# Half-Duplex APs with Dynamic TDD vs. Full-Duplex APs in Cell-Free Systems

Anubhab Chowdhury and Chandra R. Murthy  The authors are with the ECE Dept.,
Indian Institute of Science, Bangalore, 560012, India. (e-mails:{anubhabc,
<EMAIL_ADDRESS>
###### Abstract

In this paper, we present a comparative study of half-duplex (HD) access
points (APs) with dynamic time-division duplex (DTDD) and full-duplex (FD) APs
in cell-free (CF) systems. Although both DTDD and FD CF systems support
concurrent downlink transmission and uplink reception capability, the sum
spectral efficiency (SE) is limited by various cross-link interferences. We
first present a novel pilot allocation scheme that minimizes the pilot length
required to ensure no pilot contamination among the user equipments (UEs)
served by at least one common AP. Then, we derive the sum SE in closed form,
considering zero-forcing combining and precoding along with the signal-to-
interference plus noise ratio optimal weighting at the central processing
unit. We also present a provably convergent algorithm for joint uplink-
downlink power allocation and uplink/downlink mode scheduling of the APs (for
DTDD) to maximize the sum SE. Our numerical results illustrate the superiority
of the proposed algorithms over several benchmarks and show that the sum SE
with DTDD can outperform an FD CF system with similar antenna density. Thus,
DTDD combined with CF is a promising alternative to FD that attains the same
performance using HD APs, while obviating the burden of intra-AP interference
cancellation.

###### Index Terms:

Cell-free MIMO, full-duplex, dynamic TDD, pilot allocation, fractional
programming

##  I Introduction

Wireless systems for $5$G and beyond are required to serve an increasingly
large number of user-equipments (UEs) while supporting uniformly good quality-
of-service (QoS) and high spectral efficiency (SE) requirements. With this in
mind, two potential physical layer solutions have been envisioned: $(i)$ the
capability to serve both uplink (UL) and the downlink (DL) UEs using the same
time-frequency resources, thereby potentially doubling the sum UL-DL SE over
conventional time/frequency division duplexing (T/FDD) systems, and $(ii)$
distributed deployments of the remote radio units or access points (APs) for
ubiquitous connectivity and high macro-diversity as opposed to a centralized
MIMO cellular system. The key enablers of the above are: $(i)$ the use of
full-duplex (FD) APs [1] or the use of dynamic time-division duplexing (DTDD)
with half-duplex (HD) APs [2, 3, 4]; and $(ii)$ cell-free multiple-input
multiple-output (CF-MIMO) [5, 6]; respectively. This paper presents a
comparative performance analysis of FD and DTDD CF-MIMO systems.

We note that, in CF systems, both FD and DTDD can enable simultaneous
transmission (reception) to (from) HD UEs over the same time-frequency
resources. However, cross-link interferences (CLIs), i.e., the inter-AP
interference (InAI) and inter-UE interference (InUI), limit the achievable SE
in both systems. Additionally, in the FD system, the received signal at each
AP is contaminated by its own transmitted signal, called _self-interference
(SI)_ [1], which we refer to as intra-AP interference (IrAI) in the sequel.
The cancellation of IrAI demands power-hungry and expensive hardware in
addition to baseband signal processing overheads. However, if IrAI can be
effectively canceled, in an FD-CF system, _all_ the APs in the vicinity can
assist in a given UE’s transmission/reception, while in the DTDD system, only
the subset of APs operating in UL (DL) can assist in decoding (precoding) the
UE’s data signal. On the other hand, as we will see, InAI is higher in an FD
system, as _all_ the APs interfere with the received signal at any AP.
However, with DTDD, only the subset of APs operating in DL mode cause InAI at
the APs operating in UL mode. Thus, which duplexing scheme is better and under
what conditions is not clear. Answering this question via a careful
theoretical analysis is the main aim of this paper.

####  I-1 Literature review

CF, DTDD, and FD are all technologies that have received intense research
attention over the past few years. The focus of this literature review is on
studies that consider either DTDD or FD systems in the CF architecture. The
analysis of the sum SE of a DTDD CF system with fully centralized processing
was presented in [7]; however, with a pre-scheduled set of UL and DL APs. We
proposed a distributed processing-based sum SE analysis with a low complexity
algorithm for AP scheduling and fixed power allocation for UL and DL in [3].
This work significantly extends our previous work [3] by considering minimum
mean squared error (MMSE)/zero-forcing (ZF) precoding and combining at the APs
and signal-to-interference plus noise ratio (SINR)-optimal combining at the
central processing unit (CPU), pilot allocation via graph coloring, joint UL-
DL power allocation for sum SE maximization, and comparing the performance
against an FD system. Recently, the authors in [8] investigated the joint
power UL-DL power allocation problem, assuming perfect CSI, in a DTDD CF
system. The power allocation policy of [8] is derived for a maximal ratio
combiner and precoder, with equal weighting-based combining of the APs’
signals at the CPU in the UL, which is suboptimal for CF [5].

On the other hand, the performance improvement achieved by FD APs over
conventional TDD CF systems has been investigated in [9, 10, 11]. An
expository study on the interplay of the CLIs and IrAI on the achievable SEs
under DTDD and FD was conducted in [12]. However, the results obtained were
based on fixed power allocation and under the availability of perfect CSI.
Now, finding a pilot allocation scheme in a CF system is challenging because
multiple APs jointly serve the UEs in the area. Specifically, in contrast to a
cellular system where only the serving AP requires CSI from a given UE, in CF,
accurate CSI is required at all the APs in the vicinity of the UE. The works
that account for pilot allocation in DTDD CF systems consider either complex
iterative algorithm [3], random allocation of pilots [7], or assume orthogonal
pilots across all UEs [8]. Recent works on FD CF systems have also considered
orthogonal pilots [9, 13, 14] or have analyzed the effect of pilot
contamination by abstracting it as an additive channel error term in the
channel estimate [11], which does not explicitly account for the pilot length,
design or allocation across the UEs. Here, we systematically address the pilot
allocation problem in both DTDD and FD settings.

####  I-2 Contributions

To the best of our knowledge, a comprehensive study of these two duplexing
schemes accounting for practical issues such as pilot allocation, optimal
weighting at the CPU, interference canceling precoder and combiner design, AP
scheduling, and UL/DL power allocation is not available in the literature. It
is essential to account for these aspects because the critical bottleneck,
namely, the CLIs of DTDD and FD, is heavily dependent on and can be controlled
by these factors. Our key contributions are:

  1. 1.

We minimize the number of pilots required to ensure orthogonality among the
UEs in close proximity via formulating an equivalent graph coloring problem,
with the constraint that connected vertices (i.e., the UEs which are connected
to a common AP) are allotted distinct colors (orthogonal pilots). Although the
problem is NP-hard in general, it can be optimally solved for bipartite graphs
(as is the case in our problem) via a low complexity greedy algorithm (see
Algorithm 1), thereby minimizing the number of colors (i.e., the pilot
length). We empirically show that Algorithm 1 offers a substantial improvement
in the normalized mean square error (NMSE) of the estimated channels compared
to existing methods.

  2. 2.

We analyze the sum UL-DL SE considering MMSE combiners and regularized ZF
(RZF) precoders. We also derive closed-form expressions for the sum UL-DL SE
with ZF combiners and precoders (see Lemma 5 and Lemma 6). These expressions
uncover the effects of InAI, IrAI, and InUI on the UL-DL SEs, and how power
control and UL/DL scheduling of the APs (for DTDD) dictate the strengths of
these CLIs. Also, in the UL, we present an SINR optimal weighting scheme,
which ensures that the received SINR at the CPU is maximized (see Lemma 4).

  3. 3.

Next, we focus on the sum UL-DL SE maximization with set constraints on the
UL/DL APs and transmit power constraints on the APs and UEs. This problem of
joint AP scheduling and power control is non-convex and NP-hard. We decouple
it into two sub-problems.

    1. i.

We optimize the UL and DL power control coefficients for a given AP schedule.
We solve this non-convex problem using fractional programming (FP)111FP
convexifies the non-convex cost function such that the optimal solution of the
surrogate cost function and the original cost function is the same [15]. This
is in contrast with the approach adopted in [8, 10], where the convex cost
function is typically a lower bound of the original cost function and the
algorithms optimize the lower bound. that employs a series of equivalent
convex reformulations. Further, closed-form solutions for the power allocation
coefficients and associated auxiliary variables are derived using the
alternating direction method of multipliers (ADMM)222ADMM is an effective
approach for reducing the computational cost in large dimensional problems
compared to interior-point methods or general purpose solvers such as CVX or
MOSEK [16]. in the case of DL and using an augmented Lagrange multiplier in
the case of UL. Also, in the UL, we observe considerable improvement in the
proposed SINR optimal combining coupled with UL power control compared to
existing benchmarks and when either of these two schemes is applied
individually (see Fig. 4). The resulting algorithms for each sub-problem are
shown to converge to local optima (see Proposition 3 and Proposition 4).
Finally, our proposed FP-based algorithms are precoder/combiner scheme
agnostic, unlike [8], and require fewer auxiliary variables, which makes our
solutions scalable for large distributed systems.

    2. ii.

For AP scheduling, we develop a greedy AP mode (UL/DL) selection algorithm,
where, at each iteration, we select the AP and the corresponding mode such
that the incremental gain in the sum UL-DL SE is maximum. This pragmatic low-
complexity approach solves an otherwise exponentially complex scheduling
algorithm in polynomial time.

We perform extensive numerical experiments that reveal the superiority of the
proposed pilot length optimization and pilot allocation scheme, the UL/DL
power control algorithms, and AP-scheduling algorithm over several existing
schemes (see Fig. 4 and Fig. 5). Surprisingly, our results show that for the
same number of APs and antenna density, DTDD procures a better sum UL-DL SE
compared to an FD-enabled CF system (see Fig. 6). Specifically, the
$90\%$-likely sum UL-DL SE of the DTDD CF system is $21\%$ higher than that of
the FD system under similar system parameters (see Fig. 6(a)). Further, we
observe that even with double the antenna density, the performance of the FD
system can be limited by InAI and IrAI, while DTDD is more resilient to InAI
(see Fig. 7). Thus, we can obtain the benefits of FD via DTDD itself,
obviating the need for IrAI suppression at the APs.

_Notation:_ The transpose, hermitian, complex conjugation, and trace
operations are denoted by $(\cdot)^{T}$, $(\cdot)^{H}$, $(\cdot)^{*}$, and
${\tt tr}(\cdot)$, respectively. For a matrix
$\mathbf{A}\in\mathbb{C}^{M\times N}$,
$[\mathbf{A}]_{:,j}\in\mathbb{C}^{M\times 1}$ denotes its $j$th column. For a
vector $\mathbf{x}\in\mathbb{C}^{N}$, $[\mathbf{x}]_{m}$ denotes its $m$th
entry. $\mathsf{diag}(\mathbf{x})\in\mathbb{C}^{N\times N}$ is a diagonal
matrix with $m$th diagonal entry being $[\mathbf{x}]_{m}$.
${}^{\prime}|\cdot|^{\prime}$, ${}^{\prime}\backslash^{\prime}$,
${}^{\prime}\cup^{\prime}$, and ${}^{\prime}\mathsf{c}^{\prime}$ denote the
cardinality, set difference, union, and complement of sets, respectively. The
symbol $\emptyset$ denotes an empty set. $\mathbb{E}[\cdot]$ and ${\tt
var}\\{\cdot\\}$ denote the mean and variance of a random variable/vector,
respectively. $\mathbf{x}\sim\mathcal{CN}(\mathbf{0}_{N},\mathbf{R}_{N})$
indicates that $\mathbf{x}\in\mathbb{C}^{N}$ is a zero mean ($\mathbf{0}_{N}$)
circularly symmetric complex Gaussian (CSCG) random vector with covariance
matrix $\mathbf{R}_{N}\in\mathbb{C}^{N\times N}$.

##  II System Model

In the DTDD CF setup, $M$ HD-APs each equipped with $N$ antennas jointly and
coherently serving a total of $K$ single antenna UL and DL UEs using the same
time-frequency resources. Let the sets $\mathcal{U}_{\mathsf{u}}$ and
$\mathcal{U}_{\mathsf{d}}$ contain the indices of UL UEs and DL UEs,
respectively, with
$\mathcal{U}_{\mathsf{u}}\cap\mathcal{U}_{\mathsf{d}}=\emptyset$,
$\mathcal{U}_{\mathsf{u}}\cup\mathcal{U}_{\mathsf{d}}=\mathcal{U}$, and
$\lvert\mathcal{U}\rvert=K$. The UL channel from the $k$th UE to the $m$th AP
is modeled as
$\mathbf{f}_{mk}=\sqrt{\beta_{mk}}\mathbf{h}_{mk}\in\mathbb{C}^{N}$, where
$\beta_{mk}>0$ captures the effect of large scale fading and path-loss which
remain unchanged over several channel coherence intervals and are known to the
APs and the CPU [5]. The fast fading component,
$\mathbf{h}_{mk}\sim\mathcal{CN}(\mathbf{0}_{N},\mathbf{I}_{N})\in\mathbb{C}^{N}$,
is independent and identically distributed (i.i.d.) and is estimated at the
APs using pilot signals at the beginning of each coherence block.

(a) DTDD CF MIMO system: the overall system can serve UL and DL UEs
simultaneously, forming a _virtual_ FD system.

(b) FD-enabled CF system: each AP can serve both UL and DL UEs; however, the
APs suffer from IrAI.

Figure 1: The system and models for DTDD and FD CF systems, respectively.

In the FD system, each AP is equipped with $N_{\mathsf{tx}}$ transmit and
$N_{\mathsf{rx}}$ receive antennas. Let
$\mathbf{f}_{\mathsf{u},mk}=\sqrt{\beta_{\mathsf{u},mk}}\mathbf{h}_{\mathsf{u},mk}\in\mathbb{C}^{N_{\mathsf{rx}}}$
and
$\mathbf{f}_{\mathsf{d},mn}=\sqrt{\beta_{\mathsf{d},mn}}\mathbf{h}_{\mathsf{d},mn}\in\mathbb{C}^{N_{\mathsf{tx}}}$
be the UL channel between the receive antenna array of the $m$th AP to $k$th
UL UE and the DL channel between the transmit antenna array of the $m$th AP to
the $n$th DL UE, respectively. Here, $\beta_{\mathsf{i},mk}$ and
$\mathbf{h}_{\mathsf{i},mk}$, $\mathsf{i}=\\{\mathsf{u},\mathsf{d}\\}$,
corresponds to the fast and slow fading components of the UL and the DL
channels, respectively, and follow similar statistical modeling as in the DTDD
system.

The inter-AP channels remain constant for several coherence intervals and are
mostly line-of-sight (LoS), whose CSI can be made available to the CPU before
the pilot and data transmission phase. Thus, the overhead of inter-AP channel
estimation does not affect the available data transmission duration. However,
the inter-AP CSI at the CPU may be erroneous, which we model using Gaussian
distributed additive noise as per [1, 3, 9]. Specifically, the inter-AP
channel from the $j$th DL AP to the $m$th UL AP is denoted by
$\tilde{\mathbf{{G}}}_{mj}\in\mathbb{C}^{N\times N}$ whose elements are i.i.d.
$\mathcal{CN}(0,\zeta_{mj}^{\mathsf{InAP}})$. Here
$\zeta_{mj}^{\mathsf{InAP}}$ captures the effects of both the large scale
fading and the power of the residual InAI.

For FD CF, IrAI at each AP can be suppressed via active SI cancelation and
antenna isolation. It has been argued that the residual IrAI follows the
Rayleigh distribution [17]. Following this, we model $m$th AP’s residual IrAI
channel between the transmit and the receive antenna links as
$\mathbf{G}_{mm}^{\mathsf{SI}}\sim\mathcal{CN}(0,\zeta_{mm}^{\mathsf{SI}})$,
where the residual IrAI $\zeta_{mm}^{\mathsf{SI}}$ depends on the interference
suppression capability of the hardware[11, 13].

Finally, let $\mathtt{g}_{nk}$ denote the channel between $k$th UL UE and the
$n$th DL UE, modeled as $\mathcal{CN}(0,\epsilon_{nk})$, and is independent
across all UEs [1, 3]. The channel modeling discussed above is illustrated in
Fig. 1(a) and Fig. 1(b).

###  II-A Problem Statement

Having described the system model, we now present the key problems considered
in this paper. As illustrated in Fig. 1(a) and Fig. 1(b), the performance of
both DTDD and FD is affected by CLIs, viz. InAI, IrAI (for FD), and InUI. Now,
the strengths of these CLIs, as we shall see in Sec. IV, depend on the
estimated UL/DL channel statistics, choice of precoder and combiners, and,
most critically, on the UL and DL power allocation strategies. Further, in
DTDD, we get the additional flexibility to schedule the APs’ UL/DL modes,
which can reduce InAI. Keeping these in mind, our goal is to maximize the
achievable sum UL-DL SE under these two duplexing schemes.333In [6], the
authors point out that since a distributed CF system is inherently fair,
additionally requiring fairness degrades the sum SE without an appreciable
improvement in the fairness criterion. Thus, we consider the sum UL-DL SE as
the metric to be maximized. In this regard, we pose and address the following
problems: $(i)$ Allocation of pilot signals that can ensure no contamination
in the APs in the vicinity of every UE while using a minimum number of
orthogonal pilots; $(ii)$ design of combiners and precoders at the APs and the
CPU to reduce interference; $(iii)$ AP scheduling in the UL/DL modes for DTDD;
and $(iv)$ UL/DL power allocation policies for DTDD and FD to maximize the sum
UL-DL SE. We begin with channel estimation and pilot allocation in the next
section.

##  III Channel Estimation and Pilot Allocation

####  III-1 DTDD CF

During the channel estimation phase, the UL and DL UEs synchronously send UL
pilot sequences to the APs. The APs use the received pilot signals to estimate
the channels between the UEs and the APs. Now, allocating orthogonal pilot
sequences to all the UEs can incur inordinately high channel estimation
overhead. Therefore, we consider that pilot length is constrained to be
$\tau_{p}$, where $\tau_{p}\leq K$, and these pilot sequences are reused among
the UEs. Let
$\mathcal{P}=\\{\boldsymbol{\varphi}_{1},\boldsymbol{\varphi}_{2},\ldots,\boldsymbol{\varphi}_{\tau_{p}}\\}$
be the set of orthonormal pilot sequences, where
$\boldsymbol{\varphi}_{l}\in\mathbb{C}^{\tau_{p}}$. Let $l(k)$ denote the
index of the pilot used by the $k$th UE and $\mathcal{P}_{l(k)}$ denote the
set of UE indices, including the $k$th UE, that use
$\boldsymbol{\varphi}_{l(k)}$. Therefore,
$\langle\boldsymbol{\varphi}_{l(k)},\boldsymbol{\varphi}_{l(k^{\prime})}\rangle=1$,
if $k^{\prime}\in\mathcal{P}_{l(k)}$, and equals to $0$ if
$k^{\prime}\notin\mathcal{P}_{l(k)}$. Let $\mathcal{E}_{p,k}$ be the power of
the pilot signal of the $k$th UE. Then, the received signal matrix at the
$m$th AP becomes
$\mathbf{Y}_{p,m}=\sum\nolimits_{k=1}^{K}\sqrt{\tau_{p}\mathcal{E}_{p,k}}\mathbf{f}_{mk}\boldsymbol{\varphi}_{l(k)}^{T}+\mathbf{W}_{p,m}\in\mathbb{C}^{N\times\tau_{p}},$
where $\mathbf{W}_{p,m}$ is the receiver noise matrix with i.i.d. entires
distributed as $\mathcal{CN}(0,N_{0})$. Post-multiplying $\mathbf{Y}_{p,m}$ by
$\boldsymbol{\varphi}_{l(k)}^{*}$, the processed signal at the $m$th AP
becomes
$\mathbf{y}_{p,m}=\sqrt{\mathcal{E}_{p,k}\tau_{p}}\mathbf{f}_{mk}+\sum\nolimits_{n\in\mathcal{P}_{l(k)}\backslash
k}\sqrt{\mathcal{E}_{p,n}\tau_{p}}\mathbf{f}_{mn}+\mathbf{w}_{p,m},$ with
$\mathbf{w}_{p,m}=\mathbf{W}_{p,m}\boldsymbol{\varphi}_{l(k)}^{*}\sim\mathcal{CN}(\mathbf{0}_{N},N_{0}\mathbf{I}_{N})$.
We can now estimate the channel $\mathbf{f}_{mk}$ using $\mathbf{y}_{p,m}$.

######  Lemma 1.

The linear MMSE (LMMSE) estimate of $\mathbf{f}_{mk}$, denoted as
$\hat{\mathbf{f}}_{mk}$, can be evaluated as
$\hat{\mathbf{f}}_{mk}={\mathbb{E}[\mathbf{f}_{mk}\mathbf{y}_{p,m}^{H}]}({\mathbb{E}[\mathbf{y}_{p,m}\mathbf{y}_{p,m}^{H}]})^{-1}\mathbf{y}_{p,m}$,
which simplifies to
$\hat{\mathbf{f}}_{mk}=\sqrt{\tau_{p}\mathcal{E}_{p,k}}\beta_{mk}c_{mk}\mathbf{y}_{p,m}$,
with
$c_{mk}\triangleq({\tau_{p}\mathcal{E}_{p,k}\beta_{mk}+\tau_{p}\sum\nolimits_{n\in\mathcal{P}_{l(k)}\backslash
k}\mathcal{E}_{p,n}\beta_{mn}+N_{0}})^{-1}$. Further,
$\hat{\mathbf{f}}_{mk}\sim\mathcal{CN}(\mathbf{0}_{N},\alpha_{mk}^{2}\mathbf{I}_{N})$,
with $\alpha_{mk}^{2}=c_{mk}\tau_{p}\mathcal{E}_{p,k}\beta_{mk}^{2}$. The
estimation error, denoted by
$\tilde{\mathbf{f}}_{mk}\triangleq{\mathbf{f}}_{mk}-\hat{\mathbf{f}}_{mk}$, is
distributed as
$\mathcal{CN}(\mathbf{0}_{N},\bar{\alpha}_{mk}^{2}\mathbf{I}_{N})$, with
$\bar{\alpha}_{mk}\triangleq\sqrt{\beta_{mk}-\alpha_{mk}^{2}}$, and
$\tilde{\mathbf{f}}_{mk}$ is uncorrelated with $\hat{\mathbf{f}}_{mk}$ due to
orthogonality principle.

