90 IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 3, NO. 1, FEBRUARY 2014

Distributed Estimation in Homogenous Poisson Wireless Sensor NetworksAbhishek Agarwal and Aditya K. Jagannatham

Abstract—In this paper we develop a homogenous Poissonpoint process based framework to characterize the performanceof distributed estimation in wireless sensor networks. Suchstochastic geometry based approaches have been shown to berobust compared to idealistic grid/ cluster models in wirelessnetworks. We consider a fading wireless channel between thesensors and the Fusion Center (FC) and characterize the dis-tributed estimation outage in a variety of scenarios. Towardsthis end we consider optimal maximal ratio combining (MRC)based coherent sensor transmission and derive closed formexpressions for the outage probability. Further, we extend it toscenarios with fading channel phase based partial Channel StateInformation (CSI) and subsequently incorporate the effects ofvarious path-loss models in large scale wireless sensor networksto derive bounds for the outage probability. Simulation results arepresented to validate the performance of the proposed analysis.

Index Terms—Wireless sensor networks, distributed estima-tion, Poisson point process, estimation outage.

I. INTRODUCTION

W IRELESS Sensor Networks (WSN) which deployminiature sensor nodes over a wide geographical area

to sense and estimate various environmental parameters suchas temperature and humidity have gained widespread appeal[1]. Towards this end, the individual sensors transmit thesensed data to a Fusion Center (FC), which combines thesemeasurements to generate the parameter estimate. Such aparadigm of hierarchical sensor network signal processingbased distributed estimation significantly enhances the accu-racy of the noisy measurements at the energy constrainedconstituent sensor nodes. Moreover such analog parameterforwarding schemes have been demonstrated to lead to optimalscaling properties for WSNs [2]. Several works in existingliterature have explored the performance of various distributedestimation schemes in WSNs. In [3] the authors analyze theasymptotic variance of the error estimate for different fadingchannel distributions, and for different levels of Channel StateInformation (CSI) at the sensor nodes. The authors in [4]present a linear decentralized scheme for minimum meansquared error (MMSE) estimation based on complete CSI atthe sensor nodes, while in [5] an analysis is presented tocharacterize the estimation diversity and the estimation outageprobability.

The performance of such distributed estimation schemes inWSNs depends on the configuration and distance propertiesof sensor deployment. However practical WSN installations,similar to cellular networks, are often characterized by highlyirregular configurations arising due to the harsh geographic

Manuscript received September 30, 2013. The associate editor coordinatingthe review of this letter and approving it for publication was T. Q. S. Quek.

The authors are with the Department of EE, IIT Kanpur, 208016, India(e-mail: adityaj@iitk.ac.in).

Digital Object Identifier 10.1109/WCL.2013.112313.130696

conditions, which are not conducive for a planned grid stylesensor deployment. This is especially true in random de-ployment based ad hoc sensor networks [6]. A significantshortcoming of existing works such as [2], [3], [4], [5] is thatthey do not consider the statistical properties of the estimationprocess arising out of a random deployment of sensor nodes,which is frequently the case in WSNs. In this context, spatialpoint processes such as the stationary Poisson point process(PPP) based framework has been shown to be ideally suitedfor the analysis of the behavior of such networks with randomspatial configurations [7], [8], [9]. Further, the homogenousPPP modeling has been shown to yield tractable and accurateresults for SINR characterization in actual urban cellulardeployments. Even though PPP is a novel tool for analysisof networks, a majority of works in existing literature focusonly on cellular and allied approaches for SINR maximization.A unique aspect of the work in this paper is that it considersPPP based analysis for a distributed estimation scenario. In[10] the authors consider a Poisson process based placementof sensors and data collectors. However, the framework thereindoes not consider distributed sensing and considers the signalsfrom other co-located sensors as interference. The work in [11]considers a distributed detection framework in Poisson sensornetworks. However, the authors consider a simplistic modelwhich ignores the fading nature of the wireless channel andthe availability of various degrees of channel state information(CSI) at the transmitter. Therefore, in this work we presenta performance analysis of distributed estimation in WSNsemploying a stationary PPP based framework to characterizethe stochastic geometry of such networks. Since, it is basedon a random PPP analysis of WSNs, it has a good practicalutility as it can be used in scenarios with irregular placement ofsensor nodes. We demonstrate the performance of the optimalMRC scheme at the FC and derive a closed form expressionfor the probability of estimation outage. We then extend theanalysis to the scenario where partial CSI in the form ofthe phase of the fading channel coefficient is available atthe sensor nodes and derive bounds on the estimation outageprobability. Subsequently, we also derive the performancebounds incorporating large scale signal propagation effectssuch as path-loss for widely dispersed sensor nodes and finallyincorporate noisy sensor measurements. Simulation resultsare presented in the end to validate the performance of theproposed analysis.

