Optimal Direction Finding in Unknown Noise

Environments Using Antenna Arrays in Wireless

Sensor NetworksMinghui Li and Yilong LuIntelligent Systems Center

Nanyang Technological University, 50 Nanyang Drive, Singapore 637553E-mail: {EMHLI, EYLU}gntu.edu.sg

Abstract-With the advancements in wireless sensor network(WSN) platform architecture and cost-effective smart antennas, itis feasible to integrate antenna arrays on the sensor node in thesame dimensions with slightly additional cost, and someintegrated platforms have been reported. In this paper, weconsider the challenging problem of direction finding in unknownnoise fields with the onboard antenna array, arising from thedesire to better exploit the spatial diversity in the harsh WSNdeployment environments for various network-level benefits. Wepresent an optimal algorithm based on the maximum likelihood(ML) criteria, and computed using particle swarm optimization(PSO) for accurate and fast direction estimation. The MLcriterion function is derived using parameterization of noisecovariance, and the PSO is incorporated with newly introducedfeatures and properly selected parameters to enhance itsconvergence. Simulation results demonstrate that the proposedalgorithm produces excellent bearing estimates, even inunfavorable scenarios involving few antenna elements, low signal-to-noise ratios and short data samples, which are the typical WSNworking conditions due to the power, space and cost constraints.

I. INTRODUCTION

The research and application of smart antenna systems inwireless sensor networks (WSN) have not been explored untilrecently. Antenna arrays have for long been consideredunsuitable for integration in sensor nodes due to size and costconstraints. They consist of more than one antenna elementsand require multiple front ends to translate the radio frequency(RF) signals to base band signals. However, with theadvancements in cost-effective antenna arrays and nodearchitecture, the integration becomes feasible and array basedsensor node platforms have been built. A sensor node platformcontaining four-element circular antenna arrays andtransceivers is reported in [1]. Leang and Kalis [2] design aplatform integrating multiple antennas and using single frontend together with a switched parasitic element [3] for signaltranslation, and demonstrate that the integration is feasible inthe same dimensions as the counterpart with single antenna,and increase the total cost by only 300.

Besides improving the communication quality by directionaltransmission and interference suppression, antenna arraysdeliver benefits to WSN at other aspects. In [4], a sensormanagement algorithm using adaptive antenna arrays at thecentral nodes is proposed to save energy consumption and

prolong the network lifetime. Ash and Potter [5] present alocalization scheme using received signal strength (RSS) anddirection-of-arrival (DOA) measurements from the antennaarray on each sensor node, and demonstrate that sub-meterlocation accuracy is achievable. Kalis and Dimitriou [6] use thedirection measurements of the sink to route the localtransmission for each sensor node to improve efficiency andreduce energy consumption.To exploit full potential of adaptive antenna arrays in

wireless sensor networks, the bearings of the sinks orneighboring nodes are necessary to be estimated [4]-[6]. Mostpopular direction finding techniques are established with theassumption of white noise [7]. However, the sensor nodesusually reside on the ground in a "radio unfriendly"environment that is subject to various disturbances such asmultipath distortion, ground clutter, and etc., therefore thenoise fields tend to be spatially correlated due to the dominantambient noise [8]-[9].

In this paper, we consider the problem ofDOA estimation inunknown additive noise with antenna arrays, which has drawnmuch attention recently. Several algorithms have beenproposed, which can be divided into two categories based onthe problem formulation. First, use a parametric model for thenoise covariance. In [8] and [10], descriptive models arepresented, where the noise covariance is modeled as a linearcombination of known weighting matrices. And second,exploit properties or parameterization of the signals, e.g., in[11]-[12].Most aforementioned methods attempt to apply maximum

likelihood (ML) criteria to the problem due to the superiorstatistical performance, and the estimates are computed bymaximizing a multimodal nonlinear likelihood function withrespect to all unknown parameters. In general, the DOAs mustbe jointly estimated along with other variables via optimizationof a high-dimensional problem. How to tackle the likelihoodfunction accurately and efficiently is a challenging task, whichis generally as important as establishment of the ML criteriafunction.Two types of solution approaches are available. Global

optimization algorithms such as genetic algorithms (GA) [13]or simulated annealing can be designed to optimize the exact

1-4244-1 178-5/07/$25.00 §2007 IEEE.

