# Joint Minimum DL-UL Rate Maximization
for Cell-Free Massive MIMO

Bikshapathi Gouda, Italo Atzeni, and Antti Tölli
Centre for Wireless Communications, University of Oulu, Finland
Emails: {bikshapathi.gouda, italo.atzeni<EMAIL_ADDRESS>The work of B.
Gouda and A. Tölli was supported by the Academy of Finland under grant no.
318927 (6Genesis Flagship). The work of I. Atzeni was supported by the Marie
Skłodowska-Curie Actions (MSCA-IF 897938 DELIGHT).

###### Abstract

In cell-free massive multiple-input multiple-output (MIMO) systems, the
beamforming strategies at the base stations (BSs) and user equipments (UEs)
can be computed building on bi-directional training [1, 2, 3]. However, the
precoding/decoding optimization in the downlink (DL) and in the uplink (UL)
generally requires two separate bi-directional training phases, which can be
wasteful in the case of short scheduling blocks. This paper proposes a
framework to reduce the bi-directional training overhead by considering a
common beamforming training strategy for both DL and UL when the UEs to be
served in the two directions are the same. In doing so, we consider the
problem of maximizing the (weighted) minimum DL-UL rate among all the UEs.
Numerical results show that, in scenarios with short scheduling blocks, the
proposed framework outperforms the case where the DL and UL beamforming
strategies are computed individually via two separate bi-directional training
phases thanks to the reduced training overhead. Even more substantial gains
are observed with respect to the case with a single bi-directional training
phase, where the DL (resp. UL) beamforming strategies are reused in the UL
(resp. DL).

##  I Introduction

Massive multiple-input multiple-output (MIMO) is a multi-antenna technology at
the foundation of the 5G New Radio (NR) standard. It enables highly
directional beamforming and spatial multiplexing of many user equipments (UEs)
thanks to massive antenna arrays at the base stations (BSs). However, the
spectral efficiency of cell-edge UEs may still be compromised due to the high
pathloss and interference from the neighboring BSs [4]. In this context, cell-
free massive MIMO aims to eliminate the inter-cell interference and ensure
uniformly good service for all the UEs. This is realized by distributing a
large number of antenna elements across the network and serving the UEs via
coherent transmission/reception from all the BSs [5, 6, 7]. To this end, the
BSs are assumed to be connected to a central processing unit (CPU) via
backhaul links, which provides the UE-specific data and, possibly, the
beamforming strategies adopted for the coherent transmission/reception.

Cell-free massive MIMO systems have been shown to outperform their traditional
cellular massive MIMO and small-cell counterparts in several scenarios of
practical interest [8, 9]. In this context, it has been demonstrated that
cooperative beamforming at the BSs can bring substantial gains with respect to
local beamforming (such as simple matched filtering) at the expense of more
intense signaling [8, 9, 1, 2, 3]. In [8, 9], the cooperative beamforming
strategies at the BSs are designed at the CPU for the case of single-antenna
UEs. On the other hand, [1, 2, 3, 10] consider multi-antenna UEs and optimize
the cooperative beamforming strategies at the BSs together with the
beamforming vectors at the UEs in a distributed fashion building on bi-
directional training [11]. While a large part of the cell-free massive MIMO
literature (including [9, 1, 2, 3]) focuses on the sum-rate performance, other
works tackle the problem of maximizing the minimum signal-to-interference-
plus-noise ratio (SINR) among all the UEs. In this regard, [8] considers the
max-min fairness power control problem for both downlink (DL) and uplink (UL)
transmissions. Furthermore, [12] formulates the max-min SINR problem in the UL
and optimizes the beamforming strategies at the BSs using geometric
programming. A similar study is provided in [13] assuming fronthaul capacity
constraints. Lastly, [14] formulates the max-min SINR problem for the DL case
and optimizes the beamforming strategies at the BSs exploiting the quasi-
concavity of the minimum SINR objective.

The computation of the DL and UL beamforming strategies generally requires two
separate bi-directional training phases [11]. However, when the UEs to be
served in the DL and in the UL are the same and the scheduling blocks are
short, it may be convenient to devise a common beamforming strategy for both
DL and UL that involves only one bi-directional training phase. In this paper,
we tackle the joint optimization of the DL and UL beamforming strategies in
cell-free massive MIMO systems with multi-antenna UEs. The proposed framework
produces a set of power-constrained beamforming vectors that are used at the
BSs and at the UEs for both DL and UL transmissions. In doing so, we consider
the problem of maximizing the (weighted) minimum DL-UL rate among all the UEs.
Our approach allows to avoid separate bi-directional training phases for the
optimization of the DL and UL beamforming strategies, reduces the overall
computational complexity, and can be adjusted to strike the desired balance
between DL and UL performance. To facilitate the practical implementation of
our method, we also present a heuristic approach that avoids additional
signaling between the BSs and the UEs, which entails only a modest performance
loss. Numerical results show that, in scenarios with short scheduling blocks,
the proposed joint DL-UL design outperforms the case where the DL and UL
beamforming strategies are computed individually via two separate bi-
directional training phases. The observed gains are even higher with respect
to the case where the DL (resp. UL) beamforming strategies are reused in the
UL (resp. DL) direction.

##  II System Model and Problem Formulation

Consider a cell-free massive MIMO network where a set of BSs
$\mathcal{B}\triangleq\\{1,\ldots,B\\}$, each equipped with $M$ antennas,
serves a set of UEs $\mathcal{K}\triangleq\\{1,\ldots,K\\}$, each equipped
with $N$ antennas. The UEs are served in both the DL and the UL and
receive/transmit up to $S$ data streams in each direction. Assuming a time
division duplex (TDD) setting with channel reciprocity, let
$\mathbf{H}_{b,k}\in\mbox{$\mathbb{C}$}^{M\times N}$ denote the UL channel
matrix between UE $k\in\mathcal{K}$ and BS $b\in\mathcal{B}$, with
$\mathbf{H}_{k}\triangleq[\mathbf{H}_{1,k}^{\mathrm{T}},\ldots,\mathbf{H}_{B,k}^{\mathrm{T}}]^{\mathrm{T}}\in\mbox{$\mathbb{C}$}^{BM\times
N}$ representing the aggregate UL channel matrix of UE $k$. Let
$\mathbf{w}_{s,k}\in\mbox{$\mathbb{C}$}^{BM\times 1}$ be the aggregate
beamforming vector used at all the BSs to transmit and receive stream $s$ of
UE $k$. Similarly, let $\mathbf{v}_{s,k}\in\mbox{$\mathbb{C}$}^{N\times 1}$ be
the beamforming vector used at UE $k$ to transmit and receive stream $s$. With
the above definitions, the DL rate of UE $k$ across all the streams can be
written as

