Sectorized Antenna-based DoA Estimation and Localization: … · 2017. 6. 9. · WERNER et al.:...

2272 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO. 11, NOVEMBER 2015

Sectorized Antenna-based DoA Estimationand Localization: Advanced Algorithms

and MeasurementsJanis Werner, Student Member, IEEE, Jun Wang, Aki Hakkarainen, Student Member, IEEE,Nikhil Gulati, Student Member, IEEE, Damiano Patron, Student Member, IEEE, Doug Pfeil,

Kapil Dandekar, Senior Member, IEEE, Danijela Cabric, Senior Member, IEEE, andMikko Valkama, Member, IEEE

Abstract—Sectorized antennas are a promising class of anten-nas for enabling direction-of-arrival (DoA) estimation and suc-cessive transmitter localization. In contrast to antenna arrays,sectorized antennas do not require multiple transceiver branchesand can be implemented using a single RF front-end only, thusreducing the overall size and cost of the devices. However, forgood localization performance the underlying DoA estimator is ofuttermost importance. In this paper, we therefore propose a novelhigh performance DoA estimator for sectorized antennas that doesnot require cooperation between the transmitter and the localizingnetwork. The proposed DoA estimator is broadly applicable withdifferent sectorized antenna types and signal waveforms, and haslow computational complexity. Using computer simulations, weshow that our algorithm approaches the respective Cramer-Raolower bound for DoA estimation variance if the signal-to-noiseratio (SNR) is moderate to large and also outperforms the existingestimators. Moreover, we also derive analytical error models forthe underlying DoA estimation principle considering both freespace as well as multipath propagation scenarios. Furthermore,we also address the fusion of the individual DoA estimates intoa location estimate using the Stansfield algorithm and study thecorresponding localization performance in detail. Finally, we showhow to implement the localization in practical systems and demon-strate the achievable performance using indoor RF measurementsobtained with practical sectorized antenna units.

Manuscript received August 15, 2014; revised December 7, 2014; acceptedJanuary 11, 2015. Date of publication May 6, 2015; date of current versionOctober 19, 2015. This work was supported by the Doctoral Program of thePresident of Tampere University of Technology, the Tuula and Yrjö NeuvoFund, the Nokia Foundation, the Academy of Finland under the project251138 “Digitally-Enhanced RF for Cognitive Radio Devices,” and the FinnishFunding Agency for Technology and Innovation (Tekes, under the projects“Reconfigurable Antenna-based Enhancement of Dynamic Spectrum AccessAlgorithms,” “5G Networks and Device Positioning,” and “Future Small-CellNetworks using Reconfigurable Antennas”).

J. Werner, A. Hakkarainen, and M. Valkama are with the Department of Elec-tronics and Communications Engineering, Tampere University of Technology,Tampere FI-33101, Finland (e-mail: [email protected]; [email protected]; [email protected]).

J. Wang and D. Cabric are with the Department of Electrical Engineering,University of California, Los Angeles, CA 90095 USA (e-mail: [email protected]; [email protected]).

N. Gulati, D. Patron, D. Pfeil, and K. Dandekar are with the Departmentof Electrical and Computer Engineering, Drexel University, Philadelphia,PA 19104 USA (e-mail: [email protected]; [email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JSAC.2015.2430292

Index Terms—Angle-of-arrival, cognitive radio, directionalantennas, direction-of-arrival estimation, leaky-wave antennas,localization, location-awareness, measurements, reconfigurableantennas, sectorized antennas, Stansfield algorithm.

I. INTRODUCTION

TRANSMITTER (TX) localization has many applicationareas [1], [2]. In many of those areas, it is desired that

the TX location is estimated based on observing or measuringthe transmission alone, without any direct collaboration orfeedback signaling between the TX and localization network. Incognitive radio networks, as an example, the primary user (PU)cannot be expected to cooperate with the secondary network.Nevertheless, PU location information has been identified asone of the key requirements to enable several advanced func-tionalities in cognitive radio networks [2]. Other good examplesof TX localization are spectrum enforcement, surveillance, firstresponder operations as well as transportation and navigation.

The location of a non-cooperative TX can generally beestimated by measuring time-difference-of-arrival (TDoA) [3],received-signal-strength (RSS) [4], [5], direction-of-arrival(DoA) [6] or combinations thereof [7], [8]. However, TDoAmeasurements require accurate timing synchronization in thelocalizing network itself [7]. In order to avoid the increasedcomplexity associated with synchronization, most recent re-search has therefore focused on RSS [4], [5] and DoA-based lo-calization [6]–[8]. Intuitively and as shown also in, e.g., [6]–[8],the accuracy of DoA-based localization systems is stronglydetermined by the quality of DoA estimates. Therefore, thispaper will thoroughly investigate advanced DoA estimation al-gorithms and their application and performance in localizationsystems.

Traditional approaches of DoA estimation such as the pop-ular MUSIC algorithm [9] require digitally controlled antennaarrays (DCAA). For accurate DoA estimation, the DCAAs mustbe equipped with a large number of antennas, each with a com-plete receiver branch [10]. However, the associated hardwarecomplexity of such an implementation might be infeasible inhandheld devices. We have therefore recently proposed to useso-called sectorized antennas for DoA estimation [11]–[13].A sectorized antenna is an abstraction that encompasses alltypes of antennas that can receive energy selectively within anangular sector. Thereby, it is not required that different sectors

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WERNER et al.: SECTORIZED ANTENNA-BASED DoA ESTIMATION AND LOCALIZATION 2273

can receive the signal in a time-parallel manner. Sectorizedantennas can, consequently, be implemented with a single RFfront-end only and are therefore generally much less hardware-intensive than DCAAs. Practical examples of sectorized anten-nas are reconfigurable antennas such as leaky-wave antennas(LWAs) [14] or electronically steerable parasitic array radiators(ESPARs) [15] as well as switched-beam systems (SBSs) [16]or antenna arrays with a single RF front-end [17], [18].

In the literature, various DoA estimators for sectorized an-tennas have been proposed. However, most of these estimatorswere developed for a specific type of sectorized antenna andsome are moreover restricted to specific signal types or re-quire a cooperating TX. In [19] the angle-dependent frequencyresponse of an LWA is exploited to estimate the DoA of awideband pulsed signal. The algorithms in [15], [20], [21]require a cooperating TX that sends the same signal repeatedly,such that the signal is received in all sectors of an ESPAR [15]or an LWA [20], [21]. In that way, the antennas can be usedto emulate a DCAA and subsequent MUSIC DoA estimationbecomes possible. DoA estimation of multiple DS-CDMAsignals impinging on a base station equipped with an SBS isconsidered in [16]. However, the estimator in [16] requireseither the RSS to be known or an additional omnidirectionalantenna at the receiver in order to normalize the received signalsproperly. In [22], [23] different DoA estimators for antennaarrays with a single RF front-end are compared based onsimulations with advanced propagation modeling [22], [23] andopen field measurements [23]. Analog DoA estimation with anLWA is proposed in [24]. Since the estimation is performed bycontinuously changing the antenna’s beam, the algorithm is notapplicable to all sectorized antennas. In addition, the associatedmeasurement process takes longer time than a measurement insectors (i.e., in discrete steps). In [11], [22], [25] an estimatoris considered that estimates the DoA as the orientation of thesector with the maximum power measurement. In accordancewith [11], we refer to this algorithm as the maxE estimator.While maxE has very low computational complexity, its per-formance is ultimately limited by the number of sectors and isfar from the performance bounds as discussed in detail in [11]and [12].

Another DoA estimator for sectorized antennas, namelythe simplified least squares (SLS) estimator, was proposed in[12]. SLS builds on the assumption that only a few sectorsreceive the TX signal at a sufficiently high SNR, while theattenuation in the other sectors is too high to exploit the TXsignal component. Based on this assumption, SLS measuresthe powers in all sectors and discards all sector-powers exceptfor the maximum two. The ratio of the two remaining sector-powers is then used in a least-squares formulation to estimatethe DoA. SLS has been shown to outperform the estimatorsin [11], [25], and also closely approaches the Cramer-Raobound (CRB) for moderate SNRs [12]. However, for high SNRsthe performance of SLS saturates and does not achieve theCRB. In addition, our analysis in [12] assumes a somewhatoptimal beamwidth for the antenna’s main beam. This optimalbeamwidth decreases with the number of sectors. However, inpractice it is often impossible to change the beamwidth, whichis determined by the underlying antenna technology. Instead,

the only way to improve DoA estimation performance is toincrease the number of sectors while keeping the beamwidthconstant. Consequently, the assumption that all but two sectorscan be discarded in DoA estimation does not apply for a numberof practical scenarios, which severely degrades the performanceof SLS DoA estimation for all SNRs. Moreover, the estimatorsin [11], [12], [25], require a sectorized antenna where the mainbeam is equally shaped in all sectors. In practice this is notnecessarily the case as is evident, e.g., from the LWA in [14].

