Accurate placement of elements in large antenna arrays is a difficult and costly process. We explore the use of the validated exponential analysis (VEXPA) technique that was previously formulated to solve a direction-of-arrival (DOA) estimation problem, to find the antenna element positions in an array after the installation phase, so that cost-savings can be realised during placement of the antenna elements. Measurements are taken from harmonically related input signals transmitted from an Unmanned Aerial Vehicle (UAV) for which the position in the sky is known. It is shown how the UAV's zenith angle can be manipulated to generate parameters required for VEXPA's de-aliasing step. A simple simulation illustrates the functioning of the proposed method

Cuyt, Annie

der Villiers, Dirk

Knaepkens, Ferre

Lee, Wen-shin

Louw, Ridalise

Weideman, Rina-Mari

Wijnholds, Stefan

Stirling Online Research Repository (RIOXX)

URSI GASS 2021, Rome, Italy, 28 August - 4 September 2021Antenna Position Estimation Through Sub-Sampled Exponential Analysis of Harmonically RelatedInput SignalsRidalise Louw*(1), Ferre Knaepkens(2), Annie Cuyt(2)(3), Wen-shin Lee(4)(2), Stefan J. Wijnholds(5)(1), Dirk I. L. deVilliers(1), Rina-Mari Weideman(1)(1) Department of Electrical and Electronic Engineering, Stellenbosch University, Stellenbosch, South Africa(2) Department of Computer Science, University of Antwerp, Middelheimlaan 1, 2020 Antwerpen, Belgium(3) College of Mathematics and Statistics, Shenzhen University, Shenzhen, Guangdong 518060, China(4) Division of Computing Science and Mathematics, University of Stirling, Stirling FK9 4LA, Scotland (UK)(5) Netherlands Institute for Radio Astronomy (ASTRON), Dwingeloo, The NetherlandsAbstractAccurate placement of elements in large antenna arrays isa difficult and costly process. We explore the use of thevalidated exponential analysis (VEXPA) technique that waspreviously formulated to solve a direction-of-arrival (DOA)estimation problem, to find the antenna element positions inan array after the installation phase, so that cost-savings canbe realised during placement of the antenna elements. Mea-surements are taken from harmonically related input signalstransmitted from an Unmanned Aerial Vehicle (UAV) forwhich the position in the sky is known. It is shown how theUAV’s zenith angle can be manipulated to generate param-eters required for VEXPA’s de-aliasing step. A simple sim-ulation illustrates the functioning of the proposed method.1 IntroductionAccurate beamforming of an array of antennas requires ac-curate knowledge of the positions of the individual anten-nas. Establishing those antenna positions can be an elabo-rate process for large-N radio interferometers like the LowFrequency Array (LOFAR) [1] and the Square KilometreArray (SKA) [2]. In [3], a method was proposed to measurethe positions of the individual antennas using an UnmannedAerial Vehicle (UAV). As this is done after installation, itfacilitates a more cost-effective roll-out of the antennas bybeing tolerant to antenna placement errors and errors whenconnecting the antennas to the back-end. In this contribu-tion, we recognise that these UAV measurements can actu-ally be considered as an inverse-DOA estimation problemand explore the possibility to use the exponential analysismethod developed for DOA estimation in [4] for antennaposition estimation.2 Problem FormulationFigure 1 illustrates the odd harmonic signals Si(t) transmit-ted from the UAV located at zenith and azimuth angles, θand φ , respectively. The index i ∈ N is a range of nat-ural numbers that is used to distinguish between narrow-band signals at different odd harmonic frequencies ωi =2(i+ 12 )ω0, with ω0 the base frequency. At a time t theFigure 1. Illustration of signals Si(t) transmitted from theUAV towards a planar array.signals are expressed asSi(t) = si(t)exp( jωit) (1)where si(t) = ai(t)exp( jpi(t)) and the slowly varyingamplitude and phase of the signal are denoted by ai(t) andpi(t), respectively. It is assumed that this signal is strongenough so that astronomical sources