# Weighted Sum Rate Maximization of the mmWave Cell-Free MIMO Downlink Relying
on Hybrid Precoding

Author list  Chenghao Feng, Wenqian Shen, Jianping An, and Lajos Hanzo This
work was supported in part by the National Natural Science Foundation of China
(NSFC) under Grant 61901034, and in part by the Open Research Fund of the
Shaanxi Province Key Laboratory of Information Communication Network and
Security under Grant ICNS201905. L. Hanzo would like to acknowledge the
financial support of the Engineering and Physical Sciences Research Council
projects EP/N004558/1, EP/P034284/1, EP/P034284/1, EP/P003990/1 (COALESCE), of
the Royal Society’s Global Challenges Research Fund Grant as well as of the
European Research Council’s Advanced Fellow Grant QuantCom. (Corresponding
author: Wenqian Shen.) C. Feng, W. Shen, and J. An are with the School of
Information and Electronics, Beijing Institute of Technology, Beijing 100081,
China (e-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>and an@bit.edu.cn).L.
Hanzo is with the Department of Electronics and Computer Science, University
of Southampton, Southampton SO17 1BJ, UK (e-mail: lh@ecs.soton.ac.uk).

###### Abstract

The cell-free MIMO concept relying on hybrid precoding constitutes an
innovative technique capable of dramatically increasing the network capacity
of millimeter-wave (mmWave) communication systems. It dispenses with the cell
boundary of conventional multi-cell MIMO systems, while drastically reducing
the power consumption by limiting the number of radio frequency (RF) chains at
the access points (APs). In this paper, we aim for maximizing the weighted sum
rate (WSR) of mmWave cell-free MIMO systems by conceiving a low-complexity
hybrid precoding algorithm. We formulate the WSR optimization problem subject
to the transmit power constraint for each AP and the constant-modulus
constraint for the phase shifters of the analog precoders. A block coordinate
descent (BCD) algorithm is proposed for iteratively solving the problem. In
each iteration, the classic Lagrangian multiplier method and the penalty dual
decomposition (PDD) method are combined for obtaining near-optimal hybrid
analog/digital precoding matrices. Furthermore, we extend our proposed
algorithm for deriving closed-form expressions for the precoders of fully
digital cell-free MIMO systems. Moreover, we present the convergency analysis
and complexity analysis of our proposed method. Finally, our simulation
results demonstrate the superiority of the algorithms proposed for both fully
digital and hybrid precoding matrices.

###### Index Terms:

Cell-free MIMO, hybrid precoding, mmWave communication.

##  I Introduction

The next-generation wireless communication systems have an increasing demand
for network capacity [1]. The cell-free multi-input multi-output (MIMO)
concept constitutes a promising technique [2] of enhancing the capacity of
wireless communication systems due to its capability of supporting a large
number of handsets, by relying on distributed access points (APs) controlled
by a central processing unit (CPU) [3]. The key advantage of the cell-free
MIMO architecture is its user-centric (UC) nature, where each user is
collaboratively served by all APs without cell boundaries [2, 4, 5]. In fact,
typically a number of users form a UC cluster and they are supported by a
number of APs. When the user density is high, more APs join the cluster with
the objective of load balancing, which leads to amorphous clusters of
different sizes. By cancelling the inter-user interference (IUI) using
transmit precoding (TPC), cell-free MIMO significantly improves the network
capacity of wireless communications [6].

Millimeter-wave (mmWave) solutions have been identified as one of the key
techniques for future wireless communication systems as a benefit of their
abundant bandwidth [7, 8, 9, 10]. However, the extremely high path-loss of
mmWave signals has to be compensated [11]. Fortunately, thanks to the short
wavelength of mmWave carriers, a large number of antennas can be packaged in a
compact physical size [12, 13, 14, 15]. The high spatial gain of a high-
dimensional antenna array is able to overcome the severe path-loss [16, 17,
18]. Due to the sparsity of mmWave channels, hybrid TPC has been proposed for
achieving high spatial gain despite using a reduced number of radio frequency
(RF) chains, which considerably reduces the power consumption of mmWave
systems [19, 20, 21]. Therefore, the combination of hybrid TPC and cell-free
MIMO has both a high network capacity and a high energy efficiency [22, 23].

TABLE I: Boldly Contrasting Our Contributions to the Relevant Literatures | [2] | [3] | [22] | [23] | [24] | [25] | [26] | Proposed
---|---|---|---|---|---|---|---|---
Hybrid Precoding |  |  | ✓ | ✓ |  |  |  | ✓
Uplink | ✓ | ✓ | ✓ | ✓ |  |  |  |
Downlink | ✓ |  |  | ✓ | ✓ | ✓ | ✓ | ✓
Multiple antennas at each user |  |  | ✓ |  |  | ✓ |  | ✓
Multiple data streams for each user |  |  | ✓ |  |  |  |  | ✓
Sum rate |  | ✓ | ✓ | ✓ |  | ✓ | ✓ | ✓

###  I-A Literature review

Some TPC techniques designed for cell-free MIMO networks have been reported in
[2, 3, 24, 25, 26, 27, 28]. Ngo et al. [2] propose to use a maximum ratio
transmission (MRT)-based TPC at the APs in the downlink (DL) and use a matched
filter at the users on the uplink (UL) for maximizing the system throughputs.
Björnson and Sanguinetti [3] propose to maximize the sum rate of the uplink in
cell-free MIMO by using the classical weighted minimum mean square error
(WMMSE) based TPC. Nayebi et al. [24] propose to employ a zero-forcing (ZF)
TPC on the DL of cell-free MIMO. Zhang and Dai [25] propose to maximize the
weighted sum rate on the DL of cell-free MIMO by exploiting the recently
proposed fractional programming (FP) technique, where only one data stream is
considered for each user. Furthermore, some prior treatises consider user
scheduling in cell-free MIMO. Interdonato et al. [26] propose a pair of
distributed TPC schemes in cell-free MIMO, namely the local partial ZF TPC and
the local protective partial ZF TPC, which improve the spectral efficiency
with the aid of user grouping. Nguyen et al. [27] propose to jointly optimize
both the AP-user association and the AP selection for striking a trade-off
between a network’s spectral efficiency and energy efficiency for cell-free
MIMO. Wang et al. [28] consider both a ZF and a regularized ZF (RZF) TPC and
advocate a novel genetic algorithm based user scheduling strategy (GAS) for
alleviating the uplink-to-downlink interference in cell-free MIMO. Moreover,
since numerous APs are deployed in cell-free networks, the energy-efficient
hybrid precoding structures have substantial benefits in cell-free MIMO [22,
23], especially when sparse mmWave channels are considered.

Hybrid TPCs have been widely studied both in single-cell and multi-cell
networks. El-Ayach et al. [29] propose to design a hybrid TPC by using the
popular orthogonal matching pursuit (OMP)-based method. Alkhateeb and Heath
[30] conceive an efficient hybrid analog/digital codebook and propose a
codebook-based method for hybrid TPC designs in wideband mmWave systems.
Sohrabi and Yu [31, 32] conceive an iterative TPC design method for finite
resolution of phase shifters, which achieves near-optimal spectral efficiency
both in single-user and multi-user MIMO systems. Furthermore, some low-
complexity analog TPCs have also been proposed. Yu et al. [33] design a
switch-based network for dynamically changing the connection between the phase
shifters, RF chains and antennas. Gao et al. [20] propose a pioneering sub-
array-connected phase-shifter architecture and propose a successive
interference cancelation (SIC)-based hybrid TPC method attaining a near-
optimal spectral efficiency performance at a low complexity. Chen et al. [34]
conceive a randomized two-timescale hybrid TPC scheme for the DL of multicell
massive MIMO systems. Passive TPC structures have also been proposed by
researchers. Han et al. [35] employ the so-called Butler matrices for
enhancing discrete fourier transform (DFT)-based systems. They also propose a
corresponding two-step hybrid TPC method for achieving near-optimal
performance at a significantly reduced computational complexity. Applying
hybrid TPC structures in cell-free MIMO is a promising but challenging field.
The UC property of cell-free MIMO, the power constraints imposed on each AP
and the constant-modulus constraints of analog TPC design must be jointly
considered. This motivates us to fill in this knowledge-gap.

There are very few contributions investigating hybrid TPC techniques in cell-
free MIMO. Femenias et al. [36] propose to maximize the achievable max-min
per-user rate in cell-free MIMO relying on hybrid TPC structures. Nguyen et
al. [22] propose both centralized and semi-centralized hybrid TPC algorithms
for maximizing the achievable data rate in the UL of cell-free MIMO. However,
the derivation of hybrid TPC matrices in the UL of cell-free MIMO cannot be
directly applied in the DL. Alonzo et al. [23] propose to decompose the fully
digital TPCs into hybrid ones by using a subspace decomposition algorithm for
the DL of cell-free MIMO. Nevertheless, this subspace decomposition inevitably
causes some sum rate loss. Therefore, as seen from Table I, the demand for a
general framework of DL hybrid TPC with near-optimal sum rate performance in
cell-free MIMO attracts our attention.

###  I-B Contribution

Against this background, the main contributions of this paper are summarized
as follows:

  1. 1.

We propose to maximize the weighted sum rate (WSR) of the mmWave cell-free
MIMO DL relying on hybrid TPC structures, where multiple APs, multiple users
and multiple data streams per user are considered. We stipulate a pair of
constraints for our optimization problem, namely the transmit power constraint
of each AP and the constant-modulus constraint imposed on each element of the
analog TPC matrices. Both the objective function (OF) and the constant-modulus
constraint are non-convex, which make the problem formulated difficult to
solve. To deal with this issue, we first reformulate the OF into an equivalent
form by exploiting the WMMSE criterion based upon the UC property of cell-free
MIMO. Furthermore, we introduce auxiliary TPC matrices and propose a low-
complexity block coordinate descent (BCD) algorithm for iteratively designing
near-optimal hybrid TPC matrices.

  2. 2.

In each iteration of the proposed BCD algorithm, we exploit the classic
Lagrangian multiplier method combined with the penalty dual decomposition
(PDD) method for designing the hybrid analog/digital TPC matrices of the cell-
free MIMO DL. Specifically, we first derive the optimal closed-form Lagrangian
multiplier-based solutions for finding the auxiliary TPC matrices through
sophisticated matrix manipulations and transformations. Subsequently, we
optimize the penalty items by adopting the classic least square (LS) method,
which minimizes the Euclidian distance between the auxiliary TPC matrices and
the hybrid analog/digital TPC matrices.

  3. 3.

We further extend our proposed BCD algorithm to the fully digital cell-free
MIMO DL. In contrast to the hybrid TPC scenario, there is no penalty item in
the fully digital TPC design, which leads to the difficulties of low-rank
matrix inversion for obtaining the optimal Lagrangian multipliers. Therefore,
we circumvent this problem by deriving the exact ranks of the key intermediate
matrices, which are used for calculating Lagrangian multipliers. Our
simulation results show that the proposed algorithm can achieve better WSR
performance than its conventional counterparts.

The remainder of this paper is organized as follows. In Section II, our system
model, channel model and problem formulation are introduced. In Section III,
our BCD algorithm proposed for mmWave cell-free MIMO hybrid TPC design is
presented. In Section IV, we extend our proposed BCD algorithm to the mmWave
cell-free MIMO DL relying on a fully digital TPC structure. In Section V, our
numerical results are provided. Finally, our conclusions are drawn in Section
VI.

_Notation_ : Lower-case and upper-case boldface letters denote vectors and
matrices, respectively; $(\cdot)^{\rm{T}}$, $(\cdot)^{\text{H}}$,
$(\cdot)^{-1}$ and $(\cdot)^{\dagger}$ denote the transpose, conjugate
transpose, inverse and pseudo-inverse of a matrix, respectively; $\otimes$
denotes the Kronecker product; $\mathrm{Tr}(\cdot)$ presents the trace
function; $\|\cdot\|_{F}$ denotes the Frobenius norm of a matrix; $|a|$ is the
absolute value of a scalar; $|\mathbf{A}|$ is the determinant of a matrix;
$\mathbf{A}_{\left[i,:\right]}$ and $\mathbf{A}_{\left[:,j\right]}$ represent
the $i$-th row and $j$-th column of the matrix $\mathbf{A}$, respectively;
$\mathbf{A}_{\left[a:b,:\right]}$ and $\mathbf{A}_{\left[:,c:d\right]}$
represent the sub-matrices of the matrix $\mathbf{A}$ containing the $a$-th to
the $b$-th rows and the $c$-th to the $d$-th columns, respectively; Finally,
$\mathbf{I}_{P}$ denotes the identity matrix of size $P\times P$.

##  II System Model and Channel Model of Cell-Free MIMO Relying on Hybrid
Precoding Structures

In this section, we introduce the narrow-band mmWave cell-free MIMO DL model,
where hybrid TPCs are used at the APs.

###  II-A System Model of Cell-Free MIMO Relying on Hybrid Precoding
Structures

Figure 1: Illustration of the cell-free MIMO concept. Figure 2: Illustration
of the hybrid TPC structure at each AP.

As shown in Fig. 1, there are $B$ multiple-antenna APs relying on hybrid TPCs
for supporting $U$ multiple-antenna users in the cell-free MIMO network. Each
AP is equipped with $N_{t}$ transmit antennas (TAs) and $N_{\rm RF}$ RF
chains, $N_{\rm RF}\leq N_{t}$, as shown in Fig. 2. At each user, fully
digital structures are adopted, where $N_{r}$ receive antennas (RAs) are
employed for receiving $N_{s}$ data streams, $N_{s}\leq N_{r}$. We assume that
$UN_{s}\leq BN_{\rm RF}$ in our cell-free MIMO. For simplicity, we define
$\mathcal{N}_{t}=\left\\{1,2,\cdots,N_{t}\right\\}$,
$\mathcal{N}_{r}=\left\\{1,2,\cdots,N_{r}\right\\}$, $\mathcal{N}_{\rm
RF}=\left\\{1,2,\cdots,N_{\rm RF}\right\\}$,
$\mathcal{N}_{s}=\left\\{1,2,\cdots,N_{s}\right\\}$,
$\mathcal{U}=\left\\{1,2,\cdots,U\right\\}$,
$\mathcal{B}=\left\\{1,2,\cdots,B\right\\}$.

