# Collaborative Automotive Radar Sensing via Mixed-Precision Distributed Array
Completion

Arian Eamaz, Farhang Yeganegi⋆, Yunqiao Hu⋆, Mojtaba Soltanalian, and Shunqiao
Sun  Arian Eamaz, Farhang Yeganegi and Mojtaba Soltanalian are with the ECE
Department, University of Illinois at Chicago, Chicago, IL 60607 USA. Yunqiao
Hu and Shunqiao Sun are with the Department of Electrical and Computer
Engineering, University of Alabama, Tuscaloosa, AL 35487 USA.This work was
supported in part by the U.S. National Science Foundation Grants CCF-1704401,
CCF-2153386 and ECCS-2340029. (_Corresponding author: Arian Eamaz_)⋆ Equal
contribution, the names were arranged in alphabetical order.

###### Abstract

This paper investigates the effects of coarse quantization with mixed
precision on measurements obtained from sparse linear arrays, synthesized by a
collaborative automotive radar sensing strategy. The mixed quantization
precision significantly reduces the data amount that needs to be shared from
radar nodes to the fusion center for coherent processing. We utilize the low-
rank properties inherent in the constructed Hankel matrix of the mixed-
precision array, to recover azimuth angles from quantized measurements. Our
proposed approach addresses the challenge of mixed-quantized Hankel matrix
completion, allowing for accurate estimation of the azimuth angles of
interest. To evaluate the recovery performance of the proposed scheme, we
establish a quasi-isometric embedding with a high probability for mixed-
precision quantization. The effectiveness of our proposed scheme is
demonstrated through numerical results, highlighting successful
reconstruction.

###### Index Terms:

Coarse quantization, collaborative radar sensing, dithered mixed-precision
sensing, Hankel matrix completion, sparse linear array.

##  I Introduction

Millimeter wave (mmWave) automotive radars at $77$ GHz are highly reliable in
all weather environments [1]. Benefiting from multiple-input multiple-output
(MIMO) radar technology, mmWave radars can synthesize virtual arrays with
large aperture sizes using a small number of transmit and receive antennas
[1]. To reduce the hardware cost, sparse arrays synthesized by MIMO radar
technology have been widely adopted in automotive radar. Collaborative sensing
offers numerous benefits for object detection and radar imaging over single
automotive radar sensor [2]. For example, enhancing signal gain, increasing
reliability, adaptability, and greater spatial diversity[3, 4]. In automotive
sensing tasks, such as direction-of-arrival (DOA) estimation, collaborative
sensing involves coherently integrating measurements from subarrays of each
radar to synthesize an array with a larger aperture, thereby enhancing angle
resolution and increasing Signal-to-Noise Ratio (SNR) at the same time[5].
Hence, there is significant interest in developing novel radar sensing methods
within the collaborative sensing framework[2]. However, sharing the beam
vectors collected from collaborative radar sensing with the fusion center,
which contains high-resolution data, can pose practical difficulties and lead
to the wastage of communication bandwidth. To circumvent this issue,
quantizing the beam vectors appears to be a highly efficient solution.

Conventional high-resolution quantization methods typically demand many
quantization levels, resulting in heightened power consumption, elevated
manufacturing expenses, and diminished analog-to-digital converters (ADCs)
sampling rates. In search of alternative solutions, researchers have delved
into coarse quantization scenarios, such as one-bit quantization, where
signals are compared with a fixed threshold at ADCs, producing binary outputs.
This approach facilitates high-rate sampling while mitigating implementation
costs and energy consumption compared to multi-bit ADCs. One-bit ADCs hold
significant promise in MIMO systems [6].

However, utilizing a pure one-bit ADC system can encounter challenges, such as
significant rate loss in high SNR scenarios and dynamic range issues, where a
strong target may overshadow a weaker one [7]. To mitigate these concerns, a
mixed-ADC architecture has been proposed [8], where most receive antenna
outputs are sampled by one-bit ADCs, while high-resolution ADCs sample a few.
Recent studies [9] have explored the optimal arrangement of ADCs based on the
Cramér-Rao Bound (CRB) of acquired measurements, suggesting that evenly
distributing high-resolution ADCs along the edges of linear arrays enhances
the CRB for Direction of Arrival (DoA) estimation. However, even multi-bit
quantization with more than 4 bits in practical scenarios can effectively
mimic a high-resolution setting [10]. In [11], we outlined the construction of
a Hankel matrix utilizing one-bit data obtained from Sparse Linear Arrays
(SLA). Through both numerical simulations and theoretical analysis, we
demonstrated that utilizing a one-bit Hankel matrix can achieve successful
target detection performance with a high probability.

In this paper, we introduce a collaborative radar sensing system to synthesize
a distributed sparse array with large aperture where most beamvectors are
quantized as one-bit data, while the remaining data is quantized as multi-bit.
We construct a Hankel matrix using the acquired mixed-precision data, which
contains missing entries. Consequently, the problem of target azimuth
estimation is transformed into completing the Hankel matrix from quantized
data. It has been demonstrated that when the scalar parameter of uniform
dithers is designed to dominate the dynamic range of measurements, multi-bit
scalar quantization simplifies to a one-bit comparator[12]. Taking advantage
of the properties of uniform dithering, which are adept at canceling
quantization effects in expected values, the $\ell_{1}$-norm distance between
the matrix and its estimate finds an upper bound with a certain probability.

