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Joint Direction-of-Arrival and Time-of-Arrival Estimation with Ultra-wideband Elliptical Arrays

This paper presents a general technique for the joint Direction-of-Arrival (DoA) and Time-of-Arrival (ToA) estimation in multipath environments. The proposed ultra-wideband technique is based on phase-mode expansions and the use of nearly frequency-invariant elliptical arrays. New possibilities open with the present approach, as not only elliptical, but also circular and linear arrays can be considered with the same implementation. Systematic selection/rejection of signals-of-interest/signals-not-of-interest in smart wireless environments is possible, unlike with previous approaches based on circular arrays. Concentric elliptical arrays of many sizes and eccentricities can be jointly considered, with the subsequent improvement that entails in DoA and ToA detection. This leads to the realization of pseudo-random array patterns; namely, quasi-arbitrary geometries created from the superposition of multiple elliptical arrays. Some simulation and experimental tests (measurements in an anechoic chamber) are carried out for several frequency bands to check the correct performance of the method. The method is proven to give accurate estimations in all tested scenarios, and to be robust against noise, channel nonidealities, position uncertainty in sensor placement and interferences.


I. INTRODUCTION
D IRECTION-of-arrival (DoA) estimation has been one of the main lines of research in the field of wireless communications in the last decades [1]- [5]. Knowing the position (and time) at which a wave arrives is essential in mobile networks [6], [7], vehicular networks [8], [9], tracking and navigation systems such as GPS [10]- [12], radar [13], [14], sonar [15], and many other wireless systems. Even more exotic applications such as failure detection in electronic components are starting to use DoA techniques as a novel alternative [16], [17]. In fact, new communication environments are materializing these days; namely, vehicleto-everything (V2X), ship-to-ship (S2S), high speed train-totrain or UAV-to-UAV [18]- [21]. Thus, a great deal of effort is being put into characterizing their most important parameters and key performance indicators, direction of arrival (DoA) and time of arrival (ToA) among them, with the aim of improving bandwidth, latency, data rate, power consumption and reliability in present and future communication systems [22]- [28].
Conventional DoA and ToA estimation methods, i.e., delayand-sum beamforming [29] or the more classical implementations of MUSIC [1], ESPRIT [30] and maximum-likehood [31] algorithms, were originally developed to work under a narrowband assumption. Naturally, their use is not adequate for today's broadband communication channels such as the high-frequency millimeter-wave links (26,38, and 60 GHz) dedicated to 5G wireless networks. Thus, many broadband DoA methods have emerged in recent years to overcome the limitations of narrowband approaches [32]. Among the well-established broadband methods, we can find narrowband decomposition (a wideband channel is decomposed into small bins that are treated independently with narrowband techniques) [33] or the use of tapped-delay filters with adaptive coefficients [34]. Although they are simple solutions, both demand high computational resources since the number of required filters increases as the considered bandwidth does.
Alternatively, frequency-independent beamformers (FIB) [35], [36] were formulated to reduce the number of required filters in a fixed bandwidth, and thus the computational demands. Originally, FIB focused on the use of linear arrays. Then, FIB-based algorithms were smartly combined with the omnidirectional characteristics that uniform circular arrays offer to develop efficient methods for DoA and ToA estimation [37]. The outcomes of this original work were promising, as the spatial response of the filter was equalized with a remarkably reduced number of weight coefficients. Nonetheless, some important aspects of the method were later clarified in [38]. Subsequently, [37] was rapidly extended in different ways. For instance, in [39], it was demonstrated that the bandwidth of FIB circular arrays can be broadened and the precision on the DoA estimation improved by employing multiple concentric arrays instead of a single one. In [40], [41], multipath components and spherical 3-D propagation were also considered. Moreover, the approach was directly applied to the millimeter-wave frequency range.
In this document, we propose a generalization of the method successfully developed in [37]- [41] for the joint estimation of the direction-of-arrival and time-of-arrival in multipath environments. Those previous works were based on the use of wideband circular arrays. Here, we demonstrate that wideband elliptical arrays can extent the capabilities of the joint estimation in many forms: (i) Circular and linear geometries are subcases contained by more general elliptical shapes. Thus, we present one of the few methods reported in the literature that is able to deal with linear, circular and elliptical arrays in a single stroke. (ii) Elliptical arrays present new degrees of freedom compared to circular arrays. Particularly, the ellipse radius is not fixed, which provides many configurations for several arrangements based on different eccentricities and rotation angles. (iii) Highly flattened elliptical arrays (and so, linear arrays) show an excellent directivity along the direction of its semi-major axis, as well as remarkable estimation performance for many elevation angles in this direction. Thus, the present methodology allows for the efficient and systematic selection of signals-of-interest (SOI) and rejection of signals-not-of-interest (SNOI) in smart radio environments, unlike conventional FIB-DoA methods implemented with circular arrays. (iv) Concentric elliptical arrays (CEAs) can be analyzed with the present formulation. As reported by previous works, the use of concentric arrays is expected to significantly improve the accuracy in the joint estimation as well as the bandwidth [39]. Furthermore, the use of concentric elliptical arrays can lead to the implementation of pseudorandom grids, fact that will be exploited in Section III.D for DoA and ToA estimation. These pseudorandom grids have the advantage of not being limited to a certain number of sensor arrangements. (v) The method is proven to be robust to noise and interferences in DoA and ToA detection. In addition, its computational demands are comparable to state-of-art approaches.
