Localization of Objects Using Cross-Correlation of Shadow Fading Noise and Copulas


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When a radio transmitter is mobile, obstacles in the
radio path can cause temporal variation in Received Signal Strength Indicator (RSSI) measured by receivers due to multipath and shadow fading. While fading, in general, is detrimental to accurately localizing a target, fading correlation between adjacent receivers may be exploited to improve localization accuracy. However, multipath fading correlation is a short range phenomenon that rapidly falls to zero within a wavelength whereas,
shadow fading correlation is independent of signal wavelength and has longer range thereby making it suitable for localization with wireless transceivers that operate at shorter wavelength. Therefore,
this paper presents a novel wireless localization scheme that employs a combination of cross-correlation between shadow fading noise and copula technique to recursively estimate the location of a transmitter. A stochastic filter that models multipath fading as an Ornstein-Uhlenbeck process followed by a Generalized Auto Regressive Conditional Heteroskedasticity (GARCH) filtering is
proposed to extract shadow fading residuals from measured RSSI values. Subsequently, Student-T Copula function is used to create the log likelihood function, which acts as the cost function for localization, by combining spatial shadow fading correlation arising among adjacent receivers due to pedestrian traffic in the area. Maximum Likelihood Estimate (MLE) is used for position estimation as it inherits the statistical consistency and asymptotic
normality. The performance of our proposed localization method is validated over simulations and hardware experiments.

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Localization of Objects Using Cross-Correlation of Shadow Fading Noise and Copulas

  1. 1. Localization of Objects Using Cross-Correlation of Shadow Fading Noiseand Copulas Mohammed Rana Basheer, S. Jagannathan Dept. of Electrical and Computer Engineering Rolla, MO, USA {mrbxcf, sarangap}@mst.edu
  2. 2. Introduction Real Time Location Systems (RTLS) Used for locating or tracking assets in places where GPS signals are not readily available Methodologies  Time of Arrival (ToA),  Time Difference of Arrival (TDoA),  Angle of Arrival (AoA) or  Received Signal Strength Indicator (RSSI) Boeing factory floor* RSSI based localization is cheaper as it involves mostly a software updated on an existing wireless infrastructure However, accuracy and periodic radio profiling issues have limited their adoption in factory environment*http://www.ce.washington.edu/sm03/boeingtour.htm 2
  3. 3. Localization Errors Multipath fading and shadow fading noise are the primary cause for large localization error in an indoor environment Rx Rx Tx Tx Multipath Fading Shadow Fading 3
  4. 4. RSSI Profile of ERL 114  Spans 12m x 13m  Typical lab floor with tables, partitions, heavy equipments such as pumps etc.  0.6m x 0.6m gridRSSI (dB) Layout of ERL 114 4
  5. 5. Similarity in Fading Noise Statistics Fading noise depends on the radio signal propagation environment Adjacent wireless receivers will experience similar fading noise statistics Cross-correlation in fading noise between adjacent receivers may be used to Rx2 determine their relative Rx1 position to a common transmitter Tx Shadow Fading 5
  6. 6. Previous Work Cross-correlation of multipath noise signals from adjacent receivers were used by Basheer et. al.1 for localizing transmitters However, multipath cross- correlation tapers of to zero within a wavelength of radial separation Cross-correlation in shadow fading noise between adjacent receivers arising due to pedestrian or machinery traffic in their vicinity was found to span larger distance Multipath noise correlation with distance1Basheer,M.R.; Jagannathan, S.; , "Localization of objects using stochastic tunneling," Wireless Communicationsand Networking Conference (WCNC), 2011 IEEE , pp.587-592, 28-31 March 2011 6
  7. 7. Previous Work (contd.)  Non-Parametric Methods treat the localization as a dimensionality reduction problem Pi [ri1 , ri 2 ,..., riK ]T RK Xi [ xi , yi , zi ]T R3  Multi Dimensional Scaling (MDS)2  Local Linear Embedding (LLE)3  Isomap4  However, linear relationship requirement between cross-correlation and radial distance breaks rapidly at distances more than a wavelength of radial separation in wireless devices2X. Ji, and H. Zha, "Sensor positioning in wireless ad-hoc sensor networks using multidimensional scaling," 23rd Annual Joint Conf. of theIEEE Computer and Communication Society, vol.4, pp. 2652- 2661, Mar. 2004.3N. Patwari and A. O. Hero, “Manifold learning algorithms for localization in wireless sensor networks,” in Proceedings of the IEEEInternational Conference on Acoustics, Speech and Signal Processing (ICASSP), vol. 3, pp. 857-880, May 2004.4Wang C, Chen J, Sun Y, Shen X. “Wireless sensor networks localization with Isomap,” IEEE International Conference on Communications,2009. 7
