# Title of the paper

#  Fully Digital and Hybrid Beamforming Design For Millimeter-Wave MIMO-OFDM
Two-Way Relaying Systems

###### Abstract

In this work, we consider the design of hybrid analog-digital (HAD) multi-
carrier MIMO-OFDM two-way relaying systems, where the relay station is
equipped with a HAD amplify-and-forward architecture and every mobile station
is equipped with a fully-digital beamforming architecture. We propose a sub-
optimal solution by reformulating the original non-convex problem as a
constrained Tucker2 decomposition with the objective of minimizing the sum
Euclidean-norm between the HAD amplification matrices and their fully-digital
counterparts. For the fully-digital amplification matrix design, we use a
Frobenius-norm maximization of the effective channels on every subcarrier and
propose an effective solution applicable for multi-stream communication
scenarios. After that, we propose an alternating maximization (AltMax) HAD
solution by exploiting the tensor structure of the reformulated problem.
Simulation results are provided, where we show that the proposed fully-digital
and AltMax-based HAD amplification matrix designs outperform some benchmark
methods, especially for multi-stream communication scenarios.

Index Terms—  Two-way relaying, hybrid beamforming, millimeter-wave
communication, Tucker2 tensor decomposition.

##  1 Introduction

Recently, the integration of millimeter-wave (mm-wave) frequencies and massive
MIMO techniques has been recognized as one of the key solutions to improve the
spectral efficiency (SE) of 5G and beyond mobile systems [1, 2]. However, two
major practical challenges arise. On the one hand, the conventional fully-
digital beamforming architectures become impractical, due to their high-cost
and high-energy consumption, as they require a dedicated radio-frequency (RF)
chain for each antenna. On the other hand, high-frequency communications are
very sensitive to signal-blockage and signal-attenuation, which makes mm-wave-
based systems mainly applicable to short-range communications. Therefore,
hybrid analog-digital (HAD) beamforming architectures [2, 3, 4, 5, 6] and
relay-assisted communications [7, 8, 9, 10, 11, 12] have been proposed to
tackle these challenges. In HAD architectures, the number of RF chains can be
reduced significantly, compared to the number of the antenna elements.
Meanwhile, relay stations (RSs) can be used to increase the communication
range.

Among several relaying schemes, two-way relaying (TWR) is known to be more
resources efficient [8]. In TWR [7, 8, 9, 13, 10, 11, 12], the data
transmission between two mobile stations (MSs) occurs in two subsequent
transmission phases. First, both MSs transmit to the RS. Then, the RS
transmits back to both MSs, either using a decode-and-forward (DF) or an
amplify and forward (AF) relaying architecture. In this paper, we focus our
attention on the AF scheme. Moreover, it is known that TWR systems can be
transformed into two separate single-user MIMO channels if the self-
interference at every MS is subtracted [8, 9]. This implies that, if the
amplification matrix is known, the optimal decoding and encoding matrices at
the MSs are given by the dominant singular vectors of the resulting effective
channels. Thus, the remaining challenge is how to design the RS amplification
matrix. To this end, the authors of [8] proposed a method, called Algebraic
norm-maximizing (ANOMAX), which designs the RS amplification matrix that
maximizes the weighted sum of the Frobenius norms of the effective channels.
However, based on the observations, ANOMAX is mainly applicable to single-
stream transmissions. The authors of [9] extended ANOMAX by proposing rank-
restored ANOMAX (RR-ANOMAX) to increase the rank of the amplification matrix
by adjusting its singular values while preserving its subspaces. On the other
hand, the authors of [7] proposed an amplification matrix design using the
zero-forcing (ZF) and minimum mean square error (MMSE) filters. However, it
was reported in [8], using computer simulations, that the ANOMAX-based
solutions outperform the ZF-based and the MMSE-based solutions.

In this paper, we develop a new fully-digital RS amplification matrix design
for wide-band multi-carrier MIMO OFDM AF TWR systems, which extends the
results of [8] and [9]. To this end, we consider the Frobenius-norm
maximization of the effective channels on every subcarrier and propose a
different solution than [9] to further enhance the rank of the obtained fully-
digital RS amplification matrices, termed enhanced RR-ANOMAX (ERR-ANOMAX). As
the second contribution of the paper, we design HAD amplification matrix by
reformulating the original non-convex problem as a constrained Tucker2
decomposition [14, 15, 16] with the objective of minimizing Euclidean-norm
between the HAD amplification matrices and their fully-digital counterparts.
Then, we propose two HAD solutions, a non-iterative HOSVD-based solution [15]
and an iterative alternating maximization (AltMax)-based solution [5]. The
simulation results show that ERR-ANOMAX outperforms ANOMAX and RR-ANOMAX in
multi-stream communication scenarios.

