Analysis of UWB Radar Sensor Networks
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2010 proceedings
Analysis of UWB Radar Sensor Networks Stefania Bartoletti, Andrea Conti, and Andrea Giorgetti
Abstract—Radar sensor networks (RSNs) are gaining importance in the context of passive localization and tracking. The performance of RSNs is affected by disturbances, system’s parameters, network topology, and the number of radar elements. In this paper, we derive a unified analytical framework that takes all this aspects into account and allows the derivation of probability of detection and localization uncertainty. The results enable the system designer to have a clear understanding on the effects of each system parameter and the trade-off between performance and complexity. Moreover, the potential for highaccuracy passive localization of ultrawide bandwidth (UWB) systems is shown.
I. I NTRODUCTION The need for efficient security systems is increasing in the last years and this reflects into a fervent research activity worldwide. For this reason, passive indoor localization is gaining importance thanks to the wide variety of its possible applications in private and public environments supervision. In this context radar systems show good capability in terms of detection, localization, and tracking within an area of interest called surveillance area (SA). A monostatic radar is composed of a single device that emits and receives pulses, whereas in a bistatic radar (BR) transmitter and receiver are not co-located [1]. A multistatic radar (MR) is a generalization of the BR with at least one transmitter and more than one receiver [2]. Radar sensor networks (RSNs) are essentially MRs systems. Cluttered propagation environment, walls, and objects outside the SA deteriorate the system performance and increase the risk of false alarms [3]–[5]. In the last years several works deal with techniques to improve radar systems. Ultrawide bandwidth (UWB) communications [6], [7] are robust to clutter [8] and interferences. Several studies, such as [9], show that these advantages are due to UWB features (short duration of pulses, ability of penetrating obstacles of various materials, large bandwidth). The occupation of a large band, in conformity with the Federal Communications Commission (FCC), improves resolution making UWB suitable for different applications such as position estimation through different techniques [10], [11], image reconstruction [12] and object imaging such as UWB tomographic radars [13]. For these reasons this technology is intended for many applications such as surveillance, disaster relief, and tracking of moving objects [14]. S. Bartoletti and A. Conti are with ENDIF University of Ferrara and WiLAB c/o University of Bologna, Italy, e-mail: [email protected]. A. Giorgetti is with DEIS University of Bologna and WiLAB, Cesena (FC), Italy, e-mail: [email protected]. This work has been supported in part by the European Commission in the scope of the FP7 ICT integrated project CoExisting Short range Radio by Advanced Ultra-WideBand radio Technology (EUWB) Grant no. 215669.
Error sources for ranging and thus affecting localization are given by [15]: multipath propagation also known as clutter (that depend on signal bandwidth [16]), clock drifts [17] and interferences [18]. Several studies have been performed to suppress static clutter choosing an appropriate system configuration [4], [5]. To take into account these error sources when developing system model, several error bounds have been defined for the most common types of signal and for all ranges of received signal-to-noise ratio (SNR) values [19]. The above mentioned sources of ranging errors result in uncertainty of target position evaluation. Several methods for analysis of positioning accuracy have been proposed (see, e.g., [20], [21]). The network topology influences system behaviour and performance; in particular, the capability of succeeding in detecting single or multiple intruders and the precision in evaluating its location depends also on number of nodes and their placement [22], [23]1 For this reason choosing the best network topology is one of the main goals for an effective system design to achieve the target performance with minimum complexity. In [24] some configurations of MRs are analyzed to improve system performance through an adequate placement of nodes. The performance is evaluated in terms of target resolutions through shaping the ambiguity function. A different approach based on simulations is presented in [25] for circular SA and radar elements on the border. In this paper, we propose a methodology to analyze MRs through a rigorous mathematical framework. Differently from previous works, such methodology is quite general to be applied to arbitrary configurations; in addition it allows to consider the effects of several system parameters such as: signal bandwidth, transmitted power, number of receivers, SNR etc. In this work, we focus our attention on positioning accuracy and coverage area. Finally, our mathematical model is compared with numerical results in order to prove its consistency. In the remainder of the paper, Section II contains a description of configurations and assumptions, Section III describes the methodology to compare different configurations in terms of localization uncertainty area and percentage of SA covered, Section IV shows numerical results and, in V we give our conclusions. II. S YSTEM M ODEL AND A SSUMPTIONS We consider a single transmitter and multiple receivers scenario. Each transmitter/receiver pair can be considered as 1 In addition, the a priori knowledge about topology of nodes and intra-node cooperation influence localization accuracy [21].
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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2010 proceedings
a BR so we start considering BRs to build a mathematical model of MRs. A BR system is composed of a transmitter that emits a sequence of pulses and a receiver that receives the pulses: either directly from the transmitter or reflected from any obstacle (clutter) encountered during the propagation [25]. The time-of-arrival (ToA) represents the time of flight during which the pulses cover the transmitter-target-receiver path. Therefore, if ToA is known, the target is on an ellipse with foci at the transmitter and the receiver, respectively [25]. Assuming a Cartesian coordinate system, the ellipse is described by the following equation 2 2 2 2 d = (x − Tx ) +(y − Ty ) + (x − Rx ) +(y − Ry ) , (1) where (Tx , Ty ) and (Rx , Ry ) are the transmitter and receiver coordinates, respectively. If we suppose complete clutter removal the signal received is composed by only two pulses: the direct one via transmitterreceiver path and the reflected one via transmitter-targetreceiver path. Therefore a single pulse transmission causes two received pulses. The receiver can resolve paths only if they are sufficiently separated in time. The required temporal separation depends on receiver specifications and provides a minimum distance at which a target can be detected which results in a minimum ellipse [25]. If the target is inside the minimum ellipse the receiver cannot resolve the two pulses. The minimum temporal distance necessary to resolve two paths is called minimum resolvable delay γ. The minimum distance detected is given by (2) Dmin = (Rx − Tx )2 + (Ry − Ty )2 + γc . A target located in pT = (xtarget , ytarget ) is detectable if it is outside the minimum ellipse, thus Dmin < (xtarget − Tx )2 + (ytarget − Ty )2 + (xtarget − Rx )2 + (ytarget − Ry )2 . (3) Beside the minimum ellipse, the coverage depends also on the minimum signal-to-noise ratio (SNR) required at the receiver in order to detect the target. Accordingly to [25], we define the SNR as Ns PR (4) SNR N0 PRF where Ns is the number of pulses collected to perform a target detection and ToA estimation (often referred as scan period).2 Without loss of generality we choose Ns = 1 since when coherent reception is considered the transmitted power can be reduced by a factor Ns when Ns > 1 to have the same RSN performance. PRF is the pulse repetition frequency, N0 denotes the one-sided noise power spectral density (PSD) and PR /PRF represents the received energy per pulse scattered by the target. 2 As explained in [25], [26] depending on the scenario and the application, during a scan period the RSN can collect energy from many pulses in order to achieve the necessary area coverage and detection probability.