We assume that all APs serve all users. Thus, all APs synchronously transmit
data symbol $\left\\{\mathbf{s}_{u}\right\\}_{u=1}^{U}$, where
$\mathbf{s}_{u}\in\mathbb{C}^{N_{s}\times 1}$ is the data symbols intended for
the $u$-th user. The transmitted data symbols are assumed to have normalized
power, i.e., $\mathbb{E}\left\\{\mathbf{s}_{u}\mathbf{s}_{u}^{\rm
H}\right\\}=\mathbf{I}_{N_{s}}$, $\forall u\in\mathcal{U}$. The data symbols
$\mathbf{s}_{u}$ transmitted from the $b$-th AP are first digitally precoded
by $\mathbf{F}_{{\rm BB},b,u}\in\mathbb{C}^{N_{\rm RF}\times N_{s}}$. Then the
analog TPC $\mathbf{F}_{{\rm RF},b}\in\mathbb{C}^{N_{t}\times N_{\rm RF}}$
relying on phase shifters is applied. Therefore, the signals
$\mathbf{x}_{b}\in\mathbb{C}^{N_{t}\times 1}$ after hybrid precoding at the
$b$-th AP are expressed by

$\displaystyle\mathbf{x}_{b}=\mathbf{F}_{{\rm
RF},b}\sum_{u=1}^{U}\mathbf{F}_{{\rm BB},b,u}\mathbf{s}_{u}.$ (1)

We define the mmWave channel between the $b$-th AP and the $u$-th user as
$\mathbf{H}_{b,u}\in\mathbb{C}^{N_{r}\times N_{t}}$. It is assumed that the
CSI is perfectly known at all APs [37, 38, 39], noting that both the accurate
channel estimation and the robust TPC design relying on partial CSI constitute
rather different problems in cell-free MIMO systems [40, 41, 42]. Given the
strict length-limitation, these have to be set aside for future research. Then
the signals $\mathbf{y}_{u}\in\mathbb{C}^{N_{r}\times 1}$ received at the
$u$-th user are expressed by

$\displaystyle\mathbf{y}_{u}=\sum_{b=1}^{B}\mathbf{H}_{b,u}\mathbf{x}_{b}$
$\displaystyle=\underbrace{\sum_{b=1}^{B}\mathbf{H}_{b,u}\mathbf{F}_{{\rm
RF},b}\mathbf{F}_{{\rm BB},b,u}\mathbf{s}_{u}}_{\text{desired signal of the
$u$-th user}}$ $\displaystyle+\underbrace{\sum_{b=1}^{B}\sum_{j=1,j\neq
u}^{U}\mathbf{H}_{b,u}\mathbf{F}_{{\rm RF},b}\mathbf{F}_{{\rm
BB},b,j}\mathbf{s}_{j}}_{\text{IUI}}+\mathbf{n}_{u},$ (2)

where the first item represents the desired signal of the $u$-th user, the
second item stands for the IUI, and
$\mathbf{n}_{u}\sim\mathcal{CN}\left(0,\sigma^{2}\mathbf{I}_{N_{r}}\right)\in\mathbb{C}^{N_{r}\times
1}$ is the additive white Gaussian noise (AWGN). We further reformulate (II-A)
as

$\displaystyle\mathbf{y}_{u}=\mathbf{H}_{u}\mathbf{F}_{{\rm
HB},u}\mathbf{s}_{u}+\sum_{j=1,j\neq u}^{U}\mathbf{H}_{u}\mathbf{F}_{{\rm
HB},j}\mathbf{s}_{j}+\mathbf{n}_{u},$ (3)

where
$\mathbf{H}_{u}=\left[\mathbf{H}_{1,u},\mathbf{H}_{2,u},\cdots,\mathbf{H}_{B,u}\right]\in\mathbb{C}^{N_{r}\times
BN_{t}}$, $\mathbf{F}_{{\rm HB},u}=\left[\mathbf{F}_{{\rm HB},1,u}^{\rm
H},\mathbf{F}_{{\rm HB},2,u}^{\rm H},\cdots,\mathbf{F}_{{\rm HB},B,u}^{\rm
H}\right]^{\rm H}\in\mathbb{C}^{BN_{t}\times N_{s}}$, and $\mathbf{F}_{{\rm
HB},b,u}=\mathbf{F}_{{\rm RF},b}\mathbf{F}_{{\rm
BB},b,u}\in\mathbb{C}^{N_{t}\times N_{s}}$.

###  II-B Narrow-band MmWave Channel Model

In this paper, we adopt the popular physical multi-path mmWave channel model,
where uniform planar arrays (UPAs) of antennas are employed both at APs and
users [29, 32, 43]. Therefore, we can express the channel $\mathbf{H}_{b,u}$
between the $b$-th AP and the $u$-th user by

$\displaystyle\mathbf{H}_{b,u}=\sqrt{\frac{N_{t}N_{r}}{L}}\sum_{\ell=1}^{L}\alpha_{\ell,b,u}\mathbf{a}_{r}\left(\phi^{\mathrm{r}}_{\ell,b,u},\theta^{\mathrm{r}}_{\ell,b,u}\right)\mathbf{a}_{t}^{\text{H}}\left(\phi^{\mathrm{t}}_{\ell,b,u},\theta^{\mathrm{t}}_{\ell,b,u}\right),$
(4)

where $L$ denotes the number of paths,
$\alpha_{\ell,b,u}\sim\mathcal{CN}\left(0,1\right)$ is the complex gain of the
$\ell$-th path between the $b$-th AP and the $u$-th user. The vectors
$\mathbf{a}_{t}\left(\phi^{\mathrm{t}}_{\ell,b,u},\theta^{\mathrm{t}}_{\ell,b,u}\right)\in\mathbb{C}^{N_{t}\times
1}$ and
$\mathbf{a}_{r}\left(\phi^{\mathrm{r}}_{\ell,b,u},\theta^{\mathrm{r}}_{\ell,b,u}\right)\in\mathbb{C}^{N_{r}\times
1}$ represent the antenna array responses at the $b$-th AP and the $u$-th
user, which can be expressed as

$\displaystyle\mathbf{a}_{t}\left(\phi^{\mathrm{t}}_{\ell,b,u},\theta^{\mathrm{t}}_{\ell,b,u}\right)$
$\displaystyle=\frac{1}{\sqrt{N_{t}}}\left[1,\cdots,e^{j\frac{2\pi}{\lambda}d\left(m\mathrm{sin}\phi^{\mathrm{t}}_{\ell,b,u}\mathrm{sin}\theta^{\mathrm{t}}_{\ell,b,u}+n\mathrm{cos}\theta^{\mathrm{t}}_{\ell,b,u}\right)},\right.$
$\displaystyle\cdots,\left.e^{j\frac{2\pi}{\lambda}d\left(\left(W-1\right)\mathrm{sin}\phi^{\mathrm{t}}_{\ell,b,u}\mathrm{sin}\theta^{\mathrm{t}}_{\ell,b,u}+\left(H-1\right)\mathrm{cos}\theta^{\mathrm{t}}_{\ell,b,u}\right)}\right]^{\text{T}},$
(5)

where $\phi^{\mathrm{t}}_{\ell,b,u}$ and $\theta^{\mathrm{t}}_{\ell,b,u}$
denote the azimuth and elevation angle-of-departure (AoD) from the $b$-th AP
to the $u$-th user, $\lambda$ represents the wavelength, and $d$ is the
antenna spacing. Furthermore, $m\in\\{0,1,\cdots,W-1\\}$ and
$n\in\\{0,1,\cdots,H-1\\}$ with $W$ and $H$ denoting the number of antennas in
horizontal and vertical directions. Similarly,
$\mathbf{a}_{r}\left(\phi^{\mathrm{r}}_{\ell,b,u},\theta^{\mathrm{r}}_{\ell,b,u}\right)$
can be expressed in the same form as (II-B) by replacing
$\phi^{\mathrm{t}}_{\ell,b,u}$ and $\theta^{\mathrm{t}}_{\ell,b,u}$ with the
azimuth and elevation angle-of-arrival (AoA) $\phi^{\mathrm{r}}_{\ell,b,u}$
and $\theta^{\mathrm{r}}_{\ell,b,u}$.

###  II-C Problem Formulation

Again, we aim for maximizing the WSR of the mmWave cell-free MIMO DL by
designing the hybrid TPC matrices of all $B$ APs. We first present the
achievable WSR expression of

$\displaystyle
R=\sum_{u=1}^{U}\omega_{u}\log\left|\mathbf{I}_{N_{r}}+\mathbf{J}^{-1}_{u}\mathbf{H}_{u}\mathbf{F}_{{\rm
HB},u}\mathbf{F}_{{\rm HB},u}^{\mathrm{H}}\mathbf{H}_{u}^{\mathrm{H}}\right|,$
(6)

where $\mathbf{J}_{u}=\sum_{j=1,j\neq u}^{U}\mathbf{H}_{u}\mathbf{F}_{{\rm
HB},j}\mathbf{F}_{{\rm HB},j}^{\rm H}\mathbf{H}_{u}^{\rm
H}+\sigma^{2}\mathbf{I}_{N_{r}}$ denotes the interference-plus-noise item, and
$\omega_{u}$ is the weighting coefficient of the $u$-th user. The WSR
maximization problem of the mmWave cell-free MIMO DL relying on hybrid TPC is
formulated as

$\displaystyle\max_{\mathcal{F}_{\rm RF},\mathcal{F}_{\rm BB}}\quad$
$\displaystyle R$ (7a) $\displaystyle s.t.\quad\ $
$\displaystyle\sum_{u=1}^{U}\mathrm{Tr}\left(\mathbf{F}_{{\rm
RF},b}\mathbf{F}_{{\rm BB},b,u}\mathbf{F}_{{\rm BB},b,u}^{\rm
H}\mathbf{F}_{{\rm RF},b}^{\rm H}\right)\leq P_{b,\max},$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\forall
b\in\mathcal{B},$ (7b) $\displaystyle\left|\mathbf{F}_{{\rm
RF},b}\left(i,j\right)\right|=1,\forall i\in\mathcal{N}_{t},\forall
j\in\mathcal{N}_{\rm RF},\forall b\in\mathcal{B},$ (7c)

where $\mathcal{F}_{\rm RF}=\left\\{\mathbf{F}_{{\rm RF},b}|\forall
b\in\mathcal{B}\right\\}$, $\mathcal{F}_{\rm BB}=\left\\{\mathbf{F}_{{\rm
BB},b,u}|\forall b\in\mathcal{B},\forall u\in\mathcal{U}\right\\}$. The
constraint of (7b) ensures that the transmit power at the $b$-th AP is lower
than the maximum power $P_{b,\max}$. The constraint of (7c) is due to the
constant-modulus constraint of phase shifters used in the analog TPCs. We
observe that both of the OF (7a) and the constraint (7c) are non-convex. Hence
it is challenging to solve the problem (7) directly.

##  III Proposed Hybrid Precoding Algorithm for MmWave cell-free MIMO

In this section, we propose a low complexity method for solving the problem
(7). Firstly, we transform the problem into an equivalent form by exploiting
the equivalence between the sum rate and the WMMSE [44, 45, 46, 47]. Then we
propose a BCD algorithm for solving the problem in an alternating optimization
manner. The analog and digital TPCs are alternately optimized by applying the
Lagrangian multiplier method combined with the PDD method, which is inspired
by [48, 49, 50].