The document is organized as follows. Section II presents the theoretical framework for the joint DoA and ToA estimation with ultra-wideband elliptical arrays. Furthermore, we discuss on the advantages and limitations of the method. Section III presents some numerical simulation results in order to validate the present approach. Section IV presents some experimental results extracted from the measurement facilities at the University of Granada. Finally, conclusions are drawn in Section V.

II. THEORETICAL FRAMEWORK
Let us consider the situation illustrated in Fig 1. An incident spherical wave l, described by the unknown azimuth and elevation angles φ l and θ l , respectively, impinges on the P sensors (p = 0, 1, ..., P − 1) that conform the elliptical array. The frequency response at the center of the elliptical array is given by where κ l and τ l are the attenuation and propagation delay for path/wave l. Both parameters are to be determined. The frequency response at the p-th sensor, H p,l (f ), will include a phase shift with respect to the frequency response estimated at the center of the array, H l (f ). By applying trigonometric relations, we arrive to where the term d /d p, accounts for free-space path loss and with In the former expression, d l is the distance from the center of the elliptical array to the source point, r p is the distance from the center of the array to sensor p, φ p is the azimuth angle associated to sensor p (see Fig. 1), both given by Additionally, the Taylor series for ∆d p,l allows to expand this term as [41] ∆d p,l ≈ r p sin(θ l ) cos(φ l − φ p ) .
Thus, (2) can be expressed as Considering that the source l is located far from the elliptical array, the term d l /d p,l ≈ 1, so (8) can be simplified to Now, we particularize the study to the case θ l = 90 o ; namely, where the incident plane coincides with the plane where the elliptical array is located. Thus, (9) reduces to This approximation is fundamental in order to develop further steps of the method. Although it may seem prohibitive, this approach was demonstrated to work over a wide range of elevation angles different from θ l = 90 o [41].
The Jacobi-Anger identity [42] allows us to expand the former expression and decouple the frequency-dependent and phase-dependent components in two different terms as where J n,p (·) is the Bessel function of the first kind of order n associated to sensor p. At a first glance, some differences can be observed with respect to the cases shown in [38], [39], [41] for circular arrays. When considering elliptical arrays, the frequency-dependent component [J n,p (·)] varies for each sensor p. This is due to the fact that the distance between the center of the array and the position of sensor p is not constant [r p = r p (φ p ) = r], unlike in circular arrays where this distance is the same for all sensors (r p = r).
The azimuth angle φ l can be estimated by applying a phase-mode expansion, H m,l (f ), [38], [39], [41] to the former expression that includes basis functions of the form e jmφp (m being the integer order of the phase-mode) and a frequency-dependent filter W m,p (f ), leading to The infinite sum in n can be split in two different addends: the term n = m and a series for all n = m. This allows to rewrite (12) as Previous works [38], [39], [41] focused on circular arrays took advantage of the independence of the Bessel functions with respect to p [J n,p (·) = J n (·)] and the orthogonality relations of the sum P −1 p=0 e j(m−n)φp /P to cancel out the infinite series (∀n = m) in eq. (13). However, this is not exactly the situation for elliptical arrays. Thus, some approximations have to be considered in order to proceed. Under the assumption of low eccentricity e levels (ellipses that are not too flattened, e 1), the former discussion still applies since r p ≈ r (constant radius in average) and the sensors in the elliptical array are quasi-uniformly distributed (φ p ≈ 2πp/P ). As a consequence, J n,p (2πf r p /c) ≈ J n (2πf r/c) can be cleared out from the sum in p and the orthogonality relation P −1 p=0 e j(m−n)φp /P = 0 , ∀n = m, remains valid. Thus, the last double sum in eq. (13) is approximately zero under loweccentricity assumption; namely, Thus, eq. (13) is simplified to The approximation taken above, although originally derived for low eccentricities, will be demonstrated in Sections III and IV to be applicable to a wide range of elliptical arrays, even those that reach eccentricity values of e = 0.99, as long as the number of considered sensors is large. In practice, P should be selected so the sensor separation is less than λ/2, where λ stands for the wavelength. Then, we can reduce (13) to if we consider the filter W m,p (f ) to be Alternatively, we can choose W m,p (f ) as since it was proven in [41] to give more accurate results in realistic scenarios where elevation angles different from θ l = 90 o are involved. Additionally, eq. (17) involves deep nulls for some frequencies f , fact that limits the channel bandwidth. The situation is different for the filter depicted in eq. (18), whose performance is