####  III-2 FD CF

For the FD system, all UEs transmit pilots in the UL direction, and we
estimate the channels between the UEs and every AP’s _transmit_ and _receive_
antennas using these UL pilots. The DL precoders are later designed using
channel reciprocity, which obviates the need for separate DL training [1, 13,
9]. Similar to the DTDD case, the MMSE channel estimates at the APs have the
following properties.

######  Lemma 2.

The estimated UL channel $\hat{\mathbf{f}}_{\mathsf{u},mk}$ follows
$\mathcal{CN}(\mathbf{0},\alpha_{\mathsf{u},mk}^{2}\mathbf{I}_{N})$, with
$\alpha_{\mathsf{u},mk}^{2}=\tau_{p}\mathcal{E}_{p,k}\beta_{\mathsf{u},mk}^{2}c_{\mathsf{u},mk}$
and
$c_{\mathsf{u},mk}^{-1}={\tau_{p}\mathcal{E}_{p,k}\beta_{\mathsf{u},mk}+\tau_{p}\sum\nolimits_{{n\in\mathcal{P}_{l(k)}\backslash
k}}\mathcal{E}_{p,n}\beta_{\mathsf{u},mn}+N_{0}}$. The estimated DL channel
$\hat{\mathbf{f}}_{\mathsf{d},mn}$ follows
$\mathcal{CN}(\mathbf{0}_{N},\alpha_{\mathsf{d},mn}^{2}\mathbf{I}_{N})$, with
$\alpha_{\mathsf{d},mn}^{2}=\tau_{p}\mathcal{E}_{p,n}\beta_{\mathsf{d},mn}^{2}c_{\mathsf{d},mn}$
and
$c_{\mathsf{d},mn}^{-1}={\tau_{p}\mathcal{E}_{p,n}\beta_{\mathsf{d},mn}+\tau_{p}\sum\nolimits_{{n^{\prime}\in\mathcal{P}_{l(n)}\backslash
n}}\mathcal{E}_{p,n^{\prime}}\beta_{\mathsf{d},mn^{\prime}}+N_{0}}$. The UL/DL
channel estimation error $\tilde{\mathbf{f}}_{\mathsf{i},mn}$ follows
$\mathcal{CN}(\mathbf{0}_{N},\bar{\alpha}_{\mathsf{i},mn}^{2}\mathbf{I}_{N})$,
with
$\bar{\alpha}_{\mathsf{i},mn}\triangleq\sqrt{\beta_{\mathsf{i},mn}-\alpha_{\mathsf{i},mn}^{2}}$,
$\mathsf{i}\in\\{\mathsf{u},\mathsf{d}\\}$.

Next, we discuss the pilot allocation algorithm. Our goals are two-fold: to
minimize the number of orthonormal pilots, $\tau_{p}$, for channel estimation
and mitigate the effect of pilot contamination.

####  III-3 Pilot Assignment Algorithm

In CF systems, APs jointly serve the UEs. Thus, physically proximal UEs should
not reuse the same pilot sequences even if their nearest APs are different. On
the other hand, assigning pilot sequences so that the received signals at all
the APs are contamination free requires a pilot length at least equal to the
number of UEs; which in turn reduces the duration available for data
transmission. However, we note that although all the APs can serve all the
UEs, only a subset of APs within the vicinity of a UE receive a signal with
sufficient strength for decodability. In other words, at a given AP, the pilot
contamination caused by a UE located far away is minimal due to path-loss and
shadowing. Hence, we consider a UE-centric clustering and ensure that given
any UE, the received signals at the APs within its cluster are contamination-
free. We emphasize that all the APs can still participate in data processing
to/from all the UEs; we enforce orthogonality within the clusters only for the
purpose of pilot allocation. This is illustrated in Fig. 2(a), where the $k$th
UE is connected to the $m$th and $m^{\prime}$th AP. Thus, the pilot assigned
to the $k$th UE should be orthonormal to all the UEs served by the $m$th and
$m^{\prime}$th APs. Next, we discuss how to form such clusters.

We define the following sets:

$\displaystyle\mathcal{U}_{k}\triangleq\left\\{m~{}\mathrm{s.t.}~{}\|\mathbf{u}_{k}-\mathbf{a}_{m}\|\leq
r_{\mathsf{o}},\forall m\in\mathcal{A}\right\\},~{}\forall k\in\mathcal{U}$
(1a)
$\displaystyle\mathcal{A}_{m}\triangleq\left\\{k~{}\mathrm{s.t.}~{}\|\mathbf{u}_{k}-\mathbf{a}_{m}\|\leq
r_{\mathsf{o}},\forall k\in\mathcal{U}\right\\},~{}\forall m\in\mathcal{A},$
(1b)

where $\mathcal{A}$ is the set of AP indices, $\mathbf{u}_{k}$ and
$\mathbf{a}_{m}$ are the locations of the $k$th and $m$th AP,
$r_{\mathsf{o}}\triangleq\max\left\\{\max\nolimits_{k\in\mathcal{U}}d_{m_{k}k},d_{\mathsf{SNR}_{\mathsf{o}}}\right\\}$,
where $m_{k}$ is the AP index closest to the $k$th UE, i.e., if $d_{mk}$ is
the distance between the $m$th AP and the $k$th UE, then
$d_{m_{k}k}=\min\left\\{d_{mk},\forall m\in\mathcal{A}\right\\}$. Also,
$d_{\mathsf{SNR}_{\mathsf{o}}}$ is the distance from any UE where the received
SNR is at least $\gamma_{\mathrm{min}}$, i.e.,
$d_{\mathsf{SNR}_{\mathsf{o}}}=\max_{d}\left\\{\frac{N\mathcal{E}_{p}\beta(d)}{N_{0}}\geq\gamma_{\mathrm{min}}\right\\}$,
with $\beta(d)=\left(d/d_{0}\right)^{-\mathrm{PL}}$, $d_{0}$ is the reference
distance, and $\mathrm{PL}$ is the path-loss exponent. This choice of
$r_{\mathsf{o}}$ ensures that:

  1. (1)

There is no UE that is not connected to any AP. In particular, every UE is
connected to at least one AP even if the received signal strength to its
nearest AP is below $\gamma_{\mathrm{min}}$, i.e., if
$\max_{k\in\mathcal{U}}d_{m_{k}k}>d_{\mathsf{SNR}_{\mathsf{o}}}$.

  2. (2)

Every UE is connected to all APs where the received signal strength is at
least $\gamma_{\mathrm{min}}$. However, in a dense deployment where
$\max_{k\in\mathcal{U}}d_{m_{k}k}<d_{\mathsf{SNR}_{\mathsf{o}}}$, unnecessary
connections to UEs to APs where the received signal strength is below
$\gamma_{\mathrm{min}}$ are avoided.

With the clusters as defined above, our pilot length minimization problem can
be written as

$\displaystyle\min~{}$ $\displaystyle\tau_{p}$
$\displaystyle\mathrm{subject~{}to}~{}$
$\displaystyle\langle\boldsymbol{\varphi}_{l(k)},\boldsymbol{\varphi}_{l(k^{\prime})}\rangle=0,\forall
k,k^{\prime}\in\mathcal{A}_{m},\forall m\in\mathcal{U}_{k}.$ (2)

The constraint above ensures that any two UEs that are connected to a common
AP are assigned orthogonal pilot sequences. Note that the above pilot
assignment is only based on the UEs’ and APs’ locations and not on the channel
state instantiations. We recast (III-3) as a graph coloring problem. We define
a graph $\mathcal{G}=\left\\{\mathcal{V},\mathcal{E}\right\\}$, where the
vertex set $\mathcal{V}$ represents the UEs, i.e., $\mathcal{U}$, and edge set
is
$\mathcal{E}\triangleq\left\\{e_{kk^{\prime}}~{}\mathrm{s.t.}~{}\mathcal{U}_{k}\cap\mathcal{U}_{k^{\prime}}\neq\emptyset;\forall
k,k^{\prime}\in\mathcal{U}\right\\}$. Thus, if two UEs, indexed by $k$ and
$k^{\prime}$, are connected to at least one common AP, there is an edge
between them. A color assigned to a vertex represents the pilot sequence
assigned to it. Then, to satisfy the constraint in (III-3), we must ensure
that any two connected vertices have distinct colors. On the other hand, if
two vertices are not connected by an edge, they can potentially reuse the same
color (pilot sequence). Now, let $\mathcal{C}$ be the set of distinct colors,
and $\mathcal{C}(k)$ indicate the color assigned to the $k$th UE. Then, the
equivalent coloring problem becomes

$\displaystyle\min\quad$ $\displaystyle\lvert\mathcal{C}\rvert$
$\displaystyle\mathrm{subject~{}to}\quad$
$\displaystyle\mathcal{C}(k)\neq\mathcal{C}(k^{\prime}),~{}\mathrm{if}~{}e_{kk^{\prime}}\in\mathcal{E},\forall
k,k^{\prime}\in\mathcal{U}.$ (3)

From the above arguments, we have the following proposition.

######  Proposition 1.

The pilot length minimization problem in (III-3) and the coloring problem in
(III-3) are equivalent.

The coloring problem in (III-3) is NP-complete [Chapter $3$, [18]]. Therefore,
we first recast the problem in (III-3) as a bipartite graph coloring problem,
where there are two sets of nodes representing APs and UEs, and edges between
them represent the connections defined by (1). This bipartite graph coloring
problem can be solved efficiently using low complexity greedy techniques, such
as the DSATUR algorithm [19]. We summarize the solution in Algorithm 1.

The DSATUR algorithm procures _optimal_ coloring for all _bipartite graphs_
[19] in terms of minimizing the number of distinct colors. In each iteration,
Algorithm 1 performs two stages. The first stage selects the vertex among the
uncolored vertices based on the number of connected colored or uncolored
vertices. Once a vertex is chosen, it is assigned a color that has not been
assigned to the vertices connected to it and has been reused the least number
of times. When no distinct color is available, a new color is assigned, and
the set $\mathcal{C}$ is updated. Finally, we generate orthonormal pilots
based on the UE to color mapping.

Next, in Fig. 2, we illustrate the AP-UE connectivity and the pilot assignment
via an example. We take $8$ APs over a $500$ square meter area. Fig. 2(b)
illustrates the formation of the uncolored graph where two vertices (UEs) are
connected if they share at least one common AP as per Fig. 2(a). Fig. 2(c)
shows the assignment of the pilot sequences via the corresponding colors of
the UEs. Next, Fig. 2(d) demonstrates the outcome of the algorithm via a
bipartite graph, where the nodes on the left-hand-side represent APs, and the
right-hand-side nodes are for UEs. There is an edge if an AP and UE are
connected via the set $\mathcal{A}_{m}$. We observe that all the edges
emanating from an AP have distinct colors, which implies that all UEs
connected to that AP are assigned orthonormal pilots.

Input : $\mathcal{\bar{U}}=\left\\{1,2,\ldots,K\right\\}$,
$\mathcal{C}=\emptyset$

1 while  _$\mathcal{\bar{U}}\neq\emptyset$_ do

/* Stage $1$: Select the uncolored vertex */

2 Select the $k\in\mathcal{\bar{U}}$ that has the maximum number of distinct
colored vertices connected to it.

3 If there is more than one such vertex, choose the $k\in\mathcal{\bar{U}}$
within the subset of vertices with the maximum number of distinct colors with
the maximum number of vertices connected to it.

4 Choose any $k\in\mathcal{\bar{U}}$ at random if there is more than one such
vertex.

/* Stage $2$: Assign the least used color from the set of available colors */

5 Assign color $c(k)$ to $k$th vertex such that  $\displaystyle c(k)=\min\quad
c(p)\in\mathcal{C}$ $\displaystyle\mathrm{subject~{}to}~{}c(k^{\prime})\neq
c(p),$ $\displaystyle\quad\quad\quad\quad\quad\forall
k^{\prime}\in\left\\{l~{}\mathrm{s.t.}~{}e_{kl}\in\mathcal{E}\right\\}.$ (4)

6 if  _$c(k)=\emptyset$_ then

7 Assign a new color $c(k)$ to vertex $k$

8 Update: $\mathcal{C}\leftarrow\mathcal{C}\cup c(k)$

9

10

Algorithm 1 Pilot Allocation via Graph Coloring

(a) Uncolored AP-UE connections: the lines between the APs and UEs denote the
clusters formed by $r_{\mathsf{o}}$.

(b) Graph formed by connecting UEs (vertices) that share common AP(s).

(c) Colored AP-UE connections: Distinct colors correspond to distinct
orthonormal pilot sequences.

(d) All lines emerging from a UE have the same color, and all the lines
merging into an AP have distinct colors.

Figure 2: An example of the pilot assignment algorithm with
$M=8,N=10,K=20,d_{0}=20$ m, and $\mathrm{PL}=3.76$. The pilot SNR is $10$ dB
and $\gamma_{\mathrm{min}}$ is $0$ dB. We observe that to solve (III-3) we
need $5$ colors, and hence $\tau_{p}=5$.

The attractive features of Algorithm 1 are: the algorithm does not require an
exchange of instantaneous UL/DL SINRs for pilot allocation, unlike existing
algorithms [20, 21, 22]. Also, algorithms in the literature that require only
UE locations for pilot allocation [23, 24] do not consider pilot length
optimization. The main advantage of our pilot allocation technique is that, in
a DTDD and FD-based system, we can isolate the issue of pilot allocation from
AP scheduling and power control, which makes our solution easy to implement.

(a) Comparison of Algorithm 1 with existing literature in terms of NMSE.

(b) Scaling of pilot length with number of UEs ($K$).

Figure 3: Comparison of Algorithm 1 with existing literature and its benefits.

Finally, we numerically evaluate the efficacy of the proposed pilot allocation
algorithm. The system model closest to our work is [3], and thus we compare
the proposed algorithm with the greedy pilot allocation presented in [3]. We
also compare against a random pilot allocation scheme.444For both the greedy
and random allocation schemes, we take the pilot length returned by our
algorithm and assign that many pilots across the UEs. In Fig. 3(a), we compare
the performance of the pilot allocation scheme in terms of the NMSE in channel
estimation at the APs as a function of pilot SNR. The pilot allocation
returned by Algorithm 1 leads to considerably lower NMSE than the greedy
method. Also, as we increase the value of $r_{\mathsf{o}}$, the cardinality of
$\mathcal{U}_{k}$ in (1a) increases. This leads to a higher value of pilot
length, and hence, the NMSE decreases even further. Fig. 3(b) illustrates the
minimum pilot length required for given UE density and $r_{\mathsf{o}}$. Even
with a large number of UEs, say $500$, the pilot length required is only _one-
tenth_ that required for allocating fully orthonormal pilots.

In the following sections, we present the SE analysis, AP scheduling
algorithm, and UL-DL power allocation policies for the DTDD system. As we will
see in Sec. VI, our analysis easily generalizes to an FD-enabled CF system
also.

##  IV Spectral Efficiency Analysis: CF DTDD

This section presents the UL and DL signaling model and derives closed-form
expressions for the sum UL-DL SE, which we further use for AP scheduling and
power allocation. Let the sets $\mathcal{A}_{\mathsf{u}}$ and
$\mathcal{A}_{\mathsf{d}}$ contain the indices of the APs scheduled in the UL
and DL, respectively. In a DTDD system, the APs are HD. Thus,
$\mathcal{A}_{\mathsf{u}}\cap\mathcal{A}_{\mathsf{d}}=\emptyset$. Also, let
$\mathcal{A}_{\mathsf{s}}\triangleq\mathcal{A}_{\mathsf{u}}\cup\mathcal{A}_{\mathsf{d}}\subseteq\mathcal{A}$.

###  IV-A Analysis with MMSE combiner & RZF precoder

_In the UL,_ the $k$th UE ($k\in\mathcal{U}_{\mathsf{u}}$) sends the symbol
$s_{\mathsf{u},k}$ with power $\mathcal{E}_{\mathsf{u},k}$. The data symbol of
each UE is modeled as zero mean, unit variance, and
$\mathbb{E}[s_{\mathsf{u},k}s_{\mathsf{u},k^{\prime}}^{*}]=0,k^{\prime}\neq
k,\forall k,k^{\prime}\in\mathcal{U}_{\mathsf{u}}$. The signal received at the
$m$th UL AP can be expressed as
$\mathbf{y}_{\mathsf{u},m}=\sum\nolimits_{\begin{subarray}{c}n\in\mathcal{U}_{\mathsf{u}}\end{subarray}}{\textstyle\sqrt{\mathcal{E}_{\mathsf{u},n}}}\mathbf{f}_{mn}s_{\mathsf{u},n}+\sum\nolimits_{j\in\mathcal{A}_{\mathsf{d}}}\tilde{\mathbf{G}}_{mj}\mathbf{x}_{\mathsf{d},j}+\mathbf{w}_{\mathsf{u},m}\in\mathbb{C}^{N},$
where
$\mathbf{x}_{\mathsf{d},j}=\sqrt{\mathcal{E}_{\mathsf{d}}}\mathbf{P}_{j}\mathsf{diag}(\boldsymbol{\kappa}_{j})\mathbf{s}_{\mathsf{d}}=\sqrt{\mathcal{E}_{\mathsf{d}}}\left[\mathbf{p}_{j1},\ldots,\mathbf{p}_{j|\mathcal{U}_{\mathsf{d}}|}\right]\mathsf{diag}(\boldsymbol{\kappa}_{j})\mathbf{s}_{\mathsf{d}}$
is the transmitted DL data vector with $\mathcal{E}_{\mathsf{d}}$ being the
total radiated power,
$\mathbf{p}_{jn}=\left[\mathbf{P}_{j}\right]_{:,n}\in\mathbb{C}^{N}$ being the
precoding matrix for the $n$th DL UE, and
$\boldsymbol{\kappa}_{j}\triangleq[\kappa_{j1},\kappa_{j2},\ldots,\kappa_{j|\mathcal{U}_{\mathsf{d}}|}]^{T}$
being the vector of power control coefficients, all at the $j$th DL AP. Here,
$\kappa_{jn}$, i.e., the $n$th element of $\boldsymbol{\kappa}_{j}$, indicates
the fraction of power dedicated by the $j$th AP to the $n$th DL UE
($n\in\mathcal{U}_{\mathsf{d}}$). The DL signal vector
$\mathbf{s}_{\mathsf{d}}=\left[s_{\mathsf{d},1},\ldots,s_{\mathsf{d},|\mathcal{U}_{\mathsf{d}}|}\right]^{T}$
follows
$\mathbb{E}\left[\mathbf{s}_{\mathsf{d}}\mathbf{s}_{\mathsf{d}}^{H}\right]=\mathbf{I}_{|\mathcal{U}_{\mathsf{d}}|}$.
Finally,
$\mathbf{w}_{\mathsf{u},m}\sim\mathcal{CN}(\mathbf{0}_{N},N_{0}\mathbf{I}_{N})$
is the additive noise. Each AP pre-processes the received signals using local
combiners and sends them to the CPU for joint decoding, which is important for
the scalability of the overall system [5]. Let
$\mathbf{v}_{mk}\in\mathbb{C}^{N}$ be the local combining vector at the $m$th
AP for $k$th UE’s UL data stream. Then, the local estimate of the $k$th UE’s
signal at the $m$th AP becomes
$\hat{s}_{\mathsf{u},mk}=\sqrt{\mathcal{E}_{\mathsf{u},k}}\mathbf{v}_{mk}^{H}\mathbf{f}_{mk}s_{\mathsf{u},k}+\sum\nolimits_{n\in\mathcal{U}_{\mathsf{u}}\backslash
k}\sqrt{\mathcal{E}_{\mathsf{u},n}}\mathbf{v}_{mk}^{H}\mathbf{f}_{mn}s_{\mathsf{u},n}+\sqrt{\mathcal{E}_{\mathsf{d}}}\sum\nolimits_{j\in\mathcal{A}_{\mathsf{d}}}\sum\nolimits_{n\in\mathcal{U}_{\mathsf{d}}}\kappa_{jn}\mathbf{v}_{mk}^{H}\tilde{\mathbf{G}}_{mj}\mathbf{p}_{jn}s_{\mathsf{d},n}+\mathbf{v}_{mk}^{H}\mathbf{w}_{\mathsf{u},mk}$.
To design $\mathbf{v}_{mk}$, the AP utilizes the knowledge of
$\hat{\mathbf{f}}_{mk},\forall k\in\mathcal{U}_{\mathsf{u}}$, so that the MSE
of the locally estimated signal is minimized, i.e.,
$\mathbf{v}_{mk}^{\mathsf{opt.}}=\min\limits_{\mathbf{v}_{mk}\in\mathbb{C}^{N}}\mathbb{E}\left[\lvert
s_{\mathsf{u},k}-\mathbf{v}_{mk}^{H}\mathbf{y}_{\mathsf{u},m}\rvert^{2}|\hat{\mathbf{f}}_{mk}\right]$.
Thus, the optimal combiner is
$\mathbf{v}_{mk}^{\mathsf{opt.}}=\mathbf{R}_{mk}^{-1}\hat{\mathbf{f}}_{mk}$,
where
$\mathbf{R}_{mk}=\Big{(}\sum\nolimits_{k\in\mathcal{U}_{\mathsf{u}}}\mathcal{E}_{\mathsf{u},k}\hat{\mathbf{f}}_{mk}\hat{\mathbf{f}}_{mk}^{H}+\sum\nolimits_{k\in\mathcal{U}_{\mathsf{u}}}\mathcal{E}_{\mathsf{u},k}\mathbb{E}[\tilde{\mathbf{f}}_{mk}\tilde{\mathbf{f}}_{mk}^{H}]+\mathcal{E}_{\mathsf{d}}\sum\nolimits_{j\in\mathcal{A}_{\mathsf{d}}}\sum\nolimits_{n\in\mathcal{U}_{\mathsf{d}}}\kappa_{jn}^{2}\mathbb{E}[\tilde{\mathbf{G}}_{mj}\mathbf{p}_{jn}\mathbf{p}_{jn}^{H}\tilde{\mathbf{G}}_{mj}^{H}]+N_{0}\mathbf{I}_{N}\Big{)}$.
Next, these locally estimated signals relayed from the APs are linearly
combined at the CPU with combining weights being $\omega_{mk},\forall
m\in\mathcal{A}_{\mathsf{u}},\forall k\in\mathcal{U}_{\mathsf{u}}$ so that the
received SINR at the CPU for each UE is maximized. Thus, the $k$th stream of
the received signal at the CPU is
$\hat{s}_{\mathsf{u},k}=\sqrt{\mathcal{E}_{\mathsf{u},k}}\sum\nolimits_{m\in\mathcal{A}_{\mathsf{u}}}\omega_{mk}^{*}\mathbf{v}_{mk}^{H}\mathbf{f}_{mk}s_{\mathsf{u},k}+\sum\nolimits_{n\in\mathcal{U}_{\mathsf{u}}\backslash
k}\sqrt{\mathcal{E}_{\mathsf{u},n}}\sum\nolimits_{m\in\mathcal{A}_{\mathsf{u}}}\omega_{mk}^{*}\mathbf{v}_{mk}^{H}\mathbf{f}_{mn}s_{\mathsf{u},n}+\sum\nolimits_{m\in\mathcal{A}_{\mathsf{u}}}\omega_{mk}^{*}\sum\nolimits_{j\in\mathcal{A}_{\mathsf{d}}}\sqrt{\mathcal{E}_{\mathsf{d}}}\sum\nolimits_{n\in\mathcal{U}_{\mathsf{d}}}\kappa_{jn}\mathbf{v}_{mk}^{H}\tilde{\mathbf{G}}_{mj}\mathbf{p}_{jn}s_{\mathsf{d},n}+\sum\nolimits_{m\in\mathcal{A}_{\mathsf{u}}}\omega_{mk}^{*}\mathbf{v}_{mk}^{H}\mathbf{w}_{\mathsf{u},mk}$.
Then the UL SINR of the $k$th UE, denoted by $\eta_{\mathsf{u},k}$, at the CPU
is equal to [5]

