# Designing and manufacturing of interference notch filter with a single
reflection band

Mohammadreza Salehpoor  Department of Physics, University of Isfahan, Hezar
Jerib, Isfahan 81746-73441, Iran  Hossein Vahid  Department of Physics,
University of Isfahan, Hezar Jerib, Isfahan 81746-73441, Iran  Ali Heidary
Fard  Department of Physics, University of Isfahan, Hezar Jerib, Isfahan
81746-73441, Iran  Hamidreza Fallah  Department of Physics, University of
Isfahan, Hezar Jerib, Isfahan 81746-73441, Iran  Quantum Optics Research
Group, Department of Physics, University of Isfahan, Hezar Jerib, Isfahan
81746-73441, Iran  Morteza Hajimahmoodzadeh  Department of Physics, University
of Isfahan, Hezar Jerib, Isfahan 81746-73441, Iran

###### Abstract

The main idea of this work is to design a notch filter structure with a narrow
notch width and maximum reflection while reducing fabrication challenges. In
addition, using anti-reflection layers in the outermost part of the designed
structure, the pass-band ripples are reduced. In designed systems, various
quarter-wave Coefficients were used, which have less challenge in the
manufacturing part. Furthermore, the stability of the deposition conditions
and the density of the layers affect their quality and consequently the result
of environmental tests. Therefore, the sputtering method with RF and DC
sources was used to construct the designed structure.

Keywords: Notch filters, reactive magnetron sputtering, discrete layer
structure, thin films, quarter-wave.

##  I Introduction

During the last decades, notch filters (NFs) have been used in different
signal processing Minguillon et al. (2017) applications, including speech and
image processing, data transmission, seismology Lee (2013), biomedical
engineering Juan et al. (2013) and other applications like Raman and
fluorescence spectroscopy, laser systems, laser protective coatings Janicki et
al. (2005); Sprague et al. (2004), Protection of optical equipment against
high beams Verly (2007) in order to control the wavelength of light and
eliminate some frequencies Nikolić et al. (2018); Nehorai (1985); Jeong and
Mang (2018). NFs are optical instruments based on the interference phenomenon,
which rejects frequencies inside a narrow band and allows other frequencies to
pass without change Scherer et al. (2008); Zhang et al. (2012); Thelen (1971,
1993); Khazraj et al. (2017).

Single-band and multi-band are of two main kinds of NFs that have been mostly
used in industrial engineering. The main methods that have been used to
produce such filters are rugate Lyngnes and Kraus (2014), discrete layer (HL)
designs Janicki et al. (2005); Rahmlow Jr et al. (2006), and holographic
technologies Sprague et al. (2004). The rugate designs with a small difference
between refractive index of layers, which have less ripple in the transmission
regions Southwell (1989), are well known in the literature. The latter design
requires the deposition of layers by a refractive index gradient or so-called
flip-flop structures, which is very complex and difficult to produce and
economically not useful.

In the discrete layer (HL) method, materials with different refractive indices
are used to manufacture NFs. H and L indicate the quarter-wave layer with the
higher and lower refractive index, respectively. The optical thickness of
these layers is a quarter of the central wavelength Janicki et al. (2005). The
challenging parts of this design are the deposition of the layers in a
repeating order with high accuracy, reducing transmission band ripples, making
high layer quality. In addition, low level of particle defects, reasonable
stress in the layers, and less manufacturing cost per sample are among
important factors. In this method, the layers are placed on top of each other
periodically that leads to ease in the design and manufacturing, but we should
note that the transmission band ripples increase due to the sudden change of
the refractive index between the layers with high and low refractive index. on
the other hand, with fewer layers can be reached to a certain level of
reflection in the center of the reflection band Schallenberg et al. (2010);
Scherer et al. (2008).

The deposition method and selecting the appropriate materials are very
effective in the quality of the filter made. The reflectance bandwidth
decreases by reducing the difference between the refractive index of the two
materials in the structure. In this study, we use appropriate materials in the
HL structure to provide a functional design with high reproducibility for the
NFs at a center wavelength of $532$ nm. Then, using anti-reflect layers, we
reduce the transmission band ripples. We use the sputtering method with two RF
and DC sources to manufacture the designed structure. Finally, we present the
result of quality assessments conducted on the fabricated samples.

##  II Design methodology

###  II.1 Initial design

The first step to create an optical instrument is to choose a suitable design
method. An important issue that arises before selecting a design method is how
to perform the final design practically. The method of quarter-wave stack
structure, in the deposition stage, is more straightforward than other
methods, and fewer layers are used in the structure, which make it more
economical and less challenging. Furthermore, fewer layers also reduce the
stress between layers and lead to better resistance to damage from
environmental conditions.

In this method, the layers are placed alternately to form structures such as
(nHmL)s or (nLmH)s. In this notation, m and n determine the number of layers
(odd integer) of each quarter-wave, and s expresses the number of pairs of
layers used in the design. The reflection bandwidth or notch width (NW), as
one of the characteristics of NF, depends on the difference between the
refractive indices of the two materials with high and low refractive indices
or the ratio between them. Therefore, the selection of materials should be
such that the difference between the refractive indices of two materials
optimize NW and the maximum reflection to desired values. The NW is given by
Macleod (2017)

${\rm
NW}=\frac{4}{\pi}\arcsin[\frac{(\rho^{2}+2\rho\cos\gamma+1)^{\frac{1}{2}}}{\rho+1}],$
(1)

where

$\gamma=(\frac{s-1}{s})\pi\quad\mathrm{and}\quad\rho=\frac{n_{\rm H}}{n_{\rm
L}}.$ (2)

Equation (1) then, in a simpler form, becomes Macleod (2017)

${\rm NW}=2\Delta g=\frac{4}{\pi}\arcsin(\frac{n_{\rm H}-n_{\rm L}}{n_{\rm
H}+n_{\rm L}}).$ (3)

The maximum reflection depends on the number of repetitions of the main
period, s, in the structure Macleod (2017); Diehl and George (2003) and can be
calculated by Macleod (2017)

$R_{\rm max}=\frac{n_{\rm sub}-\rho^{2s}n_{\rm H}^{2}}{n_{\rm
sub}+\rho^{2s}n_{\rm H}^{2}},$ (4)

where $n_{\rm sub}$ is the the refractive index of the substrate. In this
study, we used Y2O3 and Al2O3 as the materials with high refractive index (H)
and SiO2 as the material with low refractive index (L). We used BK7 with the
refractive index of $1.52$ as the substrate, and we considered air with that
of $1$ as the incidence medium in the design. The central wavelength of design
is $\lambda=532$ nm, which is located in the center of the reflection band.
The physical thickness (PT) of the layers is such that the layers have optical
thicknesses equal to odd integer coefficients of one-quarter of the central
wavelength. Details of the designed structures with two pairs of materials,
Y2O3-SiO2 and Al2O3-SiO2, are given in Table 1.