##  2 System Model

As depicted in Fig. 1, we consider111Notation: The transpose, the Hermitian
transpose, the Moore-Penrose pseudo-inverse, the Kronecker product, and the
Khatri-Rao product are denoted by
$\bm{X}^{T},\bm{X}^{H},\bm{X}^{+},\otimes,\text{and}~{}\diamond$,
respectively. The Frobenius norm is denoted by $\|\cdot\|_{\mathrm{F}}$. The
operator $\mathrm{diag}\\{\bm{x}\\}$ returns a diagonal matrix with $\bm{x}$
as the diagonal arguments, $\text{vec}\\{\bm{X}\\}$ stacks the columns of
matrix $\bm{X}$ into a vector, and $\text{unvec}\\{\bm{x}\\}$ is the inverse
of the $\text{vec}\\{\bm{X}\\}$ operator. The following property is also used:
$\text{vec}\\{\bm{A}\mathrm{diag}(\bm{b})\bm{C}\\}=(\bm{C}^{T}\diamond\bm{A})\bm{b}$.
an AF TWR mm-wave massive MIMO-OFDM communication system, where two MSs
communicate with each other via an intermediate RS over $K$ subcarriers. It is
assumed that the RS and MS $\ell$ are equipped with $M_{\mathrm{RS}}$ and
$M_{\ell}$ antennas, respectively. For $\ell\in\\{1,2\\}$ and
$k\in\\{0,\dots,K-1\\}$, let $\bm{H}_{\ell,k}$ be $M_{\mathrm{RS}}\times
M_{\ell}$ MIMO channel between MS $\ell$ and the RS on the $k$th subcarrier.
We assume a TDD system so that the channel reciprocity holds, i.e., the
reverse (downlink) channels are the transpose of the forward (uplink)
channels. Let $\bm{F}_{\ell,k}$ and $\bm{W}_{\ell,k}$ denote the
$M_{\ell}\times N_{s}$ fully-digital precoding and decoding matrices of MS
$\ell$ on the $k$th subcarrier, respectively, where $N_{s}$ denotes the number
of data streams. Moreover, let $\bm{G}_{k}$ denote the $M_{\mathrm{RS}}\times
M_{\mathrm{RS}}$ RS amplification matrix on the $k$th subcarrier.

In TWR systems, the data transmission occurs in two phases. In phase 1, the
MSs transmit their signals to the RS simultaneously so that the received
signal at the RS on the $k$th subcarrier can be written as

$\bm{x}_{k}=\bm{H}_{1,k}\bm{F}_{1,k}\bm{s}_{1,k}+\bm{H}_{2,k}\bm{F}_{2,k}\bm{s}_{2,k}+\bm{n}_{\mathrm{RS},k},$
(1)

where $\bm{s}_{\ell,k}$ is the data vector with
$\mathbb{E}\\{\bm{s}_{\ell,k}\bm{s}^{H}_{\ell,k}\\}=\bm{I}_{N_{s}}$ and
$\bm{n}_{\mathrm{RS},k}$ contains zero-mean circularly symmetric complex
Gaussian noise with variance $\sigma^{2}_{\mathrm{RS}}$. Here, it is assumed
that
$\operatorname{\mathbb{E}}\\{\|\bm{F}_{\ell,k}\|_{2}^{2}\\}=P_{\mathrm{MS}}$,
where $P_{\mathrm{MS}}$ denotes the maximum transmit power of a MS per
subcarrier. In phase 2, the RS transmits $\bm{x}_{k}$ to both MSs so that the
received signal at MS $\ell$ on the $k$th subcarrier is given as

$\displaystyle\bm{y}_{\ell,k}=$
$\displaystyle\bm{W}^{H}_{\ell,k}{\bm{H}^{T}_{\ell,k}}\bm{G}_{k}\bm{x}_{k}+\bm{W}^{H}_{\ell,k}\tilde{\bm{n}}_{\ell,k},$
(2)

where
$\tilde{\bm{n}}_{\ell,k}={\bm{H}^{T}_{\ell,k}}\bm{G}_{k}\bm{n}_{\mathrm{RS},k}+\bm{n}_{\ell,k}$
is the total additive noise, in which $\bm{n}_{\ell,k}$ denotes the additive
white Gaussian noise at MS $\ell$ with variance $\sigma^{2}_{\mathrm{MS}}$.
Similarly, we assume that
$\operatorname{\mathbb{E}}\\{\|\bm{G}_{k}\bm{x}_{k}\|_{2}^{2}\\}=P_{\mathrm{RS}}$,
where $P_{\mathrm{RS}}$ denotes the RS maximum transmit power per subcarrier.
By expanding (2), we have