Based on the radar equation, it follows a limit on received power due to target reflection that is inversely proportional to the product of the distances between the radar antennas and target [1]. In particular, considering a transmitted signal with flat PSD over [fL , fL + B], with fL the lower spectrum frequency and B the signal bandwidth, and assuming constant antenna gains, GT and GR , and constant radar cross section (RCS), σ, over the signal bandwidth, the received power scatterd by the target can be written as [25] ST GT GR σc2 1 1 (5) − PR = (l1 l2 )2 (4π)3 fL fL + B where ST is the one-sided transmitted PSD, l1 and l2 are the distances between target and transmitter and receiver, respectively, and c is the speed of light. Note that the assumptions on frequency independent characteristics of antenna gains, RCS and transmitted PSD in (5) can be easily removed. In this paper we made such assumptions in order to simplify the notation without affecting the main contribution of the analysis developed since despite such assumptions (5) capture all the essential parameters. To ensure reliable target detection and localization accuracy a SNR at the receiver, SNRth , must be guaranteed. Starting from (4) and (5), the corresponding minimum received power, PR,th , provides a maximum distance product (l1 l2 )∗ that is a limit on BR coverage, with 1 ST GT GR σc2 1 ∗ . (6) − (l1 l2 ) PR,th (4π)3 fL fL + B We are interested to know the area covered by the considered BR, that is also that one from which signals backscattered satisfy the SNR requirement. The locus of points having a constant product of the distances measured from two fixed points is called Cassini oval and its shape depends on distances product itself [25]. Starting from Cassini oval equation, a target located in a point of coordinate pT is detectable by the considered BR if the following inequality is verified 2
(l1 l2 )∗ >[(xtarget − Tx )2 + (ytarget − Ty )2 ] ×[(xtarget − Rx )2 + (ytarget − Ry )2 ] .
(7)
Finally the coverage area of the BR is the area inside its maximum Cassini oval and outside its minimum ellipse as above defined. Let us suppose we want to employ a BR to monitor a circular area of radius R, and we assume that targets can be located only inside this SA. The detection probability represents the probability that target is inside the coverage area. If the coverage area is a portion of the area we want to supervise, the detection probability can be calculated as the ratio between coverage area and SA. Differently, if the coverage area is not completely inside the SA there is a part of the coverage area that we are not interested to supervise. The detection probability, in this case, is the ratio between the intersection of the coverage area with the SA and the SA
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2010 proceedings
itself. If S represents the set of points (then possible target positions) inside SA, then
S p s.t. x2 + y 2 < R
2
(8)
where p = (x, y) ∈ R2 . If C and E represents the area inside the maximum Cassini oval and outside the minimum ellipse, then C p s.t. [(x − Tx )2 + (y − Ty )2 ] 2 × [(x − Rx )2 + (y − Ry )2 ] − (l1 l2 )∗ < 0 (9) and 2 2 E p s.t. (x − Tx ) + (y − Ty ) 2 2 + (x − Rx ) + (y − Ry ) − Dmin > 0 .
(10)
Let us consider the characteristic function 1B (p) for a set B which has value 1 if p ∈ B, and 0 otherwise.3 Therefore, the detection probability results
1C (p) · 1E (p)dp , (11) PD = Ep[1C∩E (p)] = S m2 {S} where Ep[z] denotes the expected value of z with respect to all possible target positions p. If the target position is uniformly distributed inside the SA, (11) can be interpreted as m2 {C ∩ E ∩ S} /m2 {S} in the general case or m2 {C ∩ E} /m2 {S} when the coverage area is completely inside the SA. Since in a MR each transmitter/receiver pair can be considered as a BR its own, the coverage area of a MR coincides with the intersection between the coverage areas of all BRs components. Let us suppose N receivers detecting the target. The possible target positions are the points of intersection between the N ellipses having foci in the transmitter and each of N receivers, respectively. Thus, the target is localized if the system of equations (ellipses) has an unique solution, otherwise we have an ambiguity on target position. It follows that three receivers at least are needed to localize the target (the system of equations has a unique solution), whereas one to detect it (for detection purposes no matter how many solutions the system have). III. P ERFORMANCE M ETRICS To analyze system performance, we focus our attention on the two main aspects of RSNs: the probability to detect and localize the target, and the uncertainty associated to localization. Differently from other studies, a mathematical model to evaluate system performance is developed.
define the n-dimensional measure of a set B as mn {B} = 1B (p)dp.
3 We B
A. Probability of Detection Since a MR is a BR generalization, in order to derive the performance of a RSN it is necessary to generalize (9) and (10) to the MR case, while (8) remains the same. We consider that all N receivers of the RSN are identical except for their position in the scenario. In the RSN each transmitter/receiver pair covers a different area defined by its own Cassini oval and minimum ellipse. The maximum Cassini oval and the minimum ellipse for the (k) (k) kth receiver with coordinates (Rx , Ry ) with k ∈ {1, ..., N } are represented by the sets Ck and Ek as Ck p s.t. (x − Tx )2 + (y − Ty )2
2 (12) × (x − Rx(k) )2 + (y − Ry(k) )2 − (l1 l2 )∗ < 0 and
2 2 Ek p s.t. (x − Tx ) + (y − Ty ) (k) 2 (k) 2 + (x − Rx ) + (y − Ry ) − Dmin > 0 .
(13)
It follows that each transmitter/receiver pair has its own coverage area represented by Ak = Ck ∩ Ek ∩ S .
(14)
Since a target can be detected only if it is inside the coverage area of at least one transmitter/receiver pair, the detection probability for the RSN becomes
N 1 (p)·1 (p)dp C Ek , (p) = S k=1 k PD = Ep 1 N k=1 Ak m2 {S} (15) where stands for summation with boolean OR rule. When the actual target position is uniformly distributed in the SA, N (15) can be rewritten as m2 k=1 Ak /m2 {S}. To find an expression for the probability that a target is detectable by k receivers, it is essential to understand which part of our SA is covered by k and only k receivers, that is the area covered by k receivers at least removing the area covered by k + 1 receivers at least. The area covered by k receivers at least is the union of areas covered by each k-ple of the group composed of all transmitter/receiver pairs (if N is the number of receivers, there are Nk possible k-ples). Therefore, the area Rk covered by k receivers, results k N −k+1 N N −k+2 (16) ... Aip Rk = i1 =1
i =i +1
i =i
+1 p=1
k k−1 ⎧ 2 1 N −(k+1)+1 N −(k+1)+2 ⎨ ... ∩ NOT ⎩
i1 =1
i2 =i1 +1
N
k+1
ik+1 =ik +1 q=1
Aiq
⎫ ⎬ ⎭
.
