# Fixed-point methods for long-term power control and beamforming design in
large-scale MIMO

Lorenzo Miretti , Renato L. G. Cavalcante ,
Sławomir Stańczak L. Miretti, R. L. G. Cavalcante, and S. Stańczak are with
the Fraunhofer Institute for Telecommunications Heinrich-Hertz-Institut HHI,
Berlin 10587, Germany (email: {miretti, cavalcante,
stanczak}@hhi.fraunhofer.de). L. Miretti and S. Stańczak are also with
Technische Universität Berlin, Berlin 10587, Germany. The authors acknowledge
the financial support by the German Federal Ministry of Education and Research
(BMBF) in the programme of “Souverän. Digital. Vernetzt.” Project 6G-ANNA
(16KISK087) and Joint Project 6G-RIC (16KISK020K and 16KISK030).

###### Abstract

This study presents novel applications of fixed-point methods to solve
previously open joint power control and beamforming design problems in modern
large-scale MIMO systems, e.g., based on the cell-free massive MIMO and XL-
MIMO concepts. In particular, motivated by the need for scalable system
architectures, we revisit the classical sum power minimization and max-min
fair design criteria by considering long-term power control and beamforming
design based on channel statistics and possibly limited channel state
information (CSI) sharing across distributed processing units. This approach
is believed to mitigate the severe scalability issues of competing short-term
optimal algorithms in the literature, which must be executed for every channel
realization by a central controller endowed with global CSI, hence imposing
very demanding requirements in terms of computation and interconnection
capabilities. The obtained optimal algorithms are then illustrated and
compared against existing short-term and long-term approaches via numerical
simulations in a cell-free massive MIMO setup.

††publicationid: pubid:  This work has been submitted to the IEEE for possible
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##  I Introduction

A classical problem in wireless communication systems employing multi-user
multiple-input-multiple-output (MU-MIMO) technologies for spectrally efficient
multiple access is the joint optimization of transmit power and beamformers
according to a given design criterion. Two popular criteria are the sum power
minimization subject to signal-to-interference-to-noise (SINR) constraints,
and the maximization of the minimum weighted SINR subject to power constraints
(sometimes also referred to as the SINR balancing problem) [1, 2, 3, 4, 5]. By
focusing on the uplink of a system with $K$ single-antenna users and a base
station equipped with $M$ antennas, these two optimization problems can be
informally stated as

$\underset{\begin{subarray}{c}\bm{p},\bm{v}_{1},\ldots,\bm{v}_{K}\end{subarray}}{\text{minimize}}\;\sum_{k\in\mathcal{K}}p_{k}\text{
s.t. }(\forall
k\in\mathcal{K})~{}\mathsf{SINR}_{k}(\bm{H},\bm{v}_{k},\bm{p})\geq\gamma_{k},$
(1)

and

$\underset{\begin{subarray}{c}\bm{p},\bm{v}_{1},\ldots,\bm{v}_{K}\end{subarray}}{\text{maximize}}\;\min_{k\in\mathcal{K}}\frac{\mathsf{SINR}_{k}(\bm{H},\bm{v}_{k},\bm{p})}{\gamma_{k}}\text{
s.t. }(\forall k\in\mathcal{K})~{}p_{k}\leq P,$ (2)

where $\mathcal{K}=\\{1,\ldots,K\\}$ is the set of user indexes,
$(\gamma_{1},\ldots,\gamma_{K})\in\mathbbmss{R}_{+}^{K}$ and
$P\in\mathbbmss{R}_{+}$ are problem parameters, and $\mathsf{SINR}_{k}$
denotes the instanteous SINR of user $k$ for given MU-MIMO channel matrix
$\bm{H}\in\mathbbmss{C}^{K\times M}$, power control coefficients
$\bm{p}=(p_{1},\ldots,p_{K})\in\mathbbmss{R}_{+}^{K}$, and receive beamformers
$\bm{v}_{1},\ldots,\bm{v}_{K}\in\mathbbmss{C}^{M}$. Current literature
provides many efficient and provably convergent algorithms for solving not
only (1) and (2), but also many variants covering, for example, downlink
transmission. Notably, these problems can be solved via fixed-point methods
based on an axiomatic uplink power control framework introduced in [6] for sum
power minimization problems, and extended in [7] to weighted max-min problems.
Furthermore, although developed for the uplink, this framework can also be
applied to related downlink problems by means of uplink-donwlink duality
arguments [2, 4, 5]. Embracing this abstract framework, as opposed to solving
problems on a case-by-case basis, offers many advantages. In particular, we
not only gain valuable insights into the network behavior at a high-level [8,
9, 10], but we also gain access to powerful numerical techniques for
addressing more involved variants of (1) and (2) without the need for ad-hoc
proofs [5, 11, 12, 13]. In this study, we present a novel application of the
abstract framework in [6, 7] to non-trivial variants of (1) and (2) tailored
to large-scale MIMO systems.

###  I-A Long-term power control and beamforming design

In recent years, problems of the type in (1) and (2) have attracted
significantly renewed interest in the context of massive MIMO [14, 15], and,
in particular, of its most ambitious embodiments such as those based on the
concepts of cell-free massive MIMO [16, 17] or extremely large-scale MIMO (XL-
MIMO) [18]. Most of the research on power control and beamforming design for
these future technologies [14, 15, 16, 19, 20, 21, 22, 17, 23] is essentially
devoted to addressing the following scalability issues of the aforementioned
classical approaches:

  * •

first, (1) and (2) need to be solved for every realization of $\bm{H}$, which
imposes demanding requirements in terms of computational complexity; and

  * •

second, the solutions to (1) and (2) are functions of all entries of $\bm{H}$
in general, which results in high signaling overhead for exchanging the
required channel state information (CSI) among antennas and (potentially
distributed) processing units.

The main working assumption of this study is that the above challenges make
classical techniques impractical for large-scale MIMO systems, which calls for
alternative approaches. Leaving heuristics aside, we can circumvent the above
limitations by considering optimization problems involving ergodic rate
expressions of the type

$\mathsf{E}[\log_{2}(1+\mathsf{SINR}_{k}(\mathbbm{H},\mathbbm{v}_{k},\bm{p})],$
(3)

where $\mathbbm{H}$ is a random matrix, and where $\mathbbm{v}_{k}$ is a
function of the available CSI. This approach, which we refer to as the long-
term approach, has the potential advantage that the power vector and
beamforming design (i.e., the _functions_ mapping CSI to instantaneous
beamforming coefficients) can be optimized only once for many channel
realizations based on relatively slowly time-varying channel statistics. In
addition, it enables the explicit modeling of constraints, such as limited CSI
sharing, using the notion of information constraints proposed in [24].
Introducing these information constraints is particularly useful for
distributed beamforming design in cell-free networks in scenarios where
beamformers are computed locally at each access point (AP) based on local or
partially shared channel measurements [16, 17]. In particular, note that
common AP clustering techniques [25, 17] alone are not sufficient to model
these scenarios. The main drawback of the long-term approach is that it leads
to challenging functional optimization problems in (infinite dimensional)
Hilbert spaces, and the difficulty is further exacerbated by the presence of
an expectation outside the logarithm in the rate expressions. Therefore, as of
today, no optimal method for joint long-term power control and beamforming
design for large-scale MIMO systems is available in the literature.

###  I-B Proposed solution

The main contribution of this study is to show that, by replacing (3) with the
so-called use-and-then-forget (UatF) bound on the achievable uplink ergodic
rates [14, 15], we are able to formulate tractable variants of (1) and (2) to
consider, for the first time, long-term power control and (potentially
distributed) beamforming design. Specifically, we show that the proposed
problem formulations enable us to derive globally optimal fixed-point methods
based on the same axiomatic power control framework in [6, 7] used for solving
(1) and (2). The key step of our derivation relies on a previously unexplored
relation between the maximization of the UatF bound and a minimum mean-square
error (MMSE) problem under arbitrary information constraints [24]. We remark
that a similar relation is well-known for ergodic rate expressions of the type
in (3) [15], but it it does not lead to tractable joint long-term power
control and beamforming design problems, and it is only valid for centralized
beamforming [17].

To relate the main contribution of the present study with existing approaches,
we recall that iterative algorithms for long-term power control under fixed
beamforming design based on the UatF bound (or its related downlink version)
have been extensively studied in both the cellular and cell-free massive MIMO
literature (see, e.g., [14, 15, 16, 19, 22, 21, 17, 23]). Furthermore, the
optimal centralized beamforming design under fixed power control based on (3),
which corresponds to the classical MMSE beamforming design, is well known, but
the fact that this design also maximizes the UatF bound has been largely
overlooked in the literature, to the best of our knowledge. Moreover, the
problem of optimal distributed beamforming design based on the UatF bound with
fixed power control has been solved only recently in [24]. However, none of
these previous studies have jointly optimized beamformers and the transmit
power. The closest approaches to the problem at hand date back to the
aforementioned short-term methods addressing (1) and (2), which suffer from
scalability issues, and they do not cover distributed beamforming design.
Related results based on the long-term approach and the UatF bound have been
reported in [20]. That study derives an optimal algorithm for long-term power
control and large-scale fading decoding (LSFD) design in cell-free networks,
but the beamformers are restricted to the maximum-ratio combining beamformers,
so that study considers a simpler version of the problem described later. In
fact, we point out that the problem in [20] is a particular instance of the
framework in [7].

###  I-C Summary of contributions and organization of the paper

As a preliminary step towards the derivation of the main contribution of this
study, we provide in Section II an up-to-date introduction to the axiomatic
framework for power control in [6, 7]. We offer a concise review, uncovering
overlooked connections within the existing literature. Then, equipped with the
necessary background, in Section III we formally define the considered long-
term power control and beamforming design problems for large-scale MIMO
systems, and we establish their connection to the above axiomatic power
control framework. To illustrate the potential of the proposed approach, in
Section IV we conduct an extensive numerical study on canonical user-centric
cell-free massive MIMO networks. We investigate many approaches for power
control and beamforming design, and, as interesting byproducts of our main
contribution, we obtain the following additional results:111A preliminary
version of this study focusing on max-min fairness and distributed beamforming
was presented in [26]. This study extends [26] by including the minimum sum
power criterion and other beamforming models. Furthermore, it significantly
improves the comparison against competing approaches, and it presents a more
complete derivation and analysis of the algorithms based on an up-to-date
review of fixed-point methods. In addition, it provides novel theoretical and
experimental insights on cell-free networks.

  * •

We present the first optimal comparison of small-cells networks and cell-free
massive MIMO networks regarding their ability to deliver uniform quality of
service to all users. Specifically, our study takes into account optimal joint
power control and potentially distributed beamforming design. Notably, we
provide a straightforward explanation for performance differences using direct
and intuitive arguments. Our approach departs from previous studies such as
those in [16, 21, 17], which, relying on suboptimal schemes, frequently yield
results that are hard to interpret and may even lead to different conclusions.

