Machine-Learning-Enabled Multi-Frequency Synthesis of Space-Time-Coding Digital Metasurfaces

Abstract

Digital metasurfaces based on space-time coding have established themselves as a powerful and versatile platform for joint spatial/spectral control of electromagnetic waves. However, their advanced design remains a largely open problem with significant computational challenges. This study introduces a novel approach, based on deep neural networks, to address this challenge. The proposed technique enables the simultaneous and independent multi-frequency synthesis of scattering patterns, allowing precise tailoring of the harmonic equivalent currents (both in magnitude and phase), and enhancing spectral efficiency. These results, experimentally validated at X-band microwave frequencies, substantially broaden the capabilities of space-time coding digital metasurfaces, paving the way for advanced applications in wireless communications, sensing, and imaging.

1 Introduction

During the past twenty years, the field of metamaterials and metasurfaces has emerged as a highly dynamic and promising research area in electromagnetics (EM). These artificial materials can be engineered to overcome the inherent limitations of natural materials, thereby enabling unprecedented field manipulations and wave–matter interactions. Ref. [1] and references therein provide a recent and comprehensive overview of the state of the art and future research directions in this field.

The introduction of digital metasurfaces (DMs) in 2014 marked a significant breakthrough in the field.^[2] Unlike conventional gradient metasurfaces that rely on continuously varying properties, DMs operate based on a quantized local response, which can naturally be represented in a digital format. In the simplest architectures, a generic element (meta-atom) contains a number S of electronic switches such as diodes, which allow controlling the reflection phase response with 2^S distinct states. Thus, in the simplest binary case (S  =  1), two states like 0° and 180° are allowed, whereas in a 2-bit scenario (S  =  2), four states (e.g., 0, 90°, 180°, and 270°) can be attained. These responses can be controlled by an electronic integrated circuit such as a field-programmable gate array (FPGA), thereby rendering the system inherently programmable. This establishes a fascinating convergence between the physical realm of metasurfaces and the information domain and opens up a broad variety of potential applications.^{[3, 4]} Within the framework of upcoming (sixth-generation) wireless networks, DMs are also known as reconfigurable intelligent surfaces (RISs),^[5] and are expected to play a key role in dynamically optimizing communication links and enhancing network performance, within the overarching concept of smart radio environments.^[6]

A major advancement in the field of DMs is the introduction of space–time coding (STC),^[7] which extends beyond the conventional spatial (quantized) modulation by adding a temporal dimension on a timescale significantly longer than the EM one. This enables joint spatial/spectral field manipulations, opening up a myriad of applications. Directing the reader to Refs. ^{[8, 9]} for recent comprehensive reviews, we limit ourselves to mention harmonic beam steering/shaping,^{[7, 10]} programmable nonreciprocity,^[11] analog signal processing,^[12] direction-of-arrival estimation,^[13] frequency-modulated-continuous-wave radars,^[14] low observability,^[15] sea-clutter simulation,^[16] and wireless communications.^[17-19]

Specifically, in the context of wireless communications, STC-DMs enable sophisticated space- and frequency-division multiplexing capabilities, which allow the implementation of full-duplex system architectures^[18] as well as the optimization of network capacity while obeying fairness constraints.^[19] Moreover, they significantly streamline the complexity of hardware requirements, providing a more efficient approach to system design.^[17] However, fully harnessing this potential necessitates the independent and simultaneous synthesis of desired scattering patterns across different frequencies, which currently remains a largely open problem. Prior studies^[10] have attempted to address this challenge by employing intertwined coding sub-sequences, but have achieved only partial control, limiting the overall spectral efficiency. With the establishment of machine learning (ML) as a powerful tool for the synthesis of metasurfaces,^[20-22] recent research^[23] has explored this approach for the synthesis of STC-DMs. While noteworthy, this prior work has primarily focused on achieving harmonic beam steering, lacking comprehensive control over beam shaping. Consequently, achieving precise and computationally affordable control over the pattern shape continues to be a significant challenge in this field.