In section II we describe the statistical properties of thespatially random WSNs and derive the outage performanceof optimal combining at the FC with full and partial CSI.Subsequently, we analyze the WSN performance incorporatingpath-loss and in section III extend the analysis to the case ofnoisy sensor measurements. Simulation results are presentedin section IV followed by the conclusion in section V.

2162-2337/14$31.00 c© 2014 IEEE

AGARWAL and JAGANNATHAM: DISTRIBUTED ESTIMATION IN HOMOGENOUS POISSON WIRELESS SENSOR NETWORKS 91

II. PPP BASED WSN MODELING AND OPTIMAL FCCOMBINING

Consider a WSN with wireless sensor nodes deployedaccording to a two-dimensional stationary PPP Φ of density λ[12]. A comprehensive introduction to the properties of suchspatial point processes in the context of wireless networkscan be found in [8]. Similar to [3], we consider a multipleaccess channel between the sensor nodes and the FC. Letthe measurement of the parameter θ sensed by the node atlocation x ∈ Φ be given as θ+ ηx, where ηx is the zero-meanadditive white Gaussian noise (AWGN) with power σ2

η i.e.ηx ∼ CN(0, σ2

η). The signal y received at the FC is,

y =∑x∈Φ

αxhx(θ + ηx) + ν, (1)

where hx is the Rayleigh fading channel coefficient of averagepower gain 2σ2

h between the sensor at x ∈ Φ and the FC,while αx denotes the corresponding precoding coefficient. Thequantity ν is the AWGN at the FC, ν ∼ CN(0, σ2

ν). We beginby considering the scenario with full CSI at the sensor nodesand noiseless sensor measurements i.e. σ2

η is small. The resultsfor partial CSI and noisy sensor measurements are derived inlater sections in this work.

The Best Linear Unbiased Estimator (BLUE) for θ is givenas θ = y∑

x∈Φ αxhx[13, Chapter 3], and the variance of the cor-

responding estimation error can be derived as E

{∣∣∣θ − θ∣∣∣2} =

σ2ν

|∑x∈Φ hxαx|2 . Hence, one can now compute the probability of

estimation outage P(T ), for a given threshold T as,

P(T ) = P

(E

{∣∣∣θ − θ∣∣∣2} ≥ T

)

= P

⎛⎝∣∣∣∣∣∑x∈Φ

hxαx

∣∣∣∣∣2

≤ σ2ν

T

⎞⎠ . (2)

It can be readily seen that the optimal coefficient αx for thefull CSI based SNR maximization at the FC is given by h�

x

‖h‖ ,

where ‖h‖ =√∑

x∈Φ |hx|2 and h�x denotes the complexconjugate of hx. Thus, the corresponding estimation outagePFCSI(T ) can be computed by substituting αx =

h�x

|h| in (2).

Therefore,∣∣∑

x∈Φ hxαx∣∣2 =

|∑x∈Φ |hx|2|2‖h‖2 =

∑x∈Φ |hx|2

and,

PFCSI(T ) = P

(∑x∈Φ

|hx|2 ≤ σ2ν

T

)(3a)

= En

{P

(∑x∈Φ

|hx|2 ≤ σ2ν

T

∣∣∣∣∣ |Φ| = n

)}(3b)

= En

{P

(χ22n <

σ2ν/σ

2h

T

∣∣∣∣ |Φ| = n

)}(3c)

=

n=∞∑n=0

(λ |A|)n e−λ|A|

n!

