Power-Based Direction-of-Arrival Estimation Using a Single Multi-Mode Antenna

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1 Power-Based Direction-of-Arriva Estiation Using a Singe Muti-Mode Antenna Robert Pöhann, Siwei Zhang, Thoas Jost, Arin Daann Geran Aerospace Center (DLR) Institute of Counications and Navigation Oberpfaffenhofen, Wessing, Gerany Eai: {Robert.Poehann, Siwei.Zhang, Thoas.Jost, Arin.Daann}@dr.de arxiv: v2 [cs.it] 18 Oct 217 Abstract Phased antenna arrays are widey used for direction-of-arriva (DoA) estiation. For ow-cost appications, signa power or received signa strength indicator (RSSI) based approaches can be an aternative. However, they usuay require utipe antennas, a singe antenna that can be rotated, or switchabe antenna beas. In this paper we show how a uti-ode antenna (MMA) can be used for power-based DoA estiation. Ony a singe MMA is needed and neither rotation nor switching of antenna beas is required. We derive an estiation schee as we as theoretica bounds and vaidate the through siuations. It is found that power-based DoA estiation with an MMA is feasibe and accurate. I. INTRODUCTION The state-of-the-art approach of direction-of-arriva (DoA) estiation reies on signa phase differences (assuing narrowband signas) between the eeents of an antenna array [1]. DoA estiation ethods are we known, however using an antenna array requires a copex receiver structure. For each array eeent, a separate receiver channe is required and the receiver channes have to be coherent and the syste we caibrated. For ow-cost appications, another possibiity is to use signa power, i.e. received signa strength indicator (RSSI) easureents for DoA estiation. This requires knowedge of the antenna pattern and the possibiity to cance out or estiate the unknown path oss and transit power of the signa. In this paper we focus on power-based DoA estiation. Different techniques can be found in iterature. One approach is to use an array of directiona antennas pointing in different directions, see e.g. [2]. If ony one antenna is avaiabe, an actuator can be used to rotate the antenna and thus obtain easureents fro different anges. This approach ay be used with either onidirectiona antennas having knowedge of the nu in the pattern, see e.g. [3], or directiona antennas, e.g. [4]. Instead of rotating the antenna, it is possibe to ake a controed oveent with the whoe patfor (e.g. a quadrocopter) []. Mechanica actuators have the drawback that they increase the power consuption of the syste, possiby need aintenance and iit the update rate. Hence the authors in [6] avoid oving parts and propose a switched bea antenna. In genera, high-resoution properties can be achieved with RSSI easureents. In [7], a variant of utipe signa characterization (MUSIC) suitabe for signa power easureents is presented. The authors in [8] appy ethods known fro finite-rate-of-innovation (FRI) saping to obtain high-resoution. A ethods in the iterature that the authors are aware of have in coon that they either use utipe antennas or a singe antenna with soe sort of rotationa oveent or beas that require switching. In contrast, this paper presents a power-based DoA estiation schee with a uti-ode antenna (MMA), which does not require any oveent or switching of antenna beas. An MMA is in fact a singe antenna eeent. Copared to antenna arrays, it has the advantage of being ore copact, which can be iportant for appications with size constraints. The reainder of this paper is organised as foows. In Section II, the concept of the MMA is introduced. Section III quicky recapituates the theoretica basis of this work tered wavefied odeing. The DoA estiation schee is then deveoped in Section IV. The perforance is evauated in