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SIGNAL IEEE PROCESSING LETTERS, VOL. 12, NO. 10, OCTOBER 2005 709 DOA Estimation via a Network of Dumb Sensors Under the SENMA Paradigm Stefano Marano, Vincenzo Matta, Peter Willett, and Lang Tong Abstract Following the SENMA concept, we consider a wireless network of very dumb and cheap sensors, polled by a travelling rover. Sensors are randomly placed and isotropic: Individually, they have no ability to resolve the direction of arrival (DOA) of an acoustic wave. However, they do observe the wavefront at different times. We assume that the communication load must be as limited as possible, so that these times cannot be communicated to the rover. Notwithstanding the lack of transmission of arrival times and the lack of DOA resolution ability of the individual sensors, DOA estimation is possible and simple, and asymptotic ef ciency becomes closely approximated after a reasonable number of rover snapshots. Key features are the directionality of the rover antenna, the area it surveys, and the average number of sensors inside that area, as accorded a Poisson distribution. Index Terms Data fusion, direction of arrival (DOA), sensor network. I. INTRODUCTION A LARGE network of extremely low-complexity (a.k.a., dumb ) sensors is employed to estimate the direction of arrival (DOA) of a plane-wave (far- eld, and for concreteness, let us assume acoustic) event. The system is designed to detect the wavefront passage regardless of the signal waveform features. The sensors are isotropic: None of them has any ability at all to resolve the DOA on its own; however, each can memorize the time instant of the acoustic wavefront passage. The sensors are randomly displaced over a certain surveyed area according to a Poisson eld model, as might occur were the sensors dropped by an aircraft in an unstructured way. According to the SENMA model, a travelling rover receives (electromagnetic) signals from the sensors that lie in its eld of view. As is well known, a distinct feature of sensor networks is the tradeoff between the communication load, the requirement to fuse the data, and the accuracy of the network inference goal (e.g., detection of events, parameter estimation, etc.) (e.g. see [3]). We avoid any concern about the communication burden: All of the sensors transmit to the rover using one and the same channel. The key point is that they do not transmit bits of data but simply emit an analog periodic signal made of short pulses. In aggregate, they form a train of delta-like pulses, and this is what Manuscript received July 8, 2004; revised September 8, 2004. This work was supported in part by the Of ce of Naval Research. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Steven M. Kay. S. Marano and V. Matta are with DIIIE, Universit degli Studi di Salerno, I-84084 Fisciano, Italy (e-mail: marano@unisa.it; vmatta@unisa.it). P. Willett is with the Electrical and Computer Engineering Department, University of Connecticut, Storrs, CT 06269 USA (e-mail: willett@engr.uconn.edu). L. Tong is with the Electrical and Computer Engineering Department, Cornell University, Ithaca NY 14853 USA (e-mail: ltong@ece.cornell.edu). Digital Object Identi er 10.1109/LSP.2005.855543 Fig. 1. Addressed scenario. A travelling rover polls the remote sensors inside its eld of view. The DOA estimation procedure is based on the number of sensors that lie inside the strip of width . The separate box introduces some notations: is the sought DOA, and is the rover orientation, whose eld of view is an ellipse with axes r and h. the rover observes. The directionality of the rover antenna is key, and in fact, the more asymmetric is the antenna lobe, the more effective the estimation procedure becomes;1 however, there are limits, as discussed in the following. We nd it convenient to work with a reasonable, simple mathematical model: the rover s antenna pattern probably in practice some sort of truncated cone is modeled as an ellipse. There is no requirement for an elliptical eld of view, only that the eld of view is known; the ellipse makes analysis convenient and explicit. II. MODEL The notional scenario is depicted in Fig. 1. We consider a large , covering a cernetwork of wireless sensors, say, , tain two-dimensional region. The sensors are randomly located according to a Poisson eld probability model, with being the sensor density per unit of area: The average number of sensors, . inside an arbitrarily shaped region of area , is We assume that each is an acoustic antenna, with no directionality capabilities: Its antenna pattern is isotropic. If hit by a short-duration acoustic wavefront, coming from an arbitrary direction, sensor starts to transmit an electromagnetic periodic , where is a short pulse of signal arbitrary shape, is the time at which the acoustic wavefront 1The sensors are dumb and isotropic; the rover is not. 1070-9908/$20.00 2005 IEEE 710 IEEE SIGNAL PROCESSING LETTERS, VOL. 12, NO. 10, OCTOBER 2005 Fig. 2. Signal collected by the rover is schematically depicted in the top plot. With an appropriate observation interval (larger than twice the pulse period T ), it is possible to order the pulses: They are arranged in the same order in which the sensors have emitted them (i.e., have been hit by the acoustic wavefront); see bottom plot. Here, t is the middle point between and , and the =v-interval centered on t includes the two pulses at and . These are emitted