ML (EML) functions; however, their implementation issomewhat cumbersome due to slow convergence, thusimpractical for WSN applications. Alternatively, one mayderive an approximate ML (AML) estimator merely dependingon the DOAs using large sample assumptions and least squarecriteria, and apply a Newton-type technique to solve it, as in [8]and [10]-[12]. This strategy has several limitations: 1) AMLfunctions are multivariate and computation-extensive; 2) AMLis inferior to EML in accuracy; 3) Newton-type techniques arelocal search methods, and good initialization is crucial.

In most cases, the sensor nodes have fixed locations afterdeployment, and the inter-node directions are estimated only atthe time of network discovery and initialization. Therefore,accurate but off-line direction finding algorithms can beconsidered. In WSN, communication consumes much moreenergy than computation, e.g., the power consumed by aBerkeley Mica2dot mote to transmit 1 bit data is equivalent toroughly 2100 instruction executions [14]; and thecommunication efficiency will be significantly enhanced withaccurate direction estimates of desired transmission and maininterference [15]. Motivated by those observations, weinvestigate the feasibility of applying particle swarmoptimization (PSO) to the problem as a robust global searchalgorithm, and propose an approach by pairing an enhancedPSO with EML functions for optimal and fast DOA estimationin wireless sensor networks. PSO is a recent addition toevolutionary algorithms first introduced by Eberhart andKennedy in 1995 [16]. PSO is a population-based stochasticoptimization paradigm. As an emerging technology, PSO hasattracted a lot of attention in recent years, and has beensuccessfully applied in many fields, such as phased arraysynthesis [17], electromagnetic optimization [18], blind sourceseparation [19], and etc. Most of the applications demonstratedthat PSO could give competitive or even better results in amuch faster and cheaper way, compared to other heuristicmethods such as GA. In addition, PSO appears to be robust tocontrol parameters. These features make PSO more attractivefor WSN applications.The design of the optimization algorithm is a combination of

the problem-independent PSO kernel and some newlyintroduced problem-dependent features such as searching spacemapping, particle velocity control, and particle positionclipping. This architecture plus properly selected parametersmake the PSO algorithm highly flexible and reusable, whilebeing sufficiently specific and effective in the currentapplication. Via extensive numerical studies, we demonstratethat PSO-EML yields superior performance over Newton-AML-like schemes, especially in unfavorable scenariosinvolving low signal-to-noise ratios (SNR), small arrays, andshort data samples, which are the typical working conditions inWSN; furthermore, PSO-EML is more efficient in computation.

II. DATA MODEL AND PROBLEM FORMULATION

We consider an array ofM antenna elements arranged in anarbitrary geometry on a sensor node and N narrowband far-

field signal sources at unknown locations. The complex M-vector of array outputs is modeled by the standard equation

y(t) =A(O)s(t) + n(t), t =1, 2, ..., L (1)where 0 = [0i, - ]0,T] is the source DOA vector, and the kth

column of the complex MxN matrix A (9) is the so called

steering vector a (0k) for the DOA Ok . The ith element

ai (0k) models the gain and phase adjustments of the kthsignal at the ith sensor. Furthermore, the complex N-vector s(t)is composed of the emitter signals, and n(t) models the additivenoise.The vectors of signals and noise are assumed to be stationary,

temporally white, zero-mean complex Gaussian randomprocesses with second-order moments given by

E{s (t)sH (s)} = P8ts

E{s(t)sT (s)} = 0

E{n (t) nH (s)} = Qits(2)

E{n(t)nT (s)} = 0

where (is is the Kronecker delta, (.)H denotes complex

conjugate transpose, (.)T denotes transpose, and E[.] stands

for expectation. Assuming that the noise and signals are

independent, the data covariance matrix is given by

R =E{y(t)yH (t)} = APAH +Q . (3)

The problem addressed herein is the estimation of 0, (and ifnecessary, along with the parameters in P and Q,) from a batchofL measurements y(l), ... , y(L).Under the assumption of additive Gaussian noise and

Gaussian distributed signals, the normalized (with L) negativelog-likelihood function of the data vectors takes the form

I(6,P,Q)= logR +tr{R-lR} (4)

where tr {.} stands for trace, log, denotes the naturalI

logarithm of the determinant, and R is the covariance matrixof the measured data

L

R = E y(t)yH (t).t=l

(5)