Our numerical findings showcase the efficacy of incorporating mixed ADCs
alongside a well-suited dithering scheme. This combination efficiently
utilizes the low-resolution dither samples during the reconstruction phase. As
a result, we achieve significant outcomes, including precise detection of
target azimuth locations with enhanced resolution.

_Notation_ : Throughout this paper, we use bold lowercase and bold uppercase
letters for vectors and matrices, respectively. We represent a vector
$\mathbf{x}$ and a matrix $\mathbf{X}$ in terms of their elements as
$\mathbf{x}=[x_{i}]$ and $\mathbf{X}=[X_{i,j}]$, respectively. The sets of
real and complex numbers are denoted by $\mathbb{R}$ and $\mathbb{C}$,
respectively. The vector/matrix transpose is denoted by $(\cdot)^{\top}$.
Given a scalar $x$, we define the operator $(x)^{+}$ as
$\max\left\\{x,0\right\\}$. The nuclear norm of a matrix
$\mathbf{X}\in\mathbb{C}^{n_{1}\times n_{2}}$ is denoted by
$\left\|\mathbf{X}\right\|_{\star}=\sum^{r}_{i=1}\sigma_{i}$ where $r$ and
$\left\\{\sigma_{i}\right\\}^{r}_{i=1}$ are the rank and singular values of
$\mathbf{X}$, respectively. The Frobenius norm of a matrix
$\mathbf{X}\in\mathbb{C}^{n_{1}\times n_{2}}$ is defined as
$\|\mathbf{X}\|_{\mathrm{F}}=\sqrt{\sum^{n_{1}}_{r=1}\sum^{n_{2}}_{s=1}\left|x_{rs}\right|^{2}}$,
where $x_{rs}$ is the $(r,s)$-th entry of $\mathbf{X}$. In the real case, we
also define $\|\mathbf{X}\|_{\mathrm{max}}=\sup_{i,j}|X_{i,j}|$. The
$\ell_{1}$-norm for a matrix $\mathbf{X}$ means
$\|\mathbf{X}\|_{1}=\|\operatorname{vec}(\mathbf{X})\|_{1}$. The operator
$\operatorname{diag}\left(\mathbf{b}\right)$ denotes a diagonal matrix with
$\\{b_{i}\\}$ as its diagonal elements. The set $[n]$ is defined as
$[n]=\left\\{1,\cdots,n\right\\}$. The Hadamard (element-wise) product is
$\odot$. The notation $x\sim\mathcal{U}_{[a,b]}$ means a random variable drawn
from the uniform distribution over the interval $[a,b]$. The memoryless scalar
quantizer is denoted by
$\mathcal{Q}_{{}_{\Delta}}:\mathbb{R}\rightarrow\mathcal{A}_{K}$, where
$\Delta$ is the resolution parameter and $\mathcal{A}_{K}$ is the finite
alphabet set given by

$\mathcal{A}_{K}:=\\{\pm\frac{k\Delta}{2}:k\in[K]\\}.$ (1)

The relation between the number of bits $B$ and $K$ is given by
$B=\log_{2}(K)+1$. When we introduce a uniform dither generated as
$\tau\sim\mathcal{U}_{\left[-\frac{\Delta}{2},\frac{\Delta}{2}\right]}$, to
the input signal of the quantizer, the resulting quantization process is
termed as _uniform quantization_. This process can be defined as follows:

$\mathcal{Q}_{{}_{\Delta}}\left(x\right)=\Delta\left(\left\lfloor\frac{x+\tau}{\Delta}\right\rfloor+\frac{1}{2}\right).$
(2)

##  II Collaborative Automotive Radar Sensing with Mixed-Precision ADCs

This section briefly elucidates the collaborative radar sensing system.
Subsequently, we introduce our scheme of mixed-precision uniform quantization.
This method involves sampling certain measurements and quantizing the
remainder using a multi-bit scheme, guided by the design of scalar parameters
for uniform dithers.

###  II-A Coherent Automotive Radar Network with Distributed Sparse Array and
Mixed Quantization Precision

Typical automotive radar sensor for Advanced Driver Assistance Systems (ADAS)
functions comprises a single radar chipset with 3 transmit and 4 receive
antennas. This configuration leads to a restricted array aperture,
consequently limiting the sensor’s spatial resolution [1]. We consider an
automotive radar network consisting of multiple low-end automotive radars for
enhanced array aperture through coherent collaborative sensing. One example of
such a radar network with two automotive radars is illustrated in Fig. 1 (a)
[13]. As shown in Fig. 1 (b), the collaborative sensing consists of a
multistatic radar system consisting of two automotive radars. The two radars
are clock synchronized. The beamvectors from each radar are shared to data
fusion center for coherent processing. Each radar receives signal from both
monostatic and bistatic paths. The monostatic path denotes the line-of-sight
trajectory traversed by the radar signal from the transmitter to the target,
and back to the receiver, depicted by the solid blue and yellow lines. The
bistatic path signifies the trajectory of the signal from the transmitter of
one radar to the target, and then back to the receiver of another one,
depicted by the green and brown dotted line in Fig. 1 (b). We suppose the far-
field model still holds since the two radars are parallel and the ratio of the
separation $d_{0}$ between the two radars and the target distance satisfies
the far-field condition.

Figure 1: (a) Collaborative sensing hardware demo consisting of two MIMO FMCW
radar units [13]; (b) Simplified illustration of a collaborative sensing
system with two radar units, each with 3 TXs and 4 RXs. The two radars are
clock synchronized. The beamvectors from each radar are shared to data fusion
center for coherent processing. (c) Example of an automotive collaborative
radar with $14$ physical array elements and the synthesized virtual
distributed sparse array of $48$ elements and aperture of $76\lambda$.

