# Near-Field Intelligent Reflecting Surfaces for Millimeter Wave MIMO Full
Duplex

Chandan Kumar Sheemar, Stefano Tomasin, Dirk Slock, and Symeon Chatzinotas An
initial version of this work is part of the PhD thesis [1].Chandan Kumar
Sheemar and Symeon Chatzinotas are with the SnT department at the University
of Luxembourg (emails:chandankumar.sheemar@uni.lu,,symeon.chatzinotas@uni.lu);
Stefano Tomasin is with the DEI department at the University of Padova, Italy
(email:stefano.tomasin@unipd.it); Dirk Slock is with the Communication Systems
department at EURECOM, France<EMAIL_ADDRESS>
###### Abstract

Full duplex (FD) systems suffer from very high hardware cost and high power
consumption to mitigate the self-interference (SI) in the analog domain.
Moreover, in millimeter wave (mmWave) they rely on hybrid beamforming (HYBF)
as a signal processing tool to partially deal with the SI, which presents many
drawback, e.g., high insertion loss, high power consumption and high
computational complexity for its configuration. This article proposes the use
of near-field (NF-) IRSs for FD systems with the objective to solve the
aforementioned issues cost-efficiently. Namely, we propose to truncate the
analog stage of the mmWave FD systems and assist them with an NF-IRS, to
simultaneously and smartly control the uplink (DL) and downlink (DL) channels,
while assisting in shaping the SI channel: this to obtain very strong passive
SI cancellation. A novel joint active and passive beamforming design for the
weighted sum-rate (WSR) maximization of a NF-IRS-assisted mmWave point-to-
point FD system is presented. Results show that the proposed solution fully
reaps the benefits of the IRSs only when they operate in the NF, which leads
to considerably higher gains compared to the conventional massive MIMO (mMIMO)
mmWave FD and half duplex (HD) systems.

###### Index Terms:

Full duplex, Near-Field Intelligent Reflecting Surfaces, Passive Self-
Interference Cancellation, Millimeter Wave, Hybrid Beamforming

##  I Introduction

To achieve ubiquitous connectivity for continuously evolving wireless
networks, intelligent reflecting surfaces (IRSs) are considered as one of the
most prominent hardware technology for beyond fifth generation (B5G) and six
generation (6G) networks [1, 2], with the potential to create smart,
reconfigurable, and highly energy-efficient wireless systems. An IRS is a
meta-surface made of passive elements, which can be flexibly programmed to
reflect the impinging signals in the desired way [3].

On one hand, both its low hardware cost and the passive nature of its meta
elements enable a sustainable evolution of wireless networks [4]. On the other
hand, given the limited wireless spectrum, there has been a growing interest
in millimeter wave (mmWave) full duplex (FD) technology, which offers
simultaneous transmission and reception in the same frequency band, thus
doubling the spectral efficiency with respect to half duplex (HD) systems [5].
However, FD systems suffer from self-interference (SI), which can be from 90
to $110\leavevmode\nobreak\ $dB higher than the received signal of interest
[6]. SI cancellation (SIC) techniques are a viable tool to mitigate this
impairment [7], and they can operate both in the analog and digital domains
[8, 9, 10]. Analog SIC techniques entail high hardware cost and power
consumption to sufficiently mitigate SI and avoid the analog-to-digital
converter (ADC) saturation [11, 12]. Hybrid beamforming (HYBF) is also
considered as a promising signal processing tool to reduce SI in mmWave FD
systems [13, 14, 15, 16, 17, 18]. Although enabling the design of mmWave FD
systems with fewer radio frequency (RF) chains, the main pitfalls of HYBF
systems include high power consumption, high insertion loss [19, 20], and high
computational complexity for the phasors’ configuration, which further
enhances power consumption [21]. Such inherent characteristics of HYBF
motivates the exploration of other alternatives.

###  I-A State-of-the-Art

The mmWave FD systems aided by IRSs can potentially revolutionize the wireless
communications and pave the path towards highly spectrally efficient, energy
efficient, smart, reconfigurable, and sustainable networks [22]. Recent
studies on FD systems assisted with the IRSs, limited only to
sub-$6\leavevmode\nobreak\ $GHz bands, have been presented in [23, 24, 25, 26,
27, 28, 29, 30]. In [23], a novel beamforming design is proposed for decode-
and-forward FD relays assisted by one IRS to maximize the minimum achievable
rate. In [24], the authors proposed a joint active and passive beamforming
design to cover dead zones in a multi-user multiple-input single-output (MISO)
FD network, while suppressing the single-antenna FD user-side self and co-
channel interference. In [25], the performance of an IRS-aided FD system in
terms of outage and error probabilities and the impact of IRS in mitigating SI
on single-antenna FD system is investigated. In[26], the authors presented a
novel beamforming design for a MISO FD system with two users (one in uplink
(UL) and one in downlink (DL)) to minimize the power consumption of the FD
access point while the UL user is subject to the minimum rate constraint. In
[27], a novel beamforming design is presented, based on the Arimoto-Blahut
algorithm for IRS-assisted point-to-point FD systems, to maximize the sum-
rate. The importance of IRSs in improving the security of FD systems under
imperfect channel state information (CSI) is analyzed in [28], where a novel
beamforming design is proposed to maximize the worst-case achievable secrecy
rate under a bounded CSI error model. A mixed time-scale joint active and
passive beamforming design for an IRS-assisted multi-user FD system is
presented in [29], and by resorting to a deep neural network the computational
burden of the proposed solution is further reduced. In [30], the capability of
IRSs in lowering the SI for a single-antenna FD system is shown on a
experimental testbed. In [31], the authors evaluated the potential of an IRS
in canceling the SI by placing it close the FD node. However, such work is
limited to the sub-6 GHz band and only SI with IRS was investigated, without
considering any UL traffic. Moreover, this work is limited to a simple MISO
case.

###  I-B Motivation and Main Contributions

Although the aforementioned papers studied the IRS-aided FD systems in many
different scenarios, all results are limited to the sub-$6$ GHz band.
Moreover, the advantage of handling SI is analyzed mostly for single-antenna
FD users, with IRSs placed in the far field (FF), which can potentially limit
the impact of passive SIC, due to a smaller power gain than the SI channel. In
[31], the authors considered SI with the IRS in the near-field (NF). However,
this work is limited to a MISO case and does not consider UL traffic in their
design, which motivates further studies by considering a general case of
multiple-input multiple-output (MIMO) FD systems, and a more practical case
wherein the IRS is not used only for SIC purposes, but also to simultaneously
improve the UL and DL channel quality. Moreover, despite the drawbacks of the
mmWave massive MIMO (mMIMO) FD systems such as high hardware cost and power
consumption, a solution to overcome them has not yet been explored.

This work proposes to remove the analog stage of the mmWave FD system in both
transmission and reception by allowing each FD node to operate with few
antennas in a fully digital mode. To tackle the propagation challenges of
mmWave, we propose to assist each FD node with an NF-IRS. We remark that the
proposed idea herein is also prominent in THz FD systems. However, as the
investigation of such systems is still nascent, we restrict our work to the
mmWave band. The objective of the NF-IRSs is to simultaneously compensate for
the analog beamforming and combining gain of the mmWave FD nodes by providing
a smart control of the UL and DL channels. Moreover, since the IRSs operate in
the NF, each NF-IRS also jointly assists in shaping the MIMO SI channel,
leading to a very strong passive SIC scheme. Note that the MIMO FD nodes also
benefit from the active digital beamforming gain for SIC, which, combined with
passive SIC, leads to superior SI suppression. We consider the maximization of
the weighted sum-rate (WSR) criteria and propose a novel joint active and
passive beamforming design for the NF-IRSs-aided mmWave FD systems. Compared
to mMIMO mmWave FD, our method requires only a few active components, leading
to a significant reduction in power consumption, hardware cost, and
computational complexity. Moreover, our method also significantly reduces the
power and hardware costs of the analog SIC architecture, which strictly
depends on the number of antenna elements.

Simulation results show that the proposed method outperforms the traditional
mmWave mMIMO fully-digital HD system with only a few antenna elements.
Moreover, the NF-IRSs not only replace the analog stage but also provide
significantly larger gains than the mMIMO FD systems. Namely, a MIMO FD aided
with a NF-IRS of size $30\times 30$ can achieve $\sim 4$ times higher gain
than the mMIMO HD system while operating with $5$ times less antennas;
meanwhile, the classical mMIMO FD system could theoretically only achieve $2$
times higher gain. Finally, results also show that the MIMO FD systems can
fully reap the benefits of the IRSs only when they operate in the NF. IRSs
operating in the far field (FF) can provide only a small gain over the
conventional mMIMO FD systems.

_Paper Organization:_ The rest of the paper is organized as follows. First, we
introduce the system model of the considered system in Section II. The novel
joint active and passive beamforming design is then derived in Section III.
Finally, Sections IV and V present the numerical results and conclusions,
respectively.

_Mathematical Notations:_ Boldface lower and upper case characters denote
vectors and matrices, respectively. $\mathbb{E}\\{\cdot\\}$,
$\mbox{Tr}\\{\cdot\\}$, $\mathbf{I}$ and $\odot$ denote expectation, trace,
identity matrix, and Hadamard product, respectively. The superscripts
$(\cdot)^{T}$ and $(\cdot)^{H}$ denote transpose and conjugate-transpose
(Hermitian) operators, respectively.

Figure 1: NF-IRSs assisted mmWave point-to-point FD system. Tx and Rx denote
transmitter and receiver, respectively.