In this paper, we therefore propose, analyze and test thethree-stage SLS (TSLS) DoA estimator for sectorized antennas.TSLS does not make any assumptions about the TX signal, nordoes it require sectors with equally shaped radiation patterns.We only require that the main beam of the radiation patterncan be roughly approximated by a Gaussian-shaped curve,an approximation that has been shown feasible for a largenumber of antennas [11], [26], [27]. For the proposed estimator,we then shown that its performance closely approaches therespective CRB for moderate to large SNR. Finally, this paperdemonstrates the application of TSLS in a practical localizationsystem that computes a TX location estimate from TSLS DoAestimates obtained at multiple sensors.

In detail, our contributions in this paper are the following:

• Generalization of the model for sectorized antennas in[11] such that it is also applicable to sectorized antennaswhere the shape of the main beam varies throughout thesectors.

• We propose and analyze the universal and high-performance TSLS DoA estimator for sectorized an-tennas. TSLS computes the DoA in three stages. First,TSLS selects the sectors suitable for DoA estimation. Us-ing pairs of these sectors, TSLS subsequently estimatessector-pair DoAs (SP-DoAs) that are then fused togetherto obtain the final DoA estimate.

• We develop different methods for SP-DoA fusion.• We derive analytical error models for bias and variance

of SP-DoA estimates.• We analyze the effects of multipath on the underlying

DoA estimation principle both analytically and throughsimulations.

• We study in detail the performance of a complete local-ization system where TSLS DoA estimates from multiplesensors are combined into a TX location estimate bymeans of the modified Stansfield algorithm.

• We apply TSLS DoA estimation in practice and demon-strate the achievable localization performance with real-world RF measurements that were obtained in an indoorenvironment.

This paper is organized as follows. Section II describes thelocalization system and the sectorized antenna model. TheTSLS estimator is introduced in Section III, while the analyticalerror models for SP-DoA estimation are derived in Section IV.The Stansfield localization algorithm is briefly reviewed inSection V. A thorough performance and complexity evalua-tion of DoA estimation and localization with TSLS throughcomputer simulations and practical measurements can be foundin Section VI and Section VII, respectively. The work is


TABLE IMOST COMMONLY USED ABBREVIATIONS

Fig. 1. Considered localization system with K RX units and a single TX.

concluded in Section VIII. Details of derivations and proofs canbe found in the Appendices A, B, and C.

Notation: Throughout this paper, vectors and matrices arewritten as bold letters. The absolute value of a complex numberx is represented as |x|, the maximum of two real values x1 and x2is written as max(x1, x2), and mody(x) expresses the remainderof the division x/y. E[X] and var[X] denote expected value andvariance of the random variable X, while bias[y] = E[y] − ydenotes the bias of estimator y for a deterministic quantity y.RMSE = √

E[‖y − y‖2] denotes the root-mean squared errorof an estimator y for a deterministic vector (or scalar) y, where‖y‖ refers to the L2 norm. We use |A| to denote the cardinality ofa set A. The superscripts (·)T and (·)−1 represent transpose andmatrix inverse, respectively. The trace of a matrix is expressedas trace(M), while diag(x) denotes the diagonal matrix withthe elements of vector x on its diagonal. The symbol en is usedto express a vector with 1 in the n-th coordinate and 0s oth-erwise. Finally, X ∼ N (μx, σ

2x ) denotes a Gaussian distributed

real random variable with mean μx and variance σ 2x and Z ∼

CN (μz, σ2z ) denotes a circular symmetric complex Gaussian

distributed random variable with mean μz and variance σ 2z .

For the readers’ convenience, we have collected the mostcommonly used abbreviations in Table I.

II. SYSTEM MODEL

In this paper, we consider a localization system illustrated inFig. 1. In this system, K receivers (RXs) with known locations�k = [xk, yk]T, k = 1, . . . , K collaborate in order to estimatethe location, �P = [xP, yP]T, of a non-cooperative TX. With

Fig. 2. An illustration of the antenna model (1) for sector m.

the help of a sectorized antenna, each RX k estimates the TXsignal DoA ϕk using the algorithm we propose in Section III.Thereafter, the DoA estimates from all RXs are communicatedto a central fusion center, where they are combined into a TXlocation estimate �P = [xP, xP]T using a modified version of theStansfield algorithm [11], [28] (see Section V).

As discussed in [11], [26], [27], the main beam of manydirectional antennas can be well approximated through aGaussian-like shape. Here, we consider a generalized radiationpattern model that has more degrees of freedom compared toour earlier model (e.g., [11]). Consequently, this new model iswell suited for a broader range of practical antennas.

Assume that each sensor is capable of taking measurementsin M different sectors. The radiation pattern of sector m, m =1, . . . , M is then modeled as

ζm(ϕ) = αm exp(− [M(ϕ − ϑm)]2 /β2

m

)(1)

where βm is the beamwidth of the main beam, ϑm is theorientation and αm is the attenuation of the antenna in sectorm, and M(ϕ) = mod2π(ϕ + π) − π . An illustration of (1) canbe found in Fig. 2. The TSLS DoA estimator, proposed inSection III, estimates SP-DoAs using measurements from twosectors. Hence, for a given sector pair (i, j), i, j = 1, . . . , M,we distinguish between two cases: 1) equal beamwidth sectors(EBS): βi = βj and 2) different beamwidth sectors (DBS): βi �=βj. This distinction is necessary since EBS and DBS requiredifferent SP-DoA estimators as will be shown in Section III-B.From (1), the earlier model in [11] is obtained when αi = αj =1, βi = βj and |ϑij| = |ϑi − ϑj| = 2π

M for all sectors i, j =1, . . . , M. We will refer to this special class of antennas asequal-sector antennas (ESAs) and parameterize them via theside-sector suppression as that determines the beamwidth asβ = 2π/[M√

ln(1/as)] [11]. In practice, antennas such as theLWA [14] that we use in our measurements (see Section VII)cannot be modeled as ESAs since the beamwidth as well as theattenuation vary for different sectors. An SBS composed of acircular antenna array, in contrast, has approximately constantαm and βm and can hence be modeled as an ESA.


Initially, every RX k computes so-called sector-powers fromthe N received samples rk,m(n) in sector m according to

εk,m = 1

N

N−1∑n=0

∣∣rk,m(n)∣∣2 . (2)

In free space, the received signal samples can be modeled as [12]

rk,m(n) = ζm(ϕk)sk,m(n) + wk,m(n) (3)

where sk,m ∼ CN (0, γk) is the incoming signal impinging withDoA ϕk on RX k in sector m with RSS γk, and wk,m(n) ∼CN (0, σ 2

w) is additive noise. As shown in [12], we can thenapproximate the distribution of the sector-powers as εk,m ∼N (μk,m, σ 2

k,m) with μk,m = E[εk,m] = ρk,mγk + σ 2w and σ 2

k,m =var[εk,m] = 1

N (ρk,mγk + σ 2w)

2where ρk,m = [ζm(ϕk)]2.

In multipath scenarios, in contrast, we use the followingmodel for the received signal samples

rk,m(n) =G∑

g=1

ζk,m,gψk,gsk,m(n) + wk,m(n) (4)

where G is the number of paths, ζk,m,g = ζm(ϕk,g) is the atten-uation from the mth sector of the kth RX on the DoA of thegth path ϕk,g, and ψk,g is the power scaling of the gth path. Weassume g = 1 is the path with the greatest power (e.g., line-of-sight path), so that ψk,1 > ψk,g, ∀ g > 1. We further assumepath powers sum to unity, i.e.,

∑Gg=1 ψ2

k,g = 1. Note that in (4)we assume the sample time delays among multiple paths arenegligible. This assumption is valid when the signal bandwidthis significantly smaller than the channel coherence bandwidth,or all paths with delays greater than the sample period obtainsmall energies. More complicated multipath signal modelswith significant delays among paths will be included in ourfuture work. Notice, however, that the above model anywaytakes into account the different arriving angles of differentmultipath components. Similar to the free space scenario, wecan approximate the sector-powers as εk,m ∼ N (μk,m, σ 2

k,m),

with μk,m = η2k,mγk + σ 2

w and σ 2k,m = 1

N (η2k,mγk + σ 2

w)2, where

ηk,m = ∑Gg=1 ζk,m,gψk,g.

For ease of presentation and without loss of generality, wemoreover make the following simplifying assumptions in thispaper:

• All RXs are equipped with a similar kind of antenna.• Sector-orientations ϑm, m = 1, . . . , M are the same at all

RXs.• Sectors are numbered in ascending order of their orienta-

tion, i.e., ϑm < ϑm+1, m = 1, . . . , M.

