# Uplink Transmit Power Optimization for Distributed Massive MIMO Systems with
1-Bit ADCs

Bikshapathi Gouda, Italo Atzeni, and Antti Tölli The authors are with the
Centre for Wireless Communications, University of Oulu, Finland (e-mail:
{bikshapathi.gouda, italo.atzeni, antti.tolli}@oulu.fi). This work is
supported by the Research Council of Finland (336449 Profi6, 346208 6G
Flagship, 343586 CAMAIDE and 348396 HIGH-6G) and by the European Commission
(101095759 Hexa-X-II). Part of this work was presented at IEEE GLOBECOM 2023
[1].

###### Abstract

This paper addresses the problem of uplink transmit power optimization in
distributed massive multiple-input multiple-output systems, where remote radio
heads (RRHs) are equipped with 1-bit analog-to-digital converters (ADCs).
First, in a scenario where a single RRH serves a single user equipment (UE),
the signal-to-noise-and-distortion ratio (SNDR) is shown to be a non-monotonic
and unimodal function of the UE transmit power due to the quantization
distortion (QD). Upon the introduction of multiple RRHs, adding properly tuned
dithering at each RRH is shown to render the SNDR at the output of the joint
receiver unimodal. In a scenario with multiple RRHs and UEs, considering the
non-monotonic nature of the signal-to-interference-plus-noise-and-distortion
ratio (SINDR), both the UE transmit powers and the RRH dithering levels are
jointly optimized subject to the min-power and max-min-SINDR criteria, while
employing Bussgang-based maximum ratio combining (BMRC) and minimum mean
squared error (BMMSE) receivers. To this end, gradient and block coordinate
descent methods are introduced to tune the UE transmit powers, whereas a line
search coupled with gradient updates is used to adjust the RRH dithering
levels. Numerical results demonstrate that jointly optimizing the UE transmit
power and the RRH dithering levels can significantly enhance the system
performance, thus facilitating joint reception from multiple RRHs across a
range of scenarios. Comparing the BMMSE and BMRC receivers, the former offers
a better interference and QD alleviation while the latter has a lower
computational complexity.

##  I Introduction

Fully digital massive multiple-input multiple-output (MIMO) is widely
recognized for its ability to realize flexible beamforming and large-scale
spatial multiplexing [2]. As the next-generation wireless systems are moving
towards higher carrier frequencies, both the bandwidth and the number of
antennas at the base station (BS) are bound to increase significantly [3].
However, fully digital massive MIMO arrays with high-resolution analog-to-
digital converters (ADCs) are exceedingly complex and power intensive [4]. To
mitigate this issue, low-resolution and 1-bit ADCs can be employed, which
significantly reduces the power consumption without excessively compromising
the performance [5, 6, 7, 8, 9]. Furthermore, adopting low-resolution and
1-bit ADCs in a distributed (also known as cell-free) massive MIMO setting
allows to reduce the backhaul signaling required for the data and channel
state information (CSI) exchange among the BSs [10, 11, 12].

For a single-BS massive MIMO system with low-resolution ADCs, the spectral and
energy efficiency were characterized with respect to the transmit power of the
user equipments (UEs) in [13, 14]. On the other hand, studies such as [6, 7,
8] investigated the uplink channel estimation and the achievable signal-to-
interference-plus-noise-and-distortion ratio (SINDR) with 1-bit ADCs. All the
aforementioned works provided an asymptotic analysis of the SINDR assuming a
Gaussian approximation of the quantization distortion (QD) arising from the
1-bit ADCs. This approximation is accurate in scenarios characterized by
either a low UE transmit power or a large number of UEs [5]. The evaluation of
the error vector magnitude against the UE transmit power with few UEs
indicated inferior performance with high UE transmit powers, which can be
attributed to the non-Gaussian QD [15]. However, appropriately tuned dithering
(i.e., intentionally applied noise) at the BS can enhance the performance in
such a scenario [15]. A similar behavior was observed while examining the mean
square error (MSE) of the channel estimation and the symbol error rate (SER)
in [9, 16]. These works demonstrated that the MSE of the channel estimation
and the SER obtained with 16-QAM data symbols can be enhanced by appropriately
tuning the UE transmit powers. To this end, an adaptive transmit power
optimization mechanism tailored for a single-UE system with 1-bit ADCs was
developed in [17] to optimize the UE transmit power in order to minimize the
SER. While many previous studies have considered conventional maximum ratio
combining (MRC) or zero-forcing (ZF) receivers, it was shown in [18] that
Bussgang-based MRC, ZF, and minimum mean square error (MMSE) receivers perform
better, as they incorporate the Bussgang gain into the design.

The optimization of the uplink and downlink transmit powers enhances the
network energy efficiency and ensures that each UE receives the required
quality of service (QoS), while also mitigating the impact of near-far
scenarios. In a coordinated multi-cell setting, the joint optimization of the
UE transmit powers and the receivers at the BS under QoS constraints was
addressed in [19]. Assuming MMSE receivers, [20] focused on optimizing the UE
transmit power for both the pilot and data transmissions under per-UE QoS
constraints. A similar study was provided in [21], which also addressed the
power allocation for pilot design to minimize the pilot contamination.
Considering the complexities and scalability of global power optimization in
multi-cell systems, a simplified approach was proposed in [22] assuming
constant interference from the other cells. Furthermore, a simplified transmit
power optimization method via geometric mean considering max-min fairness in
each cell was proposed in [23]. In a multi-cell context with low-resolution
ADCs, the optimization of the uplink and downlink transmit powers and
beamformers is studied in [24, 25] while assuming the Gaussian approximation
for the QD.

In the context of distributed massive MIMO systems with joint transmission or
reception, the optimization of the UE transmit powers and the receivers at the
BSs to ensure fairness among the UEs was studied in [26]. Furthermore, the
uplink spectral efficiency was analyzed considering the UE transmit powers
with both the ZF and MRC receivers in [27]. In [28], a scalable and less
complex uplink transmit power optimization scheme was proposed under the max-
min fairness criterion considering the MRC receiver. The energy efficiency of
cell-free systems with low-resolution ADCs was analyzed in conjunction with an
uplink power control scheme for max-min fairness in [29, 30]. Additionally,
the spectral efficiency of cell-free systems with 1-bit ADCs, considering
quantization and channel estimation errors, was evaluated in [31]. It is
important to note that the works in [29, 31, 30] also employ the Gaussian
approximation for QD, which becomes increasingly accurate with higher ADC
resolution. However, this approximation does not fully capture the true
performance characteristics of 1-bit ADCs due to the non-Gaussian QD,
especially with a low-to-moderate number of UEs.

Contribution. In this paper, we examine the problem of uplink transmit power
optimization in a distributed massive MIMO system where the remote radio heads
(RRHs) are equipped with 1-bit ADCs. First, we provide insights into the
signal-to-noise-and-distortion ratio (SNDR) of a UE in single- and multi-RRH
scenarios. Our analysis reveals that the SNDR is non-monotonic and highly
dependent on the UE transmit powers and locations due to the non-Gaussian QD
of the 1-bit ADCs. Specifically, the SNDR is unimodal in the single-RRH case
and multimodal in the multi-RRH case. Nonetheless, the SNDR can be made
unimodal even in the multi-RRH scenario by tuning the dithering levels at the
RRHs.111Radio frequency (RF) circuitry akin to automatic gain control can be
used to tune the dithering levels at the RF stage before the 1-bit ADCs. The
joint optimization of the UE transmit powers and RRH dithering levels is
tackled considering the min-power and max-min-SINDR objectives in a multi-UE,
multi-RRH scenario employing the Bussgang-based MRC (BMRC) and MMSE (BMMSE)
receivers from [18]. For a given set of RRH dithering levels, the UE transmit
powers are optimized for both objectives using either the gradient or the
block coordinate descent (BCD) method. The BCD method is augmented with a
tailored termination criterion to prevent the UE transmit powers from entering
the QD-dominated region of the SINDR. Additionally, we propose a simple
approach for updating the RRH dithering levels based on a line search followed
by a fine-tuning with gradient updates. Numerical results indicate that
optimizing the UE transmit powers alongside the RRH dithering levels can
greatly enhance the system performance by enabling joint reception from
multiple RRHs across various scenarios. We further show that, if achieving the
desired target SINDR requires reception over a large virtual antenna array
resulting from aggregating multiple RRHs in combination with RRH-specific
dithering, then the UE transmit power is heavily dominated by the distance to
the farthest serving RRH.

The contributions of this paper are summarized as follows:

  * •

The UE transmit powers and RRH dithering levels are optimized for the
considered distributed MIMO system with 1-bit ADCs taking into account the
non-monotonic behavior of the SINDR due to the QD.

  * •

Gradient and BCD methods are proposed to optimize the UE transmit powers for
the min-power and max-min-SINDR objectives, considering both the BMRC and
BMMSE receivers.

  * •

A simple approach for updating the RRH dithering levels based on a line search
is provided. Additionally, the RRH dithering levels are fine-tuned using the
gradient method for the min-power and max-min-SINDR objectives.

  * •

Simulation results demonstrate that optimizing the UE transmit powers
alongside the RRH dithering levels improves the system performance by
facilitating joint reception from multiple RRHs in several scenarios.

Part of this work is included in our conference paper [1], which presented the
UE transmit power optimization with fixed RRH dithering levels.

Outline. The rest of the paper is structured as follows. Section II introduces
the distributed massive MIMO system model with RRHs equipped with 1-bit ADCs.
Section III characterizes the SINDR of a UE in both single and multi-RRH
scenarios. Based on this, Sections IV and V delve into the optimization of the
UE transmit powers and RRH dithering levels for the min-power and max-min-
SINDR designs, respectively. Finally, Section VI presents the numerical
results and Section VII provides some concluding remarks.

Notation. Lowercase and uppercase boldface letters denote vectors and
matrices, respectively. $(\cdot)^{\mathrm{T}}$ and $(\cdot)^{\mathrm{H}}$
represent the transpose and Hermitian transpose operators, respectively.
$\|\cdot\|$ denotes the Euclidean norm for vectors. $\mathrm{Re}[\cdot]$,
$\mathrm{Im}[\cdot]$, and $\mathbb{E}[\cdot]$ denote the real part, imaginary
part, and expectation operators, respectively. $\mathbf{I}_{L}$ stands for the
$L$-dimensional identity matrix. $\mathrm{Diag}(\cdot)$ produces a diagonal
matrix with the elements of the vector argument or the diagonal elements of
the square matrix argument on its diagonal. The Kronecker product is denoted
by $\otimes$. $\mathrm{sgn}(\cdot)$ is the sign function. $\succ$ denotes the
element-wise $>$ operator of a vector. $[a_{1},\ldots,a_{L}]$ denotes the
horizontal concatenation, whereas $\\{a_{1},\ldots,a_{L}\\}$ and
$\\{a_{\ell}\\}_{\ell\in\mathcal{L}}$ represent sets; the latter notation is
occasionally relaxed as $\\{a_{\ell}\\}$ for brevity.
$\mathcal{C}\mathcal{N}(0,\sigma^{2})$ represents the complex normal
distribution with zero mean and variance $\sigma^{2}$.
$\frac{\partial}{\partial x}(\cdot)$ denotes the partial derivative with
respect to $x$. $\mathcal{L}_{(\textrm{P})}(\cdot)$ represents the Lagrangian
of the optimization problem $(\textrm{P})$.

