# Pseudo-Random Quantization Based Two-Stage Detection in One-Bit Massive MIMO
Systems

Gökhan Yılmaz, , and
Ali Özgür Yılmaz The work of G. Yılmaz was supported by Vodafone Turkey within
the framework of 5G and Beyond Joint Graduate Support Program coordinated by
the Information and Communication Technologies Authority of Turkey.G. Yılmaz
is with the Department of Electrical Engineering, Eindhoven University of
Technology, Eindhoven, The Netherlands (e-mail<EMAIL_ADDRESS>and A. Ö.
Yılmaz is with the Department of Electrical and Electronics Engineering,
Middle East Technical University, Ankara, Turkey. (e-mail:
<EMAIL_ADDRESS>
###### Abstract

Utilizing low-resolution analog-to-digital converters (ADCs) in uplink massive
multiple-input multiple-output (MIMO) systems is a practical solution to
decrease power consumption. The performance gap between the low and high-
resolution systems is small at low signal-to-noise ratio (SNR) regimes.
However, at high SNR and with high modulation orders, the achievable rate
saturates after a finite SNR value due to the stochastic resonance (SR)
phenomenon. This paper proposes a novel pseudo-random quantization (PRQ)
scheme by modifying the quantization thresholds that can help compensate for
the effects of SR and makes communication with high-order modulation schemes
such as $1024$-QAM in one-bit quantized uplink massive MIMO systems possible.
Moreover, modified linear detectors for non-zero threshold quantization are
derived, and a two-stage uplink detector for single-carrier (SC) multi-user
systems is proposed. The first stage is an iterative method called Boxed
Newton Detector (BND) that utilizes Newton’s Method to maximize the log-
likelihood with box constraints. The second stage, Nearest Codeword Detector
(NCD), exploits the first stage solution and creates a small set of most
likely candidates based on sign constraints to increase detection performance.
The proposed two-stage method with PRQ outperforms the state-of-the-art
detectors from the literature with comparable complexity while supporting
high-order modulation schemes.

###### Index Terms:

Detection, massive MIMO, pseudo-random quantization (PRQ), one-bit analog-to-
digital converter (ADC), stochastic resonance (SR), dithering, multi-user.

##  I Introduction

Massive multiple-input multiple-output (MIMO) systems can increase the
spectral efficiency in multi-user settings by orders of magnitude due to much
larger multiplexing and diversity potential [1]. However, since each antenna
requires a separate radio-frequency (RF) chain, power consumption and hardware
cost can increase to unacceptable levels. Analog-to-digital converter (ADC)
units are one of the most power-consuming components in RF chains. The power
consumption of an ADC increases linearly with the sampling rate and
exponentially with the resolution. Therefore, low-resolution ADCs have become
an attractive solution [2]. Especially, one-bit ADCs have additional benefits,
such as simple implementation and operation without automatic gain control
(AGC) units. However, the received signal is severely distorted due to one-bit
quantization.

The achievable rate [3, 4, 5, 6] and the channel capacity [7, 8, 9] of one-bit
systems are well-studied. The performance gap between the one-bit and
infinite-resolution systems is small at low signal-to-noise ratio (SNR), yet
it gets more prominent as SNR increases, which is undesirable in achieving
higher rates per user. Stochastic resonance (SR) is observed when the presence
of white noise in a nonlinear system’s input increases the detectability of
the input signal [10]. Different examples of systems where SR is encountered
can be found in [11, 12, 13]. The intentional addition of noise into a
system’s input to reduce quantization errors is called dithering [14].
Dithering can be achieved by adding artificially generated noise into the
quantizer input, but it can also be achieved by just changing the quantization
thresholds [15]. Non-zero threshold quantization has been utilized in some of
the previous works in literature. In [16], deep unfolding is utilized to
reconstruct a signal from one-bit measurements, with uniformly distributed
thresholds. Threshold designs with a set partitioning scheme in [17] and an
adaptive technique in [18] are utilized to minimize channel estimation errors.
In [19], an antithetic dithering scenario is exploited to increase the
estimation performance. Changing the sampling characteristics can also be
helpful to obtain better performance, examples of which can be found in [15,
20, 21].

Linear detectors for one-bit massive MIMO systems have been studied
extensively. They can be separated into two classes: conventional and
Bussgang-based, as in [22, 23]. The conventional linear detectors [3, 5] are
quantization-unaware and constructed as if the system is infinite-resolution.
Bussgang-based methods account for quantization and can offer better
performance and lower error floors than their conventional counterparts.
Although linear detectors are easy to implement, higher rates can be achieved
with more sophisticated approaches. In [24], a Bayes-optimal joint channel
estimation and detection scheme is proposed. [22] utilizes an ADMM-based
algorithm for detection. A message passing algorithm for CP-free systems
operating in ISI channels is proposed in [25]. A recent study in [26] utilizes
a hybrid scheme with analog processing and adaptive quantization thresholds to
maximize the achievable rate.

In [27], a near maximum likelihood (ML) approach is presented with a projected
gradient descent algorithm, and [28] proposes a gradient descent algorithm
with box constraints for one-bit MIMO systems that employ orthogonal frequency
division multiplexing (OFDM). The works in [16, 23, 29] use gradient descent
with the deep unfolding technique. Other detectors that rely on machine
learning can be found in [30, 31, 32]. The detectors in [27, 30, 23] follow a
two-stage approach. The second stage in [27] improves the performance by
generating a reduced set to apply ML detection. The complexity is limited by a
nearest neighbor search algorithm in [23]. In [30], the final decisions are
made using a metric called weighted Hamming distance [33].

In this paper, we focus on a new quantization strategy for one-bit massive
MIMO systems that exploits pseudo-randomly generated non-zero quantization
thresholds by generating a dithering effect to better recover the amplitude of
received signal and to enhance detection performance. We first modify the
linear filters to work with non-zero threshold quantization. Then, we focus on
a new two-stage detection scheme. The first stage works as an equalizer to
obtain an estimate of the signal to be detected by approximating ML detection
to an unconstrained optimization problem for which Newton’s method is
utilized. Then, in the second stage, the first-stage estimate is used to
create a set of most likely candidates to finalize decision-making. This two-
stage procedure can help obtain an approximate-ML detection method that is
computationally efficient.

Next, we introduce a new quantization strategy which is a reminiscent of the
conventional dithering approach that relies on modifying the quantization
thresholds of ADCs instead of generating analog dither signals. This strategy
involves generating pseudo-random quantization thresholds and uses the
threshold information during baseband processing to increase detection
performance while not requiring additional circuitry. A critical advantage of
this approach is not requiring temporal updates for different channel
realizations and not requiring modification as long as the average SNR remains
the same by exploiting the properties of massive MIMO. We analyze the
performance of the PRQ scheme by both information theoretic and coding
theoretic approaches to show that the proposed PRQ scheme can outperform the
conventional zero-threshold quantization (ZTQ) scheme adopted by the previous
detection studies in the literature [28, 24, 23, 27, 31, 5, 30, 32, 34, 22]
without requiring additional power and processing. Also, the simulation
results show that by combining the proposed detection and quantization
schemes, one-bit massive MIMO systems can operate with high-order modulation
schemes beyond QPSK or $16$-QAM in multi-user settings.

###  I-A Contributions

We propose a novel PRQ scheme that can help mitigate the negative effects of
SR. The proposed scheme relies on changing the quantization thresholds for
dithering without requiring additional processing and circuitry. An achievable
rate analysis and a minimum Hamming distance analysis are made to demonstrate
the advantage of PRQ compared to the conventional ZTQ. Modified linear
detectors for non-zero threshold quantization are derived. With the
appropriate scaling factor, conventional linear receivers perform very closely
to their Bussgang-based counterparts. The proposed first stage, named Boxed
Newton Detector (BND), relies on Newton’s Method with box constraints to
generate an estimate of the input. It outperforms the existing gradient-based
detectors [27, 30, 23] in terms of error performance with comparable
complexity, while not requiring different hyperparameter selections for
different system setups. With a similar motivation as in [23], the proposed
second stage, named Nearest Codeword Detector (NCD), creates a set of
candidate vectors to refine the first stage solution. However, unlike [23], it
does so by taking the one-bit observations as binary codewords to find a
limited number of candidates based on the minimum Hamming distance criterion
to lower complexity and to increase detection performance. The proposed scheme
can outperform the existing detectors in the literature with much lower error
floors. Moreover, communication with high-order modulations such as $256$-QAM,
$1024$-QAM, and $4096$-QAM, whose performances were not reported by any of the
previous work in the literature, is shown to be possible with PRQ.

###  I-B Notation

Lower-case letters represent scalars, lower-case bold letters represent column
vectors, and upper-case bold letters represent matrices. $(.)^{T}$ is the
transpose, and $(.)^{H}$ is the Hermitian. The $n^{\textrm{th}}$ element of a
vector $\bm{c}$ is $c_{n}$. The $n^{\textrm{th}}$ row and $m^{\textrm{th}}$
column of a matrix $\bm{C}$ is denoted as $[\bm{C}]_{n,m}$.
$\operatorname*{diag}(.)$ creates a diagonal matrix. $\mathbb{E}[.]$ is the
expectation operator. $|.|$ denotes the absolute value of its scalar arguments
and the cardinality of its set arguments. $\mathbb{R}$ and $\mathbb{C}$ are
the set of real and complex numbers, respectively. $\Re\\{.\\}$ and
$\Im\\{.\\}$ give their arguments’ real and imaginary parts, respectively.
$\mathcal{CN}(\bm{\mu},\bm{\Sigma})$ denotes the complex Gaussian distribution
with mean vector $\bm{\mu}$ and covariance matrix $\bm{\Sigma}$.
$\phi(x)=\sqrt{\frac{1}{2\pi}}\exp(-\frac{x^{2}}{2})$ is the probability
density function (PDF) of the standard Gaussian distribution.