Figure 1(a) shows the areas Rk for k = 1, 2, 3, 4 (respectively red, orange, green and yellow area) for a particular configuration with N = 4.
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N = 3 receivers, that is the minimum number of receivers needed to succeed in localization. The area covered by one receiver at least is the area in which targets can be detected. Differently, the coverage area is the area in which three receivers (then all receivers) can detect and localize the target. For this configuration, we have R1 = A1 ∪ A2 ∪ A3 , R2 = (A1 ∩ A2 ) ∪ (A1 ∩ A3 ) ∪ (A2 ∩ A3 ) ∩ NOT {A1 ∩ A2 ∩ A3 } and R3 = A1 ∩ A2 ∩ A3 . In this case, PL results PL =
m2 {A1 ∩ A2 ∩ A3 } . m2 {S}
(20)
C. Uncertainty Area
(a) Rk , k = 1, 2, 3, 4.
(b) Uncertainty area (red spot). Fig. 1. Examples of area coverage and uncertainty with R = 1 and N = 4. The single transmitter is in the centre of the circular SA and the four receivers on its circumference.
B. Probability of Localization A target is localized when it is located inside the coverage areas of three transmitter/receiver pairs at least, that is the coverage area of the system A is given by A
N
Rk ,
(17)
k=3
where Rk is derived in (16). Thus, the localization probability PL is (18) PL Ep[1A (p)] which simplifies into PL =
m2
N k=3
Rk
m2 {S}
(19)
in the case of uniform probability of target location inside the SA. Let us consider, for example, the configuration with
An uncertainty of ToA causes an uncertainty of the distance estimate between each TX/RX pair and target. Thus, the detection ellipse becomes an annulus, whose thickness depends on uncertainty value: therefore the intersections between ellipses (that in ideal conditions provide target coordinates) become the intersections between annuluses that are no longer a single points but a set of points called uncertainty area [25]. The measure of uncertainty area depends on several factors, such as the number of receivers detecting the target, the transmitted pulse shape, the SNR and also the target position itself with respect to the system configuration adopted. Several studies have been performed to find bounds on the variance of ToA estimation for different pulse shaping and SNR in AWGN and multipath channel [15]. An example of uncertainty area for 3 out of 4 receivers detecting the target is shown in Fig. 1(b). In fact, if the target is detected by a single transmitter/receiver pair the uncertainty area corresponds to the intersection between the detection annulus area and the SA. Differently if two receivers detect the target, the uncertainty area corresponds to the intersection between two annuluses resulting in general to two distinct areas. If k is the number of transmitter/receiver pairs detecting the target and k ≥ 3, the target is localized and the uncertainty area decreases as k increases. Let us consider the target in position pT detected by the j th receiver with a ToA estimate in the uncertainty interval [τj (pT ) − /2, τj (pT ) + /2], where τj (pT ) is the ToA corresponding to the actual target position and specify the ToA error range. The error in ToA estimate leads to a detection annulus defined as the area between the two ellipses corresponding to the extremum of the uncertainty interval. Therefore, the detection annulus can be represented as the set Ej+ (pT ) ∩ Ej− (pT ) ∩ S where Ej+ (pT ) and Ej− (pT ) are defined as 2 2 + (21) Ej (pT ) p s.t. (x − Tx ) + (y − Ty ) (j) 2 (j) 2 0 . + (x − Rx ) + (y − Ry ) − c τj (pT ) − 2
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2010 proceedings
The region around pT is therefore Uk (pT ) = uncertainty + E (p ) ∩ Ej− (pT ) ∩ S and can be represented by T j∈Ok j 1Uk (pT ) (p) = j∈Ok
1E + (pT ) (p) · 1E − (pT ) (p) · 1S (p) j
j
(23)
with Ok = {k receivers out of N detecting the target}. The corresponding uncertainty area is ! 1Uk (pT ) (p)dp . (24) AU,k (pT ) = m2 {Uk (pT )} = S
IV. N UMERICAL R ESULTS We present results on detection, localization probability, and localization uncertainty, using the proposed general framework applied to UWB RSNs.4 For example, the values of system parameters chosen for a SA of radius R = 50 m are: fL = 5 GHz, B = 500 MHz, GT = GR = 1, σ = 1 m2 , ST /PR,th = 7500 Hz−1 (if not otherwise specified), and a ToA error range such tha the annuluses thickness is 0.36 m. Figure 2 shows the detection probability and the localization probability as a function of the number of receivers (from 3 to 10) and ST /PR,th = 2500 and 7500 Hz−1 for the configuration with the transmitter in the centre of the SA and all receivers on its circumference at a constant angular distance each other. The effects on PL of the transmitter position is shown in Figure 3, where the PL is reported as a function of the x coordinate of the transmitter position ranging from the center of the SA, Tx = 0, to Tx = 2R. Since the uncertainty area depends on the actual target position, we develop a statistical analysis of uncertainty area considering random target position inside the SA. We define a constant ToA estimation error and we calculate the uncertainty area for different target positions with (24). Note that, for each target position considered in the calculation, there is a different number of receivers, k, capable to detect the target depending upon the target belongs to the region Rk (16). In particular, if a target is moving along the radius of the SA from the centre to the circumference it encounters all Rk as k varies from N to 0. If we assume that SNR does not change significantly inside the coverage area of each transmitter/receiver pair, then only the number of receivers k detecting the target influences the uncertainty area. Since k is proportional to target radial coordinate in a polar coordinate system, we calculate uncertainty area of 20 target positions, each one having radial coordinate respectively equal to i 20/R with i ∈ {1, ..., 20}. In Figure 4 the cumulative distribution function (CDF) of uncertainty area that resulted very similar to that one obtained by computation of 1000 target random positions.5 To emphasize the dependence of uncertainty CDF on k we repeat the comparison descripted above between an analytical and derived CDFs after decreasing the ratio ST /PR,th from 4 All
system parameters are scaled to the radius R of the circular SA. define the whole SA as the conventional value of uncertainty if the target is not detectable, even though the real value should be infinity. 5 We
