Multi-Scale Single-Bit RP-EMS Synthesis for Advanced Propagation Manipulation through System-by-Design

A new method for synthesizing Single-Bit Reconfigurable Passive Electromagnetic Skins (1RP-EMSs) featuring advanced beam shaping capabilities is proposed. By using single-bit unit cells, the multi-scale problem of controlling 1RP-EMSs is formulated as a two-phase process. First, the macro-scale synthesis of the discrete surface current that radiates the electromagnetic (EM) field fitting user-designed requirements is performed by means of an innovative quantized version of the iterative projection method (QIPM). Successively, the meta-atom states of the 1RP-EMS are optimized with a customized implementation of the System-by-Design paradigm to yield a 1RP-EMS that supports such a feasible reference current. A representative set of numerical results is reported to assess the effectiveness of the proposed approach in designing and controlling single-bit meta-atom RP-EMSs that enable complex wave manipulations.


Introduction and Rationale
Electromagnetic Skins (EMSs) are currently the core of a theoretical, methodological, and practical revolution within the academic and industrial communities working on wireless communications [1]- [10]. Several research studies on the foundation, the modeling, the simulation, the design, and the test of EMSs are currently under development with a strong interdisciplinary effort combining chemistry, physics, metamaterial science, electromagnetic (EM) engineering, telecommunications, and signal processing expertises [1]- [3][7] [8]. As a matter of fact, starting from their early conceptualization as thin metasurfaces able to manipulate the wave propagation beyond Snell's laws [11], EMSs are considered as one of the key enabling factors of the revolutionary Smart EM Environment (SEME) paradigm in wireless communications [4]- [6][12] [13].
Certainly, a multiplicity of methodological and practical challenges [2]- [4][8] [10][11] [14] still needs to be addressed to have a full transition from traditional wireless systems to the SEMEenhanced ones. In particular, the complexity associated to the design, the fabrication, the implementation, the control, and the integration within a wireless scenario of EMSs is the main critical issue. More specifically, complexity arises at (i) the EMS design level owing to the multi-scale nature of its layout that features micro/nano-scale descriptors combined with meso/macro-scale reflection and communication properties, (ii) the SEME level due to the interactions between the EMSs and the large-scale propagation scenario, and (iii) the "propagation management" level because of the need to fruitfully integrate the EMSs in a heterogenous wireless infrastructure, which includes the base stations, the integrated access and backhaul (IAB) nodes, and the smart repeaters, as well, to yield measurable performance improvements in the overall wireless network.
Within such a framework, the design of planar artificial materials with advanced propagation management capabilities has been recently demonstrated for static passive EM skins (SP-EMS) by exploiting artificial intelligence (AI) techniques within the System-by-Design (SbD) paradigm [5][6] [15]. Such an approach leverages on the decomposition of the problem at hand into a source design phase and a subsequent optimization of the surface descriptors of the SP-EMS within the Generalized Sheet Transition Condition (GSTC) framework [5][6] [11]. Thanks to the modularity of such a synthesis tool and its multi-scale-oriented nature, the efficient de-sign of wide-aperture EMSs that enable advanced pattern shaping properties has been carried out [5] [6] despite the use of extremely simple unit cells.
