Power Allocaton for Dstrbuted BLUE Estmaton wth Full and Lmted Feedback of CSI Mohammad Fanae, Matthew C. Valent, and Natala A. Schmd Lane Department of Computer Scence and Electrcal Engneerng West Vrgna Unversty, Morgantown, WV, U.S.A. E-mal: mfanae@mx.wvu.edu, valent@eee.org, and natala.schmd@mal.wvu.edu. arxv:1309.3674v1 [cs.it] 14 Sep 013 Abstract Ths paper nvestgates the problem of adaptve power allocaton for dstrbuted best lnear unbased estmaton BLUE of a random parameter at the fuson center FC of a wreless sensor network WSN. An optmal power-allocaton scheme s proposed that mnmzes the L -norm of the vector of local transmt powers, gven a maxmum varance for the BLUE estmator. Ths scheme results n the ncreased lfetme of the WSN compared to smlar approaches that are based on the mnmzaton of the sum of the local transmt powers. The lmtaton of the proposed optmal power-allocaton scheme s that t requres the feedback of the nstantaneous channel state nformaton CSI from the FC to local sensors, whch s not practcal n most applcatons of large-scale WSNs. In ths paper, a lmted-feedback strategy s proposed that elmnates ths requrement by desgnng an optmal codebook for the FC usng the generalzed Lloyd algorthm wth modfed dstorton metrcs. Each sensor amplfes ts analog nosy observaton usng a quantzed verson of ts optmal amplfcaton gan, whch s receved by the FC and used to estmate the unknown parameter. Index Terms Lmted feedback, best lnear unbased estmator BLUE, generalzed Lloyd algorthm, L -norm, power allocaton, dstrbuted estmaton, parameter estmaton, fuson center, wreless sensor networks. I. INTRODUCTION Dstrbuted estmaton s a technology that enables a wde range of wreless sensor network WSN applcatons, such as event detecton, classfcaton, and object trackng [1 6]. In a WSN performng dstrbuted estmaton, the frst step s for the spatally dstrbuted sensors to locally process ther nosy observatons that are correlated wth an unknown parameter to be estmated. Each sensor ether transmts ts analog local observatons usng an amplfy-and-forward strategy [1 4] or sends a quantzed verson of ts local observatons to the fuson center FC [4 6]. In ths paper, we wll consder the former approach due to ts smplcty and practcal feasblty and wll concentrate on the best lnear unbased estmaton BLUE of an unknown random parameter at the FC. In order to fnd the BLUE estmator of the unknown parameter, the FC combnes lnearly processed, nosy observatons of local sensors receved through orthogonal channels corrupted by fadng and addtve Ths work was supported n part by the Offce of Naval Research under Award No. N00014 09 1 1189. The contrbutons of M. Fanae and M.C. Valent were sponsored n part by the Natonal Scence Foundaton under Award No. CNS 075081. The work of M. Fanae s sponsored n part by the Natonal Scence Foundaton under Award No. IIA-1317103. Gaussan nose. Ths paper wll address one of the man ssues n the case of analog amplfy-and-forward local processng, whch s fndng the optmal local amplfcaton gans [1 4]. The values of these gans set the nstantaneous transmt power of sensors; therefore, we refer to ther determnaton as the optmal power allocaton to sensors. Cu et al. [] have proposed an optmal power-allocaton scheme to mnmze the sum of the local transmt powers, gven a maxmum estmaton dstorton defned as the varance of the BLUE estmator of a random scalar parameter at the FC of a WSN. Although optmal wth respect to the total transmt power n the network, ths strategy could result n assgnng very hgh transmt powers to sensors wth hgh qualty observatons and less nosy channels, whle assgnng zero power to other sensors. The drect consequence of such power allocaton s that some sensors wll de quckly, whch could n turn result n a network partton, whle the remanng sensors have ether low observaton qualty or too nosy communcaton channels. In order to allevate ths drawback, we propose an adaptve power-allocaton strategy that mnmzes thel -norm of the local transmt power vector, gven a maxmum estmaton dstorton as defned above. Ths