Table 1: Details of designed structures with SiO2-Y2O3 and SiO2-Al2O3 Design number | 1 | 2
---|---|---
$\lambda$ (nm) | $532$ | $532$
Materials | SiO2-Al2O3 | SiO2-Y2O3
Number of layers | $44$ | $26$
$R_{\rm max}$ (%) | $98.84$ | $98.53$
NW (nm) | $17$ | $26$
Total PT ($\mu$m) | $14$ | $8$

Although using combination of $3$ and $5$ coefficients instead of single
quarter-wave layers increases the thickness of structures, but significantly
reduces the transmission band ripples and the number of total layers required
in the design. the transmission diagrams of both structures are given in
Figure 1.

Figure 1: Transmittance diagrams of designed structures with SiO2-Y2O3 and
SiO2-Al2O3. The information about the structures is listed in Table 1.

From Fig. 1 and the information of Table 1, it can be concluded that reducing
the difference between the refractive indices of the two materials reduces the
NW. Meanwhile, we need more layers to achieve the maximum reflection.

###  II.2 Final design optimization

We use anti-reflect layers, which are made of the same materials used in the
main structure, in the outermost layers of the design to reduce ripples more.
Regarding this method, the strong scattering of the equivalent refractive
index of the multilayer is compensated by the scattering of the anti-
reflection structure and the transmission band ripples are improved Shankar
and Kumar (2014). Table $2$ shows the results of the initial and optimized
designs.

Table 2: Comparative performance of NF with various stack. The table presents the stack formula, maximum reflection $R_{\rm max}$, average ripple reflectivity ARR, and the notch width NW. Design number | Stack formula |  $R_{\rm max}$ (%) | ARR (%) | NW (nm)
---|---|---|---|---
1 | [(5H5L)4(3H3L)15(5H5L)3] | $98.05$ | $6.11$ | $17$
1 | [(5H5L)4(3H3L)15(5H5L)3]$2$HLHL | $98.84$ | $3.02$ | $17$
2 | [(5H5L)2(3H3L)9(5H5L)2] | $97.04$ | $9.57$ | $26$
2 | [(5H5L)2(3H3L)9(5H5L)2]$2$HL | $98.60$ | $3.77$ | $26$

In addition, the optimized transmittance diagrams for both designed structures
compared to their initial designs are shown in Figures 2 and 3. Adding anti-
reflection layers increases the maximum reflection at the central wavelength.

Figure 2: Initial and optimized designs of SiO2-Al2O3. The information about
structures is listed in Table 2.  Figure 3: Initial and optimized designs of
SiO2-Y2O3. The information about structures is listed in Table 2.

In the next section, we discuss the feasibility of manufacturing the structure
containing Al2O3. This structure has a smaller NW in regards to the smaller
difference between its refractive index and SiO2. Moreover, it is more
resistant and cheaper than Y2O3.

##  III Experiment

###  III.1 Manufacturing method

The layers were deposited employing Reactive magnetron sputtering method,
which is a suitable method for dielectric materials deposition, and it can
produce dense layers with good resistance to environmental stresses.
Increasing the deposition rate by increasing the applied power, reducing
sputtering of the substrate and vacuum chamber, which prevents contamination,
are the advantages of this method. Reducing the substrate’s heating during
deposition enables the sputtering process to be used for different substrates.
Also, this method needs less gas pressure during the deposition and is
economically affordable due to its material consumption compared to other
methods.

###  III.2 SiO2 and Al2O3 single layers

In the design software, we design a single layer of SiO2 and a single layer of
Al2O3 at the center wavelength of 532 nm. In this step, we use SF6 and BK7 as
substrates for SiO2 and Al2O3 single layer, respectively. The deposition was
done using DC source for SiO2 and RF source for Al2O3 with silicon and
aluminum targets with 99.99% purity and after complete cleaning of the
substrate. Optimal conditions for deposition of SiO2 and Al2O3 were obtained
in 11 sccm and 3 sccm oxygen flow, respectively. Deposition rates of SiO2 and
Al2O3 are 6 Å/s and $1.5$ Å/s respectively. The deposition was performed for
both single layers at 100°C, chamber pressure of 3 mTorr during deposition,
the argon gas flow of 25 sccm, and the start pressure of $8\times 10^{-6}$
Torr. Figure 4 shows the spectral transmission of the fabricated single
layers.

Figure 4: The spectral transmission of the SiO2 and Al2O3 single layers.

###  III.3 (3H3L)2 stack

Optimal conditions for deposition of stacks are the same as ones used for
single layers. However, before deposition the final design structure and in
order to achieve better results, it is necessary to deposit a simpler
structure with fewer layers. Figure 5 shows the transmittance spectra of the
deposited and the designed structures are in good agreement. It indicates that
the obtained deposition conditions in the previous sections for single layers
are approximately the optimal ones.

Figure 5: Measured and designed transmittance spectra for (3H3L)2 stack.

###  III.4 NF deposition

According to design 1, we use silica and alumina materials as H and L
refractive indices, respectively. Regarding the optimized structure, see
Figure 3, the final design at the central wavelength of $532$ nm is given by
the design formula number 1 (see Table 1).

In Sec. III.3, by performing several experiments, design and fabrication of
single layers of materials in the structure, optimizing deposition conditions,
and finally designing and deposition of 4–layer stacks were done.

The deposition conditions are similar to explained ones in Sec. III.2. The
control of deposition rate and the thickness of the layers is crucial.

In Sec. III.3, the optimization was performed for samples, which are located
on the rotating segment, and the necessary corrections were made. Therefore,
we expect the samples to show good compliance with the designed structure. In
the transmittance spectra, second side of the sample has a reflection, so
first in the design software, we applied the effect of the reflection of the
second side of the samples, and then we compare design and measured
transmittance spectra. Figure 6 shows the measured and designed spectra of the
central sample.

Figure 6: Measured and designed transmittance spectra for NF (SiO2-Al2O3).

As shown in Figure 6, the final deposited structure and the final designed
structure at the location of the central wavelength and the reflectance
bandwidth are well matched. By extracting the data obtained from the
transmission spectrum, the center of the reflection band is located at $530$
nm, which is a perfect match with the design. We can see the ripples only in
some areas of the transmission band. In the transmission band, there is a
mismatch at some point between the transmittance spectrum of the deposited
structure and the designed structure. Generally, the mismatches between the
transmittance spectra of designed and deposited structures in the coating
technology are due to the mismatches between the thickness of the same layers,
the refractive indices, and the packing density in the deposited and the
designed structures. Absorption in some coating areas, contamination, and
error of spectrometer and operator are other factors. In Figure 6, although
the spectra of the designed and the deposited structures are very well matched
in many wavelengths, the control and stability of the deposition process must
be very high to achieve spectra closer to the designed structure. Some of
these mismatches are inevitable because the final designed structure of NF
consists of 52 layers with different thicknesses. In the manufactured NF, the
NW was $17$ nm, the maximum reflectance at central wavelength was more than
$95$ %, and the average transmittance was approximately $90$ %.

