solutionSolutionsolutionfile solutionfile[TwoWayCFmMIMO_OJC_2]

# Multipair Two-Way DF Relaying with Cell-Free Massive MIMO

Anastasios K. Papazafeiropoulos, Pandelis Kourtessis, Symeon Chatzinotas, John
M. Senior A. Papazafeiropoulos is with the Communications and Intelligent
Systems Research Group, University of Hertfordshire, Hatfield AL10 9AB, U. K.,
and with SnT at the University of Luxembourg, Luxembourg. P. Kourtessis and
John M. Senior are with the Communications and Intelligent Systems Research
Group, University of Hertfordshire, Hatfield AL10 9AB, U. K. S. Chatzinotas is
with the SnT at the University of Luxembourg, Luxembourg. E-mails:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider a two-way half-duplex decode-and-forward (DF) relaying system with
multiple pairs of single-antenna users assisted by a cell-free (CF) massive
multiple-input multiple-output (mMIMO) architecture with multiple-antenna
access points (APs). Under the practical constraint of imperfect channel state
information (CSI), we derive the achievable sum spectral efficiency (SE) for a
finite number of APs with maximum ratio (MR) linear processing for both
reception and transmission in closed-form. Notably, the proposed CF mMIMO
relaying architecture, exploiting the spatial diversity, and providing better
coverage, outperforms the conventional collocated mMIMO deployment. Moreover,
we shed light on the power-scaling laws maintaining a specific SE as the
number of APs grows. A thorough examination of the interplay between the
transmit powers per pilot symbol and user/APs takes place, and useful
conclusions are extracted. Finally, differently to the common approach for
power control in CF mMIMO systems, we design a power allocation scheme
maximizing the sum SE.

###### Index Terms:

Two-way relaying, cell-free massive MIMO systems, decode-and-forward, power-
scaling law, beyond 5G MIMO.

##  I Introduction

Massive multiple-input multiple-output (mMIMO) systems, where a large number
of antennas in both collocated and distributed setups serves simultaneously a
lower number of users, has become one of the key fifth-generation (5G)
physical-layer technologies towards higher throughput and energy efficiency by
means of simple linear signal processing [1, 2]. Recently, a distributed mMIMO
architecture under the name of cell-free (CF) mMIMO, enjoying the benefits of
network MIMO, has emerged by providing higher coverage probability and
exploiting the diversity against shadow fading [3]. In particular, a CF mMIMO
system includes a large number of single-antenna access points (APs) that is
connected to a central processing unit (CPU) and serves jointly all users by
means of coherent joint transmission/reception. In this direction, in [4], APs
were equipped with multiple antennas to increase the array and diversity
gains. Generally, CF mMIMO systems outperform small cells (SCs) and collocated
deployments but given that this is an emerging promising architecture at its
infancy, its study is limited [4, 5, 6, 7, 8, 9, 10, 11]. For example, authors
in [5] achieved better data rates by means of a user-centric approach, where
the APs serve a group of users instead of all of them, while in [9], the
realistic spatial randomness of the APs was taken into account by means of a
Poisson point process (PPP) to obtain the coverage probability in CF mMIMO
systems.

Multipair relaying systems, where multiple pairs of users communicate
simultaneously by means of a relay to enhance the network coverage, have been
improved further by the introduction of the mMIMO characteristics, which
enhance the spatial diversity and achieve an order of magnitude spectral
efficiency (SE) improvement [12, 13]. This technique, known as multipair mMIMO
relaying, initially considered one-way transmission by means of amplify-and-
forward (AF) as well as decode-and-forward (DF) protocols [12, 14, 13, 15].
For instance, in the case of DF relaying, the authors in [12] examined the
achievable SE in Rayleigh fading channels for both maximum-ratio (MR) and
zero-forcing (ZF) linear processing, while in [13], optimization of the energy
efficiency was performed. In reference to AF relaying, the power allocation
and max-min user selection was investigated in [14].

Unfortunately, the one-way transmission strategy incurs an SE degradation by
$50\%$ [13, 16]. To reduce this loss, the two-way mMIMO relaying, where
bidirectional simultaneous data transmission applies, has attracted
significant attention [17, 18, 19, 20, 21, 22, 23]. Hence, its main advantage
in comparison to one-way relaying is the reduction of the time required for
information exchange between the user pairs into just two time slots. For
instance, [17] and [18] characterized the power-scaling laws for half-duplex
(HD) and full-duplex (FD) transmission, respectively. Still, first works
assumed only perfect channel state information (CSI), which is highly
unrealistic since practical systems have the availability of only imperfect
CSI. Thus, a training phase with pilot transmission and minimum mean-square-
error (MMSE) estimation at the relay was considered in [19] and [20] for AF
and DF protocols, respectively. Apart from evaluating the impact of imperfect
CSI on the system performance, several other important questions arise. For
example, [21] and [22] investigated the interplay among the transmit powers of
the pilot, user, and relay for DF and AF, respectively. Interestingly,
distributed relaying for multipair two-way channels has been studied but only
for AF and perfect CSI conditions [24, 23].

###  I-A Motivation

This work relies on vital observations: i) Collocated mMIMO relaying in [21]
has not fully taken advantage of the benefits of practical distributed mMIMO
systems. In particular, CF mMIMO systems exploit the spatial diversity,
provide better coverage since the APs are closer to the users, and suppress
the inter-AP interference due to the APs cooperation111Some other differences
between the two are architectures follow. First, in CF mMIMO systems, each AP
obtains its local CSI. In this way, we achieve to alleviate the computational
burden at the CPU. Moreover, CF mMIMO systems achieve increased fairness with
respect to the users since, in CF, it is more possible to have an AP close to
a user. Also, CF achieves lower latency since the APs are closer to the users.
Actually, their advantageous layout is more attractive for mobile edge
computing and caching. Notably, in [8], being one of the references pointing
to the differences between CF mMIMO and single-cell collocated mMIMO systems,
the authors highlighted the differences in the generation of the correlation
matrices and the allocation of the pilots.. ii) Two-way relaying is more
advantageous than one-way transmission. iii) Given that AF undergoes noise
amplification, DF may perform better at a low signal-to-noise ratio (SNR)
[25]. Thus, DF is a more attractive method for mMIMO relaying normally
operating at the low power regime [1]. iv) DF is more flexible than AF in the
case of two-way relaying since it can allow power allocation in both
directions [26]. v) Although some works have studied FD for CF mMIMO systems
[27, 28, 29], FD may not be advisable for CF mMIMO systems due to the large
power level difference of the transmit/received signals in the near/far-field
while the APs might be quite close to the users. vi) The existing literature
on CF mMIMO relaying has not studied at all the insightful power-scaling laws.

###  I-B Contributions and Outcomes

The main contributions are summarized as follows.

  * •

Motivated by the above observations, we establish the theoretical framework
for a two-way CF mMIMO relaying system with imperfect CSI by employing the DF
protocol, where a large number of distributed APs plays the role of a
relay222This framework will be the ground to study other interesting topics on
CF mMIMO systems and two-way relaying such as the impact of imperfect backhaul
links [6].. Notably, the two-way design enhances the CF mMIMO relaying
performance by achieving a significant increase in the SE with comparison to
one-way communication but also CF mMIMO is expected to enhance two-way
transmission due its accompanying advantages compared to collocated mMIMO
systems.

  * •

Contrary to the existing work [21], which has studied two-way mMIMO relaying
with a large number of collocated antennas, we have accounted for the emerging
CF mMIMO architecture at the position of the large collocated deployment.
Notably, although [21] proposes asymptotic approximations of the SE for a
large number of antennas in terms of deterministic equivalents, we consider a
finite number of APs, where the analysis is more general while we also cover
the scenario, where the number of APs grows to infinity333Among other
differences between this and [21], we would like to highlight that the channel
model and the power constraints are different. In our work, the power
constraints concern each AP and the channel model describes the link from each
AP to a user. On the contrary, in [21], the power constraints concern the
single collocated relay, and the channel model describes the link from the
collocated relay to a user. Moreover, our expressions include summations over
the number of APs, corresponding to the contributions from different APs. Such
summations do not appear in [21] and result in simplified equations while in
this work, equations are more complicated and require different manipulations
which are relevant to the CF mMIMO analysis. Also, in this work, the channel
estimation takes place at each AP. On the contrary, in [21], the channel
estimation is performed at the collocated relay.. In addition, contrary to
[23], which assumed the AF protocol and the unrealistic assumption of perfect
CSI, we have considered imperfect CSI and the DF protocol, being more suitable
for CF mMIMO systems. Also, differently to [27, 28, 29] which considered FD
transmission for CF mMIMO systems, we focus on a totally different
architecture being HD two-way transmission, and we investigate the problems of
power-scaling laws and power allocation by maximizing the sum SE. Notably,
previous works on CF considered only the max-min fairness method to obtain the
power control coefficients.

  * •

We derive the achievable sum SE of a two-way CF mMIMO relaying network
employing the DF protocol with imperfect CSI and linear processing by means of
maximum ratio combiner (MRC) for the uplink as well as maximum ratio
transmission (MRT) for the downlink in closed form for a finite number of APs
and we demonstrate better performance over the collocated deployment with a
large number of antennas444The application of other linear techniques such as
zero-forcing presents certain trade-offs between complexity and performance.
For instance, they demand more backhaul, which might be prohibitive in the
case of large distributed networks (CF mMIMO systems). In particular, the
results could also be extended by incorporating the more robust MMSE
processing, as suggested recently in [8]. Given the MMSE intractability, the
extension can take place by simulations or by the deterministic equivalent
analysis [30, 31, 9]. The significance of these observations suggests them to
be an interesting topic of future research.. To the best of our knowledge, no
other prior work has obtained similar expressions under the CF mMIMO relaying
consideration.

  * •

We carry out an asymptotic analysis for this architecture to investigate the
power-scaling laws maintaining a specific SE as the number of APs increases.
Notably, this is the unique work on CF mMIMO relaying that obtains power
scaling laws, which are different compared to the scaling laws in [21] since
they include summations with respect to the number of APs and the relevant
variables correspond to different APs instead of a relay with collocated
antennas. We observe a trade-off among the transmit power of each pilot
symbol, user, and relay, and we shed light on the impact of the scaling
parameters.

  * •

We formulate an optimization problem for CF mMIMO systems, maximizing the sum
SE by keeping constant the total transmit power, in order to obtain the
necessary power control coefficients while previous works on CF were relied on
a power control by using the max-min fairness approach. Numerical results show
the improvement of the sum SE compared to uniform power allocation.

###  I-C Paper Outline

The remainder of this paper is organized as follows. Section II presents the
system model of a two-way CF mMIMO relaying system with multiple antennas APs
employing the DF protocol. This section includes also the channel estimation
and data transmission phases. Section III provides the SE analysis for a
finite number of APs. Section IV presents the power-scaling laws under
different power settings while Section V addresses the optimal power
allocation. The numerical results are discussed in Section VI, and Section VII
concludes the paper.

###  I-D Notation

Vectors and matrices are denoted by boldface lower and upper case symbols,
respectively. The symbols $(\cdot)^{\scriptscriptstyle\mathsf{T}}$,
$(\cdot)^{\scriptscriptstyle\mathsf{H}}$, and $(\cdot)^{*}$ express the
transpose, Hermitian transpose, and conjugate operators, respectively. The
expectation and variance operators are denoted by
$\mathbb{E}\left[\cdot\right]$ and ${\mathrm{var}}\left[\cdot\right]$,
respectively. Also, ${\mathbf{b}}\sim{\cal C}{\cal
N}{({\mathbf{0}},\mathbf{\Sigma})}$ represents a circularly symmetric complex
Gaussian vector with zero mean and covariance matrix $\mathbf{\Sigma}$.

##  II System Model

As illustrated in Fig. 1, we consider a CF mMIMO architecture, where a set of
APs playing the role of distributed relays, assists the exchange of
information in a multipair two-way relaying system. Specifically, we assume
that a set $\mathcal{W}=\\{1,\ldots,W\\}$ of $W=|\mathcal{W}|$ communication
user pairs, consisted of users $\mathrm{T}_{\mathrm{A},i}$ and
$\mathrm{T}_{\mathrm{B},i}$, $i=1,\ldots,W$, is served simultaneously by means
of the set $\mathcal{M}=\\{1,\ldots,M\\}$ of $M=|\mathcal{M}|$ APs in the same
time-frequency resources. Moreover, all APs connect to a CPU via perfect
backhaul links. Certain conditions such as severe shadowing do not allow the
existence of direct links between the user pairs. Also, each AP is equipped
with $N$ antennas, and each user has a single antenna. All the nodes, i.e.,
the user pairs and the APs are randomly distributed in a wide area and operate
in the HD mode.

Figure 1: A multipair two-way CF mMIMO relaying network with $M$ multi-antenna
APs and $W$ user pairs.

The system operation takes place within a coherence interval under the time
division duplex (TDD) protocol where channel reciprocity is met [2]. The data
transmission phase includes two stages, namely, the multiple-access channel
(MAC) and broadcasting (BC) stages. In the former, i.e., the MAC stage, all
users transmit to the APs. In particular, the message, sent by user $i$, is
decoded by means of joint processing at the CPU where all APs have sent their
received signals. In a similar way, during the second stage, all APs transmit
simultaneously to all user pairs.

The channel model includes both small and large-scale fading. Especially, the
large-scale fading describes the effects of shadowing and path-loss. Also, the
large-scaling fading changes slowly, i.e., it can be assumed constant for
several coherence intervals while the small-scale fading stays static during
the duration of a coherence interval but it changes from one interval to the
next. In mathematical terms, the channel vector between
$\mathrm{T}_{\mathrm{A},i}$ and $m$th AP is given by
${\mathbf{h}}_{mi}\in\mathbb{C}^{N\times
1}\sim\mathcal{CN}\left({\mathbf{0}},\alpha_{\mathrm{A},mi}{\bm{\mathrm{I}}}_{N}\right)$,
known as uncorrelated Rayleigh fading, where $\alpha_{\mathrm{A},mi}$
represents the large-scale fading of the corresponding link [3]. Similarly,
the channel between $\mathrm{T}_{\mathrm{B},i}$ and $m$th AP is denoted by
${\mathbf{g}}_{mi}\in\mathbb{C}^{N\times
1}\sim\mathcal{CN}\left({\mathbf{0}},\alpha_{\mathrm{B},mi}{\bm{\mathrm{I}}}_{N}\right)$
with $\alpha_{\mathrm{B},mi}$ being the large-scale fading of this link.

###  II-A Channel Estimation

Given that the propagation channels are piece-wise constant over a coherence
interval, both ${\mathbf{h}}_{mi}$ and ${\mathbf{g}}_{mi}$ need to be
estimated in every interval by means of pilot transmission. Hence, let
$\tau_{\mathrm{c}}$ and $\tau_{\mathrm{p}}$ be the durations of the coherence
interval and the uplink training in symbols
($\tau_{\mathrm{p}}<\tau_{\mathrm{c}}$) [3]. Both $\mathrm{T}_{\mathrm{A},i}$
and $\mathrm{T}_{\mathrm{B},i}$, $i=1\dots,W$, send simultaneously orthogonal
pilot sequences
${\bm{\varphi}}_{\mathrm{A},i}\in\mathbb{C}^{\tau_{\mathrm{p}}\times 1}$ and
${\bm{\varphi}}_{\mathrm{B},i}\in\mathbb{C}^{\tau_{\mathrm{p}}\times 1}$,
respectively555Works on CF mMIMO systems usually assume non-orthogonal pilots
among the users. However, popular works in this area exist that are based on
the assumption of orthogonal pilots e.g., [32]. In this work, which is the
first one studying the multipair two-way transmission with CF mMIMO relaying,
we have assumed orthogonal pilots among users to enable comparison with
existing works on collocated mMIMO systems. Future studies on multipair two-
way transmission could include in the design the use of non-orthogonal
pilots.. Note that this mutual orthogonality requires $\tau_{\mathrm{p}}\geq
2W$,
${\bm{\varphi}}_{\mathrm{A},i}^{{\scriptscriptstyle\mathsf{H}}}{\bm{\varphi}}_{\mathrm{B},i}=0$,
and
${\bm{\varphi}}_{\mathrm{A},i}^{{\scriptscriptstyle\mathsf{H}}}{\bm{\varphi}}_{\mathrm{A},j}={\bm{\varphi}}_{\mathrm{B},i}^{{\scriptscriptstyle\mathsf{H}}}{\bm{\varphi}}_{\mathrm{B},j}=0,~{}\forall
i\neq j$ [12]. In addition, we assume
$\|{\bm{\varphi}}_{\mathrm{A},i}\|^{2}=\|{\bm{\varphi}}_{\mathrm{B},i}\|^{2}=1$.
Thus, the $m$th AP receives

$\displaystyle{\mathbf{Y}}_{\mathrm{p},m}=\sqrt{\tau_{\mathrm{p}}p_{\mathrm{p}}}\sum_{i=1}^{W}\left({\mathbf{h}}_{mi}{\bm{\varphi}}_{\mathrm{A},i}^{{\scriptscriptstyle\mathsf{H}}}+{\mathbf{g}}_{mi}{\bm{\varphi}}_{\mathrm{B},i}^{{\scriptscriptstyle\mathsf{H}}}\right)+{\mathbf{W}}_{\mathrm{p},m},$
(1)

where ${\mathbf{W}}_{\mathrm{p},m}$ is an $N\times\tau_{\mathrm{p}}$ matrix
describing the additive white Gaussian noise (AWGN) and having i.i.d elements
distributed as $\mathcal{CN}\left(0,1\right)$. Also, $p_{\mathrm{p}}$
expresses the normalized transmit signal-to-noise ratio (SNR) of each pilot
symbol.

