# On the Mean Square Error Optimal Estimator
in One-Bit Quantized Systems

Benedikt Fesl, Michael Koller, and Wolfgang Utschick,  Benedikt Fesl and
Michael Koller are with Professur für Methoden der Signalverarbeitung,
Technische Universität München, 80333 München, Germany (e-mail:
<EMAIL_ADDRESS>michael.koller@tum.de). Wolfgang Utschick is with
Professur für Methoden der Signalverarbeitung, Technische Universität München,
80333 München, Germany (e-mail<EMAIL_ADDRESS>and with the Munich Data
Science Institute (MDSI).

###### Abstract

This paper investigates the mean square error (MSE)-optimal conditional mean
estimator (CME) in one-bit quantized systems in the context of channel
estimation with jointly Gaussian inputs. We analyze the relationship of the
generally nonlinear CME to the linear Bussgang estimator, a well-known method
based on Bussgang’s theorem. We highlight a novel observation that the
Bussgang estimator is equal to the CME for different special cases, including
the case of univariate Gaussian inputs and the case of multiple pilot signals
in the absence of additive noise prior to the quantization. For the general
cases we conduct numerical simulations to quantify the gap between the
Bussgang estimator and the CME. This gap increases for higher dimensions and
longer pilot sequences. We propose an optimal pilot sequence, motivated by
insights from the CME, and derive a novel closed-form expression of the MSE
for that case. Afterwards, we find a closed-form limit of the MSE in the
asymptotically large number of pilots regime that also holds for the Bussgang
estimator. Lastly, we present numerical experiments for various system
parameters and for different performance metrics which illuminate the behavior
of the optimal channel estimator in the quantized regime. In this context, the
well-known stochastic resonance effect that appears in quantized systems can
be quantified.

###### Index Terms:

Bussgang theorem, channel estimation, conditional mean estimation, one-bit
quantization, mean square error.

##  I Introduction

©This work has been submitted to the IEEE for possible publication. Copyright
may be transferred without notice, after which this version may no longer be
accessible.

Digital signal processing has great impact on modern communication systems, in
which the analog signals undergo quantization through analog-to-digital
converters (ADCs) . The resolution of the ADCs determines how much information
is preserved after the quantization of the analog signal. In many
applications, the extreme case of one-bit quantization is of particular
interest, e.g., in the context of lossy compression and rate distortion theory
[1, 2], wireless sensor networks [3], or audio coding [4]. In recent years,
one-bit quantization gained a lot of interest in multiple-input multiple-
output (MIMO) communication systems, where transmitter and receiver are
possibly equipped with a large number of antennas. Since the power consumption
of the ADCs, which are needed for every antenna, is growing exponentially with
their resolution, one-bit ADCs are considered as a power-efficient solution
[5]. A major drawback of a system with one-bit ADCs is the severe information
loss which diminishes the capacity [6, 7, 8, 9, 10, 11] and achievable rate
[12, 13], and causes signal processing tasks like channel estimation to become
challenging [14, 15, 16]. However, in contrast to multi-bit considerations,
the study of one-bit quantization allows for closed-form solutions in many
applications which we discuss in the following. Thereafter, the found exact
solutions can be leveraged to approximate solutions for higher resolutions.

Remarkably, it is shown in [6, 7] that the capacity is not severely reduced by
the coarse quantization at low signal-to-noise ratios (SNRs). The capacity was
further analyzed for high SNRs in [8] and for a large number of antennas in
[9]. In [12, 13], the achievable uplink throughput and rate is discussed. In
order to achieve high data rates in quantized systems, accurate estimation of
the channel state information (CSI) is particularly important. In
consideration of the highly nonlinear quantization, channel estimation is
considerably more demanding, leading to an extensive collection of approaches
in the literature, e.g., [17, 18, 19, 20, 11, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30, 31, 32]. The work in [11] studies least squares (LS) estimation. In
[17, 18, 19, 20], iterative maximum likelihood (ML) methods are proposed,
whereas in [21, 22, 23], the maximum a posteriori (MAP) estimator is analyzed.
In [24, 21] the Cramér-Rao bound (CRB) is investigated. In [25, 26, 27, 23],
compressive sensing (CS) is studied for channel estimation, and deep learning
approaches were investigated in [28, 29, 30, 31, 32].

Apart from that, the Bussgang theorem turned out to be helpful for channel
estimation [33]. The theorem, a special case of the more general Price theorem
[34], states that the auto-correlation and cross-correlation of a zero-mean
Gaussian signal before and after it has passed through a nonlinear operation
is equal up to a constant. A direct consequence of the theorem is the fact
that the signal after the quantization can be decomposed in a linear fashion
into a desired signal part and an uncorrelated distortion. This property was
investigated for various nonlinearities [35] and further extended to non-zero
quantization thresholds [36]. In the context of signal processing, the theorem
was successfully used to design channel estimators. To this end, the
linearized model is combined with the linear minimum mean square error (MMSE)
estimator [37, 38, 39, 40].

Despite this extensive variety of channel estimation algorithms, there is so
far no detailed discussion about the CME for the one-bit quantization case.
The CME is well-known to be the minimizer for the MSE loss function, which is
the most frequently considered criterion for the channel estimation
performance. More generally, the CME is the optimal predictor for all Bregman
loss functions, of which the MSE is a special case [41]. On the one hand,
there are approximations of the CME, e.g., in wideband systems with many
channel taps and antennas where the quantization noise becomes approximately
Gaussian [42], or with neural networks that are trained on unquantized data
[29]. On the other hand, the CME is known for the simple one-dimensional case
for a Gaussian input with a single pilot observation [1, Sec. 13.1], [43].
However, to our knowledge, there exists no general discussion about the
multivariate case with correlated channel entries, or with longer pilot
sequences that can not be optimally decorrelated.

In [40], a two-stage (concatenated) MMSE estimator is proposed, where in the
first stage the product of the pilot matrix and the channel is estimated with
the CME. Afterwards, it is assumed that the remaining error is Gaussian, which
is unlikely to hold in general, such that the channel can be recovered by
decorrelating with the (pseudo-)inverse pilot matrix. This corresponds to a
second stage consisting of a linear MMSE solution. Interestingly, this
approach is shown to be equal to the Bussgang estimator in the one-bit
quantization case.

Contributions: We discuss the CME in systems with one-bit quantization for the
general case of multivariate Gaussian channels in the presence of additive
white Gaussian noise (AWGN) and multiple pilot observations. We present
closed-form solutions for special cases and numerical computations for the
general case. In that respect, we discuss the optimal pilot sequence from the
perspective of the CME that was independently suggested in [29] for learning a
neural network based channel estimator. We identify cases in which the
Bussgang estimator is equivalent to the CME by proving that their closed-form
MSE expressions are equal and we find asymptotic MSE limits for the large
number of pilots regime. We then quantify the gap between the linear Bussgang
estimator and the nonlinear CME in the general case by numerical experiments
and show that the gap increases for higher dimensions and longer pilot
sequences in the presence of correlations between the channel entries. In this
context, the well-known stochastic resonance effect [44], which allows for
outperforming the asymptotic limits in the presence of noise, is numerically
quantified. We finally discuss further performance metrics such as the cosine
similarity and a lower bound on the achievable rate for a matched filter.

The remainder of this work is structured as follows. In Section II, the
problem formulation is described, in Section III, the Bussgang estimator is
summarized, and in Section IV, the findings on the CME are presented. Section
V then discusses further performance metrics, Section VI presents the
numerical results, and Section VII concludes the work.

Notation: The (element-wise) signum function is denoted by
$\operatorname{sign}(\cdot)$. The Kronecker product is denoted by $\otimes$
and the trace of matrix is written as $\operatorname{tr}(\cdot)$. The real and
imaginary part of a complex scalar/vector/matrix is stated as $\Re(\cdot)$ and
$\Im(\cdot)$, respectively. The (column-wise) vectorization of a matrix which
stacks the columns is denoted by $\operatorname{vec}(\cdot)$. We denote the
probability density function (PDF) of a continuous random variable (RV)
${\bm{x}}$ as $f_{{\bm{x}}}$ and the probability mass function (PMF) of a
discrete RV ${\bm{y}}$ as $p_{{\bm{y}}}$. We denote by
$\mathcal{N}_{\mathbb{C}}({\bm{x}};{\bm{\mu}},{\bm{C}})$ the complex Gaussian
distribution with mean ${\bm{\mu}}$ and covariance ${\bm{C}}$, evaluated at
${\bm{x}}$, which yields a circularly symmetric RV after subtracting its mean.
Similarly, the real-valued Gaussian density is denoted by
$\mathcal{N}({\bm{x}};{\bm{\mu}},{\bm{C}})$. We denote the Gaussian error
function as
$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}\exp(-t^{2}){\operatorname{d}}t$.
We denote the uniform distribution between $a$ and $b$ as $\mathcal{U}(a,b)$.
The function $\angle(\cdot)$ gives the angle in $[0,2\pi)$ of a complex
number. The $k,\ell$th entry of a matrix ${\bm{X}}$ is denoted by
$[{\bm{X}}]_{k,\ell}$.

##  II Problem Formulation

We consider the following generic system equation

${\bm{R}}=Q({\bm{Y}})=Q({\bm{h}}{\bm{a}}^{\operatorname{T}}+{\bm{N}})\in\mathbb{C}^{N\times
M}$ (1)

where ${\bm{R}}=[{\bm{r}}_{1},{\bm{r}}_{2},\dots,{\bm{r}}_{M}]$ contains $M$
quantized observations of the vector of interest ${\bm{h}}\in\mathbb{C}^{N}$
with the known pilot vector ${\bm{a}}\in\mathbb{C}^{M}$ that fulfills the
power constraint $\|{\bm{a}}\|_{2}^{2}=M$. We denote by ${\bm{Y}}$ the
unquantized received signal. Let the vector ${\bm{h}}$ be a zero-mean Gaussian
RV, i.e.,
${\bm{h}}\sim\mathcal{N}_{\mathbb{C}}({\bm{0}},{\bm{C}}_{{\bm{h}}})$. The
desired signal is distorted with additive zero-mean Gaussian noise
${\bm{N}}=[{\bm{n}}_{1},{\bm{n}}_{2},\dots,{\bm{n}}_{M}]$ where
${\bm{n}}_{i}\sim\mathcal{N}_{\mathbb{C}}({\bm{0}},{\bm{C}}_{{\bm{n}}})$ and
is afterwards quantized with the complex-valued one-bit quantization function

$Q(\cdot)=\frac{1}{\sqrt{2}}\left(\operatorname{sign}(\Re(\cdot))+{\operatorname{j}}\operatorname{sign}(\Im(\cdot))\right)$
(2)

which is applied element-wise to the input vector/matrix. The system model in
(1) can be equivalently described in its (column-wise) vectorized form as

${\bm{r}}=Q({\bm{y}})=Q({\bm{A}}{\bm{h}}+{\bm{n}})\in\mathbb{C}^{NM}$ (3)

with ${\bm{A}}={\bm{a}}\otimes\operatorname{\mathbf{I}}$,
${\bm{r}}=\operatorname{vec}({\bm{R}})$,
${\bm{y}}=\operatorname{vec}({\bm{Y}})$, and
${\bm{n}}=\operatorname{vec}({\bm{N}})$. The system model describes, e.g., a
single-input multiple-output (SIMO) wireless communication scenario where a
base station equipped with $N$ antennas receives $M$ pilot signals from a
single-antenna user terminal. The ADCs at the receiver have one-bit
resolution, which is modeled by the quantization function in (2). Although we
especially focus on the application of channel estimation in this work, the
analysis is not limited to this instance and might yield insights to various
applications such as lossy compression [2], wireless sensor networks [3],
audio coding [4], or control theory [45], to name a few.

