# Uplink Wave-Domain Combiner for Stacked Intelligent Metasurfaces Accounting
for Hardware Limitations

Maryam Rezvani1, Raviraj Adve1, Akram bin Sediq2, Amr El-Keyi2  1Dept. of
Electrical and Computer Engineering, University of Toronto, Toronto, Canada
2Ericsson Canada, Ottawa, Canada,
emails: {mrezvani<EMAIL_ADDRESS>{akram.bin.sediq, amr.el-
<EMAIL_ADDRESS>
###### Abstract

Using refractive metasurfaces (RMTSs) as part of the antenna design offers a
promising solution to the ever-increasing demand for improved energy
efficiency of wireless communications. To overcome the limitations of using
one RMTS layer, a recent proposal is to cascade multiple layers, a structure
called stacked intelligent metasurfaces (SIMs) where the desired precoder and
combiner is formed in the wave domain. However, while proposing the antenna
structure, the analysis did not account for the attendant limitations imposed
by the hardware used which has a significant impact on the performance of
SIMs. In this paper, we study the achievable sum-rate of a SIM antenna in an
uplink wireless communication scenario accounting for hardware limitations. We
begin by proposing a system model that captures both the effect of noise and
hardware limitations in these systems. We then formulate the achievable sum-
rate problem; since optimizing the rate is non-convex, we propose two
approaches: a gradient ascent algorithm and an interior point method to find a
close-to-optimum combiner. To show the efficiency of using SIMs at a
basestation, we compare the achievable sum-rate with that of a digital phased
array (DPA). We provide two comparisons: first, in Rayleigh fading and
realistic 3GPP channels and then under a constraint of an equal number of
radio frequency (RF) chains and equal physical aperture size constraint. Our
results show that SIM antennas can surpass DPA performance under an equal
number of RF chains but has inferior performance under equal aperture size.

###### Index Terms:

Holographic multiple-input multiple-output (MIMO) systems, Stacked Intelligent
Metasurfaces (SIM), Uplink wireless communications, Wave-domain combiner.

##  I Introduction

Future wireless communication networks must meet the rising demand for higher
data rates and increased throughput, while also prioritizing sustainability
and energy efficiency [1]. To achieve higher data rates, extra-large multiple-
input multiple-output (MIMO) antennas, also called extra-large digital phased
arrays (DPAs), have been introduced as an enhancement over massive MIMO
systems. By increasing the number of antennas in a DPA, the signal-to-noise
ratio (SNR) improves, reducing the power required by users. However, using
larger DPAs lead to higher power consumption at base stations (BSs) due to the
numerous power-hungry radio frequency (RF) chains.

To tackle BS power consumption, the two main approaches are to introduce low
complexity signal processing techniques such as [2] or sharing signal
processing between digital and analog/wave domains. Hybrid MIMO systems push
part of the signal processing to the analog domain using phase shifters [3]
while, metasurface (MTS) antennas share signal processing between
electromagnetic (EM) wave and digital domains [4, 5, 6]. Our focus here is on
MTS antennas.

MTSs, comprising subwavelength unit-cells, enable manipulation of its
properties via voltages applied to diodes in each unit-cell. For example,
reflective intelligent surfaces utilize MTSs to manipulate wireless channel
properties thereby enhancing channel gain. Similarly, deploying refractive
MTSs (RMTSs) at BSs provide the ability to form any desired radiation pattern
at the EM level with higher energy efficiency and lower latency [6].

The effect of the unit-cell on the impinging wave is captured as a weighting
factor. Using single-layer unit-cells, the phase and amplitude of the
weighting factor follow Lorentzian modulation, i.e., are coupled [7, 8]. This
coupling limits the radiation patterns that can be achieved by the MTS
antenna. Achieving phase-only modulation, spanning $[0,2\pi)$ interval while
keeping the amplitude constant, requires a “stacked unit-cell” design, where
each unit-cell comprises at least three reactive impedance sheets [9] where
the impedance of each layer is controlled by changing the applied voltage to
its varactor diode [10].