###  III-A Problem Transformation

On basis of the UC nature of the cell-free MIMO DL, the problem (7) can be
equivalently transformed into a WMMSE problem by introducing a set of linear
combiners $\mathcal{G}=\left\\{\mathbf{G}_{u}\in\mathbb{C}^{N_{r}\times
N_{s}}|\forall u\in\mathcal{U}\right\\}$ and a set of arbitrary positive
definite weighting matrices $\mathcal{W}=\left\\{\mathbf{W}_{u}\in\right.$
$\left.\mathbb{C}^{N_{s}\times N_{s}}|\forall u\in\mathcal{U}\right\\}$.
Specifically, the signals $\hat{\mathbf{s}}_{u}\in\mathbb{C}^{N_{s}\times 1}$
estimated by the $u$-th user acquired with the aid of $\mathbf{G}_{u}$ are
expressed as

$\displaystyle\hat{\mathbf{s}}_{u}=\mathbf{G}_{u}^{\text{H}}\mathbf{y}_{u}.$
(8)

Then, we are ready to derive the MSE covariance matrix for the $u$-th user as

$\displaystyle\mathbf{E}_{u}$
$\displaystyle=\mathbb{E}\left[\left(\hat{\mathbf{s}}_{u}-\mathbf{s}_{u}\right)\left(\hat{\mathbf{s}}_{u}-\mathbf{s}_{u}\right)^{\text{H}}\right]$
$\displaystyle=\left(\mathbf{G}_{u}^{\text{H}}\mathbf{H}_{u}\mathbf{F}_{{\rm
HB},u}-\mathbf{I}_{N_{s}}\right)\left(\mathbf{G}_{u}^{\text{H}}\mathbf{H}_{u}\mathbf{F}_{{\rm
HB},u}-\mathbf{I}_{N_{s}}\right)^{\text{H}}$
$\displaystyle+\mathbf{G}_{u}^{\text{H}}\mathbf{J}_{u}\mathbf{G}_{u}.$ (9)

The problem (7) can be equivalently transformed into the WMMSE problem, as
detailed in [44, 45, 46, 47]

$\displaystyle\max_{\mathcal{W},\mathcal{G},\mathcal{F}_{\rm
RF},\mathcal{F}_{\rm BB}}\
\sum_{u=1}^{U}\omega_{u}\left[\log\left|\mathbf{W}_{u}\right|-\mathrm{Tr}\left(\mathbf{W}_{u}\mathbf{E}_{u}\right)+\mathrm{Tr}\left(\mathbf{I}_{N_{s}}\right)\right]$
(10a) $\displaystyle\quad\quad
s.t.\quad\quad\quad\text{(\ref{ObjEE_v1}b)},\quad\text{(\ref{ObjEE_v1}c)}.$
(10b)

However, the problem (10) is still challenging to solve due to the non-convex
constraint of (7c). To deal with this problem, we introduce a set of auxiliary
TPC matrices $\mathcal{F}=\left\\{\mathbf{F}_{u}\in\mathbb{C}^{BN_{t}\times
N_{s}}|\forall u\in\mathcal{U}\right\\}$ as the approximate solution of
$\left\\{\mathbf{F}_{{\rm HB},u}|\forall u\in\mathcal{U}\right\\}$. Then,
relying on the PDD method [48, 49], we introduce a set of penalty parameters
$\left\\{\rho_{b}|\forall b\in\mathcal{B}\right\\}$ and the corresponding
penalty terms $\frac{1}{2\rho_{b}}\left\|\mathbf{F}_{b,u}-\mathbf{F}_{{\rm
HB},b,u}\right\|_{F}^{2}$, where
$\mathbf{F}_{b,u}=\mathbf{F}_{u\left[\left(b-1\right)N_{t}+1:bN_{t},:\right]}$
denotes the submatrix of $\mathbf{F}_{u}$. By neglecting the constant term
$\mathrm{Tr}\left(\mathbf{I}_{N_{s}}\right)$, the problem (10) is transformed
into

$\displaystyle\max_{\mathcal{W},\mathcal{G},\mathcal{F},\mathcal{F}_{\rm
RF},\mathcal{F}_{\rm BB}}\
\sum_{u=1}^{U}\omega_{u}\left[\log\left|\mathbf{W}_{u}\right|-\mathrm{Tr}\left(\mathbf{W}_{u}\widetilde{\mathbf{E}}_{u}\right)\right]$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad-\sum_{b=1}^{B}\sum_{u=1}^{U}\frac{1}{2\rho_{b}}\left\|\mathbf{F}_{b,u}-\mathbf{F}_{{\rm
HB},b,u}\right\|_{F}^{2}$ (11a) $\displaystyle\quad\quad
s.t.\quad\sum_{u=1}^{U}\mathrm{Tr}\left(\mathbf{F}_{b,u}\mathbf{F}_{,b,u}^{\rm
H}\right)\leq P_{b,\max},\forall
b\in\mathcal{B},\quad\text{(\ref{ObjEE_v1}c)}$ (11b)

where we have

$\displaystyle\widetilde{\mathbf{E}}_{u}$
$\displaystyle=\left(\mathbf{G}_{u}^{\text{H}}\mathbf{H}_{u}\mathbf{F}_{u}-\mathbf{I}_{N_{s}}\right)\left(\mathbf{G}_{u}^{\text{H}}\mathbf{H}_{u}\mathbf{F}_{u}-\mathbf{I}_{N_{s}}\right)^{\text{H}}$
$\displaystyle+\sum_{j=1,j\neq
u}^{U}\mathbf{G}_{u}^{\text{H}}\mathbf{H}_{u}\mathbf{F}_{j}\mathbf{F}_{j}^{\rm
H}\mathbf{H}_{u}^{\rm
H}\mathbf{G}_{u}+\sigma^{2}\mathbf{G}_{u}^{\text{H}}\mathbf{G}_{u}.$ (12)

We observe that the problem (11) is more tractable because the OF (11a) is a
concave function with respect to $\mathcal{W}$, $\mathcal{G}$, and
$\mathcal{F}$, provided that the other matrices are fixed. Hence the problem
(11) is a convex optimization problem. Moreover, the hybrid analog/digital TPC
matrices $\mathcal{F}_{\rm RF}$ and $\mathcal{F}_{\rm BB}$ are only related to
the penalty terms in (11a). Therefore, we can derive the closed-form solution
of $\mathcal{W}$, $\mathcal{G}$, $\mathcal{F}$, $\mathcal{F}_{\rm RF}$ and
$\mathcal{F}_{\rm BB}$ by exploiting the BCD algorithm. Briefly, the key idea
of the BCD algorithm is to iteratively obtain one set of variables while
keeping the others fixed.

###  III-B Optimization of $\mathcal{G}$ and $\mathcal{W}$

The closed-form solutions of $\mathcal{G}$ and $\mathcal{W}$ can be acquired
by using the partial derivative method. Specifically, after substituting
(III-A) into (10a), we derive the partial derivative of (10a) with respect to
$\mathbf{G}_{u}$ and set it to zero. Then the optimal solution of
$\mathbf{G}_{u}$ for the $u$-th user is given by

$\displaystyle\mathbf{G}_{u}^{\star}=\left(\sum_{j=1}^{U}\mathbf{H}_{u}\mathbf{F}_{j}\mathbf{F}_{j}^{\rm
H}\mathbf{H}_{u}^{\rm
H}+\sigma^{2}\mathbf{I}_{N_{r}}\right)^{-1}\mathbf{H}_{u}\mathbf{F}_{u}.$ (13)

Similarly, we can derive the optimal solution of $\mathbf{W}_{u}$ for the
$u$-th user as

$\displaystyle\mathbf{W}_{u}^{\star}=\widetilde{\mathbf{E}}_{u}^{-1}.$ (14)

###  III-C Optimization of Approximate Hybrid Precoding Matrices $\mathcal{F}$

Given both the linear combining matrices $\mathcal{G}$, as well as the
weighting matrices $\mathcal{W}$, and the analog TPC matrices
$\mathcal{F}_{\rm RF}$ and the digital TPC matrices $\mathcal{F}_{\rm BB}$,
the subproblem with respect to $\mathcal{F}$ is expressed as

$\displaystyle\min_{\mathcal{F}}\
\sum_{u=1}^{U}\omega_{u}\left[\mathrm{Tr}\left(\mathbf{W}_{u}\widetilde{\mathbf{E}}_{u}\right)\right]$
$\displaystyle\quad\quad\quad+\sum_{b=1}^{B}\sum_{u=1}^{U}\frac{1}{2\rho_{b}}\left\|\mathbf{F}_{b,u}-\mathbf{F}_{{\rm
HB},b,u}\right\|_{F}^{2}$ (15a) $\displaystyle\
s.t.\quad\sum_{u=1}^{U}\mathrm{Tr}\left(\mathbf{F}_{b,u}\mathbf{F}_{,b,u}^{\rm
H}\right)\leq P_{b,\max},\forall b\in\mathcal{B},$ (15b)

which is transformed from the problem (11) by neglecting the variables
irrelevant to $\mathcal{F}$. We can derive the closed-form solutions of the
problem (15) by applying the Lagrangian multiplier method. Firstly, we
transform the problem (15) into an unconstrained form by introducing a set of
Lagrangian multipliers $\left\\{\lambda_{b}\geq 0|\forall
b\in\mathcal{B}\right\\}$ as

$\displaystyle\min_{\left\\{\lambda_{b}|\forall
b\in\mathcal{B}\right\\},\mathcal{F}}\ $
$\displaystyle\sum_{u=1}^{U}\omega_{u}\left[\mathrm{Tr}\left(\mathbf{W}_{u}\widetilde{\mathbf{E}}_{u}\right)\right]$
$\displaystyle\quad\quad\quad+\sum_{b=1}^{B}\sum_{u=1}^{U}\frac{1}{2\rho_{b}}\left\|\mathbf{F}_{b,u}-\mathbf{F}_{{\rm
HB},b,u}\right\|_{F}^{2}$
$\displaystyle+\sum_{b=1}^{B}\lambda_{b}\left[\sum_{u=1}^{U}\mathrm{Tr}\left(\mathbf{F}_{b,u}\mathbf{F}_{,b,u}^{\rm
H}\right)-P_{b,\max}\right].$ (16)

By substituting (III-A) into (III-C), we further transform the problem (III-C)
as follows

$\displaystyle\min_{\left\\{\lambda_{b}|\forall
b\in\mathcal{B}\right\\},\mathcal{F}}\ $
$\displaystyle\sum_{u=1}^{U}\omega_{u}\left[\mathrm{Tr}\left(\mathbf{F}_{u}^{\rm
H}\mathbf{A}_{u}\mathbf{F}_{u}\right)\right]$
$\displaystyle-\sum_{u=1}^{U}\omega_{u}\left[\mathrm{Tr}\left(\mathbf{F}_{u}^{\rm
H}\mathbf{H}_{u}^{\rm H}\mathbf{G}_{u}\mathbf{W}_{u}\right)\right]$
$\displaystyle-\sum_{u=1}^{U}\omega_{u}\left[\mathrm{Tr}\left(\mathbf{W}_{u}\mathbf{G}_{u}^{\rm
H}\mathbf{H}_{u}\mathbf{F}_{u}\right)\right]$
$\displaystyle+\sum_{b=1}^{B}\sum_{u=1}^{U}\frac{1}{2\rho_{b}}\left\|\mathbf{F}_{b,u}-\mathbf{F}_{{\rm
HB},b,u}\right\|_{F}^{2}$
$\displaystyle+\sum_{b=1}^{B}\lambda_{b}\left[\sum_{u=1}^{U}\mathrm{Tr}\left(\mathbf{F}_{b,u}\mathbf{F}_{,b,u}^{\rm
H}\right)-P_{b,\max}\right],$ (17)

where $\mathbf{A}_{u}=\mathbf{H}_{u}^{\rm
H}\mathbf{G}_{u}\mathbf{W}_{u}\mathbf{G}_{u}^{\rm
H}\mathbf{H}_{u}\in\mathbb{C}^{BN_{t}\times BN_{t}}$. The derivation of the
closed-form $\mathbf{F}_{u}$ and $\left\\{\lambda_{b}\right\\}_{b=1}^{B}$ is
more elaborate in our mmWave cell-free MIMO scenario, since the last two items
of (III-C) are functions of $\mathbf{F}_{b,u}$, which are the submatrices of
$\mathbf{F}_{u}$. These two items are introduced for satisfying the transmit
power constraints of each AP and the constant-modulus constrains of phase
shifters. These two constrains of our mmWave cell-free MIMO DL relying on
hybrid TPC structures differentiate our problem formulation from that of the
existing literatures [51, 52]. For solving this challenging problem, we have
the following proposition.

Proposition 1: We propose to iteratively update
$\left\\{\mathbf{F}_{b,u}|\forall b\in\mathcal{B},\forall
u\in\mathcal{U}\right\\}$ in each iteration of the BCD algorithm. Hence we
recast problem (III-C) first. Specifically, we define
$\mathbf{A}_{u}=\mathbf{A}_{u}^{\frac{1}{2}}\left(\mathbf{A}_{u}^{\frac{1}{2}}\right)^{\rm
H}$ and
$\widetilde{\mathbf{A}}_{b,u}=\widehat{\mathbf{A}}_{b,u}\sum_{i=1,i\neq
b}^{B}\widehat{\mathbf{A}}_{b,i}^{\rm
H}\mathbf{F}_{b,i}\in\mathbb{C}^{N_{t}\times N_{s}}$, where
$\left\\{\widehat{\mathbf{A}}_{b,u}\in\mathbb{C}^{N_{t}\times BN_{t}}|\forall
b\in\mathcal{B}\right\\}$ are a set of variables introduced for constructing
$\mathbf{A}_{u}^{\frac{1}{2}}=\left[\widehat{\mathbf{A}}_{1,u}^{\rm
H},\right.$ $\left.\widehat{\mathbf{A}}_{2,u}^{\rm
H},\cdots,\widehat{\mathbf{A}}_{B,u}^{\rm H}\right]^{\rm H}$. We also
introduce $\left\\{\mathbf{C}_{b,u}\in\mathbb{C}^{N_{t}\times N_{s}}|\forall
b\in\mathcal{B}\right\\}$ for constructing
$\mathbf{C}_{u}=\omega_{u}\mathbf{H}_{u}^{\rm
H}\mathbf{G}_{u}\mathbf{W}_{u}=\left[\mathbf{C}_{1,u}^{\rm
H},\mathbf{C}_{2,u}^{\rm H},\cdots,\mathbf{C}_{B,u}^{\rm H}\right]^{\rm H}$.
Then the problem (III-C) can be characterized as

$\displaystyle\min_{\left\\{\mathbf{F}_{b,u}|\forall b\in\mathcal{B},\forall
u\in\mathcal{U}\right\\}}$
$\displaystyle\sum_{u=1}^{U}\left[\mathrm{Tr}\left(\mathbf{F}_{b,u}^{\rm
H}\left(\mathbf{Q}_{b,u}+\frac{1}{2\rho_{b}}\mathbf{I}_{N_{t}}+\lambda_{b}\mathbf{I}_{N_{t}}\right)\mathbf{F}_{b,u}\right)\right]$
$\displaystyle+\sum_{i=1,i\neq
b}^{B}\sum_{u=1}^{U}\left[\mathrm{Tr}\left(\mathbf{F}_{i,u}^{\rm
H}\left(\frac{1}{2\rho_{b}}\mathbf{I}_{N_{t}}+\lambda_{b}\mathbf{I}_{N_{t}}\right)\mathbf{F}_{i,u}\right)\right]$
$\displaystyle-\sum_{u=1}^{U}\left[\mathrm{Tr}\left(\left(\mathbf{M}_{b,u}+\frac{1}{2\rho_{b}}\mathbf{F}_{{\rm
HB},b,u}\right)^{\rm H}\mathbf{F}_{b,u}\right)\right]$
$\displaystyle-\sum_{i=1,i\neq
b}^{B}\sum_{u=1}^{U}\left[\mathrm{Tr}\left(\left(\mathbf{C}_{i,u}+\frac{1}{2\rho_{b}}\mathbf{F}_{{\rm
HB},i,u}\right)^{\rm H}\mathbf{F}_{i,u}\right)\right]$
$\displaystyle-\sum_{u=1}^{U}\left[\mathrm{Tr}\left(\mathbf{F}_{b,u}^{\rm
H}\left(\mathbf{M}_{b,u}+\frac{1}{2\rho_{b}}\mathbf{F}_{{\rm
HB},b,u}\right)\right)\right]$ $\displaystyle-\sum_{i=1,i\neq
b}^{B}\sum_{u=1}^{U}\left[\mathrm{Tr}\left(\mathbf{F}_{i,u}^{\rm
H}\left(\mathbf{C}_{i,u}+\frac{1}{2\rho_{b}}\mathbf{F}_{{\rm
HB},i,u}\right)\right)\right].$ (18)

where we have
$\mathbf{Q}_{b,u}=\omega_{u}\widehat{\mathbf{A}}_{b,u}\widehat{\mathbf{A}}_{b,u}^{\rm
H}$ and
$\mathbf{M}_{b,u}=\mathbf{C}_{b,u}-\omega_{u}\widetilde{\mathbf{A}}_{b,u}$.