In our approach, we take a distinct route, still rooted in ML but focusing on synthesizing the equivalent current (both magnitude and phase) at harmonic frequencies rather than directly targeting the scattering patterns. This strategy leverages deep neural networks (DNNs), and enables the synthesis of rather general scattering patterns by applying, independently and simultaneously, conventional array synthesis approaches. To cope with the inherent ill-posedness of the inverse problem, which usually results in slow and inefficient training of the DNN, we employ a “tandem” architecture, which has been already successfully applied to various scenarios including nanophotonics,^[24-29] plasmonics,^[30] cloaking^[31] and scattering reduction,^[32] microwave absorbers,^[33] and antennas.^[34] It should be noted that all these prior studies deal with static structures, with the objective of designing meta-atom geometries and configurations for specific reflection or scattering responses. In contrast, our research focuses on a dynamically modulated structure with the goal of developing an STC strategy to independently manipulate scattering patterns across multiple frequencies. Consequently, while the optimization tool employed may be similar, our application domain is markedly different.

To demonstrate the potential of our approach, we illustrate some representative examples of synthesis at multiple frequencies, based on both phase and magnitude tailoring, including designs such as Dolph–Chebyshev, Fourier, multi-beam, diffuse scattering, and orbital-angular-momentum (OAM) configurations. These results are validated experimentally via a 2-bit STC-DM prototype operating at microwave (X-band) frequency, demonstrating good agreement between theory and measurements.

In summary, our results significantly expand the capabilities for STC-DMs, opening avenues for advanced applications in scenarios ranging from wireless communications to sensing and imaging.

2 Proposed Approach

2.1 Problem Geometry, Modeling, and Formulation

Figure 1 provides a conceptual illustration of the proposed ML-based reflective STC-DM. In this configuration, each meta-atom (represented as a golden square) exhibits a reflection phase that can be switched among several quantized levels via embedded diodes. This modulation, which effectively encodes physical states into digital signals, is controlled in space and time by an FPGA, based on a given STC matrix (with two spatial dimensions and a temporal one) of digital symbols. While conceptually similar to our previous investigations,^{[7, 10]} this approach leverages ML to pursue a more ambitious goal, namely, the independent and simultaneous multi-frequency synthesis with enhanced capabilities in beam shaping/steering, as well as increased spectral efficiency.

Figure 1

Conceptual illustration of an STC-DM performing multi-frequency beam shaping/steering powered by a tandem DNN.

From a modeling perspective, we consider a STC-DM composed of an N × M periodic array of meta-atoms, with period d, laying in the x − y plane. For excitation, we assume a linearly polarized, normally incident time-harmonic [exp(j2πf_ct)] plane-wave, with x-directed electric field. The generic meta-atom, indexed as (n, m), undergoes a temporal periodic modulation of the reflection coefficient with a period T₀ assumed to be significantly longer than the EM timescale. This modulation can be mathematically expressed as

$$$$\begin{equation}{{\Gamma }_{nm}}\ \left( t \right){\mathrm{ = }}\mathop \sum \limits_{l = 1}^L \Gamma _{nm}^{\left( l \right)}\ {{U}^{\left( l \right)}}\left( t \right),\quad 0 < t < {{T}_{\mathrm{0}}}\end{equation}$$$$(1)

where U^(l)(t) [=  1 for (l − 1)T_s < t < lT_s, and 0 otherwise] denote rectangular pulses of duration T_s = T₀ /L (with L being the number of pulses), and $$\Gamma _{nm}^{( l )}$$ represents the reflection-coefficient states (at the carrier frequency f_c). For now, we assume unit-magnitude reflection, i.e., $$\Gamma _{nm}^{( l )} = \exp ( {j\varphi _{nm}^{( l )}} )$$, with the phase response $$\varphi _{nm}^{( l )}$$ evenly quantized on 2^S levels within the range [− 180°, 180°]. Nevertheless, we will investigate the impact of non-idealities, such as reflection losses and unevenly quantized phases.

Relying on the adiabatic approximation detailed in ref. [7] the far-field scattered field under slow modulation can be represented in the time domain as:

$$$$\begin{eqnarray} &&f \left( {\theta ,\phi ,t} \right)\nonumber\\ &&= \ E\left( {\theta ,\phi } \right)\mathop \sum \limits_{n\ = \ 1}^N \mathop \sum \limits_{m\ = \ 1}^M {{\Gamma }_{nm}}\left( t \right)\exp \left\{ {j\frac{{2{{\pi}}d}}{{{{\lambda }_c}}}\sin \theta \left[ {\left( {n - 1} \right)\cos \phi + \left( {m - 1} \right)\sin \phi } \right]} \right\}\end{eqnarray}$$$$(2)

where θ and ϕ are the elevation and azimuth angles, respectively, E(θ, ϕ) denotes the meta-atoms scattering pattern (approximated as cosθ in our implementation), and λ_c is the vacuum wavelength at the carrier frequency f_c. By substituting Equation (1) into this formulation, and applying a Fourier-series expansion, we can express the scattered field as:

$$$$\begin{equation}f\ \left( {\theta ,\phi ,t} \right) = \mathop \sum \limits_{{{\nu}} = - \infty }^\infty {{F}^{\left( {{\nu}} \right)}}\left( {\theta ,\phi } \right){\mathrm{exp}}\left[ {j2\pi \left( {{{f}_c} + {{\nu}}{{f}_0}} \right)t} \right]\end{equation}$$$$(3)

with f₀ =  1/T₀ representing the modulation frequency. This results in a harmonic expansion where each term corresponds to a distinct scattering pattern:

$$$$\begin{eqnarray}{{F}^{\left( {{\nu}} \right)}}\ \left( {\theta ,\phi } \right)\nonumber\\ = \ E\left( {\theta ,\phi } \right)\mathop \sum \limits_{n\ = \ 1}^N \mathop \sum \limits_{m\ = \ 1}^M a_{nm}^{\left( {{\nu}} \right)}\exp \left\{ {j\frac{{2{{\pi}}d}}{{{{\lambda }_c}}}\sin \theta \left[ {\left( {n - 1} \right)\cos \phi + \left( {m - 1} \right)\sin \phi } \right]} \right\}\end{eqnarray}$$$$(4)

where

$$$$\begin{equation}a_{nm}^{\left( {{\nu}} \right)} = \mathop \sum \limits_{l = 1}^L \frac{{\Gamma _{nm}^{\left( l \right)}}}{L}{\mathrm{sinc}}\left( {\frac{{{\mathrm{\pi \nu }}}}{L}} \right)\exp \left[ {\frac{{ - j{\mathrm{\pi \nu }}\left( {2l - 1} \right)}}{L}} \right]\end{equation}$$$$(5)

This approach models each harmonic scattering pattern as a 2-D array factor (multiplied by the meta-atom's pattern), with equivalent currents given by Equation (5). Remarkably, despite the inherently unit-magnitude and phase-quantized nature of $$\Gamma _{nm}^{( l )}$$, the resulting equivalent currents generally exhibit a richer and broader magnitude/phase coverage, as discussed in ref. [7] This characteristic expands the potential parameter space for the synthesis. However, Equation (5) establishes a complex and nonlinear relationship between these equivalent currents and the actual design variables at our disposal, namely, the reflection-coefficient quantized phases in the time series given in Equation (1). This inherent coupling among syntheses at different harmonic frequencies prevents the straightforward application of conventional semi-analytical array design techniques that exploit both phase and magnitude tailoring, such as the Dolph–Chebyshev or Fourier method.^[35] As a result, one is typically led to resort to brute-force optimization approaches,^[7] which may become computationally unaffordable, especially when dealing with electrically large structures and multiple harmonic frequencies.

In ref. [10], a semi-analytical method was introduced to decouple the syntheses at different frequencies by employing temporally intertwined coding sub-sequences with suitably chosen symmetry conditions. Though systematic and computationally efficient, this approach inherently fixes the magnitude of the equivalent currents, restricting it to phase-only syntheses. Additionally, it introduces some constraints between the number of controllable harmonic orders and the number of bits in the coding sequence, and naturally generates spectral replicas at harmonic frequencies outside the desired range, thereby limiting the spectral efficiency.

In ref. [23], an ML approach was put forward in the form of a physics-driven vector-quantized intelligent autoencoder. Upon training, this model takes a series of target harmonic scattering patterns as input and efficiently generates optimized STC matrices in real time. While computationally efficient and accurate, this approach has only been validated on relatively simple, single-beam steered patterns, without magnitude tailoring.

As a result, there remains an open challenge in developing a synthesis approach that is both computationally efficient and versatile, capable of tailoring both magnitude and phase distributions, across multiple harmonic frequencies. In what follows, we address this problem via an ML-based approach, employing tandem DNNs.