Since full control over the radiation pattern requires independent control
over amplitude and phase [11] of each weighting factor, using only one layer
of phase-only RMTS at BS is insufficient to form any desired radiation pattern
(e.g., having to null multiuser interference). To tackle this problem, the
work in [5, 12] introduced a multi-layer structure called stacked intelligent
metasurfaces (SIM) where several phase-only modulated RMTSs are stacked in
front of a small DPA and wrapped by absorber walls [13]. This design is based
on diffractive deep neural networks (D2NNs) [14]. In SIM antenna, the number
of elements (and RF chains) in the DPA is equal to the number of users served
by the BS. Also, using a wave-domain (WD) combiner/precoder implemented by the
SIM renders the digital combiner redundant, thereby simplifying the required
digital signal processing [5].

Since its introduction, different aspects of SIM-assisted wireless
communication have been studied. In [5], a joint power allocation and WD
precoder algorithm is introduced for downlink communications. Also, in [12] a
joint precoder and combiner is designed for single-user scenarios where both
receiver and transmitter are equipped with SIM antennas. In [15], it is shown
that using a SIM antenna, a 2D discrete Fourier transform can be implemented
over the air and hence, provide an opportunity to estimate the angle of
arrival at an EM level. Furthermore, in [16], it is shown that a BS equipped
with a SIM can form any desired beam pattern.

In all these papers, the focus is on downlink wireless communication where the
transmitter, i.e., BS, is equipped with a SIM antenna. This is while the
equally important problem of uplink communication where single-antenna users
communicate with a SIM-assisted BS is untouched. The uplink problem is a
unique one since SIM must effectively distinguish between the weak signal
suffered from channel propagation and noise, and while focusing signal on its
DPA, also mitigate noise and interference.

Furthermore, the previous works often assume SIM structures to be lossless,
i.e., a transmission coefficient of one for each RMTS layer. However, signal
power losses occur due to copper and dielectric losses, as well as the
resonant nature of each unit-cell [17], making signal detection amidst
background noise even more challenging. Also, the current approach to model
interlayer EM propagation is based on the Rayleigh-Sommerfeld relation [5, 15,
12, 14] which assumes large dimensions of implemented RMTSs compared to the
wavelength and the EM field is not evaluated close to the RMTS [18]. These
assumptions do not hold in SIM structures, leading to an unfair comparison
with other antenna designs when using the Rayleigh-Sommerfeld relation to
model the interlayer propagation. All these issues motivate us to study the
SIM-assisted BSs problem in the uplink while taking into account its realistic
hardware limitations.

Contributions: To maximize the achievable sum-rate of an uplink wireless
communication system where a BS is equipped with a SIM antenna, we propose a
system and SIM model that considers noise and EM propagation effects using
closed-form EM wave relations. Then, we formulate an optimization problem to
maximize the sum-rate with respect to the weighting factor of each unit-cell
in each RMTS layer. Since both the objective function (the achievable sum-
rate) and the feasible domain (phase-only weighting factors) are non-convex,
we employ a gradient-based ascent (GA) algorithm with adaptive learning rates
to find a close-to-optimal WD combiner. Additionally, we explore the interior-
point method (IPM), which utilizes the Hessian of the objective function to
improve update directions beyond the gradient alone.

In sum, the contributions of this paper are:

  * •

We propose a system model for a BS equipped with a SIM antenna, in the uplink,
which captures the effect of noise and hardware limitations in these systems,
and accurately models the EM propagation between RMTS layers.

  * •

To design a WD combiner that maximizes achievable sum-rate in the uplink of
the SIM antenna, we propose a GA algorithm with a variable step size. Also, we
implement an IPM approach and investigate its performance.

  * •

Our simulation results show that both in Rayleigh fading and realistic 3GPP
channels, our proposed GA algorithm and IPM can improve the performance of SIM
antennas with respect to DPA with an equal number of RF chains in single-user
and multi-user scenarios. But under equal aperture size, DPA outperforms SIM
antenna.

Notation: Lowercase letters denote scalars. We use lowercase and uppercase
boldface letters to denote vectors and matrices, respectively. $(\cdot)^{H}$
denotes conjugate transpose. We represent the vector fields using an arrow,
e.g., $\vec{\mathbf{A}}$. Furthermore, by $\mathbf{A}_{:,n}$ and
$\mathbf{A}_{m,:}$, we refer to $n$-th column and $m$-th row of $\mathbf{A}$,
respectively. Also, $\jmath$ is defined as $\sqrt{-1}$ and
$\operatorname{sinc}(x)=\frac{\sin(\pi x)}{\pi x}$.
$\mathcal{CN}(\bm{\mu},\mathbf{R})$ represents the complex Gaussian
distribution with mean $\bm{\mu}$ and covariance matrix $\mathbf{R}$.