##  II System Model

We consider a mmWave point-to-point FD communication system consisting of two
mmWave MIMO FD nodes, indicated with indices $l$ and $r$. They are assisted
with one NF-IRS each, denoted with the indices $i_{l}$, and $i_{r}$,
respectively. Both the IRSs are assumed to be placed in such a way that the
link between the SI channel and the other node is in line-of-sight (LoS). Each
IRS operates in the NF for the assisted FD node and in the FF for the other
node, as shown in Fig. 1. The MIMO FD node $l$ is assumed to be equipped with
$M_{l}$ transmit and $N_{l}$ receive antennas, while node $r$ is assumed to be
equipped with $M_{r}$ transmit and $N_{r}$ receive antennas. Let
$\mathbf{V}_{l}\in\mathbb{C}^{M_{l}\times d_{l}}$ and
$\mathbf{V}_{r}\in\mathbb{C}^{M_{r}\times d_{r}}$ denote the digital
beamformers of the two nodes, for the unitary-variance data streams
$\mathbf{s}_{l}\in\mathbb{C}^{d_{l}\times 1}$ and
$\mathbf{s}_{r}\in\mathbb{C}^{d_{r}\times 1}$, respectively. The IRS $i_{l}$
has $L_{r}\times L_{c}$ elements, and IRS $i_{r}$ has $R_{r}\times R_{c}$,
elements. Let $\bm{\phi}_{l}$ and $\bm{\phi}_{r}$ denote vectors collecting
the phase shift response of the IRSs $i_{l}$ and $i_{r}$, respectively: the
elements of $\bm{\phi}_{l}$ and $\bm{\phi}_{r}$ at position $i$ and $j$ are
phasors of the form $\bm{\phi}_{l}(i)=e^{i\theta_{i}^{l}}$ and
$\bm{\phi}_{r}(j)=e^{i\theta_{j}^{r}}$, respectively. Let
$\mathbf{\Phi}_{l}=\mbox{diag}(\bm{\phi}_{l})\in\mathbb{C}^{L_{r}L_{c}\times
L_{r}L_{c}}$ and
$\mathbf{\Phi}_{r}=\mbox{diag}(\bm{\phi}_{r})\in\mathbb{C}^{R_{r}R_{c}\times
R_{r}R_{c}}$ denote the diagonal matrices containing the phase responses of
the IRSs $i_{l}$ and $i_{r}$, respectively.

Let $\mathbf{H}_{i_{l},l}\in\mathbb{C}^{L_{r}L_{c}\times M_{l}}$ and
$\mathbf{H}_{i_{r},l}\in\mathbb{C}^{R_{r}R_{c}\times M_{l}}$ denote the direct
channels from the FD node $l$ to its NF IRS $i_{l}$ and to the other IRS
$i_{r}$, operating in FF, respectively. A similar notation holds for node
$i_{r}$. The channel matrices from IRSs $i_{l}$ and $i_{r}$ to the receive
antenna array of FD node $l$ are
$\mathbf{H}_{l,i_{l}}\in\mathbb{C}^{N_{l}\times L_{r}L_{c}}$ and
$\mathbf{H}_{l,i_{r}}\in\mathbb{C}^{N_{l}\times R_{r}R_{c}}$, respectively.
The channel matrices from the IRSs $i_{l}$ and $i_{r}$ to the receive antenna
array of node $r$ are $\mathbf{H}_{r,i_{l}}\in\mathbb{C}^{N_{r}\times
L_{r}L_{c}}$ and $\mathbf{H}_{r,i_{r}}\in\mathbb{C}^{N_{r}\times R_{r}R_{c}}$,
respectively. Let $\mathbf{H}_{i_{r},i_{l}}\in\mathbb{C}^{R_{r}R_{l}\times
L_{r}L_{c}}$ denote the channel from IRS $i_{l}$ to the IRS $i_{r}$, while the
channel from IRS $i_{r}$ to IRS $i_{l}$ is denoted with
$\mathbf{H}_{i_{r},i_{l}}\in\mathbb{C}^{R_{r}R_{c}\times L_{r}L_{c}}$. Due to
reciprocity, the channel from IRS $r$ to IRS $l$ is given by
$\mathbf{H}_{i_{l},i_{r}}=\mathbf{H}_{i_{r},i_{l}}^{T}$. The direct channels
from node $l$ to node $r$ and from node $r$ to node $l$ are denoted with
$\mathbf{H}_{r,l}\in\mathbb{C}^{N_{r}\in M_{l}}$ and
$\mathbf{H}_{l,r}\in\mathbb{C}^{N_{l}\in M_{r}}$, respectively. The SI channel
for the MIMO nodes $l$ and $r$ are denoted with
$\mathbf{H}_{l,l}\in\mathbb{C}^{N_{l}\in M_{l}}$ and
$\mathbf{H}_{r,r}\in\mathbb{C}^{N_{r}\in M_{r}}$, respectively.

###  II-A Channel Modeling

Let $\mathbf{a}_{l}(\phi_{l}^{n_{p},n_{c}})$ and
$\mathbf{a}_{r}^{T}(\theta_{r}^{n_{p},n_{c}})$ denote the receive and transmit
antenna array response at the MIMO node $l$ and $r$, respectively, with angle
of arrival (AoA) $\phi_{l}^{n_{p},n_{c}}$ and angle of departure (AoD)
$\theta_{r}^{n_{p},n_{c}}$. The direct channel $\mathbf{H}_{l,r}$ from node
$r$ to node $l$ in mmWave can be modeled as [14]

$\mathbf{H}_{l,r}=\sqrt{\beta_{l,r}}\sqrt{\frac{M_{r}N_{l}}{N_{c}N_{p}}}\sum_{n_{c}=1}^{N_{c}}\sum_{n_{p}=1}^{N_{p}}\alpha_{l,r}^{(n_{p},n_{c})}\mathbf{a}_{l}(\phi_{l}^{n_{p},n_{c}})\mathbf{a}_{r}^{T}(\theta_{r}^{n_{p},n_{c}}),$
(1)

where $N_{c}$ and $N_{p}$ are the number of clusters and rays [14, Fig. 1],
respectively, and $\alpha_{l,r}^{(n_{p},n_{c})}\sim\mathcal{CN}(0,1)$ is a
complex Gaussian random variable. Thus, the channels have amplitude and phase
distributed according to the Rayleigh and uniform distribution, respectively.
Note that, compared to [14], (1) is scaled by $\beta_{l,r}=D_{l,r}/D_{l,r}$,
which captures its dependency on the distance, with $D_{l,r}$ being a distance
between the centers of the MIMO FD nodes $l$ and $r$: this is equal to one for
the direct link (1) and will be larger than 1 for all the channels closer than
$D_{l,r}$, emphasizing the stronger channel gain. Namely, for any channel
matrix $\mathbf{H}_{m,n}$, its distance-dependent scale factor is
$\beta_{m,n}=D_{l,r}/D_{m,n}$, with $D_{m,n}\leq D_{l,r},\forall m,n.$ The
remaining FF channels $\mathbf{H}_{r,l}$, $\mathbf{H}_{i_{r},l}$,
$\mathbf{H}_{l,i_{r}}$, $\mathbf{H}_{i_{r},l}$, $\mathbf{H}_{i_{l},r}$,
$\mathbf{H}_{i_{l},i_{r}}$, and $\mathbf{H}_{i_{r},i_{l}}$ can be modeled
similarly to (1). The SI channel for node $l$ can be modeled as [14]

$\mathbf{H}_{l,l}=\sqrt{\beta_{l,l}}\Big{[}\sqrt{\frac{\kappa_{l}}{\kappa_{l}+1}}\mathbf{H}_{l}^{LoS}+\sqrt{\frac{1}{\kappa_{l}+1}}\mathbf{H}_{l}^{ref}\Big{]},$
(2)

where scalar $\kappa_{l}$ is the Rician factor, $\mathbf{H}_{l}^{LoS}$ and
$\mathbf{H}_{l}^{ref}$ denote the LoS and reflected contributions for the SI
channel, respectively, and scalar $\beta_{l,l}=D_{l,r}/D_{l,l}$ is the
distance-dependent average channel gain, with $D_{l,l}$ the distance between
the centers of the transmit and receive array of the MIMO FD node $l$. Channel
matrix $\mathbf{H}_{l}^{ref}$ can be modeled as in (1). Matrix
$\mathbf{H}_{l}^{LoS}$ considers the LoS path between the transmit and receive
antennas, which are in the NF and its modeling will be discussed after a brief
introduction to the NF concept.

Due to the NF-IRSs, channels $\mathbf{H}_{i_{l},l}$, $\mathbf{H}_{l,i_{l}}$,
$\mathbf{H}_{i_{r},r}$, and $\mathbf{H}_{r,i_{r}}$ cannot be modeled as in
(1). To shed light on the NF concept, let $D$ denote the size of the radiating
element, which could be either the transmit array of MIMO FD node
$i\in\mathcal{F}=\\{l,r\\}$, or the IRS operating in NF. The space surrounding
any transmitting device can be divided into three regions: $1)$ Reactive NF
(RE-NF), $2)$ Radiating NF (RA-NF), and $3)$ FF, as shown in Fig. 2. The
radius $R1$ and $R2$ are the boundaries between the regions and depend on both
the size $D$ and the wavelength $\lambda$. Regions RE-NF, RA-NF, and FF are
also known as inductive NF, Fresnel zone, and Fraunhofer zone, respectively.
In the RE-NF zone, the reactive fields dominate the radiation fields, and
therefore its not of interest for wireless communications. In the RA-NF zone,
the radiation fields dominate the reactive fields, however, the received wave
has a spherical wave-front. In the FF zone, the received wave-front is plane
and the receiver sees the transmitter as a point-wise source.

Figure 2: Field regions of the radiating element of size $D$.

In mmWave FD systems, to compensate for the analog stage and assist with SIC,
NF-IRSs should be placed in the RA-NF zone, i.e., at a distance $R_{1}\leq
z\leq R_{2}$. Therefore, in contrast to the FF case, we consider a spherical
wave-front for the channels $\mathbf{H}_{i_{l},l}$, $\mathbf{H}_{l,i_{l}}$,
$\mathbf{H}_{i_{r},r}$, and $\mathbf{H}_{r,i_{r}}$, which is also true for the
LoS components of the SI channels for FD nodes $l$ and $r$.

Since the LoS path between the transmit and receive antennas of the FD node
$l$ is in NF, the elements of channel matrix $\mathbf{H}_{l}^{LoS}$ can be
modelled as [14]

$\mathbf{H}_{l}^{LoS}(m,n)=\frac{\rho}{r_{m,n}}e^{-i2\pi\frac{d_{m,n}}{\lambda}},$
(3)

where $\lambda$ denotes the wavelength and the power normalization constant
$\rho$ assures that $\mathbb{E}(||\mathbf{H}_{l}^{LoS}||_{F}^{2})=M_{l}N_{l}$;
the scalar $d_{m,n}$ denotes the distance between $m$-th receive and $n$-th
transmit antenna of node $l\in\mathcal{F}$, which depends on the transmit and
receive array geometry (9) [14]. A similar modelling holds also for the LoS
component $\mathbf{H}_{r}^{LoS}$ of the SI channel $\mathbf{H}_{r,r}$.