$\displaystyle\bm{y}_{\ell,k}=\bm{y}^{\text{DS}}_{\ell,k}+\bm{y}^{\text{SI}}_{\ell,k}+{\bm{W}^{H}_{\ell,k}}\tilde{\bm{n}}_{\ell,k},$
(3)

where
$\bm{y}^{\text{SI}}_{\ell,k}={\bm{W}^{H}_{\ell,k}}\bar{\bm{H}}_{\ell,\ell,k}\bm{F}_{\ell,k}\bm{s}_{\ell,k}$
is the self-interference (SI) and
$\bm{y}^{\text{DS}}_{\ell,k}={\bm{W}^{H}_{\ell,k}}\bar{\bm{H}}_{\ell,\ell^{\prime},k}\bm{F}_{\ell^{\prime},k}\bm{s}_{\ell^{\prime},k},\ell\neq\ell^{\prime}$,
is the desired-signal (DS), in which we have defined the effective channels

$\displaystyle\bar{\bm{H}}_{\ell,\ell^{\prime},k}={\bm{H}^{T}_{\ell,k}}\bm{G}_{k}\bm{H}_{\ell^{\prime},k},~{}\ell,\ell^{\prime}\in\\{1,2\\}.$
(4)

Note that the SI appears in (3) since each MS receives back its own signal
from the RS. However, under ideal conditions, as elaborated in [8, Section
III-B], the SI term can be subtracted completely by MSs. Consequently, (3)
reduces to

$\displaystyle\bm{z}_{\ell,k}=\bm{y}_{\ell,k}-\bm{y}^{\text{SI}}_{\ell,k}=\bm{y}^{\text{DS}}_{\ell,k}+\bm{W}^{H}_{\ell,k}\tilde{\bm{n}}_{\ell,k}.$
(5)

From the above, the SE of MS $\ell$ on the $k$th subcarrier can be expressed
as

$\textrm{SE}_{\ell,k}=\frac{1}{2}\log_{2}\Big{|}\Big{(}\bm{I}_{N_{s}}+\Big{(}\bm{W}^{H}_{\ell,k}\bm{\Phi}_{\ell,k}\bm{W}_{\ell,k}\Big{)}^{-1}\bm{R}_{\ell,k}\Big{|},$
(6)

where
$\bm{\Phi}_{\ell,k}=\mathbb{E}\\{\tilde{\bm{n}}_{\ell,k}\tilde{\bm{n}}^{H}_{\ell,k}\\}={\sigma^{2}_{\mathrm{RS}}\bm{H}^{T}_{\ell,k}}\bm{G}_{k}\bm{G}^{H}_{k}{\bm{H}^{*}_{\ell,k}}+\sigma^{2}_{\mathrm{MS}}\bm{I}_{M_{\mathrm{MS}}}$,
$\bm{R}_{\ell,k}=\mathbb{E}\\{\big{(}\bm{y}^{\text{DS}}_{\ell,k}\big{)}\big{(}\bm{y}^{\text{DS}}_{\ell,k}\big{)}^{H}\\}={\bm{W}^{H}_{\ell,k}}\bar{\bm{H}}_{\ell,\ell^{\prime},k}\bm{F}_{\ell^{\prime},k}\bm{F}^{H}_{\ell^{\prime},k}\bar{\bm{H}}^{H}_{\ell,\ell^{\prime},k}{\bm{W}_{\ell,k}}$
are the effective noise and the desired signal covariance matrices,
respectively.

Fig. 1: An AF TWR system.

Our goal is to design the beamforming matrices to maximize the total SE of the
system, which can be expressed as

$\displaystyle\max_{\\{\bm{F}_{\ell,k},\bm{W}_{\ell,k},\bm{G}_{k},\forall
k\\}}\quad\textstyle\sum_{\ell=1}^{2}\sum_{k=0}^{K-1}\textrm{SE}_{\ell,k}$ (7)
s.t.
$\displaystyle\quad\|\bm{F}_{\ell,k}\|^{2}_{\mathrm{F}}=P_{\mathrm{MS}},\quad\|\bm{G}_{k}\bm{x}_{k}\|^{2}_{2}=P_{{\mathrm{RS}}}.$