##  II System and SIM Model

Figure 1: SIM antenna in uplink scenario

We consider a system where a BS equipped with a $L$-layer SIM, containing $N$
unit-cells in each layer, serves $K$ users, as depicted in Fig. 1. The SIM is
placed in front of a $M$-element DPA where each element is connected to an RF
chain. The received signal at the BS is described as

$\mathbf{x}=g(\mathbf{G}^{\text{SIM}}(\sqrt{P_{T}}\mathbf{Hs}+\mathbf{z}_{\text{ant}})+\mathbf{z}_{\text{RF}})$
(1)

where $\mathbf{x}\in\mathbb{C}^{M\times 1}$ is the received signal at BS,
$P_{T}$ is the transmitted signal power of each user assuming equal power
transmission, $\mathbf{H}\in\mathbb{C}^{N\times K}$ is the channel matrix,
$\mathbf{z}_{\text{ant}}\in\mathbb{C}^{N\times 1}$ is external noise captured
by the antenna from the environment; its entries are correlated with same the
correlation matrix as the antenna [4]. Furthermore, $g$ and
$\mathbf{z}_{\text{RF}}\in\mathbb{C}^{M\times 1}$ are gain and noise terms,
respectively, introduced by the RF chain where
$z_{\text{RF}}\sim\mathcal{CN}(0,\sigma^{2}_{\text{RF}})$. Finally,
$\mathbf{G}^{\text{SIM}}\in\mathbb{C}^{M\times N}$ is the equivalent WD
combiner introduced by the SIM structure. In SIM antennas, there is no need
for a digital combiner [5], hence, we set $M=K$.

Similar to [5, 12], we model the effect of the SIM, i.e.,
$\mathbf{G}^{\text{SIM}}$, as a product of propagation, $\mathbf{P}^{l}$, and
RMTS weighting matrices, $\mathbf{W}^{l}$, corresponding to the $l$-th RMTS
layer. In the $l$-th layer $\mathbf{W}^{l}$ is a diagonal matrix, i.e.,

$\mathbf{W}^{l}=\text{diag}(e^{\jmath\theta^{l}_{1}},e^{\jmath\theta^{l}_{2}},\dots,e^{\jmath\theta^{l}_{N}})$

where $\theta_{i}^{l}$ is the phase shift introduced by $i$-th unit cell in
$l$-th layer. Likewise, $\mathbf{P}^{l}$ captures the effect of EM propagation
between $l$-th and $(l+1)$-th layer.

To model the propagation from DPA to the first RMTS, one common practice is to
use Rayleigh-Sommerfeld diffraction theory [5]. The theory considers EM fields
as scalars and is only applicable when the diffracting aperture is large
compared to the wavelength and the diffracted field is not observed close to
the aperture [18, Ch. 3.1]. However, in SIM antennas, neither condition is
satisfied; importantly, the DPA and RMTSs are in a few wavelengths or even
subwavelength distances from each other. For example, denoting $\lambda_{0}$
as wavelength, in [5], the interlayer distance is $5\lambda_{0}/L$ for $L$ up
to $7$. On the other hand, in [14], Rayleigh-Sommerfeld theory is used to
model interlayer EM propagation for the D2NN SIM structure since the
interlayer distance is at least $5.5\lambda_{0}$, respectively. We emphasize
that accurately modeling the propagation matrix, $\mathbf{P}$, is essential to
capture variations in EM power due to propagation within the SIM structure and
compare SIM antennas with other antenna types fairly.

