# Dynamic Scattering Arrays for Simultaneous Electromagnetic Processing and
Radiation in Holographic MIMO Systems

Davide Dardari D. Dardari is with the Dipartimento di Ingegneria dell’Energia
Elettrica e dell’Informazione “Guglielmo Marconi” (DEI), WiLAB-CNIT,
University of Bologna, Cesena Campus, Cesena (FC), Italy, (e-mail:
davide.dardari@unibo.it).

###### Abstract

To meet the stringent requirements of next-generation wireless networks,
multiple-input multiple-output (MIMO) technology is expected to become massive
and pervasive. Unfortunately, this could pose scalability issues in terms of
complexity, power consumption, cost, and processing latency. Therefore, novel
technologies and design approaches, such as the recently introduced
holographic MIMO paradigm, must be investigated to make future networks
sustainable.

In this context, we propose the concept of a dynamic scattering array (DSA) as
a versatile 3D structure capable of performing joint wave-based computing and
radiation by moving the processing from the digital domain to the
electromagnetic (EM) domain. We provide a general analytical framework for
modeling DSAs, introduce specific design algorithms, and apply them to various
use cases. The examples presented in the numerical results demonstrate the
potential of DSAs to further reduce complexity and the number of
radiofrequency (RF) chains in holographic MIMO systems while achieving
enhanced EM wave processing and radiation flexibility for tasks such as
beamforming and single- and multi-user MIMO.

###### Index Terms:

Dynamic Scattering Arrays, Holographic MIMO; EM signal processing;
superdirectivity.

##  I Introduction

Next-generation wireless systems are expected to provide enhanced performance
in terms of capacity, reduced latency, and new functionalities such as
integrated sensing and communication [1, 2]. This trend is driving the
investigation of fundamental limits using physically consistent models, new
technologies, and novel design paradigms to approach them [3, 4, 5].

In this context, the recently introduced _holographic communications_ paradigm
is envisioned as a holistic way to manipulate, with unprecedented flexibility,
the electromagnetic (EM) field generated or sensed by an antenna [6]. It
involves designing innovative solutions capable of approaching the fundamental
limits imposed by the wireless channel through the massive deployment of
reconfigurable intelligent surfaces [7], extremely large antenna arrays (ELAA)
[8], and large intelligent surfaces, also known as holographic MIMO [9, 10,
11].

In particular, holographic MIMO surfaces are envisioned as an efficient
implementation of large antenna systems, advancing beyond massive MIMO and
ELAAs using simplified and reconfigurable hardware that is reduced in size,
cost and power consumption, and facilitate signal processing in the analog
domain [11]. In holographic MIMO, the density of the antenna elements is
increased, ideally approaching a continuous distribution, which allows for
highly flexible manipulation of EM waves. Additionally, exploiting higher
frequency bands, such as millimeter wave and THz, opens up the potential to
utilize the extra degrees-of-freedom (DoF) provided by the channel in the
radiating near-field propagation region, even under line-of-sight (LOS)
conditions [12, 13, 14, 15, 16, 17]. Closely spaced antenna elements (below
$\lambda/2$, with $\lambda$ being the wavelength) do not significantly
increase the exploitable DoF of the channel [18, 10]. Nevertheless, when the
elements spacing is made very small, strong coupling occurs, and the gain of a
linear array shifts from $N$, as in conventional arrays, to $N^{2}$, being $N$
the number of antenna elements, resulting in the so-called “superdirectivity”
[19]. While achieving very narrow beams using size-limited antennas is highly
desirable in many contexts, the superdirectivity effect is only obtained in
the end-fire direction of the array and can lead to extremely high driving
currents corresponding to high $Q$-factor values (and hence a lower bandwidth)
[20].

The utilization of high-frequency bands in the millimeter wave and THz ranges,
coupled with the integration of antennas with a very large number of elements,
is pushing current technology towards insurmountable barriers in hardware
complexity and power consumption. This issue, also known as _digital
bottleneck_ , poses serious challenges for the sustainability of future
wireless networks. In MIMO-based systems, hybrid digital-analog solutions have
been extensively explored to partially alleviate these issues by reducing the
number of radiofrequency (RF) chains and the digital processing burden, albeit
at the expense of flexibility [21].

A promising approach toward sustainability is to delegate part of the signal
processing directly to the EM level, known as electromagnetic signal and
information theory (ESIT) [22, 23, 24]. This can be achieved by designing
reconfigurable EM environments [25] using novel EM metamaterials devices to
perform basic processing functions (e.g., spatial first derivative) [26],
RISs, or the recently introduced stacked intelligent metasurfaces [27].
However, SIMs do not fully exploit the possibilities offered by EM phenomena
in antenna structures for processing and radiation, thereby inheriting some
limitations of standard antenna arrays as will be detailed later.

To overcome the aforementioned issues, we put forth the idea of a dynamic
scattering array (DSA) where EM processing and radiation are performed jointly
“over the air” by using a few active antenna elements surrounded by a cloud of
interacting programmable scatterers. As will be demonstrated, DSAs offer
several advantages in terms of flexibility and size reduction while minimizing
the number of RF chains required.

###  I-A Related State of the Art

The idea of introducing passive scattering elements close to an antenna to
exploit mutual coupling in the reactive near-field region and shape its
radiation properties dates back to the early days of wireless transmissions. A
pioneering work is that by H. Yagi published in 1928 that introduced the
classic Yagi-Uda array antenna. This antenna consists of a set of linear
parallel dipoles with a spacing of about $0.2-0.3\lambda$, where the first
dipole is active (i.e., fed with the signal) and the others act as passive
scatterers [28]. Over the subsequent decades, this idea has been extensively
developed in various directions within the antenna theory community.

In more recent times, the introduction of new technologies and materials has
enabled the development of new antenna structures designed to meet the demands
of higher performance, flexibility, frequency, and lower cost, as required by
new-generation wireless systems. Among these advancements, metasurface
antennas have received particular attention. They are composed of
subwavelength elements printed on a grounded dielectric slab. These antennas
exploit the interaction between a cylindrical surface wave, excited by a
monopole, and an anisotropic impedance boundary condition realized with
passive printed elements to produce an almost arbitrary aperture field [29].
The dynamic counterpart to static metasurface antennas is the dynamic
metasurface antenna (DMA), a type of traveling wave antenna with
reconfigurable subwavelength apertures. Since only one RF chain is needed for
each row of the planar array, DMAs offer a compelling solution, balancing
flexibility with a reduction in RF chains [30, 16]. Static and dynamic
metasurface antennas represent a promising avenue for the research aimed at
reducing power consumption in wireless systems, thanks to their ability to
shape signals at the EM level. Other recent technologies include fluid and
reconfigurable antennas [31, 32, 33]. However, they still lack the necessary
flexibility to completely replace their digital counterparts, especially for
complex linear processing operations.

An important step toward holographic MIMO is the introduction of SIMs or
stacked RIS [27, 34, 35, 36]. The concept of SIM originates from recent
advances in optics and microwave circuits designed to perform machine learning
tasks using layered EM surfaces to mimic a deep neural network for image
processing [37, 38]. Similarly, a SIM consists of a closed vacuum container
with several stacked metasurface layers. The first layer is an active planar
antenna array fed by a set of RF chains. The EM wave generated by the array
travels through subsequent layers composed of numerous reconfigurable meta-
atoms. The last layer is in charge of radiating the EM field outward. By
properly configuring the transmission properties of each meta-atom in the SIM,
the system can manipulate the traveling EM wave to produce a customized
waveform shape. Examples of optimization techniques for SIMs to achieve
various EM wave processing functionalities can be found in [36, 34, 27, 35].
Specifically, [27] investigates a holographic MIMO link where both the
transmitter and the receiver use a SIM. The authors show that with a 7-layer
SIM with $\lambda/2$ meta-atom spacing a good fit with the ideal singular
value decomposition (SVD) precoding and combining tasks is obtained. In [34]
and [35], optimization schemes to realize custom radiation patterns and multi-
user downlink beamforming are proposed, whereas the authors in [36]
demonstrate the capability to perform a 2D-Fourier transform in the wave
domain. Although an energy consumption model has not yet been studied, it is
expected that SIMs will be more energy efficient compared to conventional
digital transceivers.