In the following, we consider the modelling of the NF channels between the
MIMO FD nodes and their NF-IRSs. For such channels, due to very small path-
loss, the LoS components are dominating, and therefore we ignore the multi-
path between the FD nodes and NF-IRSs. The impinging wave on the NF-IRS
$i_{l}$ transmitted from node $l\in\mathcal{F}$ is spherical; therefore,
similar to (3), we consider a variation in the phase-shift due to distance
from each antenna element of FD node $l$ to each element of IRS $i_{l}$ as

$\mathbf{H}_{i_{l},l}(m,n)=\sqrt{\beta_{i_{l},l}}\Big{[}\frac{\rho_{i_{l},l}}{d_{m,n}}e^{-jkd_{m,n}}\Big{]},$
(4)

where $k=2\pi/\lambda$ is the wave number and $d_{n,m}$ denotes the distance
between the $n$-th transmit antenna and $m$-th element of the IRS, with $1\leq
n\leq M_{l}$, $1\leq m\leq L_{r}L_{c}$, $\rho_{i_{l},l}$ is a normalization
constant to assure
$\mathbb{E}[||\mathbf{H}_{i_{l},l}||_{F}^{2}]=L_{r}L_{c}M_{l}$, and
$\beta_{i_{l},l}=D_{l,r}/D_{i_{l},l}$ is the distance-dependent average
channel gain.

Similarly, for the NF channel from the IRS $i_{l}$ to the receive antenna
array of FD node $l$, we have

$\mathbf{H}_{l,i_{l}}(i,j)=\sqrt{\beta_{l,i_{l}}}\Big{[}\frac{\rho_{l,i_{l}}}{d_{i,j}}e^{-jkd_{i,j}}\Big{]},$
(5)

where $d_{i,j}$ denotes the distance from the $j$-th element of IRS $i_{l}$ to
the transmit antenna $i$ at the FD node $l$, with $1\leq j\leq L_{r}L_{c}$ and
$1\leq i\leq N_{l}$, $\rho_{l,i_{l}}$ is the power normalization constant
which assures $\mathbb{E}[||\mathbf{H}_{l,i_{l}}||_{F}^{2}]=L_{r}L_{c}N_{l}$,
and $\beta_{l,i_{l}}=D_{l,r}/D_{l,i_{l}}$ is the distance dependent average
channel gain. The elements of the channel matrices $\mathbf{H}_{i_{r},r}$ and
$\mathbf{H}_{r,i_{r}}$ can also be modeled similarly as (4) and (5),
respectively. For a concise summary of the aforementioned notations, see Table
I.

Table I: Notation

$\mathbf{V}_{l}$ |  Digital beamformer for FD node $l$
---|---
$\mathbf{V}_{r}$ |  Digital beamformer for FD node $r$
$\mathbf{\Phi}_{l}$ |  Diagonalized phase response of IRS $i_{l}$
$\mathbf{\Phi}_{r}$ |  Diagonalized phase response of IRS $i_{r}$
$\mathbf{H}_{i_{l},l}$ |  Channel matrix from FD node $l$ to IRS $i_{l}$
$\mathbf{H}_{i_{r},l}$ |  Channel matrix from FD node $l$ to IRS $i_{l}$
$\mathbf{H}_{i_{r},r}$ |  Channel matrix from FD node $r$ to IRS $i_{r}$
$\mathbf{H}_{i_{l},r}$ |  Channel matrix from FD node $r$ to IRS $i_{l}$
$\mathbf{H}_{l,i_{l}}$ |  Channel matrix from IRS $i_{l}$ to FD node $l$
$\mathbf{H}_{l,i_{r}}$ |  Channel matrix from IRS $i_{r}$ to FD node $l$
$\mathbf{H}_{r,i_{l}}$ |  Channel matrix from IRS $i_{l}$ to FD node $r$
$\mathbf{H}_{r,i_{r}}$ |  Channel matrix from IRS $i_{r}$ to FD node $r$
$\mathbf{H}_{i_{l},i_{r}}$ |  Channel matrix from IRS $i_{r}$ to IRS $i_{l}$
$\mathbf{H}_{i_{r},i_{l}}$ |  Channel matrix from IRS $i_{l}$ to IRS $i_{r}$
$\mathbf{H}_{r,l}$ |  Channel matrix from FD node $l$ to $r$
$\mathbf{H}_{l,r}$ |  Channel matrix from FD node $r$ to $l$
$\mathbf{H}_{l,l}$ |  SI channel matrix for FD node $l$
$\mathbf{H}_{r,r}$ |  SI channel matrix for FD node $r$

$\tilde{\mathbf{H}}_{l,r}=\mathbf{H}_{l,r}+\mathbf{H}_{l,i_{r}}\mathbf{\Phi}_{r}\mathbf{H}_{i_{r},r}+\mathbf{H}_{l,i_{l}}\mathbf{\Phi}_{l}\mathbf{H}_{i_{l},r}+\mathbf{H}_{l,i_{l}}\mathbf{\Phi}_{l}\mathbf{H}_{i_{l},i_{r}}\mathbf{\Phi}_{r}\mathbf{H}_{i_{r},r}$
(6a)
$\tilde{\mathbf{H}}_{r,l}=\mathbf{H}_{r,l}+\mathbf{H}_{r,i_{l}}\mathbf{\Phi}_{l}\mathbf{H}_{i_{l},l}+\mathbf{H}_{r,i_{r}}\mathbf{\Phi}_{r}\mathbf{H}_{i_{r},l}+\mathbf{H}_{r,i_{r}}\mathbf{\Phi}_{r}\mathbf{H}_{i_{r},i_{l}}\mathbf{\Phi}_{l}\mathbf{H}_{i_{l},l}$
(6b)
$\tilde{\mathbf{H}}_{l,l}=\mathbf{H}_{l,l}+\mathbf{H}_{l,i_{l}}\mathbf{\Phi}_{l}\mathbf{H}_{i_{l},l}+\mathbf{H}_{l,i_{r}}\mathbf{\Phi}_{r}\mathbf{H}_{i_{r},l}+\mathbf{H}_{l,i_{r}}\mathbf{\Phi}_{r}\mathbf{H}_{i_{r},i_{l}}\mathbf{\Phi}_{l}\mathbf{H}_{i_{l},l}$
(6c)
$\tilde{\mathbf{H}}_{r,r}=\mathbf{H}_{r,r}+\mathbf{H}_{r,i_{r}}\mathbf{\Phi}_{r}\mathbf{H}_{i_{r},r}+\mathbf{H}_{r,i_{l}}\mathbf{\Phi}_{l}\mathbf{H}_{i_{l},r}+\mathbf{H}_{r,i_{l}}\mathbf{\Phi}_{l}\mathbf{H}_{i_{l},i_{r}}\mathbf{\Phi}_{r}\mathbf{H}_{i_{r},r}.$
(6d)

In the mmWave band $30-300$ GHz, the wavelength spans the interval $1-10$mm.
For the mmWave FD systems, results have been reported with $\leq 32$ RF
chains, which allowed HYBF to achieve performance comparable to the fully
digital beamforming, [14, 32]. Assuming uniform linear arrays (ULAs) with
antenna elements separated by half-wavelength, for a typical antenna array
with 32 antenna elements, we have $D=16\lambda$. Typically, the IRS are made
of hundreds of meta elements, and therefore, the NF region of each MIMO FD
node is strictly dictated by the size of the NF-IRS. Assuming an NF-IRS
$i_{l}$ of size $50\lambda\times 50\lambda$, the boundary regions can be
identified as $R_{1}=0.62\sqrt{L_{max}^{3}/\lambda}$ and
$R_{2}=2{L_{max}}^{2}/\lambda$, where $L_{max}=max\\{L_{r},L_{c}\\}$, and in
mmWave band, typical values are reported in Table II.

Table II: Field region boundaries for an IRS of size D = $50\lambda$. $\lambda$ | $R_{1}$ | $R_{2}$
---|---|---
$1$ mm |  $0.22$ m |  $5$ m
$10$ mm |  $2.19$ m |  $50$ m

Note that, given the shorter distance between the FD nodes and IRSs, it is
possible to control to the IRS with the IRS controller connected through a
wired connection instead of a wireless connection, which can lead to enhanced
security and reliability of the control signaling.

###  II-B Problem Formulation

Let
$\tilde{\mathbf{H}}_{l,r},\tilde{\mathbf{H}}_{r,l},\tilde{\mathbf{H}}_{l,l}$
and $\tilde{\mathbf{H}}_{r,r}$ denote the effective channels
$\mathbf{H}_{l,r},\mathbf{H}_{r,l},\mathbf{H}_{l,l}$ and $\mathbf{H}_{r,r}$,
respectively, including the smartly controlled paths by the IRS response,
given as in (6). Let $\mathbf{y}_{l}$ and $\mathbf{y}_{r}$ denote the total
received signal at the FD nodes $l$ and $r$, respectively, which can be
written as

$\displaystyle\mathbf{y}_{l}=$
$\displaystyle\tilde{\mathbf{H}}_{l,r}\mathbf{V}_{r}\mathbf{s}_{r}+\tilde{\mathbf{H}}_{l,l}\mathbf{V}_{l}\mathbf{s}_{l}+\mathbf{n}_{l},$
(7a) $\displaystyle\mathbf{y}_{r}=$
$\displaystyle\tilde{\mathbf{H}}_{r,l}\mathbf{V}_{l}\mathbf{s}_{l}+\tilde{\mathbf{H}}_{r,r}\mathbf{V}_{r}\mathbf{s}_{r}+\mathbf{n}_{r},$
(7b)

with $\mathbf{n}_{l}\sim\mathcal{CN}(\bm{0},\mathbf{I})$ and
$\mathbf{n}_{r}\sim\mathcal{CN}(\bm{0},\mathbf{I})$ denoting the additive
white Gaussian noise (AWGN) vectors. Let $\overline{k}$ denote the indices in
the set $\mathcal{F}$ except the element $k$. Let ($\mathbf{R}_{l}$)
$\mathbf{R}_{\overline{l}}$ and ($\mathbf{R}_{r}$) $\mathbf{R}_{\overline{r}}$
denote the (signal and) interference plus noise covariance matrices, which,
with the effective channel matrices, can be written as

$\mathbf{R}_{l}=\underbrace{\tilde{\mathbf{H}}_{l,r}\mathbf{V}_{r}\mathbf{V}_{r}^{H}\tilde{\mathbf{H}}_{l,r}^{H}}_{\triangleq\mathbf{S}_{l}}+\tilde{\mathbf{H}}_{l,l}\mathbf{V}_{l}\mathbf{V}_{l}^{H}\tilde{\mathbf{H}}_{l,l}^{H}+\mathbf{I},$
(8a)
$\mathbf{R}_{r}=\underbrace{\tilde{\mathbf{H}}_{r,l}\mathbf{V}_{l}\mathbf{V}_{l}^{H}\tilde{\mathbf{H}}_{r,l}^{H}}_{\triangleq\mathbf{S}_{r}}+\tilde{\mathbf{H}}_{r,r}\mathbf{V}_{r}\mathbf{V}_{r}^{H}\tilde{\mathbf{H}}_{r,r}^{H}+\mathbf{I},$
(8b)
$\mathbf{R}_{\overline{l}}=\mathbf{R}_{l}-\mathbf{S}_{l},\quad\mathbf{R}_{\overline{r}}=\mathbf{R}_{r}-\mathbf{S}_{r}.$
(8c)