in the field of view ofthe array can be ignored, as in [3]. For large arrays, thesource is in the radiating near-field of the antenna, thusa curved phase front is incident on the array. A possiblesolution for this is to estimate the positions of elementsin the array incrementally using a subset of antennas toensure the far-field condition. For purposes of illustration,the far-field condition is assumed, such that Si(t) is a planewave incident on the array.The first element is chosen to coincide with the ori-gin, i.e., δ1 = (0,0). The time delay of incidence on themth antenna element at a position δm = (δmx,δmy) on the© 2021 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.x− y grid relative to δ1 is given byτm =r̂m · r̂UAVc=δmx sinθ cosφ +δmy sinθ sinφc,(2)where r̂m = δmxx̂+ δmyŷ+ (0)ẑ is the position of the mthantenna element in the (x,y)-plane, r̂UAV = sinθ cosφ x̂+sinθ sinφ ŷ+cosθ ẑ is the unit vector from the origin to theUAV’s position (θ ,φ) and c is equal to the propagation ve-locity of the signal, or in free space, the speed of light. Thenarrowband assumption that the signal does not change no-ticeably as it moves across the elements of the array givesan output at the mth antenna element at time t for frequencyi asfmi(t) = Si(t + τm)≈ si(t)exp( jωit)exp( jωiτm)= Si(t)exp(jωicr̂m · r̂UAV).(3)2.1 Element Data ModelTo extract the two-dimensional element positions(δmx,δmy), we require signals from two linearly inde-pendent directions in a plane parallel to the (x,y)-plane[5]. Let the first direction ∆1 = cosφ1x̂+ sinφ1ŷ and thesecond direction ∆2 = cosφ2x̂ + sinφ2ŷ. The commonfactor sinθ is intentionally excluded in these definitions, asit is used in the following section to model the sub-sampledexponential analysis problem. Then at a fixed time t, weuse the following shorthand notations:fmi = fmi(t),αi = si(t),βi = Si(t) = αi exp( jωit),Ψm1 = jω0csinθ(δmx cosφ1 +δmy sinφ1),Ψm2 = jω0csinθ(δmx cosφ2 +δmy sinφ2).(4)We distinguish between the coefficients αi1 and αi2 for thedifferent flight paths of the UAV along ∆1 and ∆2. Thenβi1 = αi1 exp( jωit) and βi2 = αi2 exp( jωit). The samplescollected at each element m are filtered into sub-bands, thusfor the two different directions we havefmi1 = βi1 exp(2(i+ 12)Ψm1),fmi2 = βi2 exp(2(i+ 12)Ψm2).(5)The coefficients βi depend on the different frequencies ωi.This is undesirable, so a pre-processing step is to mixall signals down to DC through digital multiplication ofexp(− jωit ′) at each corresponding sub-band, where t ′ in-dicates a time different to t. A constant phase compo-nent remains in each case, denoted as exp( jωi∆T 1) andexp( jωi∆T 2), where ∆T 1 and ∆T 2 refer to the time differ-ence t − t ′ corresponding to each case. This is dealt withby realising that the reference antenna element’s positionδ1 = (0,0) and at time t = 0:f1i(0) = si(0)exp( jωi∆T )exp(0) (6)which is some constant for both directions ∆1 and ∆2 thatcan be normalised to 1. We can therefore remove the vary-ing phase offsets by dividing all mixed down sample setsthat are collected at different time intervals with the mixeddown value of the first sample in time for the first antennaelement f1i. The samples in both directions then become:f ′mi1 =αi1 exp( jωi∆T 1)exp(2(i+ 12)Ψm1)f1i1(0)= exp(2(i+ 12)Ψm1),f ′mi2 =αi2 exp( jωi∆T 2)exp(2(i+ 12)Ψm2)f1i2(0)= exp(2(i+ 12)Ψm2).(7)It is interesting to note that these phases are equal to thephases of the correlations measured on the equivalent base-lines between element m and the element at the origin.2.2 Sub-Sampled Exponential AnalysisLet λ0 denote the wavelength corresponding to the base fre-quency ω0. Then, if the spatial Nyquist criteria∣∣sinθ(δmx cosφ1 +δmy sinφ1)∣∣< λ02 (8a)∣∣sinθ(δmx cosφ2 +δmy sinφ2)∣∣< λ02 , (8b)are not met for all antenna element positions δm, we aredealing with a sub-sampled exponential analysis problemthat can be solved by using the VEXPA technique describedin [6]. VEXPA’s de-aliasing method makes use of co-prime scale and shift parameters σ and ρ , respectively. TheUAV’s zenith angle θ can be exploited to model these pa-rameters. First