# Wireless Communications with Space-Time Modulated Metasurfaces

Marouan Mizmizi  Dario Tagliaferri  Umberto Spagnolini  M. Mizmizi, D.
Tagliaferri, U. Spagnolini are with the Department of Electronics, Information
and Bioengineering (DEIB) of Politecnico di Milano, 20133 Milan, Italy
(e-mail: [marouan.mizmizi, dario.tagliaferri<EMAIL_ADDRESS>U.
Spagnolini is Huawei Industry Chair.This paper was supported by the European
Union under the Italian National Recovery and Resilience Plan (NRRP) of
NextGenerationEU, partnership on “Telecommunications of the Future”
(PE00000001 - program “RESTART”).

###### Abstract

Space-time modulated metasurfaces (STMMs) are a newly investigated technology
for the next 6G generation wireless communication networks. An STMM augments
the spatial phase function with a time-varying one across the meta-atoms,
allowing for the conveyance of information that possibly modulates the
impinging signal. Hence, STMM represents an evolution of reconfigurable
intelligent surfaces (RIS), which only design the spatial phase pattern. STMMs
convey signals without a relevant increase in the energy budget, which is
convenient for applications where energy is a strong constraint. This paper
proposes a mathematical model for STMM-based wireless communication, that
creates the basics for two potential STMM architectures. One has excellent
design flexibility, whereas the other is more cost-effective. The model
describes STMM’s distinguishing features, such as space-time coupling, and
their impact on system performance. The proposed STMM model addresses the
design criteria of a full-duplex system architecture, in which the temporal
signal originating at the STMM generates a modulation overlapped with the
incident one. The presented numerical results demonstrate the efficacy of the
proposed model and its potential to revolutionize wireless communication.

###### Index Terms:

Space-time modulated metasurfaces, space-time phase coupling, RIS, 6G

##  I Introduction

The development of future wireless communication networks is a collective
effort that entails the integration of multiple technologies as well as
addressing several challenges. The goal of 6G is to provide ultra-high data
rates, low latency, and high reliability for a wide range of applications and
services such as autonomous vehicles, eHealth, and the Internet of Things [1,
2]. To reach these objectives, 6G links must integrate advanced technologies
with the use of high-frequency bands and energy-efficient solutions.

The emergence of electromagnetic (EM) metasurfaces, a.k.a. reconfigurable
intelligent surfaces (RIS), has opened up new possibilities in the design of
wireless propagation [3]. A RIS is typically made of quasi-passive arrays of
sub-wavelength-sized meta-atoms whose scattering properties are engineered to
manipulate the impinging EM waves and control the reflection [4, 5]. RIS
underwent tremendous conceptual and, to a minor extent, experimental
development through several studies. In particular, RISs have been proposed
for overcoming blockage [6], with specific application to vehicular scenarios
[7, 8, 9], for coverage extension [10] and, recently, as support for sensing
[11].

The RIS technology (and metamaterials in general) has experienced a surge in
interest in the last decade, following the formulation of the generalized
Snell’s law in 2011 [12]. It is based on the conservation of both the linear
momentum and the energy of reflected and refracted waves [13]. The linear
momentum is related to spatial symmetry, i.e., an EM wave impinging from an
arbitrary angle $\theta_{i}$ is reflected at the specular one
$\theta_{o}=\theta_{i}$. Setting the spatial-phase gradient across the RIS
implies breaking the spatial symmetry, thus allowing a reflection toward
$\theta_{o}\neq\theta_{i}$. Energy conservation relates to the time-reversal
symmetry or Lorentz reciprocity, i.e., if we can reflect an EM wave coming
from a particular angle $\theta_{i}$ towards a specific angle $\theta_{o}$,
the same EM wave coming back from $\theta_{o}$ will be reflected toward
$\theta_{i}$.

(a) RIS

(b) STCM

(c) Proposed STMM

Figure 1: The application of metasurfaces has evolved from (1a) RIS, which
allows control of the direction of reflection, to (1b) STCM, which allows more
advanced reflection and control of the reflected frequency, to (1c) STMM,
which can convey information superposed to the reflected signal.

In 2014, the authors in [14] proposed the universal Snell’s law of reflection
realizing efficient solar panels. The key idea is to break the energy
conservation law by adding a temporal phase gradient across the metasurface
and then combining the spatial and temporal phase gradients in the metasurface
to create anomalous nonreciprocal reflections. This principle is at the
foundation of space-time modulated metasurfaces (STMMs) and generalizes RISs,
opening up the possibility of conveying information in the reflected EM waves
[15]. Fig. 1 depicts the evolution of metasurfaces from RIS to STCM and STMM.
The authors in [16, 17, 18, 19] used a spatial-temporal gradient to create
space-time coded metasurfaces (STCMs), which can both reflect the signal in
space and shift its frequency, generating harmonics. Herein, STMMs differ from
STCMs in that they are intended to convey information through reflection,
whereas STCMs are primarily designed to synthesize time-varying reflection
patterns. In this latter direction, the work in [20] proposes an STMM-based
MIMO transmitter that operates on a sinusoidal feed and encodes data using the
harmonics of a properly designed phase-discontinuous symbol. However, the
interaction between spatial and temporal phases has never been investigated.
Nevertheless, the theoretical findings of [20] are validated by experimental
results in a $2\times 2$ MIMO setting, demonstrating a rate of 20 Mbps. The
authors of [21] discuss the realization of advanced modulation schemes by
engineering the phase response of the single STMM meta-atom. From a system-
level perspective, a wireless communication system using a time-modulated
metasurface to convey information can be conceptually considered a backscatter
communication system, where a backscatter device (tag) transmits information
to an intended receiver by reflecting the signal impinging from an external
transmitter [22]. The authors of [23] outline the potential of backscatter
communications for low-power Internet of Things applications, providing
analytical tools to evaluate the system performance for a generic
backscattering tag (with possibly amplification capabilities). The generalized
space shift keying for ambient backscatter communication systems is proposed
in [24], where the RIS switches the reflected signal between two receiving
points, encoding the information in the direction of reflection. Among the
relevant contributions using RISs, the authors of [25] propose to enhance
ambient backscatter communications with a phase-modulated RIS employing a
binary phase shift keying modulation. The work [26] proposes a RIS-based
backscatter communication system to realize computational task offloading in
energy-constrained networks, harvesting part of the energy of the impinging
signal to enable the conveyance of information through reflection. The work in
[27], instead, describes a RIS-based beamforming-plus-phase modulation system
to serve a user with information augmentation, focusing on beamforming and
detection performance.

All of the preceding works either discuss the physical realization or detail
the specific applications of STCMs/STMMs (referred to as time-modulated RISs
in backscatter communication literature), but never explicitly account for
space-time phase coupling. By generalizing the RIS principle, stemming from
the physical principle ruling the space-time modulation of metasurfaces, i.e.,
the universal Snell’s law, this paper aims to demonstrate the potential and
associated challenges of an STMM-based wireless communication system. Herein,
we consider a master unit (MU), for example, a base station (BS), and a slave
unit (SU), e.g., user equipment (UE), in a generic full-duplex system (MU
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ SU). The phase
control of STMM enables the superposition of a MU $\leftarrow$ SU signal to a
MU $\rightarrow$ SU one originating at the MU. Full-duplex technologies and
systems are of great interest because they will enable many 6G services, and
the potential applications and challenges are discussed in [28]. In the
proposed study case, the link MU $\rightarrow$ SU uses a portion of the
bandwidth $B_{d}$ in the spectrum. The SU uses the STMM to retro-reflect and
modulate the received MU $\rightarrow$ SU signal with a phase-only MU
$\leftarrow$ SU signal with a bandwidth $B_{u}$. The resulting signal (MU
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ SU) occupies a
total bandwidth of $B_{tot}=B_{d}+B_{u}$, as detailed in Section V. The SU
relies solely on the STMM to encode the information, opening up intriguing
applications in severe energy-constrained scenarios. To summarize, the main
contributions of the paper are as follows:

  * •

We propose a mathematical model for STMM-based wireless communication (MU
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ SU). The SU
terminal retro reflects the signal adding information to the MU by time-
modulating the phase of the STMM. One of the major innovations proposed in
this paper is the use of a modulated feed signal at the STMM for signal MU
$\leftarrow$ SU over retro reflections.

  * •

We analytically characterize the coupling between the spatial and temporal
phases in STMM, evaluating their practical impact on the proposed
communication model. Noteworthy, none of the current literature works consider
the coupling of temporal and spatial phase gradients. We discuss two possible
STMM implementations, one allowing the phase of each meta-atom of the STMM to
be changed in both time and space and the other, cost-effective, that only
requires a single time-varying component, that provides a temporal phase
gradient common to all the meta-atoms. Our results show that space-time
coupling strongly limits the reception angles, the size of the metasurface,
and the MU $\leftarrow$ SU communication rate, giving rise to a cut-off to
retro-reflection MU $\leftarrow$ SU bandwidth limit. Further, we also
formalize a method for space-time coupling compensation within the cut-off
region, enabling a flexible and reliable implementation of the proposed
communication model.

  * •

We analytically derive the upper-bound performance of the proposed model to
form the guidelines for proper STMM design based on the main design
parameters, i.e., transmitted power, MU-SU distance, and size of the
metasurface.

  * •

We analyze the maximum spectral efficiency achievable of MU
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ SU considering the
generic phase modulation for retro-reflection, namely continuous phase
modulation (CPM) with and without countermeasures for space-time coupling. We
also analytically and numerically evaluate the impact of channel estimation
errors on the MU $\leftarrow$ SU spectral efficiency, affecting the spatial
phase configuration of the STMM and related space-time phase decoupling.

Organization: The remainder of the paper is organized as follows: Section II
outlines the fundamental equations describing the STMM behavior, system and
channel model are presented in Section III, Section IV discusses the design
principles and challenges of the STMM, while UL signal design is presented in
Section V. Finally, numerical results and conclusions are in Section VI and
Section VII, respectively.