III. THREE-STAGE SLS DOA ESTIMATION

The underlying principle of TSLS DoA estimation is similarto that of our earlier proposed SLS DoA estimator. In fact, forESAs and L = 2, TSLS results in the same DoA estimates asSLS. However, in contrast to SLS, TSLS DoA estimation can bebased on L > 2 sectors. As depicted in Fig. 3, TSLS first selectsa subset Lk, |Lk| = L of all sectors, subsequently estimates

Fig. 3. The three stages of TSLS DoA estimation: sector selection (SSL),SP-DoA estimation (SDE) and DoA fusion (DFU) with the weighting methodsequal weighting (EW), power weighting (PW), and variance weighting (VW).

a SP-DoA, ϕk,ij, for all sector pairs (i, j), i, j ∈ Lk, i �= j, andfinally obtains the DoA estimate, ϕk, through weighted fusionof the SP-DoAs. The three stages of TSLS DoA estimation arediscussed in detail in the following sections.

A. Sector Selection (SSL)

The purpose of SSL is to find the subset Lk containing the Lsectors that are best suited to extract the DoA from their respec-tive sector-powers. Towards that end, SSL first finds the maxi-mum adjacent sector-pair (MASP), i.e., the sector-pair (q, q +1) such that εk,qεk,q+1 > εk,lεk,l+1 for all other sector-pairs (l,l + 1), l = 1, . . . , M, l �= q. Thus, when αm does not vary toostrongly with m, ϑq <ϕk <ϑq+1 with high probability. For evenL, SSL then returns the sector subset Lk ={q−L/2+1, . . . , q +L/2}. For odd L, SSL next estimates whether ϕk is closer to ϑq

or ϑq+1. This is achieved by comparing εk,q and εk,q+1. If εk,q >

εk,q+1 then SSL estimates that |ϕk − ϑq| < |ϕk − ϑq+1| andreturns Lk = {q − (L + 1)/2 + 1, . . . , q + (L − 1)/2}. Oth-erwise, Lk = {q − (L − 1)/2 + 1, . . . , q + (L + 1)/2} is re-turned. Note that for clarity we have not considered that thesector indices are circular for antennas spanning the whole 360◦range, i.e., sector 1 is also adjacent to sector M. However, theextension of the above to the circular case is trivial.

We have compared this SSL method to other methods such aspicking the L sectors with the L maximum sector-powers. Forthe ESA and LWA that we discuss in more detail in Section VI-B,we found that the above described method works best. It isimportant to note though that the best SSL method for oneantenna type might not necessarily be the best for anotherantenna type. For example, if αm varies strongly with the sectorsm, then another SSL might perform better. On the other hand, ifαl is particularly small in a sector l, then sector l is not suitablefor DoA estimation anyways, as the TX signal is severelyattenuated in that sector.

B. Sector-Pair DoA Estimation (SDE)

In TSLS a SP-DoA is estimated for all sector-pairs (i, j),i, j ∈ Lk, i �= j. Towards that end, the noise-centered sector-powers (NCSP) are first calculated as

pk,i = εk,i − σ 2w. (5)

Thereafter, NCSPs with pk,i < 0 are discarded and the re-spective sectors are removed from Lk. The SP-DoAs are then


obtained as the DoA ϕk,ij that minimizes the squared error ofthe ratio of NCSPs from sector i and j, i.e.,

ϕk,ij = arg minϕ

(pk,i

pk,j− ρk,i(ϕ)

ρk,j(ϕ)

)2

. (6)

In that way, we can estimate the DoA without estimating theRSS, which is also contained in pk,m, m = i, j but with goodapproximation not anymore in the ratio pk,i/pk,j. In order toobtain a closed-form solution to (6), we assume that the mainbeam of the antenna can be approximated through a Gaussiancurve, which is possible for many practical antennas [26], [29].This closed-form solution is different for EBS and DBS asderived in Appendix A.

1) Equal Beamwidth Sectors: The solution of (6) for EBS isstraight-forward and is given by

ϕk,ij = ϑij + κij

(ln

pk,i

pk,j− 2 ln

αi

αj

)(7)

with ϑk,ij = 12 (ϑi + ϑj) and κij = β2

4(ϑi−ϑj). For the special case

of αi = αj we obtain the equation for classical SLS that we havealready derived in [11].

2) Different Beamwidth Sectors: For DBS the minimizationin (6) results in a quadratic equation and has, hence, twosolutions of the form

ϕ[1/2]k,ij = λij ± bijg(pk,i, pk,j) (8)

with

g(p) =√

(ϑij)2 − βij lnαi

αj+ 1

2βij ln

pk,i

pk,j, (9)

where p = (pk,i, pk,j)T , λij = β2

i ϑj−β2j ϑi

βij, bij = βiβj

βij, βij =

β2i − β2

j , and ϑij = ϑi − ϑj. The ambiguity in (8) can beresolved by taking the NCSPs into account. For each of thesolutions ϕ

[l]k,ij l = 1, 2, we can estimate two RSSs as γ

[l]k,ij,m =

pk,m

ρ[l]k,ij,m

, where m = i, j and ρ[l]k,ij,m = [αm exp(−[M(ϕ

[l]k,ij −

ϑm)]2/β2m)]2 is the estimated attenuation in sector m given that

the DoA is ϕ[l]k,ij. In the case N → ∞, the two RSS estimates

per DoA solution ϕ[l]k,ij, l = 1, 2 are equal, i.e., γ

[l]k,ij,i = γ

[l]k,ij,j,

if ϕ[l]k,ij = ϕk. Therefore, we choose the solution l in (8) such

as to minimize |γ [l]k,ij,i − γ

[l]k,ij,j|. If g(p) becomes imaginary, the

respective SP-DoA estimate is discarded.

C. DoA Fusion (DFU)

SDE results in P SP-DoA estimates ϕk,ij = ϕk + δϕk,ij thathave to be fused together in order to obtain the final DoAestimate. Obviously, it is desirable to give larger weight tothose SP-DoA estimates that have a smaller error δϕk,ij. How-ever, ϕk,ij are circular random variables such that conven-tional weighted averaging is not applicable. Instead, we usea weighted fusion method for circular random variables asdiscussed in [30]:

sin ϕk

cos ϕk=

∑wk,ij sin ϕk,ij∑wk,ij cos ϕk,ij

, (10)

where we denote the weights for DoA ϕk,ij as wk,ij. Geomet-rically, (10) can be interpreted as the summation of P vectorswith magnitude wk,ij and angle ϕk,ij in a two-dimensional plane.The resulting vector then has an angle that is equal to the finalDoA estimate ϕk. Our simulations have shown that (10) hasidentical performance to conventional weighted averaging if theerror δϕk,ij is very small. However, for large δϕk,ij (10) outper-forms conventional weighted averaging significantly. Differentchoices for the weights wk,ij are discussed next.

1) Equal Weighting (EW): For equal weighting of theSP-DoAs, the weights can simply be set to wEW

k,ij = 1.2) Sector-Power Weighting (PW): Based on the discussion

in Section III-A, sector-powers are a good indication for thepotential contribution of different sectors in DoA estimation.Consequently, an intuitive and robust weighting scheme forSP-DoA ϕk,ij is the product of NCSPs i and j

wPWk,ij = pk,ipk,j. (11)

3) Variance Weighting (VW): If the variance σ 2ij =var[δϕk,ij]

of individual SP-DoA errors is known, a weighting schemewk,ij = 1/σ 2

ij can be applied. However, in practice the knowl-

edge of σ 2ij is not a realistic assumption. In Section IV-A

we derive an approximation v(1)k,ij ≈ σ 2

ij of the SP-DoA error

variance in free space. This variance can be estimated as v(1)k,ij,

simply by approximating the μk,i that is contained in (19) viaak,i as pk,i. Thus, a variance-based weighting scheme can beachieved by setting

wVWk,ij =

(v

(1)k,ij

)−1. (12)

D. Validity Check

In particular for low SNR it is possible that pk,i < 0 ∀ i =1, . . . , M or that all SP-DoAs are discarded, as discussed inSection III-B2. Then P = 0 and DFU stage cannot be executed.In that case, TSLS estimates the DoA using the modified maxEestimator introduced in [31].

IV. ANALYTICAL MODELS FOR THE

ERROR OF SP-DOA ESTIMATION

In this section we derive analytical models for the errorof the SP-DoA estimates obtained in SDE. Towards that end,we first notice that pk,m is distributed as pk,m ∼ N (μk,m,

σ 2k,m) with μk,m = E[pk,m] = ρk,mγk and σ 2

k,m = var[pk,m] =1N (ρk,mγk + σ 2

w)2. We then assume that σ 2

k,m � μk,m, whichholds for sufficiently high SNR and a moderate to large numberof samples. Given this assumption, we can approximate theSP-DoA estimate ϕk,ij through its n-th order Taylor seriesexpansion developed around the means pk,i = μk,i and pk,j =μk,j. In our derivation we will be using the following lemma:

Lemma 1: Given an L × 1 random vector X ∼ N (μ, Q) withmean vector μ = E[X] and diagonal covariance Q = E[(X −μ)(X − μ)T ] = diag[σ 2

1 , . . . , σ 2L ], and the random variable

Y = f (X), where f : RL×1 → R. Denote Y(n)(μ) as the n-thorder Taylor series of Y around μ, the gradient of Y as j =∂f

∂X1e1 + . . . + ∂f

∂XLeL and the Hessian matrix of Y as H. Then,


we obtain the expected values of the first and second orderTaylor series of Y developed around its mean as

E[Y(1)(μ)

]= f (μ) (13)

E[Y(2)(μ)

]= f (μ) + 1

2trace {H(μ)Q} (14)

and the respective variances as

var[Y(1)(μ)

]=

∑l

j2l (μ)σ 2

l (15)

var[Y(2)(μ)

]=

∑l

j2l (μ)σ 2

l + 1

2trace

{(H(μ)Q)2

}. (16)

A detailed derivation of this lemma can be found inAppendix B.