define a virtual wavelengthλv > λ0 (9)large enough to ensure that the spatial Nyquist criteria∣∣δmx cosφ1 +δmy sinφ1∣∣< λv2 (10a)∣∣δmx cosφ2 +δmy sinφ2∣∣< λv2 (10b)are met for all m antenna element positions. Then, defineΦm1 = jωvc(δmx cosφ1 +δmy sinφ1)Φm2 = jωvc(δmx cosφ2 +δmy sinφ2),(11)where the virtual frequency ωv =2πcλv.We need to collect the following samples at each an-tenna element m:f ′m(iσ)1 = exp(σ(i+ 12)Φm1)(12a)f ′m(iσ+ρ)1 = exp((σ(i+ 12)+ρ)Φm1)(12b)f ′m(iσ)2 = exp(σ(i+ 12)Φm2)(12c)f ′m(iσ+ρ)2 = exp((σ(i+ 12)+ρ)Φm2). (12d)To do this, let θσ denote the zenith angle of the UAV duringcollection of the samples in (12a) and (12c). From compar-ison with (7) we haveσ(i+ 12)ωv = 2(i+ 12)ω0 sinθσ , (13)so thatsinθσ =σωv2ω0. (14)The left hand side of (13) is the desired model and the righthand side is the description of what is physically going on.The angle θσ is calculated from the scale parameter σ ∈Q.It is not a requirement for σ (or θσ ) to have the same valuein both directions ∆1 and ∆2, but this is done to simplify theanalysis. The UAV’s zenith angles θρi during collection ofthe samples in (12b) and (12d) are calculated from the shiftparameter ρ ∈Q and once again comparing with (7):(σ(i+ 12)+ρ)ωv = 2(i+ 12)ω0 sinθρi , (15)such thatsinθρi =(σ(i+ 12)+ρ)ωv2(i+ 12)ω0. (16)The values of σ ,ρ ∈ Q must be chosen as coprime andlead to sensible values for the angles θσ ,θρi . Notice thatthe UAV’s flight path should include 2(n + 1) positionsleading to signal DOAs of (φ1,θσ ),(φ1,θρi),(φ2,θσ ), and(φ2,θρi), where n is the number of frequency harmonicstransmitted.From the samples in (12) we can solve each antennaelement’s position δm relative to δ1. We collect Nt timesamples, also known as snapshots, at different timeintervals for each base term in (12). We then calculateNt scaled base terms exp(σΦm1) and exp(σΦm2) withthe Root-MUSIC algorithm [7], which requires a timeaverage of the samples at each antenna for the differentfrequencies in order to approximate the covariance matrix.For this purpose, we use a subset of Ns < Nt randomlychosen sub-snapshots for each of the Nt evaluations ofRoot-MUSIC.Note from (12) that the base terms exp(ρΦm1) andexp(ρΦm2) are the coefficients of the shifted samples.These are solved in the least squares sense for each antennaelement at each snapshot separately in both directions fromthe respective Vandermonde systems1exp(σΦm)...exp((n−1)σΦm)exp(ρΦm) = f′m(σ+ρ)...f ′m((n−1)σ+ρ) , (17)with n the number of frequency harmonics, ωi.As a result of noise, the Nt evaluations of exp(σΦm1),exp(ρΦm1), exp(σΦm2) and exp(ρΦm2) are clusteredaround the true solution in the complex plane. To locateour best estimate of these base terms, we do a search for thedensest point among the possible Nt points. The densestpoint has a specified minimum number of points around itwithin the smallest possible radius.We then have two sets of possible solutions for eachdirection:{exp(Φm1 +j2πσl): l = 0, . . . ,σ −1},{exp(Φm1 +j2πρl): l = 0, . . . ,ρ−1} (18)and {exp(Φm2 +j2πσl): l = 0, . . . ,σ −1},{exp(Φm2 +j2πρl): l = 0, . . . ,ρ−1}.(19)Since σ and ρ are chosen to be coprime, the intersection ofthe sets in (18) contains only one root which is the uniquesolution exp(Φm1), and similarly, the intersection of thesets in (19) gives the unique solution for exp(Φm2) [8].Thus the aliasing problem is solved.Since the vectors ∆1 and ∆2 are linearly independent,the 2×2 regular linear systemjωvc[cosφ1 sinφ1cosφ2 sinφ2][δmxδmy]=[Φm1Φm2](20)is solved for each antenna element to get its position(δmx,δmy). The above procedure is repeated for every an-tenna element to find all positions δm.3 Simulation ResultsA simulation was done using parameters from a UAV flightover a LOFAR low-band antenna (LBA) outer sub-stationconsisting of 48 dipole antennas. A cross-scan flight isperformed within a 100× 100 m square at an altitude ofapproximately 100 m. From the cross-scan we can useφ1 = 45◦ and φ2 = 135◦ which leads to orthogonal direc-tions ∆1 and ∆2 when viewing the flight path from above.Figure 2 illustrates the layout of the sub-station.The UAV transmits the 5th, 7th, 9th, and 11th harmonicsof a base frequency f0 = 6.3585 MHz, thus i = [2,3,4,5].The largest absolute distances from the origin |δmx| and∣∣δmy∣∣ of all elements in the sub-station is about 60 m.The wavelength at the base frequency is λ0 = 47.1483m, therefore the conditions in (8) are not met. To allowfor recovery from aliasing, let ωv = ω0/4 and σ = 5, tomeet the conditions in (10). This gives us θσ = 38.7◦. LetFigure 2. Antenna elements in an LBA outer sub-station.