The overall goal is to estimate the signal ${\bm{h}}$ in an optimal sense
where the vector ${\bm{a}}$ is known and the channel and noise are both zero-
mean Gaussian with known covariances. The optimality depends on the metric of
choice, however, the most frequently used criterion in the context of channel
estimation is the MSE between the true channel ${\bm{h}}$ and its estimate
$\hat{{\bm{h}}}$, i.e.,

$\text{MSE}={\operatorname{E}}[\|{\bm{h}}-\hat{{\bm{h}}}\|_{2}^{2}]$ (4)

for which the CME, defined as

$\hat{{\bm{h}}}={\operatorname{E}}[{\bm{h}}|{\bm{r}}]=\int{\bm{h}}{\bm{f}}_{{\bm{h}}|{\bm{r}}}({\bm{h}}|{\bm{r}}){\operatorname{d}}{\bm{h}},$
(5)

is known to yield optimal estimates. The CME is generally not available since
it requires full knowledge of the conditional PDF $f_{{\bm{h}}|{\bm{r}}}$.
Furthermore, even if the PDF is known, the CME does generally neither have a
closed-form solution nor is it feasible to compute numerically in real-time
systems. For this reason, the CME is usually approximated by sub-optimal
approaches with a practically reasonable trade-off between performance and
complexity, e.g., using the linear MMSE estimator. In a jointly Gaussian
setting, the linear MMSE estimator is well-known to be the CME. However, in
the quantized case, this statement is no longer true in general. Nonetheless,
for a zero-mean jointly Gaussian input, the Bussgang decomposition, based on
Bussgang’s theorem [33], can be leveraged to design a linear MMSE estimator in
the quantized case. The Bussgang estimator is sub-optimal in general due to
its linearity. However, up to now there has been no analysis of the connection
between the Bussgang estimator and the CME in the case of one-bit
quantization.

In this work, we investigate the MSE optimality of the Bussgang estimator and
identify cases in which the Bussgang estimator is equal to the CME and we show
when the Bussgang estimator differs from the CME. To this end, after revising
the Bussgang estimator, we derive the CME for one-bit quantization in the
multivariate case. Simulation results based on the derived quantities verify
the discussion.

##  III Bussgang Estimator

In this section, we briefly revise the linear MMSE estimator based on the
Bussgang decomposition which is a direct consequence of Bussgang’s theorem
[33]. In particular, the Bussgang decomposition implies that the system in (3)
can be written as a linear combination of the desired signal part and an
uncorrelated distortion ${\bm{q}}$ as

${\bm{r}}=Q({\bm{y}})={\bm{B}}{\bm{y}}+{\bm{\eta}}={\bm{B}}{\bm{A}}{\bm{h}}+{\bm{q}},$
(6)

where ${\bm{B}}$ is the Bussgang gain that can be obtained from the linear
MMSE estimation of ${\bm{r}}$ from ${\bm{y}}$ as

${\bm{B}}={\bm{C}}_{{\bm{r}}{\bm{y}}}{\bm{C}}_{{\bm{y}}}^{-1}=\sqrt{\frac{2}{\pi}}\operatorname{diag}({\bm{C}}_{{\bm{y}}})^{-\frac{1}{2}},$
(7)

cf. [46, Sec. 9.2], and where the distortion term
${\bm{q}}={\bm{B}}{\bm{n}}+{\bm{\eta}}$ contains both the AWGN ${\bm{n}}$ and
the quantization noise ${\bm{\eta}}$. As the statistically equivalent model
(6) is linear, one can formulate the linear MMSE estimator

$\hat{{\bm{h}}}_{\text{buss}}={\bm{C}}_{{\bm{h}}{\bm{r}}}{\bm{C}}_{{\bm{r}}}^{-1}{\bm{r}}.$
(8)

The cross-correlation matrix between the channel and the received signal is
calculated as

${\bm{C}}_{{\bm{h}}{\bm{r}}}={\operatorname{E}}[{\bm{h}}({\bm{B}}{\bm{A}}{\bm{h}}+{\bm{q}})^{\operatorname{H}}]={\bm{C}}_{{\bm{h}}}{\bm{A}}^{\operatorname{H}}{\bm{B}}^{\operatorname{H}}$
(9)

which follows from the fact that the noise term ${\bm{q}}$ is uncorrelated
with the channel ${\bm{h}}$, see [12, Appendix A]. The auto-correlation matrix
${\bm{C}}_{{\bm{r}}}$ is calculated via the so-called arcsine law [47] as

${\bm{C}}_{{\bm{r}}}=\frac{2}{\pi}\left(\arcsin\left({\bm{\Psi}}\Re({\bm{C}}_{{\bm{y}}}){\bm{\Psi}}\right)+{\operatorname{j}}\arcsin\left({\bm{\Psi}}\Im({\bm{C}}_{{\bm{y}}}){\bm{\Psi}}\right)\right)$
(10)

where ${\bm{\Psi}}=\operatorname{diag}({\bm{C}}_{{\bm{y}}})^{-\frac{1}{2}}$.
Further reading on the Bussgang estimator can be found in [12, 37].

In [40], it is discussed that the Bussgang estimator is equal to a two-stage
estimator that first computes the CME of the product ${\bm{A}}{\bm{h}}$ of the
pilot matrix and the channel, and afterwards recovers the channel ${\bm{h}}$.
Since the resulting error after the first step is non-Gaussian, the second
step is non-optimal from the CME perspective. In the following, we discuss
cases in which the Bussgang estimator is indeed equal to the CME, and we show
the differences between the CME and the Bussgang estimator in the general
case.

##  IV Conditional Mean Estimator

In this section, we derive the CME for one-bit quantization for the general
system in (3). Afterwards, we discuss the CME for different simplifications,
i.e., with and without AWGN or multiple pilots, and for the uni- as well as
multivariate case, where these cases are differently combined to provide a
general overview. For each instance, we discuss possible closed-form solutions
of the CME and its relationship to the Bussgang estimator, where we identify
cases in which equality between these estimators holds. We start by rewriting
the CME from (5) as

$\displaystyle\operatorname{E}[{\bm{h}}|{\bm{r}}]$
$\displaystyle=\int{\bm{h}}\frac{p_{{\bm{r}}|{\bm{h}}}({\bm{r}}|{\bm{h}})f_{{\bm{h}}}({\bm{h}})}{p_{{\bm{r}}}({\bm{r}})}\operatorname{d}{\bm{h}}$
(11)
$\displaystyle=\frac{1}{p_{{\bm{r}}}({\bm{r}})}\int{\bm{h}}f_{{\bm{h}}}({\bm{h}})p_{{\bm{r}}|{\bm{h}}}({\bm{r}}|{\bm{h}})\operatorname{d}{\bm{h}}$
(12)

by making use of Bayes’ theorem. It is known that
$f_{{\bm{h}}}=\mathcal{N}_{\mathbb{C}}({\bm{0}},{\bm{C}}_{{\bm{h}}})$. In the
following, we also find expressions for the prior and conditional
probabilities. We first define the set of all vectors ${\bm{y}}$ which are
mapped onto a given observation ${\bm{r}}$ as

$\overline{Q}({\bm{r}})=\left\\{{\bm{y}}\in\mathbb{C}^{MN}:Q({\bm{y}})={\bm{r}}\right\\}.$
(13)

Integrating the PDF $f_{{\bm{y}}}$ over this set yields the prior probability
of observing the respective quantized observation, i.e.,

$\displaystyle p_{{\bm{r}}}({\bm{r}})$
$\displaystyle=\int_{\overline{Q}({\bm{r}})}f_{{\bm{y}}}({\bm{y}})\operatorname{d}{\bm{y}},$
(14)

where due to the joint Gaussianity we have
$f_{{\bm{y}}}=\mathcal{N}_{\mathbb{C}}({\bm{0}},{\bm{C}}_{{\bm{y}}})$ with
${\bm{C}}_{{\bm{y}}}={\bm{A}}{\bm{C}}_{{\bm{h}}}{\bm{A}}^{\operatorname{H}}+{\bm{C}}_{{\bm{n}}}$,
cf. (3). The conditional probability
$p_{{\bm{r}}|{\bm{h}}}({\bm{r}}|{\bm{h}})$ is found similarly, however, now
the integrand changes to
$\mathcal{N}_{\mathbb{C}}({\bm{A}}{\bm{h}},{\bm{C}}_{{\bm{n}}})$, where
${\bm{h}}$ is no longer a RV:

$\displaystyle
p_{{\bm{r}}|{\bm{h}}}({\bm{r}}|{\bm{h}})=\int_{\overline{Q}({\bm{r}})}\mathcal{N}_{\mathbb{C}}({\bm{x}};{\bm{A}}{\bm{h}},{\bm{C}}_{{\bm{n}}})\operatorname{d}{\bm{x}}.$
(15)

Plugging (14) and (15) into (12) yields a generally computable expression of
the CME for the model in (3).

###  IV-A Numerical Integration

For the most general case, the integrals in (12), (14), and (15) do neither
have a closed-form solution, nor can they be simplified. In this case, the CME
has to be found by numerical integration. However, the number of solutions for
the discrete number of possible observations ${\bm{r}}\in\mathbb{C}^{MN}$
grows exponentially with $4^{MN}$, not to mention the increasing complexity
for each individual solution. Since in (14) and (15) we have integrals over
Gaussian PDFs, there exist computationally efficient algorithms. In this
paper, the numerical integration of the Gaussian density is based on an
algorithm from [48] which first transforms the integral into an integration
over the unit hypercube by combining a Cholesky decomposition of the
covariance matrix with a priority ordering of the integration variables. The
transformed integral is then approximated based on a Monte Carlo method which
is implemented in [49]. In the following, we discuss special instances of the
general model (3) that allow for a simplified computation of the CME (12).

###  IV-B Univariate Case with a Single Observation

In the following, the case of a scalar $N=1$ system with a single pilot
observation, i.e., $M=1$ and $a=1$, is considered. Since the quantization
function in (2) is applied independently to the real and imaginary part of its
input, deriving only the real part formula of the CME is sufficient here since
the imaginary part is computed analogously. To this end we consider
$h_{\Re}\sim\mathcal{N}(0,\sigma^{2}/2)$,
$n_{\Re}\sim\mathcal{N}(0,\eta^{2}/2)$, and for the ease of notation we define
$r_{\Re}=\operatorname{sign}(h_{\Re}+n_{\Re})$.