Starting from (4), a number of concentrated ML estimatorshave been obtained [10]-[12]. In this paper, we focus on theML criteria derived using parameterization of the noisecovariance, because this assumption applies no constraints tothe signals, which may virtually model any scenarios in WSN.Based on a Fourier series expansion of the spatial noise

power density function, the noise covariance Q is assumed tobe modeled by the following linear parameterization:

J

Q(1)= 7jlj=l

where ii = [771, r1T is a vector of unknown noise Fourier

coefficients, Sj is a known function of the array geometrygiven by

Y.(j-j)/2j odd

L.j/2 jevenwhere

Y a (O) aH (0) cos (10) dOa(O) aH (0) sin (10) dO

1= 0,1,2,.... It is assumed that J is known or has beenestimated. Similar noise models appear in [8], [10].By solving for P in terms of 9 and Q (i) and substituting

back to (4), we get an exact ML function that depends on both9 and il [8]

I, (0, il) = log IQI + log GRG +H + tr{HR} (9)

where

A Q-1/2A

G AA(AHA (AH(10)R Q-1/2RQ-1/2H =I-G .

The ML estimates of 9 and il are obtained by minimizing (9).Further derivation of a function merely depending on theDOAs seems impossible. However, using the large sampleassumption and least square criteria, we can get the followingapproximation of (9)

/~~~~~~~~ ~~2

I2(0)= H(I-B(BHHB) BHH)d (11)

where

d = vec{R

B=A OA

r [vec { },.vec{j}] (12)

H=I _F(FHF) FH

where vec {.} is a concatenation of the columns of the

bracketed matrix, (.) denotes conjugate, denotes

Kronecker product, and 11-11 stands for Euclidean norm. The

DOA estimates are obtained by minimizing (11) with a searchof reduced dimension N.

III. PSO ALGORITHM FORML PARAMETER ESTIMATION

A. Particle Swarm OptimizationParticle swarm optimization is a stochastic optimization

algorithm, which mimics animal social behaviors such asflocking of birds and the methods by which they find roostingplaces or food sources. PSO starts with the initialization of apopulation of individuals in the search space and works on thesocial behavior of the particles in the swarm. Each particle isassigned a position in the problem space, which represents acandidate solution to the problem under consideration. Each ofthese particle positions is scored to obtain a scalar cost, namedfitness, based on how well it solves the problem. Theseparticles then fly through the problem space subject to bothdeterministic and stochastic update rules to new positions,which are subsequently scored. Each particle adaptivelyupdates its velocity and position according to its own flyingexperience and its companions' flying experience. With theoscillation and stochastic adjustment, particles explore regionsthroughout the problem space and eventually settle down neara good solution.

Consider a D-dimensional problem space and a swarmconsisting ofP particles. The position of the ith particle is a D-dimensional vector xi [x1l, xi2, ", XiD ]. The velocity of this

particle is represented as vi [v1l, vi2, , ViD] The bestprevious position of the ith particle, which gives the bestfitness value, is denoted as pi = [Pil, Pi2, *', PiD] The bestposition found by any particle in the swarm is represented by

Pg =[Pg,Pg2, ,PgD]. At every iteration, the velocity and

the position of each particle are updated according to thefollowing equations:

k+l k k k (k k k k k)vx = )v+ crip xi +C p+ xi

k+l k k+lxi xi+v

(13)

(14)

where 0 denotes element-wise product, i = 1, 2, ..., P I

k = 1,2, ..., indicates the iterations, 0) is a parameter called theinertia weight, c1 and c2 are positive constants referred to ascognitive and social parameters respectively, r1 and r2 are D-dimensional vectors consisting of independent randomnumbers uniformly distributed between 0 and 1.