##  IV Conclusions

HL method has fewer challenges in designing and manufacturing NFs. In this
method, the differences between the refractive indices of the materials used
in structures are the determining factor in the NW and the maximum reflection.
In addition, combined coefficients of 3 and 5 in the design of the NFs are
beneficial because these coefficients significantly reduce the transmittance
band ripples compared to the single quarter-wave structures. The ripples in
the transmittance spectra are essential challenges in the design and
fabrication of NFs. In this paper, by antireflection layers and a half-wave
layer used as the interface layer in the outermost layers of the final
structure, the transmission band ripples were reduced appropriately. Magnetron
sputtering method is a very new and efficient method with suitable economic
cost. This method produces high-quality layers. Success in environmental tests
depends directly on the manufacturing method. In this study, samples made by
the magnetron sputtering method were subjected to environmental tests, which
yielded positive results. It should be noted that the stability of the coating
rate and precise control of the inlet gases to the chamber during the coating
process has a significant impact on the formation of coatings with appropriate
stoichiometry. In addition, increasing the substrate temperature causes better
adhesion of the first layer to the surface of the substrate.

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# Uplink Power Control for Distributed Massive MIMO with 1-Bit ADCs

Bikshapathi Gouda, Italo Atzeni, and Antti Tölli  Centre for Wireless
Communications, University of Oulu, Finland
Emails: {bikshapathi.gouda, italo.atzeni<EMAIL_ADDRESS>This work is
supported by the Academy of Finland (336449 Profi6, 346208 6G Flagship, 348396
HIGH-6G, and 357504 EETCAMD) and by the European Commission (101095759 Hexa-X-
II).

###### Abstract

We consider the problem of uplink power control for distributed massive
multiple-input multiple-output systems where the base stations (BSs) are
equipped with 1-bit analog-to-digital converters (ADCs). The scenario with a
single user equipment (UE) is first considered to provide insights into the
signal-to-noise-and-distortion ratio (SNDR). With a single BS, the SNDR is a
unimodal function of the UE transmit power. With multiple BSs, the SNDR at the
output of the joint combiner can be made unimodal by adding properly tuned
dithering at each BS. As a result, the UE can be effectively served by
multiple BSs with 1-bit ADCs. Considering the signal-to-interference-plus-
noise-and-distortion ratio (SINDR) in the multi-UE scenario, we aim at
optimizing the UE transmit powers and the dithering at each BS based on the
min-power and max-min-SINDR criteria. To this end, we propose three algorithms
with different convergence and complexity properties. Numerical results show
that, if the desired SINDR can only be achieved via joint combining across
multiple BSs with properly tuned dithering, the optimal UE transmit power is
imposed by the distance to the farthest serving BS (unlike in the unquantized
case). In this context, dithering plays a crucial role in enhancing the SINDR,
especially for UEs with significant path loss disparity among the serving BSs.

††publicationid: pubid:  979-8-3503-1090-0/23/$31.00 © 2023 IEEE

##  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 [1]. As we move towards higher carrier frequencies in the
next-generation wireless systems, the number of antennas at the base station
(BS) is bound to increase significantly [2]. Fully digital massive MIMO arrays
with high-resolution analog-to-digital converters (ADCs), however, are
exceedingly complex and power hungry. Low-resolution and 1-bit ADCs can be
used to considerably decrease the power consumption without compromising the
performance [3, 4, 5, 6]. Moreover, adopting low-resolution and 1-bit ADCs in
a distributed/cell-free massive MIMO setting allows to reduce the backhaul
signaling overhead (among the BSs or between the BSs and the central
processing unit) associated with the network-wide beamforming design [7, 8].

The works [4, 5] investigated the uplink channel estimation and achievable
signal-to-interference-plus-noise-and-distortion ratio (SINDR) using 1-bit
ADCs in single-BS systems. In addition, the spectral and energy efficiency of
massive MIMO systems with low-resolution ADCs are studied as a function of the
UE transmit power in [9]. These studies employ the Gaussian approximation for
quantization distortion (QD) to evaluate system performance, which accurately
represents the behavior in scenarios with low UE transmit power or a large
number of UEs. The performance of 1-bit ADC systems with a low number of UEs
is evaluated in [10], which is inferior at high UE transmit power due to QD.
However, the introduction of properly tuned dithering at the BS enhances
performance at high transmit powers [10]. Regarding cell-free massive MIMO
systems, the performance of 1-bit ADCs is evaluated in [11], while the energy
efficiency of such systems with low-resolution ADCs is analyzed in conjunction
with an uplink power control scheme for max-min fairness in [12]. It is
important to note that the works in [11, 12] also adopt the Gaussian
approximation for QD, which is more accurate with increasing the number of ADC
bits. However, the approximation may not fully capture the true performance
characteristics of 1-bit ADC systems.

In this paper, we consider the uplink power control problem in the distributed
massive MIMO systems with 1-bit ADCs. We first provide insight into the
signal-to-noise-and-distortion ratio (SNDR) of a UE in single and multi-BS
scenarios with 1-bit ADCs. Our analysis reveals that the SNDR of the UE is
non-monotonic and highly dependent on the UE transmit powers and location.
Specifically, in the single-BS case, the SNDR is unimodal, whereas, in the
multi-BS case, the SNDR is non-unimodal. However, we show that by tuning the
Gaussian dithering (i.e., the noise level), the SNDR can be made unimodal even
in the multi-BS scenario. We also optimize the transmit powers of UEs by
considering the min-power and max-min-SINDR optimization problems in a multi-
UE scenario. To solve these optimization problems, we use gradient, fixed-
point, and block coordinate descent (BCD) methods. Our numerical results
indicate that in a single BS system, the SINDR reaches a saturation point
beyond a certain power of the UEs. Moreover, in a multi-BS scenario, if
achieving the desired target SINDR involves reception from multiple BSs using
a large number of antennas and dithering, then the UE transmit power heavily
depends on the distance to the farthest serving BS. In this context, employing
dithering at the BSs becomes crucial in enhancing the SINDR for UEs that
exhibit a substantial variation in path loss among the serving BSs.