$R^{\textnormal{\tiny{DL}}}_{k}\triangleq\sum_{s=1}^{S}\log_{2}(1+\mathrm{SINR}^{\textnormal{\tiny{DL}}}_{s,k})$
(1)

where we have defined the DL SINR corresponding to stream $s$ of UE $k$ as

$\mathrm{SINR}^{\textnormal{\tiny{DL}}}_{s,k}\triangleq\frac{|\mathbf{v}_{s,k}^{\mathrm{H}}\mathbf{H}_{k}^{\mathrm{H}}\mathbf{w}_{s,k}|^{2}}{\sum_{(\bar{k},\bar{s})\neq(k,s)}|\mathbf{v}_{s,k}^{\mathrm{H}}\mathbf{H}_{k}^{\mathrm{H}}\mathbf{w}_{\bar{s},\bar{k}}|^{2}+\sigma_{\textnormal{\tiny{UE}}}^{2}\|\mathbf{v}_{s,k}\|^{2}}$
(2)

where $\sigma_{\textnormal{\tiny{UE}}}^{2}$ is the additive white Gaussian
noise (AWGN) variance at the UEs. Similarly, the UL rate of UE $k$ across all
the streams can be written as

$R^{\textnormal{\tiny{UL}}}_{k}\triangleq\sum_{s=1}^{S}\log_{2}(1+\mathrm{SINR}^{\textnormal{\tiny{UL}}}_{s,k})$
(3)

where we have defined the UL SINR corresponding to stream $s$ of UE $k$ as

$\displaystyle\mathrm{SINR}^{\textnormal{\tiny{UL}}}_{s,k}$
$\displaystyle\triangleq\frac{|\mathbf{w}_{s,k}^{\mathrm{H}}\mathbf{H}_{k}\mathbf{v}_{s,k}|^{2}}{\sum_{(\bar{k},\bar{s})\neq(k,s)}|\mathbf{w}_{s,k}^{\mathrm{H}}\mathbf{H}_{\bar{k}}\mathbf{v}_{\bar{s},\bar{k}}|^{2}+\sigma_{\textnormal{\tiny{BS}}}^{2}\|\mathbf{w}_{s,k}\|^{2}}$
(4)

where $\sigma_{\textnormal{\tiny{BS}}}^{2}$ is the AWGN variance of the BS.

In this paper, we target the maximization of the minimum DL-UL rate using
identical beamforming strategies for transmission and reception at the BSs and
at the UEs.111Since scaling the combining vectors does not effect the SINR,
any scaled version of the optimized beamforming vectors can be used at the
receiver. Moreover, at the end of the beamforming training process, the
receiver is free to use minimum mean squared error (MMSE) combining if
required, which does not impact the beamforming training process. The
identical DL-UL beamforming design allows the joint optimization of the
precoding-decoding strategies using a single bi-directional training mechanism
and eliminates the need for separate bi-directional training phases for DL and
UL. This reduces the overall training overhead and increases the effective
data rate. The optimization problem that maximizes the minimum DL-UL rate with
maximum transmit power at the BS and at the UEs
$\rho_{\textnormal{\tiny{BS}}}$ and $\rho_{\textnormal{\tiny{UE}}}$,
respectively, can be written as

$\displaystyle\begin{array}[]{cl}\displaystyle\max_{\\{\mathbf{w}_{s,k},\mathbf{v}_{s,k}\\}}&\displaystyle\min\big{(}\alpha\min_{k}R^{\textnormal{\tiny{DL}}}_{k},(1-\alpha)\min_{k}R^{\textnormal{\tiny{UL}}}_{k}\big{)}\\\
\mathrm{s.t.}&\displaystyle\sum_{k\in\mathcal{K}}\sum_{s=1}^{S}\|\mathbf{E}_{b}\mathbf{w}_{s,k}\|^{2}\leq\rho_{\textnormal{\tiny{BS}}},\quad\forall
b\in\mathcal{B}\\\
&\displaystyle\sum_{s=1}^{S}\|\mathbf{v}_{s,k}\|^{2}\leq\rho_{\textnormal{\tiny{UE}}},\quad\forall
k\in\mathcal{K}\end{array}$ (8)

where $\mathbf{E}_{b}\in\mbox{$\mathbb{R}$}^{M\times BM}$ is a selection
matrix such that
$\mathbf{E}_{b}\mathbf{w}_{s,k}=[\mathbf{w}_{s,k}((b-1)M+1),\ldots,\mathbf{w}_{s,k}(bM)]^{\mathrm{T}}$
and $\alpha$ is the weight between DL and UL rates.

Let us introduce the auxiliary variables
$\gamma_{s,k}\triangleq\mathrm{SINR}^{\textnormal{\tiny{DL}}}_{s,k}$ and
$\bar{\gamma}_{s,k}\triangleq\mathrm{SINR}^{\textnormal{\tiny{UL}}}_{s,k}$.
Hence, (8) can be rewritten as

$\displaystyle\displaystyle\max_{\begin{subarray}{c}\\{\mathbf{w}_{s,k},\mathbf{v}_{s,k}\\},\\\
\\{\gamma_{s,k},\bar{\gamma}_{s,k}\\},R\end{subarray}}$ $\displaystyle\ R$
(9a) $\displaystyle\mathrm{s.t.}\hskip 15.649pt$ $\displaystyle\
\alpha\sum_{s=1}^{S}\log_{2}(1+\gamma_{s,k})\geq R,\quad\forall
k\in\mathcal{K}$ (9b) $\displaystyle\
(1-\alpha)\sum_{s=1}^{S}\log_{2}(1+\bar{\gamma}_{s,k})\geq R,\quad\forall
k\in\mathcal{K}$ (9c) $\displaystyle\
\eqref{eq:SINR_k_dl}\geq\gamma_{s,k},\quad\forall
k\in\mathcal{K},\leavevmode\nobreak\ s=1,\ldots,S$ (9d) $\displaystyle\
\eqref{eq:SINR_k_ul}\geq\bar{\gamma}_{s,k},\quad\forall
k\in\mathcal{K},\leavevmode\nobreak\ s=1,\ldots,S$ (9e) $\displaystyle\
\displaystyle\sum_{k\in\mathcal{K}}\sum_{s=1}^{S}\|\mathbf{E}_{b}\mathbf{w}_{s,k}\|^{2}\leq\rho_{\textnormal{\tiny{BS}}},\quad\forall
b\in\mathcal{B}$ (9f) $\displaystyle\
\sum_{s=1}^{S}\|\mathbf{v}_{k}\|^{2}\leq\rho_{\textnormal{\tiny{UE}}},\quad\forall
k\in\mathcal{K}$ (9g)

where $R$ is the minimum common rate in the DL and in the UL. We apply
alternating optimization to (9) to optimize {$\mathbf{w}_{s,k}\\}$ and
$\\{\mathbf{v}_{s,k}\\}$. Specifically, we derive the optimal BS beamformers
$\\{\mathbf{w}_{s,k}\\}$ for fixed $\\{\mathbf{v}_{s,k}\\}$ in Section III and
the optimal UE beamformers $\\{\mathbf{v}_{s,k}\\}$ for fixed
$\\{\mathbf{w}_{s,k}\\}$ in Section IV.