7500 to 2500 Hz−1 (i.e., by increasing the probability that the target is detected by k < N receivers). Figure 4 shows, in this case, two saturation levels corresponding to two impossible ranges of uncertainty area values. To find the best configuration in terms of uncertainty area, Figure 5 shows the uncertainty CDF for different RSN configurations. The first configuration is composed by the transmitter in the centre of the SA and all receivers on its circumference at a constant angular distance. The second one consists of all RSN nodes (including the transmitter) located on the circumference still at a constant angular distance. For both configurations we changed the number of receivers N ∈ {3, 4, ..., 8}. Figure 5 has been obtained keeping constant the transmitter power at the value that allows the worst case topology (three receivers and transmitter on the circumference) to cover at least the 90% of the SA. V. C ONCLUSIONS In this paper a mathematical framework for the analysis of RSNs based on MR is proposed. It accounts for the main system parameters and impairments and enables the derivation of the performance in terms of detection probability inside a surveillance area and uncertainty area around the estimated target location. The framework has been applied to study the performance of a RSN based on UWB technology varying several parameter such as: number of receivers, transmitted power to receiver sensitivity ratio, and network topology. Results, here specialized for circular SA show the superiority of the topology with the transmitter in the centre of the SA and all receivers located on its circumference at a constant angular distance each others. ACKNOWLEDGMENT The authors wish to thank M. Chiani for helpful discussions. R EFERENCES [1] M. I. Skolnik, “An analysis of bistatic radar,” IEEE Trans. on Aerospace and Navigational Electronics, pp. 19–27, 1963. [2] V. S. Chernyak, “Fundamentals of multisite radar systems,” Gordon and Breach Science, 1998, amsterdam, The Netherlands. [3] F. Ahmad and M. Amin, “Noncoherent approach to through-the-wall radar localization,” IEEE Trans. on Aerospace and Electronic Systems, vol. 42, pp. 1405–1419, 2006. [4] L. M. H. Ulander and T. Martin, “Bistatic ultra-wideband SAR for imaging of ground targets under foliage,” Proc. IEEE Int. Conf. on Ultra-Wideband (ICUWB), 2005. [5] S. Nag and M. Barnes, “A moving target detection filter for an ultrawideband radar,” Proc. IEEE Radar Conference 2003, pp. 147–153, May 2003, huntsville, Ala, USA. [6] M. Z. Win and R. A. Scholtz, “Ultra-wide bandwidth time-hopping spread-spectrum impulse radio for wireless multiple-access communications,” IEEE Trans. on Comm., vol. 48, pp. 679–691, Apr 2000. [7] R. A. Scholtz and M. Z. Win, “Impulse radio: how it works,” IEEE Commun. Letters, vol. 2, pp. 36–38, Feb 1998. [8] D. Dardari and M. Z. Win, “Threshold-based time-of-arrival estimators in UWB dense multipath channels,” Proc. IEEE Int. Conf. on Commun. (ICC), vol. 10, pp. 4723–4728, Jun 2006, istanbul, Turkey. [9] V. Chernyak, “Multisite ultra- wideband radar systems with information fusion: Some principal features,” European Radar Conference, pp. 141– 144, Oct. 03, paris. [10] H. Wymeersch, J. Lien, and M. Z. Win, “Cooperative localization in wireless networks,” Proc. of the IEEE, vol. 97, no. 2, pp. 427–450, Feb 2009.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2010 proceedings
[11] S. Gezici and H. V. Poor, “Position estimation via ultra-wideband signals,” Proc. of the IEEE, vol. 97, no. 2, pp. 386–403, Feb 2009. [12] M. Sato and K. Yoshida, “Bistatic UWB radar system,” Proc. IEEE Int. Conf. on Ultra-Wideband (ICUWB), 2007. [13] L. Jofre, A. Broquetas, J. Romeu, S. Blanch, A. P. Toda, X. Fabregas, and A. Cardama, “UWB tomographic radar imaging of penetrable and impenetrable objects,” Proc. of the IEEE, vol. 97, no. 2, pp. 451–464, Feb. 2009. [14] P. K. Dutta, A. K. Arora, and S. B. Bibyk, “Towards radar enabled sensor networks,” Proc. of the Int. Conf. on Information Proc. in Sensor Networks, 2006, new York, NY, USA. [15] D. Dardari, A. Conti, U. Ferner, A. Giorgetti, and M. Win, “Ranging with ultrawide bandwidth signals in multipath environments,” Proc. of the IEEE, vol. 97, pp. 404–426, 2009. [16] A. F. Molisch, “Ultra-wide-band propagation channels,” Proc. of the IEEE, vol. 97, no. 2, pp. 353–371, Feb. 2009. [17] F. H. Sanders, R. L. Sole, B. L. Bedford, D. Franc, and T. Pawlowitz, “Effects of RF interference on radar receivers,” NTIA Report TR-06-444, Sep. 2006, http://www.its.bldrdoc.gov/pub/ntia-rpt/06-444/. [18] A. Giorgetti, M. Chiani, and M. Z. Win, “The effect of narrowband interference on wideband wireless communication systems,” IEEE Trans. on Comm., vol. 53, no. 12, pp. 2139–2149, Dec. 2005. [19] H. L. V. Trees, K. L. Bell, and Y. Wang, “Bayesian Cram´er-Rao bounds for multistatic radar,” Proc. Int. Waveform Diversity and Design Conference, Jan. 2006, lihue, Hawaii, USA. [20] M. Aftanas, J. Rovnakova, M. Riskova, D. Kocur, and M. Drutarovsky, “An analysis of 2D target positioning accuracy for M-sequence UWB radar system under ideal conditions,” 17th Int. Conf. Radioelektronika, pp. 1–6, Apr. 07. [21] D. Dardari, A. Conti, J. Lien, and M. Z. Win, “The effect of cooperation on localization systems using UWB experimental data,” EURASIP J. on Advances in Signal Processing, vol. 2008, no. 49, Jan 2008. [22] Z. Ebrahimian and R. A. Scholtz, “Receiver sites for accurate indoor position location systems,” Int. Conf. Wireless Comm. and Appl. Computation Electromagnetics, pp. 23–26, Apr. 05. [23] H. D. Ly and Q. Liang, “Collaborative multi-target detection in radar sensor networks,” Proc. Military Comm. Conf., Oct. 2007, orlando,Fla,USA. [24] I. Bradaric, G. T. Capraro, and M. C. Wicks, “Sensor placement for improved target resolution in distributed radar systems,” Proc. IEEE Radar Conference 2008, pp. 1–6, May 2008, rome. [25] E. Paolini, A. Giorgetti, M. Chiani, R. Minutolo, and M. Montanari, “Localization capability of cooperative anti-intruder radar systems,” EURASIP J. on Advances in Signal Processing, vol. 2008, Feb. 2008. [26] M. Chiani, A. Giorgetti, M. Mazzotti, R. Minutolo, and E. Paolini, “Target detection metrics and tracking for UWB radar sensor networks,” in Proc. IEEE Int. Conf. on Ultra-Wideband (ICUWB), Vancouver, CANADA, Sep. 2009, pp. 1–6.