Because of the symmetry of the zero-mean Gaussian density, we get
$p_{r_{\Re}}(r_{\Re})=1/2$. Furthermore, the integral in (15) can be computed
in closed-form to, cf. [50],

$\displaystyle p_{r_{\Re}|h_{\Re}}(r_{\Re}=1|h_{\Re})$
$\displaystyle=\int_{0}^{\infty}\mathcal{N}(x;h_{\Re},\eta^{2}/2)\operatorname{d}x$
(16)
$\displaystyle=\frac{1}{2}\left(1+\operatorname{erf}\left(\frac{h_{\Re}}{\eta}\right)\right),$
(17) $\displaystyle p_{r_{\Re}|h_{\Re}}(r_{\Re}=-1|h_{\Re})$
$\displaystyle=\int_{-\infty}^{0}\mathcal{N}(x;h_{\Re},\eta^{2}/2)\operatorname{d}x$
(18)
$\displaystyle=\frac{1}{2}\left(1-\operatorname{erf}\left(\frac{h_{\Re}}{\eta}\right)\right).$
(19)

We can summarize (17) and (19) as

$p_{r_{\Re}|h_{\Re}}(r_{\Re}|h_{\Re})=\frac{1}{2}\left(1+r_{\Re}\operatorname{erf}\left(\frac{h_{\Re}}{\eta}\right)\right).$
(20)

Plugging this result into (12) yields, cf. [51, eq. (2.7.2.4)],

$\displaystyle{\operatorname{E}}[h_{\Re}|r_{\Re}]$
$\displaystyle=\int_{-\infty}^{\infty}hf_{h_{\Re}}(h)\left(1+r_{\Re}\operatorname{erf}\left(\frac{h}{\eta}\right)\right)\operatorname{d}h$
(21)
$\displaystyle=\frac{1}{\sqrt{\pi}}\frac{\sigma^{2}}{\sqrt{\sigma^{2}+\eta^{2}}}r_{\Re}.$
(22)

This can now easily be extended to the complex case with
$h\sim\mathcal{N}_{\mathbb{C}}(0,\sigma^{2})$,
$n\sim\mathcal{N}_{\mathbb{C}}(0,\eta^{2})$, and $r=Q(h+n)$ which reads as

$\displaystyle\operatorname{E}[h|r]$
$\displaystyle=\sqrt{\frac{2}{\pi}}\frac{\sigma^{2}}{\sqrt{\sigma^{2}+\eta^{2}}}r.$
(23)

The closed-form MSE from (4) is then computed to

$\displaystyle\operatorname{MSE}$
$\displaystyle=\sigma^{2}\left(1-\frac{2}{\pi}\frac{\sigma^{2}}{\sigma^{2}+\eta^{2}}\right).$
(24)

Interestingly, if we evaluate the Bussgang estimator in (8), we get the exact
same CME and MSE expressions. To conclude, for this case the Bussgang
estimator is indeed the CME. Note that in the multivariate case with an
identity pilot matrix ${\bm{A}}=\operatorname{\mathbf{I}}$ and uncorrelated
channels ${\bm{C}}_{{\bm{h}}}=\operatorname{diag}({\bm{c}}_{\bm{h}})$ as well
as uncorrelated noise
${\bm{C}}_{{\bm{n}}}=\operatorname{diag}({\bm{c}}_{{\bm{n}}})$, the PDFs in
(14) and (15) can be decomposed into the product of their marginals. Thus, the
CME from (23) can be applied element-wise and is therefore also equal to the
Bussgang estimator. The same expression as in (23) was found in [43], more
generally for non zero-mean channels. However, the Bussgang theorem from [33]
can not be applied for non zero-mean RVs and thus, the equality does not hold
in general.

Additionally, we can evaluate the well-known linear MMSE estimator for
unquantized observations where it is the CME. There, the closed-form MSE is
simply

$\text{MSE}=\sigma^{2}\left(1-\frac{\sigma^{2}}{\sigma^{2}+\eta^{2}}\right).$
(25)

We can now compute the necessary SNR for the unquantized system to achieve the
same MSE as the one-bit quantized system without AWGN which we will analyze
later in the numerical experiments section.

###  IV-C Multivariate Noiseless Case with a Single Observation

In this case, we consider $M=1$ with ${\bm{n}}={\bm{0}}$. Similarly as in
Section IV-B, it is sufficient to derive only the real part of the CME. To
this end, we consider ${\bm{r}}_{\Re}=\operatorname{sign}({\bm{h}}_{\Re})$
with ${\bm{h}}_{\Re}\sim\mathcal{N}({\bm{0}},{\bm{C}}_{{\bm{h}}})$ where
${\bm{C}}_{{\bm{h}}}\in\mathbb{R}^{N\times N}$ is a full matrix. We abuse the
notation for
$\overline{Q}({\bm{r}}_{\Re})=\\{{\bm{h}}_{\Re}\in\mathbb{R}^{N}:\operatorname{sign}({\bm{h}}_{\Re})={\bm{r}}_{\Re}\\}$.
In the following Theorem 1, a simplified expression of the general CME (12) is
derived whose advantage is discussed afterwards.

######  Theorem 1.

The CME for the above system can be computed element-wise as

$\displaystyle\left[\operatorname{E}[{\bm{h}}_{\Re}|{\bm{r}}_{\Re}]\right]_{i}=\frac{1}{p_{{\bm{r}}_{\Re}}({\bm{r}}_{\Re})}\sum_{n=1}^{N}[{\bm{r}}_{\Re}]_{n}[{\bm{C}}_{{\bm{h}}}]_{i,n}$
(26)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathcal{N}(0;0,[{\bm{C}}_{{\bm{h}}}]_{n,n})\int_{\overline{Q}({\bm{r}}_{\Re}^{n})}\mathcal{N}^{n}({\bm{x}};{\bm{0}},{\bm{C}}_{{\bm{h}}})\operatorname{d}{\bm{x}},$

where $\mathcal{N}^{n}({\bm{0}},{\bm{C}}_{{\bm{h}}})$ is the
($N-1$)-dimensional Gaussian PDF where the $n$th row and column of
${\bm{C}}_{{\bm{h}}}^{-1}$ is deleted. The superscript $n$ of a vector
truncates its $n$th element.

Proof: See Section -A.

The advantage of the integral in (26) is that, in contrast to the double
integral that appears in (12), cf. (15), it can be efficiently computed by
means of [48] as described in Section IV-A because there appear only integrals
over Gaussian PDFs. Note that the expression (26) simplifies to the element-
wise CME as computed in (23) for diagonal covariances since then
$[{\bm{C}}_{{\bm{h}}}]_{i,n}^{i\neq n}=0$ and the $(N-1)$-dimensional integral
equals $2^{-(N-1)}$ due to symmetry. This analysis can be interpreted as an
asymptotic SNR analysis in which the AWGN goes to zero.

The extension to the complex-valued case is straightforward by stacking the
real and imaginary parts of each complex vector and using the real-valued
representation $\check{{\bm{C}}}_{{\bm{h}}}\in\mathbb{R}^{2N\times 2N}$ of the
covariance matrix ${\bm{C}}_{{\bm{h}}}\in\mathbb{C}^{N\times N}$, cf. [52,
Sec. 15.4], given as

$\displaystyle\check{{\bm{C}}}_{{\bm{h}}}=\begin{bmatrix}\Re({\bm{C}}_{{\bm{h}}})&-\Im({\bm{C}}_{{\bm{h}}})\\\
\Im({\bm{C}}_{{\bm{h}}})&\Re({\bm{C}}_{{\bm{h}}})\end{bmatrix}.$ (27)

In contrast to the univariate case in Section IV-B, the resulting CME
expression (26) is nonlinear and no longer in closed-form, which is clearly a
difference to the linear Bussgang estimator. We will present numerical
evaluations for this case that reveal a performance gap between the
estimators.

###  IV-D Univariate Noiseless Case with Multiple Observations

In this case, we consider $N=1$ with ${\bm{n}}={\bm{0}}$, i.e., the system
from (3) simplifies to ${\bm{r}}=Q({\bm{a}}h)$ with
$h\sim\mathcal{N}_{\mathbb{C}}(0,\sigma^{2})$. Note that in this case it is
not sufficient to compute the real-valued CME because of the product of
channel and pilot vector. This was the reason for the two-stage approach in
[40]. Opposite to the unquantized case, having multiple pilot observations of
the same realization is helpful in the context of quantization as more
information can be recovered. Because the amplitude of the received signal is
lost after the one-bit quantizer, having varying amplitude in the pilot
sequence ${\bm{a}}$ has no influence on the observed signal. However, one can
introduce phase shifts in order to increase the obtained information.

In the following, we discuss the optimal pilot sequence and the resulting CME
based on it. We start by rewriting the CME in terms of the magnitude and phase
of the channel $h=\alpha{\operatorname{e}}^{{\operatorname{j}}\theta}$ where,
due to the circular symmetry, the phase is uniformly distributed as
$\theta\sim\mathcal{U}(0,2\pi)$ and is independent of the Rayleigh distributed
magnitude, cf. [53, Appendix A.1.3]. Consequently, we obtain

$\displaystyle{\operatorname{E}}[\alpha{\operatorname{e}}^{{\operatorname{j}}\theta}|{\bm{r}}]={\operatorname{E}}[\alpha|{\bm{r}}]\hskip
1.0pt{\operatorname{E}}[{\operatorname{e}}^{{\operatorname{j}}\theta}|{\bm{r}}]={\operatorname{E}}[\alpha]\hskip
1.0pt{\operatorname{E}}[{\operatorname{e}}^{{\operatorname{j}}\theta}|{\bm{r}}],$
(28)

since the magnitude information is lost due to the one-bit quantization and
its estimate is thus independent of the received signal. Therefore, we can set
the magnitude of each pilot to one without loss of generality in order to
fulfill the power constraint $\|{\bm{a}}\|_{2}^{2}=M$. It is shown in the
following that a recovery of the phase information of the channel depending on
the number of pilots $M$ up to a certain resolution can be achieved.

$\Re$$\Im$$\frac{1}{\sqrt{2}}$$\frac{1}{\sqrt{2}}$$\phi_{\text{low}}$$\phi_{\text{high}}$$q_{1}$$[{\bm{y}}]_{1}$$\Re$$\Im$$\phi_{\text{low}}$$\phi_{\text{high}}$$\frac{1}{\sqrt{2}}$$\frac{1}{\sqrt{2}}$$q_{1}$$[{\bm{y}}]_{1}$$[{\bm{y}}]_{2}$$\psi_{2}$$\psi_{2}$$\Re$$\Im$$\frac{1}{\sqrt{2}}$$\frac{1}{\sqrt{2}}$$\phi_{\text{low}}$$\phi_{\text{high}}$$q_{1}$$[{\bm{y}}]_{1}$$[{\bm{y}}]_{2}$$[{\bm{y}}]_{3}$$\psi_{3}$$\psi_{3}$

Figure 1: Illustration of the iterative restriction of the angular range of
the channel’s phase and determination of the sub-sector by means of
$\phi_{\text{low}}({\bm{r}})$ and $\phi_{\text{high}}({\bm{r}})$ for the case
of $M=3$ pilots.

The key consideration thereby is to consecutively send pilots with an
increasing phase shift and detect whether or not the quantization label is
changing, which allows to narrow down the angular range of the channel’s
phase. We first note that there exist four quantization points in the complex
plane, each representing a quadrant with an angle of $\pi/2$. We further
assume without loss of generality that the first pilot is sent without a phase
shift, which determines in which of the four quadrants with an angular range
of $\pi/2$ the channel lies. In turn, if the subsequent pilots have a phase
shift up to $\pi/2$ the quantization label will ultimately change once the
rotated channel ends up in the neighboring main quadrant. Building on this, an
optimal pilot sequence is of the form

$[{\bm{a}}]_{m}=\exp({\operatorname{j}}\psi_{m}),~{}\psi_{m}\in[0,\tfrac{\pi}{2}),~{}m=1,\dots,M,$
(29)

which we assume to be ordered as $\psi_{1}<\dots<\psi_{M}$.

