# Quasi-Newton Detection in One-Bit Pseudo-Randomly Quantized Wideband Massive
MIMO Systems

Gökhan Yılmaz, , and
Ali Özgür Yılmaz The work of G. Yılmaz is supported by Vodafone Turkey within
the framework of the 5G and Beyond Joint Graduate Support Program coordinated
by the Information and Communication Technologies Authority of Turkey.The
authors are with the Department of Electrical and Electronics Engineering,
Middle East Technical University, Ankara, Turkey. (E-mail: {yilmaz.gokhan_01,
<EMAIL_ADDRESS>
###### Abstract

Using low-resolution analog-to-digital converters (ADCs) is a valuable
solution to decrease power consumption and cost in massive MIMO systems.
Previous studies show that the performance gap between low and high-resolution
systems gets more prominent as the signal-to-noise ratio (SNR) increases since
the detection performance starts to saturate at some point due to the
stochastic resonance (SR) phenomenon. In our previous work, we proposed new
quantization and detection schemes for one-bit massive MIMO systems operating
under frequency-flat fading. This paper offers a new frequency domain
equalization (FDE) scheme that can work with the previously proposed pseudo-
random quantization (PRQ) scheme to mitigate the effects of SR to support
high-order modulation schemes such as $64$-QAM and $256$-QAM. Our equalizer is
based on a projected quasi-Newton method for one-bit uplink massive MIMO
systems applicable for orthogonal frequency division multiplexing (OFDM) and
single carrier (SC) transmission under frequency-selective fading. The
proposed low-complexity detector can outperform the benchmark detector from
the literature with very similar complexity. We analyze the effects of PRQ
under frequency-selective fading for different scenarios and show that the PRQ
scheme can close the performance gap between SC and OFDM systems by
simulations.

###### Index Terms:

Detection, massive MIMO, orthogonal frequency division multiplexing (OFDM),
pseudo-random quantization (PRQ), one-bit analog-to-digital converter (ADC),
quasi-Newton method, frequency-domain equalization (FDE), one-bit
quantization.

##  I Introduction

Power consumption of an ADC increases exponentially as its resolution
increases [1, 2]. Therefore, utilizing low-resolution ADCs is a potential
solution to deal with the high power consumption and cost in massive multiple-
input multiple-output (MIMO) systems [3]. One-bit ADCs have additional
benefits, such as having simple circuitry and not requiring automatic gain
units (AGCs). However, especially at high SNR, nonlinear distortion due to
one-bit quantization leads to significant performance degradation in terms of
achievable rate.

Unlike unquantized linear systems, channel capacity [4, 5, 6, 7] and
achievable rate [8, 9, 10] of one-bit quantized systems saturate at high SNR,
which is a consequence of the SR phenomenon [11, 12]. SR is a counter-
intuitive phenomenon encountered in nonlinear systems where the maximum
performance is achieved at a finite SNR value. Amplitude-varying modulation
schemes are more prone to the negative effects of SR [13]. Dithering is a
helpful technique to reduce quantization errors by intentionally adding noise
to the input of quantizers [12]. In [14], PRQ is shown to be a valuable
technique to work with low-resolution quantizers, and our previous work [13]
uses this approach in SC systems operating under frequency-flat fading by
generating pseudo-random quantization thresholds to deal with SR. In [13, 15],
we show that pseudo-randomly generated non-zero quantization thresholds can
help the receiver better exploit the spatial degrees of freedom to recover the
amplitude of the signal to be sensed. This technique does not require changing
the thresholds with respect to time or different channel realizations.

Utilizing non-zero quantization thresholds in one-bit massive MIMO systems for
signal estimation is studied in the literature with different approaches
focusing on threshold optimization for channel estimation, examples of which
can be found in [16, 17, 18, 19]. These require additional computations and
time-varying thresholds within the channel coherence time. In [13], we also
show that the conventional zero-threshold quantization (ZTQ) that is adopted
by the previous works in the literature results in performance loss at the
high SNR regime even when the optimal maximum likelihood (ML) detector is
employed under frequency-flat fading. Inspired by the results from our
previous work, we utilize PRQ under frequency-selective fading with both OFDM
and SC transmission. With the proposed detector based on the maximum
likelihood sequence detection (MLSD) approach, one-bit massive MIMO systems
can benefit from the merits of PRQ to improve high-SNR performance.
Unfortunately, linear methods do not benefit from the PRQ approach as was
shown in [13, 15].

Linear methods offer simple implementation capability and low complexity, but
their high SNR performance saturates quickly [20, 21]. [9] shows that
frequency-selectivity of the wireless channel favors one-bit wideband MIMO-
OFDM systems and a large delay spread helps lower the amplitude distortion.
[22] focuses on linear and iterative block decision feedback equalizers for
SC-FDE. In [23], SVM-based channel estimation and detection algorithms are
proposed for both SC and OFDM systems. An algorithm based on Bussgang
decomposition and Ungerboeck factorization for CP-free SC systems is presented
in [24]. In [25], one-bit quantization is utilized for phase-only
measurements, and in [26], massive MIMO with spatial modulation is considered.
Even though SC transmission can be favorable [27], OFDM is a widely used
scheme with many advantages regarding resource allocation and has variants
such as orthogonal frequency division multiple access (OFDMA). In [28], a
first-order projected gradient descent algorithm, 1BOX, is utilized in one-bit
OFDM systems, which is taken as the benchmark algorithm in this paper. Exact
and inexact expectation maximization (EM) algorithms are deployed in [29] and
[30], respectively. Note that 1BOX is chosen as a benchmark since exact EM
requires high complexity, and inexact EM performs very close to 1BOX but
requires many iterations to converge.

Gradient-based optimization is commonly used to obtain an equalizer based on
the log-likelihood function in one-bit massive MIMO systems [31, 21, 32, 28,
13]. For the flat-fading channels, [31] is the first to focus on such a
scheme. Then, [16, 21, 32] utilize the deep unfolding technique to train a
neural network to find suitable hyperparameters such as the step size since
[31] requires a large number of iterations when the step size is not adjusted
for different system setups and operating SNR values. Our previous work, [13],
is the first to consider second-order optimization on the log-likelihood
function to construct an equalizer for flat-fading channels, which helps
obtain fast convergence without the burden of choosing different step sizes
for different system setups. In [13], we focus on a two-stage detector for
one-bit massive MIMO systems under frequency flat fading. The first stage is
based on NM with penalty to enforce box constraint on the signal to be
equalized. The second stage tries to find a small set of possible candidates
based on sign constraints by taking one-bit observations as codewords in
space. However, the two-stage approach from [13] is not feasible under
frequency-selective fading since NM and a nearest neighbor search algorithm
would result in unaffordable complexity increases due to inter-symbol
interference (ISI). The usage of the cyclic prefix (CP) also does not help
construct the ML detector in the frequency domain (FD) since conversion to the
FD causes a possibly tractable but much-complicated conditional distribution
for the quantized observations due to the quantization noise. Hence, contrary
to the unquantized systems, the detectors proposed for flat-fading channels
are not directly applicable under frequency-selective fading in one-bit
massive MIMO systems.

###  I-A Contributions

As in the case of systems under frequency-flat fading from our previous work
[13] and [15, Chapter 3], we utilize PRQ in our system to exploit dithering by
modifying quantization thresholds to mitigate the adverse effects of SR on
detection performance in frequency-selective fading channels, instead of
relying on the conventional ZTQ scheme adopted by the existing uplink one-bit
massive MIMO detection schemes [28, 20, 33, 29, 30, 34, 32, 25, 35, 24]. We
compare the performances of PRQ and ZTQ for various scenarios and show that
when the sum interference composed of multi-user interference (MUI) and ISI is
small, frequency-selective systems can benefit more from the PRQ scheme to
support high-order modulations.

We formulate equalization in one-bit wideband massive MIMO systems as a
constrained optimization problem and propose to solve it using a second-order
optimization technique, projected NM, applicable when a non-zero threshold
quantization scheme is employed. Unlike [28], this approach does not require
selecting suitable step sizes for different system setups due to the
additional second-order derivative information. After deriving the necessary
relations using NM, we utilize two approximations to decouple equalization
among subcarriers and to avoid matrix inversion to reach the proposed
projected quasi-Newton detector (PQND), which is an essential improvement over
[13], where the proposed time domain equalizer (TDE) under flat-fading would
result in a substantial complexity increase under frequency-selective fading.
Adopting second-order optimization techniques to optimize the log-likelihood
under frequency-selective fading has not been used before. The approximations
that lower the complexity are also a novel part of this work.

By employing the PRQ scheme and the proposed PQND, we show that one-bit uplink
massive MIMO systems operating under frequency-selective fading can support
high-order modulation schemes such as $64$-QAM and $256$-QAM in multi-user
settings. The proposed detector outperforms the benchmark detector from [28]
in terms of error performance when the conventional ZTQ scheme or the PRQ
scheme is employed with very similar complexity. We extend the proposed
detector for CP-SC systems as an SC-FDE method and show that PRQ can help
lower the performance gap between OFDM and SC systems [27] when one-bit
quantization is employed.