The WSR maximization problem for the NF-IRSs assisted mmWave MIMO FD system,
under the sum-power constraint for the FD nodes and the unit-modulus
constraint for the NF-IRSs can be formally stated as

$\underset{\begin{subarray}{c}\mathbf{V}_{l},\mathbf{V}_{r},\\\
\mathbf{\Phi}_{l},\mathbf{\Phi}_{r}\end{subarray}}{\text{max}}\quad
w_{l}\mbox{ln}\big{[}\mbox{det}\big{(}\mathbf{R}_{\overline{l}}^{-1}\mathbf{R}_{l}\big{)}\big{]}+w_{r}\mbox{ln}\big{[}\mbox{det}\big{(}\mathbf{R}_{\overline{r}}^{-1}\mathbf{R}_{r}\big{)}\big{]}$
(9a)
$\text{s.t.}\quad\mbox{Tr}\big{(}\mathbf{V}_{k}\mathbf{V}_{k}^{H}\big{)}\leq
p_{k},\quad\forall k\in\mathcal{F},$ (9b)
$\quad\quad|\phi_{r}(i)|=1\;\&\;|\phi_{l}(j)|=1,\forall i,j,$ (9c)

where $w_{l}$ and $w_{r}$ denote the rate weights for node $l$ and $r$,
respectively, $p_{l}$ and $p_{r}$ denote their total sum power constraint,
respectively.

##  III Joint Active and Passive Beamforming

The WSR maximization problem stated above is not concave, and finding its
global optimum is challenging. To obtain a simpler solution, we adopt the
weighted minimum mean squared error (WMMSE) method, which solves (9) based on
alternating optimization, leading to a sub-optimal solution [33].

To proceed, we assume that the MIMO FD nodes $l$ and $r$ deploy the digital
combiners $\mathbf{F}_{l}$ and $\mathbf{F}_{r}$ to estimate the received data
streams. i.e.,

$\displaystyle\hat{\mathbf{s}}_{l}=\mathbf{F}_{r}\mathbf{y}_{l},\quad\hat{\mathbf{s}}_{r}=\mathbf{F}_{l}\mathbf{y}_{r}.$
(10)

Let $\mathbf{E}_{l}$ and $\mathbf{E}_{r}$ denote the mean squared error (MSE)
matrices for the MIMO FD nodes $l$ and $r$, respectively. Given (10), they can
be written as

$\displaystyle\mathbf{E}_{l}=$
$\displaystyle\mathbb{E}[(\mathbf{F}_{l}\mathbf{y}_{l}-\mathbf{s}_{r})(\mathbf{F}_{l}\mathbf{y}_{l}-\mathbf{s}_{r})^{H}]=\mathbf{F}_{l}\tilde{\mathbf{H}}_{l,r}\mathbf{V}_{r}\mathbf{V}_{r}^{H}\tilde{\mathbf{H}}_{l,r}^{H}\mathbf{F}_{l}^{H}$
(11a)
$\displaystyle-\mathbf{F}_{l}\tilde{\mathbf{H}}_{l,r}\mathbf{V}_{r}+\mathbf{F}_{l}\tilde{\mathbf{H}}_{l,l}\mathbf{V}_{l}\mathbf{V}_{l}^{H}\tilde{\mathbf{H}}_{l,l}^{H}\mathbf{F}_{l}^{H}+\mathbf{F}_{l}\mathbf{F}_{l}^{H}$
$\displaystyle-\mathbf{V}_{r}^{H}{\tilde{\mathbf{H}}_{l,r}}^{H}\mathbf{F}_{l}^{H}+\mathbf{I}$
$\displaystyle\mathbf{E}_{r}=$
$\displaystyle\mathbb{E}[(\mathbf{F}_{r}\mathbf{y}_{r}-\mathbf{s}_{l})(\mathbf{F}_{r}\mathbf{y}_{r}-\mathbf{s}_{l})^{H}]=\mathbf{F}_{r}\tilde{\mathbf{H}}_{r,l}\mathbf{V}_{l}\mathbf{V}_{l}^{H}\tilde{\mathbf{H}}_{r,l}^{H}\mathbf{F}_{r}^{H}$
(11b)
$\displaystyle-\mathbf{F}_{r}\tilde{\mathbf{H}}_{r,l}\mathbf{V}_{l}+\mathbf{F}_{r}\tilde{\mathbf{H}}_{r,r}\mathbf{V}_{r}\mathbf{V}_{r}^{H}\tilde{\mathbf{H}}_{r,r}^{H}\mathbf{F}_{r}^{H}+\mathbf{F}_{r}\mathbf{F}_{r}^{H}$
$\displaystyle-\mathbf{V}_{l}^{H}\tilde{\mathbf{H}}_{r,l}^{H}\mathbf{F}_{r}^{H}+\mathbf{I}.$

We assume that combiners $\mathbf{F}_{l}$ and $\mathbf{F}_{r}$ are optimized
based on the minimum MSE (MMSE) criteria, and therefore, they can obtained by
solving the following optimization problem

$\underset{\begin{subarray}{c}\mathbf{F}_{l},\mathbf{F}_{r}\end{subarray}}{\min}\quad\mbox{Tr}(\mathbf{E}_{l})+\mbox{Tr}(\mathbf{E}_{r}),$
(12)

which leads to the following expressions for the MMSE combiners

$\mathbf{F}_{l}=\mathbf{V}_{r}^{H}\tilde{\mathbf{H}}_{l,r}^{H}(\tilde{\mathbf{H}}_{l,r}\mathbf{V}_{r}\mathbf{V}_{r}^{H}\tilde{\mathbf{H}}_{l,r}^{H}+\tilde{\mathbf{H}}_{l,l}\mathbf{V}_{l}\mathbf{V}_{l}^{H}\tilde{\mathbf{H}}_{l,l}^{H}+\mathbf{I})^{-1},$
(13a)
$\mathbf{F}_{r}=\mathbf{V}_{l}^{H}\tilde{\mathbf{H}}_{r,l}^{H}(\tilde{\mathbf{H}}_{r,l}\mathbf{V}_{l}\mathbf{V}_{l}^{H}\tilde{\mathbf{H}}_{r,l}^{H}+\tilde{\mathbf{H}}_{r,r}\mathbf{V}_{r}\mathbf{V}_{r}^{H}\tilde{\mathbf{H}}_{r,r}^{H}+\mathbf{I})^{-1}.$
(13b)

Assuming the combiners to be fixed according to (13), and by plugging these
expressions in the error covariance matrices, it is immediate to show that the
error covariance matrices can be written as

$\mathbf{E}_{l}=(\mathbf{I}+\mathbf{V}_{r}^{H}\tilde{\mathbf{H}}_{l,r}^{H}\mathbf{R}_{\overline{l}}\tilde{\mathbf{H}}_{l,r}\mathbf{V}_{r})^{-1},$
(14)
$\mathbf{E}_{r}=(\mathbf{I}+\mathbf{V}_{l}^{H}\tilde{\mathbf{H}}_{r,l}^{H}\mathbf{R}_{\overline{r}}\tilde{\mathbf{H}}_{r,l}\mathbf{V}_{l})^{-1}.$
(15)

Therefore, maximizing the WSR is equivalent to minimizing the MSE error and
problem (9) can be restated with respect to the digital beamformers and IRSs’
phase response with the minimization of the MSE as

$\underset{\begin{subarray}{c}\mathbf{V}_{l},\mathbf{V}_{r},\\\
\mathbf{\Phi}_{l},\mathbf{\Phi}_{r}\end{subarray}}{\min}\quad\mbox{Tr}(\mathbf{W}_{l}\mathbf{E}_{l})+\mbox{Tr}(\mathbf{W}_{r}\mathbf{E}_{r})$
(16a)
$\text{s.t.}\quad\mbox{Tr}\big{(}\mathbf{V}_{k}\mathbf{V}_{k}^{H}\big{)}\leq
p_{k},\quad\forall k\in\mathcal{F},$ (16b)
$\quad\quad|\phi_{r}(i)|=1\;\&\;|\phi_{l}(j)|=1,\forall i,j,$ (16c)

where $\mathbf{W}_{i}$ and $\mathbf{W}_{l}$ are the constant weight matrices
associated with the MIMO FD nodes $l$ and $r$, respectively. The gradient of
the WSR and the WMMSE optimization problems turns out to be the same when the
weight matrices are chosen as [34], i.e.,

$\mathbf{W}_{l}=\frac{w_{l}}{\mbox{ln}\;2}\mathbf{E}_{l}^{-1},\quad\mathbf{W}_{r}=\frac{w_{r}}{\mbox{ln}\;2}\mathbf{E}_{r}^{-1}.$
(17)

###  III-A Active Digital Beamforming

We assume the combiners and the IRSs’ response to be fixed and consider the
optimization of the digital beamformers $\mathbf{V}_{l}$ and $\mathbf{V}_{r}$
for the MIMO FD nodes $l$ and $r$, respectively. The active digital
beamforming optimization problem can be formally stated as follows

$\underset{\begin{subarray}{c}\mathbf{V}_{l},\mathbf{V}_{r}\end{subarray}}{\min}\quad\mbox{Tr}(\mathbf{W}_{l}\mathbf{E}_{l})+\mbox{Tr}(\mathbf{W}_{r}\mathbf{E}_{r}),$
(18a)
$\text{s.t.}\quad\mbox{Tr}\big{(}\mathbf{V}_{k}\mathbf{V}_{k}^{H}\big{)}\leq
p_{k},\quad\forall k\in\mathcal{F}.$ (18b)

By taking the partial derivative of the Lagrangian function of (18) with
respect to the digital beamformers $\mathbf{V}_{l}$ and $\mathbf{V}_{r}$, we
obtain the following optimal beamformers

