#  Reducing the Control Overhead of Intelligent Reconfigurable Surfaces Via a
Tensor-Based Low-Rank Factorization Approach

Bruno Sokal, Paulo R. B. Gomes, André L. F. de Almeida, Behrooz Makki, and
Gabor Fodor Bruno Sokal, Paulo R. B. Gomes, and André L. F. de Almeida are
with the Wireless Telecom Research Group (GTEL), Department of Teleinformatics
Engineering, Federal University of Ceará, Fortaleza-CE, Brazil. E-mails:
<EMAIL_ADDRESS>Makki is with Ericsson Research,
Göteborg, Sweden . E-mail<EMAIL_ADDRESS>Fodor is with
Ericsson Research and KTH Royal Institute of Technology, Stockholm, Sweden .
E-mail<EMAIL_ADDRESS>work was supported by the Ericsson
Research, Sweden, and Ericsson Innovation Center, Brazil, under UFC.48
Technical Cooperation Contract Ericsson/UFC. This study was financed in part
by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil
(CAPES)-Finance Code 001, and CAPES/PRINT Proc. 88887.311965/2018-00. André L.
F. de Almeida acknowledges CNPq for its financial support under the grant
312491/2020-4. G. Fodor was partially supported by the Digital Futures project
PERCy. Part of this work has been submitted for possible presentation in IEEE
GLOBECOM 2022 [1].

###### Abstract

Passive intelligent reconfigurable surfaces (IRS) are becoming an attractive
component of cellular networks due to their ability of shaping the propagation
environment and thereby improving the coverage. While passive IRS nodes
incorporate a great number of phase-shifting elements and a controller entity,
the phase-shifts are typically determined by the cellular base station (BS)
due to its computational capability. Since the fine granularity control of the
large number of phase-shifters may become prohibitive in practice, it is
important to reduce the control overhead between the BS and the IRS
controller. To this end, in this paper we propose a low-rank approximation of
the near-optimal phase-shifts, which would incur prohibitively high
communication overhead on the BS-IRS controller links. The key idea is to
represent the potentially large IRS phase-shift vector using a low-rank tensor
model. This is achieved by factorizing a _tensorized_ version of the IRS
phase-shift vector, where each component is modeled as the Kronecker product
of a predefined number of factors of smaller sizes, which can be obtained
_via_ tensor decomposition algorithms. We show that the proposed low-rank
models drastically reduce the required feedback requirements associated with
the BS-IRS control links. Our simulation results indicate that the proposed
method is especially attractive in scenarios with a strong line of sight
component, in which case nearly the same spectral efficiency is reached as in
the cases with near-optimal phase-shifts, but with a drastically reduced
communication overhead.

###### Index Terms:

Reconfigurable intelligent surface (RIS), feedback overhead, control
signaling, low-rank approximation, tensor modeling, PARAFAC, Tucker.

##  I Introduction

Intelligent reconfigurable surface (IRS) is a candidate technology for beyond
fifth generation and sixth generation networks due to its ability to control
the electromagnetic properties of the radio-frequency waves by performing an
intelligent phase-shift to the desired direction [2, 3, 4, 5, 6, 7, 8, 9].
Usually, IRS is defined as a planar ($2$-D) surface with a large number of
independent reflective elements, in which they can be fully passive or with
some elements active [10, 11, 12]. IRS is connected to a smart controller that
sets the desired phase-shift for each reflective element, by applying bias
voltages at the elements e.g., PIN diodes. The main advantage of fully passive
IRSs is its full-duplex nature, i.e., no noise amplification is observed since
no signal processing is possible. However, the fully passive nature of the
IRSs makes the channel state information (CSI) acquisition process difficult,
since no pilots are processed, thus only the cascade channel can be estimated.
Nevertheless, in the case of employing a few active elements in the IRS, this
issue is suppressed and channel can be estimated using, for example,
compressed sensing tools [10]. Another advantage of an IRS with fully passive
elements is that the power consumption is concentrated at the controller. This
makes the IRS a more attractive technology in terms of energy efficiency
compared to alternative technologies, e.g., amplify-and-forward and decode-
and-forward relays [13, 14, 15].

Several works have addressed the CSI acquisition problem in IRS-assisted
networks, e.g., [16, 17, 18, 19, 20, 21, 22, 23]. The work of [16] proposes a
tensor-based method where the authors show the benefits of exploiting the
multidimensional structure of the received signal by separating the cascade
channel. The work of [18] proposes a compressed sensing approach in a multi-
user uplink multiple-input multiple-output (MIMO) scenario. In [19], a two-
timescale channel estimation framework is proposed to overcome the pilot
overhead in a multi-user IRS-aided system. Also, [20] addresses the channel
estimation problem in millimiter-wave MIMO systems. The work of [21] proposes
a channel estimation framework for milimiter-wave (mmWave) IRS-assisted MIMO
systems based on compressed sensing techniques. The authors of [22] propose a
low-complexity framework for channel estimation and passive beamforming in
MIMO IRS-assisted systems.

Although many works focus on channel estimation [16, 17, 18, 19, 20, 21, 22,
23], achievable rate maximization [23, 24, 25], energy efficiency (EE)
maximization [26, 27, 28, 29, 30], and interference mitigation problems [31,
32, 33], few works have addressed the problem of reducing the channel training
or the feedback-overhead of IRS phase-shifts to the IRS controller. The work
of [24] proposes a protocol design to maximize the transmission rate in IRS-
assisted MIMO-OFDM systems. Also, [34] proposes a framework for overhead-aware
feedback and resource allocation in IRS-assisted MIMO systems. The main idea
of [34] is to optimize the network resource such as the bandwidth and the
total power used for transmission and feedback. However, the number of phase-
shifts to be conveyed to the IRS controller can still be large and results in
high feedback signaling overhead, especially for large IRS panels.

In this work, we propose a overhead-aware model for designing the IRS phase-
shifts. Our idea is to represent the IRS phase-shift vector with a low-rank
model. This is achieved by factorizing a _tensorized_ version of the IRS
phase-shift vector, where each component is modelled as the Kronecker product
of a predefined number of factors.  These factors are estimated using tensor
decompositions such as the PARAllel FACtors (PARAFAC) [35] and Tucker [36].
After the estimation process, the phases of the factors are quantized and fed
back to the IRS controller, which can reconstruct the IRS phase-shift vector
based on the chosen low-rank tensor model. The main contributions of this work
are the following:

  1. 1.

Our proposed IRS phase-shift factorization allows to save network resources by
reducing the total IRS phase-shift feedback overhead. This allows a more
frequent IRS phase-shift feedback, for a fixed feedback load, which can
significantly improve the end-to-end latency, crucial in a fast varying
channels, high mobility scenarios and/or the cases with moderate/large sizes
of the IRS. Also, thanks to the significant reduction on the feedback
overhead, the IRS-assisted network can decide to multiplex phase-shifts
associated with a higher number of users in the same feedback channel.

  2. 2.

The proposed IRS phase-shift factorization provides a flexible feedback design
by controlling the parameters of the low-rank factorization model, such as the
number of components, the number and the size of the factors, as well as their
respective resolution. This is an important feature of our proposed feedback-
aware model, since for limited feedback control links, the low-rank model and
its factorization parameters can be efficiently adjusted to the available
capacity of the feedback link, providing more degrees of freedom to system
design.

  3. 3.

Our tensor-based factorization approach relies on the optimum IRS phase-shift
vector, which means that it can be implemented in every IRS-assisted network
and in multiple communication links, i.e., downlink or uplink, in single-input
single-output, multiple-input single-output, as well in MIMO systems.

Different from the works of [24] and [34], we aim to reduce the IRS phase-
shifts feedback overhead by conveying to the IRS controller only the factors
of our proposed low-rank model. Our approach is analytical and provides a
systematic way of controlling the feedback overhead by adjusting the
parameters of the low-rank IRS model, namely, its rank and the corresponding
number of factors of each rank-one component. Our simulations show that the
proposed low-rank model for the IRS phase-shifts can achieve the same spectral
efficiency (SE) as the state-of-the-art in line of sight (LOS) scenarios,
while the feedback payload (number of bits to be fed back) is dramatically
reduced. For example, taking an IRS with $N=1024$ elements, the feedback
duration can be $50$ times smaller than the state-of-the-art, depending on the
low-rank model parameters. Also, when taking into account the total system SE
and EE, i.e., both the IRS phase-shift feedback duration, and the channel
estimation duration, our proposed model outperforms the state-of-the-art.

The rest of the paper is organized as follows.  Section II provides an
introduction of the tensor notation and decompositions that are exploited in
this paper. The system model is described in Section III. Section IV details
our proposed feedback overhead-aware method and provides the details of the
PARAFAC-IRS and Tucker-IRS models for IRS phase-shift vector factorization.
Section V describes the phase-shift and weighting factors quantization
procedure and the reconstruction of the IRS phase-shift vector at the IRS
controller. The effects of the factorization parameters and the quantization
process are also discussed in this section. Simulation results are provided in
Section VI and the final conclusions and perspectives are discussed in Section
VII.

###  I-A Notation and Properties

Scalars are represented as non-bold lower-case letters $a$, column vectors as
lower-case boldface letters $\bm{a}$, matrices as upper-case boldface letters
$\bm{A}$, and tensors as calligraphic upper-case letters $\bm{\mathcal{A}}$.
The superscripts $\\{\cdot\\}^{\text{T}}$, $\\{\cdot\\}^{\text{*}}$,
$\\{\cdot\\}^{\text{H}}$ and $\\{\cdot\\}^{+}$ stand for transpose, conjugate,
conjugate transpose and pseudo-inverse operations, respectively. The operator
$\Arrowvert\cdot\Arrowvert_{\text{F}}$ denotes the Frobenius norm of a matrix
or tensor, and $\mathbb{E}\\{\cdot\\}$ is the expectation operator. The
operator $\text{diag}\left(\bm{a}\right)$ converts $\bm{a}$ into a diagonal
matrix, while $\text{diag}(\bm{A})$ returns a vector whose elements are the
main the diagonal of $\bm{A}$. Moreover, $\text{vec}\left(\bm{A}\right)$
converts $\bm{A}\in\mathbb{C}^{I_{1}\times R}$ to a column vector
$\bm{a}\in\mathbb{C}^{I_{1}R\times 1}$ by stacking its columns on top of each
other, while the unvec($\cdot$) operator is the inverse of the vec operation.
The symbol $\circ$ denotes the outer product operator. Also,
$\bm{a}_{r}\in\mathbb{C}^{I\times 1}$ represents the $r$-th column of
$\bm{A}\in\mathbb{C}^{I\times R}$. Let us define two matrices
$\bm{A}=\left[\bm{a}_{1},\ldots,\bm{a}_{R}\right]\in\mathbb{C}^{I_{1}\times
R}$ and
$\bm{B}=\left[\bm{b}_{1},\ldots,\bm{b}_{R}\right]\in\mathbb{C}^{I_{2}\times
R}$. The Kronecker product between them is defined by