###  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 $\bm{a}$ is $a_{n}$. The $n^{\textrm{th}}$ row and $k^{\textrm{th}}$
column of a matrix $\bm{A}$ is denoted as $[\bm{A}]_{(n,k)}$.
$\operatorname*{bcirc}\\{.\\}$ creates a block circulant matrix by taking its
argument matrices as the first block of rows. $\operatorname*{bdiag}\\{.\\}$
creates a block diagonal matrix consisting of its arguments.
$\lVert\bm{.}\rVert$ denotes the $\ell_{2}$ norm of a vector. $\mathbb{R}$ and
$\mathbb{C}$ are the set of real and complex numbers, respectively.
$\Re\\{.\\}$ and $\Im\\{.\\}$ give their arguments’ real and imaginary parts,
respectively. $\bar{\bm{F}}$ is the unitary discrete Fourier transform (DFT)
matrix of size $V\times V$, where
$[\bar{\bm{F}}]_{n,k}=\mathrm{e}^{-j2\pi(n-1)(k-1)/V}/\sqrt{V}$.
$\mathcal{F}_{v}\\{.\\}=\sum_{m=0}^{V-1}\\{.\\}\mathrm{e}^{-j2\pi
mv/V}/\sqrt{V}$ is the unitary $V$-point DFT operator and
$\mathcal{F}_{m}^{-1}\\{.\\}=\sum_{v=0}^{V-1}\\{.\\}\mathrm{e}^{+j2\pi
mv/V}/\sqrt{V}$ is the unitary $V$-point inverse DFT operator.
$\mathcal{CN}(\bm{\mu},\bm{\Sigma})$ 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. $\odot$ is the
Hadamard, and $\otimes$ is the Kronecker product.

##  II System Model

Figure 1: A block diagram that summarizes the system model. S/P is for serial-
to-parallel, and P/S is for parallel-to-serial.

We consider an $N\times K$ uplink massive MIMO-OFDM system with $V$
subcarriers, where $K$ single-antenna users are served by a base station (BS)
equipped with $N$ antennas. We extend the model to SC transmission in Section
III. This setup can be generalized to multi-antenna users easily when we
assume independent streams are being transmitted from the antennas of each
user. If uplink precoding is employed, different approaches can be followed to
improve the achievable rate, which is an open field of study. We focus on a
single-cell scenario and assume the analog frontend filters are sharp enough
so that there is no adjacent channel interference as in many existing works
from the literature [28, 20, 33, 10, 4, 29, 30, 32, 31, 21, 25, 22, 23, 13].
When the leakage from an adjacent band is very large compared to the desired
in-band signal, even unquantized systems will suffer, for which power control
is a commonly used measure. If the interference from an adjacent band is small
and it consists of many contributors, i.e., users, the resultant scenario
would cause an effective increase in the noise variance as a result of the
central limit theorem (CLT), which would cause a slight decrease in the SNR.
The source of in-band interference can be MUI or multi-cell interference
(MCI). Since we focus on a single scenario, the effects of MCI are assumed to
be negligible as in the existing works. The effects of MUI and ISI are
extensively studied in this work.

Constellation symbols to be transmitted are selected independently from an
$M$-QAM alphabet, denoted as $\bar{\mathcal{M}}$, with equal likelihood at all
subcarriers and for all users. A block diagram that summarizes the adopted
system model is shown in Fig. 1. The FD symbol of the $k^{\textrm{th}}$ user
at the $v^{\textrm{th}}$ subcarrier is shown as $\bar{x}_{k}[v]$ for
$k=1,2,\ldots,K$ and $v=0,1,\ldots,V-1$, where
$\mathbb{E}[|\bar{x}_{k}[v]|^{2}]=E_{s}=1$. We denote the vector containing
the FD symbols of all users at the $v^{\textrm{th}}$ subcarrier as
$\bar{\bm{x}}[v]=\begin{bmatrix}\bar{x}_{1}[v]&\bar{x}_{2}[v]&\ldots&\bar{x}_{K}[v]\end{bmatrix}^{T}$
for $v=0,1,\ldots,V-1$. The concatenated version of an FD vector for all
subcarriers or a time domain (TD) vector for all time indices can be obtained
as

$\underline{\bm{x}}=\begin{bmatrix}\bm{x}[0]^{T}&\bm{x}[1]^{T}&\ldots&\bm{x}[V-1]^{T}\end{bmatrix}^{T}.$
(1)

Each user takes their FD symbols’ inverse unitary DFT before transmission to
obtain their TD signals, which can be shown as

$\bar{\bm{s}}[m]=\frac{1}{\sqrt{V}}\sum_{m=0}^{V-1}\bar{\bm{x}}[v]e^{+j2\pi
mv/V},$ (2)

for $m=0,1,\ldots,V-1$. Equivalently,
$\underline{\bar{\bm{s}}}=\bar{\bm{Q}}_{K}^{H}\underline{\bar{\bm{x}}}$, where
$\bar{\bm{Q}}_{i}=\bar{\bm{F}}\otimes\bm{I}_{i}$ for $i\in\mathbb{Z}^{+}$, for
$n=0,1,\ldots,V-1$ and $k=0,1,\ldots,V-1$. Hence, $\bar{\bm{Q}}_{K}^{H}$
corresponds to each user’s unitary inverse DFT operation.

We assume the BS has perfect knowledge of the channel impulse response (CIR),
and the multipath channel has $L$ taps. Hence, each user adds a CP of length
$L_{\textrm{CP}}\geq L-1$ to the beginning of their TD signals to help
mitigate the effects of ISI at the receiver side. Then, the CP-added version
of the TD signal vector is passed through an ideal digital-to-analog converter
(DAC) and in-phase/quadrature (I/Q) modulated. The CIR at the
$\ell^{\mathrm{th}}$ tap, between the $k^{\mathrm{th}}$ user and
$n^{\mathrm{th}}$ receiver antenna is denoted as $\bar{h}_{(n,k)}[\ell]$ for
$k=1,2,\ldots,K$, $n=1,2,\ldots,N$, and $\ell=0,1,\ldots,L-1$. The
$\ell^{\textrm{th}}$ tap of CIR between all users and BS antennas can be
represented as $\bar{\bm{H}}[\ell]$, where
$[\bar{\bm{H}}[\ell]]_{(n,k)}=\bar{h}_{n,k}[\ell]$ with
$\sum_{\ell=0}^{L-1}\mathbb{E}[|\bar{h}_{(n,k)}[\ell]|^{2}]=1$. Assuming
perfect synchronization, the received signal is I/Q demodulated, symbol-rate
sampled, and CP is discarded. The unquantized received TD signal at time $m$
can be expressed as

$\bar{\bm{y}}[m]=\sum_{\ell=0}^{L-1}\bar{\bm{H}}[\ell]\bar{\bm{s}}[\langle
m-\ell\rangle_{V}]+\bar{\bm{w}}[m],$ (3)

for $m=0,1,\ldots,V-1$, where $\bar{y}_{n}[m]$ and $\bar{w}_{n}[m]$ for
$n=1,2,\ldots,N$ represent the unquantized version of the received signal and
the thermal noise sample at antenna $n$ and time $m$, respectively.
$\bar{\bm{w}}[m]\sim\mathcal{CN}(\bm{0}_{N},N_{0}\bm{I}_{N})$ for
$m=0,1,\ldots,V-1$, and there is no temporal correlation among noise samples.
$\langle.\rangle_{V}$ is the modulo $V$ operator. An equivalent fully
vectorized form of (3) can also be obtained. To do so, we define the TD block
circulant MIMO channel circular convolution matrix of size $NV\times KV$ as

$\underline{\bar{\bm{H}}}_{b}=\operatorname*{bcirc}\\{\bar{\bm{H}}[0],\bm{0},\ldots,\bm{0},\bar{\bm{H}}[L-1],\ldots,\bar{\bm{H}}[1]\\},$
(4)

which, due to CP, can be expressed as

$\underline{\bar{\bm{H}}}_{b}=\bar{\bm{Q}}_{N}^{H}\underline{\bar{\bm{\Lambda}}}_{b}\bar{\bm{Q}}_{K},$
(5)

where $\underline{\bar{\bm{\Lambda}}}_{b}$ is the block diagonal FD MIMO
channel matrix, which can be shown as

$\underline{\bar{\bm{\Lambda}}}_{b}=\operatorname*{bdiag}\\{\bar{\bm{\Lambda}}[0],\bar{\bm{\Lambda}}[1],\ldots,\bar{\bm{\Lambda}}[V-1]\\}.$
(6)

The decomposition shown in (5) utilizes the fact that the columns of the DFT
matrix are eigenvectors of any circulant matrix [36] and applies this to the
MIMO scenario as in [35]. Hence, the unquantized representation of the
received signal can be found using