$\displaystyle\mathbf{V}_{l}=$
$\displaystyle\big{(}\mathbf{X}_{l}+\mu_{l}\mathbf{I}\big{)}^{-1}\tilde{\mathbf{H}}_{r,l}^{H}\mathbf{F}_{r}^{H}\mathbf{W}_{r},$
(19a)
$\displaystyle\mathbf{V}_{r}=\big{(}\mathbf{X}_{r}+\mu_{r}\mathbf{I}\big{)}^{-1}\tilde{\mathbf{H}}_{l,r}^{H}\mathbf{F}_{l}^{H}\mathbf{W}_{l},$
(19b)

where the matrices $\mathbf{X}_{l}$ and $\mathbf{X}_{r}$ are defined as

$\mathbf{X}_{l}=\tilde{\mathbf{H}}_{r,l}^{H}\mathbf{F}_{r}^{H}\mathbf{W}_{r}\mathbf{F}_{r}\mathbf{H}_{r,l}+\mathbf{H}_{l,l}^{H}\mathbf{F}_{l}^{H}\mathbf{W}_{r}\mathbf{F}_{l}\mathbf{H}_{l,l},$
(20a)
$\mathbf{X}_{r}=\tilde{\mathbf{H}}_{l,r}^{H}\mathbf{F}_{l}^{H}\mathbf{W}_{l}\mathbf{F}_{l}\mathbf{H}_{l,r}+\mathbf{H}_{r,r}^{H}\mathbf{F}_{r}^{H}\mathbf{W}_{l}\mathbf{F}_{r}\mathbf{H}_{r,r},$
(20b)

and scalars $\mu_{l}$ and $\mu_{r}$ denote the Lagrange multipliers for the
sum-power constraint of the MIMO FD nodes $l$ and $r$, respectively. The
Lagrange multipliers can be calculated while meeting the sum-power
constraints. Namely, to find the Lagrange multiplier, we can consider the
singular value decomposition of
$\mathbf{X}_{l}=\mathbf{U}_{l}\mathbf{\Sigma}_{l}\mathbf{U}_{l}^{H}$ and
$\mathbf{X}_{r}=\mathbf{U}_{r}\mathbf{\Sigma}_{r}\mathbf{U}_{r}^{H}$ and write
the power constraints in (19), after simple steps, as

$\mbox{Tr}(\mathbf{V}_{l}\mathbf{V}_{l}^{H})=\sum_{i=i}^{N_{l}}\frac{\mathbf{G}_{l}(i,i)}{(\mu_{l}+\mathbf{\Sigma}_{l}(i,i))^{2}}=p_{l},$
(21a)
$\mbox{Tr}(\mathbf{V}_{r}\mathbf{V}_{r}^{H})=\sum_{i=i}^{N_{r}}\frac{\mathbf{G}_{r}(i,i)}{(\mu_{r}+\mathbf{\Sigma}_{r}(i,i))^{2}}=p_{r},$
(21b)

where the matrices $\mathbf{G}_{l}$ and $\mathbf{G}_{r}$ are defined as
follows

$\mathbf{G}_{l}=\mathbf{U}_{l}^{H}\tilde{\mathbf{H}}_{r,l}^{H}\mathbf{F}_{r}^{H}\mathbf{W}_{r}\mathbf{W}_{r}\mathbf{F}_{r}\tilde{\mathbf{H}}_{r,l}\mathbf{U}_{l},$
(22a)
$\mathbf{G}_{r}=\mathbf{U}_{r}^{H}\tilde{\mathbf{H}}_{l,r}^{H}\mathbf{F}_{l}^{H}\mathbf{W}_{l}\mathbf{W}_{l}\mathbf{F}_{l}\tilde{\mathbf{H}}_{l,r}\mathbf{U}_{r}.$
(22b)

To find the optimal values of $\mu_{l}$ and $\mu_{r}$, a linear search method
can be adopted and we consider finding the multipliers with the Bisection
method. If the values of the Lagrange multipliers result to be negative, then
they are set to zero.

###  III-B Passive Beamforming With IRSs

We assume the digital combiners and beamformers to be fixed and consider the
optimization of $\mathbf{\Phi}_{r}$ and $\mathbf{\Phi}_{l}$, which are the
passive beamforming response of the IRSs $i_{r}$ and $i_{l}$, respectively. As
shown in the Appendix, the optimization problem with respect to
$\mathbf{\Phi}_{r}$ and $\mathbf{\Phi}_{l}$, after some algebraic
manipulations, can be formally stated as

$\underset{\begin{subarray}{c}\mathbf{\phi}_{r}\end{subarray}}{\min}\quad\bm{\phi}_{r}^{H}\mathbf{\Sigma}_{r}\bm{\phi}_{r}+{\mathbf{s}_{r}}^{H}\bm{\phi}_{r}^{*}+{\mathbf{s}_{r}}^{T}\bm{\phi}_{l},$
(23a) $\text{s.t.}\quad|\phi_{r}(i)|=1,\quad\forall i,$ (23b)
$\underset{\begin{subarray}{c}\mathbf{\phi}_{l}\end{subarray}}{\min}\quad\bm{\phi}_{l}^{H}\mathbf{\Sigma}_{l}\bm{\phi}_{l}+{\mathbf{s}_{l}}^{H}\bm{\phi}_{l}^{*}+{\mathbf{s}_{r}}^{T}\bm{\phi}_{l},$
(23c) $\text{s.t.}\quad|\phi_{l}(i)|=1,\quad\forall i.$ (23d)

where matrices $\mathbf{\Sigma}_{r}$, $\mathbf{\Sigma}_{r}$, and vectors
$\mathbf{s}_{l},\mathbf{s}_{r}$ are defined in the Appendix. Problem (23) is
non-convex due to the unit-modulus constraint. To render a feasible solution,
we adopt the majorization-maximization optimization method [35]. Its objective
is to solve a difficult problem through a series of more tractable sub-
problems. We carry out its explanation only for $\bm{\phi}_{r}$ and a similar
reasoning follows also for $\bm{\phi}_{l}$. Let $f(\bm{\phi}_{r}^{(n)})$
denote the value of the objective function as a function of $\bm{\phi}_{r}$ at
the $n$-th iteration. We proceed by constructing its upper bound with
$g(\bm{\phi}_{r}|\bm{\phi}_{r}^{(n)})$. In particular, we construct an
approximate problem by using $g(\bm{\phi}_{r}|\bm{\phi}_{r}^{(n)})$ at the
$n+1$-th iteration. If $g(\bm{\phi}_{r}|\bm{\phi}_{r}^{(n)})$ satisfies the
following conditions

  1. 1.

$g(\bm{\phi}_{r}^{(n)}|\bm{\phi}_{r}^{(n)})=f(\bm{\phi}_{r}^{(n)})$,

  2. 2.

$\nabla_{\bm{\phi}_{r}}g(\bm{\phi}_{r}|\bm{\phi}_{r}^{(n)})|_{\bm{\phi}_{r}=\bm{\phi}_{r}^{(n)}}=\nabla_{\bm{\phi}_{r}}f(\bm{\phi}_{r}(n))|_{\bm{\phi}_{r}=\bm{\phi}_{r}^{(n)}}$,

  3. 3.

$g(\bm{\phi}_{r}|\bm{\phi}_{r}^{(n)})\geq f(\bm{\phi}_{r}^{(n)}),$

then the sequence of the solutions obtained at each iteration will result in a
monotonically decreasing objective function and converge to the actual
solution of $f(\cdot)$. The first two conditions dictate that
$g(\bm{\phi}_{r}|\bm{\phi}_{r}(n))$ and its first order gradient should be the
same, while the third condition states that it should construct an upper bound
for the original function. It has been shown in [36] that a simple upper bound
$g(\bm{\phi}_{r}|\bm{\phi}_{r}^{(n)})$ or
$g(\bm{\phi}_{l}|\bm{\phi}_{l}^{(n)})$ for $\mathbf{\Phi}_{l}$ in our problem
(see (36c)-(36d) in the Appendix) is given by

$g(\bm{\phi}_{r}|\bm{\phi}_{r}^{(n)})=2\mbox{Re}\\{{\mathbf{s}_{r}}^{H}\mathbf{q}_{r}^{(n)}\\}+o_{r},$
(24a)
$g(\bm{\phi}_{l}|\bm{\phi}_{r}^{(n)})=2\mbox{Re}\\{{\mathbf{s}_{l}}^{H}\mathbf{q}_{l}^{(n)}\\}+o_{l},$
(24b)

where $o_{r}$ and $o_{l}$ denote constant terms, and $\mathbf{q}_{r}^{(n)}$
and $\mathbf{q}_{l}^{(n)}$ are given by

$\mathbf{q}_{r}^{(n)}=(\lambda_{r}^{max}\mathbf{I}-\mathbf{\Sigma}_{r})\bm{\phi_{r}}^{(n)}-{\mathbf{s}_{r}}^{*},$
(25a)
$\mathbf{q}_{l}^{(n)}=(\lambda_{l}^{max}\mathbf{I}-\mathbf{\Sigma}_{l})\bm{\phi_{l}}^{(n)}-{\mathbf{s}_{l}}^{*},$
(25b)

and $\lambda_{r}^{max}$ and $\lambda_{l}^{max}$ denote the maximum eigenvalues
of $\mathbf{\Sigma}_{r}$ and $\mathbf{\Sigma}_{l}$, respectively. Then our
optimization problem (see (36a)-(36b) and (36c)-(36d) in the Appendix), by
ignoring all the terms which are constant, can be restated by using the upper
bounds (24a) and (24b), respectively, as

$\underset{\begin{subarray}{c}\bm{\phi}_{r}\end{subarray}}{\min}\quad
2\mbox{Re}\\{{\mathbf{s}_{r}}^{H}\mathbf{q}_{r}^{(n)}\\},$ (26a)
$\text{s.t.}\quad|\bm{\phi}_{r}(i)|=1,\quad\forall i,$ (26b)
$\underset{\begin{subarray}{c}\bm{\phi}_{l}\end{subarray}}{\min}\quad
2\mbox{Re}\\{{\mathbf{s}_{l}}^{H}\mathbf{q}_{l}^{(n)}\\},$ (26c)
$\text{s.t.}\quad|\bm{\phi}_{l}(i)|=1,\quad\forall i.$ (26d)

By solving (26)-(26b) and (26c)-(26d) we get that the optimal results at
$n+1$-th iteration as

$\bm{\phi_{r}}^{(n+1)}=e^{i\angle\mathbf{q}_{r}^{(n)}}.$ (27a)

$\bm{\phi_{l}}^{(n+1)}=e^{i\angle\mathbf{q}_{l}^{(n)}}.$ (27b)

At each iteration, when the digital beamformers are fixed, the optimization of
$\mathbf{\Phi}_{r}$ and $\mathbf{\Phi}_{l}$ consists in updating their
response iteratively until convergence by solving a series of more tractable
optimization problems (27a)-(27b). The procedure for the optimization of the
phase response of IRS $i$, where $i=i_{l}$ or $i=i_{r}$, is stated in
Algorithm $1$, which needs to be iterated for each update of the digital
beamformers and combiners. The overall procedure for joint active and passive
beamforming for NF-IRSs assisted mmWave FD system is formally stated in
Algorithm $2$.