Otherwise, reconfigurable passive EMSs (RP-EMSs) have been proposed and widely studied to dynamically control the propagation environment for adaptively improving the communication performance [1]- [3][7] [16]. Towards this end, RP-EMS unit cells needs either analog (e.g., varactors/varistor [9][14] [17] and mechanically-tuned sub-parts [10]) or digitally-controlled (e.g., p-i-n diodes [18]) components. From an applicative viewpoint, the implementation of a continuous control on each RP-EMS cell can yield to very expensive and complex architectures, thus it is generally avoided [19] and the RP-EMS analog states are often discretized using few bits, B, per cell [9] [19] or they are implemented by using binary switches [18]. Therefore, RP-EMSs are usually digitally-controlled systems [18]- [20] with relatively limited per-cell degrees-offreedom (DoFs) when compared to SP-EMSs [5] [6]. A key consequence of such a per-cell constraint, mainly when low-bit (B → 1) RP-EMS are at hand [19], turns out to be the very limited control of the shape of the reflected beam [19]. Thus, the mainstream state-of-the-art literature on RP-EMSs has been concerned with the synthesis of RP-EMSs with "simple" anomalous reflection capabilities and narrow beam focusing (i.e., pencil beam-like) [9][18]- [20]. However, demonstrating more advanced footprint control/shaping with a digital RP-EMS would be of great interest in practice since it would allow one to efficiently concentrate the reflected power in arbitrary desired areas (i.e., roads, squares, streets, buildings) and not just in spots. Unfortunately, the approach derived in [5][6] to design SP-EMSs affording shaped footprint patterns cannot be directly applied to RP-EMSs [5] [6]. Indeed, the synthesis of the reference surface current, which is performed in the first step of [5] [6] and that exploits the non-uniqueness of the associated inverse source (IS) problem to take advantage of the non-radiating currents (NRCs) [5] [21], assumes that the unit cell of the corresponding EMS allows a fine tuning of the reflection phase [5] [6]. By definition, this is actually prevented when dealing with digital RP-EMSs [19] making the design process ineffective and potentially unable to fulfil complex coverage requirements.
Dealing with RP-EMSs, the objective of this work is twofold. On the one hand, it is aimed at presenting and validating an innovative method for the synthesis (i.e., the design and the control) of high-performance holographic 1RP-EMSs. On the other hand, it is devoted to prove that minimum complexity RP-EMSs can be used in SEME scenarios to yield complex wave propagation phenomena despite the coarse tuning of the reflection phase.
Starting from the design of a meta-atom of the RP-EMS that features only a single-bit reconfiguration and by generalizing the theoretical concepts on complex large-scale EM wave manipulation systems [5][6] [22]- [25], the first step of the proposed method for the synthesis of 1RP-EMSs deals with the computation of a discrete-phase current that radiates a field distribution fitting complex footprint patterns. A digital SbD-based RP-EMS optimization is then carried out to set the 1RP-EMS configuration that supports such a reference discrete-phase current. Towards this end, suitable AI paradigms for building reliable and computationally-efficient "RP-EMS digital twins" [5][6] [22]- [25] are exploited to properly address the issues related to the multi-scale complexity of the problem at hand.
The outline of the paper is as follows. First, the 1RP-EMS synthesis problem is formulated (Sect. 2), then Sect. 3 details the proposed two-step (i.e., design and control) synthesis method.
Representative results from a wide set of numerical experiments are reported for assessment purposes, while comparisons with state-of-the-art techniques [5] [6] are considered (Sect. 4).