Following the approach in [12], we obtain the MMSE estimated channels for the
$i$th pairbased on [33] as

$\displaystyle{\mathbf{h}}_{mi}$
$\displaystyle=\hat{{\mathbf{h}}}_{mi}+\tilde{{\mathbf{h}}}_{mi}$ (2)
$\displaystyle{\mathbf{g}}_{mi}$
$\displaystyle=\hat{{\mathbf{g}}}_{mi}+\tilde{{\mathbf{g}}}_{mi},$ (3)

where
$\hat{{\mathbf{h}}}_{mi}\sim\mathcal{CN}\left({\mathbf{0}},\phi_{\mathrm{A},mi}{\bm{\mathrm{I}}}_{N}\right)$
and
$\tilde{{\mathbf{h}}}_{mi}\sim\mathcal{CN}\left({\mathbf{0}},{e}_{\mathrm{A},mi}{\bm{\mathrm{I}}}_{N}\right)$
are the estimated and estimation error channel vectors, being mutually
independent with
$\phi_{\mathrm{A},mi}=\frac{\tau_{\mathrm{p}}p_{\mathrm{p}}\alpha^{2}_{\mathrm{A},mi}}{1+\tau_{\mathrm{p}}p_{\mathrm{p}}\alpha_{\mathrm{A},mi}}$
and
$e_{\mathrm{A},mi}=\frac{\alpha_{\mathrm{A},mi}}{1+\tau_{\mathrm{p}}p_{\mathrm{p}}\alpha_{\mathrm{A},mi}}$,
respectively. Similar expressions hold for the estimated channel and
estimation error vectors in (3), i.e.,
$\hat{{\mathbf{g}}}_{mi}\sim\mathcal{CN}\left({\mathbf{0}},\phi_{\mathrm{B},mi}{\bm{\mathrm{I}}}_{N}\right)$
and
$\tilde{{\mathbf{g}}}_{mi}\sim\mathcal{CN}\left({\mathbf{0}},e_{\mathrm{B},mi}{\bm{\mathrm{I}}}_{N}\right)$
are mutually independent with
$\phi_{\mathrm{B},mi}=\frac{\tau_{\mathrm{p}}p_{\mathrm{p}}\alpha^{2}_{\mathrm{B},mi}}{1+\tau_{\mathrm{p}}p_{\mathrm{p}}\alpha_{\mathrm{B},mi}}$
and
$e_{\mathrm{B},mi}=\frac{\alpha_{\mathrm{B},mi}}{1+\tau_{\mathrm{p}}p_{\mathrm{p}}\alpha_{\mathrm{B},mi}}$,
respectively.

###  II-B Data Transmission

The communication takes place in two phases, as described next.

####  II-B1 Phase I

It is known as a MAC phase, where all user pairs simultaneously transmit their
data to all relay nodes, being the APs. In other words, the received signal by
AP $m$ from the $W$ pairs $\mathrm{T}_{\mathrm{A},i}$,
$\mathrm{T}_{\mathrm{B},i}$ is given by

$\displaystyle\\!\\!{\mathbf{y}}_{m}\\!=\\!\sqrt{p_{\mathrm{u}}}\sum_{i=1}^{W}\\!\left(\sqrt{\eta_{\mathrm{A},i}}{\mathbf{h}}_{mi}q_{\mathrm{A},i}+\sqrt{\eta_{\mathrm{B},i}}{\mathbf{g}}_{mi}q_{\mathrm{B},i}\right)+{\mathbf{n}}_{\mathrm{u},m},$
(4)

where $q_{\mathrm{A},i}$ and $q_{\mathrm{B},i}$ are the data by the $i$th user
pair weighted by the power control coefficients ${\eta_{\mathrm{A},i}}$ and
${\eta_{\mathrm{B},i}}$, respectively. Note that
$\mathbb{E}\\{|q_{\mathrm{A},i}|^{2}\\}=\mathbb{E}\\{|q_{\mathrm{B},i}|^{2}\\}=1$
while $0\leq\eta_{\mathrm{A},i}\leq 1$ and $0\leq\eta_{\mathrm{B},i}\leq 1$.
Also, $p_{\mathrm{u}}$ denotes the normalized uplink SNR and
${\mathbf{n}}_{\mathrm{u},m}\sim\mathcal{CN}\left({\mathbf{0}},{\bm{\mathrm{I}}}_{N}\right)$
is the AWGN vector at AP $m$.

By accounting for linear detection by means of the linear receiver matrix
${\mathbf{W}}_{m}=\left[{\mathbf{W}}_{m,\mathrm{A}},{\mathbf{W}}_{m,\mathrm{B}}\right]\in\mathbb{C}^{N\times
2W}$, the $m$th AP multiplies its received signal ${\mathbf{y}}_{m}$ with the
transpose of the linear detector and obtains the post-processed signals as

$\displaystyle{\mathbf{r}}_{m}=\begin{bmatrix}{\mathbf{W}}_{m,\mathrm{A}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{y}}_{m}\\\
{\mathbf{W}}_{m,\mathrm{B}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{y}}_{m}\end{bmatrix}.$
(5)

Herein, we assume application of MRC due to its low complexity. Also, it is
suggested for implementation in a distributed fashion (locally at the APs),
and it results in simplified expressions as suggested initially in [3]. Note
that the top $W$ rows of ${\mathbf{r}}_{m}$ correspond to $W$ signals from
$\mathrm{T}_{\mathrm{A}}$ while the remaining $W$ bottom rows stand for the
signals from $\mathrm{T}_{\mathrm{B},i}$ $\left(i=1,\ldots,W\right)$. Next,
all APs sent their processed signals to the CPU through perfect backhaul links
and the CPU obtains

$\displaystyle{\mathbf{r}}$ $\displaystyle=\sum_{m=1}^{M}{\mathbf{r}}_{m}$ (6)
$\displaystyle=\begin{bmatrix}\sum_{m=1}^{M}{\mathbf{W}}_{m,\mathrm{A}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{y}}_{m}\\\
\sum_{m=1}^{M}{\mathbf{W}}_{m,\mathrm{B}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{y}}_{m}\end{bmatrix}.$
(7)

Now, we focus on the detection of the transmitted symbols of the $i$th user
pair at the CPU. Specifically, we denote

$\displaystyle r_{m_{i}}$
$\displaystyle=\sum_{m=1}^{M}({\mathbf{W}}_{m,\mathrm{A}}^{{\scriptscriptstyle\mathsf{H}}})_{i}{\mathbf{y}}_{m}$
(8) $\displaystyle r_{m_{W+i}}$
$\displaystyle=\sum_{m=1}^{M}({\mathbf{W}}_{m,\mathrm{B}}^{{\scriptscriptstyle\mathsf{H}}})_{i}{\mathbf{y}}_{m}$
(9)

where $r_{m_{i}}$ and $r_{m_{W+i}}$ are the detected symbols from
$\mathrm{T}_{\mathrm{A},i}$ and $\mathrm{T}_{\mathrm{B},i}$ , respectively.
Note that $({\mathbf{X}})_{i}$ denotes the $i$th row of ${\mathbf{X}}$. Thus,
the total detected symbols at the CPU from both $\mathrm{T}_{\mathrm{A},i}$
and $\mathrm{T}_{\mathrm{A},i}$ are given by

$\displaystyle\tilde{r}_{i}$
$\displaystyle=\sum_{m=1}^{M}r_{m_{i}}+r_{m_{W+i}}$ (10)
$\displaystyle=\sum_{m=1}^{M}\left(\hat{{\mathbf{h}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}+\hat{{\mathbf{g}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}\right){\mathbf{y}}_{m}.$
(11)

####  II-B2 Phase II

This phase includes encoding of the received information and broadcasting it
to all user pairs. Specifically, AP $m$ applies linear precoding matrices in
terms of MRT given by $\hat{{\mathbf{G}}}_{m}$ and $\hat{{\mathbf{H}}}_{m}$ to
transmit the signals ${\mathbf{q}}_{\mathrm{A}}$ and
${\mathbf{q}}_{\mathrm{B}}$ to $\mathrm{T}_{\mathrm{A},i}$ and
$\mathrm{T}_{\mathrm{B},i}$ by using the power control coefficients
${\bm{\eta}}_{\mathrm{A},m}$ and ${\bm{\eta}}_{\mathrm{B},m}$, respectively.
Thus, the transmit signal is written as

$\displaystyle{\mathbf{x}}_{m}=\sqrt{p_{\mathrm{d}}}\left(\hat{{\mathbf{G}}}_{m}^{*}{\bm{\eta}}_{\mathrm{A},m}^{1/2}{\mathbf{q}}_{\mathrm{A}}+\hat{{\mathbf{H}}}_{m}^{*}{\bm{\eta}}_{\mathrm{B},m}^{1/2}{\mathbf{q}}_{\mathrm{B}}\right),$
(12)

where $p_{\mathrm{d}}$ is the maximum normalized transmit power while for
notational convenience, we have denoted
$\hat{{\mathbf{G}}}_{m}=\left[\hat{{\mathbf{g}}}_{m1},\ldots,\hat{{\mathbf{g}}}_{mW}\right]\in\mathbb{C}^{N\times
W}$,
$\hat{{\mathbf{H}}}_{m}=\left[\hat{{\mathbf{h}}}_{m1},\ldots,\hat{{\mathbf{h}}}_{mW}\right]\in\mathbb{C}^{N\times
W}$,
${\bm{\eta}}_{\mathrm{A},m}=\text{diag}\left(\right)\left(\eta_{\mathrm{A},m1},\ldots,\eta_{\mathrm{A},mW}\right)$,
${\bm{\eta}}_{\mathrm{B},m}=\text{diag}\left(\right)\left(\eta_{\mathrm{B},m1},\ldots,\eta_{\mathrm{B},mW}\right)$,
${\mathbf{q}}_{\mathrm{A}}=\left[q_{\mathrm{A},1},\ldots,q_{\mathrm{A},W}\right]^{{\scriptscriptstyle\mathsf{T}}}$,
and
${\mathbf{q}}_{\mathrm{B}}=\left[q_{\mathrm{B},1},\ldots,q_{\mathrm{B},W}\right]^{{\scriptscriptstyle\mathsf{T}}}$.
Note that the power control coefficients are chosen to satisfy the power
constraint, $\mathbb{E}\\{\|{\mathbf{x}}_{m}\|^{2}\\}\leq p_{\mathrm{d}}$,
which gives

$\displaystyle
N\sum_{i=1}^{W}\left(\eta_{\mathrm{A},mi}\phi_{\mathrm{B},mi}+\eta_{\mathrm{B},mi}\phi_{\mathrm{A},mi}\right)\leq
1.$ (13)

Also, we have a total power constraint for all APs acting as relay nodes,
i.e., $\mathbb{E}\\{\sum^{M}_{m=1}\|{\mathbf{x}}_{m}\|^{2}\\}=p_{\mathrm{r}}$,
which results in

$\displaystyle
p_{\mathrm{d}}=\frac{p_{\mathrm{r}}}{N\sum_{i=1}^{W}\sum^{M}_{m=1}\left(\eta_{\mathrm{A},mi}\phi_{\mathrm{B},mi}+\eta_{\mathrm{B},mi}\phi_{\mathrm{A},mi}\right)}.$
(14)

The received signal at $\mathrm{T}_{\mathrm{\mathrm{A}},i}$ by all APs is
written as

$\displaystyle
z_{\mathrm{A},i}=\sum_{m=1}^{M}{\mathbf{h}}_{mi}^{{\scriptscriptstyle\mathsf{T}}}{\mathbf{x}}_{m}+n_{\mathrm{A},i}$
$\displaystyle\\!=\\!\sqrt{p_{\mathrm{d}}}\\!\sum_{m=1}^{M}\\!\sum_{j=1}^{W}{\mathbf{h}}_{mi}^{{\scriptscriptstyle\mathsf{T}}}\\!\left(\\!\eta_{\mathrm{A},mj}^{1/2}\hat{{\mathbf{g}}}_{mj}^{*}q_{\mathrm{A},j}\\!+\\!\eta_{\mathrm{B},mj}^{1/2}\hat{{\mathbf{h}}}_{mj}^{*}q_{\mathrm{B},j}\\!\right)\\!+\\!n_{\mathrm{A},i}.$

Similarly, $\mathrm{T}_{\mathrm{\mathrm{B}},i}$ receives by all APs

$\displaystyle z_{\mathrm{B},i}$
$\displaystyle\\!=\\!\sqrt{p_{\mathrm{d}}}\\!\sum_{m=1}^{M}\\!\sum_{i=1}^{W}\\!{\mathbf{g}}_{mi}^{{\scriptscriptstyle\mathsf{T}}}\\!\left(\\!\eta_{\mathrm{A},mj}^{1/2}\hat{{\mathbf{g}}}_{mj}^{*}q_{\mathrm{A},j}\\!+\\!\eta_{\mathrm{B},mj}^{1/2}\hat{{\mathbf{h}}}_{mj}^{*}q_{\mathrm{B},j}\\!\right)\\!$
$\displaystyle+\\!n_{\mathrm{B},i}.$ (15)

Note that
${\mathbf{n}}_{\mathrm{X},i}\sim\mathcal{CN}\left({\mathbf{0}},{\bm{\mathrm{I}}}_{N}\right)$
is the additive noise at $\mathrm{T}_{\mathrm{\mathrm{X}},i}$
($\mathrm{X}\in\\{\mathrm{A},\mathrm{B}\\}$).

##  III SE Analysis

This section presents the SE performance analysis of the DF two-way CF mMIMO
with MRC/MRT linear processing by means of exact and closed-form expressions.