Algorithm 1 Optimization of IRS $i\in\mathcal{I}$

Given: $\bm{\phi}_{i}(0)$.
Initialize: iteration index $n=1$, accuracy $\epsilon$.
Evaluate: $f(\bm{\phi}_{i}(0))$.
Repeat until convergence

Calculate
$\mathbf{q}_{i}^{(n)}=(\lambda_{i}^{max}\mathbf{I}-\mathbf{\Sigma}_{i})\bm{\phi}_{i}^{(n)}-{\mathbf{s}_{i}}^{*}$

Update $\bm{\phi}_{i}^{(n+1)}$ with
$\bm{\phi}_{i}^{(n+1)}=\bm{\phi_{r}}^{(n+1)}=e^{i\angle\mathbf{q}_{i}^{(n)}}$.

if
$|f(\bm{\phi}_{i}^{(n+1)})-f(\bm{\phi}_{i}^{(n)})|/f(\bm{\phi}_{i}^{(n+1)})\leq\epsilon$

Stop and return $\bm{\phi}_{i}^{(n+1)}$.

else n=n+1 and repeat.

Algorithm 2 WSR maximization for MIMO FD with NF-IRSs

Given: The CSI and rate weights.
Initialize iteration index $n$, accuracy $\epsilon$, beamformers and
combiners.
Repeat until convergence

for $\forall i\in\mathcal{F}$

Update $\mathbf{F}_{i}$ with (13).

Update $\mathbf{W}_{i}$ with (17).

Update $\mathbf{V}_{i}$ with (19).

Update $\mathbf{\Phi}_{i}$ with Algorithm 1.

if convergence condition is satisfied

Stop and return the optimized variables.

else repeat.

###  III-C Convergence

The proof of convergence of Algorithm 2 is based on the proof for a more
general equivalent optimization problem, which includes the MSE weights and
receive filters as new optimization variables in addition to digital
beamformers and the IRSs’ phase response. The global optimization problem as a
function of the new variables is given as

$\underset{\begin{subarray}{c}\mathbf{W}_{l},\mathbf{W}_{r},\mathbf{F}_{l},\mathbf{F}_{r}\\\
\mathbf{V}_{r},\mathbf{V}_{l},\mathbf{\Phi}_{l},\mathbf{\Phi}_{r}\end{subarray}}{\min}\sum_{f\in\mathcal{F}}\mbox{Tr}(\mathbf{W}_{f}\textbf{MSE}_{f})-w_{f}\mbox{logdet}(\frac{\mbox{ln
2}}{w_{f}}\mathbf{W}_{f})+d_{f}\frac{w_{f}}{\mbox{ln}2}.$ (28)

By fixing the digital beamformers and the IRSs response, the weight matrices
and the combiners are chosen to be the MMSE combiners as (13). By replacing
the optimal MMSE receivers given in (28) in (11a) and $\eqref{err_r}$,
respectively, we obtain a new cost function

$\underset{\begin{subarray}{c}\mathbf{W}_{l},\mathbf{W}_{r},\mathbf{V}_{r}\mathbf{V}_{l},\\\
\mathbf{\Phi}_{l},\mathbf{\Phi}_{r}\end{subarray}}{\min}\sum_{f\in\mathcal{F}}\mbox{Tr}(\mathbf{W}_{f}\mathbf{E}_{f})-w_{f}\mbox{logdet}(\frac{\mbox{ln
2}}{w_{f}}\mathbf{W}_{f})+d_{f}\frac{w_{f}}{\mbox{ln}2}.$ (29)

The optimization of (29) with respect to the weight matrix $\mathbf{W}_{f}$
yields the solution
$\mathbf{W}_{f}=\frac{w_{f}}{\mbox{ln}\;2}\mathbf{E}_{f}^{-1}$. By plugging
the optimized weight matrices in (29), we obtain the new cost function

$\underset{\begin{subarray}{c}\mathbf{V}_{l},\mathbf{V}_{r},\\\
\mathbf{\Phi}_{l},\mathbf{\Phi}_{r}\end{subarray}}{\min}\quad\sum_{f\in\mathcal{F}}-w_{f}\mbox{ln
det}\big{(}(\mathbf{E}_{f})^{-1}\big{)}$ (30a)
$\text{s.t.}\quad\mbox{Tr}\big{(}\mathbf{V}_{k}\mathbf{V}_{k}^{H}\big{)}\leq
p_{k},\quad\forall k\in\mathcal{F}$ (30b)
$\quad\quad|\phi_{r}(i)|=1\;\&\;|\phi_{l}(j)|=1,\forall i,j,$ (30c)

which is the original WSR cost function (9). Every update of the digital
beamformers $\mathbf{V}_{l}$ and $\mathbf{V}_{r}$ and the IRSs’ phase response
by minimizing the MSE leads to a monotonic increase in the WSR, which assures
convergence of the proposed joint active and passive beamforming design.

A typical average convergence behavior of the proposed beamforming design for
the NF-IRSs assisted FD systems is shown in Fig. 3. We recall that the WMMSE
method is a sub-optimal solution for WSR maximization [33], and therefore
Algorithm 2 converges to a local optimum.

Figure 3: Average WSR per iteration for the proposed scheme at
SNR=$30\leavevmode\nobreak\ $dB with NF-IRSs of size $30\times 30$.

##  IV Numerical Results

In this section, we present simulation results to evaluate the performance of
the proposed joint active and passive beamforming design to analyze its
potential to replace HYBF and combining simultaneously for the FD systems,
while shaping the SI with NF-IRSs.

For comparison, we define the following benchmark schemes:

  * •

A mMIMO HD system with fully digital beamforming capability, $100$ transmit
and $50$ receive antennas for each MIMO HD node, which is denoted as
HD-$100\times 50$.

  * •

A mMIMO FD system with fully digital beamforming capability, $100$ transmit
and $50$ receive antennas for each MIMO FD node, denoted as FD-$100\times 50$.

Note that digital beamforming designs represent upper bounds for the HYBF
designs. Therefore, we compare our solution against the mMIMO fully digital
systems, which provide the maximum achievable gain for the HYBF designs.

We define the signal-to-noise ratio (SNR) for the NF-IRSs assisted FD system
as SNR = $p_{r}/\sigma_{r}^{2}=p_{l}/\sigma_{l}^{2}$, where $p_{i}$ and
$\sigma_{i}^{2}$ denote the total transmit power and noise variance for node
$i$, respectively, with $i=l$ or $i=r$. We assume that the considered systems
operate at the frequency of $30$ GHz, i.e., $\lambda=10$ mm. The noise
variance is set to be $\sigma_{i}^{2}=1$ and the total transmit power is
chosen to meet the desired SNR. For the NF-IRS assisted FD system, the number
of antennas is chosen to be $5$ times less, i.e., only $M_{l}=M_{r}=20$
transmit and $N_{l}=N_{r}=10$ receive antennas, respectively, and the number
of data streams is $d_{l}=d_{r}=2$. The FD nodes are assumed to be
$D_{l,r}=200$ m far. We consider the NF-IRSs of size $10\times 10$, $20\times
20$, and $30\times 30$ for both the MIMO FD nodes, with the center between two
IRS elements located at $\lambda/2$. They are placed at the minimum distance
of $3$ m from their FD nodes, which is varied up to $90$ m. We assume that
both the FD nodes are equipped with ULAs, and their transmit and the receive
arrays are assumed to be separated by a distance $D_{b}=20$ cm, with a
relative angle $\Theta_{b}=180^{\circ}$ and $r_{m,n}$ in (3) is set given
$D_{b}$ and $\Theta_{b}$ as in [14, (9)]. The Rician factor is set as
$\kappa_{b}=1$ and the rate weights are chosen as $w_{l}=w_{r}=1$. The number
of paths and number of clusters are set as $N_{c,b}=N_{p,b}=3$ and the AoA
$\theta_{r}^{n_{p},n_{c}}$ and AoD $\phi_{l}^{n_{p},n_{c}}$ are assumed to be
uniformly distributed in the interval
$\mathcal{U}\sim[-30^{\circ},30^{\circ}]$. The digital beamformers are
initialized as the dominant eigenvectors of the effective channel covariance
matrices, and the response of the NF-IRSs is initialized with random phases.
The results reported herein are averaged over $200$ channel realizations.

Let $\hat{x},\hat{y}$, and $\hat{z}$ denote the $3$ versors on the three
dimensional space and any point on the $3$ dimensional vector space can be
written as $(x\hat{x},y\hat{y},z\hat{z})$. We assume the FD nodes with ULAs to
be aligned with $\hat{x}$ direction and the first transmit antenna of the FD
nodes $l$ and $r$ are assumed to be placed in the positions $(0,0,0)$ and
$(0,D_{l,r\hat{y}},0)$, respectively, with other antennas placed in positions
$x+$. Both the NF-IRSs are assumed to be at the same distance from their FD
nodes, and placed on the x-y plane with first element of the NF-IRSs $i_{l}$
and $i_{r}$ placed in positions $(0,D_{IRS}\hat{y},0)$ and
$(0,D_{r,l}-D_{IRS}\hat{y},0)$, respectively. Note that when the IRS operate
in the NF of the FD node, joint passive SIC and intelligent control of the UL
and DL channels also depend on the orientation of the NF-IRSs. Namely, the NF-
IRSs can provide better SIC by jeopardizing the UL and DL channel control if
they are more oriented towards the receive antennas of the NF FD node, or vice
versa. The chosen orientation of the NF-IRSs in our setup provides an optimal
trade-off between the passive SIC capabilities of NF-IRSs and control over the
UL and DL channels.

Figure 4: Average WSR as a function of SNR with NF-IRSs of size $30\times 30$,
$20\times 20$, and $10\times 10$, placed at a distance of $3$ m.

Figure 5: Average WSR as a function of SNR with NF-IRSs of size $30\times 30$,
$20\times 20$, and $10\times 10$, placed at a distance of $30$ m.

Figure 6: Average WSR as a function of the distance between the FD nodes and
the NF-IRSs at SNR$=0$ dB.

Figure 7: Average WSR as a function of the distance between the FD nodes and
the NF-IRSs at SNR$=30$ dB.