B. PSO-EML DOA Estimation andParameter SelectionIn this section, we describe the formulation of the PSO

algorithm for ML estimation of source and noise parameters.The main steps are outlined in Fig. 1. In this study, thealgorithm starts by initializing a population of particles in the"normalized" search space with random positions constrainedbetween zero and one in each dimension, and randomvelocities. The D-dimensional position vector of the ith particletakes the form x ,11.N ¢1I JII where O<O j <Ki

n = 1, ,N , j = 1, ,J , N >1 , J>1, and N+J=D. Aparticle position vector is converted to a candidate solution

vector in the problem space through a suitable mapping. Thescore of the mapped vector evaluated by the likelihoodfunction I, (0, il) (9) is regarded as the fitness of the

corresponding particle. The introduction of normalized searchspace and mapping makes parameter selection and algorithmdesign less problem-dependent, and enhance the reusability.The manipulation of a particle's velocity according to (13) is

regarded as the central element of the entire optimization.Three components typically contribute to the new velocity. Thefirst part refers to the inertial effect of the movement, which isjust proportional to the old velocity and is the tendency of theparticle to proceed in the same direction it has been traveling.The inertial weight is considered critical for the convergence

behavior ofPSO [20]. A larger facilitates searching new area

and global exploration while a smaller tends to facilitatelocal exploitation in the current search area. In this study, isselected to decrease during the optimization process, thus PSOtends to have more global search ability at the beginning of therun while having more local search ability near the end of theoptimization. Given a maximum value cmax and a minimumvalue cWmil, w) is updated as follows:

k {Wmaxmin 1

max min (k-1), 1<k<[rK]rK

[rK]+l1<k<K

Setup problem: Initialize swarm:

* Define problem - * Random positions* Select PSO parameters * Random velocities

Solution is final global best position p,

Fig. 1. Flowchart illustrating the main steps of the PSO-EML technique.

(15)

where [rK] is the number of iterations with time decreasinginertial weight, 0 < r < 1 is a ratio, K is the maximum iterationnumber, and [.] is a rounding operator. Based on empiricalpractice [21] and extensive test runs, we select °max = 00.9

°)min =0.4, and r =0.4 0.8.

The second and third components of the velocity updateequation introduce stochastic tendencies to return towards theparticle's own best historical position and the group's besthistorical position. These paradigms allow particles to profitboth from their own discoveries as well as the discoveries ofthe swarm as a whole, mixing local and global informationuniquely for each particle on each iteration. Constants c1 and c2

are used to bias the particle's search towards the two bestlocations. These two parameters are not critical for theconvergence of PSO. Following common practice in theliterature [22], cl=c2=2, although these values could be fine-turned for the problem at hand.

Since there was no actual mechanism for controlling thevelocity of a particle, it is necessary to define a maximumvelocity to avoid the danger of swarm explosion anddivergence [23]. The velocity limit can be applied to vi alongeach dimension separately by

V1d={ MAX' Vid > VMAX (16)

-VM4AX' Vid < M4AX

where d I DI or for the modulus of the velocity vector by

the rule

vi = IfIVi, if |vi| >VmK4IV

(17)

Like the inertial weight, large values of VM4X or VMA encourage

global searching while small values encourage local searching.In this study, limitation along each dimension is applied andVM4X is set to the half value of the dynamic range [21], i.e.,VMA4X=0.5.The new particle position is calculated using equation (14).

If any dimension of the new position vector is less than zero or

more than one, it is clipped or adjusted to stay within this range.

It should be noted that, at any time of the optimization process,

two DOA components in a position vector are not allowed tohave the same values. For individuals with same components,one of the elements will be replaced by a valid random valuetill no collision exists.The optimization iteration will be terminated if the specified

maximum iteration number K is reached or the best particleposition of the whole swarm keeps static for a sufficiently largenumber of successive iterations. The final global best positionpg is taken as the ML estimates of the DOAs and noiseparameters.Some previous works demonstrate that the performance of

the PSO algorithm is not significantly affected by changing theswarm size P. The typical range of P is 20 to 50, which issufficient for most of the problems to achieve good results.Furthermore, PSO is not sensitive to initial particle positions;however, insertion of a good initial estimate may enhance theconvergence.

IV. SIMULATION RESULTS

Repeatfor each iteration

Repeatfor each particleMap particle position to solution in problem spaceEvaluate fitnessUpdate personal best pi and global best pgUpdate particle velocity

I~~~~~~~~~~~~~~~~~~~~~~~ IILimit particle velocityUpdate particle positionClip or adjust particle position if required