##  III BS Beamforming Design

For a fixed set of UE beamformers $\\{\mathbf{v}_{s,k}\\}$, the BS beamformers
$\\{\mathbf{w}_{s,k}\\}$ are obtained by solving

$\displaystyle\begin{array}[]{cl}\displaystyle\max_{\\{\mathbf{w}_{s,k},\gamma_{s,k},\bar{\gamma}_{s,k}\\},R}&R\\\
\mathrm{s.t.}&\textnormal{\eqref{eq:cen_rdl}--\eqref{eq:cen_bs_pwr}}.\end{array}$
(12)

The constraints in (9d) and (9e) are non-convex with respect to
$\\{\mathbf{w}_{s,k}\\}$, $\\{\gamma_{s,k}\\}$, and
$\\{\bar{\gamma}_{s,k}\\}$. Hence, we use successive convex approximation
(SCA) [15] based on the first-order approximation of (9d) and (9e).

We begin by introducing the auxiliary variables
$p_{s,k}(\mathbf{w}_{s,k},\gamma_{s,k})\triangleq\frac{1}{\gamma_{s,k}}|\mathbf{v}_{s,k}^{\mathrm{H}}\mathbf{H}_{k}^{\mathrm{H}}\mathbf{w}_{s,k}|^{2}$
and
$q_{s,k}(\mathbf{w}_{s,k},\bar{\gamma}_{s,k})\triangleq\frac{1}{\bar{\gamma}_{s,k}}|\mathbf{w}_{s,k}^{\mathrm{H}}\mathbf{H}_{k}\mathbf{v}_{s,k}|^{2}$.
The first-order approximations of $p_{s,k}$ and $q_{s,k}$ at iteration $(i+1)$
are given, respectively, by

$\displaystyle\tilde{p}_{s,k}(\mathbf{w}_{s,k},\gamma_{s,k})$
$\displaystyle\triangleq-\frac{|\mathbf{v}_{s,k}^{\mathrm{H}}\mathbf{H}_{k}^{\mathrm{H}}\mathbf{w}^{(i)}_{s,k}|^{2}}{\gamma_{s,k}^{(i)}}\frac{\gamma_{s,k}}{\gamma_{s,k}^{(i)}}$
$\displaystyle\phantom{=}\
+2\frac{{(\mathbf{w}_{s,k}^{(i)})^{\mathrm{H}}}\mathbf{H}_{k}\mathbf{v}_{s,k}\mathbf{v}_{s,k}^{\mathrm{H}}\mathbf{H}_{k}^{\mathrm{H}}\mathbf{w}_{s,k}}{\gamma_{s,k}^{(i)}}$
(13)

and

$\displaystyle\tilde{q}_{s,k}(\mathbf{w}_{s,k},\bar{\gamma}_{s,k})$
$\displaystyle=-\frac{\big{|}(\mathbf{w}_{s,k}^{(i)})^{\mathrm{H}}\mathbf{H}_{k}\mathbf{v}_{s,k}\big{|}^{2}}{\bar{\gamma}_{s,k}^{(i)}}\frac{\bar{\gamma}_{s,k}}{\bar{\gamma}_{s,k}^{(i)}}$
$\displaystyle\phantom{=}\
+2\frac{{(\mathbf{w}_{s,k}^{(i)})^{\mathrm{H}}}\mathbf{H}_{k}\mathbf{v}_{s,k}\mathbf{v}_{s,k}^{\mathrm{H}}\mathbf{H}_{k}^{\mathrm{H}}\mathbf{w}_{s,k}}{\bar{\gamma}_{s,k}^{(i)}}.$
(14)

Since $p_{s,k}$ and $q_{s,k}$ are convex, it follows that $\tilde{p}_{s,k}\leq
p_{s,k}$ and $\tilde{q}_{s,k}\leq q_{s,k}$. Therefore, we approximate (9d) and
(9e) as

$\displaystyle{\sum_{(\bar{k},\bar{s})\neq(k,s)}|\mathbf{v}_{s,k}^{\mathrm{H}}\mathbf{H}_{k}^{\mathrm{H}}\mathbf{w}_{\bar{s},\bar{k}}|^{2}+\sigma_{\textnormal{\tiny{UE}}}^{2}\|\mathbf{v}_{s,k}\|^{2}}$
$\displaystyle\leq\tilde{p}_{s,k},$ (15)
$\displaystyle{\sum_{(\bar{k},\bar{s})\neq(k,s)}|\mathbf{w}_{s,k}^{\mathrm{H}}\mathbf{H}_{\bar{k}}\mathbf{v}_{\bar{s},\bar{k}}|^{2}+\sigma_{\textnormal{\tiny{BS}}}^{2}\|\mathbf{w}_{s,k}\|^{2}}$
$\displaystyle\leq\tilde{q}_{s,k}$ (16)

respectively, which yields the following convex approximation of (12):

$\displaystyle\displaystyle\max_{\\{\mathbf{w}_{s,k},\gamma_{s,k},\bar{\gamma}_{s,k}\\},R}$
$\displaystyle\ R$ (17a) $\displaystyle\mathrm{s.t.}\hskip 25.60747pt$
(9b)–(9c) and (9f) (17b) $\displaystyle\ \eqref{eq:dlapprx_bs},\quad\forall
k\in\mathcal{K},\leavevmode\nobreak\ s=1,\ldots,S$ (17c) $\displaystyle\
\eqref{eq:ulapprx_bs},\quad\forall k\in\mathcal{K},\leavevmode\nobreak\
s=1,\ldots,S.$ (17d)

The optimal BS beamformer $\mathbf{w}_{s,k}$ can be obtained by finding the
stationary point of the Lagrangian of (17), and the optimal $\mathbf{w}_{s,k}$
is given by