$\displaystyle\gamma_{k}\approx\hat{\mathrm{\gamma}}_{k}\\!\triangleq\\!\frac{\rho_{k}|\mathbf{w}_{k}^{\mathrm{H}}\hat{\mathbf{A}}\hat{\mathbf{h}}_{k}|^{2}}{\sum_{k\in\mathcal{K}}\rho_{k}\|\mathbf{w}_{k}^{\mathrm{H}}\hat{\mathbf{A}}\mathbf{C}_{\tilde{\mathbf{h}}_{k}}^{\frac{1}{2}}\|^{2}\\!+\sum_{\bar{k}\neq
k}\rho_{\bar{k}}|\mathbf{w}_{k}^{\mathrm{H}}\hat{\mathbf{A}}\hat{\mathbf{h}}_{\bar{k}}|^{2}+\\!\|\mathbf{w}_{k}^{\mathrm{H}}\hat{\mathbf{A}}\mathbf{C}_{\mathbf{z}}^{\frac{1}{2}}\|^{2}\\!+\\!\mathbf{w}_{k}^{\mathrm{H}}\mathbf{C}_{\hat{\mathbf{q}}}\mathbf{w}_{k}}=\frac{\rho_{k}|\mathbf{w}_{k}^{\mathrm{H}}\hat{\mathbf{A}}\hat{\mathbf{h}}_{k}|^{2}}{\mathbf{w}_{k}^{\mathrm{H}}\mathbf{C}_{\hat{\mathbf{r}}}\mathbf{w}_{k}-\rho_{k}|\mathbf{w}_{k}^{\mathrm{H}}\hat{\mathbf{A}}\hat{\mathbf{h}}_{k}|^{2}}$
(14)

##  II System Model

We consider an uplink distributed massive MIMO system where a set of RRHs
$\mathcal{B}\triangleq\\{1,\ldots,B\\}$, each with $M$ antennas, serves a set
of single-antenna UEs $\mathcal{K}\triangleq\\{1,\ldots,K\\}$. Each RRH
antenna is connected to two 1-bit ADCs, which quantize the in-phase and
quadrature components of the received signal. Let
$\mathbf{h}_{b,k}\in\mbox{$\mathbb{C}$}^{M\times 1}$ denote the uplink channel
between UE $k\in\mathcal{K}$ and RRH $b\in\mathcal{B}$, where
$\mathbf{h}_{k}\triangleq[\mathbf{h}_{1,k}^{\mathrm{T}},\ldots,\mathbf{h}_{B,k}^{\mathrm{T}}]^{\mathrm{T}}\in\mbox{$\mathbb{C}$}^{BM\times
1}$ is the aggregated uplink channel of UE $k$ across all the RRHs. We use
$d_{k}\sim\mathcal{C}\mathcal{N}(0,1)$ to denote the data symbol transmitted
by UE $k$, which is independent from the data symbols of the other UEs. The
received signal at RRH $b$ prior to the 1-bit ADCs is given by

$\displaystyle\mathbf{y}_{b}\triangleq\sum_{k\in\mathcal{K}}\sqrt{\rho_{k}}\mathbf{h}_{b,k}d_{k}+\mathbf{z}_{b}\in\mbox{$\mathbb{C}$}^{M\times
1},$ (1)

where $\rho_{k}$ is the transmit power of UE $k$ and
$\mathbf{z}_{b}\sim\mathcal{C}\mathcal{N}(0,\sigma_{b}^{2}\mathbf{I}_{M})$ is
the additive white Gaussian noise (AWGN) vector at RRH $b$. Then, the received
signal $\mathbf{y}_{b}$ is quantized as

$\displaystyle\mathbf{r}_{b}\triangleq
Q(\mathbf{y}_{b})\in\mbox{$\mathbb{C}$}^{M\times 1},$ (2)

where
$Q(\mathbf{a})\triangleq\frac{1}{\sqrt{2}}\big{(}\mathrm{sgn}\big{(}\mathrm{Re}[\mathbf{a}]\big{)}+j\,\mathrm{sgn}\big{(}\mathrm{Im}[\mathbf{a}]\big{)}\big{)}$.

Let us introduce
$\mathbf{y}\triangleq[\mathbf{y}_{1}^{\mathrm{T}},\ldots,\mathbf{y}_{B}^{\mathrm{T}}]^{\mathrm{T}}\in\mbox{$\mathbb{C}$}^{BM\times
1}$ and
$\mathbf{z}\triangleq[\mathbf{z}_{1}^{\mathrm{T}},\ldots,\mathbf{z}_{B}^{\mathrm{T}}]^{\mathrm{T}}\in\mbox{$\mathbb{C}$}^{BM\times
1}$ as the aggregated received signal and AWGN vector, respectively, across
all the RRHs. Assuming perfect CSI, we can describe the 1-bit quantized output
across all the RRHs using the Bussgang decomposition as [32]

$\displaystyle\mathbf{r}\triangleq
Q(\mathbf{y})=\mathbf{A}\mathbf{y}+\mathbf{q}\in\mbox{$\mathbb{C}$}^{BM\times
1},$ (3)

where $\mathbf{q}$ is the zero-mean, non-Gaussian QD vector that is
uncorrelated with $\mathbf{y}$ and

$\displaystyle\mathbf{A}\triangleq\sqrt{\frac{2}{\pi}}\mathrm{Diag}(\mathbf{C}_{\mathbf{y}})^{-\frac{1}{2}}\in\mbox{$\mathbb{C}$}^{BM\times
BM}$ (4)

is the Bussgang gain, with
$\mathbf{C}_{\mathbf{y}}\triangleq\mathbb{E}[\mathbf{y}\mathbf{y}^{\mathrm{H}}]=\sum_{k\in\mathcal{K}}{\rho_{k}}\mathbf{h}_{k}\mathbf{h}_{k}^{\mathrm{H}}+\mathbf{C}_{\mathbf{z}}\in\mbox{$\mathbb{C}$}^{BM\times
BM}$ and
$\mathbf{C}_{\mathbf{z}}\triangleq\mathrm{Diag}\big{(}[\sigma^{2}_{1},\ldots,\sigma^{2}_{B}]\big{)}\otimes\mathbf{I}_{M}$.

The soft-detected symbol of UE $k$ is obtained by combing $\mathbf{r}$ in (3)
with the network-wide receiver $\mathbf{w}_{k}\in\mbox{$\mathbb{C}$}^{BM\times
1}$, i.e.,

$\displaystyle\hat{d}_{k}$
$\displaystyle=\mathbf{w}_{k}^{\mathrm{H}}\mathbf{r}$ (5)
$\displaystyle=\sqrt{\rho_{k}}\mathbf{w}_{k}^{\mathrm{H}}\mathbf{A}\mathbf{h}_{k}d_{k}\\!+\\!\sum_{\bar{k}\neq
k}\\!\sqrt{\rho_{\bar{k}}}\mathbf{w}_{k}^{\mathrm{H}}\mathbf{A}\mathbf{h}_{\bar{k}}d_{\bar{k}}\\!+\\!\mathbf{w}_{k}^{\mathrm{H}}\mathbf{A}\mathbf{z}\\!+\\!\mathbf{w}_{k}^{\mathrm{H}}\mathbf{q}.$
(6)

Then, the corresponding SINDR of UE $k$ is given by

$\displaystyle\mathrm{\gamma}_{k}\\!$
$\displaystyle\triangleq\\!\frac{\rho_{k}|\mathbf{w}_{k}^{\mathrm{H}}\mathbf{A}\mathbf{h}_{k}|^{2}}{\sum_{\bar{k}\neq
k}\rho_{\bar{k}}|\mathbf{w}_{k}^{\mathrm{H}}\mathbf{A}\mathbf{h}_{\bar{k}}|^{2}\\!+\\!\|\mathbf{w}_{k}^{\mathrm{H}}\mathbf{A}\mathbf{C}_{\mathbf{z}}^{\frac{1}{2}}\|^{2}\\!+\\!\mathbf{w}_{k}^{\mathrm{H}}\mathbf{C}_{\mathbf{q}}\mathbf{w}_{k}}$
(7)
$\displaystyle=\\!\frac{\rho_{k}|\mathbf{w}_{k}^{\mathrm{H}}\mathbf{A}\mathbf{h}_{k}|^{2}}{\mathbf{w}_{k}^{\mathrm{H}}\mathbf{C}_{\mathbf{r}}\mathbf{w}_{k}-\rho_{k}|\mathbf{w}_{k}^{\mathrm{H}}\mathbf{A}\mathbf{h}_{k}|^{2}},$
(8)

where
$\mathbf{C}_{\mathbf{q}}\triangleq\mathbb{E}[\mathbf{q}\mathbf{q}^{\mathrm{H}}]=\mathbf{C}_{\mathbf{r}}-\mathbf{A}\mathbf{C}_{\mathbf{y}}\mathbf{A}^{\mathrm{H}}\in\mbox{$\mathbb{C}$}^{BM\times
BM}$ and [8]

$\displaystyle\mathbf{C}_{\mathbf{r}}$
$\displaystyle\triangleq\frac{2}{\pi}\big{(}\arcsin\big{(}\mathrm{Diag}(\mathbf{C}_{\mathbf{y}})^{-\frac{1}{2}}\mathrm{Re}[\mathbf{C}_{\mathbf{y}}]\mathrm{Diag}(\mathbf{C}_{\mathbf{y}})^{-\frac{1}{2}}\big{)}$
$\displaystyle\phantom{=}\
+j\,\arcsin\big{(}\mathrm{Diag}(\mathbf{C}_{\mathbf{y}})^{-\frac{1}{2}}\mathrm{Im}[\mathbf{C}_{\mathbf{y}}]\mathrm{Diag}(\mathbf{C}_{\mathbf{y}})^{-\frac{1}{2}}\big{)}\big{)}.$
(9)

In the following section, we present an approximation of the SINDR using the
estimated channels at the RRHs. Subsequently, we explore the SNDR of a single
UE with RRHs equipped with 1-bit ADCs. Note that, in the absence of
interference from other UEs, the terms SINDR and SNDR can be used
interchangeably.

##  III Characterization of the SINDR

Let us define
$\hat{\mathbf{h}}_{k}\triangleq\mathbf{h}_{k}-\tilde{\mathbf{h}}_{k}$ as the
channel estimate of UE $k$ across all the RRHs, where $\tilde{\mathbf{h}}_{k}$
denotes the channel estimation error. We assume that $\tilde{\mathbf{h}}_{k}$
is independent of $\hat{\mathbf{h}}_{k}$, with
$\mathbb{E}[\tilde{\mathbf{h}}_{k}]=\mathbf{0}$ and
$\mathbb{E}[\tilde{\mathbf{h}}_{k}\tilde{\mathbf{h}}^{\mathrm{H}}_{k}]\triangleq\mathbf{C}_{\tilde{\mathbf{h}}_{k}}$
[33, 34]. With these definitions, the estimate of $\mathbf{C}_{\mathbf{y}}$
can be written as
$\hat{\mathbf{C}}_{\mathbf{y}}\triangleq\sum_{k\in\mathcal{K}}{\rho_{k}}(\hat{\mathbf{h}}_{k}\hat{\mathbf{h}}_{k}^{\mathrm{H}}+\mathbf{C}_{\tilde{\mathbf{h}}_{k}})+\mathbf{C}_{\mathbf{z}}$.
Consequently, the estimate of $\mathbf{A}$ is given by

$\displaystyle\hat{\mathbf{A}}\triangleq\sqrt{\frac{2}{\pi}}\mathrm{Diag}(\hat{\mathbf{C}}_{\mathbf{y}})^{-\frac{1}{2}},$
(10)

and the quantized received signal in (3) is approximated as

$\displaystyle\mathbf{r}\approx\hat{\mathbf{r}}\triangleq\hat{\mathbf{A}}{\mathbf{y}}+\hat{\mathbf{q}},$
(11)

where $\hat{\mathbf{q}}$ represents the approximated QD.

After combining $\hat{\mathbf{r}}$ in (11) with the receiver $\mathbf{w}_{k}$,
the soft-detected symbol is approximated as

$\displaystyle\hat{d}_{k}$
$\displaystyle\approx\mathbf{w}_{k}^{\mathrm{H}}\hat{\mathbf{r}}$ (12)
$\displaystyle=\sqrt{\rho_{k}}\mathbf{w}_{k}^{\mathrm{H}}\hat{\mathbf{A}}\hat{\mathbf{h}}_{k}d_{k}+\sum_{k\in\mathcal{K}}\sqrt{\rho_{k}}\mathbf{w}_{k}^{\mathrm{H}}\hat{\mathbf{A}}\tilde{\mathbf{h}}_{k}d_{k}$
$\displaystyle\phantom{=}\ +\sum_{\bar{k}\neq
k}\sqrt{\rho_{\bar{k}}}\mathbf{w}_{k}^{\mathrm{H}}\hat{\mathbf{A}}\hat{\mathbf{h}}_{\bar{k}}d_{\bar{k}}+\mathbf{w}_{k}^{\mathrm{H}}\hat{\mathbf{A}}\mathbf{z}\\!+\\!\mathbf{w}_{k}^{\mathrm{H}}\hat{\mathbf{q}}.$
(13)

Considering the channel estimation error in (12), the SINDR of UE $k$ is
approximated as shown in (14) at the top of the page, with
$\mathbf{C}_{\hat{\mathbf{q}}}\triangleq\mathbb{E}[\hat{\mathbf{q}}\hat{\mathbf{q}}^{\mathrm{H}}]=\mathbf{C}_{\hat{\mathbf{r}}}-\hat{\mathbf{A}}\hat{\mathbf{C}}_{\mathbf{y}}\hat{\mathbf{A}}^{\mathrm{H}}$
and where $\mathbf{C}_{\hat{\mathbf{r}}}$ is obtained by replacing
$\mathbf{C}_{\mathbf{y}}$ with $\hat{\mathbf{C}}_{\mathbf{y}}$ in (II). As
$\hat{\mathbf{h}}_{k}$ approaches $\mathbf{h}_{k}$, the SINDR
$\hat{\gamma}_{k}$ in (14) converges to $\gamma_{k}$ in (7).