$\displaystyle\tilde{r}_{i}$
$\displaystyle=\sqrt{p_{\mathrm{u}}}\bigg{(}\underbrace{\sqrt{\eta_{\mathrm{A},i}}\mathbb{E}\Big{\\{}\sum_{m=1}^{M}\left(\hat{{\mathbf{h}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{h}}_{mi}+\hat{{\mathbf{g}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{h}}_{mi}\right)\Big{\\}}q_{\mathrm{A},i}+\sqrt{\eta_{\mathrm{B},i}}\mathbb{E}\Big{\\{}\sum_{m=1}^{M}\left(\hat{{\mathbf{h}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{g}}_{mi}+{\mathbf{g}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}\hat{{\mathbf{g}}}_{mi}\right)\Big{\\}}q_{\mathrm{B},i}}_{\text{desired
signal}}$
$\displaystyle+\underbrace{\sqrt{\eta_{\mathrm{A},i}}\sum_{m=1}^{M}\left(\hat{{\mathbf{h}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{h}}_{mi}+\hat{{\mathbf{g}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{h}}_{mi}\right)q_{\mathrm{A},i}-\sqrt{\eta_{\mathrm{A},i}}\mathbb{E}\Big{\\{}\sum_{m=1}^{M}\left(\hat{{\mathbf{h}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{h}}_{mi}+\hat{{\mathbf{g}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{h}}_{mi}\right)\Big{\\}}q_{\mathrm{A},i}}_{\text{estimation
error,~{}A}}$
$\displaystyle+\sqrt{\eta_{\mathrm{B},i}}\underbrace{\sum_{m=1}^{M}\sum_{m=1}^{M}\left(\hat{{\mathbf{h}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{g}}_{mi}+\hat{{\mathbf{g}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{g}}_{mi}\right)q_{\mathrm{B},i}-\sqrt{\eta_{\mathrm{B},i}}\mathbb{E}\Big{\\{}\sum_{m=1}^{M}\left(\hat{{\mathbf{h}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{g}}_{mi}+\hat{{\mathbf{g}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{g}}_{mi}\right)\Big{\\}}q_{\mathrm{B},i}}_{\text{estimation
error,~{}B}}$ $\displaystyle+\underbrace{\sum_{j\neq
i}^{W}\sum_{m=1}^{M}\sqrt{\eta_{\mathrm{A},j}}\left(\hat{{\mathbf{h}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{h}}_{mj}+\hat{{\mathbf{g}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{h}}_{mj}\right)q_{\mathrm{A},j}+\sum_{j\neq
i}^{W}\sum_{m=1}^{M}\sqrt{\eta_{\mathrm{B},j}}\left(\hat{{\mathbf{h}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{g}}_{mj}+\hat{{\mathbf{g}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{g}}_{mj}\right)q_{\mathrm{B},j}}_{\text{inter-
user interference}}\bigg{)}$
$\displaystyle+\underbrace{\sum_{m=1}^{M}\left(\hat{{\mathbf{h}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}+\hat{{\mathbf{g}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}\right){\mathbf{n}}_{\mathrm{u},m}}_{\text{post-
pocessed noise}}.$ (16)

####  III-1 Phase I

The received signal, given by (11) can be written as in (16) at the top of
next page after substituting the received signal given by (4). Also, we have
used (2) and (3) since the $m$th AP has imperfect CSI and considers the
estimated channels as its true channels. Taking as a reference the $i$th user
pair, we obtain its achievable SE in the MAC phase by means of the use-and-
then-forget capacity bounding technique where the CPU uses only statistical
knowledge of the channel when performing the detection and the unknown terms
are treated as uncorrelated additive noise [2, Ch. 3], [34]. Note that, in the
case of mMIMO, this bound exploits channel hardening and becomes tighter as
the number of antennas increases. Relied on this assumption, many works in CF
mMIMO exploited that channel hardening appears in the case of a large number
of APs. However, in [35], it was shown that, in general, channel hardening is
not met in CF mMIMO systems with single-antenna APs, but it appears if multi-
antenna APs are considered. Numerical results in Section VI, relied on this
assumption, verify the tightness of this bound. Thus, the achievable SE is
given by

$\displaystyle
R_{i}^{\text{MAC}}=\frac{\tau_{\mathrm{c}}-\tau_{\mathrm{p}}}{2\tau_{\mathrm{c}}}\log_{2}\left(1+\gamma_{i}^{\text{MAC}}\right),$
(17)

where the corresponding SINR is given by

$\displaystyle\gamma_{i}^{\text{MAC}}=\frac{\mathrm{DS}_{A_{i}}^{\text{MAC}}+\mathrm{DS}_{B_{i}}^{\text{MAC}}}{\mathrm{EE}_{\mathrm{A},i}^{\text{MAC}}+\mathrm{EE}_{\mathrm{B},i}^{\text{MAC}}+\mathrm{IUI}_{i}^{\text{MAC}}+\mathrm{N}_{i}^{\text{MAC}}}$
(18)

with the various terms provided by

$\displaystyle\mathrm{DS}_{A_{i}}^{\text{MAC}}$
$\displaystyle=\eta_{\mathrm{A},i}\Big{|}\mathbb{E}\Big{\\{}\sum_{m=1}^{M}\left(\hat{{\mathbf{h}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{h}}_{mi}+\hat{{\mathbf{g}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{h}}_{mi}\right)\\!\\!\Big{\\}}\Big{|}^{2},$
(19) $\displaystyle\mathrm{DS}_{B_{i}}^{\text{MAC}}$
$\displaystyle=\eta_{\mathrm{B},i}\Big{|}\mathbb{E}\Big{\\{}\sum_{m=1}^{M}\left(\hat{{\mathbf{h}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{g}}_{mi}+\hat{{\mathbf{g}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{g}}_{mi}\right)\\!\\!\Big{\\}}\Big{|}^{2},$
(20) $\displaystyle\mathrm{EE}_{\mathrm{A},i}^{\text{MAC}}$
$\displaystyle=\eta_{\mathrm{A},i}{\mathrm{var}}\Big{\\{}\sum_{m=1}^{M}\left(\hat{{\mathbf{h}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{h}}_{mi}+\hat{{\mathbf{g}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{h}}_{mi}\right)\\!\\!\Big{\\}},$
$\displaystyle\mathrm{EE}_{\mathrm{B},i}^{\text{MAC}}$
$\displaystyle=\eta_{\mathrm{B},i}{\mathrm{var}}\Big{\\{}\sum_{m=1}^{M}\sqrt{\eta_{\mathrm{B},i}}\left(\hat{{\mathbf{h}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{g}}_{mi}+\hat{{\mathbf{g}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{g}}_{mi}\right)\\!\\!\Big{\\}},$
$\displaystyle\mathrm{IUI}_{i}^{\text{MAC}}$ $\displaystyle=\sum_{j\neq
i}^{W}\eta_{\mathrm{A},j}\mathbb{E}\Big{\\{}\Big{|}\sum_{m=1}^{M}\hat{{\mathbf{h}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{h}}_{mj}\Big{|}^{2}+\Big{|}\sum_{m=1}^{M}\hat{{\mathbf{g}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{h}}_{mj}\Big{|}^{2}\Big{\\}}$
$\displaystyle\\!\\!\\!+\sum_{j\neq
i}^{W}\eta_{\mathrm{B},j}\mathbb{E}\Big{\\{}\Big{|}\sum_{m=1}^{M}\hat{{\mathbf{h}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{g}}_{mj}\Big{|}^{2}+\Big{|}\sum_{m=1}^{M}\hat{{\mathbf{g}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{g}}_{mj}\Big{|}^{2}\Big{\\}},$
(21) $\displaystyle\mathrm{N}_{i}^{\text{MAC}}$
$\displaystyle=\frac{1}{p_{\mathrm{u}}}\mathbb{E}\Big{\\{}\|\sum_{m=1}^{M}\hat{{\mathbf{h}}}_{mi}\|^{2}+\|\sum_{m=1}^{M}\hat{{\mathbf{g}}}_{mi}\|^{2}\Big{\\}}.$
(22)

Note that $\mathrm{DS}$, $\mathrm{EE}$, $\mathrm{IUI}_{ik}$, and $\mathrm{N}$
express the desired signal (DS) part, the estimation error (EE) part, the
inter-user interference (IUI), and the thermal noise and can refer to both MAC
and BC phases as well as users $\mathrm{T}_{\mathrm{A},i}$ and
$\mathrm{T}_{\mathrm{B},i}$. Moreover, the achievable SE of the link
$T_{\mathrm{X},i}$ where $\mathrm{X}\in\\{\mathrm{A},\mathrm{B}\\}$, is
written as

$\displaystyle
R_{\mathrm{X},i}^{\text{MAC}}=\frac{\tau_{\mathrm{c}}-\tau_{\mathrm{p}}}{2\tau_{\mathrm{c}}}\log_{2}\left(1+\gamma_{\mathrm{X},i}^{\text{MAC}}\right)$
(23)

with signal-to-interference-plus-noise ratio (SINR) given by

$\displaystyle\gamma_{\mathrm{X},i}^{\text{MAC}}=\frac{\mathrm{DS}_{\mathrm{X}_{i}}^{\text{MAC}}}{\mathrm{EE}_{\mathrm{A},i}^{\text{MAC}}+\mathrm{IUI}_{i}^{\text{MAC}}+\mathrm{N}_{i}^{\text{MAC}}}.$
(24)

####  III-2 Phase II

Since in practice, the users are not aware of the instantaneous CSI, we take
advantage again of the channel hardening and the use-and-then-forget bound
from the massive MIMO (mMIMO) literature [34]. Thus, we assume that user
$\mathrm{T}_{\mathrm{A},i}$ has knowledge only of its statistics and performs
partial self-interference cancellation to obtain

$\displaystyle\hat{z}_{\mathrm{A},i}=z_{\mathrm{A},i}-\sqrt{p_{\mathrm{d}}}\mathbb{E}\Big{\\{}\sum_{m=1}^{M}\sqrt{\eta_{\mathrm{A},mi}}{\mathbf{h}}_{mi}^{{\scriptscriptstyle\mathsf{T}}}\hat{{\mathbf{g}}}_{mi}^{*}\Big{\\}}q_{\mathrm{A},i}$
(25)
$\displaystyle=\sqrt{p_{\mathrm{d}}}\mathbb{E}\Big{\\{}\sum_{m=1}^{M}\sqrt{\eta_{\mathrm{B},mi}}{\mathbf{h}}_{mi}^{{\scriptscriptstyle\mathsf{T}}}\hat{{\mathbf{h}}}_{mi}^{*}\Big{\\}}q_{\mathrm{B},i}$
$\displaystyle\\!+\\!\sqrt{p_{\mathrm{d}}}\\!\left(\sum_{m=1}^{M}\\!\\!\\!\sqrt{\eta_{\mathrm{B},mi}}{\mathbf{h}}_{mi}^{{\scriptscriptstyle\mathsf{T}}}\hat{{\mathbf{h}}}_{mi}^{*}\\!-\\!\mathbb{E}\Bigg{\\{}\\!\sum_{m=1}^{M}\\!\\!\sqrt{\eta_{\mathrm{B},mi}}{\mathbf{h}}_{mi}^{{\scriptscriptstyle\mathsf{T}}}\hat{{\mathbf{h}}}_{mi}^{*}\\!\\!\Bigg{\\}}\\!\\!\right)\\!\\!q_{\mathrm{B},i}$
$\displaystyle\\!+\\!\sqrt{p_{\mathrm{d}}}\\!\left(\sum_{m=1}^{M}\\!\\!\\!\sqrt{\eta_{\mathrm{A},mi}}{\mathbf{h}}_{mi}^{{\scriptscriptstyle\mathsf{T}}}\hat{{\mathbf{g}}}_{mi}^{*}\\!-\\!\mathbb{E}\Bigg{\\{}\\!\sum_{m=1}^{M}\\!\\!\sqrt{\eta_{\mathrm{A},mi}}{\mathbf{h}}_{mi}^{{\scriptscriptstyle\mathsf{T}}}\hat{{\mathbf{g}}}_{mi}^{*}\\!\\!\Bigg{\\}}\\!\\!\right)\\!\\!q_{\mathrm{A},i}$
$\displaystyle+\sqrt{p_{\mathrm{d}}}\sum_{j\neq
i}^{W}\\!\\!\sum_{m=1}^{M}\left(\\!\sqrt{\eta_{\mathrm{A},mj}}{\mathbf{h}}_{mi}^{{\scriptscriptstyle\mathsf{T}}}\hat{{\mathbf{g}}}_{mj}^{*}q_{\mathrm{A},j}+\sqrt{\eta_{\mathrm{B},mj}}{\mathbf{h}}_{mi}^{{\scriptscriptstyle\mathsf{T}}}\hat{{\mathbf{h}}}_{mj}^{*}q_{\mathrm{B},j}\\!\right)$
$\displaystyle+n_{\mathrm{A},i},$ (26)

where the first term expresses the desired signal, the second and third terms
express the gain uncertainty, the fourth and fifth terms represent the
residual self-interference, the sixth and seventh terms describe the inter-
pair interference while the last term denotes the noise. The achievable SE of
$\mathrm{T}_{\mathrm{A},i}$ during the BC phase is obtained by

$\displaystyle
R_{\mathrm{A},i}^{\text{BC}}=\frac{\tau_{\mathrm{c}}-\tau_{\mathrm{p}}}{2\tau_{\mathrm{c}}}\log_{2}\left(1+\gamma_{\mathrm{A},i}^{\text{BC}}\right)\\!,$
(27)

where $\gamma_{\mathrm{A},i}^{\text{BC}}$ is given by

$\displaystyle\\!\\!\\!\gamma_{\mathrm{A},i}^{\text{BC}}=\frac{\mathrm{DS}_{A_{i}}^{\text{BC}}}{\mathrm{BU}_{A_{i}}^{\text{BC}}+\mathrm{BU}_{B_{i}}^{\text{BC}}+\sum_{j\neq
i}\\!\left(\mathrm{IUI}_{A_{j}}^{\text{BC}}+\mathrm{IUI}_{B_{j}}^{\text{BC}}\right)+\frac{1}{p_{\mathrm{d}}}}$
(28)

with

$\displaystyle\mathrm{DS}_{A_{i}}^{\text{BC}}$
$\displaystyle=\Big{|}\mathbb{E}\Big{\\{}\sum_{m=1}^{M}\sqrt{\eta_{\mathrm{B},mi}}{\mathbf{h}}_{mi}^{{\scriptscriptstyle\mathsf{T}}}\hat{{\mathbf{h}}}_{mi}^{*}\Big{\\}}\Big{|}^{2},$
(29) $\displaystyle\mathrm{BU}_{A_{i}}^{\text{BC}}$
$\displaystyle={\mathrm{var}}\Big{\\{}\sum_{m=1}^{M}\sqrt{\eta_{\mathrm{B},mi}}{\mathbf{h}}_{mi}^{{\scriptscriptstyle\mathsf{T}}}\hat{{\mathbf{h}}}_{mi}^{*}\Big{\\}},$
(30) $\displaystyle\mathrm{BU}_{B_{i}}^{\text{BC}}$
$\displaystyle={\mathrm{var}}\Big{\\{}\sum_{m=1}^{M}\sqrt{\eta_{\mathrm{A},mi}}{\mathbf{h}}_{mi}^{{\scriptscriptstyle\mathsf{T}}}\hat{{\mathbf{g}}}_{mi}^{*}\Big{\\}},$
(31) $\displaystyle\mathrm{IUI}_{A_{i}}^{\text{BC}}$
$\displaystyle=\mathbb{E}\Big{\\{}\Big{|}\sum_{m=1}^{M}\sqrt{\eta_{\mathrm{A},mj}}{\mathbf{h}}_{mi}^{{\scriptscriptstyle\mathsf{T}}}\hat{{\mathbf{g}}}_{mj}^{*}\Big{|}^{2}\Big{\\}},$
(32) $\displaystyle\mathrm{IUI}_{B_{i}}^{\text{BC}}$
$\displaystyle=\mathbb{E}\Big{\\{}\Big{|}\sum_{m=1}^{M}\sqrt{\eta_{\mathrm{B},mj}}{\mathbf{h}}_{mi}^{{\scriptscriptstyle\mathsf{T}}}\hat{{\mathbf{h}}}_{mj}^{*}\Big{|}^{2}\Big{\\}}.$
(33)

Note that $\mathrm{BU}$ refers to the beamforming gain uncertainty (BU).
Similarly, we obtain the achievable SE of $\mathrm{T}_{\mathrm{B},i}$,
$R_{\mathrm{B},i}^{\text{BC}}$, after obtaining the post-processed signal at
user $\mathrm{T}_{\mathrm{B},i}$, $\hat{z}_{\mathrm{B},i}$, by means of a
similar expression to (26). Hence, the achievable SE of the $i$th pair
$\mathrm{T}_{\mathrm{A},i}$ to $\mathrm{T}_{\mathrm{B},i}$ is given by
$\min\left(R_{\mathrm{A},i}^{\text{MAC}},R_{\mathrm{B},i}^{\text{BC}}\right)$
while the SE for the opposite direction is
$\min\left(R_{\mathrm{B},i}^{\text{MAC}},R_{\mathrm{A},i}^{\text{BC}}\right)$
with the individual SEs obtained previously. As a result, the achievable SE of
of the $i$th pair during BC is given by the sum

$\displaystyle
R_{i}^{\text{BC}}=\min\left(R_{\mathrm{A},i}^{\text{MAC}},R_{\mathrm{B},i}^{\text{BC}}\right)+\min\left(R_{\mathrm{B},i}^{\text{MAC}},R_{\mathrm{A},i}^{\text{BC}}\right).$
(34)

According to [16, 26], the achievable SE of the $i$th pair over both phases is
given by (17) and (34) as

$\displaystyle
R_{i}=\min\left(R_{i}^{\text{MAC}},R_{i}^{\text{BC}}\right)\\!,$ (35)

and the achievable sum SE of a multipair two-way CF mMIMO relaying system is
given by

$\displaystyle R=\sum_{i=1}^{W}R_{i}.$ (36)

###### Theorem 1

The achievable sum SE of a multipair two-way CF mMIMO relaying system with DF
protocol and MRC/MRT linear processing, for any finite $M$ and $W$, is given
by (36) including the SEs provided by (37)-(39) at the top of the next page
with $\bar{\mathrm{X}}$ being the complement of
$\mathrm{X}\in\\{\mathrm{A},\mathrm{B}\\}$.