Note that (7) is a non-convex optimization function due to the joint
optimization of the beamforming matrices at the MSs and the amplification
matrix at the RS. To obtain a solution, we propose in the following a sub-
optimal non-iterative approach by decoupling the optimization procedure
between the beamforming matrices of the MSs and the RS amplification matrix.
By noting that for any given $\bm{G}_{k}$ on subcarrier $k$, the system model
in (5) reduces to two independent point-to-point MIMO communication systems,
one for each MS. It implies that the optimal precoding (resp. decoding)
matrix, on every subcarrier, is given by the dominant $N_{s}$ right (resp.
left) singular vectors of the respective effective channel given by (4), with
the power allocation given by the water-filling method [17]. Specifically, by
dropping the subcarrier index $k$, to ease the notation, we assume that MS
$\ell$ first calculates the noise whitening filter
$\bm{Q}_{\ell}=\bm{\Phi}^{-{1}/{2}}_{\ell}$ so that
$\bm{Q}_{\ell}\bm{\Phi}_{\ell}\bm{Q}_{\ell}=\bm{I}_{M_{\mathrm{MS}}}$. Let
$\widetilde{\bm{H}}_{\ell,\ell^{\prime}}=\bm{Q}_{\ell}\bar{\bm{H}}_{\ell,\ell^{\prime}}$
be the whitened effective channel between MS $\ell$ and MS $\ell^{\prime}$,
with $\ell\neq\ell^{\prime}$. Then, given the SVD of
$\widetilde{\bm{H}}_{\ell,\ell^{\prime}}=\widetilde{\bm{U}}_{\ell,\ell^{\prime}}\text{diag}\\{\widetilde{\bm{\lambda}}_{\ell,\ell^{\prime}}\\}\widetilde{\bm{V}}^{H}_{\ell,\ell^{\prime}}$,
the precoding matrix of MS $\ell^{\prime}$ is given as
$\bm{F}_{\ell^{\prime}}=[\widetilde{\bm{V}}_{\ell,\ell^{\prime}}]_{[1:N_{s}]}$,
while the decoding matrix of the MS $\ell$ is given as
${\bm{W}_{\ell}}=[\widetilde{\bm{U}}_{\ell,\ell^{\prime}}]_{[1:N_{s}]}$.
Accordingly, (6) can be simplified as

$\displaystyle\text{SE}_{\ell}=\textstyle\sum_{i=1}^{N_{s}}\log_{2}(1+([\widetilde{\bm{\lambda}}_{\ell,\ell^{\prime}}]_{[i]})^{2}\cdot
p_{\ell,\ell^{\prime},i}),$ (8)

where ${p_{\ell,\ell^{\prime},i}}$ is the power of the $i$th data stream given
by
${p_{\ell,\ell^{\prime},i}}=\max\big{\\{}{1}/{\mu_{\ell,\ell^{\prime}}}-{1}/{[\widetilde{\bm{\lambda}}_{\ell,\ell^{\prime}}]_{[i]}},0\big{\\}}$,
where $\mu_{\ell,\ell^{\prime}}$ is the real-valued water-level found using,
e.g., the bisection method so that
$\sum_{i=1}^{N_{s}}{p_{\ell,\ell^{\prime},i}}=P_{\mathrm{MS}}$. From the
above, the problem that we address in this paper is the design of
$\bm{G}_{k},\forall k$.

##  3 Fully-digital-based design

In this section, we drop the subcarrier index $k$, since the proposed fully-
digital design is decoupled between the $K$ subcarriers. From (8), it is clear
that the RS amplification matrix $\bm{G}$, of any subcarrier, should be
designed so that the singular values of the whitened effective channels of
both MSs, i.e., $\widetilde{\bm{\lambda}}_{1,2}$ and
$\widetilde{\bm{\lambda}}_{2,1}$, are maximized. To achieve this goal, the
authors in [8] proposed a heuristic closed-form approach, termed ANOMAX, to
design $\bm{G}$ as

$\displaystyle\bm{G}=$
$\displaystyle\max_{\bm{G}}\Big{(}\big{\|}\bar{\bm{H}}_{1,2}\big{\|}^{2}_{F}+\big{\|}\bar{\bm{H}}_{2,1}\big{\|}^{2}_{F}\Big{)}.$
(9)

After some algebraic manipulations, (9) is rewritten as [8]