PL 0.7 0.6 0.5 0.4 0.3 0.2 0.1 20
40
60
80
100
Tx
Fig. 3. Localization probability as a function of transmitter position (varying Tx coordinate in meters.) for ST /PR,th = 2500 Hz−1 (red) and ST /PR,th = 7500 Hz−1 (blue) with 6 receivers.
uu 1.0 0.8 0.6 0.4 0.2 106
105
104
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0.01
0.1
1
uS
Fig. 4. Expected (dashed) CDF vs derived (continuous) CDF of the uncertainty area. The uncertainty area is normalized with respect to the SA. ST /PR,th = 7500 Hz−1 (red), ST /PR,th = 2500 Hz−1 (blue).
uu 1.0
P D ,P L 1.0
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A
B
C
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E
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Fig. 2. Probability of detection and localization as functions of the number of receivers N for different values of ST /PR,th : A) PD with ST /PR,th = 2500 Hz−1 ; B) PD with ST /PR,th = 7500 Hz−1 ; C) PL with ST /PR,th = 7500 Hz−1 ; D) PL with ST /PR,th = 5000 Hz−1 ; E) PL with ST /PR,th = 2500 Hz−1 .
0.2
0
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2
3
4
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uS 104
Fig. 5. CDFs of uncertainty area. The uncertainty area is normalized with respect to the SA. Blue curves refer to the transmitter in the centre of SA. Red curvers refer to the transmitter and receivers on the circumference of the SA. The number of receivers range from 3 to 8.
Analysis of UWB Radar Sensor Networks Stefania Bartoletti, Andrea Conti, and Andrea Giorgetti
Abstract—Radar sensor networks (RSNs) are gaining importance in the context of passive localization and tracking. The performance of RSNs is affected by disturbances, system’s parameters, network topology, and the number of radar elements. In this paper, we derive a unified analytical framework that takes all this aspects into account and allows the derivation of probability of detection and localization uncertainty. The results enable the system designer to have a clear understanding on the effects of each system parameter and the trade-off between performance and complexity. Moreover, the potential for highaccuracy passive localization of ultrawide bandwidth (UWB) systems is shown.
I. I NTRODUCTION The need for efficient security systems is increasing in the last years and this reflects into a fervent research activity worldwide. For this reason, passive indoor localization is gaining importance thanks to the wide variety of its possible applications in private and public environments supervision. In this context radar systems show good capability in terms of detection, localization, and tracking within an area of interest called surveillance area (SA). A monostatic radar is composed of a single device that emits and receives pulses, whereas in a bistatic radar (BR) transmitter and receiver are not co-located [1]. A multistatic radar (MR) is a generalization of the BR with at least one transmitter and more than one receiver [2]. Radar sensor networks (RSNs) are essentially MRs systems. Cluttered propagation environment, walls, and objects outside the SA deteriorate the system performance and increase the risk of false alarms [3]–[5]. In the last years several works deal with techniques to improve radar systems. Ultrawide bandwidth (UWB) communications [6], [7] are robust to clutter [8] and interferences. Several studies, such as [9], show that these advantages are due to UWB features (short duration of pulses, ability of penetrating obstacles of various materials, large bandwidth). The occupation of a large band, in conformity with the Federal Communications Commission (FCC), improves resolution making UWB suitable for different applications such as position estimation through different techniques [10], [11], image reconstruction [12] and object imaging such as UWB tomographic radars [13]. For these reasons this technology is intended for many applications such as surveillance, disaster relief, and tracking of moving objects [14]. S. Bartoletti and A. Conti are with ENDIF University of Ferrara and WiLAB c/o University of Bologna, Italy, e-mail: [email protected]. A. Giorgetti is with DEIS University of Bologna and WiLAB, Cesena (FC), Italy, e-mail: [email protected]. This work has been supported in part by the European Commission in the scope of the FP7 ICT integrated project CoExisting Short range Radio by Advanced Ultra-WideBand radio Technology (EUWB) Grant no. 215669.
Error sources for ranging and thus affecting localization are given by [15]: multipath propagation also known as clutter (that depend on signal bandwidth [16]), clock drifts [17] and interferences [18]. Several studies have been performed to suppress static clutter choosing an appropriate system configuration [4], [5]. To take into account these error sources when developing system model, several error bounds have been defined for the most common types of signal and for all ranges of received signal-to-noise ratio (SNR) values [19]. The above mentioned sources of ranging errors result in uncertainty of target position evaluation. Several methods for analysis of positioning accuracy have been proposed (see, e.g., [20], [21]). The network topology influences system behaviour and performance; in particular, the capability of succeeding in detecting single or multiple intruders and the precision in evaluating its location depends also on number of nodes and their placement [22], [23]1 For this reason choosing the best network topology is one of the main goals for an effective system design to achieve the target performance with minimum complexity. In [24] some configurations of MRs are analyzed to improve system performance through an adequate placement of nodes. The performance is evaluated in terms of target resolutions through shaping the ambiguity function. A different approach based on simulations is presented in [25] for circular SA and radar elements on the border. In this paper, we propose a methodology to analyze MRs through a rigorous mathematical framework. Differently from previous works, such methodology is quite general to be applied to arbitrary configurations; in addition it allows to consider the effects of several system parameters such as: signal bandwidth, transmitted power, number of receivers, SNR etc. In this work, we focus our attention on positioning accuracy and coverage area. Finally, our mathematical model is compared with numerical results in order to prove its consistency. In the remainder of the paper, Section II contains a description of configurations and assumptions, Section III describes the methodology to compare different configurations in terms of localization uncertainty area and percentage of SA covered, Section IV shows numerical results and, in V we give our conclusions. II. S YSTEM M ODEL AND A SSUMPTIONS We consider a single transmitter and multiple receivers scenario. Each transmitter/receiver pair can be considered as 1 In addition, the a priori knowledge about topology of nodes and intra-node cooperation influence localization accuracy [21].