approach prevents the assgnment of hgh transmt powers to sensors by puttng a hgher penalty on them, whch n tself reduces the chances of those sensors dyng and the network becomng parttoned. Furthermore, the total transmt power used n the entre network stll stays bounded. As t wll be seen n the next sectons, the optmal local amplfcaton gans found based on the proposed power-allocaton scheme depend on the nstantaneous fadng coeffcents of the channels between the sensors and FC, as s the case n []. Therefore, the FC must feed the exact channel fadng gans back to sensors through nfnte-rate, error-free lnks. Ths requrement s not practcal n most WSN applcatons, especally when the number of sensors n the network s large. In the remander of ths paper, we propose a lmted-feedback strategy to allevate ths requrement. The proposed approach s based on desgnng an optmal codebook usng the generalzed Lloyd algorthm wth modfed dstorton functons, whch s used to quantze the space of the optmal power-allocaton vectors used by the sensors to set ther local amplfcaton gans. In our prevous work [7], we have addressed the same drawback of the power-allocaton scheme proposed n [].
Fg. 1: System model of a WSN n whch the FC fnds an estmate of θ. In summary, the man contrbutons of ths paper are as follows: An adaptve power-allocaton scheme s proposed to mnmze the L -norm of the local transmt power vector, gven a maxmum estmaton dstorton at the FC. Ths scheme allevates the problem of assgnng very hgh transmt powers to some sensors, whle turnng off the other ones. Furthermore, a lmted-feedback strategy s proposed to quantze the vector space of the optmal local amplfcaton gans. Approprate dstorton functons are defned for the applcaton of the generalzed Lloyd algorthm n the doman of adaptve power allocaton for dstrbuted estmaton. The rest of ths paper s organzed as follows: In Secton II, the system model of the WSN under study s descrbed. The proposed adaptve power-allocaton strategy s derved n Secton III. A bref dscusson on the motvaton for and mplementaton of the lmted feedback for the proposed power-allocaton scheme s presented n Secton IV. Detals of the mplementaton of the proposed lmted-feedback scheme are dscussed n Secton V. Secton VI provdes the numercal results to show the applcablty of the proposed schemes. Fnally, the paper s concluded n Secton VII. II. SYSTEM MODEL Consder a WSN composed of K spatally dstrbuted sensors, as depcted n Fg. 1. The goal of the WSN s to relably estmate an unknown random parameter θ at ts fuson center FC usng lnearly amplfed versons of local nosy observatons receved through parallel orthogonal coherent channels corrupted by fadng and addtve Gaussan nose. An example of the unknown parameter to be estmated can be the ntensty of the sgnal broadcast by an energy-emttng source and sensed by a set of locally dstrbuted sgnal detectors. Ths estmated varable along wth the propagaton model of the gven sgnal n the observaton envronment could then be used to estmate the locaton of the source. It s assumed that θ has zero mean and unt power, and s otherwse unknown. Suppose that the local nosy observaton at each sensor s a lnear functon of the unknown random parameter as x h θ +n, 1,,...,K, 1 where h s the fxed local observaton gan of sensor, known at the sensor and FC, and n s the spatally ndependent and dentcally dstrbuted..d. addtve observaton nose wth zero mean and known varance σo. Note that no further assumpton s made on the dstrbuton of the random parameter to be estmated and that of the observaton nose. We defne the observaton sgnal-to-nose rato SNR at sensor as β h σ, where denotes the absolute-value operaton. o We assume that there s no nter-sensor communcaton and/or collaboraton among spatally dstrbuted sensors. Each sensor uses an amplfy-and-forward scheme to amplfy ts local nosy observaton before sendng t to the FC as z a x a h θ +a n, 1,,...,K, where z s the sgnal transmtted from sensor to the FC and a s the local amplfcaton gan at sensor. Note that the nstantaneous transmt power of