# UL-DL Duality for Cell-free Massive MIMO with Per-AP Power and Information
Constraints††thanks: L. Miretti and S. Stańczak are with the Network
Information Theory group, Technische Universität Berlin, Berlin 10587,
Germany, and the Department of Wireless Communications and Networks,
Fraunhofer Institute for Telecommunications Heinrich-Hertz-Institut HHI,
Berlin 10587, Germany (email: {miretti, slawomir.stanczak}@tu-berlin.de). R.
L. G. Cavalcante is with the Department of Wireless Communications and
Networks, Fraunhofer Institute for Telecommunications Heinrich-Hertz-Institut
HHI, Berlin 10587, Germany (email: renato.cavalcante@hhi.fraunhofer.de). E.
Björnson is with the Department of Computer Science, KTH Royal Institute of
Technology, SE-100 44 Stockholm, Sweden (email: emilbjo@kth.de).

Lorenzo Miretti, , Renato L. G. Cavalcante, , Emil Björnson, , Sławomir
Stańczak

###### Abstract

We derive a novel uplink-downlink duality principle for optimal joint
precoding design under per-transmitter power and information constraints in
fading channels. The information constraints model limited sharing of channel
state information and data bearing signals across the transmitters. The main
application is to cell-free networks, where each access point (AP) must
typically satisfy an individual power constraint and form its transmit signal
using limited cooperation capabilities. Our duality principle applies to
ergodic achievable rates given by the popular hardening bound, and it can be
interpreted as a nontrivial generalization of a previous result by Yu and Lan
for deterministic channels. This generalization allows us to study involved
information constraints going beyond the simple case of cluster-wise
centralized precoding covered by previous techniques. Specifically, we show
that the optimal joint precoders are, in general, given by an extension of the
recently developed team minimum mean-square error method. As a particular yet
practical example, we then solve the problem of optimal local precoding design
in user-centric cell-free massive MIMO networks subject to per-AP power
constraints.

###### Index Terms:

Duality, cell-free, massive MIMO, distributed precoding, team decision theory,
MMSE.

††publicationid: pubid:  This work has been submitted to the IEEE for possible
publication. Copyright may be transferred without notice, after which this
version may no longer be accessible.

##  I Introduction

Cell-free massive MIMO networks have attracted significant interest for their
potential in enhancing the performance of future generation mobile access
networks. The main focus is the evolution of known coordinated multi-point
concepts (CoMP) towards practically attractive access solutions that combine
the benefits of access point (AP) cooperation and ultra-dense deployments. To
this end, considerable research effort has been devoted to the development of
scalable and possibly user-centric system architectures and algorithms
covering, for instance, power control, pilot-based channel estimation, joint
processing such as precoding and combining, fronthaul overhead, network
topology, and initial access [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15, 16, 17, 18].

Against this background, in this study we address the open problem of optimal
joint downlink precoding design in cell-free massive MIMO networks, by
considering minimum quality-of-service requirements and by assuming that each
AP is subject to an individual power constraint and an individual information
constraint. The information constraints model limited AP cooperation
capabilities, which are motivated by the need for realizing scalable cell-free
architectures with reduced fronthaul and joint processing load. More
specifically, the information constraints model:

  * •

limited sharing of channel state information (CSI), covering, for instance,
the local CSI model as in the original version of the cell-free massive MIMO
paradigm [1]; and

  * •

limited sharing of data bearing signals, for instance, only within user-
centric cooperation clusters as in [7, 8].

Each AP must form its transmit signal as a function of the CSI and data
bearing signals specified by the constraints, and no additional information
exchange between the APs is allowed. This is in contrast to related works such
as [11], which covers iterative information exchange during precoding
computation. In the cell-free massive MIMO literature, the best performing
joint precoders are typically designed from joint uplink combiners, motivated
by a known uplink-downlink duality principle for fading channels [19, Ch. 4]
[20, Ch. 6]. However, optimal joint precoders are generally unknown owing to
the following two reasons. First, until very recently, optimal joint combiners
were not known except for the relatively simple case of full CSI sharing
within each cooperation cluster, an information constraint leading to so-
called centralized combining. Second, the known uplink-downlink duality
principle for fading channels holds for a looser and somewhat less practical
sum power constraint.

Addressing the first issue is known to be quite challenging. In essence,
depending on the information constraints, each AP may need to take combining
decisions that are robust not only to channel estimation errors, but also to
the possibly unknown combining decisions taken at the other APs. A novel
method for optimally addressing this issue, called the team minimum mean-
square error (MMSE) method, has been recently proposed in [21]. Together with
the available uplink-downlink duality principle for fading channels, this
method is used in [21, 22] to derive novel distributed precoders under various
relevant information constraints, including the aforementioned local CSI
model. However, in its current form, this method is provenly optimal under a
sum power constraint only, and hence it does not provide a solution to our
problem.

A partial solution to the second issue is given by the alternative uplink-
downlink duality principle under per-antenna power constraints developed in
[23], applied for instance in [24] in the context of CoMP. However, the method
in [23] applies to deterministic channels, i.e., to fixed channel fading
realizations, which imposes many limitations. A first limitation is that
optimal schemes typically involve solving relatively complex optimization
problems for each channel realization, which may be impractical in large
systems. Perhaps the most important limitation is that the technique in [23]
applies to centralized precoding only, i.e., it does not cover optimal
distributed precoding design under limited CSI sharing. This crucial point can
be informally explained as follows.

Consider the following classical feasibility problem over a $L\times K$ MISO
broadcast channel with per-antenna power constraints:

find $\displaystyle\bm{T}\in\mathbbmss{C}^{L\times K}$ (1) subject to
$\displaystyle(\forall k\in\\{1,\ldots,K\\})~{}r_{k}(\bm{H},\bm{T})\geq R,$
$\displaystyle\textstyle(\forall
l\in\\{1,\ldots,L\\})~{}\sum_{k=1}^{K}|[\bm{T}]_{l,k}|^{2}\leq P,$

where $r_{k}(\bm{H},\bm{T})=\log(1+\mathrm{SINR}_{k}(\bm{H},\bm{T}))$ denotes
the $k$th user’s instantaneous achievable rate for a fixed channel matrix
$\bm{H}$ and precoding matrix $\bm{T}$. A solution to this problem is given in
[23], and it takes the form of a properly regularized pseudo-inverse of
$\bm{H}$, i.e., a standard MMSE precoding matrix. Now, in order to enforce a
nontrivial information constraint on $\bm{T}$ going beyond the case of
cluster-wise centralized precoding, such as letting each entry of $\bm{T}$ to
depend on different channel estimates, the above problem needs to be radically
modified. One possible approach is to formulate the problem in a statistical
sense, by considering ergodic rate optimization over fading channels111Other
formulations may include the replacement of the average constraints with
almost sure inequalities, or deterministic worst-case approaches, i.e., to
optimize each entry of $\bm{T}$ such that the rate and power constraints are
satisfied for all possible values of the other entries of $\bm{T}$. However,
these formulations are typically intractable and may be too conservative.
Hence, they are not covered by both the current literature and this study.:

find $\displaystyle\mathbbm{T}\in\mathcal{T}$ (2) subject to
$\displaystyle(\forall
k\in\\{1,\ldots,K\\})~{}\mathsf{E}[r_{k}(\mathbbm{H},\mathbbm{T})]\geq R,$
$\displaystyle\textstyle(\forall
l\in\\{1,\ldots,L\\})~{}\sum_{k=1}^{K}\mathsf{E}\left[[\mathbbm{T}]_{l,k}\right]\leq
P,$

where $\mathbbm{T}$ and $\mathbbm{H}$ are random matrices, and where the
information constraint is encoded in $\mathcal{T}$ as a certain subset of the
space of $L\times K$ random matrices (see Section III for details). However,
the technique in [23] does not cover fading channels, i.e., it cannot solve
(2).

To address the above issues, in this study222A part of the results in this
study is presented in [25] without proof. This study extends [25] by providing
complete derivations, the expressions for optimal local and centralized
precoding, and additional details on the numerical implementation of the
proposed algorithms. we derive a novel uplink-downlink duality principle for
fading channels under per-AP power constraints. More precisely, we extend the
technique in [23] to cover a variation of (2) with
$\mathsf{E}[r_{k}(\mathbbm{H},\mathbbm{T})]$ replaced by a more tractable
lower bound where the expectation is rigorously moved inside the logarithm
following the so-called hardening bounding technique [26]. As discussed in
details throughout this study, our derivation significantly departs from [23],
mostly due to the challenges introduced while moving from classical
optimization problems over the (finite dimensional) space of deterministic
matrices, as in (1), to more involved functional optimization problems over
the (infinite dimensional) space of random matrices, as in (2). Furthermore,
building on the above result, we show that optimal joint precoders are given
by solutions to properly parametrized MMSE problems under per-AP information
constraints, i.e., by properly parametrized variations of team MMSE precoders
[21]. In summary, the main contribution of this study can be interpreted as a
nontrivial extension of the method in [23] to fading channels, and of the
method in [21] to per-AP power constraints. As a concrete application of our
findings, we then solve the previously open problem of optimal local precoding
design in cell-free massive MIMO networks under per-AP power constraints.
Moreover, we provide a potentially simpler variation of the known centralized
solution under per-AP power constraints where, in contrast to previous
techniques, its parameters are not optimized for each channel realization, but
based on relatively slowly varying channel statistics.

The paper is organized as follows. Section II and Section III provide the main
definitions, mathematical tools, and modeling assumptions. Section IV presents
and studies the main optimization problem using Lagrangian duality arguments.
Building on the obtained insights, Section V derives the proposed uplink-
downlink duality principle, which is then exploited in Section VI to
characterize the optimal solution structure. Simple applications of the main
results are illustrated in Section VII by means of numerical simulations.
Finally, Section VIII summarizes the main results, and outlines some
limitations and possible future directions.

##  II Mathematical preliminaries

###  II-A Notation and definitions

We denote by $\mathbbmss{R}_{+}$ and $\mathbbmss{R}_{++}$ the sets of,
respectively, nonnegative and positive reals. The Euclidean norm in
$\mathbbmss{C}^{K}$ is denoted by $\|\cdot\|$. Let
$(\Omega,\Sigma,\mathbb{P})$ be a probability space. We denote by
$\mathcal{H}^{K}$ the set of complex-valued random vectors, i.e., $K$-tuples
of $\Sigma$-measurable functions $\Omega\to\mathbbmss{C}$ satisfying
$(\forall\mathbbm{x}\in\mathcal{H}^{K})$
$\mathsf{E}[\|\mathbbm{x}\|^{2}]<\infty$. Together with the standard
operations of addition and real scalar multiplication, we recall that
$\mathcal{H}^{K}$ is a real vector space. Given a random variable
$X\in\mathcal{H}$, we denote by $\mathsf{E}[X]$ and $\mathsf{V}(X)$ its
expected value and variance, respectively. We use
$\mathbbm{x}-\mathbbm{y}-\mathbbm{z}$ to denote a Markov chain, i.e., to
denote that the random vectors $\mathbbm{x}$ and $\mathbbm{z}$ are
conditionally independent given another random vector $\mathbbm{y}$.
Inequalities involving vectors in $\mathbbmss{R}^{K}$ should be understood
coordinate-wise. The $k$th column of the $K$-dimensional identity matrix
$\bm{I}_{K}$ is denoted by $\bm{e}_{k}$.

###  II-B Lagrangian duality in general vector spaces

The following key result is frequently invoked throughout our study, and can
be found in [27].

######  Proposition 1.

Consider the functions $f:\mathcal{X}\to\mathbbmss{R}$ and
$\bm{g}:\mathcal{X}\to\mathbbmss{R}^{N}$, where $\mathcal{X}$ is a real vector
space, and the optimization problem

$\displaystyle\underset{X\in\mathcal{X}}{\text{minimize}}$ $\displaystyle
f(X)$ subject to $\displaystyle\bm{g}(X)\leq\bm{0}.$

Define the primal optimum
$p^{\star}:=\inf\\{f(X)~{}|~{}\bm{g}(X)\leq\bm{0},X\in\mathcal{X}\\}$, and the
dual optimum
$d^{\star}:=\sup\\{d(\bm{\lambda})~{}|~{}\bm{\lambda}\in\mathbbmss{R}^{N}_{+}\\}$,
where
$d(\bm{\lambda}):=\inf\\{f(X)+\bm{\lambda}^{\mathsf{T}}\bm{g}(X)~{}|~{}X\in\mathcal{X}\\}$
for $\bm{\lambda}\in\mathbbmss{R}_{+}^{N}$. Each of the following holds:

  1. (i)

$d^{\star}\leq p^{\star}$ (weak duality);

  2. (ii)

If $f$ and $\bm{g}$ are proper convex functions [27, pp. 39], and
$\\{X\in\mathcal{X}~{}|~{}\bm{g}(X)<\bm{0}\\}\neq\emptyset$ (Slater’s
condition), then $d^{\star}=p^{\star}$ holds (strong duality). Furthermore,
there exist Lagrangian multipliers
$\bm{\lambda}^{\star}\in\mathbbmss{R}_{+}^{N}$ such that
$d(\bm{\lambda}^{\star})=d^{\star}$.