The study of SIM is still in its infancy, and the main technological and
modeling challenges have yet to be fully identified. For instance, the
distance between the layers must be several wavelengths to validate the
currently adopted cascade model. Power losses and signal distortion caused by
the multiple layers and the bounding box require further investigation. A key
limitation of SIM is that only the final layer radiates, constraining the EM
wave characteristics to those achievable with a classical planar surface or
array, while the role of the hidden layers is to perform an EM transformation.

###  I-B Our Contribution

In this paper, we propose a 3D EM antenna structure, termed DSA, that offers
significant flexibility by providing enhanced processing capabilities directly
at the EM level. Specifically, a DSA consists of a limited number of active
antenna elements, each associated with an RF chain, surrounded by a cloud of
reconfigurable passive scatterers that interact with each other in the
reactive near field.

We first characterize the input-output characteristic of the DSA as a function
of its configuration parameters and provide a fully analytical modeling using
Hertzian dipoles. Subsequently, we propose a strategy to optimize these
parameters according to the desired processing functionality and apply this
strategy to three different use cases of interest: beamforming, multi-user
multiple-input single-output (MISO), and MIMO. Numerical results for each use
case demonstrate the great flexibility of the DSA in performing various signal
processing tasks. For instance, we will show that the tight coupling of DSA
elements allows for the realization of superdirectivity behavior independently
of the beam’s direction, in contrast to standard arrays where superdirectivity
can only be achieved in the end-fire direction [39, 20].

Compared to SIMs, in a DSA, all the elements contribute jointly to the
processing and radiation processes, thereby enhancing flexibility with more
compact structures. Indeed, in a SIM, only the elements of the last layer
radiate, and thus, the maximum achievable gain is proportional to their
number, similar to a conventional antenna array. Conversely, in the numerical
results, we demonstrate that a significantly higher gain can be achieved with
a DSA compared to a standard array. Furthermore, SIMs assume only forward
propagation from one layer to the next, with no coupling between the elements
of each layer. While this simplifies modeling and optimization significantly,
it presents critical challenges in decoupling implementation and imposes size
constraints. For example, in [27] and [34], a separation of $5\lambda$ between
each layer is considered, limiting the possibility of obtaining compact
structures. In contrast, our numerical results demonstrate that with a DSA,
better performance can be achieved with a spacing between elements of
$\lambda/4$, thanks to the joint processing and radiation involving all the
elements composing the DSA.

The results obtained in this paper demonstrate the potential of DSA to realize
wave-based computation “over the air” by leveraging the coupling between
active and scattering elements, thus minimizing the number of RF chains and
baseband processing in the digital domain. This is achieved through a semi-
passive structure with minimal energy consumption and nearly zero latency, as
the processing occurs at the speed of light.

Figure 1: Principle scheme of a dynamic scattering array.

###  I-C Notation and Definitions

Lowercase bold variables denote vectors in the 3D space, i.e.,
${\bf{p}}={\hat{\bf{x}}}\,p_{x}+{\hat{\bf{y}}}\,p_{y}+{\hat{\bf{z}}}\,p_{z}$
is a vector with cartesian coordinates $(p_{x},p_{y},p_{z})$, $\hat{\bf{p}}$
is a unit vector denoting its direction, and $p=|{\bf{p}}|$ denotes its
magnitude, where ${\hat{\bf{x}}}$, ${\hat{\bf{y}}}$, and ${\hat{\bf{z}}}$
represent the unit vectors in the $x$, $y$ and $z$ directions, respectively.
The scalar product between vectors ${\bf{p}}$ and ${\bf{r}}$ is indicated with
${{\bf{p}}}\bm{\cdot}{{\bf{r}}}$. $\delta(x)$ represents the Dirac delta
distribution and
$\delta({\bf{p}})=\delta(p_{x})\,\delta(p_{y})\,\delta(p_{z})$ its 3D version.
Boldface capital letters are matrices (e.g., ${\bf{A}}$), where ${\bf{I}}_{N}$
is the identity matrix of size $N$, $\bm{0}_{N}$ is the zero matrix of size
$N$. With reference to a generic matrix ${\bf{A}}$, $a_{n,m}=[{\bf{A}}]_{n,m}$
represents its $(n,m)$th element, whereas ${\bf{A}}^{\mathsf{T}}$ and
${\bf{A}}^{\mathsf{H}}$ indicate the transpose and the conjugate transpose of
${\bf{A}}$. We denote with
${\bf{A}}^{{\dagger}}=({\bf{A}}^{\mathsf{H}}\,{\bf{A}})^{-1}{\bf{A}}^{\mathsf{H}}$
the Moore-Penrose inverse (pseudo-inverse) and with
$\left\|{\bf{A}}\right\|_{\text{F}}$ the Frobenius norm of matrix ${\bf{A}}$.
We indicate with $\Re\left\\{z\right\\}$, $\Im\left\\{z\right\\}$, and
$z^{*}$, respectively, the real part, imaginary part, and complex conjugate of
the complex number $z$, and $\jmath$ the imaginary unit. Denote with
$\eta=377\,$Ohm the free-space impedance and $c$ the speed of light.

###  I-D Paper Organization

The rest of the paper is organized as follows: Sec. II introduces and models
the proposed DSA. In Sec. III, an analytical characterization of the DSA is
developed using Hertzian dipoles. The optimization of the parameters
characterizing the DSA is addressed in general in Sec. IV, and further
declined in 3 use cases. Numerical results associated with the 3 use cases are
provided in Sec. V. Finally, the conclusions are drawn in Sec. VI.

##  II  dynamic scattering array (DSA) Modeling

We consider the transmitting system illustrated in Fig. 1, where the DSA is
composed of $N_{\text{A}}$ active antenna elements surrounded by a cloud of
$N_{\text{S}}$ reconfigurable passive scatterers. The $n$th element (active or
passive) of the DSA, with $n=1,2,\ldots N$, $N=N_{\text{A}}+N_{\text{S}}$, is
located in position ${\bf{p}}_{n}$. The $n$th scatterer, with $n=1,2,\ldots
N_{\text{S}}$, is itself an antenna element terminated by a reconfigurable
impedance load $Z_{\text{S}_{n}}\left(\theta_{n}\right)$. To avoid power
dispersion, we impose that the load is purely reactive, i.e.,
$Z_{\text{S}_{n}}\left(\theta_{n}\right)=\jmath\theta_{n}$, where
$\theta_{n}\in\mathbb{R}$ is the reconfigurable reactance.111From the
technological point of view, the reconfigurable impedance load can be
implemented using solutions similar to those applied for RISs such as PIN
diodes and varactors [40]. Denote with
$\bm{\theta}=\left[\theta_{1},\theta_{2},\ldots,\theta_{N_{\text{S}}}\right]^{\mathsf{T}}$
the set of reconfigurable reactances and with
$\mathbf{Z}_{\text{S}}(\bm{\theta})={\rm
diag}\left(Z_{\text{S}_{1}}\left(\theta_{1}\right),Z_{\text{S}_{2}}\left(\theta_{2}\right),\ldots,Z_{\text{S}_{N_{\text{S}}}}\left(\theta_{N_{\text{S}}}\right)\right)$
the corresponding set of loads.

The DSA can be modeled as a linear $N$-port network. The real-value bandpass
voltage $v^{\text{(RF)}}(t)$ present at the generic port of the network is a
modulated RF signal with carrier frequency $f_{0}$ and it can be represented
as

$v^{\text{(RF)}}(t)=\sqrt{2}\Re\left\\{v(t)\,e^{\jmath 2\pi
f_{0}\,t}\right\\}$ (1)

where $v(t)$ is the associated complex voltage envelope. Similarly, we define
the complex envelope $i(t)$ of the real-value bandpass current
$i^{\text{(RF)}}(t)$ flowing into the port. Since the modulated signal carries
information, then they have to be treated as random processes. The (average)
active power delivered to the port is
$P={\mathbb{E}}\left\\{{\Re\left\\{v^{\text{(RF)}}(t)\,i^{\text{(RF)}}(t)\right\\}}\right\\}={\mathbb{E}}\left\\{{\Re\left\\{v^{*}(t)\,i(t)\right\\}}\right\\}$,
where ${\mathbb{E}}\left\\{{\cdot}\right\\}$ is the statistical expectation
and the last equality holds thanks to the arbitrary constant $\sqrt{2}$ in
(1). We assume that the bandwidth $B$ of $v^{\text{(RF)}}(t)$ is sufficiently
small such that the network properties are constant within $B$ and can be
evaluated at the carrier frequency $f_{0}$.

