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A Bayesian Compressive Sensing Approach to Robust Near-Field Antenna Characterization

A novel probabilistic sparsity-promoting method for robust near-field (NF) antenna characterization is proposed. It leverages on the measurements-by-design (MebD) paradigm and it exploits some a-priori information on the antenna under test (AUT) to generate an over-complete representation basis. Accordingly, the problem at hand is reformulated in a compressive sensing (CS) framework as the retrieval of a maximally-sparse distribution (with respect to the overcomplete basis) from a reduced set of measured data and then it is solved by means of a Bayesian strategy. Representative numerical results are presented to, also comparatively, assess the effectiveness of the proposed approach in reducing the"burden/cost"of the acquisition process as well as to mitigate (possible) truncation errors when dealing with space-constrained probing systems.


Introduction
We are nowadays witnessing an extraordinary technological advancement in phased array technology as a key-asset to the forthcoming 5G and 6G communications standards [1] [2]. Highperformance multi-input/multi-output (MIMO), cognitive, and multi-beam architectures will undergo mass production to allow an ubiquitous implementation of the internet-of-things (IoT)based next-generation wireless environments [3]. For a fast and reliable antenna certification at the end of large-scale manufacturing processes [4] [5], over-the-air measurements clearly constitute the most time/cost-effective option. In such a framework, far-field (FF) techniques are one viable and consolidated approach. However, they intrinsically suffer from limitations imposed by outdoor sites such as the vulnerability to weather conditions as well as to reflections from (uncontrollable) environmental obstacles/scatterers. Otherwise, near-field (NF) probing methods are alternative solutions that guarantee a higher accuracy and repeatability thanks to the exploitation of fully-controlled indoor environments and the availability of efficient NF-FF transformation strategies [6]- [18], even though they are prone to the so-called "truncation error" caused by the limited extension of the scanning surface (e.g., planar [14] [17], cylindrical [11], spherical [8], conical [12]) in real anechoic chambers. Moreover, the IEEE recommended practice for NF measurements [19] states that a reliable assessment of the radiation features of an antenna under test (AUT) needs a dense probing step Δρ (i.e., Δρ ≤ λ 2 , λ being the free-space wavelength). This results in time-consuming acquisition procedures due to the huge number of scanning positions [7]. An effective recipe to avoid/mitigate such issues is to exploit the available a-priori information on the AUT. As a matter of fact, several methodologies are based on the representation of the radiation behavior of the AUT in terms of a set of known basis functions defined with accurate full-wave (FW) simulations of the CAD models of the antenna, which are typically available from previous stages of the design process [7] [8]. Accordingly, the NF recovery problem at hand is then re-formulated as the retrieval, from a reduced set of measured data, of the expansion coefficients by means of suitable matching strategies [7] or machine learning tools [8]. Within this line of reasoning, the measurements-by-design (MebD) paradigm has been proposed as an effective tool to predict the AUT features by exploiting, unlike above-mentioned state-of-the art strategies, the generation of an over-complete basis rather than a minimum-redundancy one [20]- [22]. Thanks to this, it is possible to recast the problem at hand as a sparsity-retrieval one suitable for a fruitful exploitation of the Compressive Sensing (CS) [20] [23]. As for this latter, it is worth pointing out that a reliable application of standard (deterministic) CS solvers requires a preliminary check of the restricted isometry property (RIP) of the observation operator, which rapidly becomes computationally unaffordable even for small/medium-scale problems [23]. To overcome this issue, the MebD is mathematically reformulated in this communication within a probabilistic sparsity-promoting framework and then solved by means of a customized Bayesian CS (BCS) strategy that avoids cumbersome assessments of the RIP compliancy [24]. To the best of the authors' knowledge, the main novelties of this research work lie in: (i) the formulation of the NF prediction problem within a highlyflexible Bayesian framework dependent on neither the knowledge of the nominal/gold antenna nor on a particular topology of the probing setup; (ii) a suitable customization of the BCS to yield a robust and reliable solution of the NF field estimation in a wide range of applicative scenarios.
The outline of this communication is as follows. The mathematical formulation of the NF antenna characterization problem and its BCS-based solution method are detailed in Sect. II.
Representative results are shown and discussed in Sect. III to assess the effectiveness and the potentialities of the proposed approach. Finally (Sect. IV) some conclusions are drawn.

Mathematical Formulation
To faithfully retrieve the FF pattern features of an AUT by means of NF-FF transformation rules [18], the radiated tangential electric field distribution must be accurately estimated over a sufficiently large surface Ψ (Fig. 1) to limit as much as possible the so-called "truncation error" [16]. Generally speaking, Ψ is sampled according to the Nyquist's rule by choosing Δρ = λ 2 (Δρ being the sampling rate along a generic direction of the surface Ψ) [19] for yielding the set of T locations T = {r t ∈ Ψ; t = 1, ..., T }. Let us express the NF distribution in T , According to the MebD paradigm [20], the basis A is defined by exploiting the a-priori information on the AUT through the following procedure: 1. Uncertainty Identification: Identify the set of C uncertainty factors that can cause a deviation of the AUT radiation features from the ideal/gold ones (e.g., defects of the beam- being the sampled NF distribution in Ψ for an AUT whose c-th uncertainty descriptor has a value equal to the k-th sample of its variation range (i.e., χ c = χ where U c = (d) Add the Q c left singular vectors to the basis A by letting (e) If c = C then terminate the iterative loop and output the set of B (B = P C c=1 Q c ) basis, A. Otherwise, let c ← (c + 1) and go to Step 2(a).
As for the unknown coefficient vector w in (1), it is retrieved from a limited set of NF data collected over the measurement surface Ψ 0 ⊆ Ψ (Fig. 1). Towards this end, let us assume that the field radiated by the AUT is measured over a subset of M ≪ T probing locations M = Accordingly, w can be computed by solving the following system of equations where the observation operator It is worth pointing out that in real scenarios only a very limited subset (or none) of the C uncertainties affects the measured AUT. Therefore, the solution vector w is generally intrinsically sparse (i.e., kwk 0 ≪ B, k . k 0 being the ℓ 0 -norm) and effective solutions of (5) can be obtained within the CS framework by exploiting a probabilistic Bayesian approach [24] to avoid the RIP check. More in detail, the linear system of equations (5) is first re-arranged in the following where ℜ { . } and ℑ { . } being the real and imaginary part, respectively, ω , . Accordingly, the sparsest guess of ω (e ω , {e ω b ; b = 1, ..., (2 × B)}) that maximizes the a-posteriori probability P ( ω| δ) is computed as follows [24] e In (8) starting from initial guess of the BCS noise variance, η 0 . In (9), | . | is the matrix determinant,