  * •

We contribute with novel insights to the ongoing debate on the appropriate
ergodic rate bound for studying large-scale MIMO networks [15, 27, 17, 28].
The UatF bound underestimates the true system performance [27], but we give
numerical evidence showing that it can serve as an excellent proxy for system
optimization when dealing with intractable ergodic rate expressions such as
those in (3). For the centralized beamforming case, this evidence is further
strengthened by noticing that the known optimal beamforming design derived
under an expression similar to (3) [15, 17] coincides with the optimal design
derived under the UatF bound.

  * •

We show that long-term power control and beamforming design may provide non-
negligible performance gains compared to power control with fixed beamformers
in both centralized and distributed cell-free massive MIMO networks.
Furthermore, we show that competing short-term centralized methods may not
only be impractical, but they also suffer from a non-negligible performance
loss. Moreover, we confirm that beamfoming design is the main bottleneck in
distributed cell-free networks [17].

Notation and mathematical preliminaries: The sets of nonnegative and positive
reals are denoted by, respectively, $\mathbbmss{R}_{+}$ and
$\mathbbmss{R}_{++}$. Given $K\in{\mathbb{N}}$, we denote by $x_{k}$ the $k$th
coordinate of a vector ${\bm{x}}\in\mathbbmss{R}^{K}$. The $k$th column of the
$K$-dimensional identity matrix $\bm{I}_{K}$ is denoted by $\bm{e}_{k}$.
Inequalities involving vectors should be understood coordinate-wise; i.e.,
$(\forall{\bm{x}}\in\mathbbmss{R}_{+}^{K})(\forall{\bm{y}}\in\mathbbmss{R}_{+}^{K})(\forall
k\in\\{1,\ldots,K\\})~{}x_{k}\leq y_{k}\Leftrightarrow{\bm{x}}\leq{\bm{y}}$. A
norm $\|\cdot\|$ in $\mathbbmss{R}^{K}$ is said to be _monotone_ if
$(\forall{\bm{x}}\in\mathbbmss{R}_{+}^{K})(\forall{\bm{y}}\in\mathbbmss{R}_{+}^{K})~{}{\bm{x}}\leq{\bm{y}}\Rightarrow\|{\bm{x}}\|\leq\|{\bm{y}}\|$.
We say that a sequence
$({\bm{x}}_{n})_{n\in{\mathbb{N}}}\subset\mathbbmss{R}^{K}$ converges to
${\bm{x}}^{\star}$ if $\lim_{n\to\infty}\|{\bm{x}}_{n}-{\bm{x}}^{\star}\|=0$
for some (and hence for every) norm $\|\cdot\|$ in $\mathbbmss{R}^{K}$. A set
$C\subset\mathbbmss{R}^{K}_{+}$ is said to be downward comprehensive (on
$\mathbbmss{R}^{K}_{+}$) if
$(\forall{\bm{x}}\in\mathbbmss{R}_{+}^{K})(\forall{\bm{y}}\in
C)~{}{\bm{x}}\leq{\bm{y}}\Rightarrow{\bm{x}}\in C$. By $\mathrm{conv}(C)$ we
denote the convex hull of a set $C\in\mathbbmss{R}^{K}$. Given a mapping
$T:X\to Y$, where the sets $X$ and $Y$ satisfy $Y\subset X$, we denote by
$\mathrm{Fix}(T):=\\{x\in X~{}|~{}T(x)=x\\}$ the set of fixed points of $T$.
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
$(\forall\mathbbm{x}\in\mathcal{H}^{K})$
$\mathsf{E}[\|\mathbbm{x}\|_{2}^{2}]<\infty$. Together with the standard
operations of addition and real scalar multiplication, we recall that
$\mathcal{H}^{K}$ is a real vector space.

##  II Axiomatic framework for power control

In this section we review a clear and concise abstract framework that defines
the expected characteristics of the power control problems at hand. In more
detail, we investigate networks consisting of $K$ users denoted by the set
$\mathcal{K}:=\\{1,\ldots,K\\}$. To each user $k\in\mathcal{K}$ we associate
an utility function $u_{k}:\mathbbmss{R}^{K}_{+}\to\mathbbmss{R}_{+}$, and we
assume that all functions $(u_{k})_{k\in\mathcal{K}}$ take a common argument
${\bm{p}}=(p_{1},\ldots,p_{K})\in\mathbbmss{R}_{+}^{K}$. For example, given a
vector ${\bm{p}}\in\mathbbmss{R}_{+}^{K}$, the overall utility of all users is
given by the utility vector
$(u_{1}({\bm{p}}),\ldots,u_{K}({\bm{p}}))\in\mathbbmss{R}_{+}^{K}$. To be
concrete, in the next sections, we use the coordinate
$p_{k}\in\mathbbmss{R}_{+}$ of ${\bm{p}}\in\mathbbmss{R}_{+}^{K}$ to denote
the transmit power of user $k\in\mathcal{K}$. Furthermore, we use the
coordinate $u_{k}({\bm{p}})\in\mathbbmss{R}_{+}$ of the utility vector
$(u_{1}({\bm{p}}),\ldots,u_{K}({\bm{p}}))\in\mathbbmss{R}^{K}_{+}$ to denote
the SINR or the achievable rate of user $k\in\mathcal{K}$ for a given
${\bm{p}}\in\mathbbmss{R}_{+}^{K}$.

We impose power limits on the network by constraining power vectors to sets
$C\subset\mathbbmss{R}_{+}^{K}$ of the type
$C:=\\{{\bm{p}}\in\mathbbmss{R}_{+}^{K}~{}|~{}\|{\bm{p}}\|\leq P\\}$, where
$P$ is a given constant (typically a maximum transmit power), and, as in [7],
$\|\cdot\|$ is a monotone norm on $\mathbbmss{R}^{K}$. This structure for
power constraints is common in the literature. For example, the monotone norm
is typically chosen to be either the $l_{1}$ norm (sum power constraints) or a
weighted $l_{\infty}$ norm (per-user power constraints). Nevertheless, the
structure assumed to the set $C$ can cover a much larger class of power
constraints. More precisely, let $S\subset\mathbbmss{R}^{K}_{+}$ be any set
with nonempty interior that is also convex, compact, and downward-
comprehensive. Then the Minkowski functional
$(\forall{\bm{p}}\in\mathbbmss{R}^{K})~{}\|{\bm{p}}\|:=\inf\\{\gamma>0~{}|~{}(1/\gamma){\bm{p}}\in\mathrm{conv}(-S\cup
S)\\}$ of $\mathrm{conv}(-S\cup S)$ is a monotone norm satisfying
$S=\\{{\bm{p}}\in\mathbbmss{R}_{+}^{K}~{}|~{}\|{\bm{p}}\|\leq P\\}$ [29,
Proposition 2].

To keep the utility optimization problems tractable while maintaining the
framework reasonably general, we need to restrict the class of utility
functions. To this end, we leverage the concept of standard interference
functions, defined as follows:

######  Definition 1.

[6] A function $f:\mathbbmss{R}^{K}_{+}\to\mathbbmss{R}_{++}$ is said to be a
_standard interference function_ if the following properties hold:

  1. (i)

$(\forall{\bm{p}}\in\mathbbmss{R}^{K}_{+})(\forall{\bm{q}}\in\mathbbmss{R}^{N}_{+})~{}{\bm{p}}\geq{\bm{q}}\Rightarrow
f({\bm{p}})\geq f({\bm{q}})$; and

  2. (ii)

$(\forall{\bm{p}}\in\mathbbmss{R}^{K}_{+})$ $(\forall\alpha>1)$
$\alpha{f}({\bm{p}})>f(\alpha{\bm{p}})$.

Likewise, a mapping
$T:\mathbbmss{R}_{+}^{K}\to\mathbbmss{R}_{++}^{K}:{\bm{p}}\mapsto(f_{1}({\bm{p}}),\ldots,f_{N}({\bm{p}}))$
is said to be a standard interference (SI) mapping if each coordinate function
$f_{k}:\mathbbmss{R}^{K}_{+}\to\mathbbmss{R}_{++}$ ($k=1,\ldots,K$) is a
standard interference function.

######  Remark 1.

Standard interference functions $f:\mathbbmss{R}^{K}_{+}\to\mathbbmss{R}_{++}$
are continuous on $\mathbbmss{R}_{++}^{K}$, but they are not necessarily
continuous on the boundary
$\mathbbmss{R}^{K}_{+}\backslash\mathbbmss{R}_{++}^{K}$. Nevertheless, for all
practical purposes, hereafter we can tacitly assume continuity. The reason is
that the restriction
$f|_{\mathbbmss{R}_{++}^{K}}:\mathbbmss{R}^{K}_{++}\to\mathbbmss{R}_{++}:{\bm{p}}\mapsto
f({\bm{p}})$ of $f$ to ${\mathbbmss{R}_{++}^{K}}$ can be continuously extended
to $\mathbbmss{R}_{+}^{K}$ [30, Theorem 5.1.5].

Identifying standard interference functions directly from the definition is
not necessarily easy. However, there are simple techniques that can assist us
in this task. In the next two propositions we list some of them.

######  Proposition 1.

Let $f:\mathbbmss{R}^{K}_{+}\to\mathbbmss{R}_{++}$ be a (positive) concave
function (not necessarily everywhere continuous). Then $f$ is a standard
interference function (see [9, Proposition 1] for a simple proof of this
standard result).

(The result in the next proposition is well known [6], and the proof follows
immediately from Definition 1.)

######  Proposition 2.

Denote by $\mathcal{I}$ the set of standard interference functions mapping
$\mathbbmss{R}_{+}^{K}$ to $\mathbbmss{R}_{++}$. The following operations
preserve the properties in Definition 1:

  1. (i)

(Multiplication by positive scalars)

$(\forall f\in\mathcal{I})(\forall\alpha>0)~{}\alpha f\in\mathcal{I}$

  2. (ii)

(Composition)

$(\forall f\in\mathcal{I})(\forall g\in\mathcal{I})~{}f\circ
g:\mathbbmss{R}_{+}^{K}\to\mathbbmss{R}_{++}:{\bm{p}}\mapsto
f(g({\bm{p}}))\in\mathcal{I}$

  3. (iii)

(Maximum)

$(\forall f\in\mathcal{I})(\forall
g\in\mathcal{I})~{}{\bm{p}}\mapsto\max\\{f({\bm{p}}),g({\bm{p}})\\}\in\mathcal{I}$

  4. (iv)

(Minimum)

$(\forall f\in\mathcal{I})(\forall
g\in\mathcal{I})~{}{\bm{p}}\mapsto\min\\{f({\bm{p}}),g({\bm{p}})\\}\in\mathcal{I}$

For later reference, we say that a mapping
$T:\mathbbmss{R}_{+}^{K}\to\mathbbmss{R}_{++}^{K}$ is a positive concave
mapping if it can be written as
$(\forall{\bm{p}}\in\mathbbmss{R}_{+}^{K})~{}T({\bm{p}})=(f_{1}({\bm{p}}),\ldots,f_{K}({\bm{p}}))$,
where, for every $k\in\\{1,\ldots,K\\}$,
$f_{k}:\mathbbmss{R}^{K}_{+}\to\mathbbmss{R}_{++}$ is a (positive) concave
function.

