# Optimal DistortionPo wer radeof fs in Sensor Netw orks GaussMark Random Processes Nan Liu Sennur Ulukus Department of Electrical and Computer Engineering Uni ersity of Maryland Colle ge ark MD nkanc

### Presentations text content in Optimal DistortionPo wer radeof fs in Sensor Netw orks GaussMark Random Processes Nan Liu Sennur Ulukus Department of Electrical and Computer Engineering Uni ersity of Maryland Colle ge ark MD nkanc

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Optimal Distortion-Po wer radeof fs in Sensor Netw orks: Gauss-Mark Random Processes Nan Liu Sennur Ulukus Department of Electrical and Computer Engineering Uni ersity of Maryland, Colle ge ark, MD 20742 nkancy@umd.edu ulukus@umd.edu Abstract in estigate the optimal perf ormance of dense sensor netw orks by studying the joint sour ce-channel coding pr oblem. The erall goal of the sensor netw ork is to tak measur ements fr om an underlying random pr ocess, code and transmit those measur ement samples to collector node in co- operati multiple access channel with feedback, and

econstruct the entir random pr ocess at the collector node. pr vide lo wer and upper bounds or the minimum achie able expected distortion when the underlying random pr ocess is stationary and Gaussian. In the case wher the random pr ocess is also Mark vian, we aluate the lo wer and upper bounds explicitly and sho that they ar of the same order or wide range of sum po wer constraints. Thus, or Gauss-Mark random pr ocess, under these sum po wer constraints, we determine the achie ability scheme that is order -optimal, and expr ess the minimum achie able expected distortion as function of the sum

po wer constraint. ith the recent adv ances in the hardw are technology small cheap nodes with sensing, computing and communication capabilities ha become ailable. In practical applications, it is possible to deplo lar ge number of these nodes to sense the en vironment. In this paper we in estigate the optimal perfor mance of dense sensor netw ork by studying the joint source- channel coding problem. The sensor netw ork is composed of sensors, where is ery lar ge, and single collector node. The erall goal of the sensor netw ork is to tak measurements from an underlying random process code and

transmit those measured samples to collector node in cooperati multiple access channel with feedback, and reconstruct the entire random process at the collector node. in estigate the minimum achie able xpected distortion and the corresponding achie ability scheme when the underlying random process is Gaussian and the communication channel is cooperati Gaussian multiple access channel with feedback. ollo wing the seminal paper of Gupta and umar [1], which sho wed that multi-hop wireless ad-hoc netw orks, where users transmit independent data and utilize single-user cod- ing, decoding and forw

arding techniques, do not scale up, Scaglione and Serv etto [2] in estigated the scalability of the sensor netw orks. Sensor netw orks, where the observ ed data is correlated, may scale up for tw reasons: rst, the correlation among the sampled data increases with the increasing number This ork as supported by NSF Grants CCR 03 11311 CCF 04 47613 and CCF 05 14846 and ARL/CT Grant AAD 19 01 0011 of nodes and hence, the amount of information the netw ork needs to carry does not increase as ast as in ad-hoc wireless netw orks; and second, correlated data acilitates cooperation, and may

increase the information carrying capacity of the netw ork. The goal of the sensor netw ork in [2] as that each sensor reconstructs the data measured by all of the sensors using sensor broadcasting. In this paper we focus on the case where the reconstruction is required only at the collector node. Also, in this paper the task is not the reconstruction of the data the sensors measured, ut the reconstruction of the underlying random process. Gastpar and etterli [3] studied the case where the sensors observ noisy ersion of linear combination of Gaussian random ariables with equal ariances, code

and transmit those observ ations to collector node, and the collector node reconstructs the random ariables. In [3], the xpected distortion achie ed by applying separation-based approaches as sho wn to be xponentially orse than the lo wer bound on the minimum xpected distortion. In this paper we study the case where the data of interest at the collector node is not nite number of random ariables, ut random process, which, using Karhunen-Loe xpansion, can be sho wn to be equi alent to set of innitely man random ariables with arying ariances. assume that the sensors are able to