$\displaystyle\mathbf{w}_{s,k}$
$\displaystyle\triangleq\bigg{(}\sum_{(\bar{k},\bar{s})\neq(k,s)}\big{(}\alpha\nu_{\bar{s},\bar{k}}+(1-\alpha)\mu_{s,k}\big{)}\mathbf{H}_{\bar{k}}\mathbf{v}_{\bar{s},\bar{k}}\mathbf{v}_{\bar{s},\bar{k}}^{\mathrm{H}}\mathbf{H}_{\bar{k}}^{\mathrm{H}}$
$\displaystyle\phantom{=}\
+(1-\alpha)\mu_{s,k}\sigma_{\textnormal{\tiny{BS}}}^{2}\mathbf{I}_{BM}+\sum_{b\in\mathcal{B}}\lambda_{b}\mathbf{E}^{\mathrm{H}}_{b}\mathbf{E}_{b}\bigg{)}^{-1}$
$\displaystyle\phantom{=}\
\times\bigg{(}\bigg{(}\alpha\frac{\nu_{s,k}}{\gamma_{s,k}^{(i)}}+(1-\alpha)\frac{\mu_{s,k}}{{\bar{\gamma}_{s,k}^{(i)}}}\bigg{)}\mathbf{H}_{k}\mathbf{v}_{s,k}\mathbf{v}_{s,k}^{\mathrm{H}}\mathbf{H}_{k}^{\mathrm{H}}\mathbf{w}_{s,k}^{(i)}\bigg{)}$
(18)

where $\nu_{s,k}$, $\mu_{s,k}$, and $\lambda_{b}$ are the dual variables
corresponding to (17c), (17d), and (9f), respectively. As detailed in the
Appendix, $\mu_{s,k}$ and $\nu_{s,k}$ depend on the auxiliary dual variables
$\eta_{k}$ and $\zeta_{k}$ corresponding to (9b) and (9c), respectively, and
are updated using the sub-gradient method.

###  III-A BS Beamforming Design with UL Training

Let
$\mathbf{h}_{s,k}\triangleq\mathbf{H}_{k}\mathbf{v}_{s,k}\in\mbox{$\mathbb{C}$}^{BM\times
1}$ denote the effective UL channel vector between stream $s$ of UE $k$ and
all the BSs, and let $\mathbf{p}_{s,k}\in\mbox{$\mathbb{C}$}^{\tau\times 1}$
denote the pilot assigned to stream $s$ of UE $k$, with
$\|\mathbf{p}_{s,k}\|^{2}=\tau$. During the UL pilot-aided channel estimation
phase, each UE $k$ synchronously transmits a superposition of its pilots
$\\{\mathbf{p}_{s,k}\\}_{s=1}^{S}$ precoded with the corresponding
$\\{\mathbf{v}_{s,k}\\}_{s=1}^{S}$, i.e.,

$\displaystyle\mathbf{X}_{k}^{\textnormal{\tiny{UL}}}\triangleq\sum_{s=1}^{S}\mathbf{v}_{s,k}\mathbf{p}_{s,k}^{\mathrm{H}}\in\mbox{$\mathbb{C}$}^{N\times\tau}.$
(19)

Then, the receive signal across all the BSs is given by

$\displaystyle\mathbf{Y}^{\textnormal{\tiny{UL}}}$
$\displaystyle\triangleq\sum_{k\in\mathcal{K}}\mathbf{H}_{k}\mathbf{X}_{k}^{\textnormal{\tiny{UL}}}+\mathbf{Z}^{\textnormal{\tiny{UL-1}}}$
(20)
$\displaystyle=\sum_{k\in\mathcal{K}}\sum_{s=1}^{S}\mathbf{h}_{s,k}\mathbf{p}_{s,k}^{\mathrm{H}}+\mathbf{Z}^{\textnormal{\tiny{UL-1}}}\in\mbox{$\mathbb{C}$}^{BM\times\tau}$
(21)

where $\mathbf{Z}^{\textnormal{\tiny{UL-1}}}$ is the AWGN term across all the
BSs, and the least-squares (LS) estimate of $\mathbf{h}_{s,k}$ is obtained as

$\displaystyle\hat{\mathbf{h}}_{s,k}$
$\displaystyle\triangleq\frac{1}{\tau}\mathbf{Y}^{\textnormal{\tiny{UL}}}\mathbf{p}_{s,k}$
(22)
$\displaystyle=\mathbf{h}_{s,k}+\frac{1}{\tau}{\sum_{(\bar{k},\bar{s})\neq(k,s)}}\mathbf{h}_{\bar{s},\bar{k}}\mathbf{p}_{\bar{s},\bar{k}}^{\mathrm{H}}\mathbf{p}_{s,k}+\frac{1}{\tau}\mathbf{Z}^{\textnormal{\tiny{UL-1}}}\mathbf{p}_{s,k}.$
(23)

With the estimated channel $\hat{\mathbf{h}}_{s,k}$, the BS beamformer
$\mathbf{w}_{s,k}$ can be computed at the CPU as

$\displaystyle\hat{\mathbf{w}}_{s,k}$
$\displaystyle\triangleq\bigg{(}\sum_{(\bar{k},\bar{s})\neq(k,s)}\big{(}\alpha\nu_{\bar{s},\bar{k}}+(1-\alpha)\mu_{s,k}\big{)}\hat{\mathbf{h}}_{\bar{s},\bar{k}}\hat{\mathbf{h}}_{\bar{s},\bar{k}}^{\mathrm{H}}$
$\displaystyle\phantom{=}\
+(1-\alpha)\mu_{s,k}\sigma_{\textnormal{\tiny{BS}}}^{2}\mathbf{I}_{BM}+\sum_{b\in\mathcal{B}}\lambda_{b}\mathbf{E}^{\mathrm{H}}_{b}\mathbf{E}_{b}\bigg{)}^{-1}$
$\displaystyle\phantom{=}\
\times\bigg{(}\bigg{(}\alpha\frac{\nu_{s,k}}{\gamma_{s,k}^{(i)}}+(1-\alpha)\frac{\mu_{s,k}}{{\bar{\gamma}_{s,k}^{(i)}}}\bigg{)}\hat{\mathbf{h}}_{s,k}\hat{\mathbf{h}}_{s,k}^{\mathrm{H}}\mathbf{w}_{s,k}^{(i)}\bigg{)}.$
(24)

Note that (III-A) becomes equal to (III) as $\tau\to\infty$. The dual
variables $\nu_{s,k}$ and $\mu_{s,k}$ are computed at the CPU using the
estimated effective channel in (23), i.e., by replacing
$\mathbf{H}_{k}\mathbf{v}_{s,k}$ with $\hat{\mathbf{h}}_{s,k}$ in (45)–(46)
(see the Appendix). Similarly, $\gamma_{s,k}$ and $\bar{\gamma}_{s,k}$ are
obtained by plugging the estimated effective channels into (2) and (4),
respectively. For each bi-directional training iteration,
$\hat{\mathbf{w}}_{s,k}$ is computed via the sub-gradient method until the
convergence of the auxiliary dual variables $\eta_{k}$ and $\zeta_{k}$.