$\displaystyle\bm{A}\otimes\bm{B}=\left[\begin{array}[]{ccc}a_{1,1}\bm{B}&\ldots&a_{1,R}\bm{B}\\\
\vdots&\ddots&\vdots\\\
a_{I,1}\bm{B}&&a_{I,R}\bm{B}\end{array}\right]\in\mathbb{C}^{I_{2}I_{1}\times
RR}.$

The Khatri-Rao product, also known as the column wise Kronecker product,
between two matrices, symbolized by $\diamond$, is defined as

$\displaystyle\bm{A}\diamond\bm{B}=\left[\bm{a}_{1}\otimes\bm{b}_{1},\ldots,\bm{a}_{R}\otimes\bm{b}_{R}\right]\in\mathbb{C}^{I_{2}I_{1}\times
R}.$

We make use of the following properties

$\displaystyle\text{vec}\left(\bm{A}\bm{B}\bm{C}\right)$
$\displaystyle=\left(\bm{C}^{\text{T}}\otimes\bm{A}\right)\text{vec}\left(\bm{B}\right),$
(1)
$\displaystyle\text{vec}\left(\bm{A}\text{diag}\left(\bm{b}\right)\bm{C}\right)$
$\displaystyle=\left(\bm{C}^{\text{T}}\diamond\bm{A}\right)\bm{b},$ (2)
$\displaystyle\bm{a}^{\text{T}}\diamond\bm{B}$
$\displaystyle=\bm{B}\text{diag}(\bm{a}),$ (3)
$\displaystyle\bm{a}\otimes\bm{b}$
$\displaystyle=\text{vec}\left(\bm{b}\circ\bm{a}\right),$ (4)

where the involved vectors and matrices have compatible dimensions in each
case.

##  II Tensor Pre-Requisites

In this section, tensor preliminaries are provided by focusing on the main
notation, operations and properties that will be useful in the rest of the
paper.

Consider a set of matrices $\\{\bm{X}_{i_{3}}\\}\in\mathbb{C}^{I_{1}\times
I_{2}}$, for $i_{3}=1,\ldots,I_{3}$. Concatenating all $I_{3}$ matrices, we
form the third-order tensor
$\bm{\mathcal{X}}=[\bm{X}_{1}\sqcup_{3}\bm{X}_{2}\sqcup_{3}\ldots\sqcup_{3}\bm{X}_{I_{3}}]\in\mathbb{C}^{I_{1}\times
I_{2}\times I_{3}}$, where $\sqcup_{3}$ indicates a concatenation in the third
dimension. We can interpret $\bm{X}_{i_{3}}$ as the $i_{3}$-th frontal slice
of $\bm{\mathcal{X}}$, defined as $\bm{\mathcal{X}}_{..i_{3}}=\bm{X}_{i_{3}}$
where the “$..$” indicates that the dimensions $I_{1}$ and $I_{2}$ are fixed.
The tensor $\bm{\mathcal{X}}$ can be matricized by letting one dimension vary
along the rows and the remaining two dimensions along the columns. From
$\bm{\mathcal{X}}$, we can form three different matrices, referred to as the
$n$-mode unfoldings (for $n=\\{1,2,3\\}$ in this case), which are respectively
given by

$\displaystyle\left[\bm{\mathcal{X}}\right]_{\left(1\right)}$
$\displaystyle=[\bm{\mathcal{X}}_{..1},\ldots,\bm{\mathcal{X}}_{..I_{3}}]\in\mathbb{C}^{I_{1}\times
I_{2}I_{3}},$ (5)
$\displaystyle\left[\bm{\mathcal{X}}\right]_{\left(2\right)}$
$\displaystyle=[\bm{\mathcal{X}}_{..1}^{\text{T}},\ldots,\bm{\mathcal{X}}_{..I_{3}}^{\text{T}}]\in\mathbb{C}^{I_{2}\times
I_{1}I_{3}}$ (6) $\displaystyle\left[\bm{\mathcal{X}}\right]_{\left(3\right)}$
$\displaystyle=[\text{vec}(\bm{\mathcal{X}}_{..1}),\ldots,\text{vec}(\bm{\mathcal{X}}_{..I_{3}})]^{\text{T}}\in\mathbb{C}^{I_{3}\times
I_{1}I_{2}}.$ (7)

###  II-A Tensorization

The tensorization operation consists of mapping the elements of a vector into
a high-order tensor. Let us define the vector $\bm{y}\in\mathbb{C}^{N\times
1}$, in which $N=\prod\limits_{p=1}^{P}{N_{p}}$, where $N_{p}$ is the size of
the $p$-th partition of this vector. By applying the tensorization operator,
defined as $\bm{\mathcal{T}}\\{\cdot\\}$, we can form a $P$-order tensor
$\bm{\mathcal{Y}}=\bm{\mathcal{T}}\\{\bm{y}\\}\in\mathbb{C}^{N_{1}\times
N_{2}\times\ldots\times N_{P}}$.  The mapping of elements from $\bm{y}$ to
$\bm{\mathcal{Y}}$ is defined as

$\displaystyle\bm{\mathcal{Y}}_{n_{1},n_{2},\ldots,n_{P}}=\bm{y}_{n_{1}+(n_{2}-1)N_{1}+\ldots+(n_{P}-1)N_{P-1}\cdots
N_{2}N_{1}},$ (8)

where $n_{p}=\\{1,\ldots,N_{p}\\}$, for $p=\\{1,\ldots,P\\}$.  This operator
plays a key role on the proposed method, and will be exploited to recast the
IRS phase-shift vector as a tensor, from which the low-rank factorization
schemes are proposed.

###  II-B PARAFAC Decomposition

It is known that every matrix of rank $R$ can be expressed as the summation of
its rank-one components obtained by, e.g., singular value decomposition (SVD).
In the case of tensors, a tensor of rank $R$ is given by the summation of its
rank-one tensor factors. This decomposition is called PARAFAC [35]. For a $P$
order tensor $\bm{\mathcal{Y}}\in\mathbb{C}^{I_{1}\times
I_{2}\times\ldots\times I_{P}}$, its PARAFAC decomposition is given as

$\displaystyle\bm{\mathcal{Y}}=\sum\limits_{r=1}^{R}\bm{a}^{(1)}_{r}\circ\bm{a}^{(2)}_{r}\circ\ldots\circ\bm{a}^{(P)}_{r}\in\mathbb{C}^{I_{1}\times
I_{2}\times\ldots\times I_{P}},$ (9)

where $\bm{a}^{(p)}_{r}\in\mathbb{C}^{I_{p}\times 1}$ is $r$-th column of the
$p$-th factor matrix $\bm{A}^{(p)}\in\mathbb{C}^{I_{p}\times R}$,
$p=\\{1,\ldots,P\\}$. The $p$-th mode unfolding of $\bm{\mathcal{Y}}$, defined
as $\left[\bm{\mathcal{Y}}\right]_{\left(p\right)}\in\mathbb{C}^{I_{p}\times
I_{1}\cdots I_{p-1}I_{p+1}\cdots I_{P}}$, is expressed as

$\left[\bm{\mathcal{Y}}\right]_{\left(p\right)}=\bm{A}^{(p)}\left(\bm{A}^{(P)}\diamond\ldots\diamond\bm{A}^{(p+1)}\diamond\bm{A}^{(p-1)}\diamond\ldots\diamond\bm{A}^{(1)}\right)^{\text{T}}.$
(10)

Fig. 1 illustrates a PARAFAC tensor, for $P=3$, as the summation of rank-one
tensors. In this case, it can be shown that the three-mode unfoldings can be
factorized as [37]

$\displaystyle\left[\bm{\mathcal{Y}}\right]_{\left(1\right)}$
$\displaystyle=\bm{A}^{(1)}\left(\bm{A}^{(3)}\diamond\bm{A}^{(2)}\right)^{\text{T}}\in\mathbb{C}^{I_{1}\times
I_{2}I_{3}},$ (11)
$\displaystyle\left[\bm{\mathcal{Y}}\right]_{\left(2\right)}$
$\displaystyle=\bm{A}^{(2)}\left(\bm{A}^{(3)}\diamond\bm{A}^{(1)}\right)^{\text{T}}\in\mathbb{C}^{I_{2}\times
I_{1}I_{3}},$ (12)
$\displaystyle\left[\bm{\mathcal{Y}}\right]_{\left(3\right)}$
$\displaystyle=\bm{A}^{(3)}\left(\bm{A}^{(2)}\diamond\bm{A}^{(1)}\right)^{\text{T}}\in\mathbb{C}^{I_{3}\times
I_{1}I_{2}}.$ (13)

Figure 1: Illustration of a third-order PARAFAC tensor as a sum of $R$ rank-
one tensors.

###  II-C Tucker Decomposition

The Tucker decomposition expresses a tensor as a set of factor matrices and a
core tensor. A $P$-th order tensor
$\bm{\mathcal{Q}}\in\mathbb{C}^{I_{1}\times\ldots\times I_{P}}$ that admits a
Tucker decomposition, can be written as

$\displaystyle\bm{\mathcal{Q}}=\bm{\mathcal{G}}\times_{1}\bm{B}^{(1)}\times_{2}\ldots\times_{P}\bm{B}^{(P)}\in\mathbb{C}^{I_{1}\times\ldots\times
I_{P}},$ (14)

where $\bm{B}^{(p)}\in\mathbb{C}^{I_{p}\times R_{p}}$ is the $p$-th factor
matrix, for $p=\\{1,\ldots,P\\}$, and
$\bm{\mathcal{G}}\in\mathbb{C}^{R_{1}\times\ldots\times R_{P}}$ is the core
tensor.

Figure 2: Illustration of a third-order Tucker tensor and its factor matrices
and core tensor.