γ(n,

σ2ν

2Tσ2h

)(n− 1)!

, (3d)

where |Φ| = n in eq. (3b) denotes conditioning on the numberof sensors n in the PPP Φ, |A| denotes the area of the sensingregion A, the quantity χ2

2n denotes a chi-squared random

variable with degree 2n, and γ(x, a) denotes the incompletegamma function [14]. Consider now the availability only ofpartial CSI, similar to [3], in the form of the phase ψx = ∠hxof the fading channel coefficient. For this case, employingαx = e−jψx =

h�x

|hx| with equal power as suggested in [3], onecan compute the corresponding estimation outage PPCSI(T )from (2) as,

PPCSI(T ) = P

⎛⎝(∑x∈Φ

|hx|)2

≤ σ2ν

T

⎞⎠

(a)

≥ P

(|Φ|∑x∈Φ

|hx|2 ≤ σ2ν

T

)

= En

{P

(χ22n <

σ2ν

nTσ2h

∣∣∣∣ |Φ| = n

)}

=

n=∞∑n=0

(λ |A|)n e−λ|A|

n!

γ(n,

σ2ν

2nTσ2h

)(n− 1)!

, (4)

where inequality (a) follows from the Cauchy-Schwartz in-

equality, (∑

x∈Φ |hx|)2 ≤ |Φ|(∑

x∈Φ |hx|2)

. Next we derivethe outage bounds considering path-loss in large scale WSNs.

A. Outage Bounds in Large Scale WSNs

Consider the power law path-loss function g(x) = |x|−l,for a sensor at position x ∈ Φ, where l is the path-loss expo-nent. Let the large-scale channel coefficient hx be defined ashx = hx|x|−l/2. For this scenario, the estimation outage prob-

ability in (2) becomes P(T ) = P

(∣∣∣∑x∈Φ hxαx

∣∣∣2 ≤ σ2ν

T

).

It can now be seen that the optimal full CSI combiningcoefficient αx is derived using MRC to be h�

x‖h‖ , where∥∥∥h∥∥∥ =

√∑x∈Φ

∣∣∣hx∣∣∣2. Let SA for A ⊂ Φ be defined as

SA =∑

x∈A∣∣∣hx∣∣∣2 and let Φy =

{x ∈ Φ |

∣∣∣hx∣∣∣2 > y

}. Thus

the estimation outage probability PFCSI(T ) = P(T )|αx=

h�x

‖h‖=

P(SΦ ≤ σ2

ν

T

). The result below gives the upper-bound for the

corresponding estimation outage function PFCSI(T ).

Theorem 1. The outage probability PFCSI(T ) for the WSNwith path-loss g(x) = |x|−l, as the sensing region A → R

2,can be upper-bounded as,

PFCSI

(σ2ν

T

)≤ P (ΦT = ∅) , (5)

where P (ΦT = ∅) = exp(−λπT−2/lΓ(1 + 2/l)

).

Proof: It can be seen that PFCSI

(σ2ν

T

)= P (SΦ ≤ T ) ≤

P (SΦT ≤ T ), where SΦ = SΦT + SΦcT

. Further, from thedefinition of SΦT , we have SΦT ≤ T ⇔ ΦT = ∅. Hence,PFCSI(

σ2ν

T ) ≤ P (ΦT = ∅), which can be simplified using the

92 IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 3, NO. 1, FEBRUARY 2014

PGFL of the PPP [8, Appendix A] as,

PFCSI

(σ2ν

T

)≤ EΦ

{Πx∈ΦP

(|hx|2g (x) ≤ T)}

≤ exp

(−λ∫R2

∫ ∞

T‖x‖l

exp(−h) dh dx

)

= exp(−λπT−2/lΓ(1 + 2/l)

). (6)

Thus, the above theorem characterizes the variation ofestimation outage for a given threshold T and PPP sensor nodedensity λ. Similarly, for the partial phase only CSI scenariowith path-loss, the optimal pre-coding coefficient αx =

h�x|hx| .

Let SA =∑

x∈A |hx| for A ⊂ Φ. Thus, the estimation outage

PPCSI(T ) = P

((∑x∈Φ |hx|

)2≤ σ2

ν

T

)can be upper-bounded

as given by the result below.