  8. 8. Localization Block Diagram IEEE 802.15.4 Receiver 1 Z s1 Base Station IEEE 802.15.4 Z s2 Copula Receiver 2 Optimization XT,YT Function Z sM IEEE 802.15.4 Receiver MXT,YT =Transmitter CoordinatesZsi = Shadow Fading Residual from ith receiver 8
  9. 9. Shadow Fading Extraction Block Diagram RSSI From AR(1) X (t ) Ornstein- X s (t ) IEEE 802.15.4 + Z si Uhlenbeck Filter Receiver GARCH(1,1)X(t) = RSSI at time instance tXs(t) = Shadow Fading Residual + Path LossZsi = Shadow Fading Residual 9
  10. 10. Copula Optimization Function Z s1 Build Semi- ~ Stochastic Optimization Parametric CDF F1 Student-t Copula ~ XT,YT Function Cv, P Z s2 Build Semi- F2 Parametric CDF ~ P( x, y) FM Z sM Build Semi- Compute pair-wise Parametric CDF Cross-CorrelationZsi = Shadow Fading Residual = Semi-Parameter Shadow Fading CDF Possible Transmitter iP(x,y) = MxM shadow fading cross-correlation Coordinates (x,y)Cv,p = Student-t copula function 10
  11. 11. Extracting Shadow Fading Residuals Valenzuela et. al. has shown that multipath effects can be removed without degrading shadow fading effects in RSSI by spatial averaging the received signal power over 10λ distance5 Therefore, multipath noise can be treated as a mean Shadow fading from received signal power5 reverting process In this paper multipath noise is modeled as a stochastic process called Ornstein Uhlenbeck (OU) to isolate shadow fading residuals from RSSI 5R.A.Valenzuela, O. Landron, and D.L. Jacobs, "Estimating local mean signal strength of indoor multipath propagation," IEEE Trans. on Veh. Technol., vol.46, no.1, pp. 203-212, Feb 1997. 11
  12. 12. OU Model for Multipath Noise X(t) is the received signal strength at time instance t dX(t) is a small change in RSSI for a delta increment in time dt, Xs(t) is the local mean of RSSI which is a combination of deterministic path loss and shadow fading due to pedestrian traffic, v(t) is the rate at which the multipath noise revert to the short range mean set by shadow fading noise and deterministic path loss 2 f is the variance of multipath noise dW(t) is the delta increment of a standard Brownian motion. Estimate v(t) and σf for OU model using maximum likelihood estimators66L.Valdivieso, W. Schoutens and F. Tuerlinckx, “Maximum likelihood estimation in processes of Ornstein-Uhlenbeck type,” Statistical Inference for Stochastic Processes, vol. 12, No. 1, pp. 1-19, 2009. 12
  13. 13. AR+GARCH to Isolate Shadow FadingResiduals Autoregressive Model (AR) for Xs(t) is used to separate path loss from shadow fading residuals AR(1) where μr(t) accounts for all the deterministic power loses, β is the auto-correlation between successive samples of Xs(t) and ϵs(t)=σs(t)Zs is the deviation of the shadow fading process from the AR(1) process assumption, s2 t is the shadow fading variance and Zs is the stationary zero mean unit variance shadow fading residual. Generalized Auto Regressive Conditional Heteroskedasticity (GARCH) for s2 t to account for changes in pedestrian traffic GARCH(1,1) 13
  14. 14. CDF of Shadow Fading Residuals Semi-parametric CDF is used since the derivation of a parametric distribution for Zs obtained after OU and GARCH filtering of RSSI values is very difficult FiU ( x), x U i Upper Tail (Parametric) ~ ˆ Fi N ( x) Fi N ( x), Li x Ui Mode (Empirical) Fi L ( x), x Li Lower Tail (Parametric) Regions around the mode of the residuals will be modeled using non- parametric empirical CDF ˆ N 1 N Fi ( x) I Z s k ) x ; i 1,2, , M ( N k 1 i where I(·) is the indicator function, Zsi Zs(i1) , Zs(i2) , Zs(iN ) are N shadow fading residuals from ith receiver in the localization area 14
  15. 15. CDF of Upper and Lower Tails ofShadow Fading Residuals Upper and lower tails, were sample points are sparse by definition, a parametric Generalized Pareto Distribution7 (GPD) was applied 1 x Ui FiU ( x) 1 i i i 1 x Li Fi L ( x) 1 i i i where Ui and Li are the upper and lower location parameters for a Generalized Pareto Distribution (GPD) while ζi is the shape parameter that controls the rate at which the tail of a distribution goes to zero and ϑi is the scale parameter that accounts for variance in tail data7J. R. M. Hosking and J. R. Wallis, “Parameter and quantile estimation for the Generalized ParetoDistribution,” Technometrics, Vol. 29, No. 3, pp. 339-349j, Aug 1987 15
