# Joint optimal beamforming and power control in cell-free massive MIMO
††thanks: L. Miretti, R. L. G. Cavalcante, and S. Stańczak acknowledge the
financial support by the Federal Ministry of Education and Research of Germany
in the programme of “Souverän. Digital. Vernetzt.” Joint project 6G-RIC,
project identification number: 16KISK020K and 16KISK030.

Lorenzo Miretti, Renato L. G. Cavalcante, Sławomir Stańczak  _Fraunhofer
Institute for Telecommunications Heinrich-Hertz-Institut_
_Technical University of Berlin_
Berlin, Germany
<EMAIL_ADDRESS>
###### Abstract

We derive a fast and optimal algorithm for solving practical weighted max-min
SINR problems in cell-free massive MIMO networks. For the first time, the
optimization problem jointly covers long-term power control and distributed
beamforming design under imperfect cooperation. In particular, we consider
user-centric clusters of access points cooperating on the basis of possibly
limited channel state information sharing. Our optimal algorithm merges
powerful power control tools based on interference calculus with the recently
developed team theoretic framework for distributed beamforming design. In
addition, we propose a variation that shows faster convergence in practice.

###### Index Terms:

team MMSE, cell-free massive MIMO, power control, distributed beamforming,
user-centric

††publicationid: pubid:  978-1-6654-3540-6/22 © 2022 IEEE

##  I Introduction

Cell-free massive multiple-input multiple-output (MIMO) is one of the key
research avenues for future generation sub-$6$GHz wireless access networks.
Its main focus is the study of simple and scalable access-point (AP)
cooperation architectures in ultra-dense networks, able to offer uniformly
good service in a cost-effective manner [1, 2, 3, 4].

To achieve this ambitious goal, cell-free massive MIMO has evolved from
related standardized technologies, such as 4G and 5G coordinated multi-point
(CoMP), by introducing flexible cooperation regimes with lower information
exchange and joint processing requirements. A first innovation is the adoption
of _user-centric_ designs, where multiple network components are grouped into
possibly overlapping cooperation clusters, each of them allocated to a single
user equipment (UE), without relying on the notion of a cell [2, 3]. A second
crucial aspect is the optimal distribution of signal processing tasks between
remote control units and APs such that information exchange is minimized. For
instance, in the original cell-free massive MIMO idea [1], all short-term
array processing tasks, such as channel estimation and beamforming, are
performed by the APs on the basis of only _local_ channel measurements. Then,
joint encoding/decoding and resource allocation tasks such as power control
are performed at a remote processing unit by only assuming knowledge of long-
term statistical information. Many variations of this cooperation regime are
currently subject of active research [5, 6, 7].

Against this background, this study considers the important open problem of
joint optimal long-term power control and distributed beamforming design under
imperfect channel state information (CSI) sharing across the APs. To relate
the present study with existing approaches, we recall that iterative
algorithms for long-term power control under fixed beamforming design have
been extensively studied in both the cellular and cell-free massive MIMO
literature (see, e.g., [1, 4, 8, 9]). In contrast, the problem of optimal
distributed beamforming design without power control has been solved only
recently in [10, 11] using elements of _team_ decision theory, a rigorous
control theoretical framework for decentralized decision making under
asymmetry of information. However, none of these previous studies have jointly
optimized beamformers and transmit powers. The closest approaches to the
problem at hand date back to the studies on joint beamforming design and power
control in static channels (see, e.g., [12, 13]). Although conceptually
similar, these methods only apply to centralized or purely statistical
beamforming. Furthermore, the centralized case typically considers rather
impractical short-term optimization, i.e., for each channel state realization.

Unlike previous studies, we target a certain class of weighted max-min SINR
problems where, for the first time, joint optimal long-term power control and
distributed beamforming design is performed. By tuning the problem weights,
the corresponding solutions attain all positive boundary points of the
achievable rate region predicted by the popular _use-and-then-forget_ (UatF)
ergodic capacity inner bound [8], which is particularly suitable for
distributed cell-free operations [4]. We propose two fast algorithms: for the
first algorithm, global optimality is theoretically demonstrated; for the
second algorithm, convergence to the global optimum is illustrated via
simulations. On top of handling distributed cell-free operations, the proposed
algorithms may also offer a more practical alternative to the available
approaches for centralized operations.

