copyrght 04 A. S. Arfn and T. Ohtsuk. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton cense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. We nvestgate the propertes of data collecton n wreless sensor networks, n terms of both capacty and power allocaton strategy. We consder a scenaro n whch a number of sensors observe a target beng estmated at fuson center (FC usng mnmum meansquare error (MMSE estmator. Based on the relatonshp between mutual nformaton and MMSE (I-MMSE, the capacty of data collecton n coherent and orthogonal multple access channel (MAC models s derved. Consderng power constrant, the capacty s derved under two scenaros: equal power allocaton and optmal power allocaton of both models. We provde the upper bound of capacty as a benchmark. In partcular, we show that the capacty of data collecton scales as Θ((/log( + when the number of sensors grows to nfnty. We show through smulaton results that for both coherent and orthogonal MAC models, the capacty of the optmal power s larger than that of the equal power. We also show that the capacty of coherent MAC s larger than that of orthogonal MAC, partcularly when the number of sensors slargeandthetotalpowerp s fxed.. Introducton Wreless sensor networks (WSNs consstng of a large number of nodes are usually deployed n a large regon for many applcatons, such as survellance, securty, and envronmental montorng. The goal of a sensor network s often to delver the sensng data from all sensors to a fuson center (FC and then conduct further analyss at the FC. Thus, data collecton s mportant n sensor network applcatons []. Theoretcal measure that captures the lmts of collecton processng n sensor network s the capacty of data collecton. Capacty of data collecton reflects how fast FC can collect sensng data from all sensors []. Understandng the capacty of the network s mportant for network desgners n a feasblty of a large scale network deployment [3], partcularly, to mprove the performance of WSNs []. Furthermore, such understandng s essental n the development of effcent protocols [4]. Capacty lmts of data collecton n wreless sensor networks have been studed n the lterature [ 0]. In [4, 5], they ntroduced the transport capacty of many-to-one n dense sensor networks. The authors n [6, 7] nvestgated the capacty of data collecton wth complex physcal layer technques. The capacty that nvolves multple selected sources and destnaton has been studed n [8]. The capacty of data collecton of sngle and multsnks (FC s nvestgated [9]. In [], the authors derve capacty of data collecton n arbtrary WSNs. A data collecton capacty that consders delay and compressve sensng has been, recently, nvestgated n [0]. Most of the lterature resources calculate capacty based on ether the physcal models or the protocol model. Physcal model also known as the sgnal-to-nterference-plus nose rato (SINR model, s based on practcal trancever desgns of communcaton system that treats nterference as nose. Further, capacty calculaton s based on Shannon s formula. The other model s the protocol model. The model states that a successful transmsson occurs when a sensor falls nsde the transmsson range of ts ntended transmtter and falls outsde the nterference ranges of other nonntended transmtters. However, the protocol model s relatvely naccurate, when smultaneous transmssons are allowed n the network [3, ]. Study of dstrbuted estmaton n WSNs s one of the nterestng topcs that many researchers are workng on. Some of the results are lsted n [ 3]. Some lterature

2 ISRN Sensor Networks addresses dgtal sensor transmsson, where the nosecorrupted sensor observatons are quantzed nto bts and dgtally transmtted to the FC [ 8]. In [9 3], they consder analog sensor transmsson, where the sensors amplfy andforwardtheobservatonstothefc,andtheperformance of estmaton s generally better than that of the dgtal transmsson. Dstrbuted estmaton by consderng MAC model has been consdered n [4, 5]. They reveal that dstrbuted estmaton usng the coherent MAC s more bandwdth effcent than the orthogonal MAC. Another mportant property of many WSNs s ther strngent power constrant. In such networks, sensors have only small-sze batteres whose replacement can be costly. Thus, sensor network operatons must be energy effcent to maxmze network lfetme. However, there are only a