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