###### Proof:

Please see Appendix A. ∎

Then the optimal $\left\\{\mathbf{F}_{b,u}|\forall b\in\mathcal{B},\forall
u\in\mathcal{U}\right\\}$ can be determined by deriving the partial derivative
of (III-C) with respect to $\mathbf{F}_{b,u}$ and setting it to zero. Since
$\mathbf{F}_{b,u}$ is a function of $\lambda_{b}$, the optimal
$\mathbf{F}_{b,u}\left(\lambda_{b}\right)$ is expressed as

$\displaystyle\mathbf{F}_{b,u}^{\star}\left(\lambda_{b}\right)$
$\displaystyle=\left(\mathbf{Q}_{b,u}+\frac{1}{2\rho_{b}}\mathbf{I}_{N_{t}}+\lambda_{b}\mathbf{I}_{N_{t}}\right)^{-1}\left(\mathbf{M}_{b,u}+\frac{1}{2\rho_{b}}\mathbf{F}_{{\rm
HB},b,u}\right).$ (19)

Let us now discuss the derivation of the Lagrangian multipliers
$\left\\{\lambda_{b}|\forall b\in\mathcal{B}\right\\}$. By substituting
(III-C) into the power constraints of (15b), we arrive at the following
transformation

$\displaystyle\mathrm{Tr}\left(\left(\mathbf{F}_{b,u}^{\star}\left(\lambda_{b}\right)\right)^{\rm
H}\mathbf{F}_{b,u}^{\star}\left(\lambda_{b}\right)\right)$ $\displaystyle=$
$\displaystyle\mathrm{Tr}\left(\left(\mathbf{M}_{b,u}+\frac{1}{2\rho_{b}}\mathbf{F}_{{\rm
HB},b,u}\right)^{\rm H}\right.$
$\displaystyle\left.\left(\left(\mathbf{Q}_{b,u}+\frac{1}{2\rho_{b}}\mathbf{I}_{N_{t}}+\lambda_{b}\mathbf{I}_{N_{t}}\right)^{\rm
H}\right)^{-1}\right.$
$\displaystyle\left.\cdot\left(\mathbf{Q}_{b,u}+\frac{1}{2\rho_{b}}\mathbf{I}_{N_{t}}+\lambda_{b}\mathbf{I}_{N_{t}}\right)^{-1}\left(\mathbf{M}_{b,u}+\frac{1}{2\rho_{b}}\mathbf{F}_{{\rm
HB},b,u}\right)\right)$ $\displaystyle\overset{\left(a\right)}{=}$
$\displaystyle\mathrm{Tr}\left(\left(\mathbf{M}_{b,u}+\frac{1}{2\rho_{b}}\mathbf{F}_{{\rm
HB},b,u}\right)^{\rm H}\right.$
$\displaystyle\left.\left(\left(\mathbf{U}_{b,u}\mathbf{\Sigma}_{b,u}\mathbf{U}_{b,u}^{\rm
H}+\frac{1}{2\rho_{b}}\mathbf{I}_{N_{t}}+\lambda_{b}\mathbf{I}_{N_{t}}\right)^{\rm
H}\right)^{-1}\right.$
$\displaystyle\left.\cdot\left(\mathbf{U}_{b,u}\mathbf{\Sigma}_{b,u}\mathbf{U}_{b,u}^{\rm
H}+\frac{1}{2\rho_{b}}\mathbf{I}_{N_{t}}+\lambda_{b}\mathbf{I}_{N_{t}}\right)^{-1}\right.$
$\displaystyle\left.\left(\mathbf{M}_{b,u}+\frac{1}{2\rho_{b}}\mathbf{F}_{{\rm
HB},b,u}\right)\right)$ $\displaystyle\overset{\left(b\right)}{=}$
$\displaystyle\mathrm{Tr}\left(\left(\mathbf{\Sigma}_{b,u}+\frac{1}{2\rho_{b}}\mathbf{I}_{N_{t}}+\lambda_{b}\mathbf{I}_{N_{t}}\right)^{-2}\mathbf{U}_{b,u}^{\rm
H}\right.$
$\displaystyle\left.\left(\mathbf{M}_{b,u}+\frac{1}{2\rho_{b}}\mathbf{F}_{{\rm
HB},b,u}\right)\left(\mathbf{M}_{b,u}+\frac{1}{2\rho_{b}}\mathbf{F}_{{\rm
HB},b,u}\right)^{\rm H}\mathbf{U}_{b,u}\right)$
$\displaystyle\overset{\left(c\right)}{=}$
$\displaystyle\mathrm{Tr}\left(\left(\mathbf{\Sigma}_{b,u}+\frac{1}{2\rho_{b}}\mathbf{I}_{N_{t}}+\lambda_{b}\mathbf{I}_{N_{t}}\right)^{-2}\mathbf{P}_{b,u}\right),$
(20)

where $\left(a\right)$ holds by calculating the singular value decomposition
(SVD) as
$\mathbf{Q}_{b,u}=\mathbf{U}_{b,u}\mathbf{\Sigma}_{b,u}\mathbf{U}_{b,u}^{\rm
H}$, $\left(b\right)$ is due to the fact that $\mathbf{U}_{b,u}$ is a unitary
matrix, which has the property of $\mathbf{U}_{b,u}^{-1}=\mathbf{U}_{b,u}^{\rm
H}$, and $\left(c\right)$ holds by defining
$\mathbf{P}_{b,u}=\mathbf{U}_{b,u}^{\rm
H}\left(\mathbf{M}_{b,u}+\frac{1}{2\rho_{b}}\mathbf{F}_{{\rm
HB},b,u}\right)\left(\mathbf{M}_{b,u}+\frac{1}{2\rho_{b}}\mathbf{F}_{{\rm
HB},b,u}\right)^{\rm H}\mathbf{U}_{b,u}$. For the $b$-th AP, the transmit
power is expressed as

$\displaystyle\sum_{u=1}^{U}\left(\mathrm{Tr}\left(\left(\mathbf{F}_{b,u}^{\star}\left(\lambda_{b}\right)\right)^{\rm
H}\mathbf{F}_{b,u}^{\star}\left(\lambda_{b}\right)\right)\right)$
$\displaystyle=$
$\displaystyle\sum_{u=1}^{U}\left(\mathrm{Tr}\left(\left(\mathbf{\Sigma}_{b,u}+\frac{1}{2\rho_{b}}\mathbf{I}_{N_{t}}+\lambda_{b}\mathbf{I}_{N_{t}}\right)^{-2}\mathbf{P}_{b,u}\right)\right)$
$\displaystyle=$
$\displaystyle\sum_{u=1}^{U}\sum_{n=1}^{N_{t}}\frac{\mathbf{P}_{b,u}\left(n,n\right)}{\left(\mathbf{\Sigma}_{b,u}\left(n,n\right)+\frac{1}{2\rho_{b}}+\lambda_{b}\right)^{2}}.$
(21)

We observe that the transmit power is a monotonically decreasing function with
respect to the Lagrangian multiplier $\lambda_{b}$. The upper bound on the
transmit power of the $b$-th AP can be derived by setting $\lambda_{b}=0$,
which can be expressed as

$\displaystyle\sum_{u=1}^{U}\left(\mathrm{Tr}\left(\left(\mathbf{F}_{b,u}^{\star}\left(\lambda_{b}\right)\right)^{\rm
H}\mathbf{F}_{b,u}^{\star}\left(\lambda_{b}\right)\right)\right)$
$\displaystyle\leq\sum_{u=1}^{U}\sum_{n=1}^{N_{t}}\frac{\mathbf{P}_{b,u}\left(n,n\right)}{\left(\mathbf{\Sigma}_{b,u}\left(n,n\right)+\frac{1}{2\rho_{b}}\right)^{2}}.$
(22)

For deriving the optimal $\lambda_{b}$, we discuss the following two cases:

  1. Case 1:

If the upper bound
$\sum_{u=1}^{U}\sum_{n=1}^{N_{t}}\frac{\mathbf{P}_{b,u}\left(n,n\right)}{\left(\mathbf{\Sigma}_{b,u}\left(n,n\right)+\frac{1}{2\rho_{b}}\right)^{2}}\leq
P_{b,\max}$, the power constraint (15b) for the $b$-th AP is naturally
satisfied. Therefore, we set the optimal $\lambda_{b}^{\star}$ to $0$.

  2. Case 2:

If the upper bound
$\sum_{u=1}^{U}\sum_{n=1}^{N_{t}}\frac{\mathbf{P}_{b,u}\left(n,n\right)}{\left(\mathbf{\Sigma}_{b,u}\left(n,n\right)+\frac{1}{2\rho_{b}}\right)^{2}}>P_{b,\max}$,
the bisection search method is executed for determining the optimal
$\lambda_{b}$. Specifically, $\lambda_{b}^{\star}$ is searched within the
range $\lambda_{b}^{\star}\in\left(\lambda_{\rm lb},\lambda_{\rm ub}\right)$,
where $\lambda_{\rm lb}$ is initialized as $0$ and $\lambda_{\rm ub}$ is
initialized as
$\sqrt{\frac{\sum_{u=1}^{U}\sum_{n=1}^{N_{t}}\mathbf{P}_{b,u}\left(n,n\right)}{P_{b,\max}}}$,
which is due to
$\sum_{u=1}^{U}\sum_{n=1}^{N_{t}}\frac{\mathbf{P}_{b,u}\left(n,n\right)}{\left(\mathbf{\Sigma}_{b,u}\left(n,n\right)+\frac{1}{2\rho_{b}}+\lambda_{\rm
ub}\right)^{2}}<\sum_{u=1}^{U}\sum_{n=1}^{N_{t}}\frac{\mathbf{P}_{b,u}\left(n,n\right)}{\lambda_{\rm
ub}^{2}}\overset{\Delta}{=}P_{b,\max}$.

Once $\lambda_{b}^{\star}$ is obtained, the optimal
$\mathbf{F}_{b,u}^{\star}\left(\lambda_{b}^{\star}\right)$ is expressed as

$\displaystyle\mathbf{F}_{b,u}^{\star}\left(\lambda_{b}^{\star}\right)$
$\displaystyle=\left(\mathbf{Q}_{b,u}+\frac{1}{2\rho_{b}}\mathbf{I}_{N_{t}}+\lambda_{b}^{\star}\mathbf{I}_{N_{t}}\right)^{-1}\left(\mathbf{M}_{b,u}+\frac{1}{2\rho_{b}}\mathbf{F}_{{\rm
HB},b,u}\right).$ (23)

The procedure for deriving the optimal $\mathcal{F}$ is summarized in
Algorithm 1.

Algorithm 1 Lagrangian multiplier method and PDD method for solving the
problem (15)

0: $\mathcal{W}$, $\mathcal{G}$, $\mathcal{F}_{\rm RF}$, $\mathcal{F}_{\rm
BB}$, the penalty parameters $\left\\{\rho_{b}=\frac{N_{t}}{100}|\forall
b\in\mathcal{B}\right\\}$, initial $\mathcal{F}$, Bisection search accuracy
$\epsilon$

0:

Calculate $\mathbf{Q}_{b,u}$ and $\mathbf{M}_{b,u}$, $\forall
b\in\mathcal{B},u\in\mathcal{U}$ according to Proposition 1.

