PRACTICAL LIMITS IN RSS-BASED POSITIONING Richard K. Martin∗ , Amanda Sue King, Ryan W. Thomas∗
Jason Pennington
The Air Force Institute of Technology Dept. of Elec. & Comp. Eng. WPAFB, OH 45433 {richard.martin,amanda.king,ryan.thomas}@afit.edu
Miami University Dept. of Elec. & Comp. Eng. Oxford, OH 45056
[email protected]
ABSTRACT Received signal strength (RSS) based source localization papers often ignore the practical effects of range limits in the measurements. In many devices, this results in some sensors not reporting beyond some maximum range; and in others, the RSS is still observed but may exhibit a noise floor at large ranges. This paper models these situations and demonstrates their effect on positioning algorithms. Measured data is used to validate the cooperative and non-cooperative RSS limits. The Fisher information, Cramer-Rao lower bound, and maximum likelihood estimation error are used to quantify effects on positioning. Index Terms— Source localization, received signal strength, Cramer-Rao lower bound, noise floor 1. INTRODUCTION Position awareness of wireless devices is required in a variety of applications, such as emergency response, law enforcement, military reconnaissance, location-based billing, resource allocation and tracking, and hand-held games. In source localization (also called geolocation), a Wireless Sensor Network (WSN) is used to locate the source of an Radio Frequency (RF) transmission [1], [2]. Geolocation may be accomplished through Angle of Arrival (AOA), Received Signal Strength (RSS), Time of Arrival (TOA), and/or Time Difference of Arrival (TDOA) [2] measurements. AOA and RSS are simple to obtain and use, but they require a dense network of receivers. TOA requires cooperation between the transmitter and receiver; thus, while it is quite accurate, it is not always practical. TDOA is very accurate, but requires bandwidth-intensive cross-correlation between receivers. Though each measurement type has its own merits, this paper focuses on RSS. There are two distinct scenarios in which RSS measurements can be obtained: cooperative and non-cooperative. In cooperative systems, such as cell phone handset geolocation by base stations, the reported RSS is often just the signal power, as the digital signal can be demodulated and segregated from additive noise. In noncooperative systems, such as locating emitters in a hostile environment, the RSS may be determined by integrating the observed Power Spectral Density (PSD). However, the standard model becomes invalid for both scenarios at large distances. Most papers on RSS-based positioning largely ignore the practical effects of range limitations, as well as details of how the measure∗ Funded in part by the Office of Naval Research, and the Air Force Research Labs, Sensors Directorate. The views expressed in this paper are those of the authors, and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U.S. Government.
978-1-4577-0539-7/11/$26.00 ©2011 IEEE
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ments are actually obtained. It is often assumed that all receivers are within range of the transmitter, though [2] did also consider the case where “each sensor only makes measurements to those sensors located within r = 10 m of itself.” The goal of this paper is to explore and characterize the range-limitation issue. 2. SYSTEM MODEL Throughout, (·)∗ , (·)T , (·)H , and E {·} denote complex conjugate, matrix transpose, conjugate (Hermitian) transpose, and statistical expectation, respectively. A sample average is denoted ·. The matrices 0, 1, I, contain all zeros, all ones, and the identity matrix, respectively; and when it is not clear from the context, they will be subscripted with their dimensionality. A hat (e.g. x ˆ) indicates an estimate of its argument. Assuming log-normal fading as in [3], [4], [5], the received power in dB at each sensor is Gaussian with variance σ 2 . Typically, σ ranges from 4 dB to 12 dB [5], corresponding to uncluttered environments to environments rich in shadowing and multipath. The value of σ 2 can be approximated from controlled measurements in a given environment. Due to multipath and shadowing, the path loss exponent η may deviate from its free space value of 2. It may be as large as 5 in dense urban environments [4], though some sources state that typical values of η are in the range 2 to 4, and the authors have observed even smaller values in indoor environments. The reference transmit power is P0 , in dB; this is the power that would be received at a reference distance of d0 = 1 m. P0 and η may be pre-characterized or included as nuisance parameters. First, we consider the standard, idealized model. The WSN consists of S receiver nodes at known positions (xs , ys ), for s = 1, 2, · · · , S. The transmitter is at an unknown position (x0 , y0 ), hence the transmitter-to-receiver distance is ds = (xs − x0 )2 + (ys − y0 )2 . (1) Define the following quantities: m = [m1 , · · · , mS ]T ,
(2)
ms = P0 − ηds , ds = 10 log10 (ds /d0 ) , edB = 10 log10 (e) ≈ 4.343.