In a compact description, we define
$\mathbf{v}_{\text{A}}\in\mathbb{C}^{N_{\text{A}}\times 1}$ and
$\mathbf{i}_{\text{A}}\in\mathbb{C}^{N_{\text{A}}\times 1}$ the complex
voltage and current envelopes (in the following denoted simply as voltages and
currents) at the ports of the $N_{\text{A}}$ active antennas (i.e., the input
ports of the DSA), and $\mathbf{v}_{\text{S}}\in\mathbb{C}^{N_{\text{S}}\times
1}$ and $\mathbf{i}_{\text{S}}\in\mathbb{C}^{N_{\text{S}}\times 1}$ the
voltages and currents at the ports of the $N_{\text{S}}$ scatterers.222For
notation convenience, in the following we omit the time dependence.  We
collect all the currents and voltages at the $N$ ports of the DSA in the
vectors

$\displaystyle\mathbf{i}=\left[\begin{array}[]{c}\mathbf{i}_{\text{A}}\\\
\mathbf{i}_{\text{S}}\end{array}\right]$
$\displaystyle\mathbf{v}=\left[\begin{array}[]{c}\mathbf{v}_{\text{A}}\\\
\mathbf{v}_{\text{S}}\end{array}\right]\,.$ (6)

All the interactions between the elements of the DSA are captured by the
impedance matrix $\mathbf{Z}\in\mathbb{C}^{2N\times 2N}$, which does not
depend on the reconfigurable loads, and relates the voltages and currents of
the $N$ ports as $\mathbf{v}=\mathbf{Z}\,\mathbf{i}$. In particular, the
$(n,m)$th element of $\mathbf{Z}$ represents the mutual coupling coefficient
between the $n$th and $m$th elements obtained as the ratio between the open-
circuit voltage observed at the $m$th port and the excitation current applied
to the $n$th port supposing that the remaining ports are kept open (no current
flow) [41]. It is convenient to divide the impedance matrix $\mathbf{Z}$ into
the sub-matrices $\mathbf{Z}_{\text{AA}}$, $\mathbf{Z}_{\text{AS}}$,
$\mathbf{Z}_{\text{SA}}$, and $\mathbf{Z}_{\text{SS}}$ so that we can write

$\displaystyle\left[\begin{array}[]{c}\mathbf{v}_{\text{A}}\\\
\mathbf{v}_{\text{S}}\end{array}\right]=\left[\begin{array}[]{cc}\mathbf{Z}_{\text{AA}}&\mathbf{Z}_{\text{AS}}\\\
\mathbf{Z}_{\text{SA}}&\mathbf{Z}_{\text{SS}}\end{array}\right]\cdot\left[\begin{array}[]{c}\mathbf{i}_{\text{A}}\\\
\mathbf{i}_{\text{S}}\end{array}\right]\,.$ (13)

The voltages $\mathbf{v}_{\text{A}}$ and currents $\mathbf{i}_{\text{A}}$ at
the input port of the DSA are related by the input impedance of the DSA
$\mathbf{Z}_{\text{A}}(\bm{\theta})\in\mathbb{C}^{N_{\text{A}}\times
N_{\text{A}}}$ as
$\mathbf{v}_{\text{A}}=\mathbf{Z}_{\text{A}}(\bm{\theta})\,\mathbf{i}_{\text{A}}$.
By elaborating (13) and considering that at the scatterers’ ports it is
$\mathbf{v}_{\text{S}}=-\mathbf{Z}_{\text{S}}(\bm{\theta})\,\mathbf{i}_{\text{S}}$,
we obtain

$\displaystyle\mathbf{Z}_{\text{A}}(\bm{\theta})=\mathbf{Z}_{\text{AA}}-\mathbf{Z}_{\text{AS}}\,\left(\mathbf{Z}_{\text{SS}}+\mathbf{Z}_{\text{S}}(\bm{\theta})\right)^{-1}\,\mathbf{Z}_{\text{SA}}$
(14)

which depends on parameters $\bm{\theta}$ through the scatterers’ loads
$\mathbf{Z}_{\text{S}}(\bm{\theta})$.

The $N_{\text{A}}$ active elements are connected to $N_{\text{A}}$ RF chains,
including digital-to-analog converter (DAC) and up-conversion stages, through
a $2N_{\text{A}}$-port power matching network whose purpose is to maximize the
power transfer between the RF chains and the DSA. The relationship between the
voltages and the currents present at the ports of the matching network is
given by

$\displaystyle\left[\begin{array}[]{c}\mathbf{v}_{\text{T}}\\\
\mathbf{v}_{\text{A}}\end{array}\right]=\mathbf{Z}_{\text{M}}(\bm{\theta})\,\left[\begin{array}[]{c}\mathbf{i}_{\text{T}}\\\
-\mathbf{i}_{\text{A}}\end{array}\right]$ (19)

where, assuming no losses in the power matching network,
$\mathbf{Z}_{\text{M}}(\bm{\theta})$ is equal to [39]

$\displaystyle\mathbf{Z}_{\text{M}}(\bm{\theta})=\left[\begin{array}[]{cc}\bm{0}_{N_{\text{A}}}&-\jmath\sqrt{R}\,\Re\left\\{\mathbf{Z}_{\text{A}}(\bm{\theta})\right\\}^{\frac{1}{2}}\\\
-\jmath\sqrt{R}\,\Re\left\\{\mathbf{Z}_{\text{A}}(\bm{\theta})\right\\}^{\frac{1}{2}}&-\jmath\Im\left\\{\mathbf{Z}_{\text{A}}(\bm{\theta})\right\\}\end{array}\right]$
(22)

where $R$ is the output resistance of each RF chain. As it can be noticed from
(22), the matching network depends on the DSA input impedance
$\mathbf{Z}_{\text{A}}(\bm{\theta})$ and hence on the parameters
$\bm{\theta}$. Therefore, for a given DSA configuration, the matching network
should be changed accordingly. This is possible using, for instance, adaptive
matching networks [42]. In case a fixed network is employed instead, then not
all the available power would be transferred to the DSA. With the power
matching network, it turns out that
$\mathbf{v}_{\text{T}}=\mathbf{Z}_{\text{T}}\,\mathbf{i}_{\text{T}}$, with
$\mathbf{Z}_{\text{T}}=R\,{\bf{I}}_{N_{\text{A}}}$ and, assuming lossless
antenna elements, the transmitted power corresponds to the radiated power
$P_{\text{rad}}$ given by

$\displaystyle
P_{\text{T}}=P_{\text{rad}}={\mathbb{E}}\left\\{{\mathbf{i}_{\text{A}}^{\mathsf{H}}\,\Re\left\\{\mathbf{Z}_{\text{A}}(\bm{\theta})\right\\}\,\mathbf{i}_{\text{A}}}\right\\}=\frac{{\mathbb{E}}\left\\{{\mathbf{v}_{\text{T}}^{\mathsf{H}}\,\mathbf{v}_{\text{T}}}\right\\}}{R}\,.$
(23)

The reactive power can be evaluated as well as

$\displaystyle
P_{\text{react}}={\mathbb{E}}\left\\{{\mathbf{i}_{\text{A}}^{\mathsf{H}}\,\Im\left\\{\mathbf{Z}_{\text{A}}(\bm{\theta})\right\\}\,\mathbf{i}_{\text{A}}}\right\\}$
(24)

from which the $Q$-factor of the system can be obtained as
$Q=P_{\text{react}}/P_{\text{rad}}$. The inverse of the $Q$-factor can be used
as a rough estimate of the bandwidth of the DSA when $Q\gg 1$[41].