2: for $b=1:B$ do

Calculate the upper bound on transmit power for the $b$-th AP according to
(III-C);

4: if Case 1 is satisfied then

Set $\lambda_{b}^{\star}=0$;

6: else

Initialize $\lambda_{\rm ub}$ and $\lambda_{\rm lb}$;

8: Calculate $\lambda_{b}=\frac{\lambda_{\rm ub}+\lambda_{\rm lb}}{2}$;

If
$\sum_{u=1}^{U}\left(\mathrm{Tr}\left(\left(\mathbf{F}_{b,u}^{\star}\left(\lambda_{b}\right)\right)^{\rm
H}\mathbf{F}_{b,u}^{\star}\left(\lambda_{b}\right)\right)\right)\geq
P_{b,\max}$, set $\lambda_{\rm lb}=\lambda_{b}$. Otherwise, set $\lambda_{\rm
ub}=\lambda_{b}$;

10: If $\left|\lambda_{\rm ub}-\lambda_{\rm lb}\right|<\epsilon$, set
$\lambda_{b}^{\star}=\lambda_{b}$ and go to step 12. Otherwise, go back to
step 8;

end if

12: Calculate $\mathbf{F}_{b,u}^{\star}\left(\lambda_{b}^{\star}\right)$
according to (III-C) $\forall u\in\mathcal{U}$.

end for

14: Calculate $\mathbf{F}_{u}=\left[\mathbf{F}_{1,u}^{\rm
H},\mathbf{F}_{2,u}^{\rm H},\cdots,\mathbf{F}_{B,u}^{\rm H}\right]^{\rm H}$.

###  III-D Designing the Analog and Digital Precoding Matrices
$\mathcal{F}_{\rm RF}$ and $\mathcal{F}_{\rm BB}$

We observe that only the penalty terms
$\sum_{b=1}^{B}\sum_{u=1}^{U}\frac{1}{2\rho_{b}}\left\|\mathbf{F}_{b,u}-\mathbf{F}_{{\rm
HB},b,u}\right\|_{F}^{2}$ of the OF (11a) contain the analog TPC matrices
$\mathcal{F}_{\rm RF}$ and the digital TPC matrices $\mathcal{F}_{\rm BB}$.
Moreover, we have obtained the solution of the auxiliary TPC matrices
$\mathcal{F}$. Therefore, given that the other variables are fixed, we can
derive the closed-form solution of $\mathcal{F}_{\rm RF}$ and
$\mathcal{F}_{\rm BB}$.

For deriving $\mathcal{F}_{{\rm BB}}$, we divide the problem (11) into $BU$
subproblems. For the digital TPC matrix of the $u$-th user at the $b$-th AP,
the unconstrained subproblem is formulated by

$\displaystyle\min_{\mathbf{F}_{{\rm BB},b,u}}\
\left\|\mathbf{F}_{b,u}-\mathbf{F}_{{\rm RF},b}\mathbf{F}_{{\rm
BB},b,u}\right\|_{F}^{2}.$ (24)

The optimal closed-form solution can be obtained by adopting the well-known
least square (LS) method. Hence we have

$\displaystyle\mathbf{F}_{{\rm BB},b,u}^{\star}=\mathbf{F}_{{\rm
RF},b}^{\dagger}\mathbf{F}_{b,u}.$ (25)

For deriving $\mathcal{F}_{{\rm RF}}$, we formulate the $B$ subproblems, each
of which is expressed as

$\displaystyle\min_{\mathbf{F}_{{\rm RF},b}}\
\left\|\overline{\mathbf{F}}_{b}-\mathbf{F}_{{\rm
RF},b}\overline{\mathbf{F}}_{{\rm BB},b}\right\|_{F}^{2}$ (26a)
$\displaystyle\ s.t.\quad\left|\mathbf{F}_{{\rm
RF},b}\left(i,j\right)\right|=1,\forall i\in\mathcal{N}_{t},\forall
j\in\mathcal{N}_{\rm RF},$ (26b)

where
$\overline{\mathbf{F}}_{b}=\left[\mathbf{F}_{b,1},\mathbf{F}_{b,2},\cdots,\mathbf{F}_{b,U}\right]$
and $\overline{\mathbf{F}}_{{\rm BB},b}=\left[\mathbf{F}_{{\rm
BB},b,1},\mathbf{F}_{{\rm BB},b,2},\cdots,\mathbf{F}_{{\rm BB},b,U}\right]$.
We derive the optimal closed-form update for each subproblem and we update
each element of $\mathbf{F}_{{\rm RF},b}$ relying on the classical LS method
as [53]

$\displaystyle\mathbf{F}_{{\rm
RF},b}^{\star}\left(i,j\right)=e^{j\xi_{i,j}}\mathbf{F}_{\mathrm{RF},b}\left(i,j\right),$
(27)

where
$\xi_{i,j}=\angle\left\\{\left(\mathbf{F}_{\mathrm{RF},b}\left(i,j\right)\overline{\mathbf{F}}_{\mathrm{BB},b}\left(j,:\right)\right)^{\ast}\left(\widetilde{\mathbf{F}}_{b}\left(i,:\right)\right)^{\text{T}}\right\\}$
with $\angle\left\\{\cdot\right\\}$ denotes the function which extracts the
phase, $\widetilde{\mathbf{F}}_{b}=\overline{\mathbf{F}}_{b}-\mathbf{F}_{{\rm
RF},b}^{-j,\mathrm{col}}\overline{\mathbf{F}}_{{\rm BB},b}^{-j,\mathrm{row}}$
with $\mathbf{F}_{{\rm RF},b}^{-j,\mathrm{col}}$ denoting the submatrix of
$\mathbf{F}_{\mathrm{RF},b}$ with the $j$-th column removed and
$\overline{\mathbf{F}}_{{\rm BB},b}^{-j,\mathrm{row}}$ denoting the submatrix
of $\mathbf{F}_{\mathrm{BB},b}$ with the $j$-th row removed.

Typically, infinite-resolution phase shifters with fully-connected structure
are considered in this paper. Note that the LS method can be readily extended
to other sub-connected structures [50]. For instance, the LS method of
designing analog TPCs can be easily extended to structures with other type of
connections by updating $\mathbf{F}_{{\rm RF},b}^{\star}\left(i,j\right)$ for
$\left\\{\left(i,j\right)|\forall\left(i,j\right)\in\mathcal{D}\right\\}$,
where $\mathcal{D}$ denotes the specific connection set of phase shifters,
namely those that are connected to RF chains and antennas. For finite-
resolution phase shifters, we can extend the LS method by quantizing
$\mathbf{F}_{{\rm
RF},b}^{\star}\left(i,j\right)=\mathcal{Q}\left[e^{j\xi_{i,j}}\mathbf{F}_{\mathrm{RF},b}\left(i,j\right)\right]$
during the update of analog TPCs, where $\mathcal{Q}\left(\cdot\right)$ stands
for the quantization function.

###  III-E Overall Algorithm and Convergence Analysis

In this subsection, we will describe the overall algorithm. Since our proposed
method relies on the BCD algorithm, an iterative update procedure is utilized.
Specifically, we iteratively calculate $\mathcal{G}$, $\mathcal{W}$,
$\mathcal{F}$, $\mathcal{F}_{\rm BB}$ and $\mathcal{F}_{\rm RF}$ until
convergence is realized on basis of the aforementioned derivation. Now we
present the initialization of the algorithm. The set $\mathcal{S}$ derived in
the $n$-th iteration is denoted as $\mathcal{S}^{\left(n\right)}$. Then the
initialization of $\mathcal{F}^{\left(0\right)}$, $\mathcal{F}_{\rm
BB}^{\left(0\right)}$ and $\mathcal{F}_{\rm RF}^{\left(0\right)}$ is discussed
as follows.

$\mathcal{F}^{\left(0\right)}$: We first calculate the SVD of all channels
between the $b$-th AP and the $u$-th user as
$\mathbf{H}_{b,u}=\widetilde{\mathbf{U}}_{b,u}\widetilde{\mathbf{\Sigma}}_{b,u}\widetilde{\mathbf{V}}_{b,u}^{\rm
H}$. Then $\mathbf{F}_{u}^{\left(0\right)}$ of the $u$-th user in
$\mathcal{F}^{\left(0\right)}$ is set to
$\mathbf{F}_{u}^{\left(0\right)}=\left[\left(\mathbf{F}_{1,u}^{\left(0\right)}\right)^{\rm
H},\left(\mathbf{F}_{2,u}^{\left(0\right)}\right)^{\rm H},\cdots,\right.$
$\left.\left(\mathbf{F}_{B,u}^{\left(0\right)}\right)^{\rm H}\right]^{\rm H}$,
where
$\mathbf{F}_{b,u}^{\left(0\right)}=\frac{\sqrt{P_{b,\max}}\widetilde{\mathbf{V}}_{b,u}\left(:,1:N_{s}\right)}{\sqrt{\sum_{u=1}^{U}\left\|\widetilde{\mathbf{V}}_{b,u}\left(:,1:N_{s}\right)\right\|_{F}^{2}}}$
satisfies the power constraint (15b).

$\mathcal{F}_{\rm RF}^{\left(0\right)}$: each element in $\mathbf{F}_{{\rm
RF},b}^{\left(0\right)}$ is set to $e^{j\phi_{b,i,j}}$ with $\phi_{b,i,j}$
denoting random phases chosen from $\left[-\pi,\pi\right]$, $\forall
i\in\mathcal{N}_{t},\forall j\in\mathcal{N}_{\rm RF},\forall b\in\mathcal{B}$.

$\mathcal{F}_{\rm BB}^{\left(0\right)}$: recalling (25), the initial
$\mathcal{F}_{{\rm BB},b,u}^{\left(0\right)}$ is set to $\mathcal{F}_{{\rm
BB},b,u}^{\left(0\right)}=\frac{\sqrt{P_{b,\max}}\left(\mathbf{F}_{{\rm
RF},b}^{\left(0\right)}\right)^{\dagger}\mathbf{F}_{b,u}^{\left(0\right)}}{\sqrt{\sum_{u=1}^{U}\left\|\mathbf{F}_{{\rm
RF},b}^{\left(0\right)}\left(\mathbf{F}_{{\rm
RF},b}^{\left(0\right)}\right)^{\dagger}\mathbf{F}_{b,u}^{\left(0\right)}\right\|_{F}^{2}}}$,
which satisfies the power constraint for each AP.

Note that when the convergence condition is met, $\mathcal{F}_{\rm BB}$ may
not satisfy the power constraint for each AP. Therefore, we derive the final
$\mathbf{F}_{{\rm BB},b,u}^{\star}$, $\forall
b\in\mathcal{B},u\in\mathcal{U}$, as $\mathbf{F}_{{\rm
BB},b,u}^{\star}=\frac{\sqrt{P_{b,\max}}\mathbf{F}_{{\rm
BB},b,u}^{\star}}{\sqrt{\sum_{u=1}^{U}\left\|\mathbf{F}_{{\rm
RF},b}^{\star}\mathbf{F}_{{\rm BB},b,u}^{\star}\right\|_{F}^{2}}}$ [33, 43,
53]. The overall procedure is summarized in Algorithm 2, where $\tilde{R}$
denotes the OF (11a).

Algorithm 2 Overall algorithm for hybrid precoding design in cell-free MIMO

0: Initial $\mathcal{F}_{\rm RF}^{\left(0\right)}$, $\mathcal{F}_{\rm
BB}^{\left(0\right)}$, $\mathcal{F}^{\left(0\right)}$, iteration index $n=0$,
convergency accuracy $\xi$;

0:

while
$\left|\tilde{R}^{\left(n\right)}-\tilde{R}^{\left(n-1\right)}\right|\geq\xi$
do

2: $n=n+1$;

Calculate $\mathcal{G}^{\left(n\right)}$ according to (13) with other
variables fixed;

4: Calculate $\mathcal{W}^{\left(n\right)}$ according to (14) with other
variables fixed;

Calculate $\mathcal{F}^{\left(n\right)}$ according to Algorithm 1 with other
variables fixed;

6: Calculate $\mathcal{F}_{\rm BB}^{\left(n\right)}$ according to (25) with
other variables fixed;

Calculate $\mathcal{F}_{\rm RF}^{\left(n\right)}$ according to (27) with other
variables fixed;

8: end while

If $\sum_{u=1}^{U}\left\|\mathbf{F}_{{\rm RF},b}^{\star}\mathbf{F}_{{\rm
BB},b,u}^{\star}\right\|_{F}^{2}>P_{b,\max}$, calculate $\mathbf{F}_{{\rm
BB},b,u}^{\star}$ as $\mathbf{F}_{{\rm
BB},b,u}^{\star}=\frac{\sqrt{P_{b,\max}}\mathbf{F}_{{\rm
BB},b,u}^{\star}}{\sqrt{\sum_{u=1}^{U}\left\|\mathbf{F}_{{\rm
RF},b}^{\star}\mathbf{F}_{{\rm BB},b,u}^{\star}\right\|_{F}^{2}}}$, $\forall
u\in\mathcal{U}$, $\forall b\in\mathcal{B}$.

Next, we will demonstrate the convergence of our proposed algorithm. According
to the characteristics of the BCD algorithm, when we derive one set of
variables with the others fixed, the OF (11a) increases or at least maintains
its value. Hence, the OF (11a) is a non-decreasing function in our design.

In order to demonstrate the convergence of the proposed method, we still have
to prove that the OF (11a) is upper-bounded. We consider an ideal
unconstrained cell-free MIMO scenario, where there is no IUI. The TPC matrices
of this scenario can be directly obtained by the singular value decomposition
(SVD) for each user at each AP. Therefore, (11a) can indeed achieve
convergence. The proposed algorithm converges to at least locally optimal
solutions of the transformed problem (11).