###  I-A Notation and mathematical preliminaries

Let $(\Omega,\Sigma,\mathbb{P})$ be a probability space. We denote by
$\mathcal{H}^{K}$ the space of $K$-dimensional random vectors
$\Omega\to\mathbbmss{C}^{K}$, i.e., $K$-tuples of $\Sigma$-measurable
functions $\Omega\to\mathbbmss{C}$, satisfying
$(\forall\mathbbm{h}\in\mathcal{H}^{K})$
$\mathsf{E}[\|\mathbbm{h}\|_{2}^{2}]<\infty$. We denote by $\mathbbmss{R}_{+}$
and $\mathbbmss{R}_{++}$ the sets of, respectively, nonnegative and positive
reals. 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. A norm $\|\cdot\|$ on $\mathbbmss{R}^{K}$ is said to be
_monotone_ if
$(\forall\bm{x}\in\mathbbmss{R}_{+}^{K})(\forall\bm{y}\in\mathbbmss{R}_{+}^{K})~{}\bm{x}\leq\bm{y}\Rightarrow\|\bm{x}\|\leq\|\bm{y}\|$.

######  Definition 1.

Given a nonempty set $\mathcal{R}\subset\mathbbmss{R}_{+}^{K}$, we say that
$\bm{r}^{\star}\in\mathcal{R}$ is (weakly) Pareto optimal iff
$\nexists\bm{r}\in\mathcal{R}$ such that $\bm{r}>\bm{r}^{\star}$. The set
$\partial\mathcal{R}$ of all (weakly) Pareto optimal points is called the
(weak) Pareto boundary of $\mathcal{R}$ [14].

######  Definition 2.

A function $T:\mathbbmss{R}^{K}_{+}\to\mathbbmss{R}^{K}_{++}$ is called a
_standard interference mapping_ if the following properties hold [15]:

  1. (i)

$(\forall\bm{x}\in\mathbbmss{R}^{K}_{+})(\forall\bm{y}\in\mathbbmss{R}^{K}_{+})~{}\bm{x}\geq\bm{y}\implies
T(\bm{x})\geq T(\bm{y})$;

  2. (ii)

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

For later reference, given a standard interference mapping, we call each
coordinate function of the mapping a _standard interference function_. For
simplicity, we also require continuity for a function to be called a standard
interference mapping.

##  II Model and problem statement

###  II-A Capacity of distributed cell-free uplink networks

Let us consider the uplink of a cell-free 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 conventional flat-fading synchronous channel models [16], we focus on
the ergodic capacity in the classical Shannon sense, which is approximated
with the popular _use-and-then-forget_ (UatF) inner bound [8]. In more detail,
we define $(\forall k\in\mathcal{K})$

$R_{k}(\mathbbm{v}_{k},\bm{p}):=\log_{2}(1+\mathrm{SINR}_{k}(\mathbbm{v}_{k},\bm{p})),$
$\mathrm{SINR}_{k}(\mathbbm{v}_{k},\bm{p}):=\leavevmode\resizebox{299.19885pt}{}{$\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 $\bm{p}:=[p_{1},\ldots,p_{K}]^{\mathsf{T}}\in\mathbbmss{R}_{+}^{K}$ is a
deterministic vector of transmit powers, $\mathbbm{h}_{k}\in\mathcal{H}^{NL}$
is a random channel vector modeling the fading state between UE $k$ and all
APs, and $\mathbbm{v}_{k}\in\mathcal{H}^{NL}$ is a random beamforming vector,
which is applied by the APs to process jointly the signal of UE $k$. We assume
that $\mathbbm{v}_{k}\neq\bm{0}$ almost surely (a.s.), so
$\mathsf{E}[\|\mathbbm{v}_{k}\|_{2}^{2}]\neq 0$ and the SINR is well defined.
We stress that _random_ beamforming here simply refers to the fact that
beamformers may adapt to fading realizations. The achievable rate region is
given by