Put differently, the pilots $m=2,...,M$ divide each of the four main sectors
into $M$ circular sub-sectors if $\psi_{m}\neq 0~{}\forall m=2,...,M$. After
observing a receive signal ${\bm{r}}$, we would like to determine in which of
the sub-sectors the channel lies. More precisely, we would like to associate a
sub-sector, described by the angles $\phi_{\text{low}}({\bm{r}})$ and
$\phi_{\text{high}}({\bm{r}})$, to every possible receive signal ${\bm{r}}$
such that if we receive ${\bm{r}}$, we know that the channel lies in the sub-
sector described by $\phi_{\text{low}}({\bm{r}})$ and
$\phi_{\text{high}}({\bm{r}})$.

In Fig. 1, we exemplarily show how to determine the sub-sector of interest for
$M=3$ pilots. We therefore assume that the pilots have equidistant phase
shifts in $[0,\pi/2)$ which we prove to be MSE-optimal later in Theorem 2. The
leftmost picture shows that the first unquantized receive signal
$[{\bm{y}}]_{1}=[{\bm{a}}]_{1}h$ lies in the positive quadrant and thus the
corresponding quantized receive signal is
$[{\bm{r}}]_{1}=Q([{\bm{y}}]_{1})=q_{1}$ which determines the main sector by
means of $\phi_{\text{low}}$ and $\phi_{\text{high}}$. The middle picture
shows the situation after the second pilot is sent. Since the rotated channel
$[{\bm{y}}]_{2}=[{\bm{a}}]_{2}h$ yields the same quantization label
$[{\bm{r}}]_{2}=Q([{\bm{y}}]_{2})=q_{1}$, one can restrict the sector by
reducing the angle $\phi_{\text{high}}$ by the angle of $\psi_{2}$, due to
fact that if the channel would lie in the cropped area the rotated channel
would yield the neighboring quantization label. The rightmost picture shows
that the last receive signal is again $[{\bm{r}}]_{3}=q_{1}$, which means that
the sector can be further restricted by the same argumentation. This
determines the final boundary angles $\phi_{\text{low}}$ and
$\phi_{\text{high}}$. In the same way as in the example of Fig. 1 for a
different sequence where a change of the quantization label occurs for a
certain pilot, the lower boundary angle $\phi_{\text{low}}$ is increased,
ultimately yielding a sub-sector of the same size. This procedure is
summarized formally in Algorithm 1.

Algorithm 1 Finding the boundary angles $\phi_{\text{low}}({\bm{r}})$ and
$\phi_{\text{high}}({\bm{r}})$.

1: Set the boundary angles to the quadrant boundaries:
$\phi_{\text{low~{}}}=\angle([{\bm{r}}]_{1})-\frac{\pi}{4}$,
$\phi_{\text{high}}=\phi_{\text{high}}^{\text{init}}=\angle([{\bm{r}}]_{1})+\frac{\pi}{4}$

2: for $m=2$ to $M$ do

3: if $[{\bm{r}}]_{m}==[{\bm{r}}]_{m-1}$ then

4: $\phi_{\text{high}}=\phi_{\text{high}}^{\text{init}}-\psi_{m}$

5: else

6: $\phi_{\text{low}}=\phi_{\text{high}}-\psi_{m}$

7: break

8: end

9: end

It remains to show that the equidistant pilot sequence minimizes the MSE in
combination with the CME. Therefore, the CME of the normalized channel can be
obtained as

$\displaystyle{\operatorname{E}}[{\operatorname{e}}^{{\operatorname{j}}\theta}|{\bm{r}}]$
$\displaystyle=\int_{0}^{2\pi}{\operatorname{e}}^{{\operatorname{j}}\theta}f_{\theta|{\bm{r}}}(\theta|{\bm{r}})\operatorname{d}\theta$
(30)
$\displaystyle=\frac{1}{p_{{\bm{r}}}({\bm{r}})}\int_{0}^{2\pi}{\operatorname{e}}^{{\operatorname{j}}\theta}f_{\theta}(\theta)p_{{\bm{r}}|\theta}({\bm{r}}|\theta)\operatorname{d}\theta.$
(31)

Due to the absence of AWGN, similarly as in (47), the conditional probability
$p_{{\bm{r}}|\theta}$ is an indicator function

$\displaystyle
p_{{\bm{r}}|\theta}({\bm{r}}|\theta)=\begin{cases}1&\phi_{\text{low}}({\bm{r}})\leq\theta<\phi_{\text{high}}({\bm{r}})\\\
0&\text{else}\end{cases}$ (32)

where the boundary angles $\phi_{\text{low}}({\bm{r}})$ and
$\phi_{\text{high}}({\bm{r}})$ are determined via the above discussed
procedure. Given these preparations, we can now find the MSE-optimal pilot
sequence.

######  Theorem 2.

The MSE-optimal pilot sequence for the above system contains equidistant phase
shifts $\psi_{m}=\frac{\pi(m-1)}{2M}~{}\forall m=1,...,M,$ such that

$[{\bm{a}}]_{m}=\exp\left({\operatorname{j}}\left(\frac{\pi}{2M}(m-1)\right)\right),~{}m=1,...,M.$
(33)

Proof: See Appendix -B.

For the optimal pilot sequence (33), the boundary angles from (32) can now be
compactly written as

$\phi_{\text{low}}({\bm{r}})=\frac{1}{M}\sum_{m=1}^{M}\angle([{\bm{r}}]_{m})-\frac{\pi}{4}$
(34)

and $\phi_{\text{high}}({\bm{r}})=\phi_{\text{low}}({\bm{r}})+\frac{\pi}{2M}$.
Interestingly, the same pilot sequence was proposed in [29] with the purpose
to achieve a bijective mapping between a fixed dataset of channels and
corresponding observations with long enough pilot sequences.

We can now find a closed-form solution for the CME based on the optimal pilot
sequence from Theorem 2.

######  Theorem 3.

The CME for the above system is computed to

${\operatorname{E}}[h|{\bm{r}}]=\frac{2M\sigma}{\sqrt{\pi}}\sin\left(\frac{\pi}{4M}\right)\exp\left({\operatorname{j}}\left(\frac{\pi}{4M}+\phi_{\text{low}}({\bm{r}})\right)\right).$
(35)

Proof: See Appendix -C.

The closed-form MSE for the CME from (35) is then

MSE
$\displaystyle=\sigma^{2}\left(1-\frac{4M^{2}}{\pi}\sin^{2}\left(\frac{\pi}{4M}\right)\right).$
(36)

In contrast, the MSE for the Bussgang estimator from (8) for the present
system is computed to

MSE
$\displaystyle=\sigma^{2}\left(1-\frac{2}{\pi}{\bm{a}}^{\operatorname{H}}{\bm{C}}_{{\bm{r}}}^{-1}{\bm{a}}\right).$
(37)

Although the MSE terms in (36) and (37) seem to be different, we show their
equality in the following.

######  Theorem 4.

The MSE of the Bussgang estimator from (37) is equal to the MSE of the CME
from (36) for an arbitrary number of pilot observations $M$.

Proof: See Appendix -D.

Theorem 4 surprisingly shows that the linear Bussgang estimator achieves the
same MSE as the CME and is thus an MSE-optimal estimator for that case.
Interestingly, as a byproduct of the proof in Appendix -D we have found a
closed-form expression for the inverse covariance matrix
${\bm{C}}_{{\bm{r}}}^{-1}$ in (37). Yet again, this case can be interpreted as
an asymptotically high SNR analysis. It should be noted here that a similar
proof is not possible in the presence of AWGN as shown through numerical
experiments later. With the closed-form MSE expression in (36), we can now
study the asymptotically large number of pilots regime where we find the
following result which, due to Theorem 4, also holds for the Bussgang
estimator.

######  Theorem 5.

For $M\to\infty$ the MSE from (36) is

$\lim_{M\to\infty}\text{MSE}=\sigma^{2}\left(1-\frac{\pi}{4}\right).$ (38)

###### Proof.

We compute

$\displaystyle\lim_{M\to\infty}M\sin\left(\frac{\pi}{4M}\right)$
$\displaystyle=\lim_{M\to\infty}\frac{\pi}{4}\cos\left(\frac{\pi}{4M}\right)=\frac{\pi}{4}$
(39)

where the rule of L’Hôpital was used. Since this limit exists, we can conclude
that

$\lim_{M\to\infty}M^{2}\sin^{2}\left(\frac{\pi}{4M}\right)=\frac{\pi^{2}}{16}.$
(40)

Plugging (40) into (36) verifies (38). ∎

###  IV-E Summary

Sec. |  Dim. $N$ |  AWGN |  Corr. |  Pilots $M$ |  Optimal
---|---|---|---|---|---
IV-A |  $\geq 1$ |  yes |  yes |  $\geq 1$ |  no
IV-B |  $\geq 1$ |  yes |  no |  $1$ |  yes
IV-C |  $>1$ |  no |  yes |  $1$ |  no
IV-D |  $1$ |  no |  n.a. |  $\geq 1$ |  yes
TABLE I: Overview of different system parameters and the MSE-optimality of the
Bussgang estimator.

So far, we have discussed the CME and its relationship to the Bussgang
estimator for different system instances. Remarkably, in Section IV-B and
Section IV-D the Bussgang estimator is shown to be equal to the CME which has
a closed-form. This is a novel result that has not been stated in the
literature so far. Because of the closed-form MSE expression of the CME, an
analysis of the large number of pilots regime was possible which resulted in a
closed-form limit. What is more, we have discussed in Section IV-C that the
equality might not hold for correlated channel entries, even in the absence of
AWGN. However, we have derived a computationally efficient expression for the
CME in that case.

An overview of whether the Bussgang estimator is equal to the CME or not for
the different cases can be found in Table I. Therein, we show the cases where
the Bussgang estimator is MSE-optimal with respect to the dimension (Dim.)
$N$, the presence/absence of AWGN, the correlation (Corr.) between channel
entries and the number of pilot observations $M$. For convenience, we
referenced the subsections with the corresponding derivations. The first row
refers to the general case where none of the discussed special cases applies.
In the following, after introducing different performance metrics besides the
MSE, we discuss the above results in numerical experiments which verify the
theoretical findings.

##  V Different Performance Metrics

###  V-A Cosine Similarity

The amplitude of the received signal is lost due to the one-bit quantization.
For this reason, one might solely be interested in how well an estimator can
recover the angle of the signal of interest. A measure for the difference of
the angle $\psi$ between the true and estimated channel is the cosine
similarity which we define as

$\cos\psi=\frac{\Re({\bm{h}}^{\operatorname{H}}\hat{{\bm{h}}})}{\|{\bm{h}}\|\|\hat{{\bm{h}}}\|}.$
(41)

Note that the cosine similarity lies between $-1$ (opposite direction) and $1$
(same direction).