sensor can be found as P a h +σo a σo 1+β. 3 As t can be seen n 3, the value of the local amplfcaton gan at each sensor determnes the nstantaneous transmt power allocated to that sensor. Therefore, we wll call any strategy that assgns a set of local amplfcaton gans to sensors a power-allocaton scheme. All locally processed observatons are transmtted to the FC through orthogonal fadng channels. The receved sgnal from sensor at the FC can be descrbed as y g z +w, 1,,...,K, 4 where g s the multplcatve fadng coeffcent of the channel between sensor and the FC, and w s the spatally ndependent and dentcally dstrbuted addtve Gaussan nose wth zero mean and varanceσc. We assume that the FC can relably estmate the fadng coeffcent of the channel between each sensor and tself. Note that n the above model, we have also assumed that each sensor s synchronzed wth the FC. We defne the channel sgnal-to-nose rato of the sgnal receved from sensor as γ g σ. c III. OPTIMAL POWER ALLOCATION WITH MINIMAL L -NORM OF TRANSMIT-POWER VECTOR Gven a power-allocaton scheme and a realzaton of the fadng gans, the FC combnes the set of receved sgnals from dfferent sensors to fnd the best lnear unbased estmator BLUE for the unknown parameter θ as [8, Chapter 6] K 1 h θ a g K h a g y a g σ o +σc a g σ o +σc, 5 where the correspondng estmator varance can be found as K 1 Var θ a,g h a g a g σ o +σ c K 1 β γ a σ o 1+γ a, 6 σ o
n whch a [a 1,a,...,a K ] T and g [g 1,g,...,g K ] T are column vectors contanng the set of local amplfcaton gans a and fadng coeffcents of the channels g, respectvely. A goal of ths paper s to fnd the optmal local amplfcaton gans or equvalently, the optmal power-allocaton scheme that mnmzes the L -norm of the vector of local transmt powers defned as P [P 1,P,...,P K ] T, gven a constrant on the varance of the estmate as defned n 6. Ths objectve can be formulated as the followng convex optmzaton problem: mnmze {P } K K 7 subject to Var θ a,g By replacng P and Var θ a,g from Equatons 3 and 6, respectvely, Equaton 7 s converted to the followng form, whose optmzaton varables are the local amplfcaton gans: [ mnmze a σ {a } K o 1+β ] 8 β γ a subject to σ o 1+γ a 1 σ o Let b be defned as b βγa σ o 1+γ a. The above constraned σ o optmzaton problem could be re-wrtten as b 1+β mnmze {b } K β b γ σo 9 subject to b 1 AN b < β whch s a convex optmzaton problem n terms of b. The Lagrangan functon for ths optmzaton problem s Lb,λ 0,µ P 1 b 1+β β b γ σo 1 +λ 0 b µ b, 10 where b [b 1,b,...,b K ] T s the column vector of target optmzed varables and µ [µ 1,µ,...,µ K ] T s the Lagrangan multpler vector. The Karush-Kuhn-Tucker KKT condtons for ths optmzaton problem can be wrtten as Lb,λ 0,µ β b 1+β b β b 3 λ 0 µ 0, 11a γ σ4 o b 1, 11b µ b 0, 1,,...,K, 11c µ 0 and b 0, 1,,...,K. 11d It can be shown that the cubc equaton defned n 11a only ALGORITHM I: The water-fllng-based teratve process to fnd the unque values for the number of actve sensors K 1 and the constant λ 0. Requre: K, {β } K, and {γ } K. 1. Intalzaton. for 1,,...,K do 3. δ 1+β β γ 4. end for 5. Sort the sensors based on the ascendng values of δ so that δ 1 δ δ K. 6. K 1 K 7. EndIntalzaton 8. repeat 9. Usng the gven value for K 1, fnd the value of λ 0 by solvng 14. 10. Replace the value of λ 0 n Eq. 1 and fnd the new values of b, 1,,...,K. 11. K 1 K 1 1 1. untl The values of b do not change from the prevous teraton. In partcular, b > 0 for all K 1, and b 0 for all > K 1. 13. return K 1 and λ 0. has a unque real root, whch s n the nterval 0 < b < β as β b β 1 3 δ T 1 3 β δ + λ 0 3 λ 0 T, 1 where δ 1+β β γ, 1,,...,K, T s defned as T 1+ 1+ 8β δ 7λ 0, 13 and the operator [ ] + s defned such that [x] + x f x > 0, and [x] + 0 f x 0. Note that n dervng 1, the complementary slackness requrement 11c s used based on whch µ 0 when b > 0, and b 0 when µ > 0. It should also be noted that as the observaton SNR β or channel SNR γ decreases, the value of δ ncreases, whch n turn ncreases the value of T and decreases the value of b. Therefore, f the sensors are sorted so that δ 1 δ δ K, only the frst K 1 sensors wth the least values of δ wll have a postve value for b, and b 0 for all > K 1. The values of the number of actve sensors K 1 for whch b > 0, and the equalty-constrant Lagrangan multpler λ 0 are unque and can be found by replacng b from 1 nto 11b to derve the followng relatonshp between them: K 1 3 β β δ T λ 0 1 3 3 β δ λ 0 T K 1 β 1. 