by the two sensors inside the -strip of Fig. 1. absolute times, nor it is capable of associating any pulse with its corresponding sensor position inside its eld of view. In this letter, we propose the following suboptimal approach, which is simple to understand and easy to implement. Again with reference to Fig. 2, consider the rst and the last received echoes . Then, count how many pulses and compute , , with lie in the interval being the time for covering an acoustic distance . Let be this number, where is the snapshot index; our estimation procedure is based on the observables . We understand that more sophisticated strategies are possible: One might exploit more complete ) contained in the received information (as compared to just . Examples will be offered in [2], and we note that one such . example exploits the DOA information embedded in The receiving antenna of the rover can be arbitrarily oriented, or alternatively, the rover rotates. In both cases, it may explore the whole arc, for any given position. A key assumption made here is that successive snapshots taken by the rover always involve sensors never encountered before (i.e., snapshot independence). For analysis, it is suf cient to take each sensor as having a periodic emission; for practical battery life, they would be silent unless provoked by a rover poll. III. DOA ESTIMATION AND PERFORMANCES impinges on the device, is an a priori chosen time interval, common to all sensors, and is an integer. According to the SENMA paradigm [4], a roving base station (rover in Fig. 1) travels the area. For some xed position, the rover takes a dwell: It collects, for a certain time interval,2 the signal . Such a received signal is made of the supersignals emitted by the ensemble of sensors position of the lying in its eld of view. It is worth emphasizing that for reasons of analysis, one might assume that the sensors emit their pulse trains continually; but for reasons of battery life, the emissions would remain virtual until a rover requests them via a poll. We assume that the antenna pattern is an ellipse with the main axis of length aligned to the rover and the secondary one of length . It is also assumed that all signals coming from sensors inside the ellipse are visible, while conversely, none from outside can be received; this de nes the rover s eld of view.3 A typical waveform received by the rover is schemati(larger cally illustrated in Fig. 2. Note that we set works as well), where is the speed of the acoustic wave in the is the time needed for the acoustic wave to cover medium: the main axis of the lobe. Such a choice enables the rover to , since sensors inorder the observed pulses so that side the eld of view all have a maximum time interval and of since the pulse period is twice that. This is illustrated in Fig. 2, where the zoomed time axis (bottom) gives the correct pulse ordering.4 Clearly, the sensor positions are unknown, and the pulses are unlabeled: The rover is neither able to recover the 2A be the unknown DOA and be the Let rover s (ellipse s) orientation at snapshot . Assume that , , are independent of each other, and note that this number can be approximately taken as the number of sensors that lie in the strip of width within the ellipse5 (see Fig. 1). For the sake of simplicity, such a region is taken as rectangular: One side is given by the ellipse s diameter (corresponding to DOA and rover angle ), and the other side is . The area is accordingly computed as (1) The basic idea behind the proposed DOA estimation procedure is small and is small is that if is close to , then as well; conversely, when is orthogonal to , there is a larger contains inforarea and, consequently, a larger . That is, is a Poisson random varimation about . More precisely, . Accordable whose average value is approximately ingly, the distribution of the aggregate of observables independent snapshots is known, and from that, collected in the -ML (maximum likelihood) estimation can be numerically computed6 minimum interval of 2T can be shown to be suf cient. 3Thus, again with reference to Fig. 1, the elliptical eld of view is actually the combination of the antenna pattern and of the maximum transmitting distance of the signal emitted by the sensors. 4Actually, the depicted times should be + kT for some k; we write for simplicity; we are interested only with time differences. Note also that considering acoustic DOAs avoids possible concerns about synchronization between sensors and rover. 5Should and be generated by sensors located on opposite boundaries of the rover s eld of view, this would be true. Accordingly, the greater the sensor eld density , the better the approximation works. 