##  II System Model

We consider an uplink massive MIMO system where $K$ single-antenna users are
served by a BS equipped with $N$ antennas. A simple block diagram of the
system model is shown in Fig. 1. Each user randomly selects an equally likely
symbol from an $M$-QAM alphabet denoted by $\bar{\mathcal{M}}$. The vector of
transmitted symbols is denoted as $\bar{\bm{x}}$, where
$\bar{x}_{k}\in\bar{\mathcal{M}}$, $\mathbb{E}[\bar{x}_{k}]=0$, and
$\mathbb{E}[|\bar{x}_{k}|^{2}]=E_{s}=1$ for $k=0,1,\ldots,K$. Users transmit
their signals through a Nyquist pulse-shaping filter, an ideal digital-to-
analog converter (DAC), and I/Q modulation.

Figure 1: A block diagram that summarizes the system model.

We assume that the channel impulse response (CIR) between each user and
receiver antenna is perfectly known by the BS and can be modeled with
uncorrelated Rayleigh fading. The BS performs decision-making in this uncoded
uplink scenario to detect the symbols sent from all users by using the channel
state information (CSI) knowledge and perfectly known quantization thresholds
of ADCs in each RF chain. Additional analysis about the imperfect CSI scenario
is also included in Section IX. The CIR between receiver antenna $n$ and user
$k$ has complex Gaussian distribution, i.e.,
$\bar{h}_{(n,k)}\sim\mathcal{CN}(0,1)$ as in many massive MIMO studies [28,
22, 6, 5, 8, 29, 23, 30, 32]. Hence, we investigate the effects of only small-
scale fading. When pathloss and shadowing are taken into account, their
effects can be considered as part of the SNR scaling. The equal average
received signal power per user model can be realized using the advanced power
control capabilities of the 5G networks. The resultant $N\times K$ channel
matrix is represented as $[\bar{\bm{H}}]_{n,k}=\bar{h}_{(n,k)}$. Assuming
perfect synchronization, after the received signal is I/Q demodulated, pulse
matched-filtered, and symbol-rate sampled, the unquantized discrete-time
signal is obtained as

$\bar{\bm{y}}=\bar{\bm{H}}\bar{\bm{x}}+\bar{\bm{w}},$ (1)

where $\bar{\bm{w}}\sim\mathcal{CN}(\bm{0}_{N},N_{0}\bm{I}_{N})$ is the zero-
mean circularly symmetric Gaussian noise vector. $\bm{I}_{N}$ is the identity
matrix of size $N\times N$ and $\bm{0}_{N}$ is the all-zero vector of size
$N$. The received signal is quantized by a pair of one-bit ADCs at each
receiver antenna. The unquantized samples and the quantization threshold
vector $\bar{\bm{\tau}}=\Re\\{\bar{\bm{\tau}}\\}+j\Im\\{\bar{\bm{\tau}}\\}$
can be used to obtain the quantized signal

$\bar{\bm{r}}=\operatorname*{sign}(\Re\\{\bar{\bm{y}}-\bar{\bm{\tau}}\\})+j\,\operatorname*{sign}(\Im\\{\bar{\bm{y}}-\bar{\bm{\tau}}\\}),$
(2)

where $\operatorname*{sign}(x)=+1$, if $x\geq 0$, and
$\operatorname*{sign}(x)=-1$ if $x<0$. We define the SNR as the average SNR of
each user at each receiver antenna:

$\rho=\frac{\mathbb{E}[|\bar{x}_{k}|^{2}]}{\mathbb{E}[|\bar{w}_{n}|^{2}]}=\frac{E_{s}}{N_{0}}=\frac{1}{N_{0}},$
(3)

since $\mathbb{E}[|h_{(n,k)}|^{2}]=1$ for all $k=1,\ldots,K$ and
$n=1,\ldots,N$.

The relation between a complex vector $\bar{\bm{a}}$ and its real counterpart
$\bm{a}$, and a complex matrix $\bar{\bm{A}}$ and its real counterpart
$\bm{A}$ can be obtained as

$\bm{a}=\begin{bmatrix}\Re\\{\bar{\bm{a}}\\}\\\
\Im\\{\bar{\bm{a}}\\}\end{bmatrix}\textrm{ and
}\bm{A}=\begin{bmatrix}\Re\\{\bar{\bm{A}}\\}&-\Im\\{\bar{\bm{A}}\\}\\\
\Im\\{\bar{\bm{A}}\\}&\Re\\{\bar{\bm{A}}\\}\end{bmatrix}.$ (4)

As a result, the overall input-output relation of the system can also be
expressed as

$\bm{r}=\operatorname*{sign}(\bm{H}\bm{x}+\bm{w}-\bm{\tau}).$ (5)

Note that each element of $\bm{x}$ belongs to the set of values a symbol from
the modulation alphabet can take in one dimension. We denote this set as
$\mathcal{M}$. For example, when $\bar{\mathcal{M}}$ is the $16$-QAM alphabet,
$\mathcal{M}$ can be expressed as the $4$-PAM alphabet with an average power
of $1/2$.

##  III First Stage Linear Detectors

###  III-A Bussgang-Based Linear Filters

The Bussgang decomposition [35] is valid only when the input to the quantizer
has Gaussian distribution. Despite $\bar{\bm{y}}$ not having Gaussian
distribution, especially at low SNR when the noise term is dominant or due to
the central limit theorem (CLT) when the number of users is high, the
distribution of $\bar{\bm{y}}$ is very close to Gaussian [3, 5]. Therefore,
applying the Bussgang decomposition for designing and analyzing low-resolution
systems is a common practice [35, 3, 26, 5, 36, 22, 23]. Since we focus on a
more general scenario where thresholds can take arbitrary values, an extended
version of the Bussgang theorem must be used. In [17], the Bussgang theorem is
generalized for the non-zero threshold quantization scenario by taking the
selection process of the Bussgang gain and bias terms as the problem of
finding the minimum mean square error (MMSE) estimate of the quantized
observation vector $\bar{\bm{r}}$ using the unquantized observation
$\bar{\bm{y}}$ such that

$\bar{\bm{r}}-\bar{\bm{b}}=\bar{\bm{g}}\,\bar{\odot}\,\bar{\bm{y}}+\bar{\bm{d}},$
(6)

where $\bar{\bm{b}}=\mathbb{E}[\bar{\bm{r}}\mid\bar{\bm{\tau}}]$ is the bias
vector, $\bar{\bm{g}}\in\mathbb{C}^{N}$ is the Bussgang gain vector which can
also be represented as a diagonal matrix composed of the entries of this
vector, i.e., the MMSE filter, and $\bar{\bm{d}}\in\mathbb{C}^{N}$ is the
quantization noise vector. Note that unlike [17], we adopt a notation with
complex numbers by defining a complex extension to the Hadamard product such
that

$\bar{\bm{u}}\,\bar{\odot}\,\bar{\bm{v}}=\Re\\{\bar{\bm{u}}\\}\odot\Re\\{\bar{\bm{v}}\\}+j\Im\\{\bar{\bm{u}}\\}\odot\Im\\{\bar{\bm{v}}\\}.$
(7)

Using the derivations from [17], the Bussgang gain vector and the bias vector
can be calculated respectively as

$\bar{g}_{n}=\sqrt{\frac{4}{\pi[\bar{\bm{C}}_{y}]_{(n,n)}}}\left(\exp\left(-\frac{\Re\\{\bar{\tau}_{n}\\}^{2}}{[\bar{\bm{C}}_{y}]_{(n,n)}}\right)+j\exp\left(-\frac{\Im\\{\bar{\tau}_{n}\\}^{2}}{[\bar{\bm{C}}_{y}]_{(n,n)}}\right)\right),$
(8)

$\bar{b}_{n}=2\Phi\left(-\frac{\Re\\{\bar{\tau}_{n}\\}}{\sqrt{[{\bar{\bm{C}}}_{y}]_{(n,n)}/2}}\right)-1+j\left(2\Phi\left(-\frac{\Im\\{\bar{\tau}_{n}\\}}{\sqrt{[\bar{\bm{C}}_{y}]_{(n,n)}/2}}\right)-1\right),$
(9)

where
$\bar{\bm{C}}_{y}=\mathbb{E}[\bar{\bm{y}}\bar{\bm{y}}^{H}]=\bar{\bm{H}}\bar{\bm{H}}^{H}+N_{0}\bm{I}_{N}$
is the covariance matrix of the unquantized observation vector. For ease of
notation, we define the conditionally zero-mean version of the quantized
observation vector such that

$\bar{\bm{r}}_{e}=\bar{\bm{r}}-\bar{\bm{b}}=\bar{\bm{g}}\,\bar{\odot}\,\bar{\bm{H}}\bar{\bm{x}}+\bar{\bm{g}}\,\bar{\odot}\,\bar{\bm{w}}+\bar{\bm{d}}\\\
=\bar{\bm{H}}_{e}\bar{\bm{x}}+\bar{\bm{w}}_{e},$ (10)

where $\bar{\bm{H}}_{e}\in\mathbb{C}^{N\times K}$ is the effective channel
matrix which takes the Bussgang gains into account and
$\bar{\bm{w}}_{e}\in\mathbb{C}^{N}$ is the effective noise vector which is the
combination of quantized thermal noise and quantization distortion. Now that
we defined the linearized version of the input-output relation, we can move on
to define the Bussgang-based MRC (BMRC) and ZF (BZF) filters as

$\bar{\bm{F}}_{\mathrm{BMRC}}=\bar{\bm{\lambda}}^{e}\odot\bar{\bm{H}}_{e}^{H}\textrm{
and
}\bar{\bm{F}}_{\mathrm{BZF}}=(\bar{\bm{H}}_{e}^{H}\bar{\bm{H}}_{e})^{-1}\bar{\bm{H}}_{e}^{H},$
(11)

where $\bar{\bm{\lambda}}^{e}$ is the scaling constant to make sure we obtain
a non-scaled estimate, and it is determined as
$\bar{\lambda}_{k}^{e}=\nicefrac{{1}}{{\sum_{n=1}^{N}|[\bar{\bm{H}}_{e}]_{(n,k)}|^{2}}}$,
for $k=1,2,\ldots,K$.

