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

Mohammed Rana Basheer, S. JagannathanDept. of Electrical and Computer EngineeringRolla, MO, USA{mrbxcf, sarangap}@mst.edu

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*

*http://www.ce.washington.edu/sm03/boeingtour.htm

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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

Localization Errors Multipath fading and shadow fading noise are

the primary cause for large localization error in an indoor environment

Tx

Rx

Tx

Rx

Multipath Fading Shadow Fading

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RSSI Profile of ERL 114R

SS

I (d

B)

Grid Points Grid Points

Layout of ERL 114

Spans 12m x 13m

Typical lab floor with tables, partitions, heavy equipments such as pumps etc.

0.6m x 0.6m grid

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Similarity in Fading Noise Statistics

Adjacent wireless receivers will experience similar fading noise statistics

Cross-correlation in fading noise between adjacent receivers may be used to determine their relative position to a common transmitter

Tx

Rx1

Shadow Fading

Rx2

Fading noise depends on the radio signal propagation environment

5

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

1Basheer, M.R.; Jagannathan, S.; , "Localization of objects using stochastic tunneling," Wireless Communications and Networking Conference (WCNC), 2011 IEEE , pp.587-592, 28-31 March 2011

Multipath noise correlation with distance

Previous Work (contd.)

Non-Parametric Methods treat the localization as a dimensionality reduction problem

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 devices

2X. Ji, and H. Zha, "Sensor positioning in wireless ad-hoc sensor networks using multidimensional scaling," 23rd Annual Joint Conf. of the IEEE 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 IEEE International 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.

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7

Localization Block Diagram

IEEE 802.15.4 Receiver 2

IEEE 802.15.4 Receiver M

IEEE 802.15.4 Receiver 1

CopulaOptimization

Function

1sZ

2sZ

MsZ

8

XT,YT =Transmitter CoordinatesZsi = Shadow Fading Residual from ith receiver

Base Station

XT,YT

Ornstein-Uhlenbeck Filter

AR(1)+

GARCH(1,1)

RSSI From IEEE 802.15.4

Receiveris

Z

Shadow Fading Extraction Block Diagram

9

)(tX )(tX s

X(t) = RSSI at time instance tXs(t) = Shadow Fading Residual + Path LossZsi = Shadow Fading Residual

Build Semi-Parametric CDF

Build Semi-Parametric CDF

Build Semi-Parametric CDF

1

~F

2

~F

MF~

1sZ

2sZ

MsZ

Copula Optimization Function

PvC ,

Compute pair-wise Cross-Correlation

Possible Transmitter Coordinates (x,y)

Student-t CopulaFunction

),( yxP

Stochastic Optimization

Zsi = Shadow Fading Residual 𝓕i = Semi-Parameter Shadow Fading CDFP(x,y) = MxM shadow fading cross-correlationCv,p = Student-t copula function

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XT,YT

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 reverting process

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.

In this paper multipath noise is modeled as a stochastic process called Ornstein Uhlenbeck (OU) to isolate shadow fading residuals from RSSI

Shadow fading from received signal power5

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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

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 estimators6

OU Model for Multipath Noise

6L. 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.

2f

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AR+GARCH to Isolate Shadow Fading Residuals Autoregressive Model (AR) for Xs(t) is used to separate path loss

from shadow fading residuals

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, is the shadow fading variance and Zs is the stationary zero mean unit variance shadow fading residual.

Generalized Auto Regressive Conditional Heteroskedasticity (GARCH) for to account for changes in pedestrian traffic

AR(1)

GARCH(1,1)

ts2

ts2

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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

Regions around the mode of the residuals will be modeled using non-parametric empirical CDF

where I(·) is the indicator function, are N shadow fading residuals from ith receiver in the localization area

CDF of Shadow Fading Residuals

iLi

iiNi

iUi

Ni

LxxF

UxLxF

UxxF

xF

),(

),(ˆ),(

)(~

)()2()1( ,, Nssssi iiiZZZZ

N

k

ks

Ni MixZI

NxF

i1

)( ,,2,1;1

)(ˆ

Upper Tail (Parametric)

Lower Tail (Parametric)

Mode (Empirical)

CDF of Upper and Lower Tails of Shadow Fading Residuals Upper and lower tails, were sample points are sparse by

definition, a parametric Generalized Pareto Distribution7 (GPD) was applied

i

ii

i

Ui

UxxF

1

1)(

i

ii

i

Li

LxxF

1

1)(

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 data

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7J. R. M. Hosking and J. R. Wallis, “Parameter and quantile estimation for the Generalized Pareto Distribution,” Technometrics, Vol. 29, No. 3, pp. 339-349j, Aug 1987

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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 affect signal fading if and only if its ToA satisfies

GBSBEM Wireless Channel Model8

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

mc

rt

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.

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Shadow Fading Correlation Coefficient

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

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 .

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

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12

SS

S

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, vol.58, no.9, pp.4711-4719, Sept. 2010.

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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

where 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.

1nt

10R. B. Nelsen, “An Introduction to Copulas, Lectures Notes in Statistics,” Springer Verlag, New York, 1998.

Ms

NMvs

Nvs

NvPvP ZFtZFtZFtcL

~,,

~,

~ 12

11

1, 21

Shadow Fading Correlation Simulations

r12=10mr1=10mτm=0.1μsω=1 interferer/sq. m

r1=10mr2=10mτm=0.1μsω=1 interferer/sq. m

r1=10mr2=10mr12=10mω=1 interferer/sq. m19

Simulation Scenario

r2 vs. ρ

τm vs. ρ

r12 vs. ρ

Wireless Hardware

MSP430 16-bit Microcontroller

CC2420 Radio is an IEEE 802.15.4 receiver operating at 2.45 GHz

Patch Antenna

8 bit RSSI values

Tiny OS

Z1 Mote

Mote internals

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Experimental Results

Transmitter Location

Localization Error (m)

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

T4 2.739 2.912 4.138 1.839

MethodLocalization Error (m)

Mean Median 90th Perc. Std. DevProposed Method 2.778 2.751 4.2405 1.791

MDS 12.343 15.925 25.358 6.464

Localization Errors at Various Locations

Summary of Localization Errors

Food Court Layout

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

Degree of freedom v=4, U and L for GPD set at 90th and 10th percentile were heuristically chosen to give the best localization results

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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

Questions?

Thank you!