$\displaystyle\bm{g}=$
$\displaystyle\max_{\bm{g}}\|\bm{K}\bm{g}\|^{2}_{2},\quad\text{ s.t.
}\quad\|\bm{g}\|_{2}=1,$ (10)

where
$\bm{K}=[(\bm{H}_{2}\otimes\bm{H}_{1}),(\bm{H}_{1}\otimes\bm{H}_{2})]^{T}$ and
$\bm{g}=\text{vec}\\{\bm{G}\\}$. Using the SVD of
$\bm{K}=\bm{U}_{\bm{K}}\text{diag}(\bm{\lambda}_{\bm{K}})\bm{V}^{H}_{\bm{K}}$,
the optimal solution of (10) is given as
$\bm{g}_{\text{ANOMAX}}=[\bm{V}_{\bm{K}}]_{[:,1]}$. However, it was observed
in [8] that the unfolded matrix, i.e.,
$\bm{G}_{\text{ANOMAX}}=\text{unvec}\\{\bm{g}_{\text{ANOMAX}}\\}$ often
exhibits a low-rank structure. To make it applicable for multi-stream
communications, the authors extended their framework in [9] by proposing RR-
ANOMAX. In RR-ANOMAX, the rank of $\bm{G}_{\text{ANOMAX}}$ is increased by
adjusting its singular values while preserving its subspaces. In the
following, we build on those results and propose a method that further
enhances the rank of $\bm{G}_{\text{ANOMAX}}$, termed hereafter enhanced RR-
ANOMAX (ERR-ANOMAX), by exploiting the structure of problem (10).

In contrast to ANOMAX and RR-ANOMAX, in ERR-ANOMAX, the $\bm{g}$ vector is
constructed by taking the contributions of the first $R$ vectors in
$\bm{V}_{\bm{K}}$ as

$\displaystyle\bm{g}_{\text{E-ANOMAX}}=[\bm{V}_{\bm{K}}]_{[:,1]}+\dots+[\bm{V}_{\bm{K}}]_{[:,R]},$
(11)

where $R$ is a design parameter that we investigate numerically in Section 5.
Then, for the unfolded matrix
$\bm{G}_{\text{E-ANOMAX}}=\text{unvec}\\{\bm{g}_{\text{E-ANOMAX}}\\}$, the SVD
is given as
$\bm{G}_{\text{E-ANOMAX}}=\bm{U}_{\bm{G}}\text{diag}(\bm{\lambda}_{\bm{G}})\bm{V}^{H}_{\bm{G}}$.
This implies that
$\bm{g}_{\text{E-ANOMAX}}=(\bm{V}^{*}_{\bm{G}}\diamond\bm{U}_{\bm{G}})\bm{\lambda}_{\bm{G}}$.
Here, we propose to redesign $\bm{\lambda}_{\bm{G}}$ while keeping both
$\bm{U}_{\bm{G}}$ and $\bm{V}_{\bm{G}}$ fixed, to enhance the rank of
$\bm{G}_{\text{E-ANOMAX}}$. Specifically, by substituting
$\bm{g}_{\text{E-ANOMAX}}$ into (10), we have

$\displaystyle\bm{\lambda}^{\star}_{\bm{G}}=$
$\displaystyle\max_{\bm{\lambda}_{\bm{G}}}\|\tilde{\bm{K}}\bm{\lambda}_{\bm{G}}\|^{2}_{2}\quad\text{
s.t. }\quad\|\bm{\lambda}_{\bm{G}}\|_{2}=1,$ (12)

where $\tilde{\bm{K}}=\bm{K}(\bm{V}^{*}_{\bm{G}}\diamond\bm{U}_{\bm{G}})$. In
contrast to (10), the optimization variable of (12) is a real-valued vector.
Therefore, we propose to design $\bm{\lambda}_{\bm{G}}$ using a water-filling
like method [17]. Specifically, given the SVD of
$\tilde{\bm{K}}=\bm{U}_{\tilde{\bm{K}}}\text{diag}(\bm{\lambda}_{\tilde{\bm{K}}})\bm{V}^{H}_{\tilde{\bm{K}}}$,
the $j$th entry of $\bm{\lambda}_{\bm{G}}$ is updated as
$[\bm{\lambda}^{\star}_{\bm{G}}]_{[j]}=\max\big{\\{}{1}/{\mu}-{1}/{[\bm{\lambda}_{\tilde{\bm{K}}}]_{[j]}},0\big{\\}}$,
where $\mu$ is the real-valued water-level found using, e.g., the bisection
method so that $\sum_{j=1}[\bm{\lambda}^{\star}_{\bm{G}}]_{[j]}=1$. Finally,
we calculate $\bm{G}_{\text{ERR-
ANOMAX}}=\bm{U}_{\bm{G}}\text{diag}(\bm{\lambda}^{\star}_{\bm{G}})\bm{V}^{H}_{\bm{G}}$,
which is then normalized so that $\|\bm{G}_{\text{ERR-
ANOMAX}}\bm{x}\|^{2}_{2}=P_{{\mathrm{RS}}}$.