To model the propagation matrix from the first RMTS layer to DPA, we use the
reciprocity theorem and model EM propagation in the reverse direction [19, Ch.
7.5]. In general, the propagation matrix from the DPA to the first RMTS layer
depends on the type of antenna elements and their corresponding Green’s
function [20]. Here, to provide a valid and illustrative instantiation, we
assume that the elements of the DPA are Hertzian dipoles placed in front of a
perfect magnetic conductor111This is a common practice to make the dipole
radiation more directive and use the added radiation gain due to the image
theory [19, Ch. 7.4].. We model variations in the amplitude of impinging EM
wave, by capturing relative changes in EM power. For a Hertzian dipole, the
Poynting vector, in both near-field and far-field, is described as [11]

$\begin{split}\vec{\mathbf{W}}&=\frac{\eta_{0}}{8}\absolutevalue{\frac{I_{0}l_{d}}{\lambda_{0}}}^{2}\frac{\sin^{2}{\theta}}{r^{2}}\left[1-\jmath\frac{1}{(k_{0}r)^{3}}\right]\hat{r}\\\
&\hskip
36.135pt+\jmath\eta_{0}\frac{k_{0}|I_{0}l_{d}|^{2}\cos{\theta}\sin{\theta}}{16\pi^{2}r^{3}}\left[1+\frac{1}{(k_{0}r)^{2}}\right]\hat{\theta}\end{split}$
(2)

where $k_{0}=\frac{2\pi}{\lambda_{0}}$ is the free space wavenumber, $I_{0}$
is the excitation current, $l_{d}$ is the dipole length, $\eta_{0}$ is the
free space wave impedance, and $r$ and $\theta$ are the distance and azimuth
angle between source and evaluation point, respectively. In this paper, we are
dealing with uplink communication where BS is in the far field of the users;
therefore, only the propagating term of the Poynting vector, i.e., its real
part, is of importance.

Since the relative change in EM power due to propagation is important, we only
take into account the term which depends on observation location, i.e.,
$\frac{\sin^{2}{\theta}}{r^{2}}$. Then, by normalizing this term, we make sure
that the total power on a half-sphere at distance $r$ remains equal to the
power radiated from the source222There are no propagation losses from the
dipole antenna to the first RMTS layer since the propagation environment is
equivalent to free space.. Further, by mapping the power coefficient to the
complex amplitude of $\vec{\mathbf{E}}$, we reach the following relation for
the propagation of the electric field from the $m$-th element of the DPA to
the $n$-th unit-cell on the first RMTS layer

$|\mathbf{P}^{1}_{m,n}|=\left(\frac{3\sqrt{2}}{8\pi}\frac{\sin^{2}{\theta_{m,n}}}{r^{2}_{m,n}}\right)^{\frac{1}{2}}.$
(3)

To capture the change in phase of the electric field with respect to the
source, we calculate the change in phase of the propagating component of the
electric field radiated by a Hertzian dipole [11, Ch. 4.2], i.e.,

$\measuredangle\mathbf{P}^{\text{1}}_{m,n}=\frac{\pi}{2}-k_{0}r_{m,n}+\measuredangle\left(1+\frac{1}{\jmath
k_{0}r_{m,n}}-\frac{1}{(k_{0}r_{m,n})^{2}}\right).$ (4)

Finally, the $(m,n)$-th element of the propagation matrix $\mathbf{P}^{1}$ is
given by $|\mathbf{P}_{m,n}^{1}|e^{\jmath\measuredangle\mathbf{P}_{m,n}^{1}}$.

The propagation matrix from the $n$-th unit-cell on the $l$-th layer to the
$n^{\prime}$-th unit cell on the $(l+1)$-th layer can be calculated using the
same methodology. This is because each unit-cell can be modeled as a dipole
[21, Ch. 3.3.2] and RMTSs are designed to have zero backward radiation; hence,
the sum of the transmission power coefficients on the half sphere in front of
each unit-cell should be unity.

By concatenating the propagation matrices and weighting factor of each RMTS,
the overall WD combiner of $L$-layer SIM is given by
$\mathbf{W}^{\text{SIM}}=\mathbf{P}^{1}(\prod_{l=1}^{L-1}\mathbf{W}^{l}\mathbf{P}^{l+1})\mathbf{W}^{L}$.

Insertion Loss: Although RMTSs on each layer are defined as passive and
lossless, there are losses due to the propagation of the EM field in the
copper (unit-cells) and dielectric. Also, the resonant nature of the unit-
cells adds to the losses in each RMTS layer [17]. To account for these losses,
we introduce the transmission efficiency coefficient, $T^{\text{SIM}}$, to the
SIM model. Therefore, the WD combiner for $L$-layer SIM is described as:
$\mathbf{G}^{\text{SIM}}=\sqrt{(T^{\text{SIM}})^{L}}\mathbf{W}^{\text{SIM}}$.