We also foresee an optional baseband linear processing block implemented in
the digital domain (digital precoder), described by the matrix
$\mathbf{W}_{\text{D}}\in\mathbb{C}^{N_{\text{A}}\times N_{\text{A}}}$, whose
input is represented by the information vector
${\bf{x}}\in\mathcal{C}^{N_{\text{A}}\times 1}$ to be transmitted. The matrix
$\mathbf{W}_{\text{D}}$ relates the transmitted information vector ${\bf{x}}$
and the voltages $\mathbf{v}_{\text{T}}$ at the input of the matching network,
i.e., $\mathbf{v}_{\text{T}}=\mathbf{W}_{\text{D}}\,{\bf{x}}$. In the
following, we impose the arbitrarily normalization
$\|\mathbf{W}_{\text{D}}\|_{\text{F}}=\sqrt{R\,N_{\text{A}}}$. Such a
normalization ensures that
${\mathbb{E}}\left\\{{{\bf{x}}^{\mathsf{H}}\,{\bf{x}}}\right\\}=P_{\text{T}}$,
as typically considered in the signal processing community. The reconfigurable
reactances $\bm{\theta}$ along with the elements of matrix
$\mathbf{W}_{\text{D}}$ represent the set of parameters to be optimized to
obtain the desired processing functionality, as will be described later.

From (19) and (22) it is

$\displaystyle\mathbf{i}_{\text{A}}=\frac{1}{\jmath\sqrt{R}}\,\Re\left\\{\mathbf{Z}_{\text{A}}(\bm{\theta})\right\\}^{-\frac{1}{2}}\mathbf{v}_{\text{T}}={\bf{M}}(\bm{\theta})\,\mathbf{v}_{\text{T}}\,.$
(25)

Moreover, we have

$\displaystyle\mathbf{i}=\left[\begin{array}[]{c}{\bf{I}}_{N_{\text{A}}}\\\
-\left(\mathbf{Z}_{\text{SS}}+\mathbf{Z}_{\text{S}}(\bm{\theta})\right)^{-1}\mathbf{Z}_{\text{SA}}\end{array}\right]\cdot\mathbf{i}_{\text{A}}={\bf{C}}(\bm{\theta})\,\mathbf{i}_{\text{A}}\,.$
(28)

By combining (25) and (28), we obtain a compact relationship between the
information vector ${\bf{x}}$ and the total current $\mathbf{i}$ flowing in
the DSA

$\mathbf{i}={\bf{C}}(\bm{\theta})\,{\bf{M}}(\bm{\theta})\,\mathbf{W}_{\text{D}}\,{\bf{x}}=\mathbf{W}_{\text{EM}}(\bm{\theta})\,\mathbf{W}_{\text{D}}\,{\bf{x}}$
(29)

where $\mathbf{W}_{\text{EM}}(\bm{\theta})$

$\displaystyle\mathbf{W}_{\text{EM}}(\bm{\theta})=\frac{1}{\jmath\sqrt{R}}\,\left[\begin{array}[]{c}\Re\left\\{\mathbf{Z}_{\text{A}}(\bm{\theta})\right\\}^{-\frac{1}{2}}\\\
-\left(\mathbf{Z}_{\text{SS}}+\mathbf{Z}_{\text{S}}(\bm{\theta})\right)^{-1}\mathbf{Z}_{\text{SA}}\,\Re\left\\{\mathbf{Z}_{\text{A}}(\bm{\theta})\right\\}^{-\frac{1}{2}}\end{array}\right]$
(32)

accounts for the EM-level signal processing operated by the reconfigurable DSA
as a function of the parameters $\bm{\theta}$.

Consider now $K$ test points located in positions ${\bf{t}}_{k}$, with
$k=1,2,\ldots,K$, and that in each test point a conventional receiving antenna
is used to receive the signal with gain $G_{\text{R}}$. As commonly done in
the literature, we assume that $|{\bf{t}}_{k}-{\bf{p}}_{n}|\gg\lambda$,
$\forall k,n$, that is, the test points are located in the radiative region of
the antenna structure and the receiving antennas do not affect the
transmitting DSA. As an example, the set of $K$ antennas could represent a
conventional receiving antenna array of a MIMO system. The useful component
(i.e., without noise) of the received signal at the test positions is

$\displaystyle{\bf{y}}=\mathbf{H}_{\text{c}}\,\mathbf{i}=\mathbf{H}_{\text{c}}\,\mathbf{W}_{\text{EM}}(\bm{\theta})\,\mathbf{W}_{\text{D}}\,{\bf{x}}={\bf{H}}(\bm{\theta},\mathbf{W}_{\text{D}})\,{\bf{x}}$
(33)

where $\mathbf{H}_{\text{c}}$ is the transimpedance matrix of the radio
channel accounting for the propagation effects and
${\bf{H}}(\bm{\theta},\mathbf{W}_{\text{D}})$ is the end-to-end baseband
equivalent channel matrix as commonly defined in signal processing.

##  III DSA with Hertzian Dipoles

To further develop our framework and derive some numerical examples from which
obtaining important insights about the potential of DSAs, we suppose that all
the antenna elements in the DSA, both active and scatterers, are modeled as
Hertzian dipoles. In addition to making the derivation analytically tractable,
the consideration of Hertzian dipoles can also be justified by the fact that
the discrete infinitesimal dipole approximation is a flexible technique in
computational electromagnetics for modeling and solving more complex
antenna/scatterers structures [43].

Accordingly, we model the $n$th dipole as a very short wire of length $l_{n}$
and radius $r_{n}$, with $l_{n},r_{n}\ll\lambda$ and generic orientation
$\bm{\Omega}_{n}=\left[\sin(\phi_{n})\,\cos(\psi_{n})\,,\sin(\phi_{n})\,\sin(\psi_{n})\,,\cos(\phi_{n})\right]^{\mathsf{T}}$,
where $\phi_{n}$ and $\psi_{n}$ represent, respectively, the elevation angle
and the angle with respect to axis $z$, according to the reference coordinates
system illustrated in Fig. 1. The $n$th element $i_{n}$ of vector $\mathbf{i}$
is the current present at the port of the $n$th dipole.

The total current density distribution in the DSA is given by

$\displaystyle{\mathsf{J}}({\bf{p}})=\sum_{n=1}^{N}\bm{\Omega}_{n}\,l_{n}\,i_{n}\,\delta({\bf{p}}-{\bf{p}}_{n})\,.$
(34)

Therefore, the EM field at the generic location ${\bf{p}}$ excited by the
current density ${\mathsf{J}}({\bf{p}})$ in free space can be evaluated as
[25]

$\displaystyle{\mathsf{E}}({\bf{p}})=\int_{\mathcal{V}}\mathbf{G}_{{EJ}}\left({{\bf{p}}-{\bf{s}}}\right)\,{\mathsf{J}}({\bf{s}})\,d{\bf{s}}$
(35)

where $\mathcal{V}$ is the volume in which the current flows and
$\mathbf{G}_{{EJ}}\left({{\bf{r}}}\right)$ is the Green’s dyadic given by [41]

$\displaystyle\mathbf{G}_{{EJ}}\left({{\bf{r}}}\right)=-\jmath\frac{\eta\,e^{-\jmath\kappa_{0}\,|{\bf{r}}|}}{2\lambda|{\bf{r}}|}$
$\displaystyle\quad\quad\cdot\left[\left({\bf{I}}-\hat{{\bf{r}}}\,\hat{{\bf{r}}}^{\mathsf{T}}\right)+\frac{\jmath\lambda}{2\pi|{\bf{r}}|}\left({\bf{I}}-3\hat{{\bf{r}}}\,\hat{{\bf{r}}}^{\mathsf{T}}\right)-\frac{\lambda^{2}}{(2\pi|{\bf{r}}|)^{2}}\left({\bf{I}}-3\hat{{\bf{r}}}\,\hat{{\bf{r}}}^{\mathsf{T}}\right)\right]$
(36)

with $\kappa_{0}=2\pi/\lambda$ being the wave number and where we grouped the
terms multiplying the factors $1/|{\bf{r}}|$, $1/|{\bf{r}}|^{2}$, and
$1/|{\bf{r}}|^{3}$. By substituting (34) in (35), we obtain