Mathematical Formulation
Let a single-bit RP-EMS (1RP-EMS) be centered in the origin of the local coordinate system (x, y, z) (Fig. 1). The 1RP-EMS consists of M ×N reconfigurable binary meta-atoms displaced on a regular grid of cells with sides ∆x and ∆y on a planar region . Each (m, n)-th (m = 1, ..., M; n = 1, ..., N) meta-atom is defined by a set of U geometrical/material descriptors g g (u) ; u = 1, ..., U and it features, at the t-th (t = 1, ..., T ) time step, a binary state s mn (t) ∈ {0, 1}.
The 1RP-EMS at the t-th (t = 1, ..., T ) instant can be univocally identified by the binary micro-scale state vector S (t), S (t) {s mn (t); m = 1, ..., M; n = 1, ..., N}, and the timeindependent (i.e., it is unrealistic to change the atom layout at each time step) micro-scale descriptor vector g. Otherwise, the RP-EMS can be described from an electromagnetic view-point by the micro-scale electric/magnetic surface susceptibility vector K (t) (t = 1, ..., T ) [5] [11], whose (m, n)-th entry (m = 1, ..., M; n = 1, ..., N) is the diagonal tensor of the electric/magnetic local surface susceptibility of the (m, n)-th meta-atom, According to the Generalized Sheet Transition Condition (GSTC) technique [11][26] [27], the instantaneous far field pattern, E (r, θ, ϕ; t), reflected by the RP-EMS when illuminated by a time-harmonic plane wave at frequency f impinging from the incidence direction (θ inc , ϕ inc ) and characterized by "perpendicular" and "parallel" complex-valued electric field components E inc ⊥ and E inc is a function of the surface susceptibility vector K through the macro-scale induced surface current J (i.e., E (r, θ, ϕ; t) = F J (x, y; t) ). More in detail, it turns out to that where the surface current J is given by where J o , o ∈ {e, h}, is the electric/magnetic component of the current induced on the RP-EMS, while k 0 = 2πf √ ε 0 µ 0 and η 0 = µ 0 ε 0 are the free-space wavenumber and the impedance, respectively, which depend on the free-space permeability (permittivity) µ 0 (ε 0 ).
Such a derivation points out that the t-th (t = 1, ..., T ) far-field pattern E (r, θ, ϕ; t) can be controlled by properly adjusting the M × N binary entries of S (t), once the 1RP-EMS is designed (i.e., g is set -Sect. 2.1). Accordingly, the problem at hand can be mathematically formulated as follows 1RP-EMS Synthesis Problem -Find the optimal setting of the micro-scale descriptor vector, g opt , and the optimal configuration of T binary micro-scale state vectors, is minimized at each t-th (t = 1, ..., T ) time instant [i.e., (g opt , S opt (t)) = arg In (4), ℜ { . } is the "ramp" function and F des ( x, y, z; t) is the user-defined power pattern footprint at the t-th (t = 1, ..., T ) time instant in the observation region Ψ obs , ( x, y, z) being the RP-EMS global coordinate system (Fig. 1). Moreover, the footprint pattern is a function of the reflected far-field pattern E (r, θ, ϕ; t), (i.e., F ( x, y, z; t) = H E (r, θ, ϕ; t) ) and it is given by where d is the 1RP-EMS height over the ground plane ( Fig. 1).
It is worth noticing that, unlike the case of SP-EMSs, the synthesis of a 1RP-EMS cannot be done by minimizing (4) only once since there is a different optimal configuration S opt (t) for each t-th (t = 1, ..., T ) user-defined footprint pattern, F des ( x, y, z; t), as pointed out in the following expression where the link between F des ( x, y, z; t) and S (t) {s mn (t); m = 1, ..., M; n = 1, ..., N} is made evident. On the other hand, the U geometrical/material entries of g opt must be set once as the optimal trade-off among all T propagation scenarios.
Furthermore, the problem at hand is more complex than that of a multi-bit RP-EMS and (even) much more than of a SP-EMS. Unlike the SP case, the t-th (t = 1, ..., T ) micro-scale electric/magnetic surface susceptibility vector K (t) assumes here only a quantized set of states (i.e., 2 M ×N ) instead of a continuity of values [5] [6]. Thus, the macro-scale (reflection) properties of the arising EMS turn out to be more severely constrained than those of a SP-EMS or a multi-bit RP-EMS. Consequently, the fulfilment of complex shaping requirements on the footprint power pattern, as those in [5] [6], is certainly more difficult and it may results even physically unfeasible.
Taking into account these considerations, the "1RP-EMS Synthesis Problem" (4) is then addressed with a two-step approach where, first, the "1RP-EMS Design Problem" (Sect. 2.1) is solved by identifying the U geometrical/material descriptors of the single-bit meta-atom (i.e.,