##  II System Model

We consider an uplink distributed/cell-free massive MIMO system where a set of
BSs $\mathcal{B}\triangleq\\{1,\ldots,B\\}$ with $M$ antennas serves a set of
single-antenna UEs $\mathcal{K}\triangleq\\{1,\ldots,K\\}$. Each BS antenna is
connected to two 1-bit ADCs for 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 BS
$b\in\mathcal{B}$, 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 BSs. We use
$d_{k}\sim\mathcal{C}\mathcal{N}(0,1)$ to denote the data symbol transmitted
by UE $k$. The received signal at BS $b$ 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 BS $b$. Then,
$\mathbf{y}_{b}$ 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 BSs. Based on the Bussgang decomposition [13], we can write

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

where $\mathbf{q}$ is the zero-mean, non-Gaussian quantization distortion
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
$\mathbf{C}_{\mathbf{z}}\triangleq\mathrm{Diag}(\sigma^{2}_{1},\ldots,\sigma^{2}_{B})\otimes\mathbf{I}_{M}$.

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

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

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

$\displaystyle\mathrm{SINDR}_{k}\\!\triangleq\\!\frac{\rho_{k}\|\mathbf{w}_{k}^{\mathrm{H}}\mathbf{A}\mathbf{h}_{k}\|^{2}}{\sum_{\bar{k}\neq
k}\rho_{\bar{k}}\|\mathbf{w}_{k}^{\mathrm{H}}\mathbf{A}\mathbf{h}_{\bar{k}}\|^{2}\\!+\\!\|\mathbf{w}_{k}^{\mathrm{H}}\mathbf{A}\mathbf{C}_{\mathbf{z}}^{\frac{1}{2}}\|^{2}\\!+\\!\mathbf{w}_{k}^{\mathrm{H}}\mathbf{C}_{\mathbf{q}}\mathbf{w}_{k}},$
(6)

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 [5]

$\displaystyle\mathbf{C}_{\mathbf{r}}$
$\displaystyle\triangleq\frac{2}{\pi}\big{(}\arcsin\big{(}\mathrm{Diag}(\mathbf{C}_{\mathbf{y}})^{-\frac{1}{2}}\mathrm{Re}[\mathbf{C}_{\mathbf{y}}]\mathrm{Diag}(\mathbf{C}_{\mathbf{y}})^{-\frac{1}{2}}\big{)}$
$\displaystyle\phantom{=}\
+j\,\arcsin\big{(}\mathrm{Diag}(\mathbf{C}_{\mathbf{y}})^{-\frac{1}{2}}\mathrm{Im}[\mathbf{C}_{\mathbf{y}}]\mathrm{Diag}(\mathbf{C}_{\mathbf{y}})^{-\frac{1}{2}}\big{)}\big{)}.$
(7)

Finally, the mean squared error (MSE) of UE $k$ is given by

$\displaystyle\mathrm{MSE}_{k}\triangleq\mathbb{E}\big{[}|\mathbf{w}_{k}^{\mathrm{H}}\mathbf{r}-d_{k}|^{2}\big{]}.$
(8)

The combining vector that minimizes the above MSE corresponds to the minimum
mean squared error (MMSE) combiner, i.e.,

$\displaystyle\mathbf{w}_{k}=\sqrt{\rho_{k}}\mathbf{C}_{\mathbf{r}}^{-1}\mathbf{A}\mathbf{h}_{k}.$
(9)

By plugging (9) into (6), the latter can be simplified as

$\displaystyle\mathrm{SINDR}_{k}=\frac{\rho_{k}\mathbf{h}_{k}^{\mathrm{H}}\mathbf{A}\mathbf{C}_{\mathbf{r}}^{-1}\mathbf{A}\mathbf{h}_{k}}{1-\rho_{k}\mathbf{h}_{k}^{\mathrm{H}}\mathbf{A}\mathbf{C}_{\mathbf{r}}^{-1}\mathbf{A}\mathbf{h}_{k}}.$
(10)

In the next section, we will explore the SNDR for a single UE in both single-
BS and two-BS scenarios. It is important to note that in the absence of
interference from other UEs, the term SINDR can be used interchangeably with
SNDR.

##  III SNDR in the Single-UE Case

$-20$$0$$20$$40$$60$$0$$5$$10$$15$$20$AWGN dominatesquantization distortion
dominates$\rho_{k}$ [dBm]$\mathrm{SNDR}$ Figure 1: Single-BS case: SNDR vs UE
transmit power.

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

$-20$$0$$20$$40$$60$$0$$10$$20$$30$$\rho_{k}$ [dBm]$\mathrm{SNDR}$UE at $50$
mUE at $10$ mUE at $10$ m (dithering) Figure 3: Two-BS case: SNDR vs UE
transmit power.

We consider a single UE and simple network deployments with one or two BSs to
characterize the SNDR with respect to the UE transmit power and distance from
the BSs. To this end, we provide illustrative numerical results assuming
$M=128$ antennas at each BS and employing MMSE combiner. Further, we model the
channel between UE $k$ and each BS $b$ as uncorrelated Rayleigh fading, where
the channel coefficients are denoted by $\mathbf{h}_{b,k}$. Specifically, the
entries of $\mathbf{h}_{b,k}$ are independent and identically distributed
(i.i.d.) complex Gaussian random variables with a mean of zero and variance
$\delta_{b,k}$. The large-scale fading coefficient $\delta_{b,k}$ is
determined by the distance $d_{b,k}$ between BS $b$ and UE $k$, and it is
given by $\delta_{b,k}\triangleq-61-30\log_{10}(d_{b,k})$ [dB], with the
carrier frequency being $28$ GHz and the pathloss exponent being 3.

###  III-A Single-BS Case: SNDR Versus UE Transmit Power

Let us first consider a scenario with a single BS equipped with 1-bit ADCs,
where the UE is located at a fixed distance of $30$ m from the BS. Fig. 1
illustrates the SNDR as a function of the UE transmit power. It can be
observed that the SNDR exhibits unimodal behavior, with an optimal value
achieved at a specific UE transmit power. At low UE transmit powers, the
impact of AWGN dominates, overpowering the intended signal. Conversely, at
high UE powers, the received soft symbols suffer from amplitude loss,
resulting in QD prevailing over the intended signal. Therefore, the optimal
SNDR is achieved by the appropriate amount of AWGN, which induces a beneficial
scrambling effect on the 1-bit quantized signals at the BS.

###  III-B Two-BS Case: SNDR Versus Distance UE-Reference BS

We will characterize the SNDR of a UE in a two-BS system. To begin with, the
SNDR of a UE is characterized as a function of the distance between the UE and
one of the two BSs, which is referred to as the reference BS in Fig. 2. In
this scenario, the two BSs are equipped with 1-bit ADCs and are positioned
$100$ m apart. The UE maintains a fixed transmit power and moves along the
line connecting the two BSs. At low UE transmit power of $-5$ dBm, the SNDR
decreases as the UE moves away from the left BS and increases as it gets
closer to the other BS as in the unquantized case, where SNDR is primarily
affected by AWGN. At a higher UE transmit power of $7.5$ dBm, SNDR is limited
by the QD of the closest BS and AWGN of the farthest BS. However, as the UE
moves away from the closest BS, both QD and AWGN impacts reduce, improving
SNDR. At the cell edge, i.e., at $50$ m, SNDR is dominated by AWGN at both BSs
due to the limited UE transmit power, resulting in lower SNDR. At a very high
transmit power of $20$ dBm, SNDR is limited by QD if the UE is close to any of
the BSs, and the impact of QD reduces as the UE moves towards the cell edge,
resulting in better SNDR. Note that the UE is not power-limited even at the
cell edge in this scenario.