2.2 ML-Based Synthesis

Referring to Figure 2 for an illustrative diagram, our tandem architecture comprises two blocks: a forward DNN and an inverse DNN. We assume a symmetric set of Q  =  2K + 1 consecutive harmonic orders of interest, i.e., ν  =   − K, −K + 1, …, K − 1, K, although this assumption can be relaxed to some extent (see, e.g., the discussion in Section 3.4). The synthesis is carried out for a generic meta-atom, and hence we omit the dependence on indices (n, m) for notational simplicity. In essence, for a given meta-atom, the system takes the target harmonic equivalent currents a^(ν) as input and, via an intermediate layer and a quantizer, produces the L reflection phases φ^(l) of the coding sequence in Equation (1). More specifically, the input is a real-valued vector $${\bm{a}} \in {{\mathbb{R}}^{2Q}}$$, where the first half corresponds to Re{a^(ν)} and the second half to Im {a^(ν)}. The intermediate-layer output is a real-valued vector $${\bm{p}} \in {{\mathbb{R}}^L}$$ comprising continuous normalized phases $${{p}^{( l )}} = {{\hat{\varphi }}^{( l )}}\ /{{\pi}} \in [ { - 1,1} ]$$, which are subsequently quantized within the S − bit alphabet to yield the sought discrete phases φ^(l).

Figure 2

Schematic illustration of the tandem DNN architecture.

First, the forward DNN is pre-trained on a dataset constituted by D  =  85 000 sequences of unit-magnitude reflection coefficients $${{\Gamma }^{( l )}} = {\mathrm{\ exp}}[ {j{{{\hat{\varphi }}}^{( l )}}} ]$$, l  =  1, …, L, where $${{\hat{\varphi }}^{( l )}}$$ are random phases uniformly distributed in [− π, π], and the corresponding equivalent currents a^(ν) are derived from the analytical model in Equation (5). At this stage, the forward mapping $${\bm{a\ }} = \ \mathcal{O}{\bm{p}}$$ is well-defined and can be accurately captured by the forward DNN. However, the inherent ill-posedness of the inverse problem prevents a straightforward inversion by simply swapping the input and the output vectors. To overcome this limitation, our proposed tandem scheme seeks a nonlinear operator $$\mathcal{O}^{\prime}$$ such that $$\mathcal{O}\ ( {\mathcal{O}^{\prime}{\bm{a}}} ) = \ {\bm{a}}$$, essentially aiming for an identity relationship between the input and output. This strategy is known to mitigate the one-to-many mapping issues that occur in the training of DNNs for inverse problems.^[24] To this aim, we fix the forward DNN with the pre-trained configuration and train the inverse DNN on a dataset of the same cardinality D as the previous stage. Additionally, we introduce a power constraint:

$$$$\begin{equation}{{P}_r} \le \mathop \sum \limits_{\nu \ = \ - K}^K {{\left| {{{a}^{\left( \nu \right)}}} \right|}^2} \le 1\end{equation}$$$$(6)

where the lower bound P_r represents the desired fraction of power to allocate within the spectral range of interest, whereas the upper bound stems from power conservation. The selection of the lower bound represents a balance between optimizing spectral efficiency and ensuring successful convergence during the training phase. In our subsequent syntheses, as a reasonable tradeoff, we assume P_r =  0.8, aiming at concentrating up to 80% of the scattered power within the targeted spectral range. For training the inverse DNN, we apply a rejection scheme, generating the training set similarly to the forward case, but discarding sequences that do not satisfy the power constraint in Equation (6). As a metric for training, we use the mean squared error.

It is important to mention that our forward model can be described analytically, as shown in Equation (5). This allows for the potential use of a hybrid approach, where only the inverse model is implemented using a DNN (see also the discussion in Section 3.4 and the Supporting Information). However, in our implementation, we employ a full DNN setup to streamline the training process. This approach enables the direct use of built-in metrics and routines, which simplifies the overall implementation (see the Experimental Section for more details).

Moreover, while more sophisticated tandem DNN architectures exist,^{[26, 31-33]} here we opt for a conventional fully connected implementation^[24] because it adequately demonstrates the effectiveness and advantages of our approach in synthesizing STC-DMs with a simple architecture. Indeed, there is potential for enhancements in the tandem DNN architecture that could provide more advanced control features (see our discussion in Section 3.4 and the Supporting Information).

3 Results and Discussion

3.1 Tandem-DNN Training

Figure 3 illustrates the convergence of the inverse-DNN training process. Specifically, Figure 3a shows the mean squared error for a scenario with Q  =  5 harmonic frequencies (ν  =   − 2, −1, 0, 1, 2) and sequences of length L  =  24. The plot includes the error metrics for the training, validation, and test datasets, explicitly excluding the phase quantization effects. The errors for the three datasets are hardly distinguishable and reach values as low as ≈3 · 10⁻³. Figure 3b illustrates the correlation between the optimal error and the temporal sequence length, for various numbers of target harmonic frequencies (Q  =  3, 5, 7). Intuitively, dealing with a higher number of harmonic frequencies generally necessitates longer sequences, but beyond a critical threshold, the error may actually increase, likely due to overfitting. Further increases in the number of harmonic frequencies (Q > 7) might necessitate more complex DNN architectures, with additional hidden layers and neurons.