similar to that in eq. (17) but operates over a noticeably larger bandwidth. A detailed discussion on the filter choice can be found in [40]. In eq. (18), J m,p (·) is the first derivative of the Bessel function of order n related to sensor p. The array response for the m-th phase mode is calculated by taking into account the contribution of the L incident waves as which admits a matrix-form representation H(m, f ). Finally, the joint DoA and ToA estimation is extracted by computing the 2-D fast Fourier transform (FFT) of H(m, f ), which includes a general representation of the wireless channel properties for all considered angular and time steps. Note that DoA and ToA resolution of H(φ, τ ) is given by the number of phase modes and frequency samples in H(m, f ). Concretely, DoA resolution is calculated as 360°/M , where M is the number of considered phase modes. Thus, higher DoA resolution requires of a larger number of phase modes. ToA resolution is calculated as 1/B, where B is the bandwidth of the channel. The maximum observable time in the estimation is (K − 1)/B, which is determined by B and the number of frequency samples K. The maximum number of considered phase modes M [eq. (15)] is restricted by the denominator in W m,p (f ) [eq. (18)]. This denominator may introduce numerical errors as it approaches zero. In order to avoid numerical instabilities, M must be chosen such that To fulfill the previous condition, we know that there is a number of phase modes, namely |M lim |, above which the Bessel function J m,p (·) and its derivative tend to zero [38]. Therefore, stability can be guaranteed if M is chosen below |M lim | [40]. This limit value is dependent on the argument of J m,p (·) and J m,p (·). The larger the argument (2πf r p /c), the larger |M lim |. Consequently, i) high frequencies provide higher resolution in DoA estimation, and ii) |M lim | is defined by the sensor p whose distance r p is smaller for a given frequency f . This distance turns out to be the semi-minor axis in elliptical arrays.

A. Improving the Efficiency of the Method
It can be noted from eqs. (17), (18) that each sensor p that conforms the elliptical array has a different filter W m,p (f ). Considering M phase modes in the computation, elliptical arrays will require of M × P filters for the DoA and ToA estimation. This situation is different in circular arrays, where all the sensors share the same filter for the m-th phase mode. Thus, only M × 1 filters are needed in the case of considering circular arrays. This causes that the joint estimation with ultrawideband elliptical arrays is less computationally efficient than with ultra-wideband circular arrays, despite the multiple benefits that elliptical arrays offer (generality, selectivity, pseudorandom grids, etc.) compared to circular ones. Nonetheless, we can exploit the symmetries of elliptical geometries and Bessel functions to improve the efficiency of the proposed method.
Let us name by α p an azimuth angle that is contained in the first quadrant of the elliptical array. Angles of the form 180 o − α p (second quadrant) , 180 o + α p (third quadrant), and 360 o − α p (fourth quadrant) have associated the same value of r p . Therefore, the number of required filters in the joint estimation can be reduced to M × [P/4 + 1] by exploiting the symmetries of the ellipse. Thus, when P is large, the required number of filters is approximately reduced by a factor of 4 compared to the raw processing.
We can further reduce the complexity of the problem by considering the symmetry of Bessel functions. Concretely, we can take advantage of the following expression that relates negative and positive integer orders m. Thus, the number of filters would reduce to [M/2 + 1] × [P/4 + 1]. For large values of M and P , the number of required filters is asymptotically reduced from M × P to M/2 × P/4, that is, by a factor of 8. Additionally, the use of a single average filter W m (f ) can provide good DoA and ToA estimations when low-and medium-eccentricity elliptical arrays are involved. The average filter is given by replacing all r p values in eqs. (17)-(18) by r, computed as the average between the semi-major and semi-minor axes of the ellipse. Note that in the case of considering a single average filter, the number of required filters is reduced to M × 1, as in the case of circular arrays. Therefore, a good approximation for elliptical arrays with low and medium eccentricities can be obtained, notably reducing the computational complexity of the problem. The singleaverage-filter approach is expected to give accurate results as long as the elliptical array is not highly flattened. In practice, this approach has been found to be valid up to e 0.7. Finally, note that the time required in the total propagation channel characterization process is limited by the measurement acquisition. The measurement process takes orders of magnitude longer than the time required to apply the estimation method. Therefore, the global time for taking measurements and applying the method ends up being similar in both, circular and elliptical arrays.