###  III-F Complexity Analysis

During the design of $\mathcal{F}$, the dominant complexity contribution is
that of calculating the inverse matrix
$\left(\mathbf{Q}_{b,u}+\frac{1}{2\rho_{b}}\mathbf{I}_{N_{t}}+\lambda_{b}^{\star}\mathbf{I}_{N_{t}}\right)^{-1}$,
$\forall b\in\mathcal{B},\forall u\in\mathcal{U}$, which has the complexity
order of $\mathcal{O}\left(N_{t}^{3}\right)$. The bisection search of
$\lambda_{b}^{\star}$ for the $b$-th AP has to repeat the calculation
$T_{b}=\log_{2}\left(\frac{\lambda_{\rm ub}-\lambda_{\rm
lb}}{\epsilon}\right)$ times. Therefore, the complexity order of optimizing
$\mathcal{F}$ is $\mathcal{O}\left(BUTN_{t}^{3}\right)$, where
$T=\sum_{b=1}^{B}T_{b}$. For designing $\mathcal{F}_{\rm RF}$ and
$\mathcal{F}_{\rm BB}$, the complexity order is
$\mathcal{O}\left[B\left(N_{t}N_{\rm RF}UN_{s}\right.\right.$
$\left.\left.+N_{\rm RF}^{3}\right)\right]$. If the number of iterations
required for convergence is $N_{\max}$, the total complexity order is given by
$\mathcal{O}\left[N_{\max}\left(BUTN_{t}^{3}+B\left(N_{t}N_{\rm
RF}UN_{s}+N_{\rm RF}^{3}\right)\right)\right]$.

Note that our proposed algorithm can be modified for the cell-free MIMO DL,
where fully digital TPC structures are adopted. In the next section, we will
design the fully digital TPC matrices by modifying our proposed algorithm.

##  IV Modified Algorithm for Fully Digital Precoding in cell-free MIMO

In this section, we consider a scenario where each AP in the cell-free MIMO DL
is equipped with fully digital TPC. The fully digital TPC matrices are
designed by modifying the algorithm proposed in Section III.

###  IV-A Problem Formulation

The set of fully digital TPC matrices is defined by $\mathcal{F}_{\rm
FD}=\left\\{\mathbf{F}_{{\rm FD},u}|\forall u\in\mathcal{U}\right\\}$ and
$\mathbf{F}_{{\rm FD},u}=\left[\mathbf{F}_{{\rm FD},1,u}^{\rm
H},\mathbf{F}_{{\rm FD},2,u}^{\rm H},\cdots,\mathbf{F}_{{\rm FD},B,u}^{\rm
H}\right]^{\rm H}$. The WSR maximization problem is formulated by

$\displaystyle\max_{\mathcal{F}_{\rm FD}}\quad$ $\displaystyle R_{\rm FD}$
(28a) $\displaystyle s.t.\quad\ $
$\displaystyle\sum_{u=1}^{U}\mathrm{Tr}\left(\mathbf{F}_{{\rm
FD},u}\mathbf{F}_{{\rm FD},u}^{\rm H}\right)\leq P_{b,\max},\forall
b\in\mathcal{B},$ (28b)

where $R_{\rm
FD}=\sum_{u=1}^{U}\omega_{u}\log\left|\mathbf{I}_{N_{r}}+\mathbf{J}^{-1}_{{\rm
FD},u}\mathbf{H}_{u}\mathbf{F}_{{\rm FD},u}\mathbf{F}_{{\rm
FD},u}^{\mathrm{H}}\mathbf{H}_{u}^{\mathrm{H}}\right|$ and $\mathbf{J}_{{\rm
FD},u}=\sum_{j=1,j\neq u}^{U}\mathbf{H}_{u}\mathbf{F}_{{\rm
FD},j}\mathbf{F}_{{\rm FD},j}^{\rm H}$ $\mathbf{H}_{u}^{\rm
H}+\sigma^{2}\mathbf{I}_{N_{r}}$.

###  IV-B Modified Precoding Algorithm

Similar to the problem transformation from Section III-A to Section III-C, we
can obtain the unconstrained problem by exploiting the equivalent WMMSE
transformation and BCD algorithm for the fully digital scenario. Specifically,
we iteratively update the introduced linear combiners
$\left\\{\mathbf{G}_{u}|\forall u\in\mathcal{U}\right\\}$ in $\mathcal{G}$,
the introduced weighting matrices $\left\\{\mathbf{W}_{u}|\forall
u\in\mathcal{U}\right\\}$ in $\mathcal{W}$ and the fully digital TPC matrices
$\left\\{\mathbf{F}_{{\rm FD},u}|\forall u\in\mathcal{U}\right\\}$ in
$\mathcal{F}_{\rm FD}$. During the update, $\mathbf{G}_{u}$ and
$\mathbf{W}_{u}$ can be calculated by exploiting the partial derivative based
method, which is similar to (13) and (14) in the hybrid TPC design. Then we
propose to design fully digital TPC matrices by exploiting the Lagrangian
multiplier method. The transformed unconstrained problem is expressed by

$\displaystyle\min_{\left\\{\mu_{b}|\forall
b\in\mathcal{B}\right\\},\mathcal{F}_{\rm FD}}\ $
$\displaystyle\sum_{u=1}^{U}\omega_{u}\left[\mathrm{Tr}\left(\mathbf{F}_{{\rm
FD},u}^{\rm H}\mathbf{A}_{u}\mathbf{F}_{{\rm FD},u}\right)\right]$
$\displaystyle-\sum_{u=1}^{U}\omega_{u}\left[\mathrm{Tr}\left(\mathbf{F}_{{\rm
FD},u}^{\rm H}\mathbf{H}_{u}^{\rm
H}\mathbf{G}_{u}\mathbf{W}_{u}\right)\right]$
$\displaystyle-\sum_{u=1}^{U}\omega_{u}\left[\mathrm{Tr}\left(\mathbf{W}_{u}\mathbf{G}_{u}^{\rm
H}\mathbf{H}_{u}\mathbf{F}_{{\rm FD},u}\right)\right]$
$\displaystyle+\sum_{b=1}^{B}\mu_{b}\left[\sum_{u=1}^{U}\mathrm{Tr}\left(\mathbf{F}_{{\rm
FD},b,u}\mathbf{F}_{{\rm FD},b,u}^{\rm H}\right)-P_{b,\max}\right],$ (29)

where $\left\\{\mu_{b}\geq 0|\forall b\in\mathcal{B}\right\\}$ is the set of
Lagrangian multipliers introduced. Since no inter-user interference exists in
any of the UC cluster, our problem formulation is different from that of the
existing literature [45, 46, 47]. Similar to Proposition 1, stated for
deriving the optimal $\mathcal{F}_{\rm FD}$, we have the following Proposition
2.

Proposition 2: We propose to iteratively update
$\left\\{\mathbf{F}_{\mathrm{FD},b,u}|\forall b\in\mathcal{B},\forall
u\in\mathcal{U}\right\\}$ in each iteration of the BCD algorithm. The problem
(IV-B) can be rewritten as

$\displaystyle\min_{\left\\{\mathbf{F}_{\mathrm{FD},b,u}|\forall
b\in\mathcal{B},\forall u\in\mathcal{U}\right\\}}$
$\displaystyle\sum_{u=1}^{U}\left[\mathrm{Tr}\left(\mathbf{F}_{{\rm
FD},b,u}^{\rm
H}\left(\mathbf{Q}_{b,u}+\mu_{b}\mathbf{I}_{N_{t}}\right)\mathbf{F}_{{\rm
FD}b,u}\right)\right]$ $\displaystyle+\sum_{i=1,i\neq
b}^{B}\sum_{u=1}^{U}\left[\mathrm{Tr}\left(\mu_{b}\mathbf{F}_{{\rm
FD},i,u}^{\rm H}\mathbf{F}_{{\rm FD}i,u}\right)\right]$
$\displaystyle-\sum_{u=1}^{U}\left[\mathrm{Tr}\left(\mathbf{M}_{b,u}^{\rm
H}\mathbf{F}_{b,u}\right)\right]-\sum_{u=1}^{U}\left[\mathrm{Tr}\left(\mathbf{F}_{b,u}^{\rm
H}\mathbf{M}_{b,u}\right)\right]$ $\displaystyle-\sum_{i=1,i\neq
b}^{B}\sum_{u=1}^{U}\left[\mathrm{Tr}\left(\mathbf{C}_{i,u}^{\rm
H}\mathbf{F}_{i,u}\right)\right]-\sum_{i=1,i\neq
b}^{B}\sum_{u=1}^{U}\left[\mathrm{Tr}\left(\mathbf{F}_{i,u}^{\rm
H}\mathbf{C}_{i,u}\right)\right].$ (30)

###### Proof:

The proof is similar to that in Appendix A, except that the penalty terms are
eliminated. ∎

Based on Proposition 2, the optimal closed-form
$\mathbf{F}_{\mathrm{FD},b,u}\left(\mu_{b}\right)$ can be formulated as

$\displaystyle\mathbf{F}_{{\rm
FD},b,u}^{\star}\left(\mu_{b}\right)=\left(\mathbf{Q}_{b,u}+\mu_{b}\mathbf{I}_{N_{t}}\right)^{\dagger}\mathbf{M}_{b,u}.$
(31)

The determination of the optimal Lagrangian multipliers
$\left\\{\mu_{b}|\forall b\in\mathcal{B}\right\\}$ is different from that of
the hybrid TPC scenario due to the fact that $\mathbf{Q}_{b,u}$ may be a low-
rank matrix. For obtaining the optimal Lagrangian multipliers, we have the
following Lemma concerning the rank of $\mathbf{Q}_{b,u}$.

Lemma 1: $\text{rank}\left(\mathbf{Q}_{b,u}\right)=N_{s}$.

###### Proof:

Please see Appendix B. ∎

For $N_{s}=N_{t}$, the matrix $\mathbf{Q}_{b,u}$ is of full rank, thus the
derivation for $\left\\{\mu_{b}|\forall b\in\mathcal{B}\right\\}$ is the same
as that of the hybrid TPC design. Considering $N_{s}<N_{t}$, on the basis of
Lemma 1, the transmit power is modified according to

$\displaystyle\sum_{u=1}^{U}\left(\mathrm{Tr}\left(\left(\mathbf{F}_{{\rm
FD},b,u}^{\star}\left(\mu_{b}\right)\right)^{\rm H}\mathbf{F}_{{\rm
FD},b,u}^{\star}\left(\mu_{b}\right)\right)\right)$
$\displaystyle=\left\\{\begin{aligned}
&\sum_{u=1}^{U}\sum_{n=1}^{N_{s}}\frac{\mathbf{P}_{{\rm
FD},b,u}\left(n,n\right)}{\left(\mathbf{\Sigma}_{b,u}\left(n,n\right)+\mu_{b}\right)^{2}}\\\
&+\sum_{u=1}^{U}\sum_{n=N_{s}+1}^{N_{t}}\frac{\mathbf{P}_{{\rm
FD},b,u}\left(n,n\right)}{\mu_{b}^{2}},\quad\quad\quad\quad\text{for
}\mu_{b}\neq 0,\\\ &\sum_{u=1}^{U}\sum_{n=1}^{N_{s}}\frac{\mathbf{P}_{{\rm
FD},b,u}\left(n,n\right)}{\left(\mathbf{\Sigma}_{b,u}\left(n,n\right)\right)^{2}},\
\ \quad\quad\quad\quad\quad\quad\text{for }\mu_{b}=0.\end{aligned}\right.$
(32)

In contrast to the hybrid TPC algorithm, the transmit power is not upper
bounded by $\sum_{u=1}^{U}$ $\left(\mathrm{Tr}\left(\left(\mathbf{F}_{{\rm
FD},b,u}^{\star}\left(0\right)\right)^{\rm H}\mathbf{F}_{{\rm
FD},b,u}^{\star}\left(0\right)\right)\right)$. Therefore, we adopt the
bisection search for deriving the optimal Lagrangian multipliers
$\left\\{\mu_{b}^{\star}|\forall b\in\mathcal{B}\right\\}$. According to
(IV-B), since the transmit power $\sum_{u=1}^{U}$
$\left(\mathrm{Tr}\left(\left(\mathbf{F}_{{\rm
FD},b,u}^{\star}\left(\mu_{b}\right)\right)^{\rm H}\mathbf{F}_{{\rm
FD},b,u}^{\star}\left(\mu_{b}\right)\right)\right)$ is a monotonically
decreasing function for $\mu_{b}>0$, the optimal Lagrangian multiplier
$\mu_{b}^{\star}$ is sought within the range $\mu_{b}^{\star}\in\left(\mu_{\rm
lb},\mu_{\rm ub}\right)$ through the bisection search. During the search,
$\mu_{\rm lb}$ is initialized to a small positive value and $\mu_{\rm ub}$ is
initialized as $\sqrt{\frac{\sum_{u=1}^{U}\sum_{n=1}^{N_{t}}\mathbf{P}_{{\rm
FD},b,u}\left(n,n\right)}{P_{b,\max}}}$, which is due to
$\sum_{u=1}^{U}\sum_{n=1}^{N_{t}}\frac{\mathbf{P}_{{\rm
FD},b,u}\left(n,n\right)}{\left(\mathbf{\Sigma}_{b,u}\left(n,n\right)+\mu_{\rm
ub}\right)^{2}}<\sum_{u=1}^{U}\sum_{n=1}^{N_{t}}\frac{\mathbf{P}_{{\rm
FD},b,u}\left(n,n\right)}{\mu_{\rm ub}^{2}}\overset{\Delta}{=}P_{b,\max}$.
Then the optimal fully digital TPC matrix for the $u$-th user at the $b$-th AP
$\mathbf{F}_{b,u}^{\star}\left(\mu_{b}^{\star}\right)$ is expressed as

$\displaystyle\mathbf{F}_{{\rm
FD},b,u}^{\star}\left(\mu_{b}^{\star}\right)=\left(\mathbf{Q}_{b,u}+\mu_{b}^{\star}\mathbf{I}_{N_{t}}\right)^{\dagger}\mathbf{M}_{b,u}.$
(33)

According to Section III-E, the convergence of the fully digital TPC design
can also be proved. In the next section, we will demonstrate the superiority
of our proposed algorithms through simulation results.