$\mathcal{R}:=\underset{\begin{subarray}{c}\|\bm{p}\|_{\infty}\leq P\\\
(\forall k\in\mathcal{K})~{}\mathbbm{v}_{k}\in\mathcal{V}_{k}\\\ (\forall
k\in\mathcal{K})~{}\mathsf{E}[\|\mathbbm{v}_{k}\|_{2}^{2}]\neq
0\end{subarray}}{\bigcup}\left\\{\bm{r}\in\mathbbmss{R}_{+}^{K}\middle|\begin{array}[]{c}r_{1}\leq
R_{1}(\mathbbm{v}_{1},\bm{p})\\\ \vdots\\\ r_{K}\leq
R_{K}(\mathbbm{v}_{K},\bm{p})\end{array}\right\\},$

where $0<P<\infty$ denotes the per-UE power budget, and where the set
$\mathcal{V}_{k}\subseteq\mathcal{H}^{NL}$ denotes an _information_
constraint, defined in the following section, modelling _distributed_ cell-
free operations such as in [2]. The region $\mathcal{R}$ is especially
relevant if codewords can span many realizations of a stationary and ergodic
fading process, and it relies on the following widely-used information
theoretical simplifications [8, 17]:

  1. (i)

Correlation across different fading realizations is neglected in the encoding
and decoding phase, although it can be exploited in the channel estimation
phase (the overhead of this operation is left unspecified here);

  2. (ii)

CSI is neglected in the decoding phase, but it is exploited for coherent
receive beamforming, i.e., to generate the soft estimates feeded as an input
to the channel decoder;

  3. (iii)

_Treating-interference-as-noise_ (TIN) is considered in the decoding phase.
More precisely, the message of UE $k$ is transmitted using simple coding
techniques for the memoryless point-to-point Gaussian channel, while
interference is mitigated by only using spatial processing.

######  Remark 1.

Alternative rate regions based on TIN with various forms of CSI at the decoder
are often considered (see [9, 17, 18] and references therein). These regions
may provide less pessimistic capacity approximations, but optimization
problems become challenging because of the expectation outside the logarithm
in the resulting rate expressions. To deal with this issue, most studies
perform short-term optimization, i.e., an optimization problem is solved for
each fading realization. However, this approach is hardly practical in cell-
free networks because of the overhead required to exchange instantaneous CSI
among many geographically distant network elements. In contrast, the proposed
algorithms based on $\mathcal{R}$ are readily implementable in distributed
cell-free networks such as the ones envisioned in [4]. In these networks,
short-term array processing is confined to the APs, while joint decoding and
optimization (e.g., power control) takes place at a remote processing unit
based on long-term statistical information.

###  II-B Information constraints

In distributed cell-free operation, APs joint processing capabilities are
impaired. One of the major impairment is that the APs must compute their
portion of the beamformers on the basis of imperfect CSI sharing, or even
based on only _local_ CSI [1]. However, to date, most studies on cell-free
massive MIMO do not formally model this type of impairment, so, strictly
speaking, they do not perform truly distributed beamforming optimization. To
address this limitation, as in [10], we model constraints related to imperfect
CSI sharing with the following set:

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

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 $(\forall\mathbbm{h}\in\mathcal{H}_{l}^{N})$
$\mathsf{E}[\|\mathbbm{h}\|_{2}^{2}]<\infty$, and where
$\Sigma_{l}\subseteq\Sigma$ is the sub-$\sigma$-algebra induced by the
available CSI at the $l$th AP, also called the _information subfield_ of AP
$l$ [19]. Furthermore, another practical limitation is that only a subset of
APs are used for decoding the message of a given UE. More specifically,
scalable cell-free networks are expected to implement some type of user-
centric rule [3], which allocates to each UE a cluster of serving APs. As
shown in [11], this practical limitation can be modeled by setting
$\mathcal{H}_{l}^{N}$ in (1) to the set $\\{\bm{0}_{N}~{}(\mathrm{a.s.})\\}$
if AP $l$ is not serving UE $k$. Considering both impairments, we remark that
$\mathcal{V}_{k}$ is a linear _subspace_ of $\mathcal{H}^{NL}$.