$\displaystyle\hat{\mathbf{v}}_{s,k}$
$\displaystyle\triangleq\bigg{(}\alpha\bar{\nu}_{s,k}\big{(}\mathbf{Y}_{k}^{\textnormal{\tiny{DL-1}}}(\mathbf{Y}_{k}^{\textnormal{\tiny{DL-1}}})^{\mathrm{H}}\\!-\\!\mathbf{Y}_{k}^{\textnormal{\tiny{DL-1}}}\mathbf{p}_{s,k}\mathbf{p}_{s,k}^{\mathrm{H}}(\mathbf{Y}_{k}^{\textnormal{\tiny{DL-1}}})^{\mathrm{H}}\\!-\\!\tau\sigma^{2}_{\textnormal{\tiny{UE}}}\mathbf{I}_{N}\big{)}\\!+\\!(1\\!-\\!\alpha)\big{(}\mathbf{Y}_{k}^{\textnormal{\tiny{DL-2}}}(\mathbf{Y}_{k}^{\textnormal{\tiny{DL-2}}})^{\mathrm{H}}\\!-\\!\mathbf{Y}_{k}^{\textnormal{\tiny{DL-2}}}\mathbf{p}_{s,k}\mathbf{p}_{s,k}^{\mathrm{H}}(\mathbf{Y}_{k}^{\textnormal{\tiny{DL-2}}})^{\mathrm{H}}\\!-\\!\tau\sigma^{2}_{\textnormal{\tiny{UE}}}\mathbf{I}_{N}\big{)}$
$\displaystyle\phantom{=}\
+\\!\alpha\bar{\nu}_{s,k}\sigma_{\textnormal{\tiny{UE}}}^{2}\tau\mathbf{I}_{N}\\!+\\!\bar{\lambda}_{k}\mathbf{I}_{N}\bigg{)}^{-1}\bigg{(}\alpha\frac{\bar{\nu}_{s,k}}{\gamma_{s,k}^{(i)}}\mathbf{Y}_{k}^{\textnormal{\tiny{DL-1}}}\mathbf{p}_{s,k}\mathbf{p}_{s,k}^{\mathrm{H}}(\mathbf{Y}_{k}^{\textnormal{\tiny{DL-1}}})^{\mathrm{H}}{\mathbf{v}^{(i)}_{s,k}}\\!+\\!(1\\!-\\!\alpha)\frac{1}{\bar{\gamma}_{s,k}^{(i)}}\mathbf{Y}_{k}^{\textnormal{\tiny{DL-2}}}\mathbf{p}_{s,k}\mathbf{p}_{s,k}^{\mathrm{H}}(\mathbf{Y}_{k}^{\textnormal{\tiny{DL-2}}})^{\mathrm{H}}{\mathbf{v}^{(i)}_{s,k}}\bigg{)}$
(38)

##  IV UE Beamforming Design

For a fixed set of BS beamformers $\\{\mathbf{w}_{s,k}\\}$, the UE beamformers
$\\{\mathbf{v}_{s,k}\\}$ are obtained by solving

$\displaystyle\begin{array}[]{cl}\displaystyle\max_{\\{\mathbf{v}_{s,k},\gamma_{s,k},\bar{\gamma}_{s,k}\\},R}&R\\\
\mathrm{s.t.}&\textnormal{\eqref{eq:cen_rdl}--\eqref{eq:cen_sinr_ul}\leavevmode\nobreak\
and\leavevmode\nobreak\ \eqref{eq:cen_ue_pwr}}.\end{array}$ (27)

The constraints in (9d) and (9e) are non-convex with respect to
$\\{\mathbf{v}_{s,k}\\}$, $\\{\gamma_{s,k}\\}$, and
$\\{\bar{\gamma}_{s,k}\\}$. Therefore, as done in Section III, we resort to
SCA.

We begin by introducing the auxiliary variables
$r_{s,k}(\mathbf{v}_{s,k},\gamma_{s,k})\triangleq\frac{|\mathbf{v}_{s,k}^{\mathrm{H}}\mathbf{H}_{k}^{\mathrm{H}}\mathbf{w}_{s,k}|^{2}}{\gamma_{s,k}}$
and
$t_{s,k}(\mathbf{v}_{s,k},\bar{\gamma}_{s,k})\triangleq\frac{|\mathbf{w}_{s,k}^{\mathrm{H}}\mathbf{H}_{k}\mathbf{v}_{s,k}|^{2}}{\bar{\gamma}_{s,k}}$.
The first-order approximations of $r_{s,k}$ and $t_{s,k}$ at iteration $(i+1)$
are given, respectively, by

$\displaystyle\tilde{r}_{s,k}(\mathbf{v}_{s,k},\gamma_{s,k})$
$\displaystyle\triangleq-\frac{\big{|}(\mathbf{v}_{s,k}^{(i)})^{\mathrm{H}}\mathbf{H}_{k}^{\mathrm{H}}\mathbf{w}_{s,k}\big{|}^{2}}{{\gamma_{s,k}}^{(i)}}\frac{\gamma_{s,k}}{\gamma_{s,k}^{(i)}}$
$\displaystyle\phantom{=}\
+2\frac{{(\mathbf{v}_{s,k}^{(i)})^{\mathrm{H}}}\mathbf{H}_{k}^{\mathrm{H}}\mathbf{w}_{s,k}\mathbf{w}_{s,k}^{\mathrm{H}}\mathbf{H}_{k}\mathbf{v}_{s,k}}{\gamma_{s,k}^{(i)}},$
(28) $\displaystyle\tilde{t}_{s,k}(\mathbf{v}_{s,k},\bar{\gamma}_{s,k})$
$\displaystyle\triangleq-\frac{|{\mathbf{w}_{s,k}^{\mathrm{H}}}\mathbf{H}_{k}\mathbf{v}^{(i)}_{s,k}|^{2}}{\bar{\gamma}_{s,k}^{(i)}}\frac{\bar{\gamma}_{s,k}}{{\bar{\gamma}_{s,k}^{(i)}}}$
$\displaystyle\phantom{=}\
+2\frac{{(\mathbf{v}_{s,k}^{(i)})^{\mathrm{H}}}\mathbf{H}_{k}^{\mathrm{H}}\mathbf{w}_{s,k}\mathbf{w}_{s,k}^{\mathrm{H}}\mathbf{H}_{k}\mathbf{v}_{s,k}}{{\bar{\gamma}_{s,k}^{(i)}}}.$
(29)

Since $r_{s,k}$ and $t_{s,k}$ are convex, it follows that $\tilde{r}_{s,k}\leq
r_{s,k}$ and $\tilde{t}_{s,k}\leq t_{s,k}$. Therefore, we approximate (9d) and
(9e) as

$\displaystyle{\sum_{(\bar{k},\bar{s})\neq(k,s)}|\mathbf{v}_{s,k}^{\mathrm{H}}\mathbf{H}_{k}^{\mathrm{H}}\mathbf{w}_{\bar{s},\bar{k}}|^{2}+\sigma_{\textnormal{\tiny{UE}}}^{2}\|\mathbf{v}_{s,k}\|^{2}}$
$\displaystyle\leq\tilde{r}_{s,k},$ (30)
$\displaystyle{\sum_{(\bar{k},\bar{s})\neq(k,s)}|\mathbf{w}_{s,k}^{\mathrm{H}}\mathbf{H}_{\bar{k}}\mathbf{v}_{\bar{s},\bar{k}}|^{2}+\sigma_{\textnormal{\tiny{BS}}}^{2}\|\mathbf{w}_{s,k}\|^{2}}$
$\displaystyle\leq\tilde{t}_{s,k}$ (31)

respectively, which yields the following convex approximation of (27):

