Single-Anchor Localization and Orientation Performance Limits using Massive Arrays: MIMO vs. Beamforming

Single-Anchor Localization and Orientation Performance Limits using Massive Arrays:
Mimo vs. Beamforming

Anna Guerra,  Francesco Guidi,   Davide Dardari 
Anna Guerra and Davide Dardari are with the Dipartimento di Ingegneria dell’Energia Elettrica e dell’Informazione “Guglielmo Marconi” - DEI, University of Bologna, via Venezia 52, 47521 Cesena, ITALY. (e-mail:, Francesco Guidi is with CEA, LETI, MINATEC Campus, 38054 Grenoble, France. (e-mail: research was supported in part by the XCycle project (Grant 635975) and by the IF-EF Marie-Curie project MAPS (Grant 659067). Authors would like to thank Nicoló Decarli, Raffaele D’Errico and Henk Wymeersch for fruitful discussions.

Next generation cellular networks will experience the combination of femtocells, millimeter-wave (mm-wave) communications and massive antenna arrays. Thanks to the beamforming capability as well as the high angular resolution provided by massive arrays, only one single access point (AP) acting as an anchor node could be used for localization estimation, thus avoiding over-sized infrastructures dedicated to positioning. In this context, our paper aims at investigating the localization and orientation performance limits employing massive arrays both at the AP and mobile side. Thus, we first asymptotically demonstrate the tightness of the Cramér-Rao bound (CRB) in massive array regime, and in the presence or not of multipath. Successively, we propose a comparison between MIMO and beamforming in terms of array structure, time synchronization error and multipath components. Among different array configurations, we consider also random weighting as a trade-off between the high diversity gain of MIMO and the high directivity guaranteed by phased arrays. By evaluating the CRB for the different array configurations, results show the interplay between diversity and beamforming gain as well as the benefits achievable by varying the number of array elements in terms of localization accuracy.

Position Error Bound, Orientation Error Bound, Millimeter-wave, Massive array, 3D localization, 5G.


I Introduction

The widespread use of personal devices generates new challenges while opening new appealing scenarios for future applications, such as, for example, those entailing device-to-device (D2D) interactions or Big Data management issues. To meet these new trends, different disruptive technologies have been recently proposed for the next fifth generation (5G) wireless communications networks [chin2014emerging, khaitan2011indoor]. In particular, large-scale antenna arrays at base stations or femtocells APs allow to smartly direct the power flux towards intended users thus increasing data rates, whereas mm-wave communication provides a less crowded and larger spectrum [larsson2014massive, rusek2013scaling, swindlehurst2014millimeter].

In next years, it is expected that personal devices localization and communication capabilities will play a crucial role [di2014location]: in fact, the possibility of localizing nodes in indoor environments will be an essential feature of future devices. In this context, the AP could be used as a single-anchor node, i.e., a node whose position is a-priori known, in a radio-localization perspective permitting the mobile users to be aware of their own position. Furthermore, the adoption of more than one antenna at the transmitter (Tx) and receiver (Rx), will enable the user orientation estimation at an accuracy higher than that provided by compass and gyroscopes. Such feature could play a key role in applications beyond 5G as for example augmented reality and simultaneous localization and mapping (SLAM), where trajectory errors, comprising both position and orientation estimation inaccuracies, dramatically affect the performance [guidi2016personal]. Contrarily to traditional scenarios where dedicated multiple anchor nodes are necessary to allow classic triangulation/multilateration techniques [dardari2015indoor], here the possibility to centralize both communication and localization capabilities in a single multi-antenna AP working at mm-wave frequencies is envisioned with the advantage of drastically decreasing the overall system complexity and cost. Moreover, when operating at such high frequencies, not only APs but also user terminals could adopt massive arrays thanks to the reduced wavelength [hong2014study], thus increasing even more the localization accuracy given the potential huge set of measurements [razavizadeh2014three, witrisal2016high, guerra2015position, garcia2016direct].

While at microwave frequencies the antenna array technology is quite mature, at mm-wave severe technological constraints are still present and must be taken into account when designing positioning systems. Recently, massive antennas prototypes have been proposed with electronic beamsteering capabilities. In order to reduce the complexity, they adopt simple switches and thus, the resulting non-perfect signals phasing operations could impact the array radiation characteristics [kaouach2011wideband, clemente2013wideband, GuiEtAl:J17]. In such a scenario, it becomes of great interest to understand the fundamental limits on localization error with massive antenna arrays both at AP and mobile terminals using only a single reference node.