###### Proof.

For statement (i), see [27, Theorem 2.6.1(iii)]. For statement (ii), see [27,
Theorem 2.9.3(ii)]. ∎

For $\mathcal{X}=\mathbbmss{R}^{K}$, the above proposition corresponds to the
familiar Lagrangian duality principle for optimization problems in Euclidean
spaces [28]. In this study, we exploit the more general result in Proposition
1 to address optimization problems in the real vector space of complex-valued
random vectors $\mathcal{X}=\mathcal{H}^{K}$.

##  III System model

###  III-A Downlink achievable rates

Consider the downlink of a cell-free wireless network composed of $L$ APs
indexed by $\mathcal{L}:=\\{1,\ldots,L\\}$, each of them equipped with $N$
antennas, and $K$ single-antenna UEs indexed by
$\mathcal{K}:=\\{1,\ldots,K\\}$. By assuming a standard synchronous and
frequency-flat channel model governed by an ergodic and stationary fading
process, and simple transmission techniques based on linear precoding and on
treating interference as noise, we focus on simultaneously achievable ergodic
rates in the classical Shannon sense, approximated by the popular _hardening_
inner bound [26]. In more detail, we define the downlink rates achieved by
each UE $k\in\mathcal{K}$ for a given precoding design as

$R_{k}^{\mathrm{DL}}(\mathbbm{T}):=\log\left(1+\mathrm{SINR}_{k}^{\mathrm{DL}}(\mathbbm{T})\right),$
(3)
$\mathrm{SINR}_{k}^{\mathrm{DL}}(\mathbbm{T}):=\dfrac{|\mathsf{E}[\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{t}_{k}]|^{2}}{\mathsf{V}(\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{t}_{k})+\sum_{j\neq
k}\mathsf{E}[|\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{t}_{j}|^{2}]+1},$ (4)

where $\mathbbm{h}_{k}\in\mathcal{H}^{NL}$ is a random channel vector modeling
the fading state between UE $k$ and all APs,
$\mathbbm{t}_{k}\in\mathcal{H}^{NL}$ is a joint precoding vector applied by
all APs to the coded and modulated data bearing signal for UE $k$, and
$\mathbbm{T}:=[\mathbbm{t}_{1},\ldots,\mathbbm{t}_{K}]\in\mathcal{H}^{NL\times
K}$ is the aggregate joint precoding matrix. We stress that precoders are
defined and denoted as random quantities, since they may adapt to random
fading realizations on the basis of the the available instantaneous CSI. This
aspect is treated in detail in the next sections.

###  III-B Per-AP power and information constraints

In practical cell-free wireless networks, each AP must typically satisfy an
individual power constraint. In addition, motivated by the need for realizing
scalable cell-free architectures, each AP is also typically subject to an
individual information constraint induced by limited data and instantaneous
CSI sharing, which impair its cooperation capabilities. In this work, the
above per-AP constraints are modelled as follows. Let
$[\mathbbm{t}_{1,k}^{\mathsf{T}},\ldots,\mathbbm{t}_{L,k}^{\mathsf{T}}]^{\mathsf{T}}:=\mathbbm{t}_{k}$,
where $\mathbbm{t}_{l,k}\in\mathcal{H}^{N}$ denotes the portion of the
precoder applied by AP $l\in\mathcal{L}$ to serve UE $k\in\mathcal{K}$. By
assuming unitary power data bearing signals, we consider the average power
constraints

$(\forall
l\in\mathcal{L})~{}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{l,k}\|^{2}]\leq
P_{l}<\infty.$ (5)

We choose average power constraints instead of perhaps more common
instantaneous power constraints mostly for tractability reasons. This is in
line with the related results on uplink-downlink duality for fading channels
under a sum power constraint [19]. However, we point out that average power
constraints may be in fact quite appropriate for modern wideband systems,
where power allocation over multiple subcarriers is standard practice.

For modeling impairments related to limited CSI sharing, we follow the
recently proposed approach in [21] and let

$(\forall
k\in\mathcal{K})~{}\mathbbm{t}_{k}\in\mathcal{T}_{k}:=\mathcal{H}_{1}^{N}\times\ldots\times\mathcal{H}_{L}^{N},$
(6)

where $\mathcal{H}_{l}^{N}\subseteq\mathcal{H}^{N}$ denotes the set of
$N$-tuples of $\Sigma_{l}$-measurable functions $\Omega\to\mathbbmss{C}$
satisfying $(\forall\mathbbm{x}\in\mathcal{H}_{l}^{N})$
$\mathsf{E}[\|\mathbbm{x}\|^{2}]<\infty$, and where
$\Sigma_{l}\subseteq\Sigma$ is the sub-$\sigma$-algebra induced by the
available CSI at AP $l\in\mathcal{L}$, also called the _information subfield_
of AP $l$ [29]. Informally, by letting $S_{l}$ be a given random variable
modeling the available CSI at AP $l\in\mathcal{L}$, this constraint enforces
the precoders $\\{\mathbbm{t}_{l,k}\\}_{k\in\mathcal{K}}$ of the $l$th AP to
be functions of $S_{l}$ only. Limited CSI sharing typically leads to the case
$S_{l}\neq S_{j}$ (and hence $\Sigma_{l}\neq\Sigma_{j}$) for some $j\neq l$.

######  Remark 1.

The constraint in (6) is fairly general. For example, it covers the case of
local CSI [1] (i.e., where each AP has information on only the channel between
the UEs and itself), but also more advanced cooperation structures exploiting
the peculiarities of efficient fronthauls such as in the so-called radio
stripes concept where the APs are daisy-chained [21]. More precisely, by
letting $\hat{\mathbbm{H}}_{l}$ denote the local estimate of the local channel
matrix from AP $l\in\mathcal{L}$ to all UEs, the following cases are modeled
and studied in detail using (6) in [21]:

  * •

(local CSI) $(\forall l\in\mathcal{L})~{}S_{l}=\hat{\mathbbm{H}}_{l}$;

  * •

(unidirectional CSI) $(\forall
l\in\mathcal{L})~{}S_{l}=(\hat{\mathbbm{H}}_{1},\ldots,\hat{\mathbbm{H}}_{l})$;

  * •

(centralized CSI) $(\forall
l\in\mathcal{L})~{}S_{l}=(\hat{\mathbbm{H}}_{1},\ldots,\hat{\mathbbm{H}}_{L})$.

In addition, as mentioned in [21], (6) may also cover more general cases
where, owing to fronthaul imperfections such as quantization or delay, the APs
can only obtain degraded versions of the local CSI estimates from other APs.
To keep our results general, in most of this study we do not specify the CSI
structure. However, Section VI discusses concrete applications of (6). More
precisely, it studies the cases of local CSI and centralized CSI in detail.

From a mathematical point of view, $\mathcal{H}_{l}^{N}$ is a subspace of the
real vector space $\mathcal{H}^{N}$, which in turn implies that the constraint
in $\eqref{eq:CSI}$ models limited CSI sharing by constraining
$\mathbbm{t}_{k}$ within a subspace $\mathcal{T}_{k}$ of the real vector space
$\mathcal{H}^{LN}$ [29]. Furthermore, following the well-known user-centric
network clustering approach [7, 8], we assume that each UE $k\in\mathcal{K}$
is only served by a subset $\mathcal{L}_{k}\subseteq\mathcal{L}$ of APs. As
shown in [22], this additional practical impairment can be straightforwardly
included in (6) by replacing $\mathcal{H}_{l}^{N}$ with the set
$\\{\bm{0}_{N}~{}(\mathrm{a.s.})\\}$ for each AP $l\notin\mathcal{L}_{k}$,
i.e., not serving UE $k$. Since $\\{\bm{0}_{N}~{}(\mathrm{a.s.})\\}$ is a
(trivial) subspace of $\mathcal{H}^{N}$, this replacement does not alter the
property of $\mathcal{T}_{k}$ being a real vector space (in fact, a subspace
of $\mathcal{H}^{NL}$). This key property will allow us to address the problem
of optimal joint precoding design under per-AP information constraints using
Lagrangian duality for real vector spaces (Proposition 1).

###  III-C Dual uplink rates under arbitrary noise powers

The main optimization approach developed in this work has a natural
interpretation in terms of a virtual dual uplink channel with arbitrary noise
power. More specifically, similarly to the chosen downlink model, we consider
virtual uplink ergodic rates given by the use-and-then-forget inner bound [26]

$R_{k}^{\mathrm{UL}}(\mathbbm{v}_{k},\bm{p},\bm{\sigma}):=\log\left(1+\mathrm{SINR}_{k}^{\mathrm{UL}}(\mathbbm{v}_{k},\bm{p},\bm{\sigma})\right),$
(7)
$\begin{split}&\mathrm{SINR}_{k}^{\mathrm{UL}}(\mathbbm{v}_{k},\bm{p},\bm{\sigma}):=\\\
&\dfrac{p_{k}|\mathsf{E}[\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{v}_{k}]|^{2}}{p_{k}\mathsf{V}(\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{v}_{k})+\sum_{j\neq
k}p_{j}\mathsf{E}[|\mathbbm{h}_{j}^{\mathsf{H}}\mathbbm{v}_{k}|^{2}]+\mathsf{E}[\|\mathbbm{v}_{k}\|_{\bm{\sigma}}^{2}]},\end{split}$
(8)

where
$\mathbbm{v}_{k}=[\mathbbm{v}_{1,k}^{\mathsf{T}},\ldots,\mathbbm{v}_{L,k}^{\mathsf{T}}]^{\mathsf{T}}\in\mathcal{H}^{NL}$
is a joint combiner,
$\bm{p}:=[p_{1},\ldots,p_{K}]^{\mathsf{T}}\in\mathbbmss{R}_{+}^{K}$ is a
vector of transmit powers, and where we define
$(\forall\mathbbm{v}_{k}\in\mathcal{H}^{NL})(\forall\bm{\sigma}\in\mathbbmss{R}_{++}^{L})$

$\mathsf{E}[\|\mathbbm{v}_{k}\|_{\bm{\sigma}}^{2}]:=\sum_{l=1}^{L}\sigma_{l}\mathsf{E}[\|\mathbbm{v}_{l,k}\|^{2}]$
(9)

for given
$\bm{\sigma}:=[\sigma_{1},\ldots,\sigma_{L}]^{\mathsf{T}}\in\mathbbmss{R}_{++}^{L}$.
In the above expressions, $\bm{\sigma}$ can be interpreted as a vector
collecting uplink noise powers for each AP. We remark that the term virtual
here refers to the fact that the above rates may not be achievable in the true
uplink channel, since $(\bm{p},\bm{\sigma})$ may differ from the true uplink
transmit and noise powers. The major difference between $R_{k}^{\mathrm{UL}}$
and $R_{k}^{\mathrm{DL}}$ is that the former depends only on the joint
combiner $\mathbbm{v}_{k}$ for the signal of UE $k\in\mathcal{K}$, while the
latter depends on the entire precoding matrix $\mathbbm{T}$. In fact, the
uplink achievable rates are coupled only via the transmit and noise powers
$\bm{p},\bm{\sigma}$. This known aspect makes optimization on the uplink
channel generally easier than on the downlink channel.

##  IV Problem statement and Lagrangian duality

To address the problem of optimal joint precoding design in cell-free
networks, in this section we study a certain optimization problem subject to
minimum SINR requirements and per-AP power and information constraints. In
particular, given a tuple of power constraints
$(P_{1},\ldots,P_{L})\in\mathbbmss{R}_{++}^{K}$ and of SINR requirements
$(\gamma_{1},\ldots,\gamma_{K})\in\mathbbmss{R}_{++}^{K}$, we consider the
following infinite dimensional optimization problem:

$\displaystyle\underset{\mathbbm{T}\in\mathcal{T}}{\text{minimize}}$
$\displaystyle\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|^{2}]$ (10) subject
to $\displaystyle(\forall
k\in\mathcal{K})~{}\mathrm{SINR}^{\mathrm{DL}}_{k}(\mathbbm{T})\geq\gamma_{k},$
$\displaystyle(\forall
l\in\mathcal{L})~{}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{l,k}\|^{2}]\leq
P_{l},$

where $\mathcal{T}\subset\mathcal{H}^{NL\times K}$ is a real vector space
obtained by collecting all per-AP information constraints defined in Section
III-B. We recall that these constraints accommodate both limited instantaneous
CSI sharing and user-centric network clustering. In the following, to avoid
technical digressions, we assume that strictly feasible joint precoders exist,
i.e.,

$\left\\{\mathbbm{T}\in\mathcal{T}\middle|\begin{array}[]{l}(\forall
k\in\mathcal{K})~{}\mathrm{SINR}^{\mathrm{DL}}_{k}(\mathbbm{T})>\gamma_{k}\\\
(\forall
l\in\mathcal{L})~{}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{l,k}\|^{2}]<P_{l}\end{array}\right\\}\neq\emptyset.$

Furthermore, due to both mathematical convenience and practical reasons, in
Problem (10) we focus on the subset of feasible joint precoders minimizing the
total power consumption.