Notation: Bold upper- and lower-case letters describe matrices and column
vectors. The $(i,j)$-th entry of matrix $\mathbf{A}$ is denoted by
$[\mathbf{A}]_{(i,j)}$. Matrix transposition, conjugation, conjugate
transposition, and Frobenius norm are indicated respectively as
$\mathbf{A}^{\mathrm{T}}$, $\mathbf{A}^{*}$, $\mathbf{A}^{\mathrm{H}}$ and
$\|\mathbf{A}\|_{F}$. $\otimes$ denotes the Kronecker (tensor) froduct between
matrices, $\mathrm{tr}\left(\mathbf{A}\right)$ extracts the trace of
$\mathbf{A}$. $\mathrm{diag}(\mathbf{A})$ denotes the extraction of the
diagonal of $\mathbf{A}$, while $\mathrm{diag}(\mathbf{a})$ is the diagonal
matrix given by vector $\mathbf{a}$. $\mathbf{I}_{n}$ is the identity matrix
of size $n$. With $\mathbf{a}\sim\mathcal{CN}(\boldsymbol{\mu},\mathbf{C})$ we
denote a multi-variate circularly complex Gaussian random variable
$\mathbf{a}$ with mean $\boldsymbol{\mu}$ and covariance $\mathbf{C}$.
$\mathbb{E}[\cdot]$ is the expectation operator, while $\mathbb{R}$ and
$\mathbb{C}$ stand for the set of real and complex numbers, respectively.
$\delta_{n}$ is the Kronecker delta.

##  II Anomalous Nonreciprocal Metasurface

Figure 2: 2D STMM Scheme.

This section outlines the fundamental equations describing the behavior of an
STMM. The universal Snell’s law is a generalization when dealing with both
spatial and temporal phase gradients across a metasurface. Let us consider the
scheme depicted in Fig. 2, where a plane EM wave $\mathbf{E}_{i}$ is impinging
on a 2D planar metasurface deployed on the $xy$ plane. The incident wave can
be expressed as

$\mathbf{E}_{i}(\mathbf{x},t)=\mathbf{a}_{i}\,e^{j\Psi_{i}(\mathbf{x},t)},$
(1)

where $\mathbf{a}_{i}$ is the amplitude of the wave, whereas phase
$\Psi_{i}(\mathbf{x},t)$ is fast varying in space and time. Quantities

$\displaystyle\frac{1}{2\pi}\frac{\partial\Psi_{i}(\mathbf{x},t)}{\partial t}$
$\displaystyle=f_{i}$ (2)
$\displaystyle\nabla_{\mathbf{x}}\Psi_{i}(\mathbf{x},t)$
$\displaystyle=\mathbf{k}_{i},.$ (3)

are, respectively, the frequency $f_{i}$ and the wavevector $\mathbf{k}_{i}$
of the incident wave, whose expression in the adopted reference system (Fig.
3) is

$\mathbf{k}_{i}=\frac{2\pi}{\lambda_{i}}\left[\cos\theta_{i}\cos\phi_{i},\cos\theta_{i}\sin\phi_{i},\sin\theta_{i}\right]^{T}$
(4)

for wavelength $\lambda_{i}=c/f_{i}$ ($c$ is the wave speed). Let us define
the reflection coefficient of the metasurface as

$\Gamma(\mathbf{x},t)=\alpha(\mathbf{x},t)\,e^{j\beta(\mathbf{x},t)},$ (5)

where $\alpha(\mathbf{x},t)$ and $\beta(\mathbf{x},t)$ denote the time- and
space-varying amplitude and phase of the reflection coefficient of the STMM,
respectively. Herein, the metasurface is a phase-only device, thus
$\alpha(\mathbf{x},t)=\alpha\neq 0$ on the metasurface area. Similarly to (1),
the reflected wave can be generally expressed as

$\mathbf{E}_{o}(\mathbf{x},t)=\mathbf{a}_{o}\,e^{j\Psi_{o}(\mathbf{x},t)},$
(6)

with $\mathbf{a}_{o}\approx\mathbf{a}_{i}$ under the assumption of ideal
reflection, i.e., $\alpha=1\,\forall\mathbf{x},\,t$. The reflected wave is
characterized by wavevector $\mathbf{k}_{o}$ and frequency $f_{o}$, that
follow from the universal Snell’s law as [16]:

$\displaystyle\mathbf{k}_{o}-\mathbf{k}_{i}$
$\displaystyle=\left[\frac{\partial\beta(\mathbf{x},t)}{\partial
x},\frac{\partial\beta(\mathbf{x},t)}{\partial y},0\right]^{\mathrm{T}},$ (7)
$\displaystyle f_{o}-f_{i}$
$\displaystyle=\frac{1}{2\pi}\frac{\partial\beta(\mathbf{x},t)}{\partial t}.$
(8)

As it can be observed in (8), any spatial variation of the phase
$\beta(\mathbf{x},t)$ produces a variation of the instantaneous wavevector
$\mathbf{k}_{o}$, and a time variation of the phase $\beta(\mathbf{x},t)$
induces a variation of the instantaneous frequency $f_{o}$ in the reflected
wave. The design of $\beta(\mathbf{x},t)$ to control both direction
$(\mathbf{k}_{o})$ and frequency $(f_{o})$ of the reflected wave is discussed
in Section IV.

##  III System and Channel Model

Figure 3: Proposed system model

Let us consider the reference scenario depicted in Fig. 3, where a generic MU
establishes a full-duplex link with a SU. To simplify, the MU is equipped with
two uncoupled antenna arrays of $N$ meta-atoms each, to enable full-duplex
capabilities at MU. The SU is equipped with a planar STMM of
$M_{u}=M_{u,x}\times M_{u,y}$ meta-atoms (along $x$ and $y$ respectively), and
an Rx antenna array of $M_{d}$ meta-atoms. We assume that the STMM and the Rx
array are co-located at the SU. The assumption concerning the SU is meant to
conceptually separate for simplicity the device that receives MU $\rightarrow$
SU data from the MU $\leftarrow$ SU data stream via retro-reflection. Note
that, it is possible to make a single device capable of both receiving and
transmitting at the same time, for example, by connecting some meta-atoms of
the metasurface to one (or more) RF chain [29]. In this case, the analytical
derivations would be slightly different, but the results are equivalent to
those investigated herein.

###  III-A Signal model

The time-continuous signal from the MU can be generally expressed as:

$\mathbf{x}(t)=\mathbf{f}_{d}\;s_{d}(t)e^{j2\pi f_{i}t}$ (9)

where $\mathbf{f}_{d}\in\mathbb{C}^{N\times 1}$ is the spatial precoding
vector and $s_{d}(t)$ is the MU $\rightarrow$ SU time-continuous base-band
information signal with bandwidth $B_{d}$, centered around the carrier
frequency $f_{i}$. The MU $\rightarrow$ SU signal propagates over a high-
frequency channel [30], yielding a received signal at the SU:

$\begin{split}y_{d}(t)&=\frac{1}{\sqrt{\varrho_{d}}}\,\mathbf{w}_{d}^{H}\mathbf{H}_{d}(t)*\mathbf{x}(t)+\mathbf{w}_{d}^{H}\mathbf{n}(t)\end{split}$
(10)

where $\varrho_{d}$ represents the MU $\rightarrow$ SU path loss in power, *
is the matrix convolution, $\mathbf{w}_{d}\in\mathbb{C}^{N\times 1}$ is the SU
spatial combiner, $\mathbf{H}_{d}(t)\in\mathbb{C}^{M_{d}\times N}$ is the MU
$\rightarrow$ SU MIMO channel matrix, such that
$\mathrm{E}\left[\|\mathbf{H}_{d}(t)\|_{\mathrm{F}}^{2}\right]=NM_{d}$ while
$\mathbf{n}(t)\sim\mathcal{CN}(\mathbf{0},\sigma^{2}_{n}\mathbf{I}_{M_{d}}\delta(t))$
is the additive noise, spatially and temporally uncorrelated.

The Rx signal model at the MU can be evaluated by considering the propagation
towards the STMM and its retro-reflection to the MU. The MU $\rightarrow$ SU
signal at the STMM is

$\mathbf{r}(t)=\mathbf{H}_{i}(t)*\mathbf{x}(t)$ (11)

where $\mathbf{H}_{i}(t)\in\mathbb{C}^{M_{u}\times N}$ is the forward MIMO
channel between the MU and the STMM. The signal $\mathbf{r}(t)$ undergoes the
STMM phase reflection and then the convolution with the backward (MU
$\leftarrow$ SU) MIMO channel $\mathbf{H}_{o}(t)\in\mathbb{C}^{N\times
M_{u}}$, obtaining:

$\begin{split}y_{u}(t)&=\frac{1}{\sqrt{\varrho_{u}}}\mathbf{w}_{u}^{H}\mathbf{H}_{o}(t)*\boldsymbol{\Gamma}(t)\mathbf{r}(t)+\mathbf{w}_{u}^{H}\mathbf{z}(t)=\\\
&=\frac{1}{\sqrt{\varrho_{u}}}\mathbf{w}_{u}^{H}\mathbf{H}_{o}(t)*\boldsymbol{\Gamma}(t)\left(\mathbf{H}_{i}(t)*\mathbf{x}(t)\right)+\\\
&+\mathbf{w}_{u}^{H}\mathbf{z}(t),\end{split}$ (12)

where $\varrho_{u}$ is the UL path loss in power,
$\mathbf{w}_{u}\in\mathbb{C}^{N\times 1}$ is the uplink combiner at the MU,
$\mathbf{z}(t)\sim\mathcal{CN}(\mathbf{0},\sigma^{2}_{z}\mathbf{I}_{N}\delta(t))$
is the additive noise at the MU array and
$\boldsymbol{\Gamma}(t)\in\mathbb{C}^{M_{u}\times M_{u}}$ denotes the time-
varying STMM phase pattern matrix

$\mathbf{\Gamma}(t)=\mathrm{diag}\left(\left[e^{j\beta_{0,0}(t)},\cdots,e^{j\beta_{M_{u,x}-1,M_{u,y}-1}(t)}\right]\right).$
(13)

where the time-varying phase matrix is obtained by discretizing the reflection
coefficient in (5) over the spatial dimension, i.e., the STMM is modeled as a
modulated reflectarray with regularly spaced meta-atoms. In particular, the
relative position of the $(q,v)$-th meta-atom of the STMM w.r.t. to the phase
center is $\mathbf{p}_{q,v}=[q\,d_{x},v\,d_{y},0]^{T}$, for
$q=0,...,M_{u,x}-1$, $v=0,...,M_{u,y}-1$, being $d_{x}$ and $d_{y}$ being the
meta-atoms spacing along the $x$ and $y$ axes, respectively. Notice that,
although $\mathbf{\Gamma}(t)$ is a time-varying function, the relation with
$\mathbf{r}(t)$ is multiplicative because the time-varying signal injected by
the STMM is a reflection coefficient.