Figure 3

Convergence for inverse DNN training. a) Mean squared error for a scenario with Q  =  5 harmonic frequencies (ν  =   − 2, −1, 0, 1, 2) and sequences of length L  =  24. b) Optimal error as a function of sequence length L for various numbers of target harmonic frequencies. Continuous curves are guides to the eye only.

The corresponding results for the forward DNN are presented in Figure S1 (Supporting Information), where trends appear qualitatively similar but with significantly smaller errors, reflecting the inherently lower complexity of the forward problem.

In our subsequent examples of application, we will focus on a scenario featuring Q  =  5 harmonic frequencies, choosing a coding sequence length of L  =  24 as a reasonable tradeoff between accuracy and complexity, and a 2-bit coding (S  =  2) in view of its demonstrated practical viability in several STC-DM prototypes.^{[8, 9]}

3.2 Examples of Syntheses

Figure 4 shows two illustrative cases of multi-frequency synthesis of complex-valued equivalent currents a^(ν) for a generic meta-atom, targeting Q  =  5 consecutive harmonic frequencies (ν  =   − 2, −1, 0, 1, 2). These examples utilize a 2-bit coding sequence of length L  =  24 and ideal phase jumps of 90°, and focus on synthesizing piecewise linear magnitudes (exhibiting either asymmetric or symmetric profiles) together with linear phase shifts (either increasing or decreasing). The comparison between the targeted and synthesized values, both in magnitude and phase, demonstrates a good agreement within the desired spectral domain (highlighted by a green background). Additionally, the results for higher harmonic orders (ν up to ± 5), which are not directly controlled, are also shown. These harmonics exhibit relatively small magnitudes, ensuring that the majority of the power (71% in Figure 4a,b and 74% in Figure 4c,d) remains confined within the targeted spectral range.

Figure 4

Examples of joint syntheses for the complex-valued equivalent currents a^(ν) for a generic meta-atom across different harmonic frequencies, assuming as a target Q  =  5 consecutive harmonic frequencies (ν  =   − 2, −1, 0, 1, 2), and a 2-bit coding sequence with length L  =  24. a, b) Comparison between target and synthesized magnitudes and phases, respectively, for an example featuring piecewise asymmetric linear behavior for the magnitude, and linear decreasing behavior for the phase. c, d) Same as above, but for an example featuring piecewise symmetric linear behavior for the magnitude, and linear increasing behavior for the phase. The green-shaded regions indicate the targeted spectral range. The total power fraction concentrated within the target region is 71% and 74% in the two examples. Continuous curves are guides to the eye only.

Once the problem is decoupled, we proceed to apply these results to the multi-frequency synthesis of harmonic scattering patterns, treating each harmonic frequency independently, and deriving the desired (magnitude and phase) distributions of the equivalent currents for each meta-atom. This approach enables the application of conventional semi-analytical array design techniques,^[35] thereby removing the need for computationally expensive brute-force optimization procedures.

To illustrate the capabilities of our approach, we consider an STC-DM with 16 × 16 meta-atoms, with period d  = λ_c /2 and the same parameters as previous examples. We target Q  =  5 representative 2-D scattering patterns with nontrivial magnitude and phase distributions for harmonic orders ν  =   − 2, −1, 0, 1, 2, namely:

1. Tseng–Cheng-type optimal pattern with sidelobe level (SLL) of − 15 dB;^[35]

2. OAM beam with topological charge $$\ell = \ 1$$ (i.e., constant magnitude and spiral-type phase);^[36]

3. Triangular-shaped sector beam synthesized via the Fourier method;^[35]

4. Golay–Rudin–Shapiro diffuse-scattering pattern (with constant magnitude and binary phase);^[37]

5. Cross-shaped sector beam synthesized via the Fourier method.^[35]

The targeted magnitude and phase distributions, obtained semi-analytically, are shown in Figure S2 (Supporting Information), along with the corresponding scattering patterns. We stress that this joint synthesis does not correspond to any specific application, but is rather chosen to highlight the robustness and versatility of our approach in synthesizing diverse patterns requiring either magnitude-phase or phase-only equivalent current distributions.