###  IV-A Lagrangian dual problems

Inspired by the related results in [23] based on Lagrangian duality for finite
dimensional optimization problems, in this section we apply Lagrangian duality
for infinite dimensional optimization problems to study Problem (10). For
convenience, we adopt the compact notation

$\mathrm{SINR}_{k}^{\mathrm{DL}}(\mathbbm{T})=\dfrac{|b_{k}(\mathbbm{T})|^{2}}{|c_{k}(\mathbbm{T})|^{2}-|b_{k}(\mathbbm{T})|^{2}},$
(11)

where
$b_{k}(\mathbbm{T}):=\mathsf{E}[\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{t}_{k}]$
is the useful signal term, and
$|c_{k}(\mathbbm{T})|^{2}:=\sum_{j=1}^{K}\mathsf{E}[|\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{t}_{j}|^{2}]+1$
is the interference plus noise power term. Furthermore, we rearrange the SINR
constraints using simple algebraic manipulations, leading to the following
simple property $(\forall\mathbbm{T}\in\mathcal{T})(\forall k\in\mathcal{K})$

$\mathrm{SINR}_{k}^{\mathrm{DL}}(\mathbbm{T})\geq\gamma_{k}\iff|c_{k}(\mathbbm{T})|-\nu_{k}|b_{k}(\mathbbm{T})|\leq
0,$ (12)

where $\nu_{k}:=\sqrt{1+1/\gamma_{k}}$. A Lagrangian dual problem to (10) is
then given by

$\underset{(\bm{\lambda},\bm{\mu})\in\mathbbmss{R}_{+}^{L}\times\mathbbmss{R}_{+}^{K}}{\text{maximize}}~{}d(\bm{\lambda},\bm{\mu}),$
(13)

where (recalling (9)) we define the dual function $d(\bm{\lambda},\bm{\mu}):=$

$\inf_{\mathbbm{T}\in\mathcal{T}}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{1}+\bm{\lambda}}^{2}]-\sum_{l=1}^{L}\lambda_{l}P_{l}+\sum_{k=1}^{K}\mu_{k}(|c_{k}(\mathbbm{T})|-\nu_{k}|b_{k}(\mathbbm{T})|).$

Since the primal problem in (10) is nonconvex, by Proposition 1 we can only
guarantee (a-priori) weak duality. Furthermore, guaranteeing existence of a
solution is not immediate. However, the following important result holds.

######  Proposition 2.

Problem (10) admits a solution $\mathbbm{T}^{\star}\in\mathcal{T}$.
Furthermore, denote by $p^{\star}$ and $d^{\star}$ the optimum of the primal
problem (10) and of the dual problem (13), respectively. Strong duality holds,
i.e., $d^{\star}=p^{\star}$, and there exist Lagrangian multipliers
$(\bm{\lambda}^{\star},\bm{\mu}^{\star})\in\mathbbmss{R}_{+}^{L}\times\mathbbmss{R}_{+}^{K}$
solving Problem (13).

###### Proof.

The proof follows a similar idea as in [23, 30], and it is based on
establishing connections between Problem (10) and a convex reformulation
obtained by replacing $|b_{k}(\mathbbm{T})|$ in (12) with
$\Re(b_{k}(\mathbbm{T}))$. For additional details, see Appendix -A. ∎

We conclude this section by stating a useful consequence of Proposition 2 that
will be instrumental for proving our main results based on uplink-downlink
duality. In particular, we provide an alternative version of Proposition 2
based on a _partial_ dual problem, obtained by keeping the SINR constraints
implicit.

######  Proposition 3.

Given the subset
$\mathcal{T}_{\bm{\gamma}}:=\\{\mathbbm{T}\in\mathcal{T}~{}|~{}(\forall
k\in\mathcal{K})~{}\mathrm{SINR}_{k}^{\mathrm{DL}}(\mathbbm{T})\geq\gamma_{k}\\}$
of precoders satisfying the SINR constraints in Problem (10), define the
_partial_ dual problem

$\underset{\bm{\lambda}\in\mathbbmss{R}_{+}^{L}}{\text{maximize}}~{}\tilde{d}(\bm{\lambda}):=\inf_{\mathbbm{T}\in\mathcal{T}_{\bm{\gamma}}}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{1}+\bm{\lambda}}^{2}]-\sum_{l=1}^{L}\lambda_{l}P_{l}.$
(14)

Strong duality holds, i.e., Problem (10) and Problem (14) have the same
optimum $p^{\star}$. Furthermore, there exist Lagrangian multipliers
$\bm{\lambda}^{\star}$ solving Problem (14).

###### Proof.

Consider the alternative optimization problem

$\displaystyle\underset{\mathbbm{T}\in\mathcal{T}}{\text{minimize}}$
$\displaystyle\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|^{2}]+\gamma(\mathbbm{T})$
(15) subject to $\displaystyle(\forall
l\in\mathcal{L})~{}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{l,k}\|^{2}]\leq
P_{l},$

where $\gamma(\mathbbm{T})=0$ if $\mathbbm{T}$ belongs to the set
$\mathcal{T}_{\bm{\gamma}}$, and $\gamma(\mathbbm{T})=+\infty$ otherwise.
Problem (15) is equivalent to Problem (10), in the sense that it has the same
optimum $p^{\star}$ and set of optimal solutions. Its Lagrangian dual problem
can be written as (14). By applying weak duality (see Proposition 1) to the
term
$\inf_{\mathbbm{T}\in\mathcal{T}_{\bm{\gamma}}}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{1}+\bm{\lambda}}^{2}]$
for any fixed $\bm{\lambda}$, and by rewriting $\mathcal{T}_{\bm{\gamma}}$
according to (12), we obtain
$\sup_{\bm{\mu}\in\mathbbmss{R}_{+}^{K}}d(\bm{\lambda},\bm{\mu})\leq\tilde{d}(\bm{\lambda})$.
Taking the supremum over $\bm{\lambda}$ on both sides gives

$p^{\star}=\sup_{\bm{\lambda}\in\mathbbmss{R}_{+}^{L}}\sup_{\bm{\mu}\in\mathbbmss{R}_{+}^{K}}d(\bm{\lambda},\bm{\mu})\leq\sup_{\bm{\lambda}\in\mathbbmss{R}_{+}^{L}}\tilde{d}(\bm{\lambda})\leq
p^{\star},$

where the first equality follows from Proposition 2 and the last inequality
follows from weak duality applied to Problem (15). The second part of the
statement follows from the existence of a solution
$(\bm{\lambda}^{\star},\bm{\mu}^{\star})$ to Problem (13), and
$p^{\star}=\sup_{\bm{\mu}\in\mathbbmss{R}_{+}^{K}}d(\bm{\lambda}^{\star},\bm{\mu})\leq\tilde{d}(\bm{\lambda}^{\star})\leq
p^{\star}$. ∎

In the reminder of this study, we focus on the partial dual problem (14). The
reason is that the dual problem (13), or other variations that keep the SINR
constraints augmented, do not seem tractable, mostly because of the
difficulties in solving the (infinite dimensional) inner minimization problem
over the space of precoders subject to nontrivial information constraints.
These problems are known to be very challenging, even if convex [29, Chapter
2]. In contrast, as we will see later in the manuscript, the use of the
partial dual problem leads to a tractable inner minimization problem. This is
one of the major differences with respect to the finite dimensional case in
[23].

###  IV-B Primal-dual solution methods

A key aspect of Lagrangian optimization is the possibility of recovering a
primal solution from a dual solution. However, we emphasize that this is not
always possible even if strong duality holds. Nevertheless, the following
proposition ensures that a primal solution $\mathbbm{T}^{\star}$ to Problem
(10) can be indeed recovered from a solution to the partial dual problem (14).

######  Proposition 4.

Let $\bm{\lambda}^{\star}$ be a solution to Problem (14). Then, a solution to
Problem (10) is given by any solution to

$\underset{\mathbbm{T}\in\mathcal{T}_{\bm{\gamma}}}{\text{minimize}}~{}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{1}+\bm{\lambda}^{\star}}^{2}].$
(16)

###### Proof.

The proof is given in Appendix -B, and it is based on the same convex
reformulation of the SINR constraints adopted in the proof of Proposition 2. ∎

Starting from Proposition 4, and in particular by studying and solving Problem
(16), in the next section we derive structural properties for optimal joint
precoding. However, before moving to the next section, we first complete the
discussion on recovering a primal solution from a dual solution by
illustrating a simple algorithm for solving Problem (14), which is a concave
maximization problem. In particular, we consider a standard primal-dual
iterative algorithm based on the projected subgradient method [31, 32].

######  Proposition 5.

Choose $\bm{\lambda}^{(1)}\in\mathbbmss{R}_{+}^{L}$ and a sequence
$\\{\alpha^{(i)}\\}_{i\in\mathbbmss{N}}$ such that

$(\forall
i\in\mathbbmss{N})~{}\alpha^{(i)}\in\mathbbmss{R}_{++},\quad\lim_{i\to\infty}\alpha^{(i)}=0,\quad\sum_{i\in\mathbbmss{N}}\alpha^{(i)}=\infty.$

Define the sequence $(\bm{\lambda}^{(i)})_{i\in\mathbbmss{N}}$ generated via

$(\forall
i\in\mathbbmss{N})~{}\bm{\lambda}^{(i+1)}=\max\left\\{\bm{\lambda}^{(i)}+\frac{\alpha^{(i)}}{\|\bm{g}(\bm{\lambda}^{(i)})\|}\bm{g}(\bm{\lambda}^{(i)}),\bm{0}\right\\},$

where the $l$th entry of $\bm{g}(\bm{\lambda})$ is given by
$(\forall\bm{\lambda}\in\mathbbmss{R}_{+}^{L})(\forall l\in\mathcal{L})$,

$g_{l}(\bm{\lambda}):=\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{l,k}\|^{2}]-P_{l},\quad\mathbbm{T}\in\arg\min_{\mathcal{T}_{\bm{\gamma}}}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{1}+\bm{\lambda}}^{2}].$

Then, the subsequence of $(\bm{\lambda}^{(i)})_{i\in\mathbbmss{N}}$
corresponding to the best objective after $n$ iterations
$\max_{i=1,\ldots,n}\tilde{d}(\bm{\lambda}^{(i)})$ converges to a solution
$\bm{\lambda}^{\star}$ to Problem (14).

###### Proof.

The proof is given in Appendix -C. ∎

Note that the above algorithm requires a method for solving Problem (16) for
arbitrary Lagrangian multipliers. A possible algorithm provided in the
following section.

##  V Uplink-downlink duality

Building on the above analysis based on Lagrangian duality, in this section we
present our main result, which states that the problem of optimal joint
precoding design under per-AP power and information constraints can be
reformulated as a joint combining design and long-term power control problem
in a dual uplink channel with a properly designed noise vector $\bm{\sigma}$
(see Section III-C). More precisely, we show later in Proposition 6 that an
optimal solution to Problem (10) can be recovered from a solution to

$\displaystyle\underset{\mathbbm{V}\in\mathcal{T},\bm{p}\in\mathbbmss{R}_{+}^{K}}{\text{minimize}}$
$\displaystyle\sum_{k=1}^{K}p_{k}$ (17) subject to $\displaystyle(\forall
k\in\mathcal{K})~{}\mathrm{SINR}^{\mathrm{UL}}_{k}(\mathbbm{v}_{k},\bm{p},\bm{\sigma})\geq\gamma_{k},$
$\displaystyle(\forall
k\in\mathcal{K})~{}\mathsf{E}[\|\mathbbm{v}_{k}\|_{\bm{\sigma}}^{2}]=1,$

for some $\bm{\sigma}\in\mathbbmss{R}_{++}^{L}$ (recalling (9)), and where
$\mathbbm{V}:=[\mathbbm{v}_{1},\ldots,\mathbbm{v}_{k}]$. In addition, we
present an efficient numerical method that solves the above problem.

######  Remark 2.

Our derivation differs significantly from the derivation of the related result
in [23]. In particular, [23] exploits the peculiar structure of optimal
centralized precoding in deterministic channels, its relation to a certain
Rayleigh quotient, and a series of properties from the theory of semidefinite
programming and quadratic forms. Unfortunately, these arguments do not seem
applicable to our setup, which covers distributed precoding and random
channels. Specifically, equations similar to [23, Eq. (20)], [23, Eq. (25)],
and [23, Eq. (29)] seem difficult to derive. To address this limitation, we
follow a different path. We replace the above arguments by a variation of
well-known uplink-downlink duality results under a sum power constraint,
reviewed, e.g., in [33, 19].

###  V-A Joint precoding optimization over a dual uplink channel

The desired connection between the downlink channel and its dual uplink
channel is established by studying Problem (16), the solutions of which are
optimal joint precoders solving Problem (10). The key idea lies in
interpreting
$\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{1}+\bm{\lambda}^{\star}}^{2}]$
as an unconventional weighted definition of the average sum transmit power. To
keep the discussion general and, for instance, applicable to the algorithm
given by Proposition 5, we consider arbitrary Lagrangian multipliers, i.e., we
consider the following problem:
$(\forall\bm{\sigma}\in\mathbbmss{R}_{++}^{L})$

$\displaystyle\underset{\mathbbm{T}\in\mathcal{T}}{\text{minimize}}$
$\displaystyle\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{\sigma}}^{2}]$
(18) subject to $\displaystyle(\forall
k\in\mathcal{K})~{}\mathrm{SINR}^{\mathrm{DL}}_{k}(\mathbbm{T})\geq\gamma_{k}.$

Since the SINR constraints are feasible by assumption, following the same
arguments as in the proof of Proposition 5, we observe that the above problem
always admits a solution.