A. Free Space Propagation

The approximations for bias and variance of SP-DoA estima-tion that we present in the following are derived for DBS, i.e.,the SP-DoA estimator (8). Following similar steps, we can alsoobtain bias and variance approximations for EBS, i.e., the SP-DoA estimator (7). However, the resulting EBS approximationsare in fact equal to the DBSs approximations when noting thatβi = βj and therefore replacing βij = 0 in (17)–(20) in theEBS case.

For the derivations, we first assume that the SP-DoA es-timated in the SDE stage is the correct one out of the twopossibilities in (8), which is a very reasonable assumption asdiscussed later in Section VI-B. Next, we approximate theSP-DoA through its n-th order Taylor series as ϕk,ij ≈ ϕ

(n)k,ij. Us-

ing Lemma 1 and following the derivations in Appendix C, wethen obtain an approximation of the bias E[ϕk,ij] − ϕk ≈ b(n)

k,ij =E[ϕ(n)

k,ij] − ϕk through first and second-order Taylor series as

b(1)k,ij = 0 (17)

b(2)k,ij = ± βiβj

8g(μ)

[ak,j − ak,i − βij

4g2(μ)(ak,i + ak,j)

](18)

where ak,m = 1N

(SNR−1

k,m + 1)2

, SNRk,m = μk,m

σ 2w

= ρk,mSNRk.

Using the same approach, we can also approximate the varianceof the SP-DoA estimation error through its n-th order Taylor se-

ries, i.e., var[ϕk,ij] ≈ v(n)k,ij = var

[ϕ

(n)k,ij

]. For the first and second

order Taylor approximations, the variances are then equal to

v(1)k,ij = β2

i β2j

16[g(μ)

]2 (ak,i + ak,j) (19)

v(2)k,ij = β2

i β2j

16g2(μ)

[ak,i + ak,j + 1

2

(a2

k,i + a2k,j

)

+ βij

4g2(μ)

(a2

k,i − a2k,j

)+ 2βij

32g4(μ)(ak,i + ak,j)

2

].

(20)

The derivation of (19)–(20) is again based on Lemma 1 with thedetails given in Appendix C.

B. Multipath Propagation

In this subsection we present the bias and variance of SP-DoA estimation when multipath is considered. We present onlyresults for EBS due to their mathematical conciseness. It isstraightforward to extend the derivation to DBS. FollowingLemma 1, the bias of the SP-DoA using first and second orderTaylor approximations are given by

b(1)k,ij = ϑk,ij + κij

(2 ln

ηk,i

ηk,j− ln

αi

αj

)− ϕk,1 (21)

b(2)k,ij = b(1)

k,ij − 1

2

(κijσ

2k,i

γ 2k η4

k,i

− κijσ2k,j

γ 2k η4

k,j

), (22)

respectively, where ϕk,1 is the DoA of the strongest path, i.e.,the “true” DoA. The variances are equal to

v(1)k,ij = κ2

ijσ2k,i

γ 2k η4

k,i

+ κ2ijσ

2k,j

γ 2k η4

k,j

(23)

v(2)k,ij = v

(1)k,ij + 1

2

(κ2

ijσ4k,i

γ 4k η8

k,i

+ κ2ijσ

4k,j

γ 4k η8

k,j

). (24)

The derivations of (21)–(24) are very similar to Section IV-Aand Appendix C, and are thus omitted due to space limitation.

V. LOCALIZATION

DoA estimates ϕk from individual sensors k, k = 1, . . . , K,are fused together into a location estimate using a modifiedversion of the Stansfield algorithm [11]. The original Stansfieldalgorithm was proposed in [28] as an approximation to the max-imum likelihood estimator. Its location estimate � = (xP, yP)T

is obtained as

� = (ATWA)−1

ATWb, (25)

with

A =⎡⎢⎣

sin(ϕ1) − cos(ϕ1)...

...

sin(ϕK) − cos(ϕK)

⎤⎥⎦ , (26)

b =⎡⎢⎣

x1 sin(ϕ1) − y1 cos(ϕ1)...

xK sin(ϕK) − yK cos(ϕK)

⎤⎥⎦ (27)

and a weighting matrix W. In the original Stansfield algo-rithm, the weighting matrix is dependent on both the individualTX-RX distances as well as the quality of the DoA estimates.In practice this information is not available for the DoA fusion.Therefore, we use the modified version [11] where the con-tribution of each sensor k is weighted with an estimate of thesensor’s RSS, γk. This results in a diagonal weighting matrixW = diag(γ1, γ2, . . . , γK).


Fig. 4. Analytical and empirical bias and standard deviation of sector-pairDoA estimation in free space.

VI. NUMERICAL EVALUATIONS AND ANALYSIS

As the following presentation builds heavily on previously-defined abbreviations, the reader may refer to Table I for asummary of the most commonly used ones.

A. Error of SP-DoA Estimation

We analyze the error of SP-DoA estimation in free spaceusing two sectors with αi = 1, αj = 0.9, βi = 1/2 rad, βj =1/3 rad, ϑi = 0◦, ϑj = 20◦ and an incoming signal DoA ofϕk = 10◦. These numbers reflect a realistic example scenario,though any other numerical values could be used as well.Fig. 4 depicts the bias and standard deviation of the SP-DoAestimator proposed in Section III-B. The simulated curvesare obtained empirically over 106 realizations per SNR-stepwhile the analytical curves are obtained by calculating thefirst and second order Taylor approximation models derived inSection IV-A.

We first notice that the SP-DoA estimates are slightly biased,even for large SNR. This bias is nicely described by the 2nd-order Taylor approximation from SNR ≈ −2 dB onwards. The1st-order approximation, in contrast, fails to model the biasadequately. However, the sectors considered in here are DBS.For EBS, we have βi = βj in (18) such that b(2)

k,ij → 0 for

SNR → ∞ since ak,i → ak,j → 1N . Hence, SP-DoA estimation

with EBS is asymptotically unbiased for large SNR as we haveconfirmed already for SLS in [12], [13].

With respect to the variance, both 1st- and 2nd-order Taylorapproximations model the behavior very well for low to highSNR. Only for very low SNR < 0 dB, we observe that v

(2)k,ij ap-

proximates the variance slightly more accurately. For compari-son, we have also included the estimate of the 1st-order Taylorvariance approximation, v(1)

k,ij that is calculated according to thediscussion in Section III-C3. This estimate is very accurate forSNR > 2 dB. However, for SNRs below 2 dB, the slope of thevariance estimates is very steep, resulting in overly pessimistic

Fig. 5. Analytical and empirical bias and standard deviation of sector-pairDoA estimation with multipath.

estimates in the low SNR range. This is a consequence ofak,m, m = i, j in the estimation of (19) behaving proportional to1/p2

k,m for low SNRs. Since pk,m is symmetrically distributedaround its mean, ak,m is hence on average estimated larger thanit actually is. For TSLS DoA estimation this means that a DFUweighting with an estimated 2nd order Taylor approximationvariance (20) cannot be beneficial, as v

(2)k,ij is only slightly more

accurate than v(1)k,ij for SNRs that are anyways already too low

to estimate the variance properly.We next proceed to analyze the error of SP-DoA estimation

in multipath scenarios, using two sectors with αi = 1, αj = 0.9,ϑi =0◦, ϑj =20◦. Since our results for multipath in Section IV-Bare for EBS, we calculate the common β of the two sectorsfrom a setting of M = 6 and as = 0.4. We include two pathsin our simulations: a line-of-sight path with power scalingψ2

k,1 = 0.9 and DoA ϕk,1 = 10◦, and a reflected path with

power scaling ψ2k,2 = 0.1 and DoA ϕk,2 = 15◦. Similarly to

our example of free space propagation, these numbers reflecta realistic example scenario. Naturally, other numerical valuescould be used for multipath propagation as well. Our resultsof simulated bias and standard deviation, as well as theoreticalresults using the 1st- and 2nd-order Taylor approximationsderived in Section IV-B, are shown in Fig. 5.