$\displaystyle R_{i}^{\text{MAC}}$
$\displaystyle=\frac{\tau_{\mathrm{c}}-\tau_{\mathrm{p}}}{2\tau_{\mathrm{c}}}\log_{2}\left(1+\frac{p_{\mathrm{u}}N\left(\left(\sum_{m=1}^{M}\sqrt{\eta_{\mathrm{A},i}}\phi_{\mathrm{A},mi}\right)^{2}+\left(\sum_{m=1}^{M}\sqrt{\eta_{\mathrm{B},i}}\phi_{\mathrm{B},mi}\right)^{2}\right)}{\sum_{m=1}^{M}\left(\sum_{j=1}^{W}p_{\mathrm{u}}\left(\eta_{\mathrm{A},j}\alpha_{\mathrm{A},mj}+\eta_{\mathrm{B},j}\alpha_{\mathrm{B},mj}\right)+1\right)\left(\phi_{\mathrm{A},mi}+\phi_{\mathrm{B},mi}\right)}\right)\\!,$
(37) $\displaystyle R_{\mathrm{X},i}^{\text{MAC}}$
$\displaystyle=\frac{\tau_{\mathrm{c}}-\tau_{\mathrm{p}}}{2\tau_{\mathrm{c}}}\log_{2}\left(1+\frac{p_{\mathrm{u}}\eta_{\mathrm{A},i}N\left(\sum_{m=1}^{M}\phi_{\mathrm{X},mi}\right)^{2}}{\sum_{m=1}^{M}\left(\sum_{j=1}^{W}p_{\mathrm{u}}\left(\eta_{\mathrm{A},j}\alpha_{\mathrm{A},mj}+\eta_{\mathrm{B},j}\alpha_{\mathrm{B},mj}\right)+1\right)\left(\phi_{\mathrm{A},mi}+\phi_{\mathrm{B},mi}\right)}\right)\\!,$
(38) $\displaystyle R_{\mathrm{X},i}^{\text{BC}}$
$\displaystyle=\frac{\tau_{\mathrm{c}}-\tau_{\mathrm{p}}}{2\tau_{\mathrm{c}}}\log_{2}\left(1+\frac{Np_{\mathrm{r}}\left(\sum_{m=1}^{M}\sqrt{\eta_{\mathrm{\bar{X}},mi}}\phi_{\mathrm{X},mi}\right)^{2}}{\sum_{m=1}^{M}\sum_{j=1}^{W}\left(p_{\mathrm{r}}\alpha_{\mathrm{X},mi}+1\right)\left(\eta_{\mathrm{A},mj}\phi_{\mathrm{B},mj}+\eta_{\mathrm{B},mj}\phi_{\mathrm{A},mj}\right)}\right)\\!.$
(39)

###### Proof:

See Appendix A. ∎

Notably, our procedure results in exact closed-form results while other
similar works rely on approximations. Also, regarding the dependence of the
individual SEs with respect to the transmit power, we observe that they are
interference-limited as expected [21]. Furthermore, we notice an increase of
the SEs with the number of APs by taking advantage of the CF mMIMO
architecture, which is studied in depth below.

##  IV Power Efficiency

Herein, we present a detailed study concerning the achievable power savings by
letting the number of APs grow large, i.e., $M\to\infty$. These savings are
known as power-scaling laws that allow preserving a specific SE while reducing
the transmit powers.

We assume that all the users have the same transmit power, i.e., no power
control is considered. In particular, let
$\eta_{\mathrm{A},i}=\eta_{\mathrm{B},i}=1$ for the sake of simplicity, we are
going to shed light on the impact of the transmit power per user, pilot
symbol, and relay on the separate SEs in the large APs limit, i.e.,
$M\to\infty$.

###  IV-A Scenario A: $p_{\mathrm{p}}=\frac{E_{\mathrm{p}}}{M^{\alpha}}$, and
fixed $p_{\mathrm{r}},p_{\mathrm{u}}$

This scenario concerns the study of the power efficiency in the training
phase.

###### Proposition 1

For fixed $p_{\mathrm{r}},p_{\mathrm{u}}$, and $E_{\mathrm{p}}$, when
$p_{\mathrm{p}}=\frac{E_{\mathrm{p}}}{M^{\alpha}}$ with $\alpha>0$ and
$M\to\infty$, we obtain

$\displaystyle\gamma_{i}^{\text{MAC}}$
$\displaystyle\\!=\\!\frac{p_{\mathrm{u}}N\frac{E_{\mathrm{p}}}{M^{\alpha}}\\!\\!\left(\\!\\!\left(\\!\sum_{m=1}^{M}\\!\alpha_{\mathrm{A},mi}\right)^{\\!2}\\!+\\!\left(\sum_{m=1}^{M}\\!\alpha_{\mathrm{B},mi}\right)^{\\!2}\right)}{\\!\\!\\!\displaystyle\sum_{m=1}^{M}\\!\\!\\!\left(\sum_{j=1}^{W}\\!\\!\left(\\!\alpha_{\mathrm{A},mj}\\!+\\!\alpha_{\mathrm{B},mj}\\!\right)\\!+\\!1\\!\\!\right)\\!\\!\left(\\!\alpha_{\mathrm{A},mi}\\!+\\!\alpha_{\mathrm{B},mi}\\!\right)},\\!\\!\\!$
(40) $\displaystyle\gamma_{X,i}^{\text{MAC}}$
$\displaystyle=\frac{p_{\mathrm{u}}N\frac{E_{\mathrm{p}}}{M^{\alpha}}\left(\sum_{m=1}^{M}\alpha_{\mathrm{X},mi}\right)^{2}}{\\!\\!\\!\displaystyle\sum_{m=1}^{M}\\!\\!\\!\left(\sum_{j=1}^{W}\\!\\!\left(\\!\alpha_{\mathrm{A},mj}\\!+\\!\alpha_{\mathrm{B},mj}\\!\right)\\!+\\!1\\!\\!\right)\\!\\!\left(\\!\alpha_{\mathrm{A},mi}\\!+\\!\alpha_{\mathrm{B},mi}\\!\right)},$
(41) $\displaystyle\gamma_{\mathrm{X},i}^{\text{BC}}$
$\displaystyle\\!=\\!\frac{Np_{\mathrm{r}}\frac{E_{\mathrm{p}}}{M^{\alpha}}\left(\sum_{m=1}^{M}\alpha_{\mathrm{X},mi}\right)}{\displaystyle\sum_{m=1}^{M}\sum_{j=1}^{W}\\!\left(p_{\mathrm{r}}\alpha_{\mathrm{X},mi}+1\right)\\!\left(\alpha_{\mathrm{B},mj}+\alpha_{\mathrm{A},mj}\right)}.$
(42)

Clearly, the choice of $\alpha$ defines the result. For example, in (40), the
order of $\gamma_{i}^{\text{MAC}}$ is $\mathcal{O}\left(M^{1-\alpha}\right)$ ,
which implies that $\gamma_{i}^{\text{MAC}}\to 0$, if $\alpha>1$. On the other
hand, if $0<\alpha<1$, $\gamma_{i}^{\text{MAC}}\to\infty$. However, when
$\alpha=1$, the corresponding SINRs result in a finite limit. Similar comments
holds for the SINRs given by (41) and 42. Note that the corresponding non-zero
limits are written as

$\displaystyle\gamma_{i}^{\text{MAC}}$
$\displaystyle=\frac{p_{\mathrm{u}}N{E_{\mathrm{p}}}{}\\!\\!\left(\\!\\!\left(\\!\sum_{m=1}^{M}\\!\alpha_{\mathrm{A},mi}\right)^{\\!2}\\!+\\!\left(\sum_{m=1}^{M}\\!\alpha_{\mathrm{B},mi}\right)^{\\!2}\right)}{\\!\\!\\!\displaystyle\sum_{m=1}^{M}\\!\\!\\!\left(\sum_{j=1}^{W}\\!\\!\left(\\!\alpha_{\mathrm{A},mj}\\!+\\!\alpha_{\mathrm{B},mj}\\!\right)\\!+\\!1\\!\\!\right)\\!\\!\left(\\!\alpha_{\mathrm{A},mi}\\!+\\!\alpha_{\mathrm{B},mi}\\!\right)},$
(43) $\displaystyle\gamma_{X,i}^{\text{MAC}}$
$\displaystyle=\frac{p_{\mathrm{u}}NE_{\mathrm{p}}\left(\sum_{m=1}^{M}\alpha_{\mathrm{X},mi}\right)^{2}}{\\!\\!\\!\displaystyle\sum_{m=1}^{M}\\!\\!\\!\left(\sum_{j=1}^{W}\\!\\!\left(\\!\alpha_{\mathrm{A},mj}\\!+\\!\alpha_{\mathrm{B},mj}\\!\right)\\!+\\!1\\!\\!\right)\\!\\!\left(\\!\alpha_{\mathrm{A},mi}\\!+\\!\alpha_{\mathrm{B},mi}\\!\right)},$
(44) $\displaystyle\gamma_{\mathrm{X},i}^{\text{BC}}$
$\displaystyle\\!=\\!\frac{Np_{\mathrm{r}}E_{\mathrm{p}}\left(\sum_{m=1}^{M}\alpha_{\mathrm{X},mi}\right)}{\sum_{m=1}^{M}\sum_{j=1}^{W}\left(p_{\mathrm{r}}\alpha_{\mathrm{X},mi}+1\right)\\!\left(\alpha_{\mathrm{B},mj}+\alpha_{\mathrm{A},mj}\right)}\\!.$
(45)

###  IV-B Scenario B: $p_{\mathrm{u}}=\frac{E_{\mathrm{u}}}{M^{\beta}}$ and
$p_{\mathrm{r}}=\frac{E_{\mathrm{r}}}{M^{\gamma}}$, and fixed $p_{\mathrm{p}}$

This strategy focuses on the power efficiency of solely the data transmission
phase.

###### Proposition 2

For fixed $p_{\mathrm{p}}$, $E_{\mathrm{u}}$, and $E_{\mathrm{r}}$ when
$p_{\mathrm{u}}=\frac{E_{\mathrm{u}}}{M^{\beta}}$,
$p_{\mathrm{r}}=\frac{E_{\mathrm{r}}}{M^{\gamma}}$ with $\beta\geq 0$,
$\gamma\geq 0$, and $M\to\infty$, we obtain

$\displaystyle\gamma_{i}^{\text{MAC}}$
$\displaystyle=\frac{N\frac{E_{\mathrm{u}}}{M^{\beta}}\left(\displaystyle\left(\sum_{m=1}^{M}\phi_{\mathrm{A},mi}\right)^{2}+\left(\sum_{m=1}^{M}\phi_{\mathrm{B},mi}\right)^{2}\right)}{\sum_{m=1}^{M}\left(\phi_{\mathrm{A},mi}+\phi_{\mathrm{B},mi}\right)},$
(46) $\displaystyle\gamma_{\mathrm{X},i}^{\text{MAC}}$
$\displaystyle=\frac{N\frac{E_{\mathrm{u}}}{M^{\beta}}\left(\sum_{m=1}^{M}\phi_{\mathrm{X},mi}\right)^{2}}{\sum_{m=1}^{M}\left(\phi_{\mathrm{A},mi}+\phi_{\mathrm{B},mi}\right)},$
(47) $\displaystyle\gamma_{\mathrm{X},i}^{\text{BC}}$
$\displaystyle=\frac{N\frac{E_{\mathrm{r}}}{M^{\gamma}}\left(\sum_{m=1}^{M}\phi_{\mathrm{X},mi}\right)^{2}}{\sum_{m=1}^{M}\sum_{j=1}^{W}\left(\phi_{\mathrm{A},mj}+\phi_{\mathrm{B},mj}\right)}.$
(48)

It is straightforward to show that the order of both $\gamma_{i}^{\text{MAC}}$
and $\gamma_{\mathrm{X},i}^{\text{MAC}}$ is
$\mathcal{O}\left(M^{1-\beta}\right)$ while the order of
$\gamma_{\mathrm{X},i}^{\text{BC}}$ is $\mathcal{O}\left(M^{1-\gamma}\right)$.
As before, the selection of the parameters $\beta$ and $\gamma$ affects
directly the corresponding SINRs. Especially, we observe that if
$0\leq\beta<1$ and $0\leq\gamma<1$, the SINRs grow unboundedly. Also, we
notice that under certain circumstances, being equivalent to reducing further
the transmit powers of each user or/and the relay by means of $\beta>1$ or/and
$\gamma>1$, the sum SE of the $i$th pair $R_{i}$ tends to zero because
$R_{i}^{\text{MAC}}$ or/and $R_{i}^{\text{BC}}$ tend to zero. Especially,
$R_{i}\to 0$ when one of the conditions 1) $\beta\geq 0$ and $\gamma>1$, 2)
$\beta>1$ and $\gamma\geq 0$, 3) $\beta>1$ and $\gamma>1$ is met. As a result,
in order to make $\gamma_{i}^{\text{MAC}}$ and $\gamma_{X,i}^{\text{MAC}}$
converge to a non-zero limit, we should have $\beta=1$ while when $\gamma=1$ ,
$\gamma_{X,i}^{\text{BC}}$ takes a finite value. These limits are given by

$\displaystyle\gamma_{i}^{\text{MAC}}$
$\displaystyle\\!=\\!\frac{NE_{\mathrm{u}}\\!\\!\left(\\!\\!\left(\sum_{m=1}^{M}\phi_{\mathrm{A},mi}\right)^{\\!2}\\!+\\!\left(\sum_{m=1}^{M}\phi_{\mathrm{B},mi}\right)^{\\!2}\right)}{\sum_{m=1}^{M}\left(\phi_{\mathrm{A},mi}+\phi_{\mathrm{B},mi}\right)},$
(49) $\displaystyle\gamma_{\mathrm{X},i}^{\text{MAC}}$
$\displaystyle=\frac{NE_{\mathrm{u}}\left(\sum_{m=1}^{M}\phi_{\mathrm{X},mi}\right)^{2}}{\sum_{m=1}^{M}\left(\phi_{\mathrm{A},mi}+\phi_{\mathrm{B},mi}\right)},$
(50) $\displaystyle\gamma_{\mathrm{X},i}^{\text{BC}}$
$\displaystyle=\frac{NE_{\mathrm{r}}\left(\sum_{m=1}^{M}\phi_{\mathrm{X},mi}\right)^{2}}{\sum_{m=1}^{M}\sum_{j=1}^{W}\left(\phi_{\mathrm{A},mj}+\phi_{\mathrm{B},mj}\right)}.$
(51)

It is worthwhile to mention that Proposition 2 reveals that in the large
number of APs limit, the reduction of both the transmit power of the APs and
the users proportionally to $M^{-1}$ cancels out the effects of inter-user
interference, residual interference, and estimation error. The following
corollaries show how the sum SE $R_{i}$ changes by varying $\beta$ and
$\gamma$.

###### Corollary 1

When $\beta=1$ and $0\leq\gamma<1$, the SE of the $i$th user pair as
$M\to\infty$ is written as

$\displaystyle
R_{i}=\frac{\tau_{\mathrm{c}}-\tau_{\mathrm{p}}}{2\tau_{\mathrm{c}}}$
$\displaystyle\\!\times\\!\log_{2}\\!\left(\\!\\!1\\!+\\!\frac{\displaystyle
NE_{\mathrm{u}}\\!\\!\left(\\!\\!\left(\sum_{m=1}^{M}\\!\\!\phi_{\mathrm{A},mi}\right)^{\\!\\!2}+\left(\sum_{m=1}^{M}\\!\\!\phi_{\mathrm{B},mi}\right)^{\\!\\!2}\right)}{\sum_{m=1}^{M}\left(\phi_{\mathrm{A},mi}+\phi_{\mathrm{B},mi}\right)}\right)\\!.$
(52)

According to this corollary, $R_{i}$ is equal to $R_{i}^{\text{MAC}}$ since
$R_{i}^{\text{BC}}=0$, which means that the SE of the $i$th user pair depends
only on Phase I (MAC phase). The explanation relies on the fact that since we
have reduced the transmit power per user much less compared to the transmit
power of the APs acting as relays, the MAC phase will present lower
performance. Remarkably, in this case, $R_{i}$ does not depend on the number
of users. Also, this SE does not depend on $E_{\mathrm{r}}$, but it increases
with $E_{\mathrm{u}}$.