Figure 5 illustrates the synthesized results for the five harmonic frequencies of interest. Specifically, Figure 5a–e display the synthesized magnitude distributions of the complex-valued equivalent currents $$a_{nm}^{( \nu )}$$, while Figure 5f–j show the corresponding phase distributions, and Figures 5k–o the resulting scattering patterns. The comparison with Figure S2 (Supporting Information) highlights the good agreement between the targeted and synthesized results, demonstrating the potential of the approach. Remarkably, three of these scattering patterns, which require nonuniform magnitude distributions, were previously impossible to synthesize with our semi-analytical approach,^[10] and have not been demonstrated with the previous ML-based strategy.^[23] Therefore, they represent the first examples of joint magnitude-phase multi-frequency syntheses with STC-DMs. The scattering patterns at higher harmonic orders (|ν| > 2), not shown for brevity, consistently exhibit levels below ≈− 8 dB, thereby confirming the increased spectral efficiency.

Figure 5

Numerical example of joint syntheses at Q  =  5 harmonic frequencies, for an STC-DM featuring 16 × 16 meta-atoms with period d  = λ_c /2, 2-bit coding sequence of length L  =  24, and ideal 90° phase jumps. a–e) Synthesized magnitude distributions for the complex-valued equivalent currents $$a_{nm}^{( {{\nu}} )}$$, in false-color scale, at harmonic orders ν  =   − 2, −1, 0, 1, 2, respectively. f–j) Corresponding phase distributions. k–o) Corresponding scattering patterns, with intensity normalized with respect to the maximum value at the harmonic order ν  =   − 1. The designed STC matrix is provided in the Supporting Information.

3.3 Experimental Validation

For our experimental validation, we utilize a 2-bit, X-band STC-DM prototype, similar to our previous studies.^{[10, 11]} This prototype features a layout of 16 columns and eight connected meta-atoms, as schematized in Figure 6a. Each column shares common control voltages, simplifying the electronics but restricting the beam-shaping/steering to one plane (ϕ  =  0) only. The meta-atom, illustrated in Figure 6b, consists of a hexagonal metal patch on a grounded dielectric substrate, connected via two PIN diodes to biasing lines, with a total unit cell size of 14 · 14 mm². A photograph of the fabricated prototype, measuring 224 · 140 mm², is shown in Figure 6c.

Figure 6

Microwave prototype and measurement setup. a,b) Schematics of the STC-DM and meta-atom, respectively. c,d) Photographs of the STC-DM prototype and measurement setup, respectively.

As illustrated in Figure S3 (Supporting Information), the meta-atom is numerically designed so as, at the operational frequency of 10.3 GHz, the four possible ON/OFF combinations of the two PIN diodes yield reflection phases differing approximately by 90° steps, with reflection magnitudes above ≈− 2 dB. However, from experimental characterization, we found slight discrepancies, with the actual reflection phases of 12°,  102°,  186°,  289°, and corresponding magnitudes of − 0.06, − 2.86, − 3.25, − 0.98 dB, in view of unmodeled imperfections, approximations, and fabrication tolerances.

The measurement setup, visualized as a photograph in Figure 6d, comprises an anechoic chamber housing a turntable hosting the prototype, along with a feeding antenna connected to a signal generator, and a receiving antenna connected to a spectrum analyzer. Accordingly, we implement our joint syntheses across Q  =  5 harmonic frequencies (ν  =   − 2, −1, 0, 1, 2). Each of the 16 columns, consisting of eight connected meta-atoms, is controlled with a temporal coding sequence of length L  =  24, having a period T₀ =  1 µs (modulation frequency f₀ =  1 MHz, corresponding to a diode switching rate of 24 MHz).

In this experimental scenario, considering the restrictions imposed by the prototype, we carry out 1-D syntheses:

1. Dolph–Chebyshev pattern with a SLL of − 15 dB and a steering angle of − 10°;^[35]

2. Difference pattern with cosine tapering and a steering angle of − 30°;

3. Uniform aperture pattern with a steering angle of 20°;

4. Four-beam pattern with peaks at ± 50° and ± 15°;^[38]

5. Golay–Rudin–Shapiro diffuse-scattering pattern, with a steering angle of 35°.^[37]

Also, in this case, the above patterns are synthesized in a semi-analytical fashion via conventional antenna array design methods. Figure S4 (Supporting Information) compares the target and synthesized equivalent current distributions, demonstrating once again a good agreement.