We observe that the SP-DoA estimates are more stronglybiased in a multipath scenario than in free space, as the biasincreased from 0.02◦ in Fig. 4 to 1.3◦ in Fig. 5 for SNR greaterthan 5 dB. The bias increase is primarily because of the re-flected path that is 5◦ away from the main path. Both theoreticalbias using the 1st- and 2nd-order Taylor approximations matchthe empirical simulations for SNR greater than 0 dB. The gapbetween theoretical and simulation results is smaller for the2nd-order curve compared to the 1st-order curve. The standarddeviation of the SP-DoA estimations under multipath is alsogreatly increased compared with the free space scenario, e.g.,for 10 dB the standard deviation is 1◦ in free space and 2◦ withmultipath. Theoretical results using 1st- and 2nd-order Taylor


approximations behave similarly, i.e., they match simulationsaccurately for SNR greater than 0 dB.

Note that the root-mean squared error (RMSE) of SP-DoAestimation will increase in multipath scenarios due to the in-creased bias and variance. This fact will further affect the TSLSDoA estimation and transmitter localization performance.However, due to space limitation, the following sections onlycontain simulation results without multipath. In Section VII,the impact of severe multipath on DoA estimation and local-ization accuracy will be further discussed and elaborated withpractical RF measurements.

B. TSLS DoA Estimation Performance

The performance of TSLS DoA estimation is evaluated nextusing two different antenna models. On the one hand, we studythe performance using an ESA with M = 6 sectors, each with aradiation pattern as in (1) and as = 0.4. On the other hand, weconsider a model of an actual LWA that we use in the practicalmeasurements in Section VII. The LWA is modeled using theradiation patterns according to (1) with the parameters αm =αm, βm = βm and ϑm = ϑm shown in Table III. These param-eters were obtained from a least-squares (LS) fit of the actualradiation pattern as discussed in more detail in Section VII-B.In total, the LWA has M = 12 sectors with orientations rangingfrom roughly −50◦ to 50◦, making the antenna suitable forDoAs from around −60◦ to 60◦. In contrast, the ESA coversthe entire angular range from −180◦ to 180◦. However, withrespect to TSLS DoA estimation, the main difference betweenthe two models is that the ESA consists entirely of EBS, whilethe LWA model has only DBS. Therefore, TSLS is run with theSP-DoA estimation as described in Section III-B1 for theformer, while the latter uses SP-DoA estimation as discussedin Section III-B2. It is assumed that the DoA is uniformly dis-tributed over the whole angular coverage area of the antennas.We emulate this distribution via 100 equidistant steps in the

interval ϕk ∈[0; 180◦

M

)for the ESA [12] and 120 equidistant

steps in the interval ϕk ∈ [−60◦; 60◦) for the LWA model. Foreach DoA-step we then simulate 2000 realizations, and averageover the results at each step in order to obtain the RMSE. In thefollowing, we will be using different configurations of TSLS,such as TSLS+EW. Please refer to Fig. 3 for an overview of theconfigurations and to Section III for detailed descriptions.

Fig. 6 depicts the RMSE of DoA estimation as a functionof the SNR when using the ESA. For reference, we have alsoincluded the CRB on DoA estimation with sectorized antennas[12], along with the SLS DoA estimator [12]. For TSLS, wehave determined that L = 3 provides the best performance formoderate to high SNRs in separate simulations that are notexplicitly shown due to space limitations. Note that this value isspecific to the antenna and in particular its beamwidth as will bediscussed later in this section. From the results in [12] it can beconcluded that SLS is not making efficient use of high SNRs.In contrast, TSLS with EW in the DFU stage is approachingthe CRB for large SNR ≈ 20 dB. However, the performanceof TSLS+EW degrades rapidly for lower SNRs such thatSLS outperforms TSLS+EW already for SNR ≈ 12 dB. The

Fig. 6. DoA estimation performance with equal-sector antenna and differentalgorithms. Parameters: as = 0.4, M = 6, and N = 100.

combination of TSLS and PW performs best of all algorithmsfor the low SNR region, while its performance saturates at ahigher RMSE than the CRB and the other TSLS configurations,starting from SNR ≈ 10 dB. Nevertheless, it outperforms SLSfor all SNRs. The overall best performance is achieved withTSLS+VW. For low SNR, TSLS+VW behaves like SLS andperforms therefore only slightly worse than TSLS+PW. Thisis explained by the earlier made observation that the varianceis estimated much larger than it actually is when the sectorSNRk,m is low. In the DFU stage and for overall low SNRthis therefore leads to an implicit exclusion of all SP-DoAsother than the one used in SLS. For high SNR, on the otherhand, TSLS+VW performs like TSLS+EW. This, in turn, isexplained by the variance of the SP-DoA estimates that becomeindependent of the sector for very large SNRs and ESAs sinceak,m → 1

N for SNR → ∞ in (19).The performance of the LWA model as a function of the

SNR is shown in Fig. 7. Although the CRB was only discussedfor ESAs in [12], it was derived in a generic format such thatit is also applicable for our LWA model. Therefore, we haveincluded the CRB as a reference also in this figure. Besidesthe sector parameters, αm and βm varying for different sectorsm, the LWA model also has a much bigger overlap betweenthe sectors than the ESA. As an example, the sectors 3 and4 (Table III) have α4 ≈ α3 = 1 and ϑ43 ≈ 12◦. With theseparameters an ESA would have M = 30 sectors, resulting inβ ≈ 0.22 rad if we assume the same side-sector suppressionas = 0.4 as for the above described ESA. The LWA antenna, incontrast, has an almost 8-fold larger beamwidth in the sectors3 and 4. Compared to the ESA, much more sectors of the LWAtherefore receive the TX signal at a high SNR. This, in turn,implies that L, i.e., the number of sectors used for SP-DoAestimation, should be increased for the LWA. We have foundthat parameterizing TSLS with L = 11 results in the overallbest performance. However, this is not the case over the wholeSNR range. Using, e.g., only L = 4 sectors, yields a bit better


Fig. 7. DoA estimation performance with leaky-wave antenna approximatedby Gaussian radiation pattern (1). Parameters: M = 12, N = 100, and antennaparameters in Table III.

performance for low SNRs if EW or VW is used in the DFUstage of TSLS. When using PW, on the other hand, it seemsthat the performance is best with L = 11 sectors for all SNRs.Otherwise, the behavior of TSLS and its DFU configurationsis very similar to what we have observed already for the ESA.Most notably, TSLS+VW approaches the CRB for high SNRalso for LWAs.

The SP-DoA estimator (8) yields two solutions. To verifythat our selection mechanism described in Section III-B2 worksproperly, we have run the same simulations with the actualDoA as an input to TSLS. Out of the two possible solutionsin (8), this TSLS test version then picks the one that is closerto the actual DoA. Naturally, this test version yields betterperformance than the practical implementation. However, theincrease in performance is only marginal. Therefore, we canconclude that the selection mechanism based on the RSS workswell. For clarity of presentation, we have not included these testcurves in Fig. 7.

C. Localization Performance

In this section, we evaluate the performance of localizationwith TSLS+VW DoA estimation and subsequent Stansfieldfusion (TSLS-S). For comparison, we also include the com-bination of SLS and Stansfield (SLS-S) as well as the CRBon non-cooperative TX localization using sectorized antennas[29]. In our simulation, we assume that the RXs are uni-formly distributed on a circle with radius R = 150 m centeredaround the TX. However, no RX is placed in a protectiveinner circle with radius R0 = 5 m. The TX transmit poweris PT = 20 dBm, while the measurement noise power atthe RXs is σ 2

w = −70 dBm. For the propagation, we as-sume a log-distance path loss model with path loss exponentα = 4. Overall, these settings result in an average SNR =10 log10

(2

α−2PTσ 2

w

R2−α0 −R2−α

R2−R20

)= 33 dB. The TX signal is mod-

eled as bandlimited Gaussian with a bandwidth B = 20 MHzand without oversampling at the RXs. This leads to a correlation

Fig. 8. Localization performance with an equal-sector antenna and a uniformdistribution of RXs. Parameters: as = 0.4, M = 6, and N = 100.

of sector-powers at different RXs that we have derived in detailin [29]. As we have seen in Section VI-B, the behavior of theESA and the LWA in TSLS DoA estimation is qualitativelysimilar. To keep the simulations simple, we therefore assumethat all RXs are equipped with an ESA, parameterized as inSection VI-B. In the simulations we obtain the RSS estimatesfor weighting in the modified Stansfield algorithm using theprinciple discussed in Section III-B2. Based on the sector-powers of the MASP and the estimated DoA at every RX k, weobtain two RSS estimates γk,i and γk,j, and estimate the finalRSS as γk = 1

2 (γk,i + γk,j).Fig. 8 depicts the RMSE of location estimation as a function

of the number of RXs, K. Overall, the biggest gain in perfor-mance for increasing K is achieved for K < 20 RXs as reflectedby the CRB as well as the algorithms. When estimating theDoA with TSLS instead of SLS, we observe a localization per-formance improvement of 0.5–0.8 m. This is mainly explainedby the fact that the modified Stansfield algorithm is dominatedby RXs with a large SNRs as we have concluded in [13].Since TSLS is outperforming SLS in particular for moderate tolarge SNR, we consequently also observe a strong localizationperformance improvement. Nevertheless, TSLS-S is not ableto approach the CRB. In this context it is, however, importantto note that the CRB in [29] is derived for a more generalcase where the TX location is estimated directly from the KMsector-powers in (2). In SLS-S/TSLS-S, on the other hand, wefirst estimate K DoAs from the KM sector-powers and onlythereafter we estimate the location using the DoA estimates. Aswe have already suggested in [29], this intermediate step mightdeteriorate the performance and hence it might be impossiblefor algorithms such as TSLS-S to exactly reach the CRB.