Remark 1: Precoding and combining vectors, i.e., $\mathbf{f}_{d}$ and
($\mathbf{w}_{d}$, $\mathbf{w}_{u}$), are herein based on the knowledge of
channels $\mathbf{H}_{d}(t)$ and $\mathbf{H}_{i}(t)$, $\mathbf{H}_{o}(t)$ at
both MU and SU side. Notice that channel estimation at the SU can be obtained
through conventional approaches and shared with the MU through feedback [31].

Remark 2: Decoding of the MU $\leftarrow$ SU information signal in
$\beta_{q,v}(t)$ is possible thanks to the knowledge of the MU $\rightarrow$
SU one $s_{d}(t)$, that shall be removed by the MU to obtain $\beta_{q,v}(t)$
(not analyzed here).

###  III-B Channel Model

The cluster-based multipath channel model is widely used in the design of
mmWave and sub-THz wireless communication systems because it effectively
captures the unique propagation characteristics of these frequencies [32].
This model is used here since this paper is framed for these frequencies. The
$(m,n)$-th entry of any of the channel impulse responses in (10)-(12) can be
written as

$\displaystyle[\mathbf{H}(t)]_{m,n}$
$\displaystyle=\sum_{p=1}^{P}\xi_{p}\,\delta\left(t-\tau_{p}-\Delta
t_{p,n}-\Delta t_{p,m}\right)$ (14)

where (i) $P$ is the number of paths, (ii)
$\xi_{p}\sim\mathcal{CN}\left(0,\sigma_{p}^{2}\right)$ is the scattering
amplitude of $p$-th path, such that $\sum_{p=1}^{P}\sigma_{p}^{2}=1$, (iii)
$\tau_{p}$ is the propagation delay of the $p$-th path from the phase center
of the Tx to the phase center of the Rx, (iv) $\Delta t_{p,n}$ and $\Delta
t_{p,m}$ are the residual propagation delays due to the position of the $n$-th
Tx meta-atom and the $m$-th Rx meta-atom w.r.t. to their phase centers. As
common for communication systems operating through back-scattering, the two-
way channel to/from the STMM is characterized by a single dominant path, as
for radar systems [33]. However, the technical extent of this work is not
limited by the latter assumption. Multiple bounces (i.e., multipath forward
$\mathbf{H}_{i}(t)$ and backward $\mathbf{H}_{o}(t)$ channels) can be
considered, but (i) their path loss is usually higher (or much higher) w.r.t.
the direct MU $\mathrel{\leavevmode\hbox to14.63pt{\vbox
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pure reflections in the environment (e.g., walls) and (ii) both MU and SU are
spatially selective, thus multipath components outside the main radiation lobe
of the MU (or SU) would be strongly attenuated. The combination of spatial
selectivity and increased path-loss allows us to reasonably assume $P=1$ for
both forward $\mathbf{H}_{i}(t)$ and backward $\mathbf{H}_{o}(t)$ channels.
Considering a single path yields a common propagation delay
$\tau_{p}=\tau=D/c$ between MU and SU and residual delays at the STMM that
depend on the incidence and reflection angles $\theta=\theta_{o}=\theta_{i}$,
$\phi=\phi_{o}=\phi_{i}$ (coincident for full-duplex settings such as the one
depicted in Fig. 3). The expression of the residual delay $\Delta t$ at the
STMM is shown in Section IV-A. The inter-meta-atom spacing of Tx and Rx arrays
(at both MU and SU) is $\lambda_{i}/2$, while at STMM is
$d_{x}=d_{y}=\lambda_{i}/4$. The propagation path losses in power,
$\varrho_{d}$ and $\varrho_{u}$ in (10)-(12) are modeled as [34]

$\varrho_{d}=\frac{2^{4}\pi
D^{2}}{\lambda_{i}^{2}},\quad\quad\varrho_{u}=\frac{2^{12}\pi
D^{4}}{\lambda_{i}^{4}}$ (15)

where $D$ is the MU-SU distance. The model in (15) is obtained by assuming
that the effective area of the MU/SU arrays is equal to the physical one,
which is approximated as $N(\lambda_{i}/2)^{2}$, and that the retro-reflected
cross-section of the STMM is equal to the one of a perfect metallic plate with
an area of $M_{u}(\lambda_{i}/4)^{2}$ as if it is oriented towards the MU
[34].

Remark: The MU $\leftarrow$ SU path loss $\varrho_{u}$ is here assumed as only
dependent on the wavelength of the incident signal $\lambda_{i}$. As detailed
in Section IV, this is true for STMM phases $\beta_{q,v}(t)$ considered as
stochastic, whose average time derivative is zero, i.e.,
$\mathbb{E}[\mathrm{d}\beta_{q,v}(t)/\mathrm{d}t]=0$, as common for
information-bearing signals. Differently, for
$\mathrm{d}\beta_{q,v}(t)/\mathrm{d}t\neq 0$, (e.g., a simple frequency
shifting), the denominator of $\varrho_{u}$ would be
$\lambda_{i}^{2}\lambda_{o}^{2}$, with $\lambda_{o}$ derived from (8).

##  IV Temporal Modulation

(a) Architecture A

(b) Architecture B

Figure 4: Representation of a 1D plane wave impinging on the STMM for (4a)
Architecture A, considering each meta-atom with tunable space-time
capabilities, and (4b) Architecture B, meta-atoms spatially tunable and a
single tunable component shared across the whole metasurface.

This section defines the principles of temporal modulation at the STMM and its
peculiarities. For the implementation of the STMM, we propose two potential
architectures, as depicted in Fig. 4. The first architecture, A, models each
meta-atom with space-time tunability. Alternatively, architecture B models
each meta-atom with spatial tunability and a single tunable component for
temporal control shared with all meta-atoms. The difference between A and B is
mainly the reduced cost/complexity of the second, which comes at the price of
a slightly reduced performance, as will be discussed in the following. This
dissimilarity arises from the fact that temporal control requires fast-varying
varactors to support very high data rates in the MU $\leftarrow$ SU link,
whereas spatial control can be achieved using cost-effective pin diodes. This
discrepancy in component selection is attributed to the distinct operational
demands of each control mechanism. To ease the derivation while maintaining
the generality, we consider the MU to use precoding and combining vectors
perfectly aligned with the SU. Thus, we have that $\mathbf{f}_{d}$ and
$\mathbf{w}_{u}$ are given, e.g., estimated by conventional approaches [31].

###  IV-A Principles of Temporal Modulation

Let us consider the phase (13) applied at the STMM meta-atoms (architecture B)
as the superposition of a spatial and a temporal component, such that at the
$(q,v)$-th meta-atom is

$\beta_{q,v}(t)=\varphi_{q,v}+\gamma(t),$ (16)

for $q=0,...,M_{u,x}-1$, $v=0,...,M_{u,y}-1$, where $\varphi_{q,v}$ is the
space component of the phase used to design a complete retro-reflection of the
impinging signal and $\gamma(t)$ is an information-bearing phase signal
applied at each STMM meta-atom, and common to all. Model (16), adopted in the
current literature [20, 35] can also be considered valid for architecture A
when only one data stream is generated at the SU, as considered in this paper.

Expanding (12) with (16), we obtain (17)

$\begin{split}y_{u}(t)&=\rho\sum_{q=0}^{M_{u,x}-1}\sum_{v=0}^{M_{u,y}-1}e^{-j4\pi
f_{i}\Delta t_{q,v}}e^{j\varphi_{q,v}}e^{j\gamma(t-\Delta
t_{q,v}-\tau)}s_{d}(t-2\Delta t_{q,v}-2\tau)+z(t)\overset{(a)}{\approx}\\\
&\overset{(a)}{\approx}\rho\,s_{d}(t-2\tau)\underbrace{\sum_{q=0}^{M_{u,x}-1}\sum_{v=0}^{M_{u,y}-1}e^{-j4\pi
f_{i}\Delta t_{q,v}}e^{j\varphi_{q,v}}e^{j\gamma(t-\Delta
t_{q,v}-\tau)}}_{\text{Multiplicative MU$\leftarrow$SU
channel}\,h_{u}(t)}+z(t)\overset{(b)}{\approx}\\\
&\overset{(b)}{\approx}\rho\,s_{d}(t-2\tau)\sum_{q=0}^{M_{u,x}-1}\sum_{v=0}^{M_{u,y}-1}e^{-j4\pi
f_{i}\Delta t_{q,v}}e^{j\varphi_{q,v}}e^{j\gamma(t-\tau)}+z(t)\end{split}$
(17)

where $\rho=N^{2}\xi_{i}\xi_{o}/\sqrt{\varrho_{u}}$ accounts for geometrical
energy losses, while

$\Delta
t_{q,v}=\frac{1}{c}\left(\frac{\mathbf{p}_{q,v}^{T}\mathbf{k}_{i}}{\|\mathbf{k}_{i}\|}\right)=q\Delta
t_{x}+v\Delta t_{y}$ (18)

is the residual propagation delay for the $(q,v)$-th STMM meta-atom, located
in $\mathbf{p}_{q,v}=(\lambda_{i}/4)[q,v,0]^{T}$ (local STMM coordinates), and