###  V-B Achievable Rate Lower Bound

The achievable rate is of great interest in quantized systems [12, 13]. For
our considerations, we are aiming at a comparison between the CME and the
Bussgang estimator via a lower bound on the corresponding achievable rate of a
respective data transmission system that is taking the CSI mismatch into
account. To this end, after estimating the channel with the pilot transmission
in (3), the data symbol $s$ is transmitted over the same channel, i.e.,

${\bm{r}}=Q({\bm{h}}s+{\bm{n}})={\bm{B}}{\bm{h}}s+{\bm{q}}$ (42)

where in the second equation the linearized model with Bussgang’s
decomposition is used where ${\bm{q}}={\bm{B}}{\bm{n}}+{\bm{\eta}}$. We make
the worst-case assumption that the aggregated noise is Gaussian, i.e.,
${\bm{q}}\sim\mathcal{N}_{\mathbb{C}}({\bm{0}},{\bm{C}}_{{\bm{q}}}={\bm{C}}_{{\bm{r}}}-{\bm{B}}{\bm{C}}_{{\bm{h}}}{\bm{B}}^{\operatorname{H}})$,
cf. [54], and use a matched filter

${\bm{g}}_{\text{MF}}^{\operatorname{H}}=\hat{{\bm{h}}}^{\operatorname{H}}{\bm{B}}^{\operatorname{H}}({\bm{C}}_{{\bm{r}}}-{\bm{B}}{\bm{C}}_{{\bm{h}}}{\bm{B}}^{\operatorname{H}})^{-1}$
(43)

at the receiver. Note that the variance of the data symbol $s$ is assumed to
be one without loss of generality. We further assume that the SNR is the same
during pilot and data transmission. As in [55], we split up the received data
symbol as

$\displaystyle\hat{s}=\underbrace{{\bm{g}}_{\text{MF}}^{\operatorname{H}}{\bm{B}}\hat{{\bm{h}}}s}_{\text{desired
signal}}+\underbrace{{\bm{g}}_{\text{MF}}^{\operatorname{H}}{\bm{B}}{\bm{\epsilon}}s}_{\text{estimation
error}}+\underbrace{{\bm{g}}_{\text{MF}}^{\operatorname{H}}{\bm{B}}{\bm{q}}}_{\text{noise}}$
(44)

where ${\bm{\epsilon}}={\bm{h}}-\hat{{\bm{h}}}$ denotes the channel estimation
error. Thus, we can evaluate a lower bound on the achievable rate, cf. [55,
eq. (20)], as

$\displaystyle R$
$\displaystyle={\operatorname{E}}\left[\log_{2}\left(1+\frac{|{\bm{g}}_{\text{MF}}^{\operatorname{H}}{\bm{B}}\hat{{\bm{h}}}|^{2}}{|{\bm{g}}_{\text{MF}}^{\operatorname{H}}{\bm{B}}{\bm{\epsilon}}|^{2}+{\bm{g}}_{\text{MF}}^{\operatorname{H}}{\bm{B}}{\bm{C}}_{{\bm{q}}}{\bm{B}}^{\operatorname{H}}{\bm{g}}_{\text{MF}}}\right)\right]$
(45)

with Monte Carlo simulations.

##  VI Numerical Results

In this section, we perform numerical experiments that verify our theoretical
derivations and allow for quantifying the performance gap between the Bussgang
estimator and the CME in the general case. We consider a diagonal noise
covariance ${\bm{C}}_{{\bm{n}}}=\eta^{2}\operatorname{\mathbf{I}}$. For the
case of $N=1$, we choose without loss of generality the variance of the
channel to $\sigma^{2}=1$. In the multivariate case, we randomly sample
channel covariance matrices with the following procedure, cf. [56]. We first
generate a matrix ${\bm{S}}$ where both the real and imaginary parts of all
entries are i.i.d. as $\mathcal{U}(0,1)$. Then, we compute the eigenvalue
decomposition (EVD) of
${\bm{S}}^{\operatorname{H}}{\bm{S}}={\bm{V}}{\bm{\Sigma}}{\bm{V}}^{\operatorname{H}}$.
Finally, we construct
${\bm{C}}_{{\bm{h}}}={\bm{V}}\operatorname{diag}({\bm{1}}+{\bm{\xi}}){\bm{V}}^{\operatorname{H}}$,
where $[{\bm{\xi}}]_{m}\sim\mathcal{U}(0,1)$. Afterwards, we normalize each
covariance matrix by first dividing by its trace and afterwards multiplying by
its dimension, such that $\operatorname{tr}({\bm{C}}_{{\bm{h}}})=N$. For each
covariance matrix we draw a single channel sample. This procedure averages out
all possible structural features of the covariance matrix. For all simulations
we draw a total of $10{,}000$ samples. With the property
$\|{\bm{a}}\|_{2}^{2}=M$ and ${\operatorname{E}}[\|{\bm{h}}\|^{2}]=N$ we
define the SNR as

$\text{SNR}=\frac{\|{\bm{a}}\|_{2}^{2}{\operatorname{E}}[\|{\bm{h}}\|^{2}]}{{\operatorname{E}}[\|{\bm{n}}\|^{2}]}=\frac{1}{\eta^{2}}.$
(46)

The MSE from (4) is normalized by ${\operatorname{E}}[\|{\bm{h}}\|^{2}]=N$.

###  VI-A Univariate Case with a Single Observation

Figure 2: MSE performance of the CME from Section IV-B for $N$$=$$M$$=$$1$ for
unquantized and one-bit quantized observations.

We first verify the findings from Section IV-B with $N=M=1$ in Fig. 2 where
the normalized MSE is plotted against the SNR. It can be observed that the CME
converges to the noiseless MSE from (24) for $\eta=0$, which equals to
$1-\frac{2}{\pi}$, for high SNR values. Additionally, we have plotted the
well-known linear MMSE estimator for unquantized observation, where it is the
CME, from (25). Comparing a one-bit quantized system without AWGN with an
unquantized system, we can compute their MSE intersection point by means of
(24) and (25) with the SNR definition (46) to
$\text{SNR}=\frac{2}{\pi-2}\approx 2.435$dB which is verified by the
simulation.

###  VI-B Multivariate Noiseless Case with a Single Observation

Next, we discuss the system from Section IV-C with $M=1$ and
${\bm{n}}={\bm{0}}$. As discussed above, for this case there exists no closed-
form solution, but we can use a computationally efficient algorithm from [48]
to evaluate the integrals from (14) and (26). In Fig. 3 it can be seen that
the normalized MSE decreases for larger dimensions $N$ which is a consequence
of the existing cross-correlations that are exploited for the estimation.
Furthermore, the gap between the Bussgang estimator and the CME increases for
higher dimensions. That is, the sub-optimality of the Bussgang estimator seems
to have greater impact especially in higher dimensions. Note that in the case
of $N=1$ there exists the closed-form solution from before where the Bussgang
estimator is exactly the CME.

Figure 3: MSE performance comparison between the Bussgang estimator and the
CME from Section IV-C for the multivariate noiseless case with a single pilot
observation. Figure 4: CDF curves of the MSE performance comparison between
the Bussgang estimator and the CME from Section IV-C for the multivariate
noiseless case with a single pilot observation for $N=24$ dimensions.

Figure 5: Cosine similarity (top) and achievable rate lower bound (bottom)
comparison between the Bussgang estimator and the CME from Section IV-C for
the multivariate noiseless case with a single pilot observation.

In order to further investigate the performance gap between the CME and the
Bussgang estimator in this case, in Fig. 4, we evaluate the cumulative
distribution function (CDF) curves of the MSE of both estimators for $N=24$
dimensions. The vertical axis shows the probability that the MSE of the
respective estimators is smaller than or equal to the value on the horizontal
axis for each channel sample. It can be seen that the CME estimator is
consistently better than the Bussgang estimator, especially in the region
around the mean value of the MSE which we plotted for the ease of reference.
In contrast, both estimators perform almost equally well for outliers, i.e.,
for channel samples which yield a high estimation error.

Fig. 5 (top) evaluates the cosine similarity from (41) over different
dimensions $N$. For an increasing number of dimensions it can be seen that the
cosine similarity generally decreases. However, the CME achieves a higher
cosine similarity than the Bussgang estimator with an increasing gap for
higher dimensions where a saturation is observed. This is in accordance with
the results from the MSE investigations.

In Fig. 5 (bottom), we analyze the achievable rate lower bound from (45) in
bits/s/Hz. The lower bound is very similar for both estimators which increases
from fewer than 1 bit/s/Hz for $N=1$ up to over 3 bits/s/Hz for $N>14$. A
possible explanation for this small gap is, together with the results from the
analysis of the cosine similarity, that the Bussgang estimator is able to
provide a good estimate of the channel directions which seems to be especially
important for the achievable rate. Only for higher dimensions a small gap
between the lower bounds can be observed which agrees with the results from
the MSE analysis.

###  VI-C Univariate Noiseless Case with Multiple Observations

In this subsection, we evaluate the results from Section IV-D with $N=1$ and
${\bm{n}}={\bm{0}}$. Fig. 6 shows the normalized MSE over the number of pilots
$M$. First, it can be seen that the MSE of the estimators in (8) and (35) are
indeed equal for the conducted simulation, verifying Theorem 4. Second, the
MSE converges from $1-\frac{2}{\pi}$ for $M=1$ to $1-\frac{\pi}{4}$, which is
the limit for $M\to\infty$ as shown in Theorem 5, already at a moderate number
of $M=10$ pilots.

###  VI-D General Case

Figure 6: MSE performance comparison between the Bussgang estimator and the
CME from Section IV-D for the univariate noiseless case with multiple pilot
observations. Figure 7: MSE performance comparison between the Bussgang
estimator and the CME from Section IV-A for the univariate case with multiple
pilot observations.

In the following, we investigate more general cases which are evaluated with
numerical integration as described in Section IV-A. First, we consider the
univariate case of $N=1$ for different numbers of pilot observations
($M\in\\{1,2,5,10\\}$) and with AWGN for varying SNRs in Fig. 7 where we
compare the derived optimal pilot sequence from (33), labeled “opt”, with an
all-ones pilot sequence ${\bm{a}}={\bm{1}}$, labeled “ones” which was used
occasionally in recent works [18, 24]. For convenience, we have also plotted
the closed-form solutions for the noiseless case with $M=1$ and $M\to\infty$
as dashed lines. As expected, for $M=1$, the CME equals the Bussgang estimator
and converges to the closed-form MSE $1-\frac{2}{\pi}$ for high SNR values.
Interestingly, for an increasing number of pilot observations there is an
increasing gap between the Bussgang estimator and the CME, especially in the
mid to high SNR regime. What is more, the optimal pilot sequence achieves a
lower MSE than the all-ones pilot sequence with a large gap especially for
high SNR values, validating the analysis in Section IV-D and indicating that
the optimal pilot sequence performs well even in the case of finite SNR values
and for the sub-optimal Bussgang estimator. In contrast, at low SNR values,
both estimators perform almost equally well which can be explained by the
dominating effect of the AWGN in this regime.