$\displaystyle{\mathsf{E}}({\bf{p}})=\sum_{n=1}^{N}\,l_{n}\,i_{n}\,\mathbf{G}_{{EJ}}\left({{\bf{p}}-{\bf{p}}_{n}}\right)\,\bm{\Omega}_{n}\,.$
(37)

The open-circuit voltage at the $m$th dipole caused by the current $i_{n}$ of
the $n$th dipole by assuming all the other currents are equal to zero is

$\displaystyle
V_{m}^{(\text{oc})}=l_{m}\,l_{n}\,i_{n}\,\bm{\Omega}_{m}^{\mathsf{T}}\,\mathbf{G}_{{EJ}}\left({{\bf{p}}_{m}-{\bf{p}}_{n}}\right)\,\bm{\Omega}_{n}\,.$
(38)

From (38), the generic element $Z_{m,n}=[\mathbf{Z}]_{m,n}$ of the impedance
matrix $\mathbf{Z}$ associated with the DSA can be easily obtained as

$\displaystyle
Z_{m,n}=\frac{V_{m}^{(\text{oc})}}{i_{n}}=l_{n}\,l_{m}\,\bm{\Omega}_{n}^{\mathsf{T}}\,\mathbf{G}_{{EJ}}\left({{\bf{p}}_{n}-{\bf{p}}_{m}}\right)\,\bm{\Omega}_{m}$
(39)

for $n\neq m$. For $n=m$, $Z_{n,n}$ corresponds to the self-impedance of the
Hertzian dipole which is given by [41]

$\displaystyle
Z_{n,n}=\frac{2}{3}\pi\eta\left(\frac{l_{n}}{\lambda}\right)^{2}+\frac{\ln\left(\frac{l_{n}}{r_{n}}\right)}{\jmath
f_{0}\pi^{2}\epsilon l_{n}}\,.$ (40)

In the free-space propagation condition, the channel transimpedance matrix
$\mathbf{H}_{\text{c}}$ in (33) can be easily derived. To elaborate, the
electrical field intensity at the $k$ receiving test antenna located at
${\bf{t}}_{k}$ can be computed once the currents $\mathbf{i}$, flowing into
the dipoles (active and scatterers), are known

$\displaystyle
e_{k}=\bm{\Theta}_{k}^{\mathsf{T}}\,{\mathsf{E}}({\bf{t}}_{k})=\sum_{n_{1}}^{N}l_{n}\,\bm{\Theta}_{k}^{\mathsf{T}}\,\mathbf{G}_{{EJ}}\left({{\bf{t}}_{k}-{\bf{p}}_{n}}\right)\,\bm{\Omega}_{n}\,i_{n}$
(41)

where $\bm{\Theta}_{k}$ is the polarization direction of the $k$th receiving
antenna. As a consequence, the received signal (assuming matched load and
normalized to an impedance $R_{\text{L}}=1\,$Ohm) is given by
$y_{k}=e_{k}\sqrt{\frac{A_{\text{eff}}R_{\text{L}}}{\eta}}=e_{k}\sqrt{\frac{\lambda^{2}\,G_{\text{R}}}{4\pi\eta}}$.
Finally, the $(k,n)$th element of the transimpedance matrix
$\mathbf{H}_{\text{c}}$ is given by

$[{H_{\text{c}}}]_{k,n}=\sqrt{\frac{\lambda^{2}\,G_{\text{R}}}{4\pi\eta}}l_{n}\,\bm{\Theta}_{k}^{\mathsf{T}}\,\mathbf{G}_{{EJ}}\left({{\bf{t}}_{k}-{\bf{p}}_{n}}\right)\,\bm{\Omega}_{n}\,.$
(42)

Since it is $|{\bf{r}}_{k,n}|=|{\bf{t}}_{k}-{\bf{p}}_{n}|\gg\lambda$, (42) can
be well approximated by neglecting the $1/|{\bf{r}}|^{2}$ and
$1/|{\bf{r}}|^{3}$ components (reactive field) in (III) thus leading to

$\displaystyle\mathbf{G}_{{EJ}}\left({{\bf{r}}}\right)\simeq-\jmath\frac{\eta\kappa_{0}}{4\pi|{\bf{r}}|}\,e^{-\jmath\kappa_{0}\,|{\bf{r}}|}\,\left({\bf{I}}_{3}-\hat{{\bf{r}}}_{k,n}\,\hat{{\bf{r}}}^{\mathsf{T}}_{k,n}\right)$
(43)

and

$\displaystyle[{H_{\text{c}}}]_{k,n}\simeq-\jmath\frac{\eta}{2\lambda|{\bf{r}}|}\sqrt{\frac{G_{\text{R}}}{4\pi\eta}}l_{n}\,e^{-\jmath\kappa_{0}\,|{\bf{r}}|}\bm{\Theta}_{k}^{\mathsf{T}}\left({\bf{I}}_{3}-\hat{{\bf{r}}}_{k,n}\,\hat{{\bf{r}}}^{\mathsf{T}}_{k,n}\right)\bm{\Omega}_{n}\,.$
(44)

##  IV DSA Optimization

In this section, we illustrate a possible optimization strategy of the DSA
along with its application to 3 different use cases. Numerical results
associated with each use case will be reported in Sec. V.

###  IV-A Joint Optimization of the DSA and the Digital Precoder

For a given propagation scenario characterized by the transimpedance matrix
$\mathbf{H}_{\text{c}}$ and $K$ test points, suppose we fix a specification
${\bf{H}_{\text{opt}}}$ on the desired end-to-end channel matrix. The
reconfigurable parameters of the DSA and the digital precoder can be obtained
by solving the following constrained optimization problem:

$\displaystyle\underset{{\bm{\theta},\,\mathbf{W}_{\text{D}},\,\alpha}}{\operatorname{minimize}}\left\|\alpha\,\mathbf{H}_{\text{c}}\,\mathbf{W}_{\text{EM}}(\bm{\theta})\,\mathbf{W}_{\text{D}}-{\bf{H}_{\text{opt}}}\right\|_{\text{F}}$
(45) $\displaystyle
s.t.\quad\|\mathbf{W}_{\text{D}}\|_{\text{F}}^{2}=R\,N_{\text{A}}$

where the scalar $\alpha\in\mathbb{R}$ accounts for the possible lack in the
link budget to achieve ${\bf{H}_{\text{opt}}}$ that has to be eventually coped
with increased transmitted power.

Unfortunately, in general, the constrained optimization problem in (45) is not
convex thus making its numerical solution more challenging. A possible
approach to translating it into an unconstrained optimization problem is to
resort to the following alternate optimization procedure:

  * •

STEP 0: We set the candidate vector $\hat{\bm{\theta}}$ to an initial guess
value, for instance randomly chosen or equal to zero.

  * •

STEP 1:

For a fixed $\hat{\bm{\theta}}$, it is possible to find in closed form the
values of $\mathbf{W}_{\text{D}}$ and $\alpha$ that minimize the objective
function and satisfy the constraint in (45). In particular, we solve the
following equation

$\displaystyle\alpha\,\mathbf{H}_{\text{c}}\,\mathbf{W}_{\text{EM}}(\hat{\bm{\theta}})\,\mathbf{W}_{\text{D}}={\bf{H}_{\text{opt}}}$
(46)

of which the minimum norm solution is

$\displaystyle\alpha\mathbf{W}_{\text{D}}=\left(\mathbf{H}_{\text{c}}\,\mathbf{W}_{\text{EM}}(\hat{\bm{\theta}})\right)^{{\dagger}}{\bf{H}_{\text{opt}}}\,.$
(47)

The corresponding values of $\hat{\alpha}$ and ${\hat{\mathbf{W}}}_{\text{D}}$
are, therefore,

$\displaystyle\hat{\alpha}=\frac{1}{\sqrt{R\,N_{\text{A}}}}\left\|\left(\mathbf{H}_{\text{c}}\,\mathbf{W}_{\text{EM}}(\hat{\bm{\theta}})\right)^{{\dagger}}{\bf{H}_{\text{opt}}}\right\|_{\text{F}}$
(48)

and

$\displaystyle{\hat{\mathbf{W}}}_{\text{D}}=\frac{1}{\hat{\alpha}}\left(\mathbf{H}_{\text{c}}\,\mathbf{W}_{\text{EM}}(\hat{\bm{\theta}})\right)^{{\dagger}}{\bf{H}_{\text{opt}}}\,.$
(49)

  * •

STEP 2:

Starting from the candidates $\hat{\alpha}$ and
${\hat{\mathbf{W}}}_{\text{D}}$ from STEP 1, the following unconstrained
minimization problem involving only $\bm{\theta}$ is solved numerically

$\displaystyle\hat{\bm{\theta}}=\underset{{\bm{\theta}}}{\operatorname{arg\,min}}\left\|\hat{\alpha}\,\mathbf{H}_{\text{c}}\,\mathbf{W}_{\text{EM}}(\hat{\bm{\theta}})\,\mathbf{W}_{\text{D}}-{\bf{H}_{\text{opt}}}\right\|_{\text{F}}\,.$
(50)

  * •

STEP 3: Repeat STEPS 1 and 2 for a given number $N_{\text{alt}}$ iterations or
until no significant variations of the parameters take place.