The tensor $\bm{\mathcal{Q}}$ can also be represented as the outer product of
its factors, given as

$\displaystyle\bm{\mathcal{Q}}=\sum\limits_{r_{1}=1}^{R_{1}}\ldots\sum\limits_{r_{P}=1}^{R_{P}}\bm{\mathcal{G}}_{r_{1},\ldots,r_{P}}\left(\bm{b}^{(1)}_{r_{1}}\circ\ldots\circ\bm{b}^{(P)}_{r_{P}}\right),$

where $\bm{b}^{(p)}\in\mathbb{C}^{I_{p}\times 1}$ is the $r_{p}$-th column of
the $p$-th factor matrix $\bm{B}^{(p)}\in\mathbb{C}^{I_{p}\times R_{p}}$ for
$p=\\{1,\ldots,P\\}$ and $r_{p}=\\{1,\ldots,R_{p}\\}$. The $p$-th mode
unfolding matrix of $\bm{\mathcal{Q}}$, defined as
$\left[\bm{\mathcal{Q}}\right]_{\left(p\right)}\in\mathbb{C}^{N_{p}\times
N_{1}\cdots N_{p-1}N_{p+1}\cdots N_{P}}$, is given by

$\left[\bm{\mathcal{Q}}\right]_{\left(p\right)}=\bm{B}^{(p)}\left[\bm{\mathcal{G}}\right]_{\left(p\right)}\left(\bm{B}^{(P)}\otimes\ldots\otimes\bm{B}^{(p+1)}\otimes\bm{B}^{(p-1)}\otimes\ldots\otimes\bm{B}^{(1)}\right)^{\text{T}}.$
(15)

For $P=3$, Fig. 2, illustrates the decomposition. Its three mode unfoldings
are given by

$\displaystyle\left[\bm{\mathcal{Q}}\right]_{\left(1\right)}$
$\displaystyle=\bm{B}^{(1)}\left[\bm{\mathcal{G}}\right]_{\left(1\right)}\left(\bm{B}^{(3)}\otimes\bm{B}^{(2)}\right)^{\text{T}}\in\mathbb{C}^{I_{1}\times
I_{2}I_{3}},$ (16)
$\displaystyle\left[\bm{\mathcal{Q}}\right]_{\left(2\right)}$
$\displaystyle=\bm{B}^{(2)}\left[\bm{\mathcal{G}}\right]_{\left(2\right)}\left(\bm{B}^{(3)}\otimes\bm{B}^{(1)}\right)^{\text{T}}\in\mathbb{C}^{I_{2}\times
I_{1}I_{3}},$ (17)
$\displaystyle\left[\bm{\mathcal{Q}}\right]_{\left(3\right)}$
$\displaystyle=\bm{B}^{(3)}\left[\bm{\mathcal{G}}\right]_{\left(3\right)}\left(\bm{B}^{(2)}\otimes\bm{B}^{(1)}\right)^{\text{T}}\in\mathbb{C}^{I_{3}\times
I_{1}I_{2}}.$ (18)

##  III System Model

Figure 3: System model illustration.

We consider the system illustrated in Fig. 3, where the transmitter (TX) is
equipped with a uniform linear array (ULA) with $M_{T}$ antenna elements, the
receiver (RX) is equipped with ULA with $M_{R}$ antenna elements and the IRS
has $N$ reflective elements. To simplify the discussion, let us consider a
single stream transmission, and assume that there is no direct link between
the TX and RX,  e.g., base station (BS). First, the TX sends a pilot signal to
the RX with the aid of the IRS. Since the IRS has no signal processing
capabilities, the channel estimation and the IRS phase-shifts optimization are
performed at the RX. The received signal after processing the pilots is given
by

$\displaystyle
y=\bm{w}^{\text{H}}\bm{G}\bm{S}\bm{H}\bm{q}+\bm{w}^{\text{H}}\bm{b},$ (19)

where $\bm{b}\in\mathbb{C}^{M_{R}\times 1}$ is the additive noise at the
receiver with
$\mathbb{E}[\bm{b}\bm{b}^{\text{H}}]=\sigma^{2}_{b}\bm{I}_{M_{r}}$,
$\bm{w}\in\mathbb{C}^{M_{R}\times 1}$ and $\bm{q}\in\mathbb{C}^{M_{T}\times
1}$ are the receiver and transmitter combiner and precoder, respectively.
$\bm{H}\in\mathbb{C}^{N\times M_{T}}$ and $\bm{G}\in\mathbb{C}^{M_{R}\times
N}$ are the TX-IRS and IRS-RX involved channels, and
$\bm{S}=\text{diag}(\bm{s})\in\mathbb{C}^{N\times N}$ with
$\bm{s}=[e^{j\theta_{1}},\ldots,e^{j\theta_{N}}]\in\mathbb{C}^{N\times 1}$
being the IRS phase-shift vector, and $\theta_{n}$ is the phase-shift applied
to the $n$-th IRS element.

After the channel estimation step, the precoder and the combiner (active
beamformers) vectors $\bm{w}$ and $\bm{q}$, and the IRS phase-shift vector
$\bm{s}$ (passive beamformer) are optimized. Later, the RX needs to feedback
to the IRS controller the designed phase-shifts so that the IRS controller
tunes the phase-shift for each IRS element. Considering the fact that this
feedback occurs in a limited capacity control channel and that the IRS may
contain several hundreds to thousands of reflecting elements, the feedback of
each phase-shift with a certain resolution imposes a signaling overhead. In
this regard, the work [34] models the feedback duration as

$T_{\text{F}}=\frac{Nb_{\text{F}}}{B_{\text{F}}\text{log}\left(1+\frac{p_{\text{F}}|{g}_{\text{F}}|^{2}}{B_{\text{F}}N_{0}}\right)},$
(20)

where $N$ is the total number of IRS phase-shifts to be fed back,
$B_{\text{F}}$, $p_{\text{F}}$ are the feedback bandwith and power,
respectively, $g_{\text{F}}$ is the scalar control channel used,
$b_{\text{F}}$ is the resolution of each phase-shift, and $N_{0}$ is the noise
power density. The authors of [34] focus on the problem of rate and EE
maximization, where the rate is given by

SE
$\displaystyle=\left(1-\frac{T_{E}+T_{F}}{T}\right)B\text{log}\left(1+\frac{p_{\text{TX}}|\bm{w}^{\text{H}}\bm{G}\bm{S}\bm{H}\bm{q}|^{2}}{BN_{0}}\right),$
(21)

with $T_{E}$ and $T$ being the duration of the channel estimation phase and
the total time interval, and $B$ the transmission bandwidth. The EE is given
by $\text{EE}=\text{Rate}/P_{\text{tot}}$, and the total power consumption
$P_{\text{tot}}$ can be expressed as

$\displaystyle\text{P}_{\text{tot}}=P_{\text{E}}+\frac{T-T_{\text{E}}-T_{\text{F}}}{T}\mu
p+\frac{\mu_{\text{F}}p_{\text{F}}T_{\text{F}}}{T}+P_{\text{c}},$ (22)

where $P_{\text{E}}$ is the power used for the channel estimation phase,
$1/\mu$ is the efficiency of the transmitter power amplifier, $p_{\text{F}}$
is the power used during $T_{\text{F}}$ seconds, and $\mu_{\text{F}}$ is the
efficiency of the transmit amplifier used for feedback. The work [34]
maximizes (21) and (22) by optimizing the values of the $p$, $p_{\text{F}}$,
$B$, $B_{\text{F}}$.

Based on the model provided by [34] in (20), we propose to reduce the feedback
overhead by factorizing the IRS phase-shift vector into smaller factors, as
explained in the following section.

##  IV Proposed Feedback-Aware Method

In this section, we describe the proposed tensor low-rank approximation based
feedback-aware methods that focus on reducing the feedback duration
$T_{\text{F}}$, given in (20). First, we assume that the RX has an estimate of
the involved channels $\bm{H}$ and $\bm{G}$. The $N$ phase-shifts of the IRS
can be determined based on different state-of-the-art algorithms (see, e.g.,
[34, 6]), and are represented in a vector format as
$\bm{s}=[e^{j\theta_{1}},e^{j\theta_{2}},\ldots,e^{j\theta_{N}}]\in\mathbb{C}^{N\times
1}$. Our  initial idea  consists of factorizing $\bm{s}$ as the Kronecker
product of $P$ factors, i.e.,

$\bm{s}=\bm{s}^{(P)}\otimes\ldots\otimes\bm{s}^{(1)}\in\mathbb{C}^{N_{P}\cdots
N_{1}\times 1},$ (23)

where $\bm{s}^{(p)}\in\mathbb{C}^{N_{p}\times 1}$ and
$N=\prod\limits_{p=1}^{P}N_{p}$.

Example: To get a first insight into  the impact of this factorization on the
IRS phase-shift feedback overhead, let us consider a simple scenario with
$N=1024$ phase-shifts, and we apply our factorization method by choosing $P=3$
factors. Consider, as one example, the following factors
$\bm{s}^{(1)}=[e^{j\theta^{(1)}_{1}},\ldots,e^{j\theta^{(1)}_{32}}]\in\mathbb{C}^{32\times
1}$,
$\bm{s}^{(2)}=[e^{j\theta^{(2)}_{1}},\ldots,e^{j\theta^{(2)}_{8}}]\in\mathbb{C}^{8\times
1}$ and
$\bm{s}^{(3)}=[e^{j\theta^{(3)}_{1}},\ldots,e^{j\theta^{(3)}_{4}}]\in\mathbb{C}^{4\times
1}$, i.e., $N_{1}=32$, $N_{2}=8$ and $N_{3}=4$. Note that, $N_{1}$, $N_{2}$,
$N_{3}$ can have every size as long $N_{1}\times N_{2}\times N_{3}=N=1024$. In
this scenario, instead of conveying to the IRS controller $1024$ phase-shifts,
we only need to convey the phase-shifts of the factors, i.e., $32+8+4=44$,
reducing drastically the total amount of phase-shift overhead. Physically, the
Kronecker product in (23) represents a summation of the factors phase-shifts.
It is clear that, in a general model for a large $N$, and based on the choice
of $P$, we have that
$\sum\limits_{p=1}^{P}N_{P}<<N=\prod\limits_{p=1}^{P}N_{P}$. The discussed
example is illustrated in Fig. 4. $\blacksquare$

Figure 4: (a) IRS with $N=1024$ elements without factorization, (b) IRS with
$N=1024$ elements factorized into $P=3$ factors.

In a general view, the proposed factorization consists of three steps,
illustrated in Fig. 5 (for the PARAFAC-IRS model):

  1. 1.

Rearrangement of elements: In this step, the optimum phase-shift vector
$\bm{s}\in\mathbb{C}^{N\times 1}$ is rearranged into a $P$-th order tensor
$\bm{\mathcal{S}}\in\mathbb{C}^{N_{1}\times N_{2}\times\ldots\times N_{P}}$,
with $N=\prod\limits_{p=1}^{P}N_{p}$. This is accomplished by mapping the
elements of the IRS phase-shift vector $\bm{s}$ into the tensor $\bm{S}$,
using the tensorization operator, given in (8).

  2. 2.

Low-rank approximation (LRA): In this step, the RX selects an LRA model for
$\bm{\mathcal{S}}$ based on its unfoldings matrices. For example, the RX can
approximate the tensor $\bm{\mathcal{S}}$ as a PARAFAC or a Tucker model, and
makes use of classical tensor algorithms, such as the alternating least
squares (ALS) [37] and high order singular value decomposition (HOSVD) [38],
to estimate the factors matrices (and the core tensor, in the case of the
Tucker model).