##  III Problem Formulation and Combiner Design

The achievable sum-rate, $R$, in the uplink SIM-assisted wireless
communication systems can be written as
$R=\sum_{k=1}^{K}\log_{2}(1+\gamma_{k})$ where $\gamma_{k}$ is the signal-to-
interference-plus-noise ratio for user $k$ defined as

$\frac{P_{T}\absolutevalue{\mathbf{G}^{\text{SIM}}_{k,:}\mathbf{h}_{:,k}}^{2}}{\sum_{\jmath\neq
k}^{K}P_{T}\absolutevalue{\mathbf{G}^{\text{SIM}}_{k,:}\mathbf{h}_{:,j}}^{2}+\sigma_{\text{ant}}^{2}(\mathbf{G}^{\text{SIM}}_{k,:}\mathbf{\Sigma_{\text{rx}}}(\mathbf{G}^{\text{SIM}}_{k,:})^{H})+\sigma_{\text{RF}}^{2}}.$
(5)

We note the two noise terms in the denominator; while the second is the usual
noise term in the RF chains, the first term is due to the antenna noise and is
colored by the SIM combiner. To maximize $R$, we are interested in solving the
following optimization problem:

$\displaystyle\max_{\bm{\theta}^{1},\dots,\bm{\theta}^{L}}R$ (6a) s.t.
$\displaystyle\mathbf{W}^{\text{SIM}}=\mathbf{P}^{1}\left(\prod_{l=1}^{L-1}\mathbf{W}^{l}\mathbf{P}^{l}\right)\mathbf{W}^{L}$
(6b) $\displaystyle\mathbf{W}^{l}=\text{diag}(e^{\jmath\bm{\theta}^{l}}),$
(6c)

where
$\bm{\theta}^{l}=[\theta^{l}_{1},\theta^{l}_{2},\dots,\theta^{l}_{N}]^{T}$ is
the vector of phase shifts introduced by the unit-cells.

Solving (6) is not trivial since $L>1$ and, since the noise is colored, a
simple matched-filtering solution is not available even for the single-user
case. Hence, similar to [5], we propose a GA algorithm comprising the
following steps: we begin by randomly initializing $\bm{\theta}^{l}\ \forall
l\in\\{1,2,\dots,L\\}$ and calculating gradient of (6a) with respect to
$\theta_{l}^{n}$ using:

$\displaystyle\partialderivative{R}{{\theta^{n}_{l}}}=\sum_{k=1}^{K}\text{Re}\\{\frac{1}{\log(2)}\frac{1}{1+\gamma_{k}}\partialderivative{\gamma_{k}}{{\phi^{n}_{l}}}\partialderivative{{\phi^{n}_{l}}}{{\theta^{n}_{l}}}\\}$
(7a)
$\displaystyle\partialderivative{\gamma_{k}}{{\phi^{n}_{l}}}=\frac{(\partialderivative{S_{\text{pwr,k}}}{{\phi^{n}_{l}}}Z_{\text{pwr,k}}-\partialderivative{Z_{\text{pwr,k}}}{{\phi^{n}_{l}}}S_{\text{pwr,k}})}{Z_{\text{pwr,k}}^{2}}$
(7b)
$\displaystyle\begin{split}&\partialderivative{S_{\text{pwr,k}}}{{\phi^{n}_{l}}}=\\\
&\left[\mathbf{P}^{1}_{k,:}\prod_{ll=1}^{l-1}(\mathbf{W}^{ll}\mathbf{P}^{ll})\right]_{:,n}\left[\prod_{ll=l+1}^{L}(\mathbf{P}^{ll}\mathbf{W}^{ll})\mathbf{H}_{:,k}\right]_{n,:}\mathbf{H}^{H}_{:,k}\mathbf{W}^{H}_{k,:}\end{split}$
(7c)
$\displaystyle\begin{split}&\partialderivative{Z_{\text{pwr,k}}}{{\phi^{n}_{l}}}=\\\
&\sum_{j\neq
k}^{K}\left([\mathbf{P}^{1}_{k,:}\prod_{ll=1}^{l-1}(\mathbf{W}^{ll}\mathbf{P}^{ll})]_{:,n}[\prod_{ll=l+1}^{L}(\mathbf{P}^{ll}\mathbf{W}^{ll})\mathbf{H}_{:,j}]_{n,:}\mathbf{H}^{H}_{:,j}\mathbf{W}^{H}_{k,:}\right)\\\
&+[\mathbf{P}^{1}_{k,:}\prod_{ll=1}^{l-1}(\mathbf{W}^{ll}\mathbf{P}^{ll})]_{:,n}[\prod_{ll=l+1}^{L}(\mathbf{P}^{ll}\mathbf{W}^{ll})\mathbf{U}^{H}]_{n,:}\mathbf{U}\mathbf{W}^{H}_{k,:}\end{split}$
(7d)