The inclusion of the digital precoder, while not strictly necessary, may aid
in the optimization process of the DSA by handling a portion of the processing
when $N_{\text{A}}>1$. If omitted, we set
${\hat{\mathbf{W}}}_{\text{D}}=\mathbf{W}_{\text{D}}=\sqrt{R}\,{\bf{I}}$, and
only proceed with STEP 2 once. Of course, alternative optimization problems
apart from (45) can also be considered, such as those involving the $Q$-factor
as an optimization target or constraint. In many applications (e.g., beam
steering, as seen in use case 1 below), the optimization process can be
performed offline, computed once to create a dictionary of various DSA
parameters for different steering directions of interest or, more broadly,
different beam forms. In such cases, the complexity of the optimization
algorithm is not a concern. However, in other scenarios, performing
optimization offline may not be feasible, necessitating online optimization
based, for example, on the channel state information (CSI). In this situation,
developing efficient algorithms to minimize (50), potentially considering the
structure of $\mathbf{W}_{\text{EM}}(\hat{\bm{\theta}})$, becomes pertinent.
This aspect exceeds the scope of this paper and is reserved for future
investigations. In the numerical results, the standard numerical tool based on
the quasi-Newton method [44] with $N_{\text{i}}$ iterations is utilized to
minimize (50).

Figure 2: Use case 1: Single RF chain DSA for high gain beamforming.

###  IV-B Use Case 1: Beam Forming

Suppose we want to design a single-input DSA ($N_{\text{A}}=1$, no digital
precoding) realizing a generic radiation diagram, for instance, imposing that
at the test locations ${\bf{t}}_{k}$, the channel matrix
$[{\bf{H}_{\text{opt}}}]_{k,1}$, $k=1,2,\ldots,K$, assumes the desired value
(see Fig. 2). For instance, one might define a uniform set of test points
${\bf{t}}_{k}=[d\sin\phi_{k},0,d\cos\phi_{k}]^{\mathsf{T}}$ on the $x-z$ plane
a distance $d$ in the far-field region of the DSA, with $\phi_{k}=2\pi k/K$,
and maximize the radiation diagram at the $k=\tilde{k}$th direction
$\phi_{\tilde{k}}$ (beam steering). This can be achieved by setting
$[{\bf{H}_{\text{opt}}}]_{k,1}=\sqrt{P_{\text{R}}}$, for $k=\tilde{k}$, and
zero otherwise, being $P_{\text{R}}$ the desired received power. Incidentally,
after the optimization process, the resulting parameter $\hat{\alpha}^{2}$
will represent the required transmitted power $P_{\text{T}}$ to satisfy the
link budget. It is worth noticing that, in principle, only the optimization at
the single test point ${\bf{t}}_{\tilde{k}}$, i.e., $K=1$, would suffice to
obtain the desired beam steering result. However, the addition of many other
test points set to zero helps the minimization algorithm to speed up and avoid
the convergence to bad local minima.

Figure 3: Use case 2: Multi-user downlink MISO with a DSA.

###  IV-C Use Case 2: Multi-user MISO

In this second use case, we consider a multi-user MISO downlink scenario shown
in Fig. 3, where an DSA with $N_{\text{A}}$ RF chains (one per user) has to be
designed to serve simultaneously $K=N_{\text{A}}$ single-antenna users located
in generic positions ${\bf{t}}_{k}$, $k=1,2,\ldots,K$, by maximizing the
signal-to-interference noise ratio (SINR) at each user according to the zero-
forcing criterium [45]. Users can be both in the radiating near-field and far-
field regions of the DSA.

Given the channel transimpedance matrix
$\mathbf{H}_{\text{c}}=\left[{\mathbf{h}_{\text{c}}}_{1};{\mathbf{h}_{\text{c}}}_{2};\ldots;{\mathbf{h}_{\text{c}}}_{N_{\text{A}}}\right]$,
where ${\mathbf{h}_{\text{c}}}_{k}$ is the $N\times 1$ channel vector
associated with the $k$th user, the precoding matrix is
${\bf{V}}=\beta\,\mathbf{H}_{\text{c}}^{{\dagger}}$, where $\beta$ is a
normalization factor to make ${\bf{V}}$ unitary, i.e.,
${\bf{V}}^{\mathsf{H}}{\bf{V}}={\bf{I}}_{N}$ [45]. Therefore, the DSA and the
digital precoder must be designed such that

$\mathbf{W}_{\text{EM}}(\hat{\bm{\theta}})\,\mathbf{W}_{\text{D}}={\bf{V}}=\beta\,\mathbf{H}_{\text{c}}^{{\dagger}}$
(51)

leading to

${\bf{y}}=\mathbf{H}_{\text{c}}\,\mathbf{W}_{\text{EM}}(\hat{\bm{\theta}})\,\mathbf{W}_{\text{D}}\,{\bf{x}}=\mathbf{H}_{\text{c}}\,\mathbf{H}_{\text{c}}^{{\dagger}}\,{\bf{x}}\,.$
(52)

This is equivalent to setting
${\bf{H}_{\text{opt}}}=\mathbf{H}_{\text{c}}\,\mathbf{H}_{\text{c}}^{{\dagger}}$
in the optimization problem in (45).

###  IV-D Use Case 3: Multi-layer MIMO Communication

In this use case, we consider a MIMO communication with a receiving
conventional uniform linear array (ULA) with $K$ elements spaced at
$\lambda/2$ illustrated in Fig. 4. The purpose is to design the DSA in such a
way an optimal multi-layer MIMO link is established between the transmitter
and the receiver. It is well known from MIMO theory that for a given generic
propagation scenario with channel transimpedance matrix
$\mathbf{H}_{\text{c}}$ characterized by rank $r$, up to $r$ parallel
orthogonal links (layers) can be established between the transmitter and the
receiver on which independent data stream can be transmitted [45]. To exploit
all of them, it must be $N_{\text{A}}=r$ and the DSA must implement a suitable
precoding strategy, i.e., act as an _Holographic EM precoder_.

Let us introduce the SVD of the channel transimpedance

$\mathbf{H}_{\text{c}}={\bf{U}}\,\bm{\Lambda}\,{\bf{V}}^{\mathsf{H}}$ (53)

where ${\bf{U}}$ and ${\bf{V}}$ contain the left and right eigenvectors,
respectively, and $\bm{\Lambda}$ is a diagonal matrix gathering the singular
values of the channel transimpedance matrix $\mathbf{H}_{\text{c}}$. A typical
MIMO setup, assuming the CSI is available at the transmitter, requires a
precoding operation ${\bf{V}}$ at the transmitter and a combining operation
${\bf{U}}^{\mathsf{H}}$ at the receiver such that the channel is diagonalized,
i.e., the end-to-end channel matrix is proportional to the diagonal matrix
$\bm{\Lambda}$. Assuming the receiver performs the combining operation
$\tilde{{\bf{y}}}={\bf{U}}^{\mathsf{H}}\,{\bf{y}}$, the MIMO channel is
diagonalized if we design the DSA implementing the precoding ${\bf{V}}$ by
setting

$\mathbf{W}_{\text{EM}}(\hat{\bm{\theta}})\,\mathbf{W}_{\text{D}}={\bf{V}}$
(54)

or, equivalently,

${\bf{H}_{\text{opt}}}={\bf{U}}^{\mathsf{H}}\,\bm{\Lambda}\,.$ (55)

In this manner, it results

$\tilde{{\bf{y}}}={\bf{U}}^{\mathsf{H}}{\bf{H}}(\bm{\theta},\mathbf{W}_{\text{D}})\,{\bf{x}}={\bf{U}}^{\mathsf{H}}\mathbf{H}_{\text{c}}\,\mathbf{W}_{\text{EM}}(\hat{\bm{\theta}})\,\mathbf{W}_{\text{D}}\,{\bf{x}}=\bm{\Lambda}\,{\bf{V}}^{\mathsf{H}}\,{\bf{V}}\,{\bf{x}}=\bm{\Lambda}\,{\bf{x}}$
(56)

and $N_{\text{A}}$ parallel communication layers can be established. The
designed DSA implements the optimal precoding by using no more than
$N_{\text{A}}=r$ RF chains (i.e., the minimum possible value) which is much
simpler and less energy consuming than any conventional full digital or hybrid
solution.