Fig. 5 shows the achieved average WSR as a function of the SNR for the FD
systems assisted with NF-IRSs placed at $3$ m from the transmit array. We can
see that despite using $5$ times fewer antennas than the mMIMO FD systems, the
proposed approaches significantly outperform the benchmark schemes. Namely,
with an NF-IRSs of size $30\times 30$, they can achieve $\sim 4$ times higher
gain than the traditional mMIMO HD system. Fig. 5 shows that the achievable
gains of the NF-IRSs assisted FD systems are limited only by the size of the
IRS, and having larger NF-IRSs will lead to higher gains. Fig. 5 shows the
performance of the proposed design as a function of SNR with the NF-IRSs
placed at $30$ m far from the transmit array. We can see that when the NF-IRSs
are placed far from the FD node, they achieve significantly less gain. The
reason lies in the fact that the IRSs placed far from the FD node cannot aid
in shaping the SI channel, which is handled only with the digital beamformers,
which leads to less achievable gain.

Fig. 7 shows the performance of the proposed joint active and passive
beamforming designs, as a function of the distance between the FD node and
their NF-IRSs, which is varied in the interval $3-90$ m, with the two FD nodes
operating at the distance of $200$ m at SNR$=0\leavevmode\nobreak\ $dB. From
Fig. 7 we note that the FD systems benefit the most with NF-IRSs instead of
the FF-IRSs by obtaining a significantly higher achievable WSR compared to the
traditional mMIMO FD systems. Placing the IRSs far from the FD systems leads
to small performance gains as the FD systems improve only the quality of the
direct links and cannot assist in shaping the SI channel. Fig. 7 shows the
performance as a function of the distance between the NF-IRSs from their FD
node at SNR$=30$ dB. It is clearly visible that, regardless of the SNR at
which the NF-IRS-assisted FD systems operate, the distance between the FD
nodes and their NF-IRSs dictates the maximum achievable gain.

Fig.s 9-9 show the cumulative distribution function (CDF) of the average WSR
rate for SNR of $0\leavevmode\nobreak\ $dB and $30\leavevmode\nobreak\ $dB
with the NF-IRSs at a distance of $3\leavevmode\nobreak\ $m and
$30\leavevmode\nobreak\ $m, respectively. We can see that the gains are stable
over the CDF and also consistent with the results reported in the previous
figures. Therefore, we can conclude that with very high probability the FD
systems benefit from the IRSs the most, only when they are placed in the NF.
Placing the IRSs in the FF is not desirable for the FD system, as it leads
significant less gain compared to the NF case.

Finally, Fig. 10 shows the achievable performance of MIMO FD system assisted
with FF-IRS of variable size, in comparison with the NF-IRS-assisted system
and the benchmark schemes. We can see that the FF-IRSs even with $80\times 80$
elements achieves a significantly lower gain than the NF-IRSs of size
$30\times 30$. This results showcase the importance of the NF-IRS in paving
the path towards cost-efficient and energy-efficient mmWave FD systems with
very high performance gain.

Figure 8: CDF for the WSR with SNR$=0\leavevmode\nobreak\ $dB.

Figure 9: CDF for the WSR with SNR$=30\leavevmode\nobreak\ $dB.

Figure 10: Achievable average WSR with FF-IRS assisted MIMO FD system.

##  V Conclusions

We proposed to truncate the analog stage of the mmWave FD nodes by shifting
the mMIMO FD paradigm back to MIMO FD. Each FD node is then assisted with a
NF-IRS which aims to smartly control the UL and DL channel, while performing
very strong passive SIC. Therefore, the proposed design leads to significant
reduction in the hardware cost and power consumption. A novel joint active and
passive beamforming design for the NF-IRSs assisted mmWave FD system has been
proposed to maximize the WSR. Simulation results show that the proposed method
significantly outperform the conventional HD system. Moreover, it is also
shown that the NF-IRSs not only are capable of replacing the analog stage of
the mmWave FD systems, but they yield $\leavevmode\nobreak\ 2$ times more gain
than the mMIMO FD systems with a NF-IRS of size $30\times 30$. Larger gains
can be obtained if the size of the NF-IRSs become large.

To highlight the dependence of the IRSs’ response on the WSR, we first rewrite
the effective channel responses as

$\tilde{\mathbf{H}}_{l,r}=\mathbf{A}_{1,r}+\mathbf{A}_{2,r}\mathbf{\Phi}_{r}\mathbf{H}_{i_{r},r},\quad\tilde{\mathbf{H}}_{r,l}=\mathbf{B}_{1,r}+\mathbf{H}_{r,i_{r}}\mathbf{\Phi}_{r}\mathbf{B}_{2,r},$
(31a)
$\tilde{\mathbf{H}}_{l,l}=\mathbf{C}_{1,r}+\mathbf{H}_{l,i_{r}}\mathbf{\Phi}_{r}\mathbf{C}_{2,r},\quad\tilde{\mathbf{H}}_{r,r}=\mathbf{D}_{1,r}+\mathbf{D}_{2,r}\mathbf{\Phi}_{r}\mathbf{H}_{i_{r},r},$
(31b)

to optimize $\mathbf{\Phi}_{r}$. Alternatively, we write them as

$\tilde{\mathbf{H}}_{l,r}=\mathbf{A}_{1,l}+\mathbf{H}_{l,i_{l}}\mathbf{\Phi}_{l}\mathbf{A}_{2,l},\quad\tilde{\mathbf{H}}_{r,l}=\mathbf{B}_{1,l}+\mathbf{B}_{2,l}\mathbf{\Phi}_{l}\mathbf{H}_{i_{l},l},$
(32a)
$\tilde{\mathbf{H}}_{l,l}=\mathbf{C}_{1,l}+\mathbf{C}_{2,l}\mathbf{\Phi}_{l}\mathbf{H}_{i_{l},l},\quad\tilde{\mathbf{H}}_{r,r}=\mathbf{D}_{1,l}+\mathbf{H}_{r,i_{l}}\mathbf{\Phi}_{l}\mathbf{D}_{2,l}.$
(32b)

to highlight the effective channel responses as a function of
$\mathbf{\Phi}_{l}$, where the auxiliary matrices appearing in (31)-(32) are
defined as

$\mathbf{A}_{1,r}=\mathbf{H}_{l,r}+\mathbf{H}_{l,i_{l}}\mathbf{\Phi}_{l}\mathbf{H}_{i_{l},r},\quad\mathbf{A}_{2,r}=\mathbf{H}_{l,i_{r}}+\mathbf{H}_{l,i_{l}}\mathbf{\Phi}_{l}\mathbf{H}_{i_{l},i_{r}}$
(33a)
$\mathbf{B}_{1,r}=\mathbf{H}_{r,l}+\mathbf{H}_{r,i_{l}}\mathbf{\Phi}_{l}\mathbf{H}_{i_{l},l},\quad\mathbf{B}_{2,r}=\mathbf{H}_{i_{r},l}+\mathbf{H}_{i_{r},i_{l}}\mathbf{i_{l}}\mathbf{H}_{i_{l},l},$
(33b)
$\mathbf{C}_{1,r}=\mathbf{H}_{l,l}+\mathbf{H}_{l,i_{l}}\mathbf{\Phi}_{l}\mathbf{H}_{i_{l},l},\quad\mathbf{C}_{2,r}=\mathbf{H}_{i_{r},l}+\mathbf{H}_{i_{r},i_{l}}\mathbf{\Phi}_{l}\mathbf{H}_{i_{l},l},$
(33c)
$\mathbf{D}_{1,r}=\mathbf{H}_{r,r}+\mathbf{H}_{r,i_{l}}\mathbf{\Phi}_{l}\mathbf{H}_{i_{l},r},\quad\mathbf{D}_{2,r}=\mathbf{H}_{r,i_{r}}+\mathbf{H}_{r,i_{l}}\mathbf{\Phi}_{l}\mathbf{H}_{i_{l},i_{r}}.$
(33d)
$\mathbf{A}_{1,l}=\mathbf{H}_{l,r}+\mathbf{H}_{l,i_{r}}\mathbf{\Phi}_{r}\mathbf{H}_{i_{r},r},\quad\mathbf{A}_{2,l}=\mathbf{H}_{i_{l},r}+\mathbf{H}_{i_{l},i_{r}}\mathbf{\Phi}_{r}\mathbf{H}_{i_{r},r}$
(33e)
$\mathbf{B}_{1,l}=\mathbf{H}_{r,l}+\mathbf{H}_{r,i_{r}}\mathbf{\Phi}_{r}\mathbf{H}_{i_{r},l},\quad\mathbf{B}_{2,l}=\mathbf{H}_{r,i_{l}}+\mathbf{H}_{r,i_{r}}\mathbf{\Phi}_{r}\mathbf{H}_{i_{r},i_{l}},$
(33f)
$\mathbf{C}_{1,l}=\mathbf{H}_{l,l}+\mathbf{H}_{l,i_{r}}\mathbf{\Phi}_{r}\mathbf{H}_{i_{r},l},\quad\mathbf{C}_{2,l}=\mathbf{H}_{l,i_{l}}+\mathbf{H}_{l,i_{r}}\mathbf{\Phi}_{r}\mathbf{H}_{i_{r},i_{l}},$
(33g)
$\mathbf{D}_{1,l}=\mathbf{H}_{r,r}+\mathbf{H}_{r,i_{r}}\mathbf{\Phi}_{r}\mathbf{H}_{i_{r},r},\quad\mathbf{D}_{2,l}=\mathbf{H}_{i_{l},r}+\mathbf{H}_{i_{l},i_{r}}\mathbf{\Phi}_{r}\mathbf{H}_{i_{r},r}.$
(33h)

Let $\mathbf{S}_{r},\mathbf{Z}_{r},\mathbf{T}_{r}$
$\mathbf{S}_{l},\mathbf{Z}_{l},\mathbf{T}_{l}$ denote additional auxiliary
matrices defined as