###  II-C Weighted max-min SINR optimization

The objective of the proposed algorithms is to optimize jointly the
distributed beamformers and the transmit powers, with the intent to obtain
rate tuples on the boundary $\partial\mathcal{R}$ of $\mathcal{R}$ (see
Definition 1). More precisely, under consideration is the following
optimization problem:

$\displaystyle\underset{\begin{subarray}{c}\bm{p}\in\mathbbmss{R}_{++}^{K}\\\
\mathbbm{v}_{1},\ldots,\mathbbm{v}_{K}\in\mathcal{H}^{NL}\end{subarray}}{\text{maximize}}$
$\displaystyle\quad\min_{k\in\mathcal{K}}\omega_{k}^{-1}\mathrm{SINR}_{k}(\mathbbm{v}_{k},\bm{p})$
(2) 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 $(\omega_{1},\ldots,\omega_{K})\in\mathbbmss{R}_{++}^{K}$ is a given
vector of positive weights, which can be tuned to focus on different
operational points in $\partial\mathcal{R}$. In particular, the popular _max-
min fair_ point is obtained by setting $\bm{\omega}=\bm{1}$. More generally,
for different choices of weights, we have the next result:

######  Proposition 1.

If a solution
$(\bm{p}^{\star},\mathbbm{v}_{1}^{\star},\ldots,\mathbbm{v}_{K}^{\star})$ to
Problem (2) exists, then
$\bm{r}^{\star}:=(R_{1}(\mathbbm{v}_{1}^{\star},\bm{p}^{\star}),\ldots,R_{K}(\mathbbm{v}_{K}^{\star},\bm{p}^{\star}))$
belongs to $\partial\mathcal{R}$. Furthermore, every positive point in
$\partial\mathcal{R}$ can be obtained for some
$\bm{\omega}\in\mathbbmss{R}_{++}^{K}$.

###### Proof.

(Sketch) The proof follows from standard properties of performance regions in
multi-criteria resource allocation problems (see, e.g, [20]), and the
logarithmic relation between rate and SINR. The details are omitted. ∎

######  Remark 2.

A popular alternative to weighted max-min optimization is _weighted sum-rate_
(WSR) optimization (see, e.g., [4]). One major difference between these two
approaches is that optimal WSR solutions always produce points on the _strong_
Pareto boundary of $\mathcal{R}$ (see [20] for a definition). However, unlike
weighted max-min solutions, if $\mathcal{R}$ is nonconvex, WSR solutions may
miss some (weak or strong) Pareto optimal points that are important for a
given criterion of fairness [21], [20, Remark 1.3]. Nonconvex rate regions are
common, for instance, when the network enters an interference limited regime.
User fairness can be restored through _scheduling_ , but resource allocation
becomes more complex [20].

##  III Optimal iterative solution

In this section we present a fast iterative algorithm that provably converges
to a solution to Problem (2), provided that a solution exists and that it
attains a positive optimum. As we prove later, existence and positivity of a
solution is guaranteed by a simple condition on the fading and CSI
distribution. Specifically, it is guaranteed if $(\forall
k\in\mathcal{K})~{}\mathcal{V}_{k}^{\prime}:=\\{\mathbbm{v}_{k}\in\mathcal{V}_{k}~{}|~{}\mathsf{E}[\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{v}_{k}]\neq
0\\}$ is nonempty, which holds for most fading and CSI distributions.

######  Remark 3.

In particular, $\mathcal{V}_{k}^{\prime}\neq\emptyset$ does not hold if the
APs have no CSI and, in addition, 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 one of the main drawbacks of the UatF bounding technique, which
requires either sufficient CSI _or_ a strong channel mean to produce non-
trivial capacity inner bounds.