few authors that consder power constrant n dervng the capacty of data collecton. In [6], they characterzed the transport capacty of many-to-one dense wreless networks subject to a constrant on the total power. The energy effcency and data latency are consdered n [3] fordesgnng data gatherng capacty. However, they stll do not provde how to allocate the power optmally. In ths paper, we focus on dervng capacty of data collecton for random networks under coherent and orthogonal MAC scenaro based on equalty of mutual nformatonmnmum mean-squared error estmaton (I-MMSE. We provde a new perspectve of capacty calculaton of data collecton n WSNs that can be derved from error estmaton of the target at the FC. The relatonshp between mutual nformaton and MMSE has been revealed by Guo et al. n [6]. Frst, we derve a capacty formulaton on coherent MAC model. In coherent MAC model, we assume that there s perfect synchronzaton between sensors and the fuson center so that the transmtted messages from local sensors can be coherently combned at the fuson center. Wth such an assumpton, one key desgn consderaton at local sensors and the fuson center s how to jontly process the sensed andrecevednformatonntermsofcapacty.wewrte a problem formulaton for maxmzng the capacty and then solve t through convex optmzaton technque. We derve the optmal power allocaton strategy to maxmze the capacty. The upper bound on the capacty of data collecton wth coherent MAC model s also derved as a benchmark. Second, we derve a capacty formulaton on orthogonal MAC scenaro. The motvaton for usng orthogonal multple access schemes such as Frequency Dvson Multple Access (FDMA s the removal of the requrement on the carrer level synchronzaton among sensors [5]. As the coherent MAC model, we also derve optmal power allocaton strategy for thecasewherethecapactysmaxmzedundercertanpower constrants. In the orthogonal model, the optmal power allocaton s acheved by turnng off certan sensors wth bad channels and bad observaton qualty. The upper bound on ths model s also derved and nterestngly equal to the upper bound on coherent one. The rest of the paper s organzed as follows. Secton descrbes the prelmnary theory and system model. In Secton 3, we formulate the capacty of data collecton for the upper bound, equal power allocaton, and optmal power allocaton n coherent MAC model. In Secton 4 we formulate the capacty of data collecton for the equal power allocaton, optmal power allocaton, and the upper bound on orthogonal MAC model. Secton 5 presents some smulaton results and concluson s drawn n Secton 6.. Problem Formulaton As a prelmnary, we start by explanng the relatonshp between mutual nformaton and MMSE [6]... Capacty of the Gaussan Channel Based I-MMSE Approach. An nput-output model can be wrtten as Y= snrx+n, ( where N N(0, s standard Gaussan. We note here that snr n ( concdes wth the usual notaton of sgnal-to-nose power rato (SNR only f E[X ]=. Then, we refer to snr as SNR regardless of the nput power. The MMSE of estmatng the nput X of the model gven the nosy output Y can be denoted by mmse (X, snr = mmse (X snr +N = E [(X E [X snr + N] ], where E[ ] s the expected value. The MMSE can be regarded as a functon of SNR for every gven dstrbuton P X.Partcularly,fX N(m, σ X,the MMSE can be denoted by mmse (X, snr = (3 +σx snr. Moreover, smple quanttatve connectons between MMSE andnformatonmeasuresarerevealedn[6]. One of the results s d I (X; Y = mmse (X, snr (4 dsnr for every snr 0. The correspondng capacty of the model s σ X C=max I (X; Y = snr mmse (X, snr dsnr 0 = log ( + σ X snr, whereweadoptnaturallogarthmsandusenatsastheuntof allcapactymeasures... System Model. Suppose that there are sensors, each makng observaton on a common unknown parameter s as n Fgure. The sensors observe s wth nosy observaton n that has zero mean and varance, σ n. We assume the sensor and FC communcate wth coherent MAC. When source and ( (5