$\underline{\bar{\bm{y}}}=\underline{\bar{\bm{H}}}_{b}\,\bar{\bm{Q}}_{K}^{H}\,\underline{\bar{\bm{x}}}+\underline{\bar{\bm{w}}}=\bar{\bm{Q}}_{N}^{H}\underline{\bar{\bm{\Lambda}}}_{b}\,\underline{\bar{\bm{x}}}+\underline{\bar{\bm{w}}}=\underline{\bar{\bm{G}}}\,\underline{\bar{\bm{x}}}+\underline{\bar{\bm{w}}},$
(7)

where
$\underline{\bar{\bm{G}}}=\bar{\bm{Q}}_{N}^{H}\underline{\bar{\bm{\Lambda}}}_{b}$
represents the effective channel matrix, $\underline{\bar{\bm{w}}}$ is the
spatially and temporally white thermal noise vector. Then, the unquantized
representation of the observations can be used to find the one-bit quantized
observations as

$\underline{\bar{\bm{r}}}=\operatorname*{sign}(\Re\\{\underline{\bar{\bm{y}}}-\underline{\bar{\bm{\tau}}}\\})+j\,\textrm{sign}(\Im\\{\underline{\bar{\bm{y}}}-\underline{\bar{\bm{\tau}}}\\}),$
(8)

where $\underline{\bar{\bm{\tau}}}$ is the time-invariant quantization
threshold vector with components
$\bar{\bm{\tau}}[m_{1}]=\bar{\bm{\tau}}[m_{2}]=\bar{\bm{\tau}}$ for
$m_{1},m_{2}\in\\{0,1,\ldots,V-1\\}$. These thresholds need not be updated for
different channel realizations. The detailed procedure regarding pseudo-random
threshold assignment is explained in Section IV. The quantized observation
from the $n^{\mathrm{th}}$ antenna at time $m$ can be shown as
$\bar{r}_{n}[m]$ for $n=1,2,\ldots,N$ and $m=0,1,\ldots,V-1$.

Due to the structure of the proposed detector, we need to be able to work with
real numbers only. Hence, we introduce conversions for an alternative notation
with real numbers. 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

$\bm{a}=\begin{bmatrix}\Re\\{\bar{\bm{a}}\\}\\\
\Im\\{\bar{\bm{a}}\\}\end{bmatrix},\textrm{ and
}\bm{A}=\begin{bmatrix}\Re\\{\bar{\bm{A}}\\}&-\Im\\{\bar{\bm{A}}\\}\\\
\Im\\{\bar{\bm{A}}\\}&\Re\\{\bar{\bm{A}}\\}\end{bmatrix},$ (9)

respectively. Using the notation with real numbers, (8) can be re-written as

$\underline{\bm{r}}=\operatorname*{sign}\left(\underline{\bm{G}}\,\underline{\bm{x}}-\underline{\bm{\tau}}+\underline{\bm{w}}\right).$
(10)

We define the SNR as the ratio of the average received signal power per user
to the average noise power at each antenna such that

$\rho=\frac{\mathbb{E}\left[\sum_{\ell=0}^{L-1}|\bar{h}_{(n,k)}[\ell]\bar{s}_{k}[\langle
m-\ell\rangle_{V}]|^{2}\right]}{\mathbb{E}[|\bar{w}_{n}[m]|^{2}]}=\frac{E_{s}}{N_{0}}=\frac{1}{N_{0}}.$
(11)

##  III Projected Quasi-Newton Detector (PQND)

In order to derive a low-complexity FDE scheme that can work with non-zero
thresholds, we start with MLSD. Then, we construct an equalizer using NM with
additional constraints. Since NM prevents subcarrier-wise processing and
requires matrix inversion, we move one step further with approximations to
construct a quasi-Newton method for equalization. Using a second-order
optimization technique is helpful for this scenario since fine-tuning the
algorithm step size for different system setups and operating SNR values can
become a problem with first-order optimization methods, as indicated in [28].

###  III-A Maximum Likelihood Sequence Detector (MLSD)

In the presence of independent additive white Gaussian noise (AWGN) on each
branch of a one-bit quantized channel, as in [31, 16, 28, 21], the log-
likelihood of a quantized observation vector $\underline{\bm{r}}$, given a
transmitted signal $\underline{\bm{x}}$, a MIMO channel $\underline{\bm{G}}$,
a vector of quantization thresholds $\underline{\bm{\tau}}$, can be expressed
as

$\mathcal{L}(\underline{\bm{x}})=\bm{1}_{2NV}^{T}\ln\left(\Phi\left(\sqrt{\frac{2}{N_{0}}}\underline{\bm{r}}\odot(\underline{\bm{G}}\,\underline{\bm{x}}-\underline{\bm{\tau}})\right)\right),$
(12)

where $\Phi(x)=\int_{-\infty}^{x}\phi(\tau)\,d\tau$ is the cumulative
distribution function (CDF) of the standard Gaussian random variable, and
$\bm{1}_{i}$ is the vector of size $i\in\mathbb{Z}^{+}$, whose elements are
all set to $1$. The natural logarithm $\ln(.)$ and $\Phi(.)$ are applied
element-wise on their arguments. Then, we construct MLSD for this problem as

|l|[2] x ∈M2KV L(x) ^x_MLSD=,

where $\mathcal{M}$ is the set of possible values an FD symbol can take in
either in-phase or quadrature part. Even though the MLSD approach can provide
high performance in terms of achievable rate, it is not feasible for practical
systems where $V$ is generally very large, and the search space consists of
$M^{KV}$ candidates. In [37], the Viterbi equalizer is constructed for a one-
bit CP-free SC system. However, the complexity is still large in multi-user
systems or when utilizing high modulation orders. Note that subcarrier-wise ML
detection in the FD is not applicable since conversion to the FD causes an
unknown conditional distribution for the quantized observations due to
quantization noise.

###  III-B Equalization with Newton’s Method

As in [28], in order to utilize gradient-based optimization techniques, we
relax the discrete input set constraint in (III-A) and reformulate the problem
as

|l|[2] x ∈R2KV L(x) ~x= |x_i|≤M_b, i=1,2,…,2KV,

where $M_{b}$ is the boundary of the chosen constellation in one of the I/Q
parts. For $M$-QAM, $M_{b}$ can be found as

$M_{b}=\sqrt{\frac{3(\sqrt{M}-1)^{2}}{2(M-1)}}.$ (13)

Due to the log-concavity of $\Phi(.)$, we have a concave optimization problem,
which can be solved with gradient-based optimization methods as was done with
first-order optimization methods [31, 28] and with different variants such as
the deep unfolding method [21, 32]. The convergence of concave optimization
problems of similar forms is shown in [38, 39] for a large number of BS
antennas, i.e., massive MIMO settings.

While relaxing the discrete input set constraint, we utilize a restriction.
Each element of $\underline{\bm{x}}$ is inside the boundaries of the
constellation, which is called the box constraint [28]. Also, once a solution
to this problem is obtained, the norm of $\underline{\bm{x}}$ is set to $KV$,
which is called the norm projection as in [21]. The box constraint prevents
diverging from the constellation’s boundaries during equalization, using a
priori knowledge of the chosen constellation scheme. The norm projection is
related to the law of large numbers (LLN), for which we utilize practical
values of $V$ being large in commercial systems. We assume the total number of
constellation symbols sent from all users in one OFDM symbol duration $KV$ is
large. Since DFT operations are all unitary, we have
$\mathbb{E}[\lVert\underline{\bm{x}}\rVert^{2}]\rightarrow KVE_{s}=KV$ as
$KV\rightarrow\infty$.

In order to apply constraints with gradient-based optimization, different
strategies such as the penalty method [13] or the projection method [28, 21]
can be followed. We resort to the projection method, which does not require
modifying the cost function. We start with projected NM to solve (III-B). The
update equation of NM can be written as

$\underline{\bm{x}}^{(t)}=\mathcal{P}^{(t)}\left(\underline{\bm{x}}^{(t-1)}-\alpha(\nabla^{2}\mathcal{L}^{(t-1)})^{-1}\nabla\mathcal{L}^{(t-1)}\right),$
(14)

where $\alpha\in\mathbb{R}$ is the step size, and $\mathcal{P}^{(t)}(.)$ is
the projection function at iteration $t$ which is formally addressed in the
next subsection. For now, we deal with only gradient and Hessian calculations.