$\displaystyle\displaystyle\max_{\\{\mathbf{v}_{s,k},\gamma_{s,k},\bar{\gamma}_{s,k}\\},R}$
$\displaystyle\ R$ (32a) $\displaystyle\mathrm{s.t.}\hskip 25.60747pt$
(9b)–(9c) and (9g) (32b) $\displaystyle\ \eqref{eq:dlapprx},\quad\forall
k\in\mathcal{K},\leavevmode\nobreak\ s=1,\ldots,S$ (32c) $\displaystyle\
\eqref{eq:ulapprx},\quad\forall k\in\mathcal{K},\leavevmode\nobreak\
s=1,\ldots,S.$ (32d)

The optimal UE beamformer $\mathbf{v}_{s,k}$ can be obtained by finding the
stationary point of the Lagrangian of (32), and the optimal $\mathbf{v}_{s,k}$
is given by

$\displaystyle\mathbf{v}_{s,k}$
$\displaystyle\triangleq\bigg{(}\sum_{(\bar{k},\bar{s})\neq(k,s)}\big{(}\alpha\bar{\nu}_{s,k}+(1-\alpha)\bar{\mu}_{\bar{s},\bar{k}}\big{)}\mathbf{H}_{{k}}^{\mathrm{H}}\mathbf{w}_{\bar{s},\bar{k}}\mathbf{w}_{\bar{s},\bar{k}}^{\mathrm{H}}\mathbf{H}_{k}$
$\displaystyle\phantom{=}\
+\alpha\bar{\nu}_{s,k}\sigma_{\textnormal{\tiny{UE}}}^{2}\mathbf{I}_{N}+\bar{\lambda}_{k}\mathbf{I}_{N}\bigg{)}^{-1}\bigg{(}\bigg{(}\alpha\frac{\bar{\nu}_{s,k}}{\gamma_{s,k}^{(i)}}+(1-\alpha)\frac{\bar{\mu}_{s,k}}{\bar{\gamma}_{s,k}^{(i)}}\bigg{)}$
$\displaystyle\phantom{=}\
\times\mathbf{H}_{k}^{\mathrm{H}}\mathbf{w}_{s,k}\mathbf{w}_{s,k}^{\mathrm{H}}\mathbf{H}_{k}{\mathbf{v}_{s,k}}^{(i)}\bigg{)}$
(33)

where $\bar{\nu}_{s,k}$, $\bar{\mu}_{s,k}$, and $\bar{\lambda}_{k}$ are the
dual variables corresponding to (32c), (32d), and (9g), respectively. As
detailed in the Appendix, $\bar{\mu}_{s,k}$ and $\bar{\nu}_{s,k}$ depend on
the auxiliary dual variables $\eta_{k}$ and $\zeta_{k}$ corresponding to (9b)
and (9c), respectively. Following similar steps as in Section III-A, the
latter can be updated using the sub-gradient method. However, this requires
additional DL signaling, as described next in Section IV-A. Alternatively, one
can use the heuristic approach proposed in Section IV-B to avoid both the sub-
gradient update (which may be computationally complex at the UEs) and the
additional DL signaling.

###  IV-A UE Beamforming Design with DL Training

During the DL pilot-aided channel estimation phase, each BS $b$ synchronously
transmits a superposition of the pilots $\\{\mathbf{p}_{s,k}\\}$ precoded with
the corresponding $\\{\mathbf{w}_{s,k}\\}$, i.e.,

$\displaystyle\mathbf{X}^{\textnormal{\tiny{DL-1}}}\triangleq\sum_{k\in\mathcal{K}}\sum_{s=1}^{S}\mathbf{w}_{s,k}\mathbf{p}_{s,k}^{\mathrm{H}}\in\mbox{$\mathbb{C}$}^{BM\times\tau}.$
(34)

Then, the receive signal at UE $k$ is given by

$\displaystyle\mathbf{Y}_{k}^{\textnormal{\tiny{DL-1}}}$
$\displaystyle\triangleq\mathbf{H}_{k}^{\mathrm{H}}\mathbf{X}^{\textnormal{\tiny{DL-1}}}+\mathbf{Z}_{k}^{\textnormal{\tiny{DL-1}}}$
(35)
$\displaystyle=\sum_{\bar{k}\in\mathcal{K}}\sum_{s=1}^{S}\mathbf{H}_{k}^{\mathrm{H}}\mathbf{w}_{s,\bar{k}}\mathbf{p}_{s,\bar{k}}^{\mathrm{H}}+\mathbf{Z}_{k}^{\textnormal{\tiny{DL-1}}}\in\mbox{$\mathbb{C}$}^{N\times\tau}$
(36)

where $\mathbf{Z}_{k}^{\textnormal{\tiny{DL-1}}}$ is the AWGN term at UE $k$.
To allow each UE to independently compute its beamforming strategy, the BS
needs to transmit a superposition of precoded pilots weighted with the
corresponding dual variables $\\{\bar{\mu}_{s,k}\\}$, i.e.,

$\displaystyle\mathbf{X}^{\textnormal{\tiny{DL-2}}}\triangleq\sum_{k\in\mathcal{K}}\sum_{s=1}^{S}\sqrt{\bar{\mu}_{s,k}}\mathbf{w}_{s,k}\mathbf{p}_{s,k}^{\mathrm{H}}\in\mbox{$\mathbb{C}$}^{BM\times\tau}$
(37)

and the receive signal at UE $k$ is given by

$\displaystyle\mathbf{Y}_{k}^{\textnormal{\tiny{DL-2}}}$
$\displaystyle\triangleq\mathbf{H}_{k}^{\mathrm{H}}\mathbf{X}^{\textnormal{\tiny{DL-2}}}+\mathbf{Z}_{k}^{\textnormal{\tiny{DL-2}}}$
(38)
$\displaystyle=\sum_{\bar{k}\in\mathcal{K}}\sum_{s=1}^{S}\sqrt{\bar{\mu}_{s,\bar{k}}}\mathbf{H}_{k}^{\mathrm{H}}\mathbf{w}_{s,\bar{k}}\mathbf{p}_{s,\bar{k}}^{\mathrm{H}}+\mathbf{Z}_{k}^{\textnormal{\tiny{DL-2}}}\in\mbox{$\mathbb{C}$}^{N\times\tau}$
(39)

where $\mathbf{Z}_{k}^{\textnormal{\tiny{DL-2}}}$ is the AWGN term at UE $k$.
Here, the dual variable $\bar{\mu}_{s,k}$ is computed at the CPU as described
in Section III-A.