D. Performance and Complexity in Comparison toRelated Algorithms

In this section, we compare the proposed TSLS algorithmto related works. First, we quantify the complexity of the


TABLE IIPROPOSED DOA ESTIMATOR IN COMPARISON TO OTHER ALGORITHMS

proposed TSLS DoA estimation in terms of the number ofbasic operations. As such, we define additions/subtractions andmultiplications/divisions and refer to them as ADD and MUL,respectively. In addition to such basic operations, SLS [12] aswell as TSLS rely on some standard functions that cannot beexpressed directly in terms of the basic operations. For fastprocessing in a digital signal processor these standard functionscould be implemented in form of a look-up table, which makesthem neglectable in the overall processing time. Nevertheless,we include them in our considerations and refer to the naturallogarithm as LOG, the exponential function as EXP, the square-root as SQR, sine/cosine as SIN/COS and finally as ATAN2 tothe function that calculates the final DoA estimate from the leftside of (10), which is often referred to as the atan2 function inthe literature.

Obviously, the maxE algorithm [11], [22], [25] that estimatesthe DoA as the sector with the maximum power has the lowestcomplexity since it only calculates the sector-powers accordingto (2) and finds the smallest of those sector-powers. The sector-power calculation is in fact also part of SLS as well as TSLSand has a complexity of M(2N − 1)ADD + M(2N + 1)MUL.Next, we derive the complexity of TSLS and obtain the com-plexity of SLS [12] as a by-product by setting L = 2 and notcounting the DFU stage. In the derivation, we assume an imple-mentation of TSLS that relies on pre-calculated values as muchas possible. In (7), as an example, ϑij, κij and 2 ln αi

αjdepend

only on the approximation of the radiation pattern and cantherefore be loaded as constants during runtime. Overall, thisapproach is very feasible for a practically reasonable number ofsectors and avoids lots of computations. In the SSL stage wecompute M products of sector-powers and find the maximumof those M products. The latter is comparable to finding themaximum of the M sector-powers in maxE, which we do notinclude in our complexity analysis since the compare operationis normally neglectable in comparison to ADD and MUL. Thecomplexity in the SSL stage is thus equal to MMUL. In theSDE stage, we then calculate metrics pk,i = ln pk,i, ∀ i ∈ Lk

with complexity L(ADD + LOG). Thereafter, the calculationof ln pk,i

pk,jin (7) and (8) is reduced to a single subtraction. Let

us now define NSP as the number of sector-pairs used in theSDE stage, for which it holds that NSP ≤ (L

2

). For EBS we

then obtain an overall complexity in the SDE stage equal to(2NSP + L)ADD + NSPMUL + LLOG[+NSPADD], where thesquare brackets indicate operations that are only needed for an-tennas, where αi �= αj, i, j = 1, . . . , M. And for DBS we obtainan overall complexity of (10NSP + L)ADD + 17NSPMUL +NSPSQR + 4NSPEXP + LLOG in the SDE stage. Finally, theDFU stage has a complexity of 2(NSP − 1)ADD + MUL +NSP(COS + SIN) + AT2, with an additional complexity of

3NSPMUL and (NSP + L)ADD + (4NSP + 2L)MUL for PWand VW, respectively.

The complexity and DoA estimation performance as wellas localization performance of TSLS in comparison to maxEand SLS can be found in Table II. Since SLS and maxE areestimators targeted at ESAs, we use an ESA as described inSection VI-B as the sectorized antenna, along with simulationsetups as described in Section VI-B and Section VI-C for DoAestimation and localization, respectively. For simplicity, we donot separate between the aforementioned standard functions inthe complexity metric. Instead, we count a call to one of thestandard function as a call to a look-up table (LUT). Based onthe results in Table II, we conclude that the proposed estimatorhas the best performance of all three, while its additional com-plexity is very low and mainly due to calls to standard functions,which could be handled with LUTs for fast processing.

VII. PRACTICAL RF MEASUREMENTS

A. Measurement Setup

The practical performance of the proposed TSLS DoA es-timation with subsequent Stansfield localization was evaluatedwith the help of an extensive indoor measurement campaignat the 2.4 GHz ISM band, carried out at Drexel University.In our measurements, we used LWAs as an example of asectorized antenna. However, other sectorized antennas suchas antenna arrays with a single front-end [17], [18] would alsohave been a good alternative. During the measurements, severaluncontrolled WiFi hotspots in the surroundings were active,causing substantial in-band interference. In addition, passers-bygenerated spatial and temporal variations in the measurementconditions. Both these aspects imply that the measurementenvironment was far from ideal, thus enhancing further thepractical impact of the measurements.

As illustrated in Fig. 9, we placed a TX at three differentlocations and measured the signal at six RXs. The TX andRX locations were selected such that the performance couldbe tested in different challenging estimation scenarios. The RXantennas were oriented in such a way that they can hear the TXswithin their directivity ranges in most of the cases.

Each transceiver consisted of a software defined radio plat-form, called Wireless Open-Access Research Platform (WARP)v3 [32]. Each WARP board was connected to its own antenna(s)and to a centralized controlling system. The TXs were equippedwith omni-directional antennas whereas each RX had a singletwo-port LWA [21] with two antenna ports enabling simulta-neous measurements in two sectors. The composite right/left-handed (CRLH) LWAs [14] were tuned to operate within the


Fig. 9. Illustration of the measurement setup and the localization capabilitiesof the proposed algorithms in a lobby at Drexel University.

entire 2.4 GHz WiFi band. The physical size of the antenna is156 mm × 38 mm and the antenna consists of a cascade of12 metamaterial unit cells for obtaining good directivity anda small size simultaneously. The main beam of the antenna canbe steered from broadside to backward and forward direction bychanging two control voltages. Due to the practically symmet-ric antenna structure, antenna ports have symmetric radiationproperties with respect to the broadside direction. As a trade-off between estimation complexity and estimation accuracy, wehave used 6 control voltage values for a total of 12 antennasectors.

The system was operating with a 20 MHz channel bandwidthand a carrier frequency of 2.462 GHz. This combination isheavily overlapping with the WiFi channel no. 11 which wasmeasured to have active traffic during the measurements, act-ing as direct cochannel interference. The TX power was setto +15 dBm and since our algorithms rely on the receivedsignal powers, automatic gain controls (RF and baseband) weredeactivated in the RXs and the gains were set to constant values.As a practical example, we used orthogonal frequency divisionmultiplexing (OFDM) waveforms. Only one of the transmissionlinks, i.e., TX-RX antenna pair, was active at a time and thus weneeded to ensure fairness between different transmission linksby transmitting the same data over all links. Note that realisti-cally all RXs could receive the same transmitted data only prop-agated through different channels. Therefore, the measurementarrangement matches the real world scenario well. For testingeach link, we transmitted in total 300 packets, each containing5420 binary phase shift keying (BPSK) symbols. Finally, foreach antenna sector, we calculated the received signal powerfrom the baseband signal snapshots by averaging over thepowers of all received packets observed through the consideredsector. These sector powers are then processed further using theproposed algorithms, as described in the previous sections.

B. Localization System Settings

The TSLS DoA estimator requires anapproximation ρm(ϕ) =[ζm(ϕ)]2 of the antenna’s main beam through a Gaussian curve

TABLE IIILS FIT FOR LEAKY-WAVE ANTENNA. THE LWA HAS

SYMMETRIC SECTORS AROUND 0◦ . THEREFORE, THIS

TABLE DEPICTS ONLY 6 OUT OF THE 12 SECTORS.FURTHERMORE, WE HAVE NORMALIZED THE

ATTENUATION TO αm ≤ 1, max(αm) = 1,m = 1, . . . , M

Fig. 10. Radiation pattern for two sectors (solid lines) of the leaky-waveantenna [14] used in our measurements. The main beams are well approximatedby a Gaussian curve (dotted lines).