ters of theoretica bounds and siuations. Section V concudes the paper. II. MULTI-MODE ANTENNA In this section we provide a brief introduction to MMAs. The concept of MMAs is based on the theory of characteristic odes [9], [1]. This theory is avaiabe for ore than 4 years, with significant aount of attention over the ast 1 years. Recenty the theory is ore popuar aong antenna designers and is now known to be a usefu design aid [11]. The ain idea is to express the surface current distribution of conducting bodies as a su of orthogona functions caed characteristic odes. These odes are independent of the excitation, i.e. they are defined by the shape and the size of the conductor. It is possibe to deterine odes nuericay for antennas of arbitrary shape. For eectricay sa conductors, few odes are sufficient to describe the antenna behaviour [12]. Hence, eectricay sa conductors are we suited for an appication of the theory of characteristic odes. The idea of MMAs is to excite different characteristic odes independenty. The current distribution for the particuar ode, found by the theory of characteristic odes, defines the ocations for excitation. In genera, excitation is possibe by inductive couping at the current axia, or by capacitive couping at the respective inia [13]. The coupers beonging

2 !3 / / 3 /!6 / 6 /!9 / 9 / -2dB -1dB -1dB -db db Port 1 Port 2 Port 3 Port 4 Fig. 2. MMA x-z pane power patterns for RHCP Fig Power [db] MMA power patterns for RHCP to one ode are then connected to one port of the antenna. Due to the different odes being orthogona to each other, MMAs are abe to provide sufficient isoation of the ports [14]. Copared to cassic antenna arrays MMAs are potentiay ore copact, which coud be an advantage for appications with stringent size and weight constraints. The theory of characteristic odes provides a usefu too to design chassis antennas. For exape, the case structure of a obie handset device [1] or the patfor of an unanned aeria vehice (UAV) can be used as antenna [16]. To the authors knowedge, MMAs have so far been investigated ony for counication appications, see e.g. [17]. The ai of this paper is to show how their potentia can be used for DoA estiation, enabing appications ike ocaisation and orientation estiation. The antenna that we anayse in this paper has been proposed in [14]. Figure 1 shows the power pattern of this four port MMA for right hand circuar poarisation (RHCP). The respective power pattern of the x-z pane is given in Figure 2. Obviousy the antenna gain strongy depends on the incident ange of the signa. Moreover, the antenna patterns differ significanty between the four ports. Therefore the received signa power coud be used to estiate the DoA of the signa. In the foowing, we assue that the poarisation of the received narrowband signa is purey RHCP. III. WAVEFIELD MODELLING The MMA is described in ters of spatia sapes of the antenna power pattern obtained by eectroagnetic fied (EMF) siuation or caibration easureents. Since that representation ight be reativey sparse, an interpoation strategy is needed, where we appy wavefied odeing and anifod separation [18]. In genera, the anifod of the MMA is defined by θ [ π, π) for 2D and θ [, π], φ [, 2π) for the 3D case, where θ is the co-eevation and φ the aziuth. If a function is square integrabe on this anifod, then it can be expanded in ters of an orthonora basis. Hence the power pattern of an MMA with M ports, g(θ, φ) R M 1, can be decoposed [18], such that g(θ, φ) = G Ψ(θ, φ). (1) The atrix G R M N is caed saping atrix and Ψ(θ, φ) R N 1 is the basis vector, with N being the order of the basis. For 2D, the Fourier functions, Ψ(θ) = 1 2π e jnθ, n = [ N,..., N], (2) can be used as a basis. Then G,N/2+1:N = G,1:N/2, because g(θ) ust be rea vaued. For 3D we use the rea spherica haronic functions, 2N cos(φ)p (cos(θ)) > Y (θ, φ) = N P (cos(θ)) = 2N sin( φ)p (cos(θ)) <, (3) with degree =,..., L and order =,...,. P (.) is the associated Legendre poynoia with degree and order. The noraization factor N is given by N ( )! = 4π ( + )!. (4) Using the enueration n = ( + 1) +, we can for an orthonora basis, Ψ(θ, φ) = Y n (θ, φ), n = [,..., N]. () It is known that the agnitude of G decays superexponentiay for n beyond n = kr, with k being the anguar wavenuber and r the radius of the saest sphere encosing

3 1 Port Port Magnitude [db] 1 Port Port Fig. 3. Magnitude of spherica haronic coefficients, indices and are degree and order of the spherica haronics in (3) the antenna [18]. Figure 3 shows the agnitude of G, i.e. the spherica haronic coefficients, for the MMA. As can be seen, ost of the energy is contained in the ow order coefficients. Hence the expansion can be safey truncated at a certain order. As (1) is inear, having enough spatia sapes of the antenna power pattern g(θ, φ) avaiabe fro caibration easureents or EMF siuation, it is straightforward to deterine the saping atrix G for a given basis Ψ(θ, φ). Due to the assuption of narrowband and pure poarisation, it is sufficient to consider a scaar wavefied. Nevertheess, an extension to vector fieds is possibe [19]. Having a suitabe ode of the antenna at hand, we wi now take a ook at power-based DoA estiation. IV. DIRECTION-OF-ARRIVAL ESTIMATION For sipicity, the derivations and siuations in Sections IV-A to IV-D focus on the 2D case, i.e. ony a singe ange θ is to be estiated. The extension to 3D and two anges of arriva foows in Section IV-E. A. Signa Mode The received, saped signa r (k) at port of the MMA is given by r (k) = a (θ)s(k) + n (k) (6) where k is the sape index, a (θ) is the attenuation caused by the antenna, s(k) is the transitted signa as it arrives at the receive antenna and n (k) CN (, σ 2 ) is white circuar syetric nora distributed noise with variance σ 2. Assuing stationarity, the tie-averaged received signa power over K sapes in tie can be cacuated by P r, = 1 r (k) 2 K = a (θ) 2 1 K s(k) K = g (θ)p s + 1 K n (k) 2, n (k) 2 with the antenna power pattern g (θ) and the signa power P s. We foow an RSSI based approach, hence ony power easureents are avaiabe to the receiver. The antenna power patterns, g(θ) = [g 1 (θ),..., g M (θ)], are noraized such that ax g(θ) = 1. The SNR for g (θ) = 1 is then given by SNR = P s σ 2. (8) Defining r,r (k) = Re[r (k)] and r,i (k) = I[r (k)], the su of the squared agnitude of the received signa, S r, = r (k) 2 = r,r(k)+r 2,i(k) 2 χ 2 (2K, λ, σ 2 /2) (9) foows a noncentra χ 2 distribution [2] with 2K degrees of freedo and noncentraity paraeter ( λ = E[r,r (k)] 2 + E[r,i (k)] 2) = = (a (θ)) 2 (Re[s(k)] 2 + I[s(k)] 2 ) g (θ) s(k) 2 = Kg (θ)p s. (7) (1) Its probabiity density function (PDF) is given by ( p Sr, (x) = 1 ( x ) K 1 2 σ 2 e λ+x 2 ) λx σ λ 2 I K 1 σ 2, (11) with the odified Besse function of the first kind I ν (.). Since P r, is just a scaed version of that, its distribution can be obtained by transforation p Pr, (x) = Kp Sr, (Kx). Inserting (1), we obtain p Pr, (x) = K σ 2 ( x g (θ)p s ) K 1 2 The ean and variance are given by e K(g(θ)Ps+x) σ 2 ( I K 1 µ = E[P r, ] = K 1 E[S r, ] 2K g (θ)p s x σ 2 = K 1 (Kσ 2 + λ) = g (θ)p s + σ 2, σ 2 = VAR[P r, ] = K 2 VAR[S r, ] = K 2 (Kσ 4 + 2σ 2 λ) = K 1 (σ 4 + 2g (θ)p s σ 2 ). ). (12) (13) (14)