As depicted in Fig. 1 (c), each radar features $M_{t}=3$ Tx and $M_{r}=4$ Rx
antennas, arranged on a grid with a grid size equal to $\lambda/2$, where
$\lambda$ is the wavelength. Each automotive radar knows the orthogonal slow-
time phase codes of both radars that utilize frequency-modulated continuous-
wave (FMCW). Leveraging the orthogonality of waveforms, at each automotive
radar receiver, following the MIMO radar concept, the raw two-dimensional (2D)
data (fast time $\times$ slow time) with high-precision ADCs corresponding to
each Tx waveform can be firstly extracted. The multistatic radar system first
performs local range-Doppler processing via 2D fast Fourier transform (FFT).
Each radar yields $M_{t}M_{r}=12$ virtual array elements from monostatic path,
depicted in Fig. 1 (c) as “Monostatic path 1” and “Monostatic path 2”.
Additionally, each radar yields another virtual array with $M_{t}M_{r}=12$
elements from bistatic path, as depicted in Fig. 1 (c) as “Bistatic path 1”
and “Bistatic path 2”. The spatial separation $d_{0}=25\lambda/2$ to
accomodate the two radomes. Each radar now samples these $2M_{t}M_{r}=24$
virtual elements with mixed ADCs and sends them to the fusion center, where a
distribued sparse array of $48$ elements with mixed ADCs precision is
constructed. Note the data cube of beamvector that each automotive radar sends
to the fusion center is of size $N_{\rm fast}\times N_{\rm slow}\times
2M_{t}M_{r}$. Thus, mixed ADCs help save the communication bandwidth between
each low-end radar and the data fusion center.

As Fig. 1 (c) shows, with the knowledge of all the orthogonal waveform codes
from both radars, each radar could formulate a virtual array with
$2M_{t}M_{r}$ elements, and a large distributed sparse array with
$4M_{t}M_{r}$ elements could be synthesized in the data fusion center by
combining the beamvectors sent from these two radars. In general, the element
locations of sparse linear array can be considered a subset of a uniform
linear array (ULA) antenna positions. Without loss of generality, let the
antenna positions of an $M$-element ULA be $\left\\{kd\right\\}$,
$k=0,1,\cdots,M-1$, where $d=\lambda/2$ is the element spacing with wavelength
$\lambda$. As targets are first separated in range-Doppler domain, there is
few targets that need to be resolved in angular domain [14]. Assume there are
$P$ uncorrelated far-field target sources in the same range-Doppler bin, and
$P$ is sparse. The impinging signals on the ULA antennas are corrupted by
additive white Gaussian noise with the variance of $\sigma^{2}$. For the
single-snapshot case, the received signal from a ULA as

$\displaystyle\mathbf{x}=\mathbf{A}\mathbf{s}+\mathbf{n},$ (3)

where

$\displaystyle\mathbf{x}$
$\displaystyle=\left[x_{1},x_{2},\dots,x_{M}\right]^{\top},$ (4)
$\displaystyle\mathbf{A}$
$\displaystyle=\left[\mathbf{a}\left(\theta_{1}\right),\mathbf{a}\left(\theta_{2}\right)\dots,\mathbf{a}\left(\theta_{P}\right)\right]^{\top},$

with
$\mathbf{a}\left(\theta_{k}\right)=\left[1,e^{\mathrm{j}2\pi\frac{d\sin\left(\theta_{k}\right)}{\lambda}},\dots,e^{\mathrm{j}2\pi\frac{\left(M-1\right)d\sin\left(\theta_{k}\right)}{\lambda}}\right]^{\top}$,
for $k=1,\cdots,P$ and
$\mathbf{n}=\left[n_{1},n_{2},\dots,n_{M}\right]^{\top}$. We utilize a 1D
distributed virtual SLA synthesized by collaborative radar sensing techniques
[13] with $M_{t}$ transmit antennas and $M_{r}$ receive for each radar. The
SLA has $4M_{t}M_{r}<M$ elements while retaining the same aperture as ULA.
Denote the array element indices of ULA as the complete set $\left[M\right]$,
the array element indices of SLA can be expressed as a subset
${\Omega^{\prime}}\subset\left[M\right]$ with
$\left|\Omega^{\prime}\right|=4M_{t}M_{r}$. Thus, the signals received by the
SLA can be viewed as partial observations of $\mathbf{x}$, and can be
expressed as $\bar{\mathbf{x}}={\bf{m}}_{\Omega^{\prime}}\odot\mathbf{x}$,
where
${{\bf{m}}_{\Omega^{\prime}}}={\left[{{m_{1}},{m_{2}},\cdots,{m_{M}}}\right]^{\top}}$
is a masking vector with $m_{i}=1$, if $i\in\Omega^{\prime}$ or $m_{i}=0$ if
$i\notin\Omega^{\prime}$.