From the estimated channels, the BMRC receiver of UE $k$ is defined as

$\displaystyle\mathbf{w}_{k}^{\textnormal{\tiny{BMRC}}}\triangleq\hat{\mathbf{A}}\hat{\mathbf{h}}_{k},$
(15)

while the BMMSE receiver of UE $k$ is obtained by minimizing its MSE, defined
as
$\mathrm{MSE}_{k}\triangleq\mathbb{E}\big{[}|(\mathbf{w}_{k}^{\textnormal{\tiny{BMMSE}}})^{\mathrm{H}}\hat{\mathbf{r}}-d_{k}|^{2}\big{]}$,
which yields

$\displaystyle\mathbf{w}_{k}^{\textnormal{\tiny{BMMSE}}}=\sqrt{\rho_{k}}\mathbf{C}_{\hat{\mathbf{r}}}^{-1}\hat{\mathbf{A}}\hat{\mathbf{h}}_{k}.$
(16)

Furthermore, the SINDR in (14) is rewritten for the BMRC and BMMSE receivers
as

$\displaystyle\hat{\mathrm{\gamma}}^{\textnormal{\tiny{BMRC}}}_{k}$
$\displaystyle\triangleq\frac{\rho_{k}|\hat{\mathbf{h}}_{k}^{\mathrm{H}}\hat{\mathbf{A}}\hat{\mathbf{A}}\hat{\mathbf{h}}_{k}|^{2}}{\hat{\mathbf{h}}_{k}^{\mathrm{H}}\hat{\mathbf{A}}\mathbf{C}_{\hat{\mathbf{r}}}\hat{\mathbf{A}}\hat{\mathbf{h}}_{k}-\rho_{k}|\hat{\mathbf{h}}_{k}^{\mathrm{H}}\hat{\mathbf{A}}\hat{\mathbf{A}}\hat{\mathbf{h}}_{k}|^{2}},$
(17) $\displaystyle\hat{\mathrm{\gamma}}^{\textnormal{\tiny{BMMSE}}}_{k}$
$\displaystyle\triangleq\frac{\rho_{k}|\hat{\mathbf{h}}_{k}^{\mathrm{H}}\hat{\mathbf{A}}\mathbf{C}^{-1}_{\hat{\mathbf{r}}}\hat{\mathbf{A}}\hat{\mathbf{h}}_{k}|^{2}}{\hat{\mathbf{h}}_{k}^{\mathrm{H}}\hat{\mathbf{A}}\mathbf{C}^{-1}_{\hat{\mathbf{r}}}\hat{\mathbf{A}}\hat{\mathbf{h}}_{k}-\rho_{k}|\hat{\mathbf{h}}_{k}^{\mathrm{H}}\hat{\mathbf{A}}\mathbf{C}^{-1}_{\hat{\mathbf{r}}}\hat{\mathbf{A}}\hat{\mathbf{h}}_{k}|^{2}},$
(18)

respectively.

###  III-A SNDR in the Single-UE Case

In this section, we explore the SNDR behavior of a single UE in single- and
two-RRHs scenarios. For the SNDR analysis, we assume $M=128$ antennas and the
BMMSE receiver with perfect CSI at each RRH.222A similar SNDR behavior can be
observed with the BMRC receiver. It is important to note that, even for a
single UE, the BMMSE receiver is not a scaled version of the BMRC receiver due
to the non-linear terms in $\mathbf{C}_{\mathbf{r}}$ in (II), which restricts
the use of the Woodbury matrix identity as in the unquantized case.
Furthermore, we model the channel between UE $k$ and RRH $b$, denoted by
$\mathbf{h}_{b,k}\in\mbox{$\mathbb{C}$}^{M\times 1}$, using uncorrelated
Rayleigh fading, i.e.,
$\mathbf{h}_{b,k}\sim\mathcal{C}\mathcal{N}(0,\delta_{b,k}\mathbf{I}_{M})$.
The large-scale fading coefficient $\delta_{b,k}$ is determined by the
distance $d_{b,k}$ between UE $k$ and RRH $b$, given by
$\delta_{b,k}\triangleq-61-30\log_{10}(d_{b,k})$ [dB], which assumes a $28$
GHz carrier frequency and a pathloss exponent of $3$.

####  III-A1 Single-RRH case: SNDR versus UE transmit power.

Figure 1: Single-RRH case: SNDR versus UE transmit power.

Let us begin with a scenario featuring a single RRH equipped with 1-bit ADCs,
where the UE is positioned at a fixed distance of $30$ m from the RRH. Fig. 1
illustrates the SNDR as a function of the UE transmit power. The SNDR exhibits
a non-monotonic yet unimodal behavior, with an optimal value achieved at a
specific UE transmit power (around 0 dBm). For lower UE transmit powers, the
impact of the AWGN dominates, overpowering the intended signal. Conversely,
for higher UE transmit powers, the soft-estimated symbols suffer from
amplitude loss due to severe clipping, resulting in the QD prevailing over the
intended signal [9]. Therefore, achieving the optimal SNDR requires an
appropriate amount of UE transmit power that minimizes the combined effect of
both AWGN and QD. In the general case, the non-monotonic behavior of the SINDR
for UE $k$ due to QD is described in the following definition.

######  Definition 1.

The uplink SINDR of UE $k$, denoted by $\gamma_{k}$, exhibits non-monotonic
behavior with respect to the UE transmit power due to the QD if there exists a
UE transmit power $\rho_{k}>\rho^{\prime}_{k}$ for which
$\gamma_{k}(\rho_{k})<\gamma_{k}(\rho^{\prime}_{k})$ for fixed interference
and AWGN at the RRHs.

####  III-A2 Two-RRH case: SNDR versus distance to the reference RRH.

$0$$20$$40$$60$$80$$100$$0$$5$$10$$15$$20$$25$$30$Distance to the reference BS
[m]$\mathrm{SNDR}$$\rho_{k}=-5$ dBm$\rho_{k}=7.5$ dBm$\rho_{k}=20$ dBm Figure
2: Two-RRH case: SNDR versus distance to the reference RRH.

The SNDR of a UE in a two-RRH scenario can be characterized as a function of
the distance between the UE and one of the two RRHs, which is referred to as
the reference RRH in Fig. 2. In this scenario, the two RRHs are positioned
$100$ m apart. The UE maintains a fixed transmit power and moves along the
line connecting the two RRHs. For a low UE transmit power of $-5$ dBm, the
SNDR decreases as the UE moves away from the left RRH and increases as it gets
closer to the other RRH as in the unquantized case, where the SNDR is
primarily affected by the AWGN. For a high UE transmit power of $7.5$ dBm, the
SNDR is limited by the QD of the closest RRH and the AWGN of the farthest RRH.
However, as the UE moves away from the closest RRH, the detrimental impact of
both the QD and AWGN is reduced, thereby improving the SNDR. At the cell edge,
i.e., at $50$ m, the SNDR is dominated by the AWGN at both RRHs due to the
limited UE transmit power, resulting in a lower SNDR. At a very high UE
transmit power of $20$ dBm, the SNDR is limited by the QD if the UE is close
to any of the RRHs, and the impact of the QD reduces as the UE moves towards
the cell edge, resulting in a better SNDR. Note that the UE is not power-
limited even at the cell edge in this scenario.

####  III-A3 Two-RRH case: SNDR versus UE transmit power.

$-20$$0$$20$$40$$60$$0$$5$$10$$15$$20$$25$$30$$\rho_{k}$
[dBm]$\mathrm{SNDR}$UE at $50$ m (no dithering)UE at $10$ m (no dithering)UE
at $10$ m (dithering) Figure 3: Two-RRH case: SNDR versus UE transmit power.

Let us analyze the impact of the UE transmit power on the SNDR for a single UE
at a fixed distance from both RRHs. The impact of dithering (i.e.,
intentionally applied noise) at each RRH is also evaluated. If the UE is
equidistant from both RRHs, specifically located $50$ m away from the
reference RRH, the impact of the AWGN and QD on the SNDR is the same at both
RRHs for a fixed UE transmit power. Similar to the single-RRH case, the
optimal performance at both the RRHs is achieved at the same UE transmit
power, leading to a unimodal SNDR behavior, as shown in Fig. 3. Moreover, the
SNDR in the two-RRH scenario is higher than in the single-RRH case (see Fig.
1) due to the combining gain from both RRHs. On the other hand, if the UE is
closer to one of the RRHs (say $10$ m), the optimal performance at the closest
RRH is achieved with a very low UE transmit power compared with the farthest
RRH. This results in a bimodal SNDR behavior with two local maxima. Therefore,
to enhance the overall SNDR, the dominance of the QD at the closest RRH can be
reduced by adjusting either the dithering level or both the signal and
dithering levels at each RRH [35, 36]. For simplicity, we consider adjusting
only the dithering level at each RRH throughout the paper. As a result,
$\sigma_{b}$ becomes a controllable parameter referred to as the dithering
level, consisting of a superposition of both the AWGN and the additional
Gaussian dithering signal introduced at each RRH. By adjusting the dithering
level, a fixed QD level can be maintained (as mentioned in Remark 1),
resulting in an improved SNDR with unimodal behavior, as shown in Fig. 3. In
scenarios involving over two RRHs and a UE with unlimited transmit power, the
dithering levels can be applied to all the RRHs except the farthest one
(constrained by the AWGN) to improve the SNDR. In practice, the optimal
dithering level at each RRH depends on its pathloss ratio to the farthest RRH.

######  Remark 1.

By adjusting both the signal and dithering levels at each RRH, the QD can be
kept constant regardless of variations in the UE transmit powers.

Considering a single UE and neglecting the UE index, the quantized received
signal at RRH $b$ given in (2) can be simplified as

$\displaystyle\mathbf{r}_{b}=Q\bigg{(}\sqrt{\rho\delta_{b}}\bigg{(}\mathbf{h}^{\prime}_{b}d+\frac{\sigma_{b}}{\sqrt{\rho\delta_{b}}}\mathbf{z}_{b}^{\prime}\bigg{)}\bigg{)},$
(19)

with
$\mathbf{h}^{\prime}_{b},\mathbf{z}_{b}^{\prime}\sim\mathcal{C}\mathcal{N}(0,\mathbf{I}_{M})$
and where $\rho$ is the UE transmit power.

It is clear from (19) that, as long as the ratio
$\frac{{\sigma_{b}}}{\sqrt{\rho\delta_{b}}}$ remains constant, the output of
the 1-bit ADC is the same. Therefore, in general, $\mathbf{C}_{\mathbf{r}}$ in
(II) remains unchanged. Moreover, it can be shown that $\mathbf{A}\mathbf{y}$
in (3) does not change if ${\sigma_{b}}$ and $\sqrt{\rho\delta_{b}}$ vary with
a fixed ratio. Therefore, by fixing
$\frac{{\sigma_{b}}}{\sqrt{\rho\delta_{b}}}$ for different input signal
powers, $\mathbf{C}_{\mathbf{q}}$ (i.e., the QD) is ensured to remain
constant. We recall that the value of
$\frac{{\sigma_{b}}}{\sqrt{\rho\delta_{b}}}$ can be maintained by adjusting
either the RRH dithering level ${\sigma_{b}}$ or both the signal power $\rho$
and the RRH dithering level ${\sigma_{b}}$.

##  IV Uplink Transmit Power Optimization Based on Min-Power

In this section, we discuss the optimization of UE transmit powers to achieve
a fixed target SINDR. Additionally, we can also optimize the dithering levels
at the RRHs to attain a higher target SINDR, as discussed in Section III. For
a given target SINDR $\gamma_{k}^{\textnormal{\tiny{Tar}}}$ of UE $k$, the
min-power optimization problem can be formulated as

$\displaystyle\begin{array}[]{cl}\displaystyle\underset{\\{\rho_{k},\sigma_{b}\\}}{\mathrm{minimize}}&\displaystyle\sum_{k\in\mathcal{K}}\rho_{k}\\\
\mathrm{s.t.}&\hat{\mathrm{\gamma}}_{k}\geq\gamma_{k}^{\textnormal{\tiny{Tar}}},\quad\forall
k\in\mathcal{K}\\\ &\sigma_{b}\geq\sigma_{\min},\quad\forall
b\in\mathcal{B},\end{array}$ (23)

where $\bar{\mathrm{\gamma}}_{k}$ is given in (14) and $\sigma^{2}_{\min}$ is
the minimum dithering level corresponding to the AWGN power. Problem (23) is
not convex with respect to $\rho_{k}$ or $\sigma_{b}$.333Note that the
optimization of $\sigma_{b}$ is not required for the single RRH case. However,
considering the SINDR properties discussed in Section III, the optimization of
both the RRH dithering levels and the UE transmit powers in a multi-RRH
scenario can be achieved, as discussed in the following. First, the
optimization of the UE transmit powers is considered for a fixed set of
dithering levels across all the RRHs, as discussed next.

