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Path Loss Exponent Estimationin Large Wireless Networks

Sunil Srinivasa and Martin Haenggi

Network Communications and Information Processing (NCIP) LabDepartment of Electrical Engineering

University of Notre Dame

Aug 13, 2009

Sunil Srinivasa and Martin Haenggi ()

University of Notre Dame IT School 2009 1 / 22
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Motivation

Large-scale path loss law: signal strength attenuates withdistance d as d

S

dd 0

.

Though it is typically assumed in analysis and design problemsthat the path loss exponent (PLE) is known a priori, it is oftennot the case.

The PLE has a strong impact on the quality of links, andtherefore needs to be accurately estimated for the efficientdesign and operation of systems.

Sunil Srinivasa and Martin Haenggi ()

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Motivation (contd.)

Example 1: The information-theoretic capacity of large randomad hoc networks scales as

n2 / 2

for 2 < 3 n for 3.Depending on the value of , different routing strategies arerequired to be implemented.

A. Ozgur, O. Leveque and D. Tse, Hierarchical Cooperation AchievesOptimal Capacity Scaling in Ad Hoc Networks, IEE E T ra n s. I nfo . Th . , 2007.Sunil Srinivasa and Martin Haenggi ()

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Motivation (contd.)

Example 2: Outage probability in a planar Poisson pointprocess with Rayleigh fading.

2 2.5 3 3.5 4 4.5 50.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

= 1, p = 0.05, m = 1, N 0 = 25 dBm

O u

t a g e p r o

b a

b i l i t y

= 20 dB = 10 dB = 5 dB = 0 dB = 5 dB

The system performance critically depends on .Sunil Srinivasa and Martin Haenggi () University of Notre Dame IT School 2009 4 / 22
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System Model

Receiver

Transmitter

Filled circles: transmitters.Empty circles: receivers.

An innite Poisson pointprocess (PPP) on R 2 withdensity .

Channel access scheme isALOHA. p is the ALOHA contentionparameter. Therefore, theset of transmitters forms aPPP with density p .No synchronization.

Sunil Srinivasa and Martin Haenggi () University of Notre Dame IT School 2009 5 / 22
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System Model (contd.)

Attenuation in the channel: product of large-scale path loss, with PLE .small-scale fading (m -Nakagami).m = 1: Rayleigh fading ; m : no fading.Noise is AWGN with variance N 0.

All the transmit powers are equal to unity (no power control).

Problem: How do you accurately estimate the PLE at each node inthe network in a completely distributed manner?

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What Makes Estimating the PLE Complicated?

The large-scale path loss is commonly taken to be deterministicwhile the small-scale fading is modeled as a stochastic process.This distinction, however, does not hold when the nodesthemselves are randomly arranged. So, we need to considerthe distance and fading ambiguities jointly .Moreover, PLE estimation needs to be performed during theinitialization of the network. During this phase, the system istypically interference-limited due to the presence of uncoordinated transmissions.

Purely RSS-based estimators cannot be used in these situations.

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Overview

Propose three distributed algorithms for estimating the PLE inlarge random wireless networks that explicitly take into account

the uncertainty in the locations of the nodes.the uncertainty in the fading gains across links.the interference in the network.

Provide simulation results to demonstrate the performance of the algorithms and quantify the estimation errors.



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The Big Picture

The PLE estimation problem is essentially tackled by equatingthe empirical (observed) values of certain networkcharacteristics to their theoretically established values .

By obtaining measurements over several time slots, the PLE canbe estimated at each node in a distributed fashion.The three PLE algorithms are each based on a specicnetwork characteristic :

the mean interference.the outage probability.connectivity properties of a node.



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Simulation Details

We use 50, 000 different realizations of the PPP to analyze themean error performance of the algorithms, which is characterized

using the relative MSE, dened as E

( )2

/ .We used p = 0.05 since it was suitable. Note the tradeoffs.

Large p: results in few quasi-different realizations of thetransmitter PPP.Small p : takes long for the algorithms to convergence.



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Algo. 1: Using the Mean Interference

This algorithm assumes that the network density is known.In theory, the mean interference is given by

= 2pA 2 0 / ( 2) , (1)

where A0 is the near-eld radius of the antenna.Implementation

Nodes simply need to record the mean strength of the receivedpower, , averaged over several time slots.Equating to , and using the known values of p , A0, and , is found from a look-up table.

J. Venkataraman, M. Haenggi and O. Collins, Shot Noise Models for Outage

and Throughput Analyses in Wireless Ad Hoc Networks, MI L CO M , 2006.Sunil Srinivasa and Martin Haenggi () University of Notre Dame IT School 2009 11 / 22
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Algo. 1: Using the Mean Interference (contd.)

Relative MSE of versus the number of time slots.