All $4M_{t}M_{r}$ antennas are equipped with low-resolution ADCs. Therefore,
the received measurements undergo uniform scalar quantization (scalar
quantization with uniform dithering, i.e.,
$\bm{\uptau}=\left[\tau_{i}\sim\mathcal{U}_{\left[-\frac{\Delta}{2},\frac{\Delta}{2}\right]}+\mathrm{j}\mathcal{U}_{\left[-\frac{\Delta}{2},\frac{\Delta}{2}\right]}\right]$)
as follows:

$\widehat{x}_{i}=\mathcal{Q}_{{}_{\Delta}}\left(\operatorname{Re}\left(\bar{x}_{i}\right)\right)+\mathrm{j}\mathcal{Q}_{{}_{\Delta}}\left(\operatorname{Im}\left(\bar{x}_{i}\right)\right),\quad
i\in\Omega^{\prime}.$ (5)

Considering that the quantization operator is applied solely to entries
following $i\in\Omega^{\prime}$, we can express (5) in vector form as:

$\widehat{\mathbf{x}}=\mathcal{Q}_{{}_{\Delta}}\left(\operatorname{Re}\left(\bar{\mathbf{x}}\right)\right)+\mathrm{j}\mathcal{Q}_{{}_{\Delta}}\left(\operatorname{Im}\left(\bar{\mathbf{x}}\right)\right),$
(6)

where
$\widehat{\mathbf{x}}\in\mathcal{A}^{M}_{K}+\mathrm{j}\mathcal{A}^{M}_{K}$.
Interestingly in (6), the uniform quantizer becomes a one-bit quantizer by
restricting the measurement’s dynamic range to the scale parameter of uniform
dithers as follows [12]:

$\displaystyle\widehat{\mathbf{x}}=\frac{\Delta}{2}\left[\operatorname{sgn}\left(\operatorname{Re}\left(\bar{\mathbf{x}}+\bm{\uptau}\right)\right)+\mathrm{j}\operatorname{sgn}\left(\operatorname{Im}\left(\bar{\mathbf{x}}+\bm{\uptau}\right)\right)\right],$
(7)
$\displaystyle\sup_{i\in[M_{r}M_{t}]}\left|\operatorname{Re}\left(\bar{x}_{i}\right)\right|,\sup_{i\in[M_{r}M_{t}]}\left|\operatorname{Im}\left(\bar{x}_{i}\right)\right|\leq\frac{\Delta}{2},$

where
$\widehat{\mathbf{x}}\in\left\\{-\frac{\Delta}{2},0,\frac{\Delta}{2}\right\\}^{M}+\mathrm{j}\left\\{-\frac{\Delta}{2},0,\frac{\Delta}{2}\right\\}^{M}$.
In this paper, we partition the measurements with $i\in\Omega^{\prime}$, into
two disjoint sets, $\Omega^{\prime}_{1}$ with
$\left|\Omega^{\prime}_{1}\right|=M_{o}$ and $\Omega^{\prime}_{2}$ with
$\left|\Omega^{\prime}_{2}\right|=M_{q}$ such that $M_{o}+M_{q}=4M_{t}M_{r}$.
Each measurement within the set $\Omega^{\prime}_{1}$ undergoes quantization
into one-bit data using uniform dithering with a scale parameter $\Delta_{1}$,
where $\Delta_{1}$ dominates the dynamic range of measurements associated with
$i\in\Omega^{\prime}_{1}$. Each measurement within the set
$\Omega^{\prime}_{2}$ undergoes quantization using uniform dithering with a
scale parameter $\Delta_{2}$, where $\Delta_{2}$ is unable to dominate the
dynamic range of the measurements associated with $i\in\Omega^{\prime}_{2}$.
Hence, we have established a mixed-precision scenario in which $M_{o}$ entries
are one-bit, while the rest are multi-bit. More generally, we define an
indicator vector $\bm{\delta}=[\delta_{1},\cdots,\delta_{M}]\in\\{0,1\\}^{M}$,
where for $i\in\Omega^{\prime}$, $\delta_{i}=1$ signifies that the $i$-th
antenna is equipped with a high-precision ADC. Therefore, the mixed-precision
output can be represented as

$\mathbf{y}=\operatorname{diag}\left(\bm{\delta}\right)\widehat{\mathbf{x}}+\left(\mathbf{I}-\operatorname{diag}\left(\bm{\delta}\right)\right)\frac{\Delta_{1}}{2}\mathbf{r},$
(8)

where

$\displaystyle\widehat{\mathbf{x}}$
$\displaystyle=\mathcal{Q}_{{}_{\Delta_{2}}}\left(\operatorname{Re}\left(\bar{\mathbf{x}}\right)\right)+\mathrm{j}\mathcal{Q}_{{}_{\Delta_{2}}}\left(\operatorname{Im}\left(\bar{\mathbf{x}}\right)\right),$
(9) $\displaystyle\Delta_{2}$
$\displaystyle<2\min\left(\sup_{i\in\Omega^{\prime}_{2}}|\operatorname{Re}\left(\bar{x}_{i}\right)|,\sup_{i\in\Omega^{\prime}_{2}}|\operatorname{Im}\left(\bar{x}_{i}\right)|\right),$

and

$\displaystyle\mathbf{r}$
$\displaystyle=\operatorname{sgn}\left(\operatorname{Re}\left(\bar{\mathbf{x}}-\bm{\uptau}\right)\right)+\mathrm{j}\operatorname{sgn}\left(\operatorname{Im}\left(\bar{\mathbf{x}}-\bm{\uptau}\right)\right),$
(10) $\displaystyle\Delta_{1}$ $\displaystyle\geq
2\max\left(\sup_{i\in\Omega^{\prime}_{1}}|\operatorname{Re}\left(\bar{x}_{i}\right)|,\sup_{i\in\Omega^{\prime}_{1}}|\operatorname{Im}\left(\bar{x}_{i}\right)|\right).$

In numerous applications, the upper bound of the dynamic range of measurements
is known [15]. Therefore, based on this information, we can easily design the
scale parameter of uniform dithers.