$\displaystyle\frac{\mathcal{E}_{\mathsf{u},k}\lvert\boldsymbol{\omega_{k}}^{H}\mathbb{E}[\mathbf{u}_{kk}]\rvert^{2}}{\boldsymbol{\omega_{k}}^{H}\begin{pmatrix}\sum\nolimits_{i\in\mathcal{U}_{\mathsf{u}}}\mathcal{E}_{\mathsf{u},i}\mathbb{E}[\mathbf{u}_{ki}\mathbf{u}_{ki}^{H}]-\mathcal{E}_{\mathsf{u},k}\mathbb{E}\left[\mathbf{u}_{kk}\right]\mathbb{E}[\mathbf{u}_{kk}^{H}]&\\\
+\sum\nolimits_{n\in\mathcal{U}_{\mathsf{d}}}\mathbb{E}[\mathbf{a}_{kn}\mathbf{a}_{kn}^{H}]+\mathbf{N}_{\mathsf{eff.}}\end{pmatrix}\boldsymbol{\omega_{k}}},$
(5)

where
$\boldsymbol{\omega_{k}}\triangleq[\omega_{1k},\omega_{2k},\ldots,\omega_{|\mathcal{A}_{\mathsf{u}}|k}]^{T}\in\mathbb{C}^{|\mathcal{A}_{\mathsf{u}}|}$,
$\mathbf{u}_{ki}\triangleq[\mathbf{v}_{1k}^{H}\mathbf{f}_{1i},\mathbf{v}_{2k}^{H}\mathbf{f}_{2i},\ldots,\mathbf{v}_{|\mathcal{A}_{\mathsf{u}}|k}^{H}\mathbf{f}_{|\mathcal{A}_{\mathsf{u}}|i}]^{T}\in\mathbb{C}^{|\mathcal{A}_{\mathsf{u}}|}$,
$[\mathbf{a}_{kn}]_{m}$
$=\sum\nolimits_{j\in\mathcal{A}_{\mathsf{d}}}\sqrt{\mathcal{E}_{\mathsf{d}}}\kappa_{jn}\mathbf{v}_{mk}^{H}\tilde{\mathbf{G}}_{mj}\mathbf{p}_{jn}$,
and $\mathbf{N}_{\mathsf{eff.}}=N_{0}{\tt
diag}(\mathbb{E}[\|\mathbf{v}_{1k}\|^{2}],\ldots,\mathbb{E}[\|\mathbf{v}_{|\mathcal{A}_{\mathsf{u}}|k}\|^{2}])\in\mathbb{C}^{|\mathcal{A}_{\mathsf{u}}|\times|\mathcal{A}_{\mathsf{u}}|}$.
We evaluate the optimal weights using the following lemma.

######  Lemma 3.

The SINR of the $k$th ($k\in\mathcal{U}_{\mathsf{u}}$) UE is maximized by
$\boldsymbol{\omega}_{k}^{\mathsf{opt.}}=\mathsf{c}_{k}\sqrt{\mathcal{E}_{\mathsf{u},k}}\mathbf{R}_{\omega,k}^{-1}\mathbb{E}\left[\mathbf{u}_{kk}\right]$,
where555We use the fact that for any given vector
$\mathbf{x}\in\mathbb{C}^{N}$ and a positive definite matrix $\mathbf{A}$,
$\max_{\mathbf{y}\in\mathbb{C}^{N}}[\lvert\mathbf{y}^{H}\mathbf{x}\rvert^{2}/(\mathbf{y}^{H}\mathbf{A}\mathbf{y})]=\mathbf{x}^{H}\mathbf{A}\mathbf{x}$,
and $\mathbf{y}^{\mathsf{opt.}}=\mathbf{A}^{-1}\mathbf{x}$.

$\displaystyle\mathbf{R}_{\omega,k}\triangleq$
$\displaystyle(\textstyle{\sum\nolimits_{i\in\mathcal{U}_{\mathsf{u}}}\mathcal{E}_{\mathsf{u},i}\mathbb{E}[\mathbf{u}_{ki}\mathbf{u}_{ki}^{H}]-\mathcal{E}_{\mathsf{u},k}\mathbb{E}[\mathbf{u}_{kk}]\mathbb{E}[\mathbf{u}_{kk}^{H}]}$
$\displaystyle\textstyle{+\sum\nolimits_{n\in\mathcal{U}_{\mathsf{d}}}\mathbb{E}[\mathbf{a}_{kn}\mathbf{a}_{kn}^{H}]+\mathbf{N}_{\mathsf{eff.}})},$
(6)

and $\mathsf{c}_{k}$ is a scaling constant and can be taken as
$\sqrt{\mathcal{E}_{\mathsf{u},k}}$ to make the weights dimensionless. Also,
the maximum SINR is
$\mathcal{E}_{\mathsf{u},k}\mathbb{E}[{\mathbf{u}}_{kk}^{H}]\mathbf{R}_{\omega,k}^{-1}\mathbb{E}[{\mathbf{u}}_{kk}]$.

_In the DL,_ the counterpart of the MMSE combiner is the RZF precoder, which
is
$\mathbf{p}_{jn}=\mathcal{E}_{\mathsf{d},n}\big{(}\sum\nolimits_{n^{\prime}\in\mathcal{U}_{\mathsf{d}}}\mathcal{E}_{\mathsf{d},n^{\prime}}\hat{\mathbf{f}}_{jn^{\prime}}\hat{\mathbf{f}}_{jn^{\prime}}^{H}+\sum\nolimits_{n^{\prime}\in\mathcal{U}_{\mathsf{d}}}\mathcal{E}_{\mathsf{d},n^{\prime}}\mathbb{E}\left[\tilde{\mathbf{f}}_{jn^{\prime}}\tilde{\mathbf{f}}_{jn^{\prime}}^{H}\right]+N_{0}\mathbf{I}_{N}\big{)}^{-1}\hat{\mathbf{f}}_{jn}$.
Assuming channel reciprocity, the signal received at the $n$th
($n\in\mathcal{U}_{\mathsf{d}}$) DL UE can be written as
$r_{\mathsf{d},n}=\sum\nolimits_{j\in\mathcal{A}_{\mathsf{d}}}\kappa_{jn}\sqrt{{\mathcal{E}_{\mathsf{d}}}}\mathbf{f}_{jn}^{T}\mathbf{p}_{jn}s_{\mathsf{d},n}+\sum\nolimits_{k\in\mathcal{U}_{\mathsf{u}}}\sqrt{\mathcal{E}_{\mathsf{u},k}}\mathtt{g}_{nk}s_{\mathsf{u},k}+\sum\nolimits_{j\in\mathcal{A}_{\mathsf{d}}}\sum\nolimits_{\begin{subarray}{c}q\in\mathcal{U}_{\mathsf{d}}\backslash
n\end{subarray}}\kappa_{jq}\sqrt{{\mathcal{E}_{\mathsf{d}}}}\mathbf{f}_{jn}^{T}\mathbf{p}_{jq}s_{\mathsf{d},q}+w_{\mathsf{d},n}$.
The AWGN $w_{\mathsf{d},n}$ follows $\mathcal{CN}(0,N_{0})$. Now, the
corresponding DL SINR, denoted by $\eta_{\mathsf{d},n}$, becomes

$\displaystyle\frac{\mathcal{E}_{\mathsf{d}}|\boldsymbol{\kappa}_{n}^{T}\mathbb{E}[\mathbf{d}_{nn}]|^{2}}{\begin{pmatrix}\textstyle{\mathcal{E}_{\mathsf{d}}{\tt
var}\big{\\{}\boldsymbol{\kappa}_{n}^{T}\mathbf{d}_{nn}\\}+\mathcal{E}_{\mathsf{d}}\sum\nolimits_{q\in\mathcal{U}_{\mathsf{d}}\backslash
n}\mathbb{E}[\lvert\boldsymbol{\kappa}_{q}^{T}\mathbf{d}_{nq}\rvert^{2}]}&\\\
\textstyle{+\sum\nolimits_{k\in\mathcal{U}_{\mathsf{u}}}\mathcal{E}_{\mathsf{u},k}\mathbb{E}|\mathtt{g}_{nk}|^{2}+N_{0}}\end{pmatrix}},$
(7)

where
$\boldsymbol{\kappa}_{n}\triangleq[\kappa_{1n},\kappa_{2n},\ldots,\kappa_{|\mathcal{A}_{\mathsf{d}}|n}]^{T}$
and
$\mathbf{d}_{nq}\triangleq[\mathbf{f}_{1n}^{T}\mathbf{p}_{1q},\mathbf{f}_{2n}^{T}\mathbf{p}_{2q},\ldots,\mathbf{f}_{\lvert\mathcal{A}_{\mathsf{d}}\rvert
n}^{T}\mathbf{p}_{\lvert\mathcal{A}_{\mathsf{d}}\rvert q}]^{T}$. The DL power
control coefficients, $\boldsymbol{\kappa}_{n}$, are the weights assigned to
the signal transmitted to each UE from every AP, which can be controlled to
maximize the sum SE.

We use results from random matrix theory to derive closed-form expressions for
the SINRs. We present the expressions for the case of ZF precoders and
combiners.

###  IV-B Analysis with ZF combiner & precoder

We first construct the ZF combining and precoding matrices. As $\tau_{p}<K$,
the concatenated estimated channel matrix at $m$th UL AP, denoted by
$\mathbf{\hat{F}}_{\mathsf{u},m}=[\mathbf{\hat{f}}_{m1},\ldots,\mathbf{\hat{f}}_{m|\mathcal{U}_{\mathsf{u}}|}]\in\mathbb{C}^{N\times|\mathcal{U}_{\mathsf{u}}|}$,
may not have full column rank, i.e.,
$\mathbf{\hat{F}}_{\mathsf{u},m}^{H}\mathbf{\hat{F}}_{\mathsf{u},m}$ may not
be invertible. Thus, we construct a full rank matrix as
$\mathbf{Z}_{\mathsf{u},m}\triangleq\mathbf{Y}_{p,m}\boldsymbol{\Phi}\in\mathbb{C}^{N\times\tau_{p}}$,
where $\boldsymbol{\Phi}$ is the pilot matrix with its $l$th column being
$\boldsymbol{\varphi}_{l}$ and $\mathbf{Y}_{p,m}$. We observe that we can
compute $\hat{\mathbf{f}}_{mk}$ from $\mathbf{Z}_{\mathsf{u},m}$ using the
relation
$\hat{\mathbf{f}}_{mk}=c_{mk}\sqrt{\tau_{p}\mathcal{E}_{p,k}}\beta_{mk}\mathbf{Z}_{\mathsf{u},m}\mathbf{e}_{l(k)}$,
where $\mathbf{e}_{l(k)}\in\mathbb{C}^{\tau_{p}}$ is the standard basis with
$l(k)$th coordinate being $1$.666Here, we recapitulate that $k$th UE uses the
pilot sequence $\boldsymbol{\varphi}_{l(k)}$, and the set $\mathcal{P}_{l(k)}$
contains all the UE indices, including the $k$th UE, which use the pilot
sequence $\boldsymbol{\varphi}_{l(k)}$. Thus, multiplying
$\mathbf{Z}_{\mathsf{u},m}$ by $\mathbf{e}_{l(k)}$, we obtain a common
estimate for the channels of all UEs which use the pilot sequence
$\boldsymbol{\varphi}_{l(k)}$.  Then, the ZF combining vector intended for the
$k$th UE at the $m$th AP becomes
$\mathbf{v}_{mk}=\gamma_{mk}\mathbf{Z}_{\mathsf{u},m}(\mathbf{Z}_{\mathsf{u},m}^{H}\mathbf{Z}_{\mathsf{u},m})^{-1}\mathbf{e}_{l(k)}$,
where we set $\gamma_{mk}=\sqrt{\tau_{p}\mathcal{E}_{p,k}}\beta_{mk}$ to
ensure that
$\mathbb{E}[\mathbf{v}_{mk}^{H}\hat{\mathbf{f}}_{mk}]=\alpha_{mk}^{2}$ by
Lemma 1. In the DL, we again construct a full rank matrix
$\mathbf{Z}_{\mathsf{d},j}=\mathbf{Y}_{p,j}\boldsymbol{\Phi}\in\mathbb{C}^{N\times\tau_{p}},~{}\forall
j\in\mathcal{A}_{\mathsf{d}}$. We can obtain $\hat{\mathbf{f}}_{jn}$ from
$\mathbf{Z}_{\mathsf{d},j}$ as
$\hat{\mathbf{f}}_{jn}=c_{jn}\sqrt{\tau_{p}\mathcal{E}_{p,j}}\beta_{jn}\mathbf{Z}_{\mathsf{d},j}\mathbf{e}_{l(n)}$.
Next, the ZF precoding vector (unit normalized) intended for the $n$th DL UE
at the $j$th AP is
$\mathbf{p}_{jn}=\frac{\mathbf{Z}_{\mathsf{d},j}\left(\mathbf{Z}_{\mathsf{d},j}^{H}\mathbf{Z}_{\mathsf{d},j}\right)^{-1}\mathbf{e}_{l(n)}}{\sqrt{\mathbb{E}[\|\mathbf{Z}_{\mathsf{d},j}\left(\mathbf{Z}_{\mathsf{d},j}^{H}\mathbf{Z}_{\mathsf{d},j}\right)^{-1}\mathbf{e}_{l(n)}\|^{2}]}}$.
Using [25, Lemma $6$], we can show that
$\mathbb{E}[\|\mathbf{Z}_{\mathsf{d},j}(\mathbf{Z}_{\mathsf{d},j}^{H}\mathbf{Z}_{\mathsf{d},j})^{-1}\mathbf{e}_{l(n)}\|^{2}]=\frac{c_{jn}}{N-\tau_{p}}$.
For a detailed proof see Appendix A.777Thus, the condition $N\geq(\tau_{p}+1)$
has to be satisfied. Each AP has $\tau_{p}$ combining and precoding vectors,
and the same vector is utilized for all UEs sharing the same pilot sequence.
Next, we present closed-form expressions for the optimal weights and UL-DL
SINRs with ZF combiners and precoders.

######  Lemma 4.

The optimal weights for the $k$th UE’s data stream at the CPU for
$\lvert\mathcal{A}_{\mathsf{u}}\rvert$ UL APs is
$\boldsymbol{\omega}_{k}^{\mathsf{opt}}=\mathcal{E}_{\mathsf{u},k}\mathbf{R}_{\omega,k}^{-1}\bar{\mathbf{u}}_{k}$,
with
$\bar{\mathbf{u}}_{k}=[\alpha_{1k}^{2},\alpha_{2k}^{2},\ldots,\alpha_{|\mathcal{A}_{\mathsf{u}}|k}^{2}]^{T}$
and
$\mathbf{R}_{\omega,k}=\sum\nolimits_{k^{\prime}\in\mathcal{P}_{l(k)}\backslash
k}\mathcal{E}_{\mathsf{u},k^{\prime}}\bar{\mathbf{u}}_{k^{\prime}}\bar{\mathbf{u}}_{k^{\prime}}^{H}+\frac{1}{N-\tau_{p}}\mathbf{\dot{R}}_{\omega,k}+\frac{\mathcal{E}_{\mathsf{d}}}{N-\tau_{p}}\mathbf{\ddot{R}}_{\omega,k}$,
where $\mathbf{\dot{R}}_{\omega,k}$ and
$\mathbf{\ddot{R}}_{\omega,k}\in\mathbb{C}^{\lvert\mathcal{A}_{\mathsf{u}}\rvert\times\lvert\mathcal{A}_{\mathsf{u}}\rvert}$
are two diagonal matrices with $m$th diagonal entries

$\left[\mathbf{\dot{R}}_{\omega,k}\right]_{m}=\sum\nolimits_{k^{\prime}\in\mathcal{U}_{\mathsf{u}}}\mathcal{E}_{\mathsf{u},k^{\prime}}\alpha_{mk}^{2}(\beta_{mk^{\prime}}-\alpha_{mk^{\prime}}^{2})+N_{0}\alpha_{mk}^{2},$

and

$\left[\mathbf{\ddot{R}}_{\omega,k}\right]_{m}=\sum\nolimits_{j\in\mathcal{A}_{\mathsf{d}}}N\kappa_{jn}^{2}\zeta_{mj}^{\mathsf{InAP}}\alpha_{mk}^{2}.$

###### Proof.

See Appendix B. ∎

Later, in Fig. 4, we illustrate the correctness of the above lemma by matching
the UL SE obtained using the weights computed from the above lemma with the
sum UL SE obtained from Lemma 3 by averaging the expectations over channel
realizations. We next provide an explicit closed-form expression for the UL
SE, applicable for any choice of weights.

######  Lemma 5.

The UL SINR of the $k$th UE with ZF-combining can be written as

$\displaystyle\eta_{\mathsf{u},k}=\frac{(N-\tau_{p})\mathcal{E}_{\mathsf{u},k}(\sum\nolimits_{m\in\mathcal{A}_{\mathsf{u}}}\omega_{mk}^{*}\alpha_{mk}^{2})^{2}}{{\tt
EST}_{\mathsf{u},k}+{\tt MUI}_{\mathsf{u},k}+{\tt
IAP}_{k}+N_{0}\sum\limits_{m\in\mathcal{A}_{\mathsf{u}}}|\omega_{mk}^{*}|^{2}\alpha_{mk}^{2}},$

where ${\tt EST}_{\mathsf{u},k}$, ${\tt MUI}_{\mathsf{u},k}$ and ${\tt
IAP}_{k}$ capture the error due to channel estimation, multi-UE interference
from the pilot sharing UEs, and inter-AP CLI. These are respectively evaluated
as

$\displaystyle{\tt
EST}_{\mathsf{u},k}=\sum\limits_{k^{\prime}\in\mathcal{U}_{\mathsf{u}}}\mathcal{E}_{\mathsf{u},k^{\prime}}\sum\limits_{m\in\mathcal{A}_{\mathsf{u}}}|\omega_{mk}^{*}|^{2}\alpha_{mk}^{2}(\beta_{mk^{\prime}}-\alpha_{mk^{\prime}}^{2}),$
(8a) $\displaystyle{\tt
MUI}_{\mathsf{u},k}=(N-\tau_{p})\sum\limits_{i\in\mathcal{P}_{l(k)}\backslash
k}\mathcal{E}_{\mathsf{u},i}\left(\sum\nolimits_{m\in\mathcal{A}_{\mathsf{u}}}\omega_{mk}^{*}\alpha_{mi}^{2}\right)^{2},$
(8b) $\displaystyle{\tt
IAP}_{k}=N\sum\limits_{n\in\mathcal{U}_{\mathsf{d}}}\sum\limits_{m\in\mathcal{A}_{\mathsf{u}}}\sum\limits_{j\in\mathcal{A}_{\mathsf{d}}}\mathcal{E}_{\mathsf{d}}\kappa_{jn}^{2}\zeta_{mj}^{\mathsf{InAP}}|\omega_{mk}^{*}|^{2}\alpha_{mk}^{2}.$
(8c)

###### Proof.

See Appendix C. ∎

Note that the above result is also valid when equal weights are applied at the
CPU, i.e., $\omega_{mk}=\frac{1}{\lvert\mathcal{A}_{\mathsf{u}}\rvert},\forall
k,\forall m$. However, as will be shown in Fig. 4, optimal weighting yields
significantly improved sum SE compared to equal weighting. Next, we present
the DL SINR with ZF precoding.

######  Lemma 6.

The DL SINR of the $n$th UE with ZF precoding is

$\displaystyle\eta_{\mathsf{d},n}=\frac{(N-\tau_{p})\mathcal{E}_{\mathsf{d}}\left(\sum\nolimits_{j\in\mathcal{A}_{\mathsf{d}}}\alpha_{jn}\kappa_{jn}\right)^{2}}{{\tt
EST}_{\mathsf{d},n}+{\tt MUI}_{\mathsf{d},n}+{\tt IUE}_{n}+N_{0}},$ (9)

where ${\tt EST}_{\mathsf{d},n}$, ${\tt MUI}_{\mathsf{d},n}$, and ${\tt
IUE}_{n}$ capture the interference due to channel estimation error, the DL
multi-UE interference from pilot sharing UEs, and the UL UE to DL UE CLI.
These are respectively evaluated as

$\displaystyle{\tt
EST}_{\mathsf{d},n}=\sum\nolimits_{q\in\mathcal{U}_{\mathsf{d}}}\sum\nolimits_{j\in\mathcal{A}_{\mathsf{d}}}\mathcal{E}_{\mathsf{d}}\kappa_{jq}^{2}(\beta_{jn}-\alpha_{jn}^{2}),$
(10a) $\displaystyle{\tt
MUI}_{\mathsf{d},n}=(N-\tau_{p})\mathcal{E}_{\mathsf{d}}\sum\nolimits_{q\in\mathcal{P}_{l(n)}\backslash
n}\left(\sum\nolimits_{j\in\mathcal{A}_{\mathsf{d}}}\kappa_{jq}\alpha_{jn}\right)^{2},$
(10b) $\displaystyle{\tt
IUE}_{n}=\sum\nolimits_{{k\in\mathcal{U}_{\mathsf{u}}}}\mathcal{E}_{\mathsf{u},k}\epsilon_{nk}.$
(10c)

###### Proof.

See Appendix D. ∎

Lemma 5 and Lemma 6 explicitly capture the dependence of the SINRs on the UL-
DL power control coefficients and underlying UL-DL AP sets. Thus, the sum UL-
DL SE, given by (11) below, too, depends on the choice of these parameters.