where $\mathbf{U}$ is the Cholesky decomposition of the antenna correlation
matrix $\mathbf{\Sigma}_{rx}$,
$\gamma_{k}=\frac{S_{\text{pwr,k}}}{Z_{\text{pwr,k}}}$, and
$\partialderivative{{\phi^{n}_{l}}}{{\theta^{n}_{l}}}=\jmath
e^{\jmath\theta_{l}^{n}}$.

Then by normalizing the gradient of each layer, we prevent the vanishing or
exploding gradient effect, i.e.,
$\partialderivative{R}{{\theta^{n}_{l}}}\leftarrow\frac{1}{\rho_{l}}\partialderivative{R}{{\theta^{n}_{l}}}$
where $\rho_{l}$ is the maximum gradient length in the $l$-th layer.
Furthermore, the step size $\mu$ is calculated using the backtracking line
search algorithm [22, Ch. 9.2] with $\alpha$ as the initial guess for the step
size in each iteration. We look for a proper step length in both directions of
the gradient to minimize the chance of ending up in a local optimum. Finally,
all phase shifts are updated simultaneously using:
$\theta_{l}^{n}\leftarrow\theta_{l}^{n}+\mu\partialderivative{R}{{\theta^{n}_{l}}}$.
This process is repeated until some criteria is met.

As both (6a) and (6c) are non-convex functions, the GA algorithm can converge
to a local optimum; at attempt to mitigate the issue we use IPM. The direction
of update in each step in IPM is calculated by considering both Hessian and
gradient of (6a) [22] and hence, a better update direction is chosen. We use
Matlab’s fmincon [23] to find a sub-optimal WD combiner using IPM with
approximate Hessian and gradient. We emphasize that using IPM does not
eliminate the possibility of converging to a local optimum.

##  IV Numerical Results

In this section, we document the effectiveness of the proposed GA algorithm
and IPM in calculating WD combiner for SIM antenna in single-user and multi-
user uplink scenarios where the BS is in the far-field of user(s). Also, as
upper and lower performance limits, we report the achievable sum-rate of DPA
with an equal physical aperture area as SIM antenna, i.e., the averaged users’
signal power which reaches both antennas are the same, and with equal number
of RF chains, i.e., equal power consumption333We ignore the power consumption
of varactor diodes and driving circuitry of each RMTS unit-cell as being
comparatively negligible..

We consider an outdoor wireless communication problem, where $K$ users
communicate with a BS equipped with an antenna of size $4\lambda_{0}\times
4\lambda_{0}$ at $3$ GHz carrier frequency with $20$ MHz bandwidth. As BS is
in the far-field of users, the transmitted power, $P_{T}$, is defined as the
amplitude of the Poynting vector of the users’ transmitted signal, i.e.,
$P_{T}=\frac{\absolutevalue{\vec{\mathbf{E}}_{\text{user}}}^{2}}{2\eta_{0}}$
$[\frac{W}{m^{2}}]$. The SIM thickness, $t_{\text{SIM}}$ is $5\lambda_{0}$,
i.e., the interlayer distance is $5\lambda_{0}/L$. Furthermore, the RF chains
are assumed to be at room temperature, i.e., $T_{\text{BS}}=290\degree\ [K]$.
Finally, $g$ and noise figure of RF chain are set to be $12.5$ [dB] and $18.8$
[dB]444Calculated by considering a low pass filter [24] and first three
elements in the RF chain, i.e., low noise amplifier [25], mixer [26], and in
phase/quadrature demodulator [27]., respectively. We repeat the simulation for
10 instances of user placements and 100 channel realizations per instance.
Also, we allow a maximum of 100 and 500 iterations for the IPM and GA
algorithms, respectively.