Following our model in (8), a Hankel matrix denoted by
${\mathcal{H}}\left(\mathbf{y}\right)\in{\mathbb{C}}^{n_{1}\times n_{2}}$ can
be constructed as $[{\mathcal{H}}\left(\mathbf{y}\right)]_{i,j}=y_{i+j-1}$,
where ${n_{1}+n_{2}}={M}+1$. Specifically, in this paper, we adopt
$n_{1}=n_{2}=\left({\frac{{M+1}}{2}}\right)$ if $M$ is odd, and
$n_{1}=n_{2}-1=\left({\frac{{M}}{2}}\right)$ if $M$ is even. In [14], it has
been demonstrated that the Hankel matrix possesses a low-rank property, with
its rank equivalent to the number of targets. For the sake of simplicity in
notation, hereafter, we denote the Hankel matrix of virtual sparse array
response as ${\mathcal{H}}\left(\mathbf{x}\right)=\mathbf{X}$ and quantized
data as ${\mathcal{H}}\left(\mathbf{y}\right)=\mathbf{Q}$. In the rest of the
paper, we will discuss the process of recovering $\mathbf{X}$ from
$\mathbf{Q}$.

###  II-B Mixed-Quantized Hankel Matrix Completion

Assume $\Omega$ denotes the set of observed entries in $\mathbf{X}$ denoted by
$\mathcal{P}_{\Omega}\left(\mathbf{X}\right)$, which is obtained by the array
element indices of SLA. In this section, we aim to obtain an upper bound for
the recovery error in the mixed quantized complex matrix completion problem
with high probability. Define the set of real low-rank matrices as

$\bar{\mathcal{K}}_{r}=\left\\{\mathbf{X}^{\prime}\in\mathbb{R}^{n_{1}\times
n_{2}}\mid\operatorname{rank}(\mathbf{X}^{\prime})\leq
r,\|\mathbf{X}^{\prime}\|_{\mathrm{max}}\leq\alpha\right\\}\subset\mathbb{R}^{n_{1}\times
n_{2}}.$ (11)

Initially, we assume that $\mathbf{X}\in\bar{\mathcal{K}}_{r}$, and later, we
extend the result to the complex scenario. We will use the following
definition in our theorem:

######  Definition 1.

Define a low-rank matrix as $\mathbf{X}=[X_{i,j}]\in\bar{\mathcal{K}}_{r}$.
The consistency property of uniform quantization over the pair
$(\mathbf{X},\mathbf{Y})\in\bar{\mathcal{K}}_{r}$, is given by

$\mathcal{Q}_{{}_{\Delta}}\left(\mathcal{P}_{\Omega}\left(\mathbf{X}\right)\right)=\mathcal{Q}_{{}_{\Delta}}\left(\mathcal{P}_{\Omega}\left(\mathbf{Y}\right)\right).$
(12)

The concept of consistent reconstruction, as defined in Definition 1, has
played a central role in providing theoretical guarantees in the field of one-
bit sensing [16, 17]. Before presenting our main recovery result, we establish
an embedding between the metric spaces
$\left(\bar{\mathcal{K}}_{r}\subset\mathbb{R}^{n_{1}\times
n_{2}},\ell_{1}\right)$ and
$\left(\mathcal{Q}_{{}_{\Delta}}\left(\mathcal{P}_{\Omega}\left(\bar{\mathcal{K}}_{r}\right)\right)\subset\mathcal{A}^{n_{1}\times
n_{2}}_{K},\ell_{1}\right)$, where
$\mathcal{Q}_{{}_{\Delta}}\left(\mathcal{P}_{\Omega}\left(\bar{\mathcal{K}}_{r}\right)\right)$
represents the space of quantized matrix completion. Specifically, the
subsequent lemma establishes that the quantized mapping
$\mathcal{Q}_{{}_{\Delta}}\left(\mathcal{P}_{\Omega}\left(\mathbf{X}\right)\right)$
is a quasi-isometric embedding between
$\left(\bar{\mathcal{K}}_{r},\ell_{1}\right)$ and
$\left(\mathcal{Q}_{{}_{\Delta}}\left(\mathcal{P}_{\Omega}\left(\bar{\mathcal{K}}_{r}\right)\right),\ell_{1}\right)$:

######  Lemma 1.

For a pair $\left(\mathbf{X},\mathbf{Y}\right)\in\bar{\mathcal{K}}_{r}$,
define the following distance:

$D_{{}_{\Delta}}\left(\mathbf{X},\mathbf{Y}\right)=\frac{1}{m^{\prime}}\left\|\mathcal{Q}_{{}_{\Delta}}\left(\mathcal{P}_{\Omega}\left(\mathbf{X}\right)\right)-\mathcal{Q}_{{}_{\Delta}}\left(\mathcal{P}_{\Omega}\left(\mathbf{Y}\right)\right)\right\|_{1}.$
(13)

Then, with a positive constant $\rho$, for all pairs
$\left(\mathbf{X},\mathbf{Y}\right)\in\bar{\mathcal{K}}_{r}$, we have the
following embedding:

$\operatorname{Pr}\left(\sup_{\mathbf{X},\mathbf{Y}\in\bar{\mathcal{K}}_{r}}\left|D_{{}_{\Delta}}\left(\mathbf{X},\mathbf{Y}\right)-\frac{1}{n_{1}n_{2}}\left\|\mathbf{X}-\mathbf{Y}\right\|_{1}\right|\leq\varepsilon\right)\leq
2e^{-\frac{\epsilon^{2}m^{\prime}}{K^{2}\Delta^{2}}},$ (14)

as long as
$m^{\prime}\gtrsim\varepsilon^{-2}r\left(n_{1}+n_{2}\right)\log\left(1+\frac{\|\bar{\mathcal{K}}_{r}\|}{\rho}\right)$.