In (14),
$\nabla\triangleq\begin{bmatrix}\frac{\partial}{\partial\underline{x}_{1}}&\ldots&\frac{\partial}{\partial\underline{x}_{2KV}}\end{bmatrix}^{T}$
can be expressed as a vector of size $2KV$, and $\nabla^{2}$ can be expressed
as the outer product of $\nabla$ operator with itself:
$\nabla^{2}=\nabla\,\nabla^{T}$. Hence, the $\nabla^{2}$ operator can be
considered a $2KV\times 2KV$ matrix. $\nabla\mathcal{L}^{(t)}$ is the gradient
of the log-likelihood function $\mathcal{L}$ with respect to
$\underline{\bm{x}}$ calculated at iteration $t$. Similarly,
$\nabla^{2}\mathcal{L}^{(t)}$ is the Hessian of the log-likelihood function
$\mathcal{L}$ with respect to $\underline{\bm{x}}$ at iteration $t$. The
gradient and Hessian are functions of $\underline{\bm{x}}$, and we drop the
argument for ease of notation. Also, we define a new vector

$\underline{\bm{u}}=\sqrt{\frac{2}{N_{0}}}\underline{\bm{r}}\odot(\underline{\bm{G}}\,\underline{\bm{x}}-\underline{\bm{\tau}}),$
(15)

for compactness. Then, the gradient of the log-likelihood function can be
found as

$\nabla\mathcal{L}=\sqrt{\frac{2}{N_{0}}}\underline{\bm{G}}^{T}\left(\underline{\bm{r}}\odot\varphi\left(\underline{\bm{u}}\right)\right),$
(16)

where $\varphi(x)\triangleq\frac{d}{dx}\ln(\Phi(x))=\phi(x)/\Phi(x)$, and it
is applied element-wise on its arguments. Since PDF and CDF of any random
variable always take non-negative values, $\varphi(x)\geq 0$ for all
$x\in\mathbb{R}$. The Hessian matrix can be found as

$\nabla^{2}\mathcal{L}=\frac{2}{N_{0}}\underline{\bm{G}}^{T}\operatorname*{diag}\left(\psi\left(\underline{\bm{u}}\right)\right)\underline{\bm{G}},$
(17)

where
$\psi(x)\triangleq\frac{d^{2}}{dx^{2}}\ln(\Phi(x))=-x\varphi(x)-\varphi^{2}(x)$,
and it is applied element-wise on its arguments. Note that $\psi(x)\leq 0$ for
all $x\in\mathbb{R}$, which shows the log-concavity of $\Phi(.)$. Since
$\Phi(x)$ can approach zero exponentially fast as $x$ gets large, the
computations of $\ln(\Phi(x))$, $\varphi(x)$, and $\psi(x)$ in finite
precision can cause problems such as divergent behavior and uncertainties. A
solution to this issue is proposed in [13, Appendix B] and is also used
herein.

###  III-C Equalization with the Proposed Quasi-Newton Method

NM can provide complexity reduction compared to the MLSD approach. However, it
does not allow subcarrier-level equalization and requires a large matrix
inversion. In this subsection, we attempt to solve these problems using
approximations to obtain an equalization method with lower complexity. We
define the step
$\Delta\underline{\bm{x}}=(\nabla^{2}\mathcal{L})^{-1}\nabla\mathcal{L}$,
which can be written as

$\Delta\underline{\bm{x}}=\left(\frac{2}{N_{0}}\underline{\bm{G}}^{T}\operatorname*{diag}\left(\psi\left(\underline{\bm{u}}\right)\right)\underline{\bm{G}}\right)^{-1}\left(\sqrt{\frac{2}{N_{0}}}\underline{\bm{G}}^{T}\left(\underline{\bm{r}}\odot\varphi\left(\underline{\bm{u}}\right)\right)\right).$
(18)

Since $\underline{\bm{G}}=\bm{Q}_{N}^{T}\underline{\bm{\Lambda}}_{b}$, a more
explicit form can be obtained as

$\Delta\underline{\bm{x}}=\sqrt{\frac{N_{0}}{2}}\left(\underline{\bm{\Lambda}}_{b}^{T}\bm{Q}_{N}\operatorname*{diag}\left(\psi\left(\underline{\bm{u}}\right)\right)\bm{Q}_{N}^{T}\underline{\bm{\Lambda}}_{b}\right)^{-1}\left(\underline{\bm{\Lambda}}_{b}^{T}\bm{Q}_{N}\left(\underline{\bm{r}}\odot\varphi\left(\underline{\bm{u}}\right)\right)\right).$
(19)

The matrix to be inverted in (19) is a $2KV\times 2KV$ matrix without any
additional property for simplification. In order to simplify the relations, we
first aim to approximate $\operatorname*{diag}(\psi(\underline{\bm{u}}))$ to a
multiple of the identity matrix $\gamma\bm{I}_{2NV}$ such that

$\gamma=\frac{1}{2NV}\sum_{n=1}^{2NV}\psi(\underline{u}_{n}).$ (20)

This approximation is valid for the low SNR regime when $N_{0}$ is large so
that the elements of $\underline{\bar{\bm{u}}}$ are very close to zero. For
the high SNR regime, it is very likely that all elements of
$\underline{\bar{\bm{u}}}$ are positive and large values. Since the rate of
change of $\psi(.)$ is very low for large positive arguments, this
approximation is also helpful for high SNR.

While $\operatorname*{diag}(\psi(\underline{\bm{u}}))$ helps obtain the
second-order derivative information in each branch, we replace it with the
sample mean to obtain a form that helps simplify the matrix multiplications
and not deviate much from NM to obtain fast convergence. A similar
approximation was made in [40] to calculate feed-forward and feedback filters
in an iterative equalizer for SC systems to obtain block diagonal form in the
FD. The resultant expression for $\Delta\underline{\bm{x}}$ can be obtained as

$\Delta\underline{\bm{x}}\cong\frac{1}{\gamma}\sqrt{\frac{N_{0}}{2}}\left(\underline{\bm{\Lambda}}_{b}^{T}\underline{\bm{\Lambda}}_{b}\right)^{-1}\underline{\bm{\Lambda}}_{b}^{T}\bm{Q}_{N}\left(\underline{\bm{r}}\odot\varphi\left(\underline{\bm{u}}\right)\right).$
(21)

Note that with this approximation, decoupling between the real and imaginary
parts due to second-order derivative calculation can also be avoided. We can
now safely go back to the notation with complex numbers. The complex
counterpart of (14) for the quasi-Newton approach can be obtained as

$\underline{\bar{\bm{x}}}^{(t)}=\mathcal{P}^{(t)}\left(\underline{\bar{\bm{x}}}^{(t-1)}-\alpha\Delta\underline{\bar{\bm{x}}}^{(t-1)}\right),$
(22)

where, using (21), $\Delta\underline{\bar{\bm{x}}}$ can be written as

$\Delta\underline{\bar{\bm{x}}}=\frac{1}{\gamma}\sqrt{\frac{N_{0}}{2}}\left(\bar{\underline{\bm{\Lambda}}}_{b}^{H}\bar{\underline{\bm{\Lambda}}}_{b}\right)^{-1}\bar{\underline{\bm{\Lambda}}}_{b}^{H}\bar{\bm{Q}}_{N}\left(\bar{\underline{\bm{r}}}\,\bar{\odot}\,\bar{\varphi}\left(\bar{\underline{\bm{u}}}\right)\right),$
(23)

by defining $\bar{\odot}$ product of two vectors $\bar{\bm{a}}$ and
$\bar{\bm{b}}$, and $\bar{\varphi}(\bar{\bm{a}})$ as

$\bar{\bm{a}}\,\bar{\odot}\,\bar{\bm{b}}=\Re\\{\bar{\bm{a}}\\}\odot\Re\\{\bar{\bm{b}}\\}+j\,\Im\\{\bar{\bm{a}}\\}\odot\Im\\{\bar{\bm{b}}\\},$
(24)

$\bar{\varphi}(\bar{\bm{a}})=\varphi(\Re\\{\bar{\bm{a}}\\})+j\,\varphi(\Im\\{\bar{\bm{a}}\\}),$
(25)

respectively. Note that $\underline{\bar{\bm{u}}}$ from its each component
$\\{\bar{\bm{u}}[m]\\}_{m=0}^{V-1}$ can be found with the complex notation as

$\bar{\bm{u}}[m]=\sqrt{\frac{2}{N_{0}}}\bar{\bm{r}}[m]\,\bar{\odot}\,\left(\mathcal{F}_{m}^{-1}\\{\bar{\bm{\Lambda}}[v]\bar{\bm{x}}[v]\\}-\bar{\bm{\tau}}\right),$
(26)

Since $\underline{\bar{\bm{\Lambda}}}_{b}$ is a block diagonal matrix,
subcarrier-level processing is now applicable. (23) can be written for each
subcarrier $v=0,1,\ldots,V-1$ separately as

$\Delta\bar{\bm{x}}[v]=\frac{1}{\gamma}\sqrt{\frac{N_{0}}{2}}\left(\bar{\bm{\Lambda}}[v]^{H}\bar{\bm{\Lambda}}[v]\right)^{-1}\bar{\bm{\Lambda}}[v]^{H}\mathcal{F}_{v}\left\\{\bar{\bm{r}}[m]\,\bar{\odot}\,\bar{\varphi}\left(\bar{\bm{u}}[m]\right)\right\\}.$
(27)