  3. 3.

Normalization: The factors outputs of the LRA are normalized due to the
unitary modulus constraint of the phase-shift vector. In other words, the RX
convey to the IRS controller only the angles of the computed factors.

Figure 5: Proposed method for the IRS phase-shift vector factorization based
on the PARAFAC-IRS model.

In the next section, we generalize the concept of (23) by factorizing the IRS
phase-shift vector $\bm{s}\in\mathbb{C}^{N\times 1}$ based on the PARAFAC and
Tucker LRA models.

###  IV-A PARAFAC-IRS Low-Rank Approximation

In the tensorization step, a rearranging of elements from the IRS phase-shift
vector $\bm{s}$ to a tensor is performed, i.e.,
$\bm{\mathcal{S}}=\mathcal{T}\\{\bm{s}\\}$. Then, the RX will approximate the
optimum phase-shift tensor $\bm{\mathcal{S}}$ using a PARAFAC model, i.e.,

$\bm{\mathcal{S}}\approx\sum\limits_{r=1}^{R}\bm{s}^{(1)}_{r}\circ\ldots\circ\bm{s}^{(P)}_{r}\in\mathbb{C}^{N_{1}\times\cdots\times
N_{P}},$ (24)

where $R$ is the number of components and
$\bm{s}^{(p)}_{r}\in\mathbb{C}^{N_{p}\times r}$ is the $r$-th column of the
$p$-th factor matrix
$\bm{S}^{(p)}=[\bm{s}^{(1)}_{1},\ldots,\bm{s}^{(p)}_{R}]\in\mathbb{C}^{N_{p}\times
R}$, for $p=\\{1,\ldots,P\\}$.

Note that, applying (4) into (23), (24) is a straight-forward generalization
where we have $R$ components, and the approximation comes from the fact that
an independent phase-shift is fitted  as a combination of $P\times R$ sets of
phase-shifts, thus an approximation error is expected. However, as it will be
explained in Section VI, for scenarios with moderate/strong LOS components
(approximated rank-one channels) the effect of the fitting error on the SE
performance is negligible.

The RX estimates the factor components solving the following problem

$\displaystyle\underset{\text{for}\hskip
2.84544ptr=1,\ldots,R}{\left[\bm{\hat{s}}^{(1)}_{r},\ldots,\bm{\hat{s}}^{(P)}_{r}\right]}=\underset{\bm{s}^{(1)}_{r},\ldots,\bm{s}^{(P)}_{r}}{\text{argmin}}\left|\left|\bm{\mathcal{S}}-\sum\limits_{r=1}^{R}\bm{s}^{(1)}_{r}\circ\ldots\circ\bm{s}^{(P)}_{r}\right|\right|^{2}_{\text{F}},$
(25)

where $\bm{s}^{(p)}_{r}\in\mathbb{C}^{N_{p}\times 1}$ is the $p$-th factor
component. Let us define
$\bm{S}^{(p)}=\left[\bm{s}^{(p)}_{1},\ldots,\bm{s}^{(p)}_{R}\right]\in\mathbb{C}^{N_{p}\times
R}$ as the $p$-th factor matrix, for $p=\\{1,\ldots,P\\}$. From (10), the
$p$-mode unfolding of $\bm{\mathcal{S}}$, defined as
$\left[\bm{\mathcal{S}}\right]_{\left(p\right)}\in\mathbb{C}^{N_{p}\times
N_{1}\cdots N_{p-1}N_{p+1}\cdots N_{P}}$, is given as

$\left[\bm{\mathcal{S}}\right]_{\left(p\right)}\approx\bm{S}^{(p)}\left(\bm{S}^{(P)}\diamond\ldots\diamond\bm{S}^{(p+1)}\diamond\bm{S}^{(p-1)}\diamond\ldots\diamond\bm{S}^{(1)}\right)^{\text{T}}.$
(26)

1:Inputs: Tensor $\bm{\mathcal{S}}\in\mathbb{C}^{N_{1}\times\cdots\times
N_{P}}$, the number of components $R$, and maximum number of iterations $I$.

2:Randomly initialize the factors $\bm{\hat{S}}^{(2)}_{0}$, $\ldots$,
$\bm{\hat{S}}^{(P)}_{0}$. Iteration ${i}=0$.

3:Define a maximum number of iteration $I$.

4:for $i=1:I$ do

5: for $p=1:P$ do

6: Compute an estimate of the $p$-th factor $\bm{S}^{(p)}_{\text{i}}$ as
$\displaystyle\bm{\hat{S}}^{(p)}_{i}=\left[\bm{\mathcal{S}}\right]_{\left(p\right)}\left(\left(\bm{\hat{S}}^{(P)}_{i-1}\diamond\ldots\diamond\bm{\hat{S}}^{(p+1)}_{i-1}\diamond\bm{\hat{S}}^{(p-1)}_{i-1}\diamond\ldots\diamond\bm{\hat{S}}^{(1)}_{i-1}\right)^{\text{T}}\right)^{+}$

7: for $r=1:R$ do

8: Normalize the $r$-th column of $\bm{\hat{S}}^{(p)}_{(i)}$, defined as
$\bm{s}^{(p)}_{r,(i)}$, and store its norm as the $r$-th element of the vector
$\bm{\lambda}^{(p)}\in\mathbb{R}^{R\times 1}$ $\displaystyle\lambda^{(p)}_{r}$
$\displaystyle=\left|\left|\bm{\hat{s}}^{(p)}_{r,(i)}\right|\right|_{2},\,\,$
$\displaystyle\bm{\hat{s}}^{(p)}_{r,(i)}=\frac{\bm{\hat{s}}^{(p)}_{r,(i)}}{\lambda^{(p)}_{r}}.$

9: end for

10: end for

11: Define the weighting vector
$\bm{\lambda}=\bm{\lambda}^{(1)}\odot\cdots\odot\bm{\lambda}^{(P)}\in\mathbb{C}^{R\times
1}$.

12: $i=i+1$

13:end for

14:Return $\bm{\hat{S}}^{(1)}$, $\ldots$, $\bm{\hat{S}}^{(P)}$ and
$\bm{\lambda}$.

Algorithm 1 PARAFAC-IRS ALS

To solve the problem in (25), the RX can use the ALS algorithm [37], described
in Algorithm 1. Basically, the ALS algorithm contains $I$ iterations, where,
in each iteration, $P$ LS problems are solved. The $p$-th LS problem is
defined as

$\bm{\hat{S}}^{(p)}=\underset{\bm{S}^{(p)}}{\text{argmin}}\left|\left|\begin{split}\bm{\hat{S}}^{(p)}-\bm{S}^{(p)}\big{(}\bm{S}^{(P)}&\diamond\ldots\diamond\bm{S}^{(p+1)}\diamond\\\
\bm{S}^{(p-1)}&\diamond\ldots\diamond\bm{S}^{(1)}\big{)}^{\text{T}}\end{split}\right|\right|^{2}_{\text{F}},$
(27)

where its solution is given by

$\bm{\hat{S}}^{(p)}=\left[\bm{\mathcal{S}}\right]_{\left(p\right)}\left(\left(\bm{S}^{(P)}\diamond\ldots\diamond\bm{S}^{(p+1)}\diamond\bm{S}^{(p-1)}\diamond\ldots\diamond\bm{S}^{(1)}\right)^{\text{T}}\right)^{+}.$
(28)

In the first iteration, the first step is to estimate
$\left[\bm{\mathcal{S}}\right]_{\left(1\right)}$ based on (28), for $p=1$.
Then, its $R$ columns are normalized to unit norm and stored in as elements of
the vector $\bm{\lambda}^{(1)}\in\mathbb{R}^{R\times 1}$. After the
normalization, the estimated
$\left[\bm{\mathcal{\hat{S}}}\right]_{\left(1\right)}$ is plugged in the LS
solution (28) for $p=2$. Likewise, the columns of the estimated factor
$\left[\bm{\mathcal{\hat{S}}}\right]_{\left(2\right)}$ are normalized and
stored in a vector defined $\bm{\lambda}^{(2)}\in\mathbb{R}^{R\times 1}$, and
then, the normalized estimations
$\left[\bm{\mathcal{\hat{S}}}\right]_{\left(1\right)}$ and
$\left[\bm{\mathcal{\hat{S}}}\right]_{\left(2\right)}$ are plugged into  the
LS solution (28) for $p=3$. This process continues for the $P-3$ remaining LS
problems. Then, we compute the weighting vector
$\bm{\lambda}\in\mathbb{R}^{R\times 1}$ as the Hadamard product of all $P$
factors norms, i.e.,
$\bm{\lambda}=\bm{\lambda}^{(1)}\odot\bm{\lambda}^{(2)}\odot\cdots\odot\bm{\lambda}^{(P)}$,
finalizing the first iteration of the ALS. Then, the process repeats for all
$I$ iterations or until reaching a convergence threshold by checking the
normalized mean square error (NMSE) of the reconstructed tensor in a window of
consecutive iterations. The NMSE at the $i$-th iteration is given as

$\displaystyle\text{e}_{(i)}=\frac{\left|\left|\left[\bm{\mathcal{S}}\right]_{(1),(i)}-\left[\hat{\bm{\mathcal{S}}}\right]_{(1),{(i)}}\right|\right|^{2}_{\text{F}}}{\left|\left|\left[\bm{\mathcal{S}}\right]_{(1),(i)}\right|\right|^{2}_{\text{F}}},$

where $\left[\hat{\bm{\mathcal{S}}}\right]_{(1),(i)}$ is the reconstructed
$1$-mode unfolding at the $i$-th ALS iteration, given by

$\left[\hat{\bm{\mathcal{S}}}\right]_{(1),(i)}=\hat{\bm{S}}^{(1)}\text{diag}\left(\bm{\lambda}\right)\left(\hat{\bm{S}}^{(P)}\diamond\ldots\diamond\hat{\bm{S}}^{(2)}\right)^{\text{T}}.$
(29)

If $|\text{e}_{(i)}-\text{e}_{(i-1)}|\leq\epsilon$, where $\epsilon$ is a pre-
defined threshold, the algorithm stops [37]. In this paper, we consider
$\epsilon=10^{-6}$.