In the mixed-quantized complex Hankel matrix completion approach, the
observation of the partial matrix $\mathcal{P}_{\Omega}(\mathbf{X})$ is made
through $m^{\prime}=m^{\prime}_{1}+m^{\prime}_{2}$ low-resolution samples,
where $m^{\prime}_{1}$ are one-bit samples and $m^{\prime}_{2}$ are multi-bit
samples. Here, $m^{\prime}$ is significantly smaller than $n_{1}n_{2}$. The
mixed-precision Hankel matrix, denoted by $\mathbf{Q}$, is composed of entries
at indices $(i,j)\in\Omega_{1}$—which are derived from $\Omega^{\prime}_{1}$
of the SLA data—containing one-bit data, and entries at indices
$(i,j)\in\Omega_{2}$—also constructed based on $\Omega^{\prime}_{2}$ of SLA
data—containing multi-bit data. Note that the union of $\Omega_{1}$ and
$\Omega_{2}$ forms the complete set $\Omega$ of observed indices. The mixed-
quantized matrix can be expressed as follows:

$\mathbf{Q}=\mathcal{Q}_{{}_{\Delta_{1}}}\left(\operatorname{Re}\left(\mathcal{P}_{\Omega_{1}}\left(\mathbf{X}\right)\right)\right)+\mathcal{Q}_{{}_{\Delta_{2}}}\left(\operatorname{Re}\left(\mathcal{P}_{\Omega_{2}}\left(\mathbf{X}\right)\right)\right)+\mathrm{j}\left(\mathcal{Q}_{{}_{\Delta_{1}}}\left(\operatorname{Im}\left(\mathcal{P}_{\Omega_{1}}\left(\mathbf{X}\right)\right)\right)+\mathcal{Q}_{{}_{\Delta_{2}}}\left(\operatorname{Im}\left(\mathcal{P}_{\Omega_{2}}\left(\mathbf{X}\right)\right)\right)\right).$
(15)

In this paper, we focus on the complex matrix completion problem. For the real
part, it is straightforward to verify that the distance defined in Lemma 1, in
the mixed quantization scheme, is given by

$\displaystyle
D_{{}_{\Delta_{1},\Delta_{2}}}^{r}\left(\mathbf{X},\mathbf{Y}\right)$
$\displaystyle=\frac{\Delta_{1}}{2m^{\prime}_{1}}\left\|\operatorname{sgn}\left(\operatorname{Re}\left(\mathcal{P}_{\Omega_{1}}\left(\mathbf{X}\right)\right)\right)-\operatorname{sgn}\left(\operatorname{Re}\left(\mathcal{P}_{\Omega_{1}}\left(\mathbf{Y}\right)\right)\right)\right\|_{1}$
(16)
$\displaystyle+\frac{1}{m^{\prime}_{2}}\left\|\mathcal{Q}_{{}_{\Delta_{2}}}\left(\operatorname{Re}\left(\mathcal{P}_{\Omega_{2}}\left(\mathbf{X}\right)\right)\right)-\mathcal{Q}_{{}_{\Delta_{2}}}\left(\operatorname{Re}\left(\mathcal{P}_{\Omega_{2}}\left(\mathbf{Y}\right)\right)\right)\right\|_{1}.$

Similarly, for the imaginary part, we have

$\displaystyle
D_{{}_{\Delta_{1},\Delta_{2}}}^{i}\left(\mathbf{X},\mathbf{Y}\right)$
$\displaystyle=\frac{\Delta_{1}}{2m^{\prime}_{1}}\left\|\operatorname{sgn}\left(\operatorname{Im}\left(\mathcal{P}_{\Omega_{1}}\left(\mathbf{X}\right)\right)\right)-\operatorname{sgn}\left(\operatorname{Im}\left(\mathcal{P}_{\Omega_{1}}\left(\mathbf{Y}\right)\right)\right)\right\|_{1}$
(17)
$\displaystyle+\frac{1}{m^{\prime}_{2}}\left\|\mathcal{Q}_{{}_{\Delta_{2}}}\left(\operatorname{Im}\left(\mathcal{P}_{\Omega_{2}}\left(\mathbf{X}\right)\right)\right)-\mathcal{Q}_{{}_{\Delta_{2}}}\left(\operatorname{Im}\left(\mathcal{P}_{\Omega_{2}}\left(\mathbf{Y}\right)\right)\right)\right\|_{1}.$

Following Lemma 1, we determine an embedding for each real and imaginary part
with associated probabilities. Additionally, the triangle inequality implies
that the error of a complex value can be bounded by the sum of its real and
imaginary parts. Furthermore, it is worth mentioning that the statistical
properties of both real and imaginary values are identical. Consequently,
employing the union bound, the resulting failure probability is doubled
compared to the real part alone. The subsequent theorem provides an upper-
bound for the recovery performance in mixed-precision matrix completion:

######  Theorem 1.

Let $\mathcal{K}_{r}$ denote the set of low-rank matrices defined as

$\mathcal{K}_{r}=\left\\{\mathbf{X}^{\prime}\in\mathbb{C}^{n_{1}\times
n_{2}}\mid\operatorname{rank}(\mathbf{X}^{\prime})\leq
r,\beta\leq\alpha\right\\}\subset\mathbb{C}^{n_{1}\times n_{2}},$ (18)

where
$\beta=\max\left(\|\operatorname{Re}\left(\mathbf{X}^{\prime}\right)\|_{\mathrm{max}},\|\operatorname{Im}\left(\mathbf{X}^{\prime}\right)\|_{\mathrm{max}}\right)$.
Consider a matrix $\mathbf{X}\in\mathcal{K}_{r}$. Assume that
$m^{\prime}=m^{\prime}_{1}+m^{\prime}_{2}$ entries of $\mathbf{X}$, randomly
selected with uniform sampling, undergo mixed-precision scalar quantization
followed by (8) with dither values following
$\mathcal{U}_{[-\frac{\Delta_{1}}{2},\frac{\Delta_{1}}{2}]}+\mathrm{j}\mathcal{U}_{[-\frac{\Delta_{1}}{2},\frac{\Delta_{1}}{2}]}$
for $m^{\prime}_{1}$ one-bit data, and
$\mathcal{U}_{[-\frac{\Delta_{2}}{2},\frac{\Delta_{2}}{2}]}+\mathrm{j}\mathcal{U}_{[-\frac{\Delta_{2}}{2},\frac{\Delta_{2}}{2}]}$
for $m^{\prime}_{2}$ multi-bit data. If the pair
$\left(\mathbf{X},\mathbf{Y}\right)\in\mathcal{K}_{r}$ satisfies the
consistency property in Definition 1 for each real and imaginary part, with
constants $c$, $\left(\varepsilon_{1},\varepsilon_{2}\right)\in(0,1)$, it can
be asserted that the recovery error between $\mathbf{X}$ and $\mathbf{Y}$ is
bounded with a probability of at least
$1-4\max\left(e^{-\frac{\varepsilon^{2}_{1}m^{\prime}_{1}}{\Delta^{2}_{1}}},e^{-\frac{\varepsilon^{2}_{2}m^{\prime}_{2}}{K^{2}\Delta^{2}_{2}}}\right)$
as

$\left\|\mathbf{X}-\mathbf{Y}\right\|_{1}\leq
2n_{1}n_{2}\left(\varepsilon_{1}+\varepsilon_{2}\right),$ (19)

for all $\left(\mathbf{X},\mathbf{Y}\right)\in\mathcal{K}_{r}$ when the
required number of samples must satisfy

$m^{\prime}\gtrsim\min\left(\varepsilon_{1},\varepsilon_{2}\right)^{-2}r\left(n_{1}+n_{2}\right)\log\left(1+\frac{\|\mathcal{K}_{r}\|}{\rho}\right),$
(20)

with $\rho\leq 2n_{1}n_{2}\left(\varepsilon_{1}+\varepsilon_{2}\right)$.

The proof of Theorem 1 is presented in Appendix A.

To recover the matrix $\mathbf{X}$ from the observed mixed-quantized matrix
$\mathbf{Q}$, we employ the nuclear norm minimization problem:

$\displaystyle\underset{\mathbf{X}}{\textrm{minimize}}\quad\left\|\mathbf{X}\right\|_{\star}$
(21) $\displaystyle\text{subject
to}\quad\left\|\mathcal{P}_{\Omega}(\mathbf{X})-\mathbf{Q}\right\|_{\mathrm{F}}\leq
q,$

where the parameter $q$ denotes the impact of the quantization process.

Denote a linear transformation by $\mathcal{M}:\mathbb{C}^{n_{1}\times
n_{2}}\rightarrow\mathbb{C}^{m^{\prime}}$ and its adjoint operator as
$\mathcal{M}^{\star}:\mathbb{C}^{m^{\prime}}\rightarrow\mathbb{C}^{n_{1}\times
n_{2}}$. We employ the singular value thresholding (SVT) algorithm to address
the nuclear norm minimization problem in mixed-precision Hankel matrix
completion. If we consider the singular value decomposition (SVD) of
$\mathbf{X}$ as $\mathbf{X}=\mathbf{U}\bm{\Sigma}\mathbf{V}^{\top}$ and
$\\{\sigma_{i}\\}$ as its singular values, the SVT uses the singular value
shrinkage operator which applies the partial SVD to achieve the low-rank
matrix structure as
$\mathcal{D}_{\tau}(\mathbf{X})=\mathbf{U}\mathcal{D}_{\tau}(\bm{\Sigma})\mathbf{V}^{\top},\leavevmode\nobreak\
\mathcal{D}_{\tau}(\bm{\Sigma})=\operatorname{diag}\left(\left(\sigma_{i}-\tau\right)^{+}\right)$,
where $\tau\geq 0$ is the predefined threshold. The key distinction lies in
our approach: rather than utilizing high-resolution partial measurements at
each iteration, we employ mixed-precision data in the following manner:

$\left\\{\begin{array}[]{l}\mathbf{X}^{(k)}=\mathcal{D}_{\tau}\left(\mathcal{M}^{\star}\left(\bm{y}^{(k-1)}\right)\right),\\\
\bm{y}^{(k)}=\bm{y}^{(k-1)}+\delta_{k}\left(\mathbf{b}-\mathcal{M}\left(\mathbf{X}^{(k)}\right)\right),\end{array}\right.$
(22)

where $\left\\{\delta_{k}\right\\}$ are step sizes and
$\mathbf{b}=\operatorname{vec}\left([Q_{i,j}],\leavevmode\nobreak\
{(i,j)\in\Omega}\right)$.