######  Remark 2.

Properties (ii)-(iv) in Proposition 2 can be naturally extended to consider a
finite family of standard interference functions, but extending those results
to an infinite family is not possible in general. For example, for each
$u\in\mathbbmss{R}_{++}$, the constant function
$f_{u}:\mathbbmss{R}_{+}\to\mathbbmss{R}_{++}:x\mapsto u$ is a standard
interference function, but $x\mapsto\inf_{u\in\mathbbmss{R}_{++}}f_{u}(x)=0$
is not a standard interference function.

We now have in place all the required definitions to describe the utility
functions covered by the framework.

######  Assumption 1.

The utility function $u_{k}$ of each user $k\in\mathcal{K}$ takes the form

$\displaystyle
u_{k}:\mathbbmss{R}_{+}^{K}\to\mathbbmss{R}_{+}:{\bm{p}}\mapsto\dfrac{p_{k}}{f_{k}({\bm{p}})},$
(4)

where $f_{k}:\mathbbmss{R}_{+}^{K}\to\mathbbmss{R}_{++}$ is a _continuous_
standard interference function.

The utility functions described above are designed to take only power vectors
as arguments. This fact raises the question of whether the framework is
limited to power optimization problems. However, we can include additional
network parameters, such as receive beamformers, by implicitly incorporating
them in the definition of the utility functions. This approach allows for more
flexibility in the framework, and it is illustrated below.

######  Example 1.

(Short-term power control and beamforming) Consider the uplink of a system
with $K$ single-antenna users and a base station equipped with $M$ antennas.
For each user $k\in\mathcal{K}$, we denote by ${\bm{v}}_{k}\in\mathbb{C}^{M}$
and ${\bm{h}}_{k}\in\mathbb{C}^{M}\backslash\\{{\bm{0}}\\}$, respectively, the
receive beamformer and the channel of user $k$. By defining as the utility the
SINR with optimal receive beamformers for a given power vector, we have
$(\forall k\in\mathcal{K})$

$u_{k}:\mathbbmss{R}_{+}^{K}\to\mathbbmss{R}_{+}:{\bm{p}}\mapsto\sup_{{\bm{v}}_{k}\in\mathcal{V}_{k}}\dfrac{|{\bm{h}}_{k}^{H}{\bm{v}}_{k}|^{2}p_{k}}{\sum\limits_{j\in\mathcal{K}\backslash\\{k\\}}|{\bm{h}}_{j}^{H}{\bm{v}}_{k}|^{2}p_{j}+1},$
(5)

where
$\mathcal{V}_{k}:=\\{{\bm{v}}\in\mathbb{C}^{M}~{}|~{}\|\bm{v}\|_{2}^{2}=1\\}$
is the set of allowed beamformers. By assuming that the supremum in (5) is
attained by some beamformer in $\mathcal{V}_{k}$, we verify that the functions
$(f_{k})_{k\in\mathcal{K}}$ in Assumption 1 are given by $(\forall
k\in\mathcal{K})$

$f_{k}:\mathbbmss{R}_{+}^{K}\to\mathbbmss{R}_{++}:{\bm{p}}\mapsto\inf\limits_{{\bm{v}}_{k}\in\mathcal{S}_{k}}\dfrac{\sum\limits_{j\in\mathcal{K}\backslash\\{k\\}}|{\bm{h}}_{j}^{H}{\bm{v}}_{k}|^{2}p_{j}+1}{|{\bm{h}}_{k}^{H}{\bm{v}}_{k}|^{2}},$
(6)

where
$\mathcal{S}_{k}:=\mathcal{V}_{k}\backslash\\{{\bm{v}}\in\mathbbmss{C}^{M}|{\bm{h}}_{k}^{H}{\bm{v}}=0\\}$.
For every $k\in\mathcal{K}$, the function $f_{k}$ defined above is not only
positive, but also concave because it is the supremum of affine functions. As
a result, in light of Proposition 1, we conclude that $f_{k}$ is a standard
interference function. Clearly, short-term uplink power control and
beamforming problems of the type in (1) and (2) can be readily mapped to
optimization problems using the utilities in (5), where the choice of
beamformers is implicit in the definition of the utility.

Next, we proceed to discuss two important problems that can be solved with
provably convergent fixed point algorithms.

######  Remark 3.

The algorithmic approaches for solving the utility optimization problems
described in the next sections require a simple computational means for
evaluating the functions $(f_{k})_{k\in\mathcal{K}}$, which is not necessarily
easy. This fact can be seen from Example 1. More specifically, the functions
in (6) require the solution to an optimization problem that depends on the
structure of the sets $\mathcal{V}_{1},\ldots,\mathcal{V}_{K}$ of beamformers.
In that particular example, the solution can be obtained directly from
generalized Rayleigh quotients. However, if the sets become more complex, as
in the long-term power control and beamforming design problems for large-scale
MIMO systems introduced later in this study, then we need to use alternative
techniques for evaluating $(f_{k})_{k\in\mathcal{K}}$.

###  II-A Minimization of the sum transmit power

In this section, we describe the utility optimization problems first
considered in the wireless domain in the seminal study in [6]. We also
summarize recent results and connections to fixed point theory in Thompson
metric spaces. These connections seem to be largely ignored in the wireless
literature, but, as shown in [7, 31], their knowledge enable us to (i) gain
powerful mathematical machinery to answer many questions related to power
control problems in a unified way and to (ii) give alternative proofs of some
results in [6].

In more detail, the first power control problem we describe has the objective
of minimizing the sum transmit power subject to minimum utility requirements.
More precisely, we consider the following problem:

$\displaystyle\underset{\begin{subarray}{c}\bm{p}\in\mathbbmss{R}_{+}^{K}\end{subarray}}{\text{minimize}}$
$\displaystyle\quad\|\bm{p}\|_{1}$ (7) subject to $\displaystyle\quad(\forall
k\in\mathcal{K})~{}u_{k}(\bm{p})\geq\gamma_{k}$

where $(\gamma_{1},\ldots,\gamma_{K})\in\mathbbmss{R}_{++}^{K}$ are the given
minimum requirements for the utility functions $(u_{k})_{k\in\mathcal{K}}$
satisfying Assumption 1. We now proceed to review answers to three important
questions related to the optimization problem in (7):

  1. (Q1)

Does a solution exist?

  2. (Q2)

Is the solution unique?

  3. (Q3)

Are there simple algorithms able to compute the solution?

To answer the above questions, we first reformulate (7) as a fixed point
problem. To this end, we can use well-known properties of the utility
functions satisfying Assumption 1 (see, for example, [32, Lemma 1]) and
standard arguments in the literature to verify that, if a solution to (7)
exists, all constraints have to be satisfied with equality. Therefore,
assuming ${\bm{p}}^{\star}=(p_{1}^{\star},\ldots,p_{K}^{\star})$ to be a
solution to (7), we deduce from Assumption 1:

$(\forall
k\in\mathcal{K})~{}u_{k}({\bm{p}}^{\star})=\gamma_{k}\Leftrightarrow(\forall
k\in\mathcal{K})~{}p_{k}^{\star}=\gamma_{k}f_{k}({\bm{p}}^{\star})\\\
\Leftrightarrow{\bm{p}}^{\star}\in\mathrm{Fix}(T),$

where
$T:\mathbbmss{R}_{+}^{K}\to\mathbbmss{R}_{++}^{K}:{\bm{x}}\mapsto(\gamma_{1}f_{1}({\bm{x}}),\ldots,\gamma_{K}f_{K}({\bm{x}}))$
is a standard interference mapping as an implication of Proposition 2(i).
These relations reveal that properties of solutions to (7) are connected to
properties of the fixed point set of the standard interference mapping $T$.

Recalling that standard interference mappings are only guaranteed to be
contractive in the cone $\mathbbmss{R}_{++}^{K}$ with respect to Thompson’s
metric [30, Lemma 2.1.7][7] (but not necessarily Lipschitz contractions), we
conclude that $\mathrm{Fix}(T)$ is either a singleton or the empty set [33,
Ch. 4.1], so we now have a complete answer to questions (Q1) and (Q2): the
solution to the problem in (7) is unique _if it exists_. Furthermore, this
answer enables us to reformulate (7) equivalently as follows:

$\displaystyle\text{Find }{\bm{p}}^{\star}\in\mathbbmss{R}_{++}^{K}\text{ such
that }{\bm{p}}^{\star}\in\mathrm{Fix}(T),$ (8)

and we refer readers to [10, Proposition 4] for a necessary and sufficient
condition for the existence of the fixed point of $T$, and, hence, for the
existence of the necessarily unique solution to (7). This result in [10,
Proposition 4] can frequently simplify the analysis of existence of solutions
to (7) [10].

An additional advantage of the reformulation in (8), which is a standard fixed
point problem involving contractive mappings in Thompson metric space, is that
we may conjecture that the sequence $({\bm{p}}_{n})_{n\in{\mathbb{N}}}$
generated via

$\displaystyle{\bm{p}}_{n+1}=T({\bm{p}}_{n}),$ (9)

converges for any starting point ${\bm{p}}_{1}\in\mathbbmss{R}_{+}^{K}$
(assuming that $\mathrm{Fix}(T)\neq\emptyset$), and this conjecture can be
settled by using basic arguments in analysis [6] or by using arguments based
on fixed point theory in metric spaces [7]. For later reference, we summarize
the above results below.

######  Proposition 3.

[6, 7] Let $T:\mathbbmss{R}_{+}^{K}\to\mathbbmss{R}_{++}^{K}$ be a standard
interference mapping. Then $\mathrm{Fix}(T)$ is a singleton if not empty.
Furthermore, if $\emptyset\neq\mathrm{Fix}(T)\ni\\{{\bm{p}}^{\star}\\}$, then
the sequence $({\bm{p}}_{n})_{n\in{\mathbb{N}}}$ generated via (9) converges
in norm to ${\bm{p}}^{\star}$ for any starting point
${\bm{p}}_{1}\in\mathbbmss{R}_{+}^{K}$.

One potential limitation of the fixed point iteration in (9) is that it can be
slow. Indeed, [34, Example 2] shows that convergence can be sublinear in
general. Nevertheless, if the standard interference functions
$(f_{k})_{k\in\mathcal{K}}$ are further restricted to be positive and concave,
as common in the resource allocation problems described later, then the
convergence is guaranteed to be geometric [31] (in the sense of [31,
Definition 1]) and no better than geometric in general [35, Proposition 5]. In
this case, we can further improve the convergence speed with, for example, the
acceleration technique described in [9] or with the more computationally
intensive but faster approach in [36, Theorem 11]. (For geometric convergence
of fixed point iterations of standard interference mappings that are not
necessarily concave, we refer the readers to [35].)