6We would like to stress that different antenna patterns would simply lead to different formulas for A( ; ). Clearly, the proposed method is applicable to different patterns, with only some (presumably minor) numerical difference in the correspondent performances: The key is not the exact shape of the pattern but rather its eccentricity. MARANO et al.: DOA ESTIMATION VIA NETWORK OF DUMB SENSORS 711 As grows, the well-known asymptotic properties of ML estimation [5] become met. In particular, , and VAR , where is the -snapshot Fisher information with respect to . As we shortly show, numerical investigations con rm that such performances are, in practice, (see below). Thus, attained for moderately large values of is relevant and is now in order. computation of is additive for independent observaFirst, note that , where is the Fisher tions i.e., information from snapshot . Thus, de ning , as a shortcut for the Poisson distribution and using with argument and mean , one gets The following approximation is justi ed by a standard Monte Carlo integration approach, amounting to replacing the arithmetic mean by the statistical expectation with respect to , this latter assumed uniformly distributed in (0, ): large Fig. 3. Variance of compared to the inverse of Fisher information versus the total number of snapshots taken M . Four combinations of the relevant parameters are addressed. The arrows on the horizontal axis denote the points after which bias in the estimate becomes negligible, i.e., E[ ] . In the limit as and the opposite extreme of (2) , and the elIn the last equality, we have de ned lipse s aspect ratio has been denoted as . The function is expressible in terms of complete elliptic integrals of the and , respectively (see [1, form. rst and second kind, 17.3.1, 17.3.3] for the de nitions). In fact From (2), we see that is constant with , grows linearly with and with , and further depends upon . Some comments follow. Asymptotically, all the values can be estimated with the same accuracy. represents the effective area of the visible region; that is to say, it is the larger area available for the clustered is the avsensors counting process. Accordingly, erage number of sensors inside such region (the effective ). number decreases in , implying that the more eccentric the rover s eld of view (ellipse), the more effective the estimation of the DOA. , Ideally, for a prescribed , one would have with the product held xed. However, cannot increase without bound for obvious reasons ( is proportional to the maximum transmitting distance of sensors), nor can ; that is, and cannot be assigned we have independent of each other. The Fisher proxy is reasonable, provided that attains its large-sample optimality: and VAR . Also, recall that there are approximations in the proposed model: One is the way we computed the area in (1). We have simulated, with the double aim of checking the the approximations and of investigating at what values of asymptotic performances seem to be attained. In the simulations, an ellipsoidal eld of view is used, for simplicity and to correspond to the explicit bounds. In Fig. 3, the variance is compared to the inverse of Fisher of the estimator information, as given in (2). We see that within a reasonable number of snapshots, the asymptotic performances are met.7 The down-arrows on the horizontal axis denote the point after . which the absolute value of the estimator bias stays below and , we have To check the approximations, for given run simulations using different combinations of the relevant parameters , , , and . Qualitatively, the results are close to those given in Fig. 3. For instance, in this way, we have veri ed that the speed of convergence of the variance to its asymptote : In Fig. 3, we have is essentially insensitive to the ratio , but doubling this value basically yields chosen the same results.8 In summary, the simulations corroborate the analysis and validate the analytical relationships. 7In judging the practical impact of M , recall that the number of different rover locations is just M divided by the number of snapshots taken in a xed position. 8Clearly, for h, the analytical approximation behind (1) fails. 712 IEEE SIGNAL PROCESSING LETTERS, VOL. 12, NO. 10, OCTOBER 2005 IV. SUMMARY We have investigated the DOA estimation by a network of isotropic sensors polled by a travelling rover, with the system design based upon the SENMA paradigm. The novelty is that the sensors are unusually dumb in that they have individually no DOA capability; indeed, they have no capabilities at all except that of emitting a periodic signal following their encounter with the wavefront whose DOA is sought. The sensors are inexpensive and randomly located, they do not communicate with each other, and their positions are unknown both to them and to the rover. The idea is that they send a periodic train of short pulses that starts at the time instant that the sensor is hit by an acoustic wave of short duration. (Actually, the physical transmission to the rover is virtual until the rover polls the sensor.) The key point is that the rover s eld of view is eccentric (taken here as elliptical, but that is only for ease of analysis). The DOA information is contained in the number of sensors within a stripe in the rover s eld of view and oriented orthogonal to the DOA. This number is taken as Poisson distributed. The results are remarkably good, and asymptotically ef cient performance is obtained with a reasonable number of snapshots taken by the rover. REFERENCES [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York: Dover, 1964. [2] S. Marano, V. Matta, P. Willett, and L. Tong, Support-based and ML approaches to DOA estimation in a dumb sensor network, IEEE Trans. Signal Process., submitted for publication. [3] P. Willett and L. Tong, One aspect to cross-layer design in sensor networks, in Proc. MILCOM, Monterrey, CA, Oct. 2004, pp. 688 693. [4] L. Tong, Q. Zhao, and S. Adireddy, Sensor networks with mobile agents, in Proc. MILCOM, Boston, MA, Oct. 2003. [5] H. L. Van Trees, Detection, Estimation and Modulation Theory. New York: Wiley, 1968, pt. 1.

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