###  IV-A Optimization of the UE Transmit Powers

The SINDR constraint in (23) can be expressed using (17) or (18) as

$\displaystyle\xi^{\textnormal{\tiny{BMRC}}}_{k}$
$\displaystyle\\!\triangleq\\!(\gamma_{k}^{\textnormal{\tiny{Tar}}}\\!+\\!1){\rho_{k}|\hat{\mathbf{h}}_{k}^{\mathrm{H}}\hat{\mathbf{A}}\hat{\mathbf{A}}\hat{\mathbf{h}}_{k}|^{2}}-\\!\gamma_{k}^{\textnormal{\tiny{Tar}}}{\hat{\mathbf{h}}_{k}^{\mathrm{H}}\hat{\mathbf{A}}\mathbf{C}_{\hat{\mathbf{r}}}\hat{\mathbf{A}}\hat{\mathbf{h}}_{k}}\\!\geq
0,$ (24) $\displaystyle\xi^{\textnormal{\tiny{BMMSE}}}_{k}$
$\displaystyle\triangleq(\gamma_{k}^{\textnormal{\tiny{Tar}}}+1)\rho_{k}\hat{\mathbf{h}}_{k}^{\mathrm{H}}\hat{\mathbf{A}}\mathbf{C}_{\hat{\mathbf{r}}}^{-1}\hat{\mathbf{A}}\hat{\mathbf{h}}_{k}-{\gamma_{k}^{\textnormal{\tiny{Tar}}}}\geq
0,$ (25)

respectively. To further simplify the notation, we define
$\mathrm{RX}\triangleq\\{\mathrm{BMRC},\mathrm{BMMSE}\\}$ to represent the use
of either the BMRC or the BMMSE receiver. For fixed RRH dithering levels
$\\{\sigma_{b}\\}$, the SINDR of a UE with 1-bit quantization is a non-
monotonic function of $\rho_{k}$ and inherently bounded, unlike the signal-to-
interference-plus-noise ratio (SINR) in the unquantized case, even without
interference from other UEs. Therefore, existing approaches such as [19, 37]
cannot be directly applied to optimize the UE transmit powers. Instead, we
propose to use either the gradient or the BCD method for optimizing the UE
transmit powers.

Gradient method. For a fixed set of RRH dithering levels $\\{\sigma_{b}\\}$,
the Lagrangian of (23) for a given receiver is given by

$\displaystyle\mathcal{L}^{\textnormal{\tiny{RX}}}_{\eqref{eq:p1}}\big{(}\\{\rho_{k},\mu_{k}^{\textnormal{\tiny{RX}}}\\}\big{)}$
$\displaystyle\triangleq\sum_{k\in\mathcal{K}}\big{(}\rho_{k}-\mu_{k}^{\textnormal{\tiny{RX}}}\xi_{k}^{\textnormal{\tiny{RX}}}\big{)},$
(26)

where $\mu_{k}^{\textnormal{\tiny{RX}}}\geq 0$ is the Lagrangian dual variable
associated with the SINDR constraint in (23) for a given BMRC or BMMSE
receiver. The transmit power of UE $k$ is then updated iteratively in the
direction of the negative gradient of the Lagrangian for a given receiver as

$\displaystyle\rho_{k}^{(i)}=\rho_{k}^{(i-1)}-\zeta_{\textnormal{\tiny{$\rho$}}}\frac{\partial}{\partial\rho_{k}}\mathcal{L}^{\textnormal{\tiny{RX}}}_{\eqref{eq:p1}}\big{(}\\{\rho_{k},\mu_{k}^{\textnormal{\tiny{RX}}}\\}\big{)},$
(27)

where $\zeta_{\textnormal{\tiny{$\rho$}}}\geq 0$ is the step size and $i$ is
the iteration number. For the initial iteration, $\\{\rho_{k}\\}$ are
initialized with positive random numbers. The gradient of the Lagrangian with
respect to $\rho_{k}$ is given in (IV-A) and (IV-A) at the top of the next
page with the BMRC and BMMSE receivers, respectively, whereas the partial
derivatives $\frac{\partial}{\partial\rho_{k}}\hat{\mathbf{A}}$ and
$\frac{\partial}{\partial\rho_{k}}\mathbf{C}_{\hat{\mathbf{r}}}$ are provided
in Appendix A. The Lagrangian dual variable $\mu_{k}^{\textnormal{\tiny{RX}}}$
is updated at iteration $i$ as

$\displaystyle\mu_{k}^{\textnormal{\tiny{RX}}(i)}$
$\displaystyle=\mathrm{max}\big{(}0,\mu_{k}^{\textnormal{\tiny{RX}}(i-1)}-\nu\xi_{k}^{\textnormal{\tiny{RX}}}\big{)},$
(27)

where $\nu\geq 0$ is the step size. For the initial iteration,
$\\{\mu_{k}^{\textnormal{\tiny{RX}}}\\}$ are initialized with positive random
numbers.

$\displaystyle\frac{\partial}{\partial\rho_{k}}\mathcal{L}^{\textnormal{\tiny{BMRC}}}_{\eqref{eq:p1}}\big{(}\\{\rho_{k},\mu_{k}^{\textnormal{\tiny{BMRC}}}\\}\big{)}$
$\displaystyle=1-\mu_{k}^{\textnormal{\tiny{BMRC}}}(\gamma_{k}^{\textnormal{\tiny{Tar}}}+1)|\hat{\mathbf{h}}_{k}^{\mathrm{H}}\hat{\mathbf{A}}\hat{\mathbf{A}}\hat{\mathbf{h}}_{k}|^{2}-\sum_{\bar{k}\in\mathcal{K}}\mu_{\bar{k}}^{\textnormal{\tiny{BMRC}}}\bigg{(}4\rho_{\bar{k}}(\gamma_{\bar{k}}^{\textnormal{\tiny{Tar}}}+1)\hat{\mathbf{h}}_{\bar{k}}^{\mathrm{H}}\hat{\mathbf{A}}\hat{\mathbf{A}}\hat{\mathbf{h}}_{\bar{k}}\mathrm{Re}\bigg{[}\hat{\mathbf{h}}_{\bar{k}}^{\mathrm{H}}\bigg{(}\frac{\partial}{\partial\rho_{k}}\hat{\mathbf{A}}\bigg{)}\hat{\mathbf{A}}\hat{\mathbf{h}}_{\bar{k}}\bigg{]}$
$\displaystyle\phantom{=}\
-2\gamma_{k}^{\textnormal{\tiny{Tar}}}\mathrm{Re}\bigg{[}\hat{\mathbf{h}}_{\bar{k}}^{\mathrm{H}}\bigg{(}\frac{\partial}{\partial\rho_{k}}\hat{\mathbf{A}}\bigg{)}\mathbf{C}_{\hat{\mathbf{r}}}\hat{\mathbf{A}}\hat{\mathbf{h}}_{\bar{k}}\bigg{]}-\gamma_{k}^{\textnormal{\tiny{Tar}}}\hat{\mathbf{h}}_{\bar{k}}^{\mathrm{H}}\hat{\mathbf{A}}\bigg{(}\frac{\partial}{\partial\rho_{k}}\mathbf{C}_{\hat{\mathbf{r}}}\bigg{)}\hat{\mathbf{A}}\hat{\mathbf{h}}_{\bar{k}}\bigg{)}$
(25)

$\displaystyle\frac{\partial}{\partial\rho_{k}}\mathcal{L}^{\textnormal{\tiny{BMMSE}}}_{\eqref{eq:p1}}\big{(}\\{\rho_{k},\mu_{k}^{\textnormal{\tiny{BMMSE}}}\\}\big{)}$
$\displaystyle=1-(\gamma_{k}^{\textnormal{\tiny{Tar}}}+1)\mu_{k}^{\textnormal{\tiny{BMMSE}}}\hat{\mathbf{h}}_{k}^{\mathrm{H}}\hat{\mathbf{A}}\mathbf{C}_{\hat{\mathbf{r}}}^{-1}\hat{\mathbf{A}}\hat{\mathbf{h}}_{k}-\sum_{\bar{k}\in\mathcal{K}}(\gamma_{\bar{k}}^{\textnormal{\tiny{Tar}}}+1)\mu_{\bar{k}}^{\textnormal{\tiny{BMMSE}}}\rho_{\bar{k}}\bigg{(}\mathrm{Re}\bigg{[}\hat{\mathbf{h}}_{\bar{k}}^{\mathrm{H}}\bigg{(}2\frac{\partial}{\partial\rho_{k}}\hat{\mathbf{A}}$
$\displaystyle\phantom{=}\
-\hat{\mathbf{A}}\mathbf{C}_{\hat{\mathbf{r}}}^{-1}\frac{\partial}{\partial\rho_{k}}\mathbf{C}_{\hat{\mathbf{r}}}\bigg{)}\mathbf{C}_{\hat{\mathbf{r}}}^{-1}\hat{\mathbf{A}}\hat{\mathbf{h}}_{\bar{k}}\bigg{]}\bigg{)}$
(26)

Note that the gradient method in (27) terminates when feasible target SINDRs
are met for all the UEs. However, to address the possibility of infeasible
SINDR targets, a fixed number of iterations for the gradient method is imposed
to guarantee a finite computation time. The convergence speed of this method
depends on the slope of the non-monotonic SINDR function at different UE
transmit powers. Moreover, in the case of multiple RRHs, this method may
converge to a local optimum due to the presence of multiple peaks in the SINDR
function, as discussed in Section III.

As an alternative solution to find an appropriate set of UE transmit powers to
satisfy the SINDR constraints of (23), a fixed-point power update method akin
to [19, 37] can be devised. This approach can be especially effective for low
UE transmit powers, where the SINDR resembles the SINR of an unquantized
system. Based on this, in the following, we propose a modified BCD method to
optimize the UE transmit powers for the min-power objective.

BCD method. In this method, the transmit power of a UE is updated with a
scaling factor while the transmit powers of the other UEs are fixed in each
iteration. For a given iteration $i$, a UE $k$ with the largest deviation from
the SINDR target is selected to update its transmit power:

$\displaystyle
k=\operatornamewithlimits{argmax}_{\bar{k}\in\mathcal{K}}{\gamma_{\bar{k}}^{\textnormal{\tiny{Tar}}}}/{\hat{\mathrm{\gamma}}_{\bar{k}}^{(i-1)}}.$
(28)

Then, the transmit power of the chosen UE is scaled to satisfy the target
SINDR constraint $\gamma_{k}^{\textnormal{\tiny{Tar}}}$ as

$\displaystyle\rho_{k}^{(i)}=\min\big{(}\alpha,{\gamma_{k}^{\textnormal{\tiny{Tar}}}}/{\hat{\mathrm{\gamma}}_{k}^{(i-1)}}\big{)}\rho_{k}^{(i-1)},$
(29)

where $\hat{\mathrm{\gamma}}_{k}^{(i-1)}$ can be obtained using either the
BMRC or BMMSE receiver as in (17) or (18), respectively, given fixed transmit
powers $\\{\rho_{k}^{(i-1)}\\}_{k\in\mathcal{K}}$. The parameter $\alpha>1$ is
used to limit the maximum power increase within a single iteration to prevent
the UE transmit power from entering too far into the QD-dominated region of
the SINDR function. It is important to note that the values of the UE transmit
powers must be initialized such that the initial values of the SINDR are in
the AWGN-dominated region. As the SINDR is non-monotonic, unlike in
unquantized systems, we impose a termination criterion that stops the process
if the SINDR of a UE starts decreasing when its transmit power is increased
while the other UE transmit powers are fixed (i.e.,
$\gamma_{k}^{(i)}<\gamma_{k}^{(i-1)}$ if $\rho^{(i)}>\rho^{(i-1)}$), as
described in Definition 1. Additionally, the process is terminated if the
SINDR of all the UEs decreases when all the UE transmit powers are increased
($\gamma_{k}(\boldsymbol{\rho})<\gamma_{k}(\boldsymbol{\rho^{\prime}}),\leavevmode\nobreak\
\forall k\in\mathcal{K}$ if
$\boldsymbol{\rho}\succeq\boldsymbol{\rho^{\prime}}$), as stated in Theorem 1.
The full procedure to optimize the UE transmit powers via the BCD method is
summarized in Algorithm 1.

######  Theorem 1.

The transmit power of a UE is in the QD-dominated region of the non-monotonic
SINDR function if there exists a UE transmit power vector
$\boldsymbol{\rho}\succeq\boldsymbol{\rho^{\prime}}$ such that
$\gamma_{k}(\boldsymbol{\rho})<\gamma_{k}(\boldsymbol{\rho^{\prime}}),\leavevmode\nobreak\
\forall k\in\mathcal{K}$. Here,
$\boldsymbol{\rho}=[\rho_{1},\ldots,\rho_{K}]^{\mathrm{T}}$ and
$\boldsymbol{\rho^{\prime}}=[\rho^{\prime}_{1},\ldots,\rho^{\prime}_{K}]^{\mathrm{T}}$.