(3) (4)
The vector m is the expected RSS, assuming no noise and infinite range limits. This m can be thought of as the RSS before the effects of fading. The received power after fading is typically modelled as Gaussian in the log domain (or log-normal), p = [p1 , · · · , pS ]T ∼ N m, σ 2 I . (5)
ICASSP 2011
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Fig. 1. Contours of the PDFs of the standard model (5), the true model, and the proposed approximate model (8), assuming noncooperative RSS measurement and noise at -23 dBm.
where the parameter d0 is a short reference distance from the receiver (typically 1 m). In a cooperative system, the range limits may be imposed via truncation: ps , ps ≥ τcoop (6) pcoop,s = NaN, ps < τcoop where “NaN” means “not a number,” indicating that no RSS is reported for that node. The dB power threshold τcoop roughly corresponds to a maximum range of dmax ≈ 10(P0 −τcoop )/(10η) ,
(7)
though longer ranges are possible due to the positive tail of the lognormal fading. In a non-cooperative system, there will be a noise floor. The noise power is additive in the linear domain, hence we model the total non-cooperative RSS, including background noise and fading, as pnc ∼ N mnc , σ 2 I mnc,s = 10 log10 10ms /10 + 10τnc /10 .
(11)
3. INFORMATION DEGRADATION
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ignoring constant terms that will cancel due to differentiation. We will ignore any “NaN” terms by excluding them from summations.
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Lcoop = ln f (pcoop |x0 , y0 ) 1 = − (pcoop − m)T (pcoop − m) 2σ 2 1 (pnc − mnc )T (pnc − mnc ) , Lnc = − 2σ 2
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Note that the non-cooperative power threshold τnc is different than the cooperative power threshold τcoop . That is because τnc is the power of the background noise, whereas τcoop is the lowest signal RSS that can be detected in the presence of noise. Eq. (8) is a slight approximation because it would be more physically accurate to compute the Probability Density Function (PDF) of pnc by convolving the log-normal PDF of the signal power with the chi-squared PDF of the noise power. However, that approach is analytically intractable. The model in (8) is a reasonable analytic approximation, as shown in Fig. 1; and it agrees with measured data taken by the authors. This model, while not exact, does provide some valuable intuition.