${\mathcal{R}_{\mathsf{s}}(\mathcal{A}_{\mathsf{s}},\boldsymbol{\kappa},\boldsymbol{\mathcal{E}}_{\mathsf{u}})={\frac{\tau-\tau_{p}}{\tau}\Big{[}\sum\nolimits_{k\in\mathcal{U}_{\mathsf{u}}}\log[1+\eta_{\mathsf{u},k}]}}\\\
{+{\sum\nolimits_{n\in\mathcal{U}_{\mathsf{d}}}\log[1+\eta_{\mathsf{d},n}]\Big{]}}},$
(11)

where
$\boldsymbol{\kappa}\triangleq[\boldsymbol{\kappa}_{1}^{T},\ldots,\boldsymbol{\kappa}_{n}^{T},\ldots,\boldsymbol{\kappa}_{|\mathcal{U}|_{\mathsf{d}}}^{T}]^{T}\in\mathbb{C}^{|\mathcal{U}_{\mathsf{d}}||\mathcal{A}_{\mathsf{d}}|}$
(recall
$\boldsymbol{\kappa}_{n}=[\kappa_{1n},\kappa_{2n},\ldots,\kappa_{|\mathcal{A}_{\mathsf{d}}|n}]^{T}$),
$\boldsymbol{\mathcal{E}}_{\mathsf{u}}\triangleq\left[\mathcal{E}_{\mathsf{u},1},\ldots,\mathcal{E}_{\mathsf{u},|\mathcal{U}_{\mathsf{u}}|}\right]\in\mathbb{C}^{|\mathcal{U}_{\mathsf{u}}|}$,
and
$\mathcal{A}_{\mathsf{s}}=\mathcal{A}_{\mathsf{u}}\cup\mathcal{A}_{\mathsf{d}}\subseteq\mathcal{A}$.
Next, we optimize $\boldsymbol{\kappa}$,
$\boldsymbol{\mathcal{E}}_{\mathsf{u}}$, and find a schedule for UL/DL APs in
$\mathcal{A}_{\mathsf{s}}$ to maximize $\mathcal{R}_{\mathsf{s}}(.)$.

##  V Joint Scheduling and Power Control

Here, we aim to solve the following problem:

$\displaystyle\underset{\left\\{\mathcal{A}_{\mathsf{s}},\kappa_{jn},\mathcal{E}_{\mathsf{u},k}\right\\}}{\max}\quad$
$\displaystyle\mathcal{R}_{\mathsf{s}}\left(\mathcal{A}_{\mathsf{s}},\boldsymbol{\kappa},\boldsymbol{\mathcal{E}}_{\mathsf{u}}\right)$
(12a) $\displaystyle\mathrm{subject~{}to}\quad$
$\displaystyle\mathcal{A}_{\mathsf{u}}\cup\mathcal{A}_{\mathsf{d}}=\mathcal{A}_{\mathsf{s}}\subseteq\mathcal{A};\mathcal{A}_{\mathsf{u}}\cap\mathcal{A}_{\mathsf{d}}=\emptyset;$
(12b) $\displaystyle
0\leq\mathcal{E}_{\mathsf{u},k}\leq\mathcal{E}_{\mathsf{u}},\forall
k\in\mathcal{U}_{\mathsf{u}};$ (12c)
$\displaystyle\sum\nolimits_{\begin{subarray}{c}q\in\mathcal{U}_{\mathsf{d}}\end{subarray}}\kappa_{jq}^{2}\leq
1,\forall j\in\mathcal{A}_{\mathsf{d}}.$ (12d)

With $\mathcal{R}_{\mathsf{s}}(\cdot)$ given by (11), the above problem is
non-convex and NP-hard. We next present our solution.

####  V-1 AP-scheduling

Let us define
$\overline{\mathcal{R}}_{\mathsf{s}}\left(\mathcal{A}_{\mathsf{s}}\right)=\max\limits_{\kappa_{jn},\mathcal{E}_{\mathsf{u},k}}\quad\mathcal{R}_{\mathsf{s}}\left(\mathcal{A}_{\mathsf{s}},\boldsymbol{\kappa},\boldsymbol{\mathcal{E}}_{\mathsf{u}}\right)$
as the sum UL-DL SE attained via optimizing the power allocation coefficients
when the underlying set of APs is
$\mathcal{A}_{\mathsf{s}}=\mathcal{A}_{\mathsf{u}}\cup\mathcal{A}_{\mathsf{d}}$.
Now, the UL and DL power control coefficients for a sub-set of APs,
$\mathcal{A}_{\mathsf{s}}$, may not be optimal when we add one more AP in
either UL or DL, say
$\\{j_{\mathsf{mode}}\\},\mathsf{mode}\in\\{\mathsf{u},\mathsf{d}\\}$, to
$\mathcal{A}_{\mathsf{s}}$. Thus, we need to reoptimize the power control
coefficients for $\mathcal{A}_{\mathsf{s}}\cup\\{j_{\mathsf{mode}}\\}$. The
following proposition holds when we optimize the power control coefficients
for both $\mathcal{A}_{\mathsf{s}}$ and
$\mathcal{A}_{\mathsf{s}}\cup\\{j_{\mathsf{mode}}\\}$.

######  Proposition 2.

For two sets of scheduled UL and DL APs, $\mathcal{A}_{\mathsf{s}}$ and
$\mathcal{A}_{\mathsf{t}}$, where
$\mathcal{A}_{\mathsf{s}}\subseteq\mathcal{A}_{\mathsf{t}}$,
$\overline{\mathcal{R}}_{\mathsf{s}}\left(\mathcal{A}_{\mathsf{s}}\right)\leq\overline{\mathcal{R}}_{\mathsf{s}}\left(\mathcal{A}_{\mathsf{t}}\right)$,
where, for both $\mathcal{A}_{\mathsf{s}}$ and $\mathcal{A}_{\mathsf{t}}$, the
UL and DL power control coefficients are optimized to maximize the sum UL-DL
SE.

Thus,
$\overline{\mathcal{R}}_{\mathsf{s}}\left(\mathcal{A}_{\mathsf{s}}\right)$ is
a monotonically non-decreasing function of the underlying scheduled AP set.
Motivated by this, we schedule APs one-by-one, where, at each iteration, we
schedule an AP and its associated mode such that incremental gain in the
(optimized) sum UL-DL SE is maximized. The recipe is sketched in Algorithm 2.
Our algorithm procures the AP schedule in polynomial time ($\mathcal{O}(M)$),
whereas exhaustive search-based AP scheduling requires optimization of the sum
UL-DL SEs over all $2^{M}$ AP configurations, which is not a scalable
approach. Such greedy approaches have been previously used in the antenna
selection literature [26].

Input : $\mathcal{A}$: the set of all AP indices

Initialize : $\mathcal{A}_{\mathsf{u}}=\mathcal{A}_{\mathsf{d}}=\emptyset$
$\mathcal{A}_{\mathsf{s}}=\mathcal{A}_{\mathsf{u}}\cup\mathcal{A}_{\mathsf{d}}$

1

2while  _ $\mathcal{A}_{\mathsf{s}}^{\mathsf{c}}\neq\emptyset$ _ do

3 Evaluate:
$\displaystyle\begin{cases}i^{\star}_{\mathsf{u}}=\arg\max\limits_{i\in\mathcal{A}_{s}^{\mathsf{c}}}\overline{\mathcal{R}}_{\mathsf{s}}(\mathcal{A}_{\mathsf{u}}\cup\\{i_{\mathsf{u}}\\})\\\
i^{\star}_{\mathsf{d}}=\arg\max\limits_{i\in\mathcal{A}_{\mathsf{s}}^{\mathsf{c}}}\overline{\mathcal{R}}_{\mathsf{s}}(\mathcal{A}_{\mathsf{d}}\cup\\{i_{\mathsf{d}}\\})\end{cases}$

4 if
_$\overline{\mathcal{R}}_{\mathsf{s}}(\mathcal{A}_{\mathsf{u}}\cup\\{i_{\mathsf{u}}^{\star}\\})\geq\overline{\mathcal{R}}_{\mathsf{s}}(\mathcal{A}_{\mathsf{d}}\cup\\{i_{\mathsf{d}}^{\star}\\})$_
then

5 Update:
$\mathcal{A}_{\mathsf{s}}\leftarrow\mathcal{A}_{\mathsf{u}}=\mathcal{A}_{\mathsf{u}}\cup\\{i_{\mathsf{u}}^{\star}\\}$

6 else

7 Update:
$\mathcal{A}_{\mathsf{s}}\leftarrow\mathcal{A}_{\mathsf{d}}=\mathcal{A}_{\mathsf{d}}\cup\\{i_{\mathsf{d}}^{\star}\\}\;$

8

9

Algorithm 2 UL and DL Mode Selection

We next address the power control problem given an AP schedule
$\mathcal{A}_{\mathsf{s}}$. Now, as the UL power control coefficients affect
the DL SE via inter-UE CLI and the DL power control coefficients affect the UL
SE via inter-AP CLI, the joint optimization of UL-DL sum SE is still a non-
convex and prohibitively complex task. We present an alternating optimization
approach in the following subsection.

####  V-2 Uplink power control

First, we note that the UL power control coefficients influence the sum UL-DL
SE predominantly via the sum UL SE. This is because, although the InUE term in
the sum DL SE depends on UL transmit powers $\mathcal{E}_{u,k},\forall
k\in\mathcal{U}_{\mathsf{u}}$, it does not scale with the number of antennas
(see (10c)). Thus, the DL multi-UE and DL coherent interference terms dominate
the InUE term in the sum DL SE. Hence, for mathematical tractability, we
consider the sum UL SE maximization to optimize $\mathcal{E}_{u,k},\forall
k\in\mathcal{U}_{u}$. This simplifies the joint UL-DL SE maximization and
comes with closed-form update equations as well as convergence guarantees for
the sum UL SE maximization sub-problem. We use a similar approach for DL power
control, by maximizing the DL SE.

Second, we note that, with SINR-optimal weighting at the CPU, since the
weights maximize the received SINR at the CPU for each UE, they also maximize
the sum UL SE. However, UEs’ transmit powers can be optimized to improve the
sum UL SE. Also, the recipe for power control presented here can be applied to
any choice of weights at the CPU.

For convenience, we rewrite the problem as follows:

$\displaystyle\max_{\boldsymbol{\mathcal{E}}_{\mathsf{u}}}\quad$
$\displaystyle\sum\nolimits_{k\in\mathcal{U}_{\mathsf{u}}}\log\left(1+\frac{\mathsf{G}_{\mathsf{u},k}\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}}\right)}{\mathsf{I}_{\mathsf{u},k}\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}}\right)}\right),$
(13a) $\displaystyle\mathrm{subject~{}to}\quad$ $\displaystyle
0\leq\mathcal{E}_{\mathsf{u},k}\leq\mathcal{E}_{\mathsf{u}},\forall
k\in\mathcal{U}_{\mathsf{u}},$ (13b)

where, with the help of Lemma 5, we define
$\mathsf{G}_{\mathsf{u},k}(\boldsymbol{\mathcal{E}}_{\mathsf{u}})=(N-\tau_{p})\mathcal{E}_{\mathsf{u},k}\left(\sum\nolimits_{m\in\mathcal{A}_{\mathsf{u}}}\omega_{mk}^{*}\alpha_{mk}^{2}\right)^{2}$
and $\mathsf{I}_{\mathsf{u},k}(\boldsymbol{\mathcal{E}}_{\mathsf{u}})={\tt
EST}_{\mathsf{u},k}+{\tt
MUI}_{\mathsf{u},k}+\sigma_{\mathsf{eff.},\mathsf{u},k}^{2}$, with
$\sigma_{\mathsf{eff.},\mathsf{u},k}^{2}\triangleq\left({\tt
IAP}_{k}+N_{0}\sum\nolimits_{m\in\mathcal{A}_{\mathsf{u}}}|\omega_{mk}^{*}|^{2}\alpha_{mk}^{2}\right)$
being the effective noise that is independent of the UL transmit powers. Now,
we recognize that (13a) is a sum of the logarithm of ratios which can be
converted into a convex problem via FP using a few auxiliary variables that
can be iteratively solved in closed form. Further, FP guarantees that the
optimal value of the objective and the variables that attain the optimum are
the same for both the original objective function and the transformed
surrogate objective function. This makes FP an excellent choice for the
problem at hand. We next present the recipe in the context of our work.

To this end, we first introduce a set of auxiliary variables
$\boldsymbol{\varpi}_{\mathsf{u}}\triangleq\left\\{\varpi_{\mathsf{u},1},\ldots,\varpi_{\mathsf{u},\lvert\mathcal{U}_{\mathsf{u}}\rvert}\right\\}$,
and formulate an _equivalent_ problem of
$\max\nolimits_{\boldsymbol{\mathcal{E}}_{\mathsf{u}},\boldsymbol{\varpi}_{\mathsf{u}}}\quad
f\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}},\boldsymbol{\varpi}_{\mathsf{u}}\right)$,
with

$\displaystyle
f\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}},\boldsymbol{\varpi}_{\mathsf{u}}\right)\triangleq$
$\displaystyle\sum\nolimits_{k\in\mathcal{U}_{\mathsf{u}}}\ln\left(1+\varpi_{\mathsf{u},k}\right)-\sum\nolimits_{k\in\mathcal{U}_{\mathsf{u}}}\varpi_{\mathsf{u},k}$
$\displaystyle+\sum\nolimits_{k\in\mathcal{U}_{\mathsf{u}}}\frac{(1+\varpi_{\mathsf{u},k})\mathsf{G}_{\mathsf{u},k}\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}}\right)}{\mathsf{G}_{\mathsf{u},k}\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}}\right)+\mathsf{I}_{\mathsf{u},k}\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}}\right)}.$
(14)

and the same constraints are same as in (13a). Using results available in
[15], we can show that the above two problems are equivalent in the sense that
$\mathcal{E}_{u,k},\forall k,$ are the solution to (13) if and only if they
are also the solution to (V-2), and further, the maximum value of the
objective in (V-2) and (13a) are the same. Now, to maximize
$f\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}},\boldsymbol{\varpi}_{\mathsf{u}}\right)$,
alternately optimize $\boldsymbol{\mathcal{E}}_{\mathsf{u}}$ and
$\boldsymbol{\varpi}_{\mathsf{u}}$, while keeping the other variable constant.
We observe that
$f\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}},\boldsymbol{\varpi}_{\mathsf{u}}\right)$
is a concave differentiable function over $\boldsymbol{\varpi}_{\mathsf{u}}$
when $\boldsymbol{\mathcal{E}}_{\mathsf{u}}$ is fixed, say
$\boldsymbol{\mathcal{E}}_{\mathsf{u}}^{\tt iter}$. Thus,
$\boldsymbol{\varpi}^{{\tt
iter}+1}_{\mathsf{u}}=\arg\max\limits_{\boldsymbol{\varpi}_{\mathsf{u}}}f\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}}^{\tt
iter},\boldsymbol{\varpi}_{\mathsf{u}}\right)$ can be optimally determined by
setting $\frac{\partial
f\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}},\boldsymbol{\varpi}_{\mathsf{u}}\right)}{\partial\varpi_{\mathsf{u},k}}=0$
for each $\varpi_{\mathsf{u},k}$. This yields $\varpi_{\mathsf{u},k}^{{\tt
iter}+1}=\frac{\mathsf{G}_{\mathsf{u},k}\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}}^{\tt
iter}\right)}{\mathsf{I}_{\mathsf{u},k}\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}}^{\tt
iter}\right)},\forall k\in\mathcal{U}_{\mathsf{u}}$. Now, for fixed
$\boldsymbol{\varpi}_{\mathsf{u}}$, the first and the second term of (V-2) are
constants. The third term, i.e.,
$\sum\limits_{k\in\mathcal{U}_{\mathsf{u}}}\dfrac{(1+\varpi_{\mathsf{u},k})\mathsf{G}_{\mathsf{u},k}\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}}\right)}{\mathsf{G}_{\mathsf{u},k}\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}}\right)+\mathsf{I}_{\mathsf{u},k}\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}}\right)}$,
is in the sum of ratios form, which can be reformulated using a quadratic
transform as
$\max_{\boldsymbol{\mathcal{E}}_{\mathsf{u}},\boldsymbol{\tilde{\varpi}}}f(\boldsymbol{\mathcal{E}}_{\mathsf{u}},\boldsymbol{\tilde{\varpi}}_{\mathsf{u}})$,
where

$\displaystyle
f(\boldsymbol{\mathcal{E}}_{\mathsf{u}},\boldsymbol{\tilde{\varpi}}_{\mathsf{u}})\triangleq$
$\displaystyle\sum\nolimits_{k\in\mathcal{U}_{\mathsf{u}}}2\tilde{\varpi}_{\mathsf{u},k}\sqrt{(1+\varpi_{\mathsf{u},k})\mathsf{G}_{\mathsf{u},k}\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}}\right)}$
$\displaystyle-\sum\nolimits_{k\in\mathcal{U}_{\mathsf{u}}}\tilde{\varpi}_{\mathsf{u},k}^{2}\left(\mathsf{G}_{\mathsf{u},k}\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}}\right)+\mathsf{I}_{\mathsf{u},k}\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}}\right)\right)+\mathsf{c}_{\boldsymbol{\varpi}_{\mathsf{u}}},$
(15)

and
$\boldsymbol{\tilde{\varpi}}_{\mathsf{u}}\triangleq\left\\{\tilde{\varpi}_{\mathsf{u},1},\ldots,\tilde{\varpi}_{\mathsf{u},\lvert\mathcal{U}_{\mathsf{u}}\rvert}\right\\}$
being a new set of auxiliary variables and
$\mathsf{c}_{\boldsymbol{\varpi}_{\mathsf{u}}}$ is a constant dependent only
on $\boldsymbol{\varpi}_{\mathsf{u}}$. We can solve for
$\tilde{\varpi}_{\mathsf{u},k}$ and $\boldsymbol{\mathcal{E}}_{\mathsf{u}}$
via partial differentiation of (V-2), for fixed $\varpi_{\mathsf{u},k}$. We
summarize the overall recipe in Algorithm 3. The derivation of updates for
$\tilde{\varpi}_{\mathsf{u},{k}}^{{\tt iter}+1}$ and
$\mathcal{E}_{\mathsf{u},k}^{{\tt iter}in+1}$ (16) use simple algebraic
manipulations, and are omitted for brevity.

######  Proposition 3.

The UL power control Algorithm 3 is convergent in the objective since
$f\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}},\boldsymbol{\varpi}_{\mathsf{u}}\right)$
in (V-2) is bounded above and monotonically non-decreasing after each
iteration.

###### Proof.

See Appendix E.

∎

######  Remark 1.

Once the UL transmit powers of all the UEs are decided, the CPU can use this
information to refine the SINR optimal weights. Specifically, the CPU
initially finds $\boldsymbol{\omega}_{k}^{\mathsf{opt.}}$ considering equal
power allocation. Once the UL transmit powers are optimized, CPU can
reoptimize $\boldsymbol{\omega}_{k}^{\mathsf{opt.}}$ based on the received
SINRs with $\mathcal{E}_{\mathsf{u},k}$ given by Algorithm 3. This process can
be repeated until convergence of the sum UL SE.

######  Remark 2.

We observe that lines 3, 3, and 3 in Algorithm 3 are independent of the
combining scheme, and are applicable even under maximal ratio combining,
unlike [8] where the power control algorithm is tied to the specific combining
scheme used. More importantly, the FP approach guarantees that the maximum
objective of the surrogate and original objective functions are the same,
unlike the lower-bound surrogate-based optimization in [8].