###### Proof:

The theorem is proven by contradiction. Consider an unquantized system
following the properties of standard interference functions [38]. Let us begin
with $\boldsymbol{\rho}\succeq\boldsymbol{\rho^{\prime}}$, which implies that
there exists a UE $k$ for which
$a\triangleq\frac{\rho_{k}}{\rho^{\prime}_{k}}\geq\frac{\rho_{\bar{k}}}{\rho^{\prime}_{\bar{k}}}\geq
1,\leavevmode\nobreak\ \forall\bar{k}\in\mathcal{K}\setminus{k}$. This further
implies $a\boldsymbol{\rho^{\prime}}\succeq\boldsymbol{\rho}$. Consequently,
given a fixed receiver, the following condition holds for the SINR of UE $k$:

$\displaystyle\gamma_{k}(\boldsymbol{\rho^{\prime}})$
$\displaystyle=\frac{G_{k}(\rho^{\prime}_{k})}{I_{k}(\boldsymbol{\rho^{\prime}})}\leq
a\frac{G_{k}(\rho^{\prime}_{k})}{I_{k}(a\boldsymbol{\rho^{\prime}})}\leq\frac{G_{k}(\rho_{k})}{I_{k}(\boldsymbol{\rho})}=\gamma_{k}(\boldsymbol{\rho}),$
(30)

where $G_{k}$ and $I_{k}$ are the signal and interference powers,
respectively, of UE $k$. The first and second inequalities follow from the
scalability and monotonicity of the standard interference function [38].
Therefore, if $\boldsymbol{\rho}\succeq\boldsymbol{\rho^{\prime}}$ , then
$\gamma_{k}(\boldsymbol{\rho})\geq\gamma_{k}(\boldsymbol{\rho}^{\prime})$ for
at least one UE. Consequently, our initial hypothesis of
$\gamma_{k}(\boldsymbol{\rho})<\gamma_{k}(\boldsymbol{\rho^{\prime}}),\leavevmode\nobreak\
\forall k\in\mathcal{K}$ if
$\boldsymbol{\rho}\succeq\boldsymbol{\rho^{\prime}}$, necessitates the non-
monotonic SINR behavior of a UE as described in Definition 1, attributable to
the effects of the QD. ∎

Data: $\\{\hat{\mathbf{h}}_{k}\\}$,
$\\{\mathbf{C}_{\tilde{\mathbf{h}}_{k}}\\}$,
$\\{\gamma_{k}^{\textnormal{\tiny{Tar}}}\\}$, $\\{\sigma_{b}\\}$.
Initialization: Iteration $i=0$, $\\{\rho_{k}^{(0)}\\}$.

  * •

Compute $\\{\hat{\mathrm{\gamma}}_{k}^{(0)}\\}$ as in (17) or (18).

while
_$\underset{k\in\mathcal{K}}{\min}(\hat{\mathrm{\gamma}}_{k}^{(i)}-\gamma_{k}^{\textnormal{\tiny{Tar}}})
<0$_ do

  * •

$i=i+1$.

  * •

Select UE $k$ according to (28) to scale its transmit power.

  * •

Update the UE transmit power as in (29).

  * •

Compute $\\{\hat{\mathrm{\gamma}}_{k}^{(i)}\\}$ as in (17) or (18).

if  _
$\hat{\mathrm{\gamma}}_{k}^{(i)}\\!<\\!\hat{\mathrm{\gamma}}_{k}^{(i-1)}$ or
$\underset{j\triangleq\\{1,\ldots,i-1\\}}{\max}\underset{\bar{k}\in\mathcal{K}}{\min}(\hat{\gamma}_{\bar{k}}^{(i)}-\hat{\gamma}_{\bar{k}}^{(j)})<0$_
then

break % To avoid the QD-dominated region of SINDR.

end if

end while

Algorithm 1 Optimization of the UE transmit powers via the BCD method

The proposed BCD method is particularly effective in the AWGN-dominated
region, where the SINDR increases steadily and monotonically with the UE
transmit powers. Additionally, the convergence speed is similar to the
unquantized case and does not depend on the (fixed) step size, unlike the
gradient method. However, when the optimization of RRH-specific dithering
levels is added to the problem, a unique solution may not be always guaranteed
due to the non-standard properties of interference and quantization distortion
[38]. In the following, we discuss the optimization of the dithering levels at
the RRHs.

###  IV-B Optimization of the RRH Dithering Levels

As discussed in Remark 1, the QD remains constant for a fixed
$\frac{{\sigma_{b}}}{\sqrt{\rho\delta_{b}}}$ ratio. Therefore, when
considering a fixed set of $\\{\sigma_{b}\\}$ values, solving the UE transmit
power optimization problem (23) may lead to a suboptimal or infeasible
solution. In principle, we can approximately solve (23) by considering
discretized sets of $\\{\sigma_{b}\\}$ values and choosing the best
combination among them. However, the number of possible combinations of
$\\{\sigma_{b}\\}$ values grows exponentially with the increasing number of
RRHs. In the following, we first present a heuristic approach to provide a
coarse initial estimate of $\\{\sigma_{b}\\}$ using a simple line search.
Then, a further refinement step is introduced, where the local optimal values
close to the initial estimates are found using the gradient method.

Coarse initialization of the RRH dithering levels. The SINDR gain of a UE with
joint reception from multiple RRHs increases as the ratios
$\frac{{\sigma_{b}}}{\sqrt{\rho\delta_{b}}}$ become closer to each other.
Consequently, the SINDR gain is maximum when all the RRHs have the same
$\frac{{\sigma_{b}}}{\sqrt{\rho\delta_{b}}}$ (i.e., the QD is fixed at all the
RRHs) as explained in Section III. In the multi-UE scenario, a fixed QD for
all the UEs cannot be guaranteed at all the RRHs simultaneously. Moreover, in
a distributed RRH scenario, the SINDR of a UE is predominantly influenced by
its nearest RRH, and consequently, the dithering level at the RRH mostly
affects its closest UE. Therefore, the proposed criteria to tune the RRH
dithering levels is based on the nearest UE. To find an initial compromise set
of RRH dithering levels based on the nearest UE at each RRH, we define
$\delta_{b}^{\max}\triangleq\max_{k\in\mathcal{K}}\delta_{b,k}$ to denote the
maximum path gain (minimum path loss) between RRH $b$ and its closest UE in
$\mathcal{K}$. Furthermore, if the nearest UE of an RRH is closer than the UEs
near other RRHs, that RRH saturates more quickly, necessitating the addition
of dithering earlier than other RRHs. As a result,
$\frac{\sigma_{b}}{\sqrt{\delta_{b}^{\max}}}$ becomes more similar among RRHs,
improving the performance of the joint reception. Consequently, the RRH
dithering levels $\\{\sigma_{b}\\}$ are updated using a single parameter
$\theta$ as

$\displaystyle\sigma_{b}=\sigma_{\min}\max\big{(}1,\sqrt{{\delta_{b}^{\max}}}\
\theta\big{)}$ (31)

where the search variable $\theta$ lies within the range of
$[1/\max_{\bar{b}\in\mathcal{B}}\sqrt{\delta_{b}^{\max}},1/\min_{\bar{b}\in\mathcal{B}}\sqrt{\delta_{b}^{\max}}]$.
Clearly, if $\theta=1/\max_{\bar{b}\in\mathcal{B}}\sqrt{\delta_{b}^{\max}}$,
dithering is set to $\sigma_{b}=\sigma_{\min}\ \forall\ b$. Conversely, if
$\theta=1/\min_{\bar{b}\in\mathcal{B}}\sqrt{\delta_{b}^{\max}}$, dithering is
increased at all the RRHs except the one farthest from any UE in
$\mathcal{K}$. In general, provided that there are no limits on the UE
transmit powers, increasing the RRH dithering improves the feasible range of
SINDR values. However, for a fixed feasible SINDR target, the min-power may
exhibit a unimodal behavior with respect to the value of $\theta$. Therefore,
we use a line search method to find the optimal $\theta$ value corresponding
to the minimum sumpower. Specifically, we employ a ternary search method,
evaluating the UE transmit power for two different values of $\theta$ within
its defined limits [39]. These two points are selected to divide the entire
range of $\theta$ into three equal intervals. Subsequently, we compute the sum
power at these two values of $\theta$ and adjust the range of $\theta$ values
in the next iteration, such that the value of $\theta$ corresponding to the
minimum sum power lies in the middle of the updated range, as detailed in
Algorithm 2.

Data: $\\{\hat{\mathbf{h}}_{k}\\}$,
$\\{\mathbf{C}_{\tilde{\mathbf{h}}_{k}}\\}$,
$\\{\gamma_{k}^{\textnormal{\tiny{Tar}}}\\}$, $\rho^{\max}$,
$\epsilon_{\sigma}>0$, $\epsilon_{\rho}>0$.
Initialization: Iteration $i=0$,
$\theta^{\textnormal{\tiny{low}}}=1/\max_{b\in\mathcal{B}}\sqrt{\delta_{b}^{\max}}$,
$\theta^{\textnormal{\tiny{high}}}=1/\min_{b\in\mathcal{B}}\sqrt{\delta_{b}^{\max}}$.
%Finding the coarse RRH dithering levels.

while  _$(1)$_ do

%Evaluate the min-power at two points. for  _$l=1:2$_ do

  * •

$\theta^{(l)}\triangleq{\big{(}(3-l)\theta^{\textnormal{\tiny{low}}}+l\theta^{\textnormal{\tiny{high}}}\big{)}}/{3}$

  * •

Compute $\\{\sigma_{b}\\}$ using (31).

  * •

Compute $\\{\rho_{k}^{(l)}\\}$ using (27) or Algorithm 1.

if
_$\min_{k\in\mathcal{K}}(\hat{\gamma}_{k}-\gamma^{\textnormal{\tiny{Tar}}}_{k})\leq
0$_ then

  * •

$\rho_{k}^{(l)}=\rho^{\max}/(\theta^{(l)}),\leavevmode\nobreak\ \forall
k\in\mathcal{K}.$

end if

end for

if  _$\sum_{k\in\mathcal{K}}\rho^{(1)}_{k}
>\sum_{k\in\mathcal{K}}\rho^{(2)}_{k}$_ then

  * •

$\theta^{\textnormal{\tiny{low}}}={\big{(}2\theta^{\textnormal{\tiny{low}}}+\theta^{\textnormal{\tiny{high}}}\big{)}}/{3}$.

else

  * •

$\theta^{\textnormal{\tiny{high}}}={\big{(}\theta^{\textnormal{\tiny{low}}}+2\theta^{\textnormal{\tiny{high}}}\big{)}}/{3}.$

end if

if
_$(\theta^{\textnormal{\tiny{low}}}-\theta^{\textnormal{\tiny{high}}})\leq\epsilon_{\sigma}$_
then

break

end if

end while

% Fine-tuning the RRH dithering and UE transmit powers. while  _
$\big{(}\sum_{k\in\mathcal{K}}\rho^{(i)}_{k}-\sum_{k\in\mathcal{K}}\rho^{(i-1)}_{k}\big{)}\leq\epsilon_{\rho}$_
do

  * •

$i=i+1$

  * •

Compute $\\{\sigma_{b}\\}$ using (33).

  * •

Compute $\\{\rho_{k}^{(i)}\\}$ using (27) or (29).

end while

Algorithm 2 Optimization of the UE transmit powers and RRH dithering levels
with the min-power objective

$\displaystyle\frac{\partial}{\partial\sigma_{b}}\mathcal{L}_{\eqref{eq:p1}}^{\textnormal{\tiny{BMRC}}}\big{(}\\{\sigma_{b},\mu_{k}^{\textnormal{\tiny{BMRC}}},\upsilon_{b}\\}\big{)}$
$\displaystyle=-\sum_{\bar{k}\in\mathcal{K}}\mu_{\bar{k}}^{\textnormal{\tiny{BMRC}}}\bigg{(}4\rho_{k}(\gamma_{\bar{k}}^{\textnormal{\tiny{Tar}}}+1)\hat{\mathbf{h}}_{\bar{k}}^{\mathrm{H}}\hat{\mathbf{A}}\hat{\mathbf{A}}\hat{\mathbf{h}}_{\bar{k}}\mathrm{Re}\bigg{[}\hat{\mathbf{h}}_{\bar{k}}^{\mathrm{H}}\Big{(}\frac{\partial}{\partial\sigma_{b}}\hat{\mathbf{A}}\Big{)}\hat{\mathbf{A}}\hat{\mathbf{h}}_{\bar{k}}\bigg{]}$
$\displaystyle\phantom{=}\
-2\gamma_{k}^{\textnormal{\tiny{Tar}}}\mathrm{Re}\bigg{[}\hat{\mathbf{h}}_{\bar{k}}^{\mathrm{H}}\Big{(}\frac{\partial}{\partial\sigma_{b}}\hat{\mathbf{A}}\Big{)}\mathbf{C}_{\hat{\mathbf{r}}}\hat{\mathbf{A}}\hat{\mathbf{h}}_{\bar{k}}\bigg{]}-\gamma_{k}^{\textnormal{\tiny{Tar}}}\hat{\mathbf{h}}_{\bar{k}}^{\mathrm{H}}\hat{\mathbf{A}}\Big{(}\frac{\partial}{\partial\sigma_{b}}\mathbf{C}_{\hat{\mathbf{r}}}\Big{)}\hat{\mathbf{A}}\hat{\mathbf{h}}_{\bar{k}}\bigg{)}-\upsilon_{b}$
(34)