3 ISRN Sensor Networks 3 Source s Sensors n a x n a x. n a x g g g Fuson center y ŝ avalable n Appendx A. Applyng (5, we can express the capacty of data collecton as follows: C upp = SNR tot J 0 dγ 0 = SNR tot 0 +γ dγ ( Fgure : System model of the capacty of data collecton n WSNs. observaton are scalars, the observaton model can be wrtten as x =s+n,. (6 Suppose that the correspondng analog amplfyng and forwardng scheme s used; we have a power amplfcaton factor a of th sensor. The average transmt power of sensor s P =a (σ s +σ n =a ( + α, whereweassumethatσ s =. α =/σ n s SNR observaton of sensor. After amplfcaton, sgnals are transmtted to the FC. The receved sgnal at FC s y= = g a x + V g a s+ g a n + V, where g and V are channel gan and channel nose, respectvely. Smlarly, V sassumedtohavezeromeanandunt varance, σ V. The lnear MMSE estmator of s from y s s = (E[sy]/E[y ]y,whchhasmse,j,satsfyng[4] J =+(+ g a ( 3. Capacty of Data Collecton n Coherent MAC Model (7 (8 g a α. (9 3.. Upper Bound on Capacty of Data Collecton. Note that when all sensor observatons x = [x,...,x ] T are drectly avalable to the FC, a centralzed estmator s 0 = (E[sx]/E[x ]x,achevesanmse,j 0,as α =+ J 0 =+γ, (0 where α and γ are SNR observaton of sensor and the total SNR observaton, respectvely. Analytcal proof s also = log ( + SNR tot. Because of the randomness of sensors deployment, we assume that nosy observaton becomes..d., wth σ n,...,σ n = σ n. Then, the upper bound of the capacty of data collecton can be expressed as C upp = log ( + σ n. ( Wthout loss of generalty, we can express the capacty of the network scaled by Θ((/ log( + as the number of sensors becomes nfnty,. 3.. Capacty of Data Collecton for Equal Power Allocaton. Suppose all sensors use the same transmt power, P =P/, where P s the total transmt power. From (7, we get a = P/(α +.etj u (P denote the acheved MSE wth equal transmt power. From (9, J eq (P satsfes J eq (P =+( P + g α + ( α α + g. (3 Wth the same analogy n (0, we defne (/P + (g /(α + ( (α /(α + g as a total SNR of the system. Therefore, we can express the capacty of equal transmt power as follows: C eq (P = log ( + ( P + g α + ( α α For P, we can wrte the capacty as + g. C eq ( = log ( + ( g α + C upp. ( α α + g (4 (5

4 4 ISRN Sensor Networks We can summarze the results on (, (4, and (5 where each sensor uses exactly the same transmt power of P/.We can express for every fnte P as C upp C eq ( C eq (P. ( Capacty of Data Collecton for Optmal Power Allocaton. Here, we consder an optmal power allocaton whereby the transmt power s optmally allocated among the sensors to acheve the maxmum capacty. From the rght hand sde (RHS of (9, we denote by β = ( + g a ( g a α a total SNR. Therefore, we can easly express the capacty as follows: C= log ( + ( + g a ( = log ( + β. g a α (7 et C(P,...,P denote the capacty acheved by optmally assgnng P to sensor. Maxmzng the capacty under a sum power constrant can be wrtten as max P ; s.t. C(P,...,P a (α + P,. (8 Maxmzng capacty n (8 s equvalent to maxmzng the total SNR as follows: max β(a a ;,...,a s.t. a (α + P,. (9 Wth the ad of Appendx B that follows the soluton n [4], we get the best achevable total SNR as β opt = +(α +/(g (0 P. The optmal power allocaton achevng the optmal total SNR s P opt =c P,, ( where c =c g α (α + (α ++g, P c=( α g α (α + (α ++g P. ( Implementng optmal power allocaton, we need the FC to broadcast the constant c and P tothesensors.thesensorsuse c, P, and two local parameters, g and α, to determne ther ndvdual transmt power. Therefore, we can express the optmal capacty of data collecton as C opt = log ( + β opt = log ( + +(α 4. Capacty of Data Collecton n Orthogonal MAC Model α +/(g P. (3 In ths secton, we adopt orthogonal channels between the sensors and the FC. We assume that the observed sgnal s analog and the observaton noses are uncorrelated across sensors. In addton, we assume that the second moments of the sgnal and nose are known to the correspondng sensor and the FC. The FC deploys the MMSE estmator to generate estmates of the unknown sgnal. In ths settng, we use an analog transmsson system where observatons are amplfed andforwardedtothefc. Suppose that the receved sgnal of orthogonal MAC from sensor tofccanbewrttenas y =g a s+g a n + V, (4 where V and g are the channel nose wth zero mean and unt varance of channel and channel gan, respectvely. For MMSE estmaton, we can get an MSE, J,[4]as J =+ g a α +g a. (5 4.. Capacty of Data Collecton for Equal Power Allocaton. For equal power method, P =P/;thuswehavea =P /( +.Bychangngtheformof(5, we get α J =+ α +(+α /(g P. (6 Followng the expresson of (5 and (α /( + ( + α /(g P asatotalsnrofthesystem,wecanwrtethe capacty as C eq = log ( + α +(+α /(g P. (7

