Spatial Fidelity And Estimation in Sensor Networks

Transcription

1 Satial Fidelity And Estimation in Senso Netwoks Ameesh Pandya, Huiyu Luo, and Geg Pottie Deatment of Electical Engineeing Univesity of Califonia, Los Angeles {ameesh, huiyu, Abstact We conside satial fidelity in senso netwoks and show that many oblems in such netwoks ae not well defined without it. We conside senso netwok gatheing data on multile oint souces, o a distibuted souce such as bandlimited o non-bandlimited field. The field could eithe be deteministic o andom. Fo these netwoks, we deive the conditions leading to cooeation between the owe constained senso nodes fo the data fusion uoses and analyze the senso density unde satial fidelity and cooeation. We also oose the heuistics fo samling citeion in case of distibuted senso netwok. I. INTRODUCTION Unlike ad hoc netwok nodes, senso nodes ae deloyed to gathe data fom eithe oint o distibuted henomena. This leads to a cluste of sensos having nealy the same infomation, leading to coelation among themselves. Individual nodes will have some combination of sensing, signal ocessing and communications caability and may self-oganize fo a vaiety of cooeative sensing and communication tasks, subject to esouce constaints such as enegy and bandwidth [1, 2]. Fo uoses of this ae, the senso netwok oblem is fo some end use to extact infomation concening some souce o set of souces to some desied level of fidelity, subject to esouce constaints. Fidelity encomasses such concets as satial o temoal esolution, misidentification obability o othe accuacy measues, and netwok quality of sevice elated measues such as latency fom initial obsevation. Resouce constaints can include signal ocessing cycles, enegy consumtion, and infomation ate. Conside a bounded egion, R, with a oint souce, X sensed by a gou of L sensos, {Y i } L i=1. This can be mathematically modelled as: Y i = X + N i (1) whee, i anges fom 1,..., L and N i is zeo mean indeendent Gaussian noise with vaiance σ 2 N i. If the data available at the sensos is encoded seaately and decoded jointly, then (1) eesents a well known CEO system [3 5]. Suose R consists of m unifomly distibuted multile oint souces, {X i } m i=1, then modelling the netwok as CEO system is not obvious due to the intefeence between the souces. The scenaio whee we can aly CEO esult is when the egion R is samled with infinite sensos i.e. numbe of sensos should be much geate than the numbe of souces. In such a case, we have M indeendent CEO systems. Howeve, in actice, such a scenaio is highly unlikely. Geneally we have finite senso density and the souces could be vey nea to each othe, in some cases even ovelaing, to have any effective sensing o eoduction. Because of this eason, we conside the notion of satial fidelity fo senso netwoks in this ae. The satial fidelity, fo oint souces, could be boadly defined as the distance seaating the oint souces in Euclidean sace. The eoduction o detection decision deends on the satial fidelity. The senso netwoks, in geneal, ae densely deloyed and same souce is obseved by gou of sensos athe than one. The data available at sensos is the noisy vesion of the actual souce. Hence, sensos cooeate (data fusion) among themselves to enhance the detection obability o eoduction quality of the souce at fusion cente. We calculate the numbe of sensos equied to cooeate and the total numbe of sensos needed to deloy as a function of satial fidelity. The souce equied to be sensed by sensos could also be distibuted such as band-limited field 1 [6, 7] o non-bandlimited [8]. The distibuted ocess howeve could be modelled using coelated oint souces. In actice, distibuted continuous ocesses ae, howeve, neve fully obsevable. A tyical aoach in sensing is to samle the ocesses in time and sace, in which the distibuted henomena ae easonably modelled as sets of coelated oint souces. The satial fidelity citeion fo distibuted henomena tansfoms into a oblem of samling esolution. In othe wods, it dictates the ate at which the field should be samled with the sensos. Obviously, fo bette eoduction of the field the esolution should be as fine as ossible which leads to dense samling of the field. But this, howeve, is highly unlikely in a eal time situation. Hence, based on the field gadient o change in the senso eadings, the sensos could be eithe tuned on o off esulting in enegy consevation. Fo examle, sensos deloyed on highly smooth field [9, 10] such as C 0 o C 1 need only a few sensos functioning. On the othe hand, an abutly o fequently changing field needs a lage numbe of sensos to have the desied eoduction quality. The est of the ae is oganized as follows: In Section II satial fidelity fo the oint souces is consideed. This section also calculates the equied numbe of sensos fo the given souce density, satial fidelity, and desied distotion in the final eoduction o detection. Section III extends the idea of satial fidelity to distibuted souces leading to heuistics fo samling citeia. The ae concludes in Section IV. 1 By band-limited field, it means that the owe sectal density of the field is band-limited.