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The effects of the range limits of the previous section will cause a loss of information in the WSN. In this section, we quantify this using the Fisher information, and its inverse, the Cramer-Rao Lower Bound (CRLB) [6]. While the unknowns are x0 and y0 , all of the information provided by ps is available in the distance ds . Thus, for simplicity, in this section we will first consider the information about the scalar ds contained in the scalar ps , though later we will consider the overall information on positioning accuracy. The Fisher information about ds contained in an RSS observation ps is
2 2 1 ∂L ∂ L = ··· = 2 . (12) Js (ps |ds ) = −E ∂d2s σ ∂ms In the usual idealized case, Js =
η edB σ ds
2 ,
(13)
with units of meters−2 . In the non-cooperative case, the Fisher information becomes
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2 η edB 10ms /10 Jsnc = · (14) σ ds 10ms /10 + 10τnc /10 2 rs
In the cooperative case, we can use the Modified CRLB (MCRLB) [7] to approximate the CRLB, with a potentially looser bound. In the MCRLB, a modified Fisher information is obtained by conditioning on and later averaging over unknown nuisance parameters. In our case, we will treat random outages (occurrences of “NaN”) as nuisance parameters. From (6), the probability of an outage is P [outage] = Q P0 − ηds − τcoop /σ , (15) where the Q function Q (·) is the integral of a unit Gaussian above its argument. When there is no outage, the Fisher information is given by (13); and when there is an outage, there is no Fisher information conveyed. (In Shannon’s sense, there is information, insofar as we suspect the distance is large when there are outages; but the Fisher information only deals with the local curvature of the loglikelihood.) Thus, the modified Fisher information is
2 η edB Jscoop = · 1 − Q P0 − ηds − τcoop /σ . (16) σ ds qs
Even in the idealized case of (13), the utility of RSS measurements drops with distance. However, when considering either outages or a noise floor, the drop off is much more drastic beyond
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As we are concerned with potential divergence of the series, consider large distances. Observe that (P0 −τnc )/10 −η lim ri = 10 ·di ,
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Fig. 2. The effects of modelling a noise floor in the RSS. Distance is linear in (a), (c) and logarithmic in (b), (d). Plots (a), (b) show mean RSS, and (c), (d) show Fisher information about range per RSS.
the cut-off region. Specifically, in both (14) and (16), the idealized Fisher information of (13) is degraded by factor that drops monotonically from one to zero as the distance ranges from zero to infinity. Fig. 2 plots this effect. The top two subfigures show the RSS for the idealized and non-cooperative cases. The bottom two subfigures plot the ideal and degraded Fisher information. These plots used η = 2, σ = 4 dB, P0 = 0 dBm, τcoop = −30 dBm, and τnc = −30 dBm. Now we return to the full positioning problem. For purposes of gaining intuition, consider an infinite grid of sensors, in concentric rings about the true source location, as in Fig. 3. The spacing between rings is Δ meters, with approximately the same spacing between sensors on each ring. (This is an approximation of a regular square grid with one sensor per Δ2 meters.) Noting that ∂ds s = − sin θs and ∂d = − cos θs , we have ∂x0 ∂y0 Jideal =
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We can replace the sum over s by a dual sum, first along each ring and then over the rings. Per ring, there are Nring ≈ 2πdring /Δ sensors, and the sum of the sin2 and cos2 terms become Nring /2. Thus, 2 π d−1 I2 σ Δ ring ring
2 √ ∞ η π edB 1 I2 = σΔ i i=1
Jideal ≈
η e
though at finite distances, ri is smaller than this. The first factor is the margin between P0 and the noise floor, converted to linear scale. Applying this to (19), √
2 ∞ η π edB margin i−1−2η I2 . (21) Jnc σ Δ1+η i=1 Now the summation is the P-series, also called the Riemann Zeta function, which converges for all η > 0. For η = 1 (at the very low end of typical numbers), the summation is ≈ 1.202, and it drops monotonically to one as η increases. Thus, the square root of the CRLB (in order to get units of distance) is √
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This formula is useful because it gives an idea of how the sensor spacing Δ affects positioning performance in a very large grid. In the cooperative case, the sums in (17) and (18) get additional factors of qi as in (16):
2 √ ∞ η π edB 1 Jcoop ≈ qi I2 . (23) σΔ i i=1 It is fairly easy to bound the summation in (23) to prove that it converges. However, the authors have not been able to find a bound that is tight enough to provide any meaningful intuition.