$\Delta
t_{x}=\frac{\lambda_{i}}{4}\frac{\cos\theta\cos\phi}{c},\,\,\,\,\,\Delta
t_{y}=\frac{\lambda_{i}}{4}\frac{\cos\theta\sin\phi}{c}$ (19)

are the propagation delays of the wavefront across two adjacent STMM meta-
atoms displaced by $\lambda_{i}/4$ along $x$ and $y$. In (17), the first term
is the complete expression of the Rx signal for MU $\rightarrow$ SU $s_{d}(t)$
and MU $\leftarrow$ SU $\gamma(t)$ signals. Approximation $(a)$ implies that
there are no spatially wideband effects at the STMM due to the MU
$\rightarrow$ SU signal, namely $s_{d}(t-2\Delta T)\approx s_{d}(t)$, where
$\Delta T=M_{u,x}\Delta t_{x}+M_{u,y}\Delta t_{y}$ is the maximum delay across
the STMM meta-atoms. This is equivalent to asserting that:

$2\Delta T<<T_{d},$ (20)

where $T_{d}=1/B_{d}$ is the symbol period of the MU $\rightarrow$ SU
information signal $s_{d}(t)$. In this setting, unwanted beam squinting from
spatially wideband effects due to $s_{d}(t)$ can be avoided (see [36] for
details). With approximation $(a)$, the MU $\rightarrow$ SU signal undergoes a
multiplicative channel $h_{u}(t)$ that encodes the MU $\leftarrow$ SU phase
signal $\gamma(t)$, whose structure is discussed in Section IV-B. Similarly,
approximation $(b)$ subtends that the MU $\leftarrow$ SU phase-only signal
does not suffer the relative propagation delay across the STMM, thus
$\gamma(t-\Delta T)\approx\gamma(t)$, or, equivalently

$\Delta T\ll T_{u},$ (21)

where $T_{u}$ is the symbol period of the MU $\leftarrow$ SU phase signal
$\gamma(t)$. Approximation $(b)$ in (17) has the further implication that the
spatial phase component $\varphi_{q,v}$ is not affected by the temporal one
$\gamma(t)$, i.e., they can be separated and, consequently, the reflection
gain is maximized by a phase-only pattern (not a function of time) across the
STMM:

$\displaystyle\varphi_{q,v}=\pi(q\cos\theta\cos\phi+v\cos\theta\sin\phi),$
(22)

leading to the multiplicative model [20]

$\begin{split}y_{u}(t)\approx\rho\,M_{u}\,s_{u}(t-\tau)\,s_{d}(t-2\tau)+z(t),\end{split}$
(23)

where $s_{u}(t-\tau)=e^{j\gamma(t-\tau)}$ denotes the MU $\leftarrow$ SU
signal injected by the STMM modulating the MU signal $s_{d}(t-2\tau)$.

However, model (23) that is typically assumed in the existing literature on
STMM is valid in restricted settings (e.g., $\theta=\pi/2$ and $\phi=0$, as in
[20]). Generally, neither approximation $(a)$ nor approximation $(b)$ hold for
large-bandwidth systems and/or comparatively large STMM. Noteworthy, while the
beam squinting due to $s_{d}(t)$ cannot be compensated (except by reducing the
MU $\rightarrow$ SU bandwidth $B_{d}$), the coupling between spatial and
temporal phase patterns for $\theta\neq\pi/2$ is peculiar to the STMM when
involving wireless systems. The detrimental effect analyzed herein is the
coupling between the spatial and temporal phases. Therefore, solutions to
space-time phase coupling are detailed in Section IV-B and proposing
countermeasure for wideband MU $\leftarrow$ SU system is in Section IV-C.

###  IV-B Space-Time Phase Coupling

The space-time phase coupling can be readily explained considering Fig. 4. A
plane wave impinging on the $(q,v)$-th meta-atom of the STMM with angle
$(\theta,\phi)$, as depicted in Fig. 4, will be retro-reflected with the
following phase:

$\widetilde{\beta}_{q,v}(t)=\varphi_{q,v}+\gamma(t-\Delta t_{q,v}),$ (24)

thus the phase $\gamma(t)$ experiences a meta-atom-specific time delay that
depends on the incidence angles $(\theta,\phi)$, which induces a spurious, and
undesired, spatial phase gradient. This phenomenon leads to an unwanted
decrease in the reflection gain. In other words, assuming the conventional
spatial phase gradient detailed in (22), we have that the following theorem
holds:

######  Theorem 1.

Let $h_{u}(t)$ be the channel defined as in (17) and let the phase be defined
as (16), with $\varphi_{q,v}=\pi(q\cos\theta\cos\phi+v\cos\theta\sin\phi)$,
$u=0,...,M_{u,x}-1$, $v=0,...,M_{u,y}-1$. Then, the following inequality
holds:

$\mathbb{E}_{\gamma}\left[\lvert h_{u}(t)\rvert^{2}\right]\leq M_{u}^{2}.$
(25)

###### Proof.

The proof can be readily carried out by inspection of the structure of
$h_{u}(t)$, which yields

$\mathbb{E}_{\gamma}\left[\bigg{\lvert}\sum_{q=0}^{M_{u,x}-1}\sum_{v=0}^{M_{u,y}-1}e^{j\gamma(t-\Delta
t_{q,v}-\tau)}\bigg{\rvert}^{2}\right]\leq M_{u}^{2}.$ (26)

To have equality in (26), the arguments of the complex exponential shall
neither be a function of $q$ nor of $v$ (spatial and temporal phase
decoupling). Therefore, the sufficient conditions are
$\gamma(t)=\overline{\gamma}$ (straightforward condition) or $\Delta
t_{x}=\Delta t_{y}=0$ ($\theta=\pi/2$). ∎

Theorem 1 implies that any phase-encoded modulation signal $\gamma(t)$ leads
to a strict inequality for (26), except for very low MU $\leftarrow$ SU signal
bandwidth, fulfilling (21), or the case in which the impinging wave is
perfectly orthogonal to the STMM plane, i.e., $\theta=\pi/2$. To evaluate the
space-time phase coupling effect, let us consider a generic non-linear
$\gamma(t):[a,b]\longrightarrow\mathbb{R}$, infinitely differentiable for
$t_{0}\in(a,b)$. We can use the Taylor expansion around $t=t_{0}$, truncated
at the $n$-th term

$\begin{split}\gamma(t_{0}-\Delta
t_{q,v})&\approx\gamma(t_{0})+\sum_{i=1}^{n-1}\frac{(-\Delta
t_{q,v})^{i}}{i!}\gamma^{(i)}(t_{0}).\end{split}$ (27)

where $\gamma^{(i)}(t_{0})$ denotes the $i$-th time derivative of $\gamma(t)$
evaluated in $t=t_{0}$. Considering only the first order ($n=1$) derivative
only, namely

$\begin{split}\gamma(t_{0}-\Delta
t_{q,v})\approx\gamma(t_{0})-\gamma^{(1)}(t_{0})\Delta t_{q,v}\end{split}$
(28)

the second term is a linear phase shift with the meta-atom index that results
in the so-called beam squinting, i.e., the angular shift of the reflected beam
w.r.t. to the desired direction $(\theta,\phi)$. This effect causes a
reduction of the reflection gain that can be analytically evaluated from the
STMM’s array factor that adds up to a possible beam squinting due to a large
bandwidth $s_{d}(t)$ (herein not considered).

Figure 5: Normalized reflection gain in the presence of space-time phase
coupling. The blue line represents the predicted reflection performance by
state-of-the-art works for any incident angle (e.g., [20]). Due to the
coupling, the performance for $\theta_{i}\neq 90$ deg is in red and purple
instead. With the proposed compensation (detailed in Section IV-C), we can
attain the blue line for any value of $\theta_{i}$.

To quantify the beam squinting due to (28), let us consider a linear temporal
phase signal (or piecewise linear, e.g., obtained with frequency shift keying
(FSK) modulation) $\gamma(t)=2\pi f_{s}t$, where $f_{s}=f_{i}-f_{o}$ is the
frequency shift determined by the temporal component of the phase
$\beta(\mathbf{x},t)$ as in (8). Let us express the frequency shift as
$f_{s}=\kappa f_{i}$, where $\kappa\in\mathbb{R}$ is a coefficient that
describes the frequency up- or down-conversion, or equivalently
$f_{o}=f_{i}(1+\kappa)$. By substituting the expression of $\gamma(t)$ in (26)
and removing the expectation, we obtain

$AF(\theta,\phi,\kappa)=\frac{1}{M_{u}}\sum_{q=0}^{M_{u,x}-1}\sum_{v=0}^{M_{u,y}-1}e^{j2\pi
f_{i}\kappa(t-\Delta t_{q,v})},$ (29)

with the corresponding normalized reflection gain in (30).

$|AF(\theta,\phi,\kappa)|^{2}=\left|\frac{1}{M_{u}}\frac{\sin((\pi/4)M_{u,x}\kappa\cos\theta\cos\phi)}{\sin((\pi/4)\kappa\cos\theta\cos\phi)}\frac{\sin((\pi/4)M_{u,y}\kappa\cos\theta\sin\phi)}{\sin((\pi/4)\kappa\cos\theta\sin\phi)}\right|^{2}.$
(30)

Considering for instance a 1D case (i.e., either a 1D STMM or $\phi=0$),
replacing $\Delta t_{q,v}$ in (18) with $\Delta t_{x}$ in (19) ($\phi=0$), the
normalized reflection gain simplifies as

Figure 6: Angle of maximum reflection $\tilde{\theta}_{o}$, for different
incidence angles $\theta_{i}$ and $M_{u}=100$.