###  II-B Weighted max-min fair power control

A drawback of the optimization problem in (7) is that it is not guaranteed to
have a solution in general. Moreover, even if a solution exists, users may be
required to transmit with more power than physically available or allowed by
law, or both. To avoid these potential limitations, we can consider
optimization problems that have the objective of maximizing the minimum
weighted utility across all users subject to power constraints, or, more
formally, we solve the following power control problem (see [32] and the
references therein):

$\underset{\|\bm{p}\|\leq
P}{\text{maximize}}\min_{k\in\mathcal{K}}\gamma_{k}^{-1}u_{k}(\bm{p}),$ (10)

where $(\gamma_{1},\ldots,\gamma_{K})\in\mathbbmss{R}_{++}^{K}$ are weights
(or priorities) assigned to the utilities $(u_{1},\ldots,u_{K})$ satisfying
Assumption 1, $\|\cdot\|$ is a given monotone norm, and $P$ is the maximum
allowed transmit power (with respect to the choice of the norm $\|\cdot\|$).

As shown in the next proposition, the optimization problem in (10) always has
a solution that can be obtained with fixed point iterations involving mappings
that are contractive in a (compact) Hilbert-projective metric space. To the
best of our knowledge, in the wireless domain, this connection to fixed point
theory has been first established in [7], and, with slightly different
assumptions on the utility functions and/or the power constraints, these fixed
point algorithms have also been independently proposed in [5, 37, 38] [39, Ch.
4], which are studies building upon the results in [40]. We emphasize that,
unlike the properties imposed on the utility functions in some of these
previous studies, the properties in Definition 1 are not enough to guarantee
uniqueness of the solution to (10), so the characterization in Proposition
4(iii) is important. We omit the proof of Proposition 4 because it is
immediate from [32, Proposition 2], which builds upon the results in [7], and
it can be also obtained with the results in [40].

######  Proposition 4.

Consider the problem in (10), and recall that $(\forall
k\in\mathcal{K})(\forall{\bm{p}}\in\mathbbmss{R}^{K}_{+})~{}u_{k}({\bm{p}})=p_{k}/f_{k}({\bm{p}})$
(Assumption 1). Let $T:\mathbbmss{R}_{+}^{K}\to\mathbbmss{R}_{++}^{K}$ be the
standard interference mappings defined by
$(\forall{\bm{p}}\in\mathbbmss{R}_{++}^{K})~{}T({\bm{p}}):=(\gamma_{1}f_{1}(\bm{p}),\ldots,\gamma_{K}f_{K}(\bm{p}))$.
Then each of the following holds:

  1. (i)

The set $\mathrm{Fix}(\tilde{T})$ of fixed points of the mapping

$\tilde{T}:\mathbbmss{R}_{+}^{K}\to\mathbbmss{R}_{++}^{K}:\bm{p}\mapsto\frac{P}{\|T(\bm{p})\|}T(\bm{p})$

is a singleton, and its unique fixed point
$\bm{p}^{\star}\in\mathrm{Fix}(\tilde{T})$ is a (not necessarily unique)
solution to (10).

  2. (ii)

For every $\bm{p}_{1}\in\mathbbmss{R}_{+}^{K}$, the sequence
$(\bm{p}_{n})_{n\in\mathbbmss{N}}$ generated via

$(\forall n\in\mathbbmss{N})~{}\bm{p}_{n+1}=\tilde{T}(\bm{p}_{n})$ (11)

converges in norm to the fixed point
$\bm{p}^{\star}\in\mathrm{Fix}(\tilde{T})$.

  3. (iii)

Denote by $\mathcal{P}\subset\mathbbmss{R}_{++}^{K}$ the nonempty set of
solutions to the problem in (10). Then the fixed point
$\bm{p}^{\star}\in\mathrm{Fix}(\tilde{T})$ satisfies
$(\forall\bm{p}\in\mathcal{P})~{}\bm{p}\geq\bm{p}^{\star}$; i.e.,
$\bm{p}^{\star}$ is the least element of $\mathcal{P}$ with respect to the
partial ordering induced by the nonnegative cone.

Similarly to the fixed point iteration in (9), the fixed point algorithm in
(11) also converges geometrically (in the sense of [31, Definition 1]) if the
mapping $T:\mathbbmss{R}^{K}_{+}\to\mathbbmss{R}_{++}^{K}$ in (11) is a
(positive) concave mapping [33, Remark 2.1.12].

##  III Application to large-scale MIMO systems

Equipped with the necessary background, we now proceed with the main
contribution of this study. We consider the uplink of a large-scale MIMO
system composed by $K$ single-antenna users indexed by
$\mathcal{K}:=\\{1,\ldots,K\\}$, and a radio access infrastructure equipped
with a total number of antennas $M$ indexed by
$\mathcal{M}:=\\{1,\ldots,M\\}$. The $M$ infrastructure antennas may be split
among many geographically distributed access points equipped with a relatively
low number of antennas as in the cell-free massive MIMO concept [16, 17], or
colocated in a single array as in the XL-MIMO concept [18]. By assuming for
each time-frequency resource a standard synchronous narrowband MIMO channel
model governed by a stationary ergodic fading process, and simple transmission
techniques based on linear receiver processing and on treating interference as
noise, we focus on simultaneously achievable ergodic rates in the classical
Shannon sense, given by the popular _use-and-then-forget_ (UatF) inner bound
on the ergodic capacity region [14]. In more detail, we define the uplink
rates achieved by each user $k\in\mathcal{K}$ for a given power control policy
and beamforming design as

$R_{k}(\mathbbm{v}_{k},\bm{p}):=\log_{2}(1+\mathsf{SINR}_{k}(\mathbbm{v}_{k},\bm{p})),$
(12)
$\mathsf{SINR}_{k}(\mathbbm{v}_{k},\bm{p}):=\leavevmode\resizebox{299.19885pt}{}{$\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})+\underset{j\neq
k}{\sum}p_{j}\mathsf{E}[|\mathbbm{h}_{j}^{\mathsf{H}}\mathbbm{v}_{k}|^{2}]+\mathsf{E}[\|\mathbbm{v}_{k}\|_{2}^{2}]}$},$

where $\bm{p}:=(p_{1},\ldots,p_{K})\in\mathbbmss{R}_{+}^{K}$ is a
deterministic vector of transmit powers, $\mathbbm{h}_{k}\in\mathcal{H}^{M}$
is a random channel vector modeling the fading state between user $k$ and all
$M$ infrastructure antennas, and $\mathbbm{v}_{k}\in\mathcal{H}^{M}$ models
the beamforming vector which is applied by the infrastructure to process the
received signals of potentially all $M$ antennas to obtain a soft estimate of
the transmit signal of user $k$. We stress that $\mathbbm{v}_{k}$ is generally
a function of the instantaneous CSI realizations, and hence it is denoted as a
random vector. For convenience, we also define the global channel matrix
$\mathbbm{H}:=[\mathbbm{h}_{1},\ldots,\mathbbm{h}_{K}]$.

###  III-A Long-term joint power control and beamforming design

Based on the above model, as a non-trivial application of the axiomatic power
control framework in Section II, our objective is to solve the following two
optimization problems:

$\displaystyle\underset{\begin{subarray}{c}\bm{p}\in\mathbbmss{R}_{+}^{K}\\\
\mathbbm{v}_{1},\ldots,\mathbbm{v}_{K}\in\mathcal{H}^{M}\end{subarray}}{\text{minimize}}$
$\displaystyle\quad\|\bm{p}\|_{1}$ (13) subject to $\displaystyle\quad(\forall
k\in\mathcal{K})~{}\mathsf{SINR}(\mathbbm{v}_{k},\bm{p})\geq\gamma_{k}$
$\displaystyle\quad(\forall
k\in\mathcal{K})~{}\mathbbm{v}_{k}\in\mathcal{V}_{k}$
$\displaystyle\quad(\forall
k\in\mathcal{K})~{}\mathsf{E}[\|\mathbbm{v}_{k}\|_{2}^{2}]\neq 0;$

and

$\displaystyle\underset{\begin{subarray}{c}\bm{p}\in\mathbbmss{R}_{+}^{K}\\\
\mathbbm{v}_{1},\ldots,\mathbbm{v}_{K}\in\mathcal{H}^{M}\end{subarray}}{\text{maximize}}$
$\displaystyle\quad\min_{k\in\mathcal{K}}\gamma_{k}^{-1}\mathsf{SINR}_{k}(\mathbbm{v}_{k},\bm{p})$
(14) subject to $\displaystyle\quad\|\bm{p}\|_{\infty}\leq P$
$\displaystyle\quad(\forall
k\in\mathcal{K})~{}\mathbbm{v}_{k}\in\mathcal{V}_{k}$
$\displaystyle\quad(\forall
k\in\mathcal{K})~{}\mathsf{E}[\|\mathbbm{v}_{k}\|_{2}^{2}]\neq 0,$

where $(\gamma_{1},\ldots,\gamma_{K})\in\mathbbmss{R}_{++}^{K}$ is a given
vector of positive coefficients denoting either minimum SINR requirements or
user weights, $P\in\mathbbmss{R}_{++}$ is a given power budget, and the sets
$\mathcal{V}_{1},\ldots,\mathcal{V}_{K}\subseteq\mathcal{H}^{M}$ are given
information constraints (formally defined in Section III-B) that are imposed
to limit the joint processing and information sharing burden, thus enabling
scalable designs for large-scale MIMO systems. For example, as we will see,
the set $\mathcal{V}_{k}$ can be used to limit the dependency of the entries
of $\mathbbm{v}_{k}$ to portions of the CSI available at the infrastructure
side, as well as the number of antennas that are serving user $k$. Note that
letting $\mathcal{V}_{k}=\mathcal{H}^{M}$ allows $\mathbbm{v}_{k}$ to be any
(measurable) function of the global channel $\mathbbm{H}$.

The proposed problems share the following features:

  * •

the power control vector $\bm{p}$ is kept fixed for the whole duration of the
transmission, i.e., it does not depend on the small-scale fading realizations,
and it is optimized based on channel statistics; and

  * •

the beamformers $(\mathbbm{v}_{1},\ldots,\mathbbm{v}_{K})$ adapt to small-
scale fading based on the CSI specified by the information constraints.
However, their design, i.e., the function mapping CSI realizations to the
corresponding beamforming coefficients, is optimized based on channel
statistics as for the power control vector. Note that this leads to functional
optimization problems, for which standard solvers in Euclidean spaces (e.g.,
cvx) are not directly applicable.