Input : $\kappa_{jn},\forall
j\in\mathcal{A}_{\mathsf{d}},n\in\mathcal{U}_{\mathsf{d}}$

Initialize : $\boldsymbol{\mathcal{E}}_{\mathsf{u}}^{0}$, ${\tt iter}=0$

1 while  _$\lvert
f\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}},\boldsymbol{\varpi}_{\mathsf{u}}\right)^{{\tt
iter}+1}-f\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}},\boldsymbol{\varpi}_{\mathsf{u}}\right)^{\tt
iter}\rvert\geq\delta_{\mathsf{u}}$_ do

2 Evaluate: $\varpi_{\mathsf{u},k}^{{\tt
iter}+1}=\frac{\mathsf{G}_{\mathsf{u},k}\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}}\right)}{\mathsf{I}_{\mathsf{u},k}\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}}\right)}\lvert_{\boldsymbol{\mathcal{E}}_{\mathsf{u}}=\boldsymbol{\mathcal{E}}_{\mathsf{u}}^{\tt
iter}}$

3 Evaluate: $\frac{\partial
f(\boldsymbol{\mathcal{E}}_{\mathsf{u}},\boldsymbol{\tilde{\varpi}}_{\mathsf{u}})}{\partial\tilde{\varpi}_{\mathsf{u},k}}\Bigg{\lvert}_{\begin{subarray}{c}\boldsymbol{\mathcal{E}}_{\mathsf{u}}=\boldsymbol{\mathcal{E}}_{\mathsf{u}}^{\tt
iter},\\\ \varpi_{\mathsf{u},k}=\varpi_{\mathsf{u},k}^{\tt iter+1},\forall
k\in\mathcal{U}_{\mathsf{u}}\end{subarray}}=0\Rightarrow$
$\tilde{\varpi}_{\mathsf{u},{k}}^{{\tt
iter}+1}=\dfrac{\sqrt{\left(1+\varpi_{\mathsf{u},k}^{{\tt
iter}+1}\right)\mathsf{G}_{\mathsf{u},k}\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}}^{\tt
iter}\right)}}{\mathsf{G}_{\mathsf{u},k}\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}}^{\tt
iter}\right)+\mathsf{I}_{\mathsf{u},k}\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}}^{\tt
iter}\right)}$

4 Set: $\frac{\partial
f(\boldsymbol{\mathcal{E}}_{\mathsf{u}},\boldsymbol{\tilde{\varpi}}_{\mathsf{u}})}{\partial\mathcal{E}_{\mathsf{u},k}}\Bigg{\lvert}_{\begin{subarray}{c}\varpi_{\mathsf{u},k}=\varpi_{\mathsf{u},k}^{\tt
iter+1},\\\
\tilde{\varpi}_{\mathsf{u},{k}}=\tilde{\varpi}_{\mathsf{u},{k}}^{{\tt
iter}+1}\end{subarray}}=0$ to obtain (16)

5

6 Update: $\mathcal{E}_{\mathsf{u},k}^{{\tt
iter}+1}=\min\left\\{\eqref{eq:update_E},\mathcal{E}_{\mathsf{u}}\right\\},\forall
k\in\mathcal{U}_{\mathsf{u}}$

7 Update: ${\tt iter}={\tt iter}+1$

8

Algorithm 3 Uplink Power Control
$\displaystyle\mathcal{E}_{\mathsf{u},k}^{{\tt
iter}+1}=\frac{\left(\tilde{\varpi}_{\mathsf{u},k}^{{\tt
iter}+1}\sqrt{\left(1+\varpi_{\mathsf{u},k}^{{\tt
iter}+1}\right)}\right)^{2}(N-\tau_{p})\left(\sum\nolimits_{m\in\mathcal{A}_{\mathsf{u}}}\omega^{*}_{mk}\alpha_{mk}^{2}\right)^{2}}{\left[\sum\nolimits_{n\in\mathcal{U}_{\mathsf{u}}}(\tilde{\varpi}_{\mathsf{u},n}^{{\tt
iter}+1})^{2}\left\\{(N-\tau_{p})\left(\sum\nolimits_{m\in\mathcal{A}_{\mathsf{u}}}\omega^{*}_{mk}\alpha_{mk}^{2}\right)^{2}+\sum\nolimits_{m\in\mathcal{A}_{\mathsf{u}}}\lvert\omega^{*}_{mn}\rvert^{2}\alpha_{mn}^{2}(\beta_{mk}-\alpha_{mk}^{2})\right\\}\right]^{2}},\forall
k\in\mathcal{U}_{\mathsf{u}}.$ (16)

####  V-3 DL power control

Now, in DL also, we can apply FP to optimize the DL power control
coefficients. To this end, we first introduce some useful mathematical
notation. Let
$\mathbf{g}_{\mathsf{d},n}\triangleq\left[\mathbb{E}\left[\mathbf{f}_{1n}^{T}\mathbf{p}_{1n}\right],\mathbb{E}\left[\mathbf{f}_{2n}^{T}\mathbf{p}_{2n}\right],\ldots,\mathbb{E}\left[\mathbf{f}_{|\mathcal{A}_{\mathsf{d}}|n}^{T}\mathbf{p}_{|\mathcal{A}_{\mathsf{d}}|n}\right]\right]^{T}$
(which equals
$(N-\tau_{p})\left[\alpha_{1n},\alpha_{2n},\ldots,\alpha_{|\mathcal{A}_{\mathsf{d}}|n}\right]^{T}$
under ZF precoding.) Also, let
$\left[\mathbf{I}_{nq}\right]_{jj^{\prime}}\triangleq\mathbb{E}\left[\mathbf{f}_{jn}^{T}\mathbf{p}_{jq}\mathbf{p}_{j^{\prime}q}^{H}\mathbf{f}_{j^{\prime}n}^{*}\right],\forall
j,j^{\prime}\in\mathcal{A}_{\mathsf{d}}$. With ZF precoding, $\mathbf{I}_{nq}$
has the following structure:

$\displaystyle[\mathbf{I}_{nq}]_{jj^{\prime}}=\begin{cases}(N-\tau_{p})\alpha^{2}_{jn}\lvert\langle\boldsymbol{\varphi}_{l(n)},\boldsymbol{\varphi}_{l(q)}\rangle\rvert^{2}\\\
\hskip 28.45274pt+(\beta_{jn}-\alpha^{2}_{jn}),\text{if}~{}j=j^{\prime}\\\
(N-\tau_{p})\alpha_{j^{\prime}n}\alpha_{jn}\lvert\langle\boldsymbol{\varphi}_{l(n)},\boldsymbol{\varphi}_{l(q)}\rangle\rvert^{2},\text{otherwise}.\end{cases}$
(17)

Then,
$\mathbb{E}[|\sum\nolimits_{j\in\mathcal{A}_{\mathsf{d}}}\kappa_{jq}\mathbf{f}_{jn}^{T}\mathbf{p}_{jq}|^{2}]$
can be expressed as
$\sum\limits_{j\in\mathcal{A}_{\mathsf{d}}}\sum\limits_{j^{\prime}\in\mathcal{A}_{\mathsf{d}}}\kappa_{jq}\mathbb{E}\left[\mathbf{f}_{jn}^{T}\mathbf{p}_{jq}\mathbf{p}_{j^{\prime}q}^{H}\mathbf{f}_{j^{\prime}n}^{*}\right]\kappa_{j^{\prime}q}=\boldsymbol{\kappa}_{q}^{T}\mathbf{I}_{nq}\boldsymbol{\kappa}_{q}$.
The effective DL noise, independent of DL power control coefficients, is
defined as $\sigma_{\mathsf{eff.},\mathsf{d},n}^{2}={\tt IUE}_{n}+N_{0}$. Thus
DL SINR becomes
$\eta_{\mathsf{d},n}(\mathcal{A}_{\mathsf{s}})=\frac{\mathsf{G}_{\mathsf{d},n}\left(\boldsymbol{\kappa}\right)}{\mathsf{I}_{\mathsf{d},n}\left(\boldsymbol{\kappa}\right)}$,
with
$\mathsf{G}_{\mathsf{d},n}\left(\boldsymbol{\kappa}\right)=\left(\boldsymbol{\kappa}_{n}^{T}\mathbf{g}_{\mathsf{d},n}\right)^{2}$
and
$\mathsf{I}_{\mathsf{d},n}\left(\boldsymbol{\kappa}\right)\triangleq\sum\nolimits_{q\in\mathcal{U}_{\mathsf{d}}}\boldsymbol{\kappa}_{q}^{T}\mathbf{I}_{nq}\boldsymbol{\kappa}_{q}-\left(\boldsymbol{\kappa}_{q}^{T}\mathbf{g}_{\mathsf{d},n}\right)^{2}+\frac{1}{\mathcal{E}_{\mathsf{d}}}\sigma_{\mathsf{eff.},\mathsf{d},n}^{2}$.
Observe that, substituting $\mathbf{g}_{\mathsf{d},n}$ and (17), we obtain
Lemma 6. Now, the sum DL SE maximization problem becomes:

$\displaystyle\max_{\kappa_{jn}}\quad$
$\displaystyle\sum\nolimits_{n\in\mathcal{U}_{\mathsf{d}}}\log\left(1+\frac{\mathsf{G}_{\mathsf{d},n}\left(\boldsymbol{\kappa}\right)}{\mathsf{I}_{\mathsf{d},n}\left(\boldsymbol{\kappa}\right)}\right)$
(18a) $\displaystyle\mathrm{subject~{}to}\quad$
$\displaystyle\sum\nolimits_{\begin{subarray}{c}q\in\mathcal{U}_{\mathsf{d}}\end{subarray}}\kappa_{jq}^{2}\leq
1,\forall j\in\mathcal{A}_{\mathsf{d}}.$ (18b)

We apply the Lagrange-dual transform [15] with auxiliary variables
$\boldsymbol{\varpi}_{\mathsf{d}}=\left[\varpi_{\mathsf{d},1},\varpi_{\mathsf{d},2},\ldots,\varpi_{\mathsf{d},|\mathcal{U}_{\mathsf{d}}|}\right]$
to obtain:

$\displaystyle\max_{\kappa_{jn},\boldsymbol{\varpi}_{\mathsf{d}}}\,f(\boldsymbol{\kappa},\boldsymbol{\varpi}_{\mathsf{d}})\triangleq$
$\displaystyle\sum\nolimits_{n\in\mathcal{U}_{\mathsf{d}}}\\!\\!\\!\ln\left(1+\varpi_{\mathsf{d},n}\right)-\sum\nolimits_{n\in\mathcal{U}_{\mathsf{d}}}\varpi_{\mathsf{d},n}$
$\displaystyle+\sum\nolimits_{n\in\mathcal{U}_{\mathsf{d}}}\frac{(1+\varpi_{\mathsf{d},n})\mathsf{G}_{\mathsf{d},n}\left(\boldsymbol{\kappa}\right)}{\mathsf{G}_{\mathsf{d},n}\left(\boldsymbol{\kappa}\right)+\mathsf{I}_{\mathsf{d},n}\left(\boldsymbol{\kappa}\right)}$
(19a) $\displaystyle\mathrm{subject~{}to}\quad$
$\displaystyle\sum\nolimits_{\begin{subarray}{c}q\in\mathcal{U}_{\mathsf{d}}\end{subarray}}\kappa_{jq}^{2}\leq
1;\forall j\in\mathcal{A}_{\mathsf{d}}.$ (19b)

The problems in (18a) and in (19) are equivalent in the sense that
$\boldsymbol{\kappa}$ is a solution of (18a) if and only if it is the solution
of (19) [15]. We now alternately optimize $\boldsymbol{\kappa}$ and
$\boldsymbol{\varpi}_{\mathsf{d}}$. For fixed $\boldsymbol{\kappa}$,
$f(\boldsymbol{\kappa},\boldsymbol{\varpi}_{\mathsf{d}})$ is a concave
differentiable function over $\boldsymbol{\varpi}_{\mathsf{d}}$. Thus,
$\frac{\partial
f(\boldsymbol{\kappa},\boldsymbol{\varpi}_{\mathsf{d}})}{\partial\varpi_{\mathsf{d},n}}=0$
yields
$\varpi_{\mathsf{d},n}^{\mathsf{opt.}}=\frac{\mathsf{G}_{\mathsf{d},n}\left(\boldsymbol{\kappa}\right)}{\mathsf{I}_{\mathsf{d},n}\left(\boldsymbol{\kappa}\right)}$.
For fixed $\boldsymbol{\varpi}_{\mathsf{d}}$, the first and the second terms
of $f(\boldsymbol{\kappa},\boldsymbol{\varpi}_{\mathsf{d}})$ are constants.
Hence, to optimize $\boldsymbol{\kappa}$, we need to solve
$\max_{\boldsymbol{\kappa}}\sum\nolimits_{n\in\mathcal{U}_{\mathsf{d}}}\frac{(1+\varpi_{\mathsf{d},n})\mathsf{G}_{\mathsf{d},n}\left(\boldsymbol{\kappa}\right)}{\mathsf{G}_{\mathsf{d},n}\left(\boldsymbol{\kappa}\right)+\mathsf{I}_{\mathsf{d},n}\left(\boldsymbol{\kappa}\right)}$,
for which we use FP. To do so, we define
$\mathsf{\bar{G}}_{\mathsf{d},n}\left(\boldsymbol{\kappa}\right)\triangleq(1+\varpi_{\mathsf{d},n})\mathsf{G}_{\mathsf{d},n}\left(\boldsymbol{\kappa}\right),$
and
$\mathsf{\bar{I}}_{\mathsf{d},n}\left(\boldsymbol{\kappa}\right)\triangleq\mathsf{G}_{\mathsf{d},n}\left(\boldsymbol{\kappa}\right)+\mathsf{I}_{\mathsf{d},n}\left(\boldsymbol{\kappa}\right).$
Then, the equivalent problem is to maximize
$\sum\nolimits_{n\in\mathcal{U}_{\mathsf{d}}}\frac{\mathsf{\bar{G}}_{\mathsf{d},n}\left(\boldsymbol{\kappa}\right)}{\mathsf{\bar{I}}_{\mathsf{d},n}\left(\boldsymbol{\kappa}\right)}$
subject to
$\sum\nolimits_{\begin{subarray}{c}q\in\mathcal{U}_{\mathsf{d}}\end{subarray}}\kappa_{jq}^{2}\leq
1.$ Now, the dual of
$\sum\nolimits_{n\in\mathcal{U}_{\mathsf{d}}}\frac{\mathsf{\bar{G}}_{\mathsf{d},n}\left(\boldsymbol{\kappa}\right)}{\mathsf{\bar{I}}_{\mathsf{d},n}(\boldsymbol{\kappa})}$
is
$\sum\nolimits_{n\in\mathcal{U}_{\mathsf{d}}}(2\tilde{\varpi}_{\mathsf{d},n}\sqrt{\mathsf{\bar{G}}_{\mathsf{d},n}(\boldsymbol{\kappa})}-\tilde{\varpi}_{\mathsf{d},n}^{2}\mathsf{\bar{I}}_{\mathsf{d},n}(\boldsymbol{\kappa}))$
with new auxiliary variables
$\boldsymbol{\tilde{\varpi}}_{\mathsf{d}}=\left[\tilde{\varpi}_{\mathsf{d},1},\tilde{\varpi}_{\mathsf{d},2},\ldots,\tilde{\varpi}_{\mathsf{d},|\mathcal{U}_{\mathsf{d}}|}\right]^{T}$
[15, Corollary 1]. Substituting, we obtain

$\displaystyle\max_{\boldsymbol{\kappa},\boldsymbol{\tilde{\varpi}}_{\mathsf{d}}}\,f(\boldsymbol{\kappa},\boldsymbol{\tilde{\varpi}}_{\mathsf{d}})\triangleq$
$\displaystyle\sum\nolimits_{n\in\mathcal{U}_{\mathsf{d}}}\Big{(}2\tilde{\varpi}_{\mathsf{d},n}\sqrt{(1+\varpi_{\mathsf{d},n})\left(\boldsymbol{\kappa}_{q}^{T}\mathbf{g}_{\mathsf{d},n}\right)^{2}}$
$\displaystyle-\tilde{\varpi}_{\mathsf{d},n}^{2}\left(\sum\nolimits_{q\in\mathcal{U}_{\mathsf{d}}}\boldsymbol{\kappa}_{q}^{T}\mathbf{I}_{nq}\boldsymbol{\kappa}_{q}+\sigma_{\mathsf{eff.},\mathsf{d},n}^{2}\right)\Big{)},$
(20a) $\displaystyle\mathrm{subject~{}to}\quad$
$\displaystyle\sum\nolimits_{\begin{subarray}{c}q\in\mathcal{U}_{\mathsf{d}}\end{subarray}}\kappa_{jq}^{2}\leq
1,\forall j\in\mathcal{A}_{\mathsf{d}}.$ (20b)

Observe that $\boldsymbol{\varpi}_{\mathsf{d}}$ is already fixed. Then, for
fixed $\boldsymbol{\kappa}$,
$f(\boldsymbol{\kappa},\boldsymbol{\tilde{\varpi}}_{\mathsf{d}})$ is strongly
concave with respect to $\boldsymbol{\tilde{\varpi}}_{\mathsf{d}}$ and thus,
we can set $\frac{\partial
f(\boldsymbol{\kappa},\boldsymbol{\tilde{\varpi}}_{\mathsf{d}})}{\partial\tilde{\varpi}_{\mathsf{d},n}}=0$,
leading to
$\tilde{\varpi}_{\mathsf{d},n}^{\text{opt.}}=\frac{\sqrt{1+\varpi_{\mathsf{d},n}}\mathbf{g}_{\mathsf{d},n}^{T}\boldsymbol{\kappa}_{n}}{\sum\nolimits_{q\in\mathcal{U}_{\mathsf{d}}}\boldsymbol{\kappa}_{q}^{T}\mathbf{I}_{nq}\boldsymbol{\kappa}_{q}+\sigma_{\mathsf{eff.},\mathsf{d},n}^{2}}$;
$\forall n\in\mathcal{U}_{\mathsf{d}}$. Finally, we obtain
$\boldsymbol{\kappa}^{\text{opt.}}$ given $\boldsymbol{\varpi}_{\mathsf{d}}$
and $\boldsymbol{\tilde{\varpi}}_{\mathsf{d}}$ by solving

$\displaystyle\max_{\boldsymbol{\kappa},\boldsymbol{\tilde{\varpi}}_{\mathsf{d}}}\quad$
$\displaystyle\sum\nolimits_{n\in\mathcal{U}_{\mathsf{d}}}\left(2\tilde{\varpi}_{\mathsf{d},n}\sqrt{(1+\varpi_{\mathsf{d},n})\left(\boldsymbol{\kappa}_{n}^{T}\mathbf{g}_{\mathsf{d},n}\right)^{2}}\right.$
$\displaystyle\left.-\tilde{\varpi}_{\mathsf{d},n}^{2}\left(\sum\nolimits_{q\in\mathcal{U}_{\mathsf{d}}}\boldsymbol{\kappa}_{q}^{T}\mathbf{I}_{nq}\boldsymbol{\kappa}_{q}+\sigma_{\mathsf{eff.},\mathsf{d},n}^{2}\right)\right),$
(21a) $\displaystyle\mathrm{subject~{}to}\quad$
$\displaystyle\sum\nolimits_{\begin{subarray}{c}q\in\mathcal{U}_{\mathsf{d}}\end{subarray}}\kappa_{jq}^{2}\leq
1,\forall j\in\mathcal{A}_{\mathsf{d}}.$ (21b)

We observe that matrix $\mathbf{I}_{nq}$ is positive semi-definite. Hence,
$f(\boldsymbol{\kappa})\triangleq\sum\nolimits_{n\in\mathcal{U}_{\mathsf{d}}}(2\tilde{\varpi}_{\mathsf{d},n}\sqrt{(1+\varpi_{\mathsf{d},n})\left(\boldsymbol{\kappa}_{n}^{T}\mathbf{g}_{\mathsf{d},n}\right)^{2}}-\tilde{\varpi}_{\mathsf{d},n}^{2}(\sum\nolimits_{q\in\mathcal{U}_{\mathsf{d}}}\boldsymbol{\kappa}_{q}^{T}\mathbf{I}_{nq}\boldsymbol{\kappa}_{q}+\sigma_{\text{eff.}}^{2}))$
is concave with respect to $\boldsymbol{\kappa}$. The above problem is a QCQP,
which can be _optimally_ solved via ADMM (see [16, Chapter $5$]), yielding a
closed form update for $\kappa_{jn}$. This is presented next.

We reformulate (21a) with the help of new auxiliary variables
$\boldsymbol{\pi}=\left[\pi_{jn}\right]_{j\in\mathcal{A}_{d},n\in\mathcal{U}_{d}}$
as

$\displaystyle\min_{\boldsymbol{\kappa},\boldsymbol{\pi}}\quad$
$\displaystyle\mathds{1}_{\boldsymbol{\kappa}}(\boldsymbol{\pi})-f(\boldsymbol{\kappa}),$
(22a) $\displaystyle\mathrm{subject~{}to}\quad$
$\displaystyle\kappa_{jn}=\pi_{jn},\forall
j\in\mathcal{A}_{\mathsf{d}},n\in\mathcal{U}_{\mathsf{d}},$ (22b)

where $\mathds{1}_{\boldsymbol{\kappa}}(\boldsymbol{\pi})$ is defined as

$\displaystyle\mathds{1}_{\boldsymbol{\kappa}}(\boldsymbol{\pi})=\begin{cases}0,\quad\mathrm{if}\quad\|\boldsymbol{\pi}_{j,:}\|^{2}\leq
1,\forall j\in\mathcal{A}_{\mathsf{d}}\\\
\infty,\quad\mathrm{otherwise},\end{cases}$ (23)

where
$\boldsymbol{\pi}_{j,:}\triangleq\left[\pi_{j1},\pi_{j2},\ldots,\pi_{j\lvert\mathcal{U}_{\mathsf{d}}\rvert}\right]^{T}$.
Here, $\mathds{1}_{\boldsymbol{\kappa}}(\boldsymbol{\pi})$ is an indicator
whether the auxiliary variables satisfy the feasibility constraint
$\|\boldsymbol{\pi}_{j,:}\|^{2}\leq 1$ corresponding to $\boldsymbol{\kappa}$.
Essentially, $\boldsymbol{\pi}$ is a copy of the main optimization variable
$\boldsymbol{\kappa}$ and should satisfy the same constraint. Let
$\bar{\pi}_{jn}$ denote the scaled dual variables888In the ADMM terminology,
the variables $\boldsymbol{\kappa}$ and $\boldsymbol{\pi}$ can be considered
as the first and second blocks of primal variables, respectively. In ADMM, a
set of dual variables are introduced for the equality constraint in (22a), and
they are to be updated in each iteration with the primal variables.
corresponding to the equality constraints $\pi_{jn}=\kappa_{jn},\forall
j\in\mathcal{A}_{\mathsf{d}},n\in\mathcal{U}_{\mathsf{d}}$. Then, the
augmented Lagrangian can be written as
$\mathcal{L}\left(\boldsymbol{\kappa},\boldsymbol{\pi},\bar{\boldsymbol{\pi}}\right)=\mathds{1}_{\boldsymbol{\kappa}}(\boldsymbol{\pi})-f(\boldsymbol{\kappa})+\frac{\delta_{\mathsf{p}}}{2}\sum\nolimits_{n\in\mathcal{U}_{\mathsf{d}}}\sum\nolimits_{j\in\mathcal{A}_{\mathsf{d}}}\left(\pi_{jn}-\kappa_{jn}+\bar{\pi}_{jn}\right)^{2}$,
where $\delta_{\mathsf{p}}$ is a penalty parameter. We now update
$\boldsymbol{\kappa}_{n},\forall n\in\mathcal{U}_{\mathsf{d}},$ as the
solution of

$\displaystyle\arg\min_{\boldsymbol{\kappa}}-f(\boldsymbol{\kappa})+\frac{\delta_{\mathsf{p}}}{2}\sum\limits_{n\in\mathcal{U}_{\mathsf{d}}}\sum\limits_{j\in\mathcal{A}_{\mathsf{d}}}\left(\pi_{jn}-\kappa_{jn}+\bar{\pi}_{jn}\right)^{2}.$
(24)

Upon substituting for $f(\boldsymbol{\kappa})$ in (24), we get (25), from
which we obtain $\boldsymbol{\kappa}_{n}^{\mathsf{opt.}}$ as given in (26).