Building on (36) and (39), the UE beamformer $\mathbf{v}_{s,k}$ can be
computed as in (III-A) at the top of the page. Note that (III-A) becomes equal
to (IV) as $\tau\to\infty$. The dual variable $\bar{\nu}_{s,k}$ is computed at
UE $k$ by replacing $\mathbf{H}_{k}\mathbf{w}_{s,k}$ with
$\frac{1}{\tau}\mathbf{Y}_{k}^{\textnormal{\tiny{DL-1}}}\mathbf{p}_{s,k}$ in
(47) (see the Appendix). For $\alpha=1$ and no transmit power constraint at UE
$k$ (i.e., $\bar{\lambda}_{k}=0$), $\hat{\mathbf{v}}_{s,k}$ in (III-A) is
equivalent to the MMSE receiver, which can be implemented based solely on
$\mathbf{Y}_{k}^{\textnormal{\tiny{DL-1}}}$ (i.e., without
$\mathbf{Y}_{k}^{\textnormal{\tiny{DL-2}}}$). Similarly, for $\alpha=0$,
$\hat{\mathbf{v}}_{s,k}$ in (III-A) is equivalent to UL max-min rate transmit
beamforming, which can implemented based solely on
$\mathbf{Y}_{k}^{\textnormal{\tiny{DL-2}}}$ (i.e., without
$\mathbf{Y}_{k}^{\textnormal{\tiny{DL-1}}}$).

###  IV-B Heuristic UE Beamforming Design

Let us consider the ideal UE beamforming design given in (IV) and let us
define
$a_{\bar{s},\bar{k}}(s,k)\triangleq{\alpha\bar{\nu}_{s,k}+(1-\alpha)\bar{\mu}_{\bar{s},\bar{k}}}$
and $b_{s,k}\triangleq{\alpha\bar{\nu}_{s,k}}$. Building on these definitions,
(IV) can be equivalently written as

$\displaystyle\mathbf{v}_{s,k}$
$\displaystyle=\bigg{(}\sum_{(\bar{k},\bar{s})}a_{\bar{s},\bar{k}}(s,k)\mathbf{H}_{{k}}^{\mathrm{H}}\mathbf{w}_{\bar{s},\bar{k}}\mathbf{w}_{\bar{s},\bar{k}}^{\mathrm{H}}\mathbf{H}_{k}$
$\displaystyle\phantom{=}\
+(b_{s,k}\sigma_{\textnormal{\tiny{UE}}}^{2}+\bar{\lambda}_{k})\mathbf{I}_{N}\bigg{)}^{-1}\mathbf{H}_{k}\mathbf{w}_{s,k}.$
(41)

As in Section IV-A, the practical implementation of (IV-B) requires additional
DL signaling, i.e., $\mathbf{Y}_{k}^{\textnormal{\tiny{DL-1}}}$ and
$\mathbf{Y}_{k}^{\textnormal{\tiny{DL-2}}}$. Furthermore, the sub-gradient
update of the auxiliary dual variables slows down the overall convergence.
First, to avoid the additional DL signaling, we consider a heuristic approach
for the UE beamformer design whereby the values of $a_{\bar{s},\bar{k}}(s,k)$
are assumed to be the same for all the UE beamformers and streams, i.e.,
$a_{\bar{s},\bar{k}}(s,k)=a_{\bar{s},\bar{k}},$ $\forall s,k$. In this
setting, the heuristic UE beamforming design can be implemented based on
$\mathbf{Y}_{k}^{\textnormal{\tiny{DL-2}}}$ (obtained by replacing
$\bar{\mu}_{s,k}$ with $a_{s,k}$ in (37)) as

$\displaystyle\hat{\mathbf{v}}_{s,k}\\!$
$\displaystyle=\\!\big{(}\mathbf{Y}_{k}^{\textnormal{\tiny{DL-2}}}(\mathbf{Y}_{k}^{\textnormal{\tiny{DL-2}}})^{\mathrm{H}}\\!-\\!\tau(1\\!-\\!b_{s,k})\sigma^{2}_{\textnormal{\tiny{UE}}}\mathbf{I}_{N}\\!+\\!\bar{\lambda}_{k}\mathbf{I}_{N}\big{)}^{-1}\frac{\mathbf{Y}_{k}^{\textnormal{\tiny{DL-2}}}\mathbf{p}_{s,k}}{\sqrt{a_{s,k}}}.$
(42)

Second, to avoid the sub-gradient update of the auxiliary dual variables at
the UEs, $a_{s,k}$ and $b_{s,k}$ are heuristically updated at the BSs (CPU)
and the UEs. This leads to a faster convergence of the heuristic method to a
sub-optimal point. As an example, a simple heuristic UE beamformer can be
obtained by setting $\\{a_{s,k}\\!=\\!1\\}$ and $\\{b_{s,k}\\!=\\!0\\}$.
Otherwise, one can tune $\\{a_{s,k}\\}$ and $\\{b_{s,k}\\}$ for faster and
better convergence based on the statistics of the network parameters such as
the number of the BSs/UEs, the transmit power at the BSs/UEs, and the channel
conditions.

##  V Numerical Results

We consider a cell-free massive MIMO scenario where $B=25$ BSs, each equipped
with $M=4$ antennas, are placed on a square grid with distance between
neighboring BSs of $100$ m. Furthermore, $K=16$ UEs, each equipped with $N=2$
antennas and served with $S=2$ data streams, are randomly dropped in the same
area. Assuming uncorrelated Rayleigh fading, each channel is generated
according to
$\mathrm{vec}(\mathbf{H}_{b,k})\sim\mathcal{C}\mathcal{N}(0,\delta_{b,k}\mathbf{I}_{MN})$,
where
$\delta_{b,k}\triangleq-61.3-30\log_{10}(d_{b,k})-20\log_{10}\big{(}f_{c}\leavevmode\nobreak\
\textrm{[GHz]}\big{)}$ [dB] is the large-scale fading coefficient, $d_{b,k}$
is the distance between BS $b$ and UE $k$, and $f_{c}=28$ GHz is the carrier
frequency [16]. The maximum transmit power for both the data and the pilot
transmission is $\rho_{\textnormal{\tiny{BS}}}=30$ dBm at the BSs and
$\rho_{\textnormal{\tiny{UE}}}=20$ dBm at the UEs. The AWGN power at the BSs
and at the UEs is fixed to
$\sigma_{\textnormal{\tiny{BS}}}^{2}=\sigma_{\textnormal{\tiny{UE}}}^{2}=-95$
dBm. Lastly, we set $\alpha=0.5$ for equal weight between DL and UL rates (see
(8)).