Figure 7 presents a comparison between measured and simulated scattering patterns, considering both ideal and imperfect phase responses. Generally, good agreement is observed, particularly when phase imperfections are taken into account. Minor discrepancies, such as slightly broader beams and higher sidelobes in the measured results, are attributed to inevitable non-idealities and modeling approximations. At the center frequency (ν  =  0, Figure 7c,h), moderately larger deviations are observable. Specifically, the effects around the main lobe are mainly due to blockage from the feeding antenna, whereas the speckle at larger angles is attributable to the interference between the STC-DM reflection and the backward radiation from the feeding antenna. Simulated and measured higher-order scattering patterns are shown in Figure S5 (Supporting Information), consistently exhibiting levels ≈10 dB below the maximum.

Figure 7

Experimental validation of joint syntheses at Q  =  5 harmonic frequencies, for an STC-DM featuring 16 × 8 meta-atoms with period d  =  0.48λ_c, and 2-bit (S  =  2) the coding sequence of length L  =  24. a–e) Numerically computed scattering patterns at harmonic orders ν  =   − 2, −1, 0, 1, 2, respectively. Orange-solid and purple-dashed curves pertain to the ideal (90°-step) and imperfect phase responses, respectively. f–j) Corresponding measured results at frequencies 10.298, 10.299, 10.3, 10.301, and 10.302 GHz, respectively. The designed STC matrix is provided in the Supporting Information.

These outcomes validate the proposed approach experimentally, confirming its practical feasibility.

3.4 Some Remarks

Although the examples presented so far focus on targeting consecutive harmonic frequencies, our approach can be extended to handle more general configurations. For instance, Figure S6 (Supporting Information) illustrates numerical results for the same target patterns as in Figure 7, but now assuming non-consecutive harmonic frequencies of interest, specifically ν  =   − 2, −1, 0, 1, 3. Results show comparable performance to the consecutive case. Additionally, Figure S7 (Supporting Information) confirms that the scattering patterns at other harmonic orders, including the interior value ν  =  2, maintain levels ≈10 dB below the maximum. Once again, this indicates increased flexibility in the design, by comparison with our previous semi-analytical approach.^[10] Nevertheless, there are limitations due to the inherent sinc-type decay of the equivalent currents in Equation (5). In particular, targeting an excessive power concentration in higher harmonic orders would likely result in an ill-conditioned synthesis with poor convergence of the tandem DNN.

A key advantage of our synthesis approach is its computational efficiency, which stems from synthesizing the equivalent currents across harmonic frequencies for individual meta-atoms. This renders the computational complexity of the tandem DNN essentially independent of the size of the STC-DM. Pattern synthesis is then efficiently handled using conventional array design methods. Although our examples have focused on semi-analytical cases, numerical methods^[39] could be applied as well for each single-frequency synthesis, in view of the problem decoupling, remaining computationally feasible. This stands in contrast to previous ML-based methods that target direct scattering pattern synthesis,^[23] which results in a computational complexity that scales unfavorably with the STC-DM size. As a further important difference with these ML-based methods, our DNN training depends only on the targeted harmonic frequencies and coding-sequence length and hence does not need to be repeated if there are changes in the target scattering patterns and/or STC-DM size.

Lastly, while our primary aim was to demonstrate the practical viability and potential of our synthesis approach using a relatively simple tandem DNN architecture, there exist avenues for improvement. For example, employing tensor or convolutional layers instead of (or in addition to) fully connected ones could offer more efficient modeling of the nonlinear input–output relationship,^{[25, 26]} and therefore help increase the number of controllable harmonics. Additionally, constrained implementations^[33] may enable enhanced control over spectral efficiency and other system-level metrics.

In the Supporting Information, we present results obtained using an alternative tandem DNN scheme, which implements the forward model analytically and employs a more complex inverse DNN with both convolutional and fully connected layers (Figure S8, Supporting Information). For a scenario with Q  =  5 harmonic frequencies, the synthesis results are comparable to those shown in Figure 2, particularly when phase quantization is applied (Figures S9–S12, Supporting Information).

For practical applications, especially considering the deployment of DNNs on edge computing devices like field-programmable gate arrays, it is crucial to design a DNN-based encoding scheme that can be implemented in real time. To achieve this, the DNN structure should be as simple as possible while still meeting accuracy requirements. Thus, the fully connected architecture proposed in Figure 2 may represent a reasonable compromise.