2 Non-seaable Souces (not eoducable) Seaable Souces c Cooeative Cicle R s Fig. 1. Seaable and non-seaable oint souces. Cooeating Sensos Realay Sensos Point Souce II. SPATIAL FIDELITY AND COLLABORATIVE PROCESSING FOR POINT PHENOMENON Conside the oint souces laced too close to each othe to have any eliable distinction between the souces as in Figue 1. In some cases the souces may even ovela. Hence, thee would be significant intefeence at a senso, making difficult fo any effective sensing and thus eoduction of eithe of the souces. Note that the souces and sensos consideed in this ae ae homogeneous i.e. of same natue. This show that the senso netwok design should incooate these situations. Fo this uose the notion of satial fidelity is consideed. A. Satial Fidelity Assume that the owe decays with the distance accoding to an exonent α 2. In ode to have any eliable sensing of the oint souces thee should be a minimum seaation between souces in Euclidean sace. This will eithe comletely o atially suess the intefeence fom othe souces. Based on this, we have the following definition of satial fidelity: Definition II.1 Satial Fidelity fo Point Phenomenon: Satial fidelity, δ ij, is the minimum seaation between any two homogenous souces i and j, i j, in Euclidean sace to attain the desied detection obability o eoduction quality. Thus, X i X j δ ij, i, j (i j). (2) whee denotes Euclidean nom. Hence, satial fidelity, analogous to image ocessing, eesents esolution. The value of δ is use defined and deends on the alications o the desied distotion in the eoduced souce. Howeve, we assume the same value of satial fidelity fo any two homogeneous souces, δ ij = δ, i, j, and i j. In senso netwok alications, if the souces do not satisfy satial fidelity constaints then they ae consideed to be non eoducible. Fo instance, if two heat souces ae less than some distance away to have any eliable eading, the souces ae consideed to be indistinguishable. Howeve, this does not hold fo the mixtue of heteogeneous souces and sensos. B. Collaboative ocessing The equied numbe of sensos fo the desied detection o eoduction at the fusion cente deends on the numbe of souces. Since a cluste o gou of sensos is involved in gatheing data fom the same souce, the sensos can cooeate by Fig. 2. Data Gatheing system multile oint souces and thei esective cooeative egion. locally fusing thei data, hence limiting the numbe of tansmitting sensos and also exloiting satial coelation. Hee, the oblem of equied senso density fo given satial fidelity, numbe of souces, and desied distotion is consideed. Also, the numbe of sensos needed fo local fusion is also evaluated. 1) Density of Locally Cooeating Sensos: Conside a cicula egion of adius R s containing m andomly located oint souces. Also deloyed ae unifomly distibuted sensos as shown in Figue 2. Assume that the numbe of souces and sensos along with the satial fidelity is known. Fo the uose of this ae, assume that the numbe of sensos e unit aea, ρ, emains constant. Mathematically, ρ, is defined as: ρ = πr 2 s. (3) Since each souce has a gou of sensos obseving it athe than one dedicated senso, the multile coies of the same infomation is available. Howeve, based on the distance between souce and senso, the obsevation quality at each senso will diffe. Fo conseving esouces, it is advisable to have only one senso e souce to tansmitting the infomation. Ideally, the senso with the best (o desied) signal-to-intefeence noise atio (SINR) should be consideed as the tansmitting senso. Howeve, if no senso meets the desied SINR then the nodes can cooeate locally. Local fusion is also encouaged fo eliable detection [11]. Fo the uose of local fusion, it could be shown that thee exists a cooeating cicula egion of adius c aound each souce. The adius of this egion cetainly vaies fom souce to souce, but fo the sake of simlicity assume it to be the same fo all souces. Also, assume that the souce localization has been aleady efomed and hence the sensos ae awae of the elative distance between them and any given souce. This assumtion is only fo analytical tactability. Late, this assumtion will be elaxed fo calculating total numbe of sensos equied to be deloyed. The cooeating egions could eithe be non-ovelaing o ovelaing fo any aticula souce. Conside a souce X m being obseved by q sensos lying within a cooeative egion as shown in Figue 3. The intefeence at the sensos is assumed only due to souces and not the communication intefeence fom the othe sensos. This is a valid assumtion as the aoiate choice of medium access contol