Concerning the ultimate localization performance evaluation, a rich literature has been produced for the analysis of wideband multiple anchors systems. Specifically, in [shen2010accuracy, shen2010fundamental] authors explore the localization accuracy for a wideband sensors network composed of several independent anchors. Their results are further discussed in [han2016performance] where a more realistic Rx architecture able to exploit the carrier phases information has been taken in consideration while deriving the localization performance. Differently from these works, where anchors send orthogonal waveforms, we consider a signal model dependent on the particular arrays architecture chosen where both orthogonal and non-orthogonal waveforms can be transmitted. Moreover, our work is not focused on a specific Rx structure, as in [han2016performance], but it aims to compare different Tx array architectures. In [shahmansoori20155g, mallat2009crbs], a joint delay-angle estimation is reported considering different array technologies and frequency bandwidths. Nevertheless, these works analyze the performance in terms of CRB on delay and angular information rather than directly on localization, and neither a comparison between different array schemes, nor the time synchronization issue and the impact of multipath are treated. In our previous work [guerra2015position, guerra2016position_c], some preliminary results on positioning accuracy considering only beamforming strategies have been presented, but the comparison with multiple-input multiple-output (MIMO), as well as the impact of multipath components, was not considered.

Stimulated by this framework, in this paper we conduct a CRB-based analysis of a localization system exploiting the next 5G technologies potentialities. Differently from the state-of-the-art, we adopt a 1-step approach in which the Tx position and orientation are directly inferred from the received signals and, thus, without the need of estimating intermediate parameters (e.g., time-of-arrival (TOA)- direction-of-arrival (DOA)) or applying geometrical methods which do not ensure the optimality of the approach [cover2012elements_c].

The main contributions of this work can be summarized as follows:

  • Derivation of the theoretical performance limits on the localization and orientation error for different array configurations in a single-anchor scenario;

  • Proposal of a signal model valid for any antenna array geometry, configuration (i.e., MIMO, phased, timed arrays), and frequency band. As a case study, in the numerical results the focus is on the adoption of mm-wave massive arrays due to their expected attractiveness in next 5G applications;

  • Introduction of low-complexity random weighting approach, i.e., randomly chosen beamforming weights, and analysis of its performance compared to that of classical beamforming and MIMO solutions;

  • Investigation of the CRB tightness in massive array regime (i.e., letting the number of antennas ) for any signal-to-noise ratio (SNR) condition;

  • Analysis of the trade-off between SNR enhancement obtained via beamforming and diversity gain of MIMO considering the impact of different types of uncertainties, as, for example, the MPCs, beamforming weights and time synchronization errors;

  • Demonstration that in the massive array regime (i.e., array antennas ), the effect of multipath can be made negligible on average.

The rest of the paper is organized as follows. Sec. II describes the geometry of the localization system. Then, Sec. III introduces the signal model taking into account different array structures. In Sec. LABEL:sec:posbound the localization performance limits derivation is reported. Sec. LABEL:sec:crbtight analyzes the asymptotic conditions for which the CRB can be considered a tight bound. Sec. LABEL:sec:idealscenario derived compact formulas for a ideal free-space case. The multipath impact on localization performance are investigated in Sec. LABEL:sec:mp_loc_ac. Finally, Sec. LABEL:sec:numerical presents the localization performance results and Sec. LABEL:sec:conclusions concludes the work.


Lower case and capital letters in bold denote vectors and matrices, respectively. The subscripts , and indicate the transpose, the conjugate and the Hermitian operators. is the Euclidean norm, indicates that the matrix is non-negative definite, and represents the diagonal operator. The subscripts and refer to quantities related to the transmitting and receiving array, respectively, while the subscript to elements that can be referred to both the Tx and the Rx. indicates the free-space scenario. denotes the Fourier transform operation, a uniform distribution in the interval , and a circularly symmetric Gaussian distribution with mean and variance .

The notations of frequently-used symbols are listed as follows.


Number of Tx-Rx array antennas

Number of MPCs

Area of the Tx-Rx array

Tx centroid position and orientation

Rx centroid position and orientation

Distance between Tx-Rx centroids

Ratio between the Rx array area and the squared inter-array distance

Tx/Rx antenna position

Inter-antenna spacing

Direction cosine

Multipath parameters vector

Direct path wave direction

th path wave direction

Steering direction

Propagation delay relative to the direct and th path between the th Tx-th Rx antenna

Propagation delay relative to the direct and th path between centroids

Direct path amplitude and th complex channel coefficient

Inter-antenna delay between the th Tx antenna and the relative array centroid

Inter-antenna delay between the th Rx antenna and the relative array centroid

Transmitted signal carrier frequency, bandwidth, baseband effective bandwidth

Observation interval

Total and normalized energy at each antenna element

Single-side noise power spectral density

Receiver noise figure

SNR relative to the direct path

Equivalent low-pass signal at the th Tx antenna and transmitted signals vector in the frequency domain

Equivalent low-pass unitary-energy signal at the th Tx antenna in the frequency domain

Received signal at the th Rx antenna and received signals vector in the frequency domain

Useful Rx signal at the th Rx antenna and useful Rx signals vector in the frequency domain

Noise component at the th Rx antenna and noise vector in the frequency domain

Beamforming weight and matrix

Beamforming phase, TDL and random weight

Beamforming phase and TDL errors

Beamforming weight and matrix with errors

Time synchronization error

Estimation parameter vector

Bayesian FIM, FIM relative to data, a-priori FIM

CRB on position and orientation

Single-antenna CRB on ranging error

Ii Antenna Array Geometric Configuration

Ii-a Geometric Relationships

We consider a 3D localization scenario, as the one reported in Fig. 1, consisting of a single AP acting as reference receiving node equipped with an antenna array, with antennas, and a transmitting mobile terminal with a -antenna array. The localization process aims at directly inferring:111 As previously stated, we consider the Tx position and orientation with respect to the relative centroid (see (1)-(3) in the following) as we adopt a 1-step approach in which the Tx position and orientation are directly inferred from the received signals. Thus, we do not estimate neither the DOA (i.e., angle between arrays centroids) nor the direct path TOA.