tak noiseless samples, ut that each sensor observ es only its wn sample. Our upper bound is also de eloped by using separation-based approach, ut it is sho wn to be of the same order as the lo wer bound, for wide range of po wer constraints for Gauss-Mark random process. El Gamal [4] studied the capacity of dense sensor netw orks and found that all spatially band-limited Gaussian processes can be estimated at the collector node, subject to an non-zero constraint on the mean squared distortion. In this paper we study the minimum achie able xpected distortion for space- limited, and thus, not

band-limited, random processes, and we sho that the minimum achie able xpected distortion decreases to zero as the number of nodes increases, unless the sum po wer constraint is unusually small. rst pro vide lo wer and upper bounds for the minimum achie able xpected distortion for arbitrary stationary Gaussian random processes. Then, we focus on the case where the Gaussian random process is also Mark vian, aluate the lo wer and upper bounds xplicitly and sho that the are of the same order for wide range of po wer constraints.

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Thus, for Gauss-Mark random process, under

wide range of po wer constraints, we determine an order -optimal achie ability scheme, and identify the minimum achie able xpected distortion. Our order -optimal achie ability scheme is separation-based. It is well-kno wn [5], [6] that in multi-user channels with correlated sources, the source-channel separa- tion principle does not hold in general, and separation-based achie ability schemes may be strictly suboptimal. Ho we er in this instance, where we ha multi-user channel with correlated sources, for wide range of po wer constraints, we sho that separation-based achie ability scheme is

order optimal, when the number of nodes goes to innity The results of this paper pro vide insights for the design of lar ge sensor netw orks that aim at reconstructing the underlying random process at collector node. Our results pro vide the order -optimal scheme for the operation of the sensor nodes, the number of nodes needed to be deplo yed and the po wer constraint needed to be emplo yed, in order to achie certain erall distortion. Although, we constrain ourselv es to Gauss-Mark pro- cesses in this paper we belie that our methods can be xtended to more general Gaussian random

processes. The collector node wishes to reconstruct random process for where denotes the spatial position; is assumed to be Gaussian and stationary with autocorrelation function The sensor nodes are placed at positions and observ samples )) or simplicity and to oid irre gular cases, we assume that the sensors are equally spaced. The distortion measure is the squared error )) )) dt (1) Each sensor node and the collector node, denoted as node 0, is equipped with one transmit and one recei antenna. At an time instant, let and denote the signals transmitted by and recei ed at, node and let denote

the channel gain from node to node Then, the recei ed signal at node can be written as, =0 ;j (2) where =0 is ector of independent and identically distrib uted, zero-mean, unit-v ariance Gaussian random ari- ables. Therefore, the channel model of the netw ork is such that all nodes hear linear combination of the signals transmitted by all other nodes at that time instant. assume that ij is determined by the distance between nodes and denoted as ij as ij = ij for i; and is the path-loss xponent, which is typically between and [7]. or simplicity we assume that the collector node is at an

equal distance ay from all of the sensor nodes, i.e., for where is some constant, independent of The results can be generalized straightforw ardly to the case where are non-identical constants. assume that all sensors share sum po wer constraint of which is function of or the discussion of distortion-po wer tradeof fs of the Gauss-Mark processes, we di vide into re gions. ery lar is lar ger than Lar is between and Medium is between 1+ and Small is between and 1+ ery small is no lar ger than The reason why we di vide into these re gions will be apparent in Sections IV and V. The tw most

interesting cases for the sum po wer constraint are ind where each sensor has its indi vidual po wer constraint ind and tot where all sensors share constant total po wer constraint tot Both of these tw cases lie in the medium sum po wer constraint re gion. Our goal is to determine the scheme that achie es the minimum xpected distortion at the collector node for gi en total transmit po wer constraint and also to determine the rate at which this distortion goes to zero as function of the number of sensor nodes and the po wer constraint. In this paper we seek to understand the beha vior of the

minimum achie able xpected distortion when the number of sensor nodes is ery lar ge. introduce the big-O and big- notations. say that is O( ), if there xist constants and such that for all we say that is ( if there xist constants and such that for all All logarithms are base Due to space limitations, all proofs are omitted here and can be found in [8]. AU Gauss-Mark process, also kno wn as the Ornstein- Uhlenbeck process [9], [10], is dened as random process that is stationary Gaussian, Mark vian, and continuous in probability It is kno wn that the autocorrelation function of

this process is [11]–[13] (3) The Karhunen-Loe xpansion [14] of the Gauss-Mark process yields the eigenfunctions =0 cos sin (4) where =0 are the corresponding eigen alues and are positi constants chosen such that the eigenfunctions ha unit ener gy Ev en though it is not possible to xpress =0 in closed form, the can be bounded as 00 (5)