$\begin{split}|AF_{1D}(\theta,\kappa)|^{2}&=\left|\frac{1}{M_{u}}\frac{\sin((\pi/2)M_{u}\kappa\cos\theta)}{\sin((\pi/2)\kappa\cos\theta)}\right|^{2}.\end{split}$
(31)

This allows gaining insight into the effective reflection gain of the STMM.
Fig. 5 depicts the impact of the space-time phase coupling on the normalized
reflection gain, and for $\theta\neq\pi/2$ and $\kappa\neq 0$, the system
experiences a diminished reflection gain. For instance, at $\theta=\pi/6$ we
observe a loss $\geq 3$ dB ($M_{u}=100\times 100$) and $\geq 10$ dB
($M_{u}=200\times 200$) for $\kappa>2$ % (e.g., for a frequency shift of
$f_{s}>0.6$ GHz around $f_{i}=30$ GHz). The loss w.r.t. the maximum reflection
gain can be justified by the tilting of the direction of maximum reflection,
which can be evaluated as

$\tilde{\theta}_{o}=\arccos\left[(1+\kappa)\cos\theta\right]$ (32)

where $\tilde{\theta}_{o}$ is the angle of maximum reflection, different from
the true one $\theta=\theta_{o}=\theta_{i}$ due to the space-time phase
gradient coupling. The trend of $\tilde{\theta}_{o}$ with the frequency shift
$\kappa$ is shown in Fig. 6. The result in (32) is obtained by substituting
(28) and (22) in (17). Note that for $|\cos\theta(1+\kappa)|>1$ in (32) we
have evanescent waves, as depicted in Fig.7.

Figure 7: Angle of maximum reflection $\tilde{\theta}_{o}$ vs frequency shift
$\kappa$ and incidence angle $\theta_{i}$ ($M_{u}=100$). For grazing angles,
depending on $\kappa$, evanescent waves are excited along the surface.

Whenever the phase $\gamma(t)$ is periodic, or cyclo-stationary, the
aforementioned approaches are specialized to the harmonics of the Fourier
series decomposition of $\gamma(t)$ as experimentally observed in [17, 18]. To
implement a wireless communication system using STMM technology, the spatial
phase must be decoupled from the temporal one, such that the former can be
used to retro-reflect the impinging signal toward the MU and the latter to
convey information.

###  IV-C Space-Time Phase Decoupling

Decoupling the spatial phase from the temporal phase can be achieved
differently depending on the specific STMM architecture.

####  IV-C1 Architecture A

Decoupling can be achieved by delaying the temporal phase of each meta-atom to
compensate for the propagation delay caused by the wavefront. Specifically,
the phase of the $(q,v)$-th meta-atom must be set as:

$\beta_{q,v}(t)=\varphi_{q,v}+\gamma(t+\Delta t_{q,v}).$ (33)

To achieve perfect decoupling, assumed herein, the STMM meta-atoms must be
accurately time-synchronized using the knowledge of the incident angles as
input, e.g., after a proper estimation.

####  IV-C2 Architecture B

A straightforward yet effective countermeasure to space-time phase coupling is
to pre-compensate the spatial phase by considering the induced effect of the
temporal phase. In particular, the phase of the $(q,v)$-th meta-atom is

$\beta_{q,v}(t)=\varphi_{q,v}+\gamma(t)+\gamma^{(1)}(t)\Delta t_{q,v}.$ (34)

where the spatial phase $\varphi_{q,v}$ now incorporates the meta-atom-
dependent delay $\Delta t_{q,v}$. As demonstrated in Section VI, architecture
B allows for a cost reduction (only a single tunable component is used to
control the temporal phase) compared to architecture A, at the price of
performance reduction for large bandwidth UL signals. The design of
$\gamma(t)$, the requirements, and the achievable performance of the proposed
communication system are discussed in the following Section V.

##  V UL Signal Design

The MU $\rightarrow$ SU signal $y_{d}(t)$ can be designed following
conventional methods (e.g., linear modulation or orthogonal frequency division
multiplexing). In this section, we propose the criteria to design the phase
shape $\gamma(t)$ for controlled retro-reflection by the STMM.

###  V-A Continuous Phase Modulation

Let us consider that $\gamma(t)$ is a CPM signal for MU $\leftarrow$ SU:

$\gamma(t)=2\pi h\sum_{n}z_{n}q(t-nT_{u})$ (35)

where $h$ is the modulation index, $z_{n}\in\\{2b-1\\}$,
$b\in\\{-(M/2),...,(M/2)-1\\}$, is the $n$-th $M$-ary information symbol,
$T_{u}$ is the MU $\leftarrow$ SU symbol duration, and $q(t)$ is the phase
pulse with the following properties

$q(t)=\begin{cases}0\quad t\leq 0\\\ \frac{1}{2}\quad t\geq
L\,T_{u}\end{cases}$ (36)

where $L$ represents the modulation memory. The phase pulse is usually defined
as

$q(t)=\int_{-\infty}^{t}p(\tau)d\tau$ (37)

with $p(t)$ being the pulse shaping filter (PSF). The reason behind choosing
CPM is that it represents the most general phase modulation for $\gamma(t)$.
Indeed, depending on the parameters set $M,h,L$, and $p(t)$, the CPM can
degenerate in the other phase modulations, such as phase shift keying (PSK),
frequency shift keying (FSK), minimum shift keying (MSK), and Gaussian MSK
(GMSK) [37]. The same reference can be used as an introduction to CPM.

###  V-B Bandwidth Occupancy

The bandwidth occupied by the MU $\rightarrow$ SU received signal $y_{d}(t)$
in (10), assuming a frequency-flat channel, is the bandwidth of the
transmitted signal $s_{d}(t)$, i.e., the support of the power spectral density
(PSD)
$P_{d}(f)=\mathbb{E}\left[\left|\mathcal{F}\left\\{s_{d}(t)\right\\}\right|^{2}\right]$,
where $\mathcal{F}\left\\{.\right\\}$ denotes the Fourier Transform operator.
Under the same channel assumption, the bandwidth occupied by the MU
$\leftarrow$ SU signal $y_{u}(t)$ in (12), is the support of its PSD, defined
as

$\begin{split}P_{u}(f)&=\mathbb{E}\left[\left|\mathcal{F}\left\\{s_{d}(t)s_{u}(t)\right\\}\right|^{2}\right]=\\\
&=\mathbb{E}\left[\left|\mathcal{F}\left\\{s_{d}(t)\right\\}\right|^{2}\right]*\mathbb{E}\left[\left|\mathcal{F}\left\\{s_{u}(t)\right\\}\right|^{2}\right],\end{split}$
(38)

and thus, according to the properties of the convolution, the total bandwidth
of the system is

$B_{tot}=B_{d}+B_{u}$ (39)

with $B_{u}$ being the support of
$\mathbb{E}\left[\left|\mathcal{F}\left\\{s_{u}(t)\right\\}\right|^{2}\right]$.
Notice that the resulting spectrum $P_{u}(f)$ is centered around the carrier
frequency $f_{i}$. This follows from the fact that the information-bearing
phase signal $\gamma(t)$ has the time-derivative

$\gamma^{(1)}(t)=2\pi h\sum_{n}z_{n}p(t-nT_{u})$ (40)

that is zero on the ensemble average, i.e., $\mathbb{E}[\gamma^{(1)}(t)]=0$,
thus the average net frequency shift is zero. For a generic CPM signal, the
occupied bandwidth is infinite, but, the $99\%$ of the signal energy is within
[38]

$B_{u}=\frac{h}{T_{u}}\sqrt{\frac{g(M^{2}-1)}{3L}}+\frac{g}{T_{u}L}=\frac{\epsilon}{T_{u}}$
(41)

where $\epsilon$ is the time-bandwidth product of the specific CPM
implementation. In (41), the term $g$ depends on the PSF $p(t)$ used, e.g.,
$g=1$ for rectangular and $g=1.5$ for raised cosine. The bandwidth occupation
is inversely proportional to the CPM memory $L$. For $M=2$, $g=1$ (rectangular
PSF), the bandwidth is $B_{u}=(h\sqrt{L}+1)/(LT_{u})$.

It is worth underlining that the proposed communication system is full-duplex,
as MU $\rightarrow$ SU and MU $\leftarrow$ SU communications simultaneously
occur on the same spectrum portion $B_{tot}$. However, the spectral efficiency
is equivalent to an FDD system, with $B_{d}$ for MU $\rightarrow$ SU and
$B_{u}$ for MU $\leftarrow$ SU, as detailed in the following.

###  V-C Spectral Efficiency

The performance of the proposed system can be assessed in terms of spectral
efficiency (SE), considering both MU $\rightarrow$ SU and MU $\leftarrow$ SU
links. In particular, the SE upper bound given by Shannon’s unconstrained SE
is:

$\begin{split}\eta=\underbrace{(1-\mu)\log_{2}(1+\Upsilon_{d})}_{\eta_{d}}+\underbrace{\mu\log_{2}(1+\Upsilon_{u})}_{\eta_{u}}\end{split}$
(42)

where the first term ($\eta_{d}$) is the SE of the MU $\rightarrow$ SU and the
second ($\eta_{u}$) is the SE of the MU $\leftarrow$ SU. In (42),
$\mu=B_{u}/B_{tot}$ is the fractional bandwidth occupied by the MU
$\leftarrow$ SU, and $1-\mu=B_{d}/B_{tot}$ the MU $\rightarrow$ SU
counterpart, while $\Upsilon_{d}$ and $\Upsilon_{u}$ are the MU $\rightarrow$
SU and MU $\leftarrow$ SU signal-to-noise ratios (SNRs) at the decision
variable.

The MU $\rightarrow$ SU SNR is

$\Upsilon_{d}\leq\frac{\sigma_{s,d}^{2}NM_{d}}{\varrho_{d}\sigma_{z}^{2}}$
(43)

where the equality holds when perfect CSI is available and optimal
precoding/combing is employed, both providing the maximum MIMO gain, i.e.,
$NM_{d}$.