1RP-EMS Design Problem
As for the 1RP-EMS unit cell design, a key challenge and preparatory step to enable the footprint pattern control (i.e., F → F des ) is the choice of a meta-atom structure whose reflection properties can be suitably modified when its logical state is changed [11]. In principle, an optimal trade-off should be found by minimizing (6) with respect to g across all T user-requirements {F des ( x, y, z; t); t = 1, ..., T }, while, in this paper, a "worst case"-strategy is adopted to yield a more general and flexible implementation. The design is then carried out by requiring that the 1RP-EMS meta-atom supports the widest possible reflection variation to account not only the T propagation scenarios at hand, but more in general the largest range of admissible condi- Mathematically, this means to minimize the following cost function to yield the optimal set of the geometrical/material descriptors of the single-bit meta-atom, ]. In (7), f 0 is the central working frequency, ∠· stands for the phase of the complex argument, and Γ ⊥⊥

1RP-EMS Control Problem
Once the 1RP-EMS has been designed by setting g opt , the computation of S opt (t) should be performed by minimizing the constrained (g ≡ g opt ) version of (4) [i.e., S opt (t) = arg (min S [Φ (g opt , S (t))]), which directly relates the state vector S (t) with the footprint target F des ( x, y, z; t). However, when dealing with aperiodic wave manipulation devices [5][28]- [31], such a single-phase solution approach is usually avoided in favour of splitting the problem at hand into two parts. The former phase ("Reference Current Computation") addresses a macro-scale objective that consists in the computation of an ideal equivalent surface current J opt (x, y; t) that affords the desired footprint pattern F des ( x, y, z; t), which is coded into the following macro-scale cost function to be minimized The second (microscale) phase ("1RP-EMS Configuration") [5][28]- [31] is devoted to choose the meta-atoms configuration S opt (t) that supports the reference current J opt (x, y; t) by solving the following optimization problem where This two-phase process exploits the fast Fourier relation between currents and patterns (1), which results in very efficient implementations for large apertures [5][28]- [31], as well. Moreover, the arising currents can be re-used to design EMS arrangements with different unit cells [31]. Furthermore, the micro-scale synthesis step does not involve here the optimization of K mn to achieve ideal susceptibility distributions (which may yield, even in the SP-EMS case [5][6], to non-feasible anisotropy requirements on the cell), but it is aimed at setting the (m, On the other hand, it has to be noticed that in principle the problem at hand, whatever the solution approach (direct or two-phases), requires the phase of the wave reflected by the metaatoms to vary over continuous intervals [5][28]- [31]. This is clearly not true when dealing with B-bits RP-EMSs, since each meta-atom can only assume 2 B states for each t-th (t = 1, ..., T ) time instant. Such a limitation is even more critical for 1RP-EMSs (B = 1). Moreover, despite the two-phase decomposition, the multi-scale and quantized nature of the 1RP-EMS Control Problem still yields to a solution space with a size (i.e., 2 M ×N ) that grows exponentially with the RP-EMS aperture.
To take into account these pros & cons, a dedicated strategy needs to be implemented (Sect. 3).

Solution Method
While the "1RP-EMS Design" problem (Sect. 2.1) is a quite standard real-variable optimization problem to be addressed with a standard optimization tool, the "1RP-EMS Control" one (Sect. 2.2) turns out to be a new challenge. As a matter of fact, the most intuitive strategy for solving this latter would be that of exploiting the methodology discussed in [5] by simply replacing the model of the local susceptibility dyadics of the SP-EMS with that of the reconfigurable singlebit meta-atom at hand. However, such an approach has a fundamental drawback when applied to the 1RP-EMS control. By ignoring the quantized nature of the 1RP-EMS surface currents in the "Reference Current Computation" (10), there may not to be an implementable current distribu- , that approximates the synthesized reference current J opt , regardless of the approach to configure the 1RP-EMS (11). Therefore, an innovative method is proposed (Sect. 3.1) to compute a "feasible" ideal equivalent surface current J opt that affords the desired footprint pattern F des , while the approach used in [5] for the design of an SP-EMS is customized here to control the 1RP-EMS (3.2).