(1) in all sectors m, m = 1, . . . , M. In order to obtain theapproximation, we first measured the LWA’s radiation patternin each sector m as ρm(kϕ) = [ζm(kϕ)]2, k = 1, . . . , 360with a step-size ϕ = 1◦. We then found an approximationthrough a LS fit of (1) to ρm(kϕ). However, only the mainbeam of the antenna can be well approximated through (1).More importantly, due to the SSL process TSLS will makeuse of the approximated radiation patterns only around themain beam. In order to obtain a good approximation, the LSfit was consequently calculated only around ϑm = μϕ, μ =arg maxk ρm(kϕ), i.e., the initial estimate for the orientationof the main beam in sector m. We used an interval of � = 20◦on both sides of ϑm and obtain ρm with the parameters αm, βm

and ϑm that minimize∑�/ϕ

k=−�/ϕ[ρm(ϑm + kϕ) − ρm(ϑm +kϕ)]2. For the LWA used in our measurements, this resultsin the values shown in Table III. The radiation patterns of twosectors along with their approximation are shown in Fig. 10. Asevident, the approximation fits very well for the main beam.

TSLS is furthermore parameterized to take into account thestrong multipath environment. For heavy multipath, sectors farfrom the line-of-sight direction to the TX can be expected to bedominated by multipath components. Hence, we chose to useonly L = 3 sectors in TSLS DoA estimation. Moreover, PW ischosen as the DoA fusion method since it is more robust thanVW, which has been derived for a free space propagation, whilegenerally having better performance than EW. For simplicity,


TABLE IVPRACTICAL MEASUREMENT RESULTS

we estimate the RSSs for Stansfield weighting simply as themaximum NCSP of the MASP (see Section III-B and A) at RXk, k = 1, . . . , K, i.e., γk = max(pk,q, pk,q+1).

C. Measured Performance

Table IV(a) presents the DoA estimation errors stemmingfrom the processing of the measurements with the abovedescribed TSLS configuration. Based on these results, we rec-ognize clear differences in the DoA estimation accuracy at dif-ferent RXs. RX3 performs very well with all TXs and reachesthe lowest RMSE due to its good coverage and interference-free location. In contrast to that, RX2 yields a relatively highDoA estimation RMSE. There is no obvious geometrical reasonfor that. However, one of the active WiFi hotspots was locatedright behind the wall close to RX2 and may thus have negativelyinfluenced the estimation performance. The highest individualestimation error of 30.2◦ was obtained for the TX3-RX6 com-bination, which might have been caused by the rich scatteringconditions due to the stairs (made of metal, concrete and glass)in proximity of RX6.

When analyzing the results in Table IV(a) row-wise, wesee that the DoA of TX3 is the most difficult to estimate, asexpected due to TX3’s relatively isolated location. Moreover,TX3 was the node closest to the walls with metal doors oneither side. We suspect that this caused strong multipath, con-tributing further to the large DoA estimation error. Surprisingly,TX2 turns out to have the most accurate DoA estimates eventhough it is located close to the stairs which might act as localscatterers. Actually, only two out of six DoA estimation errorsof TX2 are in excess of 10◦, meaning that TSLS provides veryaccurate estimates in this case. Overall, these results reveal thatthe TSLS algorithm is capable of estimating DoAs with fairlyhigh accuracy even in a challenging indoor environment withsevere multipath.

Fig. 9 depicts the TX location estimates resulting fromthe above described combination of TSLS DoA estimationwith subsequent Stansfield localization. A detailed summaryof the localization results is shown in Table IV(b). Based onthese results, TX1 and TX2 can be localized very preciselyas the location error is around 60 cm for both. This is anexpected result due to their central locations with respect to theRXs. Although some of the individual DoA estimates have a

comparably large error, location estimation is fairly accurate,which is mainly explained by the Stansfield weighting schemethat gives larger weight to the DoA estimates of those RXsthat have large sector-powers. In case of TX3, the majorityof the DoA estimates are relatively inaccurate with four outof six estimation errors being larger than 20◦. Consequently,the location estimate is also more inaccurate (2.4 m). However,this is explained by the location of TX3 which was on purposeplaced at the boarder of the localization system’s coverage area.

The total localization RMSE is 1.5 m, which corresponds toa circle with an area of 2.1% of the whole measurement area.In our earlier work [31], we have used the same measurementdata to test the performance of the modified maxE DoA es-timator [31]. Using the same Stansfield fusion algorithm, butin combination with simple maxE DoA estimates resulted in aRMSE of 2.8 m, which is an almost two-fold RMSE comparedto the localization based on TSLS DoA estimates. Hence thealgorithms proposed in this article are clearly outperforming theexisting state-of-the-art.

VIII. CONCLUSION

In this paper, we have substantially extended our earlierwork on DoA estimation and TX localization using sectorizedantennas. We have introduced a modified antenna model withmore degrees of freedom and have shown that it can be usedto model a broader range of practical antennas. Based on thenew model we have proposed the novel TSLS DoA estimator.In contrast to existing DoA estimators, TSLS is applicable toall antennas that can be described by our modified antennamodel, independent of the TX signal type and without therequirement for cooperation between the TX and the localizingnetwork. Besides being more universal, we have shown thatTSLS has also better performance than existing algorithms andthat it approaches the CRB on signal DoA estimation of a non-cooperative TX if the SNR is moderate to large. In order tobetter understand the performance of TSLS, we have further-more derived analytical error models for the underlying DoAestimation principle of TSLS DoA estimation considering freespace as well as multipath propagation. We have then analyzedthe performance of localization with the Stansfield estimatorthat estimates the TX location by fusing the TSLS DoA esti-mates. As expected, the improved DoA estimation performanceof TSLS also results in a localization performance improvementcompared to the localization using previously published DoAestimators. Finally, we have shown how to configure the TSLSDoA estimator to work with a practical sectorized antenna,namely a leaky-wave antenna. Based on that configurationwe have then demonstrated the achievable performance of asectorized antenna-based localization system using real-worldmeasurements obtained in an indoor environment. Overall, webelieve that the results in this paper are crucial for the practicalimplementation of low complexity DoA estimation and TXlocalization using sectorized antenna systems. In particular thisis the case when the TX is non-cooperative or when dedicatedsignaling between TX and localization network is otherwiseunfeasible.


APPENDIX ADERIVATION OF DOA ESTIMATORS

The criterion (6) is clearly minimized for pk,i/pk,j − ρk,i/

ρk,i = 0. Recalling the Gaussian approximation ρk,m =[ζm(ϕk)]2, m = i, j in (1), we can write

0 = lnpk,i

pk,j− 2 ln

αi

αj+ 2

[M(ϕk,ij − ϑi)

]2

β2i

− 2[M(ϕk,ij − ϑj)

]2

β2j

. (28)

The above equation contains the function M(ϕ), which isdifficult to handle mathematically. However, the mapping of theangles to [−π; π) is arbitrary. Hence, for practically relevantcases we can always find a mapping for the angles ϑi andϑj such that we can write M(ϕk − ϑm) = ϕk,ij − ϑm, wherem = i, j and ϕk,ij is the mapped SP-DoA estimate. One suchmapping could be ϑi = 0 and ϑj = ϑj − ϑi. We can then solve(28) to obtain (7) and (8). For presentation simplicity, we havenot included the mapping in (7) and (8).

APPENDIX BPROOF OF LEMMA 1

In the following we will derive the mean (14) and variance (16)for the 2nd order Taylor series expansion. The 2nd order Taylorseries of Y in Lemma 1 around its mean μ can be written as

Y(2) = f + jT(X − μ) + 1

2(X − μ)TH(X − μ) (29)

where we have omitted the dependence of f , j and H on μ fornotational simplicity. According to [33, p. 53], the moments ofa quadratic form V ′ = XTAX for normally distributed randomvector X ∼ N (μ, Q) with symmetric matrix A are given as

E[V ′] = trace(AQ) + μTAμ (30)

E[V ′2] = 2[trace(AQ)2 + 2μTAQAμ

](31)

+ [trace(AQ) + μTAμ

]2. (32)

In (29), we have V = XTHX with X = X − μ ∼N (0, Q) and the Hermitian matrix H that is consequentlysymmetric. We can therefore write the moments of V as

E[V] = trace(HQ) (33)

E[V2] = 2trace(HQ)2 + [trace(HQ)]2 . (34)

Hence, the first moment of Y is obtained as in (14). The varianceof Y can be written as

var[Y(2)

]= E

[(Y(2)

)2]

− E[Y(2)

]2

= E[f 2] + E[(jTX)2

]+ 1

4E[V2]

+ 2f jTE[X] + f E[V] + jTE[XV]− f 2 − f trace{HQ} − 1

4[trace(HQ)]2 . (35)

We notice that XV is a vector with elements composedof products Xp

kXql Xr

m, where Xk, Xl and Xm areindependent and at least one of the powers p, q or r is uneven.Therefore, we have E[XV] = 0 and obtain (16) by using (33),(34) in (35). The corresponding mean (13) and variance (15)of the 1st order Taylor series are not derived explicitly, due tospace constraints. However, the derivation is very similar.