$\displaystyle\arg\min_{\boldsymbol{\kappa}}\Big{(}\sum_{n\in\mathcal{U}_{\mathsf{d}}}\boldsymbol{\kappa}_{n}^{T}\Big{(}\sum_{q\in\mathcal{U}_{\mathsf{d}}}\tilde{\varpi}_{\mathsf{d},n}^{2}\mathbf{I}_{nq}+\frac{\delta_{\mathsf{p}}}{2}\boldsymbol{I}_{\lvert\mathcal{A}_{\mathsf{d}}\rvert}\Big{)}\Big{)}\boldsymbol{\kappa}_{n}-2\sum_{n\in\mathcal{U}_{\mathsf{d}}}(\tilde{\varpi}_{\mathsf{d},n}\sqrt{(1+\varpi_{\mathsf{d},n})}\mathbf{g}_{\mathsf{d},n}+\frac{\delta_{\mathsf{p}}}{2}(\boldsymbol{\pi}_{:,n}+\boldsymbol{\bar{\pi}}_{:,n}))^{T}\boldsymbol{\kappa_{n}}.$
(25)
$\displaystyle\boldsymbol{\kappa}_{n}^{\mathsf{opt.}}=(\sum\nolimits_{q\in\mathcal{U}_{\mathsf{d}}}\tilde{\varpi}_{\mathsf{d},n}^{2}\mathbf{I}_{nq}+\frac{\delta_{\mathsf{p}}}{2}\boldsymbol{I}_{\lvert\mathcal{A}_{\mathsf{d}}\rvert})^{-1}\left(\tilde{\varpi}_{\mathsf{d},n}\sqrt{(1+\varpi_{\mathsf{d},n})}\mathbf{g}_{\mathsf{d},n}+\frac{\delta_{\mathsf{p}}}{2}\left(\boldsymbol{\pi}_{:,n}+\boldsymbol{\bar{\pi}}_{:,n}\right)\right)\in\mathbb{C}^{\lvert\mathcal{A}_{\mathsf{d}}\rvert\times
1},\forall n\in\mathcal{U}_{\mathsf{d}}.$ (26)

Here,
$\boldsymbol{\pi}_{:,n}\triangleq\left[\pi_{1,n},\pi_{2,n},\ldots,\pi_{\lvert\mathcal{A}_{\mathsf{d}\rvert,n}}\right]^{T}$
and
$\boldsymbol{\bar{\pi}}_{:,n}\triangleq\left[\bar{\pi}_{1,n},\bar{\pi}_{2,n},\ldots,\bar{\pi}_{\lvert\mathcal{A}_{\mathsf{d}\rvert,n}}\right]^{T}$.
Next, we can find the optimal update of the second block of primal variables
as

$\arg\min_{\boldsymbol{\pi}}\mathds{1}_{\boldsymbol{\kappa}}(\boldsymbol{\pi})+\frac{\delta_{\mathsf{p}}}{2}\sum\nolimits_{n\in\mathcal{U}_{\mathsf{d}}}\sum\nolimits_{j\in\mathcal{A}_{\mathsf{d}}}\left(\pi_{jn}-\kappa_{jn}+\bar{\pi}_{jn}\right)^{2}.$

Equivalently, $\pi_{jn}^{\mathsf{opt.}}$ is the solution of

$\displaystyle\arg\min_{\pi_{jn}}\quad\quad\quad$
$\displaystyle\frac{\delta_{\mathsf{p}}}{2}\sum\nolimits_{n\in\mathcal{U}_{\mathsf{d}}}\sum\nolimits_{j\in\mathcal{A}_{\mathsf{d}}}\left(\pi_{jn}-\kappa_{jn}+\bar{\pi}_{jn}\right)^{2},$
(27a) $\displaystyle\mathrm{subject~{}to}\quad\quad$
$\displaystyle\sum\nolimits_{n\in\mathcal{U}_{\mathsf{d}}}\pi_{jn}^{2}\leq
1,\forall j\in\mathcal{A}_{\mathsf{d}}.$ (27b)

Now, (27a) can be solved independently for each AP index by evaluating
$\arg\min_{\pi_{jn}}\quad\frac{\delta_{\mathsf{p}}}{2}\sum\nolimits_{n\in\mathcal{U}_{\mathsf{d}}}\left(\pi_{jn}-\kappa_{jn}+\bar{\pi}_{jn}\right)^{2}$
for each $j\in\mathcal{A}_{\mathsf{d}}$ subject to the per-AP power constraint
$\sum_{n\in\mathcal{U}_{\mathsf{d}}}\pi_{jn}^{2}\leq 1$. Using the KKT
condition, we can show that the optimal solution corresponding to the $j$th DL
AP, $\boldsymbol{\pi}_{j}^{\mathsf{opt.}}$, is

$\displaystyle\boldsymbol{\pi}_{j}^{\mathsf{opt.}}=\min\left\\{1,\sqrt{\frac{1}{\|\boldsymbol{\dot{\pi}}_{j}\|^{2}}}\right\\}\boldsymbol{\dot{\pi}}_{j},\forall
j\in\mathcal{A}_{\mathsf{d}},$ (28)

with
$\boldsymbol{\dot{\pi}}_{j}\triangleq\left[(\kappa_{j1}-\bar{\pi}_{j1}),(\kappa_{j2}-\bar{\pi}_{j2}),\ldots,(\kappa_{j\lvert\mathcal{U}_{\mathsf{d}}\rvert}-\bar{\pi}_{j\lvert\mathcal{U}_{\mathsf{d}}\rvert})\right]^{T}$.
We summarize the iterative recipe for solving our original problem (21a) via
ADMM approach in Algorithm 4. We note that the $\\{\kappa_{jn}\\}$ yielded by
Algorithm 4 are _globally optimal_. We present the overall DL power control
recipe in Algorithm 5. The stopping criterion of the algorithm is decided by
the threshold $\delta_{\mathsf{d}}$. Also, with ZF precoding, the closed-form
update equation for $\tilde{\varpi}^{\tt iter+1}_{n}$ is presented in (29). To
update $\boldsymbol{\kappa}$, $\mathbf{I}_{nq}$ needs to be substituted in
(26) from (17).

$\displaystyle\tilde{\varpi}^{\tt
iter+1}_{\mathsf{d},n}=\frac{\sqrt{1+\varpi_{\mathsf{d},n}^{{\tt
iter}+1}}\sqrt{(N-\tau_{p})}\sqrt{\mathcal{E}_{\mathsf{d}}}\left(\sum\nolimits_{j\in\mathcal{A}_{\mathsf{d}}}\alpha_{jn}\kappa_{jn}^{\tt
iter}\right)}{\sum\limits_{q\in\mathcal{U}_{\mathsf{d}}}\sum\limits_{j\in\mathcal{A}_{\mathsf{d}}}\mathcal{E}_{\mathsf{d}}(\kappa_{jq}^{\tt
iter})^{2}(\beta_{jn}-\alpha_{jn}^{2})+(N-\tau_{p})\sum\limits_{q\in\mathcal{U}_{\mathsf{d}}}\mathcal{E}_{\mathsf{d}}\left(\sum\nolimits_{j\in\mathcal{A}_{\mathsf{d}}}\kappa_{jq}^{\tt
iter}\alpha_{jn}\right)^{2}\lvert\langle\boldsymbol{\varphi}_{l(n)},\boldsymbol{\varphi}_{l(q)}\rangle\rvert^{2}+\sigma_{\mathsf{eff.},\mathsf{d},n}^{2}}.$
(29)

Input : $\pi_{jn}^{0},\forall
j\in\mathcal{A}_{\mathsf{d}},n\in\mathcal{U}_{\mathsf{d}}$,
$\delta_{\mathsf{ADMM}}>0$

Initialize : $\bar{\pi}_{jn}^{0}=0,\forall
j\in\mathcal{A}_{\mathsf{d}},n\in\mathcal{U}_{\mathsf{d}}$, ${\tt iter}=0$

1 while
_$\|\boldsymbol{\kappa_{n}}-\boldsymbol{\pi}_{:,n}\|\geq\delta_{\mathsf{ADMM}}$_
do

2 Evaluate: $\boldsymbol{\kappa}_{n}^{{\tt iter}+1}$ using (26) with
$\boldsymbol{\pi}_{:,n}^{\tt iter}$ and $\boldsymbol{\bar{\pi}}_{:,n}^{\tt
iter}$

3 Evaluate: $\boldsymbol{\pi}_{:,n}^{{\tt iter}+1}$ using (28) with
$\boldsymbol{\dot{\pi}}_{j}\triangleq\left[(\kappa_{j1}^{{\tt
iter}+1}-\bar{\pi}_{j1}^{{\tt
iter}}),\ldots,(\kappa_{j\lvert\mathcal{U}_{\mathsf{d}}\rvert}^{{\tt
iter}+1}-\bar{\pi}_{j\lvert\mathcal{U}_{\mathsf{d}}\rvert}^{{\tt
iter}})\right]^{T}$

4 Update: $\bar{\pi}_{jn}^{{\tt iter}+1}=\pi_{jn}^{\tt iter}-\kappa_{jn}^{{\tt
iter}+1}+\bar{\pi}_{jn}^{\tt iter}$

5 Update: ${\tt iter}={\tt iter}+1$

Algorithm 4 Solving (21a) via ADMM

Input : $\mathcal{E}_{\mathsf{u},k},\forall k\in\mathcal{U}_{\mathsf{u}}$

Initialize : $\kappa_{jn}^{0},\forall
j\in\mathcal{A}_{\mathsf{d}},n\in\mathcal{U}_{\mathsf{d}}$, ${\tt iter}=0$

1 while  _$\lvert
f(\boldsymbol{\kappa},\boldsymbol{\varpi}_{\mathsf{d}})^{{\tt
iter}+1}-f(\boldsymbol{\kappa},\boldsymbol{\varpi}_{\mathsf{d}})^{\tt
iter}\rvert\geq\delta_{\mathsf{d}}$_ do

2 Evaluate: $\varpi_{\mathsf{d},n}^{\tt
iter+1}=\dfrac{\left((\boldsymbol{\kappa}_{q}^{\tt
iter})^{T}\mathbf{g}_{\mathsf{d},n}\right)^{2}}{\sum\limits_{q\in\mathcal{U}_{\mathsf{d}}}(\boldsymbol{\kappa}_{q}^{\tt
iter})^{T}\mathbf{I}_{nq}\boldsymbol{\kappa}_{q}^{\tt
iter}-\left((\boldsymbol{\kappa}_{q}^{\tt
iter})^{T}\mathbf{g}_{\mathsf{d},n}\right)^{2}+\sigma_{\text{eff.}}^{2}}$

3 Evaluate: $\tilde{\varpi}^{\tt
iter+1}_{n}=\dfrac{\sqrt{1+\varpi_{\mathsf{d},n}^{\tt
iter+1}}\mathbf{g}_{\mathsf{d},n}^{T}\boldsymbol{\kappa}_{n}^{\tt
iter}}{\sum\limits_{q\in\mathcal{U}_{\mathsf{d}}}(\boldsymbol{\kappa}_{q}^{\tt
iter})^{T}\mathbf{I}_{nq}\boldsymbol{\kappa}_{q}^{\tt
iter}+\sigma_{\text{eff.}}^{2}}$

4 Evaluate: $\boldsymbol{\kappa}_{q}^{{\tt iter}+1}$ via solving (21a) with
$\varpi_{\mathsf{d},n}=\varpi_{\mathsf{d},n}^{\tt iter+1}$ and
$\tilde{\varpi}_{\mathsf{d},n}=\tilde{\varpi}^{\tt iter+1}_{\mathsf{d},n}$

5 Update: ${\tt iter}={\tt iter}+1$

6

Algorithm 5 Downlink Power Control

######  Proposition 4.

The DL power allocation algorithm is convergent in objective since
$f(\boldsymbol{\kappa},\boldsymbol{\varpi}_{\mathsf{d}})$ is bounded above and
monotonically non-decreasing after each iteration.

We now highlight the key benefits of our DL power control scheme. First of
all, the DL power allocation algorithm is precoder and combiner agnostic
(lines 5, 5 in Algorithm 5 and lines 4, 4, and 4 in Algorithm 4 apply to any
choice of precoder), similar to our UL power allocation algorithm. This makes
our algorithms more widely applicable compared to [8]. Also, as mentioned for
UL, most of the works in literature lower bound the original cost function
(e.g., the sum SE optimization in [8], [14]) by a series of surrogate convex
functions and use available general-purpose convex solvers. In contrast, we
provide closed-form update equations for all the auxiliary variables, and,
thanks to FP, our algorithm directly optimizes the original cost function.

Numerical experiments related to the convergence of iterates, the cost
functions of the above UL and DL power control algorithms, and the choice of
penalty parameters are provided in Appendix F.

##  VI FD CF MIMO System

The analysis for the FD system is similar to that of the DTDD system presented
in Sec. IV, except that all the APs are now capable of concurrent transmission
and reception. Hence,
$\mathcal{A}_{\mathsf{u}}=\mathcal{A}_{\mathsf{d}}=\mathcal{A}$ with
$\lvert\mathcal{A}\rvert=M$. Also, each AP suffers from IrAI. Hence, to avoid
repetition, we present the final results related to the optimal CPU combining
weights, the UL and DL SE expressions, and the power control algorithms, for
the FD system without detailed proofs.

######  Corollary 1.

The optimal weighting vector for the $k$th UL UE at the CPU is
$\boldsymbol{\alpha}_{k}^{\mathsf{opt.}}=\mathcal{E}_{\mathsf{u},k}\mathbf{R}_{\alpha}^{-1}\bar{\mathbf{u}}_{k}$,
with
$\bar{\mathbf{u}}_{k}$$=[\alpha_{\mathsf{u},1k}^{2},\alpha_{\mathsf{u},2k}^{2},\ldots,\alpha_{\mathsf{u},Mk}^{2}]^{T}$
and $\mathbf{R}_{\alpha}=\sum\limits_{k^{\prime}\in\mathcal{P}_{l(k)\backslash
k}}\mathcal{E}_{\mathsf{u},k^{\prime}}\bar{\mathbf{u}}_{k^{\prime}}\bar{\mathbf{u}}_{k^{\prime}}^{H}+\frac{1}{N_{\mathsf{rx}}-\tau_{p}}\mathbf{\dot{R}}_{\alpha}+\frac{\mathcal{E}_{\mathsf{d}}}{N_{\mathsf{rx}}-\tau_{p}}\mathbf{\dot{R}}_{\alpha}$,
where $\mathbf{\dot{R}}_{\alpha}$ and $\mathbf{\ddot{R}}_{\alpha}$ are
diagonal matrices with $m$th diagonal entry being

$\displaystyle[\mathbf{\dot{R}}_{\alpha}]_{m}=\sum\nolimits_{k^{\prime}\in\mathcal{U}_{\mathsf{u}}}\mathcal{E}_{\mathsf{u},k^{\prime}}\alpha_{\mathsf{u},mk}^{2}(\beta_{\mathsf{u},mk^{\prime}}-\alpha_{\mathsf{u},mk^{\prime}}^{2})+N_{0}\alpha_{\mathsf{u},mk^{\prime}}^{2},$
$\displaystyle[\mathbf{\ddot{R}}_{\alpha}]_{m}=\sum\nolimits_{j=1,j\neq
m}^{M}N_{\mathsf{tx}}\kappa_{jn}^{2}\zeta_{mj}^{\mathsf{InAP}}\alpha_{\mathsf{u},mk}^{2}+N_{\mathsf{tx}}\kappa_{mn}^{2}\alpha_{\mathsf{u},mk}^{2}\zeta_{mm}^{\mathsf{SI}},$

respectively, for $m=1,2,\ldots,M$.

######  Corollary 2.

The UL and DL SINRs of the FD CF MIMO system are, respectively,

$\displaystyle\eta_{\mathsf{u},k}(\boldsymbol{\kappa},\boldsymbol{\mathcal{E}}_{\mathsf{u}})=\frac{(N_{\mathsf{rx}}-\tau_{p})\mathcal{E}_{\mathsf{u},k}(\sum\nolimits_{m=1}^{M}\omega_{mk}^{*}\alpha_{\mathsf{u},mk}^{2})^{2}}{\begin{pmatrix}{\tt
EST}_{\mathsf{u},k}+{\tt MUI}_{\mathsf{u},k}\\\ +{\tt
IAP}_{k}+N_{0}\sum\nolimits_{m=1}^{M}|\omega_{mk}^{*}|^{2}\alpha_{\mathsf{u},mk}^{2}\end{pmatrix}},$
(31)

and

$\displaystyle\eta_{\mathsf{d},n}(\boldsymbol{\kappa},\boldsymbol{\mathcal{E}}_{\mathsf{u}})=\frac{(N_{\mathsf{tx}}-\tau_{p})(\sum\nolimits_{j=1}^{M}\sqrt{\mathcal{E}_{\mathsf{d}}}\alpha_{\mathsf{d},jn}\kappa_{jn})^{2}}{{\tt
EST}_{\mathsf{d},n}+{\tt MUI}_{\mathsf{d},n}+{\tt IUE}_{n}+N_{0}},$ (32)

where ${\tt EST}_{\mathsf{u},k}$, ${\tt MUI}_{\mathsf{u},k}$ and ${\tt
IAP}_{k}$ correspond to interferences caused by the channel estimation error,
multi-UE interference, and inter-AP interference. For DL, ${\tt
EST}_{\mathsf{d},n}$, ${\tt MUI}^{d}_{n}$ and ${\tt IUE}_{n}$ represent error
due to channel estimation, the DL multi-UE interference and the UL UE to DL UE
CLI, respectively. These terms can be evaluated as

$\displaystyle{\tt
EST}_{\mathsf{u},k}=\sum\nolimits_{k^{\prime}\in\mathcal{U}_{\mathsf{u}}}\mathcal{E}_{\mathsf{u},k^{\prime}}\sum\nolimits_{m=1}^{M}|\omega_{mk}^{*}|^{2}\alpha_{\mathsf{u},mk}^{2}(\beta_{\mathsf{u},mk^{\prime}}-\alpha_{\mathsf{u},mk^{\prime}}^{2}),$
$\displaystyle{\tt
MUI}_{\mathsf{u},k}=(N_{\mathsf{rx}}-\tau_{p})\sum\nolimits_{i\in\mathcal{P}_{l(k)}\backslash
k}\mathcal{E}_{\mathsf{u},i}\Big{(}\sum\nolimits_{m=1}^{M}\omega_{mk}^{*}\alpha_{\mathsf{u},mi}^{2}\Big{)}^{2},$
$\displaystyle{\tt
IAP}_{k}=N_{\mathsf{tx}}\mathcal{E}_{\mathsf{d}}\sum\limits_{n\in\mathcal{U}_{\mathsf{d}}}\sum\limits_{m=1}^{M}\Big{(}\sum\limits_{\begin{subarray}{c}j=1,j\neq
m\end{subarray}}^{M}\kappa_{jn}^{2}\zeta_{mj}^{\mathsf{InAP}}|\omega_{mk}^{*}|^{2}\alpha_{\mathsf{u},mk}^{2}$
$\displaystyle\hskip
48.36958pt+\kappa_{mn}^{2}|\omega_{mk}^{*}|^{2}\alpha_{\mathsf{u},mk}^{2}\zeta_{mm}^{\mathsf{SI}}\Big{)},$
$\displaystyle{\tt
EST}_{\mathsf{d},n}=\sum\nolimits_{q\in\mathcal{U}_{\mathsf{d}}}\sum\nolimits_{j=1}^{M}\mathcal{E}_{\mathsf{d}}\kappa_{jq}^{2}(\beta_{\mathsf{d},jn}-\alpha_{\mathsf{d},jn}^{2}),$
$\displaystyle{\tt
MUI}^{d}_{n}=(N_{\mathsf{tx}}-\tau_{p})\mathcal{E}_{\mathsf{d}}\sum\nolimits_{q\in\mathcal{P}_{l(n)}\backslash
n}\Big{(}\sum\nolimits_{j=1}^{M}\kappa_{jq}\alpha_{\mathsf{d},jn}\Big{)}^{2},$
$\displaystyle{\tt
IUE}_{n}=\sum\nolimits_{{k\in\mathcal{U}_{\mathsf{u}}}}\mathcal{E}_{\mathsf{u},k}\epsilon_{nk}.$

Thus, the sum UL-DL SE of the FD enabled CF-system can be expressed as
$\mathcal{R}_{\mathsf{s}}(\boldsymbol{\kappa},\boldsymbol{\mathcal{E}}_{\mathsf{u}})=\frac{\tau-\tau_{p}}{\tau}[\sum\nolimits_{k\in\mathcal{U}_{\mathsf{u}}}\log(1+\eta_{\mathsf{u},k}(\boldsymbol{\kappa},\boldsymbol{\mathcal{E}}_{\mathsf{u}}))+\sum\nolimits_{n\in\mathcal{U}_{\mathsf{d}}}\log(1+\eta_{\mathsf{d},n}(\boldsymbol{\kappa},\boldsymbol{\mathcal{E}}_{\mathsf{u}}))]$.

From the expression for ${\tt IAP}_{k}$, we see that the DL signals from _all_
the APs interfere with the UL signals of any AP, unlike DTDD. On the other
hand, there is no need for scheduling APs in UL/DL in the FD system, and the
power allocation algorithms presented in Sec. V-3 and Sec. V-2 can be directly
applied. We omit the details for brevity. Further, our SE expressions match
with those in the FD literature under special cases such as perfect channel
estimation [9] and equal combining at the CPU [9, 8].

##  VII Numerical Results

We consider that the UEs are dropped uniformly at random locations in a $1$
km2 square area. We take $5,000$ random channel instantiations for Monte Carlo
averaging. We consider $50\%$ of the UEs to have UL data demands. The APs are
deployed on a uniform rectangular grid for better coverage. The large scale
fading between the $m$th AP and the $k$th UE is modeled as
$\beta_{mk}=10^{\frac{\mathsf{PL}_{mk}+\sigma_{\mathsf{sh.}}z_{mk}}{10}}$,
where the path-loss $\mathsf{PL}_{mk}$ follows the three-slope model in [9],
$\sigma_{\mathsf{sh.}}=6$ dB, and $z_{mk}\sim\mathcal{N}(0,1)$. The system
bandwidth and noise figure are taken as $20$ MHz and $9$ dB, respectively,
which gives a noise variance of $-92$ dBm. The coherence interval consists of
$200$ channel uses [5]. The pilot SNR is taken as $20$ dB. We set the
algorithm parameters $\delta_{\mathsf{u}}$, $\delta_{\mathsf{d}}$,
$\delta_{\mathsf{ADMM}}$ and $\delta_{\mathsf{p}}$ to $0.001$. Other
parameters, such as the number of APs, UEs, UL/DL data transmit powers, etc.,
are mentioned in the plots.