######  Proposition 6.

For all $\bm{\sigma}\in\mathbbmss{R}_{++}^{L}$, Problem (18) and Problem (17)
have the same optimum. Furthermore, a solution to Problem (18) is given by

$(\forall
k\in\mathcal{K})~{}\mathbbm{t}_{k}^{\star}=\sqrt{q^{\star}_{k}}\mathbbm{v}_{k}^{\star},$

where
$(\mathbbm{V}^{\star},\bm{p}^{\star})\in\mathcal{T}\times\mathbbmss{R}_{++}^{K}$
is a solution to Problem (17), and
$\bm{q}^{\star}:=(q_{1}^{\star},\ldots,q^{\star}_{K})\in\mathbbmss{R}_{++}^{K}$
is given by

$\bm{q}^{\star}=(\bm{D}-\bm{B})^{-1}(\bm{D}-\bm{B}^{\mathsf{T}})\bm{p}^{\star},$

where

$\bm{B}:=\begin{bmatrix}\mathsf{E}[|\mathbbm{h}_{1}^{\mathsf{H}}\mathbbm{v}^{\star}_{1}|^{2}]&\ldots&\mathsf{E}[|\mathbbm{h}_{1}^{\mathsf{H}}\mathbbm{v}^{\star}_{K}|^{2}]\\\
\vdots&\ddots&\vdots\\\
\mathsf{E}[|\mathbbm{h}_{K}^{\mathsf{H}}\mathbbm{v}^{\star}_{1}|^{2}]&\ldots&\mathsf{E}[|\mathbbm{h}_{K}^{\mathsf{H}}\mathbbm{v}^{\star}_{K}|^{2}]\end{bmatrix},\quad\bm{D}:=\mathrm{diag}(\bm{d}),$
$\bm{d}:=\left((1+\gamma_{1}^{-1})|\mathsf{E}[\mathbbm{h}^{\mathsf{H}}_{1}\mathbbm{v}^{\star}_{1}]|^{2},\ldots,(1+\gamma_{K}^{-1})|\mathsf{E}[\mathbbm{h}^{\mathsf{H}}_{K}\mathbbm{v}^{\star}_{K}]|^{2}\right).$

###### Proof.

The function $\mathsf{E}[\|\cdot\|_{\bm{\sigma}}^{2}]$ is a valid norm in
$\mathcal{T}_{k}$. Hence, we can rewrite
$\inf_{\mathbbm{T}\in\mathcal{T}_{\bm{\gamma}}}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{\sigma}}^{2}]$
in a normalized form as the following optimization problem:

$\displaystyle\underset{\mathbbm{V}\in\mathcal{T},\bm{q}\in\mathbbmss{R}_{+}^{K}}{\text{minimize}}$
$\displaystyle\sum_{k=1}^{K}q_{k}$ (19) subject to $\displaystyle(\forall
k\in\mathcal{K})~{}\mathrm{SINR}^{\mathrm{DL}}_{k}(\mathbbm{V}\mathrm{diag}(\bm{q})^{\frac{1}{2}})\geq\gamma_{k},$
$\displaystyle(\forall
k\in\mathcal{K})~{}\mathsf{E}[\|\mathbbm{v}_{k}\|_{\bm{\sigma}}^{2}]=1,$

where we used the change of variables $(\forall k\in\mathcal{K})$
$\mathbbm{t}_{k}=:\sqrt{q_{k}}\mathbbm{v}_{k}$. The vector $\bm{q}$ can be
interpreted as a downlink power control vector, by (unconventionally)
measuring the power of each $\mathbbm{t}_{k}$ in terms of its norm
$\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{\sigma}}^{2}]=q_{k}$. For any choice of
$\mathbbm{V}\in\mathcal{T}$ with normalized columns, i.e., such that $(\forall
k\in\mathcal{K})~{}\mathsf{E}[\|\mathbbm{v}_{k}\|_{\bm{\sigma}}^{2}]=1$,
consider now the following downlink power control problem:

$\displaystyle\underset{\bm{q}\in\mathbbmss{R}_{+}^{K}}{\text{minimize}}$
$\displaystyle\sum_{k=1}^{K}q_{k}$ (20) subject to $\displaystyle(\forall
k\in\mathcal{K})~{}\mathrm{SINR}^{\mathrm{DL}}_{k}(\mathbbm{V}\mathrm{diag}(\bm{q})^{\frac{1}{2}})\geq\gamma_{k}.$

From known sum power duality arguments in the power control literature
(reviewed, e.g., in [33, 19]), it follows that Problem (20) is feasible if and
only if the following uplink power control problem is feasible, for the same
choice of $\mathbbm{V}$:

$\displaystyle\underset{\bm{p}\in\mathbbmss{R}_{+}^{K}}{\text{minimize}}$
$\displaystyle\sum_{k=1}^{K}p_{k}$ (21) subject to $\displaystyle(\forall
k\in\mathcal{K})~{}\mathrm{SINR}^{\mathrm{UL}}_{k}(\mathbbm{v}_{k},\bm{p},\bm{\sigma})\geq\gamma_{k}.$

When feasible, Problem (20) and Problem (21) are known to have unique and
positive solutions meeting the SINR constraints with equality, and to attain
the same optimum. The solutions are related by rearranging the constraints as
full rank linear systems $(\bm{D}-\bm{B})\bm{q}^{\star}=\bm{1}$ and
$(\bm{D}-\bm{B}^{\mathsf{T}})\bm{p}^{\star}=\bm{1}$, respectively. When not
feasible, we say that the two optima equal $\infty$. By taking the infimum of
both optima over the set of $\mathbbm{V}\in\mathcal{T}$ such that $(\forall
k\in\mathcal{K})~{}\mathsf{E}[\|\mathbbm{v}_{k}\|_{\bm{\sigma}}^{2}]=1$, we
obtain that Problem (17) and Problem (18) have the same optimum. Finally, the
proof is completed by recalling that Problem (18) always has a solution. ∎

The key message of the uplink-downlink duality principle in Proposition 6 is
that optimal joint precoders solving Problem (10) are given by a scaled
version of optimal joint combiners solving the dual uplink problem (17) with
the noise vector $\bm{\sigma}=\bm{1}+\bm{\lambda}^{\star}$, where
$\bm{\lambda}^{\star}$ are Lagrangian multipliers solving Problem (14).

###  V-B Dual uplink power control with implicit optimal combining

We now focus on the solution to the dual uplink problem (17). By exploiting
the property that the uplink SINR constraints are only coupled via the power
vector $\bm{p}$, we observe that the optimum to Problem (17) is equivalently
given by the optimum to the following power control problem:

$\underset{\bm{p}\in\mathbbmss{R}_{+}^{K}}{\text{minimize}}\>\sum_{k=1}^{K}p_{k}\text{
subject to }(\forall
k\in\mathcal{K})~{}u_{k}(\bm{p},\bm{\sigma})\geq\gamma_{k},$ (22)

where the optimization of the combiners is left implicit in the definition of
$(\forall
k\in\mathcal{K})(\forall\bm{p}\in\mathbbmss{R}_{+}^{K})(\forall\bm{\sigma}\in\mathbbmss{R}_{++}^{L})$

$u_{k}(\bm{p},\bm{\sigma}):=\sup_{\begin{subarray}{c}\mathbbm{v}_{k}\in\mathcal{T}_{k}\\\
\mathsf{E}[\|\mathbbm{v}_{k}\|_{\bm{\sigma}}^{2}]\neq
0\end{subarray}}\mathrm{SINR}^{\mathrm{UL}}_{k}(\mathbbm{v}_{k},\bm{p},\bm{\sigma}).$
(23)

A solution $(\mathbbm{V}^{\star},\bm{p}^{\star})$ to Problem (17) is related
to a solution to Problem (22) in the sense that $\bm{p}^{\star}$ is also a
solution to Problem (22), and that the columns of $\mathbbm{V}^{\star}$ attain
the suprema in (23) for $\bm{p}=\bm{p}^{\star}$. Note that, without loss of
generality and for mathematical convenience in later steps, we removed in (23)
the unit-norm constraint, because the value of
$\mathsf{E}[\|\mathbbm{v}_{k}\|_{\bm{\sigma}}^{2}]$ does not change the uplink
SINR, as long as it is non-zero.

The main implication of the above discussion is that optimal joint combiners
(and hence, by Propostion 6, optimal joint precoders) solving Problem (17) can
be obtained by solving a set of disjoint uplink SINR maximization problems
under per-AP information constraints, i.e., by evaluating all
$u_{k}(\bm{p},\bm{\sigma})$ in (23) for some coefficients
$(\bm{p},\bm{\sigma})\in\mathbbmss{R}_{++}^{K}\times\mathbbmss{R}_{++}^{L}$.
The next sections discuss challenges and solutions related to this crucial
step, and they reveal a useful solution structure. However, before moving to
the next section, we first present a numerical method for computing
$\bm{p}^{\star}$, i.e., for solving Problem (22), assuming that
$u_{k}(\bm{p},\bm{\sigma})$ in (23) can be indeed evaluated. Specifically, we
apply the celebrated framework of interference calculus for power control [34,
35], and obtain:

######  Proposition 7.

Fix $\bm{\sigma}\in\mathbbmss{R}_{++}^{L}$. For every
$\bm{p}^{(1)}\in\mathbbmss{R}_{++}^{K}$, the sequence
$(\bm{p}^{(i)})_{i\in\mathbbmss{N}}$ generated via the fixed-point iterations
$(\forall i\in\mathbbmss{N})~{}\bm{p}^{(i+1)}=T_{\bm{\sigma}}(\bm{p}^{(i)})$,
where

$(\forall\bm{p}\in\mathbbmss{R}_{++}^{K})~{}T_{\bm{\sigma}}(\bm{p}):=\begin{bmatrix}\frac{\gamma_{1}p_{1}}{u_{1}(\bm{p},\bm{\sigma})}&\ldots&\frac{\gamma_{K}p_{K}}{u_{K}(\bm{p},\bm{\sigma})}\end{bmatrix}^{\mathsf{T}},$
(24)

converges geometrically in norm to the unique solution $\bm{p}^{\star}$ to
Problem (22).

###### Proof.

(Sketch) A simple contradiction argument proves that a solution
$\bm{p}^{\star}$ must satisfy $(\forall
k\in\mathcal{K})~{}u_{k}(\bm{p}^{\star},\bm{\sigma})=\gamma_{k}$. By trivially
extending the arguments in [36, Proposition 3] for $\bm{\sigma}=\bm{1}$ to an
arbitrary $\bm{\sigma}\in\mathbbmss{R}_{++}^{L}$, it follows that this
condition can be equivalently expressed as the fixed point equation
$\bm{p}^{\star}=[\gamma_{1}f_{1,\bm{\sigma}}(\bm{p}^{\star}),\ldots,\gamma_{K}f_{K,\bm{\sigma}}(\bm{p}^{\star})]^{\mathsf{T}}$,
where $(\forall
k\in\mathcal{K})~{}f_{k,\bm{\sigma}}:\mathbbmss{R}_{+}^{K}\to\mathbbmss{R}_{++}$
is a given positive concave function (and hence a standard interference
function, see, e.g., [36, Definition 2]) satisfying $(\forall
k\in\mathcal{K})(\forall\bm{p}\in\mathbbmss{R}_{+}^{K})~{}u_{k}(\bm{p},\bm{\sigma})=p_{k}/f_{k,\bm{\sigma}}(\bm{p})$.
For the above arguments to hold, an important requirement is the property
$(\forall
k\in\mathcal{K})(\forall\bm{p}\in\mathbbmss{R}_{++}^{K})~{}u_{k}(\bm{p},\bm{\sigma})>0$.
However, this property is guaranteed by recalling that Problem (18) admits a
solution, which implies that $\exists\mathbbm{T}\in\mathcal{T}$ such that
$(\forall
k\in\mathcal{K})~{}|\mathsf{E}[\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{t}_{k}]|^{2}>0$,
and hence the existence of some $\mathbbm{V}\in\mathcal{T}$ such that all
uplink SINRs are strictly positive for any $\bm{p}\in\mathbbmss{R}_{++}^{K}$.
The proof is concluded by invoking known properties of fixed points of
standard interference mappings [34] and positive concave mappings [37]. In
particular, the results in [34] ensure that there exists a unique fixed point
$\bm{p}^{\star}$, and that fixed-point iterations converge in norm to
$\bm{p}^{\star}$. Furthermore, geometric convergence follows from [37]. ∎

The above proposition shows that optimal joint combiners solving Problem (17)
can be obtained via the iterative evaluation of $T_{\bm{\sigma}}(\bm{p})$ in
(24), which in turn involves solving the aforementioned uplink SINR
maximization problems in (23).

##  VI Optimal joint precoding structure

By exploiting the obtained uplink-downlink duality principle, we now derive
the structure of an optimal solution to Problem (10). Specifically, we show
that it suffices to consider properly scaled and regularized variations of the
so-called team MMSE precoders [21], parametrized by a set of coefficients
$(\bm{p},\bm{\sigma})\in\mathbbmss{R}_{++}^{K}\times\mathbbmss{R}_{++}^{L}$.