Test termination criterion

TABLE ITHE SELECTED PSO PARAMETERS

Parameter Value

cl 2.0

C2 2.0p 20K 200

VMAX 0.5

Cmax 0.9

0min 0.4

r 0.5

This section provides numerical examples to validate themerits of pairing PSO with EML, and to examine itsperformance against AML-based techniques. The selected PSOparameters are summarized in Table I, which are empiricallydetermined based on adequate test runs. The PSO algorithmstarts with a random initialization, and is terminated if themaximum iteration number K is reached or the global bestparticle position is not updated in 20 successive iterations. Thealternating projection (AP) technique [24] is used to computethe AML estimates, which is robust to initialization andapplicable to general array structures. We have performed 500Monte Carlo experiments for each point of the plot.We consider the data model (1) where the noise covariance

is modeled by a linear combination of known matrices as in (6).J=3, and the noise parameters are il = [1,1 / 4,1/ 9]. We assumethat two equal-power, uncorrelated signals impinge on a half-wavelength radius uniform circular array (UCA) with eightsensors from 60 and 640. The number of snapshots is 30. Theassumption is reasonable because the space on each node isconstrained and the UCA is compact in geometry. Thesituation is very challenging, since the separation of emitters isabout 0.1 beamwidth (the conventional resolution limit).

Fig. 2 depicts the DOA estimation root-mean-squared errors(RMSE) obtained using PSO-EML and AP-AML as a functionof SNR, and compares them with the corresponding Cramer-Rao bound (CRB) (theoretically best performance for efficientestimators). Fig. 3 shows the resolution probabilities for thesame methods. Two sources are considered to be resolved in anexperiment if both DOA estimation errors are less than the halfof their angular separation.As can be seen from Fig. 2 and Fig. 3, PSO-EML yields

significantly superior performance over AP-AML as a whole,by demonstrating lower DOA estimation RMSE and higherresolution probabilities. PSO-EML produces excellentestimates with RMSE approaching and asymptoticallyattaining the CRB. The accurate DOA estimates are observedbecause (1) EML criterion functions are statistically optimalalthough computation-extensive, and (2) the designed PSO is arobust and reliable global optimization algorithm. AP-AML,on the other hand, produces far less accurate estimates due toboth the intrinsic suboptimal nature of AML functions and

o02

10a,

100

1015

PSO- EM L -L ",

AP AMLCRB

20 25 30 35 40SNR (dB)

Fig. 2. DOA estimation RMSE values ofPSO-EML and AP-AML versus SNR.The dashdot line represents the theoretic CRB.

PSO0 EMLAP - AML

0.8

, 0.6

75 0.4 -

0.2

0:,10 15 20 25

SNR (dB)30 35 40

Fig. 3. Resolution probabilities ofPSO-EML and AP-AML versus SNR.

local convergence of the AP approach. And it cannot yet attainthe CRB at the SNR of 40dB.

Fig. 4 depicts the fitness progress curves of PSO-EMLobtained with random initialization and AML-basedinitialization respectively. The curves, which are plots of thefitness values of the global best particles versus the iterationnumber, are obtained over an average of 300 runs. AML-basedinitialization means that a priori DOA estimates attained bycomputing the AML function with one-step AP [24] isincorporated into the initial population. As can be seen fromFig. 4, PSO achieves fast convergence with introduced featuresand selected parameters, and the smooth dropping curve meansthat the global best particle is continuously updated as theswarm evolves. Furthermore, PSO is not sensitive to initialparticle positions. In this example, the PSO achievesconvergence with an average iteration number of 130, and

55Random InitializationAML - Based Initialization50

45

40

, 35 35U)a)

,30 0

25

20 f

15

10 '<0 50 100 150 200 250 300 350 400 450 500

Iterations

Fig. 4. Fitness progress curves ofPSO-EML obtained with randominitialization and AML-based initialization.

PSO-EML is approximately 10 times efficient than AP-AMLin terms of the computing time.

V. CONCLUSIONS

In this paper, we consider the challenging problem of DOAestimation in colored noise fields, arising from the desire tobetter exploit the spatial diversity in the deploymentenvironments of wireless sensor networks. Estimation accuracy

is more demanding than real-time requirements in a fixedWSN. In addition, communication is much more energy-

expensive than computation, which can be significantlyenhanced via accurate estimation of transmission andinterference directions. Motivated by those observations, we

propose an optimal and fast DOA estimator by pairing the PSOalgorithm with EML functions. Simulation results demonstratethat PSO-EML yields superior perfornance over AML-basedalgorithms, especially in unfavorable scenarios involving smallarrays, low SNRs, short data samples, and clustered sensor

nodes, which are the typical WSN working environments dueto the constraints of cost, power and space.

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