$\displaystyle\mathbf{S}_{r}=$
$\displaystyle\mathbf{H}_{i_{r},r}\mathbf{V}_{r}\mathbf{V}_{r}^{H}{\mathbf{A}_{1,r}}^{H}\mathbf{F}_{l}^{H}\mathbf{W}_{l}\mathbf{F}_{l}\mathbf{A}_{2,r}-\mathbf{H}_{i_{r},r}\mathbf{V}_{r}\mathbf{W}_{l}\mathbf{F}_{l}\mathbf{A}_{2,r}$
(34a)
$\displaystyle+\mathbf{C}_{2,r}\mathbf{V}_{l}\mathbf{V}_{l}^{H}{\mathbf{C}_{1,r}}^{H}\mathbf{F}_{l}^{H}\mathbf{W}_{l}\mathbf{F}_{l}\mathbf{H}_{l,i_{r}}$
$\displaystyle-\mathbf{B}_{2,r}\mathbf{V}_{l}\mathbf{W}_{r}\mathbf{F}_{r}\mathbf{H}_{r,i_{r}}+\mathbf{B}_{2,r}\mathbf{V}_{l}\mathbf{V}_{l}^{H}{\mathbf{B}_{1,r}}^{H}\mathbf{F}_{r}^{H}\mathbf{W}_{r}\mathbf{F}_{r}\mathbf{H}_{r,i_{r}}$
$\displaystyle+\mathbf{H}_{i_{r},r}\mathbf{V}_{r}\mathbf{V}_{r}^{H}{\mathbf{D}_{1,r}}^{H}\mathbf{F}_{r}^{H}\mathbf{W}_{r}\mathbf{F}_{r}\mathbf{D}_{2,r},$
$\displaystyle\mathbf{Z}_{r}=$
$\displaystyle\mathbf{H}_{r,i_{r}}^{H}\mathbf{F}_{r}^{H}\mathbf{W}_{r}\mathbf{F}_{r}\mathbf{H}_{r,i_{r}}+{\mathbf{D}_{2,r}}^{H}{\mathbf{A}_{2,r}}^{H}\mathbf{W}_{r}\mathbf{F}_{r}\mathbf{D}_{2,r}$
(34b)
$\displaystyle+{\mathbf{A}_{2,r}}^{H}\mathbf{F}_{l}^{H}\mathbf{W}_{l}\mathbf{F}_{l}\mathbf{A}_{2,r}+\mathbf{H}_{l,i_{r}}^{H}\mathbf{F}_{l}^{H}\mathbf{W}_{l}\mathbf{F}_{l}\mathbf{H}_{l,i_{r}},$
$\displaystyle\mathbf{T}_{r}=$
$\displaystyle\mathbf{B}_{2,r}\mathbf{V}_{l}\mathbf{V}_{l}^{H}{\mathbf{B}_{2,r}}^{H}+\mathbf{H}_{i_{r},r}\mathbf{V}_{r}\mathbf{V}_{r}^{H}\mathbf{H}_{i_{r},r}^{H}+\mathbf{H}_{i_{r}}\mathbf{V}_{r}\mathbf{V}_{r}^{H}\mathbf{H}_{i_{r},r}^{H}$
(34c)
$\displaystyle+\mathbf{C}_{2,r}\mathbf{V}_{l}\mathbf{V}_{l}^{H}{\mathbf{C}_{2,r}}^{H},$
$\displaystyle\mathbf{S}_{l}=$
$\displaystyle\mathbf{A}_{2,l}\mathbf{V}_{r}\mathbf{V}_{r}^{H}\mathbf{A}_{1,l}^{H}\mathbf{F}_{l}^{H}\mathbf{W}_{l}\mathbf{F}_{l}\mathbf{H}_{l,i_{l}}+\mathbf{H}_{i,l}\mathbf{V}_{l}\mathbf{V}_{l}^{H}\mathbf{C}_{1,l}^{H}\mathbf{A}_{l}^{H}\mathbf{W}_{l}\mathbf{A}_{l}\mathbf{C}_{2,l}$
(35a)
$\displaystyle-\mathbf{A}_{2,l}\mathbf{V}_{r}\mathbf{W}_{l}\mathbf{A}_{l}\mathbf{H}_{l,i_{l}}+\mathbf{H}_{i_{l},l}\mathbf{V}_{l}\mathbf{V}_{l}^{H}\mathbf{B}_{1,l}^{H}\mathbf{A}_{l}^{H}\mathbf{W}_{r}\mathbf{A}_{r}\mathbf{B}_{2,l}$
$\displaystyle+\mathbf{D}_{2,l}\mathbf{V}_{r}\mathbf{V}_{r}^{H}\mathbf{D}_{1,l}^{H}\mathbf{A}_{r}^{H}\mathbf{W}_{r}\mathbf{A}_{r}\mathbf{H}_{r,i_{l}}-\mathbf{H}_{i_{l},l}\mathbf{V}_{l}\mathbf{W}_{r}\mathbf{A}_{r}\mathbf{B}_{2,l},$
$\displaystyle\mathbf{Z}_{l}=$
$\displaystyle\mathbf{H}_{l,i_{l}}^{H}\mathbf{A}_{l}^{H}\mathbf{W}_{l}\mathbf{A}_{l}\mathbf{H}_{l,i_{l}}+\mathbf{C}_{2,l}^{H}\mathbf{A}_{l}^{H}\mathbf{W}_{l}\mathbf{A}_{l}\mathbf{C}_{2,l}+\mathbf{A}_{l}^{H}\mathbf{W}_{r}\mathbf{A}_{r}\mathbf{B}_{2,l}$
(35b)
$\displaystyle+\mathbf{H}_{r,i_{l}}^{H}\mathbf{A}_{r}^{H}\mathbf{W}_{r}\mathbf{H}_{r,i_{l}},$
$\displaystyle\mathbf{T}_{l}=$
$\displaystyle\mathbf{A}_{2,l}\mathbf{V}_{r}\mathbf{V}_{r}^{H}\mathbf{A}_{2,l}^{H}+\mathbf{H}_{i_{l},l}\mathbf{V}_{l}\mathbf{V}_{l}^{H}\mathbf{H}_{i_{l},l}^{H}+\mathbf{H}_{i_{l},l}\mathbf{V}_{l}\mathbf{V}_{l}^{H}\mathbf{H}_{i_{l},l}^{H}$
(35c)
$\displaystyle+\mathbf{D}_{2,l}\mathbf{V}_{r}\mathbf{V}_{r}^{H}\mathbf{D}_{2,l}^{H}.$

By substituting (34) and (35) in the expressions of the error covariance
matrices, the minimization of the MSE optimization problem with respect to
$\mathbf{\Phi}_{r}$ and $\mathbf{\Phi}_{l}$, can be formally stated as

$\underset{\begin{subarray}{c}\mathbf{\Phi}_{r}\end{subarray}}{\min}\quad\mbox{Tr}(\mathbf{\Phi}_{r}^{H}\mathbf{Z}_{r}\mathbf{\Phi}_{r}\mathbf{T}_{r})+\mbox{Tr}(\mathbf{\Phi}_{r}^{H}\mathbf{S}_{r}^{H})+\mbox{Tr}(\mathbf{\Phi}_{r}\mathbf{S}_{r})+c_{r}$
(36a) $\text{s.t.}\quad\quad\quad|\phi_{r}(i)|=1,\quad\forall i,$ (36b)
$\underset{\begin{subarray}{c}\mathbf{\Phi}_{l}\end{subarray}}{\min}\quad\mbox{Tr}(\mathbf{\Phi}_{l}^{H}\mathbf{Z}_{l}\mathbf{\Phi}_{l}\mathbf{T}_{l})+\mbox{Tr}(\mathbf{\Phi}_{l}^{H}\mathbf{S}_{l}^{H})+\mbox{Tr}(\mathbf{\Phi}_{l}\mathbf{S}_{l})+c_{l}$
(36c) $\text{s.t.}\quad\quad\quad|\phi_{l}(j)|=1,\quad\forall j,$ (36d)

where $c_{l}$ and $c_{r}$ denote constant terms which are independent of
$\mathbf{\Phi}_{l}$ and $\mathbf{\Phi}_{r}$, respectively. We remark that,
when solving (36a)-(36b), $\mathbf{\Phi}_{l}$ is assumed to fixed, and
similarly, when solving (36c)-(36d), $\mathbf{\Phi}_{r}$ will be assumed to
fixed.

Recall that $\bm{\phi}_{l}$ and $\bm{\phi}_{r}$ are vectors containing the
diagonal elements of $\mathbf{\Phi}_{l}$ and $\mathbf{\Phi}_{r}$,
respectively. As the off-diagonal elements of the $\mathbf{\Phi}_{r}$ are
zero, we wish to maximize the WSR or minimize the MSE with respect to
$\bm{\phi}_{l}$ and $\bm{\phi}_{r}$. For such purpose, from [37, identity
1.10.6] we restate the first trace term appearing in (36a) and (36c) as

$\mbox{Tr}(\mathbf{\Phi}_{r}^{H}\mathbf{Z}_{r}\mathbf{\Phi}_{r}\mathbf{T}_{r})=\bm{\phi}_{r}^{H}\mathbf{\Sigma}_{r}\bm{\phi}_{r},\quad\mbox{where}\;\mathbf{\Sigma}_{r}=\mathbf{Z}_{r}\odot\mathbf{T}_{r}^{T},$
(37a)
$\mbox{Tr}(\mathbf{\Phi}_{l}^{H}\mathbf{Z}_{l}\mathbf{\Phi}_{l}\mathbf{T}_{l})=\bm{\phi}_{l}^{H}\mathbf{\Sigma}_{l}\bm{\phi}_{l},\quad\mbox{where}\;\mathbf{\Sigma}_{l}=\mathbf{Z}_{l}\odot\mathbf{T}_{l}^{T}.$
(37b)

Let $\mathbf{s}_{r}$ and $\mathbf{s}_{l}$ be the vectors containing the
diagonal elements of the matrices $\mathbf{S}_{r}$ (34a) and $\mathbf{S}_{l}$
(35a), respectively, given as
$\mathbf{s}_{r}=[\mathbf{S}_{r}(1,1),...,\mathbf{S}_{r}(R_{r},R_{c})]^{T}$ and
$\mathbf{s}_{l}=[\mathbf{S}_{l}(1,1),...,\mathbf{S}_{l}(R_{r},R_{c})]^{T}$.
The second and third terms appearing in (36a) and (36c) can be restated as

$\mbox{Tr}(\mathbf{\Phi}_{r}^{H}\mathbf{S}_{r}^{H})={\mathbf{s}_{r}}^{H}\bm{\phi}_{r}^{*},\quad\mbox{Tr}(\mathbf{\Phi}_{r}\mathbf{S}_{r})={\mathbf{s}_{r}}^{T}\bm{\phi}_{r}.$
(38a)
$\mbox{Tr}(\mathbf{\Phi}_{l}^{H}\mathbf{S}_{l}^{H})={\mathbf{s}_{l}}^{H}\bm{\phi}_{l}^{*},\quad\mbox{Tr}(\mathbf{\Phi}_{l}\mathbf{S}_{l})={\mathbf{s}_{l}}^{T}\bm{\phi}_{l}.$
(38b)

By using the aforementioned properties, the optimization problems (36a) and
(36c) to optimize $\bm{\phi}_{l}$ and $\bm{\phi}_{r}$, respectively, can be
written as (23).

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