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4. SIMULATIONS AND EXPERIMENTS (18)
The harmonic series i i−1 diverges, hence an infinite grid analysis is not possible in the idealized case. However, if we consider the effects of the noise floor in the non-cooperative case, the sums in (17) and (18) get additional factors of ri2 as in (14): Jnc ≈
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First, we use experimental data to verify the models in Section 2. Fig. 4 shows non-cooperatively measured data. The transmitter was a WARP FPGA board [8] and a WiSpy [9] was used as the receiver. At distances beyond 3 m, the FPGA power is below the noise floor, but the WiSpy can still measure a PSD, and the WSN does not know if the resulting RSS is due to signal, noise, or both. The trend line was obtained by fitting P0 , η, and τnc in (9). Fig. 5 shows cooperatively measured, obtained from IEEE 802.11 packets [10]. Each data point is the average of 2000 measurements at a given location. The dashed lines at −20 dBm and −95 dBm indicate the largest and smallest RSS values seen in all
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Fig. 6. RMSE vs. number of sensors for (left) the geometry in Fig. 3 and (right) a network of comparable density but with square spacing. The number of sensors was increased by successively appending rings or boxes. 5. REFERENCES [1] D. Blatt and A. O. Hero, III, “Energy-Based Sensor Network Source Localization via Projection Onto Convex Sets,” IEEE Trans. Signal Processing, vol. 54, no. 9, pp. 3614–3619, Sept. 2006. [2] N. Patwari, J. N. Ash, S. Kyperountas, A. O. Hero, III, R. L. Moses, and N. S. Correal, “Locating the nodes: cooperative localization in wireless sensor networks,” IEEE Signal Processing Mag., vol. 22, no. 4, pp. 54–69, July 2005. [3] M. Kieffer and E. Walter, “Centralized and Distributed Source Localization by a Network of Sensors Using Guaranteed Set Estimation,” in Proc. Int. Conf. Acoustics, Speech, & Signal Proc., Toulouse, France, May 2006, vol. 4, pp. 977–980.
5 10 15 10*log10( distance (m) )
Fig. 5. RSS data (cooperative) from IEEE 802.11a packets [10].
[4] A. J. Weiss, “On the Accuracy of a Cellular Location System Based on RSS Measurements,” IEEE Transactions on Vehicular Technology, vol. 52, no. 6, pp. 1508–1518, Nov. 2003.
1.7 million data points (before averaging). As the distance increases, packets falling below −95 dBm are lost. Using the geometries in Fig. 3, simulations were used to determine Root Mean Squared Error (RMSE) vs. number of sensors for three cases: (i) standard data model, (ii) cooperative model of (6), and non-cooperative model of (8). In each case, a Maximum Likelihood (ML) algorithm was derived to match the data model. As shown in Fig. 6, in the cooperative case, the RMSE levels off around 4 rings/boxes (about 64 sensors). Based on Section 3, we expect the standard model to eventually approach 0 RMSE but the noncooperative case to level off at some point. However, these asymptotic effects do not show up with even 400 sensors. This may be because the harmonic series diverges so slowly that extremely large networks are needed to see the effects. Thus, in terms of RMSE, all three data models effectively have limits. If the data is measured in a cooperative sense, it reaches its actual limits much sooner than the standard data model would suggest.
[5] T. S. Rappaport, Wireless Communications: Principles and Practice, Prentice-Hall, Englewood Cliffs, NJ, 1996.
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[6] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation theory, Prentice Hall, 1993. [7] F. Gini, R. Reggiannini, and U. Mengali, “The Modified CramerRao Bound in Vector Parameter Estimation,” IEEE Trans. on Comm., vol. 46, pp. 52–60, Jan. 1998. [8] “WARP FPGA Board,” Rice University. [Online.] Available: http://warp.rice.edu/trac/wiki/FPGA%20Board . [9] “Wi-Spy 2.4i Spectrum Analyzer,” MetaGeek. [Online.] Available: http://www.metageek.net/products/wi-spy-24i . [10] K. Bauer, D. McCoy, E. W. Anderson, D. Grunwald, and D. C. Sicker, “CRAWDAD trace cu/rssi/text/omni 16dbm (v. 2009-05-28),” Downloaded from http://crawdad.cs.dartmouth.edu/cu/rssi/text/omni 16dbm, July 2010.