###  VI-A MMSE precoding under information constraints

As discussed in the previous sections, an optimal solution to Problem (10) can
be obtained by solving a set of uplink SINR maximization problems of the type
in (23). Solving these problems appears quite challenging for the following
reasons: (i) the non-convex fractional utility involving expectations in both
the numerator and denominator, and (ii) the information constraints. However,
the next proposition shows that (i) is not the main challenge, because we can
consider an alternative and simpler convex utility, in the same spirit of the
known relation between SINR and MMSE in deterministic channels [38].

######  Proposition 8.

For given $k\in\mathcal{K}$, $\bm{p}\in\mathbbmss{R}_{++}^{K}$, and
$\bm{\sigma}\in\mathbbmss{R}_{++}^{L}$, consider the optimization problem
(recalling (9))

$\underset{\mathbbm{v}_{k}\in\mathcal{T}_{k}}{\emph{minimize}}~{}\mathsf{E}\left[\|\bm{P}^{\frac{1}{2}}\mathbbm{H}^{\mathsf{H}}\mathbbm{v}_{k}-\bm{e}_{k}\|^{2}\right]+\mathsf{E}\left[\|\mathbbm{v}_{k}\|_{\bm{\sigma}}^{2}\right],$
(25)

where $\bm{P}:=\mathrm{diag}(\bm{p})$, and
$\mathbbm{H}:=[\mathbbm{h}_{1},\ldots,\mathbbm{h}_{K}]$. Problem (25) has a
unique solution $\mathbbm{v}_{k}^{\star}\in\mathcal{T}_{k}$, and this solution
satisfies

$u_{k}(\bm{p},\bm{\sigma})=\mathrm{SINR}_{k}\left(\mathbbm{v}_{k}^{\star},\bm{p},\bm{\sigma}\right).$

###### Proof.

The proof follows readily by extending the arguments in [36, Proposition 4]
for $\bm{\sigma}=\bm{1}$ to an arbitrary
$\bm{\sigma}\in\mathbbmss{R}_{++}^{L}$. Note that the assumption
$u_{k}(\bm{p},\bm{\sigma})>0$, required for the proof, is always satisfied in
this work, as already discussed in the proof of Proposition 7. ∎

######  Remark 3.

Together with Proposition 6 and the discussion in Section V-B, Proposition 8
shows that optimal joint precoders solving Problem (10) can be obtained as
solutions to MMSE problems under information constraints, i.e., they are given
by (properly scaled) solutions to Problem (25), for some parameters
$(\bm{p},\bm{\sigma})\in\mathbbmss{R}_{++}^{K}\times\mathbbmss{R}_{++}^{L}$.

In the particular case where all APs have complete knowledge of the channel
$\mathbbm{H}$, and fully share UEs’ data, a solution to Problem (25) is simply
given by a variation of the well-known regularized zero-forcing solution

$\mathbbm{V}=\left(\mathbbm{H}\bm{P}\mathbbm{H}^{\mathsf{H}}+\bm{\Sigma}\right)^{-1}\mathbbm{H}\bm{P}^{\frac{1}{2}},$
(26)

where, similar to [23], the typical regularization of the matrix inversion
stage via a scaled identity matrix is replaced by a more general diagonal
regularization matrix
$\bm{\Sigma}:=\mathrm{diag}(\sigma_{1}\bm{I}_{N},\ldots,\sigma_{L}\bm{I}_{N})$
parametrized by $\bm{\sigma}$ which takes into account the per-AP power
constraints.

######  Remark 4.

A major difference between the solution in [23] and (26) is that the former
considers short-term optimization of its parameters, i.e., for every channel
realization. In contrast, our work considers long-term optimization of the
parameters
$(\bm{p},\bm{\sigma})\in\mathbbmss{R}_{++}^{K}\times\mathbbmss{R}_{++}^{L}$
based on channel statistics.

For more general and nontrivial information constraints, the solution to
Problem (25) can be interpreted as the best distributed approximation of
regularized channel inversion, and it can be obtained via a minor variation of
the recently developed _team_ MMSE precoding method given in [21]. In the next
sections we discuss this aspect in detail.

###  VI-B Uplink channel estimation and CSI sharing

To provide concrete examples of solutions to Problem (25), we first slightly
restrict the model in Section III, while still covering most scenarios studied
in the (cell-free) massive MIMO literature. For all AP $l\in\mathcal{L}$ and
UE $k\in\mathcal{K}$, we let each sub-vector $\mathbbm{h}_{l,k}$ of
$\mathbbm{h}_{k}^{\mathsf{H}}=:[\mathbbm{h}_{1,k}^{\mathsf{H}},\ldots\mathbbm{h}_{L,k}^{\mathsf{H}}]$
be independently distributed as
$\mathbbm{h}_{l,k}\sim\mathcal{CN}\left(\bm{\mu}_{l,k},\bm{K}_{l,k}\right)$
for some channel mean $\bm{\mu}_{l,k}\in\mathbbmss{C}^{N}$ and covariance
matrix $\bm{K}_{l,k}\in\mathbbmss{C}^{N\times N}$. Independence can be easily
motivated by assuming that UEs and APs are not colocated. As customary in the
literature, we further assume that the APs (or, more generally, the processing
units controlling the APs) perform pilot based over-the-uplink MMSE channel
estimation, based on the channel reciprocity property of time division duplex
systems. Specifically, we assume each AP $l\in\mathcal{L}$ to acquire local
estimates
$\hat{\mathbbm{H}}_{l}:=[\hat{\mathbbm{h}}_{l,1},\ldots,\hat{\mathbbm{h}}_{l,K}]$
of the local channel
$\mathbbm{H}_{l}:=[\mathbbm{h}_{l,1},\ldots,\mathbbm{h}_{l,K}]$ with error
$\mathbbm{Z}_{l}:=[\mathbbm{z}_{l,1},\ldots,\mathbbm{z}_{l,K}]:=\mathbbm{H}_{l}-\hat{\mathbbm{H}}_{l}$
independent from $\hat{\mathbbm{H}}_{l}$, and satisfying $(\forall
l\in\mathcal{L})(\forall k\in\mathcal{K})$
$\mathbbm{z}_{l,k}\sim\mathcal{CN}(\bm{0},\bm{\Psi}_{l,k})$ for some error
covariance $\bm{\Psi}_{l,k}\in\mathbbmss{C}^{N\times N}$. Moreover, again
motivated by the geographical separation of the APs, we assume that
$(\hat{\mathbbm{H}}_{l},\mathbbm{Z}_{l})$ is independent from
$(\hat{\mathbbm{H}}_{j},\mathbbm{Z}_{j})$ for all $(l,j)\in\mathcal{L}^{2}$
such that $l\neq j$.

After this local channel estimation step, we assume that the APs may acquire
additional information of the global channel matrix $\mathbbm{H}$ via some CSI
sharing step. More specifically, we assume that each AP $l\in\mathcal{L}$ must
form its precoders based on some side information
$S_{l}:=(\hat{\mathbbm{H}}_{l},\bar{S}_{l})$, where $\bar{S}_{l}$ denotes
additional channel information defined by the chosen CSI sharing pattern.
Overall, we model this two steps channel acquisition scheme (local channel
estimation followed by CSI sharing) by assuming the following Markov chain:

$(\forall l\in\mathcal{L})(\forall
j\in\mathcal{L})\quad\mathbbm{H}_{l}-\hat{\mathbbm{H}}_{l}-S_{l}-S_{j}-\hat{\mathbbm{H}}_{j}-\mathbbm{H}_{j}.$

The first and last part of this Markov chain correspond to the local channel
estimation step, while the middle part corresponds to the subsequent CSI
sharing step. Essentially, this Markov chain ensures that each AP
$l\in\mathcal{L}$ can only acquire a degraded version of the local estimate
$\hat{\mathbbm{H}}_{j}$ of the local channel $\mathbbm{H}_{j}$ contained
within the CSI $S_{j}$ of another AP $j\neq l$. In other words, AP $l\neq j$
cannot have a better knowledge of $\mathbbm{H}_{j}$ than AP $j$, and vice
versa333Note that, while being natural for general time division duplex
systems with reciprocity-based channel estimation, this assumption may not
hold for frequency division duplex systems with nontrivial CSI feedback
patterns. Since less attractive from both a practical and mathematical
perspective, we do not consider the latter systems. However, we remark that
these systems may also be covered by a variation of the team MMSE method in
[21].. We map the above assumptions to information constraints
$\mathcal{T}_{1},\ldots,\mathcal{T}_{K}$ in Problem (10) by letting the
information subfield $\Sigma_{l}$ of each AP $l\in\mathcal{L}$, defining the
subspace $\mathcal{H}_{l}^{N}$ in (6), be the sub-$\sigma$-algebra generated
by its available CSI $S_{l}$ on $\Omega$.

###  VI-C Centralized precoding with per-AP power constraints

A similar expression to (26) covering user-centric network clustering and
channel estimation errors can be easily obtained, provided that imperfect
channel estimates are perfectly shared within each cluster of APs. This
corresponds to a variation of the known centralized MMSE precoding solution in
[8, 20].

######  Proposition 9.

For given $k\in\mathcal{K}$, $\bm{p}\in\mathbbmss{R}_{++}^{K}$, and
$\bm{\sigma}\in\mathbbmss{R}_{++}^{L}$, and under the model in Section VI-B
with $(\forall
l\in\mathcal{L})~{}S_{l}=(\hat{\mathbbm{H}}_{1},\ldots,\hat{\mathbbm{H}}_{L})$
(centralized CSI), the unique solution to Problem (25) is given by

$\mathbbm{v}_{k}=\left(\bm{C}_{k}\hat{\mathbbm{H}}\bm{P}\hat{\mathbbm{H}}^{\mathsf{H}}\bm{C}_{k}+\bm{C}_{k}\bm{\Psi}\bm{C}_{k}+\bm{\Sigma}\right)^{-1}\hat{\mathbbm{H}}^{\mathsf{H}}\bm{C}_{k}\bm{P}^{\frac{1}{2}},$
(27)

where
$\hat{\mathbbm{H}}^{\mathsf{H}}:=[\hat{\mathbbm{H}}_{1}^{\mathsf{H}},\ldots,\hat{\mathbbm{H}}_{L}^{\mathsf{H}}]$,
$\bm{\Sigma}:=\mathrm{diag}(\sigma_{1}\bm{I}_{N},\ldots,\sigma_{L}\bm{I}_{N})$,
$\bm{\Psi}:=\sum_{k=1}^{K}p_{k}\mathrm{diag}(\bm{\Psi}_{1,k},\ldots,\bm{\Psi}_{L,k})$,
and $\bm{C}_{k}:=\mathrm{diag}(\bm{C}_{1,k},\ldots,\bm{C}_{L,k})$ is a block-
diagonal matrix satisfying

$(\forall l\in\mathcal{L})~{}\bm{C}_{l,k}=\begin{cases}\bm{I}_{N}&\text{if
}l\in\mathcal{L}_{k},\\\ \bm{0}_{N\times N}&\text{otherwise}.\end{cases}$

###### Proof.

(Sketch) We equivalently model user-centric network clustering by replacing
$\mathbbm{v}_{k}$ with $\bm{C}_{k}\mathbbm{v}_{k}$ as in [8], instead of
modifying the sets $\mathcal{T}_{k}$ in (6) as described in Section III-B.
Then, since all APs share the same CSI $\hat{\mathbbm{H}}$, optimal precoders
can be obtained as solutions to disjoint, unconstrained, and finite
dimensional conditional MMSE problems, one for each CSI realization:

$\mathbbm{v}_{k}=\arg\min_{\bm{v}_{k}\in\mathbbmss{C}^{LN}}\mathsf{E}\left[\|\bm{P}^{\frac{1}{2}}\mathbbm{H}^{\mathsf{H}}\bm{C}_{k}\bm{v}_{k}-\bm{e}_{k}\|^{2}+\|\bm{C}_{k}\bm{v}_{k}\|_{\bm{\sigma}}^{2}\middle|\hat{\mathbbm{H}}\right].$
(28)

The rest of the proof follows by evaluating the conditional expectations using
the CSI error model, and by standard results on unconstrained minimization of
quadratic forms. ∎

Although derived by assuming full CSI sharing, we observe that the computation
of the optimal precoder (27) for UE $k\in\mathcal{K}$ only requires knowledge
of $(\hat{\mathbbm{H}}_{l})_{l\in\mathcal{L}_{k}}$, i.e., only the channel
estimates of the APs serving UE $k$. Furthermore, similarly to the discussion
in [20], the computation of the inverse in (27) only involves the inversion of
a submatrix of size $N|\mathcal{L}_{k}|$.

######  Remark 5.

The special case of (27) with $\bm{\sigma}=\bm{1}$ was already proposed in [8,
20] as a good heuristic under a sum power constraint. In particular, it was
motivated by first observing that (27) maximizes a coherent uplink rate bound,
different than (7), and then by invoking uplink-downlink duality between (7)
and (3) under a sum power constraint. Although [8, 20] observed that (27) also
solves Problem (28), the connection with the maximization of (7) given by
Propostion 8, and hence the formal optimality of (27) in terms of downlink
rates in (3), was not reported.