It can also be seen that for higher numbers of pilots the MSE has a minimum at
a certain SNR after which it increases again. What is more, the asymptotic
limit of $M\to\infty$ for infinite SNR (${\bm{n}}\to{\bm{0}}$) can be
outperformed with a finite number of pilots and finite SNR. These
observations, seeming counterintuitive at first, are a consequence of the
stochastic resonance effect [44]. This is a well-known effect related to
quantization where noise can improve the performance of a system with
quantization. Hence, this analysis allows for a numerical quantification of
the stochastic resonance effect, which is the reason why the asymptotic limit
can be outperformed and thus also describes the effect of the increasing MSE
after a certain SNR value. Interestingly, the influence of the stochastic
resonance increases for higher numbers of pilot observations and the optimal
pilot sequence from (33) is less prone to the effect of an increasing MSE in
the high SNR regime than the all-ones sequence.

Figure 8: MSE performance comparison between the Bussgang estimator and the
CME from Section IV-A for the univariate case with SNR = 10dB.

Next, in Fig. 8, we analyze the behavior of the Bussgang estimator and the CME
for $N=1$ and an increasing number of pilots $M$ at a fixed SNR of 10dB.
Similar as described before, the performance gap increases for higher numbers
of pilots. Whereas the MSE of the Bussgang estimator saturates with a high
error floor, the MSE of the CME decreases further for higher numbers of
pilots.

Finally, in Fig. 9, we depict the MSE for the case of multiple dimensions with
AWGN over the SNR for correlated channel entries. For convenience, we have
also plotted the closed-form solutions for $N=1$ with and without AWGN. Due to
the exponentially increasing complexity of the numerical integration, we were
only able to plot until $N=3$ to show the tendency of the results. First, an
offset of the normalized MSE for higher dimensions can be observed which seems
to decrease with the number of dimensions. Moreover, the Bussgang estimator is
very close to the CME which is in accordance with the prior results for low
dimensions. The curve for $N=2$ ($N=3$) intersects with the noiseless case of
$N=1$ at about 15dB (12dB) SNR.

Figure 9: MSE performance comparison between the Bussgang estimator and the
CME from Section IV-A for the multivariate case with a single pilot
observation.

##  VII Conclusion and Future Work

In this work, we have investigated the CME for one-bit quantized systems in a
jointly Gaussian setting. We derived a novel closed-form solution for multiple
pilot observations and a transformation into a computationally efficient
expression where only Gaussian integrals have to be evaluated for the case
without AWGN. Additionally, we have proposed an MSE-optimal pilot sequence
which we motivated by the structure of the CME. It turned out that a
successive shift of the transmit signals’ angle is optimal which was already
proposed in prior work with a different motivation in the context of deep
learning.

We analyzed the well-known Bussgang estimator and showed the equivalence of
the Bussgang estimator and the CME in the univariate case with a single pilot
observation, which extends to the multivariate case when there is no
correlation between the channel or noise entries. In addition, we showed that
the closed-form MSE terms of the Bussgang estimator and the CME are equal in
the case of multiple pilots without AWGN. To this end, we found a closed-form
expression for the inverse of the covariance matrix of the quantized receive
signal which is derived with the arcsine law. For the general case where
numerical integration is necessary, we found that the performance gap between
the Bussgang estimator and the CME increases for higher dimensions, longer
pilot sequences, or higher SNR values. The findings of this work may
facilitate extensions to more general input distributions, e.g., Gaussian
mixture models (GMMs), different nonlinearities such as the clipper function
instead of the one-bit quantization, or to quantizers with higher resolutions.

$\displaystyle[{\bm{T}}]_{n,n}=\left.\begin{cases}M-\frac{M}{2}(2-\frac{2}{M})&1<n<M\\\
M-\frac{M}{2}(1-\frac{1}{M}+{\operatorname{j}}\frac{1}{M})+{\operatorname{j}}\frac{M}{2}(1-\frac{M-1}{M}+{\operatorname{j}}\frac{M-1}{M})&n=1\\\
M-\frac{M}{2}(1-\frac{1}{M}-{\operatorname{j}}\frac{1}{M})-{\operatorname{j}}\frac{M}{2}(1-\frac{M-1}{M}+{\operatorname{j}}\frac{1-M}{M})&n=M\end{cases}\right\\}=1.$
(67) $\displaystyle[{\bm{T}}]_{m,n}^{m\neq
n}=\left.\begin{cases}M(1-\frac{|m-n|}{M}+{\operatorname{j}}\frac{m-n}{M})-\frac{M}{2}(2-\frac{|m-n-1|+|m-n+1|}{M}+2{\operatorname{j}}\frac{m-n}{M})&1<n<M\\\
M(1-\frac{|1-n|}{M}+{\operatorname{j}}\frac{1-n}{M})-\frac{M}{2}(1-\frac{|2-n|}{M}+{\operatorname{j}}\frac{2-n}{M})+{\operatorname{j}}\frac{M}{2}(1-\frac{|M-n|}{M}+{\operatorname{j}}\frac{M-n}{M})&n=1\\\
M(1-\frac{|M-n|}{M}+{\operatorname{j}}\frac{M-n}{M})-\frac{M}{2}(1-\frac{|M-n-1|}{M}+{\operatorname{j}}\frac{M-n-1}{M})-{\operatorname{j}}\frac{M}{1}(1-\frac{|1-n|}{M}+{\operatorname{j}}\frac{1-n}{M})&n=M\end{cases}\right\\}=0.$
(68)

###  -A Proof of Theorem 1

###### Proof.

First, it can be observed that due to the absence of noise the expression in
(15) simplifies to an indicator function

$p_{{\bm{r}}_{\Re}|{\bm{h}}_{\Re}}({\bm{r}}_{\Re}|{\bm{h}}_{\Re})=\begin{cases}1,~{}~{}{\bm{h}}_{\Re}\in\overline{Q}({\bm{r}}_{\Re})\\\
0,~{}~{}{\bm{h}}_{\Re}\not\in\overline{Q}({\bm{r}}_{\Re})\end{cases}$ (47)

which means that the probability of receiving the quantized observation
${\bm{r}}_{\Re}$ is one if the underlying channel lies in
$\overline{Q}({\bm{r}}_{\Re})$ since the quantization is deterministic. With
this, we get

$\displaystyle\operatorname{E}[{\bm{h}}_{\Re}|{\bm{r}}_{\Re}]$
$\displaystyle=\frac{1}{p_{{\bm{r}}_{\Re}}({\bm{r}}_{\Re})}\int_{\overline{Q}({\bm{r}}_{\Re})}{\bm{h}}_{\Re}f_{{\bm{h}}_{\Re}}({\bm{h}}_{\Re})\operatorname{d}{\bm{h}}_{\Re}.$
(48)

Second, we can interpret the integral in (48) as the (scaled) mean of a
truncated multivariate Gaussian, i.e.,

$\displaystyle\int_{\overline{Q}({\bm{r}}_{\Re})}{\bm{h}}_{\Re}f_{{\bm{h}}_{\Re}}({\bm{h}}_{\Re})\operatorname{d}{\bm{h}}_{\Re}=p_{{\bm{r}}_{\Re}}({\bm{r}}_{\Re}){\operatorname{E}}[\tilde{{\bm{h}}}_{\Re}]$
(49)

where $\tilde{{\bm{h}}}_{\Re}\sim f_{\tilde{{\bm{h}}}_{\Re}}$ is a truncated
Gaussian RV with PDF

$f_{\tilde{{\bm{h}}}_{\Re}}(\tilde{{\bm{h}}}_{\Re})=\begin{cases}\frac{1}{p_{{\bm{r}}_{\Re}}({\bm{r}}_{\Re})}\mathcal{N}(\tilde{{\bm{h}}}_{\Re};{\bm{0}},{\bm{C}}_{{\bm{h}}})&\tilde{{\bm{h}}}_{\Re}\in\overline{Q}({\bm{r}}_{\Re})\\\
0&\tilde{{\bm{h}}}_{\Re}\not\in\overline{Q}({\bm{r}}_{\Re})\end{cases}$ (50)

and $p_{{\bm{r}}_{\Re}}({\bm{r}}_{\Re})$ is the normalization such that
$f_{\tilde{{\bm{h}}}_{\Re}}$ is a valid PDF that integrates to one. In [57,
58], the mean of a one-sided truncated Gaussian RV is derived, which in our
case is computed to

$\displaystyle
p_{{\bm{r}}_{\Re}}({\bm{r}}_{\Re}){\operatorname{E}}[[\tilde{{\bm{h}}}_{\Re}]_{i}]=$
(51)
$\displaystyle\sum_{n=1}^{N}[{\bm{r}}_{\Re}]_{n}[{\bm{C}}_{{\bm{h}}}]_{i,n}\mathcal{N}(0;0,[{\bm{C}}_{{\bm{h}}}]_{n,n})\int_{\overline{Q}({\bm{r}}_{\Re}^{n})}\mathcal{N}^{n}({\bm{x}};{\bm{0}},{\bm{C}}_{{\bm{h}}})\operatorname{d}{\bm{x}}.$

Plugging (51) into (49) together with (48) yields the expression in (26) and
thus finishes the proof. ∎

###  -B Proof of Theorem 2

###### Proof.

We first observe that with the procedure in Algorithm 1 the angle domain can
be split into at most $4M$ regions by choosing $\psi_{m}\neq 0$ $\forall
m=2,...,M$ because any of the four quadrants can be divided into $M$ distinct
regions which are identified via (32). Now, we write the CME as
$\hat{h}=\hat{\alpha}{\operatorname{e}}^{{\operatorname{j}}\hat{\theta}}$ such
that we can reformulate the MSE as

$\displaystyle{\operatorname{E}}[|\alpha{\operatorname{e}}^{{\operatorname{j}}\theta}-\hat{\alpha}{\operatorname{e}}^{{\operatorname{j}}\hat{\theta}}|^{2}]$
$\displaystyle={\operatorname{E}}[\alpha^{2}+\hat{\alpha}^{2}-2\alpha\hat{\alpha}\cos(\theta-\hat{\theta})]$
(52)

by using the cosine law. Since we are solely interested in the phase estimate
for the design of the optimal pilot sequence and the cosine is monotonically
decreasing between $[0,\pi/2)$, the minimization of the MSE between the phases
${\operatorname{E}}[(\theta-\hat{\theta})^{2}]$ is equivalent to the
minimization of the MSE from (52) for fixed magnitudes. Note that because of
the uniform distribution $f_{\theta}$ the CME $\hat{\theta}$ is exactly the
midpoint between the boundary angles which can also be seen later in Theorem
3. The optimization of the pilot sequence is thus given as

$\displaystyle\operatorname*{arg\,min}_{{\bm{a}}}~{}{\operatorname{E}}[(\theta-\hat{\theta}({\bm{a}}))^{2}]$
(53)
$\displaystyle=\operatorname*{arg\,min}_{{\bm{a}}}\int_{0}^{2\pi}(\theta-\hat{\theta}({\bm{a}}))^{2}f_{\theta}(\theta)\operatorname{d}\theta$
(54)
$\displaystyle=\operatorname*{arg\,min}_{{\bm{a}}}\sum_{i=1}^{4M-1}\int_{\phi_{i}}^{\phi_{i+1}}(\theta-\hat{\theta}({\bm{a}}))^{2}f_{\theta}(\theta)\operatorname{d}\theta$
(55)

where in (55) the integral over the whole angle domain is split up into the
$4M$ subsets that are defined via the boundary angles from (32) which fulfill
$\phi_{1}<\dots<\phi_{4M}$ where $\phi_{1}=0$ and $\phi_{4M}=2\pi$. The
optimization problem in (55) is identical to the well known problem of finding
the quantization regions of a non-uniform quantizer which minimize the
distortion [59]. Because the angles are uniformly distributed, the optimal
solution is found by equidistant integration regions, cf. [59, Sec. 5.6],
which is achieved by equidistant phase shifts in $[0,\frac{\pi}{2})$, i.e.,
$\psi_{m}=\frac{\pi(m-1)}{2M}~{}\forall m=1,...,M$. Plugging this solution
into (29) results in (33) and finishes the proof. ∎