Numerical Validation
The objective of this Section is two-fold. On the one hand, representative results from an exhaustive numerical study are reported to assess the effectiveness of the proposed BCS-based approach also in comparison with a previously-published state-of-the-art CS approach based on the orthogonal matching pursuit (OMP) [20]. On the other hand, the Section is aimed at providing the interested readers/users with some useful guidelines for its optimal application.
As for the results, besides a pictorial representation of the NF field reconstructions, the NF integral error Ξ , is computed to provide a quantitative index of the "solution quality".
In the benchmark scenario, the reference AUT is a linearly-polarized (i.e., planar phased array arranged on the (x, y) plane and it is composed by N = (N x × N y ) = A preliminary calibration of the BCS setup has been carried out by assuming as reference an AUT affected by a partial failure on the excitation coefficient of the s = 3-rd row (i.e., |ζ 3 | = 0.45 and ∠ζ 3 = π 3 [rad]). Towards this end, the initial guess of the BCS noise variance, η 0 , for the RVM-based maximization of (9) has been varied within the range 10 −7 ≤ η 0 ≤ 10 and the value of the NF integral error (11) has been computed for different noise levels (SN R ∈ [20,50] [dB]). The outcomes of this analysis are summarized in Fig. 2. As expected, the optimal value of η 0 depends on the SNR (e.g., η opt since, by definition, the larger is the noise variances, the lower is the SNR. On the other hand, no a-priori accurate information on the noise level is available in several practical cases, thus an optimal trade-off value has been chosen as η opt and it has been set here to η opt 0 = 10 −2 (Fig. 2). To assess the reliability of such a calibration in general operative conditions, the normalized error . max rt |E (r t )|, r t ∈ T , t = 1, ..., T ) is reported in Fig. 3  , Since the basis set is overcomplete, a NF field prediction needs at least one component for each uncertainty factor. In this example, there are two non-null entries for each of the C = 2 uncertainty factors. Therefore, one entry for each factor is mandatory. u = sin θ cos φ and v = sin θ sin φ being the direction cosines [(u 2 + v 2 ) ≤ 1], while F { . } stands for the NF-FF operator. As a representative example, Figures 6(a)-6(b) refer to the case with SN R = 20 [dB]. As expected, the error of the BCS is smaller than that the OMP as a direct consequence of the more accurate NF reconstruction [ Fig. 3(g) vs. Fig. 3(h) ⇒ Fig. 6(a) vs. Fig. 6(b)]. Quantitatively, the maximum FF deviation 6(a)] and Δ e P (u, v) Fig. 6(b)], respectively. Such outcomes are further confirmed by the comparison between the actual and the retrieved power patterns along the most critical cut (i.e., u = 0) in Fig. 6(c). Indeed, the OMP curve clearly shows larger and non-negligible distortions of the pattern outside the main lobe region.
Finally, let us investigate on the effectiveness of the proposed method in mitigating the "truncation error". Towards this end, the side of the measurement plane Ψ 0 has been shrinked to Fig. 7(a)

Conclusions
A novel antenna characterization technique, which leverages on the MebD paradigm and a suitable implementation of the BCS, has been proposed to faithfully recover the NF field distribution generated by an AUT starting from a limited set of measurements. More specifically, the NF reconstruction problem has been reformulated within a probabilistic sparsity-promoting framework to bypass the cumbersome check of the RIP compliancy of the measurement oper- ator. An analysis of the dependence of the BCS performance on its calibration setup has been carried out to derive some practical guidelines for its reliable application. The potentialities and limitations of the proposed method have been assessed in a comparative numerical validation under different operative conditions. The main outcomes from the numerical analyses are: • the proposed BCS-based method faithfully retrieves the NF behavior of the AUT with a remarkable robustness to the noise. This implies a reliable characterization of the FF radiation pattern of the AUT in the whole visible range; • unlike other state-of-the-art CS alternative methods, it not only yields an accurate NF/FF prediction of the radiated field, but it also provides information on which uncertainties are causing the deviations of the AUT pattern from the gold one; • it exhibits excellent performance in mitigating the truncation error caused by limitedextension probing area.
Finally, it is worth pointing out that the proposed method is general and it is extendable to any type of antenna or field distribution. Moreover, it can handle arbitrarily-shaped non-planar characterization setups and different measurement topologies (e.g., spiral scanning [13]).

FIGURE CAPTIONS
• Figure 1. Pictorial sketch of the NF antenna characterization scenario. •