B. Rotated Elliptical Arrays
A simple modification can be applied to the present formulation for the convenient use of rotated elliptical arrays in DoA and ToA estimation. Actually, this will serve as the basis for the analysis of advanced scenarios (concentric arrays, pseudorandom grids) in next sections. Considering that the elliptical array is azimuthally rotated counterclockwise an angle α [see Fig. 2(a)], the values of x p and y p in eq. (5) should be simply replaced by Alternatively, it could be convenient to work with the elliptic coordinate system, formed by the radial coordinate ξ ∈ [0, ∞) and the angular coordinate η ∈ [0, 2π), when positioning the sensors that conform the rotated elliptical array. In that case, the p-th sensor will be located at where a and b are the semi-major and semi-minor axes of the elliptical array, respectively. Both parameters are related through the eccentricity value e = 1 − b 2 /a 2 . Note that an uniform angular placement for the sensors in elliptical coordinates, η, does not necessarily imply an uniform angular placement in polar coordinates, φ p .

C. Concentric Elliptical Arrays
As an additional step, the joint estimation of the directionof-arrival and time of arrival can be generalized to the case where multiple concentric elliptical arrays are considered [see Fig. 2(b)]. As pointed out in previous works [39], this is expected to increase the accuracy in the estimation as well as the bandwidth. Additionally, the use of ultra-wideband concentric elliptical arrays will lead to advanced functionalities in DoA and ToA detection. These advanced functionalities will be explored in detail in Sections III and IV of the present manuscript.
Let us consider Ψ concentric elliptical arrays (ψ = 0, 1, ..., Ψ − 1), each of them constituted by P sensors. The phase-mode expansion for the ψ-th elliptical array will be now Thus, the phase-mode expansion of the whole system, H m,l (f ), will be of the form which is simply the average of the individual phase-mode contributions of each elliptical array. In fact, notice that the resulting expression for H m,l (f ) in eq. (26) is identical to eq. (16), despite the computation of the filters W m,p,ψ (f ) is different in this case for each elliptical array. Naturally, the computational complexity increases when considering concentric arrays. Now, the system is formed by P × Ψ sensors, so the number of required filters for DoA and ToA estimation increases to M × P × Ψ. Nonetheless, by following the recommendations given in Sec. II.A, the number of filters can be similarly reduced by a factor of 8.

III. SIMULATIONS
This Section shows and discusses some results obtained through the formulation derived in Sec. II for DoA and ToA estimation in ultra-wideband elliptical arrays. Regarding the organization, Sec. III.A analyzes the accuracy of the method for single elliptical arrays and the influence of eccentricity e and rotation angle α in the estimation. Sec. III.B introduces concentric elliptical arrays and their benefits compared to single elliptical arrays. Sec. III.C presents the performance of the joint estimation for elevation angles different than 90°. Finally, Sec. III.D explores the use of pseudorandom grids based on concentric elliptical arrays.
In order to validate the estimation for elliptical arrays, it must be ensured that DoA and ToA estimation matches φ l and τ l for the incident wave, as well as the artifacts (nondesired values expected to be several orders of magnitude below the real path of the incident wave) do not disguise the correct angles and delays. As an example, Fig. 3 shows the joint angular and delay domain of a generic H(φ, τ ), with a single incident wave at φ l = 90°and τ l = 30 ns. Both parameters are correctly estimated. However, some artifacts can be observed. If they were large compared to the real incident wave, they could misled the real path. Thus, we define ∆ as the ratio between the correct estimation and the largest artifact in the azimuth and delay domain. The larger the ∆, the better the estimation. Throughout this document, this ratio will be considered as a metric.

A. Single Elliptical Arrays
The main benefits of elliptical arrays compared to circular arrays are the new degrees of freedom related to the sensor position. While in circular arrays we only have the control of the radius r, three different independent parameters can be modified in elliptical arrays: i) semi-major axis a, ii) eccentricity e, and iii) rotation angle α. This fact allows for a wide variety of sensor arrangements.