3 Fig. 3. X m c im θ ik i X Souce-senso ositions fo SINR calculations. k If, then: λk lim P(q = k) = k! e λ (10) whee λ = ρπ 2 c. Hence, asymtotically the numbe of sensos in the cooeative egion follows a Poisson distibution with intensity ρπ 2 c. Suose X m to be a zeo-mean Gaussian with the vaiance σ 2 X m. Conside the q sensos obseving X m within the cooeative egion. The obsevation available at each senso could be modelled as: Y i = γ i X m + Z i (11) (MAC) laye otocol can always event the communication intefeence. The SINR at senso i lying in the cooeative egion of X m is then given by: P Xm ( im / 0 ) α SINR i = σn 2 i + m 1 k=1 P X k ( ik / 0 ) α (4) whee α is the ath-loss coefficient, ij is the Euclidean distance between the senso i and souce j, 0 is the constant of ootionality, P Xj is the tansmitted owe by souce X j and σn 2 i is the vaiance of zeo-mean Gaussian noise, N i, at senso i. The denominato of (4) is the intefeence at senso i: m 1 I i = σn 2 i + P Xk ( ik / 0 ) α (5) k=1 Fom geomety (Figue 3), the distance between the senso i and souce k, k = 1, 2,..., m 1, can be calculated to be: ik = 2 X mx k + 2 im 2 X mx k im cos θ (6) whee XmX k is the distance between the souce X m and X k, and θ is the measue of ix m X k. If the satial fidelity constaint, δ, is satisfied by the souce X k with X m, then XmX k δ. If the constaint is not being satisfied then both the souces X m and X k would not be consideed fo the eoduction at the fusion cente. Hence, it is clea that the satial fidelity will influence SINR i. The lage the souce seaation, the highe the SINR at the senso. Fistly, conside the statistical aoach. The obability that the senso node will be in adius c is: P c = π2 c πr 2 s = 2 c R 2 s Using (3), the above equation can be ewitten as: P c = ρπ2 c Theefoe, the obability that the numbe of sensos, q, in the cooeative egion of X m is k could be calculated as: ( ) P(q = k) = P k k c (1 P c ) k (9) (7) (8) whee i anges fom 1,..., q and SINR i = γ2 i σ2 Xm. Hee, Z σz 2 i i also consides the intefeence fom othe souces. The intefeence, I i, in (5) is cetainly not Gaussian fo a finite numbe of souces. But, if m, then fom the cental limit theoem m 1 k=1 P X k ( ik / 0 ) α tends to Gaussian. γ i s ae the inhomogeneous coefficients, modelling the actical scenaio of not being identical. Based on this, the mean squaed estimate of X m is given by [12]: [ (Γσ ) 2 1 ] ˆX m = Γ Xm + R Z Γσ 2 Xm Y (12) whee Y denotes column vecto {Y i } q i=1, Γ is a column vecto {γ i } q i=1 and R Z denotes the covaiance matix of noise {Z i } nc i=1. Since Z i s ae indeendent of each othe R Z is a diagonal matix. If Z i s ae also identical with vaiance σz 2, then the distotion in eoduction of souce X m is given by: D = σ 2 X m σ 2 Z ( Γ Γσ 2 X m + σ 2 Z) (13) The distotion, D, clealy deends on SINR which is a function of δ. This imlies that D = f(δ). Using this and (13), the numbe of sensos needed fo cooeation can be calculated in tems of satial fidelity. Fo instance, assume Γ to be a vecto of identical elements γ. In this case, Γ Γ = qγ 2. Hence fom (13), [ q = σ2 Z σ 2 ]+ Xm γ 2 σx 2 m D 1 (14) whee x + = max(0, x). The above consideed the case of one souce having q sensos locally cooeating. If m souces ae esent satisfying the fidelity constaint, then the senso netwok will have a total of mq sensos locally cooeating fo those m souces within thei assigned cooeative egion. 2) Calculating Senso Density: The analysis fo calculating the senso density involving m oint souces is highly comlicated and to some extent non-tactable. Fo instance, assuming the localization of souces as in the evious discussion esults in a non-actical scenaio and even that does not hel in simlifying things. This is because calculating the cooeating sensos could be done but evaluating the total numbe of elays is highly taffic deendent. Even the statistical analysis is equally difficult. Fo examle, conside the comutation of P(distance between any two souces δ). Although the senso nodes ae