4 Conclusion

We have put forward a novel ML-based method for the multi-frequency synthesis of STC-DMs, which utilizes a tandem-DNN architecture to map desired equivalent-current distributions at different harmonic frequencies directly onto the coding sequences of a generic meta-atom. This effectively decouples the multi-frequency synthesis problem, allowing for the application of conventional semi-analytical array design techniques for harmonic beam shaping/steering, with the possibility to precisely tailor the magnitude and phase distributions. Within the emerging framework of STC-DMs, our approach stands out as a remarkable integration of ML-based and semi-analytical techniques. It enables the synthesis of complex scattering patterns that were previously unattainable with existing design methodologies while maintaining a computational complexity that is essentially independent of the STC-DM electrical size.

The promising results of our approach have been experimentally validated using a 2-bit, X-band STC-DM prototype, paving the way for significant advancements in applications such as smart radio environments for upcoming wireless networks,^{[5, 6]} as well as in sensing and imaging technologies.^{[40, 41]} Within this framework, the proposed ML-based approach can be seamlessly integrated with system-level, artificial-intelligence-driven strategies tailored for smart radio environments.^{[42, 43]} Also of great interest in this context is the investigation of rapid switching mechanisms, like those leveraging graphene or vanadium dioxide, to enable operations at higher frequencies, such as millimeter-wave and terahertz bands,^[44-46] where the use of PIN diodes is not viable.

5 Experimental Section

Modeling

The numerical implementation of the analytic modeling in Equations (2-5) was carried out using a MATLAB^[47] script. The design and modeling of the 2-bit meta-atom in the X-band microwave prototype were performed by means of full-wave numerical simulations relying on the commercial software package CST Microwave Studio.^[48] In these simulations, a normally incident plane–wave excitation was assumed, port-type terminations along the z-direction were employed, and periodic boundary conditions in the x − y plane were implemented. The PIN diodes were modeled as equivalent circuits, namely an RL circuit for the ON state (with R  =  2  Ω,  L  =  200 pH), and an RLC circuit for the OFF state (with R  =  3.3  Ω,  L  =  200 pH,  C  =  52 fF).

Tandem DNN Implementation

The fully connected tandem DNN was implemented using the Fitnet library in MATLAB.^[49] Both the forward and inverse DNN consisted of two hidden layers. Specifically, the first hidden layer of the forward DNN comprised 10L neurons, while the second hidden layer contained 20Q neurons. Similarly, for the inverse network, both hidden layers consisted of 20Q neurons. Radial basis functions were employed as activation functions for both DNNs, while the output layers utilized the Elliot sigmoid function.^[50] For the training algorithm, the resilient backpropagation method^[51] was applied, which is particularly suited to sigmoid-type activation functions. The mean squared error was computed via the function mse.^[49] For both forward and inverse DNNs, the datasets were allocated as follows: 70% for training, 15% for validation, and 15% for testing purposes. The training process typically required 3 h on a personal computer equipped with an Intel Core i7-8550U CPU (@ 1.80–2.00 GHz) and 8 GB of access memory. In the final phase quantization step, a nearest-neighbor search was applied, leveraging the degree of freedom in the reference phase to minimize errors. The alternative tandem DNN implementation is detailed in the Supporting Information.

Prototype Realization

The fabrication of the STC-DM prototype in Figure 6c was carried out by means of standard printed-circuit-board technology, utilizing a 1.5 mm thick F4B substrate with a dielectric constant of 2.65 and a loss tangent of 0.001. Each meta-atom integrates two embedded PIN diodes (M/A-COM MADP-000907-14020x) connected to the biasing lines. To implement dynamic biasing, an FPGA hardware control board (ALTERA Cyclone IV) was employed. This board was preloaded with a computer code designed to generate control signals based on the desired STC matrices.

Experiments

The experimental setup shown in Figure 6d comprised a linearly polarized X-band horn antenna, employed as a transmitter, and a signal generator (Keysight E8257D) providing the carrier microwave signal. The transmitter and STC-DM prototype were coaxially mounted on rotary support at a distance of 0.85  m, enabling a 360° rotation in the horizontal plane with a precision of 0.1°. To measure the harmonic scattering patterns, an identical horn antenna was utilized as a receiver and connected to a spectrum analyzer (Keysight E4447A). The characterization of the meta-atom's reflection response was performed using a vector network analyzer (Keysight N5230C).

Acknowledgements

G.C. and V.G. acknowledge partial support from the FRA 2023 Program of the University of Sannio. L. Z. and T.J.C. acknowledge the support from the National Natural Science Foundation of China (62288101 and 62101123), and the 111 Project (111-2-05).

Conflict of Interest

The authors declare no conflict of interest.

Open Research

Data Availability Statement

The data that support the findings of this study are available from the corresponding authors upon reasonable request.