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 {m1,…, mN} are monitors that

measure received power {pi}

Goal – given {pi} and {(xi,yi)} (monitor

locations), estimate unknown P (and

also unknown location (x,y))

m2

P (x,y)

m3

mN

pN

(xN,yN)p3

(x3,y3)p2

(x2,y2)

p1

(x1,y1)m1

T

Problem Synopsis

Sensor Networks – Event detection

– {m1,…, mN} 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

– {m1,…, mN} monitor a wireless network

– Goal – detect maximum transmit power violation; i.e. detect misbehaving/mis-configured nodes, signal jamming

m2

P (x,y)

m3

mN

pN

(xN,yN)p3

(x3,y3)p2

(x2,y2)

p1

(x1,y1)m1

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

Pi = received power at monitor i

di = distance between the transmitter and monitor i

α = attenuation factor, (α > 1) k = normalizing constantHi = lognormal random variable

iii WdkPP )ln()ln(ln

i

ii dkPHP r.v. lognormal;iW

i eH

Wi – unknown to the monitor – represents the aggregated effect of randomness in the environment; eg: multi-path fading

di

Pi

T

mi

P

Deterministic Case

dkPPr Power propagation model:

T 1

Monitor 1

P P1

d1

best estimate of transmit power:

P* ≥ P1

Single monitor measurement

(no fading/random noise in power measurements)

Deterministic Case

Monitor 2

Note: d1, d2 are unknown

Monitor 1

P

P1

P2d12

d1 d2

2

T

1

Simple Cooperation: P* ≥ max(P1, P2)

Q: Can we do better?

Locus of T, constant)(,/1

1

2

2

1 cP

P

d

d

Two monitor scenario

1

1 dPkP Eqn (1)

2

2 dPkP Eqn (2)

Equation of a circle

cyyxx

yyxx

22

22

21

21

)()(

)()(

Deterministic Case

Two monitor scenario

cos1cos)1(

1 222

121*

cc

dP

kP

P achieves lower bound,

/1

2

/1

1

12*

11

1

PP

d

kP

21

T

(x1, y1) (x2, y2)

P1

P2T

T

T

T

T

(x, y)

xθ

cyyxx

yyxx

22

22

21

21

)()(

)()( /1

1

2

P

Pcwhere,

center of circle

0,

)1(2

)1(2

122

c

dc

Deterministic Case

Multiple monitor scenario

;)( 1

/1

1

2

2

1 cP

P

d

d

;)( 2

/1

2

3

3

2 cP

P

d

d

)( 1

/1

1

1

N

N

N

N

N cP

P

d

d

• 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(xr, yr)

dr,1

dr,2

T(x, y)

3

4

d1

d2

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 d1/d2=c1, …, dN-1/dN = cN-1

• The actual location (xr, yr) is one solution; thus dr,1/dr,2=c1, …, dr,N-1/dr,N = cN-1

• There exists another solution at (x, y) if and only if, dr,1/dr,2 = d1/d2 , …; equivalently,

T

Deterministic Case

1

2(xr, yr)

dr,1

dr,2

T(x, y)

3

4

d1

d2

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

1 2

Corollary 1: Two monitors always has multiple solutions

Deterministic Case

Multiple monitor scenario

1 3

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

m1

P (x,y)

m2 mN

pN

(xN,yN)p2

(x2,y2)p1

(x1,y1)

Let zi = ln(pi), Let Z = ln(P), and ),,( yxZ

ML estimate (Z*,x*,y*) is the value that maximizes the joint probability density function

);(maxarg*)*,*,(

zfyxZ

The joint probability density function

Maximum Likelihood Estimate

iii WdkPP )ln()ln(ln 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(pidiα)}

22 )()( yyxxd iii

22* *)(*)( yyxxd iii

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 {pi(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 PN* 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 -37.47 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 = 3.92.

– 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

])log10*log10[( 21010 PPEdBError K

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)