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where =1 is dened as +1) (6) with and is the lar gest inte ger smaller than or equal to also, may be ne gati e, in which case, the rst line in (6) should be disre garded. 00 =1 is dened as 00 1) (7) Rate-distortion

functions are easier to calculate with =0 and 00 =0 and the tw sequences will be used in place of =0 to de elop lo wer and upper bounds on the minimum achie able xpected distortion. RT A. Arbitr ary Stationary Gaussian Random Pr ocesses Let be the minimum achie able xpected distortion at the collector node for gi en total transmit po wer constraint In this section, we will de elop tw lo wer bounds on obtain our rst lo wer bound by assuming that the communication links from the sensor nodes to the collector node are noise and interference free. Let be the MMSE (minimum mean squared

error) when the collector node es- timates the underlying random process by using the xact alues of all of the samples tak en by the sensors. Then, it is straightforw ard to see that, (8) Since the random process is Gaussian, calculating is Gaussian MMSE estimation problem. It suf ces to consider the linear MMSE estimator and the resulting xpected distor tion is (0) ) dt (9) where (10) and (0) (0) (0) (11) obtain our second lo wer bound by assuming that all of the sensors kno the random process xactly and, the sensor netw ork forms an -transmit 1-recei antenna point- to-point system to

transmit the random process to the collector node. Let be the capacity of this point-to-point system and be the distortion-rate function of the random process [15]. In this point-to-point system, the separation principle holds and feedback does not increase the capacity and therefore (12) aluate we rst nd the rate distortion function, of [15, Section 4.5] as, =0 max log (13) and =0 min (14) It can be seen that the function is strictly decreasing function of when Hence, in this re gion, the in erse function of xists, which we will call Ne xt, we nd the capacity of the

-transmit 1-recei antenna point-to-point system [16] as, log (15) Then, we ha (16) By combining the tw lo wer bounds described abo e, we see that, for arbitrary stationary Gaussian random processes, lo wer bound on the minimum achie able xpected distortion is max (17) B. The Gauss-Mark Pr ocess note that and in (17) both depend on the autocorrelation function Unless we put more struc- ture on it seems dif cult to continue with an xact aluation. Hence, we constrain ourselv es to special class of Gaussian random processes, the Gauss-Mark random processes, whose autocorrelation function

is gi en in (3), in order to continue with our analysis of the distortion. First, we aluate Using (3) and the Mark vian prop- erty of it is straightforw ard to sho that [8], (18) Hence, for the Gauss-Mark process when the random pro- cess is estimated from its samples, the estimation error decays as Ne xt, we aluate for the Gauss-Mark process. Let be the distortion obtained from (16) when =0 in (13) and (14) are replaced by =0 which we dened in (5) and (6). Then, (19)

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because for all and it is more dif cult to estimate sequence of random ariables each with lar

ger ariance. Since we seek lo wer bound on the minimum achie able xpected distortion, the aluation of suf ces. Hence, for the rest of this section, we concentrate on the aluation of and gi en in (13) and (14), respecti ely for =0 will di vide our discussion into tw separate cases based on the sum po wer constraint. or the rst case, is such that lim !1 )) (20) is satised. This includes all sum po wer constraint re gions dened in Section II xcept the very small sum po wer con- straint. The cases where ind and tot are included in satisfying (20). Note from (15)

that, in this case, increases monotonically in Since we are interested in the number we consider the re gion when is ery lar ge for the function Lemma or lar enough we have (21) bound for small enough in the ne xt lemma. Lemma or small enough we have (22) are no ready to calculate the distortion. When is lar ge enough, using (15), (21) and (22), we ha log( ))) (23) conclude, based on (17), (18), (19) and (23), that when the sum po wer constraint satises (20), lo wer bound on the minimum achie able xpected distortion is max (log( ))) (24) or the second case, is such that (20) is not