###  III-C Two-BS Case: SNDR Versus UE Transmit Power

Lastly, let us consider the same setup as discussed in Section III-B to
analyze the SNDR with the UE transmit power at a fixed location. Additionally,
we also evaluate the impact of dithering at each Base Station (BS). If the UE
is equidistant from both BSs, specifically located $50$ m away from the
reference BS, the impact of AWGN and QD on the SNDR is the same at both BSs
for a fixed UE transmit power level. As a result, the optimal performance at
both the BSs is achieved at the same UE transmit power, leading to a unimodal
SNDR behavior, as shown in Fig. 3, which is similar to the single-BS case.
However, the SNDR in the two-BS scenario is higher than in the single-BS case
(see Fig 1) due to the combining gain of both the BSs. On the other hand, if
the UE is close to one of the BSs (say $10$ m), the optimal performance at the
closest BS is achieved with a very low UE transmit power compared to the
farthest BS. This results in two peaks in the SNDR behavior, making it non-
unimodal. Additionally, adjusting the UE transmit power does not make the SNDR
behavior unimodal. Therefore, to enhance the overall SNDR, we can modify the
dominance of QD at the closest BS by introducing a dithering signal. In this
context, we consider a Gaussian dithering signal, which can be achieved by
increasing the power of the AWGN. For simplicity, we consider the parameter
$\sigma_{b}$ as a controllable parameter referred to as the noise level
parameter. It allows us to adjust the power of the dithering signal. By
adjusting the noise level, we can maintain a fixed QD (as mentioned in Remark
1), resulting in an improved SNDR with unimodal behaviour, as shown in Fig. 3.
In scenarios involving over two BSs and a UE with unlimited power, dithering
levels are applied to all BSs except the farthest one. The level of dithering
at each BS depends on its pathloss ratio to the farthest BS.

######  Remark 1.

By adjusting the noise level of the 1-bit ADC BS, the QD can be made the same
for varying UE transmit powers.

To provide a better understanding of Remark 1, we consider a system with a
single 1-bit ADC. Let $\sqrt{\rho_{d}}d$ and $\sigma_{z}z$ be the input signal
and noise to the 1-bit ADC, where $\rho_{d}$ and $\sigma_{z}^{2}$ are the
signal power and noise power, respectively. The output of the 1-bit ADC can be
expressed as:

$\displaystyle
r\triangleq\mathrm{sgn}(\sqrt{\rho_{d}}d+{\sigma_{z}}z)=\mathrm{sgn}\bigg{(}\sqrt{\rho_{d}}\bigg{(}d+\frac{{\sigma_{z}}}{\sqrt{\rho_{d}}}z\bigg{)}\bigg{)}.$
(11)

It is clear from (11) that as long as the ratio
$\frac{{\sigma_{z}}}{\sqrt{\rho_{d}}}$ is constant, the output of the 1-bit
ADC remains the same. Moreover, it can be shown that $\mathbf{A}\mathbf{y}$ in
(3), does not change if ${\sigma_{z}}$ and $\sqrt{\rho_{d}}$ vary with a fixed
ratio. Therefore, by adjusting the noise level, i.e., $\sigma_{z}$, we can fix
the $\frac{{\sigma_{z}}}{\sqrt{\rho_{d}}}$ ratio for different input signal
powers, ensuring that the QD remains the same.

##  IV Uplink Power Control

In this section, we discuss the optimization of the UE transmit powers to
achieve a target SINDR and maximize the minimum SINDR. In addition, we can
optimize the noise level at the BSs to achieve higher SINDR. Let us begin with
the optimization of UE transmit powers using the min-power approach for a
fixed target SINDR.

###  IV-A Power Control Based on Min-Power

For a given target SINDR $\gamma_{k}^{\textnormal{\tiny{target}}}$ for UE $k$,
the min-power optimization problem can be formulated as follows:

$\displaystyle\begin{array}[]{cl}\displaystyle\underset{\\{\rho_{k},\sigma_{b}\\}}{\mathrm{minimize}}&\displaystyle\sum_{k\in\mathcal{K}}\rho_{k}\\\
\mathrm{s.t.}&\mathrm{SINDR}_{k}\geq\gamma_{k}^{\textnormal{\tiny{target}}},\quad\forall
k\in\mathcal{K}\\\ &\sigma_{b}\geq\sigma_{\min},\quad\forall
b\in\mathcal{B}\end{array}$ (15)

where $\mathrm{SINDR}_{k}$ is given in (10), and $\sigma^{2}_{\min}$ is the
minimum noise level corresponding to the AWGN power. It is to be noted that
the optimization of $\sigma_{b}$ is not required for the single BS case. In
the multi-BS case, the noise level at each BS and the UE transmit powers can
be tuned in an alternating manner. However, for simplicity, we will first fix
the noise levels at the BSs and optimize the UE transmit powers with the
following simplified optimization problem:111For fixed UE transmit power, a
similar optimization problem can be written for noise level optimization in
the multi-BS scenario. However, we ignore the noise level optimization for the
sake of simplicity and due to space constraints.

$\displaystyle\begin{array}[]{cl}\displaystyle\underset{\\{\rho_{k}\\}}{\mathrm{minimize}}&\displaystyle\sum_{k\in\mathcal{K}}\rho_{k}\\\
\mathrm{s.t.}&\displaystyle\rho_{k}\mathbf{h}_{k}^{\mathrm{H}}\mathbf{A}\mathbf{C}_{\mathbf{r}}^{-1}\mathbf{A}\mathbf{h}_{k}\geq\frac{\gamma_{k}^{\textnormal{\tiny{target}}}}{\gamma_{k}^{\textnormal{\tiny{target}}}+1},\quad\forall
k\in\mathcal{K}.\end{array}$ (18)

The constraint in (18) on the left-hand side exhibits a unimodal behavior
similar to that of $\mathrm{SINDR}_{k}$ in (10). This is because the
reformulation of the constraint involves only the addition of a constant value
of 1.

The SINDR of a UE with 1-bit is non-monotonic 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
cannot be used as is to optimize UE transmit powers [14]. To address this, we
propose the gradient, fixed-point, and BCD methods to optimize the powers of
(18).