In order to avoid matrix inversion, we introduce one last approximation using
the properties of massive MIMO systems. Notice that
$\left(\bar{\bm{\Lambda}}[v]^{H}\bar{\bm{\Lambda}}[v]\right)^{-1}\bar{\bm{\Lambda}}[v]^{H}$
from (27) is the zero-forcing (ZF) filter. By exploiting the large number of
BS antennas, we assume $\bar{\bm{\Lambda}}[v]^{H}\bar{\bm{\Lambda}}[v]$ is
diagonally-dominant, and
$\left(\bar{\bm{\Lambda}}[v]^{H}\bar{\bm{\Lambda}}[v]\right)^{-1}$ can be
approximated by a diagonal matrix $\operatorname*{diag}(\bm{\lambda}[v])$ such
that

$\lambda_{k}[v]=\frac{1}{\sum_{n=1}^{N}|[\bar{\bm{\Lambda}}[v]]_{(n,k)}|^{2}},$
(28)

for $k=1,2,\ldots,K$ and $v=0,1,\ldots,V-1$. Hence, we replace the ZF filter
expression during the step calculations with the maximum ratio combining (MRC)
filter by relying on massive MIMO, i.e., $N\gg K$, to obtain the final form

$\Delta\bar{\bm{x}}[v]\cong\frac{1}{\gamma}\sqrt{\frac{N_{0}}{2}}\,\bm{\lambda}[v]\odot\left(\bar{\bm{\Lambda}}[v]^{H}\mathcal{F}_{v}\left\\{\bar{\bm{r}}[m]\,\bar{\odot}\,\bar{\varphi}\left(\bar{\bm{u}}[m]\right)\right\\}\right).$
(29)

The projection function at iteration $t$, $\mathcal{P}^{(t)}(.)$ can be
expressed as

$\mathcal{P}^{(t)}\left(\underline{\bar{\bm{x}}}\right)=\left\\{\begin{array}[]{ll}\mathcal{P}_{\mathrm{box}}\left(\underline{\bar{\bm{x}}}\right),&\quad
1\leq t<T\\\
\mathcal{P}_{\mathrm{norm}}\left(\underline{\bar{\bm{x}}}\right),&\quad
t=T,\end{array}\right.$ (30)

where $T$ denotes the total number of iterations, and
$\mathcal{P}_{\mathrm{box}}(.)$ can be found as

$\mathcal{P}_{\mathrm{box}}(\bar{x})=\mathcal{P}_{\mathrm{box}}^{I}\left(\Re\\{\bar{x}\\}\right)+j\mathcal{P}_{\mathrm{box}}^{Q}\left(\Im\\{\bar{x}\\}\right),$
(31)

where $\mathcal{P}_{\mathrm{box}}^{I}(.)=\mathcal{P}_{\mathrm{box}}^{Q}(.)$
and it is defined as

$\mathcal{P}_{\mathrm{box}}^{I}(x)=\operatorname*{sign}(x)\min\\{|x|,M_{b}\\},$
(32)

and each function is applied element-wise on its arguments. We define the norm
projection function $\mathcal{P}_{\mathrm{norm}}(.)$ as

$\mathcal{P}_{\mathrm{norm}}(\underline{\bar{\bm{x}}})=\frac{\sqrt{KV}}{\lVert\underline{\bar{\bm{x}}}\rVert}\underline{\bar{\bm{x}}}.$
(33)

Note that utilizing box projection is redundant at the last iteration $T$
since minimum distance mapping is conducted on each element of the estimate
after equalization is complete. Also, utilizing the norm projection function
at each iteration may cause problems during convergence. As in [21], norm
projection is applied only at the last iteration. Before starting the
iterative updates, an initial solution should be found, preferably not far
from the optimum. MRC estimate is a suitable choice with its low complexity,
which can be found as

$\bar{\bm{x}}^{(0)}[v]=\sqrt{\frac{\pi\sigma_{y}^{2}}{4}}\,\bm{\lambda}[v]\odot\left(\bar{\bm{\Lambda}}[v]^{H}\mathcal{F}_{v}\left\\{\bar{\bm{r}}[m]\right\\}\right),$
(34)

for $v=0,1,\ldots,V-1$, where $\sigma_{y}^{2}=K+N_{0}+\sigma_{\tau}^{2}$, and
$\sigma_{\tau}^{2}$ is the variance of the quantization thresholds which act
as an additional form of AWGN by taking the quantization thresholds to be
random. The scaling factor $\sqrt{\pi\sigma_{y}^{2}/4}$ is added to compensate
for the missing quantization label for linear processing as in [13], which is
obtained using the Bussgang decomposition and assuming an effective diagonal
covariance matrix $\sigma_{y}^{2}\bm{I}_{NV}$ for the unquantized observation
vector.

At high SNR, singular Hessian matrices can be encountered during the iterative
updates since $N_{0}$ takes very small values and $\psi(x)\rightarrow 0$ as
$x\rightarrow\infty$. To avoid singularities in such circumstances, we define
a damping factor as in [13] such that

$\zeta=\max\\{1,\rho/\rho_{d}\\},$ (35)

where $\rho_{d}$ is the damping SNR and the actual $N_{0}$ is multiplied with
$\zeta$ before starting the iterative updates to operate as if we are at the
damping SNR once the actual SNR exceeds this value.

After the iterative updates are complete, symbol-by-symbol minimum distance
mapping is applied on the estimate for $k=1,\ldots,K$ and $v=0,\ldots,V-1$ to
obtain the final decisions

|l|[2] ¯x ∈¯M |¯x_k^(T)[v] - ¯x| ^¯x_k[v]=.

Algorithm 1 Projected Quasi-Newton Detector (PQND)

0:
$\\{\bar{\bm{r}}[m]\\}_{m=0}^{V-1},\\{\bar{\bm{\Lambda}}[v]\\}_{v=0}^{V-1},\bar{\bm{\tau}},\alpha,T,N_{0},\zeta$

0: $\\{\hat{\bar{\bm{x}}}[v]\\}_{v=0}^{V-1}$

1: Calculate $\\{\bm{\lambda}[v]\\}_{v=0}^{V-1}$ as in (28)

2: Set the initial solution
$\\{\tilde{\bar{\bm{x}}}[v]\leftarrow\bar{\bm{x}}^{0}[v]\\}_{v=0}^{V-1}$ by
using (34)

3: Apply the damping factor $N_{0}\leftarrow\zeta N_{0}$ (35)

4: for $t=1$ to $T$ do

5: Calculate $\\{\bar{\bm{u}}[m]\\}_{m=0}^{V-1}$ using (26)

6: Calculate $\gamma$ as in (20)

7: Calculate the step $\\{\Delta\bar{\bm{x}}[v]\\}_{v=0}^{V-1}$ using (29)

8: Update
$\\{\tilde{\bar{\bm{x}}}[v]\leftarrow\tilde{\bar{\bm{x}}}[v]-\alpha\Delta\bar{\bm{x}}[v]\\}_{v=0}^{V-1}$
as part of (22)

9: if $t<T$ then

10: Apply box projection as part of (22) using (31)
$\\{\tilde{\bar{\bm{x}}}[v]\\}_{v=0}^{V-1}\leftarrow\mathcal{P}_{\mathrm{box}}\left(\\{\tilde{\bar{\bm{x}}}[v]\\}_{v=0}^{V-1}\right)$

11: else

12: Apply norm projection as part of (22) using (33)
$\\{\tilde{\bar{\bm{x}}}[v]\\}_{v=0}^{V-1}\leftarrow\mathcal{P}_{\mathrm{norm}}\left(\\{\tilde{\bar{\bm{x}}}[v]\\}_{v=0}^{V-1}\right)$

13: end if

14: end for

15: Apply minimum distance mapping as in (III-C)
$\\{\\{\hat{\bar{x}}_{k}[v]\leftarrow\operatorname*{arg\>min}_{\bar{x}\in\bar{\mathcal{M}}}|\tilde{\bar{x}}_{k}[v]-\bar{x}|\\}_{k=1}^{K}\\}_{v=0}^{V-1}$

16: return $\\{\hat{\bar{\bm{x}}}[v]\\}_{v=0}^{V-1}$

The complete procedure for the proposed PQND is summarized in Algorithm 1. We
started with the update equation of NM (14) and utilized two approximations to
obtain a low-complexity equalization scheme. The first is related to the
identity approximation of the diagonal matrix
$\operatorname*{diag}(\psi(\underline{\bm{u}}))\cong\gamma\bm{I}_{NV}$ as in
(21). The second is the diagonally dominant approximation for
$\left(\bar{\bm{\Lambda}}[v]^{H}\bar{\bm{\Lambda}}[v]\right)^{-1}\cong\operatorname*{diag}(\bm{\lambda}[v])$
as in (29). Therefore, a projected quasi-Newton method is constructed that
does not require matrix inversion. Note that equalization can easily be
modified for channel estimation, which is left as future work.