Compared to traditional short-term approaches that solve an optimization
problem such as (1) and (2) for each channel (or global CSI) realization, the
long-term approach in this study is particularly attractive for large-scale
MIMO systems due to the following reasons:

  * •

first, the power control vector and beamforming design can be updated
following large-scale fading variations, which are typically several order of
magnitude slower than small-scale fading variations; and

  * •

second, the beamforming design space covers distributed implementations where
the computation of the beamformers can be split among multiple distributed
processing units with limited CSI sharing capabilities, each controlling a
subset of the $M$ infrastructure antennas.

######  Remark 4.

A well-known challenge in long-term approaches is the need to identify,
estimate, and track some form of channel statistics that can be used to
produce effective solutions. In contrast, short-term approaches can be often
implemented based on minimal or even no statistical information. However, the
main working assumption of this study is that short-term approaches are not
feasible in large-scale MIMO systems, thus making long-term approaches not
only an option but rather a necessity.

######  Remark 5.

The rate expression given by (12) assumes that CSI is neglected in the
decoding phase. Alternative rate expressions of the type in (3) based on
treating interference as noise with various forms of CSI at the decoder are
often considered (see [15, 27, 17, 28] and references therein). These
expressions may provide more realistic and less pessimistic capacity
approximations, but long-term power control and beamforming design becomes
challenging because of the expectation outside the logarithm in the resulting
rate expressions. In contrast, in this study we show that (12) leads to
tractable problem formulations. In addition, we will also give numerical
evidence showing that (12) can serve as an excellent proxy for system
optimization according to intractable ergodic rate expressions such as those
in (3).

###  III-B Information constraints

Large-scale MIMO systems are often impaired by practical limitations in the
interconnections among antennas and processing units. In distributed
beamforming implementations, this may lead to imperfect CSI sharing and hence
that the distributed processing units must form their portion of the
beamformers on the basis of different CSI. As an extreme example, in
distributed cell-free massive MIMO networks the access points are typically
assumed to operate on the basis of only _local_ CSI [16], i.e., without any
CSI sharing. However, to date, most related studies do not formally model this
type of constraint, so, strictly speaking, they do not truly consider
distributed beamforming optimization. To address this limitation, we model
constraints related to imperfect CSI sharing following the approach introduced
in [24]. We first partition the $M=NL$ infrastructure antennas into $L$ groups
of $N$ antennas, and assume that the corresponding portions of the beamformers
are computed by $L$ separate processing units. Given an arbitrarily
distributed tuple $(\mathbbm{H},S_{1},\ldots,S_{L})$, where $S_{l}$ is the
available CSI at the $l$th processing unit, we then let

$\mathcal{V}_{k}:=\mathcal{H}_{1}^{N}\times\ldots\times\mathcal{H}_{L}^{N},$
(15)

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 $(\forall\mathbbm{x}\in\mathcal{H}_{l}^{N})$
$\mathsf{E}[\|\mathbbm{x}\|_{2}^{2}]<\infty$, and where
$\Sigma_{l}\subseteq\Sigma$ is the sub-$\sigma$-algebra induced by the CSI
$S_{l}$, which is also called the _information subfield_ of the $l$th
processing unit. The interested reader is referred to [41] for an introduction
to the measure theoretical notions used in the above definitions. However, we
stress that these notions are by no means required for understanding the key
results of this study. The crucial point is that, informally, the constraint
set $\mathcal{V}_{k}$ enforces the portion of $\mathbbm{v}_{k}$ computed by
the $l$th processing unit to be a function of $S_{l}$ only.

Furthermore, another typical constraint for enabling scalable large-scale MIMO
designs is that only a subset of all infrastructure antennas should be used
for decoding the message of a given user. For example, cell-free massive MIMO
networks are expected to implement some type of user-centric rule [25], which
allocates to each user a cluster of serving access points. As shown in [42,
43], this constraint can be modeled by replacing $\mathcal{H}_{l}^{N}$ in (15)
to the set $\\{\bm{0}_{N}~{}(\text{a.s.})\\}$ if the $l$th processing unit is
not serving the $k$th user. Considering also this replacement, we remark that
$\mathcal{V}_{k}$ is a subspace of the real vector space $\mathcal{H}^{M}$
[43].

###  III-C Connection to the axiomatic power control framework

We now map Problem (13) and Problem (14) to an equivalent power control
problem of the type covered in Section II, where the optimization of the
beamformers is implicit in the definition of the utilities
$(u_{1},\ldots,u_{K})$. More precisely, by exploiting the fact that uplink
SINRs are only coupled through the power vector $\bm{p}$, we obtain:

######  Proposition 5.

For each user $k\in\mathcal{K}$, define the utility

$(\forall\bm{p}\in\mathbbmss{R}_{+}^{K})~{}u_{k}(\bm{p}):=\sup_{\begin{subarray}{c}\mathbbm{v}_{k}\in\mathcal{V}_{k}\\\
\mathsf{E}[\|\mathbbm{v}_{k}\|_{2}^{2}]\neq
0\end{subarray}}\mathsf{SINR}_{k}(\mathbbm{v}_{k},\bm{p}),$ (16)

and assume that $(\forall
k~{}\in\mathcal{K})~{}\mathcal{V}^{\prime}_{k}:=\\{\mathbbm{v}_{k}\in\mathcal{V}_{k}~{}|~{}\mathsf{E}[\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{v}_{k}]\neq
0\\}\neq\emptyset$. Then, each of the following holds:

  1. (i)

The utility (16) satisfies $(\forall
k\in\mathcal{K})(\forall\bm{p}\in\mathbbmss{R}_{++}^{K})$

$u_{k}(\bm{p})=\sup_{\mathbbm{v}_{k}\in\mathcal{V}^{\prime}_{k}}\mathsf{SINR}_{k}(\mathbbm{v}_{k},\bm{p})>0;$

  2. (ii)

The utility (16) satisfies (4) with the standard interference function
$(\forall
k\in\mathcal{K})(\forall\bm{p}\in\mathbbmss{R}_{+}^{K})~{}f_{k}(\bm{p}):=$

$\inf_{\mathbbm{v}_{k}\in\mathcal{V}^{\prime}_{k}}\dfrac{p_{k}\mathsf{V}(\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{v}_{k})+\sum_{j\neq
k}p_{j}\mathsf{E}[|\mathbbm{h}_{j}^{\mathsf{H}}\mathbbm{v}_{k}|^{2}]+\mathsf{E}[\|\mathbbm{v}_{k}\|_{2}^{2}]}{|\mathsf{E}[\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{v}_{k}]|^{2}};$

  3. (iii)

Let $\bm{p}^{\star}$ be a solution to Problem (7) (resp. Problem (10)) with
utilities in (16) and monotone norm $\|\cdot\|=\|\cdot\|_{1}$ (resp.,
$\|\cdot\|=\|\cdot\|_{\infty}$). If $(\forall k~{}\in\mathcal{K})$
$\exists\mathbbm{v}^{\star}_{k}\in\mathcal{V}^{\prime}_{k}$ such that
$u_{k}(\bm{p}^{\star})=\mathsf{SINR}_{k}(\mathbbm{v}^{\star}_{k},\bm{p}^{\star})$
(i.e., the $\sup$ is attained), then
$(\bm{p}^{\star},\mathbbm{v}_{1}^{\star},\ldots,\mathbbm{v}_{K}^{\star})$
solves Problem (13) (resp. Problem (14)).

###### Proof.

  1. (i)

Since
$\mathcal{V}^{\prime}_{k}\subset\\{\mathbbm{v}_{k}\in\mathcal{V}_{k}~{}|~{}\mathsf{E}[\|\mathbbm{v}_{k}\|_{2}^{2}]\neq
0\\}$, we have
$(\forall\bm{p}\in\mathbbmss{R}_{+}^{K})~{}u_{k}(\bm{p})\geq\sup_{\mathbbm{v}_{k}\in\mathcal{V}^{\prime}_{k}}\mathsf{SINR}_{k}(\mathbbm{v}_{k},\bm{p})$.
By definition of $\mathcal{V}^{\prime}_{k}$,
$\sup_{\mathbbm{v}_{k}\in\mathcal{V}^{\prime}_{k}}\mathsf{SINR}_{k}(\mathbbm{v}_{k},\bm{p})>0$
for $p_{k}>0$. This also shows that beamformers such that
$\mathsf{E}[\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{v}_{k}]=0$ can be omitted
when evaluating (16).

  2. (ii)

To confirm that $f_{k}$ is indeed a standard interference function, recall
that $f_{k}$ is concave because it is constructed by taking the point-wise
infimum of affine (and hence concave) functions. Furthermore, by definition of
$\mathcal{V}_{k}^{\prime}$, which also implies
$(\forall\mathbbm{v}_{k}\in\mathcal{V}_{k}^{\prime})~{}\mathsf{E}[\|\mathbbm{v}_{k}\|_{2}^{2}]\neq
0$,

$\displaystyle(\forall\bm{p}\in\mathbbmss{R}_{+}^{K})~{}f_{k}(\bm{p})\geq\inf_{\mathbbm{v}_{k}\in\mathcal{V}^{\prime}_{k}}\dfrac{\mathsf{E}[\|\mathbbm{v}_{k}\|_{2}^{2}]}{|\mathsf{E}[\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{v}_{k}]|^{2}}>0.$

Therefore, we conclude that $f_{k}$ is not only concave but also positive, and
hence, according to Proposition 1, it is a standard interference function.

  3. (iii)

Immediate, hence omitted.

∎

The main implication of Proposition 5 is that, if the mild technical condition
$(\forall k\in\mathcal{K})$ $\mathcal{V}_{k}^{\prime}\neq\emptyset$ holds, an
optimal power vector $\bm{p}^{\star}$ for Problem (13) and Problem (14) can be
obtained via the (normalized) fixed-point iterations in Section II, where each
$i$th step involves the evaluation of the standard interference mapping

$T(\bm{p})=\begin{bmatrix}\dfrac{\gamma_{1}p_{1}}{u_{1}(\bm{p})}&\ldots&\dfrac{\gamma_{K}p_{K}}{u_{K}(\bm{p})}\end{bmatrix}$

for $\bm{p}=\bm{p}_{i}\in\mathbbmss{R}_{++}^{K}$, where $u_{k}(\bm{p})$ is
given by (16). The main challenge of this approach lies in the computation of
$u_{k}(\bm{p})$, which requires solving a SINR maximization problem over the
set of feasible beamformers. The following section discusses how to perform
this step, and it shows that the supremum in (16) is attained for all
$\bm{p}\in\mathbbmss{R}_{++}^{K}$ under the technical condition
$\mathcal{V}_{k}^{\prime}\neq\emptyset$, which is typically satisfied in
practice.