##  4 HAD-based design

As mentioned in [18], the fully-digital system is impractical in mm-wave
massive MIMO systems due to the large number of RF chains. Therefore, we
consider a HAD AF structure with $N_{\mathrm{RS}}<M_{\mathrm{RS}}$ RF chains,
so that the amplification matrix $\bm{G}_{k}$ has a structure given as
$\bm{G}_{k}\overset{\textrm{def}}{=}\bm{A}_{\mathrm{T}}\bm{B}_{k}\bm{A}^{T}_{\mathrm{R}}$,
where $\bm{B}_{k}$ is the $N_{\mathrm{RS}}\times N_{\mathrm{RS}}$ baseband
beamforming matrix for the $k$th subcarrier, while $\bm{A}_{\mathrm{T}}$ and
$\bm{A}_{\mathrm{R}}$ are the common $M_{\mathrm{RS}}\times N_{\mathrm{RS}}$
transmit and received analog beamforming matrices, respectively, implemented
using phase-shifting networks. Therefore, we have that
$|[\bm{A}_{\mathrm{T}}]_{[i,j]}|=1,\forall i,j,$ and
$|[\bm{A}_{\mathrm{R}}]_{[i,j]}|=1,\forall i,j$. Considering the HAD
amplification matrix given above, the maximization of the total SE of the
system in (7) is more challenging due to the constant modulus constraints of
the analog beamforming matrices. Therefore, we propose a heuristic solution to
(7), similarly to [19], as

$\displaystyle{\min_{\bm{A}_{\mathrm{T}},\bm{A}_{\mathrm{R}},\bm{B}_{k},\forall
k}}$
$\displaystyle\textstyle\sum_{k=0}^{K-1}\Big{\|}\bm{G}_{\mathrm{FD},k}-\bm{A}_{\mathrm{T}}\bm{B}_{k}\bm{A}^{T}_{\mathrm{R}}\Big{\|}_{F}^{2}$
(13) $\displaystyle|[\bm{A}_{\mathrm{R}}]_{[i,j]}|=1,\forall
i,j,|[\bm{A}_{\mathrm{T}}]_{[i,j]}|=1,\forall i,j,$

where $\bm{G}_{\mathrm{FD},k}$ is a given unconstrained fully-digital RS
amplification matrix on the $k$th subcarrier. For the known
$\bm{G}_{\mathrm{FD},k},\forall k$, we form a 3-way tensor
$\bm{\mathcal{G}}_{\mathrm{FD}}$ by concatenating
$\bm{G}_{\mathrm{FD},k},\forall k$, along the third mode, as

$\bm{\mathcal{G}}_{\mathrm{FD}}=[\bm{G}_{\mathrm{FD},0}\sqcup_{3}\dots\sqcup_{3}\bm{G}_{\mathrm{FD},K-1}]\in\mathbb{C}^{M_{\mathrm{RS}}\times
M_{\mathrm{RS}}\times K}.$ (14)

Here, we note that the tensor $\bm{\mathcal{G}}_{\mathrm{FD}}$ in (14) admits
a Tucker2 decomposition [14, 15, 16] given as

$\displaystyle\bm{\mathcal{G}}_{\mathrm{FD}}=\bm{\mathcal{B}}\times_{1}\bm{A}_{\mathrm{T}}\times_{2}\bm{A}_{\mathrm{R}}\times_{3}\bm{I}_{K},$
(15)

where $\bm{\mathcal{B}}\in\mathbb{C}^{N_{\mathrm{RS}}\times
N_{\mathrm{RS}}\times K}$ is the core tensor formed as
$\bm{\mathcal{B}}=[\bm{B}_{0}\sqcup_{3}\dots\sqcup_{3}\bm{B}_{K-1}]$.
Utilizing (15), (13) can be rewritten as a constrained Tucker2 decomposition
as

$\displaystyle{\min_{\bm{A}_{\mathrm{T}},\bm{A}_{\mathrm{R}},\bm{\mathcal{B}}}}$
$\displaystyle\quad\|\bm{\mathcal{G}}_{\mathrm{FD}}-\bm{\mathcal{B}}\times_{1}\bm{A}_{\mathrm{T}}\times_{2}\bm{A}_{\mathrm{R}}\times_{3}\bm{I}_{K}\|_{\mathrm{F}}^{2}$
(16) $\displaystyle|[\bm{A}_{\mathrm{R}}]_{[i,j]}|=1,\forall i,j,\text{ and
}|[\bm{A}_{\mathrm{T}}]_{[i,j]}|=1,\forall i,j.$

HOSVD-based solution: A direct solution to (16) can be obtained from the HOSVD
[15] of the tensor $\bm{\mathcal{G}}_{\mathrm{FD}}$. Specifically, let the
HOSVD of $\bm{\mathcal{G}}_{\mathrm{FD}}$ be written as