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a BR so we start considering BRs to build a mathematical model of MRs. A BR system is composed of a transmitter that emits a sequence of pulses and a receiver that receives the pulses: either directly from the transmitter or reflected from any obstacle (clutter) encountered during the propagation [25]. The time-of-arrival (ToA) represents the time of flight during which the pulses cover the transmitter-target-receiver path. Therefore, if ToA is known, the target is on an ellipse with foci at the transmitter and the receiver, respectively [25]. Assuming a Cartesian coordinate system, the ellipse is described by the following equation 2 2 2 2 d = (x − Tx ) +(y − Ty ) + (x − Rx ) +(y − Ry ) , (1) where (Tx , Ty ) and (Rx , Ry ) are the transmitter and receiver coordinates, respectively. If we suppose complete clutter removal the signal received is composed by only two pulses: the direct one via transmitterreceiver path and the reflected one via transmitter-targetreceiver path. Therefore a single pulse transmission causes two received pulses. The receiver can resolve paths only if they are sufficiently separated in time. The required temporal separation depends on receiver specifications and provides a minimum distance at which a target can be detected which results in a minimum ellipse [25]. If the target is inside the minimum ellipse the receiver cannot resolve the two pulses. The minimum temporal distance necessary to resolve two paths is called minimum resolvable delay γ. The minimum distance detected is given by (2) Dmin = (Rx − Tx )2 + (Ry − Ty )2 + γc . A target located in pT = (xtarget , ytarget ) is detectable if it is outside the minimum ellipse, thus Dmin < (xtarget − Tx )2 + (ytarget − Ty )2 + (xtarget − Rx )2 + (ytarget − Ry )2 . (3) Beside the minimum ellipse, the coverage depends also on the minimum signal-to-noise ratio (SNR) required at the receiver in order to detect the target. Accordingly to [25], we define the SNR as Ns PR (4) SNR N0 PRF where Ns is the number of pulses collected to perform a target detection and ToA estimation (often referred as scan period).2 Without loss of generality we choose Ns = 1 since when coherent reception is considered the transmitted power can be reduced by a factor Ns when Ns > 1 to have the same RSN performance. PRF is the pulse repetition frequency, N0 denotes the one-sided noise power spectral density (PSD) and PR /PRF represents the received energy per pulse scattered by the target. 2 As explained in [25], [26] depending on the scenario and the application, during a scan period the RSN can collect energy from many pulses in order to achieve the necessary area coverage and detection probability.
Based on the radar equation, it follows a limit on received power due to target reflection that is inversely proportional to the product of the distances between the radar antennas and target [1]. In particular, considering a transmitted signal with flat PSD over [fL , fL + B], with fL the lower spectrum frequency and B the signal bandwidth, and assuming constant antenna gains, GT and GR , and constant radar cross section (RCS), σ, over the signal bandwidth, the received power scatterd by the target can be written as [25] ST GT GR σc2 1 1 (5) − PR = (l1 l2 )2 (4π)3 fL fL + B where ST is the one-sided transmitted PSD, l1 and l2 are the distances between target and transmitter and receiver, respectively, and c is the speed of light. Note that the assumptions on frequency independent characteristics of antenna gains, RCS and transmitted PSD in (5) can be easily removed. In this paper we made such assumptions in order to simplify the notation without affecting the main contribution of the analysis developed since despite such assumptions (5) capture all the essential parameters. To ensure reliable target detection and localization accuracy a SNR at the receiver, SNRth , must be guaranteed. Starting from (4) and (5), the corresponding minimum received power, PR,th , provides a maximum distance product (l1 l2 )∗ that is a limit on BR coverage, with 1 ST GT GR σc2 1 ∗ . (6) − (l1 l2 ) PR,th (4π)3 fL fL + B We are interested to know the area covered by the considered BR, that is also that one from which signals backscattered satisfy the SNR requirement. The locus of points having a constant product of the distances measured from two fixed points is called Cassini oval and its shape depends on distances product itself [25]. Starting from Cassini oval equation, a target located in a point of coordinate pT is detectable by the considered BR if the following inequality is verified 2
(l1 l2 )∗ >[(xtarget − Tx )2 + (ytarget − Ty )2 ] ×[(xtarget − Rx )2 + (ytarget − Ry )2 ] .
(7)
Finally the coverage area of the BR is the area inside its maximum Cassini oval and outside its minimum ellipse as above defined. Let us suppose we want to employ a BR to monitor a circular area of radius R, and we assume that targets can be located only inside this SA. The detection probability represents the probability that target is inside the coverage area. If the coverage area is a portion of the area we want to supervise, the detection probability can be calculated as the ratio between coverage area and SA. Differently, if the coverage area is not completely inside the SA there is a part of the coverage area that we are not interested to supervise. The detection probability, in this case, is the ratio between the intersection of the coverage area with the SA and the SA
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itself. If S represents the set of points (then possible target positions) inside SA, then
S p s.t. x2 + y 2 < R
2
(8)
where p = (x, y) ∈ R2 . If C and E represents the area inside the maximum Cassini oval and outside the minimum ellipse, then C p s.t. [(x − Tx )2 + (y − Ty )2 ] 2 × [(x − Rx )2 + (y − Ry )2 ] − (l1 l2 )∗ < 0 (9) and 2 2 E p s.t. (x − Tx ) + (y − Ty ) 2 2 + (x − Rx ) + (y − Ry ) − Dmin > 0 .
(10)
Let us consider the characteristic function 1B (p) for a set B which has value 1 if p ∈ B, and 0 otherwise.3 Therefore, the detection probability results
1C (p) · 1E (p)dp , (11) PD = Ep[1C∩E (p)] = S m2 {S} where Ep[z] denotes the expected value of z with respect to all possible target positions p. If the target position is uniformly distributed inside the SA, (11) can be interpreted as m2 {C ∩ E ∩ S} /m2 {S} in the general case or m2 {C ∩ E} /m2 {S} when the coverage area is completely inside the SA. Since in a MR each transmitter/receiver pair can be considered as a BR its own, the coverage area of a MR coincides with the intersection between the coverage areas of all BRs components. Let us suppose N receivers detecting the target. The possible target positions are the points of intersection between the N ellipses having foci in the transmitter and each of N receivers, respectively. Thus, the target is localized if the system of equations (ellipses) has an unique solution, otherwise we have an ambiguity on target position. It follows that three receivers at least are needed to localize the target (the system of equations has a unique solution), whereas one to detect it (for detection purposes no matter how many solutions the system have). III. P ERFORMANCE M ETRICS To analyze system performance, we focus our attention on the two main aspects of RSNs: the probability to detect and localize the target, and the uncertainty associated to localization. Differently from other studies, a mathematical model to evaluate system performance is developed.
define the n-dimensional measure of a set B as mn {B} = 1B (p)dp.