Once a filter $\bar{\bm{F}}$ to be applied on the conditionally zero-mean
observation vector $\bar{\bm{r}}_{e}$ is selected, the estimates can be
obtained as

$\tilde{\bar{\bm{x}}}=\bar{\bm{F}}\bar{\bm{r}}_{e}.$ (12)

Finally, if no further processing is to be applied, decisions can be obtained
by symbol-by-symbol detection, i.e., element-wise minimum distance mapping.

###  III-B Conventional Linear Filters

Since one-bit measurement causes the loss of amplitude information, inserting
appropriate labels to our quantized observations is a helpful way to work with
variable-amplitude modulation schemes. To do so, we continue with the Bussgang
decomposition and look for approximations that make all entries of
$\bar{\bm{g}}_{n}$ the same in both the I and Q parts. To do so, we first
assume the threshold values are unknown and
$\bar{\bm{\tau}}\sim\mathcal{CN}(\bm{0}_{N},\sigma^{2}_{\tau})$. This can be a
useful assumption if the empirical distribution of the thresholds is close to
Gaussian. Then randomly selected thresholds can be seen as an additional form
of noise over additive white Gaussian noise (AWGN). Therefore the effective
noise variance becomes $\sigma^{2}_{e}=N_{0}+\sigma^{2}_{\tau}$ since the
thermal noise and threshold selections are independent. With this selection,
we can also act as if $\bar{\bm{\tau}}=\bm{0}_{N}$. Now, the effective
unquantized observation covariance matrix becomes
$\bar{\bm{C}}_{y,e}=\bar{\bm{H}}\bar{\bm{H}}^{H}+(N_{0}+\sigma^{2}_{\tau})\bm{I}_{N}$.

Finally, the wireless channel’s effect must be considered to obtain a label
that is the same for all branches. At low SNR or when the number of users is
large, the unquantized observation covariance matrix can be approximated by a
diagonal matrix such that
$\bar{\bm{C}}_{y,e}\cong(K+N_{0}+\sigma^{2}_{\tau})\bm{I}_{N}$. By using these
approximations, the bias vector becomes $\bar{\bm{b}}=\bm{0}_{N}$ and each
element of the Bussgang gain vector can be approximated as
$\bar{g}_{n}\cong\sqrt{\frac{4}{\pi(K+N_{0}+\sigma^{2}_{\tau})}}(1+j)$. Since
the I and Q parts are now the same, the input-output relation can be
simplified as

$\bar{\bm{r}}_{e}=\bar{\bm{r}}=\sqrt{\frac{4}{\pi(K+N_{0}+\sigma^{2}_{\tau})}}\bar{\bm{y}}+\bar{\bm{d}}.$
(13)

By defining the quantization label as
$\ell_{q}=\sqrt{\nicefrac{{\pi(K+N_{0}+\sigma_{\tau}^{2})}}{{4}}}$ and
multiplying both sides of (13) with $\ell_{q}$, we can get

$\ell_{q}\bar{\bm{r}}=\bar{\bm{y}}+\ell_{q}\bar{\bm{d}}=\bar{\bm{H}}\bar{\bm{x}}+\bar{\bm{w}}+\ell_{q}\bar{\bm{d}}.$
(14)

Using this quantization label, a non-scaled estimate of the $\bar{\bm{x}}$
vector can now be obtained when conventional linear filters are used. Finding
the conventional MRC and ZF filters is now straightforward. The MRC and ZF
filters can be calculated as

$\bar{\bm{F}}_{\mathrm{MRC}}=\bar{\bm{\lambda}}\odot\bar{\bm{H}}^{H}\textrm{
and
}\bar{\bm{F}}_{\mathrm{ZF}}=(\bar{\bm{H}}^{H}\bar{\bm{H}})^{-1}\bar{\bm{H}}^{H},$
(15)

where the elements of scaling vector $\bar{\bm{\lambda}}$ for $k=1,2,\ldots,K$
can be found as
$\bar{\lambda}_{k}[v]=\nicefrac{{1}}{{\sum_{n=1}^{N}|[\bar{\bm{H}}_{(n,k)}|^{2}}}$.

Once a filter of choice $\bar{\bm{F}}$ is selected, it is applied on the
quantized observation vector $\bar{\bm{r}}$ to obtain the estimates as

$\tilde{\bar{\bm{x}}}=\ell_{q}\bar{\bm{F}}\bar{\bm{r}}.$ (16)

Lastly, if no further processing is applied, decisions can be obtained using
element-wise minimum distance mapping.

##  IV Maximum Likelihood (ML) Detector

Likelihood-based detectors have an important advantage against linear
detectors in terms of bit error rate (BER) performance, especially at high SNR
[23]. As explained in different works such as [27, 16, 23], the conditional
probability mass function (PMF) of the quantized observations can be expressed
as

$\operatorname*{p}(\bm{r}|\bm{x},\bm{\tau},\bm{H})=\prod_{n=1}^{2N}\Phi\left(\frac{r_{n}(\bm{h}_{n}^{T}\bm{x}-\tau_{n})}{\sqrt{N_{0}/2}}\right),$
(17)

where $\bm{h}_{n}^{T}$ is the $n^{\textrm{th}}$ row of the channel matrix
$\bm{H}$, i.e.,
$\bm{H}=\begin{bmatrix}\bm{h}_{1}&\bm{h}_{2}&\ldots&\bm{h}_{2N}\end{bmatrix}^{T}$,
and $\Phi(\alpha)=\int_{-\infty}^{\alpha}\phi(\tau)\,d\tau$ is the cumulative
distribution function (CDF) of the standard Gaussian random variable. Then, by
utilizing the log-likelihood, we can construct the ML detector as

$\hat{\bm{x}}_{\textrm{ML}}=\operatorname*{argmax}_{\bm{x}\in\mathcal{M}^{2K}}\left\\{\bm{1}_{N}\ln\left(\Phi\left(\sqrt{\frac{2}{N_{0}}}\bm{r}\odot(\bm{H}\bm{x}-\bm{\tau})\right)\right)\right\\},$
(18)

where the natural logarithm $\ln(.)$ and $\Phi(.)$ are applied element-wise on
their arguments.

##  V Proposed First Stage: Boxed Newton Detector (BND)

When the modulation order $M$ or the number of users $K$ is large, complexity
becomes infeasible for ML detection. Different works in the literature focus
on a gradient-based approach to maximize the log-likelihood function [27, 16,
23, 29]. In this section, a similar procedure is followed with a new iterative
receiver that uses a modified cost function and the Hessian information for
faster convergence via Newton’s method. For compactness, we denote the log-
likelihood as

$\mathcal{L}(\bm{x})=\bm{1}_{N}\ln\left(\Phi\left(\sqrt{\frac{2}{N_{0}}}\bm{r}\odot(\bm{H}\bm{x}-\bm{\tau})\right)\right),$
(19)

where $\bm{1}_{N}$ is a vector of size $N$ whose elements are all set to $1$.
We approximate the ML detection problem as a new optimization problem where we
relax the constraint that $\bm{x}$ belongs to a discrete input set and add a
penalty function to reflect the box constraints related to the boundary of the
chosen constellation scheme to compensate for the relaxation and obtain
favorable convergence. The penalty method is useful for keeping the
formulation as an unconstrained optimization problem while using the
constellation boundaries as a priori information. The boundary of an $M$-QAM
type constellation with unit average power on a single dimension can be
calculated as

$M_{b}=\sqrt{\frac{3(\sqrt{M}-1)^{2}}{2(M-1)}}.$ (20)

The new unconstrained optimization problem formulation with the penalty
function $\mathcal{P}(.)$ can be expressed as

$\tilde{\bm{x}}=\operatorname*{argmax}_{\bm{x}\in\mathbb{R}^{2K}}\\{\mathcal{L}(\bm{x})-\mathcal{P}(\bm{x})\\},$
(21)

where the penalty function reflects how much deviation occurs from the
boundaries of the constellation scheme which can be expressed as

$\mathcal{P}(\bm{x})=\frac{\theta}{2}\sum_{k=1}^{2K}\operatorname*{R}(|x_{k}|-M_{b})^{2},$
(22)

where $\theta\in\mathbb{R}$ is a constant that determines the strength of box
constraints, and $\operatorname*{R}(x)=\max\\{0,x\\}$ is the unit-ramp
function. Gradient-based optimization can be utilized for (21) due to the log-
concavity of $\Phi(.)$ and the negative of the penalty function being concave
as well. By denoting the argument of (21) as our cost function
$\mathcal{J}(\bm{x})=\mathcal{L}(\bm{x})-\mathcal{P}(\bm{x})$, the iterative
update equation of Newton’s Method becomes

$\bm{x}^{k+1}=\bm{x}^{k}-(\nabla^{2}\mathcal{J}(\bm{x}^{k}))^{-1}\nabla\mathcal{J}(\bm{x}^{k}),$
(23)

where $\nabla=\begin{bmatrix}\frac{\partial}{\partial
x_{1}}&\ldots&\frac{\partial}{\partial x_{2K}}\end{bmatrix}^{T}$ can be
expressed as a vector of size $2K$, and $\nabla^{2}$ can be expressed as the
outer product of the $\nabla$ operator with itself:
$\nabla^{2}=\nabla\,\nabla^{T}$. Hence, the $\nabla^{2}$ operator can be
considered a $2K\times 2K$ matrix. $\nabla\mathcal{J}(\bm{x}^{k})$ is the
gradient of the cost function $\mathcal{J}$ with respect to $\bm{x}$
calculated at $\bm{x}^{k}$, i.e., the estimate of $\bm{x}$ at iteration $k$.
Similarly, $\nabla^{2}\mathcal{J}(\bm{x}^{k})$ is the Hessian of the cost
function $\mathcal{J}$ with respect to $\bm{x}$ calculated at $\bm{x}^{k}$.