ρ = 7 so that θρi = [77.2◦,61◦,55◦,51.6◦] at the differentfrequency harmonics ωi.The effect of noise on the system was evaluated from100 Monte Carlo runs for each noise level from 15 – 50 dBin steps of 5 dB. The amount of snapshots Nt = 80 and thesub-snapshots used to evaluate Root-MUSIC is Ns = 0.5Nt .The RMS errors calculated for the x− and y−positions ofthe antenna elements are shown in Figure 3.Figure 3. Root mean square errors of element positions ina LOFAR LBA outer sub-station for different noise levels.Nt = 80, Ns = 0.5Nt .At an SNR of 15 dB, the RMS error is close to 30 cm forboth x− and y−positions. This quickly drops down to er-rors around 5 cm at 30 dB. The results of Figure 3 provethat the proposed method works in the case of a controlledsimulation. Further experiments are being done to under-stand the influence that various practical effects can haveon the results.4 ConclusionIn this paper, it is shown how the validated exponentialanalysis technique, VEXPA, can be used to estimate the po-sitions of elements in an antenna array using harmonicallyrelated input signals transmitted from a UAV. The UAV’sposition in the sky is exploited to create a model that allowsrecovery from aliasing. The effectiveness of the method hasbeen illustrated by evaluation of the results for varied noiselevels. The next step is to prove that the method workswhen using practical data from the field. Work in this re-gard is currently ongoing.5 AcknowledgementsThis work is based on the research supported in part bythe National Research Foundation of South Africa (GrantNumber 75322). It is also supported by the NetherlandsOrganisation for Scientific Research.References[1] M.P. van Haarlem et al., "LOFAR: The Low Fre-quency Array", Astronomy & Astrophysics, 556,A2, August 2013, pp. 1–53, doi: 10.1051/0004-6361/201220873[2] P. E. Dewdney, P. J. Hall, R. T. Schilizzi, and T. J.L. W. Lazio, "The Square Kilometre Array," Proceed-ings of the IEEE, 97, 8, August 2009, pp. 1482–1496,doi: 10.1109/JPROC.2009.2021005.[3] S. J. Wijnholds, G. Pupillo, P. Bolli, and G.Virone, "UAV-Aided Calibration for Commission-ing of Phased Array Radio Telescopes," in URSIAsia-Pacific Radio Science Conference (URSI AP-RASC), 2016, pp. 228–231, doi: 10.1109/URSIAP-RASC.2016.7601375.[4] F. Knaepkens, A. Cuyt, W-s. Lee and D. I. L. de Vil-liers, "Regular Sparse Array Direction of Arrival Es-timation in One Dimension," in IEEE Transactionson Antennas and Propagation, 68, 5, May 2020, pp.3997–4006, doi: 10.1109/TAP.2019.2963618.[5] A. Cuyt, Y. Hou, F. Knaepkens and W-s. Lee, "SparseMultidimensional Exponential Analysis with an Ap-plication to Radar Imaging," SIAM Journal on Sci-entific Computing, 42, 3, 2020, pp. B675-B695,doi:10.1137/19M1278004.[6] M. Briani, A. Cuyt, F. Knaepkens, and W-s. Lee,"VEXPA: Validated EXPonential Analysis throughRegular Sub-Sampling," Signal Processing, 177, De-cember 2020, doi:10.1016/j.sigpro.2020.107722.[7] A. Vesa, "Direction of Arrival Estimation Using MU-SIC and Root-MUSIC algorithm," in 18th Telecom-munications Forum, 2010, pp. 582–585[8] A. Cuyt and W-s. Lee, "How to get high resolution re-sults from sparse and coarsely sampled data," Appliedand Computational Harmonic Analysis, 48, 3, 2020,pp. 1066-1087, doi: 10.1016/j.acha.2018.10.001.

Antenna position estimation through sub-sampled exponential analysis of harmonically related input signals

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Antenna position estimation through sub-sampled exponential analysis of harmonically related input signals

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