$\displaystyle\frac{\partial}{\partial\sigma_{b}}\mathcal{L}_{\eqref{eq:p1}}^{\textnormal{\tiny{BMMSE}}}\big{(}\\{\sigma_{b},\mu_{k}^{\textnormal{\tiny{BMMSE}}},\upsilon_{b}\\}\big{)}$
$\displaystyle=-\sum_{\bar{k}\in\mathcal{K}}\mu_{\bar{k}}^{\textnormal{\tiny{BMMSE}}}(\gamma_{\bar{k}}^{\textnormal{\tiny{Tar}}}+1)\rho_{\bar{k}}\bigg{(}2\mathrm{Re}\bigg{[}\hat{\mathbf{h}}_{\bar{k}}^{\mathrm{H}}\Big{(}\frac{\partial}{\partial\sigma_{b}}\hat{\mathbf{A}}\Big{)}\mathbf{C}_{\hat{\mathbf{r}}}^{-1}\hat{\mathbf{A}}\hat{\mathbf{h}}_{\bar{k}}\bigg{]}-\hat{\mathbf{h}}_{\bar{k}}^{\mathrm{H}}\hat{\mathbf{A}}\mathbf{C}_{\hat{\mathbf{r}}}^{-1}\Big{(}\frac{\partial}{\partial\sigma_{b}}\mathbf{C}_{\hat{\mathbf{r}}}\Big{)}\mathbf{C}_{\hat{\mathbf{r}}}^{-1}\hat{\mathbf{A}}\hat{\mathbf{h}}_{\bar{k}}\bigg{)}\\!-\\!\upsilon_{b}$
(35)

After optimizing the UE transmit powers in (23) with the coarse dithering
levels from (31), both the UE transmit powers and RRH dithering levels can be
fine-tuned using the gradient method in an alternating manner. For fixed RRH
dithering levels, the UE transmit powers are optimized with the gradient
method described in Section IV-A. For fixed UE transmit powers, the coarse RRH
dithering levels are fine-tuned using the gradient method. To do this, for a
fixed $\\{\rho_{k}\\}$, we construct the Lagrangian of (23) for a given
receiver as

$\displaystyle\mathcal{L}_{\eqref{eq:p1}}^{\textnormal{\tiny{RX}}}\big{(}\\{\sigma_{b},\mu_{k}^{\textnormal{\tiny{RX}}},\upsilon_{b}\\}\big{)}$
$\displaystyle\triangleq\sum_{k\in\mathcal{K}}(\rho_{k}-\mu_{k}^{\textnormal{\tiny{RX}}}\xi_{k}^{\textnormal{\tiny{RX}}})\\!-\\!\sum_{b\in\mathcal{B}}\upsilon_{b}(\sigma_{b}-\sigma_{\min}),$
(32)

where $\upsilon_{b}\geq 0$ is the Lagrangian dual variable corresponding to
the minimum noise (dithering) level constraint in (23). The dithering level
$\sigma_{b}$ is then updated iteratively in the direction of the negative
gradient of the Lagrangian as

$\displaystyle\sigma_{b}^{(i)}=\max\big{(}\sigma_{\min},\sigma_{b}^{(i-1)}-\zeta_{\textnormal{\tiny{$\sigma$}}}\frac{\partial}{\partial\sigma_{b}}\mathcal{L}_{\eqref{eq:p1}}^{\textnormal{\tiny{RX}}}\big{(}\\{\sigma_{b},\mu_{k}^{\textnormal{\tiny{RX}}},\upsilon_{b}\\}\big{)}\big{)},$
(33)

where $\zeta_{\textnormal{\tiny{$\sigma$}}}\geq 0$ is the step size and $i$ is
the iteration number. For the initial iteration, the values of ${\sigma_{b}}$
are obtained from (31). The gradient of the Lagrangian with respect to
$\sigma_{b}$ is given in (IV-B) and (35) at the top of the next page with the
BMRC and BMMSE receivers, respectively, whereas the partial derivative of
$\hat{\mathbf{A}}$ and $\mathbf{C}_{\hat{\mathbf{r}}}$ with respect to
$\sigma_{b}$ are provided in Appendix B. The dual variable
$\mu_{k}^{\textnormal{\tiny{RX}}}$ is updated as in (27). Similarly, the dual
variable $\upsilon_{b}$ is updated in the same way for both the BMRC and BMMSE
receivers:

$\displaystyle\upsilon_{b}^{(i)}=\max\big{(}0,\upsilon_{b}^{(i-1)}-\kappa(\sigma_{b}-\sigma_{\min})\big{)},$
(36)

where $\kappa\geq 0$ is the steps size. For the initial iteration,
$\\{\upsilon_{b}\\}$ are initialized with positive random values.

The fine-tuning of the UE transmit powers and RRH dithering levels involves a
single gradient update in each iteration. This process continues until there
is no significant improvement in the sum power. The full procedure to optimize
the UE transmit powers and the RRH dithering levels with the min-power
objective is summarized in Algorithm 2.

##  V Uplink Transmit Power Optimization Based on Max-Min-SINDR

In this section, we consider the maximization of the minimum SINDR across all
the UEs subject to UE-specific UL transmit power constraints
$\rho_{\textnormal{\tiny{UE}}}$. Similar to Section IV, both the UE transmit
powers and RRH dithering levels are treated as optimization variables. Thus,
the epigraph form of the max-min-SINDR optimization problem is given by

$\displaystyle\begin{array}[]{cl}\underset{\\{\rho_{k},\sigma_{b}\\},\gamma}{\mathrm{maximize}}&\gamma\\\
\mathrm{s.t.}&\hat{\gamma}_{k}\geq\gamma,\quad\forall k\in\mathcal{K}\\\
&\rho_{k}\leq\rho_{\textnormal{\tiny{UE}}},\quad\forall k\in\mathcal{K}\\\
&\sigma_{b}\geq\sigma_{\min}.\quad\forall b\in\mathcal{B},\end{array}$ (41)

where $\gamma$ is the optimization variable representing the minimum rate
across all the UEs. The UE-specific SINDR value $\hat{\gamma}_{k}$ can be
obtained using either BMRC or BMMSE receivers given in (17) or (18),
respectively. Problem (41) can be solved using min-power objective as in (23)
by fixing $\gamma$ and employing the bisection method to determine the optimal
value of $\gamma$. However, in the subsequent discussion, we explore
alternative techniques to solve (41) without relying on the bisection search
for $\gamma$. As (41) is also not convex with respect to $\rho_{k}$ and
$\sigma_{b}$, we first discuss the optimization of the UE transmit powers for
fixed RRH dithering levels as in Section IV. Then, we proceed to discuss the
optimization of the RRH dithering levels.

###  V-A Optimization of the UE Transmit Powers

Data: $\\{\hat{\mathbf{h}}_{k}\\}$,
$\\{\mathbf{C}_{\tilde{\mathbf{h}}_{k}}\\}$, $\rho_{\textnormal{\tiny{UE}}}$,
$\beta$.
Initialization: Iteration $i=0$, $\\{\rho_{k}^{(0)}\\}$.

  * •

Compute $\\{\hat{\mathrm{\gamma}}_{k}^{(0)}\\}$ as in (17) or (18).

  * •

$\gamma^{(0)}\triangleq\min_{k\in\mathcal{K}}\hat{\mathrm{\gamma}}_{k}^{(0)}$.

while  _
$\max_{k\in\mathcal{K}}{\rho_{k}^{(i)}}\leq\rho_{\textnormal{\tiny{UE}}}$_ do

  * •

$\bar{k}=\operatornamewithlimits{argmin}_{k\in\mathcal{K}}\hat{\gamma}_{k}^{(i)}$.

  * •

$i=i+1$.

  * •

$\rho^{(i)}_{\bar{k}}=\beta\rho^{(i-1)}_{\bar{k}}$, and
$\rho^{(i)}_{k}=\rho^{(i-1)}_{k},\quad\forall
k\in\mathcal{K}\setminus\\{\bar{k}\\}.$

  * •

Compute $\\{\hat{\mathrm{\gamma}}_{k}^{(i)}\\}$ as in (17) or (18).

  * •

$\gamma^{(i)}\triangleq\min_{k\in\mathcal{K}}\hat{\mathrm{\gamma}}_{k}^{(i)}$.

if  _
$\hat{\gamma}_{\bar{k}}^{(i)}\\!<\\!\hat{\gamma}_{\bar{k}}^{(i-1)}\textnormal{{or}}\
\max_{k\in\mathcal{K}}\hat{\mathrm{\gamma}}_{k}^{(i)}\\!<\\!\max\limits_{j=\\{0,\ldots,i\\}}\gamma^{(j)}\
$_ then

break

end if

end while

  * •

The optimal SINDR $\mathrm{\gamma}=\max_{j=\\{0,\ldots,i\\}}\gamma^{(j)}$.

Algorithm 3 Optimization of the UE transmit powers with the max-min-SINDR
objective via the BCD method

For a fixed set of RRH dithering levels, we may choose to optimize the UE
transmit powers of problem (41) using the gradient method similar to that in
Section IV. However, the gradient update of (41) becomes more complex and slow
due to the extra variable $\gamma$. Therefore, we employ a BCD method akin to
the fixed-point power update, while considering the impact QD on the SINDR of
UEs as in Section IV.

BCD method. The BCD method for the UE transmit power optimization, described
in Section IV for the min-power objective, can be adapted for the max-min
SINDR objective. The target SINDR objective in (41) is the same for all the
UEs. Thus, the user selection in (28) simplifies to finding a UE with the
minimum SINDR, i.e.,
$k=\operatornamewithlimits{argmin}_{\bar{k}\in\mathcal{K}}{\hat{\mathrm{\gamma}}_{\bar{k}}^{(i-1)}}$.
Since the target SINDR is a variable and not known, only the fixed scaling
factor $\alpha>1$ in (29) is used to update the transmit power of the selected
UE, while keeping the transmit powers of the other UEs unchanged.
Consequently, the minimum SINDR is increased in every iteration as long as the
SINDR values remain in the AWGN-dominated region for an appropriate choice of
$\beta$. The BCD method is terminated if any UE reaches its maximum power or
if the SINDR of any UE decreases despite increasing its transmit power, while
keeping the other UE transmit powers fixed, (i.e.,
$\gamma_{k}^{(i)}<\gamma_{k}^{(i-1)}$ if $\rho^{(i)}>\rho^{(i-1)}$) as defined
in Definition 1. Additionally, the process is terminated if the SINDR of all
the UEs decreases with increasing their transmit powers (i.e.,
$\max_{k\in\mathcal{K}}\gamma_{k}(\boldsymbol{\rho})<\gamma(\boldsymbol{\rho^{\prime}})$
if $\boldsymbol{\rho}\succeq\boldsymbol{\rho^{\prime}}$), as stated in Lemma
1. At the end of the BCD method, the maximum of minimum SINDRs over the
iterations is considered as the max-min SINDR, i.e.,
$\mathrm{\gamma}=\max_{j=\\{0,\ldots,i\\}}\gamma^{(j)}$. The full procedure to
optimize the UE transmit powers with the max-min-SINDR objective via the BCD
method is summarized in Algorithm 3.

######  Lemma 1.

The transmit power of a UE is in the QD-dominated region of the non-monotonic
SINDR function if there exists a UE transmit power vector
$\boldsymbol{\rho}\succeq\boldsymbol{\rho^{\prime}}$ such that
$\max_{k\in\mathcal{K}}\gamma_{k}(\boldsymbol{\rho})<\gamma(\boldsymbol{\rho^{\prime}})$,
where
$\gamma(\boldsymbol{\rho^{\prime}})=\min_{k\in\mathcal{K}}\gamma_{k}(\boldsymbol{\rho^{\prime}})$.

###### Proof:

The condition
$\max_{k\in\mathcal{K}}\gamma_{k}(\boldsymbol{\rho})<\gamma(\boldsymbol{\rho^{\prime}})$
can be rewritten as
$\gamma_{k}(\boldsymbol{\rho})<\gamma_{k}(\boldsymbol{\rho^{\prime}}),\leavevmode\nobreak\
\forall k\in\mathcal{K}$. Then, the rest of the proof follows similar steps to
the proof of Theorem 1. ∎

###  V-B Optimization of the RRH Dithering Levels

To find the jointly (locally) optimal solution for (41), the UE transmit
powers need to be optimized for different values of RRH dithering levels. To
avoid the complexity of searching for all the possible RRH dithering levels,
we choose to employ the same heuristic technique as discussed in Section IV to
obtain coarse initial RRH dithering values of $\\{\sigma_{b}\\}$ from a single
variable $\theta$ as in (31). Considering the limitation on the UE transmit
powers, the SINDR is a unimodal function with respect to $\theta$ or RRH
dithering levels. Therefore, we can employ the line search method similar to
the min-power design (as in Algorithm 2) to find the suitable value of theta
$\theta$. However, in this design, the range of $\theta$ values is updated
based on maximizing the minimum SINDR across all the UEs in each iteration, as
detailed in Algorithm 4.