In our numerical investigations, we discovered that a more robust optimization
is obtained by solving the following equivalent problem instead of (50)

$\displaystyle\hat{\bm{\theta}}=\underset{{\bm{\theta}}}{\operatorname{arg\,min}}\left\|\hat{\alpha}\,{\bf{U}}^{\mathsf{H}}\mathbf{H}_{\text{c}}\,\mathbf{W}_{\text{EM}}(\hat{\bm{\theta}})\,\mathbf{W}_{\text{D}}-\bm{\Lambda}\right\|_{\text{F}}$
(57)

that consists in multiplying both terms in (50) by ${\bf{U}}^{\mathsf{H}}$.

##  V Numerical Results

In this section, we present some numerical results related to the use cases
described in Sec. IV with the purpose of validating and investigating the
potential of the proposed DSA with respect to classical array structures. The
following parameters have been considered in the numerical evaluations if not
otherwise specified: $f_{0}=28\,$GHz, $G_{\text{R}}=0\,$dB, $R=50\,$Ohm,
$l_{n}=\lambda/50$, $r_{n}=10^{-2}\lambda$, $\forall n$, vertical polarization
($\bm{\Omega}_{n}=[0,1,0]^{\mathsf{T}}$, $\forall n$).

Figure 4: Use case 3: Multi-layer MIMO with the DSA acting as an holographic
precoder. Figure 5: Radiation diagram of a disk-shape DSA ($L_{\text{R}}=1$)
with $\Delta_{L}$ and $L=5$. Figure 6: 3D pattern (linear scale) obtained with
the DSA ($\phi=0^{\circ}$). Same configuration of Fig. 5. Figure 7: Radiation
pattern of a conventional ULA with $N_{\text{A}}=6$ active elements. Figure 8:
Radiation pattern of a conventional UCA with $N_{\text{A}}=36$ active
elements.

###  V-A Beam Forming and Superdirectivity

We first investigate the beam forming capabilities of the DSA using one RF
chain ($N_{\text{A}}=1$ active antenna element) and the deployment of the
scatterers according to $L$ concentric cylinders with incremental radius of
step $\Delta_{L}\,$ and scatterers separated of $\lambda/2$ within each
cylinder along the circle and the $y$ axis (see Fig. 2). We denote with
$L_{\text{R}}$ the number of vertical rings composing each cylinder. The
cylindric structure appears very appealing especially for its use in base
stations or access points thanks to its circular symmetry. When
$L_{\text{R}}=1$, the cylinder degenerates into a disk. Obviously, other
structures can be considered as well depending on the specific application.

The target end-to-end channel matrix ${\bf{H}_{\text{opt}}}$ with $K=108$ test
points deployed on the $x-z$ plane at distance $d=100\,$m (far field)
according to the use case in Sec. IV-B was considered for 4 different steering
angles, $0^{\circ}$, $30^{\circ}$, $60^{\circ}$, and $90^{\circ}$,
respectively. Only the optimization Step 2 was performed with
$N_{\text{i}}=1500$ iterations and setting $\mathbf{W}_{\text{D}}=\sqrt{R}$
(in this case $\mathbf{W}_{\text{D}}$ is a scalar and no digital processing
takes place).

In Fig. 5, the radiation diagrams for the 4 steering angles obtained with
$\Delta_{L}=\lambda/4$, $L=5$, $L_{\text{R}}=1$ (disk shape), corresponding to
$N_{\text{S}}=121$ scatterers are shown. As it can be observed, the gain of
the DSA is independent of the angle and it is about $18.6\,$dB. In addition,
limited back radiation is obtained without the need to insert a ground plane
that would impede the steering in the angle range $[90^{\circ}-270^{\circ}]$.
An example of the corresponding 3D radiation diagram for steering angle
$0^{\circ}$ is illustrated in Fig. 6.

As pointed out in the Introduction, the performance of any SIM is constrained
by the final layer surface of the structure, therefore it is upper bounded by
that one can obtain by replacing the SIM with a full digital planar or linear
array with the same layout and number of elements. Therefore, for the sake of
comparison, we consider the radiation diagrams obtained using a standard full-
digital ULA and a uniform circolar array (UCA) with the same aperture of the
DSA ($3.2\,$cm diameter) in the $x-z$ plane. The corresponding radiation
diagrams are reported in Figs. 7 and 8, respectively. These arrays require
$N_{\text{A}}=6$ and $N_{\text{A}}=36$ active antennas compared to only one
active antenna of the DSA. As well-known, in ULAs the gain degrades when
approaching $90^{\circ}$ from a maximum value of $10.6\,$dB (about
$10\log_{10}(N_{\text{A}}\,G_{\text{d}})$, with $G_{\text{d}}=3/2$ the gain of
the Hertzian dipole) and symmetric back radiation is present. Instead, with
the UCA the gain ($15.6\,$dB) is insensitive to the steering direction at the
expense of higher side lobes. Nevertheless, in both cases the proposed DSA
exhibits superdirectivity capability with an additional gain of $8\,$dB and
$3\,$dB, respectively. Contrarily to standard arrays, where superdirectivity
is obtained only in the end-fire direction [19], here notably superdirectivity
is equally obtained in all steering directions.

TABLE I: Cylindric shape. $L=5$. Configuration | Gain (dB) | $N_{\text{S}}$ |  $Q$-factor
---|---|---|---
$\Delta_{L}=\lambda/8$, $L_{\text{R}}=1$ | 15 | 89 | 1.1
$\Delta_{L}=\lambda/6$, $L_{\text{R}}=1$ | 16.7 | 100 | 1.25
$\Delta_{L}=\lambda/4$, $L_{\text{R}}=1$ | 18.6 | 121 | 0.91
$\Delta_{L}=\lambda/2$, $L_{\text{R}}=1$ | 16 | 184 | 0.24
$\Delta_{L}=\lambda$, $L_{\text{R}}=1$ | 15.7 | 310 | 0.24
$\Delta_{L}=\lambda/4$, $L_{\text{R}}=3$ | 20.4 | 363 | 0.3
TABLE II: Random shape. $L_{\text{R}}=1$ $N_{\text{S}}$ | 60 | 121 | 242 | 484
---|---|---|---|---
Gain (dB) | 12.3 | 17 | 18.8 | 18
$Q$-factor | 0.29 | 1 | 1.55 | 3

In Table I, the impact of the cylinders spacing $\Delta_{L}\,$ and the number
of rings $L_{\text{R}}$ is investigated. As it can be noticed, the maximum
gain is obtained for $\Delta_{L}=\lambda/4$ confirming that superdirectivity
can be achieved only by exploiting the mutual coupling between the antenna
elements but, at the same time, that there is an optimum spacing value to be
found. In fact, on one hand, higher spacing reduces the coupling and hence the
contribution to radiation of those scatterers located far away from the active
element. On the other hand, very close spacing tightens the coupling between
the elements and reduces the DoF available for optimization. A higher number
of rings ($L_{\text{R}}=3$) further increases the directivity by thinning the
radiation diagram in the $y$ direction. The corresponding $Q$-factor is in
general relatively small denoting a wideband characteristic of the DSA under
the considered setting and there is no need to generate very high currents as
happens in conventional superdirective end-fire arrays [39].