###  III-D Extension to SC-FDE

Both OFDM and SC systems can benefit from the circular convolution properties
in situations where the system employs CP. Hence, the OFDM model can easily be
extended for SC-FDE. For OFDM transmission, the transmitted TD signal is the
unitary inverse DFT of the constellation symbols as shown in (2). For SC-FDE,
the transmitted TD signal directly corresponds to the constellation symbols.
Hence, for SC-FDE, (7) can be re-written as

$\underline{\bar{\bm{y}}}=\underline{\bar{\bm{H}}}_{b}\,\underline{\bar{\bm{x}}}+\underline{\bar{\bm{w}}}=\bar{\bm{Q}}_{N}^{H}\underline{\bar{\bm{\Lambda}}}_{b}\,\bar{\bm{Q}}_{K}\,\underline{\bar{\bm{x}}}+\underline{\bar{\bm{w}}}=\underline{\bar{\bm{G}}}\,\underline{\bar{\bm{a}}}+\underline{\bar{\bm{w}}},$
(36)

where
$\underline{\bar{\bm{G}}}=\bar{\bm{Q}}_{N}^{H}\underline{\bar{\bm{\Lambda}}}_{b}$
again, and
$\underline{\bar{\bm{a}}}=\bar{\bm{Q}}_{K}\,\underline{\bar{\bm{x}}}$. Hence,
the signal to be equalized is the unitary DFT of the transmitted constellation
symbols. The whole procedure is the same as that of the OFDM system, except
before applying the box constraints, the FD signal must be converted to the
TD, and after the projection, it must be converted back to the FD to continue
the iterative updates.

##  IV Pseudo-Random Quantization (PRQ) Scheme

One-bit massive MIMO systems are better at detecting phase than amplitude
since the cardinality of the amplitude alphabet of the quantized observations
is $1$, but the cardinality of the phase alphabet is $4$ [25]. One-bit
quantization causes a black/white or -1/+1 situation where the message to be
decoded is a gray tone or a number between -1 and 1. Suppose we have a real-
valued AWGN channel scenario at a very high SNR to decode a message that can
take values $\\{\pm 1.0,\pm 0.5\\}$, and we have independent observations to
decode the message. If the message $0.5$ is sent, all we can deduce in a one-
bit quantized scenario where the thresholds are set to zero is that the
message is most likely greater than zero. However, suppose the thresholds are
changed to uniformly sample the region between $-1$ and $+1$. Then, we can
better understand the signal’s amplitude rather than just its sign
information. Thermal noise can significantly help understand the signal’s
amplitude in such a coarsely quantized single-input multiple-output (SIMO)
scenario, which is related to the SR phenomenon.

SR occurs in nonlinear systems where the maximum performance is achieved at a
finite SNR value in contrast to linear systems where higher SNR leads to
better performance [11]. When the signal to be detected is not distinguishable
by a sensor, the presence of white noise in a nonlinear system’s input can
help increase the detectability of the input signal [12]. We utilize the PRQ
scheme to mitigate the effects of the SR phenomenon under frequency-selective
fading, which is discussed thoroughly for the flat-fading scenario in [13,
15]. In the PRQ scheme, we rely on changing the quantization thresholds by
assigning them pseudo-randomly generated numbers. Randomizing the quantization
process can be a handy tool to mitigate quantization noise in low-resolution
systems [14]. Randomized quantization exploits dithering, where a randomly
generated analog signal is added to/subtracted from the input to the
quantizer. Generating analog signals requires additional hardware, which may
not be desirable. Therefore, a preferable alternative is to change the
quantization threshold of each ADC unit, which will result in the same
dithering effect as the conventional scenario. We change the domain of
dithering from temporal (time) to spatial (antennas) with this approach.

In [13, 15], we analyze this issue with a coding theoretic analysis by
measuring the average of the minimum Hamming distances of the binary space
codes obtained by one-bit quantization at infinite SNR using ZTQ and PRQ. Our
analysis shows that the average minimum Hamming distance of the space code
with variable amplitude modulation schemes becomes zero with ZTQ. Hence, the
resultant code is not uniquely decodable. On the other hand, by employing PRQ,
the minimum Hamming distance takes non-zero values for different scenarios,
which suggests that the performance will surely increase when PRQ is used.
Also, in [15, Section 3], we show that the achievable rate, even in low
dimensional SIMO systems with high-order modulation schemes, increases with
PRQ. Hence, ZTQ is not optimal when interference is small, and PRQ can help
obtain favorable performance, especially at high SNR, due to better amplitude
recovery.

Each pair of ADCs in the BS RF chains have a complex-valued threshold that
separately represents the I/Q parts. The thresholds are generated from the
Gaussian distribution, i.e.,
$\bar{\bm{\tau}}\sim\mathcal{CN}(\bm{0}_{N},\sigma_{\tau}^{2}\bm{I}_{N})$ as
in the conventional dithering scenario. Since we are trying to detect zero-
mean signals and our unquantized observations are also 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. From many trials and observations, we saw that the variance
selection depends on the system parameters such as $N$, $K$, and $\rho$. If
the variance is too small, the performance is not affected. If it is selected
as too large, the performance gets worse since the threshold values become too
large to differentiate relatively small amplitude differences.

Frequency-selectivity is essential in one-bit massive MIMO systems that help
deal with SR [9]. Hence, different from the frequency-flat fading scenario
[13], the statistics of the wireless channel, in addition to the number of BS
antennas ($N$) and the number of users ($K$), must be used to determine the
threshold variance. Increasing the number of BS antennas increases the spatial
degrees of freedom. Hence, using thresholds with larger variance is more
suitable. However, increasing $K$ that affects the magnitude of the MUI or $L$
that affects the magnitude of the ISI on the received signal seems to provide
the desired dithering effect to operate with variable-amplitude modulation
schemes. Hence, a lower threshold variance is required when MUI and ISI sum
interference is high. How interference can sometimes be advantageous when the
number of users is small is investigated in [13, 15]. 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 $\rho_{\mathrm{t}}$ is an appropriate technique. By
relying on many observations, in order to avoid selecting different variances
for different scenarios, we parameterize the threshold SNR value as

$\rho_{\mathrm{t}}=\left\\{\begin{array}[]{ll}0.15K^{2}+L_{s}-2.50\log_{2}(N)+14\textrm{
dB},&\textrm{ for OFDM}\\\ 0.15K^{2}+L_{s}-2.50\log_{2}(N)+16\textrm{
dB},&\textrm{ for SC-FDE}\end{array}\right.$ (37)

where $L_{s}$ is the number of strong, i.e., high-power channel taps, which is
determined as

$L_{s}=\sum_{\ell=0}^{L-1}\frac{\operatorname*{sign}(p_{d}[\ell]-1/L)+1}{2},$
(38)

and $p_{d}[\ell]=\mathbb{E}[|\bar{h}_{n,k}[\ell]|^{2}]$ is the power delay
profile (PDP) of the channel. Hence, the number of strong taps is equal to
those whose power is greater than the power distributed between the taps of a
uniform PDP channel. This selection may not be optimal and relies only on
empirical findings and observations. The threshold SNR for SC-FDE is $2$ dB
higher than OFDM, which proved more helpful during our trials. This can be
related to the higher amplitude variation in OFDM signals, which requires a
larger variance in the dither signal, i.e., the quantization thresholds. Now
that the threshold SNR value is selected, the variance of the quantization
thresholds can be found accordingly using

$\sigma_{\tau}^{2}=\operatorname*{R}\left(\frac{E_{s}}{\rho_{\mathrm{t}}}-N_{0}\right)=\operatorname*{R}\left(\frac{1}{\rho_{\mathrm{t}}}-N_{0}\right).$
(39)

where $\operatorname*{R}(x)=\max\\{0,x\\}$ is the unit ramp function. The
effective nonlinear channel between the users and the receiver antennas
changes when the threshold information is used during baseband processing. If
the information were not used, the effective nonlinear channel would have
remained the same, and only a decrease in the SNR would occur since the
thresholds would act as an additional form of noise over AWGN. The PRQ
application in this paper does not actually lower the SNR since the thresholds
are perfectly known, but it changes the effective channel and provides higher
achievable rates [15, Chapter 3]. In addition to quantization, changing the
sampling characteristics can also be helpful to obtain better performance as
in [41, 5, 35]. However, changing the sampling characteristics complicates the
system model due to correlation and increases the complexity of baseband
processing methods.

##  V Computational Complexity Analysis

TABLE I: Computational Complexity Comparison of the Proposed PQND with the
Existing Methods

Detector | Number of Flops
---|---
GEC-SR [33] | $\mathcal{O}(TNV\log_{2}(NV))+\mathcal{O}(TN^{2}V)$
EM [29] | $\mathcal{O}(TNKV^{2})$
IEM [30] | $\mathcal{O}(TVN\log_{2}(V))+\mathcal{O}(TVNK)$
GAMP [29] | $\mathcal{O}(TNKV^{2})$
1BOX [28] | $\mathcal{O}(TVN\log_{2}(V))+\mathcal{O}(TVNK)$
PQND | $\mathcal{O}(TVN\log_{2}(V))+\mathcal{O}(TVNK)$
NM | $\mathcal{O}(TVNK^{2}V^{3})$

The proposed PRQ scheme does not require digital processing except for
generating pseudo-random numbers assigned as threshold values of the ADCs. The
thresholds are kept the same for different channel realizations as long as the
average SNR remains the same. If the SNR changes, scaling the thresholds is
sufficient, and there is no need for regenerating the pseudo-random numbers.
The proposed PQND algorithm has many advantages. PQND can be adapted to
different non-zero quantization threshold scenarios while making likelihood
function-based FDE possible. Hence, a large channel delay spread is not a
significant source of complexity.