5 ISRN Sensor Networks 5 For P,wecanwrteanupperboundonthe capacty of data collecton n orthogonal MAC as C upp = log ( + α = log ( + σ n =C upp. (8 Interestngly,wecanseethattheupperboundoncapactyof datacollectonfororthogonalmacandthatforthecoherent MAC are equal. 4.. Capacty of Data Collecton for Optmal Power Allocaton. To maxmze the capacty of data collecton on orthogonal MAC under optmal power method, frst, we need to mnmze the MSE under total power constrant, P. TheMSEfor optmal power method of the orthogonal MAC for the case of scalar source and observatons s gven n [4] Capacty (nats/s/hz P (db Upper bound coherent MAC Equal power coherent MAC (unlmted power Equal power coherent MAC Optmal power coherent MAC J =+ α ( +α c 0 g, (9 where c 0 and arethethresholdofg /( + α /c 0 whetherasensortransmtsorkeepsslentandthenumber of actve sensors, respectvely. The threshold c 0 s defned by c 0 = ( m= (α m (α m +/g m +P. ( m= (α m α m +/ g m (30 Followng (5, we can express thecapacty as, C opt = log ( + α ( +α c 0 g. (3 We note that the optmal power method for orthogonal MACwllallocatemostofpowertosensorsthathavegood observaton and channel qualtes. Hence, the actve sensors are sensors that have good observaton and channel qualtes. 5. Smulaton Results In Fgure, we plot the curves of capacty of data collecton for coherent MAC model versus total transmt power P n db (relatve to the channel nose power wth the number of sensors =0. In the smulaton, sensor observaton nose varance s set as σ n = 0.5. Thechannelgans,g,aretaken as c g d ε where d s unformly taken from real nterval [, 0] andε s a path loss parameter that we assume ε=. Parameter c g s a normalzaton constant to make E(g =. Smulatons are averaged over 5000 realzatons. Those parameters are also used n all smulatons. For coherent MAC model n Fgure, wecanseethatwhenp ncreases, Fgure : Capacty of data collecton for equal power method versus optmal power method n coherent MAC as P ncreases. Note that power P s taken relatve to the channel nose power. Snce we assume that the channel nose has untary varance, thus we label the total transmt power n unt of db. equalpowermethodandoptmalpowermethodconvergeto two dfferent lmts that are C eq ( and C upp,respectvely. Thssbecausetheoptmalpowermethodallocatespowerby takng nto account channel gan sensor observaton whle theequalpowermethoddoesnot.moreover,thelmtof the equal power method, C eq (, s due to nhomogeneous sensng envronment. We can see n Fgure 3 for orthogonal MAC model that the capacty of the optmal power method s larger than that oftheequalpowermethod.thssbecausetheoptmalpower method allocates most of power to sensors that have good observaton and channel qualtes. Moreover, as P ncreases, both the optmal method and the equal one converge to the upper bound. In hgh power regme, each sensor has a redundant power to transmt the sensng data and can easly combat the channel nose. In Fgure 4, wecomparetheoptmalpowermethodfor both MAC models. We can see that the optmal power method for coherent MAC outperforms the orthogonal MAC. Ths s a consequence of usng orthogonal lnks that have dfferent channel noses. We also compare the equal method of both MAC n Fgure 5.Inhghpowerregme(P> 5 db, the equal method for orthogonal has larger capacty because the coherent MAC s lmted by the fnte number of sensor observatons. We smulate the capacty versus the number of sensors wth the total power beng constant at P = 0dB (relatve to channel nose varance for both models n Fgure 6. The capacty of both models ncreases as the total number of

6 6 ISRN Sensor Networks Capacty (nats/s/hz Capacty (nats/s/hz P (db Upper bound orthogonal MAC Equal power orthogonal MAC Optmal power orthogonal MAC Fgure 3: Capacty of data collecton for equal power method versus optmalpowermethodnorthogonalmacasp ncreases. Note that power P s taken relatve to the channel nose power. Snce we assume that the channel nose has untary varance, thus we label the total transmt power n unt of db. Capacty (nats/s/hz P (db Optmal power coherent MAC Optmal power orthogonal MAC Fgure 4: Comparson between the capacty of data collecton n coherent MAC and orthogonal MAC for optmal power allocaton method P (db Equal power coherent MAC Equal power orthogonal MAC Fgure 5: Comparson between the capacty of data collecton n coherent