14 The values of K 1 and λ 0 can be found through the waterfllng-based teratve process summarzed n Algorthm I. It can be shown that the soluton of the above teratve algorthm n terms of K 1 and λ 0 always exsts and s unque. Havng found b through the above process, the local amplfcaton gan a can be found as follows: λ 3 0 1 β δ a γ σ T o 1, K 1 1 3 β δ. 15 3 λ 0 T 0, > K 1 The above power-allocaton strategy assgns a zero amplfca-
ton gan or equvalently, zero transmt power to the sensors for whch δ s large, because ether the sensor s observaton SNR or ts channel SNR s too low. The assgned nstantaneous transmt power to other sensors s non-zero and based on the value of δ for each sensor. Note that based on the above power-allocaton scheme, there s a unque one-to-one mappng between g and a that could be denoted as a f g. IV. LIMITED FEEDBACK FOR POWER ALLOCATION The optmal power-allocaton scheme proposed n the prevous secton s based on the assumpton that the complete forward channel state nformaton CSI s avalable at local sensors. In other words, Equaton 15 shows that the optmal value of the local amplfcaton gan at sensor s a functon of ts channel SNR γ, whch n tself s a functon of the nstantaneous fadng coeffcent of the channel between sensor and the FC. Therefore, n order to acheve the mnmum L -norm of the vector of local transmt powers, the FC must feed the nstantaneous amplfcaton gan a back to each sensor. 1 Ths requrement s not practcal n most applcatons, especally n large-scale WSNs, snce the feedback nformaton s typcally transmtted through fnte-rate dgtal feedback lnks. In the rest of ths paper, we propose a lmted-feedback strategy to allevate the above-mentoned requrement for nfnte-rate dgtal feedback lnks from the FC to the local sensors. For each channel realzaton, the FC frst fnds the optmal power-allocaton scheme usng the approach proposed n the prevous secton. Note that the FC has access to the perfect backward CSI;.e., the nstantaneous fadng gan of the channel between each sensor and tself. Therefore, t can fnd the exact power-allocaton strategy of the entre network based on 15, gven any channel realzaton. In the next step, the FC sends back the ndex of the quantzed verson of the optmzed power-allocaton vector to all sensors. In the lmted-feedback strategy summarzed above, the FC and local sensors must agree on a codebook of the local amplfcaton gans or equvalently, a codebook of possble power-allocaton schemes. The optmal codebook can be desgned offlne by quantzng the space of the optmzed powerallocaton vectors usng the generalzed Lloyd algorthm [9] wth modfed dstorton metrcs. Let L be the number of feedback bts that the FC uses to quantze the space of the optmal local power-allocaton vectors nto L dsjont regons. Note that L s the total number of feedback bts broadcast by the FC, and not the number of bts fed back to each sensor. A codeword s chosen n each quantzaton regon. The length of each codeword s K, and ts th entry s a real-valued number representng a quantzed verson of the 1 Note that nstead of feedng a back to each sensor, the FC could send back the fadng coeffcent of the channel between each sensor and the FC. However, the knowledge of g alone s not enough for sensor to compute the optmal value of ts local amplfcaton gan a. The sensor must also know whether t needs to transmt or stay slent. There are two ways that the extra data can be fed back to the sensors: Ths nformaton could be encoded n an extra one-bt command nstructng each sensor to transmt or stay slent, or the sensor could lsten for the entre vector of g sent by the FC over a broadcast channel. Sendng back each value of a avods ths extra communcaton. optmal