The computational complexity analysis of the proposed PQND is now analyzed in
the order expressed in Algorithm 1. Line 2 is the computation of the MRC
estimate as an initial solution. Then, the remaining parts are repeated for
$T$ iterations as part of PQND. The arguments
$\\{\bar{\bm{u}}[m]\\}_{m=0}^{N-1}$ of the nonlinear functions are calculated
in Line 5. The sample mean of the average second-order derivative information
$\gamma$ is computed in Line 6. Note that the nonlinear function $\psi(x)$ can
be computed using only $\varphi(x)$. Therefore, computing $\varphi(.)$ is
sufficient, which can easily be handled using a look-up table. The step
$\\{\Delta\bar{\bm{x}}[v]\\}_{v=0}^{V-1}$ towards the optimum for a given
iteration is calculated in Line 7. Since $N\gg K$ for massive MIMO, the
complexity-dominant parts of the algorithm are Lines 5 and 7 both with
complexity $\mathcal{O}(TVN\log_{2}(V))+\mathcal{O}(TVNK)$. Hence, we can
state that the complexity of PQND increases linearly with the number of BS
antennas and users and super-linearly with the number of subcarriers due to
the usage of the fast Fourier transform (FFT) algorithm and does not depend on
the modulation order.

The complexities of the proposed detector and the existing works from the
literature that are applicable under frequency-selective fading are shown in
Table I. The generalized expectation consistent signal recovery (GEC-SR)
method [33] requires a quadratic complexity increase in the number of BS
antennas. EM and GAMP methods, explained in detail in [29], require a
quadratic complexity increase in the data block length. Inexact EM (IEM) [30]
and 1BOX [28] methods have comparable complexity with the proposed PQND
method. However, the performance of the IEM method is very similar to the 1BOX
detector, and the number of iterations for the algorithm to converge is
reportedly very large [30]. NM has the largest complexity, making operations
within commercial systems unfeasible. The benchmark algorithm to compare with
the proposed PQND is the 1BOX detector from [28], which also utilizes FDE
tools based on the likelihood function. However, it does so by utilizing
first-order optimization. PQND and 1BOX have comparable complexity with
similar computation steps. However, since PQND involves a second-order
optimization technique, achieving a target error rate requires fewer
iterations.

##  VI Simulation Results

In this section, the error performance of the proposed detector is
investigated for different scenarios. The proposed quasi-Newton optimization
technique’s step size is $\alpha=0.7$ for all scenarios. NM can generally work
with $\alpha=1$. Due to approximations used while obtaining PQND, relatively
smaller step size is used to avoid divergence. $T=6$ iterations are considered
for all scenarios. The damping SNR for PQND is determined as
$\rho_{d}=20-150\frac{K}{N}$ dB. The variance of quantization thresholds is
determined as in (39) when PRQ is utilized. The number of subcarriers is
chosen as $V=1024$ unless otherwise stated. The frequency-selective wireless
channel is modeled with Rayleigh fading, i.e.,
$\bar{h}_{n,k}[\ell]\sim\mathcal{CN}(0,p[\ell])$, and each coefficient is
statistically independent both in space and time. BER performance is
investigated with two PDPs throughout the simulations. The power of each tap
in an exponential PDP channel can be calculated as

$p[\ell]=\frac{\exp(-\mu\ell)}{\sum_{m=0}^{L-1}\exp(-\mu m)},$ (40)

where $\mu$ is the power decay rate and the sum in the denominator is added to
satisfy average unit power. To model a small delay spread (SDS) channel, we
use exponential PDP where $L=8$ and $\mu=1$, which results in $L_{s}=3$. To
model a large delay spread (LDS) channel, we utilize TDL-A delay profile from
[42], which is a tapped delay line (TDL) channel model introduced by 3GPP for
link level simulations of 5G systems, for which $L=23$ and $L_{s}=7$.

###  VI-A Convergence of the Algorithms

(a) Log-Likelihood

(b) Error Rate

Figure 2: Convergence analysis of the original Newton’s Method (NM) with and
without box constraints, box-constrained one-step approximated quasi-Newton
approach, proposed two-stage approximated quasi-Newton approach (PQND), and
gradient-descent (1BOX) algorithms with OFDM transmission. The negative log-
likelihood is plotted with respect to the number of iterations for a
$128\times 10$ system employing ZTQ and $16$-QAM constellation in the SDS
channel. The average is obtained over $400$ channel realizations. The step
size, $\alpha$, is taken as $1$ for NM, $0.7$ for the one-stage approximated
version, $0.7$ for the two-stage approximated version, and $0.009$ for the
gradient descent algorithm. The number of subcarriers is chosen as $V=32$.

We rely on the concavity of the log-likelihood to utilize gradient-based
optimization. In [38, 39], the convergence of boxed constrained gradient
descent algorithm in the one-bit massive MIMO setup is shown. For our
purposes, the negative of the log-likelihood function (a) and the error-rate
analysis (b) with respect to the number of iterations using the original NM
with and without box constraints, constrained NM with only the first-step
approximation, and the final two-stage approximation along with the first-
order approach, 1BOX from [28], are shown in Fig. 2. A $128\times 8$ system
employing $16$-QAM modulation and OFDM in the SDS channel is used to obtain
the results. As indicated in the caption, different step size selections are
made to obtain fast and favorable convergence. The initial solution is used as
the MRC estimate for all algorithms, and the same projection functions are
applied for all methods for fairness.

The plots show that even though the original ML without constraints can
quickly converge to a solution in the continuous space in the log-likelihood
sense, the optimum does not reflect the result we seek. The box-constrained
version of the original ML moves in the right direction for a few iterations
before diverging. This can relate to the SNR damping we used to avoid
numerical problems. However, calculating the nonlinear functions of interest
also requires high accuracy since the numbers of interest take very small
values. Therefore, in addition to complexity concerns, the original ML may not
be feasible due to these issues. Maximizing the log-likelihood in the
continuous space does not always mean getting the best error performance since
the transmitted symbols come from a finite cardinality alphabet. If NM is used
with approximations, it can easily be seen that we can obtain convergence.
Also, the PQND method performs very closely to the one-step approximated
method, which is a rational tradeoff to avoid matrix inversion. Finally, even
after careful adjustments of the step size for the 1BOX method, it fails to
converge within $10$ iterations as opposed to the quasi-Newton approaches.
Hence, the proposed PQND is very advantageous in terms of both convergence and
complexity.

###  VI-B Detector Performance Comparison with ZTQ

The BER performances of different detectors from the literature are compared
in Fig. 3 in both SDS and LDS channels with ZTQ. The results are obtained for
$K=10$ users for two setups where $N=128$ with $16$-QAM for the first and
$N=256$ with $64$-QAM for the second. Linear detectors MRC and ZF are used for
comparison along with 1BOX from [28] and the proposed PQND. The MRC estimate
is found as in (34), and similarly, the ZF estimate can be found as

(a) SDS Channel

(b) LDS Channel

Figure 3: Comparison of BER performances of MRC, ZF, 1BOX [28], and PQND
methods in SDS (a) and LDS (b) channel models with respect to SNR for a
$128\times 10$ system with $16$-QAM and a $256\times 10$ system with $64$-QAM
using OFDM.

$\tilde{\bar{\bm{x}}}[v]=\sqrt{\frac{\pi\sigma_{y}^{2}}{4}}\,\left(\bar{\bm{\Lambda}}[v]^{H}\bar{\bm{\Lambda}}[v]\right)^{-1}\bar{\bm{\Lambda}}[v]^{H}\mathcal{F}_{v}\left\\{\bar{\bm{r}}[m]\right\\},$
(41)

for $v=0,1,\ldots,V-1$, which are followed by symbol-by-symbol detection as
shown in (III-C). A detailed explanation regarding linear filtering methods
for the one-bit non-zero threshold quantization scenario can be found in [15,
Chapter 4]. 1BOX involves only box projections at each iteration, though for
fairness during comparison, we utilize the same projection function for 1BOX
as PQND as described in (30). Hence, different than [28], norm projection is
used at the final iteration of 1BOX, which leads to better performance.
Regarding the damping constant, a similar discussion about acting as if the
SNR is lower is made in [28]. From several trials, we have seen that choosing
$\rho_{d}=15$ dB for 1BOX is suitable for both system setups. $T=6$ iterations
of 1BOX are utilized, the same as PQND. The step size is an issue for first-
order optimization methods, especially for varying SNR. Again, by observing
the performance for different setups, we select the step size of 1BOX as
$0.007$ (defined as $\kappa$ in [28]).