Received signal power: The power captured by each antenna/unit-cell is equal
to $P_{T}A_{\text{eff}}$ where $A_{\text{eff}}$ is the effective area of the
antenna [11, Ch. 2.15]. In the SIM, each unit-cell is approximated by a
continuous current sheet, hence, $A_{\text{eff}}$ is equal to the physical
area of each unit-cell $A_{p}$. Considering unit-cells of size
$\frac{\lambda_{0}}{4}\times\frac{\lambda_{0}}{4}$ at $3$ GHz,
$A_{\text{eff,UC}}=0.0025$. For massive MIMO, assuming patch antenna as the
elements of DPA, $A_{\text{eff}}$ is calculated by expressions in [11, Ch.
2.16] that results in $A_{\text{eff,PA}}$ at $3$ GHz to be $0.0026$.

Antenna noise power: The noise power captured by the antenna is defined by
$P_{z_{\text{ant}}}=KT_{E}B\frac{A_{\text{eff}}}{\lambda_{0}^{2}}\Delta\Omega$
[28, Ch. 13.7.1] where $K$ is the Boltzmann constant, $T_{E}$ is the
environment temperature well approximated by $290\degree\ K$ [29], and
$\Delta\Omega$ is the solid angle of noise source at the antenna equal to
$2\pi$.

Channel model: In rich scattering environments, we use the Kronecker Rayleigh
fading model where
$\mathbf{H}=\mathbf{\Sigma}_{rx}\tilde{\mathbf{H}}\mathbf{\Sigma}_{tx}$ and
$\mathbf{\Sigma}_{tx}$ and $\mathbf{\Sigma}_{rx}$ are the users’ and antennas’
correlation matrices, respectively. The entries of $\tilde{\mathbf{H}}$ are
independent identically distributed (i.i.d.) random variables drawn from a
standard normal distribution, i.e.,
$\tilde{\mathbf{H}}\sim\mathcal{CN}(\mathbf{0},\mathbf{I}_{NK})$. We assume
users are independent of each other hence,
$\mathbf{\Sigma}_{tx}=\text{diag}(\beta_{1},\dots,\beta_{K})$ where
$\beta_{k}$ is the path loss of the $k$-th user, here computed using the
Winner II urban microcell non-line of sight scenario [30]. To calculate
$\mathbf{\Sigma}_{rx}$, we assume scatterers are distributed in 3D space and
hence,
$(\mathbf{\Sigma}_{\text{rx}})_{(n,n^{\prime})}=\operatorname{sinc}\left(\frac{2d_{n,n^{\prime}}}{\lambda_{0}}\right)$
[31] where $d_{n,n^{\prime}}$ is the distance between the $n$-th and
$n^{\prime}$-th unit-cells.

In both Rayleigh and 3GPP channels [32], since the ground occupies whole view
of the antenna, the external noise correlation matrix is equivalent to the
antenna correlation matrix in a rich scattering environment, i.e.,
$\mathbf{z}_{\text{ant}}\sim\mathcal{(}0,P_{z_{\text{ant}}}\mathbf{\Sigma}_{\text{rx}})$.
Also, whenever the antenna’s correlation matrix is needed,
$\mathbf{\Sigma}_{rx}$ is calculated as described above.

Finally, we assume the DPA is formed by an array of patch antennas with an
efficiency of $0.9$ [33] and the efficiency of each RMTS in SIM,
$T^{\text{SIM}}$, is set to $0.7$ [10]. For the sake of comparison, we report
a simple matched-filtering combiner, where each $\theta_{i}^{l}$ compensates
for the phase of the equivalent channel in front of $l$-th layer and
$[\mathbf{P}^{l}]_{n,n}$.

(a)

(b)

Figure 2: SIM (a) Efficiency of the proposed GA algorithm $P_{T}=20\
[\text{dBm}/\text{m}^{2}]$, $L=4$ and $K=1$, (b) SE vs. Number of layers
$P_{T}=20\ [\text{dBm}/\text{m}^{2}]$.