Theorem 2. The estimation outage PPCSI(T ) for the WSNwith path-loss g(x) = |x|−l, as the sensing region A → R

2,can be upper-bounded as,

PPCSI

(σ2ν

T

)≤ P (ΦT = ∅) , (7)

where P (ΦT = ∅) = exp(−λπT−2/lΓ(1 + 2/l)

).

Proof: Similar to the procedure for the complete CSIscenario above, we have, PPCSI

(σ2ν

T

)≤ P

(SΦT ≤ √

T)=

P (ΦT = ∅). Hence, the above upper-bound is identical to the

bound derived in (6) for PFCSI

(σ2ν

T

).

III. OUTAGE BOUNDS WITH NOISY SENSOR

MEASUREMENTS

In this section we derive the corresponding outage boundsfor noisy sensor measurements. From the system model in (1),θ, the BLUE of θ is,

θ = θ +ν∑

x∈Φ hxαx+

∑x∈Φ ηxhxαx∑x∈Φ hxαx

.

Thus the variance in the estimate θ is,

E

{∣∣∣θ − θ∣∣∣2} =

σ2v∣∣∑

x∈Φ hxαx∣∣2 +

∑x∈Φ σ

2η|hxαx|2∣∣∑

x∈Φ hxαx∣∣2 . (8)

The bounds for the corresponding estimation outage PFCSI(T )and PPCSI(T ) for the complete and partial phase only CSIscenarios respectively are given by the result below.

Theorem 3. The outage probabilities PFCSI(T ) and PPCSI(T )for the WSN with noisy measurements described above can belower-bounded as,

PFCSI(T ) ≥∑∞n=0 e

−λ|A| (λ|A|)nn! F+

χ22n

(σ2ν

(T−σ2η/n)σh

2

)(9)

PPCSI(T ) ≥∑∞

n=0 e−λ|A| (λ|A|)n

n! F+χ22n

(σ2ν

(nT−σ2η)σh

2

),

(10)

where F+χ2n(x) = 1 for x < 0 and F+

χ2n(x) = Fχ2

n(x)

otherwise, with Fχ2n(x) =

γ(n2 ,x)

Γ(n2 )

denoting the cumulative

probability density for the χ2n distribution.

Proof: From eq. (8), the outage probability PT is,

PT = P

(|∑x∈Φ

hxαx|2 ≤ σ2ν

T+σ2η

T

∑x∈Φ

|hxαx|2). (11)

Using the Cauchy-Schwartz inequality, we have,|∑x∈Φ hxαx|2 ≤ |Φ|∑x∈Φ |hxαx|2. Thus, with αx =

h�x

‖h‖for the full CSI scenario, it follows that,

PFCSI(T ) = P(∑

x∈Φ |hx|2 ≤ σ2ν

T +σ2η

T

∑x∈Φ |hx|4∑x∈Φ |hx|2

)(a)

≥P(∑

x∈Φ |hx|2 ≤ σ2ν

T +σ2η

T1|Φ|∑x∈Φ |hx|2

)=En

{P(σh

2χ22n ≤ σ2

ν

T +σ2η

nT σh2χ2

2n

)∣∣∣ |Φ| = n}

=∑∞

n=0 e−λ|A| (λ|A|)n

n! F+χ22n

(σ2ν

(T−σ2η/n)σh

2

), (12)

where inequality (a) follows from the above Cauchy-Schwartzinequality for αx =

h�x

‖h‖ . Similarly, with αx =h�x

|hx| for thepartial phase only CSI scenario, using the Cauchy-Schwartzinequality we have,

PPCSI(T ) ≥P

(|Φ|∑x∈Φ |hxαx|2 ≤ σ2

νT

+σ2η

T

∑x∈Φ |hxαx|2

)

=En

{P

((n− σ2

νT)σ2

hχ22n ≤ σ2

η

T

)∣∣∣∣ |Φ| = n

}

=∑∞

n=0 e−λ|A| (λ|A|)n

n!F+

χ22n

(σ2ν

(nT−σ2η)σh

2

). (13)

Thus, the above theorem yields a bound on the estimationoutage for noisy sensor measurements in a PPP distributedWSN with full and partial CSI at the sensor nodes.