Utilizing a second-order optimization technique such as Newton’s method for
this problem helps for getting fast convergence and avoids the burden of
selecting different step-sizes for different system setups as opposed to [27,
28]. Machine learning tools have been used to avoid this issue [23, 29].
However, these methods require an offline learning stage and larger number of
iterations compared to Newton’s method, making these approaches
computationally comparable. We should go over the log-likelihood and penalty
functions’ first and second-order derivatives before expressing the gradient
and Hessian information. The first-order derivatives of the log-likelihood
function and the penalty function can be expressed as

$\frac{d}{dx_{k}}\mathcal{L}(\bm{x})=\sqrt{\frac{2}{N_{0}}}\sum_{n=1}^{2N}r_{n}[\bm{H}]_{(n,k)}\varphi\left(u_{n}\right),$
(24)

$\frac{d}{dx_{k}}\mathcal{P}(\bm{x})=\theta\operatorname*{sign}(x_{k})\operatorname*{R}(|x_{k}|-M_{b}),$
(25)

respectively, for $k=1,2,\ldots,2K$, where
$\bm{u}=\sqrt{\frac{2}{N_{0}}}\bm{r}\odot(\bm{H}\bm{x}-\bm{\tau})$, and
$\varphi(x)=\frac{d}{dx}\ln(\Phi(x))=\frac{\phi(x)}{\Phi(x)}$. Then the
second-order derivatives can be obtained as

$\frac{d^{2}}{dx_{k}dx_{m}}\mathcal{L}(\bm{x})=\frac{2}{N_{0}}\sum_{n=1}^{2N}[\bm{H}]_{(n,k)}[\bm{H}]_{(n,m)}\psi\left(u_{n}\right),$
(26)

$\frac{d^{2}}{dx_{k}dx_{m}}\mathcal{P}(\bm{x})=\theta\operatorname*{U}(|x_{k}|-M_{b})\delta[k-m],$
(27)

for $k=1,2,\ldots,2K$ and $m=1,2,\ldots,2K$, where
$\psi(x)=\frac{d^{2}}{dx^{2}}\ln(\Phi(x))=-x\varphi(x)-\varphi^{2}(x)$,
$\operatorname*{U}(x)=\max\\{0,\operatorname*{sign}(x)\\}$ is the unit-step
function, and $\delta[.]$ is the discrete unit impulse function.

Due to linearity of differentiation, we have
$\nabla\mathcal{J}(\bm{x})=\nabla\mathcal{L}(\bm{x})-\nabla\mathcal{P}(\bm{x})$.
As in [16], the gradient of the log-likelihood function can be found as

$\nabla\mathcal{L}(\bm{x})=\sqrt{\frac{2}{N_{0}}}\bm{H}^{T}\left(\bm{r}\odot\varphi\left(\bm{u}\right)\right),$
(28)

where $\odot$ denotes the Hadamard product, and $\varphi(.)$ is applied
element-wise on its arguments. Then, the gradient of the penalty function can
be found as

$\nabla\mathcal{P}(\bm{x})=\theta\operatorname*{sign}(\bm{x})\odot\operatorname*{R}(|\bm{x}|-M_{b}\bm{1}_{2K}),$
(29)

where $\operatorname*{R}(.)$ is applied element-wise on its arguments.
Similarly, we have
$\nabla^{2}\mathcal{J}(x)=\nabla^{2}\mathcal{L}(x)-\nabla^{2}\mathcal{P}(x)$.
The Hessian of the log-likelihood function can be calculated as

$\nabla^{2}\mathcal{L}(\bm{x})=\frac{2}{N_{0}}\bm{H}^{T}\operatorname*{diag}\left(\psi\left(\bm{u}\right)\right)\bm{H},$
(30)

where $\psi(.)$ is applied element-wise on its arguments. Since $\Phi(x)$ can
approach zero exponentially fast, computations of $\ln(\Phi(x))$,
$\varphi(x)$, and $\psi(x)$ in finite precision can cause problems such as
divergent behavior or uncertainties. This problem is solved in Appendix A.
Finally, the Hessian of the penalty function is

$\nabla^{2}\mathcal{P}(\bm{x})=\theta\operatorname*{diag}(\operatorname*{U}(|\bm{x}|-M_{b}\bm{1}_{2K})),$
(31)

where $\operatorname*{U}(.)$ is applied element-wise on its arguments.

Algorithm 1 Boxed Newton Detector (BND)

0:
$\bm{r},\bm{H},\bm{\tau},\theta,\epsilon,M_{b},T_{\textrm{max}},N_{0},\zeta$

0: $\tilde{\bm{x}}$

1: Set the initial solution to the MRC estimate
$\tilde{\bm{x}}\leftarrow\bm{x}^{0}$ using (15)

2: Apply the damping factor $N_{0}\leftarrow\zeta N_{0}$

3: for $i=1$ to $T_{\textrm{max}}$ do

4: Calculate the gradient $\nabla\mathcal{J}(\tilde{\bm{x}})$ using (28) and
(29)

5: Calculate the Hessian $\nabla^{2}\mathcal{J}(\tilde{\bm{x}})$ using (30)
and (31)

6: Calculate the step
$\Delta\bm{x}\leftarrow(\nabla^{2}J(\tilde{\bm{x}}))^{-1}\nabla
J(\tilde{\bm{x}})$

7: Iterative update $\tilde{\bm{x}}\leftarrow\tilde{\bm{x}}-\Delta\bm{x}$ as
in (23)

8: if $\lVert\Delta\bm{x}\rVert^{2}<\epsilon\,\lVert\bm{x}\rVert^{2}$ then

9: return $\tilde{\bm{x}}$

10: end if

11: end for

12: return $\tilde{\bm{x}}$

Now that the update rule is wholly defined, an initial solution $\bm{x}^{0}$
that is preferably not far away from the final solution should be found. The
MRC estimate is a suitable selection for this purpose, and it can easily be
found using (15). Note that at high SNR, when $N_{0}$ is very small, the
Hessian matrix can become very close to singular since $\psi(x)\rightarrow 0$
as $x\rightarrow\infty$. To avoid such behavior, we define a damping factor
$\zeta$ such that

$\zeta=\max\\{1,\rho/\rho_{d}\\},$ (32)

where $\rho_{d}$ is the damping SNR. Before starting the iterative updates,
$N_{0}$ is multiplied with this term to provide numeric stability. The
complete procedure for BND is summarized in Algorithm 1. Once the iterative
updates start, the algorithm is terminated if the maximum number of iterations
$T_{\mathrm{max}}$ is reached or if further iterations do not cause
significant changes, which is determined by the termination threshold
$\epsilon$. The output of the algorithm $\tilde{\bm{x}}$ is the first-stage
solution, and if a one-stage approach is to be followed, then symbol-by-symbol
detection is applied on the estimate. If not, the estimate is supplied to the
second stage for further processing.

##  VI Proposed Second Stage: Nearest Codeword Detector (NCD)

After finding an estimate using a first-stage method, NCD can be utilized to
make more accurate decisions compared to symbol-by-symbol detection. In this
part, the first step is to decide on the reliability of each element of the
first-stage estimate. Then, a set of candidate vectors is formed based on the
reliability information of each element, very similar to the ideas from [27,
30, 23]. The candidate set is then narrowed down based on the minimum Hamming
distance criterion in the codeword domain. Finally, ML detection is conducted
on the smaller candidate set to make the final decisions. The complete summary
of the proposed second stage detector is made in Algorithm 2, and the detailed
explanation of the whole procedure is made in the following parts.