###  III-A Power control using interference calculus

As a preliminary step, we first revisit known results for fairly general power
control problems of the type

$\underset{\|\bm{p}\|\leq
P}{\text{maximize}}\min_{k\in\mathcal{K}}\omega_{k}^{-1}u_{k}(\bm{p}),$ (3)

where $\|\cdot\|$ is a monotone norm on $\mathbbmss{R}^{K}$, and where
$u_{k}:\mathbbmss{R}_{+}^{K}\to\mathbbmss{R}_{+}$ denotes the utility of UE
$k$ satisfying

$(\forall\bm{p}\in\mathbbmss{R}_{+}^{K})~{}u_{k}(\bm{p})=\dfrac{p_{k}}{f_{k}(\bm{p})}$
(4)

for some standard interference function
$f_{k}:\mathbbmss{R}_{+}^{K}\to\mathbbmss{R}_{++}$ (see Definition 2). This
structure covers many common power control problems in wireless networks,
including an equivalent formulation of Problem (2). An important consequence
of the above structure is the following result:

######  Proposition 2.

Define
$T:\mathbbmss{R}_{+}^{K}\to\mathbbmss{R}_{++}^{K}:\bm{p}\mapsto[\omega_{1}f_{1}(\bm{p}),\ldots,\omega_{K}f_{K}(\bm{p})]^{\mathsf{T}}$.
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 solution to Problem (3).

  2. (ii)

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

$(\forall i\in\mathbbmss{N})~{}\bm{p}_{i+1}=\tilde{T}(\bm{p}_{i})$ (5)

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 Problem (3). 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 $\mathcal{P}$ w.r.t. the partial
ordering induced by the nonnegative cone.

###### Proof.

(Sketch) (i) The proof that the mapping $\tilde{T}$ has a unique fixed point
follows directly from [22, Theorem 3.2]. We can use arguments similar to those
in [23, Sect. III] to show that the fixed point of $\tilde{T}$ is a solution
to Problem 3. However, some care has to be taken in following those arguments
because, unlike the results in [23], utilities with the structure in (4) do
not necessarily satisfy all properties in [23, Assumption 1], so, in
particular, the solution to Problem (3) is not necessarily unique. We omit the
full proof because of the space limitation.

(ii) Use the same arguments shown in the proof of [22, Theorem 3.2].

(iii) Omitted because of the space limitation. ∎

######  Remark 4.

From a practical perspective, Proposition 2 shows that Problem (3) always has
a solution. Furthermore, if the solution is not unique, then the recursion in
(5) converges in norm to the solution with minimum sum transmit power.

###  III-B Equivalent problem reformulation

We now map Problem (2) to an equivalent power control problem of the type in
(3), where the optimization of the beamformers is implicit in the definition
of the utilities $u_{k}$. More precisely, by exploiting the fact that the
SINRs are only coupled through the power vector $\bm{p}$, we obtain:

######  Proposition 3.

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

$(\forall\bm{p}\in\mathbbmss{R}_{+}^{K})~{}u_{k}(\bm{p}):=\sup_{\begin{subarray}{c}\mathbbm{v}_{k}\in\mathcal{V}_{k}\\\
\mathsf{E}[\|\mathbbm{v}_{k}\|_{2}^{2}]\neq
0\end{subarray}}\mathrm{SINR}_{k}(\mathbbm{v}_{k},\bm{p}),$ (6)

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

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

  2. (ii)

The utility (6) 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 (3) with utilities in (6) and
monotone norm $\|\cdot\|=\|\cdot\|_{\infty}$. If $(\forall
k~{}\in\mathcal{K})$
$\exists\mathbbm{v}^{\star}_{k}\in\mathcal{V}^{\prime}_{k}$ such that
$u_{k}(\bm{p}^{\star})=\mathrm{SINR}_{k}(\mathbbm{v}^{\star}_{k},\bm{p}^{\star})$
(i.e., the $\sup$ is attained), then
$(\bm{p}^{\star},\mathbbm{v}_{1}^{\star},\ldots,\mathbbm{v}_{K}^{\star})$
solves Problem (2).