  16. 16. Shadow Fading Wireless Propagation Model Geometrically Based Single Bounce Elliptical Model (GBSBEM) Wireless Channel Model8 is assumed under shadow fading Any radio signal that reaches the receiver after bouncing off of a scatterer in the localization region can GBSBEM Wireless Channel Model8 affect signal fading if and only if its ToA satisfies r t m c where r is the radial separation between the transmitter and receiver, c is the speed of radio waves, r/c is the ToA of LoS signal and τm is the signal integration time at the reciever 8J.C.Liberti, and T.S. Rappaport, "A geometrically based model for line-of-sight multipath radio channels," Vehicular Tech. Conf., 1996. Mobile Tech. for the Human Race., IEEE 46th , vol.2, pp.844-848, May 1996. 16
  17. 17. Shadow Fading Correlation Coefficient IEEE 802.15.4 receivers computes RSSI as the squared sum of incoming signal amplitude arriving within an RSSI integration time9 Radio signal attenuation for scatterers are assumed to be Normally distributed while Poisson distribution is assumed for pedestrian traffic in the localization area Theorem 1: Shadow fading noise correlation coefficient (ρ) between two IEEE 802.15.4 receivers R1 and R2 separated by radial distances r1 and r2 respectively from a common transmitter is given by S12 S1 S 2 Overlapping of scattering regions causing cross-correlation in shadow fading where |·| is the area operator, S1 and S2 are the elliptical scatterer regions surrounding receivers R1 and R2 respectively, S12 is overlapping region between scattering regions S1 and S2 . 9Hyeon-Jin Jeon, T. Demeechai, Woo-Geun Lee, Dong-Hwan Kim and Tae-Gyu Chang, "IEEE 802.15.4 BPSK Receiver Architecture Based on a New Efficient Detection Scheme," IEEE Trans. on Signal Processing, 17 vol.58, no.9, pp.4711-4719, Sept. 2010.
  18. 18. Likelihood Function from Student-t Copula Copula10 function helps to create joint distributions from marginal CDFs and their inter-dependency  Gaussian and Student-t Copula models linear dependency  Gumbel, Frank and Clayton Copulas model tail dependency Theorem 2: The likelihood function (LP) for estimating the position of a transmitter from N shadow fading residuals measured by M IEEE 802.15.4 receivers is given by 1 ~N 1 ~N 1 ~N LP cv, P tv F1 Z s1 , tv F2 Z s2 ,, tv FM Z sM 1 where tn is the inverse CDF or quantile function vector of a student-t distribution with degree of freedom v, cv,P {•} is an M-variate student-t copula density with v degree of freedom, P is an MxM correlation coefficient matrix given by Ρ={ρkl}; k,l ϵ {1,2,…,M} and ρkl is the correlation coefficient between receiver k and l.10R. B. Nelsen, “An Introduction to Copulas, Lectures Notes in Statistics,” Springer Verlag, New York, 1998. 18
  19. 19. Shadow Fading Correlation Simulations r2 vs. ρ r12=10m r1=10m τm=0.1μs ω=1 interferer/sq. m Simulation Scenario τm vs. ρ r1=10m r1=10m r2=10m r2=10m r12=10m τm=0.1μs ω=1 interferer/sq. m r12 vs. ρ ω=1 interferer/sq. m 19
  20. 20. Wireless Hardware MSP430 16-bit Microcontroller CC2420 Radio is an IEEE 802.15.4 receiver operating at 2.45 GHz Z1 Mote Patch Antenna 8 bit RSSI values Tiny OS Mote internals 20
  21. 21. Experimental Results Localization area approx. 1250 sq. m with an average of 1000 people moving in this area during peak lunch hour traffic on a weekend between of 10AM and 1PM 8 Receivers R1 through R8 localizing a transmitter Localization Errors at Various Locations Transmitter Localization Error (m) Location Mean Median 90th Perc. Std. Dev T1 2.458 2.329 3.962 1.727 T2 2.378 2.267 3.628 1.221 T3 3.537 3.496 5.234 2.377 Food Court Layout T4 2.739 2.912 4.138 1.839  Degree of freedom v=4, U and L for Summary of Localization Errors GPD set at 90th and 10th percentile Localization Error (m) were heuristically chosen to give the Method best localization results Mean Median 90th Perc. Std. Dev Proposed Method 2.778 2.751 4.2405 1.791 MDS 12.343 15.925 25.358 6.464 21
  22. 22. Summary Extended the operating frequency range of cross- correlation based localization from 10MHz to 2.45GHz Copula likelihood function was found to be a better cost function for cross-correlation based localization than MDS as it adapts to LoS conditions between receiver and transmitter Cross-correlation based localization method is particularly suited for fading rich environment such as factory floor, malls etc. where there is a high pedestrian or machinery traffic 22
  23. 23. Questions? Thank you!