The first simulation aims to determine the influence of the eccentricity on the joint DoA and ToA estimation. For this purpose, four ellipses with e = 0, 0.7, 0.95 and 0.99 are simulated for an incident azimuth range φ l = [−90°, 90°] and τ l = 30 ns. The frequency band is chosen to be from 28 GHz to 30 GHz (B = 2 GHz) for K = 100 frequency samples. These frequencies are part of the band n257 defined by the 3rd Generation Partnership Project (3GPP) and are expected to be fundamental in the deployment of 5G New Radio (5G NR) [43]. The angular spacing in elliptical coordinates is 0.5°, which results in P = 720 sensors. The considered semi-major axis a = 0.5 m and the ellipse is placed with its semi-major axis oriented along the horizontal direction (α = 0°). Given the highest frequency f = 30 GHz, we can ensure that the largest separation between sensors is 0.437λ for any value of eccentricity. Therefore, spatial Nyquist theorem is fullfiled. Finally, the elevation incident angle is θ l = 90°, i.e., matching the plane of the elliptical array sensors. Fig. 4(a) shows the metric ∆ for several azimuth angles φ l and eccentricity values e. The circular array (e = 0) shows a constant value for ∆ along the whole azimuth range. When the eccentricity increases, the ellipse tends to flatten on the semi-minor axis. This implies an increase in the directivity of the estimation in the direction of the semi-major axis. For e = 0.7, the estimation is correct in the whole range. For e = 0.95, ∆ > 0 for azimuth values up to ±50°, where the largest artifact begins to mislead the correct incident angle. The same behavior can be found for e = 0.99, where the estimation is valid up to ±15°. The reasoning behind these results is that for very high eccentricities, the angular response of an elliptical array closely resembles that of a linear array in the direction of the semi-major axis. Therefore, when the incident wave is perpendicular to the semi-major axis of the elliptical array, an uncertainty/artifact appears at φ l + 180°, since the near-zero curvature of the array only exploits half of the angular domain, i.e., 180°. This evolution can be seen as the eccentricity increases and the array flattens out in Fig. 4(a). Note that the formulation derived in Sec. II is valid for circular, elliptical and linear arrays. Although it is not explicitly shown, due to the array symmetry, the estimation is similar in the range φ l = [90°, 270°]. Also notice that the larger the eccentricity, the lower the number of considered phase modes M , as detailed in Sec. II.
At the beginning of this Section, three degrees of freedom related to ellipses were discussed: semi-major axis, eccentricity and rotation angle. As Fig.4(a) has shown, high flattened ellipses are angularly selective along direction of the semimajor axis. In order to tune the angular response, Fig.4(b) analyses the effect of a rotation angle α in the ellipse geometry. Particularly, it includes a rotation angle α = 90°compared to the geometry of Fig.4(a). The direct consequence is the 90°shift on the azimuth angle estimation. Now, the semimajor axis is located in the vertical axis, providing excellent estimation for very flattened ellipses when the incident angle is nearby ±90°. However, incident angles in the horizontal axis (0°and 180°) get poor estimations if eccentricities are large, contrary to what is observed in Fig.4(a).
In summary, the proposed formulation performs proper estimation of DoA and ToA for the whole azimuth range for eccentricity values up to 0.7. Above this value, the estimation shows a directional behavior in the direction where the semimajor axis is located. By rotating the ellipses, the angular response can be tuned. More concretely, the angular selectivity must be considered an inherent property of the chosen (elliptical) geometry, not of the method itself. This behavior could be of potential application for the selection (suppression) of signals-of-interest (signals-not-of-interest) in smart wireless environments.

B. Concentric Elliptical Arrays
Sec. II.C introduced the analysis of concentric elliptical arrays. Since previous subsection has shown the eccentricity effect on the estimation, we can take advantage of the concentric arrays in order to improve the estimation. By combining H m,l (f ) from the ellipses shown in Figs. 4(a) and 4(b) [see eq. (26)], a new ellipse arrangement can be formed. Fig. 4(c) presents the metric ∆ for the joint DoA and ToA estimation when two concentric elliptical arrays are considered. For high eccentricities, an interesting behavior can be found. ∆ is maximized for all those DoA coinciding with the semi-major axes of the concentric ellipses (i.e., φ l = 0°and φ l = 90°). A clear example can be seen for e = 0.95. In Figs. 4(a) and 4(b), ∆ indicates good estimations for approximately an 80°a zimuth range. However, in Fig. 4(c), ∆ > 8 dB in the whole range. Therefore, concentric ellipses with different rotation angles can provide different ∆ patterns. This fact opens up the possibility of implementing pseudorandom grids contained within elliptical geometries, which will be discussed in later sections.