After the ALS algorithm, the phase-shifts of each factor and the weighting
vector $\bm{\lambda}$ are quantized to be conveyed to the IRS controller. In
this case, the feedback duration is given by

$T_{\text{F}}^{(\text{PARAFAC})}=\frac{\text{T}_{\text{PR}}+R\sum\limits_{p=1}^{P}N_{p}\cdot
b^{(p)}_{\text{F}}+(R-1)\cdot
b^{(\text{w})}_{\text{F}}}{B_{\text{F}}\text{log}\left(1+\frac{p_{\text{F}}|{g}_{\text{F}}|^{2}}{B_{\text{F}}N_{0}}\right)},$
(30)

where $\text{T}_{\text{PR}}$ is the number of bits required for a preamble of
the frame, in order to inform the IRS controller the factorization parameters,
such as $P$, $R$ and the quantization bits $b^{(p)}_{\text{F}}$ and
$b^{(\text{w})}_{\text{F}}$, where the $b^{(p)}_{\text{F}}$ is the number of
bits used for quantize the phase-shifts of the $p$-th factor, while
$b^{(\text{w})}_{\text{F}}$ is the number of bits for quantizing the elements
of the weighting vector $\bm{\lambda}$.

As one example, Fig. 6 illustrates the ratio between state-of-the-art
approach, where the $N$ IRS phase-shifts are fed back, with the proposed
PARAFAC-IRS approach, i.e., $N/(R\sum\limits_{p=1}^{P}N_{P})$, not taking into
account the preamble $\text{T}_{\text{PR}}$ and the resolution. Let us define
a vector $\bm{N}_{\text{P}}=\left[N_{1}\ldots
N_{P}\right]\in\mathbb{R}^{P\times 1}$ that contains the factor’s size for a
certain $P$. We can observe that, for $R=1$, the feedback duration of the
proposed approach for the case of $P=2,N_{1}=256,N_{2}=4$, is almost five
times smaller than the state-of-the-art, while, when we increase the number of
factors $P$, the size of the factors can be reduced, thus decreasing the
feedback duration, such that, when we have $P=10$ and $N_{p}=2$, for
$p=\\{1,\ldots,P\\}$, the feedback overhead of the proposed approach is
approximately fifty times smaller than the state-of-the art [34]. As noticed,
with increasing $P$, the feedback duration of our proposed approach decreases,
but also, as it will be discussed in Section VI, the SE in non-line of sight
(NLOS) scenarios. Thus, to overcome this loss the RX can increase the number
of components $R$ at the cost of a higher feedback overhead. In this way, the
proposed overhead-aware method shows off a trade-off between SE and feedback
overhead.

Figure 6: Feedback payload ratio for $N=1024$.

###  IV-B Tucker-IRS Low-Rank Approximation

Let us consider that the RX node opts to fit the phase-shift tensor
$\bm{\mathcal{S}}$ as a Tucker model, i.e.,

$\bm{\mathcal{S}}\approx\sum\limits_{r_{1}=1}^{R_{1}}\ldots\sum\limits_{r_{P}=1}^{R_{P}}\bm{\mathcal{G}}_{r_{1},\ldots,r_{P}}\left(\bm{s}^{(1)}_{r_{1}}\circ\ldots\circ\bm{s}^{(P)}_{r_{P}}\right)\in\mathbb{C}^{N_{1}\times\ldots\times
N_{P}}.$ (31)

where $\bm{\mathcal{G}}\in\mathbb{C}^{R_{1}\times\ldots\times R_{P}}$ is the
$P$-th order core tensor and $\bm{s}^{(p)}_{r_{p}}\in\mathbb{C}^{N_{p}\times}$
is the $r_{p}$-th column of the $p$-th factor matrix
$\bm{S}^{(p)}\in\mathbb{C}^{N_{p}\times R_{p}}$, for $p=\\{1,\ldots,P\\}$ and
$r_{p}=\\{1,\ldots,R_{p}\\}$. According to (32), the $p$-th mode unfolding of
$\bm{\mathcal{S}}$ is given by

$\left[\bm{\mathcal{S}}\right]_{\left(p\right)}=\bm{S}^{(p)}\left[\bm{\mathcal{G}}\right]_{\left(p\right)}\left(\bm{S}^{(P)}\otimes\ldots\otimes\bm{S}^{(p+1)}\otimes\bm{S}^{(p-1)}\otimes\ldots\otimes\bm{S}^{(1)}\right)^{\text{T}}$
(32)

Based on the Tucker-IRS model, the RX estimates each factor matrix
$\bm{S}^{(p)}\in\mathbb{C}^{N_{p}\times R}$, for $p=\\{1,\ldots,P\\}$, and the
core tensor $\bm{\mathcal{G}}$. This estimation procedure can be performed
using, e.g., the HOSVD algorithm [38] given in Algorithm 2, which, in this
case, consists of the RX estimating the factors matrices by computing the SVD
of all $P$-mode unfolding matrices of $\bm{\mathcal{S}}$ independently.
Defining the SVD of $\left[\bm{\mathcal{S}}\right]_{\left(p\right)}$ as
$\bm{U}^{(p)}\bm{\Sigma}^{(p)}\bm{V}^{\text{H}}$, an estimate of
$\bm{S}^{(p)}$ is given by

$\bm{\hat{S}}^{(p)}=\bm{U}^{(p)}_{.1:R_{p}}\in\mathbb{C}^{N_{p}\times R_{p}},$
(33)

which is the truncation of the left singular matrix $\bm{U}^{(p)}$ to its
first $R_{P}$ columns, $p=\\{1,\ldots,P\\}$. The diagonal of the truncated
singular matrix $\bm{\Sigma^{(p)}}$, defined as
$\bm{\sigma}^{(p)}=\text{diag}(\bm{\Sigma}^{(p)}_{1:R_{p},1:R_{p}})\in\mathbb{C}^{R_{p}\times
1}$, is stored to provide the weights to the $R_{p}$ components in the
quantization procedure. Once the $P$ factor matrices are estimated, the RX
obtains an estimate of the core tensor $\bm{\mathcal{G}}$ as

$\bm{\hat{g}}=\left(\bm{\hat{S}}^{(P)}\otimes\ldots\otimes\bm{\hat{S}}^{(1)}\right)^{\text{H}}\bm{s},\in\mathbb{C}^{R_{1}\cdots
R_{P}\times 1},$ (34)

where $\bm{\hat{g}}=\text{vec}\left(\hat{\bm{\mathcal{G}}}\right)$ and
$\bm{s}=\text{vec}\left(\bm{\mathcal{S}}\right)$ are the vectorization of the
core tensor and the IRS phase-shift tensor, respectively.

The feedback duration of the Tucker-IRS model is given as

$T_{\text{F}}^{(\text{Tucker})}=\frac{\text{T}_{\text{PR}}+\left(\sum\limits_{p=1}^{P}R_{p}N_{p}b^{(p)}_{\text{F}}\right)+\prod\limits_{p=1}^{P}R_{p}+b^{(\text{w})}_{\text{F}}\prod\limits_{p=1}^{P}(R_{p}-1)}{B_{\text{F}}\text{log}\left(1+\frac{p_{\text{F}}|{g}_{\text{F}}|^{2}}{B_{\text{F}}N_{0}}\right)},$
(35)

where $\text{T}_{\text{PR}}$ is the preamble duration that informs to the IRS
controller the chosen LRA model, the number of factors $P$, and the number of
components $R_{p}$, for $p=\\{1,\ldots,P\\}$. The term
$\sum\limits_{p=1}^{P}R_{p}N_{p}b^{(p)}_{\text{F}}$ represents the cost, in
bits, of the conveyed phase-shifts, $\prod\limits_{p=1}^{P}R_{p}$ is the cost
of the phase-shifts of the core tensor, and
$b^{(\text{w})}_{\text{F}}\prod\limits_{p=1}^{P}(R_{p}-1)$ is the term related
to the cost of the weighting factors.

##  V Discussion on Quantization, Reconstruction and Parameter Choices

###  V-A Phase-shift Quantization

After estimating the factors in Algorithms 1 or 2, the RX quantizes the phase-
shifts of each factor with $b_{F}^{(p)}$ bits. Let us define
$\bm{\tilde{a}}=\mathcal{Q}\\{\bm{a},b\\}$ as the quantization operation,
which quantizes a phase-shift vector $\bm{a}$ with $b$ bits. For the PARAFAC-
IRS model, we have the following quantized factors
$\bm{\tilde{s}}_{r}=\mathcal{Q}\left\\{\bm{\hat{s}}^{(p)}_{r},b_{\text{F}}^{(p)}\right\\}$
for $p=\\{1,\ldots,P\\}$ and $r=\\{1,\ldots,R\\}$. In addition, for the
Tucker-IRS model, we have the following quantized factors and core tensor
$\bm{\tilde{s}}_{r_{p}}=\mathcal{Q}\left\\{\bm{\hat{s}}^{(p)}_{r_{p}},b_{\text{F}}^{(p)}\right\\}$
and
$\tilde{\bm{\mathcal{G}}}_{r_{1},\ldots,r_{P}}=\mathcal{Q}\left\\{\hat{\bm{\mathcal{G}}}_{r_{1},\ldots,r_{P}},b_{\text{F}}^{(p)}\right\\}$,
for $p=\\{1,\ldots,P\\}$ and $r_{p}=\\{1,\ldots,R_{p}\\}$. For the phase-shift
quantization of the $p$-th factor, we use the following codebook

$\displaystyle\mathcal{C}_{\phi}^{(p)}=\left\\{-\pi+\frac{2\pi}{2^{b^{(p)}_{\text{F}}}},\hskip
5.69046pt-\pi+\frac{4\pi}{2^{b^{(p)}_{\text{F}}}},\hskip
5.69046pt\ldots,\hskip 5.69046pt\pi\right\\}.$

###  V-B Weighting Factor Quantization

For the PARAFAC-IRS model, let us define $\lambda_{\text{max}}$ as the largest
element of $\bm{\lambda}$. Then, we define a new weighting vector
$\bm{\lambda}^{\prime}=\bm{\lambda}/\lambda_{\text{max}}\in\mathbb{R}^{R\times
1}$. Since the largest element of $\bm{\lambda}^{\prime}$ is one, we do not
need to quantize this element. Hence, we define a new vector
$\bar{\bm{\lambda}}\in\mathbb{R}^{R-1\times 1}$ that contains all elements of
$\bm{\lambda}^{\prime}$, with the exception the of the largest one. Then, we
quantize the weighting vector by defining
$\tilde{\bar{\bm{\lambda}}}=\mathcal{Q}\left\\{\bar{\bm{\lambda}},b^{(\text{w})}_{\text{F}}\right\\}$.
Finally, we define $\tilde{\bm{\lambda}}\in\mathbb{R}^{R\times 1}$ as the
quantized weighting vector by inserting in the correct position the largest
element of $\bm{\lambda}^{\prime}$ (one) in
$\tilde{\bar{\bm{\lambda}}}\in\mathbb{R}^{R-1\times 1}$. At the end, the
weighting vector quantization cost is $(R-1)b^{\text{w}}_{\text{F}}$ bits.