IV. NUMERICAL RESULTS AND DISCUSSION

The results obtained in the previous sections for a ho-mogenous PPP based WSN are verified using simulations.The AWGN noise variance at the receiver σ2

ν = 1, and thevariance of the Rayleigh fading channel σ2

h = 1/2. Figure 1ashows the estimation outage probability, PFCSI(T ) versus 1/T ,for different values of λ|A| ∈ {40, 80, 120} for the full CSInoiseless measurement scenario. The corresponding analyticalvalues of the outage probability obtained using the closed formexpression in eq. (3) are also plotted therein, which coincidewith the results obtained through simulations. Similarly, theoutage probability, PPCSI(T ) versus 1/T , for the noiselesspartial phase only CSI and the corresponding lower-boundfrom eq. (4) are given in fig. 1b. We can observe from thefigure that the bounds are tight for low values of the number ofsensors |Φ(A)| since the Cauchy-Schwartz inequality is tightfor lower |Φ(A)|.

The outage probability PFCSI(T ) versus 1/T for the powerlaw path-loss with full CSI and the corresponding lowerbounds from Theorem 1 for different values of λ ∈ {3, 5}are given in fig. 1c. The bounds can be seen to be close fordifferent values of λ. Figure 1d plots the simulated outageprobability PPCSI(T ) and the bounds in Theorem 2 for thepartial phase only CSI. Finally, figs. 1e and 1f give the plotsfor the outage probabilities PFCSI(T ) and PPCSI(T ) versus 1/Tfor the full and partial CSI scenarios respectively for the noisysensor measurement scenario considered in Section III. Thecorresponding bounds from Theorem 3 are also given therein.

AGARWAL and JAGANNATHAM: DISTRIBUTED ESTIMATION IN HOMOGENOUS POISSON WIRELESS SENSOR NETWORKS 93

0 50 100 150 200 2500

0.2

0.4

0.6

0.8

1

1/T

P FCSI(T)

sim; λ |A| = 40exp ; λ |A| = 40sim ; λ |A| = 80exp ; λ |A| = 80sim ; λ |A| = 120exp ; λ |A| = 120

(a) PFCSI(T ) versus 1/T

0 0.5 1 1.5 2

x 104

0

0.2

0.4

0.6

0.8

1

1/T

PFCSI(T)

sim : λ‖A‖ = 40lb : λ‖A‖ = 40

lb : λ‖A‖ = 60sim : λ‖A‖ = 80lb : λ‖A‖ = 80

(b) PPCSI(T ) versus 1/T

0 2000 4000 6000 8000 10000 120000

0.2

0.4

0.6

0.8

1/T

PFCSI(T

)

sim : λ = 3ub : λ = 3sim : λ = 5ub : λ = 5

(c) PFCSI(T ) versus 1/T

0 2 4 6 8 10 12 14 16

x 104

−0.5

0

0.5

1

1/T

PFCSI(T)

sim : lambda = 5ub : lambda = 5

(d) PPCSI(T ) versus 1/T

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1/T

PFCSI(T)

sim : λ‖A‖ = 10lb : λ‖A‖ = 10sim : λ‖A‖ = 15lb : λ‖A‖ = 15sim : λ‖A‖ = 20lb : λ‖A‖ = 20

(e) PFCSI(T ) versus 1/T

0 20 40 60 800

0.2

0.4

0.6

0.8

1

1/T

PPCSI(T)

sim: λ‖A‖ = 10lb: λ‖A‖ = 10sim: λ‖A‖ = 15lb: λ‖A‖ = 15sim: λ‖A‖ = 20lb: λ‖A‖ = 20

(f) PPCSI(T ) versus 1/T

Fig. 1: Simulated and analytical plots of the estimation outage probability. Here sim denotes simulated values, exp closed formexpression, and ub and lb denote the upper and lower bounds respectively.

The simulated values agree closely with the bounds for lowervalues of λ|A| similar to the previous scenarios.