C. Technique performance for θ l = 90 o Up to this point, simulations have been carried out for a fixed elevation angle of θ l = 90°. In [40], it was demonstrated that the filter from eq. (18) provides accurate estimations for θ l = 90°. With the purpose of validating the estimation for different elevation angles, Fig. 5 shows the metric ∆ for different values of θ l . Here, the incident azimuth angle is assumed to be φ l = 0°. The configuration parameters are the same as those shown in Sec. III.A. Although the analytical framework discussed in Sec. II was originally derived under the assumption of θ l = 90°, it can be observed in Fig. 5 that the accuracy in DoA estimation is still satisfactory in a wide range of elevation angles. This is a remarkable feature, as elevation/tilt angles typically vary between 90 and 75 degrees in real deployments [44], [45]. In this range, ∆ > 13 dB for all the considered eccentricities. Concerning ToA, it is estimated as 30 ns for θ l = 90°, matching τ l . When θ l moves toward 30°, ToA estimation suffers a slight variation of +0.5 ns. This effect was also noticed for circular arrays in [41]. Finally, it is worth noting that elliptical arrays outperform DoA and ToA estimation compared to circular arrays for several elevation angles.

D. Pseudorandom Grids
Previous subsections have shown that elliptical arrays present directive behaviour for DoA estimation. Fig. 6 shows some examples of generic geometries formed through the superposition of concentric ellipses. In contrast to concentric circular arrays where only concentric ring shapes can be obtained [39], concentric ellipses provide a wide range of possible sensor arrangements. Hence, some of these geometries may be approximated as pseudorandom grids and the joint DoA and ToA estimation could be performed with sensors located in pseudorandom positions. In real deployments, the use of circular arrays may not be feasible due to space limitations. Conversely, elliptical arrays take up a smaller area and can better adapt to the geometry of any structure. Additionally, we can take advantage of the directivity property in the direction of the semi-major axis to steer the elliptical arrays in a specific range of angles.
As a study case, Fig. 7(a) shows a superposition of concentric ellipses to form a pseudorandom grid. Throughout this subsection, it will be studied in depth to analyze its performance compared to single and concentric circular arrays. This geometry consists of 8 ellipses with semi-major axis   [43]. 720 sensors are considered per ellipse, which gives a maximum separation between consecutive sensors of 3 mm (0.437λ at f = 43.5 GHz). The number of considered frequency samples is K = 200 and the number of phase modes is fixed to M = 250.
By applying the formulation for concentric elliptical arrays, Figs. 7(b) and 7(c) present two cases of joint estimation for this geometry. Particularly, Fig. 7(b) shows the estimation of the angular-delay domain for an incident wave at φ l = 45°a nd τ l = 20 ns. Fig. 7(c) illustrates the same domain for φ l = 250°and τ l = 40 ns. In both cases, the estimation is clearly maximized around ToA and DoA. In addition to these cases, the geometry has been tested for incident waves in the entire range of φ l and τ l , showing excellent results with artifacts below 20 dB in the worst case.
In order to compare the performance of concentric elliptical arrays with other array arrangements, Fig. 8 shows the angulardelay estimation for three different arrangements. Fig. 8(a) is obtained from an uniform circular array (UCA) with radius r = 34.5 cm and P = 720. Fig. 8(b) represents the estimation for a nine ring uniform concentric circular array (UCCA) whose outer circle is equal to the one shown in Fig. 8(a). The inner circle has radius r = 15 cm and all nine rings are equidistant. Finally, Fig. 8(c) uses the geometry presented in Fig. 7(a), i.e., a concentric elliptical array (CEA). Two main conclusions can be drawn from this figure: (i) Sidelobes appear in the UCA estimation 15 dB below the real path estimation and (ii) UCCA and CEA estimations obtain almost similar performance with sidelobes below 25 dB in both cases. These are virtually hidden for a dynamic range of 35 dB. Therefore, UCCA and CEA outperform UCA array because they better map the spatial region. The main advantage of CEA is that this mapping is not limited to concentric rings, but can be performed with several arrangements.
Previously, the degrees of freedom of the position of the sensors in an ellipse have been discussed. Actually, it is possible to go one step further. The theoretical framework developed in Sec. II has demonstrated that non constant r p values are feasible for the joint estimation. In order to achieve a real pseudorandom grid, an independent and uncorrelated bivariate normal distribution N 2 (0, σ 2 ), with σ x = σ y = σ/ √ 2, is included in the sensor positions x p and y p [eq. (24)]. This distribution adds noise in the sensor position which randomizes the grid. Figs. 9(a)-(d) shows the angular-delay domain estimation for arrangements with different σ values. Simulation parameters are equal to those ones shown for Fig. 7 with incident wave at φ l = 45°and τ l = 20 ns. Standard deviation σ is chosen in terms of wavelength and it ranges from 0.5λ to 5λ. The maximum artifacts from Fig. 9(a) to Fig. 9(d) are found to be 23.9 dB, 23.5 dB, 21.4 dB and 18.2 dB below the correct estimation, respectively. Although artifacts increase as the position noise does, they are still low enough to correctly detect the DoA and ToA. A remarkable result is that of Fig. 9(d), where the elevated value σ directly hides the elliptical shape of the CEA. Even in this case, the joint estimation is remarkably good. For a dense enough sensor arrangement, most of the sensors satisfy the spatial Nyquist theorem. If, on average, the spatial Nyquist theorem is fulfilled, a good estimation is expected. Therefore, it can be concluded that, if a dense pseudorandom mesh is approximated by a set of concentric ellipses, the joint estimation of DoA and ToA can be properly performed. Finally, note that the pseudorandom grids shown in Fig. 9 cannot be rigorously analyzed with previous approaches [37]- [41], as sensors with different radius r p must be considered.