$\displaystyle\frac{\partial}{\partial\rho_{k}}\mathcal{L}(\\{\rho_{k},\mu_{k}\\})$
$\displaystyle=1-\mu_{k}\mathbf{h}_{k}^{\mathrm{H}}\mathbf{A}\mathbf{C}_{\mathbf{r}}^{-1}\mathbf{A}\mathbf{h}_{k}-\sum_{\bar{k}\in\mathcal{K}}\mu_{\bar{k}}\rho_{\bar{k}}\frac{\partial}{\partial\rho_{k}}\bigg{(}\mathbf{h}_{\bar{k}}^{\mathrm{H}}\mathbf{A}\mathbf{C}_{\mathbf{r}}^{-1}\mathbf{A}\mathbf{h}_{\bar{k}}\bigg{)},$
(21)
$\displaystyle\frac{\partial}{\partial\rho_{k}}\mathbf{h}_{\bar{k}}^{\mathrm{H}}\mathbf{A}\mathbf{C}_{\mathbf{r}}^{-1}\mathbf{A}\mathbf{h}_{\bar{k}}$
$\displaystyle=2\mathrm{Re}\bigg{[}\mathbf{h}_{\bar{k}}^{\mathrm{H}}\bigg{(}\frac{\partial}{\partial\rho_{k}}\mathbf{A}\bigg{)}\mathbf{C}_{\mathbf{r}}^{-1}\mathbf{A}\mathbf{h}_{\bar{k}}\bigg{]}-\mathbf{h}_{\bar{k}}^{\mathrm{H}}\mathbf{A}\mathbf{C}_{\mathbf{r}}^{-1}\bigg{(}\frac{\partial}{\partial\rho_{k}}\mathbf{C}_{\mathbf{r}}\bigg{)}\mathbf{C}_{\mathbf{r}}^{-1}\mathbf{A}\mathbf{h}_{\bar{k}}$
(22)

$\displaystyle\frac{\partial}{\partial\rho_{k}}\mathrm{Re}[\mathbf{C}_{\mathbf{r}}(i,j)]$
$\displaystyle=\frac{\mathbf{A}(i,i)\mathbf{A}(j,j)}{\sqrt{1-|q_{i,j}|^{2}}}\bigg{(}\mathrm{Re}\big{[}\mathbf{h}_{k}(i)\mathbf{h}_{k}(j)\big{]}-\frac{1}{2}\mathrm{Re}\big{[}\mathbf{C}_{\mathbf{y}}(i,j)\big{]}\big{(}|\mathbf{h}_{k}(i)|^{2}\mathbf{A}^{2}(i,i)+|\mathbf{h}_{k}(j)|^{2}\mathbf{A}^{2}(j,j)\big{)}\bigg{)}$
(24)

1) Gradient Update. Let us begin by writing the Lagrangian of the optimization
problem (18) as

$\displaystyle\mathcal{L}({\rho_{k},\mu_{k}})\\!\triangleq\\!\sum_{k\in\mathcal{K}}\bigg{(}\\!\rho_{k}-\mu_{k}\bigg{(}\\!\rho_{k}\mathbf{h}_{k}^{\mathrm{H}}\mathbf{A}\mathbf{C}_{\mathbf{r}}^{-1}\mathbf{A}\mathbf{h}_{k}\\!-\frac{\gamma_{k}^{\textnormal{\tiny{target}}}}{\gamma_{k}^{\textnormal{\tiny{target}}}+1}\bigg{)}\bigg{)},$
(19)

where $\mu_{k}\geq 0$ is the Lagrangian dual variable associated with the
constraint. Similar to Section III, it can be observed that
$\mathcal{L}({\rho_{k},\mu_{k}})$ is unimodal with respect to
$\\{\rho_{k}\\}_{k\in\mathcal{K}}$ for any $\mu_{k}\in\mathbb{R}_{++}$ in a
single BS scenario. Therefore, we can use a gradient approach to compute the
powers of the UEs to satisfy the SINDR constraint. Finally, the power of UE
$k$ is updated iteratively in the negative direction of the gradient as

$\displaystyle\rho_{k}^{(i)}=\rho_{k}^{(i-1)}-\zeta\left.\frac{\partial\mathcal{L}({\rho_{k},\mu_{k}})}{\partial\rho_{k}}\right|_{\\{\rho_{\bar{k}}=\rho_{\bar{k}}^{(i-1)},\mu_{\bar{k}}=\mu_{\bar{k}}^{(i-1)}\\}_{\bar{k}\in\mathcal{K}}},$
(20)

where $\zeta\geq 0$ is the step size. The gradient of the Lagrangian with
respect to $\rho_{k}$ is given in (21)–(22) at the top of the next page. To
compute this gradient, we need to evaluate the gradient of the matrix
$\mathbf{A}$ with respect to $\rho_{k}$, which is given by

$\displaystyle\frac{\partial}{\partial\rho_{k}}\mathbf{A}=-\sqrt{\frac{1}{2\pi}}\mathrm{Diag}(\mathbf{C}_{\mathbf{y}})^{-\frac{3}{2}}\mathrm{Diag}(\mathbf{h}_{k}\mathbf{h}_{k}^{\mathrm{H}}).$
(23)

Moreover, the gradient of the $(i,j)$th element of the real part of
$\mathbf{C}_{\mathbf{r}}$ is given in (24) at the top of the next page, where
we denote $q_{i,j}\triangleq\mathbf{C}_{\mathbf{r}}(i,j)$. The gradient for
the imaginary part of $\mathbf{C}_{\mathbf{r}}$ can be obtained by replacing
$\mathrm{Re}$ with $\mathrm{Im}$ in (24). Finally, we have
$\frac{\partial}{\partial\rho_{k}}\mathbf{C}_{r}(i,i)=0,\leavevmode\nobreak\
\forall i$. The dual variable $\mu_{k}$ is updated as

$\displaystyle\mu_{k}^{(i)}=\mathrm{max}\bigg{(}0,\mu_{k}^{(i-1)}-\eta\bigg{(}\\!\rho_{k}\mathbf{h}_{k}^{\mathrm{H}}\mathbf{A}\mathbf{C}_{\mathbf{r}}^{-1}\mathbf{A}\mathbf{h}_{k}\\!-\\!\frac{\gamma_{k}^{\textnormal{\tiny{target}}}}{\gamma_{k}^{\textnormal{\tiny{target}}}+1}\bigg{)}\bigg{)},$
(25)

where $\eta\geq 0$ is the step size and $i$ is the iteration index.

Note that the gradient update in (20) is terminated when the SINDR constraint
is met for all UEs. However, the convergence speed of this method depends on
the slope of the unimodal function at different UE transmit powers. Moreover,
in the case of multiple BSs, this method may converge to a local optimum due
to the presence of multiple peaks in the SINDR with respect to UE transmit
power, as discussed in Section III.

To meet the SINDR constraints of (18), an appropriate UE transmit power can be
determined using the standard fixed-point power update method [14]. This
method is effective at lower powers, where the SINDR resembles the SINR of an
unquantized system.