###  -C Proof of Theorem 3

###### Proof.

We use the simplified representation of (28) where the mean of the Rayleigh
distributed magnitudes is known to be, cf. [53, Appendix A.1.3],

${\operatorname{E}}[\alpha]=\frac{\sqrt{\pi}}{2}\sigma.$ (56)

Furthermore, due to the equidistant phase shifts which divide the input space
into $4M$ distinct regions, cf. Appendix -B, and the symmetry of the Gaussian
PDF, we have the prior probability $p_{{\bm{r}}}({\bm{r}})=\frac{1}{4M}$ in
this case. Plugging this result and the definition of the uniform distribution
$f_{\theta}$ into (31), the CME of the normalized channel is computed to

$\displaystyle{\operatorname{E}}[{\operatorname{e}}^{{\operatorname{j}}\theta}|{\bm{r}}]$
$\displaystyle=\frac{4M}{2\pi}\int_{\phi_{\text{low}}({\bm{r}})}^{\phi_{\text{low}}({\bm{r}})+\frac{\pi}{2M}}{\operatorname{e}}^{{\operatorname{j}}\theta}\operatorname{d}\theta.$
(57)

It is sufficient to solve the integral for the first angle sector
$[0,\frac{\pi}{2M})$ and afterwards shift the solution by the angle of
$\phi_{\text{low}}({\bm{r}})$. We thus compute

$\displaystyle\int_{0}^{\frac{\pi}{2M}}{\operatorname{e}}^{{\operatorname{j}}\theta}\operatorname{d}\theta$
$\displaystyle=\sin\left(\frac{\pi}{2M}\right)+{\operatorname{j}}\left(1-\cos\left(\frac{\pi}{2M}\right)\right)$
(58)
$\displaystyle=2\sin\left(\frac{\pi}{4M}\right)\exp\left({\operatorname{j}}\frac{\pi}{4M}\right).$
(59)

After plugging (56) together with the generalized solution from (57) and (59)
into (28) yields (35) which finishes the proof. ∎

###  -D Proof of Theorem 4

###### Proof.

With the vector in (33), its outer product is computed as

$[{\bm{a}}{\bm{a}}^{\operatorname{H}}]_{m,n}=\exp\left({\operatorname{j}}\frac{\pi}{2M}(m-n)\right)$
(60)

and thus

$\displaystyle[\Re({\bm{a}}{\bm{a}}^{\operatorname{H}})]_{m,n}$
$\displaystyle=\cos\left(\frac{\pi}{2M}(m-n)\right),$ (61)
$\displaystyle[\Im({\bm{a}}{\bm{a}}^{\operatorname{H}})]_{m,n}$
$\displaystyle=\sin\left(\frac{\pi}{2M}(m-n)\right).$ (62)

For the considered case, the unquantized observation is given as
${\bm{y}}={\bm{a}}h$, and thus its covariance matrix can be written as
${\bm{C}}_{{\bm{y}}}=\sigma^{2}{\bm{a}}{\bm{a}}^{\operatorname{H}}$.
Furthermore, from the property that $|[{\bm{a}}]_{m}|^{2}=1$, it follows that
$\operatorname{diag}({\bm{C}}_{{\bm{y}}})=\sigma^{2}\operatorname{\mathbf{I}}$.
Consequently, the expression for the covariance matrix from (10) simplifies to

$\displaystyle{\bm{C}}_{{\bm{r}}}$
$\displaystyle=\frac{2}{\pi}\left[\arcsin(\Re({\bm{a}}{\bm{a}}^{\operatorname{H}}))+{\operatorname{j}}\arcsin(\Im({\bm{a}}{\bm{a}}^{\operatorname{H}}))\right].$
(63)

After plugging in (61) and (62) we get

$\displaystyle[{\bm{C}}_{{\bm{r}}}]_{m,n}$
$\displaystyle=\frac{2}{\pi}\left[\arcsin\left(\cos\left(\frac{\pi}{2M}(m-n)\right)\right)\right.$
(64)
$\displaystyle~{}~{}~{}\left.+{\operatorname{j}}\arcsin\left(\sin\left(\frac{\pi}{2M}(m-n)\right)\right)\right]$
$\displaystyle=1-\frac{|m-n|}{M}+{\operatorname{j}}\frac{m-n}{M},$ (65)

where the equation in (65) holds since the arguments in (64) satisfy
$-\frac{\pi}{2}<\frac{\pi}{2M}(m-n)<\frac{\pi}{2}$. The following Lemma
computes ${\bm{C}}_{{\bm{r}}}^{-1}$ in closed form.

######  Lemma 1.

The inverse of ${\bm{C}}_{{\bm{r}}}$ from (65) is given as

$[{\bm{C}}_{{\bm{r}}}^{-1}]_{m,n}=\begin{cases}M&m=n\\\
-\frac{M}{2}&|m-n|=1\\\ {\operatorname{j}}\frac{M(n-m)}{2(M-1)}&|m-n|=M-1\\\
0&\text{otherwise}\end{cases}.$ (66)

###### Proof.

To prove that (66) is the inverse of (63) we define the matrix
${\bm{T}}={\bm{C}}_{{\bm{r}}}^{-1}{\bm{C}}_{{\bm{r}}}$. If we show that
${\bm{T}}$ is the identity matrix, i.e., ${\bm{T}}=\operatorname{\mathbf{I}}$,
then Lemma 1 holds. The $m,n$th element of ${\bm{T}}$ is given as
$[{\bm{T}}]_{m,n}=\sum_{l=1}^{M}[{\bm{C}}_{{\bm{r}}}^{-1}]_{m,l}[{\bm{C}}_{{\bm{r}}}]_{l,n}$.
Due to the sparsity of ${\bm{C}}_{{\bm{r}}}^{-1}$, only three summands are
non-zero. For example, we can compute $[{\bm{T}}]_{m,n}$ for $m=n=1$:

$\displaystyle[{\bm{T}}]_{1,1}=\sum_{l=1}^{M}[{\bm{C}}_{{\bm{r}}}^{-1}]_{1,l}[{\bm{C}}_{{\bm{r}}}]_{l,1}$
$\displaystyle=[{\bm{C}}_{{\bm{r}}}^{-1}]_{1,1}[{\bm{C}}_{{\bm{r}}}]_{1,1}+[{\bm{C}}_{{\bm{r}}}^{-1}]_{1,2}[{\bm{C}}_{{\bm{r}}}]_{2,1}+[{\bm{C}}_{{\bm{r}}}^{-1}]_{1,M}[{\bm{C}}_{{\bm{r}}}]_{M,1}$
$\displaystyle=M\cdot
1+\left(-\frac{M}{2}\right)\cdot\left(1-\frac{1}{M}+{\operatorname{j}}\frac{1}{M}\right)$
$\displaystyle\quad+{\operatorname{j}}\frac{M}{2}\cdot\left(1-\frac{M-1}{M}+{\operatorname{j}}\frac{M-1}{M}\right)=1.$

We show that the diagonal elements of ${\bm{T}}$, i.e., $m=n$, are one in (67)
and that all off-diagonal elements of ${\bm{T}}$, i.e., $m\neq n$, are zero in
(68). ∎

With Lemma 1, we can now find the expression of

$\displaystyle{\bm{a}}^{\operatorname{H}}{\bm{C}}_{{\bm{r}}}^{-1}{\bm{a}}$
$\displaystyle=\sum_{m=1}^{M}\sum_{n=1}^{M}[{\bm{a}}^{\operatorname{H}}]_{m}[{\bm{C}}_{{\bm{r}}}^{-1}]_{m,n}[{\bm{a}}]_{n}$
(69)
$\displaystyle=\sum_{m=1}^{M}M-\frac{M}{2}\sum_{m=1}^{M}\left(\exp(-{\operatorname{j}}\frac{\pi}{2M})+\exp({\operatorname{j}}\frac{\pi}{2M})\right)$
(70) $\displaystyle=M^{2}-M^{2}\cos\left(\frac{\pi}{2M}\right)$ (71)
$\displaystyle=2M^{2}\sin^{2}\left(\frac{\pi}{4M}\right).$ (72)

Plugging this into (37) results in the same MSE term as in (36) and thus
finishes the proof. ∎