###### Corollary 2

When $0\leq\beta<1$ and $\gamma=1$, the SE of the $i$th user pair as
$M\to\infty$ is written as

$\displaystyle
R_{i}=\frac{\tau_{\mathrm{c}}-\tau_{\mathrm{p}}}{2\tau_{\mathrm{c}}}\log_{2}\left(1+\frac{NE_{\mathrm{r}}\left(\sum_{m=1}^{M}\phi_{\mathrm{A},mi}\right)^{2}}{\displaystyle\sum_{m=1}^{M}\sum_{j=1}^{W}\left(\phi_{\mathrm{A},mj}+\phi_{\mathrm{B},mj}\right)}\right)$
$\displaystyle+\frac{\tau_{\mathrm{c}}-\tau_{\mathrm{p}}}{2\tau_{\mathrm{c}}}\log_{2}\left(1+\frac{NE_{\mathrm{r}}\left(\sum_{m=1}^{M}\phi_{\mathrm{B},mi}\right)^{2}}{\displaystyle\sum_{m=1}^{M}\sum_{j=1}^{W}\left(\phi_{\mathrm{A},mj}+\phi_{\mathrm{B},mj}\right)}\right)\\!.$
(53)

Corollary 2 denotes that the SE of the $i$th user pair appears a bottleneck in
the BC phase because the transmit power of the APs during this phase has been
cut down more than the transmit power of each user. Herein, we notice that
$R_{i}$ decreases with the number of user pairs $W$ while it increases with
$E_{\mathrm{r}}$ and it is independent of $E_{\mathrm{u}}$.

###### Corollary 3

When $\beta=\gamma=1$, the SE of the $i$th user pair as $M\to\infty$ is
written as

$\displaystyle R_{i}=\min\left(R_{i}^{\text{MAC}},R_{i}^{\text{BC}}\right),$
(54)

where

$\displaystyle
R_{i}^{\text{MAC}}=\frac{\tau_{\mathrm{c}}-\tau_{\mathrm{p}}}{2\tau_{\mathrm{c}}}$
$\displaystyle\\!\times\\!\log_{2}\\!\left(\\!\\!1\\!+\\!\frac{\displaystyle
NE_{\mathrm{u}}\\!\\!\left(\\!\\!\left(\sum_{m=1}^{M}\\!\\!\phi_{\mathrm{A},mi}\\!\right)^{\\!\\!2}\\!+\\!\left(\sum_{m=1}^{M}\\!\\!\phi_{\mathrm{B},mi}\\!\right)^{\\!\\!2}\right)}{\sum_{m=1}^{M}\left(\phi_{\mathrm{A},mi}+\phi_{\mathrm{B},mi}\right)}\right)$
(55) $\displaystyle
R_{i}^{\text{BC}}=\min\left(R_{\mathrm{A},i}^{\text{MAC}},R_{\mathrm{B},i}^{\text{BC}}\right)+\min\left(R_{\mathrm{B},i}^{\text{MAC}},R_{\mathrm{A},i}^{\text{BC}}\right),$
(56)

with $R_{\mathrm{X},i}^{\text{MAC}}$ and $R_{\mathrm{X},i}^{\text{BC}}$ given
by using (50) and (51), respectively.

This corollary indicates that if we reduce the transmit power of the APs and
users simultaneously and equally to $1/M$, both the MAC and BC affect $R_{i}$.

###  IV-C Scenario C: $p_{\mathrm{p}}=\frac{E_{\mathrm{p}}}{M^{\alpha}}$,
$p_{\mathrm{u}}=\frac{E_{\mathrm{u}}}{M^{\beta}}$, and
$p_{\mathrm{r}}=\frac{E_{\mathrm{r}}}{M^{\gamma}}$

Such a scenario is the most general, where we can achieve power savings in
both training and data transmission phases.

###### Proposition 3

When $p_{\mathrm{p}}=\frac{E_{\mathrm{p}}}{M^{\alpha}}$
$p_{\mathrm{u}}=\frac{E_{\mathrm{u}}}{M^{\beta}}$, and
$p_{\mathrm{r}}=\frac{E_{\mathrm{r}}}{M^{\gamma}}$ with $\alpha\geq 0$,
$\beta\geq 0$, $\gamma\geq 0$, and $E_{\mathrm{p}}$, $E_{\mathrm{u}}$,
$E_{\mathrm{r}}$ constants, as $M\to\infty$, we obtain

$\displaystyle\gamma_{i}^{\text{MAC}}$
$\displaystyle\\!=\\!\frac{\displaystyle
N\frac{E_{\mathrm{p}}E_{\mathrm{u}}}{M^{\alpha+\beta}}\\!\\!\left(\left(\sum_{m=1}^{M}\\!\\!\alpha_{\mathrm{A},mi}\right)^{\\!\\!2}+\left(\sum_{m=1}^{M}\\!\\!\alpha_{\mathrm{B},mi}\right)^{\\!\\!2}\right)}{\sum_{m=1}^{M}\left(\alpha_{\mathrm{A},mi}+\alpha_{\mathrm{B},mi}\right)},$
(57) $\displaystyle\gamma_{\mathrm{X},i}^{\text{MAC}}$
$\displaystyle=\frac{N\frac{E_{\mathrm{p}}E_{\mathrm{u}}}{M^{\alpha+\beta}}\left(\sum_{m=1}^{M}\alpha_{\mathrm{X},mi}\right)^{2}}{\sum_{m=1}^{M}\left(\alpha_{\mathrm{A},mi}+\alpha_{\mathrm{B},mi}\right)},$
(58) $\displaystyle\gamma_{\mathrm{X},i}^{\text{BC}}$
$\displaystyle=\frac{N\frac{E_{\mathrm{p}}E_{\mathrm{r}}}{M^{\alpha+\gamma}}\left(\sum_{m=1}^{M}\alpha_{\mathrm{X},mi}\right)^{2}}{\sum_{m=1}^{M}\sum_{j=1}^{W}\left(\alpha_{\mathrm{A},mj}+\alpha_{\mathrm{B},mj}\right)}.$
(59)

Following the same procedure as before, we observe that the orders of the
SINRs in the MAC and BC phases are
$\mathcal{O}\left(M^{1-\alpha-\beta}\right)$ and
$\mathcal{O}\left(M^{1-\alpha-\gamma}\right)$, respectively. Hence, the
corresponding SINRs converge to non-zero limits only when $\alpha+\beta=1$ and
$\alpha+\gamma=1$. Otherwise, they can grow unboundedly or tend to zero. These
limits are given by

$\displaystyle\gamma_{i}^{\text{MAC}}$
$\displaystyle\\!=\\!\frac{\displaystyle
NE_{\mathrm{p}}E_{\mathrm{u}}\\!\\!\left(\\!\\!\left(\sum_{m=1}^{M}\\!\\!\alpha_{\mathrm{A},mi}\right)^{\\!\\!2}+\left(\sum_{m=1}^{M}\\!\\!\alpha_{\mathrm{B},mi}\right)^{\\!\\!2}\right)}{\sum_{m=1}^{M}\left(\alpha_{\mathrm{A},mi}+\alpha_{\mathrm{B},mi}\right)},$
(60) $\displaystyle\gamma_{\mathrm{X},i}^{\text{MAC}}$
$\displaystyle=\frac{NE_{\mathrm{p}}E_{\mathrm{u}}\left(\sum_{m=1}^{M}\alpha_{\mathrm{X},mi}\right)^{2}}{\sum_{m=1}^{M}\left(\alpha_{\mathrm{A},mi}+\alpha_{\mathrm{B},mi}\right)},$
(61) $\displaystyle\gamma_{\mathrm{X},i}^{\text{BC}}$
$\displaystyle=\frac{NE_{\mathrm{p}}E_{\mathrm{r}}\left(\sum_{m=1}^{M}\alpha_{\mathrm{X},mi}\right)^{2}}{\sum_{m=1}^{M}\sum_{j=1}^{W}\left(\alpha_{\mathrm{A},mj}+\alpha_{\mathrm{B},mj}\right)}.$
(62)

The following corollaries present trade-offs between the transmit powers of
the pilot symbols and the APs and/or users.

###### Corollary 4

When $\alpha+\beta=1$ and $\beta>\gamma\geq 0$, the SE of the $i$th user pair
as $M\to\infty$ is written as

$\displaystyle
R_{i}=\frac{\tau_{\mathrm{c}}-\tau_{\mathrm{p}}}{2\tau_{\mathrm{c}}}$
$\displaystyle\\!\\!\\!\\!\times\\!\log_{2}\\!\\!\left(\\!\\!1\\!+\\!\frac{\displaystyle
NE_{\mathrm{p}}E_{\mathrm{u}}\\!\\!\left(\\!\\!\\!\left(\sum_{m=1}^{M}\\!\\!\alpha_{\mathrm{A},mi}\\!\right)^{\\!\\!2}\\!\\!+\\!\left(\sum_{m=1}^{M}\\!\\!\alpha_{\mathrm{B},mi}\\!\right)^{\\!\\!2}\right)}{\sum_{m=1}^{M}\left(\alpha_{\mathrm{A},mi}+\alpha_{\mathrm{B},mi}\right)}\\!\\!\right)\\!.\\!$
(63)

The inequality implies that $\alpha+\gamma<1$. Hence,
$\gamma_{\mathrm{X},i}^{\text{BC}}\to 0$, and the SE of the $i$th user pair is
determined only by the MAC Phase. Clearly, $R_{i}$ does not depend on the
number of user pairs, which results in the increase with $W$ of the sum SE
given by (36).

###### Corollary 5

When $\alpha+\gamma=1$ and $\gamma>\beta\geq 0$, the SE of the $i$th user pair
as $M\to\infty$ is written as

$\displaystyle\\!\\!\\!\\!R_{i}$
$\displaystyle=\frac{\tau_{\mathrm{c}}-\tau_{\mathrm{p}}}{2\tau_{\mathrm{c}}}\log_{2}\left(1+\frac{NE_{\mathrm{p}}E_{\mathrm{r}}\left(\sum_{m=1}^{M}\alpha_{\mathrm{A},mi}\right)^{2}}{\displaystyle\sum_{m=1}^{M}\sum_{j=1}^{W}\left(\alpha_{\mathrm{A},mj}+\alpha_{\mathrm{B},mj}\right)}\right)$
$\displaystyle\\!\\!\\!\\!\\!+\\!\frac{\tau_{\mathrm{c}}-\tau_{\mathrm{p}}}{2\tau_{\mathrm{c}}}\log_{2}\left(1+\frac{NE_{\mathrm{p}}E_{\mathrm{r}}\left(\sum_{m=1}^{M}\alpha_{\mathrm{B},mi}\right)^{2}}{\displaystyle\sum_{m=1}^{M}\sum_{j=1}^{W}\left(\alpha_{\mathrm{A},mj}+\alpha_{\mathrm{B},mj}\right)}\right)\\!.$
(64)

Herein, the inequality suggests that $\alpha+\beta<1$, which means that
$\gamma_{\mathrm{X},i}^{\text{MAC}}\to 0$, and only the BC phase defines the
SE of the $i$th user pair.

###### Corollary 6

When $\alpha+\beta=1$ and $\beta=\gamma\geq 0$, the SE of the $i$th user pair
as $M\to\infty$ is written as

$\displaystyle R_{i}=\min\left(R_{i}^{\text{MAC}},R_{i}^{\text{BC}}\right),$
(65)

where

$\displaystyle\\!\\!\\!R_{i}=\frac{\tau_{\mathrm{c}}-\tau_{\mathrm{p}}}{2\tau_{\mathrm{c}}}$
$\displaystyle\\!\\!\\!\\!\times\\!\log_{2}\\!\\!\left(\\!\\!1+\frac{\displaystyle
NE_{\mathrm{u}}\\!\\!\left(\\!\\!\left(\sum_{m=1}^{M}\phi_{\mathrm{A},mi}\right)^{\\!\\!2}+\left(\sum_{m=1}^{M}\phi_{\mathrm{B},mi}\right)^{\\!\\!2}\right)}{\sum_{m=1}^{M}\left(\phi_{\mathrm{A},mi}+\phi_{\mathrm{B},mi}\right)}\\!\\!\right)\\!\\!$
(66) $\displaystyle
R_{i}^{\text{BC}}=\min\left(R_{\mathrm{A},i}^{\text{MAC}},R_{\mathrm{B},i}^{\text{BC}}\right)+\min\left(R_{\mathrm{B},i}^{\text{MAC}},R_{\mathrm{A},i}^{\text{BC}}\right),$
(67)

with $R_{\mathrm{X},i}^{\text{MAC}}$ and $R_{\mathrm{X},i}^{\text{BC}}$ given
in terms of (61) and (62), respectively.

In other words, both conditions $\alpha+\beta=1$ and $\alpha+\gamma=1$ are
fulfilled. By shedding further light on this corollary, we observe an
interplay appearing among the transmit powers. For example, a reduction of the
pilot transmit power would result in a degradation of the estimated channel
that could be balanced by an increase of the transmit power of the users/APs
during the transmission phase to preserve the performance with respect to the
SE.

###### Remark 1

In all corollaries above, the corresponding SE could be boosted by increasing
the involved $E_{\mathrm{i}}$, $i\mathrm{=p,u,r}$. For example, Corollary 2
suggests that $R_{i}$ can be increased with $E_{\mathrm{r}}$ by increasing the
transmit power of the APs.

##  V Power Allocation

Different from the previous section, where the transmit powers of all users
were assumed equal for the sake of exposition of the scaling laws, in this
section, we elaborate on the optimal power allocation among the users and the
APs during both MAC and BC phases, respectively. We assume that the power
allocation takes place during the data transmission phase while the power
design of the training phase in terms of $p_{\mathrm{p}}$ has previously being
determined. In particular, we follow the procedure in [12, 21] and adapt it
according to our system architecture having $M$ distributed APs as relay
nodes.