QIPM-Based Reference Current Computation
In order to define a "feasible" reference current, a quantized version of the iterative projection method (QIPM) is derived.
Let C be the "1RP-EMS Current Space" composed by the whole set of the 1RP-EMS admissible surface currents having the following mathematical form where ι denotes the current polarization, while α mn (t) [α mn (t) = A {s mn (t)}] and χ mn (t) Starting from a random initialization of the discrete coefficients α  N), whose values are randomly drawn from A and X , the QIPM generates a succession of P trial current distributions, {J (p) ; p = 1, ..., p}. First, the footprint pattern . It is then projected into the corresponding feasibility space through the projection operator The QIPM convergence is checked and the iterations are stopped if either p = P or if the index If this holds true, the reference current is set to the p-th estimate, J opt = J (p) .
Otherwise, the minimum norm current, J The quantization of the minimum norm current is subsequently carried out by approximating it . More in detail, the amplitude and the phase coefficients of J (p+1) are determined by minimizing the mismatch cost function they are then substituted in (13) to yield J (p+1) . The iteration index is then updated (p ← p + 1) and the entire QIPM process is restarted from the footprint pattern computation.
It is worth pointing out that, unlike state-of-the-art approaches [5][6], the operation in (16) outputs an estimated current J (p+1) that fulfils the feasibility condition, thus it is assured that the current distribution determined at the convergence, J opt , can be surely implemented with a 1RP-EMS layout.

1RP-EMS Configuration Method
By following the guidelines in [5], but here customized to a binary control problem, a SbDbased optimization is carried out to identify the 1RP-EMS discrete micro-scale status S opt (t) of M × N binary entries. Towards this end, a set of L trial 1RP-EMS configurations is iteratively processed until either the number of SbD iterations reaches the maximum value I ψ th being a user-defined convergence threshold].
Starting from a random initial configuration, consists of the following operations: • 1RP-EMS Surrogate Modeling -The set of L micro-scale electric/magnetic surface sus-

Numerical Results
This section is aimed at illustrating the synthesis process of 1RP-EMSs described in Sect. 3 as well as at demonstrating its effectiveness and potentialities. Towards this end, the design of the single-bit meta-atom is first presented along with the full-wave validation of its properties

Single-Bit Meta-Atom Design and Validation
Since a key objective of this work is to prove that it is possible to achieve advanced beam shaping properties with minimum-complexity RP-EMSs, the design of the single-bit meta-atom has been carried out according to Sect. 2.1 by also taking into account the following constraints: (i) the meta-atom features a single-layer geometry to minimize the fabrication complexity; (ii) the single-bit (B = 1) reconfigurability of the RP-EMS unit cell is obtained by applying a single bias voltage; (iii) the shape of the layout of the printed cell is very regular to keep its EM behavior independent on the accuracy of the fabrication process; (iv) the 1RP-EMS structure works whatever the polarization of the incident field.
The unit cell in [36] has been then considered as reference model. It consists of a simple square patch ( Fig. 2) with two edges connected to the ground plane through two p-i-n diodes [green rectangles - Fig. 2(a)] and two vias [yellow circles - Fig. 2(a) Fig. 2(b) and Fig. 2(c), respectively.
The reflection performance of the optimized meta-atom are illustrated in Fig. 3 for the broadside incidence. More in detail, the plots of the phase [ Fig. 3(a)] and the magnitude [ Fig. 3 It is finally worthwhile to remark that, while the successive control step (Sect. 2.2) has been performed in this paper with the single-bit cell in Fig. 2, the proposed approach for configuring the 1RP-EMS can be adopted regardless of the working frequency, the number of bits per cell, B, and the meta-atom complexity [38].

Single-Bit RP-EMS Control
To assess the features and the potentialities of the 1RP-EMS control method in Sects where W Ψ 1 2η 0 Ψ F ( x, y, z; t) d xd yd z is the power reflected in the Ψ region and Ψ ext = Ψ obs − Ψ cov , which is equal to γ QIP M ≈ 4.3 × 10 −1 , while γ IP M ≈ 3.6 × 10 −1 [ Fig. 10(a)].
In order to give the interested readers a more exhaustive picture of the advantages of using the QIPM approach instead of the IPM one when dealing with discrete RP-EMSs, Figure 10 cov , where the entrance to the Uffizi museum is located [ Fig. 12(a)].
The 1RP-EMS, which has been assumed to be placed at d = 15 [m] on the building in Fig. 12(b), is requested to switch between the "Signoria+Uffizi" coverage (i.e., F des ( x, y, z; t) t=t 1 as in (18) by setting Ψ cov = Ψ (1) cov ) and the "Uffizi" coverage (i.e., F des ( x, y, z; t) t=t 2 as in (18) Fig. 12(a)]. Such a behavior is not unexpected due to both the strong non-linearity of the control problem at hand (11) and the binary nature of the control