###### 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
$(\forall\bm{p}\in\mathbbmss{R}_{+}^{K})~{}u_{k}(\bm{p})\geq\sup_{\mathbbm{v}_{k}\in\mathcal{V}^{\prime}_{k}}\mathrm{SINR}_{k}(\mathbbm{v}_{k},\bm{p})$.
By definition of $\mathcal{V}^{\prime}_{k}$,
$\sup_{\mathbbm{v}_{k}\in\mathcal{V}^{\prime}_{k}}\mathrm{SINR}_{k}(\mathbbm{v}_{k},\bm{p})>0$
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 (6).

  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 it is a standard interference function [24, Proposition 1].

  3. (iii)

Immediate, hence omitted.

∎

The main implication of Proposition 3 is that an optimal power vector
$\bm{p}^{\star}$ for Problem (2) can be obtained via the fixed-point
iterations in (5), where each $i$th step involves the evaluation of the
standard interference mapping
$T(\bm{p})=[\frac{\omega_{1}p_{1}}{u_{1}(\bm{p})},\ldots,\frac{\omega_{K}p_{K}}{u_{K}(\bm{p})}]^{\mathsf{T}}$
for $\bm{p}=\bm{p}_{i}$, and where $u_{k}(\bm{p})$ is given by (6). 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
distributed beamformers. The following section discusses how to perform this
step, and it shows that the supremum in (6) is always attained in practice.

###  III-C Team MMSE beamforming

The following proposition states that optimal distributed beamformers under
fixed power control, i.e., beamformers that attain the supremum in (6) for a
given $\bm{p}$, can be obtained by solving a minimum mean squared error (MMSE)
problem under information constraints. For convenience, we define the random
channel matrix $\mathbbm{H}:=[\mathbbm{h}_{1},\ldots,\mathbbm{h}_{K}]$.

######  Proposition 4.

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

$\underset{\mathbbm{v}_{k}\in\mathcal{V}_{k}}{\emph{minimize}}~{}\mathrm{MSE}_{k}(\mathbbm{v}_{k},\bm{p}),$
(7)

where we define $(\forall\mathbbm{v}_{k}\in\mathcal{H}^{NL})$

$\mathrm{MSE}_{k}(\mathbbm{v}_{k},\bm{p}):=\mathsf{E}\left[\|\mathrm{diag}(\bm{p})^{\frac{1}{2}}\mathbbm{H}^{\mathsf{H}}\mathbbm{v}_{k}-\bm{e}_{k}\|_{2}^{2}\right]+\mathsf{E}\left[\|\mathbbm{v}_{k}\|_{2}^{2}\right].$

Problem (7) has a unique solution $\mathbbm{v}_{k}^{\star}\in\mathcal{V}_{k}$.
Furthermore, if $\mathcal{V}_{k}^{\prime}\neq\emptyset$, then
$u_{k}(\bm{p})=\mathrm{SINR}_{k}\left(\mathbbm{v}_{k}^{\star},\bm{p}\right)$
holds.

###### Proof.

Existence and uniqueness of the solution follows from [10, Lemma 4]. For the
second statement, we observe that

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

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., [8, 9, 10]), and $(c)$ follows from
$\mathcal{V}_{k}^{\prime}\neq\emptyset$ and Proposition 3(i). If
$\mathbbm{v}_{k}^{\star}\neq\bm{0}$ a.s., the identity
$u_{k}(\bm{p})=\mathrm{SINR}_{k}\left(\mathbbm{v}_{k}^{\star},\bm{p}\right)$
follows readily from the above chain of equalities. The proof is concluded by
observing that
$\inf_{\mathbbm{v}_{k}\in\mathcal{V}_{k}}\mathrm{MSE}_{k}(\mathbbm{v}_{k},\bm{p})<1$
implies $\mathbbm{v}_{k}^{\star}\neq\bm{0}$ a.s.. ∎