MAC and orthogonal MAC for equal power allocaton method. Capacty (nats/s/hz Upper bound Equal power coherent MAC Optmal power coherent MAC Equal power orthogonal MAC Optmal power orthogonal MAC Fgure 6: Capacty of data collecton of equal power method versus optmal power method as ncreases for ether coherent and orthogonal MAC n a fnte power budget, P =0dB. sensors ncreases. Ths s because as the number of sensors ncreases the total SNR also ncreases. However, we can see that, wth ths fnte total power and a large number of sensors, the capacty of the coherent MAC s larger than that of the orthogonal MAC for both methods, equal and optmal power. Ths s because the corrupted channels n orthogonal MACcannotbeelmnatedevenwhen goes to nfnty. However, n the corehent MAC model, channel nose ncurs only once per recepton at FC. In Fgure 7, we plot the percentage of actve sensors versus the total transmsson power, where we set = 00 n the smulaton for optmal method n orthogonal MAC. We note that the number of actve sensors s less than

7 ISRN Sensor Networks 7 Actve sensors (% P (db Fgure 7: Percentage of actve sensors as P ncreases for optmal power method n orthogonal MAC. when the total power budget s small. Ths confrms that the optmal power allocaton for orthogonal MAC allocates most of power to only the sensors that have good observaton and channel qualtes. Actvatng only the sensors that have good observaton and channel qualtes can be used to conserve energy of the sensors and extend sensor s lfetme. 6. Concluson We studed the capacty of data collecton n wreless sensor networks by consderng power allocaton strategy. We consdered a scenaro n whch a number of sensors observe a target beng estmated at fuson center (FC usng mnmum mean-square error (MMSE estmator. Based on the relatonshp between mutual nformaton and mnmum mean-square error (I-MMSE, we derved the capacty of data collecton n both coherent MAC model and orthogonal MAC model. Consderng power constrant, we derved the capacty under two scenaros: equal power allocaton and optmal power allocaton of both models. We also provded the upper bound of capacty as a benchmark. In partcular, we showed that the capacty of data collecton scaled as Θ((/ log( + when the number of sensors grows to nfnty. We verfed the capacty calculaton by smulaton results as follows. ( For coherent MAC model, we derved the optmal power allocaton strategy that maxmzes the capacty. The capacty of the optmal power s larger than that of the equal power because the optmal power method takesntoaccountthesnrobservatonandchannelganto determne ther ndvdual transmt power. ( For orthogonal MAC model, we derved the optmal power allocaton strategy that maxmzes the capacty. The capacty of the optmal power s larger than that of the equal power because theoptmalpowermethodallocatesmostofpowertoonly the sensors that have good SNR observaton and channel qualtes, whle the sensors wth bad observaton and bad channel qualtes wll be turned off. Turnng off the sensors wth bad observaton and bad channel qualtes can be used to conserve energy of the sensors and extend sensor s lfetme. Moreover, we showed that the capacty of coherent MAC s larger than that of orthogonal MAC, partcularly when the number of sensors s large and the total power P s fxed. Thssconsequenceofusngorthogonallnkfromthesensors to FC where the corrupted channel cannot be elmnated even when goes to nfnty. Appendces A. Dervaton of the ower Bound of MMSE Instead of ntutve assumpton, we provde analytcal dervatonofthelowerboundonmmseestmatonthatcanbe acheved when the total power P and the channel gan g =.Asg =,wecanwrte(9as J =+(+ a ( a α. (A. Based on Cauchy-Schwarz nequalty [7] that( n k l ( n k (n l,wecanrewrte(a.as J +(+ a ( We have power amplfcaton factor, a then we have a ( α. J +(+P (α + (A. = P/(α +; (P (α + ( α. (A.3 Thus, as P, (P (/((α + /( + P (/((α +. Then, the lower bound of MMSE s =+ α J. 0 B. Dervaton of the Optmal SNR (A.4 We know that (9 snotaconvexproblem,butwecan transform t nto an equvalent convex form. et us suppose the optmal soluton of (9sβ opt (P;then/β opt (P should be monotoncally decreasng as P ncreases. Then, we reform the objectve functon n (9 as mnmzng the total power consumpton subject to a gven nverse of total SNR constrant as follows: mn P s.t. P= P β(p,...,p β. (B.