Algorithm 2 Nearest Codeword Detector (NCD)

0: $\tilde{\bm{x}},\bm{r},\bm{H},\bm{\tau},P,\gamma,U_{\mathrm{max}}$

0: $\hat{\bm{x}}$

1: Find the nearest decision boundaries $\bm{t}$ as in (34)

2: Obtain the sets of reliable indices $\mathcal{R}$ as in (35) and unreliable
indices $\mathcal{U}$ as in (36)

3: while $|\mathcal{U}|>U_{\mathrm{max}}$ do

4: Decrease the size of the unreliable region as $\gamma\leftarrow 0.95\gamma$

5: Reobtain the sets $\mathcal{R}$ and $\mathcal{U}$ according to the new
$\gamma$

6: end while

7: Find the candidate element sets $(\mathcal{X}_{k})_{k=1}^{2K}$ using (37)

8: Generate the candidate vector set $\mathcal{X}$ using (38)

9: if $|\mathcal{X}|>P$ then

10: Apply symbol-by-symbol detection to get
$\check{x}_{k}\leftarrow\operatorname*{argmin}_{x\in\mathcal{M}}|\tilde{x}_{k}-x|$
for $k=1,2,\ldots,2K$

11: Remove the symbol-by-symbol detected vector from the set
$\mathcal{X}\leftarrow\mathcal{X}\setminus\\{\check{\bm{x}}\\}$

12: if $|\mathcal{X}|>1$ then

13: Sort $\mathcal{X}$ according to the $\mathscr{H}(.)$ metric as in (41)

14: Discard all elements of $\mathcal{X}$ except the first $P-1$

15: end if

16: Add $\check{\bm{x}}$ back to the set to obtain the result from (42)
$\mathcal{X}\leftarrow\mathcal{X}\cup\\{\check{\bm{x}}\\}$

17: end if

18: Apply ML detection on the set $\mathcal{X}$

19: return $\tilde{\bm{x}}$

The set of decision boundaries utilized during symbol-by-symbol detection for
$M$-QAM constellations is found as

$\mathcal{T}=\left\\{\pm
n\sqrt{\frac{6}{M-1}}\,\bigg{|}\,n\in\left\\{0,1,\ldots,\frac{\sqrt{M}-2}{2}\right\\}\right\\}.$
(33)

For example, the decision boundary set for QPSK constellation is $\\{0\\}$,
and for $16$-QAM constellation, it is $\left\\{0,\pm
2\sqrt{\frac{1}{10}}\right\\}$. Then, the closest decision boundary to the
$k^{\textrm{th}}$ element of the estimate $\tilde{\bm{x}}$ is defined as

$t_{k}=\operatorname*{argmin}_{t\in\mathcal{T}}|\tilde{x}_{k}-t|,$ (34)

for $k=1,2,\ldots,2K$. Then, the set of reliable and unreliable indices can be
found as

$\mathcal{R}=\\{k\mid
k\in\\{1,2,\ldots,2K\\},\,|\tilde{x}_{k}-t_{k}|>\gamma\\},$ (35)

$\mathcal{U}=\\{k\mid k\in\\{1,2,\ldots,2K\\}\setminus\mathcal{R}\\},$ (36)

respectively. If the $k^{\textrm{th}}$ element of the estimate is within a
certain margin of its closest decision boundary defined by the hyperparameter
$\gamma\in\mathbb{R}$, then it is not reliable to conduct symbol-by-symbol
detection on $\tilde{x}_{k}$, for $k=1,2,\ldots,2K$. Now, the set of possible
assignments for each element of the estimate can be found as

$\tilde{\mathcal{X}}_{k}=\left\\{\begin{array}[]{ll}\left\\{\operatorname*{argmin}_{x\in\mathcal{M}}|x_{k}-x|\right\\},&\quad
k\in\mathcal{R}\\\ \\\ \left\\{t_{k}\pm\sqrt{\frac{3}{2(M-1)}}\right\\},&\quad
k\in\mathcal{U}\end{array}\right.$ (37)

for $k=1,2,\ldots,2K$. If the $k^{\textrm{th}}$ element of the estimate is
reliable, then there is only one possible assignment to this element obtained
by the minimum distance rule. If it is unreliable, the possible assignment is
a set composed of two elements that are the neighbors to the closest decision
boundary. Hence, each set’s cardinality can be $1$ or $2$. The resultant set
of candidate vectors from the combinations of each candidate element set can
be obtained as

$\tilde{\mathcal{X}}=\tilde{\mathcal{X}}_{1}\times\tilde{\mathcal{X}}_{2}\times\ldots\times\tilde{\mathcal{X}}_{2K},$
(38)

where $\times$ denotes the Cartesian product operator. Examples of the set
formation procedures shown in (37) and (38) can be found in [23].

The cardinality of the candidate vector set $|\tilde{\mathcal{X}}|$ is at most
$2^{2K}$, which means the set can grow exponentially as the number of users
increases, which can cause two problems. First, the size of the set may get
too large when $|\mathcal{U}|$ is large, which can cause memory problems and
an undesirable complexity increase for ML detection at the final step. The
first problem is dealt with adaptively changing $\gamma$, which can adjust the
size of the unreliable region. If the size of $\mathcal{U}$ is large, i.e., it
is greater than $U_{\mathrm{max}}$, then decreasing $\gamma$ can help us
obtain a smaller unreliable region, hence a smaller set size for
$\tilde{\mathcal{X}}$. For the second problem, similar to the idea from [23],
the aim is to limit the search complexity by finding a subset $\mathcal{X}$ of
$\tilde{\mathcal{X}}$ that includes the most likely candidates. In [27] and
[30], the search complexity is not limited in the second stage, and in [23], a
limited number of nearest neighbors to the estimate vector from the candidate
set $\tilde{\mathcal{X}}$ are found. However, especially at high SNR,
detection performance can benefit from the sign constraints imposed by one-bit
quantization. Since quantized observations take binary values, $\bm{r}$ can
easily be seen as a codeword in the spatial domain. Hence, the size of the set
of candidate vectors can be limited by finding their codeword representations,
then ordering them according to their Hamming distance to the actual quantized
observation vector. The spatial codeword representation $c(.)$ of a candidate
vector $\acute{\bm{x}}\in\tilde{\mathcal{X}}$ is defined as

$c(\acute{\bm{x}})\triangleq\operatorname*{sign}\left(\mathbb{E}[\,\acute{\bm{y}}-\bm{\tau}\mid\acute{\bm{x}},\bm{H},\bm{\tau}\,]\right)=\operatorname*{sign}\left(\bm{H}\acute{\bm{x}}-\bm{\tau}\right),$
(39)

where $\acute{\bm{y}}$ is the unquantized observation vector obtained when
$\acute{\bm{x}}$ is sent. The following is defined for ease of notation:

$\mathscr{H}(\acute{\bm{x}})\triangleq\operatorname*{d_{H}}(\bm{r},c(\acute{\bm{x}})),$
(40)

where $\operatorname*{d_{H}}\\{\,.\,,\,.\,\\}$ denotes the Hamming distance
between its arguments. Now, the aim is to create a set
$\mathcal{X}\subseteq\tilde{\mathcal{X}}$ whose cardinality is a
hyperparameter denoted as $P$ and it consists the symbol-by-symbol detected
vector $\check{\bm{x}}$ whose $k^{\textrm{th}}$ element is
$\check{x}_{k}=\operatorname*{argmin}_{x\in\mathcal{M}}|\tilde{x}_{k}-x|$ for
$k=1,2,\ldots,2K$, and the $P-1$ candidate vectors from $\tilde{\mathcal{X}}$
such that the Hamming distance between their spatial codeword representations
and $\bm{r}$ are the smallest. Including the symbol-by-symbol detection result
as a candidate is enforced due to favorable performance, especially at low
SNR. Next, in order to sort the candidate vectors according to the Hamming
distance between their spatial codeword representations and the quantized
observation vector, an ordered vector sequence is defined as

$(\acute{\bm{x}}_{i})_{i=1}^{|\mathcal{A}|}\in\tilde{\mathcal{X}}\setminus\\{\check{\bm{x}}\\}\>\textrm{such
that}\>\mathscr{H}(\acute{\bm{x}}_{i})\leq\mathscr{H}(\acute{\bm{x}}_{i+1}).$
(41)

Sequence indexing is a useful way to find the first $P-1$ candidates. Indexing
may not be unique and encountering a situation such as
$\mathscr{H}(\acute{\bm{x}}_{P-1})=\mathscr{H}(\acute{\bm{x}}_{P})$ is
possible. However, the performance gain obtained by the second stage increases
as SNR increases, and for large enough $K$, a situation such as
$\mathscr{H}(\bm{x})=\mathscr{H}(\acute{\bm{x}}_{P-1})$ is not likely. The
final set $\mathcal{X}$ can be found as

$\mathcal{X}=\\{\acute{\bm{x}_{1}},\acute{\bm{x}}_{2},\ldots,\acute{\bm{x}}_{P-1}\\}\cup\\{\check{\bm{x}}\\}.$
(42)

Finally, ML detection (18) is applied on the reduced set $\mathcal{X}$ instead
of $\mathcal{M}^{2K}$ to obtain the second-stage solution.

##  VII Pseudo-Random Quantization (PRQ) Scheme

###  VII-A Proposed Quantization Strategy

Generating artificial noise with a particular distribution is one way of
dithering the input signal. However, the same dithering effect can be achieved
by shifting the quantization thresholds. Dithering can also be called
randomized quantization, with both random and pseudo-random [15] applications.
In Fig. 2a, the conventional version of dithering is illustrated where a
randomly generated analog signal is subtracted from the incoming analog signal
with the help of an additional DAC. PRNG is a pseudo-random number generator
and helps obtain random sequences from a selected probability distribution. As
seen in the figure, the advantage of PRQ is that we can utilize the knowledge
of the dither signal/threshold during the detection operation. However,
employing an additional DAC in the RF chain can be undesirable.

(a) Conventional Dithering

(b) Proposed Dithering

Figure 2: Illustrations of random (without the dashed line connection) and
pseudo-random (with the dashed line connection) quantization schemes with the
conventional (a) and the proposed (b) dithering architectures.