The MU $\leftarrow$ SU Rx signal undergoes a matched filtering over $T_{u}$
with the Tx signal $s_{d}(t-2\tau)$, which is clearly known at MU. Assuming
the delay $\tau$ is known, the SNR can be analytically evaluated as in
Appendix A, depending on whether the STMM exhibits a wideband behavior w.r.t.
$\gamma(t)$ or not. A suitable upper bound on the SNR is obtained by assuming
that the wideband regime involves the same UL symbol, i.e., $\Delta T<T_{u}$,
yielding

$\Upsilon_{u}\leq\frac{\sigma_{s,d}^{2}N^{2}M_{u}^{2}}{\varrho_{u}\sigma^{2}_{z}}|AF(\theta,\phi,\kappa)|^{2}T_{u}B_{tot}$
(44)

where $|AF(\theta,\phi,\kappa)|^{2}\leq 1$ is the loss due to the space-time
phase coupling, defined in (30), and $T_{u}B_{tot}$ is the processing gain of
the matched filter. The latter is motivated by the fact that, while the MU
$\leftarrow$ SU noise $z(t)$ occupies the whole bandwidth $B_{tot}$, the
signal only occupies a fraction of it $\mu B_{tot}$. Thus, the matched filter
makes the MU $\leftarrow$ SU SNR inversely proportional to the fractional
bandwidth occupation, $\Upsilon_{u}\propto 1/\mu$. The equality in (44) holds
under the following conditions: (i) perfect CSI available at both MU and SU,
(ii) known delay $\tau$ at the MU, and (iii) constant phase
$\gamma(t)\in[0,T_{u})$.

The MU $\rightarrow$ SU and MU $\leftarrow$ SU capacity in (42) has a global
optimum with respect to the unconstrained UL bandwidth allocation $\mu$. From
the derivative of $\eta$ w.r.t. $\mu$

$\begin{split}\frac{\mathrm{d}\eta}{\mathrm{d}\mu}&=-\log_{2}(1+\Upsilon_{d})-\frac{\widetilde{\Upsilon}_{u}\log_{2}e}{\widetilde{\Upsilon}_{u}+\mu}+\\\
&+\log_{2}\left(1+\frac{\widetilde{\Upsilon}_{u}}{\mu}\right)=0\end{split}$
(45)

whose root provides the optimal duplexing allocation. In (45), we denote with
$\widetilde{\Upsilon}_{u}$ the MU $\leftarrow$ SU SNR terms that do not depend
on the allocation $\mu$. The total capacity is analyzed in Section VI.

##  VI Numerical Results

(a)

(b)

(c)

Figure 8: Spectral efficiency considering different MU arrays configuration
($N=16\times 4,\;32\times 16$) as a function of various design parameters:
(8a) SNR at the antenna of the SU in MU $\rightarrow$ SU, (8b) MU
$\leftrightarrow$ SU distance, and (8c) STMM size.

This section presents numerical and analytical results for the proposed STMM-
based wireless communication system, aimed at gaining insight into upper
achievable performance, operating conditions, and possible limitations. In all
the following, the frequency of operation is $f_{0}=30$ GHz. We consider a CPM
modulation with a rectangular PSF, $L=1$, and $M$-ary$=2$, where
$\kappa=h/(2T_{u})$. This coincides with CP-FSK for $h=1$ (herein considered)
and with MSK for $h=1/2$. The trends and considerations are the same for a
generic CPM or any phase/frequency modulation, but the absolute values must be
derived case by case.

###  VI-A Impact of system parameters

The first set of results, summarized in Fig. 8, shows the achievable SE (both
MU $\rightarrow$ SU and MU $\leftarrow$ SU) varying system parameters, in the
utter performance assumption of $|AF(\theta,\phi,\kappa)|^{2}=1$, namely with
(44), which is commonly assumed in the state of the art on STMM [20]. These
results can be attained for (i) architecture A with space-time decoupling
(33), (ii) architecture B with space-time decoupling (34) and limited
bandwidth $B_{u}$, and (iii) normal incidence, i.e., $\theta_{i}=90$ deg. In
all cases, we have $\mu=0.5$, thus the same bandwidth for MU $\rightarrow$ SU
and MU $\leftarrow$ SU. Fig. 8a shows the SE varying the MU $\rightarrow$ SU
equivalent single-input single-output (SISO) SNR, namely

$\Upsilon_{0}=\frac{\Upsilon_{d}}{NM_{d}}=\frac{\sigma^{2}_{s,d}}{\varrho_{d}\,\sigma_{z}^{2}},$
(46)

as well as the number of MU antennas $N$ and STMM meta-atoms $M_{u}$. The MU
$\mathrel{\leavevmode\hbox to14.63pt{\vbox
to7.37pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-3.68529pt\hbox
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}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}} {}{{}}{} {{}{}}{{}}
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}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.86603}{0.5}{-0.5}{-0.86603}{0.39836pt}{-0.94135pt}\pgfsys@invoke{
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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ SU link length is
fixed to $D=100$ m and the number of SU Rx meta-atoms $M_{d}=8\times 4$. We
notice the dramatic impact of the number of STMM meta-atoms $M_{u}$ that
allows boosting the MU $\leftarrow$ SU SE (up to be higher than the MU
$\rightarrow$ SU one for $M_{u}=200\times 200$). A similar effect, though to a
lesser extent, is obtained by increasing $N$. The trend of the SE varying the
MU $\mathrel{\leavevmode\hbox to14.63pt{\vbox
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}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.86603}{-0.5}{0.5}{0.86603}{13.82802pt}{0.94135pt}\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}} {}{{}}{} {{}{}}{{}}
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}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.86603}{0.5}{-0.5}{-0.86603}{0.39836pt}{-0.94135pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ SU link length $D$
is instead shown in Fig. 8b. For fixed MU $\rightarrow$ SU performance
$\eta_{d}$, the UL one rapidly decays with $D$, still being higher than the
former one for massive MU arrays and STMM (i.e., a comparatively high number
of STMM meta-atoms). As can be expected, when $N$ is fixed, e.g., by
technological limitations, the overall MU $\leftarrow$ SU SE $\eta_{u}$ is
mostly ruled by the number of STMM meta-atoms $M_{u}$, as confirmed by Fig.
8c, where the SE is let vary as a function of the STMM size (side number of
meta-atoms $\sqrt{M_{u}}$), for $D=50$ m and $D=100$ m. The results in Fig. 8
show that with an appropriate design, the STMM can provide excellent MU
$\leftarrow$ SU performance even at relatively large distances, comparable
with the cell radius in current mmWave 5G systems. Remarkably, most of the
energy employed to achieve $\eta_{d}$ and $\eta_{u}$ is spent at the MU, while
the SU does not need a power amplifier. However, wideband effects due to the
MU $\leftarrow$ SU signal $\gamma(t)$ practically limit the performance and,
mostly, the MU $\leftarrow$ SU bandwidth $B_{u}$, as discussed in the
following.

(a) $M_{u}=100\times 100$

(b) $M_{u}=200\times 200$

Figure 9: Spectral efficiency for 9a) $M_{u}=100\times 100$ and 9b)
$M_{u}=200\times 200$, and considering different incident angles varying the
net frequency shift $\kappa$, which is proportional to the MU $\leftarrow$ SU
bandwidth $B_{u}$. The red line represents the results obtained according to
[20], which neglects the compensation. The blue and black lines represent the
achievable spectral efficiency $\eta_{u}$ when compensating the coupling in
Arch. B and A, respectively.

###  VI-B Impact of space-time phase coupling and decoupling

The wideband effects on the STMM operation are investigated in Fig. 9,
reporting the MU $\leftarrow$ SU SE $\eta_{u}$ varying $\kappa$, i.e., the
equivalent linear frequency shift due to the modulating signal $\gamma(t)$. We
assume (i) $B_{u}\gg B_{d}$, such that to ignore any wideband effect due to
the MU $\rightarrow$ SU signal $s_{d}(t)$, that is the approximation (a) in
(17) and (ii) perfect channel state information (CSI) is available at the SU,
namely the incidence/reflection angles $(\theta,\phi)$ are perfectly known.
The distance is $D=100$ m and $N=32\times 16$. The aim of these results is to
quantitatively examine the impact of the space-time phase coupling at the
STMM, while the impact on an imperfect CSI is discussed in the next set of
results. The analysis in Section IV-B has shown that the space-time coupling
is not influenced by transmitted power or distance, but rather by MU
$\leftarrow$ SU bandwidth $B_{u}$, angle of incidence/reflection
$(\theta,\phi)$, and the size of the STMM $M_{u}$. Fig. 9 considers both
architectures A and B, with ideal compensations (33)-(34), respectively, and
two incidence angles, $\theta=30,60$ deg ($\phi=0$ deg for simplicity). The
non-compensation case is shown as a benchmark. architecture A has the best
performance, independent of $\kappa$, thanks to the possibility of
compensating for the two-way delay at each meta-atom of the STMM. Considering
architecture B, instead, where only one time-varying component is present (and
common to all the STMM meta-atoms), the UL SE $\eta_{u}$ progressively
degrades for increasing grazing incidence angles ($\theta\rightarrow 0$ deg)
and STMM meta-atoms $M_{u}$. In both Fig. 9a and 9b, we can distinguish three
operation regions, outlined in Appendix A and corresponding to (i) $\Delta
T\ll T_{u}$, i.e., ideal narrowband STMM operation; (ii) $\Delta T\leq T_{u}$,
i.e., wideband STMM operation without ISI; (iii) $\Delta T>T_{u}$, i.e.,
wideband STMM operation with ISI. The first operation region occurs for very
low modulation bandwidths (e.g., $\kappa\leq 1\%$ for $M_{u}=100\times 100$,
$\theta=30$ deg and $\kappa\leq 0.5\%$ for $M_{u}=200\times 200$, $\theta=30$
deg), limiting the effective MU $\leftarrow$ SU throughput. The second
operation region happens for larger modulation bandwidths, up to a cut-off
frequency $\overline{\kappa}$ (e.g., $\kappa\leq\overline{\kappa}=2.5\%$ for
$M_{u}=100\times 100$, $\theta=30$ deg and $\kappa\leq\overline{\kappa}=1.2\%$
for $M_{u}=200\times 200$, $\theta=30$ deg). In this regime, the STMM benefits
from the space-time decoupling (34) to improve the SE $\eta_{u}$ up to
$\approx 3$ bit/s/Hz (for $\kappa\rightarrow\overline{\kappa}$ in Fig. 9b).
Above the cut-off limit $\overline{\kappa}$, the STMM realized with
architecture B does not work properly: the SNR after the matched filter
$\Upsilon_{u}$ is abated by the presence of the ISI from past modulation
symbols in $\gamma(t)$ (see Appendix A). The performance rapidly drops to
zero. It is worth remarking that past symbols represent ISI as far as the
simple matched filter approach (before trellis decoding) is adopted. In
principle, the Rx can be designed to account for the unwanted additional delay
due to the STMM reflection, attaining a similar performance to architecture A;
however, this would increase the complexity of the implementation and it is
not considered here. Alternatively, a viable solution may be to consider an
STMM that combines the advantages of architectures A and B, e.g., by allowing
the delay tuning at different subsets of meta-atoms.