######  Remark 6.

The condition $\mathcal{V}_{k}^{\prime}\neq\emptyset$ holds for most fading
and CSI distributions of practical interest. However, note that
$\mathcal{V}_{k}^{\prime}\neq\emptyset$ does not hold if the infrastructure
has no CSI and, in addition, if the channels have zero mean. In this case,
$(\forall\mathbbm{v}_{k}\in\mathcal{V}_{k})~{}\mathsf{E}[\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{v}_{k}]=\mathsf{E}[\mathbbm{h}_{k}]^{\mathsf{H}}\mathsf{E}[\mathbbm{v}_{k}]=0$
holds. This is related to one of the main drawbacks of the UatF bounding
technique, which requires either sufficient CSI or a strong channel mean to
produce nonzero rate bounds.

###  III-D SINR maximization via MSE minimization

Solving the SINR maximization problems in (16) 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 MSE for fixed
channel matrices. Let us consider for all $k\in\mathcal{K}$ the MSE between
the signal $x_{k}\sim\mathcal{CN}(0,1)$ of user $k$ and its soft estimate
$\hat{x}_{k}:=\mathbbm{v}_{k}^{\mathsf{H}}\mathbbm{y}$ obtained by processing
the output of the uplink channel
$\mathbbm{y}:=\sum_{k\in\mathcal{K}}\sqrt{p_{k}}\mathbbm{h}_{k}x_{k}+\mathbbm{n}$
with noise $\mathbbm{n}\sim\mathcal{CN}(\bm{0},\bm{I})$, where the noise and
all user signals are mutually independent and independent of
$(\mathbbm{H},\mathbbm{v}_{1},\ldots,\mathbbm{v}_{K})$. Specifically, let
$(\forall
k\in\mathcal{K})(\forall\mathbbm{v}_{k}\in\mathcal{V}_{k})(\forall\bm{p}\in\mathbbmss{R}_{+}^{K})$

$\begin{split}\mathsf{MSE}_{k}(\mathbbm{v}_{k},\bm{p}):=&\;\mathsf{E}[|x_{k}-\hat{x}_{k}|^{2}]\\\
=&\;\mathsf{E}\left[\|\bm{P}^{\frac{1}{2}}\mathbbm{H}^{\mathsf{H}}\mathbbm{v}_{k}-\bm{e}_{k}\|_{2}^{2}\right]+\mathsf{E}\left[\|\mathbbm{v}_{k}\|_{2}^{2}\right],\end{split}$

where $\bm{P}:=\mathrm{diag}(\bm{p})$, and where the second equality can be
verified via simple manipulations. We then have:

######  Proposition 6.

For given $k\in\mathcal{K}$ and $\bm{p}\in\mathbbmss{R}_{++}^{K}$, consider
the optimization problem

$\underset{\mathbbm{v}_{k}\in\mathcal{V}_{k}}{\emph{minimize}}~{}\mathsf{MSE}_{k}(\mathbbm{v}_{k},\bm{p}).$
(17)

Problem (17) has a unique solution
$\mathbbm{v}_{k}^{\star}\in\mathcal{V}_{k}$. Furthermore, if
$\mathcal{V}_{k}^{\prime}\neq\emptyset$, the solution
$\mathbbm{v}_{k}^{\star}$ satisfies

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

###### Proof.

Existence and uniqueness of the solution follows from [24, Lemma 4]. For the
second statement, we observe that

$\displaystyle\inf_{\mathbbm{v}_{k}\in\mathcal{V}_{k}}\mathsf{MSE}_{k}(\mathbbm{v}_{k},\bm{p})$
$\displaystyle\overset{(a)}{=}\inf_{\begin{subarray}{c}\mathbbm{v}_{k}\in\mathcal{V}_{k}\\\
\mathsf{E}[\|\mathbbm{v}_{k}\|_{2}^{2}]\neq
0\end{subarray}}\inf_{\beta\in\mathbbmss{C}}\mathsf{MSE}_{k}(\beta\mathbbm{v}_{k},\bm{p})$
$\displaystyle\overset{(b)}{=}\inf_{\begin{subarray}{c}\mathbbm{v}_{k}\in\mathcal{V}_{k}\\\
\mathsf{E}[\|\mathbbm{v}_{k}\|_{2}^{2}]\neq
0\end{subarray}}1-\dfrac{p_{k}|\mathsf{E}[\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{v}_{k}]|^{2}}{\sum_{j\in\mathcal{K}}p_{j}\mathsf{E}[|\mathbbm{h}_{j}^{\mathsf{H}}\mathbbm{v}_{k}|^{2}]+\mathsf{E}\left[\|\mathbbm{v}_{k}\|_{2}^{2}\right]}$
$\displaystyle=\inf_{\begin{subarray}{c}\mathbbm{v}_{k}\in\mathcal{V}_{k}\\\
\mathsf{E}[\|\mathbbm{v}_{k}\|_{2}^{2}]\neq
0\end{subarray}}\dfrac{1}{1+\mathsf{SINR}_{k}(\mathbbm{v}_{k},\bm{p})}=\dfrac{1}{1+u_{k}(\bm{p})}\overset{(c)}{<}1,$

where $(a)$ follows from
$(\forall\beta\in\mathbbmss{C})(\forall\mathbbm{v}_{k}\in\mathcal{V}_{k})$
$\beta\mathbbm{v}_{k}\in\mathcal{V}_{k}$, $(b)$ follows from standard
minimization of scalar quadratic forms as in the proof of the UatF bound (see,
e.g., [14, 15]), and $(c)$ follows from
$\mathcal{V}_{k}^{\prime}\neq\emptyset$ and Proposition 5(i). If
$\mathbbm{v}_{k}^{\star}\neq\bm{0}$ a.s., the identity
$u_{k}(\bm{p})=\mathsf{SINR}_{k}\left(\mathbbm{v}_{k}^{\star},\bm{p}\right)$
follows readily from the above chain of equalities. The proof is concluded by
observing that
$\inf_{\mathbbm{v}_{k}\in\mathcal{V}_{k}}\mathsf{MSE}_{k}(\mathbbm{v}_{k},\bm{p})<1$
implies $\mathbbm{v}_{k}^{\star}\neq\bm{0}$ a.s.. ∎

In the particular case where all users are served by all antennas based on
perfect knowledge of the channel $\mathbbm{H}$ (trivial information
constraints), the solution to Problem (17) is simply given by the well-known
MMSE beamforming solution

$(\forall
k\in\mathcal{K})~{}\mathbbm{v}_{k}=\left(\mathbbm{H}\bm{P}\mathbbm{H}^{\mathsf{H}}+\bm{I}\right)^{-1}\mathbbm{h}_{k}\sqrt{p_{k}}.$
(18)

Variations of (18) are readily obtained also for the case of imperfect yet
perfectly shared CSI among all processing units serving each user, under mild
assumptions on the channel estimation error model (see, e.g., [15, Corollary
4.3] or the example in Section IV).

For more general and nontrivial information constraints, the solution to
Problem (17) can be interpreted as the best distributed approximation of
regularized channel inversion, and it can be obtained via the recently
developed _team_ MMSE beamforming method given in [24].222Although presented
in the context of downlink beamforming, the method in [24] directly applies to
the uplink case. This method states that the solution to (17) corresponds to
the unique solution to a linear feasibility problem, which, depending on the
information constraints, can be solved in closed-form or approximately via
efficient numerical techniques. For example, in the context of user-centric
cell-free massive MIMO networks, this method is used in [24] [43] to provide
the first explicit solution to the problem of optimal local beamforming
design. Furthermore, it can be applied to more advanced information
constraints involving delayed CSI sharing [44], or exploiting the
peculiarities of serial fronthauls such as in the so-called _radio stripe_
concept [24] [42].

######  Remark 7.

The connection between the MMSE solution in (18) (or its variations) and more
traditional uplink ergodic rate bounds of the type in (3), is well-known (see,
e.g., [15, Corollary 4.2]). However, the connection between Problem (17) and
the UatF bound in (12) given by Proposition 6 seems largely overlooked in the
literature. A known advantage of working with (12) is that it makes long-term
power control tractable [14, 15]. This study shows that this tractability also
extends to joint long-term power control and beamforming design, and it even
allows to cover distributed beamforming design under nontrivial information
cosntraints. In addition, (12) can be rigorously connected to a similar
downlink ergodic rate bound (the so-called hardening bound) by means of
uplink-downlink duality arguments under sum [15] and per-antenna power
constraints [43], hence providing optimal downlink designs.

###  III-E Proposed algorithms

We now have all the elements for deriving an algorithmic solution to both
Problem (13) and Problem (14), i.e., to perform joint optimal long-term power
control and beamforming design according to the considered criteria. To keep
the proposed algorithms general, in the following section we do not specify
the setup (i.e., $\mathbbm{H}$ and the information constraints). For
particular applications to cell-free massive MIMO networks, we refer to our
numerical examples in Section IV. Specifically, we propose to solve Problem
(13) via the fixed-point iterations in (9), as summarized below:

$\bm{p}\in\mathbbmss{R}_{++}^{K}$

repeat

for $k\in\mathcal{K}$ do

$\mathbbm{v}_{k}\leftarrow\arg\min_{\mathbbm{v}_{k}\in\mathcal{V}_{k}}\mathsf{MSE}_{k}(\mathbbm{v}_{k},\bm{p})$

end for

$\bm{p}\leftarrow\begin{bmatrix}\frac{\gamma_{1}p_{1}}{\mathsf{SINR}_{1}(\mathbbm{v}_{1},\bm{p})}&\ldots&\frac{\gamma_{K}p_{K}}{\mathsf{SINR}_{K}(\mathbbm{v}_{K},\bm{p})}\end{bmatrix}$

until no significant progress is observed.

Similarly, we propose to solve Problem (14) via the fixed-point iterations in
(11), as summarized below:

$\bm{p}\in\mathbbmss{R}_{++}^{K}$

repeat

for $k\in\mathcal{K}$ do

$\mathbbm{v}_{k}\leftarrow\arg\min_{\mathbbm{v}_{k}\in\mathcal{V}_{k}}\mathsf{MSE}_{k}(\mathbbm{v}_{k},\bm{p})$

end for

$\bm{t}\leftarrow\begin{bmatrix}\frac{\gamma_{1}p_{1}}{\mathsf{SINR}_{1}(\mathbbm{v}_{1},\bm{p})}&\ldots&\frac{\gamma_{K}p_{K}}{\mathsf{SINR}_{K}(\mathbbm{v}_{K},\bm{p})}\end{bmatrix}$

$\bm{p}\leftarrow\frac{P}{\|\bm{t}\|_{\infty}}\bm{t}$

until no significant progress is observed.

Asymptotic convergence to an optimal solution can be established for both
algorithms by using Proposition 3, Proposition 5, and Proposition 6. Clearly,
the above algorithms relies on solving Problem (17) at each iteration, which,
as already mentioned, can be done in many practical cases following [24]. We
remark once more that $\mathbbm{v}_{k}$ is a function of the CSI, and not a
deterministic vector. In practice, as we will see in the numerical examples,
storing $\mathbbm{v}_{k}$ during the algorithm execution means storing its
long-term parameters (e.g., the regularization factor in a regularized channel
inversion block).