IV. MEASUREMENTS
In order to validate the simulations and theoretical framework, multiple measurements have been carried out at the facilities of the University of Granada. These facilities consist of a semi-anechoic and semi-reverberation chamber whose dimensions are 5 × 3.5 × 3.5 meters (61.25 m 3 ). The measurements are performed in the semi-anechoic part, where multiple absorbers are found in the walls in order to avoid any reflection. Therefore, the Line-of-Sight (LoS) can be analyzed in the propagation channel between a transmitter (TX) and a receiver (RX). To recreate the simulation setup, TX is placed at a distance d l and azimuth angle φ l with respect to the center of a certain ellipse, which is formed by a virtual array at RX. Fig. 10 shows the measurement setup in the semi-anechoic chamber. The transmitter antenna is fixed, while the receiver antenna, located in the measurement system, can move in the XY plane positioned at the same height (z = 179 cm) as the transmitter antenna. The measuring system allows a In the first experiment, TX is placed at distance d l = 195 cm and angle φ l = 270°. RX forms an ellipse with a = 24.2 cm, e = 0.95 and α = 90°. P = 720 sensors are considered, for a maximum separation of 2.1 mm between sensors (0.437λ for f = 62 GHz). Finally, the number of phase modes M is set to 180. Figs. 11(a) and 11(b) show the DoA and ToA estimation for the simulated and measured cases, respectively. As expected, the estimation is maximized for τ l = 6.5 ns (d l = 195 cm) and φ l = 270°, with an artifact located in φ l ± 180°. In Fig. 11(b), the estimation of DoA and ToA is slightly more spreaded due to measurement imperfections with respect to simulation. However, the maximum is still found at τ l = 7 ns (d l = 210 cm) and φ l = 270°. The variation of 15 cm is mainly due to the calibration process; namely, horn length (7.6 cm) and monopole transition (2 cm) are not initially considered. The distance resolution is 7.5 cm, thus the estimated value of d l is within the expected error band in measurement. Finally, note that two reflections due to back propagation in the semi-reverberation chamber can be found at 34.25 ns and 42 ns in Fig. 11(b). The second experiment proposes the DoA and ToA joint estimation through an elliptical array with parameters: semimajor axis a = 24.2 cm, eccentricity e = 0.7 and null rotation (α = 0°). The number of sensors P remains as 720 and M is set to 250. TX is placed at d l = 120 cm (τ l = 4 ns) from the TX antenna aperture to the ellipse center, and φ l = 330°. Figs. 12(a) and 12(b) present a remarkable agreement between simulations and measurements. As in Fig. 11(b), the estimated τ l in the measurement is slightly larger, 4.25 ns, due to the length of the horn antenna and waveguides which are not included in the path.
Figs. 11 and 12 have validated the theoretical framework and the simulations proposed for this estimation method. As explained in Sec. II, this technique is based on a correct modeling of the incident signal on the P sensors, i.e., H p,l . Thus, phase-mode expansion H m,l and 2-D FFT lead to the joint DoA and ToA estimation. A proper modeling of the phase is fundamental in H p,l . As an example, Fig. 13 illustrates the H p,l phase distribution for every sensor and frequency in the setup previously shown in Fig. 12. High similarity of the phase distribution in simulations and measurements can be observed. This fact demonstrates the proper modeling of the incident wave for elliptical cases in real measurements, thus yielding good joint DoA and ToA estimations.