local amplfcaton gan for sensor. The proposed quantzaton scheme could then be thought of as a mappng from the space of channel state nformaton to a dscrete set of L length-k real-valued power-allocaton vectors. Detals of ths quantzaton method are descrbed n the next secton. V. CODEBOOK DESIGN USING LLOYD ALGORITHM Let C [a 1 a a L] T be a L K codebook matrx of the optmal local amplfcaton gans, where [C] l, denotes ts element n row l and column as the optmal gan of sensor n codeword l. Note that each a l, l 1,,..., L s assocated wth a realzaton of the fadng coeffcents of the channels between local sensors and the FC. We apply the generalzed Lloyd algorthm wth modfed dstorton metrcs to solve the problem of vector quantzaton n the space of the optmal local amplfcaton gans. Ths algorthm desgns the optmal codebook C n an teratve process, as explaned n the followng dscussons. In order to mplement the generalzed Lloyd algorthm, a dstorton metrc must be defned for the codebook and for each codeword. Let D B C denote the average dstorton for codebook C defned as D B C E a [ mn l {1,,..., L } ] D W a l,a, 16 where E a [ ] denotes the expectaton operaton wth respect to the optmal vector of local amplfcaton gans and D W a l,a represents the dstance between codeword a l and the optmal power-allocaton vector a, defned as D W a l,a J a l J a, 17 where J s the optmzaton cost of the power-allocaton vector. Let P l and P be the vectors of local transmt powers, when the vector of local amplfcaton gans s a l and a, respectvely. The cost functonja s defned as the L -norm of the correspondng vector of transmt powers P,.e., K 1 K 1 J a P [ a σ o 1+β ]. 18 Let A R K+ be the K-dmensonal vector space of the optmal local amplfcaton gans, whose entres are chosen from the set of real-valued non-negatve numbers. Gven the dstorton functon for the codebook C and that for each one of ts codewords defned n Equatons 16 and 17, respectvely, the two man condtons of the generalzed Lloyd algorthm could be reformulated for our vector-quantzaton problem as follows [9, Chapter 11]: Nearest Neghbor Condton: Ths condton fnds the optmal Vorono cells of the vector space to be quantzed, gven a fxed codebook. Based on ths condton, gven a codebook C, the space A of optmzed power-allocaton vectors s dvded nto L dsjont quantzaton regons or Vorono cells wth the lth regon represented by codeword a l C and defned as A l {a A : D W a l,a D W a k,a, k l}. 19
ALGORITHM II: The process of optmal codebook desgn based on the generalzed Lloyd algorthm wth modfed dstorton functons. Requre: K and L. Requre: Fadng model of the channel between local sensors and the FC. Requre: M. M s the number of tranng vectors n space A. Requre: ǫ. ǫ s the dstorton threshold to stop the teratons. 1. Intalzaton. G s A set of M length-k vectors of channel-fadng realzatons based on the gven fadng model of the channels between local sensors and the FC. M L. 3. A s The set of optmal local power-allocaton vectors assocated wth the channel fadng vectors n G s, found by applyng Eq. 15. A s s the set of tranng vectors, and A s A. { } 4. a 0 L l l1 Randomly select L optmal power-allocaton vectors from the set A s as the ntal set of codewords. [ ] T 5. C 0 a 0 1 a0 a0 C 0 s the ntal codebook. L 6. NewCost D B C 0 and j 0. The average dstorton of codebook s found usng Eq. 16. 7. EndIntalzaton 8. repeat 9. j j +1 and OldCost NewCost. 10. Gven codebook C j 1, optmally partton the set A s nto L dsjont subsets based on the Nearest-Neghbor Condton usng Eq. 19. Denote the resulted optmal parttons by A j 1 l, l 1,,..., L. 11. for all A j 1 l, l 1,,..., L do 1. a j l Optmal codeword assocated wth partton Aj 1 l 13. end for 14. C j found based on the Centrod Condton usng Eq. 0. [ ] a j T 1 aj aj L 15. NewCost D B C j 16. untl OldCost NewCost ǫ 17. return C OPT C j. C j s the new codebook. Centrod Condton: Ths condton fnds the optmal codebook, gven a specfc parttonng of the vector space to be quantzed. Based on ths condton, gven a specfc parttonng of the space of the optmzed powerallocaton