##  IV Simulated case study

###  IV-A Simulation scenario

Figure 1: Pictorial representation of the simulated setup: $K=64$ users
uniformly distributed within a squared service area of size $500\times
500~{}\text{m}^{2}$, and $L=16$ regularly spaced APs with $N=8$ antennas each.

We consider the large-scale MIMO network depicted in Figure 1, where $K=64$
users are uniformly distributed within a squared service area of size
$500\times 500~{}\text{m}^{2}$, and $L=16$ regularly spaced APs with $N=8$
antennas each. By neglecting for simplicity spatial correlation, 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
$\mathbbm{h}_{l,k}\sim\mathcal{CN}\left(\bm{0},\beta_{l,k}\bm{I}_{N}\right)$,
where $\beta_{l,k}>0$ denotes the channel gain between AP $l$ and user $k$. We
follow the same 3GPP-like path-loss model adopted in [17] for a $2$ GHz
carrier frequency:

$\beta_{l,k}=-36.7\log_{10}\left(D_{l,k}/1\;\mathrm{m}\right)-30.5+Z_{l,k}-\sigma^{2}\quad\text{[dB]},$

where $D_{l,k}$ is the distance between AP $l$ and user $k$ 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 $\rho=4$. The shadow fading is
correlated as
$\mathsf{E}[Z_{l,k}Z_{j,i}]=\rho^{2}2^{-\frac{\delta_{k,i}}{9\text{ [m]}}}$
for all $l=j$ and zero otherwise, where $\delta_{k,i}$ is the distance between
user $k$ and user $i$. The noise power is $\sigma^{2}=-174+10\log_{10}(B)+F$
[dBm], where $B=20$ MHz is the bandwidth, and $F=7$ dB is the noise figure.
Under this setup, in the following sections we focus on the power minimization
problem (13) with SINR constraints corresponding to a minimum per-user rate of
$2.5$ bit/s/Hz, and on the max-min fair problem (14) with unitary weights and
per-user power budget $P=20$ dBm.

###  IV-B Information constraints

For the choice of information constraints, we specialize the model in Section
III-B by considering the three representative scenarios described below.
First, we assume that each user $k\in\mathcal{K}$ is served only by its $Q$
strongest APs, that is, 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 $\beta_{l,k}$ and by
keeping only the first $Q$ elements. Furthermore, we assume each AP $l$ to
acquire (and possibly exchange) local CSI
$\hat{\mathbbm{H}}_{l}:=[\hat{\mathbbm{h}}_{l,1},\ldots,\hat{\mathbbm{h}}_{l,K}]$,

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

This model reflects the canonical cell-free massive MIMO implementation with
pilot-based local channel estimation, and possible CSI sharing through the
fronthaul; we neglect for simplicity estimation noise and pilot contamination
effects, and we focus on a simple model where small-scale fading coefficients
are either perfectly known at some APs or completely unknown. In contrast, we
assume that the relevant long-term statistical information is perfectly shared
within the network. We then study the following information constraints,
corresponding to different solutions to the subproblem (17) solved at each
iteration of the proposed algorithms.

####  IV-B1 Small cells

We assume each user to be served only by its strongest AP, i.e., $Q=1$, and no
CSI sharing, i.e., $(\forall l\in\mathcal{L})~{}S_{l}=\hat{\mathbbm{H}}_{l}$.
For all $\bm{p}\in\mathbbmss{R}_{++}^{K}$ and $k\in\mathcal{K}$, the $l$th
subvector of the optimal beamformer $\mathbbm{v}_{k}$ corresponding to the
unique AP $l\in\mathcal{L}_{k}$ serving user $k$ is given by the known MMSE
solution [15]

$\mathbbm{v}_{l,k}=\mathbbm{V}^{\mathsf{LMMSE}}_{l}\bm{e}_{k},$

where $(\forall
l\in\mathcal{L})~{}\mathbbm{V}^{\mathsf{LMMSE}}_{l}:=\left(\hat{\mathbbm{H}}_{l}\bm{P}\hat{\mathbbm{H}}_{l}^{\mathsf{H}}+\bm{\Sigma}_{l}\right)^{-1}\hat{\mathbbm{H}}_{l}\bm{P}^{\frac{1}{2}}$,
and
$\bm{\Sigma}_{l}:=\left(1+\sum_{i\in\\{k\in\mathcal{K}|l\notin\mathcal{L}_{k}\\}}\gamma_{l,i}p_{i}\right)\bm{I}_{N}$
is the local CSI error plus noise covariance matrix.

####  IV-B2 Distributed cell-free

We assume each user to be jointly served by its $Q=4$ strongest APs, and no
CSI sharing, i.e., $(\forall l\in\mathcal{L})~{}S_{l}=\hat{\mathbbm{H}}_{l}$.
For all $\bm{p}\in\mathbbmss{R}_{++}^{K}$ and $k\in\mathcal{K}$, the $l$th
subvector of the optimal beamformer $\mathbbm{v}_{k}$ corresponding to AP
$l\in\mathcal{L}$ is the so-called local team MMSE solution [24, 42]

$(\forall
l\in\mathcal{L})~{}\mathbbm{v}_{l,k}=\mathbbm{V}^{\mathsf{LMMSE}}_{l}\bm{c}_{l,k},$
(19)

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

####  IV-B3 Centralized cell-free

We assume each user to be jointly served by its $Q=4$ strongest APs, and
perfect CSI sharing, i.e., $(\forall
l\in\mathcal{L})~{}S_{l}=\hat{\mathbbm{H}}^{\mathsf{H}}:=[\hat{\mathbbm{H}}_{1}^{\mathsf{H}}\ldots,\hat{\mathbbm{H}}_{L}^{\mathsf{H}}]$.
For all $\bm{p}\in\mathbbmss{R}_{++}^{K}$ and $k\in\mathcal{K}$, the optimal
beamformer $\mathbbm{v}_{k}$ is given by the known centralized MMSE solution
[17]

$\mathbbm{v}_{k}=\left(\bm{D}_{k}\hat{\mathbbm{H}}\bm{P}\hat{\mathbbm{H}}^{\mathsf{H}}\bm{D}_{k}+\bm{D}_{k}\bm{\Sigma}\bm{D}_{k}\right)^{-1}\hat{\mathbbm{H}}^{\mathsf{H}}\bm{D}_{k}\bm{P}^{\frac{1}{2}},$
(20)

where $\bm{\Sigma}:=\mathrm{diag}(\bm{\Sigma}_{1},\ldots,\bm{\Sigma}_{L})$ is
the global CSI error plus noise covariance matrix, and
$\bm{D}_{k}:=\mathrm{diag}(\bm{D}_{1,k},\ldots,\bm{D}_{L,k})$ is a block-
diagonal matrix satisfying

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

Although neglected for simplicity, we remark that spatial correlation, channel
estimation noise, pilot contamination effects, and different clustering rules
could be readily incorporated in the above models following [24, 43]. We
further point out that, as shown in [24], for the particular channel and local
CSI error model considered here, (19) boils down to the known local MMSE
solution with optimal LSFD design [17].

(a)

(b)

Figure 2: Convergence behaviour of (a) the fixed-point iterations for
computing a solution to Problem (13); and (b) the normalized fixed-point
iterations for computing a solution to Problem (14) under different
information constraints.

###  IV-C Convergence behavior of the proposed algorithms

We run several steps of the proposed fixed-point algorithms used for computing
an optimal solution to Problem (13) and Problem (14), for the information
constraints described above and a single random user drop. In both cases, we
monitor convergence to an optimal solution (if it exists) in terms of the
distance $\|\bm{p}_{i}-\bm{p}^{\star}\|_{2}$, where $\bm{p}_{i}$ denotes the
output of the $i$th step of the considered fixed-point algorithm, and where
$\bm{p}^{\star}$ denotes the optimal power vector. The initialization is set
to $\bm{p}_{1}=P\bm{1}$, and all expectations are approximated using a fixed
training set of $N_{\mathsf{sim}}=100$ channel realizations. It is worthwhile
underlying that the implemented algorithms operate only by updating
deterministic coefficients using long-term information. For example, when it
comes to optimal distributed beamforming design, the implemented routines
update only the long-term parameters
$(\bm{c}_{l,1},\ldots,\bm{c}_{l,K},\bm{\Sigma}_{l})_{l\in\mathcal{L}}$.

Figure 2a reports the convergence behavior of the fixed-point iterations for
computing an optimal solution to Problem (13). Due to unfeasible minimum rate
constraints, the curve related to the small-cells network shows a divergent
behaviour and hence it is omitted. In contrast, since the minimum rate
constraints are feasible for both the centralized and distributed cell-free
networks, the corresponding fixed-point iterations converge to an optimal
solution, as expected. Furthermore, we observe that convergence is geometric
in both cases, as predicted by the discussion in Section II-A. However, we
notice a faster convergence for the centralized case. This can be explained
using the results in [35], which show that the convergence speed can be
directly connected to a well-defined notion of spectral radius for non-linear
mappings. Informally, in our setup, this notion corresponds to achievable
rates in the interference limited regime. Therefore, the different convergence
speed can be explained by the fact that, due to better interference management
capabilities, the rates in the centralized case saturate at a larger value
than in the distributed case as the SNR grows large. The interested reader is
referred to [35] for additional details.

Figure 2b reports the convergence behavior of the normalized fixed-point
iterations for computing an optimal solution to Problem (14). In contrast to
the unnormalized fixed-point iterations, the normalized fixed-point iterations
always converge. Similarly to Figure 2a, we observe geometric convergence as
predicted by the discussion in Section II-B. Furthermore, we observe a similar
difference in convergence speed across the different information constraints.
This difference seems to depend again on the different interference management
capabilities as in Figure 2a. A formal estimation of the convergence speed
could be obtained, for example, using the bounds in [33, Remark 2.1.12].
However, we point out that, in contrast to the unnormalized fixed-point
iterations, a rigorous connection between convergence speed and interference
management capabilities is not reported in the literature and requires further
analysis.

###  IV-D Comparison among information constraints

We now show how our theoretical results can be applied to revisit the
available studies comparing cell-free networks and small-cells networks in
delivering uniformly good quality of service to all users (i.e, in the max-min
sense), such as [16, 21, 17]. One main limitation of the available studies is
that, being based on either suboptimal power control or suboptimal beamforming
design, they do not truly characterize the achievable max-min fair performance
of the considered networks and hence they may lead to misleading conclusions.