3 We B
A. Probability of Detection Since a MR is a BR generalization, in order to derive the performance of a RSN it is necessary to generalize (9) and (10) to the MR case, while (8) remains the same. We consider that all N receivers of the RSN are identical except for their position in the scenario. In the RSN each transmitter/receiver pair covers a different area defined by its own Cassini oval and minimum ellipse. The maximum Cassini oval and the minimum ellipse for the (k) (k) kth receiver with coordinates (Rx , Ry ) with k ∈ {1, ..., N } are represented by the sets Ck and Ek as Ck p s.t. (x − Tx )2 + (y − Ty )2
2 (12) × (x − Rx(k) )2 + (y − Ry(k) )2 − (l1 l2 )∗ < 0 and
2 2 Ek p s.t. (x − Tx ) + (y − Ty ) (k) 2 (k) 2 + (x − Rx ) + (y − Ry ) − Dmin > 0 .
(13)
It follows that each transmitter/receiver pair has its own coverage area represented by Ak = Ck ∩ Ek ∩ S .
(14)
Since a target can be detected only if it is inside the coverage area of at least one transmitter/receiver pair, the detection probability for the RSN becomes
N 1 (p)·1 (p)dp C Ek , (p) = S k=1 k PD = Ep 1 N k=1 Ak m2 {S} (15) where stands for summation with boolean OR rule. When the actual target position is uniformly distributed in the SA, N (15) can be rewritten as m2 k=1 Ak /m2 {S}. To find an expression for the probability that a target is detectable by k receivers, it is essential to understand which part of our SA is covered by k and only k receivers, that is the area covered by k receivers at least removing the area covered by k + 1 receivers at least. The area covered by k receivers at least is the union of areas covered by each k-ple of the group composed of all transmitter/receiver pairs (if N is the number of receivers, there are Nk possible k-ples). Therefore, the area Rk covered by k receivers, results k N −k+1 N N −k+2 (16) ... Aip Rk = i1 =1
i =i +1
i =i
+1 p=1
k k−1 ⎧ 2 1 N −(k+1)+1 N −(k+1)+2 ⎨ ... ∩ NOT ⎩
i1 =1
i2 =i1 +1
N
k+1
ik+1 =ik +1 q=1
Aiq
⎫ ⎬ ⎭
.
Figure 1(a) shows the areas Rk for k = 1, 2, 3, 4 (respectively red, orange, green and yellow area) for a particular configuration with N = 4.
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N = 3 receivers, that is the minimum number of receivers needed to succeed in localization. The area covered by one receiver at least is the area in which targets can be detected. Differently, the coverage area is the area in which three receivers (then all receivers) can detect and localize the target. For this configuration, we have R1 = A1 ∪ A2 ∪ A3 , R2 = (A1 ∩ A2 ) ∪ (A1 ∩ A3 ) ∪ (A2 ∩ A3 ) ∩ NOT {A1 ∩ A2 ∩ A3 } and R3 = A1 ∩ A2 ∩ A3 . In this case, PL results PL =
m2 {A1 ∩ A2 ∩ A3 } . m2 {S}
(20)
C. Uncertainty Area
(a) Rk , k = 1, 2, 3, 4.
(b) Uncertainty area (red spot). Fig. 1. Examples of area coverage and uncertainty with R = 1 and N = 4. The single transmitter is in the centre of the circular SA and the four receivers on its circumference.
B. Probability of Localization A target is localized when it is located inside the coverage areas of three transmitter/receiver pairs at least, that is the coverage area of the system A is given by A
N
Rk ,
(17)
k=3
where Rk is derived in (16). Thus, the localization probability PL is (18) PL Ep[1A (p)] which simplifies into PL =
m2
N k=3
Rk
m2 {S}
(19)
in the case of uniform probability of target location inside the SA. Let us consider, for example, the configuration with
An uncertainty of ToA causes an uncertainty of the distance estimate between each TX/RX pair and target. Thus, the detection ellipse becomes an annulus, whose thickness depends on uncertainty value: therefore the intersections between ellipses (that in ideal conditions provide target coordinates) become the intersections between annuluses that are no longer a single points but a set of points called uncertainty area [25]. The measure of uncertainty area depends on several factors, such as the number of receivers detecting the target, the transmitted pulse shape, the SNR and also the target position itself with respect to the system configuration adopted. Several studies have been performed to find bounds on the variance of ToA estimation for different pulse shaping and SNR in AWGN and multipath channel [15]. An example of uncertainty area for 3 out of 4 receivers detecting the target is shown in Fig. 1(b). In fact, if the target is detected by a single transmitter/receiver pair the uncertainty area corresponds to the intersection between the detection annulus area and the SA. Differently if two receivers detect the target, the uncertainty area corresponds to the intersection between two annuluses resulting in general to two distinct areas. If k is the number of transmitter/receiver pairs detecting the target and k ≥ 3, the target is localized and the uncertainty area decreases as k increases. Let us consider the target in position pT detected by the j th receiver with a ToA estimate in the uncertainty interval [τj (pT ) − /2, τj (pT ) + /2], where τj (pT ) is the ToA corresponding to the actual target position and specify the ToA error range. The error in ToA estimate leads to a detection annulus defined as the area between the two ellipses corresponding to the extremum of the uncertainty interval. Therefore, the detection annulus can be represented as the set Ej+ (pT ) ∩ Ej− (pT ) ∩ S where Ej+ (pT ) and Ej− (pT ) are defined as 2 2 + (21) Ej (pT ) p s.t. (x − Tx ) + (y − Ty ) (j) 2 (j) 2 0 . + (x − Rx ) + (y − Ry ) − c τj (pT ) − 2
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The region around pT is therefore Uk (pT ) = uncertainty + E (p ) ∩ Ej− (pT ) ∩ S and can be represented by T j∈Ok j 1Uk (pT ) (p) = j∈Ok
1E + (pT ) (p) · 1E − (pT ) (p) · 1S (p) j
j
(23)
with Ok = {k receivers out of N detecting the target}. The corresponding uncertainty area is ! 1Uk (pT ) (p)dp . (24) AU,k (pT ) = m2 {Uk (pT )} = S