Note that even though the gradient expression involves the $\sqrt{N_{0}/2}$
constant at the beginning, as can be seen in (16), first-order optimization-
based methods generally discard the term and take it as part of the step size,
examples of which can be found in [31, 21]. However, for the proposed PQND,
the SNR-dependent scaling is not discarded and becomes useful with the second-
order derivative information to select a fixed step size for all scenarios.
The MLSD method is not included in the comparison since it can be simulated
only if $V$ and $K$ are chosen to be very small as the complexity grows as
$M^{KV}$. Unlike unquantized systems, the ML detector cannot be employed in
the FD for one-bit systems for each subcarrier separately since the
conditional distribution of the quantized observations’ FD representation
takes a complicated form. The original NM’s complexity for this scenario is
still very large due to the inversion of a $2KV\times 2KV$ matrix.

Again, by comparing the plots, we can see that the performance obtained in the
LDS channel is better than in that of the SDS channel. Increased frequency-
selectivity helps obtain larger multipath diversity to reduce amplitude
distortions. Similar findings were also reported in [43], where the error
performance of an unquantized system is studied. Linear detectors MRC and ZF
perform much poorly compared to the more sophisticated likelihood-based
methods 1BOX and PQND. For a fixed iteration number, the proposed PQND
outperforms 1BOX in all scenarios, as expected, since second-order techniques
converge faster than the first-order methods [44].

###  VI-C Comparison of ZTQ and PRQ Performances with PQND and 1BOX

(a) PQND

(b) 1BOX

Figure 4: BER performances with PQND (a) and 1BOX (b) detectors in the SDS
channel with respect to SNR ($\rho$) when $K=2$ for $N=64$ with $16$-QAM,
$N=128$ with $64$-QAM, and $N=256$ and $256$-QAM obtained with ZTQ and PRQ
schemes using OFDM.

In Fig. 4, the performances of systems with two users are investigated when
ZTQ and PRQ schemes are utilized with high-order modulations in the SDS
channel using PQND and 1BOX methods. The two-user scenario is chosen because
when the number of users is small, the merits of the PRQ scheme are more
evident. An exhaustive search is not possible in such a setup since the ML
detector cannot be written for each subcarrier separately in the FD as in the
case of unquantized systems since the conditional distribution of the FD
observations is not easily tractable. We can think of one channel use as a
data block of length $V$ symbols representing $V\log_{2}(M)$ bits sent from
all $K$ users and a total of $2NV$ bits of information being received at the
BS. Hence, we can define a load similar to a code rate such that
$R_{c}=\frac{K\log_{2}(M)}{2N}$. Hence, for a two-user and $64$-antenna system
transmitting $16$-QAM symbols, the load can be found as $\frac{1}{16}$. As the
load increases, there is less need for PRQ, and the system’s performance
decreases to levels that are not operable due to 1-bit quantization. However,
when the load is small, the merits of PRQ compared to ZTQ are more visible
when variable-amplitude modulation schemes are employed.

The results are obtained for $N=64$ with $M=16$, $N=128$ with $M=64$, and
$N=256$ with $M=256$. The high SNR performance generally starts to saturate
with ZTQ after some point. The spatial degrees of freedom are exploited much
better with PRQ, and high SNR error floors are decreased to relatively much
lower levels. Most proposed methods for frequency-flat and frequency-selective
channel scenarios focus on QPSK and 16-QAM constellations. According to Fig.
4, higher-order modulation schemes can also be applicable in one-bit massive
MIMO systems with the PRQ scheme. It can also be seen that both methods can
benefit from the PRQ approach. In [28], the 1BOX method is proposed with ZTQ.
However, it can easily be modified for the PRQ scenario by updating the log-
likelihood function according to the non-zero quantization thresholds. As in
[13], a larger number of BS antennas increases the performance gains obtained
with PRQ.

###  VI-D Effect of Changing the Number of Users using ZTQ and PRQ with OFDM
and SC-FDE

(a) SDS Channel and OFDM

(b) LDS Channel and OFDM

(c) SDS Channel and SC-FDE

(d) LDS Channel and SC-FDE

Figure 5: BER performances in SDS (a-c) and LDS (b-d) channel models using
OFDM (a-b) and SC-FDE (c-d) schemes with respect to the number of users ($K$)
at $\rho=30$ dB of SNR for $N=64$ with $16$-QAM, $N=128$ with $64$-QAM, and
$N=256$ with $256$-QAM obtained with ZTQ and PRQ schemes.

In Fig. 5, the error performances in SDS and LDS channels with respect to the
number of users are plotted at $\rho=30$ dB of SNR again for $N=64$ with
$16$-QAM, $N=128$ with $64$-QAM, and $N=256$ with $256$-QAM both using ZTQ and
PRQ when OFDM and SC-FDE schemes are employed. Since the high SNR performance
is the limiting factor for high-order modulations, $\rho=30$ dB is selected to
better understand the high SNR behavior of ZTQ and PRQ. Similar to the
findings from [13, 15], starting with the single-user scenario, increasing the
number of users results in better performance at the beginning, which are
implications of the SR phenomenon and MUI serving as a source of dither. The
performance gain obtained with PRQ is larger when the number of users is
small. The LDS channel exhibits a higher ISI level than the SDS channel. With
the increased ISI and multipath diversity, performance in the LDS channel is
again better compared to the SDS channel. Again, starting with $K=1$, the
number of users for which ZTQ and PRQ start to perform the same is smaller for
the LDS channel. These results suggest that at high SNR, up to a certain point
of sum interference composed of MUI and ISI, massive MIMO systems can benefit
from PRQ to achieve higher rates per user by employing higher modulation
orders as opposed to the conventional ZTQ scheme.

Comparing the OFDM and SC-FDE performances, as discussed in [27], the SC
transmission scheme is more tolerant to hardware impairments than the OFDM
scheme. The performance of the proposed detector verifies these expectations
as the SC-FDE scheme outperforms OFDM, especially when the number of users is
small. However, the performance gap between OFDM and SC-FDE for the small
number of users scenario almost vanishes when the proposed PRQ scheme is
employed. Hence, the larger amplitude variation of OFDM is tolerated with the
better amplitude recovery of PRQ. Also, when the number of users gets large,
the distribution of the unquantized received signal will approach the Gaussian
distribution due to the CLT. Since the distribution of the received signal for
a large number of users is very similar for OFDM and SC-FDE, their
performances are very similar, as expected, when the number of users is large.

###  VI-E Effect of Changing the Number of BS Antennas using ZTQ and PRQ

(a) SDS Channel

(b) LDS Channel

Figure 6: BER performance of a SIMO system in SDS (a) and LDS (b) channel
models with respect to the logarithm of the number of antennas ($\log_{2}(N)$)
at $\rho=30$ dB of SNR with $64$-QAM, $256$-QAM, and $1024$-QAM obtained with
ZTQ and PRQ schemes using OFDM.

In Fig. 6, we check the $64$-QAM, $256$-QAM, and $1024$-QAM error performances
in a SIMO system with respect to the number of antennas at $\rho=30$ dB both
in SDS and LDS channels. As can be seen, the SDS channel does not allow the
usage of these high-order modulation schemes with the conventional ZTQ, even
when the number of antennas is very large. Only $64$-QAM transmission seems
viable among the chosen modulation schemes in the LDS channel. Hence, in the
SIMO scenario with no MUI, ISI helps lower the amplitude distortion as
suggested in [9] to support high-order modulation schemes with variable
amplitudes. However, even in the LDS channel, employing PRQ is necessary to
work with $256$-QAM and $1024$-QAM by better exploiting the spatial degrees of
freedom.

##  VII Conclusion

This paper proposes a new detection method, PQND, that operates with PRQ for
one-bit wideband massive MIMO systems. PQND is derived based on NM with
additional approximations to obtain a quasi-Newton method to optimize the log-
likelihood function by operating at the subcarrier level in the FD. By
utilizing PQND and PRQ, one-bit massive MIMO-OFDM and CP-SC-MIMO systems can
support high-order modulation schemes and benefit from higher rates per user
at high SNR. The proposed detector outperforms the first-order optimization-
based benchmark method 1BOX with comparable complexity. Other existing methods
from the literature offer comparatively larger complexity due to either not
being applicable for subcarrier-level equalization or requiring a large number
of iterations. The amplitude recovering capabilities of PRQ is also helpful
for reducing the performance gap between SC and OFDM systems.

Multi-bit ADCs and oversampling effects are also interesting topics to be
considered with PRQ and PQND for future research. It would be easy to modify
the proposed PQND method for multi-bit ADCs since only the likelihood function
needs to be updated to consider the probability of an interval between the
quantization thresholds. However, oversampling would complicate the structure
due to the temporal correlation among noise samples. In such a scenario,
resorting to machine learning tools would be preferable. It would also be very
interesting to consider PRQ or time-varying threshold ideas as in [19, 18]
within such systems. For future work, the energy efficiency and spectral
efficiency tradeoff of low-resolution multi-bit or hybrid systems could be
beneficial to analyze the merits of low-resolution quantization in massive
MIMO systems.

## Acknowledgments

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

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