4 i.i.d. in location, the Euclidean distance between them is deendent. Hence, calculating this obability fo m souces itself is comlicated. Hence, simulation is efomed to evaluate the senso density fo a given satial fidelity, numbe of souces, and desied mean squaed distotion. The simulation setu assumes a unit aea cicula egion with m Gaussian oint souces andomly deloyed. Fo the uose of the simulations, the aea of the cicula egion is ket constant. Fistly, the simulation is caied out to calculate the obability of the souces that do not satisfy satial fidelity citeia. That is, the numbe of souces ae calculated that ae saced at distance less than δ. Fo this uose, m is vaied fom 10 to 200. The simulation fo each value of m is executed fo 1000 iteations and the value of obability is calculated aveaging ove those values. The obability lot fo diffeent values of δ is shown in Figue 4. Pob(numbe of souces not satisfying δ constaint) δ = 0.04 δ = 0.05 δ = 0.07 δ = Total numbe of Souces Fig. 4. Pobability lot fo the numbe of souces not satisfying satial fidelity constaints. With the incease in numbe of souces, the satial seaation between them will cetainly educe as the aea of the egion is constant. Hence at highe node density, the esolution should be fine fo the desied eoduction. As seen in Figue 4, at highe souce density, the obability of souces disobeying satial fidelity constaint, δ, inceases. Next, simulation fo the senso density is consideed. Fo this uose the numbe of souces, m, is fixed. The goal in this simulation is to know the mean squaed distotion in the estimates. This will give the elation between the equied numbe of sensos and distotion. The tansmission owe is assumed to decay in second ode with the distance. The sensos obseving the souce ae locally fused (assuming coheent souces) until the desied signal stength is obtained. The simulation lot is given in Figue 5. Simila to the obability simulation, each oint in the lot is the ensemble aveage ove 1000 iteations. Fo any aticula δ, with the incease in numbe of sensos, the distotion should decease. This is obvious as the dense deloyment of sensos ovides bette signal stength at the senso along with high satial coelation and senso divesity. Now, as the δ inceases, then the numbe of souces that satisfy the satial fidelity constaint deceases. Hence, the total numbe Mean Squaed Distotion, D δ = 0.1 δ = 0.2 δ = 0.3 δ = 0.4 δ = Numbe of Sensos, Fig. 5. Simulation Plot deicting the elation between the equied numbe of sensos and mean squaed distotion fo diffeent values of δ. The ath-loss coefficient is assumed to be 2. of souces to conside fo eoduction deceases. This leads to the deloyment of fewe sensos fo highe δ as seen fom Figue 5. III. SPATIAL FIDELITY FOR DISTRIBUTED PHENOMENON This section consides the sensing of distibuted henomena such as field. The samling of these fields is one of the most challenging aeas of eseach. In an ideal case, samling above the Nyquist ate avoids aliasing effect. The assumtion of sue Nyquist samling also holds fo this ae. Howeve, detemining whethe the field is unde samled, ove samled, o citically samled still oses an inteesting oblem. These samling issues have been consideed in [6, 14]. Hee, samling of the henomenon is consideed in a diffeent context. Note that while collectively sensos ae obseving a distibuted souce, each of them is collecting data fom a oint souce. All togethe, these highly coelated oint souces fom the distibuted henomena. Section II discussed the notion of satial fidelity fo oint henomena. We now extend that definition fo distibuted henomena. The idea behind the satial fidelity, howeve, emains the same. That is, it eesents esolution of the field. Definition III.1 Satial Fidelity fo Distibuted Phenomenon: Satial fidelity, δ, fo the distibuted henomenon eesents the cut-off o samling ate. It is govened by the gadient o the change in the amlitude of the field with esect to time. Conside the distibuted field as in Figue 6. Suose the field is samled at evey δ 0 units. The satial samling fequency of the field is then 1 δ 0. If change in the amlitude is eesented by, then the sloe of a cuve within a samle is given by (Figue 6): ζ = lim l 0 l = X l = X δ Howeve, this fequency may not esult into tue eoduction of the field. Conside the cuve EF. It abutly changes fom E to F and hence, it is desiable to have moe samles (o highe