Data: $\mathbf{h}_{k}$,
$\\{\gamma_{k}^{\textnormal{\tiny{target}}}\\}_{\\{k\in\mathcal{K}\\}}$,
$\epsilon>0$.
Initialization: UE transmit power $\rho_{k}^{(0)},\forall k$, iteration $i=0$.

  * •

Compute $\mathrm{SINDR}_{k}^{(0)}$ as in (10).

while
_$|\underset{k\in\mathcal{K}}{\min}(\mathrm{SINDR}_{k}^{(i)})-\gamma_{k}^{\textnormal{\tiny{target}}}|\geq\epsilon$_
do

  * •

$i=i+1$.

  * •

Update all the UE transmit powers as in (26).

  * •

Compute the $\mathrm{SINDR}_{k}^{(i)}$ as in (10).

if  _
$\underset{k\in\mathcal{K}}{\min}(\mathrm{SINDR}_{k}^{(i)}\\!-\\!\mathrm{SINDR}_{k}^{(i-1)})\\!\leq\\!0$_
then

break % $\gamma_{k}^{\textnormal{\tiny{target}}}$ is not achievable.

end if

end while

Algorithm 1 (Fixed-point)

2) Fixed-Point Update. The transmit powers of the UEs are updated in parallel
at each iteration. Specifically, at iteration $i$, the transmit power of UE
$k$ is scaled to satisfy the SINDR constraint given by (18) using the fixed-
point iterative power update method [14]:

$\displaystyle\begin{aligned}
\rho_{k}^{(i)}=\frac{\gamma_{k}^{\textnormal{\tiny{target}}}}{\mathrm{SINDR}_{k}^{(i-1)}}\rho_{k}^{(i-1)},\end{aligned}$
(26)

where $\mathrm{SINDR}_{k}^{(i-1)}$ is the SINDR of UE $k$ computed using the
transmit powers $\\{\rho_{k}^{(i-1)}\\}_{k\in\mathcal{K}}$ according to (10).
It is important to note that the power values must be initialized to the
smallest possible value and cannot be arbitrarily large to ensure the
feasibility of this method. As the SNDR is unimodal, unlike the unquantized
system, we implement a termination criterion to terminate the power update
process if the SINDR starts decreasing with the increase in transmit power.
The fixed-point iterative power update method for computing the UE transmit
powers is summarized in Algorithm 1.

The fixed-point power update algorithm is more effective at lower powers where
the SINDR increases linearly with the power values. However, at the optimal
SINDR value, the increase in SINDR with respect to the UE transmit power is
not linear, which may result in slower convergence. Additionally, since the
interference and QD of 1-bit ADC do not follow the properties of a standard
interference function, this algorithm may not have a unique solution in such
cases.

3) BCD Update. In the single BS case, the SINDR of a UE is unimodal with
respect to the UE transmit power while keeping all other UE transmit powers
fixed. Therefore, we employ a BCD method to update the UE transmit powers. In
this method, we iteratively search for the required power of each UE to meet
its target SINDR while keeping the other UE transmit powers fixed. The powers
of all UEs are updated cyclically until the target SINDR is achieved by all
UEs simultaneously, as described in Algorithm 2. We use a simple line search
with a UE transmit power resolution of $\Delta\rho$ and maximum power of
$\rho_{\textnormal{\tiny{UE}}},\forall k$, to optimize the power values.
However, a faster update of the power values can be achieved by using a
backtracking line search since the SINDR function is unimodal. Moreover, the
optimal power values in each iteration can be used as the initial values for
the next iteration to expedite the updates.

Similar to the fixed-point algorithm, the BCD algorithm also terminates if the
target SINDR is not achievable within the power limits or if the UE SINDR
decreases with increasing the transmit power. This method can be extended to
multi-BS scenarios, where the feasibility of noise levels at the BS needs to
be ensured to achieve a better target SINDR. Since the SINDR of a UE is non-
unimodal for any noise level at the BS, we evaluate the algorithm for
different noise levels at each BS to find the first peak in the SINDR. Then,
we select the optimal SINDR value among all the computed noise levels.

###  IV-B Power Control Based on Max-Min-SINDR

In this section, we aim to optimize the UE transmit powers subject to a power
constraint while maximizing the minimum SINDR objective, which can be
formulated as

$\displaystyle\begin{array}[]{cl}\underset{\\{\rho_{k},\sigma_{b}\\}}{\mathrm{maximize}}&\underset{k\in\mathcal{K}}{\min}\
\mathrm{SINDR}_{k}\\\
\mathrm{s.t.}&\rho_{k}\leq\rho_{\textnormal{\tiny{UE}}},\quad\forall k\\\
&\sigma_{b}\geq\sigma_{\min}.\quad\forall b\end{array}$ (30)

Continuing with a similar approach as outlined in Section IV-A, the
optimization of UE transmit powers, given fixed
$\\{\sigma_{b}\\}_{b\in\mathcal{B}}$ values, can be achieved through the
following simplified epigraph optimization problem:note1

$\displaystyle\begin{array}[]{cl}\underset{\\{\rho_{k}\\},\gamma}{\mathrm{maximize}}&\quad\gamma\\\
\mathrm{s.t.}&\quad\rho_{k}\mathbf{h}_{k}^{\mathrm{H}}\mathbf{A}\mathbf{C}_{\mathbf{r}}^{-1}\mathbf{A}\mathbf{h}_{k}\geq\frac{\gamma}{\gamma+1},\quad\forall
k\\\ &\quad\rho_{k}\leq\rho_{\textnormal{\tiny{UE}}},\quad\forall
k\end{array}$ (34)

where $\gamma\triangleq\underset{k}{\min}\quad\\!\\!\\!\mathrm{SINDR}_{k}$.
The above problem can be solved using a bisection algorithm over $\gamma$. For
a fixed value of $\gamma$ (the same for all the UEs), we can solve (18) to
check the feasibility of (34). Therefore, we can use the same algorithms, such
as the gradient, fixed-point, and BCD, to optimize the UE transmit powers for
the max-min SINDR problem. Note that at the optimal $\gamma$ value, it is
possible that none of the power constraints meet equality. However, in the
unquantized system, at least one of the power constraints meets equality at
the optimal value of $\gamma$.

Data: $\mathbf{h}_{k}$, $\rho_{\textnormal{\tiny{UE}}}$,
$\gamma_{k}^{\textnormal{\tiny{target}}}$, $\epsilon>0$, $\Delta\rho$.
Initialization: UE transmit power $\rho_{k}^{(0)},\forall k$, iteration $j=0$,
$p=0$.
while
_$|\underset{k\in\mathcal{K}}{\min}(\mathrm{SINDR}_{k}^{(j)})-\gamma_{k}^{\textnormal{\tiny{target}}}|\geq\epsilon$_
do

  * •

$j=j+1$.

for  _$k=1,\ldots,K$_ do

  * •

$p=0$, $\mathrm{SINDR}_{k}^{(0)}=0$.

for  _$\rho=\rho_{k}^{(j-1)}:\Delta\rho:\rho^{\textnormal{\tiny{UE}}}$_ do

  * •

$p=p+1$.