In Fig. 4, we compare the cumulative distribution function (CDF) of the
achievable UL SE with different power allocation schemes. We observe that FP-
based power control added with weighted combing at the CPU (see
$\mathsf{WC+FP}$) uniformly outperforms only FP-based power control (see
$\mathsf{FP}$ without $\mathsf{WC}$) and only weighted combing at the CPU (see
$\mathsf{WC}$). This underlines the need for weighted combing along with UL
power control rather than applying each individually. Further, we compare the
proposed $\mathsf{WC+FP}$ scheme with the estimated channel variance and
large-scale fading-dependent power control scheme proposed by Nikbakht et al.
in [27, 28], and we observe almost $4$-fold improvement in $90\%$-likely UL SE
rendered by our algorithm. Also, we verify the correctness of our derived
closed form expression for weighted combing given in Lemma 4 (see
$\mathsf{WC:}$ Lemma $5$) with that of Lemma 3 (see $\mathsf{WC:}$ Lemma $5$),
and see that SE achieved by derived weights matches the theoretical SE.

Figure 4: UL sum SE under the proposed power control algorithm and comparison
with existing approaches [27, 28]. Optimal weighting at the CPU along with FP-
based power control yields the best performance. Figure 5: DL sum SE under the
proposed power control algorithm and comparison with an existing approach
[29]. This figure illustrates the improvement in DL sum SE that can be
attained via our algorithm.

In Fig. 5, we compare the CDFs of the achievable DL SE with our proposed FP
and ADMM-based algorithm with equal power allocation and the scalable DL power
allocation algorithm proposed by Interdonato et al. in [29]. We observe almost
$43\%$ improvement in the $90\%$ sum DL SE attained by our proposed algorithm
compared to the method in [29]. Further, compared to the equal power
allocation scheme for DL outlined by Vu et al. in [9], Algorithm 5 procures
almost $5$-fold improvement in the SE. Also, as we increase the maximum DL
transmit power budget per AP, the DL SE uniformly improves for all schemes.

We now compare the performances of DTDD and FD CF MIMO. In particular, we
assume the InAI and IrAI are well suppressed, at $-40$ dB. Figure 6(a)
compares the sum SEs under the two duplexing schemes considering different AP
and antenna densities. We observe that an FD system with
$(N_{\mathsf{tx}}=N_{\mathsf{rx}}=N,M=64)$ offers only $6\%$ improvement in
the $90\%$-likely sum SE compared to a DTDD enabled CF-system with $N$
antennas per $64$ APs. However, the former system has double the antenna
density compared to the DTDD CF system. If we consider the same antenna
density in the two systems, then the $90\%$-likely sum SE of the DTDD CF
system is $21\%$ more than that of the FD system (see
$(N_{\mathsf{tx}}=N_{\mathsf{rx}}=N/2,M=64)$). This is because, in DTDD, the
APs are scheduled based on the local UL/DL load in its vicinity, and hence, if
there is more UL load near to one or a set of APs, those APs are scheduled in
UL, which in turn leads to a beamforming gain in UL that scales with $N$. On
the other hand, in the FD system, the beamforming gain scales with $N/2$.
Recall that although in the FD system, all the APs are FD enabled, APs far
away from the UEs contribute minimally to the overall sum SE. Thus, scheduling
APs based on the localized traffic load is more beneficial.

In Fig. 6(b), we illustrate the effect of IrAI on the performance of the FD CF
system and contrast it with the DTDD CF system. When the IrAI is $-20$ dB, FD
uniformly outperforms the DTDD system even when the FD system has double the
antenna density. However, as the IrAI strength increases, the sum SE of FD
starts to deteriorate, achieving a $40\%$ lower sum SE than DTDD when IrAI is
$10$ dB. Thus, the performance of FD is highly dependent on the level of IrAI
suppression, while DTDD completely obviates the need for IrAI suppression and
offers similar performance to that of an FD system having double the antenna
density and low IrAI.

In Fig. 6(c), we further inspect the variation of the sum SE over a wide range
of IrAI for the FD system and compare the performance to the DTDD system. Even
with double antenna density, the FD system can perform very poorly when IrAI
strength becomes more pronounced (see the shaded region).

(a) Performance comparison of DTDD and FD systems with various antenna and AP
densities. We consider $K=40$. Each HD AP is equipped with $N=8$ antennas.
InAI and IrAI strengths are taken as $-40$ dB.

(b) Effect of intra-AP interference (IrAI) on the performance of the FD system
while InAI is maintained the same for both DTDD and FD systems.

(c) Sum UL-DL SE as a function of IrAI. DTDD can outperform FD even though the
latter has double the antenna density

Figure 6: Performance of DTDD compared to FD system for different antenna and
AP densities and IrAI strengths.

Finally, in Fig. 7, we illustrate the effect of InAI on the sum SE. An FD
system with $(M=64,N_{\mathsf{tx}}=N_{\mathsf{rx}}=N=8)$ with IrAI $-20$ dB
outperforms DTDD with $(M=64,N=8)$ (i.e., half the antenna density compared to
the FD system) when the InAI is no more than $\approx 11$ dB above the noise
floor. However, beyond an InAI of $11$ dB, the sum SE of the FD system
degrades compared to the DTDD system. This is because, in the FD system, all
APs cause InAI, while in DTDD, only the DL-scheduled APs cause inter-AP
interference.

Figure 7: Effects of inter-AP interference (InAI) on the sum UL-DL SE. We
observe that DTDD is more resilient to InAI.

##  VIII Conclusion

This paper presented a comparative study of DTDD and FD in CF systems. First,
we developed a novel graph coloring-based pilot allocation algorithm that
ensures no contamination in the received signals at the APs in close vicinity
to every UE while minimizing the required pilot length. Then, we optimized the
UL and DL power allocation and AP scheduling for DTDD to maximize the sum UL-
DL SE. We solved this NP-hard and non-convex problem by decoupling it into AP-
scheduling, UL, and DL power allocation sub-problems. We developed FP-based
UL/DL power allocation algorithms and proved the convergence of the sub-
problems to local optima. Further, we provided closed-form update equations
using the Lagrange dual transform and ADMM for the sub-problems, making them
easy to implement. We numerically illustrated the superiority of our proposed
algorithms over existing methods. Finally, we saw that DTDD outperforms FD
when the two systems have a similar antenna density. This happens because DTDD
can schedule the APs in UL or DL based on the localized traffic load and
achieve better array gain for a given antenna density and InAI suppression.
Thus, we conclude that although both DTDD and FD enable the CF system to serve
UL and DL UEs concurrently, DTDD is preferable because it can meet and even
outperform FD without requiring the use of IrAI cancellation hardware.

##  Appendix A Comments on ZF Precoding

For ZF precoding we use a scaling of
$\frac{1}{\sqrt{\mathbb{E}\left[\left\|\mathbf{Z}_{\mathsf{d},j}\left(\mathbf{Z}_{\mathsf{d},j}^{H}\mathbf{Z}_{\mathsf{d},j}\right)^{-1}\mathbf{e}_{l(n)}\right\|^{2}\right]}}$
to normalize the precoding vector. Here, we show that
$\mathbb{E}\left[\left\|\mathbf{Z}_{\mathsf{d},j}(\mathbf{Z}_{\mathsf{d},j}^{H}\mathbf{Z}_{\mathsf{d},j})^{-1}\mathbf{e}_{l(n)}\right\|^{2}\right]=\frac{c_{jn}}{N-\tau_{p}}$.

$\displaystyle\mathbb{E}\left[\left\|\mathbf{Z}_{\mathsf{d},j}(\mathbf{Z}_{\mathsf{d},j}^{H}\mathbf{Z}_{\mathsf{d},j})^{-1}\mathbf{e}_{l(n)}\right\|^{2}\right]$
$\displaystyle=\mathbb{E}\left[\left[(\mathbf{Z}_{\mathsf{d},j}^{H}\mathbf{Z}_{\mathsf{d},j})^{-1}\right]_{l(n)}\right]$
$\displaystyle{(a)\atop=}\frac{1}{(\tau_{p}\sum\nolimits_{n\in\mathcal{P}_{l(n)}}\mathcal{E}_{p,n}\beta_{jn}+N_{0})}\mathbb{E}\left[\left[(\mathbf{\dot{Z}}_{\mathsf{d},j}^{H}\mathbf{\dot{Z}}_{\mathsf{d},j})^{-1}\right]_{l(n)}\right]$
$\displaystyle{(b)\atop=}\frac{c_{jn}}{\tau_{p}}\mathbb{E}\left[{\tt
tr}\left[(\mathbf{\dot{Z}}_{\mathsf{d},j}^{H}\mathbf{\dot{Z}}_{\mathsf{d},j})^{-1}\right]\right]{(c)\atop=}\frac{c_{jn}}{N-\tau_{p}},$
(34)

where, use the following relations:

  * •

$(a):$
$\mathbf{Z}_{\mathsf{d},j}\mathbf{e}_{l(n)}=\sqrt{(\tau_{p}\sum\limits_{n\in\mathcal{P}_{l(n)}}\mathcal{E}_{p,n}\beta_{jn}+N_{0})}[\mathbf{\dot{Z}}_{\mathsf{d},j}\mathbf{e}_{l(n)}]$,
where $\left[\mathbf{\dot{Z}}_{\mathsf{d},j}\mathbf{e}_{l(n)}\right]$ has
i.i.d. $\mathcal{CN}(0,1)$ entries. Also,
$\left[(\mathbf{Z}_{\mathsf{d},j}^{H}\mathbf{Z}_{\mathsf{d},j})^{-1}\right]_{l(n)}$
denotes the $(l(n),l(n))$th diagonal entry of
$\left[(\mathbf{Z}_{\mathsf{d},j}^{H}\mathbf{Z}_{\mathsf{d},j})^{-1}\right]$.

  * •

$(b):$ Uses the definition
$c_{jn}=(\tau_{p}\sum\nolimits_{{\color[rgb]{0,0,0}k}\in\mathcal{P}_{l(n)}}\mathcal{E}_{p,{\color[rgb]{0,0,0}k}}\beta_{j{\color[rgb]{0,0,0}k}}+N_{0})^{-1}$
as per Lemma 1 and the fact that all the diagonal entries of
$\left[(\mathbf{\dot{Z}}_{\mathsf{d},j}^{H}\mathbf{\dot{Z}}_{\mathsf{d},j})^{-1}\right]$
have the same expected value.

  * •

$(c):$ Uses the fact that $\mathbb{E}\left[{\tt
tr}(\mathbf{W}^{-1})\right]=\frac{m}{n-m}$, where
$\mathbf{W}\in\mathbb{C}^{m\times m}$ is a complex central Wishart matrix with
$n$ degrees of freedom.

This completes the proof.

##  Appendix B Optimal Weights: Proof of Lemma 4

###### Proof.

The $m$th term of $\mathbb{E}\left[\mathbf{u}_{kk}\right]$ in (5) is
$\mathbb{E}\left[\mathbf{v}_{mk}^{H}\mathbf{f}_{mk}\right]$, which with ZF
combing becomes

$\displaystyle\gamma_{mk}\mathbb{E}\left[\mathbf{e}_{l(k)}^{T}\left(\mathbf{Z}_{\mathsf{u},m}^{H}\mathbf{Z}_{\mathsf{u},m}\right)^{-1}\mathbf{Z}_{\mathsf{u},m}^{H}\mathbf{f}_{mk}\right]$
$\displaystyle{(a)\atop=}c_{mk}\gamma_{mk}^{2}\mathbb{E}\left[\mathbf{e}_{l(k)}^{T}\left(\mathbf{Z}_{\mathsf{u},m}^{H}\mathbf{Z}_{\mathsf{u},m}\right)^{-1}\mathbf{Z}_{\mathsf{u},m}^{H}\mathbf{Z}_{\mathsf{u},m}\mathbf{e}_{l(k)}\right]$
$\displaystyle+\gamma_{mk}\mathbb{E}\left[\mathbf{e}_{l(k)}^{T}\left(\mathbf{Z}_{\mathsf{u},m}^{H}\mathbf{Z}_{\mathsf{u},m}\right)^{-1}\mathbf{Z}_{\mathsf{u},m}^{H}\mathbf{\tilde{f}}_{mk}\right]=\alpha_{mk}^{2},$

where in $(a)$ we used the fact that
$\mathbf{f}_{mk}=\hat{\mathbf{f}}_{mk}+\tilde{\mathbf{f}}_{mk}$, and
substituted for $\hat{\mathbf{f}}_{mk}$. The final step follows as
$c_{mk}\gamma_{mk}^{2}=\tau_{p}\mathcal{E}_{p,k}\beta_{mk}^{2}c_{mk}=\alpha_{mk}^{2}$,
and the second term in $(a)$ goes to zero as the combiner vector and the
estimation error are independent and each has zero mean. Next, we evaluate
$\mathbb{E}[\mathbf{u}_{ki}\mathbf{u}_{ki}^{H}]\in\mathbb{C}^{\lvert\mathcal{A}_{\mathsf{u}}\rvert\times\lvert\mathcal{A}_{\mathsf{u}}\rvert}$
in closed form. Let us first consider the UEs that share the same pilot
($\boldsymbol{\varphi}_{l(k)}$) as the $k$th UE, i.e., the UEs indexed by
$i\in\mathcal{P}_{l(k)},i\neq k$. In this case, the $m$th diagonal entry of
$\mathbb{E}[\mathbf{u}_{ki}\mathbf{u}_{ki}^{H}]$ is
$\mathbb{E}\left[\mathbf{v}_{mk}^{H}\mathbf{f}_{mi}\mathbf{f}_{mi}^{H}\mathbf{v}_{mk}\right]$,
which can be expanded as

$\displaystyle\mathbb{E}\left[\mathbf{v}_{mk}^{H}\hat{\mathbf{f}}_{mi}\hat{\mathbf{f}}_{mi}^{H}\mathbf{v}_{mk}\right]+\mathbb{E}\left[\mathbf{v}_{mk}^{H}\tilde{\mathbf{f}}_{mi}\tilde{\mathbf{f}}_{mi}^{H}\mathbf{v}_{mk}\right]$
$\displaystyle{(b)\atop=}\alpha^{4}_{mk}+\mathbb{E}\left[\mathbf{v}_{mk}^{H}\mathbb{E}\left[\tilde{\mathbf{f}}_{mi}\tilde{\mathbf{f}}_{mi}^{H}\right]\mathbf{v}_{mk}\right],$
(35)

where, in $(b)$, we use the fact that for ZF combining

$\displaystyle\mathbb{E}\left[\mathbf{v}_{mk}^{H}\hat{\mathbf{f}}_{mi}\hat{\mathbf{f}}_{mi}^{H}\mathbf{v}_{mk}\right]=\begin{cases}0,~{}\text{if}~{}i\notin\mathcal{P}_{l(k)}\\\
\alpha^{4}_{mk},~{}\text{if}~{}i\in\mathcal{P}_{l(k)}\end{cases},$ (36)

which is easy to verify. Now, since
$\tilde{\mathbf{f}}_{mi}\sim\mathcal{CN}(\mathbf{0},(\beta_{mi}-\alpha_{mi}^{2})\mathbf{I}_{N})$,
we can show that

$\mathbb{E}\left[\mathbf{v}_{mk}^{H}\mathbf{v}_{mk}\right]=\gamma_{mk}^{2}\mathbb{E}\left[\left[\left(\mathbf{Z}_{\mathsf{u},m}^{H}\mathbf{Z}_{\mathsf{u},m}\right)^{-1}\right]_{l(k)}\right]=\frac{\alpha_{mk}^{2}}{N-\tau_{p}}.$

Thus, the $m$th diagonal entry becomes

$\displaystyle\left[\mathbb{E}\left[\mathbf{u}_{ki}\mathbf{u}_{ki}\right]\right]_{m}=\alpha_{mk}^{4}+\frac{1}{N-\tau_{p}}\alpha_{mk}^{2}(\beta_{mi}-\alpha_{mi}^{2}).$
(37)

Now, the $(m,n)$th element of
$\mathbb{E}\left[\mathbf{u}_{ki}\mathbf{u}_{ki}^{H}\right]$ can be calculated
as

$\displaystyle\left[\mathbb{E}\left[\mathbf{u}_{ki}\mathbf{u}_{ki}^{H}\right]\right]_{m,n}=\mathbb{E}\left[\mathbf{v}_{mk}^{H}\hat{\mathbf{f}}_{mi}\hat{\mathbf{f}}_{ni}^{H}\mathbf{v}_{nk}\right]+\mathbb{E}\left[\mathbf{v}_{mk}^{H}\tilde{\mathbf{f}}_{mi}\tilde{\mathbf{f}}_{ni}^{H}\mathbf{v}_{nk}\right]$
$\displaystyle=\mathbb{E}\left[\mathbf{v}_{mk}^{H}\hat{\mathbf{f}}_{mi}\right]\mathbb{E}\left[\hat{\mathbf{f}}_{ni}^{H}\mathbf{v}_{nk}\right]+\mathbb{E}\left[\mathbf{v}_{mk}^{H}\mathbb{E}\left[\tilde{\mathbf{f}}_{mi}\tilde{\mathbf{f}}_{ni}^{H}\right]\mathbf{v}_{nk}\right]$
$\displaystyle=\alpha_{mk}^{2}\alpha_{ni}^{2}.$ (38)

Next, for the UEs that do not share the same pilot as the $k$th UE, it is easy
to see from (B) that the off-diagonal entries of
$\left[\mathbb{E}\left[\mathbf{u}_{ki}\mathbf{u}_{ki}^{H}\right]\right]$ are
zero. The $m$th diagonal entry is

$\displaystyle\mathbb{E}\left[\mathbf{v}_{mk}^{H}\mathbf{f}_{mi}\mathbf{f}_{mi}^{H}\mathbf{v}_{mk}\right]$
$\displaystyle=\mathbb{E}\left[\mathbf{v}_{mk}^{H}\mathbb{E}\left[\tilde{\mathbf{f}}_{mi}\tilde{\mathbf{f}}_{mi}^{H}\right]\mathbf{v}_{mk}\right]$
$\displaystyle=\frac{\alpha_{mk}^{2}(\beta_{mi}-\alpha_{mi}^{2})}{N-\tau_{p}}.$
(39)

We can next evaluate the interference caused by the DL APs. The $m$th diagonal
entry of $\mathbb{E}\left[\mathbf{a}_{kn}\mathbf{a}_{kn}^{H}\right]$ can be
written as
$\sum\nolimits_{j\in\mathcal{A}_{\mathsf{d}}}\mathcal{E}_{\mathsf{d}}\kappa_{jn}^{2}\mathbb{E}\left[\left|\mathbf{v}_{mk}^{H}\tilde{\mathbf{G}}_{mj}\mathbf{p}_{jn}\right|^{2}\right]$,
where the expectation is

$\displaystyle\mathbb{E}\left[\left|\mathbf{v}_{mk}^{H}\tilde{\mathbf{G}}_{mj}\mathbf{p}_{jn}\right|^{2}\right]=\mathbb{E}\left[\mathbf{v}_{mk}^{H}\mathbb{E}\left[\tilde{\mathbf{G}}_{mj}\mathbf{p}_{jn}\mathbf{p}_{jn}^{H}\tilde{\mathbf{G}}_{mj}^{H}\right]\mathbf{v}_{mk}\right]$
$\displaystyle{=}N\zeta_{mj}^{\mathsf{InAP}}\mathbb{E}\left[\|\mathbf{v}_{mk}\|^{2}\right]=\frac{N\zeta_{mj}^{\mathsf{InAP}}\alpha_{mk}^{2}}{N-\tau_{p}}.$
(40)

Now, it is easy to show that due to the independence of the channel vectors
involved, the off-diagonal entries of
$\mathbb{E}\left[\mathbf{d}_{kn}\mathbf{d}_{kn}^{H}\right]$ are zero. Finally,
$\mathbf{N}_{\mathsf{eff.}}=N_{0}{\tt
diag}\left(\frac{\alpha_{1k}^{2}}{N-\tau_{p}},\ldots,\frac{\alpha_{|\mathcal{A}_{\mathsf{u}}|k}^{2}}{N-\tau_{p}}\right).$
This completes the steps in the derivation of
$\boldsymbol{\omega}_{k}^{\mathsf{opt.}}$ and $\mathbf{R}_{\omega,k}$. With a
little algebraic manipulation, we arrive at the expressions in the Lemma 4. ∎

##  Appendix C UL SE for DTDD: Proof of Lemma 5

###### Proof.

Here, we sketch the derivation of the UL SE with zero-forcing combining. To
this end, we first derive the following terms in closed form.