  • the position of the Tx centroid ;

  • the orientation of the Tx

when the Rx centroid position and orientation are known.222Without loss of generality, the Rx is assumed located at the origin of the coordinates system. With reference to Fig. 1, indicates the position of the th transmitting antenna relative to the Tx geometric center and dependent on the Tx orientation, and the position of the th receiving antenna relative to the Rx geometric center. Considering spherical coordinates, we have


with the direction cosine is expressed as


and and being the distance and the couple of angles between the considered array antenna from the correspondent array centroid.333Note that the elevation angle in all the text is indicated with and it can assume values in the interval . Contrarily the azimuthal angle is denoted with and it ranges between .

Fig. 1: Multi-antenna system configuration.

The rotational matrix is given by


where and define the counter-clock wise rotation around the -axis and the clock wise rotation around the -axis, respectively. Finally designates the angle of incidence between arrays centroids (direct path) and represents the intended pointing direction of the steering process when applied.

The diameter of the transmitting and receiving arrays is assumed much smaller than the inter-array distance , i.e., . Note that this hypothesis is especially verified at mm-wave where the array dimensions are very small thanks to the reduced wavelength. Moreover the arrays are supposed to be sufficiently far from the surrounding scatterers thus obtaining identical angles of incidence for both direct and MPCs at each antenna element.

We take MPCs into consideration as nuisance parameters in the localization process and the first path is assumed always experiencing a line-of-sight (LOS) propagation condition. For what the MPCs parameters are concerned, we follow the same notation introduced in [han2016performance]. In particular, let and , with , indicate the angles of departure from the transmitting array and of incidence at the Rx side of the th path, respectively. The angular biases and are the displacement with respect to the direct path at the Tx and Rx side. Obviously, it is , when direct path is considered.

Let and being the propagation delay related to the direct path between the transmitting and receiving centroids and between the th and th antenna, respectively, where is the speed of light. Considering the multipath, the th propagation delay between array centroids is defined as where is the non-negative delay bias of the th path with [han2016performance]. According to the geometric assumption previously described, the TOA and amplitude between each couple of transmitting and receiving antennas can be expressed using the following approximations [mallat2009crbs, han2016performance]


where is the amplitude of the th path between the th receiving and the th transmitting antenna, and and are, respectively, the receiving and transmitting inter-antenna propagation delays defined as


Fig. 2: Array geometric configuration.

Fig. 2 reports a graphical explanation of the considered system and delays. As it can be seen, the approximation in (4) permits to write the TOA as the summation of the inter-antenna delays and the delay between the array centroids.

Fig. 3: From the left to the right: Phased, timed, MIMO and random weighting array schemes.

Ii-B Special Case: Planar Array Geometry

Planar array configurations appear to be the most suitable when considering the integration of massive arrays in portable devices or in small spaces. For this reason, in addition to the general analysis valid for any geometric configuration (i.e., any antennas spatial deployment and arrays orientation), some compact specialized equations will be derived in the following sections for squared arrays of area (), with the antennas equally spaced of . Both arrays are considered lying on the -plane and being located one in front of the other with and with , so that and thus . In this case, the antenna coordinates in (1) becomes


where are the antenna indexes along the and axis respectively, and indicates the number of antennas.

We assume for now a free-space propagation condition so that and . Consequently it is possible to specialize (5) as


Note that, in the special case in which the Rx and Tx orientation is , the inter-antenna delays are zeros, i.e. , as the antennas are aligned to the array centroids, thus the incident wave impinges simultaneously at all the antennas.

Iii Antenna Array Schemes and Signal Model

In this section, different types of antenna array schemes are analyzed starting from a unified signal model with the purpose to highlight their beamforming and diversity gain properties. Specifically, the four array structures reported in Fig. 3 will be analysed from a signal processing point-of-view and by focusing on how the different signaling schemes translate into different localization capabilities. Table III reports a comparison in terms of arrays complexity, capabilities and cost.

Signal Design Array Implementation Complexity Cost Beamforming capabilities Timed Low: same signal for all antenna branches High: TDLs needed when High Yes Phased Low: same signal for all antenna branches Medium: only PSs Medium Yes MIMO High: one different signal for all antenna branches High: a RF chain for each branch High No Random Low: same signal for all antenna branches Low: only PSs Low No
TABLE I: Array schemes comparison.
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