The solution to (7), which can be interpreted as the best distributed
approximation of a regularized zero-forcing beamformer, can be obtained via
the recently developed _team_ MMSE beamforming framework given by [10].
Although presented in the context of downlink beamforming with perfect message
sharing, the method in [10] directly applies to the uplink case and to user-
centric clustering [11]. This method states that the solution to (7)
corresponds to the unique solution to a linear feasibility problem, which,
depending on the CSI structure, can be solved explicitly or approximately via
efficient numerical techniques. We remark that this method can be applied to
fairly general CSI structures, for instance involving quantized or delayed
information sharing, or exploiting the peculiarities of efficient fronthauls
such as in the so-called _radio stripe_ concept [10].

To keep the proposed algorithms general, in the following section we do not
specify the setup (i.e., $\mathbbm{H}$ and the information structure
$\mathcal{V}_{1},\ldots,\mathcal{V}_{K}$), and we simply assume that a
solution to (7) is available. However, as an example, our numerical results
consider the particular case of local CSI as in [1], together with a simple
user-centric clustering rule.

###  III-D Proposed algorithms

We now have all the elements for deriving an algorithmic solution to Problem
(2), i.e., to perform joint optimal long-term power control and distributed
beamforming design. Specifically, we propose to solve Problem (2) via the
fixed-point iterations in (5), as summarized below:

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

repeat

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

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

end for

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

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

until no significant progress is observed.

Asymptotic convergence to an optimal solution is guaranteed by Proposition 2,
Proposition 3, and Proposition 4. Clearly, the above algorithm relies on
solving Problem (7) at each iteration, which, as already mentioned, can be
done following [10, 11]. Note 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).

An intuitive variant of the first algorithm proposed above alternates between
distributed beamforming optimization and power control, and it is summarized
below:

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

repeat

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

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

end for

$\bm{p}\leftarrow\arg\max_{\|\bm{p}\|_{\infty}\leq
P}\min_{k\in\mathcal{K}}\omega_{k}^{-1}\mathrm{SINR}_{k}(\mathbbm{v}_{k},\bm{p})$

until no significant progress is observed.

In addition to solving Problem (7), this algorithm requires a solution to
$\max_{\|\bm{p}\|_{\infty}\leq
P}\min_{k\in\mathcal{K}}\omega_{k}^{-1}\mathrm{SINR}_{k}(\mathbbm{v}_{k},\bm{p})$
for given beamformers $(\mathbbm{v}_{1},\ldots,\mathbbm{v}_{K})$, which is
obtained in closed form in [25]. The advantage is that this power control step
maximizes the improvement in utility. The disadvantage is that performing this
power control step is more complex than performing a fixed point iteration.
However, this increase in complexity is in practice negligible when compared
to the complexity of solving Problem (7). Convergence properties of this
algorithm will be studied in a future work, and they are here only tested
using numerical experiments.

##  IV Numerical experiments and final remarks

###  IV-A Simulation setup

Figure 1: Pictorial representation of the simulated setup: $K=64$ UEs
uniformly distributed within a squared service area of size $1\times
1~{}\text{km}^{2}$, and $L=16$ regularly spaced APs with $N=8$ antennas each.
Each UE is jointly served by a cluster of $Q=4$ APs offering the strongest
channel gains.

We consider the network depicted in Figure 1, where $K=64$ UEs are uniformly
distributed within a squared service area of size $1\times 1~{}\text{km}^{2}$,
and $L=16$ regularly spaced APs with $N=8$ antennas each. By neglecting for
simplicity channel 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
$\mathbbm{h}_{l,k}\sim\mathcal{CN}\left(\bm{0},\gamma_{l,k}\bm{I}_{N}\right)$,
where $\gamma_{l,k}>0$ denotes the channel gain between AP $l$ and UE $k$. We
follow the same 3GPP-like path-loss model adopted in [4] for a $2$ GHz carrier
frequency:

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

where $D_{l,k}$ is the distance between AP $l$ and 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 $\rho=4$. 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
UE $k$ and UE $i$. 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.
The power budget is set to $P=20$ dBm.