To obtain a more efficient system, we can rely on variable-threshold
quantizers to help get the same effect as generating the dither signal and
subtracting it from the received signal. The resulting setup is shown in Fig.
2b. As seen in the figure, we rely on changing the quantization threshold
instead of using an additional DAC. However, in practice, the response time of
the ADC can become a burden if the threshold is varied at every symbol period
or even every frame. Hence, keeping the thresholds constant during channel
coherence time would be preferable. Since we deal with massive MIMO with large
antenna arrays, the distribution of the dither signals, i.e., quantization
thresholds is important rather than their different realizations due to the
law of large numbers (LLN). Hence, we change the domain of dithering from
temporal (time) to spatial (antennas) with this approach. From the circuit
design perspective, since one-bit ADCs operate with basic comparator units
that compare two voltage values to output a digital value, it should not be a
challenging issue to compare the incoming voltage to a non-zero voltage value
that can be set easily. Moreover, this is a more challenging issue for high-
resolution ADCs where the quantization thresholds are adjusted continuously
with the help of AGCs depending on the received signal power.

We propose a new quantization scheme for uplink one-bit massive MIMO systems
to overcome the adverse effects of the SR phenomenon at high SNR, where either
the performance peaks at a unique SNR value or a performance saturation occurs
after a finite SNR. When a dithering scheme is applied in the system, the
distribution of the dither signal, the threshold values for our scenario, must
be selected. By observing different scenarios, we saw that selecting the
Gaussian distribution is a useful technique. Then, the mean and variance of
the thresholds must also be selected. Since we are trying to detect signals
with zero-mean and our unquantized observations are zero-mean, the intuitive
selection is to generate the threshold with zero-mean. The selection of the
variance parameter is more challenging than the mean since it is hard to
analytically track the effects of the empirical distribution of a pseudo-
random selection. If the variance is too small, the performance is not
affected. If it is too large, the performance gets worse since the threshold
values become too large to differentiate relatively small amplitude
differences. After many trials and observations, we saw that increasing $N$
requires the variance to be increased, whereas increasing $K$ requires the
variance to be decreased. Also, the PRQ approach does not yield any benefits
and causes performance degradation below a particular SNR value. Hence,
gradually increasing the threshold variance after a threshold SNR value
$\rho_{t}$ is an appropriate technique. In order to avoid selecting different
variances for different scenarios, we parameterize the threshold SNR value as

$\rho_{t}=5\log_{10}\left(100\frac{K}{N}\right)\textrm{ dB}.$ (43)

(a) Threshold Variance

(b) SNR

Figure 3: BER performance of the ML detector with respect to threshold
variance at $\rho=20$ dB (a) and SNR with ZTQ and PRQ performances according
to the chosen threshold SNR from (43).

This selection may not be optimal and relies only on empirical findings and
observations. Now that the threshold SNR value is selected, the quantization
threshold variance can be found accordingly using

$\sigma_{\tau}^{2}=\operatorname*{R}\left(\frac{E_{s}}{\rho_{t}}-N_{0}\right)=\operatorname*{R}\left(\frac{1}{\rho_{t}}-N_{0}\right).$
(44)

Upon generation, rescaling the thresholds to strictly satisfy
$\lVert\bm{\tau}\rVert^{2}=N\sigma_{\tau}^{2}$ is helpful to obtain more
stable results. The proposed PRQ scheme does not require updates for different
channel realizations and depends only on the SNR. In Fig. 3a the effect of
changing the threshold variance is investigated for different system setups
using the ML detector in Rayleigh fading channel at a fixed SNR value. As can
be seen, the threshold variance becomes an important factor for which a very
high or a very low selection yields poor performance. Note that even though
the selection $\sigma^{2}_{\tau}=1$ seems to be yielding the best performance,
the difference between different scenarios become more evident when smaller
SNR steps are used during simulations and when the number of users is selected
to be higher. Note that from this point on, the threshold variance selections
are made according to (44). In Fig. 3b, the BER curves obtained with different
system setups are shown both with ZTQ and PRQ in the Rayleigh fading channel.
The peak performance is achieved, i.e., SR occurs, between 5-10 dB of SNR for
each system with ZTQ. The two-user performance with $16$-QAM is better than
that of the single-user at high SNR with ZTQ, which shows how multi-user
interference (MUI) can act as a dither source. With PRQ, we not only stop the
performance degradation but also obtain superior performance while approaching
and at the SR point.

###  VII-B Achievable Rate Analysis in One-Bit SIMO

We have seen that the thresholds should be optimized according to the given
channel realization for the conventional MIMO systems, i.e., when the number
of antennas is small. Thankfully, massive MIMO systems can benefit from a more
robust approach due to a large number of antennas, and important performance
gains can be obtained without optimizing the thresholds and utilizing PRQ.
Threshold optimization is also tricky since the mutual information or the
error probability should be optimized. For our setup, the analytical
expression for the conditional mutual information can be written as

$\mathcal{I}(\bm{r};\bm{x}\mid\bm{H},\bm{\tau})=\frac{-1}{M^{K}}\sum_{\bm{x}_{1}\in\mathcal{M}^{2K}}\sum_{\bm{r}\in\\{\pm
1\\}^{2N}}\operatorname*{p}(\bm{r}\mid\bm{x}_{1},\bm{H},\bm{\tau})\log_{2}\left(\frac{1}{M^{K}}\sum_{\bm{x}_{2}\in\mathcal{M}^{2K}}\operatorname*{p}(\bm{r}\mid\bm{x}_{2},\bm{H},\bm{\tau})\right)\\\
-\frac{1}{M^{K}}\sum_{\bm{x}_{3}\in\mathcal{M}^{2K}}\sum_{n=1}^{2N}\mathcal{H}_{b}\left(\Phi\left(\sqrt{2\rho}(\bm{h}_{n}^{T}\bm{x}_{3}-\tau_{n})\right)\right),$
(45)

where $\mathcal{H}_{b}(.)$ is the binary entropy function. A detailed
derivation of mutual information is given in Appendix B. Finding the optimal
thresholds is difficult since the solution is not necessarily unique, and the
dimensionality is very high for the massive MIMO setup. To observe the effects
of PRQ, we resort to random sampling by utilizing the Monte Carlo method. For
the simulations, we aim to find the achievable rate with ZTQ and PRQ in a SIMO
system where a BS equipped with $4$ antennas is serving a single user in AWGN
and Rayleigh fading channels. The conditional mutual information for any given
channel realization is calculated for $500$ random realizations of the
thresholds for the PRQ scheme and $1$ with zero thresholds for ZTQ. The
average rate and the maximum rate obtained during trials for each channel are
recorded with PRQ. Since the AWGN channel is deterministic, only a single
channel realization is utilized. For the Rayleigh channel, the average is
calculated over $1000$ channel realizations.

(a) AWGN Channel

(b) Rayleigh Channel

Figure 4: Mutual information plotted against SNR for both AWGN (a) and
Rayleigh (b) channels where $N=4$, $K=1$, and $M=4,16,64$ with one-bit ZTQ
(lines) and PRQ (markers) schemes. The average and the maximum rate obtained
with the PRQ scheme are recorded.

The mutual information plots with respect to SNR are shown in Fig. 4a for the
AWGN channel and in Fig. 4b for the Rayleigh fading channel. The performances
in both channels exhibit no significant difference when the QPSK constellation
is used. This is an intuitive result since each pair of ADCs from each antenna
divides the 2D space into four regions, the cardinality of the QPSK alphabet
is also four. However, the behavior changes drastically for 16-QAM and 64-QAM.
The SR phenomenon is visible in the AWGN channel with $16$-QAM and $64$-QAM
constellations. We can no longer observe SR in this $4$-antenna scenario for
the Rayleigh fading channel. This seems to be due to the amplitude and phase-
varying nature of the Rayleigh fading channel, which helps obtain different
instantaneous SNR values at each branch. Nonetheless, the PRQ scheme
outperforms the ZTQ scheme for both channel types, even on average. If the
thresholds are optimized with respect to the given channel realization, the
rate can be increased further in this low-dimensional setup.

###  VII-C Minimum Hamming Distance Analysis

Calculating mutual information for the massive MIMO setup is not feasible due
to very large dimensionality. Therefore, another approach is followed to grasp
how the PRQ scheme works for massive MIMO systems using an intuition from
coding theory [33]. Quantized observations can be seen as binary codewords in
space for each given channel realization. We can only intervene in the
encoding process by changing the thresholds. The resultant spatial channel
code will determine our error performance. A measure of the performance of a
code is its minimum Hamming distance, which determines the diversity order of
a code in fading channels [37]. In this part, we will investigate the average
of the minimum Hamming distance of the space code obtained in the Rayleigh
fading channel via Monte Carlo simulations at infinite SNR with both ZTQ and
PRQ schemes. The infinite SNR scenario is chosen to understand the high SNR
behavior, which is a critical part of our purposes. For a given channel and a
given threshold realization, the minimum Hamming distance of the codebook is
calculated for the infinite-SNR scenario as

$d_{\mathrm{H}}^{\,\mathrm{min}}=\min_{\bm{x}_{1},\bm{x}_{2}\in\mathcal{M}^{2K}}\operatorname*{d_{H}}(c(\bm{x}_{1}),c(\bm{x}_{2})),$
(46)

where $c(.)$ is the spatial codeword representation of its argument vector, as
in (39).