4 For arge λ or arge K, (12) is approxiatey Gaussian distributed P r, N ( µ, σ 2 ). B. ML Estiator The paraeters to be estiated are given by Γ = [θ, P s, σ 2 ]. (1) p MSE(3) CRB ^! ML We consider the case of unknown signa power P s and noise variance σ 2. Using the Gaussian approxiation, i.e. K is arge, the og-ikeihood function for the signa power easureents P r = [P r,1,..., P r,m ] can be written as n p(p r ; Γ) = 1 2 n(2π σ2 ) =1 1 2 σ 2 ( Pr, (g (θ)p s + σ 2 ) ) 2. (16) The corresponding axiu ikeihood (ML) estiator can then be derived as ˆΓ ML = arg ax n p(p r; Γ) Γ = arg in n ( 2πK 1 (σ 4 + 2g (θ)p s σ 2 ) ) Γ =1 + K(P r, P s g (θ) σ 2 ) 2 σ 4 + 2P s g (θ)σ 2. A sipified version of the estiator, ˆΓ S = arg in Γ =1 (17) (P r, P s g (θ) σ 2 ) 2 P s g (θ) + 2σ 2, (18) can be obtained by negecting the ogarithic ter. C. CRB Derivation A ower bound on the variance of any unbiased estiator is the Craér-Rao bound (CRB) [21]. For a given set of unknowns Γ = [θ, P s, σ 2 ], it is defined as the inverse of the Fisher inforation atrix (FIM) I(Γ), VAR[ˆΓ] I(Γ) 1. (19) Foowing the Gaussian assuption in (16), the eeents of the FIM I(Γ) R 3 3 can be cacuated as [21] [I(Γ)] i,j = 1 σ 2 =1 µ µ + 1 σ 2 σ 2 Γ i Γ j 2 σ 4. (2) Γ i Γ j Cacuation of the partia derivatives of (13) and (14) requires the derivative of (1) and (2), which is given by g(θ) = G jn 2π e jnθ, n = [ N,..., N]. (21) Finay we obtain the CRB for the estiation of θ, VAR[θ] [I(Γ) 1 ] 1,1 = CRB(θ). (22) p MSE(3) Fig. 4. 1! RMSE of the siuation and CRB for SNR = 2 db CRB ^! ML 9 / FOV ^! ML 18 / FOV ^! S 9 / FOV ^! S 18 / FOV 1! SNR [db] Fig.. Siuated RMSE of ML and sipified estiator and CRB vs. SNR for fu (18 ) and iited (9 ) FOV D. Siuation Resuts In order to evauate the feasibiity of the proposed DoA estiation approach, we have used EMF siuation data of the MMA prototype, visuaized in Figure 2. The siuated root-ean-square error (RMSE) and the CRB depending on θ are shown in Figure 4. The pot indicates that the achievabe accuracy depends on the incident ange, with a RMSE spread of ore than one order of agnitude over the anifod. Nevertheess, the ML estiator is abe to attain the CRB for high signa-to-noise ratio (SNR). In Figure the ean over the anifod is potted in dependence of the SNR present at the receiver. We have seen earier, in Figure 2, that the antenna patterns are reativey syetric around. For that reason, we copare the perforance with a iited fied of view (FOV) of 9 to the fu FOV of 18. It can be seen that for SNR 18 db, the ML estiator ˆΓ ML defined in (17) is efficient, i.e. it attains the CRB. For ower SNR, the RMSE is significanty bigger for 18 due to abiguities caused by the antenna pattern. The CRB is cacuated independent of the a-priori inforation regarding FOV iitation, hence it is not an accurate ower bound in the ow SNR region. Finay the pot indicates that the sipified estiator ˆΓ S given by (18) is sufficienty accurate. Ony for high SNR, a sight increase in RMSE copared to the ML

5 Fig. 6. CRB(θ) (eft) and CRB(φ) (right) for SNR = 2 db Fig. 8. CRB(θ) (eft) and CRB(φ) (right) for SNR = 3 db Fig. 7. MSE(θ) (eft) and MSE(φ) (right) for SNR = 2 db Fig. 9. MSE(θ) (eft) and MSE(φ) (right) for SNR = 3 db estiator is visibe. E. Extension to 3D Having studied the perforance in 2D, we now extend the proposed DoA estiation schee to the ore practica case of 3D, i.e. two unknown anges of arriva. The extension of the signa ode to 3D, r (k) = a (θ, φ)s(k) + n (k), (23) is straight forward. One ore paraeter, φ, has to be estiated, so we have Γ = [θ, φ, P s, σ 2 ]. The FIM then grows to I(Γ) R 4 4. Appying wavefied odeing with spherica haronics as described in Section III, the partia derivatives of the antenna power pattern are g(θ, φ) = G Y n(θ, φ), (24a) g(θ, φ) = G φ φ Y n(θ, φ), (24b) using the enueration n = ( + 1) +. The partia derivative of (3) with respect to θ (for θ ) is given by 2N cos(φ) P (cos(θ)) > Y (θ, φ) = N P (cos(θ)) = P 2N (cos(θ)) sin( φ) <. (2) The