For the Tucker model, a similar approach is made, with the difference that
there are $P$ weighting vectors sorted by their largest value due to the SVD
procedure. Considering the $p$-th weighting vector
$\bm{\sigma}^{(p)}\in\mathbb{R}^{R_{p}\times 1}$, we normalize it by the first
element, yielding
$\bm{\sigma}^{(p)\prime}=\bm{\sigma}^{(p)}/\sigma^{(p)}_{1}$. For the
quantization, we define a vector
$\bar{\bm{\sigma}}^{(p)}\in\mathbb{R}^{R_{p}-1\times 1}$ that contains all
elements of $\bm{\sigma}^{(p)\prime}$ with exception of the first one. Then,
we define the quantized $p$-th weighting factor as
$\tilde{\bar{\bm{\sigma}}}^{(p)}=\mathcal{Q}\left\\{\bar{\bm{\sigma}}^{(p)},b^{(\text{w})}_{\text{F}}\right\\}\in\mathbb{R}^{R_{p}-1\times
1}$. Finally, for the $p$-th quantized vector, we define the quantized vector
$\tilde{\bm{\sigma}}^{(p)}=[1,\tilde{\bm{\sigma}}^{(p)\prime}]\in\mathbb{R}^{R_{p}\times
1}$. At the end, the quantization of the $P$ weighting factors for the Tucker
model costs
$b^{\text{w}}_{\text{F}}\cdot\prod\limits_{p=1}^{P}\left(R_{p}-1\right)$ bits.

For both PARAFAC and Tucker models, we define the following amplitude codebook

$\displaystyle\mathcal{C}_{\text{w}}=\left\\{0.01,\hskip
5.69046pt0.01+l,\hskip 5.69046pt0.01+2l,\hskip 5.69046pt\ldots,\hskip
5.69046pt1\right\\},$ (36)

where $l=\frac{1-0.01}{2^{b^{(\text{w})}_{\text{F}}}-1}$ is the pre-defined
step. For simplicity, the values of the amplitudes in (36) are rounded to the
second decimal point.

###  V-C IRS Phase-shift Vector Reconstruction

After quantization, the RX conveys the factors to the IRS controller. Then,
the phase-shift vector is reconstructed as

$\displaystyle\bm{s}=e^{j\angle\bm{\hat{s}}}\in\mathbb{C}^{N\times 1},$ (37)

where $\bm{\hat{s}}$ is given by

$\displaystyle\bm{\hat{s}}=\sum\limits_{r=1}^{R}\tilde{\lambda}_{r}\left(\bm{\tilde{s}}^{(P)}_{r}\otimes\ldots\otimes\bm{\tilde{s}}^{(1)}_{r}\right),$
(38)

for the PARAFAC-IRS model, while for the Tucker-IRS model, $\bm{\hat{s}}$ is
factorized as

$\displaystyle\bm{\hat{s}}=\sum\limits_{r_{1}=1}^{R_{1}}\ldots\sum\limits_{r_{P}=1}^{R_{P}}\tilde{\bm{\mathcal{G}}}_{r_{1},\ldots,r_{P}}\left(\tilde{\sigma}^{(P)}_{r_{P}}\bm{\tilde{s}}^{(P)}_{r_{P}}\right)\otimes\ldots\otimes\left(\tilde{\sigma}^{(1)}_{r_{1}}\bm{\tilde{s}}^{(1)}_{r_{P}}\right).$
(39)

1:Inputs: Tensor $\bm{\mathcal{S}}$, the number of components $R_{p}$, for
$p=\\{1,\ldots,P\\}$.

2:for $p=1:P$ do

3: Compute the SVD of the $p$-mode unfolding of $\bm{\mathcal{S}}$ as
$\displaystyle\left[\bm{\mathcal{S}}\right]_{\left(p\right)}=\bm{U}^{(p)}\bm{\Sigma}^{(p)}\bm{V}^{(p)\text{H}}.$

4: Store the diagonal of the truncated singular matrix defined as
$\bm{\sigma}^{(p)}=\text{diag}\left(\bm{\Sigma}^{(p)}_{1:R_{p},1:R_{p}}\right)\in\mathbb{C}^{R_{p}\times
1}$.

5: Set an estimation of $\bm{S}^{(p)}$ by truncating the left singular matrix
to its first $R$ columns
$\displaystyle\bm{\hat{S}}^{(p)}=\bm{U}^{(p)}_{.1:R_{p}}.$

6:end for

7:Compute an estimate of the core tensor
$\bm{g}=\text{vec}\left(\bm{\mathcal{G}}\right)$ as
$\displaystyle\text{vec}\left(\hat{\bm{g}}\right)=\left(\bm{\hat{S}}^{(P)\text{H}}\otimes\ldots\otimes\bm{\hat{S}}^{(1)\text{H}}\right)\text{vec}\left(\bm{\mathcal{S}}\right).$

8:Define
$\hat{\bm{\mathcal{G}}}=\bm{\mathcal{T}}\\{\text{vec}\left(\hat{\bm{g}}\right)\\}$.

9:Return $\bm{\hat{S}}^{(1)},\ldots,\bm{\hat{S}}^{(P)}$ and
$\hat{\bm{\mathcal{G}}}$.

Algorithm 2 Tucker-IRS HOSVD

###  V-D On the Effect of the Factorization Parameters

In this section, we discuss the choice of the factorization parameters and the
system performance implications.

  * •

Number of factors $P$: This parameter defines the total number of factors used
in the LRA. Its minimum value for the proposed factorization is $P=2$, i.e.,
$P=1$ means that no factorization is employed, its maximum value is
$\text{log}_{2}(N)$, for the case where all the factors have size $N_{p}=2$
for $p=\\{1,\ldots,P\\}$. By increasing the value of $P$, the number of
factors of increases, allowing to reduce the size of the factor components
$N_{p}$. Consequently, increasing $P$ reduces the phase-shift feedback
overhead. Nevertheless, by selecting the minimum value of $P$, the size of
each factor component increases, which increases the spectral efficiency at
the cost of a higher feedback overhead.

  * •

Number of components: For the PARAFAC model, we have $R$ components, while for
the Tucker model we have $P\cdot\sum\limits_{p=1}^{P}R_{p}$ components. For
both models, the number of components is a performance indicator since when
increases, the approximation error of the LRA in (24) and (31) decreases. The
RX selects its value based on the estimated channels. For example, if the
channels have low-rank, or in the presence of a moderate/strong LOS component,
the RX may choose $R=1$. Also, $R=1$ (PARAFAC model) or $R_{p}=1$ (for the
Tucker model) are the choices that minimizes the feedback overhead. On the
other hand, by increasing $R$ (or $R_{p}$), the SE increases at the cost of a
higher feedback load.

  * •

Size of factor components $N_{p}$: The size of the factor components indicates
the total number of independent phase-shifts in the proposed solution, which
it also affects the performance. For example, for $N=256$, $P=2$ and $R=1$ for
the PARAFAC-IRS model, two possible configurations are $(N_{1}=128$,
$N_{2}=2)$ and $(N_{1}=N_{2}=16)$. For the first choice, the system has more
independent phase-shifts ($130$), thus a higher SE. However, its feedback
overhead is higher than that of the second configuration that requires only
$32$ phase-shifts to be reported in the feedback channel.

(a) No quantization and feedback.

(b) Comparison of the feedback payload (F.P.) in bits.

Figure 7: Comparison between the state-of-the-art [34], PARAFAC-IRS and
Tucker-IRS models with different numbers of components. $N=1024$, $P=3$, with
$N_{1}=64$, $N_{2}=4$.

###  V-E On the Effect of the Phase-shift and Weighting Factor Quantization

After the factorization step, the phase-shifts of the factor matrices
$\bm{S}^{(p)}$ are quantized before being conveyed to the IRS controller. From
the fact that the proposed method factorizes the IRS phase-shift vector into
$P$ smaller factors, we can select different numbers of bits for the
quantization of each factor, unlike the conventional IRS-assisted systems,
where the RX (or TX) conveys the $N$ phase-shifts with the same quantization
resolution of $b_{\text{F}}$ bits.  The proposed method allows the system to
adapt the phase-shift resolution of the factors to the available control link
capacity, i.e., for each factor we may have a different resolution of
$b^{(p)}_{\text{F}}$ in bits, $p=\\{1,\ldots,P\\}$, providing more flexibility
to the system design. Regarding the weighting factors, they play a more
important role when the number of components $R>1$ (in the PARAFAC model), or
$R_{p}>1$, $p=\\{1,\ldots,P\\}$ (in the Tucker model), since they control the
importance of the rank-one components in each model.

##  VI Simulation Results

In this section, we evaluate the performance of the proposed IRS phase-shift
overhead-aware feedback model in terms of feedback duration, achievable data
rate, SE and EE. The channels in (19) are modeled as

$\displaystyle\bm{H}$
$\displaystyle=\sqrt{\alpha_{H}\frac{K_{H}}{K_{H}+1}}\bm{H}_{\text{LOS}}+\sqrt{\alpha_{H}\frac{1}{K_{H}+1}}\bm{H}_{\text{NLOS}},$
(40) $\displaystyle\bm{G}$
$\displaystyle=\sqrt{\alpha_{G}\frac{K_{G}}{K_{G}+1}}\bm{G}_{\text{LOS}}+\sqrt{\alpha_{G}\frac{1}{K_{G}+1}}\bm{G}_{\text{NLOS}},$
(41)

where $\alpha_{H}$ and $\alpha_{G}$ are the path-loss components of the TX-IRS
and IRS-RX links, respectively. The matrices $K_{H}$ and $K_{G}$ are the
Rician factors associated with $\bm{H}$ and $\bm{G}$, respectively.
$\bm{H}_{\text{LOS}}$, $\bm{G}_{\text{LOS}}$ follow a geometric-based channel
model, while the entries of $\bm{H}_{\text{NLOS}}$, $\bm{G}_{\text{NLOS}}$ are
modeled as circularly symmetric complex Gaussian random variables, with zero
mean an unit variance, i.e.,
$\bm{H}_{\text{NLOS}}\sim\mathcal{CN}(0,\bm{I}_{M_{T}})$ and
$\bm{G}_{\text{NLOS}}\sim\mathcal{CN}(0,\bm{I}_{M_{R}})$. More details of (40)
and (41) are given in Appendix A.