Following similar arguments as in Remark 4, we further notice the following
difference between (27) and similar expressions obtained using the technique
in [23], reported, for instance, in [24]. Although both (27) and these
expressions involve a form of regularized channel inversion based on
instantaneous CSI, the regularization parameters $(\bm{p},\bm{\sigma})$ in
(27) are selected based on channel statistics, instead of instantaneous CSI as
in [23, 24]. Despite some potential performance degradation, this feature may
simplify practical implementation, since it avoids performing complex
parameter optimization (essentially, power control) for each channel
realization.

###  VI-D Distributed precoding with per-AP power constraints

When the APs have different CSI, it is not possible to decompose the problem
into disjoint conditional MMSE problems as in the proof of (27), and more
advanced methods must be used [21]. In this section we illustrate the
extension of some key results in [21] to the case of per-AP power constraints.
We start with the following general result:

######  Proposition 10.

For given $k\in\mathcal{K}$, $\bm{p}\in\mathbbmss{R}_{++}^{K}$, and
$\bm{\sigma}\in\mathbbmss{R}_{++}^{L}$, and under the model in Section VI-B,
the unique solution to Problem (25) is also the unique
$\mathbbm{v}_{k}\in\mathcal{T}_{k}$ satisfying $(\forall l\in\mathcal{L}_{k})$

$\mathbbm{v}_{l,k}=\mathbbm{V}_{l}\left(\bm{e}_{k}-\sum_{j\in\mathcal{L}_{k}\backslash\\{l\\}}\bm{P}^{\frac{1}{2}}\mathsf{E}\left[\hat{\mathbbm{H}}_{j}^{\mathsf{H}}\mathbbm{v}_{j,k}\Big{|}S_{l}\right]\right)\quad\mathrm{a.s.},$
(29)

where
$\mathbbm{V}_{l}:=\left(\hat{\mathbbm{H}}_{l}\bm{P}\hat{\mathbbm{H}}_{l}^{\mathsf{H}}+\sum_{k\in\mathcal{K}}p_{k}\bm{\Psi}_{l,k}+\sigma_{l}\bm{I}_{N}\right)^{-1}\hat{\mathbbm{H}}_{l}\bm{P}^{\frac{1}{2}}$.

###### Proof.

The proof follows readily by replacing $\bm{\sigma}=\bm{1}$ with an arbitrary
$\bm{\sigma}\in\mathbbmss{R}_{++}^{L}$ in the proofs of [21, Lemma 2] and [22,
Lemma 1]. Informally, (29) is obtained by minimizing the objective in (25)
with respect to a subvector $\mathbbm{v}_{l,k}$, by conditioning on $S_{l}$,
and by fixing the other subvectors $\mathbbm{v}_{j,k}$ for $j\neq l$. This
readily gives a set of necessary optimality conditions. The key step of the
proof shows that these conditions are also sufficient. ∎

Proposition 10 shows that optimal distributed precoding is composed by a local
MMSE precoding stage, followed by a corrective stage taking into account the
possibly unknown effect of the other APs based on the available CSI. From an
optimization point of view, Proposition 10 states that the solution to (25) is
the unique solution to an infinite dimensional linear feasibility problem,
that can be solved in closed form for many nontrivial yet relevant setups, or
via efficient approximation techniques when the expectations in (29) cannot be
easily evaluated. We note that the difference between Proposition 10 and the
related results given in [21, 22] for a sum power constraint is the
introduction of a more flexible regularization of the local MMSE stage, which
now involves the selection of a potentially different parameter $\sigma_{l}$
for each AP $l\in\mathcal{L}$.

As for the results in [21, 22], we remark that Proposition 10 applies to
fairly general CSI sharing patterns (see Remark 1). Due to space limitations,
in the remainder of this study we focus on the relatively simple case of local
precoding with user-centric network clustering, and leave the study of more
complex setups for future work. However, we point out that the same approach
can be readily applied to extend the results on sequential precoding and
undirectional CSI sharing given by [21, 22] to the case of per-AP power
constraints.

######  Proposition 11.

For given $k\in\mathcal{K}$, $\bm{p}\in\mathbbmss{R}_{++}^{K}$, and
$\bm{\sigma}\in\mathbbmss{R}_{++}^{L}$, and under the model in Section VI-B
with $(\forall l\in\mathcal{L})~{}S_{l}=\hat{\mathbbm{H}}_{l}$ (local CSI),
the unique solution to Problem (25) is given by

$(\forall l\in\mathcal{L})~{}\mathbbm{v}_{l,k}=\mathbbm{V}_{l}\bm{c}_{l,k},$
(30)

where $\mathbbm{V}_{l}$ is a local MMSE stage as in Proposition 10, and
$\bm{c}_{l,k}\in\mathbbmss{C}^{K}$ is a statistical precoding stage given by
the unique solution to the linear system of equations

$\begin{cases}\bm{c}_{l,k}+\sum_{j\in\mathcal{L}_{k}\backslash\\{l\\}}\bm{\Pi}_{j}\bm{c}_{j,k}=\bm{e}_{k}&\forall
l\in\mathcal{L}_{k},\\\ \bm{c}_{l,k}=\bm{0}_{K\times
1}&\text{otherwise,}\end{cases}$

where $(\forall
l\in\mathcal{L})~{}\bm{\Pi}_{l}:=\mathsf{E}\left[\bm{P}^{\frac{1}{2}}\hat{\mathbbm{H}}_{l}^{\mathsf{H}}\mathbbm{V}_{l}\right]$.

###### Proof.

The proof follows by verifying that (30) satisfies the optimality conditions
(29). The details are similar to the full data sharing and sum power
constraint case in [21, Theorem 4], hence they are omitted. ∎

The computation of the local MMSE stage in (30) requires local instantaneous
CSI, local statistical CSI, and the parameters $(\bm{p},\bm{\sigma})$ selected
based on global statistical CSI as for (27). The computation of the
statistical precoding stage, whose role is to optimally enhance the local MMSE
precoding decisions by exploiting additional statistical features of the
channel and CSI of the other APs, requires global statistical CSI. A possible
implementation could be to let a central processing unit perform the
computation of the statistical precoding stages and the selection of
$(\bm{p},\bm{\sigma})$, and to let the APs perform the remaining computations
locally, without any instantaneous CSI sharing. Interestingly, we observe that
the matrix $\bm{\Pi}_{l}$ for $l\in\mathcal{L}$ takes the form of a $K\times
K$ covariance matrix that can be locally estimated at the $l$th AP using local
CSI only, and then shared on a long-term basis for the computation of the
statistical precoding stages. As observed in [21] for the simpler case of a
sum power constraint, (30) may provide significant performance gain over the
local MMSE stage alone (where, for each $l\in\mathcal{L}$ and
$k\in\mathcal{K}$, $\bm{c}_{l,k}$ is replaced by $c_{l,k}\bm{e}_{k}$ for a
single tunable large-scale fading coefficient $c_{l,k}\in\mathbbmss{R}_{+}$
[20]) for non-zero mean channel models and/or in the presence of pilot
contamination.

##  VII Numerical examples

In this section we illustrate some applications of our results by simulating
the performance of optimal joint precoding in a simple example of user-centric
cell-free massive MIMO network under either centralized or local CSI. Note
that due to the theoretical nature and space limitations of this paper, the
numerical results that follow are for illustrative purposes only and should by
no means be understood as an exhaustive evaluation of the performance of cell-
free networks.

###  VII-A Simulation setup

Figure 1: Pictorial representation of the simulated setup: $K=16$ UEs
uniformly distributed within a squared service area of size $1\times
1~{}\text{km}^{2}$, and $L=16$ regularly spaced APs with $N=4$ antennas each.
Each UE is jointly served by a cluster of $Q=4$ APs offering the strongest
channel gains.

We consider the downlink of the network depicted in Figure 1, where $K=16$ UEs
are independently and uniformly distributed within a squared service area of
size $1\times 1~{}\text{km}^{2}$, and served by $L=16$ regularly spaced APs
with $N=4$ antennas each. For all $(l,k)\in\mathcal{L}\times\mathcal{K}$, we
assume for simplicity zero mean uncorrelated channels, i.e.,
$\bm{\mu}_{l,k}=\bm{0}$ and $\bm{K}_{l,k}=\kappa_{l,k}\bm{I}_{N}$, where
$\kappa_{l,k}$ denotes the channel gain between AP $l$ and UE $k$. We adopt
the following 3GPP-like channel gain model, suitable for an urban microcell
scenario at $3.7$ GHz carrier frequency [39, Table 7.4.1-1]

$\kappa_{l,k}=-35.3\log_{10}\left(\Delta_{l,k}/1\;\mathrm{m}\right)-34.5+Z_{l,k}-P_{\mathrm{noise}}\quad\text{[dB]},$

where $\Delta_{l,k}$ is the distance between AP $l$ and UE $k$ including a
difference in height of $10$ m, and $Z_{l,k}\sim\mathcal{CN}(0,\rho^{2})$ [dB]
are shadow fading terms with deviation $\rho=7.82$. The shadow fading is
correlated as
$\mathsf{E}[Z_{l,k}Z_{j,i}]=\rho^{2}2^{-\frac{\delta_{k,i}}{13\text{ [m]}}}$
for all $l=j$ and zero otherwise, where $\delta_{k,i}$ is the distance between
UE $k$ and UE $i$. The noise power is
$P_{\mathrm{noise}}=-174+10\log_{10}(B)+F$ [dBm], where $B=100$ MHz is the
bandwidth, and $F=7$ dB is the noise figure. The per-AP power constraints are
set to $(\forall l\in\mathcal{L})~{}P_{l}=30$ dBm.

We assume that each UE $k\in\mathcal{K}$ is served by its $Q=4$ strongest APs
only, i.e., by the subset of APs indexed by
$\mathcal{L}_{k}\subseteq\mathcal{L}$, where each set $\mathcal{L}_{k}$ is
formed by ordering $\mathcal{L}$ w.r.t. decreasing $\kappa_{l,k}$ and by
keeping only the first $Q$ elements. Finally, we assume that the APs estimate
the small-scale fading channel coefficients of the served UEs only. In
particular, we consider the following simple model where the channel
coefficients are either perfectly known or completely unknown:

$(\forall
k\in\mathcal{K})~{}\hat{\mathbbm{h}}_{l,k}:=\begin{cases}\mathbbm{h}_{l,k}&\text{if
}l\in\mathcal{L}_{k},\\\
\mathsf{E}[\mathbbm{h}_{l,k}]&\text{otherwise}.\end{cases}$

###  VII-B Probability of feasibility

Figure 2: Probability of feasibility of different minimum rate requirements,
under different information constraints and a per-AP power constraint
$P_{l}=30$ dBm. Remarkably, our feasibility analysis covers optimal joint
precoding under user-centric network clustering and either local CSI (local
precoding) or centralized CSI (centralized precoding). The performance of the
corresponding solutions under a sum power constraint $\sum_{l=1}^{L}P_{l}$ are
also evaluated. As expected, due to the more restrictive information
constraint, local precoding offers worse performance than centralized
precoding. Similarly, a per-AP power constraints offers worse performance than
a sum power constraint. Nevertheless, we observe that the local precoding with
per-AP power constraint still robustly supports fairly high rates around $2.5$
b/s/Hz to all UEs.

Figure 2 plots the probability that the feasible set of Problem (10) is
nonempty, by letting $(\forall k\in\mathcal{K})~{}\gamma_{k}=\gamma$ for
different choices of $\gamma$, corresponding to minimum rate requirements
within $1$-$4$ b/s/Hz for all UEs. We consider both centralized precoding as
in Proposition 9 and local precoding as in Proposition 11. As a baseline, we
also consider the corresponding optimal precoders subject to a sum power
constraint $P_{\text{sum}}=\sum_{l=1}^{L}P_{l}$.

The probability of feasibility is approximated by solving $100$ instances of
Problem (10) for $100$ independent UE drops. Each instance is solved using the
numerical methods given by Proposition 5 and Proposition 7, combined as nested
loops. In particular, building on Proposition 6, the subgradients
$\bm{g}(\bm{\lambda})$ in Proposition 5 are computed by solving the downlink
problem
$\inf_{\mathcal{T}_{\bm{\gamma}}}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{1}+\bm{\lambda}}^{2}]$
through its dual uplink problem (17) with $\bm{\sigma}=\bm{1}+\bm{\lambda}$,
using the fixed-point algorithm in Proposition 7. For each UEs drop, we draw a
sample set of $100$ independent CSI realizations for approximating the
expectations via empirical averages. Although developed for producing a
solution to Problem (10) under the assumption of strict feasibility, we use
the same algorithm for testing feasibility. Note that, for the considered
setup, the event of having a feasible set with empty interior has zero
probability mass, hence we test feasibility by detecting the events of strict
feasibility and infeasibility only. Additional details on how the algorithm
performs this test are given in Section VII-C.

(a)  (b)

Figure 3: Example of convergence behavior of the proposed algorithm for
different step size constants $\alpha$. We consider an arbitrary UE drop,
local precoding, and a minimum rate requirement of $3.5$ b/s/Hz. For each
iteration $i\in\mathbbmss{N}$ of the outer loop, we plot (a) the dual
objective $\tilde{d}(\bm{\lambda}^{(i)})$, and (b) the maximum transmit power
over all APs $\max_{l\in\mathcal{L}}g_{l}(\bm{\lambda}^{(i)})+P_{l}$. The non-
monotonic convergence is a common feature of projected subgradient methods. We
observe that, despite a seemingly slower convergence in the first iterations,
the more aggressive step size choice $\alpha=17$ produces a feasible solution
satisfying the per-AP power constraint $(\forall l\in\mathcal{L})~{}P_{l}=30$
dBm after $20$ iterations only. This is enough to declare feasibility, since,
for each outer iteration, the inner loops ensure that the SINR constraints are
always satisfied.