An example using a random deployment of the scatterers within a disk as a
function of their density is provided in Table II for a fixed diameter of
$3.2\,$cm. As it can be seen, the achievable performance is similar to that of
the regular cylindrical deployment when compared with the same number of
scatterers even though in general higher values of $Q$-factor are obtained
because of the possibility that couples of scatterers are very close to each
other. The analysis and optimization of scatterers deployment strategies is a
topic that deserves future investigations.

An interesting aspect worth consideration is the sensitivity of the
performance to parameter mismatch that might arise in practical
implementations. We evaluated the gain degradation for the configuration in
Fig. 5 when the actual implemented $n$th parameter of the DSA,
$\hat{\theta}_{n}$, is affected by an error $d_{n}$ modeled as a zero-mean
Gaussian random variable with standard deviation $\sigma_{n}$, i.e.,
$\hat{\theta}_{n}=\theta_{n}+d_{n}$. The corresponding gains obtained for
$\sigma_{n}=0$, $\sigma_{n}=1\%$ and $\sigma_{n}=10\%$ of the nominal absolute
value $|\theta_{n}|$ are, respectively, $G=18.6\,$dB, $G=15.8\,$dB, and
$G=14.5\,$dB, showcasing a $4\,$dB loss with a $10\%$ relative error.

Figure 9: Radiation diagram of a zero-forcing 4-user MISO system based on a
DSA.

###  V-B Multi-user MISO

We consider now the scenario of use case in Sec. IV-C illustrated in Fig. 3,
where a DSA with $\Delta_{L}=\lambda/4$, $L_{\text{R}}=1$, $L=5$,
$N_{\text{S}}=121$, $N_{\text{A}}=4$ active antennas and $N_{\text{A}}$ RF
chains is used to communicate simultaneously with $K=N_{\text{A}}$ single-
antenna users located at a distance of $d=100\,$m and angles $-40^{\circ}$,
$20^{\circ}$, $80^{\circ}$, and $200^{\circ}$ in the $x-z$ plane. The DSA has
been optimized according to (51) (zero forcing criterium) with
$N_{\text{i}}=50$ and $N_{\text{alt}}=10$. The corresponding radiation
diagrams are shown in Fig. 9. The coupling between users is impressive
($<-100\,$dB) at the expense of increased leakage in unwanted directions and a
reduction of the gain of some dBs compared to the pattern in Fig. 5
corresponding to a single user.

Alternatively, one could use the approach in use case 1 of Sec. IV-B to reduce
the leakage in unwanted directions and increase the gain. However, our
numerical investigations have revealed that the corresponding coupling between
users would increase significantly. Therefore, the choice of the optimization
method strongly depends on the specific application requirements in terms of
gain versus coupling.

###  V-C Multi-layer MIMO

Finally, we consider the scenario of use case 3 in Sec. IV-D shown in Fig. 4,
where a multi-layer MIMO link with a user located at $d=10\,$m equipped with a
standard ULA with $K=20$ elements spaced of $\lambda/2$ has to be established
in non-line-of-sight (NLOS) condition. The DSA is equipped with
$N_{\text{A}}=4$ active antennas and $N_{\text{A}}$ RF chains to allow the
transmission of up to 4 data streams (layers) according to channel
characteristics. The simulated NLOS channel consists of 5 multi-paths caused
by the reflection of 5 scatterers located at angles
$\phi=\\{-43^{\circ},-14^{\circ},14^{\circ},43^{\circ},72^{\circ}\\}$ and
distance $5\,$m. The corresponding strongest singular values of the channel
are $\bm{\Lambda}={\rm
diag}\left(-35.6{\,\text{dB}},-42.4{\,\text{dB}},-43.7{\,\text{dB}},-46.3{\,\text{dB}},-82{\,\text{dB}},\ldots\right)\,$,
which has clearly rank $r=4$. The DSA has the following parameters
$\Delta_{L}=\lambda/4$, $L_{\text{R}}=1$, $N_{\text{S}}=121$, and it is
optimized according to the criterium in (54) with $N_{\text{i}}=50$ and
$N_{\text{alt}}=10$.

In Table III, the resulting coefficients $\hat{\bm{\Lambda}}$ of the end-to-
end channel is reported. The comparison between the diagonal and off-diagonal
values indicates that the coupling between different layers is completely
negligible, i.e., the channel is almost perfectly diagonalized. Compared to
the singular values of the channel in $\bm{\Lambda}$, the diagonal elements of
$\hat{\bm{\Lambda}}$ exhibit the same behavior with a loss of about $6\,$dB
that can be ascribed to the fact that the antenna is not ideal. The effect of
implementation errors is reported in the bottom part of the table for $1\%$
and $10\%$ relative error standard deviation, respectively. It can be seen
that the effect of implementation errors becomes no longer negligible
(coupling above $-10\,$dB) with $10\%$ relative error.

TABLE III: ${\left[\hat{\Lambda}\right]}_{n,k}$ in dB, for $n,k=1,2,3,4$.
Perfect implementation
---
-41.6 | -177 | -190 | -175
-170 | -48 | -185 | -170
-183 | -184 | -49.7 | -182
-164 | -166 | -180 | -52.3
$1\%$ relative implementation error
-41.9 | -90.6 | -80 | -80.5
-89 | -48.7 | -98.5 | -86.7
-87 | -111 | -50 | -84.9
-120 | -101 | -89.7 | -52.5
$10\%$ relative implementation error
-47 | -56.7 | -59.4 | -62.2
-66.6 | -48.3 | -61.6 | -65.3
-67 | -61.4 | -52.4 | -64.9
-57.2 | -74.7 | -62.8 | -55.3

##  VI Conclusion

In this paper, we have introduced the concept of the dynamic scattering array
(DSA) as an appealing technology aimed at shifting part of the signal
processing from the digital domain to the EM domain. This is achieved through
the joint optimization of EM processing and radiation of active and
reconfigurable scattering elements interacting in the reactive near field. We
have shown that the DSA provides unprecedented flexibility in managing the EM
field by reducing or circumventing digital processing at the baseband, thereby
surpassing the flexibility offered by classical MIMO and recent SIM
implementations. Consequently, the reduction in RF chains and digital
processing facilitates the realization of energy-efficient, low-latency, and
cost-effective holographic MIMO systems, then addressing the sustainability
concerns of future wireless networks.

Although numerous implementation challenges must be addressed before DSAs can
become a reality, such as the design of flexible reconfigurable scatterers, we
hope that our promising theoretical results will inspire further research into
technologies for the efficient implementation of DSAs as a key enabler of
holographic MIMO within the ESIT paradigm.

## Acknowledgment

This work was supported by the European Union under the Italian National
Recovery and Resilience Plan (NRRP) of NextGeneration EU, partnership on
“Telecommunications of the Future” (PE00000001 - program “RESTART”), and by
the HORIZON-JU-SNS-2022-STREAM-B-01-03 6G-SHINE project (Grant Agreement No.
101095738).

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  *[MIMO]: multiple-input multiple-output
  *[DSA]: dynamic scattering array
  *[EM]: electromagnetic
  *[DSAs]: dynamic scattering array
  *[RF]: radiofrequency
  *[ELAA]: extremely large antenna arrays
  *[ELAAs]: extremely large antenna arrays
  *[DoF]: degrees-of-freedom
  *[LOS]: line-of-sight
  *[ESIT]: electromagnetic signal and information theory
  *[RISs]: reconfigurable intelligent surface
  *[SIMs]: stacked intelligent metasurface
  *[DMA]: dynamic metasurface antenna
  *[DMAs]: dynamic metasurface antenna
  *[RIS]: reconfigurable intelligent surface
  *[SIM]: stacked intelligent metasurface
  *[SVD]: singular value decomposition
  *[MISO]: multiple-input single-output
  *[DAC]: digital-to-analog converter
  *[CSI]: channel state information
  *[SINR]: signal-to-interference noise ratio
  *[ULA]: uniform linear array
  *[ EM]: electromagnetic
  *[UCA]: uniform circolar array
  *[ULAs]: uniform linear array
  *[NLOS]: non-line-of-sight