vectors {A 1,A,...,A L}, the optmal codeword assocated wth each Vorono cell A l A s the centrod of that cell wth respect to the dstance functon defned n 17 as a l arg mn a l A l E a Al [D W a l,a], 0 where the expectaton operaton s performed over the set of members of partton A l The optmal codebook s desgned offlne by the FC usng the above two condtons. It can be shown that the average codebook dstorton defned n 16 wll monotoncally decrease through the teratve usage of the Centrod Condton and the Nearest-Neghbor Condton [9, Chapter 11]. Detals of the codebook-desgn process are summarzed n Algorthm II. The optmal codebook s stored n the FC and all sensors. Upon observng a realzaton of the channel fadng vector g, the FC fnds ts assocated optmal power-allocaton vector a OPT, usng 15 to calculate each one of ts elements. It then dentfes the closest codeword n the optmal codebook C to a OPT wth respect to the dstance metrc defned n 17. Fnally, the FC broadcasts the L-bt ndex of that codeword over an error-free dgtal feedback channel to all sensors as l arg mn D W ak,a OPT. 1 k {1,,..., L },a k C Upon recepton of the ndex l, each sensor knows ts quantzed local amplfcaton gan or equvalently, ts powerallocaton weght as [C] l,, where l and are the row and column ndexes of the codebook C, respectvely. VI. NUMERICAL RESULTS In ths secton, numercal results are provded to assess the performance of the optmal power-allocaton scheme proposed n Secton III and to verfy the effectveness of the lmtedfeedback strategy proposed n Secton V n achevng the energy effcency close to that of a WSN wth full CSI feedback. In ths paper, the energy effcency of a powerallocaton scheme s defned as the L -norm of the vector of local transmt powers formulated n 18. In our smulatons, the local observaton gans are randomly chosen from a Gaussan dstrbuton wth unt mean and varance 0.09. In all smulatons, the average power of h across all sensors s set to be 1.. The observaton and channel nose varances are set to σo 10 dbm and σc 90 dbm, respectvely. The followng fadng model s consdered for the channels between local sensors and the FC: g η 0 d d 0 α f, 1,,...,K, where η 0 30 db s the nomnal fadng gan at the reference dstance set to be d 0 1 meter, d s the dstance between sensor and the FC n meters, α s the path-loss exponent, and f s the ndependent and dentcally dstrbuted..d. Raylegh-fadng random varable wth unt varance. The dstance between sensors and the FC s unformly dstrbuted between 50 and 150 meters. The sze of the tranng set n the optmal codebook-desgn process descrbed n Algorthm II s set to M 5,000, and the codebookdstorton threshold for stoppng the teratve algorthm s assumed to beǫ 10 4. The results are obtaned by averagng over 10,000 Monte Carlo smulatons. Fgure llustrates the energy effcency of the adaptve power-allocaton scheme proposed n Secton III. The fgure depcts the average L -norm of the vector of local transmt powers versus the maxmum dstorton target for dfferent values of the number of sensors n the network K. The energy effcency for the case of equal power allocaton,.e., the mnmum transmt power requred to acheve the gven target dstorton at the FC, s also shown wth dotted lne as a benchmark. As t can be seen n ths fgure, the energy effcency of the network mproves as the number of sensors ncreases. Ths s due to the fact that when there are fewer sensors n the network, each one of them needs to transmt wth a hgher power n order for the FC to acheve the same estmaton dstorton. Note that n our analyss, there s no constrant on the total transmt power consumed n the entre network. Another observaton from Fg. s that the proposed adaptve power allocaton scheme acheves a hgher energy