###  VII-C Comments on numerical implementation

In this section we give some additional details on the numerical
implementation of the algorithm producing Figure 2.

We first test feasibility under a sum power constraint, i.e., we test if
$\inf_{\mathcal{T}_{\bm{\gamma}}}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|^{2}]\leq
P_{\text{sum}}$ holds. To this end, we initialize the outer loop with
$\bm{\lambda}^{(1)}=\bm{0}$, and the inner loop with $(\forall
k\in\mathcal{K})~{}p^{(1)}_{k}=\gamma_{k}/\sum_{l\in\mathcal{L}}\kappa_{l,k}$.
This choice ensures that $\bm{p}^{(1)}\leq T_{\bm{1}}(\bm{p}^{(1)})$ holds,
and hence, by known properties of the considered fixed-point algorithm [40,
Fact 4], $(\sum_{k\in\mathcal{K}}p_{k}^{(i)})_{i\in\mathbbmss{N}}$ is a
monotonically increasing sequence converging to
$\inf_{\mathcal{T}_{\bm{\gamma}}}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|^{2}]$
if $\mathcal{T}_{\bm{\gamma}}\neq\emptyset$, or diverging to $+\infty$ if
$\mathcal{T}_{\bm{\gamma}}=\emptyset$ (unfeasible SINR requirements). The
inner loop is terminated at some step $i\in\mathbbmss{N}$ if no significant
progress is observed, in which case we declare the problem feasible under a
sum power constraint, or if the early stop condition
$\sum_{k\in\mathcal{K}}p_{k}^{(i)}>P_{\mathrm{sum}}$ is met, in which case we
declare infeasibility under a sum power constraint.

If this initial feasibility test is passed, we continue with the outer loop,
using the step size rule $(\forall
i\in\mathbbmss{N})~{}\alpha^{(i)}=\alpha/\sqrt{i}$, $\alpha=10$. Since the
condition $\mathcal{T}_{\bm{\gamma}}\neq\emptyset$ was already detected, the
inner loops are guaranteed to compute the partial dual function
$\tilde{d}(\bm{\lambda})=\inf_{\mathcal{T}_{\bm{\gamma}}}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{1}+\bm{\lambda}}^{2}]-\sum_{l=1}^{L}\lambda_{l}P_{l}$
and its subgradient $\bm{g}(\bm{\lambda})$ for all
$\bm{\lambda}\in\mathbbmss{R}_{+}^{L}$, up to some numerical tolerance. The
outer loop is terminated at some step $i\in\mathbbmss{N}$ if no significant
progress is observed, or if the early stop condition
$\tilde{d}(\bm{\lambda}^{(i)})>P_{\mathrm{sum}}$ is met. In the former case,
and if the obtained solution is feasible up to some numerical tolerance, we
declare feasibility. In all other cases, we declare infeasibility. In fact, if
feasible, the obtained precoders are also a solution to Problem (10), i.e., a
minimum sum power solution. In practice, when the interest is to test only
feasibility, we terminate the algorithm as soon as a feasible solution is
detected, i.e., if the additional early stop condition
$\bm{g}(\bm{\lambda}^{(i)})\leq\bm{0}$ is met at some step
$i\in\mathbbmss{N}$. Mostly because of the heuristic step size constant
$\alpha$, we remark that the algorithm might have slow convergence for some UE
drops (see Figure 3 for an illustration of the convergence behavior for a
particular UE drop). Hence, we also introduce a maximum number of iterations
and a maximum number of algorithm restarts with different constants $\alpha$.
If a conclusion is not reached within these thresholds, we remove the
corresponding UEs drop.

##  VIII Conclusion and future directions

This study marks a step forward in the process of extending known analytic
tools for deterministic channels and instantaneous rate expressions to fading
channels and ergodic rate expressions. Such extensions are often advocated in
the cellular and cell-free massive MIMO literature because they allow for a
more refined analysis of modern networks covering practical aspects such as
imperfect CSI and system optimization based on long-term channel statistics.
Just as coding over fading realizations is an efficient way to manage fading
dips, system optimization based on the long-term perspective typically lead,
in our opinion, to more efficient solutions.

As first main result, this study advances the current understanding of
distributed cell-free networks by showing that the recently introduced team
MMSE precoding method provides joint precoders that are optimal not only under
a sum power constraint, as stated in [21], but also under a per-AP power
constraint. For illustration purposes, this study derives the structure of
optimal local precoding under a per-AP power constraint. Although not
explicitly covered in this study, an optimal structure can be derived also for
other examples of distributed precoding such as the ones based on sequential
information sharing over radio stripes [21, 22]. An interesting future
direction is thus to revisit the available studies on performance evaluation
of distributed cell-free downlink implementations, in light of the above
results.

As second main result, this study provides an alternative tool to [23] for
designing centralized cell-free networks subject to per-AP power constraints,
by demonstrating optimality of centralized MMSE precoding with parameters
tuned only once for many channel realizations, instead of for each channel
realization as in the studies based on [23]. This is a consequence of
considering the hardening bound as figure of merit. Although the hardening
bound generally gives more pessimistic capacity estimates than alternative
ergodic rate bounds based on coherent or semi-coherent decoding [41], in our
opinion this drawback is counterbalanced by a significantly enhanced
tractability for precoding optimization.

A key aspect of our results is the compact parametrization of optimal joint
precoding in terms of a set of coefficients $(\bm{p},\bm{\sigma})$,
interpreted as virtual uplink transmit and noise powers. One limitation of
this study is that the presented algorithm for tuning these parameters can be
quite slow, and it is designed to solve SINR feasibility problems only (more
specifically, Problem (10)). Thus, another interesting future direction is the
development of efficient algorithms, perhaps based on heuristics such as
neural networks or other statistical learning tools, for tuning
$(\bm{p},\bm{\sigma})$ in network utility maximization problems such as sum
rate or minimum rate maximization.

Another subtle limitation of this work lies in the assumption of strict
feasibility for the constraints of Problem (10), which is used for ensuring
Slater’s condition to hold in Proposition 1. Using an information-theoretical
perspective, this means that our results do not cover optimal joint precoding
for rate tuples lying on the boundary of the achivable rate region, but only
on its interior. However, from an engineering perspective, this limitation is
practically irrelevant, since rate tuples arbitrarily close to the boundary
are still covered. Nevertheless, resolving this limitation is an interesting
yet involved future direction for theoretical analysis.

###  -A Convex reformulation

Consider the reformulation of Problem (10) obtained by replacing each
(rearranged) SINR constraint in (12) with

$(\forall
k\in\mathcal{K})~{}|c_{k}(\mathbbm{T})|-\nu_{k}\Re\left(b_{k}(\mathbbm{T})\right)\leq
0.$ (31)

More precisely, consider

$\displaystyle\underset{\mathbbm{T}\in\mathcal{T}}{\text{minimize}}$
$\displaystyle\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|^{2}]$ (32) subject
to $\displaystyle(\forall
k\in\mathcal{K})~{}|c_{k}(\mathbbm{T})|-\nu_{k}\Re\left(b_{k}(\mathbbm{T})\right)\leq
0$ $\displaystyle(\forall
l\in\mathcal{L})~{}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{l,k}\|^{2}]\leq
P_{l}$

and its Lagrangian dual problem

$\underset{(\bm{\lambda},\bm{\mu})\in\mathbbmss{R}_{+}^{L}\times\mathbbmss{R}_{+}^{K}}{\text{maximize}}~{}d^{\prime}(\bm{\lambda},\bm{\mu}),$
(33)

where we define the dual function
$d^{\prime}(\bm{\lambda},\bm{\mu}):=\inf_{\mathbbm{T}\in\mathcal{T}}\Lambda^{\prime}(\mathbbm{T},\bm{\lambda},\bm{\mu})$
and the Lagrangian $\Lambda^{\prime}(\mathbbm{T},\bm{\lambda},\bm{\mu}):=$

$\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{1}+\bm{\lambda}}^{2}]-\sum_{l=1}^{L}\lambda_{l}P_{l}+\sum_{k=1}^{K}\mu_{k}(|c_{k}(\mathbbm{T})|-\nu_{k}\Re(b_{k}(\mathbbm{T}))).$

The main advantage of the above reformulation is that it gives a convex
optimization problem, as shown next.

######  Lemma 1.

The objective and all constraints of Problem (32) are proper convex functions.

###### Proof.

Consider the norm $\|\cdot\|_{\star}:=\sqrt{\langle\cdot,\cdot\rangle}$ on
$\mathcal{H}^{K+1}$ induced by the inner product
$(\forall\mathbbm{x},\mathbbm{y}\in\mathcal{H}^{K+1})~{}\langle\mathbbm{x},\mathbbm{y}\rangle:=\Re\\{\mathsf{E}[\mathbbm{y}^{\mathsf{H}}\mathbbm{x}]\\}$.
We note that
$|c_{k}(\mathbbm{T})|=\sqrt{\sum_{j=1}^{K}\mathsf{E}[|\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{t}_{j}|^{2}]+1}$
is given by the composition of $\|\cdot\|_{\star}$ with an affine map
$\mathcal{T}\to\mathcal{H}^{K+1}:\mathbbm{T}\mapsto[\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{t}_{1},\ldots,\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{t}_{K},1]^{\mathsf{T}}$,
hence it is convex. Furthermore, since
$\Re\left(b_{k}(\mathbbm{T})\right)=\Re\left(\mathsf{E}[\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{t}_{k}]\right)$
is linear, convexity of the reformulated SINR constraints readily follows. We
omit the proof for the convexity of the objective and power constraints, since
it is trivial. Finally, repeated applications of Cauchy-Schwarz inequality
prove that all the aforementioned functions are also proper functions. ∎

The next simple lemma can be used to relate Problem (10) to Problem (32),
following a similar idea in [30, 23].

######  Lemma 2.

Consider an arbitrary $\mathbbm{T}\in\mathcal{T}$. Then, there exists
$\mathbbm{T}^{\prime}\in\mathcal{T}$ such that $(\forall
k\in\mathcal{K})(\forall l\in\mathcal{L})$

$\begin{split}|b_{k}(\mathbbm{T})|&=\Re\left(b_{k}(\mathbbm{T}^{\prime})\right),\\\
|c_{k}(\mathbbm{T})|&=|c_{k}(\mathbbm{T}^{\prime})|,\\\
\mathsf{E}[\|\mathbbm{t}_{l,k}\|^{2}]&=\mathsf{E}[\|\mathbbm{t}_{l,k}^{\prime}\|^{2}].\end{split}$
(34)

###### Proof.

Observe that, $(\forall k\in\mathcal{K})(\forall l\in\mathcal{L})$, the terms
$|b_{k}(\mathbbm{T})|$, $|c_{k}(\mathbbm{T})|$, and
$\mathsf{E}[\|\mathbbm{t}_{l,k}\|^{2}]$ are invariant to columnwise phase
rotations of the argument, i.e., they do not vary if we replace $\mathbbm{T}$
with $[\mathbbm{t}_{1}e^{j\theta_{1}},\ldots,\mathbbm{t}_{K}e^{j\theta_{K}}]$
for any $(\theta_{1},\ldots,\theta_{K})\in[0,2\pi]^{K}$. In particular, we can
always pick $(\theta_{1},\ldots,\theta_{K})\in[0,2\pi]^{K}$ such that
$|b_{k}(\mathbbm{T})|=\Re(b_{k}(\mathbbm{T}))$ holds. ∎

######  Lemma 3.

Let $p^{\star}$ and $r^{\star}$ be the optimum of Problem (10) and Problem
(32), respectively. The two problems have the same optimum, i.e.,
$p^{\star}=r^{\star}$.

###### Proof.

The simple property $(\forall x\in\mathbbmss{C})~{}\Re(x)\leq|x|$ shows that
$(\forall k\in\mathcal{K})(\forall\mathbbm{T}\in\mathcal{T})$

$|c_{k}(\mathbbm{T})|-\nu_{k}|b_{k}(\mathbbm{T})|\leq|c_{k}(\mathbbm{T})|-\nu_{k}\Re\left(b_{k}(\mathbbm{T})\right),$
(35)

from which the inequality $r^{\star}\geq p^{\star}$ readily follows (recall
also (12)). Then, consider a minimizing sequence for Problem (10), i.e., a
(not necessarily convergent) sequence
$(\mathbbm{T}^{(n)})_{n\in\mathbbmss{N}}$ such that $(\forall
n\in\mathbbmss{N})$ $\mathbbm{T}^{(n)}$ satisfies all constraints of Problem
(10), and
$\lim_{n\to\infty}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}^{(n)}_{k}\|^{2}]=p^{\star}$
[42, Definition 1.8]. By Lemma 2, we can define another sequence
$(\mathbbm{T}^{(n)^{\prime}})_{n\in\mathbbmss{N}}$ such that $(\forall
n\in\mathbbmss{N})$ $\mathbbm{T}^{(n)^{\prime}}$ satisfies all constraints of
Problem (32), attains the same objective
$\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}^{(n)^{\prime}}_{k}\|^{2}]=\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}^{(n)}_{k}\|^{2}]\geq
r^{\star}$, and hence satisfies
$p^{\star}=\lim_{n\to\infty}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}^{(n)^{\prime}}_{k}\|^{2}]\geq
r^{\star}$. Combining both inequalities $p^{\star}\leq r^{\star}$ and
$p^{\star}\geq r^{\star}$ completes the proof. ∎

######  Lemma 4.

The dual functions $d(\bm{\lambda},\bm{\mu})$ and
$d^{\prime}(\bm{\lambda},\bm{\mu})$ in Problem (13) and Problem (33),
respectively, satisfy
$(\forall(\bm{\lambda},\bm{\mu})\in\mathbbmss{R}_{+}^{L}\times\mathbbmss{R}_{+}^{K})$
$d(\bm{\lambda},\bm{\mu})=d^{\prime}(\bm{\lambda},\bm{\mu})$.