For a fair comparison between the state-of-the-art [34] and the proposed
PARAFAC-IRS and Tucker-IRS models, we optimize the precoder ($\bm{q}$),
combiner ($\bm{w}$), and the IRS phase-shifts ($\bm{s}$) using the upper-bound
solution of [34]. In this case, they are given as

$\displaystyle\bm{w}=\bm{U}^{(\text{G})}_{.1},\hskip
2.84544pt\bm{q}=\bm{V}^{(\text{H})}_{.1},\hskip
2.84544pt\bm{s}^{(\text{opt})}_{n}=e^{-\angle\left(\bm{V}^{\text{G})}_{n,1}\cdot\bm{U}^{(\text{H})}_{n,1}\right)},$

with $n=\\{1,\ldots,N\\}$, and
$\bm{U}^{(\text{G})}_{.1}\in\mathbb{C}^{M_{R}\times 1}$,
$\bm{V}^{(\text{G})}_{.1}\in\mathbb{C}^{N\times 1}$ are the dominant left and
right singular vectors of $\bm{G}$, while
$\bm{U}^{(\text{H})}_{.1}\in\mathbb{C}^{N\times 1}$,
$\bm{V}^{(\text{H})}_{.1}\in\mathbb{C}^{M_{T}\times 1}$ are the dominant left
and right singular vectors of $\bm{H}$.

In Figs. 7-9, we set $\alpha_{H}=\alpha_{G}=1$, and consider $K_{H}=K_{G}=K$
to simplify the presentation of the figures. However, we have tested the
results for a broad range of channel models and parameter settings and
observed the same qualitative conclusions as those presented.

###  VI-A PARAFAC-IRS vs Tucker-IRS

As a first experiment, we compare, in terms of achievable data rate, the two
proposed strategies, PARAFAC-IRS and Tucker-IRS, with the state-of-the-art
method [34], where the IRS phase-shift vector is not factorized. The
achievable data rate is given by

$\displaystyle\text{log}_{2}\left(1+\frac{|\bm{w}^{\text{H}}\bm{G}\bm{S}\bm{H}\bm{q}|^{2}}{\sigma^{2}_{b}}\right)\hskip
4.26773pt,\text{in bits/s/Hz},$ (42)

where $\bm{S}=\text{diag}(\bm{s}^{(\text{opt})})\in\mathbb{C}^{N\times N}$ is
the diagonal matrix containing the optimum IRS phase-shifts, which are given
in (38) for the PARAFAC-IRS model and in (39) for the Tucker model.

In Fig. 7, we assume $P=3$ for the proposed IRS factorization models, with
$N_{1}=64$, and $N_{2}=N_{3}=4$. As expected, in this scenario the state-of-
the-art solution [34], provides the performance upper bound, since no
factorization is applied.

In Fig. 7 (a), we compare the models in an ideal scenario with continuous
phase-shift and continuous values for the weighting factors. For simplicity,
let us define for the Tucker model, the vector
$\bm{R}_{\text{P}}=\left[R_{1},R_{2},\ldots
R_{P},\right]\in\mathbb{R}^{P\times 1}$ that contains the number of components
for each factor for a certain $P$ (with $P=3$ in this case). As expected, when
the number of components $R$ or $\bm{R}_{(\text{3})}$ increases, the
achievable data rate also increases, and we can observe that for the PARAFAC-
IRS model with $R=16$ and for the Tucker-IRS model with $\bm{R}_{3}=[16,4,4]$,
the proposed models achieves the optimum performance of the benchmark method
[34].

In practice, both the phase-shift and the weighting factors have to be
quantized, as illustrated in Fig. 7 (b), there is an optimal point for the
PARAFAC ($R=4$, for this case) since when $R>4$ the performance degrades due
to overfitting. For the Tucker model, when the number of components of
$\bm{R}_{3}$ increases, the performance in the NLOS region ($K<-5$ dB) also
improves at the cost of a higher feedback overhead. Note that, for the
moderate/strong LOS scenario ($K>5$ dB), the number of components for both
models does not give a noticeable performance enhancement. In this way, a
proper model for a NLOS scenario would be the Tucker one, while PARAFAC is
preferable in moderate/strong LOS cases, since it leads to the best
performance with the lowest feedback cost, which can be explained by the fact
the the channel matrices have low rank and the contributions of the components
compared to the largest one are negligible.

In the following, we consider the PARAFAC-IRS model, due its simplicity and
lower phase-shift and weight feedback cost. However, we have tested the
results for the cases with Tucker method and observed the same qualitative
conclusions as those presented.

Figure 8: For an IRS with $N=1024$, TX and RX with $M_{R}=M_{T}=2$ and
$b^{(p)}_{\text{F}}=b_{\text{F}}=3$ bits, for the IRS phase-shift quantization
resolution, for $p=\\{1,\ldots,P\\}$. Figure 9: Feedback payload for the
PARAFAC-IRS model with $R=1$, varying the number of IRS elements. Figure 10:
Performance of the PARAFAC-IRS method by varying the resolution
$b^{(p)}_{\text{F}}$ per factor, for fixed control link of $1024$ bits.

###  VI-B On the Effect of the Number of Factors $P$

In Fig. 8, we compare the achievable data rate of the PARAFAC-IRS model with
$R=1$, by varying the number of factors. We can observe that, for the NLOS
region ($K<-5$ dB), increasing $P$ leads to a degradation on the performance.
This is due to the fact that, for a larger $P$, we have less independents
phase-shifts. For example, for $P=10$, the phase-shifts of the IRS elements
are given by the sum of $10$ factorized phase-shifts. However, when the Rician
factor $K$ increases, the performance gap between our proposed model and the
state-of-the-art [34] reduces. This is explained by the fact that, the IRS
phase-shift optimization is based on the channel estimation, thus when $K$
increases, the LOS components become stronger, and we have a better
approximation of the PARAFAC-IRS model for $R=1$. In terms of feedback
overhead, when $P=2$ and for the $K<-5$ dB region, our proposed method has a
data rate loss of approximately $1$bps/Hz. However, the feedback overhead is
$50\%$ less than that of the benchmark solution [34]. On the other hand, when
the scenario changes to $K>5$ dB, the proper parameter choice is $P=10$, since
this configuration has a negligible performance loss compared to the state-of-
the-art one, while having a lower feedback cost compared to the other proposed
configurations ($P=2,3,4$).

Physically, the results illustrated on Fig. 8 can be interpreted as a
performance adaptation in the NLOS case, i.e., the RX can properly choose the
factorization parameters to meet a required data rate performance or feedback
saving. For instance, in this example, by choosing $P=10$, the RX can afford
more often feedback than configurations with smaller values of $P$.

For a better understanding of the merits of the proposed method, Fig.9 shows
the feedback payload in bits by varying the number of IRS elements. As shown,
for different methods the payload increases linearly with the number of IRS
elements. For a given $P$, we may have different sets of factor sizes defined
by $\bm{N}_{\text{P}}=[N_{1},\ldots,N_{P}]^{\text{T}}\in\mathbb{R}^{P\times
1}$, where the values of $P$ are set to $P=2,3,4$. We select the size
configuration that leads to the better performance, which is the one that has
the maximum possible number of independent phase-shifts. For example, assuming
$N=1000$, the size configuration is $\bm{N}_{2}=[500,2]$ for $P=2$,
$\bm{N}_{3}=[250,2,2]$ for $P=3$, and $\bm{N}_{4}=[125,2,2,2]$ for $P=4$.
Thus, it becomes clear that increasing $P$ drastically reduces the feedback
overhead.

###  VI-C On the Effect of the Factor Quantization

Here, we evaluate the performance of the proposed method in a limited feedback
channel, i.e., we assume that the feedback control link has a maximum capacity
of $1024$ bits. In this case, traditional quantization applied to the
unconstrained IRS phase shift vector (without factorization) is limited to a
one bit resolution. We assume this challenging scenario to observe the
performance impact of the proposed method when the resolution of the
individual factors are adapted. To this end, we assume $N\cdot
b_{\text{F}}\geq\bm{N}_{\text{P}}\cdot\bm{b}^{(\text{P})\text{T}}_{\text{F}}$.
In Fig. 10, different sets of size configurations for $P=2$ are evaluated,
with different resolutions per factor. The configuration $\bm{N}_{2}=[512,2]$
has the worst performance due to the fact that the first factor ($512$
elements) can only be quantized with $1$ bit. However, the size of the factors
is reduced, the resolution per factor can be increased accordingly to meet the
limited control link capacity limit. For instance, when $\bm{N}_{2}=[256,4]$
and $\bm{b}_{\text{F}}^{(p)}=[3,16]$, the total number of bits is $256\cdot
3+4\cdot 16=832$. We can observe that, by increasing the resolution of the
factors, the performance gets closer to that of the state-of-the-art phase
shift quantization (solid curve). In particular, note that for $K>-5$ dB, our
approach provides the best results. Thus, the proposed method can not only
reduce the feedback overhead, as illustrated in Figs. 8 and 9, but also it
effectively provides higher data rates than traditional quantization over the
unconstrained IRS phase shifts, approaching the continuous phase-shift case.

(a) SE performance of the state-of-the-art and the proposed method..

(b) EE performance of the state-of-the-art and the proposed method.

Figure 11: SE and EE performance of the proposed method varying the feedback
bandwidth, with $N=1024$, $M_{R}=M_{T}=16$,
$b^{(p)}_{\text{F}}=b_{\text{F}}=3$ bits, for $p=2,3,10$, for a Rician factor
$K=10$ dB.

(a) SE performance of the state-of-the-art and the proposed method..

(b) EE performance of the state-of-the-art and the proposed method.

Figure 12: SE and EE performance of the proposed method varying the feedback
bandwidth, with $N=1024$, $b^{(p)}_{\text{F}}=b_{\text{F}}=4$ bits, for
$p=2,3,10$, for a Rician factor $K=10$ dB.

(a) SE performance of the state-of-the-art and the proposed method.

(b) EE performance of the state-of-the-art and the proposed method.

Figure 13: EE and SE performance of the proposed method varying the feedback
power, with $N=1024$, $M_{R}=M_{T}=2$, $b^{(p)}_{\text{F}}=b_{\text{F}}=3$
bits, for $p=2,3,10$, Rician factor $K=10$ dB.

###  VI-D Total System SE and EE Evaluation

In this section, we evaluate the performance, in terms of SE and EE, of the
proposed method by considering the total system rate, i.e., taking into
account the channel estimation procedure duration and the IRS phase-shift
feedback duration. To this end, we make use of the expressions given in (21)
and (22). The channel estimation period, in (21), is given as
$\text{T}_{E}=(M_{T}N+1)T_{0}$, where $T_{0}=0.8\mu$ seconds denotes the
duration of the pilot tones [34]. The frame duration is given by
$T=\text{T}_{PD}+\text{T}_{\text{F}}$, where
$\text{T}_{PD}=\text{T}_{E}+\text{T}_{D}$, is divided into $30\%$ for pilot
transmissions ($\text{T}_{E}$) and $70\%$ for data transmission
$\text{T}_{D}$. Regarding the power parameters of (22), we have
$P_{\text{E}}=P_{0}(1+NM_{T})T_{0}$, where $P_{0}=0.8$ mW is the pilot tone
power. Other parameter definitions can be found in Table I. The feedback
channel $g_{\text{F}}$ is generated from a circular symmetric complex Gaussian
distribution, normalized by
$\sqrt{\beta_{F}}=\sqrt{\alpha_{H}}=\sqrt{\alpha_{G}}$ to account for the
effects of pathloss and shadowing, as given in Table I. In our next
experiments, we assume $K=10$ dB, $N=1024$. For the proposed method, we
consider the PARAFAC-IRS model with $R=1$. As for the number of factors, we
study three configurations, with $P=2,(\bm{N}_{2}=[512,2]$), $P=3$
($\bm{N}_{3}=[256,2,2]$) and $P=10$
($\bm{N}_{10}=[2,\ldots,2]\in\mathbb{R}^{10\times 1}$.