Average L Norm of Transmt Power Vector 0.7 0.6 0.5 0.4 0.3 0. 0.1 Equal Power Allocaton, K10 Optmal Power Allocaton, K10 Equal Power Allocaton, K50 Optmal Power Allocaton, K50 Average L Norm of Transmt Power Vector 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.0 0.01 L L 3 L 4 Full Feedback 0 0.01 0.0 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Maxmum Dstorton Target Fg. : Average energy effcency versus the target estmaton dstorton for the proposed adaptve power-allocaton scheme and the equal power-allocaton strategy. effcency than the equal power-allocaton strategy. As the maxmum dstorton constrant at the FC s relaxed,.e., the value of s ncreased, the gan n the energy effcency decreases slghtly. Fgure 3 llustrates the effect of L as the number of feedback bts from the FC to local sensors on the energy effcency of the proposed power-allocaton scheme. It should be emphaszed that L s the total number of feedback bts broadcast by the FC, and not the number of bts fed back to each sensor. Ths fgure depcts the average L -norm of the vector of local transmt powers versus the maxmum dstorton target for dfferent values of the number of feedback bts L, when there are K 50 sensors n the network. As t can be seen n ths fgure, the energy effcency of the proposed adaptve power allocaton wth lmted feedback s close to that wth full feedback, and gets closer to t as the number of feedback bts s ncreased. VII. CONCLUSIONS In ths paper, an adaptve power-allocaton scheme was proposed that mnmzes the L -norm of the vector of local transmt powers n a WSN, gven a maxmum varance for the BLUE estmator of a random scalar parameter at the FC. Ths approach results n an ncrease n the lfetme of the network at the expense of a potental slght ncrease n the sum total transmt power of all sensors. The next contrbuton of ths paper was to propose a lmted-feedback strategy to elmnate the requrement of nfnte-rate feedback of the nstantaneous forward CSI from the FC to local sensors. Ths scheme desgns an optmal codebook by quantzng the vector space of the optmal local amplfcaton gans usng the generalzed Lloyd algorthm wth modfed dstorton functons. Numercal results showed that the proposed adaptve power-allocaton scheme acheves a hgh energy effcency, and that even wth a lmted number of feedback bts small codebook, ts average energy effcency based on the proposed lmted-feedback strategy s close to that of a WSN wth full CSI feedback. 0 0.01 0.0 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Maxmum Dstorton Target Fg. 3: Average energy effcency of the proposed power allocaton scheme versus the target estmaton dstorton for dfferent values of the number of feedback bts L, when there are K 50 sensors n the network. REFERENCES [1] J.-J. Xao, S. Cu, Z.-Q. Luo, and A. J. Goldsmth, Lnear coherent decentralzed estmaton, IEEE Transactons on Sgnal Processng, vol. 56, no., pp. 757 770, February 008. [] S. Cu, J.-J. Xao, A. J. Goldsmth, Z.-Q. Luo, and H. V. Poor, Estmaton dversty and energy effcency n dstrbuted sensng, IEEE Transactons on Sgnal Processng, vol. 55, no. 9, pp. 4683 4695, Sep. 007. [3] M. K. Banavar, C. Tepedelenloğlu, and A. Spanas, Estmaton over fadng channels wth lmted feedback usng dstrbuted sensng, IEEE Transactons on Sgnal Processng, vol. 58, no. 1, pp. 414 45, January 010. [4] M. Fanae, M. C. Valent, N. A. Schmd, and M. M. Alkhweld, Dstrbuted parameter estmaton n wreless sensor networks usng fused local observatons, n Proceedngs of SPIE Wreless Sensng, Localzaton, and Processng VII, vol. 8404, Baltmore, MD, May 01. [5] J.-J. Xao, S. Cu, Z.-Q. Luo, and A. J. Goldsmth, Power schedulng of unversal decentralzed estmaton n sensor networks, IEEE Transactons on Sgnal Processng, vol. 54, no., pp. 413 4, February 006. [6] A. Rbero and G. B. Gannaks, Bandwdth-constraned dstrbuted estmaton for wreless sensor networks part I: Gaussan case, IEEE Transactons on Sgnal Processng, vol. 54, no. 3, pp. 1131 1143, March 006. [7] M. Fanae, M. C. Valent, and N. A. Schmd, Lmtedfeedback-based channel-aware power allocaton for lnear dstrbuted estmaton, n Proceedngs of Aslomar Conference on Sgnals, Systems, and Computers, Pacfc Grove, CA, November 013. [8] S. M. Kay, Fundamentals of Statstcal Sgnal Processng: Estmaton Theory, 1st ed. NJ: Prentce Hall, 1993. [9] A. Gersho and R. M. Gray, Vector Quantzaton and Sgnal Compresson. Boston: Kluwer Academc Publshers, 199.