We begin by considering the ideal scenario where the CPU and each UE have
global knowledge of the channels and of the previously computed beamforming
strategies. The performance of the proposed joint DL-UL design, termed as DL-
UL opt., is compared with the case where the DL and UL beamforming strategies
are computed individually via two separate bi-directional training phases,
termed as Separate DL/UL opt. We also provide a comparison with the cases
where the DL (resp. UL) beamforming strategies are computed via a single bi-
directional training phase and are reused in the UL (resp. DL) direction,
termed as DL opt. (resp. UL opt.). The practical versions of these schemes
that avoid the sub-gradient updates at the UEs and the additional DL signaling
are referred to as DL-UL heur., Separate DL/UL heur., and UL heur.,
respectively (see Section IV-B).

As shown in Figure 1, the minimum DL-UL rate of the proposed DL-UL opt.
approaches that of the Separate DL/UL opt. However, the latter requires two
separate bi-directional training phases for the DL and UL beamforming design.
Furthermore, the DL-UL opt. outperforms the DL opt. and the UL opt., where all
these methods are characterized by the same training requirements. A similar
trend can be also observed for the practical heuristic UE beamforming design.
In this respect, the proposed DL-UL heur. outperforms the DL opt., the UL
heur., and the Separate DL/UL heur.

Figure 2 plots the effective minimum DL-UL rate taking into account the bi-
directional training overhead with scheduling block size of 4 slots. While the
rate improves with the number of bi-directional training iterations, the
fraction of time left for the actual data transmission is reduced, giving rise
to a clear performance trade-off. At the optimal number of iterations, the
effective minimum DL-UL rate of the proposed DL-UL opt. is 25% higher than
that of the Separate DL/UL opt., since the latter involves two separate bi-
directional training phases for the DL and UL beamforming design. As expected,
the effective minimum DL-UL rates produced by the DL opt. and the UL opt. are
lower than that of the DL-UL opt., where all these methods are characterized
by the same training requirements. This behavior can be also seen in the
practical heuristic UE beamforming design, where the proposed DL-UL heur.
improves the effective minimum DL-UL rate by 17% with respect to the DL opt.
More significant gains are observed when comparing to the UL heur. and the
Separate DL/UL heur.

Figure 1: Minimum DL-UL rate versus the number of bi-directional training
iterations. Figure 2: Effective minimum DL-UL rate versus the number of bi-
directional training iterations with scheduling block size of 4 slots.

In Figure 3, the optimal effective minimum DL-UL rate is plotted as a function
of the scheduling block size. For short scheduling blocks, the proposed DL-UL
opt. outperforms the Separate DL/UL opt., the DL opt., and the UL opt.
However, for large scheduling blocks (i.e., larger than 12 slots), the
Separate DL/UL opt. is superior to our joint DL-UL design, since the impact of
the bi-directional training reduces with the increasing scheduling block size.
Even in the practical heuristic UE beamforming design, the proposed DL-UL
heur. surpasses the DL opt., the UL heur., and the Separate DL/UL heur., as
well as the Separate DL/UL opt. for scheduling blocks shorter than 6 slots.
Moreover, the effective minimum DL-UL rate of the DL opt. is better than that
of the Separate DL/UL heur. In fact, the two schemes have the same minimum DL-
UL rate (as seen in Figure 1) but the latter requires more training.

Figure 3: Effective minimum DL-UL rate versus the scheduling block size.

##  VI Conclusions

The computation of the DL and UL beamforming strategies in cell-free massive
MIMO systems generally requires two separate bi-directional training
phases.Considering the scenario where the UEs to be served in the DL and in
the UL are the same and the scheduling blocks are short, we jointly optimize
the DL and UL beamforming strategies using only one bi-directional training
phase. The proposed framework maximizes the (weighted) minimum DL-UL rate
among all the UEs and produces a set of power-constrained beamforming vectors
that are used at the BSs and at the UEs for both DL and UL transmissions.
Moreover, we present a heuristic approach that avoids additional signaling
between the BSs and the UEs, which entails only a modest performance loss. For
short scheduling blocks, the reduced training overhead of the proposed joint
DL-UL optimization brings substantial gains (up to 25%) with respect to the
case where the DL and UL beamforming strategies are computed individually via
two separate bi-directional training phases. Even higher gains are observed
with respect to the case where the DL (resp. UL) beamforming strategies are
reused in the UL (resp. DL) direction.

## Appendix

The dual variables corresponding to (9b) and (9c) are given by

$\displaystyle\eta_{k}$
$\displaystyle\triangleq\eta_{k}^{(i)}-\delta(\alpha\sum_{s=1}^{S}\log_{2}(1+\gamma_{s,k}^{(i)})-R),$
(43) $\displaystyle\zeta_{k}$
$\displaystyle\triangleq\zeta_{k}^{(i)}-\delta((1-\alpha)\sum_{s=1}^{S}\log_{2}(1+\bar{\gamma}_{s,k}^{(i)})-R)$
(44)

respectively, where $\delta$ is the sub-gradient step size. In addition, the
dual variables corresponding to (17c) and (17d) are defined as

$\displaystyle\nu_{s,k}$
$\displaystyle\triangleq\frac{\eta_{k}\alpha(\gamma_{s,k}^{(i)})^{2}\log(2)}{({\gamma_{s,k}^{(i)}}+1)|\mathbf{v}_{s,k}^{\mathrm{H}}\mathbf{H}_{k}^{\mathrm{H}}\mathbf{w}^{(i)}_{s,k}|^{2}}$
(45)

and

$\displaystyle\mu_{s,k}$
$\displaystyle\triangleq\frac{\zeta_{k}(1-\alpha)(\bar{\gamma}_{s,k}^{(i)})^{2}\log(2)}{({\bar{\gamma}_{s,k}^{(i)}}+1)\big{|}{(\mathbf{w}_{s,k}^{(i)})^{\mathrm{H}}}\mathbf{H}_{k}\mathbf{v}_{s,k}\big{|}^{2}}$
(46)

respectively. Lastly, the dual variables corresponding to (32c) and (32d) are
defined as

$\displaystyle\bar{\nu}_{s,k}$
$\displaystyle\triangleq\frac{\eta_{k}\alpha(\gamma_{s,k}^{(i)})^{2}\log(2)}{({\gamma_{s,k}^{(i)}}+1)\big{|}{(\mathbf{v}_{s,k}^{(i)})^{\mathrm{H}}}\mathbf{H}_{k}^{\mathrm{H}}\mathbf{w}_{s,k}\big{|}^{2}},$
(47) $\displaystyle\bar{\mu}_{s,k}$
$\displaystyle\triangleq\frac{\zeta_{k}(1-\alpha)(\bar{\gamma}_{s,k}^{(i)})^{2}\log(2)}{({\bar{\gamma}_{s,k}^{(i)}}+1)|\mathbf{w}^{\mathrm{H}}_{s,k}\mathbf{H}_{k}\mathbf{v}_{s,k}^{{(i)}}|^{2}}$
(48)

respectively.

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