8 8 ISRN Sensor Networks In terms of power amplfcaton factor a and straghtforward modfcaton of (B., we get mn a,...,a s.t. (+ a ( + α g a ( g a α β. (B. We know that (B.sstllconvexntermsofa. Therefore, we use a slack varable t= g a α as mn a,...,a s.t. (+ a ( + α g a βt g a α t=0. The agrangan functon for (B.3s (a,t,μ,η= a ( + α +μ(+ +η(t g a βt g a α, (B.3 (B.4 where η Rand μ 0. From the agrangan functon, we can derve soluton based on KTT condton [8]: = μβt + η = 0, t a =(+α +μg a g α η=0,, μ(+ g a βt =0, g a α =0. From the second KKT condtons; we have (B.5 g α η a = (+α +μg (B.6. We use the slack varable t= g a α and (B.6ntothe frst KTT condtons, we get μg α +μg =β. +α (B.7 Now, we plug n the frst KKT condtons, t = ((β η/μ, and (B.6nto thethrd KTT condtonsas η=( β μ g 4 α ( + α +μg /. (B.8 Thus, the optmal power allocaton of sensor n terms of μ and η s where we have P =a (+α = η 4 P= P g α ( + α, (B.9 ( + α +μg = η g α ( + α 4 ( + α +μg (a =μ. =c (B.0 By drect calculaton that nvolves (B.7, (B.8, and (B.9, (a s held. From (B.6, we obtan that the optmal SNR, β opt,sa functon of the total power, P,as Conflct of Interests α β opt = +(α +/(g (B. P. The authors declare that there s no conflct of nterests regardng the publcaton of ths paper. References []C.Wang,C.Jang,Y.u,X.,andS.Tang, Aggregaton capacty of wreless sensor networks: extended network case, IEEE Transactons on Computers, no. 99, pp. 0, 0. [] S. Chen, M. Huang, S. Tang, and Y. Wang, Capacty of data collecton n arbtrary wreless sensor networks, IEEE Transactons on Parallel and Dstrbuted Systems,vol.3,no.,pp.5 60, 0. [3] P. Sant, On the data gatherng capacty and latency n wreless sensor networks, IEEE Journal on Selected Areas n Communcatons,vol.8,no.7,pp.,00. [4]D.Marco,E.J.Duarte-Melo,M.u,andD..Neuhoff, On the many-to-one transport capacty of a dense wreless sensor network and the compressblty of ts data, ecture Notes n Computer Scence,vol.634,pp. 6,003. [5] E.J.Duarte-MeloandM.u, Data-gatherngwrelesssensor networks: organzaton and capacty, Computer Networks, vol. 43,no.4,pp ,003. [6] H. El Gamal, On the scalng laws of dense wreless sensor networks: the data gatherng channel, IEEE Transactons on Informaton Theory,vol.5,no.3,pp.9 34,005.

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