The work in the paper above considers a simplistic modelfor distributed sensing in wireless sensor networks. However, itdescribes a new approach to analyze the estimation propertiesand behavior of random deployment based wireless sensornetworks. In that sense, the work presented in the paperemploys a model similar to [3]. This approach can however beextended to more realistic and practical scenarios readily. Forinstance, in [4], the authors consider a simplistic correlationmodel θ = as+n, where a represents the degree of correlationamongst the sensor measurements. Also, in works such as[15], the authors use a power exponential model, which theydescribe is suited for wireless sensor network applications.Such models can be incorporated in future PPP based analysisof wireless sensor networks.

V. CONCLUSION

A framework has been presented in this paper to char-acterize the estimation performance of distributed sensingin WSNs. A unique aspect of the above work is that theproposed theory is based on the stationary Poisson pointprocess framework and thus is robust to the stochastic spa-tial configurations occurring in practical WSN deployments.Closed form expressions and bounds for the probability ofestimation outage have been derived for a variety of scenarioswith full and partial phase only CSI incorporating path-lossand noisy sensor measurements. Simulation results have beenpresented to support the analysis.

REFERENCES

[1] I. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, “A surveyon sensor networks,” IEEE Commun. Mag., vol. 40, no. 8, pp. 102–114,2002.

[2] M. Gastpar and M. Vetterli, “Source-channel communication in sensornetworks,” in Information Processing in Sensor Networks. Springer,2003, pp. 162–177.

[3] M. Banavar, C. Tepedelenlioglu, and A. Spanias, “Estimation overfading channels with limited feedback using distributed sensing,” IEEETrans. Signal Process., vol. 58, no. 1, pp. 414–425, 2010.

[4] J.-J. Xiao, S. Cui, Z.-Q. Luo, and A. Goldsmith, “Linear coherentdecentralized estimation,” IEEE Trans. Signal Process., vol. 56, no. 2,pp. 757–770, 2008.

[5] S. Cui, J.-J. Xiao, A. Goldsmith, Z.-Q. Luo, and H. Poor, “Estimationdiversity and energy efficiency in distributed sensing,” IEEE Trans.Signal Process., vol. 55, no. 9, pp. 4683–4695, 2007.

[6] R. Shorey, A. Ananda, M. Chan, and W. Ooi, Coverage and ConnectivityIssues in Wireless Sensor Networks. Wiley-IEEE Press, 2009.

[7] R. Ganti and M. Haenggi, “Interference and outage in clustered wirelessad hoc networks,” IEEE Trans. Inf. Theory, vol. 55, no. 9, pp. 4067–4086, 2009.

[8] M. Haenggi and R. K. Ganti, “Interference in large wireless networks,”2009. Available: http://www.nd.edu/~mhaenggi/pubs/now.pdf

[9] R. K. Ganti and M. Haenggi, “Interference and outage in clusteredwireless ad hoc networks,” IEEE Trans. Inf. Theory, vol. 55, no. 9,p. 4067, 2009.

[10] T. Kwon and J. Cioffi, “Random deployment of data collectors forserving randomly-located sensors,” IEEE Trans. Wireless Commun.,vol. 12, no. 6, pp. 2556–2565, 2013.

[11] T. Quek, D. Dardari, and M. Win, “Energy efficiency of dense wirelesssensor networks: to cooperate or not to cooperate,” IEEE J. Sel. AreasCommun., vol. 25, no. 2, pp. 459–470, 2007.

[12] D. J. Daley and D. Vere-Jones, An Introduction to the Theory of PointProcesses, Volume II: General Theory and Structure. Springer, 2007.

[13] D. Tse and P. Viswanath, Fundamentals of Wireless Communication.Cambridge University Press, 2005.

[14] M. Abramowitz and I. Stegun, “Handbook of mathematical functions:with formulas, graphs and mathematical tables,” Applied MathematicalSeries. National Bureau of Standards, no. 55, 1965.

[15] M. C. Vuran and I. F. Akyildiz, “Spatial correlation-based collaborativemedium access control in wireless sensor networks,” IEEE/ACM Trans.Networking, vol. 14, no. 2, pp. 316–329, 2006.