##  V Numerical Results

In our simulations, unless otherwise stated, there are $B=2$ APs and $U=4$
users in the cell-free MIMO DL considered. Each AP has $N_{t}=8\times 8$ TAs
and each user has $N_{r}=2\times 2$ RAs. The number of RF chains at each AP is
set to $N_{\rm RF}=8$. The number of data streams for each user is set to
$N_{s}=2$. The weighting coefficients for the users are set to
$\omega_{1}=\cdots=\omega_{U}=1$. The maximum transmit power for the APs is
$P_{1,\max}=\cdots=P_{B,\max}=P_{\max}=30$dBm. The penalty parameters
introduced are set to $\rho_{1}=\cdots=\rho_{B}=\frac{100}{N_{t}}$ [50]. The
thresholds for Algorithm 1 and Algorithm 2 are set to $\epsilon=10^{-6}$ and
$\xi=10^{-4}$. The narrow-band mmWave channels have $L=8$ paths. The azimuth
AoA $\phi^{\mathrm{r}}_{b,u}$ and AoD $\phi^{\mathrm{t}}_{b,u}$ are uniformly
distributed in the interval $\left(-\pi,\pi\right]$, while the elevation AoA
$\theta^{\mathrm{r}}_{b,u}$ and AoD $\theta^{\mathrm{t}}_{b,u}$ are uniformly
distributed in the interval $\left(-\frac{\pi}{2},\frac{\pi}{2}\right]$ [54].
The noise power is set to $\sigma^{2}=1$.

Figure 3: WSR v.s. the number of iterations.

In Fig. 3, we evaluate the validity of our convergence analysis for both the
hybrid and for the fully digital TPC design. It is shown that $N_{\max}=100$
iterations are enough for achieving near-perfect convergence for both the
hybrid and for the fully digital TPC. This observation confirms the low
complexity of the proposed algorithm.

Figure 4: WSR v.s. transmit power at each AP.

In Fig. 4, we characterize both the fully digital and the hybrid TPC matrix
design in the cell-free MIMO DL vs. the maximum transmit power. Observe in
Fig. 4 that the system’s WSR is increased, when the transmit power increases
from 10dBm to 40dBm. We also observe that as the the number of RF chains is
increased, the WSR also increases. When the number of RF chains is $N_{\rm
RF}=8$, the cell-free MIMO relying on hybrid TPC achieves a WSR very close to
that of fully digital TPC, which verifies the benefits of hybrid TPCs in the
cell-free MIMO DL. For verifying the superiority of our proposed algorithm, we
compare it to the algorithm of [23], where the hybrid TPC matrices are
obtained through decomposing the fully digital TPC matrices derived into
hybrid ones through subspace decomposition. As shown in Fig. 4, the proposed
algorithm outperforms the algorithm of [23] in terms of the WSR performance.
Moreover, the algorithm of [23] has higher computational complexity primarily
owing to the iterative update of the analog and digital TPC matrices during
the hybrid TPC design.

Figure 5: WSR v.s. the number of RF chains.

In order to investigate the performance of hybrid TPCs, we increase the number
of RF chains from $N_{\rm RF}=2$ to $N_{\rm RF}=64$. Observe in Fig. 5 that
the WSR of hybrid TPCs increases with $N_{\rm RF}$ and a near-optimal WSR is
attained for $N_{\rm RF}\geq 16$. We also observe that when the number of RF
chains in the hybrid TPC is equal to the number of antennas in the fully
digital TPC, the WSR of hybrid TPC becomes significantly better when the
number of RF chains is limited. This further demonstrates the superiority of
employing hybrid TPCs in the cell-free MIMO DL.

Figure 6: WSR v.s. the number of APs. Figure 7: WSR v.s. the number of users.

In Fig. 6, we investigate the WSR vs. the number of APs in the cell-free MIMO
DL. In addition to the conclusion drawn from the aforementioned simulations,
we observe that the WSR increases as the number of APs increases from $2$ to
$6$. This means that a large number of APs deployed in the cell-free MIMO DL
has the potential of drastically improving the WSR performance.

Fig. 7 portrays the WSR vs. the number of users. Note that the subspace
decomposition algorithm cannot be executed when $UN_{s}<N_{\rm RF}$, hence we
do not compare our proposed algorithm to it in those scenarios. As illustrated
in Fig. 7, when the number of users increases from $2$ to $8$, the WSR
increases first and then decreases for $N_{\rm RF}=8$. This is due to the IUI,
which is increased with the number of users, while the number of RF chains is
low.

Figure 8: WSR v.s. the number of TAs.

We then portray the WSR vs. the number of TAs in Fig. 8. As the number of TAs
increases from $N_{t}=4\times 4=16$ to $N_{t}=10\times 10=100$, the WSR is
nearly doubled. This is because more TAs are able to attain a higher TPC gain
and antenna diversity for the cell-free MIMO DL.

Figure 9: WSR v.s. the number of APs for the fully digital TPC scenario.
Figure 10: WSR v.s. the number of users for the fully digital TPC scenario.

As illustrated in Fig. 9, our proposed fully digital TPC shows a better WSR
performance compared to its conventional counterparts as the number of APs
increases from 2 to 6. For explicitly showing the relationships between our
proposed algorithm and its counterparts, we set $N_{t}=4\times 4=16$ in this
figure. Furthermore, we set the number of data streams for each user to
$N_{s}=1$, because the problem transformation in [25] using FP is only
suitable for such a case. We set $N_{r}=N_{s}=1$, when we compare our proposed
algorithm to the traditional ZF algorithm and to the MRT algorithm [26]. We
observe that as the number of APs increases, the WSR advantage of the proposed
algorithm becomes higher.

We also show the performance comparisons of fully digital TPC vs. the number
of users in Fig. 10. We increase $U$ from $2$ to $8$. It is shown in Fig.10
that the FP [25] method with $N_{r}=4$, $N_{s}=1$ outperforms the proposed
fully digital TPC design when the number of users exceeds 6.

##  VI Summary and Conclusion

A powerful hybrid TPC was conceived for the mmWave cell-free MIMO DL. A WSR
maximization problem was formulated subject to a realistic power constraint
for each AP and the constant-modulus constraint for the phase shifters of the
analog TPCs. Then we transformed it into an equivalent WMMSE problem by
exploiting the UC nature of the cell-free MIMO DL. We proposed a low
complexity BCD algorithm for iteratively solving the transformed problem. We
then also further extended our proposed algorithm to the fully digital cell-
free MIMO scenario. Additionally, the convergence and computational complexity
analysis of our proposed algorithm were presented. Our simulation results
demonstrated the superiority of the proposed algorithms both in hybrid and in
fully digital TPCs.

##  Appendix A Proof of Proposition 1

In order to complete the proof, we rewrite the terms in (III-C) as follows.
The expression in the $\mathrm{Tr}\left(\cdot\right)$ function of the first
term in (III-C) can be rewritten as

$\displaystyle\mathbf{F}_{u}^{\rm H}\mathbf{A}_{u}\mathbf{F}_{u}=$
$\displaystyle\mathbf{F}_{u}^{\rm
H}\mathbf{A}_{u}^{\frac{1}{2}}\left(\mathbf{A}_{u}^{\frac{1}{2}}\right)^{\rm
H}\mathbf{F}_{u}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\mathbf{F}_{1,u}^{\rm H}&\mathbf{F}_{2,u}^{\rm
H}&\cdots\mathbf{F}_{B,u}^{\rm
H}\end{bmatrix}\begin{bmatrix}\widehat{\mathbf{A}}_{1,u}\\\
\widehat{\mathbf{A}}_{2,u}\\\ \vdots\\\
\widehat{\mathbf{A}}_{B,u}\end{bmatrix}$
$\displaystyle\begin{bmatrix}\widehat{\mathbf{A}}_{1,u}^{\rm
H}&\widehat{\mathbf{A}}_{2,u}^{\rm H}&\cdots\widehat{\mathbf{A}}_{B,u}^{\rm
H}\end{bmatrix}\begin{bmatrix}\mathbf{F}_{1,u}\\\ \mathbf{F}_{2,u}^{\rm H}\\\
\vdots\\\ \mathbf{F}_{B,u}\end{bmatrix}$ $\displaystyle=$
$\displaystyle\left(\sum_{b=1}^{B}\mathbf{F}_{b,u}^{\rm
H}\widehat{\mathbf{A}}_{b,u}\right)\left(\sum_{b=1}^{B}\mathbf{F}_{b,u}^{\rm
H}\widehat{\mathbf{A}}_{b,u}\right)^{\rm H}$ $\displaystyle=$
$\displaystyle\mathbf{F}_{b,u}^{\rm
H}\widehat{\mathbf{A}}_{b,u}\widehat{\mathbf{A}}_{b,u}^{\rm
H}\mathbf{F}_{b,u}+\mathbf{F}_{b,u}^{\rm
H}\widehat{\mathbf{A}}_{b,u}\sum_{i=1,i\neq
b}^{B}\widehat{\mathbf{A}}_{i,u}^{\rm H}\mathbf{F}_{i,u}$
$\displaystyle+\sum_{i=1,i\neq b}^{B}\mathbf{F}_{i,u}^{\rm
H}\widehat{\mathbf{A}}_{i,u}\widehat{\mathbf{A}}_{b,u}^{\rm
H}\mathbf{F}_{b,u}$ $\displaystyle+\sum_{i=1,i\neq b}^{B}\mathbf{F}_{i,u}^{\rm
H}\widehat{\mathbf{A}}_{i,u}\sum_{i=1,i\neq
b}^{B}\widehat{\mathbf{A}}_{i,u}^{\rm H}\mathbf{F}_{i,u}.$ (34)

Similar to (A), the expression in the $\mathrm{Tr}\left(\cdot\right)$ function
of the second term in (III-C) can be rewritten as

$\displaystyle\mathbf{F}_{u}^{\rm H}\mathbf{H}_{u}^{\rm
H}\mathbf{G}_{u}\mathbf{W}_{u}=\frac{1}{\omega_{u}}\mathbf{F}_{u}^{\rm
H}\mathbf{C}_{u}=\frac{1}{\omega_{u}}\sum_{b=1}^{B}\mathbf{F}_{b,u}^{\rm
H}\mathbf{C}_{b,u}.$ (35)

The third term in (III-C) can be readily derived by calculating the conjugate
transpose of (35). Furthermore, we expand the Frobenius norm in the fourth
term in (III-C) as

$\displaystyle\left\|\mathbf{F}_{b,u}-\mathbf{F}_{{\rm
HB},b,u}\right\|_{F}^{2}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}\left(\left(\mathbf{F}_{b,u}-\mathbf{F}_{{\rm
HB},b,u}\right)^{\rm H}\left(\mathbf{F}_{b,u}-\mathbf{F}_{{\rm
HB},b,u}\right)\right)$ $\displaystyle=$
$\displaystyle\mathrm{Tr}\left(\mathbf{F}_{b,u}^{\rm
H}\mathbf{F}_{b,u}-\mathbf{F}_{b,u}^{\rm H}\mathbf{F}_{{\rm
HB},b,u}-\mathbf{F}_{{\rm HB},b,u}^{\rm H}\mathbf{F}_{b,u}\right.$
$\displaystyle\left.+\mathbf{F}_{{\rm HB},b,u}^{\rm H}\mathbf{F}_{{\rm
HB},b,u}\right).$ (36)

Substituting (A)-(A) into (III-C) and neglecting the irrelevant terms, we can
prove Proposition 1.

##  Appendix B Proof of Lemma 1

Recalling the definition of
$\mathbf{Q}_{b,u}=\omega_{u}\widehat{\mathbf{A}}_{b,u}\widehat{\mathbf{A}}_{b,u}^{\rm
H}$ in Proposition 1, the rank of $\mathbf{Q}_{b,u}$ depends on
$\widehat{\mathbf{A}}_{b,u}$, which is the submatrix of
$\mathbf{A}_{u}^{\frac{1}{2}}$. Therefore, we first investigate the rank of
$\mathbf{A}_{u}$. We decompose $\mathbf{W}_{u}$ as
$\mathbf{W}_{u}=\mathbf{W}_{u}^{\frac{1}{2}}\left(\mathbf{W}_{u}^{\frac{1}{2}}\right)^{\rm
H}$ and define the SVD of
$\mathbf{H}_{u}^{\text{H}}\mathbf{G}_{u}\mathbf{W}_{u}^{\frac{1}{2}}$ as
$\mathbf{H}_{u}^{\text{H}}\mathbf{G}_{u}\mathbf{W}_{u}^{\frac{1}{2}}=\widehat{\mathbf{U}}_{u}\widehat{\mathbf{\Sigma}}_{u}\widehat{\mathbf{V}}_{u}^{\rm
H}=\widehat{\mathbf{U}}_{u}\begin{bmatrix}\overline{\mathbf{\Sigma}}_{u}\\\
\mathbf{0}_{\left(BN_{t}-N_{s}\right)\times
N_{s}}\end{bmatrix}\widehat{\mathbf{V}}_{u}^{\rm H}$, where
$\overline{\mathbf{\Sigma}}_{u}=\text{diag}\left(\eta_{u,1},\eta_{u,2},\cdots,\eta_{u,N_{s}}\right)\in\mathbb{C}^{N_{s}\times
N_{s}}$ is composed of all non-zero singular values of
$\mathbf{H}_{u}^{\text{H}}\mathbf{G}_{u}\mathbf{W}_{u}^{\frac{1}{2}}$ since
$\mathbf{W}_{u}^{\frac{1}{2}}$ is of size $N_{s}\times N_{s}$. Then we have