APPENDIX CDERIVATION OF SP-DOA ESTIMATION

BIAS AND VARIANCE

For SP-DoA estimation with DBS, we have ϕij = f (p) =λij ± bijg(p) with p = (pk,i, pk,j)

T . This leads to the 2 × 1gradient vector j with elements

j1 = [f ]pk,i(μ) = ± βiβj

4pk,ig(μ)(36)

j2 = [f ]pk,i(μ) = ∓ βiβj

4pk,jg(μ)(37)

and the 2 × 2 Hessian matrix H with elements

H11 = [f ]pk,ipk,i(μ) = ∓βiβj4

[g(μ)

]2 + βij

16p2k,i

[g(μ)

]3 (38)

H22 = [f ]pk,jpk,j(μ) = ±βiβj4

[g(μ)

]2 − βij

16p2k,j

[g(μ)

]3(39)

H12 = H21 = [f ]pk,ipk,j(μ) = ± βiβjβij

16pk,ipk,j[g(μ)

]3 (40)

both evaluated at p = μ. We furthermore obtain

trace(HQ) = H11σ21 + H22σ

22 (41)

trace(HQ)2 =(

H11σ21

)2 + 2H212σ

21 σ 2

2 +(

H22σ22

)2. (42)

since Q is diagonal. Inserting (38)–(40) in (41) and using (41)in (14) yields E[ϕ(2)

k,ij] and, after subtraction of the DoA ϕk, thebias (18). (19) results from (15) and (36)–(37). Finally, inserting(42) in (16) and using the results from (19) leads to (20).

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Janis Werner (S’14) was born in Berlin, Germany,in 1986. He received the Dipl.Ing. degree in electricalengineering from Dresden University of Technology(TUD), Germany, in 2011. Currently, he is work-ing towards the Ph.D. degree at Tampere Univer-sity of Technology (TUT), Finland, where he is aResearcher with the Department of Electronics andCommunications Engineering. His main researchinterests are localization with an emphasis on direc-tional antenna-based systems as well as the utiliza-tion and further processing of location information

in future generation mobile networks.

Jun Wang received the B.Eng. degree in electronicinformation engineering from Beijing University ofTechnology and the M.Phil. degree in electrical engi-neering from City University of Hong Kong, in 2006and 2009, respectively. He received the Ph.D. degreein electrical engineering from the University of Cal-ifornia, Los Angeles, CA, USA, in 2014, advised byProf. Danijela Cabric. He joined Broadcom Corpora-tion as a Research Scientist in wireless communica-tion systems design in 2014. His research interestsinclude acquisition and utilization of primary-user

location information and various issues related to spectrum sensing in cognitiveradio networks.

Aki Hakkarainen (S’14) received the M.Sc. degree(with honors) in communication electronics fromTampere University of Technology (TUT), Finland,in 2007. From 2007 to 2009, he was a RF DesignEngineer with Nokia, Salo. From 2009 to 2011, hewas a Radio System Specialist with Elisa, Tampere.Currently, he is working towards the Ph.D. degree atTUT. He is a Researcher and Doctoral Student withthe Department of Electronics and CommunicationsEngineering, TUT. His research interests includedigital signal processing methods for RF impairment

mitigation and digital signal processing for localization.

Nikhil Gulati (S’07) received the B.Eng. degreein electronics instrumentation and control from theUniversity of Rajasthan, India, in 2005 and theM.S. degree in electrical engineering with a ma-jor in systems and control from Drexel University,Philadelphia, PA, USA, in 2007, where he conductedresearch on sensor networks, autonomous systemsand real-time control. He is currently pursuing thePh.D. degree in electrical engineering at Drexel Uni-versity. He worked as a Software Engineer from2007–2010. His research is focused on developing

adaptive learning algorithms for cognitive radios employing electrical recon-figurable antennas. He also works on developing distributed algorithms forinterference management and heterogeneous wireless networks.


Damiano Patron (S’15) received the B.S. degreein electronics engineering from University of Padua,Italy, in 2010 and the M.S. degree in electricaland computer engineering from Drexel University,Philadelphia, PA, USA, in 2013, where he is cur-rently pursuing the Ph.D. degree in telecommuni-cation engineering. From 2002 to 2010, he workedwith Euro-Link S.r.l. as a System Integrator of RFIDsystems. In 2010, he joined the start-up companyAdant Inc. as an RF and Antenna Engineer, de-signing reconfigurable antennas for WiFi and RFID

applications. His research focuses on the design of reconfigurable antennas forthroughput maximization and DoA estimation in wireless networks. He alsoworks in the development of wearable sensors and power harvesting systems.He is the author and coauthor of several patents in the field of reconfigurableantennas, antennas miniaturization and wearable technologies. He receivedseveral awards including the Drexel Frank and Agnes Seaman FellowshipAward in 2013 and the Young Scientist Best Paper Award at the ICEAA—IEEEAPWC—EMS 2013 Conference.

Doug Pfeil received the B.S. and M.S. degrees incomputer engineering in 2007 from Drexel Univer-sity, Philadelphia, PA, USA. He is currently pursu-ing the Ph.D. degree from Drexel University wherehe is also a member of the research staff in theWireless Systems Lab. His current research interestsinclude software defined radio, free space opticalcommunication, physical layer design, and FPGAbaseband processing with an emphasis on the prac-tical implementation of novel hybrid RF and opticalcommunication techniques.

Kapil Dandekar (S’95–M’01–SM’07) received theB.S. degree in electrical engineering from the Uni-versity of Virginia in 1997, and the M.S. and Ph.D.degrees in electrical and computer engineering fromthe University of Texas at Austin in 1998 and 2001,respectively. In 1992, he worked at the U.S. NavalObservatory and from 1993-1997 he worked at theU.S. Naval Research Laboratory.

In 2001, he joined the Electrical and Com-puter Engineering Department, Drexel University,Philadelphia, PA, USA. He is currently a Professor

of Electrical and Computer Engineering at Drexel University; the Director ofthe Drexel Wireless Systems Laboratory (DWSL); Associate Dean for Researchand Graduate Studies in the Drexel University College of Engineering. DWSLhas been supported by the U.S. National Science Foundation, Army CERDEC,National Security Agency, Office of Naval Research, and private industry.His current research interests and publications involve wireless, ultrasonic,and optical communications, reconfigurable antennas, and smart textiles. In-tellectual property from DWSL has been licensed by external companies forcommercialization. He is a member of the IEEE Educational Activities Boardand co-founder of the EPICS-in-IEEE program.

Danijela Cabric (S’02–M’07–SM’14) received theDipl. Ing. degree from the University of Belgrade,Serbia, in 1998, and the M.Sc. degree in electricalengineering from the University of California, LosAngeles, CA, USA, in 2001. She received the Ph.D.degree in electrical engineering from the Universityof California, Berkeley, CA, USA, in 2007, whereshe was a member of the Berkeley Wireless Re-search Center. In 2008, she joined the faculty ofthe Electrical Engineering Department, University ofCalifornia, Los Angeles, where she is now Associate

Professor. She received the Samueli Fellowship in 2008, the Okawa FoundationResearch Grant in 2009, Hellman Fellowship in 2012 and the National ScienceFoundation Faculty Early Career Development (CAREER) Award in 2012. Sheserves as an Associate Editor for the IEEE JOURNAL ON SELECTED AREAS INCOMMUNICATIONS (Cognitive Radio series) and IEEE COMMUNICATIONS

LETTERS, and TPC Co-Chair of 8th International Conference on Cognitive Ra-dio Oriented Wireless Networks (CROWNCOM) 2013. Her research interestsinclude novel radio architecture, signal processing, and networking techniquesto implement spectrum sensing functionality in cognitive radios.

Mikko Valkama (S’00–M’01) was born in Pirkkala,Finland, on November 27, 1975. He received theM.Sc. and Ph.D. degrees (both with honors) in elec-trical engineering (EE) from Tampere Universityof Technology (TUT), Finland, in 2000 and 2001,respectively. In 2002, he received the Best Ph.D.Thesis award by the Finnish Academy of Science andLetters for his dissertation entitled “Advanced I/Qsignal processing for wideband receivers: Modelsand algorithms”. In 2003, he was working as a Vis-iting Researcher with the Communications Systems

and Signal Processing Institute, SDSU, San Diego, CA, USA. Currently, he isa Full Professor and Department Vice-Head at the Department of Electronicsand Communications Engineering, TUT, Finland. He has been involved inorganizing conferences, like the IEEE SPAWC’07 (Publications Chair) held inHelsinki, Finland. His general research interests include communications signalprocessing, estimation and detection techniques, signal processing algorithmsfor software defined flexible radios, cognitive radio, full-duplex radio, radiolocalization, 5G mobile cellular radio, digital transmission techniques such asdifferent variants of multicarrier modulation methods and OFDM, and radioresource management for ad-hoc and mobile networks.

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