IV. N UMERICAL R ESULTS We present results on detection, localization probability, and localization uncertainty, using the proposed general framework applied to UWB RSNs.4 For example, the values of system parameters chosen for a SA of radius R = 50 m are: fL = 5 GHz, B = 500 MHz, GT = GR = 1, σ = 1 m2 , ST /PR,th = 7500 Hz−1 (if not otherwise specified), and a ToA error range such tha the annuluses thickness is 0.36 m. Figure 2 shows the detection probability and the localization probability as a function of the number of receivers (from 3 to 10) and ST /PR,th = 2500 and 7500 Hz−1 for the configuration with the transmitter in the centre of the SA and all receivers on its circumference at a constant angular distance each other. The effects on PL of the transmitter position is shown in Figure 3, where the PL is reported as a function of the x coordinate of the transmitter position ranging from the center of the SA, Tx = 0, to Tx = 2R. Since the uncertainty area depends on the actual target position, we develop a statistical analysis of uncertainty area considering random target position inside the SA. We define a constant ToA estimation error and we calculate the uncertainty area for different target positions with (24). Note that, for each target position considered in the calculation, there is a different number of receivers, k, capable to detect the target depending upon the target belongs to the region Rk (16). In particular, if a target is moving along the radius of the SA from the centre to the circumference it encounters all Rk as k varies from N to 0. If we assume that SNR does not change significantly inside the coverage area of each transmitter/receiver pair, then only the number of receivers k detecting the target influences the uncertainty area. Since k is proportional to target radial coordinate in a polar coordinate system, we calculate uncertainty area of 20 target positions, each one having radial coordinate respectively equal to i 20/R with i ∈ {1, ..., 20}. In Figure 4 the cumulative distribution function (CDF) of uncertainty area that resulted very similar to that one obtained by computation of 1000 target random positions.5 To emphasize the dependence of uncertainty CDF on k we repeat the comparison descripted above between an analytical and derived CDFs after decreasing the ratio ST /PR,th from 4 All
system parameters are scaled to the radius R of the circular SA. define the whole SA as the conventional value of uncertainty if the target is not detectable, even though the real value should be infinity. 5 We
7500 to 2500 Hz−1 (i.e., by increasing the probability that the target is detected by k < N receivers). Figure 4 shows, in this case, two saturation levels corresponding to two impossible ranges of uncertainty area values. To find the best configuration in terms of uncertainty area, Figure 5 shows the uncertainty CDF for different RSN configurations. The first configuration is composed by the transmitter in the centre of the SA and all receivers on its circumference at a constant angular distance. The second one consists of all RSN nodes (including the transmitter) located on the circumference still at a constant angular distance. For both configurations we changed the number of receivers N ∈ {3, 4, ..., 8}. Figure 5 has been obtained keeping constant the transmitter power at the value that allows the worst case topology (three receivers and transmitter on the circumference) to cover at least the 90% of the SA. V. C ONCLUSIONS In this paper a mathematical framework for the analysis of RSNs based on MR is proposed. It accounts for the main system parameters and impairments and enables the derivation of the performance in terms of detection probability inside a surveillance area and uncertainty area around the estimated target location. The framework has been applied to study the performance of a RSN based on UWB technology varying several parameter such as: number of receivers, transmitted power to receiver sensitivity ratio, and network topology. Results, here specialized for circular SA show the superiority of the topology with the transmitter in the centre of the SA and all receivers located on its circumference at a constant angular distance each others. ACKNOWLEDGMENT The authors wish to thank M. Chiani for helpful discussions. R EFERENCES [1] M. I. Skolnik, “An analysis of bistatic radar,” IEEE Trans. on Aerospace and Navigational Electronics, pp. 19–27, 1963. [2] V. S. Chernyak, “Fundamentals of multisite radar systems,” Gordon and Breach Science, 1998, amsterdam, The Netherlands. [3] F. Ahmad and M. Amin, “Noncoherent approach to through-the-wall radar localization,” IEEE Trans. on Aerospace and Electronic Systems, vol. 42, pp. 1405–1419, 2006. [4] L. M. H. Ulander and T. Martin, “Bistatic ultra-wideband SAR for imaging of ground targets under foliage,” Proc. IEEE Int. Conf. on Ultra-Wideband (ICUWB), 2005. [5] S. Nag and M. Barnes, “A moving target detection filter for an ultrawideband radar,” Proc. IEEE Radar Conference 2003, pp. 147–153, May 2003, huntsville, Ala, USA. [6] M. Z. Win and R. A. Scholtz, “Ultra-wide bandwidth time-hopping spread-spectrum impulse radio for wireless multiple-access communications,” IEEE Trans. on Comm., vol. 48, pp. 679–691, Apr 2000. [7] R. A. Scholtz and M. Z. Win, “Impulse radio: how it works,” IEEE Commun. Letters, vol. 2, pp. 36–38, Feb 1998. [8] D. Dardari and M. Z. Win, “Threshold-based time-of-arrival estimators in UWB dense multipath channels,” Proc. IEEE Int. Conf. on Commun. (ICC), vol. 10, pp. 4723–4728, Jun 2006, istanbul, Turkey. [9] V. Chernyak, “Multisite ultra- wideband radar systems with information fusion: Some principal features,” European Radar Conference, pp. 141– 144, Oct. 03, paris. [10] H. Wymeersch, J. Lien, and M. Z. Win, “Cooperative localization in wireless networks,” Proc. of the IEEE, vol. 97, no. 2, pp. 427–450, Feb 2009.
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PL 0.7 0.6 0.5 0.4 0.3 0.2 0.1 20
40
60
80
100
Tx
Fig. 3. Localization probability as a function of transmitter position (varying Tx coordinate in meters.) for ST /PR,th = 2500 Hz−1 (red) and ST /PR,th = 7500 Hz−1 (blue) with 6 receivers.
uu 1.0 0.8 0.6 0.4 0.2 106
105
104
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0.01
0.1
1
uS
Fig. 4. Expected (dashed) CDF vs derived (continuous) CDF of the uncertainty area. The uncertainty area is normalized with respect to the SA. ST /PR,th = 7500 Hz−1 (red), ST /PR,th = 2500 Hz−1 (blue).
uu 1.0
P D ,P L 1.0
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A
B
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E
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Fig. 2. Probability of detection and localization as functions of the number of receivers N for different values of ST /PR,th : A) PD with ST /PR,th = 2500 Hz−1 ; B) PD with ST /PR,th = 7500 Hz−1 ; C) PL with ST /PR,th = 7500 Hz−1 ; D) PL with ST /PR,th = 5000 Hz−1 ; E) PL with ST /PR,th = 2500 Hz−1 .
0.2
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3
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uS 104
Fig. 5. CDFs of uncertainty area. The uncertainty area is normalized with respect to the SA. Blue curves refer to the transmitter in the centre of SA. Red curvers refer to the transmitter and receivers on the circumference of the SA. The number of receivers range from 3 to 8.