  * •

Compute the $\mathrm{SINDR}_{k}^{(p)}$ of UE $k$ as in (10)
with the UE transmit powers of
$[\rho_{1}^{(j)},\ldots,\rho_{k-1}^{(j)},\rho,\rho_{k+1}^{(j-1)},\ldots,\rho_{K}^{(j-1)}]$.

if
_$|\mathrm{SINDR}_{k}^{(p)}-\gamma_{k}^{\textnormal{\tiny{target}}}|\leq\epsilon$
or $\mathrm{SINDR}_{k}^{(p)}\leq\mathrm{SINDR}_{k}^{(p-1)}$_ then

break

end if

end for

  * •

$\rho_{k}^{(j)}=\rho$.

if  _$|\mathrm{SINDR}_{k}^{(p)}-\gamma_{k}^{\textnormal{\tiny{target}}}|
>\epsilon$_ then

break

end if

end for

if  _$|\mathrm{SINDR}_{k}^{(p)}-\gamma_{k}^{\textnormal{\tiny{target}}}|
>\epsilon$ _ then

break % $\gamma_{k}^{\textnormal{\tiny{target}}}$ is not achievable.

end if

  * •

Compute the $\mathrm{SINDR}_{k}^{(j)}$ of all the UEs with the updated power
values, i.e., $\\{\rho_{k}^{(j)}\\}_{k\in\mathcal{K}}$.

end while

Algorithm 2 (BCD)

##  V Numerical Results

$0$$5$$10$$15$$5$$10$$15$$20$$25$Target $\mathrm{SINDR}$Min-power
[dBm]GradientFixed-pointBCD Figure 4: Single-BS case: min-power vs target
SINDR.

$0$$20$$40$$60$$80$$100$$0$$5$$10$$15$$20$$25$Distance UE-reference BS [m]Min-
power [dBm]Target $\mathrm{SINDR}=13$Target $\mathrm{SINDR}=20$ Figure 5: Two-
BS case: min-power vs distance UE-reference BS.

$0$$20$$40$$60$$80$$100$$0$$5$$10$$15$$20$$25$Distance UE-reference BS [m]Max-
min-SINDR$\rho_{k}=-5$ dBm$\rho_{k}=7.5$ dBm$\rho_{k}=20$ dBm$\rho_{k}=20$ dBm
(dithering) Figure 6: Two-BS case: max-min-SINDR vs distance UE-reference BS.

To evaluate the performance of the system we consider the similar setup as
discussed in Section III, with four single-antenna UEs placed equidistant from
both the BSs. The AWGN power at the BSs is set to $\sigma^{2}_{\min}=-95$ dBm,
and dithering (noise level) can be added on top of it if needed to improve the
performance. Fig.4 illustrates the min-power as a function of the target SINDR
in a single-BS case. The Gradient, Fixed-point, and BCD update methods perform
similarly. However, due to the QD of the 1-bit ADC BS, the maximum feasible
target SINDR is limited to around $15$. The computational complexity of the
Gradient method in each iteration is determined by (21), which is higher
compared to the Fixed-point, and BCD methods. Convergence is also influenced
by the step sizes, represented as $\zeta$ and $\eta$ in (20) and (25),
respectively. On the other hand, the Fixed-point method has lower
computational complexity than the BCD method. However, when approaching the
optimal SINDR, the BCD method may exhibit superior performance owing to the
fixed increment of power in each step, i.e., $\Delta\rho$. Consequently, in
the subsequent scenarios involving two-BSs, we employ the BCD method to
characterize the min-power and max-min SINDR designs.

Fig. 5 shows the min-power required to achieve a given target SINDR as a
function of the distance between the UE and the reference BS. In this
simulation, two BSs are located $100$ m apart, and the UEs move along the line
connecting both BSs. If the target SINDR is achieved by a single BS, such as
SINDR of $13$, the required power increases as the UE moves away from the BS,
as expected in the unquantized system. Therefore, the maximum min-power is
required when the UE is located between the two BSs. However, if the target
SINDR requires joint reception from both BSs with dithering, as in the case of
SINDR of $20$, the min-power required to achieve the target SINDR is maximum
when the UE is closer to one of the BSs and minimum at the cell edge, which is
$50$ m in this case. This result differs from the unquantized system.

Considering the same setup used for Fig. 5, in Fig. 6, we characterized the
max-min-SINDR as a function of the UE distance from the reference BS for
various UE transmit powers. When the UE transmit power is too low, such as
$-5$ dBm, the achievable SINDR decreases as the UEs move further from the BS,
and it is lowest at the cell edge ($50$ m), as expected in the unquantized
system. If we increase the UE transmit power constraint to $7.5$ dBm, the
SINDR is lower when the UEs are closer to a BS since there is no combining
gain from the other BS. The SINDR is maximum at $15$ m, where both the UE
transmit power and combining gains are optimal, and it decreases as the
distance from the BSs increases due to the limited UE transmit power. In the
aforementioned scenarios with limited UE transmit power, the presence of
dithering at the BSs may not improve the SINDR. However, when the UE transmit
power is higher, such as $20$ dBm, the UE is not constrained by transmit power
even at the cell edge. This allows for the maximum achievable SINDR by
combining across the BSs, irrespective of the presence of dithering. As UEs
move away from the cell edge, the SINDR decreases without dithering.
Nevertheless, incorporating dithering significantly improves and maintains the
maximum SINDR, as long as the UEs have sufficient power to reach the farthest
serving BS, as depicted in Fig. 6. However, UEs located near a BS still lack
the necessary power to reach other BS, resulting in a decrease in the
achievable SINDR.

##  VI Conclusions

We investigated the uplink SNDR of a UE in both single-BS and multi-BS
scenarios with 1-bit ADCs. In a single-BS case, the SNDR is unimodal, while it
is non-unimodal in a multi-BS case. However, by adjusting the dithering at the
BSs, the SNDR can be optimized and made unimodal. We optimized the UE transmit
powers using the min-power and max-min-SINDR criteria and employed gradient,
fixed-point, and BCD methods to solve the optimal UE transmit powers. The
numerical analysis shows that in a single-BS system, the SINDR saturates
beyond a specific UE transmit power. Meanwhile, in a multi-BS system, when the
target SINDR is attainable through reception from multiple BSs with dithering,
the UE transmit power depends on the farthest serving BS. In this context, we
have demonstrated that incorporating dithering at the nearest BSs can
effectively enhance the SINDR for UEs with a large disparity in path loss
among the serving BSs.

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