(a)

(b)

Figure 10: MU$\leftarrow$SU spectral efficiency varying the equivalent SISO
SNR $\Upsilon_{0}$, for (10a) $M_{u}=100\times 100$ and (10b) $M_{u}=200\times
200$ and considering both perfect and imperfect CSI as well as space-time
phase coupling (with the proposed compensation and without compensation [20]).
As a benchmark, green curves denote the spectral efficiency for a OOK-
modulated backscatter communication system subject to CSI errors and without
compensation (e.g., [22]).

###  VI-C Impact of imperfect CSI

In practice, incidence/reflection angles $(\theta,\phi)$ are estimated at the
SU Rx array, leading to unavoidable errors affecting both the spatial phase
configuration of the STMM (imperfect pointing) as well as the space-time phase
decoupling (imperfect compensation). The combination of the latter two effects
has the consequence of diminishing the amount of Rx signal energy at the MU
side, decreasing the SE w.r.t. the ideal case (see Appendix B for the
analytical derivations). Fig. 10 shows the MU$\leftarrow$SU SE $\eta_{u}$
varying the equivalent SISO SNR $\Upsilon_{0}$ (46), with and without space-
time decoupling (as in [20]) and in presence of errors on the estimation of
the elevation angle $\theta$, whose variance is quantified herein with the
Cramér-Rao lower bound (CRLB) [39] (Appendix B). We consider the very same
simulation parameters used for Fig. 8a, namely $N=16\times 4$ Tx antennas at
the MU, $M_{d}=8\times 4$ Rx antennas at the SU, $\theta=30$ deg, $\phi=0$ deg
(known), $\kappa=1\%$ (Fig. 10a), $\kappa=2\%$ (Fig. 10b). With the current
settings, the standard deviation of the error on $\theta$ ranges from $8.7$
deg at $\Upsilon_{0}=-40$ dB to $0.087$ deg at $\Upsilon_{0}=0$ dB. As a
further benchmark, we show the constrained SE of a backscatter communication
system where the STMM is OOK-modulated, with no space-time phase decoupling
and imperfect CSI as well [22]. In the low SNR region, the error on the
estimation of $\theta$ (ruled by $\Upsilon_{0}$) jeopardizes any space-time
phase decoupling effort. Namely, the imperfect pointing caused by a wrong
setting of the spatial STMM phase (red curves) is much larger than the effect
of the space-time phase coupling counterpart (yellow curves). Notice that
yellow curves represent the performance of state-of-the-art works on STMM-
based wireless systems [20], where $\theta$ is assumed to be perfectly known
but the space-time coupling is not compensated. However, as $\Upsilon_{0}$
grows large, the performance trend swaps: the effect of the imperfect CSI
vanishes (red curves attain the blue ones asymptotically) and only an
uncompensated space-time phase coupling affects the performance. Of course, as
the spatial selectivity of the STMM increases ($M_{u}$), the impact of
imperfect CSI and coupling gets worse. Interestingly, we observe a threshold
SNR $\overline{\Upsilon}_{0}$ value (depending on the size of of the STMM
$M_{u}$) that determines two operating regions: for
$\Upsilon_{0}\leq\overline{\Upsilon}_{0}$, the SE $\eta_{u}$ is dominated by
the effect of CSI errors (requiring proper countermeasures, e.g., increasing
the Rx array size $M_{d}$), while for $\Upsilon_{0}>\overline{\Upsilon}_{0}$,
the performance is dominated by the space-time phase coupling, confirming the
outcomes of Fig. 9 and requiring the suitable compensation proposed in this
work. In any case, the proposed STMM-based wireless communication system
outperforms the OOK-modulated one, which only employs a two-level amplitude
modulation (yielding a -3 dB matched filter gain at the MU) and does not
operate any compensation for the space-time phase coupling.

Figure 11: Total spectral efficiency considering different metasurface sizes,
i.e., $M_{u}=100\times 100$ and $M_{u}=200\times 200$, and different distances
varying the bandwidth ratio $\mu$.

###  VI-D Impact of the bandwidth ratio $\mu$

Concerning the optimal choice of $\mu$, namely the bandwidth splitting between
MU $\rightarrow$ SU and MU $\leftarrow$ SU, it depends on the size of the STMM
and the distance between the MU and SU. Fig. 11 shows the MU $\rightarrow$ SU,
MU $\leftarrow$ SU, and total SE $\eta_{d}$, $\eta_{u}$ and $\eta$
respectively, varying $\mu$ for $D=100$ m, $N=32\times 16$, and assuming ideal
narrowband STMM behavior (upper-performance limit). The total SE shows a peak,
as predicted by (45), that depends on the number of STMM meta-atoms $M_{u}$.
This result can have two readings: one can maximize the spectral efficiency
$\eta$ over the entire bandwidth $B_{tot}$, and thus the fraction $\mu$
allocated depends on the specific setup, or one can allocate the fraction
$\mu$ based on the MU and SU requirements, but the spectral efficiency $\eta$
over the entire band may not be optimal. In other words, the method used to
allocate the fraction $\mu$ of the band can be based on either maximizing
overall spectral efficiency $\eta$ or on specific MU and SU requirements.
While maximizing overall spectral efficiency $\eta$ is desirable in some
cases, it may not be the best approach if the MU and SU requirements must be
met. In such cases, one may have to compromise on the overall spectral
efficiency $\eta$ to meet the requirements of the MU and SU.

##  VII Conclusions and Open Challenges

This paper proposes a mathematical model for the design of STMMs in wireless
communication systems, where the spatial component of the modulation controls
the direction in which the signal is reflected, and the temporal component
conveys information in the designated direction. The model examines two
potential implementations, Architectures A and B, with Architecture A having
greater design flexibility than Architecture B, which is more cost-effective.
The model describes the unique features of the STMMs, including the space-time
phase coupling and how it affects system performance. Analytical descriptions
of these effects are provided, along with suggested countermeasures.
Additionally, the proposed model addresses the design criteria and challenges
of STMM-based wireless communications, specifically in the context of a full-
duplex system architecture, which is one of the promising solutions where
STMMs can be beneficial. The full-duplex system architecture comprises a
master unit (MU) that transmits a downlink MU $\rightarrow$ SU signal and a
slave unit (SU) that can receive, reflect, and modulate the MU $\rightarrow$
SU using STMM technology. In order to transmit information in uplink MU
$\leftarrow$ SU, the SU modulates the temporal phase gradient applied to the
metasurface meta-atoms. To validate the analytical findings, numerical results
are presented, demonstrating the efficacy of the proposed model, even in the
presence of imperfect CSI acquisition. Nonetheless, the proposed mathematical
model only serves as an initial step for designing and analyzing STMM-based
wireless communication systems. Further research is imperative to fully
explore the capabilities and limitations of STMM technology. This should
include investigations into multi-user scenarios, the impact of mobility on
system performance, and experimental validations to bridge the gap between
theory and practical implementation. Addressing these challenges will pave the
way for harnessing the full potential of STMMs in future wireless
communication applications.

## Appendix A

The Rx signal at the MU side undergoes matched filtering and phase estimation,
before being fed to the sequence detector, e.g., Viterbi. We derive the SNR
after the matched filter at the MU in the assumption that the Rx signal is the
one expressed by (17), approximation $(a)$. The estimated phase at the $n$-th
MU $\leftarrow$ SU symbol is provided by

$\hat{\gamma}(nT_{u})=\arg\left\\{\int_{nT_{u}}^{(n+1)T_{u}}y_{u}(t)s_{d}^{*}(t-2\tau)dt\right\\}.$
(47)

The SNR is obtained from $r_{n}$ with a double expectation over the Tx signal
$s_{d}(t)$ and the modulating phase $\gamma(t)$. The power of the useful
signal part is in (48),

$\begin{split}\sigma^{2}_{signal}&=\mathbb{E}_{s,\gamma}\left[\bigg{\lvert}\int_{T_{u}}\rho\lvert
s_{d}(t-2\tau)\rvert^{2}\sum_{u,v}e^{j\gamma(t-\Delta
t_{u,v}-\tau)}\bigg{\rvert}^{2}dt\right]\leq\\\
&\leq\mathbb{E}_{s}\left[\int_{T_{u}}|\rho|^{2}\lvert
s_{d}(t-2\tau)\rvert^{4}dt\right]\mathbb{E}_{\gamma}\left[\int_{T_{u}}\bigg{\lvert}\sum_{u,v}e^{j\gamma(t-\Delta
t_{u,v}-\tau)}\bigg{\rvert}^{2}dt\right]\approx\\\
&\approx|\rho|^{2}\sigma^{4}_{s,d}T_{u}\,\,\mathbb{E}_{\gamma}\left[\int_{T_{u}}\bigg{\lvert}\sum_{u,v}e^{j\gamma(t-\Delta
t_{u,v}-\tau)}\bigg{\rvert}^{2}dt\right],\end{split}$ (48)

where we made use of the Schwartz inequality and assume, for simplicity of
exposition, that the MU $\rightarrow$ SU signal $s_{d}(t)$ has a constant
power $\sigma^{2}_{s,d}$ within the integration interval $T_{u}$. According to
the size of the STMM $M_{u}=M_{u,x}\times M_{u,y}$ compared to the MU
$\leftarrow$ SU bandwidth $B_{u}$ (related to the symbol time through (41)),
we can distinguish three cases, as detailed in the following.