As in Section IV, after the coarse optimization of the UE transmit powers and
RRH dithering levels. We can further fine-tune them via alternating
optimization. For fixed values of $\\{\sigma_{b}\\}$, the UE transmit powers
can be optimized using the BCD method (in Algorithm 3). For a fixed UE
transmit power, we can further adjust the RRH dithering levels
$\\{\sigma_{b}\\}$ using the gradient method to improve the value of $\gamma$.
To do this, for a fixed $\\{\rho_{k}\\}$, we construct the Lagrangian of (41)
for a given receiver as

$\displaystyle\mathcal{L}^{\textnormal{\tiny{RX}}}_{\eqref{eq:p5}}\big{(}\\{\sigma_{b},\gamma,\eta_{k}^{\textnormal{\tiny{RX}}},\upsilon_{b}\\}\big{)}\\!$
$\displaystyle\triangleq\\!\gamma\\!-\\!\sum_{k\in\mathcal{K}}\\!\eta_{k}^{\textnormal{\tiny{RX}}}\chi_{k}^{\textnormal{\tiny{RX}}}\\!-\\!\sum_{b\in\mathcal{B}}\\!\upsilon_{b}(\sigma_{b}\\!-\\!\sigma_{\min}),$
(42)

where $\chi_{k}^{\textnormal{\tiny{RX}}}$ represents the SINDR constraint in
(41), obtained by replacing $\gamma_{k}^{\textnormal{\tiny{Tar}}}$ with
$\gamma$ in $\eqref{eq:sindr_con_bmrc}$ and $\eqref{eq:sindr_con_bmmse}$
corresponding to the BMRC and BMMSE receivers, respectively.
$\eta_{k}^{\textnormal{\tiny{RX}}}\geq 0$ are the Lagrangian dual variable
associated with the SINDR constraint in (41). Lastly, $\upsilon_{b}\geq 0$ is
the Lagrangian dual variable corresponding to the minimum dithering (noise)
level in (41). Subsequently, for a given receiver, the dithering level at RRH
$b$ is iteratively updated in the direction of the negative gradient of the
Lagrangian as follows:

$\sigma_{b}^{(i)}=\max\Big{(}\sigma_{\min},\sigma_{b}^{(i-1)}-\zeta_{\textnormal{\tiny{$\sigma$}}}\frac{\partial}{\partial\sigma_{b}}\mathcal{L}_{\eqref{eq:p5}}^{\textnormal{\tiny{RX}}}\big{(}\\{\sigma_{b},\gamma,\eta_{k}^{\textnormal{\tiny{RX}}},\upsilon_{b}\\}\big{)}\Big{)}.$
(43)

The gradient of the Lagrangian can be obtained by replacing
$\gamma_{k}^{\textnormal{\tiny{Tar}}}$ with $\gamma,\leavevmode\nobreak\
\forall k\in\mathcal{K}$, in (IV-B) and (35) with the BMRC and BMMSE
receivers, respectively. For the initial iteration, the values of $\sigma_{b}$
are obtained from (31). The dual variable $\eta_{k}^{\textnormal{\tiny{RX}}}$
is then updated as in (27), replacing $\gamma_{k}^{\textnormal{\tiny{Tar}}}$
with $\gamma,\leavevmode\nobreak\ \forall k\in\mathcal{K}$, and the dual
variable $\upsilon_{b}$ is updated as in (36). To simplify the computation of
$\gamma$, let us define
$a_{k}^{\textnormal{\tiny{BMRC}}}\triangleq\rho_{k}\left|\hat{\mathbf{h}}_{k}^{\mathrm{H}}\hat{\mathbf{A}}\hat{\mathbf{A}}\hat{\mathbf{h}}_{k}\right|^{2}$,
$b_{k}^{\textnormal{\tiny{BMRC}}}\triangleq\hat{\mathbf{h}}_{k}^{\mathrm{H}}\hat{\mathbf{A}}\mathbf{C}{\hat{\mathbf{r}}}\hat{\mathbf{A}}\hat{\mathbf{h}}_{k}$,
$a_{k}^{\textnormal{\tiny{BMMSE}}}\triangleq\rho_{k}\hat{\mathbf{h}}_{k}^{\mathrm{H}}\hat{\mathbf{A}}\mathbf{C}{\hat{\mathbf{r}}}^{-1}\hat{\mathbf{A}}\hat{\mathbf{h}}_{k}$,
$b_{k}^{\textnormal{\tiny{BMRC}}}\triangleq 1$. With these definitions, for a
given receiver, the complementary slackness condition of the SINDR constraint
in (41), for all the UEs is written as [40]

$\displaystyle\sum_{k\in\mathcal{K}}\eta_{k}^{\textnormal{\tiny{RX}}}\big{(}(\gamma+1)a_{k}^{\textnormal{\tiny{RX}}}-\gamma
b_{k}^{\textnormal{\tiny{RX}}}\big{)}=0.$ (44)

As a result, the value of $\gamma$ can be updated as

$\displaystyle\gamma=\frac{\sum_{k\in\mathcal{K}}\eta_{k}^{\textnormal{\tiny{RX}}}a_{k}^{\textnormal{\tiny{RX}}}}{\sum_{k\in\mathcal{K}}\eta_{k}^{\textnormal{\tiny{RX}}}(b_{k}^{\textnormal{\tiny{RX}}}-a_{k}^{\textnormal{\tiny{RX}}})}.$
(45)

However, at the end of the iterative process, we have
$\gamma\triangleq\min_{k\in\mathcal{K}}\hat{\mathrm{\gamma}}_{k}$. For the
fine-tuning a single gradient update of the RRH dithering levels is used for
every update of the UE transmit powers. Finally, the fine-tuning of the UE
transmit powers and RRH dithering levels terminates if there is no significant
improvement in the minimum SINDR. The full procedure to optimize the UE
transmit powers and the RRH dithering levels with the max-min-SINDR objective
is summarized in Algorithm 4.

Data: $\\{\hat{\mathbf{h}}_{k}\\}$,
$\\{\mathbf{C}_{\tilde{\mathbf{h}}_{k}}\\}$, $\epsilon_{\sigma}>0$,
$\epsilon_{\gamma}>0$.
Initialization: Iteration $i=0$,
$\theta^{\textnormal{\tiny{low}}}=1/\max_{b\in\mathcal{B}}\sqrt{\delta_{b}^{\max}}$,
$\theta^{\textnormal{\tiny{high}}}=1/\min_{b\in\mathcal{B}}\sqrt{\delta_{b}^{\max}}$.
%Coarse RRH dithering level optimization.

while  _$(1)$_ do

%Evaluate the max-min-SINDR at two points. for  _$l=1:2$_ do

  * •

$\theta^{(l)}\triangleq{\big{(}(3-l)\theta^{\textnormal{\tiny{low}}}+l\theta^{\textnormal{\tiny{high}}}\big{)}}/{3}$.

  * •

Compute $\\{\sigma_{b}\\}$ using (31).

  * •

Compute $\gamma^{(l)}$ using Algorithm 3.

end for

if  _$\gamma^{(1)} >\gamma^{(2)}$_ then

  * •

$\theta^{\textnormal{\tiny{low}}}={\big{(}2\theta^{\textnormal{\tiny{low}}}+\theta^{\textnormal{\tiny{high}}}\big{)}}/{3}$.

else

  * •

$\theta^{\textnormal{\tiny{high}}}={\big{(}\theta^{\textnormal{\tiny{low}}}+2\theta^{\textnormal{\tiny{high}}}\big{)}}/{3}.$

end if

if
_$(\theta^{\textnormal{\tiny{low}}}-\theta^{\textnormal{\tiny{high}}})\leq\epsilon_{\sigma}$_
then

break

end if

end while

% Fine-tuning the RRH dithering and UE transmit powers. while  _
$(\gamma^{(i)}-\gamma^{(i-1)})\leq\epsilon_{\sigma}$_ do

  * •

$i=i+1$

  * •

Compute $\\{\sigma_{b}\\}$ using (43).

  * •

Compute $\\{\rho_{k}^{(i)}\\}$ using Algorithm 3.

end while

Algorithm 4 Optimization of the UE transmit powers and RRH dithering levels
with the max-min-SINDR objective

##  VI Numerical Results

To evaluate the performance of the distributed massive MIMO system with 1-bit
ADCs, we consider $B=4$ RRHs, each equipped with $M=256$ antennas, positioned
at the corners of a square grid with distance to the closest neighboring RRH
of $100$ m, and the height of each RRH is $5$ m. All the RRHs are connected to
the central processing unit (CPU). Unless otherwise stated, we consider $K=4$
single-antenna UEs placed at the corners of a square cluster of size $b\times
b$ m2, as depicted in Fig. 4. The minimum distance between the closest RRH and
the UE cluster is denoted by $d_{\textrm{ref}}$. The AWGN power at the RRH is
set to $\sigma^{2}_{\min}=-95$ dBm and additional dithering (i.i.d noise) can
be added if necessary to enhance the performance. Details of the channel model
are provided in Section III. We employ orthogonal time resources for the
channel estimation of each UE with a pilot length of $128$. Additionally, this
estimation utilizes the maximum UE transmit powers and adjusts the RRH
dithering levels based on the pathloss or the received signal power. Finally,
the UE transmit power for the min-power design is optimized using the Gradient
and BCD methods, while for the max-min-SINDR, it is optimized using the BCD
method, as discussed in Sections IV and V, respectively.

CPU$b$$b$$d_{\textrm{ref}}$RRH Figure 4: Distributed massive MIMO system
considered in the numerical analysis.
$5$$10$$15$$20$$25$$30$$35$$-15$$-5$$5$$15$$25$Target $\mathrm{SINDR}$Min-
power [dBm]BMRC (gradient)BMRC (BCD)BMMSE (gradient)BMMSE (BCD) Figure 5: Min-
power versus target SINDR with dithering.

Fig. 5 illustrates the relationship between the min-power and the target SINDR
with both the BMRC and BMMSE receivers with $d_{\text{ref}}=0$ and $b=10$ m.
To achieve higher SINDR targets through joint reception from multiple RRHs,
dithering is introduced at all the RRHs except the farthest one. The Gradient
and BCD methods exhibit comparable performance. The computational complexity
of the Gradient method in each iteration with BMRC and BMMSE receivers,
determined by (IV-A) and (IV-A), respectively, is higher compared to the BCD
method. Convergence is also influenced by the step sizes, denoted by
$\zeta_{\rho}$ (in (27)) and $\nu$ (in (27)). Furthermore, in both the
Gradient and BCD methods, the computational complexity with the BMMSE receiver
exceeds that of the BMRC receiver, as the former involves matrix inversion
operations. However, to achieve the same target SINDR, the BMRC receiver
requires slightly more transmit power compared to the BMMSE receiver at high
SINDR, as the latter effectively reduces the impact of the QD and interference
between the UEs. This reduction in QD and interference with the BMMSE receiver
also facilitates achieving a higher target SINDR, with or without adding
dithering at the RRHs, for a given min-power. Dithering at the RRH is not
required to achieve target SINDRs of $15$ or less for BMRC receivers and $20$
or less for BMMSE receivers. However, the joint reception from multiple RRHs
and dithering at the closest RRHs are necessary to achieve the target SINDRs
of $20$ and $25$ or more with BMRC and BMMSE receivers, respectively, while
minimizing the sum of UE transmit powers. Consequently, a sharp increase in
the min-power is observed, which is required to combat the pathloss to the
faraway RRHs, resulting in a significant power difference between the BMRC and
BMMSE receivers at a target SINDR of $20$.

$-5$$0$$5$$10$$15$$20$$25$$0$$10$$20$$30$$40$$50$$\rho_{\textnormal{\tiny{UE}}}$
[dBm]Max-min-SINDRBMRC (no dithering)BMRC (dithering)BMMSE (no dithering)BMMSE
(dithering)Heuristic (no dithering)Heuristic (dithering) Figure 6: Max-min-
SINDR versus maximum UE transmit power.
$-5$$0$$5$$10$$15$$20$$25$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$\rho_{\textnormal{\tiny{UE}}}$
[dBm]SERBMRC (no dithering)BMRC (dithering)BMMSE (no dithering)BMMSE
(dithering)Heuristic (no dithering)Heuristic (dithering) Figure 7: SER versus
maximum UE transmit power with 16-QAM data symbols.

Fig. 6 illustrates the achievable max-min-SINDR for a given maximum UE
transmit power $\rho_{\textnormal{\tiny{UE}}}$ with $d_{\text{ref}}=0$ and
$b=10$ m. As expected, the BMMSE receiver outperforms the BMRC receiver in
both scenarios, with and without dithering. Without dithering, the maximum
achievable SINDR is around $17$ for the BMRC receiver and $20$ for the BMMSE
receiver, respectively, while with joint reception from multiple RRHs and
dithering, the maximum achievable SINDR increases to $40$ and $42$ for BMRC
and BMMSE receivers, respectively. The impact of the joint reception from
multiple RRHs with dithering is notable when the UE transmit power exceeds $5$
dBm, as it implies that the SINDR is in the QD-dominated region with respect
to the closest RRH and the UE has enough power to reach the other RRHs. To
reduce the computational complexity of UE transmit power and dithering
optimizations, we propose using the BMRC receiver for solving the max-min-
SINDR objective and then switching to the BMMSE receiver for the data
reception. This method is referred to as Heuristic. While the performance of
Heuristic is inferior to that of the BMMSE receiver, it is better than the
performance of the case where BMRC receiver is used also for data detection,
with and without dithering. Considering the same setup as in Fig. 6, we plot
the maximum SER of a UE in Fig. 7 with 16-QAM data symbols. The SER exhibits a
similar behavior to the max-min-SINDR. The lowest SER without dithering is
about $10^{-2}$, whereas with joint reception and dithering, it is around
$2\times 10^{-4}$ at the UE transmit power of $25$ dBm. Additionally, The
performance of the Heuristic method surpasses that of the BMRC receiver.