Algorithm Efficiency: We begin by studying the efficiency of the proposed GA
algorithm. Fig. 2(a) plots the results averaged over 1000 and for 10 randomly
selected Rayleigh fading channel realizations. As shown, in all cases the
major improvement is during the first 200 iterations and all cases converge to
at least a local optimum. Also, Fig. 2(b) plots the achievable sum-rate of SIM
using the GA and IPM algorithms for different numbers of layers. As shown, by
increasing the number of layers, the achievable sum-rate of both GA and IPM
algorithms experience an initial improvement and eventually saturates or
experiences a drop. As the number of layers increases, the number of variables
increases as well which results in a higher possibility for GA algorithm to
converge to a poor local optimum. IPM, using the Hessian of the objective
function, can find a better update direction toward the global optimum and
escape smaller local optima. Also, when $L$ increases, the interlayer distance
decreases, leading to poor interconnectivity and reduced achievable sum-rate.

Single-User Performance For a single-user scenario, depicted in Fig. 3 is the
achievable sum-rate of a 5-layer SIM, using both IPM and GA algorithms, and
DPA with a physical aperture size equal to $4\lambda_{0}\times 4\lambda_{0}$.
Also, for comparison, the achievable sum-rate of a DPA with $M=K=1$ RF chains
is reported, as is the performance of the SIM with a simple MF combiner. We
set $\alpha$ to $1.8$ in the GA algorithm. As shown, using IPM and GA
algorithm, we gain a significant improvement in performance compared to using
a simple MF combiner. Interestingly, using GA provides a higher sum-rate than
using IPM in both Rayleigh fading and realistic 3GPP channels. In this case it
appears that the flexibility in choosing the learning rate in each iteration
helped GA algorithm in single-user cases to skip local optimums and find a
closer-to-optimum combiner than IPM. Also, the achievable sum-rate of SIM
antenna is lower than that of the DPA using maximum ratio combiner due to
higher losses in the system (in comparison with DPA under equal aperture size
constraint). However, when the number of RF chains is made equal, putting a
SIM in front of a single antenna improves the uplink sum-rate of the user.
Here, the SIM acts as a lens that focuses the received signal power by the
user on its own receiver antenna.

(a)

(b)

Figure 3: Achievable sum-rate in (a) Rayleigh fading (b) realistic 3GPP
channels. $K=M=1,L=5$.

(a)

(b)

Figure 4: Achievable sum-rate in (a) Rayleigh fading (b) realistic 3GPP
channels. $K=M=2,L=5$.

Multiuser Scenario: Finally, we consider a multi-user scenario with $K=2$,
depicted in Fig. 4. In this case, DPA uses zero-forcing to combine the
received signal, $\alpha$ is set to $2.2$ in the GA algorithm, and we choose a
random user to form the MF combiner. As show, in both Rayleigh fading and
realistic 3GPP channels, the WD combiner calculated by IPM achieves a higher
sum-rate in comparison with GA and MF combiners. The more complex optimization
problem due to inter-user interference, increases the possibility of GA
algorithm ending up early in a local optimum. It is worth noting that due to
the non-convexity of objective function and constraints of (6), performance of
GA algorithm is highly dependent on the initial guess of the backtracking
algorithm, $\alpha$. A more sophisticated algorithm for choosing step length
may result in a better performance of GA algorithm.

##  V Conclusion

In this paper, we study the possibility of using SIM antennas at the BS for
uplink wireless communications. We begin by proposing a system model that
captures the effect of different noise sources and hardware limitations in SIM
antennas. Also, we model the needed interlayer propagation matrices using EM
theory to accurately capture variations in signal power. Then we propose a GA
algorithm and also apply IPM to design a WD combiner to maximize the
achievable sum-rate in uplink communications. Our simulation results show that
in both Rayleigh fading and realistic 3GPP channels using SIM antenna improves
achievable sum-rate in uplink in comparison with DPA under an equal number of
RF chains. On the other hand, DPA achieves higher sum-rates than SIM when both
antennas have equal aperture area.

## Acknowledgments

This work was supported by the Ericsson Canada. Maryam Rezvani gratefully
acknowledges insightful discussions with Parham Abbasloo.

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