## References

  * [1] T. Cover and J. Thomas, _Elements of Information Theory_. John Wiley & Sons, Ltd, 2005.
  * [2] A. Kipnis, Y. C. Eldar, and A. J. Goldsmith, “Fundamental Distortion Limits of Analog-to-Digital Compression,” _IEEE Trans. Inf. Theory_ , vol. 64, no. 9, pp. 6013–6033, 2018.
  * [3] S. Khobahi and M. Soltanalian, “Signal Recovery From 1-Bit Quantized Noisy Samples via Adaptive Thresholding,” in _52nd Asilomar Conf. Signals, Syst., Comput._ , 2018, pp. 1757–1761.
  * [4] Y. You, _Audio Coding: Theory and Applications_. Springer NY, 2010.
  * [5] R. Walden, “Analog-to-Digital Converter Survey and Analysis,” _IEEE J. Sel. Areas Commun._ , vol. 17, no. 4, pp. 539–550, 1999.
  * [6] A. Mezghani and J. A. Nossek, “Analysis of Rayleigh-Fading Channels with 1-Bit Quantized Output,” in _IEEE Int. Symp. Inf. Theory_ , 2008, pp. 260–264.
  * [7] J. A. Nossek and M. T. Ivrlač, “Capacity and Coding for Quantized MIMO Systems,” in _Proc. Int. Conf. Wireless Commun. Mobile Comput._ , 2006, p. 1387–1392.
  * [8] J. Mo and R. W. Heath, “High SNR Capacity of Millimeter Wave MIMO Systems with One-Bit Quantization,” in _Inf. Theory Appl. Workshop (ITA)_ , 2014\.
  * [9] A. Bazrafkan and N. Zlatanov, “Asymptotic Capacity of Massive MIMO With 1-Bit ADCs and 1-Bit DACs at the Receiver and at the Transmitter,” _IEEE Access_ , vol. 8, pp. 152 837–152 850, 2020.
  * [10] A. Mezghani and J. Nossek, “Capacity Lower Bound of MIMO Channels with Output Quantization and Correlated Noise,” in _Int. Symp. Inf. Theory_ , 2012.
  * [11] S. Jacobsson, G. Durisi, M. Coldrey, U. Gustavsson, and C. Studer, “One-Bit Massive MIMO: Channel Estimation and High-Order Modulations,” in _IEEE Int. Conf. Commun. Workshop (ICCW)_ , 2015, pp. 1304–1309.
  * [12] Y. Li, C. Tao, G. Seco-Granados, A. Mezghani, A. L. Swindlehurst, and L. Liu, “Channel Estimation and Performance Analysis of One-Bit Massive MIMO Systems,” _IEEE Trans. Signal Process._ , vol. 65, no. 15, pp. 4075–4089, 2017.
  * [13] S. Jacobsson, G. Durisi, M. Coldrey, U. Gustavsson, and C. Studer, “Throughput Analysis of Massive MIMO Uplink With Low-Resolution ADCs,” _IEEE Trans. Wireless Commun._ , vol. 16, no. 6, pp. 4038–4051, 2017.
  * [14] M. Ivrlac and J. Nossek, “On MIMO Channel Estimation with Single-Bit Signal-Quantization,” in _Int. ITG Workshop Smart Antennas (WSA)_ , 2007\.
  * [15] M. S. Stein, S. Bar, J. A. Nossek, and J. Tabrikian, “Performance Analysis for Channel Estimation With 1-Bit ADC and Unknown Quantization Threshold,” _IEEE Trans. Signal Process._ , vol. 66, no. 10, pp. 2557–2571, 2018.
  * [16] J. Choi, J. Mo, and R. W. Heath, “Near Maximum-Likelihood Detector and Channel Estimator for Uplink Multiuser Massive MIMO Systems With One-Bit ADCs,” _IEEE Trans. Commun._ , vol. 64, no. 5, pp. 2005–2018, 2016.
  * [17] T. Lok and V.-W. Wei, “Channel Estimation with Quantized Observations,” in _IEEE Int. Symp. Inf. Theory_ , 1998, p. 333.
  * [18] R. P. David and J. Cal-Braz, “Feedback-Controlled Channel Estimation with Low-resolution ADCs in Multiuser MIMO Systems,” in _IEEE Int. Conf. Acoust., Speech, Signal Process. (ICASSP)_ , 2019, pp. 4674–4678.
  * [19] F. Liu, X. Shang, Y. Cheng, and G. Zhang, “Computationally Efficient Maximum Likelihood Channel Estimation for Coarsely Quantized Massive MIMO Systems,” _IEEE Wireless Commun. Lett._ , vol. 26, no. 2, pp. 444–448, 2022.
  * [20] F. Liu, X. Shang, and H. Zhu, “Efficient Majorization-Minimization-Based Channel Estimation for One-Bit Massive MIMO Systems,” _IEEE Trans. Wireless Commun._ , vol. 20, no. 6, pp. 3444–3457, 2021.
  * [21] A. Mezghani, F. Antreich, and J. A. Nossek, “Multiple Parameter Estimation with Quantized Channel Output,” in _Int. ITG Workshop Smart Antennas (WSA)_ , 2010, pp. 143–150.
  * [22] C. Studer and G. Durisi, “Quantized Massive MU-MIMO-OFDM Uplink,” _IEEE Trans. Commun._ , vol. 64, no. 6, pp. 2387–2399, 2016.
  * [23] C. Stöckle, J. Munir, A. Mezghani, and J. A. Nossek, “Channel Estimation in Massive MIMO Systems using 1-Bit Quantization,” in _IEEE 17th Int. Workshop Signal Process. Advances Wireless Commun. (SPAWC)_ , 2016.
  * [24] F. Wang, J. Fang, H. Li, Z. Chen, and S. Li, “One-Bit Quantization Design and Channel Estimation for Massive MIMO Systems,” _IEEE Trans. Veh. Technol._ , vol. 67, no. 11, pp. 10 921–10 934, 2018.
  * [25] N. J. Myers, K. N. Tran, and R. W. Heath, “Low-Rank mmWave MIMO Channel Estimation in One-Bit Receivers,” in _IEEE Int. Conf. Acoust., Speech Signal Process. (ICASSP)_ , 2020, pp. 5005–5009.
  * [26] L. Xu, F. Gao, and C. Qian, “Gridless Angular Domain Channel Estimation for mmWave Massive MIMO System with One-Bit Quantization via Approximate Message Passing,” in _Proc. IEEE Global Commun. Conf. (GLOBECOM)_ , 2019.
  * [27] J. Mo, P. Schniter, and R. W. Heath, “Channel Estimation in Broadband Millimeter Wave MIMO Systems With Few-Bit ADCs,” _IEEE Trans. Signal Process._ , vol. 66, no. 5, pp. 1141–1154, 2018.
  * [28] Y. Dong, H. Wang, and Y.-D. Yao, “Channel Estimation for One-Bit Multiuser Massive MIMO Using Conditional GAN,” _IEEE Commun. Lett._ , vol. 25, no. 3, pp. 854–858, 2021.
  * [29] Y. Zhang, M. Alrabeiah, and A. Alkhateeb, “Deep Learning for Massive MIMO With 1-Bit ADCs: When More Antennas Need Fewer Pilots,” _IEEE Wireless Commun. Lett._ , vol. 9, no. 8, pp. 1273–1277, 2020.
  * [30] R. Zhu and G. Zhang, “A Segment-Average Based Channel Estimation Scheme for One-Bit Massive MIMO Systems with Deep Neural Network,” in _Proc. IEEE Int. Conf. Commun. Technol. (ICCT)_ , 2019, pp. 81–86.
  * [31] L. V. Nguyen, A. L. Swindlehurst, and D. H. N. Nguyen, “SVM-Based Channel Estimation and Data Detection for One-Bit Massive MIMO Systems,” _IEEE Trans. Signal Process._ , vol. 69, pp. 2086–2099, 2021.
  * [32] B. Fesl, M. Koller, N. Turan, and W. Utschick, “Learning a Low-Complexity Channel Estimator for One-Bit Quantization,” in _54th Asilomar Conf. Signals, Syst., Comput._ , 2020, pp. 393–397.
  * [33] J. J. Bussgang, “Crosscorrelation Functions of Amplitude-Distorted Gaussian Signals,” MIT Research Lab. Electronics, Tech. Rep. 216, 1952.
  * [34] R. Price, “A Useful Theorem for Nonlinear Devices Having Gaussian Inputs,” _IRE Trans. Inf. Theory_ , pp. 69–72, 1958.
  * [35] H. E. Rowe, “Memoryless Nonlinearities With Gaussian Inputs: Elementary Results,” _Bell Syst. Tech. J._ , pp. 1519–1525, 1982.
  * [36] Q. Wan, J. Fang, H. Duan, Z. Chen, and H. Li, “Generalized Bussgang LMMSE Channel Estimation for One-Bit Massive MIMO Systems,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 6, pp. 4234–4246, 2020.
  * [37] O. T. Demir and E. Bjornson, “The Bussgang Decomposition of Nonlinear Systems: Basic Theory and MIMO Extensions [Lecture Notes],” _IEEE Signal Process. Magazine_ , vol. 38, pp. 131–136, 2021.
  * [38] H. Kim and J. Choi, “Channel Estimation for One-Bit Massive MIMO Systems Exploiting Spatio-Temporal Correlations,” in _Proc. IEEE Global Commun. Conf. (GLOBECOM)_ , 2018.
  * [39] S. Rao, G. Seco-Granados, H. Pirzadeh, J. A. Nossek, and A. L. Swindlehurst, “Massive MIMO Channel Estimation With Low-Resolution Spatial Sigma-Delta ADCs,” _IEEE Access_ , vol. 9, pp. 109 320–109 334, 2021.
  * [40] H. Lee, Y.-S. Jeon, H. Do, and N. Lee, “Concatenated MMSE Estimation for Quantized OFDM Systems,” in _IEEE Int. Conf. Commun. (ICC)_ , 2019.
  * [41] A. Banerjee, X. Guo, and H. Wang, “On the Optimality of Conditional Expectation as a Bregman Predictor,” _IEEE Trans. Inf. Theory_ , vol. 51, no. 7, pp. 2664–2669, 2005.
  * [42] C. Mollén, J. Choi, E. G. Larsson, and R. W. Heath, “Uplink Performance of Wideband Massive MIMO With One-Bit ADCs,” _IEEE Trans. Wireless Commun._ , vol. 16, no. 1, pp. 87–100, 2017.
  * [43] C.-K. Wen, C.-J. Wang, S. Jin, K.-K. Wong, and P. Ting, “Bayes-Optimal Joint Channel-and-Data Estimation for Massive MIMO With Low-Precision ADCs,” _IEEE Trans. Signal Process._ , vol. 64, no. 10, pp. 2541–2556, 2016.
  * [44] M. D. McDonnell, N. G. Stocks, C. E. M. Pearce, and D. Abbott, _Stochastic Resonance: From Suprathreshold Stochastic Resonance to Stochastic Signal Quantization_. Cambridge University Press, 2008.
  * [45] R. E. Curry, _Estimation and Control with Quantized Measurements_. MIT Press, 1970.
  * [46] A. Papoulis and S. U. Pillai, _Probability, Random Variables and Stochastic Processes_. McGraw-Hill Education, 2002.
  * [47] G. Jacovitti and A. Neri, “Estimation of the Autocorrelation Function of Complex Gaussian Stationary Processes by Amplitude Clipped Signals,” _IEEE Trans. Inf. Theory_ , vol. 40, no. 1, pp. 239–245, 1994.
  * [48] A. Genz, “Numerical Computation of Multivariate Normal Probabilities,” _J. Comput. Graphical Statist._ , vol. 1, no. 2, pp. 141–149, 1992.
  * [49] P. Virtanen _et al._ , “SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python,” _Nature Methods_ , vol. 17, pp. 261–272, 2020.
  * [50] D. B. Owen, “A Table of Normal Integrals,” _Commun. Statist. - Simul. Comput._ , vol. 9, pp. 389–419, 1980.
  * [51] N. Korotkov and A. Korotkov, _Integrals Related to the Error Function_. Chapman and Hall/CRC, 2020\.
  * [52] S. M. Kay, _Fundamentals of Statistical Signal Processing: Estimation Theory_. Prentice Hall, 1993.
  * [53] D. Tse and P. Viswanath, _Fundamentals of Wireless Communication_. Cambridge University Press, 2005.
  * [54] B. Hassibi and B. Hochwald, “How Much Training is Needed in Multiple-Antenna Wireless Links?” _IEEE Trans. Inf. Theory_ , vol. 49, no. 4, pp. 951–963, 2003.
  * [55] Y. Li, C. Tao, L. Liu, G. Seco-Granados, and A. L. Swindlehurst, “Channel Estimation and Uplink Achievable Rates in One-Bit Massive MIMO Systems,” in _IEEE Sensor Array Multichannel Signal Process. Workshop (SAM)_ , 2016.
  * [56] F. Pedregosa _et al._ , “Scikit-learn: Machine Learning in Python,” _J. Mach. Learn. Research_ , vol. 12, pp. 2825–2830, 2011.
  * [57] G. M. Tallis, “The Moment Generating Function of the Truncated Multi-Normal Distribution,” _J. R. Stat. Soc. Series B Stat. Methodol._ , pp. 223–229, 1961.
  * [58] J. Cartinhour, “One-Dimensional Marginal Density Functions of a Truncated Multivariate Normal Density Function,” _Commun. in Statist. - Theory and Methods_ , vol. 19, no. 1, pp. 197–203, 1990.
  * [59] A. Gersho and R. M. Gray, _Vector Quantization and Signal Compression_. Springer NY, 1992.