Cooperative Transmit Power Estimation under Wireless Fading Murtaza Zafer (IBM US), Bongjun Ko (IBM US), Ivan W. Ho (Imperial College, UK) and Chatschik Bisdikian (IBM US)

Problem Synopsis Node T is a wireless transmitter with unknown Tx power P, and unknown location (x,y) Nodes {m 1,…, m N } are monitors that measure received power {p i } Goal – given {p i } and {(x i,y i )} (monitor locations), estimate unknown P (and also unknown location (x,y)) m2m2 P (x,y) m3m3 mNmN p N (x N,y N ) p 3 (x 3,y 3 ) p 2 (x 2,y 2 ) p 1 (x 1,y 1 ) m1m1 T

Problem Synopsis Sensor Networks – Event detection – {m 1,…, m N } are sensors, and T is the source point of an event – Goal – detect important events, eg: bomb explosion, based on measured power Wireless Ad-hoc Networks – physical layer monitoring – {m 1,…, m N } monitor a wireless network – Goal – detect maximum transmit power violation; i.e. detect misbehaving/mis- configured nodes, signal jamming m2m2 P (x,y) m3m3 mNmN p N (x N,y N ) p 3 (x 3,y 3 ) p 2 (x 2,y 2 ) p 1 (x 1,y 1 ) m1m1 T Applications Blind estimation – no prior knowledge (statistical or otherwise) of the location or transmission power of T

Talk Overview Power propagation model – Lognormal fading Deterministic Case – geometrical insights Single/two monitor scenario Multiple monitor scenario Stochastic Case Maximum Likelihood (ML) estimate Asymptotic optimality of ML estimate Numerical Results Conclusion

Power Propagation model Lognormal fading P i = received power at monitor i d i = distance between the transmitter and monitor i α = attenuation factor, (α > 1) k = normalizing constant H i = lognormal random variable W i – unknown to the monitor – represents the aggregated effect of randomness in the environment; eg: multi-path fading didi PiPi T mimi P

Deterministic Case Power propagation model: T 1 Monitor 1 P P1P1 d1d1 best estimate of transmit power: P* P 1 Single monitor measurement (no fading/random noise in power measurements)

Deterministic Case Monitor 2 Note: d 1, d 2 are unknown Monitor 1 P P1P1 P2P2 d 12 d1d1 d2d2 2 T 1 Simple Cooperation: P* max(P 1, P 2 ) Q: Can we do better? Locus of T, Two monitor scenario Eqn (1) Eqn (2) Equation of a circle

Deterministic Case Two monitor scenario P achieves lower bound, 2 1 T (x 1, y 1 )(x 2, y 2 ) P1P1 P2P2 T T T T T (x, y) x θ where, center of circle

Deterministic Case Multiple monitor scenario With multiple monitors – diversity in measurements System of equations with unknowns (x,y,P) We should be able to solve these equations to obtain exact P ? Answer: Yes and No !!

Deterministic Case 1 2 (x r, y r ) d r,1 d r,2 T (x, y) 3 4 d1d1 d2d2 Theorem: There is a unique solution (P*, x*, y*) except when the monitors are placed on an arc of a circle or a straight line that does not pass through the actual transmitter location. Proof: A location (x, y) is a solution if and only if it satisfies d 1 /d 2 =c 1, …, d N-1 /d N = c N-1 The actual location (x r, y r ) is one solution; thus d r,1 /d r,2 =c 1, …, d r,N-1 /d r,N = c N-1 There exists another solution at (x, y) if and only if, d r,1 /d r,2 = d 1 /d 2, …; equivalently, T

Deterministic Case 1 2 (x r, y r ) d r,1 d r,2 T (x, y) 3 4 d1d1 d2d2 Observation: Without transmit power information, and if monitors lie on an arc of a circle, even with infinite monitors and no fading, the transmission power (and transmitter location) cannot be uniquely determined. T Theorem: There is a unique solution (P*, x*, y*) except when the monitors are placed on an arc of a circle or a straight line that does not pass through the actual transmitter location.

Deterministic Case Multiple monitor scenario 12 Corollary 1: Two monitors always has multiple solutions

Deterministic Case Multiple monitor scenario 13 Corollary 1: Two monitors always has multiple solutions Counter-intuitive Insight: For any regular polygon placement of monitors the transmission power cannot be uniquely determined !! Corollary 2: Three monitors as a triangle always has multiple solutions 2 Conversely: For all non-circular placement of monitors, transmission power can be uniquely determined.

Talk Overview Power propagation model – Lognormal fading Deterministic Case – geometrical insights Single/two monitor scenario Multiple monitor scenario Stochastic Case ML estimate Asymptotic optimality of ML estimate Numerical Results Conclusion

Stochastic Case m1m1 P (x,y) m2m2 mNmN p N (x N,y N ) p 2 (x 2,y 2 ) p 1 (x 1,y 1 ) Let z i = ln(p i ), Let Z = ln(P), and ML estimate (Z*,x*,y*) is the value that maximizes the joint probability density function The joint probability density function Maximum Likelihood Estimate T Power attenuation model

Stochastic Case Theorem: The ML estimate for N monitor case is given as, (x*,y*) is the solution to the minimization above, where the objective function is sample variance of {ln(p i d i α )} distance between some location (x,y) and monitor i distance between estimated Tx. location (x*,y*) and monitor i P* is proportional to the geometric mean of {p i (d* i ) α }

Stochastic Case What happens when N increases ? more number of measurements of received power increase in the spatial diversity of measurements Does the transmission power estimate improve ? Answer: Yes !! ; Estimator is asymptotically optimal

Stochastic Case Asymptotic optimality as N increases Random Monitor Placement N monitors placed i.i.d. randomly in a bounded region Г Each monitor makes an independent measurement of the received power Random placement is such that it is not a distribution over an arc of a circle Let P N * be the estimated transmit power using the results presented earlier Theorem: As N increases the estimated transmit power converges to the actual power P almost surely,

Numerical Results Synthetic data set –N = 2 to 20 monitors placed uniformly at random in a disk of radius R = 40. –Received power is generated by i.i.d. lognormal fading model for each monitor. –Performance measured: averaged over estimation for 1000 transmitter locations. Empirical data set –Sensor network measurement data by N. Patwari. –Total 44 wireless devices; each device transmits at dBm; received powers are measured between all pairs of devices –The data is statistically shown to fit well to the lognormal fading model = 2.3, and dB = –Randomly chosen N=3,4,…,10 monitors out of 44 devices.

Numerical Results Performance metric –The above metric measures the average mean-square dB error Estimators –MLE-Coop-fmin ML estimate with fminsearch in MATLAB for location estimation –MLE-Coop-grid ML estimation with location estimation by dividing region into grid points –MLE-ideal ML estimate by assuming that the transmitter location is magically known –MLE-Pair ML estimate is obtained by considering only monitor pairs Average taken over all the pair-wise estimates

Numerical Results Synthetic data set Empirical data set (MLE-Coop-grid)

Conclusion Blind estimation of transmission power – Studied estimators for deterministic and stochastic signal propagation – Utilized spatial diversity in measurements – Obtained asymptotically optimal ML estimate – Presented numerical results quantifying the performance Geometrical insights – Two-monitor estimation was equivalent to locating the transmitter on a certain unique circle – If monitors are placed on a arc of a circle, the transmission power cannot be determined with full accuracy (even with infinite monitors)