DoFs.
Concerning the distribution of the radiated power pattern, Figure 14 cov , is a more challenging problem than that of the "Uffizi" site, Ψ  Fig. 15(b)] proves the reliability of the EMS configuration S opt (t)⌋ t=T =1 in Fig. 15(a) to match the coverage requirements on a complex region Ψ cov . The readers are suggested to notice that this test case has been already successfully addressed in [5] with SP-EMSs, but here the DoFs are far less than those available in the SP-EMS case [5].
Finally, the numerical assessment ends with a test on the performance of the proposed 1RP-EMS synthesis method in a scenario that needs a Dynamic Multi-Beam (C = 3) Reconfigurability.
More in detail, the problem at hand is that of C = 3 users, each occupying a coverage region

Conclusions
An innovative method for the synthesis of RP-EMSs, based on single-bit meta-atoms able to support advanced propagation manipulation features in SEME scenarios, has been proposed.
The arising multi-scale optimization problem has been addressed by means of a two-step approach starting from the design of a meta-atom that features only a single-bit reconfiguration.
First, a discrete-phase current, which radiates a field distribution fitting complex user-defined requirements on the footprint pattern, has been computed. Then, a digital SbD-based optimization has been carried out to set the binary configuration of the RP-EMS atoms that supports such a reference discrete-phase current.
To the best of the authors' knowledge on the state-of-the art literature, the main theoretical and methodological advancements of this work lie in: • the assessment that RP-EMS architectures featuring single-bit meta-atoms can allow complex wave manipulations without the need of continuous phase variations [5][6]; • the derivation of an approach for the control of the RP-EMS to afford complex footprint shapes and not only pencil beams [9]; • the non-trivial extension of the synthesis paradigm, adopted so far to synthesize static reflectarrays and SP-EMSs [5][6] [21], to minimum-complexity RP-EMSs by deriving a computationally effective reconfiguration method.
From the numerical validation, the following outcomes can be drawn: • the QIPM-based approach for the definition of the reference currents significantly improves the coverage efficiency with respect to state-of-the-art techniques [5] regardless of the 1RP-EMS aperture at hand [ Fig. 10(a)]; • despite the minimum complexity of the meta-atoms (B = 1), the synthesized RP-EMSs feature advanced wave manipulation properties in realistic scenarios (Fig. 14) as well as in very complex "demonstrative" cases (e.g., Fig. 15); • the proposed method turns out to be an enabling tool for multi-beam reconfiguration and/or independent user-tracking through 1RP-EMS layouts (Fig. 16).
Future works, beyond the scope of this manuscript, will be aimed at assessing the performance of the proposed method when using multi-bit meta-atoms and or different meta-atom geometries.

Appendix
Expression of E mn (t) and H mn (t) The surface averaged fields E mn (t) and H mn (t) can be expressed as [5] and where k inc is the incident wave vector (k inc −k 0 [sin (θ inc ) cos (ϕ inc ) x + sin (θ inc ) sin (ϕ inc ) y + cos (θ inc ) z]), k ref is the corresponding reflected wave vector according to standard plane wave theory [11], and e ⊥ = k inc × ν |k inc × ν| and e = e ⊥ ×k inc | e ⊥ ×k inc | are the "perpendicular" and "parallel" unit vectors (i.e., TE and TM modes) [5][6] [11].

FIGURE CAPTIONS
• Figure 1. Problem geometry. Sketch of the smart EM environment scenario.