$P_{\text{max}}$/$P_{c,0}$/$P_{c,n}$ | $B_{\text{max}}$ | $N_{0}$ |  $\alpha_{H}$/$\alpha_{G}$ |  $\mu$/$\mu_{\text{F}}$
---|---|---|---|---
$45$/$45$/$10$ dBm |  $100$ MHz |  $-174$ dBm/Hz |  $110$/$110$ dB |  $1$/$1$

TABLE I:

Figs.11 and 12, we analyze the total SE and EE of the proposed method with the
state-of-the-art [34], by varying the feedback bandwidth
$B_{\text{F}}=B_{\text{max}}-B$, where $B_{\text{max}}$ is the total available
bandwidth given in Table I. As shown, in Figs. 11(a) and 11(b), when the
number of factors increases the feedback duration reduction pays off in the
total system SE and EE. The proposed method achieves a gain in the SE of
$32\%$ for $P=10$, $20\%$ for $P=3$, and $14\%$ for $P=2$, over the state-of-
the-art, considering the $B_{\text{F}}=200$ kHz, with a similar gain in the
EE.

In Figs. 12(a) and 12(b) we compare the proposed PARAFAC-IRS model under $R=1$
and $P=10$ with the state-of-the-art by varying the number of antennas. In
this case, we observe that, for a feedback bandwidth $B_{\text{F}}\leq 200$
kHz, the proposed factorization with a $2\times 2$ setup outperforms (in terms
of SE and EE) the state-of-the-art one under the $4\times 4$ setup, while
presenting the same performance than the state-of-the-art one under the
$16\times 16$ setup. Finally, Figs. 13(a) and 13(b) show the SE and EE
performances of the proposed method as a function of the feedback power
$p_{\text{F}}$, with $p_{\text{F}}=P_{\text{max}}-p_{\text{TX}}$. We notice
that the proposed configurations provide the best results in all scenarios.

To summarize the results illustrated in Figs.11-13, we conclude that the
proposed tensor-based LRA IRS phase-shift factorization models allows to
reduce the number of phase-shifts to be conveyed to the IRS-controller, which
significantly reduces the feedback overhead, resulting in SE and EE
performance enhancements. In addition, our approach reaches similar
performance to the non-factorized IRS, especially in moderate/strong LOS
scenarios, as it can be seen in Figs. 7-10.  From a system-level viewpoint,
the network can resort to the proposed overhead-aware IRS model to increase
the feedback periodicity, i.e., by providing more frequent feedback, which is
crucial in fast time-varying channels, where the IRS should be reconfigured
more frequently to follow the environment changes. Moreover, the proposed IRS
factorization methods allow the network to multiplex more IRS phase-shifts in
the same feedback channel, which is useful to accommodate multi-user IRS-
assisted communications.

##  VII Conclusions and perspectives

In this paper, we proposed two IRS phase-shift feedback overhead-aware methods
based on tensor signal processing, namely, PARAFAC-IRS and Tucker-IRS. We
showed that the proposed methods significantly reduce the IRS phase-shift
feedback overhead, compared to the state-of-the-art approach, where the IRS
phase shifts are not factorized. The PARAFAC-IRS method is preferable in the
case of moderate/strong LOS scenarios, achieving a spectral efficiency that is
close to that of the state-of-the-art, while providing a feedback overhead
reduction. Moreover, in NLOS scenarios, the Tucker-IRS model achieves a higher
data rate than the PARAFAC-IRS model at the expense of a higher feedback
overhead. By controlling the factorization parameters, we showed how to trade
off data rate for feedback-overhead, allowing the network controller to adapt
the IRS factorization parameters to meet a determined quality of service.

##  Appendix A Channel Model

We provide details on the channel models for $\bm{H}$ and $\bm{G}$, given in
(40) and (41), respectively. As mentioned, the NLOS components of $\bm{H}$ and
$\bm{G}$ are modeled as random channels with
$\mathbb{E}[\bm{H_{\text{NLOS}}}^{\text{H}}\bm{H}_{\text{NLOS}}]=\bm{I}_{M_{T}}$
and
$\mathbb{E}[\bm{G}_{\text{NLOS}}\bm{G}^{\text{H}}_{\text{NLOS}}]=\bm{I}_{M_{R}}$.
Nonetheless, the LOS components are given as

$\displaystyle\bm{H}_{\text{LOS}}$
$\displaystyle=\alpha_{H}\bm{b}_{\text{IRS}}\cdot\bm{a}^{\text{H}}_{\text{TX}}\in\mathbb{C}^{N\times
M_{T}},$ $\displaystyle\bm{G}_{\text{LOS}}$
$\displaystyle=\alpha_{G}\bm{b}_{\text{RX}}\cdot\bm{a}^{\text{H}}_{\text{IRS}}\in\mathbb{C}^{M_{R}\times
N},$

where $\alpha_{H}$ and $\alpha_{G}$ are the path-loss components of the TX-IRS
and IRS-RX links, respectively. Assuming that the TX and the RX are equipped
with ULAs with half-wavelength inter-element spacing, their steering vectors
can be written as

$\displaystyle\bm{a}_{\text{TX}}=\left[1,e^{j\pi\text{sin}\theta_{\text{TX}}},\ldots,e^{j\pi(M_{T}-1)\text{sin}\theta_{\text{TX}}}\right]^{\text{T}}\in\mathbb{C}^{M_{T}\times
1},$ (43)
$\displaystyle\bm{b}_{\text{RX}}=\left[1,e^{j\pi\text{sin}\theta_{\text{RX}}},\ldots,e^{j\pi(M_{R}-1)\text{sin}\theta_{\text{RX}}}\right]^{\text{T}}\in\mathbb{C}^{M_{R}\times
1},$ (44)

where $\theta_{\text{TX}}$ and $\theta_{\text{RX}}$ are the TX and RX angle of
departure (AOD) and angle of arrival (AOA), respectively, which are generated
from a uniform random distribution with
$\\{\theta_{\text{TX}},\theta_{\text{RX}}\\}\in[-\pi,\pi]$. Since the IRS is a
$2$-D panel, the steering vectors associated with arrival and departure angles
can be factorized as the Kronecker product of horizontal and vertical
component vectors, respectively, as follows:

$\displaystyle\bm{b}_{\text{IRS}}$
$\displaystyle=\bm{b}^{(v)}_{\text{IRS}}\otimes\bm{b}^{(h)}_{\text{IRS}}\in\mathbb{C}^{N_{h}N_{v}\times
1},$ (45) $\displaystyle\bm{a}_{\text{IRS}}$
$\displaystyle=\bm{a}^{(v)}_{\text{IRS}}\otimes\bm{a}^{(h)}_{\text{IRS}}\in\mathbb{C}^{N_{h}N_{v}\times
1},$ (46)

where $N=N_{h}N_{v}$, $\bm{b}^{(h)}_{\text{IRS}}\in\mathbb{C}^{N_{h}\times 1}$
and $\bm{b}^{(v)}_{\text{IRS}}\in\mathbb{C}^{N_{v}\times 1}$ are the AOA
steering vectors in the azimuth and elevation directions, respectively.
Likewise, $\bm{a}^{(h)}_{\text{IRS}}\in\mathbb{C}^{N_{h}\times 1}$ and
$\bm{a}^{(v)}_{\text{IRS}}\in\mathbb{C}^{N_{v}\times 1}$ are the AOD steering
vectors in the azimuth and elevation directions, respectively.

$\displaystyle\bm{b}^{(h)}_{\text{IRS}}$
$\displaystyle=[1,e^{j\pi\text{sin}\psi^{\text{AOA}}_{\text{IRS}}\text{cos}\phi^{\text{AOA}}_{\text{IRS}}},\ldots,e^{j\pi(N_{h}-1)\text{sin}\psi^{\text{AOA}}_{\text{IRS}}\text{cos}\phi^{\text{AOA}}_{\text{IRS}}}],$
$\displaystyle\bm{b}^{(v)}_{\text{IRS}}$
$\displaystyle=[1,e^{j\pi\text{cos}\phi^{\text{AOA}}_{\text{IRS}}},\ldots,e^{j\pi(N_{h}-1)\text{cos}\phi^{\text{AOA}}_{\text{IRS}}}],$
$\displaystyle\bm{a}^{(h)}_{\text{IRS}}$
$\displaystyle=[1,e^{j\pi\text{sin}\psi^{\text{AOD}}_{\text{IRS}}\text{cos}\phi^{\text{AOD}}_{\text{IRS}}},\ldots,e^{j\pi(N_{h}-1)\text{sin}\psi^{\text{AOD}}_{\text{IRS}}\text{cos}\phi^{\text{AOD}}_{\text{IRS}}}],$
$\displaystyle\bm{a}^{(v)}_{\text{IRS}}$
$\displaystyle=[1,e^{j\pi\text{cos}\phi^{\text{AOD}}_{\text{IRS}}},\ldots,e^{j\pi(N_{h}-1)\text{cos}\phi^{\text{AOD}}_{\text{IRS}}}],$

where $\phi^{\text{AOA}}_{\text{IRS}}$ and $\phi^{\text{AOD}}_{\text{IRS}}$
are the elevation angles of arrival and departure, while
$\psi^{\text{AOA}}_{\text{IRS}}$ and $\psi^{\text{AOD}}_{\text{IRS}}$ are the
azimuth angles of arrival and departure. The azimuth angles
$\psi^{\text{AOA}}_{\text{IRS}}$ and $\psi^{\text{AOD}}_{\text{IRS}}$ are
generated from a uniform random distribution with
$\\{\psi^{\text{AOA}}_{\text{IRS}},\psi^{\text{AOD}}_{\text{IRS}}\\}\in[-\pi,\pi]$,
while the elevation angles $\phi^{\text{AOA}}_{\text{IRS}}$ and
$\phi^{\text{AOD}}_{\text{IRS}}$ are generated from an uniform random
distribution with
$\\{\phi^{\text{AOA}}_{\text{IRS}},\phi^{\text{AOD}}_{\text{IRS}}\\}\in[0,\pi/2]$.

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