###### Proof.

Fix
$(\bm{\lambda},\bm{\mu})\in\mathbbmss{R}_{+}^{L}\times\mathbbmss{R}_{+}^{K}$
and define the Lagrangian of Problem (10)
$\Lambda(\mathbbm{T},\bm{\lambda},\bm{\mu}):=$

$\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{1}+\bm{\lambda}}^{2}]-\sum_{l=1}^{L}\lambda_{l}P_{l}+\sum_{k=1}^{K}\mu_{k}(|c_{k}(\mathbbm{T})|-\nu_{k}|b_{k}(\mathbbm{T})|),$

which satisfies
$d(\bm{\lambda},\bm{\mu})=\inf_{\mathbbm{T}\in\mathcal{T}}\Lambda(\mathbbm{T},\bm{\lambda},\bm{\mu})$.
The property $(\forall x\in\mathbbmss{C})~{}\Re(x)\leq|x|$ readily gives
$\Lambda(\mathbbm{T},\bm{\lambda},\bm{\mu})\leq\Lambda^{\prime}(\mathbbm{T},\bm{\lambda},\bm{\mu})$,
and hence $d(\bm{\lambda},\bm{\mu})\leq d^{\prime}(\bm{\lambda},\bm{\mu})$.
Then, consider a minimizing sequence $(\mathbbm{T}^{(n)})_{n\in\mathbbmss{N}}$
such that $(\forall n\in\mathbbmss{N})$ $\mathbbm{T}^{(n)}\in\mathcal{T}$ and
$\lim_{n\to\infty}\Lambda(\mathbbm{T}^{(n)},\bm{\lambda},\bm{\mu})=d(\bm{\lambda},\bm{\mu})$.
By Lemma 2, we can define another sequence
$(\mathbbm{T}^{(n)^{\prime}})_{n\in\mathbbmss{N}}$ such that $(\forall
n\in\mathbbmss{N})$ $\mathbbm{T}^{(n)^{\prime}}\in\mathcal{T}$ and
$\Lambda^{\prime}(\mathbbm{T}^{(n)^{\prime}},\bm{\lambda},\bm{\mu})=\Lambda(\mathbbm{T}^{(n)},\bm{\lambda},\bm{\mu})\geq
d^{\prime}(\bm{\lambda},\bm{\mu})$, hence satisfying
$d(\bm{\lambda},\bm{\mu})=\lim_{n\to\infty}\Lambda^{\prime}(\mathbbm{T}^{(n)^{\prime}},\bm{\lambda},\bm{\mu})\geq
d^{\prime}(\bm{\lambda},\bm{\mu})$. Combining both inequalities
$d(\bm{\lambda},\bm{\mu})\leq d^{\prime}(\bm{\lambda},\bm{\mu})$ and
$d(\bm{\lambda},\bm{\mu})\geq d^{\prime}(\bm{\lambda},\bm{\mu})$ completes the
proof. ∎

######  Lemma 5.

Problem (32) admits a unique solution $\mathbbm{T}^{\prime}\in\mathcal{T}$.
Furthermore, strong duality holds for Problem (32), i.e., Problem (33) and
Problem (32) have the same optimum, and there exist Lagrangian multipliers
$(\bm{\lambda}^{\prime},\bm{\mu}^{\prime})$ solving Problem (33).

###### Proof.

From Lemma 2, it immediately follows that since Slater’s condition is assumed
to hold for Problem (10), then it holds also for Problem (32). Then, strong
duality and existence of Lagrangian multipliers follow by recalling
Proposition 1 and Lemma 1. Existence of a unique solution
$\mathbbm{T}^{\prime}$ to Problem (32) follows by the Hilbert projection
theorem (see, e.g., [43]), since the objective
$\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|^{2}]$ is the norm induced by the
inner product
$(\forall\mathbbm{T},\mathbbm{V}\in\mathcal{T})~{}\langle\mathbbm{T},\mathbbm{V}\rangle:=\sum_{k=1}^{K}\Re\\{\mathsf{E}[\mathbbm{v}_{k}^{\mathsf{H}}\mathbbm{t}_{k}]\\}$
on $\mathcal{T}$, and the constraints define a nonempty closed convex subset
of $\mathcal{T}$ (closedness follows by continuity of the functions defining
each constraint). In other words, in this Hilbert space, the solution to the
problem is the projection of the zero vector onto the closed convex set
defined by the constraints. ∎

The proof of Proposition 2 is readily given by combining Lemma 3, Lemma 4,
Lemma 5, and by noticing that the unique solution $\mathbbm{T}^{\prime}$ to
Problem (32) is also a solution to Problem (10) (note: the converse does not
hold in general).

###  -B Recovering a primal solution from a dual solution

Starting from the strong duality property in Proposition 3, we obtain that a
primal-dual pair $(\mathbbm{T}^{\star},\bm{\lambda}^{\star})$ jointly solving
Problem (10) and Problem (14) must satisfy

$\begin{split}p^{\star}&=\inf_{\mathbbm{T}\in\mathcal{T}_{\bm{\gamma}}}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{1}+\bm{\lambda}^{\star}}^{2}]-\sum_{l=1}^{L}\lambda^{\star}_{l}P_{l}\\\
&\leq\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}^{\star}_{k}\|_{\bm{1}+\bm{\lambda}^{\star}}^{2}]-\sum_{l=1}^{L}\lambda^{\star}_{l}P_{l}\\\
&=\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}^{\star}_{k}\|^{2}]+\sum_{l=1}^{L}\lambda^{\star}_{l}\left(\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}^{\star}_{l,k}\|^{2}]-P_{l}\right)\\\
&\leq\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}^{\star}_{k}\|^{2}]=p^{\star},\end{split}$

where the first inequality follows by the definition of infimum, and the last
inequality follows since $(\mathbbm{T}^{\star},\bm{\lambda}^{\star})$ satisfy
the primal and dual constraints. The above chain of inequalities shows that
$\mathbbm{T}^{\star}$ attains the infimum. If there was a unique
$\mathbbm{T}\in\mathcal{T}_{\bm{\gamma}}$ attaining the infimum, then it would
also be the unique solution $\mathbbm{T}^{\star}$ to Problem (10). However, a
similar property does not hold in general whenever the infimum is attained by
multiple elements of $\mathcal{T}_{\bm{\gamma}}$. In particular, the infimum
may be attained not only by $\mathbbm{T}^{\star}$, but also by some other
$\mathbbm{T}\in\mathcal{T}_{\bm{\gamma}}$ violating the power constraints.
Nevertheless, we now prove that this case can be here excluded, by using the
next two lemmas. For convenience, we define the set
$\mathcal{T}^{\prime}_{\bm{\gamma}}\subseteq\mathcal{T}$ of precoders
satisfying the reformulated SINR constraints (31).

######  Lemma 6.

For every $\bm{\sigma}\in\mathbbmss{R}_{++}^{L}$, there exists a unique
$\mathbbm{T}^{\prime}\in\mathcal{T}^{\prime}_{\bm{\gamma}}$ attaining
$\inf_{\mathbbm{T}\in\mathcal{T}^{\prime}_{\bm{\gamma}}}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{\sigma}}^{2}]$.

###### Proof.

The proof follows by the Hilbert projection theorem (see, e.g., [43]), since
the objective
$\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{\sigma}}^{2}]$ is the norm
induced by the inner product
$(\forall\mathbbm{T},\mathbbm{V}\in\mathcal{T})~{}\langle\mathbbm{T},\mathbbm{V}\rangle:=\sum_{k=1}^{K}\sum_{l=1}^{L}\sigma_{l}\Re\\{\mathsf{E}[\mathbbm{v}_{l,k}^{\mathsf{H}}\mathbbm{t}_{l,k}]\\}$
on $\mathcal{T}$, and the constraints define a nonempty closed convex subset
of $\mathcal{T}$. Specifically, in this Hilbert space, the infimum is attained
by the projection of the zero vector onto the closed convex set
$\mathcal{T}^{\prime}_{\bm{\gamma}}$. ∎

######  Lemma 7.

For every $\bm{\sigma}\in\mathbbmss{R}_{++}^{L}$, if $\mathbbm{T}$ attains
$\inf_{\mathbbm{T}\in\mathcal{T}_{\bm{\gamma}}}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{\sigma}}^{2}]$,
then $\exists\mathbbm{T}^{\prime}$ attaining
$\inf_{\mathbbm{T}\in\mathcal{T}_{\bm{\gamma}}^{\prime}}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{\sigma}}^{2}]$
with the same power consumption.

###### Proof.

Following similar lines as for Lemma 3, we can show that
$\inf_{\mathbbm{T}\in\mathcal{T}_{\bm{\gamma}}}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{\sigma}}^{2}]=\inf_{\mathbbm{T}\in\mathcal{T}^{\prime}_{\bm{\gamma}}}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{\sigma}}^{2}]$
holds. The rest of the proof is a simple application of Lemma 2. ∎

Informally, the rest of the proof exploits the power-preserving mapping
between
$\inf_{\mathbbm{T}\in\mathcal{T}_{\bm{\gamma}}}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{\sigma}}^{2}]$
and
$\inf_{\mathbbm{T}\in\mathcal{T}^{\prime}_{\bm{\gamma}}}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{\sigma}}^{2}]$
given by Lemma 7, and the fact that
$\inf_{\mathbbm{T}\in\mathcal{T}^{\prime}_{\bm{\gamma}}}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{\sigma}}^{2}]$
is a singleton by Lemma 6, to exclude that
$\inf_{\mathbbm{T}\in\mathcal{T}_{\bm{\gamma}}}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{\sigma}}^{2}]$
is attained by points with different power consumption. More precisely, since
the optimum $\mathbbm{T}^{\star}$ to the original Problem (10) attains
$\inf_{\mathbbm{T}\in\mathcal{T}_{\bm{\gamma}}}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{1}+\bm{\lambda}^{\star}}^{2}]$
and satisfies the power constraints, then, by Lemma 7,
$\exists\mathbbm{T}^{\prime}$ attaining
$\inf_{\mathbbm{T}\in\mathcal{T}_{\bm{\gamma}}^{\prime}}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{1}+\bm{\lambda}^{\star}}^{2}]$
and satisfying the power constraints. Since, by Lemma 6, this
$\mathbbm{T}^{\prime}$ must be unique, there cannot be some
$\mathbbm{T}\neq\mathbbm{T}^{\star}$ attaining
$\inf_{\mathbbm{T}\in\mathcal{T}_{\bm{\gamma}}}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{1}+\bm{\lambda}^{\star}}^{2}]$
and violating the power constraints.

###  -C Convergence of the projected subgradient method

We apply [32, Algorithm 3.2.8] to the the minimization of $-\tilde{d}$ over
$\bm{\lambda}\in\mathbbmss{R}_{+}^{L}$. From known subgradient calculus rules,
since $-\tilde{d}$ is the supremum of a family of affine functions indexed by
$\mathbbm{T}\in\mathcal{T}_{\bm{\gamma}}$, a subgradient at
$\bm{\lambda}\in\mathbbmss{R}^{L}$ is given by the gradient of any of these
function attaining the supremum. This leads to the proposed algorithm. For all
$\bm{\lambda}\in\mathbbmss{R}_{+}^{L}$, nonemptineess of
$\arg\min_{\mathcal{T}_{\bm{\gamma}}}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{1}+\bm{\lambda}}^{2}]$
follows by combining: (i)
$\inf_{\mathbbm{T}\in\mathcal{T}_{\bm{\gamma}}}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{\sigma}}^{2}]=\inf_{\mathbbm{T}\in\mathcal{T}^{\prime}_{\bm{\gamma}}}\sum_{k=1}^{K}\mathsf{E}[\|\mathbbm{t}_{k}\|_{\bm{\sigma}}^{2}]$,
similarly to Lemma 3; (ii) Lemma 6; (iii)
$\mathcal{T}_{\bm{\gamma}}^{\prime}\subseteq\mathcal{T}_{\bm{\gamma}}$.
Convergence of the best objective to the optimum
$\tilde{d}(\bm{\lambda}^{\star})$ for $n\to\infty$ follows from [32, Lemma
3.2.1] and the proof of [32, Theorem 3.2.2], without using the Lipschitz
continuity assumption. Furthermore, convergence of the corresponding argument
follows since $\tilde{d}:\mathbbmss{R}^{L}\to\mathbbmss{R}$ is concave and
hence continuous.

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