We focus on the maximization of the sum SE constrained to a total power $P$,
i.e.,
$p_{\mathrm{u}}\sum_{i=1}^{W}\left(\eta_{\mathrm{A},i}+\eta_{\mathrm{B},i}\right)+p_{\mathrm{r}}\leq
P$. In particular, the formulation of the power allocation optimization is
described by

$\displaystyle\max_{{\bm{\eta}}_{\mathrm{A}},{\bm{\eta}}_{\mathrm{B}},p_{\mathrm{r}}}$
$\displaystyle\sum_{i=1}^{W}{R}_{i}$ (68a)
$\displaystyle\mathrm{subject}~{}\mathrm{to}$ $\displaystyle
p_{\mathrm{u}}\sum_{i=1}^{W}\left(\eta_{\mathrm{A},i}+\eta_{\mathrm{B},i}\right)+p_{\mathrm{r}}\leq
P$
$\displaystyle{\bm{\eta}}_{\mathrm{A}}\geq{\mathbf{0}},{\bm{\eta}}_{\mathrm{B}}\geq{\mathbf{0}},p_{\mathrm{d}}\geq
0,p_{\mathrm{r}}\geq 0$ $\displaystyle{R}_{i}\geq
R_{\mathrm{min}},i\in\mathcal{W}$

where we have denoted
${\bm{\eta}}_{\mathrm{A}}=\left[{\bm{\eta}}_{\mathrm{A},1},\ldots,{\bm{\eta}}_{\mathrm{A},W}\right]^{{\scriptscriptstyle\mathsf{T}}}$
and
${\bm{\eta}}_{\mathrm{B}}=\left[{\bm{\eta}}_{\mathrm{B},1},\ldots,{\bm{\eta}}_{\mathrm{B},W}\right]^{{\scriptscriptstyle\mathsf{T}}}$,
while $R_{\mathrm{min}}$ expresses the minimum SE of the $i$th pair. Given
that the logarithm is an increasing function, the optimization, given by
(68a), can be written as

$\displaystyle\min_{\begin{subarray}{c}\tilde{{\bm{\eta}}}_{\mathrm{A}},\tilde{{\bm{\eta}}}_{\mathrm{B}},p_{\mathrm{r}}\\\
\gamma_{i},\gamma_{\mathrm{A},i},\gamma_{\mathrm{B},i}\end{subarray}}$
$\displaystyle\prod_{i=1}^{W}\left(1+{\gamma}_{i}\right)^{-1}$ (69a)
$\displaystyle\mathrm{subject}~{}\mathrm{to}$
$\displaystyle{\gamma}_{i}\leq\frac{a_{1,i}\tilde{\eta}_{\mathrm{A},i}+a_{2,i}\tilde{\eta}_{\mathrm{B},i}}{\sum_{j=1}^{W}\left(a_{3,ij}\tilde{\eta}_{\mathrm{A},j}+a_{4,ij}\tilde{\eta}_{\mathrm{B},j}\right)+1}$
(69b)
$\displaystyle\gamma_{\mathrm{A},i}\leq\min\\{\frac{a_{1,i}\tilde{\eta}_{\mathrm{A},i}}{c_{i}},\frac{p_{\mathrm{r}}}{p_{\mathrm{r}}b_{B,i}+c_{B,i}}\\},i\in\mathcal{W}$
(69c)
$\displaystyle\gamma_{\mathrm{B},i}\leq\min\\{\frac{a_{2,i}\tilde{\eta}_{\mathrm{B},i}}{c_{i}},\frac{p_{\mathrm{r}}}{p_{\mathrm{r}}b_{A,i}+c_{A,i}}\\},i\in\mathcal{W}$
(69d)
$\displaystyle\gamma_{i}\leq\gamma_{\mathrm{A},i}+\gamma_{\mathrm{B},i}+\gamma_{\mathrm{A},i}\gamma_{\mathrm{B},i},i\in\mathcal{W}$
(69e)
$\displaystyle\sum_{i=1}^{W}\left(\tilde{\eta}_{\mathrm{A},i}+\tilde{\eta}_{\mathrm{B},i}\right)+p_{\mathrm{r}}\leq
P$ (69f)
$\displaystyle\tilde{{\bm{\eta}}}_{\mathrm{A}}\geq{\mathbf{0}},\tilde{{\bm{\eta}}}_{\mathrm{B}}\geq{\mathbf{0}},p_{\mathrm{d}}\geq
0,p_{\mathrm{r}}\geq 0$ (69g)
$\displaystyle\gamma_{i}^{-1}\left(2^{\frac{2\tau_{\mathrm{c}}R_{\mathrm{min}}}{\tau_{\mathrm{c}}-\tau_{\mathrm{p}}}}-1\right)\leq
1,i\in\mathcal{W}$ (69h)

where we have defined
$\tilde{\eta}_{\mathrm{A},i}=p_{\mathrm{u}}\eta_{\mathrm{A},i}$,
$\tilde{\eta}_{\mathrm{B},i}=p_{\mathrm{u}}\eta_{\mathrm{B},i}$,
$a_{1,i}=\frac{N\left(\sum_{m=1}^{M}\phi_{\mathrm{A},mi}\right)^{2}}{\sum_{m=1}^{M}\left(\phi_{\mathrm{A},mi}+\phi_{\mathrm{B},mi}\right)}$,
$a_{2,i}=\frac{N\left(\sum_{m=1}^{M}\phi_{\mathrm{B},mi}\right)^{2}}{\sum_{m=1}^{M}\left(\phi_{\mathrm{A},mi}+\phi_{\mathrm{B},mi}\right)}$,
$a_{3,ij}=\frac{\sum_{m=1}^{M}\alpha_{\mathrm{A},mj}}{\sum_{m=1}^{M}\left(\phi_{\mathrm{A},mi}+\phi_{\mathrm{B},mi}\right)}$,
$a_{4,ij}=\frac{\sum_{m=1}^{M}\alpha_{\mathrm{B},mj}}{\sum_{m=1}^{M}\left(\phi_{\mathrm{A},mi}+\phi_{\mathrm{B},mi}\right)}$,
$c_{i}=\sum_{j=1}^{W}\left(a_{3,ij}\tilde{\eta}_{\mathrm{A},j}+a_{4,ij}\tilde{\eta}_{\mathrm{B},j}\right)+1$,
$b_{X,i}=\frac{\sum_{m=1}^{M}\sum_{j=1}^{W}\alpha_{\mathrm{X},mi}\left(\eta_{\mathrm{A},mj}\phi_{\mathrm{B},mj}+\eta_{\mathrm{B},mj}\phi_{\mathrm{A},mj}\right)}{N\left(\sum_{m=1}^{M}\sqrt{\eta_{\mathrm{\bar{X}},mi}}\phi_{\mathrm{X},mi}\right)^{2}}$,
and
$c_{X,i}=\frac{\sum_{m=1}^{M}\sum_{j=1}^{W}\left(\eta_{\mathrm{A},mj}\phi_{\mathrm{B},mj}+\eta_{\mathrm{B},mj}\phi_{\mathrm{A},mj}\right)}{N\left(\sum_{m=1}^{M}\sqrt{\eta_{\mathrm{\bar{X}},mi}}\phi_{\mathrm{X},mi}\right)^{2}}$.
Moreover, $\gamma_{i}$, $\gamma_{\mathrm{A},i}$, and $\gamma_{\mathrm{B},i}$
correspond to the SINRs of $R_{i}$,
$\min\left(R_{\mathrm{A},i}^{\text{MAC}},R_{\mathrm{B},i}^{\text{BC}}\right)$
and
$\min\left(R_{\mathrm{B},i}^{\text{MAC}},R_{\mathrm{A},i}^{\text{BC}}\right)$,
respectively.

The latter optimization problem is nonconvex since it falls to the category of
complementary geometric programming (CGP). However, its solution can be
obtained by solving a sequence of convex GP problems [36, 37, 12, 21].
Initially, as in [37, Lem. 1], we approximate the objective function
$\left(1+{\gamma}_{i}\right)$ by the monomial function
$\delta_{i}{\gamma}_{i}^{\mu_{i}}$, where
$\delta_{i}=\left({\gamma}_{i}^{\circ}\right)^{-\mu_{i}}\left(1+{\gamma}_{i}^{\circ}\right)$
and $\mu_{i}=\frac{{\gamma}_{i}^{\circ}}{{\gamma}_{i}^{\circ}+1}$ [12]. Next,
we transform the two inequalities described by (69b) and (69e) into
posynomials, in order to result in a GP problem. Specifically, the first
inequality is written as

$\displaystyle\\!\\!a_{1,i}\tilde{\eta}_{\mathrm{A},i}\\!+\\!a_{2,i}\tilde{\eta}_{\mathrm{B},i}\\!\geq\\!\left(\frac{a_{1,i}\tilde{\eta}_{\mathrm{A},i}}{\phi_{\mathrm{A},i}}\right)^{\\!\phi_{\mathrm{A},i}}\\!\\!\left(\frac{a_{2,i}\tilde{\eta}_{\mathrm{B},i}}{\phi_{\mathrm{B},i}}\right)^{\\!\phi_{\mathrm{B},i}},$
(70)

where
$\phi_{\mathrm{A},i}=\frac{a_{1,i}\tilde{\eta}^{\circ}_{\mathrm{A},i}}{a_{1,i}\tilde{\eta}^{\circ}_{\mathrm{A},i}+a_{2,i}\tilde{\eta}^{\circ}_{\mathrm{B},i}}$,
$\phi_{\mathrm{B},i}=\frac{a_{2,i}\tilde{\eta}^{\circ}_{\mathrm{B},i}}{a_{1,i}\tilde{\eta}^{\circ}_{\mathrm{A},i}+a_{2,i}\tilde{\eta}^{\circ}_{\mathrm{B},i}}$
with $\tilde{\eta}^{\circ}_{\mathrm{A},i}$ and
$\tilde{\eta}^{\circ}_{\mathrm{B},i}$ denoting the initialization values. In
(70), we have applied a known property, expressing that, for any set of
positive numbers, the geometric mean is no larger than the arithmetic mean
[38]. Hence, substitution of (70) into (69b) gives

$\displaystyle{\gamma}_{i}\leq\frac{\left(\frac{a_{1,i}\tilde{\eta}_{\mathrm{A},i}}{q_{\mathrm{A},i}}\right)^{q_{\mathrm{A},i}}\left(\frac{a_{2,i}\tilde{\eta}_{\mathrm{B},i}}{q_{\mathrm{B},i}}\right)^{q_{\mathrm{B},i}}}{\sum_{j=1}^{W}\left(a_{3,ij}\tilde{\eta}_{\mathrm{A},j}+a_{4,ij}\tilde{\eta}_{\mathrm{B},j}\right)+1},i\in\mathcal{W}.$
(71)

For the second inequality, given by (69e), we follow the procedure in [21] to
approximate $f\left(x,y\right)=x+y+xy$ near an arbitrary point
$x^{\circ},y^{\circ}>0$ by means of the monomial function
$g\left(x,y\right)=\zeta x^{\lambda_{1}}y^{\lambda_{2}}$ in terms of $\zeta$,
$\lambda_{1}$, and $\lambda_{2}$. These parameters are obtained in [21] as
$\lambda_{1}=\frac{x^{\circ}\left(1+y^{\circ}\right)}{x^{\circ}+y^{\circ}+x^{\circ}y^{\circ}}$,
$\lambda_{2}=\frac{y^{\circ}\left(1+x^{\circ}\right)}{x^{\circ}+y^{\circ}+x^{\circ}y^{\circ}}$,
$\zeta=\left(x^{\circ}+y^{\circ}+x^{\circ}y^{\circ}\right)\left(x^{\circ}\right)^{-\lambda_{1}}\left(y^{\circ}\right)^{-\lambda_{2}}$.
Thus, we have

$\displaystyle\gamma_{i}\leq\zeta_{i}\gamma_{\mathrm{A},i}^{\lambda_{\mathrm{A},i}}\gamma_{\mathrm{B},i}^{\lambda_{\mathrm{B},i}},i\in\mathcal{W}$
(72)

where
$\zeta_{i}=\left(\gamma_{\mathrm{A},i}^{\circ}+\gamma_{\mathrm{B},i}^{\circ}+\gamma_{\mathrm{A},i}^{\circ}\gamma_{\mathrm{B},i}^{\circ}\right)\left(\gamma_{\mathrm{A},i}^{\circ}\right)^{-\lambda_{\mathrm{A},i}}\left(\gamma_{\mathrm{B},i}^{\circ}\right)^{-\lambda_{\mathrm{B},i}}$,
$\lambda_{\mathrm{A},i}=\frac{\gamma_{\mathrm{A},i}^{\circ}\left(1+\gamma_{\mathrm{B},i}^{\circ}\right)}{\gamma_{\mathrm{A},i}^{\circ}+\gamma_{\mathrm{B},i}^{\circ}+\gamma_{\mathrm{A},i}^{\circ}\gamma_{\mathrm{B},i}^{\circ}}$,
and
$\lambda_{\mathrm{B},i}=\frac{\gamma_{\mathrm{B},i}^{\circ}\left(1+\gamma_{\mathrm{A},i}^{\circ}\right)}{\gamma_{\mathrm{A},i}^{\circ}+\gamma_{\mathrm{B},i}^{\circ}+\gamma_{\mathrm{A},i}^{\circ}\gamma_{\mathrm{B},i}^{\circ}}$
with $\gamma_{\mathrm{A},i}^{\circ}$, $\gamma_{\mathrm{B},i}^{\circ}$ being
the initialization values. The algorithm steps are provided by Algorithm 1,
where the parameter $\theta>1$ defines the desired accuracy as a trade-off
with convergence speed. Especially, as $\theta$ approaches $1$, we result in
better accuracy while the convergence speed is slow.

Algorithm 1 Successive approximation algorithm

1\. Initialisation: Define the parameter $\theta$ and the tolerance
$\epsilon$. Set $k=1$,
$\tilde{\eta}_{\mathrm{A},i}=\tilde{\eta}_{\mathrm{B},i}=\frac{P}{4W}$,
$p_{\mathrm{r}}=\frac{P}{2}$ while $\gamma_{i}^{\circ}$,
$\gamma_{\mathrm{A},i}^{\circ}$, and $\gamma_{\mathrm{B},i}^{\circ}$ are
chosen by means of Theorem 1.
2\. Iteration $k$: Evaluate
$\mu_{i}=\frac{\gamma_{i}^{\circ}}{\gamma_{i}^{\circ}+1}$,
$\phi_{\mathrm{A},i}=\frac{a_{1,i}\tilde{\eta}^{\circ}_{\mathrm{A},i}}{a_{1,i}\tilde{\eta}^{\circ}_{\mathrm{A},i}+a_{2,i}\tilde{\eta}^{\circ}_{\mathrm{B},i}}$,
$\phi_{\mathrm{B},i}=\frac{a_{2,i}\tilde{\eta}^{\circ}_{\mathrm{B},i}}{a_{1,i}\tilde{\eta}^{\circ}_{\mathrm{A},i}+a_{2,i}\tilde{\eta}^{\circ}_{\mathrm{B},i}}$,
$\zeta_{i}=\left(\gamma_{\mathrm{A},i}^{\circ}+\gamma_{\mathrm{B},i}^{\circ}+\gamma_{\mathrm{A},i}^{\circ}\gamma_{\mathrm{B},i}^{\circ}\right)\left(\gamma_{\mathrm{A},i}^{\circ}\right)^{-\lambda_{\mathrm{A},i}}\left(\gamma_{\mathrm{B},i}^{\circ}\right)^{-\lambda_{\mathrm{B},i}}$,
$\lambda_{\mathrm{A},i}=\frac{\gamma_{\mathrm{A},i}^{\circ}\left(1+\gamma_{\mathrm{B},i}^{\circ}\right)}{\gamma_{\mathrm{A},i}^{\circ}+\gamma_{\mathrm{B},i}^{\circ}+\gamma_{\mathrm{A},i}^{\circ}\gamma_{\mathrm{B},i}^{\circ}}$,
$\lambda_{\mathrm{B},i}=\frac{\gamma_{\mathrm{B},i}^{\circ}\left(1+\gamma_{\mathrm{A},i}^{\circ}\right)}{\gamma_{\mathrm{A},i}^{\circ}+\gamma_{\mathrm{B},i}^{\circ}+\gamma_{\mathrm{A},i}^{\circ}\gamma_{\mathrm{B},i}^{\circ}}$.
Then, solve the GP problem:

$\displaystyle\min_{\begin{subarray}{c}\tilde{{\bm{\eta}}}_{\mathrm{A}},\tilde{{\bm{\eta}}}_{\mathrm{B}},p_{\mathrm{r}}\\\
\gamma_{i},\gamma_{\mathrm{A},i},\gamma_{\mathrm{B},i}\end{subarray}}$
$\displaystyle\prod_{i=1}^{W}{\gamma}_{i}^{-\mu_{i}}$ (73a)
$\displaystyle\mathrm{subject}~{}\mathrm{to}$
$\displaystyle\theta^{-1}\tilde{\eta}_{\mathrm{A},i}\leq\tilde{\eta}_{\mathrm{A},i}\leq\theta\tilde{\eta}_{\mathrm{A},i},i\in\mathcal{W}$
(73b)
$\displaystyle\theta^{-1}\tilde{\eta}_{\mathrm{B},i}\leq\tilde{\eta}_{\mathrm{B},i}\leq\theta\tilde{\eta}_{\mathrm{B},i},i\in\mathcal{W}$
(73c)
$\displaystyle\theta^{-1}\gamma_{i}^{\circ}\leq\gamma_{i}\leq\theta\gamma_{i}^{\circ},i\in\mathcal{W}$
(73d)
$\displaystyle\theta^{-1}\gamma_{\mathrm{A},i}^{\circ}\leq\gamma_{\mathrm{A},i}\leq\theta\gamma_{\mathrm{A},i}^{\circ},i\in\mathcal{W}$
(73e)
$\displaystyle\theta^{-1}\gamma_{\mathrm{B},i}^{\circ}\leq\gamma_{\mathrm{B},i}\leq\theta\gamma_{\mathrm{B},i}^{\circ},i\in\mathcal{W}$
(73f) $\displaystyle
c_{i}{\gamma}_{i}\\!\left(\\!\frac{a_{1,i}\tilde{\eta}_{\mathrm{A},i}}{\phi_{\mathrm{A},i}}\\!\right)^{\\!-\phi_{\mathrm{A},i}}\\!\\!\left(\\!\frac{a_{2,i}\tilde{\eta}_{\mathrm{B},i}}{\phi_{\mathrm{B},i}}\\!\right)^{\\!\\!-\phi_{\mathrm{B},i}}\\!\\!\leq
1,i\in\mathcal{W}$ (73g)
$\displaystyle\gamma_{i}\zeta_{i}^{-1}\gamma_{\mathrm{A},i}^{-\lambda_{\mathrm{A},i}}\gamma_{\mathrm{B},i}^{-\lambda_{\mathrm{B},i}}\leq
1,i\in\mathcal{W}$ (73h)
$\displaystyle\gamma_{\mathrm{A},i}c_{i}\left(a_{1,i}\tilde{\eta}_{\mathrm{A},i}\right)^{-1}\leq
1i\in\mathcal{W}$ (73i)
$\displaystyle\gamma_{\mathrm{B},i}c_{i}\left(a_{2,i}\tilde{\eta}_{\mathrm{B},i}\right)^{-1}\leq
1i\in\mathcal{W}$ (73j)
$\displaystyle\gamma_{\mathrm{A},i}p_{\mathrm{r}}^{-1}\left(p_{\mathrm{r}}b_{B,i}+c_{B,i}\right)\leq
1,i\in\mathcal{W}$ (73k)
$\displaystyle\gamma_{\mathrm{B},i}p_{\mathrm{r}}^{-1}\left(p_{\mathrm{r}}b_{A,i}+c_{A,i}\right)\leq
1,i\in\mathcal{W}$ (73l)
$\displaystyle\sum_{i=1}^{W}\left(\tilde{\eta}_{\mathrm{A},i}+\tilde{\eta}_{\mathrm{B},i}\right)+p_{\mathrm{r}}\leq
P$ (73m)
$\displaystyle\tilde{{\bm{\eta}}}_{\mathrm{A}}\geq{\mathbf{0}},\tilde{{\bm{\eta}}}_{\mathrm{B}}\geq{\mathbf{0}},p_{\mathrm{d}}\geq
0,p_{\mathrm{r}}\geq 0$ (73n)
$\displaystyle\gamma_{i}^{-1}\left(2^{\frac{2\tau_{\mathrm{c}}R_{\mathrm{min}}}{\tau_{\mathrm{c}}-\tau_{\mathrm{p}}}}-1\right)\leq
1,i\in\mathcal{W}$ (73o)

Let $\tilde{\eta}_{\mathrm{A},i}^{\star}$,
$\tilde{\eta}_{\mathrm{B},i}^{\star}$, $\gamma_{i}^{\star}$,
$\gamma_{\mathrm{A},i}^{\star}$, $\gamma_{\mathrm{B},i}^{\star}$,
$i\in\mathcal{W}$.
3\. If
$\max_{i}|\tilde{\eta}_{\mathrm{A},i}^{\star}-\tilde{\eta}_{\mathrm{A},i}^{\circ}|<\epsilon$
and/or
$\max_{i}|\tilde{\eta}_{\mathrm{B},i}^{\star}-\tilde{\eta}_{\mathrm{B},i}^{\circ}|<\epsilon$
and/or $\max_{i}|\gamma_{i}^{\star}-\gamma_{i}^{\circ}|<\epsilon$ and/or
$\max_{i}|\gamma_{\mathrm{A},i}^{\star}-\gamma_{\mathrm{A},i}^{\circ}|<\epsilon$
and/or
$\max_{i}|\gamma_{\mathrm{B},i}^{\star}-\gamma_{\mathrm{B},i}^{\circ}|<\epsilon$
$\rightarrow$ Stop. Otherwise, go to step $4$.
4\. Update initial values. Set $k=k+1$,
$\tilde{\eta}_{\mathrm{A},i}^{\circ}=\tilde{\eta}_{\mathrm{A},i}^{\star}$,
$\tilde{\eta}_{\mathrm{B},i}^{\circ}=\tilde{\eta}_{\mathrm{B},i}^{\star}$,
$\gamma_{i}^{\circ}=\gamma_{i}^{\star}$,
$\gamma_{\mathrm{A},i}^{\circ}=\gamma_{\mathrm{A},i}^{\star}$,
$\gamma_{\mathrm{B},i}^{\circ}=\gamma_{\mathrm{B},i}^{\star}$, and go to step
$2$.

##  VI Numerical Results

This section depicts the analytical results provided by means of Theorem 1 and
Propositions 1-3 that illustrate the performance of a multi-pair two-way CF
mMIMO system. For the sake of comparison, we have accounted for a conventional
two-way collocated massive MIMO architecture employing DF as described by
[21]. Also, our analytical results are accompanied by Monte Carlo simulations
by means of $10^{3}$ independent channel realizations, in order to verify them
and show their tightness.

TABLE I: Parameters Values for Numerical Results  Description | Values
---|---
Number of APs | $M=200$
Number of Antennas/AP | $N=3$
Number user pairs | $W=5$
Carrier frequency | $f_{0}=2~{}\mathrm{GHz}$
Power per pilot symbol | $\bar{p}_{\mathrm{p}}=100~{}\mathrm{mW}$
Uplink transmit power | ${\bar{p}_{\mathrm{u}}}=100~{}\mathrm{mW}$
Path loss exponent | $\alpha=4$
Communication bandwidth | $W_{\mathrm{c}}=20~{}\mathrm{MHz}$
Coherence bandwidth | $B_{\mathrm{c}}=200~{}\mathrm{KHz}$
Coherence time | $T_{\mathrm{c}}=1~{}\mathrm{ms}$
Duration of uplink training |  $\tau_{\mathrm{p}}=10$ samples
Boltzmann constant | $\kappa_{\mathrm{B}}=1.381\times 10^{-23}~{}\mathrm{J/K}$
Noise temperature | $T_{0}=290~{}\mathrm{K}$
Noise figure | ${N_{\mathrm{F}}}=9~{}\mathrm{dB}$

###  VI-A Simulation Setup

Unless otherwise stated, the following set of parameters is used during the
simulations. In particular, we consider $M=200$ APs and $W=5$ user pairs
uniformly distributed in an area of $1\times 1~{}\mathrm{km}^{2}$. Each AP is
equipped with $N=3$ antennas. Note that the area edges are wrapped around to
avoid the boundary effects. Also, we assume that the coherence time and
bandwidth are $T_{\mathrm{c}}=1~{}\mathrm{ms}$ and
$B_{\mathrm{c}}=200~{}\mathrm{kHz}$, respectively, which means that the
coherence block consists of $200$ channel uses. The orthogonality among pilots
requires at least $\tau_{\mathrm{p}}=2W=10$. Moreover, we assume that
${p}_{\mathrm{p}}$, ${{p}_{\mathrm{u}}}$, and ${{p}_{\mathrm{r}}}$ correspond
to the normalized powers, obtained by dividing
$\bar{p}_{\mathrm{p}}=\bar{p}_{\mathrm{u}}$, and $\bar{p}_{\mathrm{r}}$ by the
noise power ${N_{\mathrm{P}}}=W_{\mathrm{c}}$ $\times$ $\kappa_{\mathrm{B}}$
$\times$ $T_{0}$ $\times$ ${N_{\mathrm{F}}}$. The various parameters are found
in Table I. Power control in terms of maximizing the sum SE is considered only
in Subsection VI-D. Hence, without any power control, we assume that in the
uplink, all users transmit with full power, i.e.,
$\eta_{\mathrm{A},i}=\eta_{\mathrm{B},i}=1,\forall i$. Similarly, in the
downlink, all APs transmit with full power, which means
$\eta_{mi}=\left(N\sum_{i=1}^{W}\left(\phi_{\mathrm{B},mi}+\phi_{\mathrm{A},mi}\right)\right)^{-1}$
by satisfying (13). Also, the transmit powers during the MAC and BC phases are
assumed equal, i.e., $2W{p}_{\mathrm{u}}={p}_{\mathrm{r}}$.

We take into account for [8, Remark 4], and thus, we consider the 3GPP Urban
Microcell model in [39, Table B.1.2.1-1] as a more appropriate benchmark for
CF mMIMO systems than the established model presented initially in [3] because
of two main reasons: i) although CF mMIMO systems are more likely suggested
for shorter distances, the model in [3] assumes that shadowing is met for
users found $50\mathrm{m}$ further from an AP, and ii) the COST-Hata model,
used in [3] is suitable for macro-cells with APs being at least $1\mathrm{km}$
far from the users and at least $30\mathrm{m}$ above the ground while the CF
setting suggests APs found at lower height and being very close to the users.
Specifically, the large-scale fading coefficient, described by this mode for a
$2~{}\mathrm{GHz}$ carrier frequency, is given by

$\displaystyle\alpha_{\mathrm{X},mk}[dB]=-30.5-36.7\log_{10}\left(\frac{d_{\mathrm{X},mk}}{1~{}\mathrm{m}}\right)+F_{\mathrm{X},mk},$

where $d_{\mathrm{X},mk}$ expresses the distance between AP $m$ and user $k$
while $F_{\mathrm{X},mk}\sim\mathcal{CN}\left(0,4^{2}\right)$ describes the
shadow fading. In addition, the shadowing terms between different users are
assumed to be correlated as
$\mathbb{E}\\{F_{\mathrm{X},mk}F_{\mathrm{X},ij}\\}=4^{2}2^{-\delta_{\mathrm{X},kj}/9}$
only when $m=i$, where $\delta_{\mathrm{X},kj}$ is the distance between users
$k$ and $j$.

###  VI-B Demonstration of basic properties

Initially, we assume that $p_{\mathrm{p}}=p_{\mathrm{u}}$ as well as
$p_{\mathrm{r}}=2Wp_{\mathrm{u}}$, which mean that users transmit equal power
during the training and data transmission phases.

Fig. 2 presents the sum SE versus $p_{\mathrm{u}}$ with varying number of APs
along with Monte-Carlo simulations verifying the analytical expressions since
the lines almost coincide. Also, we notice that SE saturates at high SNR due
to the inter-user interference, as expected. Moreover, SE increases with the
increasing number of APs $M$.

Fig 3 depicts the sum SE versus the number of APs $M$. Also, we have
considered the scenario of genie receivers at the users during the BC phase.
In other words, we have assumed that the corresponding receivers are aware of
instantaneous CSI and not just its statistics. Since the gap between the two
lines is small, the downlink channel hardens and no extra training is
required. In addition, for the sake of comparison, we have included the
collocated scenario with a base station at the center of the area being the
relay node and having $MN$ antennas,
$\alpha_{\mathrm{X},mk}=\alpha_{\mathrm{X}k}$, $\eta_{k}=M\eta_{mk}$, $\forall
m$, and in general, all the parameters equal across the index $m$ [27]. It can
be seen that the CF mMIMO relay setting outperforms the collocated layout
because the diversity against path-loss and shadow fading is exploited.
Moreover, we have considered the conventional orthogonal scheme where the
transmission of each pair takes place at different time slots or frequency
bands. As the number of APs increases, the two-way CF MIMO performs better
because the effect from the inter-user interference decreases. Thus, when $M$
is low, the orthogonal scheme performs better, but the mMIMO system behaves
better as $M$ increases and channels become orthogonal, which means a
mitigation of the interference. Compared to Fig. 3, which assumes $W=5$ user
pairs, Fig. 3 shows the sum SE in the case of $W=20$ user pairs. In the latter
figure, the outperformance of the CF mMIMO setting over the collocated
scenario is more pronounced because of its concomitant advantages. Hence, we
observe that the performance gap at $M=400$ AP is $8\%$ and $39\%$, when $W=5$
and $W=20$, respectively.

###  VI-C Power-scaling laws

Herein, we verify Propositions 1-3 providing the power-scaling laws, also
denoted as asymptotic results that correspond to the scenarios A-C mentioned
earlier. In addition, we elaborate on the resultant power savings with
comparison to the analytical exact results provided by Theorem 1.

Figure 2: Sum SE per versus the uplink transmit power
$p_{\mathrm{p}}=p_{\mathrm{u}}$ for varying number of APs $M$ with validation
by Monte-Carlo simulations ($N=3$, $W=5$, and
$p_{\mathrm{r}}=2Wp_{\mathrm{u}}$).

Figure 3: Sum SE versus the number of APs $M$ for different scenarios ($N=3$,
$p_{\mathrm{p}}=p_{\mathrm{u}}$, and $p_{\mathrm{r}}=2Wp_{\mathrm{u}}$), when
(a) $W=5$ and (b) $W=20$ user pairs.

Fig 4 sheds light into Scenario A by depicting the sum SE with respect to the
number of APs $M$ for varying scaling in terms of the parameter $\alpha$. In
general, we observe that the asymptotic results approach the exact curves as
$M\to\infty$. When $\alpha=0.7\left(0<\alpha<1\right)$, $R_{i}\to\infty$,
while if $\alpha=1$, the asymptotic SE saturates and approaches the analytical
result. The third group of curves corresponds to $\alpha=1.4>1$. In such case,
$R_{i}$ tends to zero.

Figs. 5 and 6 illustrate the properties regarding the power savings of
Scenario B described by Proposition 2. Especially, in any of the cases i)
$\beta=1$ and $0<\gamma<1$, ii) $0<\beta<1$ and $\gamma=1$, and iii)
$\beta=\gamma=1$, we show in Fig. 5 that the asymptotic results converge to
specific values and approach the exact results in the large number of APs
regime ($M\to\infty$) as described by Corollaries 3, 4. In the upper set of
plots of Fig. 6, we observe that when $\beta$ or $\gamma$ is greater than one,
which means that the transmit power of the MAC or BC phase is cut down too
much, the sum SE approaches zero as $M\to\infty$. In fact, the larger the
parameter, being greater than one, the faster the decrease of the SE to zero.
Furthermore, when both the transmit powers of users and APs are reduced
tolerably such that $\beta<1$ and $\gamma<1$, the sum SE $R_{i}$ increases
without bound as the lower set of lines of Fig. 6 reveals.

Figure 4: Sum SE versus the number of APs $M$ by means of asymptotic (Scenario
A) and exact analysis (Theorem 1) for $N=3$, $W=5$,
$p_{\mathrm{p}}={E_{\mathrm{p}}}/{M^{\alpha}}$ with
$E_{\mathrm{p}}=10~{}\mathrm{dB}$.

Fig. 7 represents Scenario C describing the interplay between the pilot symbol
power and and the users/relay powers. The sums $\alpha+\beta$ and
$\alpha+\gamma$ determine the behavior of the sum SE in the large number of
APs limit. Hence, if we set $\alpha=1.1$, $\beta=1.2$, $\gamma=0.4$ and
$\alpha=0.9$, $\beta=1.4$, $\gamma=0.6$, we observe that both lines converge
to zero as $M\to\infty$ since, in both cases, we have $\alpha+\beta=2.3$ and
$\alpha+\gamma=1.5$. Notably, the two lines converge to each other (their gap
decreases) as $M$ increases, i.e., the asymptotic sum SE is the same because
the two sums are kept identical. Furthermore, the line with $\alpha=0.9$
provides better SE for finite number of APs although the transmit relay power
is cut down more because the channel is estimated with higher quality. The
middle set of lines demonstrates that the sum SE grows without bound when both
$\alpha+\beta<1$ and $\alpha+\gamma<1$ simultaneously. The third subfigure
illustrates that if any of the two following conditions are satisfied, the sum
SE approaches a non-zero limit. Specifically, if i) $\alpha+\beta=1$ and
$\beta>\gamma\geq 0$ or ii) $\alpha+\gamma=1$ and $\gamma>\beta\geq 0$, then
$R_{i}$ saturates.