$0$$20$$40$$60$$80$$100$$120$$140$$-10$$-5$$0$$5$$10$$15$$20$$25$$d_{\textrm{ref}}+\frac{b}{\sqrt{2}}$
[m]Min-power [dBm]BMRC (target $\mathrm{SINDR}=15$)BMRC (target
$\mathrm{SINDR}=35$)BMMSE (target $\mathrm{SINDR}=15$)BMMSE (target
$\mathrm{SINDR}=35$) Figure 8: Min-power versus distance to the reference RRH.

Fig. 8 illustrates the minimum power required to achieve a given target SINDR
as a function of the distance between the center of the UE cluster and the
bottom-left RRH (also referred to as the reference RRH), while the center of
the UE cluster moves along the dotted lines from the bottom-left RRH to the
top-right RRH, as shown in Fig. 4 with $b=10$ m. If the target SINDR is
achieved by a single RRH, such as a SINDR of $15$, the required transmit power
increases as the center of the UE cluster moves away from the RRH, as expected
in the unquantized system. Therefore, the maximum min-power is required when
the center of the UE cluster is located in the middle of the bottom-left and
the top-right RRHs. However, if the target SINDR requires joint reception from
all the RRHs with dithering, as in the case of a SINDR of $35$, the min-power
required to achieve the target SINDR is maximum when the center of the UE
cluster is closer to one of the RRHs and minimum when it is equidistant from
all the RRHs. This is because when the UEs are close to an RRH, they require
more transmit power to reach the far-away RRHs for joint reception with
dithering. This result differs from the unquantized system. As expected, the
min-power required at any distance with the BMMSE receiver is less than with
the BMRC receiver in both scenarios. However, at high SINDR, the performance
of the BMMSE receiver is much better than the BMRC receiver compared to the
low SINDR.

$0$$20$$40$$60$$80$$100$$120$$140$$15$$20$$25$$30$$35$$40$$45$$50$$d_{\textrm{ref}}+\frac{b}{\sqrt{2}}$
[m]Max-min-SINDRBMRC (no dithering)BMRC (dithering)BMMSE (no dithering)BMMSE
(dithering) Figure 9: Max-min-SINDR versus distance to the reference RRH.

In Fig. 9, we characterize the max-min-SINDR as a function of the distance to
the center of the UE cluster to the reference RRH with $b=10$ m, considering a
maximum UE transmit power of $25$ dBm. When dithering is not added, the SINDR
is primarily determined by the closest RRH to the UE cluster, due to the SINDR
transitioning into the QD-dominated region with respect to the closest RRH
with increasing UE transmit power. However, as the UE cluster moves away from
the RRH, contributions from other RRHs gradually augment the achievable SINDR.
Ultimately, when the UE cluster is equidistant from all the RRHs, each RRH
contributes to the achievable SINDR in the same way, maximizing the achievable
SINDR without dithering. Incorporating dithering significantly enhances UE
performance, especially in proximity to any RRH, ensuring sustained maximum
SINDR. However, adequate UE transmit power is essential to reach the farthest
serving RRH. Consequently, the level of dithering required varies; more
dithering is necessary when the UE cluster is close to any RRH, while no
dithering is required when the UE cluster is centrally located. As expected,
the max-min-SINDR with the BMMSE receiver exceeds that of the BMRC receiver in
all the cases. However, in the case without dithering, the achievable max-min-
SINDR is quickly improved with the BMMSE receiver compared to the BMRC
receiver as the UE cluster moves to the center of the RRHs, since the BMMSE
receiver better alleviates the QD compared to the BMRC receiver.

$5$$10$$15$$20$$25$$30$$0$$10$$20$$30$$40$$50$$b$ [m]Max-min-SINDRBMRC (no
dithering)BMRC (dithering)BMMSE (no dithering)BMMSE (dithering) Figure 10:
Max-min-SINDR versus area of the served UEs.

In Fig. 10, the max-min-SINDR is plotted against the UE cluster size $b$, with
$d_{\textrm{ref}}=0$ and a maximum UE transmit power of $25$ dBm. As the UE
cluster size increases, the effective received signal at the RRH is mostly
dominated by a single UE. This results in: i) a reduced SINDR gain from the
joint reception across the RRHs with dithering, and ii) an increase in the
SINDR without dithering, due to reduced interference as shown in Fig. 10.
However, even with a cluster size of $30$ m, dithering provides considerable
performance gains with both the BMRC and BMMSE receivers.

$2$$4$$6$$8$$10$$0$$10$$20$$30$$40$$50$$60$$K$Max-min-SINDRBMRC (no
dithering)BMRC (dithering)BMMSE (no dithering)BMMSE (dithering) Figure 11:
Max-min-SINDR versus number of UEs.

In Fig. 11, the max-min-SINDR is plotted against the number of UEs for
$d_{\textrm{ref}}=0$ with a maximum UE transmit power of $25$ dBm. The UEs are
placed on the perimeter of a circle with a radius of $10$ m, and all the UEs
are equally spaced. With dithering, as the number of UEs increases, the max-
min-SINDR decreases, mirroring the behavior observed in the unquantized
systems. Furthermore, the BMMSE receiver demonstrates superior interference
mitigation compared to the BMRC receiver, resulting in better performance
gains as the number of UEs increases. In the case without dithering, the max-
min-SINDR is not significantly impacted by increasing the number of UEs. This
is because while the interference increases, it is observed that the
correlated QD decreases with increasing the UEs.

$2^{6}$$2^{7}$$2^{8}$$2^{9}$$2^{10}$$0$$20$$40$$60$$80$$100$$120$$M$Max-min-
SINDRBMRC (no dithering)BMRC (dithering)BMMSE (no dithering)BMMSE (dithering)
Figure 12: Max-min-SINDR versus number of RRH antennas.

In Fig. 12, the max-min-SINDR is plotted as a function of the number of
antennas at each RRH for $b=10$ and a UE transmit power of $25$ dBm. The max-
min-SINDR increases with the increasing number of antennas at each RRH, both
with and without dithering, similar to the behavior observed in the
unquantized systems. It is observed that increasing the number of antennas at
each RRH results in a decrease in the UE transmit power required to achieve
the optimal max-min-SINDR. Additionally, with the increase in the number of
antennas at each RRH, the gain from dithering is slightly better than in cases
without dithering, since the QD increases with the number of antennas. Even
with a high number of antennas at each RRH, there is still a considerable
performance gap between the BMMSE receiver and the BMRC receiver due to the
presence of the QD.

$\displaystyle\frac{\partial}{\partial\rho_{k}}\mathrm{Re}[\mathbf{C}_{\hat{\mathbf{r}}}(i,j)]$
$\displaystyle=\frac{\hat{\mathbf{A}}(i,i)\hat{\mathbf{A}}(j,j)}{\sqrt{1-|q_{i,j}|^{2}}}\bigg{(}\mathrm{Re}\big{[}\hat{\mathbf{h}}_{k}(i)\hat{\mathbf{h}}_{k}(j)+\mathbf{C}_{\tilde{\mathbf{h}}_{k}}(i,j)\big{]}-\frac{\pi}{4}\mathrm{Re}\big{[}\hat{\mathbf{C}}_{\mathbf{y}}(i,j)\big{]}\sum_{l\in\\{i,j\\}}\big{(}\big{|}\hat{\mathbf{h}}_{k}(l)\big{|}^{2}+\mathbf{C}_{\tilde{\mathbf{h}}_{k}}(l,l)\big{)}\hat{\mathbf{A}}^{2}(l,l)\bigg{)}$
(43)

$\displaystyle\frac{\partial}{\partial\sigma_{b}}\mathrm{Re}[\mathbf{C}_{\hat{\mathbf{r}}}(i,j)]=\frac{-\pi\sigma_{b}\hat{\mathbf{A}}(i,i)\hat{\mathbf{A}}(j,j)}{2\sqrt{1-|q_{i,j}|^{2}}}\Big{(}\mathrm{Re}\big{[}\hat{\mathbf{C}}_{\mathbf{y}}(i,j)\big{]}\big{(}\hat{\mathbf{A}}^{2}(i,i)\mathbf{E}_{b}(i,i)+\hat{\mathbf{A}}^{2}(j,j)\mathbf{E}_{b}(j,j)\big{)}\Big{)}$
(45)

##  VII Conclusions

We investigated the uplink SNDR of a UE in both single-RRH and multi-RRH
scenarios with 1-bit ADCs. In a single-RRH case, the SNDR exhibits unimodal
behavior with respect to the UE transmit power, whereas it becomes multimodal
in a multi-RRH scenario. However, by adjusting the dithering at the RRHs, the
SNDR can be optimized and made unimodal. In a multi-UE and multi-RRH scenario,
considering the non-monotonic behavior of SINDR, we jointly optimized the UE
transmit powers and RRH dithering levels for the min-power and max-min-SINDR
objectives, employing both the BMRC and BMMSE receivers. For solving the UE
transmit powers, we employed the gradient and BCD methods, while the line
search method was utilized to determine the RRH dithering levels alongside the
gradient method. Numerical results demonstrate that updating the UE transmit
powers considering the RRH dithering enhances the system performance by
enabling joint reception from multiple RRHs across several scenarios.
Moreover, if achieving the desired target SNDR involves the reception from
multiple RRHs using a large number of antennas and dithering, in such cases,
the UE transmit power heavily depends on the distance to the farthest serving
RRH.

##  Appendix A Partial Derivative of $\hat{\mathbf{A}}$ and
$\mathbf{C}_{\hat{\mathbf{r}}}$ with Respect to $\rho_{k}$

The gradient of $\hat{\mathbf{A}}$ with respect to $\rho_{k}$ is given by

$\displaystyle\frac{\partial}{\partial\rho_{k}}\hat{\mathbf{A}}=-\sqrt{\frac{1}{2\pi}}\mathrm{Diag}(\hat{\mathbf{C}}_{\mathbf{y}})^{-\frac{3}{2}}\mathrm{Diag}(\hat{\mathbf{h}}_{k}\hat{\mathbf{h}}_{k}^{\mathrm{H}}+\mathbf{C}_{\tilde{\mathbf{h}}_{k}}).$
(42)

The gradient of the $(i,j)$th element ($i\neq j$) of the real part of
$\mathbf{C}_{\hat{\mathbf{r}}}$ is given in (43) at the top of the next page,
with $q_{i,j}\triangleq\frac{\pi}{2}\mathbf{C}_{\hat{\mathbf{r}}}(i,j)$. The
gradient of the imaginary part of $\mathbf{C}_{\hat{\mathbf{r}}}$ can be
obtained by replacing $\mathrm{Re}[\cdot]$ with $\mathrm{Im}[\cdot]$ in (43),
and
$\frac{\partial}{\partial\rho_{k}}\mathbf{C}_{\hat{\mathbf{r}}}(i,j)=0,\leavevmode\nobreak\
\forall i=j$.

##  Appendix B Partial Derivative of $\hat{\mathbf{A}}$ and
$\mathbf{C}_{\hat{\mathbf{r}}}$ with Respect to $\sigma_{b}$

The gradient of $\hat{\mathbf{A}}$ with respect to $\sigma_{b}$ is given by

$\displaystyle\frac{\partial}{\partial\sigma_{b}}\hat{\mathbf{A}}=-2\sigma_{b}\sqrt{\frac{1}{2\pi}}\mathrm{Diag}(\hat{\mathbf{C}}_{\mathbf{y}})^{-\frac{3}{2}}\mathbf{E}_{b},$
(44)

where
$\mathbf{E}_{b}=\mathrm{Diag}\big{(}[c_{1},\ldots,c_{B}]\big{)}\otimes\mathbf{I}_{M}\in\mbox{$\mathbb{C}$}^{BM\times
BM}$ with $c_{\bar{b}}=1$ if $\bar{b}=b$, otherwise $c_{\bar{b}}=0$. The
gradient of the $(i,j)$th element ($i\neq j$) of the real part of
$\mathbf{C}_{\hat{\mathbf{r}}}$ is given in (45) at the top of the next page.
The gradient of the imaginary part of $\mathbf{C}_{\hat{\mathbf{r}}}$ can be
obtained by replacing $\mathrm{Re}[\cdot]$ with $\mathrm{Im}[\cdot]$ in (45).
Finally, we have
$\frac{\partial}{\partial\sigma_{b}}\mathbf{C}_{\hat{\mathbf{r}}}(i,j)=0,\leavevmode\nobreak\
\forall i=j$.

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