TABLE I: Average Minimum Hamming Distance of the Space Code in the Rayleigh
Fading Channel

$N$ | $K$ | $M$ | with ZTQ | with PRQ
---|---|---|---|---
64 | 1 | 4 | 64.00 | 47.59
64 | 2 | 4 | 35.41 | 28.98
64 | 1 | 16 | 0.00 | 11.92
64 | 2 | 16 | 0.00 | 5.03
64 | 1 | 64 | 0.00 | 1.85
128 | 1 | 16 | 0.00 | 25.63
128 | 1 | 64 | 0.00 | 6.39

The average of the minimum Hamming distance in the Rayleigh fading channel is
obtained for both ZTQ and PRQ scenarios where $10^{4}$ randomly generated
channel realizations are matched with randomly generated quantization
thresholds. Table I shows the results for different scenarios. An interesting
result obtained from these simulations is that when ZTQ is employed with
$M\geq 16$, the minimum Hamming distance of the space code is $0$, which means
that the code is not uniquely decodable. Errors will indeed occur during the
decoding operation due to the SR phenomenon; however, if we utilize the PRQ
scheme, which gives us a chance to intervene in the codebook design, the
minimum Hamming distance of the code increases. Hence, a superior performance
compared to the ZTQ scheme will be obtained. QPSK modulation does not benefit
from PRQ, as was the case during the mutual information calculations. Also, a
larger alphabet size leads to a smaller minimum Hamming distance, whereas a
larger number of antennas, i.e., larger codeword length, leads to an increased
minimum Hamming distance for the space code with PRQ. Since the rate of the
code is $R_{c}=\frac{K\log_{2}(M)}{2N}$, increasing $K$ or $M$ results in a
lower, and increasing $N$ leads to a larger minimum Hamming distance.

##  VIII Computational Complexity Analysis

Even though the gradient descent does not require matrix inversion, Newton’s
method requires much fewer iterations to reach the optimum [38, 39]. The part
that dominates the complexity of BND is the calculation of the step towards
the optimum. Calculating the Hessian matrix dominates the complexity with
$\mathcal{O}(NK^{2})$ since $N\gg K$ for massive MIMO systems. Since this
procedure is repeated until the termination of the algorithm, the resulting
complexity is $\mathcal{O}(NK^{2}T)$, where $T$ denotes the number of
iterations. Note that computation of the nonlinear function $\psi(.)$ can be
made using only $\varphi(.)$ since they share the same arguments. The average
number of iterations of BND is around $3-5$, and the number of unreliable
indices for NCD is generally small at high SNR. Therefore, the complexity-
dominant part of NCD is the last part where ML detection is applied on the
reduced set with cardinality $P$. Even though we let $P$ get as large as $2K$
during simulations, the average cardinality of the candidate set is small at
high SNR, resulting in a complexity of $\mathcal{O}(NK^{2})$ also for NCD.

TABLE II: Computational Complexity Analysis

Method | Preprocessing | Stage 1 | Stage 2
---|---|---|---
BND-NCD | - | $\mathcal{O}(NK^{2}T)$ | $\mathcal{O}(NK^{2})$
MRC/BMRC | $\mathcal{O}(NK)$ | $\mathcal{O}(NK)$ | -
ZF/BZF | $\mathcal{O}(NK^{2})$ | $\mathcal{O}(NK)$ | -
ML | $\mathcal{O}(NK|\mathcal{M}|^{2K})$ | $\mathcal{O}(N|\mathcal{M}|^{2K})$ | -
SVM-based | - | $\mathcal{O}(NK\kappa(N))$ | -
OBMNet-NNS | Offline Training | $\mathcal{O}(NKL)$ | $\mathcal{O}(\max(M,N)KM)$

The complexities of the proposed BND-NCD and the existing detectors from the
literature are shown in Table II. MRC-based detectors offer the lowest
complexity with poor error performance. Then comes ZF-based detectors that
offer better but not adequate performance and with higher computational
complexity due to matrix inversion. The ML detector has the highest
complexity, which is not feasible to operate in commercial systems. SVM-based
detector from [30] reportedly has complexity $\mathcal{O}(NK\kappa(N))$, where
$\kappa(.)$ is a super-linear function, and large values of $N$ may lead to
unaffordable complexity. OBMNet from [23], reportedly has complexity
$\mathcal{O}(NKL)$ in its first stage, where $L$ is the number of layers of
the proposed deep neural network (DNN) architecture, and complexity of
$\mathcal{O}(\max(M,N)KM)$ in its second stage nearest neighbor search (NNS)
algorithm. Even though OBMNet does not require matrix inversion, the neural
network needs an offline training stage, and a much fewer number of iterations
is required by BND, which makes the complexities of the detectors comparable.
Also, the algorithm can be terminated early when BND’s stopping criterion is
satisfied. In contrast, the number of layers defined as part of the deep
unfolded network architecture is a constant for OBMNet. Finally, the second
stage NNS algorithm of [23] is a recursive algorithm with many for-loops. NCD
omits for-loops and calculates the candidate set efficiently.

##  IX Simulation Results

In this section, the error performance of the proposed methods is
investigated. For BND, the maximum number of iterations $T_{\mathrm{max}}=15$,
the damping factor $\zeta$ is calculated by taking
$\rho_{d}=20-160\frac{K}{N}$ dB, the penalty control parameter $\theta=20$,
and the termination threshold $\epsilon=\frac{10^{-4}}{\log_{2}(M)}$. For NCD,
the maximum size of the set of nearest spatial codewords $P=2K$, to adjust the
size of the unreliable region, $\gamma=d/2$ with $d=\sqrt{\frac{3}{2(M-1)}}$,
and the maximum size of set of unreliable indices $U_{\mathrm{max}}=K$.

###  IX-A Verification of the Convergence of BND

(a) BER

(b) Cost Function (21)

Figure 5: Empirical convergence analysis using the BER performance (a) and
negative of the cost function (b) to be optimized with respect to number of
iterations of the proposed first stage detector BND at $\rho=30$ dB with
different system setups where solid lines reflect the results obtained with
ZTQ and dashed lines are for PRQ.

We start by verifying the convergence of the proposed first stage, BND. In
Fig. 5, the error performance (a) and the negative of the cost function (b)
from (21) with respect to the number of iterations are plotted for different
system setups with both ZTQ and PRQ. As can be seen in the plots, the
convergence time depends on the system setup. However, the algorithm converges
within 2-4 iterations for each scenario.

###  IX-B One and Two-Stage ZTQ and PRQ Performances

(a) ZTQ

(b) PRQ

Figure 6: One and two-stage BER performances of the linear and the proposed
detectors in a $128\times 4$ system with $16$-QAM both with ZTQ (a) and PRQ
(b) schemes, where linear detectors are also matched with the proposed NCD for
the second stage.

We now examine he one-stage and two-stage performances of the linear detectors
and the proposed BND-NCD. The linear detectors are also matched with the
proposed NCD at the second stage. The resultant BER curves obtained by ZTQ and
the proposed PRQ schemes are shown in Fig. 6. BND outperforms the linear
detectors, ZF-type receivers top their MRC-type counterparts, and Bussgang-
based receivers perform slightly better than the conventional linear
receivers. The performance gap between the conventional and Bussgang-based
linear detectors is smaller than the previously reported works from the
literature [22, 23] due to the quantization label $\ell_{q}$. Substantial
gains can be obtained by utilizing a two-stage approach for all scenarios. The
linear detectors do not benefit much from PRQ, and only the second stage
performance of BZF differs where a slightly lower error floor is obtained. SR
is visible for all detectors with ZTQ, whereas it is no longer observed for
BND and BND-NCD with PRQ.

###  IX-C Effect of Changing the Number of Users and the Number of BS antennas

(a) Number of Users

(b) Number of BS Antennas

Figure 7: BER performance of BND-NCD obtained by ZTQ and PRQ with respect to
the number of users (a) and the number of BS antennas for a SIMO system (b)
where $\rho=30$ dB with different system setups. For the second plot (b) only,
$U_{\mathrm{max}}=2$ and $P=\log_{2}(M)/2$.

Next, we investigate the effect of the number of users on the system’s
performance at a fixed high-SNR scenario with the proposed BND-NCD by
utilizing both ZTQ and PRQ in Fig. 7a. Starting with $K=1$, increasing the
number of users results in better performance up to a certain point with ZTQ,
which also shows how the multi-user scenario helps obtain dithering. As the
number of users increases, the performance gap between the ZTQ and PRQ schemes
decreases and they perform the same after some point. As $N$ increases, the
starting point where ZTQ and PRQ schemes perform the same is delayed to a
larger value of $K$, which means that the merits of the PRQ scheme get more
evident as the number of BS antennas increases.

The performance of SIMO systems that employ ML and BND-NCD with various high-
order modulations are obtained with the ZTQ and PRQ schemes against the
logarithm of the number of BS antennas in Fig. 7b. Based on this and the
previous results, we can state that when $\nicefrac{{K}}{{N}}$ is small, the
spatial diversity of the system is not adequately utilized with the ZTQ
scheme. Moreover, we can also see that modulation orders such as $256$,
$1024$, and $4096$ can be used in one-bit quantized systems with PRQ if a
sufficient number of BS antennas are utilized. The proposed two-stage detector
can perform very close to ML with much lower computational complexity.

###  IX-D Performance Comparison of the Proposed Method with the Existing
Methods

Figure 8: BER performance comparison of the proposed detector with OBMNet [23]
and SVM-based [30] detectors from the literature in a $128\times 8$ system
that employs $16$-QAM. NNS is the second-stage detector from [23]. The maximum
cardinality of the set of nearest neighbors for NNS is chosen as $2K=16$, the
same as NCD.