###  IV-B Local CSI and path loss based user-centric clustering

We solve Problem (2) by focusing on the following simple user-centric
distributed information structure with no short-term CSI sharing. We assume
that each UE $k$ is served only by its $Q=4$ strongest APs, 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
$\gamma_{l,k}$ and by keeping only the first $Q$ elements. Then, we assume
each AP $l$ to acquire 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 mirrors the fully distributed implementation proposed in [2]. For
simplicity, we neglect estimation noise, and we focus on a model where small-
scale fading coefficients are either perfectly known at some APs or completely
unknown. Under this model, the optimal solution
$[\mathbbm{v}^{\mathsf{H}}_{1,k},\ldots,\mathbbm{v}^{\mathsf{H}}_{L,k}]:=\mathbbm{v}^{\mathsf{H}}_{k}\in\mathcal{V}_{k}$
to Problem (7) is given by [10, 11]

$(l\in\mathcal{L})(k\in\mathcal{K})~{}\mathbbm{v}_{l,k}=\mathbbm{V}_{l}\bm{c}_{l,k},$
(8)

where:

  * •

$\mathbbm{V}_{l}:=\left(\hat{\mathbbm{H}}_{l}\mathrm{diag}(\bm{p})\hat{\mathbbm{H}}_{l}^{\mathsf{H}}+\bm{\Psi}_{l}\right)^{-1}\hat{\mathbbm{H}}_{l}\mathrm{diag}(\bm{p})^{\frac{1}{2}}$
is a local MMSE beamforming stage [2] with augmented noise covariance
$\bm{\Psi}_{l}:=(1+\sum_{\\{k\in\mathcal{K}|l\notin\mathcal{L}_{k}\\}}p_{k}\gamma_{l,k})\bm{I}_{N}$;

  * •

$\bm{c}_{l,k}\in\mathbbmss{C}^{K}$ is a statistical beamforming 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
$\bm{\Pi}_{l}:=\mathsf{E}\left[\mathrm{diag}(\bm{p})^{\frac{1}{2}}\hat{\mathbbm{H}}_{l}^{\mathsf{H}}\mathbbm{V}_{l}\right]$.

The main difference between (8) and the best known heuristic, called local
MMSE beamforming with optimal large-scale fading decoding (see, e.g., [4]), is
that the latter considers only a scalar statistical stage, i.e., only one
component of $\bm{c}_{l,k}$ is different than zero. We refer to [10] for
additional comparisons against known heuristics, and for the extension to
noisy CSI.

###  IV-C Illustration of the proposed algorithms

Figure 2: Minimum achievable rate for a given step of the proposed fixed point
(FP) and alternate optimization (AO) algorithms. The odd steps perform optimal
distributed beamforming design, while the even steps update the long-term
power control coefficients. Both algorithms attain the optimal max-min fair
point with few iterations. As a baseline, we plot the performance of classical
matched filter (MF) beamforming with optimal power control [4].

Figure 2 illustrates several steps of the proposed algorithms, applied to the
computation of the optimal max-min fair point of $\mathcal{R}$, i.e., the
solution to Problem (2) for $\bm{\omega}=\bm{1}$. The initialization is the
full power case $\bm{p}=P\bm{1}$, which turns out to be highly suboptimal.
Both algorithms require very few iterations for approaching the optimum, and,
not surprisingly, alternate optimization outperforms fixed point iterations,
at the price of a slight increase in complexity. We point out that, although
not shown in this paper, fast convergence has been observed also under many
different simulation parameters.

It is worthwhile underlying that the implemented algorithms operate only by
updating deterministic coefficients using long-term information. In
particular, when it comes to optimal distributed beamforming design, the
implemented routines update only the long-term parameters $\bm{c}_{l,k}$ and
$\bm{\Psi}_{l}$ in (8), based on a fixed training set of
$N_{\mathrm{sim}}=1000$ channel realizations and channel gain information.

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