$\displaystyle\mathbf{A}_{u}=$ $\displaystyle\mathbf{H}_{u}^{\rm
H}\mathbf{G}_{u}\mathbf{W}_{u}\mathbf{G}_{u}^{\rm H}\mathbf{H}_{u}$
$\displaystyle=$ $\displaystyle\mathbf{H}_{u}^{\rm
H}\mathbf{G}_{u}\mathbf{W}_{u}^{\frac{1}{2}}\left(\mathbf{W}_{u}^{\frac{1}{2}}\right)^{\text{H}}\mathbf{G}_{u}^{\rm
H}\mathbf{H}_{u}$ $\displaystyle=$
$\displaystyle\widehat{\mathbf{U}}_{u}\widehat{\mathbf{\Sigma}}_{u}\widehat{\mathbf{\Sigma}}_{u}^{\text{T}}\widehat{\mathbf{U}}_{u}^{\rm
H}$ $\displaystyle=$
$\displaystyle\widehat{\mathbf{U}}_{u}\begin{bmatrix}\overline{\mathbf{\Sigma}}_{u}\overline{\mathbf{\Sigma}}_{u}^{\text{T}}&\mathbf{0}_{N_{s}\times\left(BN_{t}-N_{s}\right)}\\\
\mathbf{0}_{\left(BN_{t}-N_{s}\right)\times
N_{s}}&\mathbf{0}_{\left(BN_{t}-N_{s}\right)\times\left(BN_{t}-N_{s}\right)}\end{bmatrix}\widehat{\mathbf{U}}_{u}^{\rm
H},$ (37)

whose rank is $N_{s}$. Afterwards, we have

$\displaystyle\mathbf{A}_{u}^{\frac{1}{2}}=\widehat{\mathbf{U}}_{u}\begin{bmatrix}\ddot{\mathbf{\Sigma}}_{u}&\mathbf{0}_{N_{s}\times\left(BN_{t}-N_{s}\right)}\\\
\mathbf{0}_{\left(BN_{t}-N_{s}\right)\times
N_{s}}&\mathbf{0}_{\left(BN_{t}-N_{s}\right)\times\left(BN_{t}-N_{s}\right)}\end{bmatrix}\widehat{\mathbf{U}}_{u}^{\text{H}},$
(38)

where
$\ddot{\mathbf{\Sigma}}_{u}=\text{diag}\left(\sqrt{\eta_{u,1}^{2}},\sqrt{\eta_{u,2}^{2}},\cdots,\sqrt{\eta_{u,N_{s}}^{2}}\right)\in\mathbb{C}^{N_{s}\times
N_{s}}$. Furthermore, we re-express (38) as

$\displaystyle\mathbf{A}_{u}^{\frac{1}{2}}=$
$\displaystyle\widehat{\mathbf{U}}_{u}\begin{bmatrix}\begin{bmatrix}\ddot{\mathbf{\Sigma}}_{u}&\mathbf{0}_{N_{s}\times\left(N_{t}-N_{s}\right)}\\\
\mathbf{0}_{\left(N_{t}-N_{s}\right)\times
N_{s}}&\mathbf{0}_{\left(N_{t}-N_{s}\right)\times\left(N_{t}-N_{s}\right)}\end{bmatrix}&\mathbf{0}_{N_{t}\times\left(B-1\right)N_{t}}\\\
\mathbf{0}_{\left(B-1\right)N_{t}\times
N_{t}}&\mathbf{0}_{\left(B-1\right)N_{t}\times\left(B-1\right)N_{t}}\end{bmatrix}$
$\displaystyle\cdot\widehat{\mathbf{U}}_{u}^{\text{H}},$ (39)

and represent $\widehat{\mathbf{U}}_{u}$ as

$\displaystyle\widehat{\mathbf{U}}_{u}=\begin{bmatrix}\widehat{\mathbf{U}}_{u}^{\left(1,1\right)}&\widehat{\mathbf{U}}_{u}^{\left(1,2\right)}&\cdots&\widehat{\mathbf{U}}_{u}^{\left(1,B\right)}\\\
\vdots&\vdots&\ddots&\vdots\\\
\widehat{\mathbf{U}}_{u}^{\left(B,1\right)}&\widehat{\mathbf{U}}_{u}^{\left(B,2\right)}&\cdots&\widehat{\mathbf{U}}_{u}^{\left(B,B\right)}\end{bmatrix},$
(40)

where $\widehat{\mathbf{U}}_{u}^{\left(i,j\right)}\in\mathbb{C}^{N_{t}\times
N_{t}}$. Upon combining (B) and (40), we have

$\displaystyle\mathbf{A}_{u}^{\frac{1}{2}}=\begin{bmatrix}\mathbf{A}_{u}^{\frac{1}{2}\left(1,1\right)}&\mathbf{A}_{u}^{\frac{1}{2}\left(1,2\right)}&\cdots&\mathbf{A}_{u}^{\frac{1}{2}\left(1,B\right)}\\\
\vdots&\vdots&\ddots&\vdots\\\
\mathbf{A}_{u}^{\frac{1}{2}\left(B,1\right)}&\mathbf{A}_{u}^{\frac{1}{2}\left(B,2\right)}&\cdots&\mathbf{A}_{u}^{\frac{1}{2}\left(B,B\right)}\end{bmatrix},$
(41)

where
$\mathbf{A}_{u}^{\frac{1}{2}\left(i,j\right)}=\widehat{\mathbf{U}}_{u}^{\left(i,1\right)}\begin{bmatrix}\ddot{\mathbf{\Sigma}}_{u}&\mathbf{0}_{N_{s}\times\left(N_{t}-N_{s}\right)}\\\
\mathbf{0}_{\left(N_{t}-N_{s}\right)\times
N_{s}}&\mathbf{0}_{\left(N_{t}-N_{s}\right)\times\left(N_{t}-N_{s}\right)}\end{bmatrix}$
$\left(\widehat{\mathbf{U}}_{u}^{\left(j,1\right)}\right)^{\text{H}}$.
Recalling the definition of $\widehat{\mathbf{A}}_{b,u}$, we can acquire the
expression of $\widehat{\mathbf{A}}_{b,u}$ from (41) as

$\displaystyle\widehat{\mathbf{A}}_{b,u}=$
$\displaystyle\begin{bmatrix}\mathbf{A}_{u}^{\frac{1}{2}\left(b,1\right)}&\mathbf{A}_{u}^{\frac{1}{2}\left(b,2\right)}\cdots\mathbf{A}_{u}^{\frac{1}{2}\left(b,B\right)}\end{bmatrix}$
$\displaystyle=$
$\displaystyle\widehat{\mathbf{U}}_{u}^{\left(b,1\right)}\begin{bmatrix}\ddot{\mathbf{\Sigma}}_{u}&\mathbf{0}_{N_{s}\times\left(N_{t}-N_{s}\right)}\\\
\mathbf{0}_{\left(N_{t}-N_{s}\right)\times
N_{s}}&\mathbf{0}_{\left(N_{t}-N_{s}\right)\times\left(N_{t}-N_{s}\right)}\end{bmatrix}$
$\displaystyle\begin{bmatrix}\left(\widehat{\mathbf{U}}_{u}^{\left(1,1\right)}\right)^{\text{H}}&\left(\widehat{\mathbf{U}}_{u}^{\left(2,1\right)}\right)^{\text{H}}&\cdots&\left(\widehat{\mathbf{U}}_{u}^{\left(B,1\right)}\right)^{\text{H}}\end{bmatrix}$
$\displaystyle\overset{\Delta}{=}$
$\displaystyle\widetilde{\mathbf{A}}_{1}\widetilde{\mathbf{A}}_{2}\widetilde{\mathbf{A}}_{3},$
(42)

where we define
$\widetilde{\mathbf{A}}_{1}=\widehat{\mathbf{U}}_{u}^{\left(b,1\right)}$,
$\widetilde{\mathbf{A}}_{2}=\begin{bmatrix}\ddot{\mathbf{\Sigma}}_{u}&\mathbf{0}_{N_{s}\times\left(N_{t}-N_{s}\right)}\\\
\mathbf{0}_{\left(N_{t}-N_{s}\right)\times
N_{s}}&\mathbf{0}_{\left(N_{t}-N_{s}\right)\times\left(N_{t}-N_{s}\right)}\end{bmatrix}$
and
$\widetilde{\mathbf{A}}_{3}=\begin{bmatrix}\left(\widehat{\mathbf{U}}_{u}^{\left(1,1\right)}\right)^{\text{H}}&\left(\widehat{\mathbf{U}}_{u}^{\left(2,1\right)}\right)^{\text{H}}&\cdots&\left(\widehat{\mathbf{U}}_{u}^{\left(B,1\right)}\right)^{\text{H}}\end{bmatrix}$.
Adopting the Sylvester inequality, i.e.,
$\text{rank}\left(\mathbf{X}\right)+\text{rank}\left(\mathbf{Y}\right)-x_{\rm
col}\leq\text{rank}\left(\mathbf{X}\mathbf{Y}\right)\leq\min\left\\{\text{rank}\left(\mathbf{X}\right),\text{rank}\left(\mathbf{Y}\right)\right\\}$
with $x_{\rm col}$ denoting the number of columns in $\mathbf{X}$ [55], we
have
$\text{rank}\left(\widehat{\mathbf{A}}_{b,u}\right)\leq\text{rank}\left(\widetilde{\mathbf{A}}_{2}\right)=N_{s}$.
Moreover, we have

$\displaystyle\text{rank}\left(\widehat{\mathbf{A}}_{b,u}\right)\geq$
$\displaystyle\text{rank}\left(\widetilde{\mathbf{A}}_{1}\right)+\text{rank}\left(\widetilde{\mathbf{A}}_{2}\widetilde{\mathbf{A}}_{3}\right)-N_{t}$
$\displaystyle=$ $\displaystyle
N_{t}+\text{rank}\left(\widetilde{\mathbf{A}}_{2}\widetilde{\mathbf{A}}_{3}\right)-N_{t}$
$\displaystyle\geq$
$\displaystyle\text{rank}\left(\widetilde{\mathbf{A}}_{2}\right)+\text{rank}\left(\widetilde{\mathbf{A}}_{3}\right)-N_{t}$
$\displaystyle=$ $\displaystyle N_{s}+N_{t}-N_{t}$ $\displaystyle=$
$\displaystyle N_{s}.$ (43)

Then we have $\text{rank}\left(\widehat{\mathbf{A}}_{b,u}\right)=N_{s}$.
Finally, by adopting the property that
$\text{rank}\left(\mathbf{X}\mathbf{X}^{\rm
H}\right)=\text{rank}\left(\mathbf{X}\right)$ [55], we have

$\displaystyle\text{rank}\left(\mathbf{Q}_{b,u}\right)=\text{rank}\left(\widehat{\mathbf{A}}_{b,u}\widehat{\mathbf{A}}_{b,u}^{\rm
H}\right)=\text{rank}\left(\widehat{\mathbf{A}}_{b,u}\right)=N_{s}.$ (44)

Hence we can prove the Lemma.

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|  Chenghao Feng received the B.E. degree from the Beijing Institute of
Technology, Beijing, China, in 2017, where he is currently pursuing the Ph.D.
degree with the School of Information and Electronics. His current research
interests include massive MIMO, mmWave/THz communications, energy-efficient
communications, intelligent reflecting surface and networks.
---|---
|  Wenqian Shen received the B.S. degree from Xi’an Jiaotong University,
Shaanxi, China in 2013 and the Ph.D. degree from Tsinghua University, Beijing,
China in 2018. She is currently an associate professor with the School of
Information and Electronics, Beijing Institute of Technology, Beijing, China.
Her research interests include massive MIMO and mmWave/THz communications. She
has published several journal and conference papers in IEEE Transaction on
Signal Processing, IEEE Transaction on Communications, IEEE Transaction on
Vehicular Technology, IEEE ICC, etc. She has won the IEEE Best Paper Award at
the IEEE ICC 2017.
---|---
|  Jianping An (M’08) received the B.E. degree from Information Engineering
University in 1987, and the M.S. and Ph.D. degrees from Beijing Institute of
Technology, in 1992 and 1996, respectively. Since 1996, he has been with the
School of Information and Electronics, Beijing Institute of Technology, where
he now holds the post of Full Professor. From 2010 to 2011, he was a Visiting
Professor at University of California, San Diego. He has published more than
150 journal and conference articles and holds (or co-holds) more than 50
patents. He has received various awards for his academic achievements and the
resultant industrial influences, including the National Award for Scientific
and Technological Progress of China (1997) and the Excellent Young Teacher
Award by the China’s Ministry of Education (2000). Since 2010, he has been
serving as a Chief Reviewing Expert for the Information Technology Division,
National Scientific Foundation of China. Prof. An’s current research interest
is focused on digital signal processing theory and algorithms for
communication systems.
---|---
|  Lajos Hanzo (http://www-mobile.ecs.soton.ac.uk,
https://en.wikipedia.org/wiki/Lajos_Hanzo) (FIEEE’04) received his Master
degree and Doctorate in 1976 and 1983, respectively from the Technical
University (TU) of Budapest. He was also awarded the Doctor of Sciences (DSc)
degree by the University of Southampton (2004) and Honorary Doctorates by the
TU of Budapest (2009) and by the University of Edinburgh (2015). He is a
Foreign Member of the Hungarian Academy of Sciences and a former Editor-in-
Chief of the IEEE Press. He has served several terms as Governor of both IEEE
ComSoc and of VTS. He has published 1970 contributions at IEEE Xplore, 19
Wiley-IEEE Press books and has helped the fast-track career of 123 PhD
students. Over 40 of them are Professors at various stages of their careers in
academia and many of them are leading scientists in the wireless industry. He
is also a Fellow of the Royal Academy of Engineering (FREng), of the IET and
of EURASIP.
---|---