One of the features observed in the simulation is the high angular selectivity offered by highly flattened elliptical arrays. One way to explain this behavior is based on understanding how the wave reaches the array. For this purpose, two different setups have been measured. The first one has the TX posi- tioned at a distance of d l = 195 cm and an angle φ l = 270°. The RX is a virtual array with parameters: a = 24.2 cm, e = 0.99, α = 90°and P = 720. The second setup is similar, except for the rotation angle α = 0°. Fig. 14 shows the normalized Power Delay Profile (PDP) for every sensor p in RX. This PDP is calculated as the square of the IFFT of the channel frequency response H p,l for each sensor. On the left panel, it can be seen that the wave impinges in the direction of the semi-major axis. Therefore, this wave travels sensor by sensor through the entire array. This fact generates the curvature observed in the PDP for the different sensor positions. On the right panel, the wave impinges in the direction of the semi-minor axis. In this case, for high   eccentricities, the wave simultaneously reaches all the sensors of the array. This results in no curvature in the PDP, i.e, the measured channel for the different sensors is similar. In the first case, the curvature is unambiguous, since only this DoA generates such curvature. In the second case, the practical absence of curvature causes ambiguity in the estimation, since φ l = 90°would generate exactly the same PDP. Hence, it can be concluded by the measurement study that this technique takes advantage of the curvature of the elliptical array, thus justifying the curves simulated in Fig. 4. As it was noted in Sec. III.A, the selectivity is an attribute of the sensor arrangement.
Sec. III.C presented the technique performance for elevation angles θ l = 90°. In order to validate previous simulations, a setup with θ l = 70°is measured. TX antenna is placed at height z = 120 cm with φ l = 270°, and the elliptical array is kept at z = 179 cm. The distance in the XY plane between TX and the center of the elliptical array is 160 cm. Therefore, by applying basic trigonometry, d l = 170 cm (τ l = 5.66 ns) and θ l turns out to be 70°. RX parameters are a = 24.2 cm, e = 0.7, α = 90°, P = 720 and M = 250. Fig. 15 shows the simulated and measured joint estimation of the DoA and ToA for the previous scenario. The measured scenario matches the simulation with 270°DoA and 6 ns ToA estimation. The artifacts appear 16 dB below the estimation, showing a good agreement with the simulation prediction (see e = 0.7 in Fig. 5).
For the sake of completeness of the experimental study, it will be shown that the use of this technique is valid for i) multipath environments, and ii) concentric elliptical arrays. For that purpose, i) two measured rays are combined in the semi-anechoic chamber as the sum of the frequency response of each of the rays incident at the p-th sensor [see eq. (19)]. The first ray reaches the center of the array from 330°and 4 ns (120 cm) delay, while the second ray impinges the ellipse from 300°azimuth angle and 8 ns (240 cm) delay. ii) Two ellipses are combined to form a concentric elliptical array. The phase-mode response from the sum of the two previous rays is summed to create the new arrangement [see eq. (26)]. The parameters for the first elliptical array are a = 24.2 cm, e = 0.7, α = 0°, P = 720 and M = 250, while the second array is characterized by the same parameters except for α = 90°. Fig. 16 shows the joint estimation for the multipath scenario and the superposition of ellipses. Both rays can be clearly depicted in the simulations and measurements for the correct values of DoA and ToA. A good estimation is seen, which validates the proposed approach and the experimental setup for usage in multipath environments in wireless communication links.

V. CONCLUSIONS
This work proposes a technique for joint DoA and ToA estimation based on ultra-wideband elliptical arrays. The theoretical framework introduces a generalization of frequencyindependent beamformers based on uniform circular arrays. This new approach is a generalization not only valid for elliptical arrays, but also feasible for circular and linear arrays. The geometry of the elliptical array provides new degrees of freedom compared to the circle, where only the radius can be modified. The ellipses, besides tuning the semi-major axis, also allow to adjust their eccentricity or rotation angle. These degrees of freedom, together with the superposition of concentric ellipses, result in pseudo-random sensor geometries. This novel concept avoids the problem of being limited to a specific set of geometries, being able to adapt the array to several arrangements in a real deployment.
The analysis of the technique through simulations has shown good joint DoA and ToA estimations, with artifacts appearing orders of magnitude below the main component.
The study has been performed for the whole range of azimuth angles φ l , as well as for multiple time-of-arrival τ l , and three frequency bands within the mmWave range: 28-30 GHz, 39.5-43.5 GHz and 58-62 GHz. Additionally, the effect of modifying the rotation angle α, eccentricity e and elevation angles θ l on the joint estimation has been analyzed. Through these degrees of freedom, arrangements of nine geometries with pseudo-random array distribution have been simulated. One step further, a complex normal distribution is included in the sensor position in order to recreate a random grid. The results show that even including this random distribution, the estimation can be properly performed.
Finally, the simulations have been validated through multiple measurements in a semi-anechoic chamber at 58-62 GHz frequency band. DoA and ToA estimation has been initially carried out for single ellipses and single-path environments. Then, this has been extended to multipath scenarios and concentric elliptical arrays, simultaneously. Both the measured frequency response, phase distributions and the joint estimation of the DoA and ToA agree with those predicted by the simulations. These experimental results validate the theoretical framework, thus providing a method which gives accurate estimations for a large number of sensor arrangements and arbitrary geometries.