satised. is either constant independent of or goes to zero as goes to innity Examining (13), we see that is bounded belo by constant independent of and hence, is constant and does not go to zero as increases. Therefore, for all possible po wer constraints lo wer bound on the minimum achie able xpected distortion is max min log( ))) (25) which can also be xpressed “order -wise as if lim !1 if lim !1 (log( ))) otherwise (26) The rst case in (26) corresponds to the very lar sum po wer constraint dened in Section II. This is the scenario where the sum po wer

constraint gro ws almost xponentially with the number of nodes. The transmission po wer is so lar ge that the communication channels between the sensors and the collector node are as if the are perfect, and we are left with the “una oidable distortion of which we ha in reconstructing the random process from the “perfect kno wledge of its samples. Ev en though this pro vides the best performance among all three cases, it is impractical since sensor nodes are lo ener gy de vices and it is often dif cult, if not impossible, to replenish their batteries. The second case in (26) corresponds

to the very small sum po wer constraint dened in Section II. The transmission po wer is so lo that the communication channels between the sensors and the collector node are as if the do not xist. The estimation error is on the order of 1, which is equi alent to the collector node blindly estimating for all [0 Ev en though the consumed po wer is ery lo in this case, the performance of the sensor netw ork is unacceptable; en the lo wer bound on the minimum achie able xpected distortion does not decrease to zero with the increasing number of nodes. Hence, the meaningful sum po wer

constraints for the sensor nodes should be in the “otherwise case in (26), which includes the lar medium and small sum po wer constraints dened in Section II. The corresponding lo wer bound on the minimum achie able xpected distortion as function of the po wer constraint is (log ))) (27) The tw practically meaningful cases of ind and tot are in this “otherwise case. In both of these cases, the lo wer bound on the minimum achie able xpected distortion decays to zero at the rate of (log RT A. Arbitr ary Stationary Gaussian Random Pr ocesses An distortion found by using an achie ability

scheme will serv as an upper bound for the minimum achie able xpected distortion. consider the follo wing separation- based achie able scheme: First, we perform distrib uted rate- distortion coding at all sensor nodes using [17, Theorem 1]. After obtaining the indices of the rate-distortion codes, we transmit the indices as independent messages using the antenna sharing method introduced in [4]. The distortion obtained using this scheme will be denoted as apply [17, Theorem 1], generalized to sensor nodes in [18, Theorem 1], to obtain an achie able rate-distortion point. will consider the case

when all sensor nodes transmit their data at identical rates, and this rate is determined by the ratio of the sum rate and ha the follo wing theorem. Theor em The following sum ate and distortion ar ac hie v-

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able (0) dt (28) =0 log (29) wher and ar the eig en values of further aluate in the ne xt lemma. Lemma or all stationary Gaussian andom pr ocesses whose autocorr elation functions satisfy the Lipsc hitz condition in the interval and have nite right derivatives at we have max =1 (30) Lemma tells us that the xpected distortion achie ed by using the separation-based

scheme is upper bounded by the maximum of tw types of distortion. The rst distortion is of size and the size of the second distortion depends on the achie able rate of the channel through dene the second distortion as =1 (31) No we determine an achie able rate for the communi- cation channel from the sensor nodes to the collector node. The channel in its nature is multiple access channel with potential cooperation between the transmitters and feedback from the collector node. The capacity re gion for this channel is not kno wn. get an achie able sum rate for this channel by

using the idea presented in [4]. The follo wing theorem is generalization of [4, Theorem 1] from constant po wer constraint to more general po wer constraint. Theor em When the sum power constr aint and the path-loss xponent satisfy lim !1 1+ (32) the following ate is ac hie vable log( )) (33) wher is positive constant dened as lim !1 log log )) (34) otherwise appr oac hes non-ne gative constant as Theorem sho ws that when the sum po wer constraint is very lar lar or medium as dened in Section II, the achie able rate increases with Otherwise, the achie able rate is either