200 400 600 800 1000 1200 1400 1600 1800 20001000.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

Number of time slots N

R e

l a t i v e

M S E

= 1, p = 0.05, m = 1, N 0 = 25 dBm, A 0 = 1

= 2.5 = 3 = 3.5 = 4

= 4.5

The estimates are fairly accurate over a wide range of parameters.

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Algo 2: Based on (Virtual) Outage Probabilities

This algorithm does not require the knowledge of or m .In a PPP, when the signal power is exponentially distributed, theprobability of a successful transmission p s is

p s = P (SIR > ) = exp( c2/

) , (2)

where c = p m + 2 1 2 (m )m

2/ .

Nodes can determine the SIR, and consequently p s by

computing the ratio of the power of the signal (which arrivesfrom a virtual transmitter, and is assumed to be exponentiallydistributed) and the total received power.

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Algo 2: Based on (Virtual) Outage Prob. (contd.)

Implementation : A differential method.Obtain a histogram of the observed SIR values measured overseveral time slots.

The empirical success probabilities ( p

s ,i = P (

SIR >

i),i = 1, 2) are obtained at two different threshold values.

Inverting (2), an estimate of is obtained as

= 2 ln(1/ 2)

ln (ln p s ,1 / ln p s ,2) . (3)

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200 400 600 800 1000 1200 1400 1600 1800 20001000

0.05

0.1

0.15

0.2

0.25

= 1, p = 0.05, m = 1, N 0 = 25 dBm, 1 = 10 dB, 2 = 0 dB

R e

l a t i v e

M S E


= 2.5 = 3 = 3.5 = 4 = 4.5

The estimation error increases with larger .

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Relative MSE of versus the Nakagami parameter.

10 1000.5 10.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Nakagami parameter m

R e

l a t i v e

M S E

= 1, p = 0.05, N 0 = 25 dBm, 1 = 10 dB, 2 = 0 dB, N = 10000

= 2.5 = 3 = 3.5 = 4 = 4.5

This algorithm performs more accurately at lower values of m .

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Algo 3: Based on the Cardinality of the Tx Set

This algorithm also does not require to know m or .Transmitter node y is in receiver node xs transmitting set , T(x)if they are connected, i.e., the SIR at x due to ys signal is > .We prove that under the conditions of m

N ,

E |T(x) | = N T = (m ) 1 2

m

(m + 2 ) (2 2 )

2/ . (4)

We see that N T is inversely proportional to 2/ , and surmisethat this behavior holds at arbitrary m

R + .

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Algo 3: Based on the Card. of the Tx Set (contd.

ImplementationFor a known threshold 1 1, at time slot i, 1 i N , set

N T ,1 ( i ) = 1 if the node can decode a packet0 otherwise.

Evaluate N T ,1 and N T ,2 at two different threshold values 1 and2 respectively.In theory, we obtain N T ,1 / N T ,2 = (2/ 1)

2/ .

Inverting this, we have

= (2 ln(2/ 1)) / ln(N T ,1 / N T ,2) . (5)



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200 400 600 800 1000 1200 1400 1600 1800 20001000.12

0.14

0.16

0.18

0.2

0.22

0.24


R e

l a t i v e

M S E

= 1, p = 0.05, m = 1, N 0 = 25 dBm, 1 = 10 dB, 2 = 0 dB

= 2.5 = 3 = 3.5 = 4 = 4.5

In contrast to Algo. 1 and 2, the relative MSE decreases with .



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Relative MSE of versus the Nakagami parameter.

1 10 1000.50.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Nakagami parameter m

R e

l a t i v e

M S E

= 1, p = 0.05, N 0 = 25 dBm, 1 = 10 dB, 2 = 0 dB, N = 10000

= 2.5 = 3 = 3.5 = 4 = 4.5

The estimates are more accurate at lower m .


d
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Summary and Discussion

We have addressed the PLE estimation problem in the presenceof node location uncertainties, m -Nakagami fading and mostimportantly, interference!

Each of the algorithms are fully distributed and do not requireany information on the location of other nodes or the value of m .Based on the relative MSE values, we conclude that at lowvalues of , Algo. 1 performs the best (though it requires the

density to be known), while when is high, Algo. 3 is preferred.


S d Di i ( d )
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Summary and Discussion (contd.)

Each of our algorithms work by equating empirical values withtheir corresponding theoretically established ones.The caveat is that the theoretical results are for an averagenetwork while in practice, we have only a single realization of

the node distribution at hand. Thus, in general, the estimateswe obtain are biased.This also explains the fact that the performance of Algorithms 2and 3 is better at lower m .

We remark that the bias (and the MSE) can be signicantlylowered if nodes have access to several independent realizationsof the point process or if they are allowed to communicate.

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