In particular, we solve the considered max-min fair problem (14) with unitary
weights for $N_{\mathsf{CDF}}=100$ independent random user drops. Then, we
report in Figure 3 the empirical cumulative density function (CDF) of the
resulting per-user rates given by the UatF bound (12). As we can see, for the
considered setup, cell-free networks significantly outperform small-cell
networks for all percentiles. In addition, although to a lesser extent, the
centralized information structure outperforms the distributed one for all
percentiles. In contrast, most of the available studies focusing on max-min
fairness show similar CDF curves crossing at some given percentile. However,
our results show that this is an artifact of the chosen power control and
beamforming design, which are not jointly optimized for max-min fairness. Note
that a similar observation holds also by setting arbitrary weights in Problem
(14). In fact, by the principle that adding information (in our context,
replacing an information constraint $\mathcal{V}_{k}$ with a superset
$\mathcal{V}^{\prime}_{k}\supseteq\mathcal{V}_{k}$) cannot hurt, the
achievable rate region of cell-free networks is a superset of the cellular
one.

As already discussed in Remark 5, one potential limitation of the above
analysis is that it is based on the UatF bound which, despite offering
tractable problem formulations and hence leading to the proposed algorithms,
may underestimate the true system performance. However, we now demonstrate
experimentally that the UatF bound can be a quite effective proxy for
optimizing the system according to less pessimistic but intractable
performance metrics. More specifically, we report in Figure 3 the performance
of the proposed solution to Problem (14) in terms of the following classical
ergodic rate expressions based on coherent decoding: $(\forall
k\in\mathcal{K})$

$R_{k}^{\mathsf{coh}}(\mathbbm{v}_{k},\bm{p}):=\mathsf{E}\left[\log_{2}\left(1+\dfrac{p_{k}|\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{v}_{k}|^{2}}{\textstyle\sum_{j\neq
k}p_{j}|\mathbbm{h}_{j}^{\mathsf{H}}\mathbbm{v}_{k}|^{2}+\|\mathbbm{v}_{k}\|_{2}^{2}}\right)\right].$
(21)

Interestingly, we observe that the general trend of the curves in Figure 3a is
maintained in Figure 3b. In paricular, we notice that the gap between the CDFs
in Figure 3a and the corresponding ones in Figure 3b is below $0.3-0.4$ b/s/Hz
for nearly all percentiles.

(a)

(b)

Figure 3: Empirical CDF of the ergodic rates achieved by the proposed solution
to Problem (14) under multiple user drops and different information
constraints, evaluated using (a) the UatF bound in (12); and (b) its upper
bound (21) based on coherent decoding.

###  IV-E Comparison of centralized implementations

In this section we compare different methods for max-min fair system design in
centralized cell-free networks. In particular, we compare the performance of
the proposed algorithm for computing an optimal solution to Problem (14) under
a centralized information constraint against the following classical
techniques:

####  IV-E1 Short-term joint power control and beamforming design

Following a similar approach as the one in [2] for cellular networks with
perfect CSI, for each realization
$\hat{\bm{H}}=:[\hat{\bm{h}}_{1},\ldots,\hat{\bm{h}}_{K}]$ of the global CSI
$\hat{\mathbbm{H}}$, we solve

$\displaystyle\underset{\begin{subarray}{c}\bm{p}\in\mathbbmss{R}_{+}^{K}\\\
\bm{v}_{1},\ldots,\bm{v}_{K}\in\mathbbmss{C}^{M}\end{subarray}}{\text{maximize}}$
$\displaystyle\quad\min_{k\in\mathcal{K}}\dfrac{p_{k}|\hat{\bm{h}}_{k}^{\mathsf{H}}\bm{D}_{k}\bm{v}_{k}|^{2}}{\textstyle\sum_{j\neq
k}p_{j}|\hat{\bm{h}}_{j}^{\mathsf{H}}\bm{D}_{k}\bm{v}_{k}|^{2}+\bm{v}_{k}^{\mathsf{H}}\bm{D}_{k}\bm{\Sigma}\bm{D}_{k}\bm{v}_{k}}$
subject to $\displaystyle\quad\|\bm{p}\|_{\infty}\leq P$
$\displaystyle\quad(\forall k\in\mathcal{K})~{}\|\mathbbm{v}_{k}\|_{2}^{2}\neq
0,$

where the parameters $\bm{\Sigma}$ and $\bm{D}_{k}$ are the same as in (20).
The above objective corresponds to the instantaneous SINR for a variant of the
ergodic rate expression in (21) that takes into account channel estimation
errors [17, Theorem 5.1]. The optimal beamformers take the same form of (20)
(maximization of a Rayleigh quotient), with the difference that $\bm{p}$ is a
function of the instantaneous channel realization $\hat{\bm{H}}$. The solution
can be computed by adapting the block-coordinate ascent method in [2], or via
the normalized-fixed point iterations in Section II-B.

####  IV-E2 Long-term power control with fixed beamforming design

In this case, we solve

$\displaystyle\underset{\begin{subarray}{c}\bm{p}\in\mathbbmss{R}_{+}^{K}\end{subarray}}{\text{maximize}}$
$\displaystyle\quad\min_{k\in\mathcal{K}}\mathsf{SINR}_{k}(\bar{\mathbbm{v}}_{k},\bm{p})$
subject to $\displaystyle\quad\|\bm{p}\|_{\infty}\leq P,$

where $\bar{\mathbbm{v}}_{k}$ denotes a fixed beamforming design for each user
$k\in\mathcal{K}$. As done in [17], we choose the centralized MMSE solution
(20) by assuming full power transmission, i.e., we choose the solution to (16)
for $\bm{p}=\bm{1}P$. The above problem can be solved via the normalized-fixed
point iterations in Section II-B, as in [17], or in closed-form using [45].

Figure 4a reports the empirical CDF over $N_{\mathsf{CDF}}=100$ independent
user drops of the resulting per-user rates given by (21). As we can see, joint
power control and beamforming design may provide non-negligible performance
gains with respect to power control only. Furthermore, we similarly observe
that long-term optimization may perform better than short-term optimization.
The reason is that the power constraint, and the implicit constraint that all
users should have the same rate at the optimum, need not be satisfied for all
channel realizations but just on average in the long-term approach. Notably,
long-term optimization is less sensitive to channel realizations with a single
deep-fade event, which instead would severely limit the corresponding
instantaneous performance in the short-term approach. In addition to the above
gains in terms of achievable rates, we recall that the long-term approach may
also offer a significant gain in terms of computational complexity, since the
optimization problem need not be solved for each channel realization as in the
short term approach.

###  IV-F Comparison of distributed implementations

(a)

(b)

Figure 4: Empirical CDF of the ergodic rates (21) achieved by the proposed
solution to Problem (14) and by the competing techniques described in Section
IV-E and Section IV-F for the case of (a) centralized; and (b) distributed
combining.

Finally, we compare different methods for long-term max-min fair system design
in distributed cell-free networks. The short-term approach is not considered
since, to the best of our knowledge, it cannot be easily generalized to
optimal beamforming design under distributed information constraints. In
particular, we compare the performance of the proposed algorithm for computing
an optimal solution to Problem (14) under a distributed information constraint
against the following state-of-the-art techniques:

####  IV-F1 Long-term power control with fixed beamforming design

Similarly to the centralized information case, we use the normalized fixed-
point iterations in Section II-B as in [17], or the closed-form expression in
[45], to solve

$\displaystyle\underset{\begin{subarray}{c}\bm{p}\in\mathbbmss{R}_{+}^{K}\end{subarray}}{\text{maximize}}$
$\displaystyle\quad\min_{k\in\mathcal{K}}\mathsf{SINR}_{k}(\bar{\mathbbm{v}}_{k},\bm{p})$
subject to $\displaystyle\quad\|\bm{p}\|_{\infty}\leq P,$

where $\bar{\mathbbm{v}}_{k}$ denotes a fixed beamforming design for each user
$k\in\mathcal{K}$ satisfying the distributed information constraints. We
choose the optimal solution to (16) for $\bm{p}=\bm{1}P$, i.e., we choose the
local team MMSE solution (19) assuming full power transmission.

####  IV-F2 Long-term joint power control and LSFD design

Following the approach in [20], we solve

$\displaystyle\underset{\begin{subarray}{c}\bm{p}\in\mathbbmss{R}_{+}^{K}\\\
\bm{a}_{1},\ldots,\bm{a}_{K}\in\mathbbmss{C}^{L}\end{subarray}}{\text{maximize}}$
$\displaystyle\quad\min_{k\in\mathcal{K}}\mathsf{SINR}_{k}(\mathrm{diag}(a_{1,k}\bm{I}_{N},\ldots,a_{L,k}\bm{I}_{N})\bar{\mathbbm{v}}_{k},\bm{p})$
subject to $\displaystyle\quad\|\bm{p}\|_{\infty}\leq P,$

where $\bar{\mathbbm{v}}_{k}$ denotes a fixed beamforming design for each user
$k\in\mathcal{K}$, and $\bm{a}_{k}$ denotes the corresponding LSFD
coefficients as defined in [17]. As in [20], we choose maximum ratio combining
(MRC) $\bar{\mathbbm{v}}_{k}=\hat{\mathbbm{h}}_{k}$. Under this choice of
beamformers, [20] proves that the above problem can be solved via a block
coordinate ascent algorithm, where the LSFD coefficients are updated in
closed-form by maximizing a Rayleigh quotient, and where the power control
vector is updated by solving a long-term power control problem with fixed
beamforming design. However, we remark that the normalized fixed-point
iterations in Section II-B can also be used to solve the above problem as well
as more general versions not limited to MRC.

Figure 4b reports the performance of the considered methods for distributed
cell-free networks. Similarly to Figure 4a, we observe that joint power
control and beamforming design may provide non-negligible gains with respect
to power control only, although the gap appears to be smaller than in the
centralized case. Furthermore, we notice that beamforming design is clearly
the main limiting factor, since considering optimal LSFD design seems not
sufficient to recover the large performance loss of MRC with respect to the
local team MMSE scheme in (19).

##  V Conclusion and future directions

This study demonstrates the applicability of the axiomatic power control
framework in [6] [7] to challenging (infinite dimensional) long-term versions
of the classical joint power control and beamforming problems (1) and (2),
yielding novel optimal algorithms with favorable properties for modern large-
scale MIMO systems.

Due to the generality of our approach, the proposed algorithms could be
extended in future works to more involved problems, e.g., by incorporating
optimal user association as in [12, 13] or innovative information constraints
as in [24, 42, 44]. Furthermore, another interesting line of research is to
use the results in this study to revisit available studies on the performance
of future-generation wireless networks without resorting to heuristics.
Finally, a point left open by this study is the evaluation of the convergence
properties of alternative methods for the max-min fair criterion based on
block-coordinate ascent similar to [2] and [20]. In fact, Proposition 6
readily enables the derivation of a novel block-coordinate ascent algorithm
for (14). However, despite some promising experimental evidence [26],
establishing its optimality may be non-trivial, since the arguments in [2] and
[20] seem not directly applicable.

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