$\displaystyle\left\lvert\sum\nolimits_{m\in\mathcal{A}_{\mathsf{u}}}\omega_{mk}^{*}\mathbb{E}\left[\mathbf{v}_{mk}^{H}\mathbf{f}_{mk}\right]\right\rvert^{2}=\left\lvert\sum\nolimits_{m\in\mathcal{A}_{\mathsf{u}}}\omega_{mk}^{*}\alpha_{mk}^{2}\right\rvert^{2},$
(41)

which follows because

$\displaystyle\mathbb{E}\left[\mathbf{v}_{mk}^{H}\mathbf{f}_{mk}\right]=\mathbb{E}\left[\mathbf{v}_{mk}^{H}(\hat{\mathbf{f}}_{mk}+\tilde{\mathbf{f}}_{mk})\right]$
$\displaystyle=\gamma_{mk}\mathbb{E}\left[\mathbf{e}_{l(k)}^{T}\left(\mathbf{Z}_{\mathsf{u},m}^{H}\mathbf{Z}_{\mathsf{u},m}\right)^{-1}\mathbf{Z}_{\mathsf{u},m}^{H}\hat{\mathbf{f}}_{mk}\right]+\mathbb{E}\left[\mathbf{v}_{mk}^{H}\tilde{\mathbf{f}}_{mk}\right]$
$\displaystyle=\gamma_{mk}c_{mk}\sqrt{\tau_{p}\mathcal{E}_{p,k}}\beta_{mk}\mathbb{E}\left[\mathbf{e}_{l(k)}^{T}\left(\mathbf{Z}_{\mathsf{u},m}^{H}\mathbf{Z}_{\mathsf{u},m}\right)^{-1}\mathbf{Z}_{\mathsf{u},m}^{H}\mathbf{Z}_{\mathsf{u},m}\mathbf{e}_{l(k)}\right]$
$\displaystyle=\beta_{mk}^{2}c_{mk}\tau_{p}\mathcal{E}_{p,k}=\alpha_{mk}^{2}.$
(42)

Next, we evaluate
$\mathbb{E}\left[\left\lvert\sum\nolimits_{m\in\mathcal{A}_{\mathsf{u}}}\omega_{mk}^{*}\mathbf{v}_{mk}^{H}\mathbf{f}_{mk}\right\rvert^{2}\right]$.
Expanding the term, we can write,

$\displaystyle\mathbb{E}\left[\left\lvert\sum\nolimits_{m\in\mathcal{A}_{\mathsf{u}}}\omega_{mk}^{*}\mathbf{v}_{mk}^{H}\mathbf{f}_{mk}\right\rvert^{2}\right]$
$\displaystyle=\sum\limits_{m\in\mathcal{A}_{\mathsf{u}}}\sum\limits_{m^{\prime}\in\mathcal{A}_{\mathsf{u}}}\omega_{mk}^{*}\omega_{m^{\prime}k}^{*}\left(\mathbb{E}\left[\mathbf{v}_{mk}^{H}\hat{\mathbf{f}}_{mk}\hat{\mathbf{f}}_{m^{\prime}k}^{H}\mathbf{v}_{m^{\prime}k}\right]\right.$
$\displaystyle+\left.\mathbb{E}\left[\mathbf{v}_{mk}^{H}\tilde{\mathbf{f}}_{mk}\tilde{\mathbf{f}}_{m^{\prime}k}^{H}\mathbf{v}_{m^{\prime}k}\right]\right).$
(43)

It is easy to show that
$\mathbb{E}\left[\mathbf{v}_{mk}^{H}\hat{\mathbf{f}}_{mk}\hat{\mathbf{f}}_{m^{\prime}k}^{H}\mathbf{v}_{m^{\prime}k}\right]=\alpha_{mk}^{2}\alpha_{m^{\prime}k}^{2}$.
The term due to estimation error can be evaluated as

$\displaystyle\mathbb{E}\left[\mathbf{v}_{mk}^{H}\tilde{\mathbf{f}}_{mk}\tilde{\mathbf{f}}_{m^{\prime}k}^{H}\mathbf{v}_{m^{\prime}k}\right]=\gamma_{mk}\gamma_{m^{\prime}k}\mathbb{E}\left[\tilde{\mathbf{f}}_{mk}^{H}\mathbf{Z}_{\mathsf{u},m}\right.$
$\displaystyle\left.\left(\mathbf{Z}_{\mathsf{u},m}^{H}\mathbf{Z}_{\mathsf{u},m}\right)^{-1}\mathbf{e}_{l(k)}\mathbf{e}_{l(k)}^{T}\left(\mathbf{Z}_{\mathsf{u},m^{\prime}}^{H}\mathbf{Z}_{\mathsf{u},m^{\prime}}\right)^{-1}\mathbf{Z}_{\mathsf{u},m^{\prime}}^{H}\tilde{\mathbf{f}}_{m^{\prime}k}\right]$
$\displaystyle=\gamma_{mk}^{2}(\beta_{mk}-\alpha_{mk}^{2})\mathbb{E}\left[\left\|\mathbf{Z}_{\mathsf{u},m}\left(\mathbf{Z}_{\mathsf{u},m}^{H}\mathbf{Z}_{\mathsf{u},m}\right)^{-1}\mathbf{e}_{l(k)}\right\|^{2}\right]$
$\displaystyle=\dfrac{\gamma_{mk}^{2}(\beta_{mk}-\alpha_{mk}^{2})c_{mk}}{N-\tau_{p}}.$
(44)

Finally, substituting
$\gamma_{mk}=\sqrt{\tau_{p}\mathcal{E}_{p,k}}\beta_{mk}$, we can write
$\mathbb{E}\left[\mathbf{v}_{mk}^{H}\tilde{\mathbf{f}}_{mk}\tilde{\mathbf{f}}_{m^{\prime}k}^{H}\mathbf{v}_{m^{\prime}k}\right]=\dfrac{\alpha_{mk}^{2}(\beta_{mk}-\alpha_{mk}^{2})}{N-\tau_{p}}$.
Thus, using the derived results in (C), we obtain

$\displaystyle\mathbb{E}\left[\left\lvert\sum\nolimits_{m\in\mathcal{A}_{\mathsf{u}}}\omega_{mk}^{*}\mathbf{v}_{mk}^{H}\mathbf{f}_{mk}\right\rvert^{2}\right]$
$\displaystyle=\left(\sum\nolimits_{m\in\mathcal{A}_{\mathsf{u}}}\omega_{mk}^{*}\alpha_{mk}\right)^{2}$
$\displaystyle+\sum\nolimits_{m\in\mathcal{A}_{\mathsf{u}}}\lvert\omega_{mk}^{*}\rvert^{2}\dfrac{\alpha_{mk}^{2}(\beta_{mk}-\alpha_{mk}^{2})}{N-\tau_{p}}.$
(45)

Next, we consider the UE that reuses the same pilot signal as the $k$th UE.
Let the index be $i$, where $i\in\mathcal{P}_{l(k)}$. Then we have

$\displaystyle\mathbb{E}\left[\left\lvert\sum\nolimits_{m\in\mathcal{A}_{\mathsf{u}}}\omega_{mk}^{*}\mathbf{v}_{mk}^{H}\mathbf{f}_{mi}\right\rvert^{2}\right]$
$\displaystyle=\mathbb{E}\left[\left\lvert\sum\nolimits_{m\in\mathcal{A}_{\mathsf{u}}}\omega_{mk}^{*}\mathbf{v}_{mk}^{H}(\hat{\mathbf{f}}_{mi}+\tilde{\mathbf{f}}_{mi})\right\rvert^{2}\right]$
$\displaystyle=\left(\sum\limits_{m\in\mathcal{A}_{\mathsf{u}}}\omega_{mk}^{*}\alpha_{mi}\right)^{2}+\sum\limits_{m\in\mathcal{A}_{\mathsf{u}}}\lvert\omega_{mk}^{*}\rvert^{2}\dfrac{\alpha_{mk}^{2}(\beta_{mi}-\alpha_{mi}^{2})}{N-\tau_{p}}.$
(46)

Similarly, we can show that if $i^{\prime}\notin\mathcal{P}_{l(k)}$,

$\displaystyle\mathbb{E}\left[\left\lvert\sum\limits_{m\in\mathcal{A}_{\mathsf{u}}}\omega_{mk}^{*}\mathbf{v}_{mk}^{H}\mathbf{f}_{mi^{\prime}}\right\rvert^{2}\right]=\sum\limits_{m\in\mathcal{A}_{\mathsf{u}}}\lvert\omega_{mk}^{*}\rvert^{2}\dfrac{\alpha_{mk}^{2}(\beta_{mi^{\prime}}-\alpha_{mi^{\prime}}^{2})}{N-\tau_{p}},$
(47)

which follows from the fact that
$\mathbb{E}\left[\mathbf{v}_{mk}^{H}\hat{\mathbf{f}}_{mi^{\prime}}\right]=0$,
if $i^{\prime}\notin\mathcal{P}_{l(k)}$. Using the above results and the InAI
strength derived in (B), we can get the SINR expression as in Lemma 5. ∎

##  Appendix D DL SE for DTDD: Proof of Lemma 6

###### Proof.

Here, sketch the proof of the DL SE. First, we derive an expression for
beamforming gain:

$\displaystyle\left|\sum\limits_{j\in\mathcal{A}_{\mathsf{d}}}\kappa_{jn}\sqrt{{\mathcal{E}_{\mathsf{d}}}}\mathbf{f}_{jn}^{H}\mathbf{p}_{jn}\right|^{2}\mathbb{E}\left[\left|s_{\mathsf{d},n}\right|^{2}\right]$
$\displaystyle{(a)\atop{=}}\mathcal{E}_{\mathsf{d}}\left|\sum\limits_{j\in\mathcal{A}_{\mathsf{d}}}\kappa_{jn}\sqrt{\frac{N-\tau_{p}}{c_{jn}}}\hat{\mathbf{f}}_{jn}^{H}\mathbf{Z}_{\mathsf{d},j}\left(\mathbf{Z}_{\mathsf{d},j}^{H}\mathbf{Z}_{\mathsf{d},j}\right)^{-1}\mathbf{e}_{l(n)}\right|^{2}$
$\displaystyle{(b)\atop{=}}(N-\tau_{p})\mathcal{E}_{\mathsf{d}}\left|\sum\limits_{j\in\mathcal{A}_{\mathsf{d}}}\kappa_{jn}\frac{1}{\sqrt{c_{jn}}}c_{jn}\sqrt{\tau_{p}\mathcal{E}_{p,j}}\beta_{jn}\mathbf{e}_{l(n)}^{T}\mathbf{Z}_{\mathsf{d},j}^{H}\right.$
$\displaystyle\left.\quad\quad\quad\quad\quad\quad\times\mathbf{Z}_{\mathsf{d},j}\left(\mathbf{Z}_{\mathsf{d},j}^{H}\mathbf{Z}_{\mathsf{d},j}\right)^{-1}\mathbf{e}_{l(n)}\right|^{2}$
$\displaystyle=(N-\tau_{p})\mathcal{E}_{\mathsf{d}}\left(\sum\nolimits_{j\in\mathcal{A}_{\mathsf{d}}}\kappa_{jn}\alpha_{jn}\right)^{2},$
(48)

where in $(a)$ we substitute for the DL ZF precoder
$\mathbf{p}_{jn}={\mathbf{Z}_{\mathsf{d},j}\left(\mathbf{Z}_{\mathsf{d},j}^{H}\mathbf{Z}_{\mathsf{d},j}\right)^{-1}\mathbf{e}_{l(n)}}/{(\mathbb{E}[\|\mathbf{Z}_{\mathsf{d},j}\left(\mathbf{Z}_{\mathsf{d},j}^{H}\mathbf{Z}_{\mathsf{d},j}\right)^{-1}\mathbf{e}_{l(n)}\|^{2}])}$
with
$\mathbb{E}[\|\mathbf{Z}_{\mathsf{d},j}\left(\mathbf{Z}_{\mathsf{d},j}^{H}\mathbf{Z}_{\mathsf{d},j}\right)^{-1}\mathbf{e}_{l(n)}\|^{2}]$
being $\frac{c_{jn}}{N-\tau_{p}}$ as derived in Sec. A, and in $(b)$ we use
the fact that
$\hat{\mathbf{f}}_{jn}=c_{jn}\sqrt{\tau_{p}\mathcal{E}_{p,j}}\beta_{jn}\mathbf{Z}_{\mathsf{d},j}\mathbf{e}_{l(n)}$.

Next, we derive the multi-UE interference. First, observe that

$\mathcal{E}_{\mathsf{d}}{\tt
var}\big{\\{}\boldsymbol{\kappa}_{n}^{T}\mathbf{d}_{nn}\\}+\mathcal{E}_{\mathsf{d}}\sum\nolimits_{q\in\mathcal{U}_{\mathsf{d}}\backslash
n}\mathbb{E}[\lvert\boldsymbol{\kappa}_{q}^{T}\mathbf{d}_{nq}\rvert^{2}]\\\
=\mathcal{E}_{\mathsf{d}}\sum\nolimits_{q\in\mathcal{U}_{\mathsf{d}}}\mathbb{E}[\lvert\boldsymbol{\kappa}_{q}^{T}\mathbf{d}_{nq}\rvert^{2}]-\mathcal{E}_{\mathsf{d}}\left|\mathbb{E}\left[\boldsymbol{\kappa}_{n}^{T}\mathbf{d}_{nn}\right]\right|^{2}$
(49)

Using the uncorrelatedness of the channels across UEs, we can express the
multi-UE interference as shown in (D).

$\displaystyle\sum\nolimits_{\begin{subarray}{c}q\in\mathcal{U}_{\mathsf{d}}\end{subarray}}\left|\sum\nolimits_{j\in\mathcal{A}_{\mathsf{d}}}\kappa_{jq}\sqrt{{\mathcal{E}_{\mathsf{d}}}}\mathbf{f}_{jn}^{T}\mathbf{p}_{jq}\right|^{2}\mathbb{E}\left[\left|s_{\mathsf{d},q}\right|^{2}\right]$
$\displaystyle{(a)\atop{=}}\mathcal{E}_{\mathsf{d}}\sum\nolimits_{\begin{subarray}{c}q\in\mathcal{U}_{\mathsf{d}}\end{subarray}}\mathbb{E}\left|\sum\nolimits_{j\in\mathcal{A}_{\mathsf{d}}}\kappa_{jq}\left(\sqrt{N-\tau_{p}}\alpha_{jn}\left|\langle\boldsymbol{\varphi}_{l(n)},\boldsymbol{\varphi}_{l(q)}\rangle\right|+\tilde{\mathbf{f}}_{jn}^{T}\mathbf{p}_{jq}\right)\right|^{2}$
$\displaystyle=\mathcal{E}_{\mathsf{d}}\sum\nolimits_{{q\in\mathcal{U}_{\mathsf{d}}}}\sum\nolimits_{j\in\mathcal{A}_{\mathsf{d}}}\sum\nolimits_{j^{\prime}\in\mathcal{A}_{\mathsf{d}}}\kappa_{jq}\kappa_{j^{\prime}q}\left(\left(N-\tau_{p}\right)\alpha_{jn}\alpha_{j^{\prime}n}\left|\langle\boldsymbol{\varphi}_{l(n)},\boldsymbol{\varphi}_{l(q)}\rangle\right|^{2}+\mathbb{E}\left[\mathbf{p}_{jq}^{H}\tilde{\mathbf{f}}_{jn}^{*}\tilde{\mathbf{f}}_{jn}^{T}\mathbf{p}_{jq}\right]\right)$
$\displaystyle=\left(N-\tau_{p}\right)\mathcal{E}_{\mathsf{d}}\sum\nolimits_{{q\in\mathcal{U}_{\mathsf{d}}}}\left(\sum\nolimits_{j\in\mathcal{A}_{\mathsf{d}}}\kappa_{jq}\alpha_{jn}\right)^{2}\left|\langle\boldsymbol{\varphi}_{l(n)},\boldsymbol{\varphi}_{l(q)}\rangle\right|^{2}+\mathcal{E}_{\mathsf{d}}\sum\nolimits_{q\in\mathcal{U}_{\mathsf{d}}}\sum\nolimits_{j\in\mathcal{A}_{\mathsf{d}}}\kappa_{jq}\left(\beta_{jn}-\alpha_{jn}^{2}\right).$
(50)

Here, $(a)$ follows because

$\displaystyle\hat{\mathbf{f}}_{jn}^{H}\mathbf{p}_{jq}$
$\displaystyle=\begin{cases}\sqrt{N-\tau_{p}}\alpha_{jn},\quad\text{if}\quad
q\in\mathcal{P}_{l(n)}~{}\\\ 0,\quad\text{if}\quad
q\notin\mathcal{P}_{l(n)}\end{cases}$
$\displaystyle=\sqrt{N-\tau_{p}}\alpha_{jn}\left|\langle\boldsymbol{\varphi}_{l(n)},\boldsymbol{\varphi}_{l(q)}\rangle\right|.$
(51)

The result in (D) combined with (49) completes the proof. ∎

##  Appendix E Proof of Proposition 3

###### Proof.

First, we note that optimizing
$f\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}},\boldsymbol{\varpi}_{\mathsf{u}}\right)$
equals optimizing
$f\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}},\boldsymbol{\varpi}_{\mathsf{u}},\boldsymbol{\tilde{\varpi}}_{\mathsf{u}}\right)$,
where

$\displaystyle
f\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}},\boldsymbol{\varpi}_{\mathsf{u}},\boldsymbol{\tilde{\varpi}}_{\mathsf{u}}\right)$
$\displaystyle=\sum\limits_{k\in\mathcal{U}_{\mathsf{u}}}\ln\left(1+\varpi_{\mathsf{u},k}\right)-\sum\limits_{k\in\mathcal{U}_{\mathsf{u}}}\varpi_{\mathsf{u},k}+f(\boldsymbol{\mathcal{E}}_{\mathsf{u}},\boldsymbol{\tilde{\varpi}}_{\mathsf{u}}).$
(52)

This is due to the equivalence of
$\sum\nolimits_{k\in\mathcal{U}_{\mathsf{u}}}\frac{(1+\varpi_{\mathsf{u},k})\mathsf{G}_{\mathsf{u},k}\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}}\right)}{\mathsf{G}_{\mathsf{u},k}\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}}\right)+\mathsf{I}_{\mathsf{u},k}\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}}\right)}$
and
$f(\boldsymbol{\mathcal{E}}_{\mathsf{u}},\boldsymbol{\tilde{\varpi}}_{\mathsf{u}})$.
Next, we observe that

$\displaystyle f\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}}^{{\tt
iter}+1},\boldsymbol{\varpi}_{\mathsf{u}}^{{\tt
iter}+1},\boldsymbol{\tilde{\varpi}}_{\mathsf{u}}^{{\tt iter}+1}\right)$
$\displaystyle{(a)\atop\geq}f\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}}^{{\tt
iter}+1},\boldsymbol{\varpi}_{\mathsf{u}}^{{\tt
iter}+1},\boldsymbol{\tilde{\varpi}}_{\mathsf{u}}^{\tt iter}\right)$
$\displaystyle{(b)\atop\geq}f\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}}^{{\tt
iter}+1},\boldsymbol{\varpi}_{\mathsf{u}}^{{\tt
iter}},\boldsymbol{\tilde{\varpi}}_{\mathsf{u}}^{\tt iter}\right)$
$\displaystyle{(c)\atop\geq}f\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}}^{\tt
iter},\boldsymbol{\varpi}_{\mathsf{u}}^{\tt
iter},\boldsymbol{\tilde{\varpi}}_{\mathsf{u}}^{\tt iter}\right),$ (53)

where $(a)$, $(b)$ and $(c)$ follow because, at iteration indexed by ${\tt
iter}+1$, each of these variables optimally solves the equivalent _convex
surrogate_ $f(\cdot)$ while keeping the other two variables fixed. Since
$f\left(\boldsymbol{\mathcal{E}}_{\mathsf{u}},\boldsymbol{\varpi}_{\mathsf{u}},\boldsymbol{\tilde{\varpi}}_{\mathsf{u}}\right)$
has a finite upper bound, the algorithm is convergent. ∎

##  Appendix F On the convergence of the power allocation algorithms

In Fig. 8, we numerically validate the rate of convergence (we define this as
the difference in the iterates/cost function between two consecutive
iterations of the algorithms) of the iterates (i.e., $\boldsymbol{\kappa}$ and
$\boldsymbol{\mathcal{E}}_{u}$) as a function of the iteration count. We
evaluate the UL and DL power allocation algorithms with different
initializations of $\boldsymbol{\kappa}$ and
$\boldsymbol{\mathcal{E}}_{\mathsf{u}}$, and observe convergence within $6/7$
iterations while ensuring the gap between two consecutive iterates no more
than $0.001$ upon convergence. Specifically, Fig. 9 illustrates the
convergence claimed in Proposition 3. Similar results can be generated for DL
also, but we omit it to avoid repetition.

(a) Rate of convergence of UL power control coefficients

(b) Rate of convergence of the equivalent cost given in (V-2)

(c) Rate of convergence of DL power control coefficients

(d) Rate of convergence of the equivalent cost, i.e.,
$f(\boldsymbol{\kappa},\boldsymbol{\varpi}_{\mathsf{d}})$.

Figure 8: Convergence of UL and DL power allocation algorithms. Figure 9:
Convergence of the UL cost function for two different initializations. The
legends indicate the initial power allocation. For equal power allocation, the
algorithm allocates equal power to all the UL UEs as their initial values.

## References

  * [1] J. Bai and A. Sabharwal, “Asymptotic analysis of MIMO multi-cell full-duplex networks,” _IEEE Trans. Wireless Commun._ , vol. 16, no. 4, pp. 2168–2180, 2017.
  * [2] S. Fukue, H. Iimori, G. T. F. De Abreu, and K. Ishibashi, “Joint access configuration and beamforming for cell-free massive MIMO systems with dynamic TDD,” _IEEE Access_ , vol. 10, pp. 40 130–40 149, 2022.
  * [3] A. Chowdhury, R. Chopra, and C. R. Murthy, “Can dynamic TDD enabled half-duplex cell-free massive MIMO outperform full-duplex cellular massive MIMO?” _IEEE Trans. Commun._ , vol. 70, no. 7, pp. 4867–4883, 2022.
  * [4] H. Kim, J. Kim, and D. Hong, “Dynamic TDD systems for 5G and beyond: A survey of cross-link interference mitigation,” _IEEE Commun. Surveys & Tutorials_, vol. 22, no. 4, pp. 2315–2348, 2020.
  * [5] E. Björnson and L. Sanguinetti, “Making cell-free massive MIMO competitive with MMSE processing and centralized implementation,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 1, pp. 77–90, 2020.
  * [6] G. Interdonato, E. Björnson, H. Quoc Ngo, P. Frenger, and E. G. Larsson, “Ubiquitous cell-free massive mimo communications,” _EURASIP Journal on Wireless Communications and Networking_ , vol. 2019, no. 1, p. 197, 2019.
  * [7] Y. Hu, H. Ge, H. Wang, and D. Wang, “Spectral efficiency of network-assisted full-duplex for cell-free massive MIMO system under pilot contamination,” _IEEE Access_ , vol. 9, pp. 110 826–110 841, 2021.
  * [8] M. Mohammadi, T. T. Vu, B. Naderi Beni, H. Q. Ngo, and M. Matthaiou, “Virtually full-duplex cell-free massive MIMO with access point mode assignment,” in _Proc. IEEE SPAWC_ , 2022, pp. 1–5.
  * [9] T. T. Vu, D. T. Ngo, H. Q. Ngo, and T. Le-Ngoc, “Full-duplex cell-free massive MIMO,” in _Proc. IEEE ICC_ , 2019, pp. 1–6.