derivative of the Legendre poynoia P (cos(θ)), P (cos(θ)) = 1 + P sin(θ) +1(cos(θ)) + 1 tan(θ) P (cos(θ)), (26) can be cacuated with the hep of [22]. The corresponding partia derivative of (3) with respect to φ is given by 2N ( ) sin(φ)p φ Y (cos(θ)) > (θ, φ) = = 2N ( ) cos(φ)p (cos(θ)) <. (27) Finay we obtain the CRBs for the estiation of θ and φ in the 3D case, VAR[ˆθ] [I(Γ) 1 ] 1,1 = CRB(θ), VAR[ ˆφ] [I(Γ) 1 ] 2,2 = CRB(φ). (28a) (28b) In order to confir the expected DoA estiation perforance, we perfored siuations for the 3D case. Figure 6 shows the CRB in θ- and φ-doain for SNR = 2 db. The corresponding siuation resut can be seen in Figure 7. Apparenty, the CRB is not attained on the whoe anifod. Especiay for ow eevations, an excessive estiation error can be observed. Taking another ook at the antenna power pattern in Figure 1, it is obvious that at ow eevations the antenna gain is very sa. This eads to a degradation of the DoA estiation perforance. For id and high eevations, the CRB is usuay attained, except for a few directions which appear as dark dots in Figure 7. This is ost ikey caused by estiation abiguities due to the syetry of the antenna pattern. In practice a-priori inforation, i.e. a rough knowedge of the direction, is often avaiabe, which ay hep to sove the abiguity. Next we take a ook at Figures 8 and 9 showing the CRB and siuation RMSE at SNR = 3 db. To aow coparison, Figures 6 to 9 use the sae scaing. It can be seen that by increasing the SNR by 1 db, both perturbing effects, i.e. ow gain at ow eevation anges and estiation abiguities, are strongy reduced. Ony for ow eevations, an increased error can sti be observed. The higher RMSE at high eevation for the φ-doain is ess probeatic, since at the poe this transates to a saer directiona error.

6 V. CONCLUSION A suitabe ode for MMAs, buiding on the concept of wavefied odeing, was introduced. Based on that ode, a power-based axiu ikeihood DoA estiation schee has been introduced. Siuations have shown that in genera, power-based DoA estiation with MMAs is feasibe. However, for certain incident anges, abiguities in the antenna pattern cause an increased estiation variance. For the investigated MMA prototype, ow eevations are probeatic due to ow antenna gain. Atogether it can be concuded that power-based DoA estiation is possibe with MMAs, but for accurate estiates a reativey high SNR is required. Further work wi be perfored by using signa poarisations as we as investigations into coherent receivers, that are abe to obtain phase inforation fro the different ports of the MMA. ACKNOWLEDGMENT This work has been funded by the Geran Research Foundation (DFG) under contract no. HO 2226/17-1. The authors are gratefu for the constructive cooperation with Sai Akubti Aasri, Nikas Doose and Prof. Peter A. Hoeher fro the University of Kie. Aso, the authors woud ike to thank Prof. Dirk Manteuffe and his tea for providing the antenna pattern of the MMA prototype investigated in this paper. REFERENCES [1] T. E. Tuncer and B. Friedander, Cassica and Modern Direction-of- Arriva Estiation. Acadeic Press, Ju. 29. [2] J. Ash and L. Potter, Sensor network ocaization via received signa strength easureents with directiona antennas, in Proceedings of the 24 Aerton Conference on Counication, Contro, and Coputing, 24, pp [3] M. Maajner, P. Paninsic, and D. Geich, Ange of Arriva Estiation Using RSSI and Onidirectiona Rotatabe Antennas, IEEE Sensors Journa, vo. 12, no. 6, pp , Jun [4] B. N. Hood and P. Barooah, Estiating DoA Fro Radio-Frequency RSSI Measureents Using an Actuated Refector, IEEE Sensors Journa, vo. 11, no. 2, pp , Feb [] J. T. Isaacs, F. Quitin, L. R. G. Carrio, U. 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