5 deived fo oint souces could also be extended fo distibuted henomena. A δ 0 B l 0 l 1 C D E F Ideal Samling Fig. 6. Distibuted field with samling inteval, δ 0 (samling fequency = 1/δ 0. The dashed lines between oints E and F eesent an ideal situation. samling ate) fo the bette eoduction of the field. This situation is an examle of unde samling. In contast, the egion P R is smooth and flat. Fo this egion, the samling ate could be low, that is the samle at oint Q is not equied. With the eadings at P and R, the egion could be accuately eoduced. This situation coesonds to ove samling. The citically samled egion is AB in Figue 6. All these contibute to enegy cost in senso netwoks. Hence, the samling should be adative. This adative natue of samling is eesented by satial fidelity. To summaize, let δ 0 be the use secified satial fidelity citeion. If i and j eesent two oints on the field, then the following thee conditions fo satial fidelity, δ, could be ossible: Unde Samling δ > ɛ and δ < δ 0 (15) Citical Samling δ [0, ɛ 1] and δ = δ 0 (16) Ove Samling δ P δ Q (ɛ 1, ɛ] and δ > δ 0 (17) whee, ɛ and ɛ 1 ae ositive quantities. The above thee conditions define satial fidelity fo distibuted souces. Note, that the above samling citeia is oosed as heuistics. In most cases, the use ovides the lowe bound on the satial fidelity. If the lowe bound is δ 0, then δ δ 0. With this constaint then we can eoduce the field with the desied distotion. The eo in econstuction may occu fom the situations such as the egion between C and D in Figue 6. The eadings at both these oints ae the same but thee is change in the eadings in between which cannot be catued. Howeve, this eo does not oagate and is limited to local egion. Since the distibuted souce could be obseved as the collection of coelated oint souces, the souce-senso elations R H IV. CONCLUSIONS Fo senso netwoks, we believe that the main objective is infomation extaction to some level of fidelity. The notion of satial fidelity is consideed to chaacteize the senso netwok. In a boad sense, satial fidelity is the desied seaation between the oint souces. Fo a distibuted henomenon, it tanslates into desied and/o adative samling fequency. This is analogous to the esolution of ixels in image ocessing. Fo eliability, the data is consideed to be fused locally. Based on the satial fidelity and SINR available at the sensos, equied numbe of sensos fo local cooeation is calculated. Futhe, the elation between numbe of sensos and desied distotion is evaluated, assuming the knowledge of souces and satial fidelity. Satial fidelity also lead to the heuistics fo samling citeion in case of distibuted henomenon. REFERENCES [1] G Pottie and W Kaise. Wieless integated netwok sensos. Communications of the ACM, 43(5):51 58, [2] D Estin, L Giod, G Pottie, and M Sivastava. Instumenting the wold with wieless senso netwoks. In Poceedings of Intenational Confeence on Acoustics, Seech, and Signal Pocessing (ICASSP), Salt Lake City, UT, June [3] T Bege, Z Zhang, and H Vishwanathan. The CEO oblem. IEEE Tansactions on Infomation Theoy, 42(3): , May [4] H Viswanathan and T Bege. The quadatic gaussian ceo oblem. IEEE Tansactions on Infomation Theoy, 43(5), Setembe [5] Y Oohama. The ate-distotion function fo the quadatic gauasian CEO oblem. IEEE Tansactions on Infomation Theoy, 44(3): , May [6] P Ishwa, A Kuma, and K Ramchandan. Distibuted samling fo dense senso netwoks: A bit-consevation incile. In Poceedings of the Intenational Woksho on Infomation Pocessing in Senso Netwoks, Ail [7] A Scaglione and S Sevetto. On the intedeendence of outing and data comession in multi-ho senso netwoks. In Poceedings of the 8th annual intenational confeence on Mobile comuting and netwoking, ages ACM Pess, [8] D Maco, E Duate-Melo, M Liu, and D Neuhoff. On the many-to-one tansot caacity of a dense wieless senso netwok and the comessibility of its data. In Poceedings of the Intenational Woksho on Infomation Pocessing in Senso Netwoks (IPSN), Ail [9] R Nowak and U Mita. Bounday estimation in senso netwoks: Theoy and methods. In 2nd Intenational Woksho on Infomation Pocessing in Senso Netwoks, volume 20, Palo Alto, CA, [10] R Willett, A Matin, and R Nowak. Backcasting: adative samling fo senso netwoks. In Poceedings of the thid intenational symosium on Infomation ocessing in senso netwoks, ages , Bekeley, CA, ACM Pess. [11] P Vashney. Distibuted Detection and Data Fusion. New Yok: Singe- Velag, [12] A Sayed. Fundamentals of Adative Filteing. Wiley-IEEE Comute Society Pess, [13] D Ganesan, S Ratnasamy, H Wang, and D Estin. Coing with iegula satio-temoal samling in senso netwoks. SIGCOMM Comute Communication Review, 34(1): , [14] A Kuma, P Ishwa, and K Ramchandan. On distibuted samling of smooth non-bandlimited fields. In Poceedings of the thid intenational symosium on Infomation ocessing in senso netwoks, ages 89 98, 2004.