Figure 5: Sum SE versus the number of APs $M$ by means of asymptotic (Scenario
B) and exact analysis (Theorem 1) for $N=3$, $W=5$,
$p_{\mathrm{u}}={E_{\mathrm{u}}}/{M^{\beta}}$ with
$E_{\mathrm{u}}=10~{}\mathrm{dB}$, and
$p_{\mathrm{r}}={E_{\mathrm{r}}}/{M^{\gamma}}$ with
$E_{\mathrm{r}}=10~{}\mathrm{dB}$ (finite limits). Figure 6: Sum SE versus the
number of APs $M$ by means of asymptotic (Scenario B) and exact analysis
(Theorem 1) for $N=3$, $W=5$, $p_{\mathrm{u}}={E_{\mathrm{u}}}/{M^{\beta}}$
with $E_{\mathrm{u}}=10~{}\mathrm{dB}$, and
$p_{\mathrm{r}}={E_{\mathrm{r}}}/{M^{\gamma}}$ with
$E_{\mathrm{r}}=10~{}\mathrm{dB}$ (zero and unbounded limits). Figure 7: Sum
SE versus the number of APs $M$ by means of asymptotic (Scenario C) and exact
analysis (Theorem 1) for $N=3$, $W=5$,
$p_{\mathrm{p}}={E_{\mathrm{p}}}/{M^{\alpha}}$ with
$E_{\mathrm{p}}=10~{}\mathrm{dB}$,
$p_{\mathrm{u}}={E_{\mathrm{u}}}/{M^{\beta}}$ with
$E_{\mathrm{u}}=10~{}\mathrm{dB}$, and
$p_{\mathrm{r}}={E_{\mathrm{r}}}/{M^{\gamma}}$ with
$E_{\mathrm{r}}=10~{}\mathrm{dB}$ (zero, unbounded, and finite limits). Figure
8: Sum SE versus the number of APs $M$ by means of uniform (Theorem 1) and
optimal (Algorithm 1) power allocations for $p_{\mathrm{p}}=10~{}\mathrm{dB}$
(blue lines) and $p_{\mathrm{p}}=15~{}\mathrm{dB}$ (red lines).

###  VI-D Power allocation

Fig. 8 shows the performance of the two-way CF mMIMO system with optimal power
allocation using Algorithm 1 when $M=200$ APs and with varying channel
estimation accuracy by means of $p_{\mathrm{p}}$ while $\theta=1.1$ to achieve
good accuracy with reasonable convergence time. Also, we assume that the total
power budget is $P=10~{}\mathrm{dB}$. Moreover, we consider uniform power
allocation by using Theorem 1 for the sake of comparison. Notably, the optimal
power allocation performs better than uniform power allocation, and it results
in an improvement of $63.2\%$ and $59.3\%$ when $p_{\mathrm{p}}=10~{}dB$ and
$p_{\mathrm{p}}=15~{}dB$, respectively. In other words, a better channel
estimation barely affects the sum SE.

##  VII Conclusion

This paper investigated the sum SE of a multipair two-way HD relaying system
assisted by a CF mMIMO architecture employing MR processing and accounting for
imperfect CSI. Contrary to the common collocated mMIMO layout, the distributed
CF mMIMO design achieves higher performance. Furthermore, power-scaling laws,
achieving to scale the transmit powers of the users and APs while maintaining
the desired SE, were obtained. Also, the trade-offs regarding these laws was
examined. Finally, we performed an optimal power allocation concerning the
transmit powers of the APs and users during the data transmission phase
towards the improvement of the SE with comparison to uniform power allocation.

##  Appendix A Proof of Theorem 1

We start with the derivation of $\gamma_{i}^{\text{MAC}}$. The desired signals
of $\mathrm{T}_{\mathrm{A},i}$ and $\mathrm{T}_{\mathrm{B},i}$ in (18) are
written as

$\displaystyle\mathbb{E}\Big{\\{}\sum_{m=1}^{M}\\!\\!\left(\\!\hat{{\mathbf{h}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{h}}_{mi}+\hat{{\mathbf{g}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{h}}_{mi}\\!\right)\\!\\!\\!\Big{\\}}$
$\displaystyle\\!=\\!N\\!\\!\sum_{m=1}^{M}\\!\\!\phi_{\mathrm{A},mi},$ (74)
$\displaystyle\mathbb{E}\Big{\\{}\sum_{m=1}^{M}\\!\\!\left(\\!\hat{{\mathbf{h}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{g}}_{mi}+\hat{{\mathbf{g}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{g}}_{mi}\\!\right)\\!\\!\\!\Big{\\}}$
$\displaystyle\\!=\\!N\\!\\!\sum_{m=1}^{M}\\!\\!\phi_{\mathrm{B},mi},$ (75)

since ${\mathbf{h}}_{mi}$ and ${\mathbf{g}}_{mi}$ are independent. We continue
with the derivations of $\mathrm{EE}_{\mathrm{A},i}^{\text{MAC}}$,
$\mathrm{EE}_{\mathrm{B},i}^{\text{MAC}}$, $\mathrm{IUI}_{i}^{\text{MAC}}$,
and $\mathrm{N}_{i}^{\text{MAC}}$. Specifically, we have

$\displaystyle\mathrm{EE}_{\mathrm{A},i}^{\text{MAC}}$
$\displaystyle=\eta_{\mathrm{A},i}\\!\left(\\!{\mathrm{var}}\Big{\\{}\sum_{m=1}^{M}\hat{{\mathbf{h}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{h}}_{mi}\Big{\\}}+{\mathrm{var}}\Big{\\{}\sum_{m=1}^{M}\hat{{\mathbf{g}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{h}}_{mi}\Big{\\}}\\!\\!\right)\\!,$
(76)

where (76) is obtained because the variance of a sum of independent RVs is
equal to the sum of the variances. The first term of (76) is obtained as

$\displaystyle{\mathrm{var}}\Big{\\{}\sum_{m=1}^{M}\hat{{\mathbf{h}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{h}}_{mi}\Big{\\}}=\sum_{m=1}^{M}\mathbb{E}\Big{\\{}\Big{|}\hat{{\mathbf{h}}}_{mi}^{{\scriptscriptstyle\mathsf{H}}}{{\mathbf{h}}}_{mi}-\mathbb{E}\Big{\\{}\hat{{\mathbf{h}}}_{mi}^{{\scriptscriptstyle\mathsf{H}}}{{\mathbf{h}}}_{mi}\Big{|}^{2}\Big{\\}}\Big{\\}}$
(77)
$\displaystyle=\sum_{m=1}^{M}\left(\mathbb{E}\Big{\\{}\Big{|}\hat{{\mathbf{h}}}_{mi}^{{\scriptscriptstyle\mathsf{H}}}{{\mathbf{h}}}_{mi}\Big{|}^{2}\Big{\\}}-\Big{|}\mathbb{E}\\{\hat{{\mathbf{h}}}_{mi}^{{\scriptscriptstyle\mathsf{H}}}\hat{{\mathbf{h}}}_{mi}\\}\Big{|}^{2}\right)$
$\displaystyle=\sum_{m=1}^{M}\left(\mathbb{E}\Big{\\{}\Big{|}\|\hat{{\mathbf{h}}}_{mi}\|^{2}+\hat{{\mathbf{h}}}_{mi}^{{\scriptscriptstyle\mathsf{H}}}\tilde{{\mathbf{h}}}_{mi}\Big{|}^{2}\Big{\\}}-N^{2}\phi_{\mathrm{A},mi}^{2}\right)$
(78)
$\displaystyle=\\!\sum_{m=1}^{M}\\!\\!\left(\mathbb{E}\Big{\\{}\|\hat{{\mathbf{h}}}_{mi}\|^{4}\Big{\\}}\\!+\\!\mathbb{E}\Big{\\{}|\hat{{\mathbf{h}}}_{mi}^{{\scriptscriptstyle\mathsf{H}}}\tilde{{\mathbf{h}}}_{mi}|^{2}\Big{\\}}\\!-\\!N^{2}\phi_{\mathrm{A},mi}^{2}\right)$
(79)
$\displaystyle=\\!N\\!\sum_{m=1}^{M}\\!\\!\left(\left(N+1\right)\phi^{2}_{\mathrm{A},mi}\\!+\\!\phi_{\mathrm{A},mi}{e}_{\mathrm{A},mi}\\!-\\!N^{2}\phi_{\mathrm{A},mi}^{2}\right)$
(80)
$\displaystyle=N\sum_{m=1}^{M}\alpha_{\mathrm{A},mi}\phi_{\mathrm{A},mi},$
(81)

where (77) follows again because the variance of a sum of independent RVs is
equal to the sum of the variances. In (78), we have considered that
$\tilde{{\mathbf{h}}}_{mi}$ is independent of $\hat{{\mathbf{h}}}_{mi}$ and
has zero mean. The identity
$\mathbb{E}\\{\|\hat{{\mathbf{h}}}_{mi}\|^{4}\\}=N\left(N+1\right)\phi^{2}_{\mathrm{A},mi}$
has been used in 80, and (81) follows after some algebraic manipulations since
$e_{\mathrm{A},mi}=\alpha_{\mathrm{A},mi}-\phi_{\mathrm{A},mi}$. More easily,
the second term of (76) is given by

$\displaystyle{\mathrm{var}}\Big{\\{}\sum_{m=1}^{M}\hat{{\mathbf{g}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{h}}_{mi}\Big{\\}}=N\sum_{m=1}^{M}\alpha_{\mathrm{A},mi}\phi_{\mathrm{B},mi}.$
(82)

Hence, $\mathrm{EE}_{\mathrm{A},i}^{\text{MAC}}$ becomes my means of (81) and
(82)

$\displaystyle\mathrm{EE}_{\mathrm{A},i}^{\text{MAC}}=N\eta_{\mathrm{A},i}\sum_{m=1}^{M}\alpha_{\mathrm{A},mi}\left(\phi_{\mathrm{A},mi}+\phi_{\mathrm{B},mi}\right).$
(83)

In the same way, we derive

$\displaystyle\mathrm{EE}_{\mathrm{B},i}^{\text{MAC}}=N\eta_{\mathrm{B},i}\sum_{m=1}^{M}\alpha_{\mathrm{B},mi}\left(\phi_{\mathrm{A},mi}+\phi_{\mathrm{B},mi}\right).$
(84)

The term, describing the inter-user interference, is obtained as

$\displaystyle\mathrm{IUI}_{i}^{\text{MAC}}\\!=\\!\sum_{j\neq
i}^{W}\\!\\!\left(\\!\eta_{\mathrm{A},j}\sum_{m=1}^{M}\\!\\!\left(\mathbb{E}\\{|\hat{{\mathbf{h}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{h}}_{mj}|^{2}\\}+\mathbb{E}\\{|\hat{{\mathbf{g}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{h}}_{mj}|^{2}\\}\\!\right)\\!\\!\right)$
$\displaystyle+\sum_{j\neq
i}^{W}\\!\\!\left(\eta_{\mathrm{B},j}\sum_{m=1}^{M}\left(\mathbb{E}\\{|\hat{{\mathbf{h}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{g}}_{mj}|^{2}\\}+\mathbb{E}\\{|\sum_{m=1}^{M}\hat{{\mathbf{g}}}^{{\scriptscriptstyle\mathsf{H}}}_{mi}{\mathbf{g}}_{mj}|^{2}\\}\\!\right)\\!\\!\right)$
$\displaystyle=\sum_{j\neq
i}^{W}\left(\eta_{\mathrm{A},j}\sum_{m=1}^{M}\left(\phi_{\mathrm{A},mi}\alpha_{\mathrm{A},mj}+\phi_{\mathrm{B},mi}\alpha_{\mathrm{B},mj}\right)\right)$
$\displaystyle+\sum_{j\neq
i}^{W}\left(\eta_{\mathrm{B},j}\sum_{m=1}^{M}\left(\phi_{\mathrm{A},mi}\alpha_{\mathrm{B},mj}+\phi_{\mathrm{B},mi}\alpha_{\mathrm{A},mi}\right)\right)$
$\displaystyle=\\!N\\!\sum_{j\neq
i}^{W}\\!\sum_{m=1}^{M}\\!\\!\left(\eta_{\mathrm{A},j}\alpha_{\mathrm{A},mj}\\!+\\!\eta_{\mathrm{B},j}\alpha_{\mathrm{B},mj}\right)\\!\\!\left(\phi_{\mathrm{A},mi}\\!+\\!\phi_{\mathrm{B},mi}\right)\\!,\\!\\!\\!$
(85)

where we have applied the property
$\mathbb{E}\\{|X+Y|^{2}\\}=\mathbb{E}\\{|X|^{2}\\}+\mathbb{E}\\{|Y|^{2}\\}$,
holding between two independent random variables $X$ and $Y$ with
$\mathbb{E}\\{X\\}=0$. Next, the noise term becomes

$\displaystyle\mathrm{N}_{i}^{\text{MAC}}$
$\displaystyle=\frac{1}{p_{\mathrm{u}}}\sum_{m=1}^{M}\left(\phi_{\mathrm{A},mi}+\phi_{\mathrm{B},mi}\right).$
(86)

Substitution of (74), (75),(83), (84), (85), and (86) into (18) and (23)
provides $R_{i}^{\text{MAC}}$ as well as $R_{\mathrm{X},i}^{\text{MAC}}$ for
$\mathrm{X}\in\\{\mathrm{A},\mathrm{B}\\}$.
The proof continues with the derivation of $R_{\mathrm{A},i}^{\text{BC}}$ by
means of the computation of (29)-(33). Specifically, regarding the desired
signal, we have

$\displaystyle\mathbb{E}\Big{\\{}\sum_{m=1}^{M}\\!\\!\sqrt{p_{\mathrm{d},m}\eta_{\mathrm{B},mi}}{\mathbf{h}}_{mi}^{{\scriptscriptstyle\mathsf{T}}}\hat{{\mathbf{h}}}_{mi}^{*}\Big{\\}}\\!=\\!\mathbb{E}\Big{\\{}\\!\sum_{m=1}^{M}\\!\\!\sqrt{p_{\mathrm{d},m}\eta_{\mathrm{B},mi}}\|\hat{{\mathbf{h}}}_{mi}\|^{2}\Big{\\}}$
$\displaystyle\\!+\\!\mathbb{E}\Big{\\{}\sum_{m=1}^{M}\sqrt{p_{\mathrm{d},m}\eta_{\mathrm{B},mi}}\tilde{{\mathbf{h}}}_{mi}^{{\scriptscriptstyle\mathsf{T}}}\hat{{\mathbf{h}}}_{mi}^{*}\Big{\\}}$
$\displaystyle\\!=\\!N\sum_{m=1}^{M}\sqrt{p_{\mathrm{d},m}\eta_{\mathrm{B},mi}}\phi_{\mathrm{A},mi},$
(87)

where we have used the independence between $\hat{{\mathbf{h}}}_{mi}$ and
$\tilde{{\mathbf{h}}}_{mi}$. Also, $\mathrm{BU}_{A_{i}}^{\text{BC}}$ and
$\mathrm{BU}_{B_{i}}^{\text{BC}}$ are obtained by following the same procedure
with (81). In particular, we have

$\displaystyle\mathrm{BU}_{A_{i}}^{\text{BC}}$
$\displaystyle=N\sum_{m=1}^{M}\eta_{\mathrm{B},mi}\alpha_{\mathrm{A},mi}\phi_{\mathrm{A},mi},$
(88) $\displaystyle\mathrm{BU}_{B_{i}}^{\text{BC}}$
$\displaystyle=N\sum_{m=1}^{M}\eta_{\mathrm{A},mi}\alpha_{\mathrm{A},mi}\phi_{\mathrm{B},mi}.$
(89)

Next, regarding $\mathrm{IUI}_{A_{i}}^{\text{BC}}$, we have

$\displaystyle\mathbb{E}\Big{\\{}\Big{|}\sum_{m=1}^{M}\eta_{\mathrm{A},mj}^{1/2}{\mathbf{h}}_{mi}^{{\scriptscriptstyle\mathsf{T}}}\hat{{\mathbf{g}}}_{mj}^{*}\Big{|}^{2}\Big{\\}}=\sum_{m=1}^{M}\eta_{\mathrm{A},mj}\mathbb{E}\\{|{\mathbf{h}}_{mi}^{{\scriptscriptstyle\mathsf{T}}}\hat{{\mathbf{g}}}_{mj}^{*}|^{2}\\}$
$\displaystyle=N\sum_{m=1}^{M}\eta_{\mathrm{A},mj}\alpha_{\mathrm{A},mi}\phi_{\mathrm{B},mj},$
(90)

where in the first equation, we have used again that
$\mathbb{E}\\{|X+Y|^{2}\\}=\mathbb{E}\\{|X|^{2}\\}+\mathbb{E}\\{|Y|^{2}\\}$.
The last term, $\mathrm{IUI}_{B_{i}}^{\text{BC}}$, is obtained similarly as

$\displaystyle\mathrm{IUI}_{B_{i}}^{\text{BC}}=N\sum_{m=1}^{M}\eta_{\mathrm{B},mj}\alpha_{\mathrm{A},mi}\phi_{\mathrm{A},mj}.$
(91)

By using (87)-(91) and (14), we obtain $R_{\mathrm{A},i}^{\text{BC}}$ and
conclude the proof since $R_{\mathrm{B},i}^{\text{BC}}$ can be derived in the
same fashion.

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