####  A-1 $\Delta T\ll T_{u}$

In this case, the STMM exhibits a narrowband behavior w.r.t. to the MU
$\leftarrow$ SU signal, namely the variation of $\gamma(t)$ across the STMM
can be neglected. Therefore, we have

$\mathbb{E}_{\gamma}\left[\int_{T_{u}}\bigg{\lvert}\sum_{u,v}e^{j\gamma(t-\Delta
t_{u,v}-\tau)}\bigg{\rvert}^{2}dt\right]\approx M_{u}^{2}T_{u}$ (49)

and the reflection gain is maximized under the assumption of the perfect
knowledge of the channel (i.e., perfect spatial phase configuration in (41)).
The power of the noise after the matched filter is

$\sigma^{2}_{noise}\leq N_{0}\sigma^{2}_{s,d}T_{u}$ (50)

(including the beamforming gain at the MU), thus the SNR is upper-bounded by

$\Upsilon_{u}=\frac{\sigma^{2}_{signal}}{\sigma^{2}_{noise}}\leq\frac{|\rho|^{2}M_{u}^{2}\sigma_{s,d}^{2}}{\sigma^{2}_{z}}T_{u}B_{tot}$
(51)

from which follows (44) by expanding $\rho$.

####  A-2 $\Delta T\leq T_{u}$

In this case, the STMM exhibits a wideband behavior w.r.t. to the MU
$\leftarrow$ SU signal, namely the variation of $\gamma(t)$ across the STMM
cannot be ignored in (48), but it is less then the symbol duration. Now, we
have

$\begin{split}\mathbb{E}_{\gamma}\left[\int_{T_{u}}\bigg{\lvert}\sum_{u,v}e^{j\gamma(t-\Delta
t_{u,v}-\tau)}\bigg{\rvert}^{2}dt\right]\approx\\\ \approx
M_{u}^{2}|AF(\theta,\phi,\kappa)|^{2}T_{u}\end{split}$ (52)

it yields the case explored in Section IV with the beam squinting effect. The
SNR upper bound is

$\Upsilon_{u}\leq\frac{|\rho|^{2}M_{u}^{2}\sigma_{s,d}^{2}}{\sigma^{2}_{z}}|AF(\theta,\phi,\kappa)|^{2}T_{u}B_{tot}.$
(53)

The previous expression tends to be the narrowband one when applying
compensation techniques in Section IV-C.

####  A-3 $\Delta T>T_{u}$

In this case, the STMM exhibits a wideband behavior w.r.t. to the MU
$\leftarrow$ SU signal and the variation of $\gamma(t)$ across the STMM
exceeds the symbol duration. This means that the MU receives a signal with
multiple superposed symbols, and the STMM meta-atoms can be clustered
according to the specific MU $\leftarrow$ SU symbol they are reflecting at a
given instant of time. Considering for simplicity a linear STMM, we have
$C=\lceil M_{u}\Delta t/T_{u}\rceil$ spanned symbols, where $\Delta t$ is the
propagation delay across two adjacent meta-atoms, $\Delta
t=(\lambda_{i}/4)(\cos\theta/c)$. Considering a 2D STMM again, we can define
the following set of STMM meta-atoms:

$\begin{split}\mathcal{M}^{\prime}_{u}&=\\{(u,v)\lvert\Delta
t_{u,v}<T_{u}\\}\\\ \mathcal{M}^{\prime\prime}_{u}&=\\{(u,v)\lvert\Delta
t_{u,v}\geq T_{u}\\}\end{split}$ (54)

for $u=0,...,M_{u,x}-1$, $v=0,...,M_{u,y}-1$, where the first one indicates
the symbol of interest and the second all the remaining $C-1$ ones. Notice
that estimating the phase with the simple matched filter in (47) implies that
the $C-1$ symbols other than the $n$-th of interest act as inter-symbol
interference (ISI). Therefore, we have the relation in (55)

$\mathbb{E}_{\gamma}\left[\int_{T_{u}}\bigg{\lvert}\sum_{u,v}e^{j\gamma(t-\Delta
t_{u,v}-\tau)}\bigg{\rvert}^{2}dt\right]\approx\underbrace{\lvert\mathcal{M}^{\prime}_{u}\rvert^{2}|AF(\theta,\phi,\kappa)|^{2}T_{u}}_{\text{useful
signal}}+\underbrace{\mathbb{E}_{\gamma}\left[\int_{T_{u}}\bigg{\lvert}\sum_{(u,v)\in\mathcal{M}^{\prime\prime}_{u}}e^{j\gamma(t-\Delta
t_{u,v}-\tau)}\bigg{\rvert}^{2}dt\right]}_{\text{ISI}}$ (55)

and thus the useful signal power is

$\sigma^{2}_{signal}\approx|\rho|^{2}\sigma^{4}_{s,d}\lvert\mathcal{M}^{\prime}_{u}\rvert^{2}|AF(\theta,\phi,\kappa)|^{2}T^{2}_{u}$
(56)

while the ISI power $\sigma^{2}_{ISI}$ comes from the integration across
different symbols, with detrimental effects on system performance. The SNR
turns into SINR as

$\begin{split}\Upsilon_{u}=\frac{|\rho|^{2}\sigma^{4}_{s,d}\lvert\mathcal{M}^{\prime}_{u}\rvert^{2}|AF(\theta,\phi,\kappa)|^{2}T^{2}_{u}}{\sigma^{2}_{ISI}+N_{0}\sigma^{2}_{s,d}T_{u}}.\end{split}$
(57)

## Appendix B

This Appendix derives the model for the characterization of the system
performance in case of imperfect CSI, i.e., when the incident elevation angle
$\theta$ is estimated with unavoidable error at the SU side,
$\widehat{\theta}=\theta+\delta\theta$. Extension to the case in which $\phi$
is subject to estimation errors is possible but covered here, thus we consider
$\phi=0$ (and $\widehat{\phi}=0$) to ease the derivations. We herein consider,
for simplicity, that (i) the estimated angle $\widehat{\theta}$ is Gaussian-
distributed, i.e.,
$\widehat{\theta}\sim\mathcal{N}(\theta,\sigma^{2}_{\theta})$ and (ii) the
estimation attains the CRLB, hence the variance of the error $\delta\theta$ is
[39]:

$\sigma^{2}_{\theta}=\frac{96}{\pi^{2}\sin^{2}\theta\,\Upsilon_{0}\,N^{2}M_{d}(M_{d}^{2}-1)}.$
(58)

Let us consider the Rx signal (17) $(a)$, where we apply the following space-
time phase at the STMM (with decoupling as for (33))

$\widehat{\beta}_{q}(t)=4\pi f_{i}\widehat{\Delta
t}_{q}+\gamma(t+\widehat{\Delta t}_{q})$ (59)

where

$\begin{split}\widehat{\Delta
t}_{q}=\frac{\lambda_{i}}{4c}q\cos\widehat{\theta}&\approx\frac{\lambda_{i}}{4c}q\left(\cos\theta-\delta\theta\sin\theta\right)\end{split}$
(60)

(the approximation is for small $\delta\theta$). We obtain from (17):

$\begin{split}y_{u}(t)&=M_{u,y}\,\rho\sum_{q=0}^{M_{u,x}-1}e^{-j4\pi
f_{i}(\Delta t_{q}-\widehat{\Delta t}_{q})}e^{j\gamma(t-(\Delta
t_{q}-\widehat{\Delta t}_{q})-\tau)}s_{d}(t-2\tau)+z(t)\\\
&\overset{(i)}{=}M_{u,y}\,\rho\,s_{u}(t-\tau)s_{d}(t-2\tau)\sum_{q=0}^{M_{u,x}-1}\underbrace{e^{-j4\pi
f_{i}(\Delta t_{q}-\widehat{\Delta t}_{q})}}_{\text{imperfect
pointing}}\underbrace{e^{-j2\pi\kappa f_{i}(\Delta t_{q}-\widehat{\Delta
t}_{q})}}_{\text{imperfect decoupling}}+z(t)\\\
&\overset{(ii)}{\approx}M_{u,y}\,\rho\,s_{u}(t-\tau)s_{d}(t-2\tau)\sum_{q=0}^{M_{u,x}-1}e^{j\frac{\pi}{2}(2+\kappa)q\delta\theta\sin\theta}+z(t)\end{split}$
(61)

where equality $(i)$ holds by considering a linearly time-varying phase
$\gamma(t)=2\pi\kappa f_{i}t$ (or piece-wise linear) and approximation $(ii)$
is for small $\delta\theta$. Notice that (61) resembles the ideal
multiplicative model (23), except for the residual summation, representing the
loss due to imperfect estimation of $\theta$. We can distinguish between two
effects: the imperfect pointing, due to errors in the spatial phase
configuration, and the imperfect decoupling, due to errors in the compensation
of the residual delay at the STMM. The ensemble average of the energy loss at
the MU side due to imperfect CSI can be evaluated from (61) $(ii)$ as follows:

$\begin{split}\frac{1}{M_{u,x}^{2}}\mathbb{E}_{\widehat{\theta}}\left[\bigg{\lvert}\frac{\sin\left(M_{u,x}\,(\pi/4)(2+\kappa)\delta\theta\sin\theta\right)}{\sin\left((\pi/4)(2+\kappa)\delta\theta\sin\theta\right)}\bigg{\rvert}^{2}\right]\end{split}$
(62)

where the expectation is carried out over the distribution of
$\widehat{\theta}$, and the overall effects are shown in Section VI-C.

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