positi constant or decreases to zero, which will result in poor estimation performance at the collector node. The function is strictly decreasing function of thus, the in erse function xists, which we will denote as Hence, to nd we rst nd and then, (35) will perform this calculation when the underlying random process is Gauss-Mark B. The Gauss-Mark Pr ocess The autocorrelation function of the Gauss-Mark process gi en in (3) satises the conditions of Lemma 3. Hence, (30) is alid, and max (36) It remains to aluate rst dene tw sequences and which

satisfy lim !1 lim !1 (37) Lemma or lar enough and in the interval of (38) we have (39) Hence, for all that satisfy lim !1 (40) and (32), we ha in the interv al of (38) and our result of (39) is applicable. No we upper bound Lemma or for lar enough we may upper bound as 12 (41) The proofs of Lemma and use the act that con er ges to when is lar ge [2]. Since the con er gence of to is not uniform in the results of Lemma and are alid only when satises (40). Hence, when is such that (40) and (32) are satised, using (39), (41) and the act that when is in the interv al of (38), is in

we ha log( ))) (42)

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Therefore, from (36), an upper bound on the minimum achie v- able xpected distortion is log( ))) (43) This upper bound on the minimum achie able xpected dis- tortion coincides with the lo wer bound described in the “oth- erwise case in (26). Ho we er it should be noted that, the “otherwise case in (26) corresponds to the lar medium and small sum po wer constraints dened in Section II, whereas (40) and (32) are satised only for the medium sum po wer constraint. AU No we compare the upper bound in (43) and the lo wer bound in (26). In the very

lar and lar sum po wer constraint re gions, our methods do not apply e.g. (43) is not alid, and we ha not sho wn whether the lo wer and upper bounds meet. Ho we er in this re gion is lar ger than and this re gion is not of practical interest. In the medium sum po wer constraint re gion, is in the wide range of 1+ to and our lo wer and upper bounds do meet and the minimum achie able xpected distortion is log( ))) (44) The order -optimal achie ability scheme is separation-based scheme, which uses distrib uted rate-distortion coding as de- scribed in [17] and optimal single-user channel coding

with antenna sharing method as described in [4]. The practically interesting cases of ind and tot all into this re gion. In both of these cases, the minimum achie able xpected distortion decreases to zero at the rate of (log (45) Hence, the po wer constraint tot performs as well as ind “order -wise”, and therefore, in practice we may prefer to choose tot In the small sum po wer constraint re gion where ranges from to 1+ our lo wer and upper bounds do not meet. The lo wer bound decreases to zero as (log ut the upper bound is non-zero constant. The main discrepanc between the lo wer and upper

bounds comes from the gap between the lo wer and upper bounds on the sum capacity and for cooperati multiple access channel with feedback. This re gion should be of practical interest because in this re gion, the sum po wer constraint is quite lo and yet the lo wer bound on the distortion is of the same order as an which increases polynomially with Hence, from the results of the lo wer bound, it seems that this re gion potentially has good performance. Ho we er our separation-based upper bound does not meet the lo wer bound, and whether the lo wer bound can be achie ed remains an open problem.

In the very small sum po wer constraint re gion, is less than and our lo wer and upper bounds meet and the minimum achie able xpected distortion is constant that does not decrease to zero with increasing This case is not of practical interest because of the unacceptable distortion. in estigate the performance of dense sensor netw orks by studying the joint source-channel coding problem. pro vide lo wer and upper bounds for the minimum achie able xpected distortion when the underlying random process is stationary and Gaussian. When the random process is also Mark vian, we aluate the lo wer and

upper bounds, and sho that the are both of order log( ))) for wide range of sum po wer constraints ranging from 1+ to In the most interesting cases when the sum po wer gro ws linearly with or is constant, the minimum achie able xpected distortion decreases to zero at the rate of log or Gauss- Mark process, under these po wer constraints, we ha found that an order -optimal scheme is separation-based scheme, that is composed of distrib uted rate-distortion coding [17] and antenna sharing method for cooperati multiple access channels [4]. xpect our results to be generalizable to more general

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## Optimal DistortionPo wer radeof fs in Sensor Netw orks GaussMark Random Processes Nan Liu Sennur Ulukus Department of Electrical and Computer Engineering Uni ersity of Maryland Colle ge ark MD nkanc

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