$\bm{\mathcal{G}}_{\mathrm{FD}}=\bm{\mathcal{S}}\times_{1}\bm{U}_{1}\times_{2}\bm{U}_{2}\times_{3}\bm{U}_{3},$
(17)

where $\bm{U}_{1}$, $\bm{U}_{2}$, and $\bm{U}_{3}$ are the unitary factor
matrices, while $\bm{\mathcal{S}}$ is the core tensor. Then, by utilizing the
fact that $\bm{A}_{\mathrm{T}}$ (resp. $\bm{A}_{\mathrm{R}}$) spans the same
subspace as $\bm{U}_{1}$ (resp. $\bm{U}_{2}$), the analog matrices and the
baseband matrix of the $k$th subcarrier can be obtained as
${\mathring{\bm{A}}_{\mathrm{T}}}=\bm{\Pi}([\bm{U}_{1}]_{[1:N_{\mathrm{RS}}]})$,
${\mathring{\bm{A}}_{\mathrm{R}}}=\bm{\Pi}([\bm{U}_{2}]_{[1:N_{\mathrm{RS}}]})$,
and
$\mathring{\bm{B}}_{k}=[\bm{\mathcal{S}}]_{[1:N_{\mathrm{RS}},1:N_{\mathrm{RS}},k]}$,
where $\bm{\Pi}(\cdot)$ is an element-wise projection function defined as
$\bm{\Pi}(u)=u/|u|$. Therefore, the $k$th HAD RS amplification matrix is given
as
$\bm{G}_{\textrm{HOSVD},k}={\mathring{\bm{A}}_{\mathrm{T}}}{\mathring{\bm{B}}_{k}}{\mathring{\bm{A}}^{T}_{\mathrm{R}}}$,
which is then normalized so that
$\|\bm{G}_{\textrm{HOSVD},k}\bm{x}_{k}\|^{2}_{2}=P_{{\mathrm{RS}}}$.

AltMax-based solution: Another solution to (16) can be obtained using our
recently proposed AltMax approach in [5]. Specifically, let
$[\bm{\mathcal{G}}_{\mathrm{FD}}]_{(1)}=\bm{A}_{\mathrm{T}}[\bm{\mathcal{B}}]_{(1)}(\bm{I}_{K}\otimes\bm{A}_{\mathrm{R}})^{T}$
denote the one-mode unfolding of $\bm{\mathcal{G}}_{\mathrm{FD}}$, in which
$[\bm{\mathcal{B}}]_{(1)}$ is formed according to the forward cyclical
ordering [20]. Then, a solution to $\bm{A}_{\mathrm{T}}$ can be obtained as

$\displaystyle\max_{\bm{A}_{\mathrm{T}}}$
$\displaystyle\quad\|\bm{A}^{H}_{\mathrm{T}}[\bm{\mathcal{G}}_{\mathrm{FD}}]_{(1)}\|^{2}_{F}=\max_{\bm{A}_{\mathrm{T}}}\text{trace}(\bm{A}^{H}_{\mathrm{T}}\bm{G}_{(1)}\bm{A}_{\mathrm{T}})$
(18) s.t. $\displaystyle\quad|[\bm{A}_{\mathrm{T}}]_{[i,j]}|=1,\forall i,j,$

where
$\bm{G}_{(1)}=[\bm{\mathcal{G}}_{\mathrm{FD}}]_{(1)}[\bm{\mathcal{G}}_{\mathrm{FD}}]^{H}_{(1)}$.
Our proposed AltMax technique in [5] can be used to solve (18), which updates
$\bm{A}_{\mathrm{T}}$ column-wise until a convergence is reached. Similarly to
(18), the $\bm{A}_{\mathrm{R}}$ matrix can be calculated utilizing the two-
mode unfolding
$[\bm{\mathcal{G}}_{\mathrm{FD}}]_{(2)}=\bm{A}_{\mathrm{R}}[\bm{\mathcal{B}}]_{(2)}(\bm{I}_{K}\otimes\bm{A}_{\mathrm{T}})^{T}$.