The performances of the proposed detector and the existing detectors from the
literature, namely, SVM-based from [30], and OBMNet from [23] are compared in
Fig. 8. The second stage for OBMNet is the nearest neighbor search (NNS)
algorithm from [23], and the SVM-based detector is utilized as a single-stage
detector. The cardinality of the set of nearest neighbors to conduct ML
detection for the second stage detector of OBMNet is also chosen as $2K=16$.
The plots show that the high-SNR error floor of the one-stage BND coincides
with that of the two-stage OBMNet-NNS, the most recently proposed high-order
modulation supporting detection scheme from the literature. Even when ZTQ is
employed, BND-NCD outperforms both machine learning based approaches. Also,
unlike any of the detectors included for comparison, the proposed two-stage
BND-NCD with PRQ reduces the error floor below $10^{-6}$. BND requires $4.5$
iterations on average, whereas the number of layers of OBMNet, which is a DNN-
based detector, is $15$.

###  IX-E Error Performance and Complexity Analysis of the Proposed Methods in
Large-Scale Systems

(a) BER Performance

(b) Computational Complexity

Figure 9: BER performance (a) and complexity-related measurements (b) against
SNR obtained with BND-NCD when $N=512$ for $K=4$ with $1024$-QAM, $K=8$ with
$256$-QAM, $K=12$ with $64$-QAM. BER plots are obtained with ZTQ and PRQ,
whereas complexity-related measurements are recorded with PRQ.
$T_{\mathrm{avg}}$ is the average number of iterations of BND, and
$|\mathcal{X}|_{\mathrm{avg}}$ is the average size of the reduced set in NCD.

The error performances of several high-order modulation schemes in multi-user
settings are examined. Additionally, complexity-related measurements regarding
the average number of iterations of BND and the average size of the reduced
set $\mathcal{X}$ are also made. BER performances (a) of different system
setups with high-order modulations using ZTQ and PRQ, and complexity-related
measurements (b) are plotted in Fig. 9b using PRQ. The proposed PRQ scheme can
mitigate the effects of SR and lower the error floors. Hence, the sources of
diversity are better exploited by PRQ. For complexity, we can see that $3-5$
iterations are required for BND, and the number of iterations increases as $K$
increases. $|\mathcal{X}|_{\mathrm{avg}}$ decreases as SNR increases, which
means that employing only the first-stage BND is sufficient at low SNR since
the performance gaps between the one and two-stage approaches are very small
at low SNR, which was shown in Fig. 6. At high SNR, even though the number of
users is the largest for $64$-QAM, it also has the smallest
$|\mathcal{X}|_{\mathrm{avg}}$. Therefore, as $M$ increases, a more accurate
estimation of the amplitudes is required. Note that the termination threshold
$\epsilon$ of BND and the maximum size of the set of candidates $P$ of NCD can
be changed to obtain a trade-off between error performance and computational
complexity.

###  IX-F Effect Imperfect CSI on the Performance of PRQ

(a) ZTQ

(b) PRQ

Figure 10: BER performance of the proposed detector in one and two stages for
the imperfect CSI scenario in a $128\times 4$ system with $16$-QAM with ZTQ
(a) and PRQ (b), where solid lines are for the one-stage BND and the dashed
lines represent the two-stage BND-NCD approach.

In the final part, we investigate the effect of imperfect CSI knowledge at the
BS on the detection performance. We model the channel estimate as the sum of
the original channel $\bar{\bm{H}}$ and the estimation error $\bar{\bm{E}}$
such that $\tilde{\bar{\bm{H}}}=\bar{\bm{H}}+\bar{\bm{E}}$. We assume that
each element of the estimation error matrix
$[\bar{\bm{E}}]_{(n,k)}\sim\mathcal{CN}(0,\sigma^{2}_{e})$, to simply model
the distortion. This model can be valid when PRQ is also used for channel
estimation due to better amplitude recovery where the estimate can be obtained
with a Gaussian distributed error [5], which is left as future work. In Fig.
10, the BER performances obtained with ZTQ and PRQ for various estimation
error variance scenarios are plotted using BND and BND-NCD with both ZTQ (a)
and PRQ (b). As can be seen when the estimation error is not large, the
performance is not affected much with PRQ. But even when the estimation error
is large, PRQ still outperforms ZTQ, which shows the robustness of the PRQ and
BND-NCD methods for the imperfect CSI scenario.

##  X Conclusion

This paper proposes a new PRQ scheme, modified linear detectors with PRQ, and
a new two-stage detector for uplink one-bit massive MIMO systems. In the first
stage, BND is utilized to estimate the transmitted signal using the log-
likelihood function via Newton’s Method with penalty. The second-stage
detector, NCD, is used to refine the estimate from the first stage to enhance
detection performance, which creates a small set of candidates based on the
sign constraints imposed by one-bit quantization. The proposed PRQ scheme
relies on changing the quantization thresholds to help mitigate the
detrimental effects of SR on the performance. With the proposed two-stage
detector and PRQ, one-bit multi-user massive MIMO systems can operate with
high-order modulation schemes such as $256$-QAM or $1024$-QAM, and superior
error performance to the existing detectors can be obtained with comparable
complexity. We extend our work to frequency-selective channels in [40] to
analyze the effect of inter-symbol interference (ISI) with PRQ, and our
detailed work on detection in massive MIMO with PRQ can be found in [41].

## Acknowledgments

This study is partially supported by Vodafone Turkey within the framework of
5G and Beyond Joint Graduate Support Program coordinated by Information and
Communication Technologies Authority of Turkey.

##  Appendix A Computations of the Nonlinear Functions

Computation of $\ln(\Phi(x))$ can be a problem when $x$ is small, and
divergence towards $-\infty$ for small arguments in finite precision can be
encountered. We can approximate $\ln(\Phi(x))$ using its first-order
derivative to avoid possible divergent behavior as

$\ln(\Phi(x))\cong\ln(\Phi(c))+\varphi(c)(x-c),$ (47)

for $x<c\in\mathbb{R}$, where
$\varphi(x)=\frac{d}{dx}\ln(\Phi(x))=\frac{\phi(x)}{\Phi(x)}$ is the first
order derivative of $\ln(\Phi(x))$. $c=-38.2$ is a suitable choice for
MATLAB’s arithmetic calculations since divergent behavior is observed for
smaller values. However, the first and second-order derivatives $\varphi(x)$
and $\psi(x)$ also show divergent behavior around the same point $c$. The
asymptotic behavior of $\varphi(x)$ towards $-\infty$ can be found as

$\lim_{x\to-\infty}\frac{\phi(x)}{\Phi(x)}=\lim_{x\to-\infty}-\frac{x\phi(x)}{\phi(x)}=\lim_{x\to-\infty}-x,$
(48)

which means that for $x<c$, $\varphi(x)=-x$ and
$\frac{d^{2}}{dx^{2}}\ln(\Phi(x))=\psi(x)=-1$ can be utilized in practice.
Note that a look-up table can be generated to calculate nonlinear functions to
decrease the complexity.

##  Appendix B Derivation of the Conditional Mutual Information

Starting with the definition of conditional mutual information:

$\mathcal{I}(\bm{r};\bm{x}\mid\bm{H},\bm{\tau})=\mathcal{H}(\bm{r}\mid\bm{H},\bm{\tau})-\mathcal{H}(\bm{r}\mid\bm{x},\bm{H},\bm{\tau}),$
(49)

where $\mathcal{H}(.)$ is the entropy function. The first term in (49) can be
calculated by the definition of conditional entropy as

$\mathcal{H}(\bm{r}\mid\bm{H},\bm{\tau})=-\sum_{\bm{r}\in\\{\pm
1\\}^{2N}}\operatorname*{p}(\bm{r}\mid\bm{H},\bm{\tau})\log_{2}(\operatorname*{p}(\bm{r}\mid\bm{H},\bm{\tau})),$
(50)

where $p(\bm{r}\mid\bm{H},\bm{\tau})$ can be found by averaging the likelihood
function
$p(\bm{r}\mid\bm{x},\bm{H},\bm{\tau})=\prod_{n=1}^{2N}\Phi\left(\frac{r_{n}(\bm{h}_{n}^{T}\bm{x}-\tau_{n})}{\sqrt{N_{0}/2}}\right)$
over all possible $\bm{x}$ vectors such that

$\operatorname*{p}(\bm{r}\mid\bm{H},\bm{\tau})=\frac{1}{M^{K}}\sum_{\bm{x}^{\prime}\in\mathcal{M}^{2K}}p(\bm{r}\mid\bm{x}=\bm{x}^{\prime},\bm{H},\bm{\tau}).$
(51)

Now that the first term in (49) is found, we can move on to the second term.
Due to the conditional independence of the quantized observations given the
input vector, the channel matrix, and the quantization thresholds, we can
write

$\mathcal{H}(\bm{r}\mid\bm{x},\bm{H},\bm{\tau})=\sum_{n=1}^{2N}\mathcal{H}(r_{n}\mid\bm{x},\bm{H},\tau_{n})=\frac{1}{M^{K}}\sum_{n=1}^{2N}\sum_{\bm{x}^{\prime}\in\mathcal{M}^{2K}}\mathcal{H}(r_{n}\mid\bm{x}=\bm{x}^{\prime},\bm{H},\tau_{n}).$
(52)

Again, by definition, and since each $r_{n}$ is a binary random variable,

$\mathcal{H}(r_{n}\mid\bm{x},\bm{H},\tau_{n})=\sum_{r_{n}\in\\{\pm
1\\}}p(\bm{r}\mid\bm{x},\bm{H},\bm{\tau})\log_{2}(p(\bm{r}\mid\bm{x},\bm{H},\bm{\tau}))=\mathcal{H}_{b}\left(\Phi\left(\frac{\bm{h}_{n}^{T}\bm{x}-\tau_{n}}{\sqrt{N_{0}/2}}\right)\right).$
(53)

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