From the obtained $\dot{\bm{A}}_{\mathrm{T}}$ and $\dot{\bm{A}}_{\mathrm{R}}$,
the core tensor can be calculated as
$[\dot{\bm{\mathcal{B}}}]_{(3)}=[\bm{\mathcal{G}}_{\mathrm{FD}}]_{(3)}[(\dot{\bm{A}}_{\mathrm{R}}\otimes\dot{\bm{A}}_{\mathrm{T}})^{T}]^{+}$,
where
$[\bm{\mathcal{G}}_{\mathrm{FD}}]_{(3)}=[\bm{\mathcal{B}}]_{(3)}(\bm{A}_{\mathrm{R}}\otimes\bm{A}_{\mathrm{T}})^{T}$.
Thus, the $k$th HAD RS amplification matrix is given as
$\bm{G}_{\textrm{AltMax},k}={\dot{\bm{A}}_{\mathrm{T}}}{\dot{\bm{B}}_{k}}{\dot{\bm{A}}^{T}_{\mathrm{R}}}$,
where ${\dot{\bm{B}}_{k}}$ is the $k$th 3-mode slice of
$\dot{\bm{\mathcal{B}}}$. Finally, we normalize $\bm{G}_{\textrm{AltMax},k}$
so that $\|\bm{G}_{\textrm{AltMax},k}\bm{x}_{k}\|^{2}_{2}=P_{{\mathrm{RS}}}$.

(a) SE vs. SNR [$R=2$]

(b) SE vs. $R$ [SNR = 25dB]

Fig. 2: Example 1: The fully-digital case [$K=1$]

(a) SE vs. SNR [$N_{\mathrm{RS}}=8,K=32$]

(b) SE vs. SNR

Fig. 3: Example 2: The HAD case [$R=2,N_{s}=4$]

##  5 Simulation Results

Similarly to [5], we assume that each channel matrix is generated using the
classical Saleh-Valenzuela model [21], where we fix the number of channel
paths $L=6$. We define the signal-to-noise ratio as
$\text{SNR}=\frac{1}{\sigma^{2}}$, where we assume that
$P_{\mathrm{RS}}=P_{\mathrm{MS}}=1$ and
$\sigma^{2}_{\mathrm{RS}}=\sigma^{2}_{\mathrm{MS}}=\sigma^{2}$. In all the
simulation scenarios, we use $M_{\mathrm{RS}}=64$ and $M_{1}=M_{2}=4$.

Example 1: The fully-digital case: Here, we compare the performance of ANOMAX
[8], RR-ANOMAX [9], and the proposed ERR-ANOMAX assuming that $K=1$ and the RS
is equipped with a fully-digital beamforming structure.

From Fig. 2(a), we can see that when $N_{s}=1$, the three methods achieve the
same performance. However, the rank restoration methods, i.e., RR-ANOMAX and
ERR-ANOMAX have a significant performance gain in the $N_{s}>1$ scenarios when
compared to ANOMAX. For $N_{s}>1$, ERR-ANOMAX outperforms the RR-ANOMAX of
[9], specially for $N_{s}=4$. Here, in Fig. 2(a), we have assumed that $R=2$
for ERR-ANOMAX. Using computer simulations, as reported in Fig. 2(b), we have
observed that setting $R=2$ provides the best performance.

Example 2: The HAD case: Here, we compare RR-ANOMAX and the proposed ERR-
ANOMAX assuming that the RS is equipped with a HAD beamforming structure. Fig.
3 shows the SE versus the SNR.

From Fig. 3(a), as expected, we can observe that ERR-ANOMAX maintains its
advantages over RR-ANOMAX in the HAD beamforming scenarios as well, where the
AltMax method is shown to outperform the HOSVD method, agreeing with our
results reported in [5]. However, from Fig. 3(b), we can see that when
$N_{\mathrm{RS}}$ reduces from $8$ to $4$, the performance of the HAD methods
reduces significantly, where both HAD methods seem to have an equal
performance. This implies that, for the HAD architectures, the number of RF
chains $N_{\mathrm{RS}}$ at the RS should be equal or larger than the total
number of data streams, i.e., $N_{\mathrm{RS}}\geq 2N_{s}$ to maintain a close
performance to that achieved by the fully-digital counterpart. Finally, Fig.
3(b) also shows that when the number of system subcarriers $K$ increases from
$32$ to $64$, the performance of the HAD methods slightly decreases, which is
more evident with the HOSVD method and less with the AltMax method.

##  6 Conclusions

In this work, we have enhanced and extended the RR-ANOMAX scheme, originally
proposed in [9] for single-carrier FD AF MIMO TWR systems, to multi-carrier
HAD AF MIMO-OFDM TWR systems. Specifically, we have proposed the ERR-ANOMAX
scheme that outperforms RR-ANOMAX in multi-stream communications. Moreover, we
have shown that the HAD amplification matrix design can be formulated as a
constrained Tucker2 decomposition, for which two solutions are proposed, an
HOSVD-based solution and an AltMax-based solution. Simulation results show
that ERR-ANOMAX AltMax outperforms the other methods.

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