##  III Numerical Investigation

In this section, we numerically scrutinize the performance of the one-
dimensional sparse array completion that jointly utilizes sparse spectrum and
sparse arrays with mixed-precision measurements for radar collaborative
sensing in automotive applications.

To attain high azimuth angular resolution, we cascade multiple automotive
radar transceivers to synthesize a large sparse array in azimuth. We focus on
the same physical array illustrated in Fig. 1 (c), where there are $M_{t}=3$
transmit antennas, and $M_{r}=4$ receive antennas for each radar arranged
along the horizontal direction at:

$\displaystyle l_{\mathrm{TX_{1}}}=[1,9,25]\lambda/2,$ (23) $\displaystyle
l_{\mathrm{TX_{2}}}=[51,67,75]\lambda/2,$ $\displaystyle
l_{\mathrm{RX_{1}}}=[1,6,7,8]\lambda/2,$ $\displaystyle
l_{\mathrm{RX_{2}}}=[68,69,70,75]\lambda/2.$

Here, $l_{\mathrm{TX_{1}}}=[1,9,25]$ and $l_{\mathrm{RX_{1}}}=[1,6,7,8]$
denote the Tx and Rx array element positions for radar 1, while
$l_{\mathrm{TX_{2}}}=[51,67,75]$ and $l_{\mathrm{RX_{2}}}=[68,69,70,75]$
represent the Tx and Rx array element positions for radar 2. In this
configuration, the first Tx and Rx antennas of the radar share the same
position, resulting in the virtual array size of $4MtMr$. In addition, two
virtual array elements overlap, which makes the final array element number
$4MtMr-1$. Therefore, a virtual array with a total of $47$ elements is
synthesized. The transmit and receive antennas and the virtual array are
plotted in Fig. 1 (c). The azimuth angles are $\theta_{1}=-34^{\circ}$ and
$\theta_{2}=18^{\circ}$. Our simulations set the SNR as $20$ $\mathrm{dB}$.
Initially, the two targets can be first separated in range-Doppler. The
complex peak values in the range-Doppler spectrum, corresponding to each
virtual sparse array, constitute an array snapshot for azimuth angle
determination.

Following Section II-A, we have considered the mixed-precision scenario with
$44$ one-bit ADCs and varying placements of four 10-bit ADCs. Specifically,
three different configurations have been considered for the $10$-bit ADCs: (a)
four multi-bit ADCs are uniformly positioned along the edge of linear arrays,
(b) the four multi-bit ADCs are allocated to the last four antennas in the
sequence, and (c) the initial four antennas in the sequence are equipped with
the multi-bit ADCs. For each configuration, a Hankel matrix
$\mathbf{Q}\in\mathbb{C}^{75\times 75}$ is constructed based on the array
response of a ULA with $150$ elements and sampled by the SLA shown in Fig. 1
(c). In the constructed Hankel matrix, $33$ percent of elements are non-zero
while $1871$ non-zero elements are one-bit and $22$ of non-zero elements are
$10$-bit values. Hence, the mixed-precision rate, defined as the ratio of
multi-bit data to one-bit data, is $0.0118$.

Figure 2: Comparison between the spectrum of two targets with azimuth angles
of $\\{\theta_{1},\theta_{2}\\}=\\{-34^{\circ},18^{\circ}\\}$ under MIMO
sparse array and fully completed array from mixed-quantized Hankel matrix
employing 44 one-bit ADCs, along with varying placements of four 10-bit ADCs,
is explored in three distinct configurations: (a) Four multi-bit ADCs are
uniformly positioned along the edge of linear arrays, with two located at the
beginning and two at the end of the antenna sequence. (b) The four multi-bit
ADCs are allocated to the last four antennas in the sequence. (c) The initial
four antennas in the sequence are equipped with the multi-bit ADCs.

Following Section II-B, the rank-$2$ Hankel matrix $\mathbf{X}$ is completed
via the proposed algorithm in (22). Let $\bar{\mathbf{X}}$ denote the
completed Hankel matrix. Then, the full ULA response can be reconstructed by
taking the average of the anti-diagonal elements of the matrix
$\bar{\mathbf{X}}$. The completed full array has an aperture size of
$76\lambda$.

Fig. 2 depicts the angle spectrum for the two targets in each configuration.
The azimuth angle spectra are derived by applying FFT to the original SLA with
the holes filled with zeros and the full array completed via mixed-quantized
matrix completion, respectively. The FFT of the SLA produces two peaks
corresponding to the correct azimuth directions but with high sidelobes,
making it challenging to detect the two targets accurately in azimuth. In
contrast, the completed full array from mixed-precision samples exhibits two
distinct peaks corresponding to the correct azimuth locations in the angle
spectrum, with significantly suppressed sidelobes. As can be seen in Fig. 2,
the ADCs arrangement does not contribute significantly to the accuracy of the
recovered azimuth angles following the mixed-quantized Hankel matrix
completion approach.

Figure 3: The spectrum of (a) three targets with azimuth angles of
$\\{\theta_{1},\theta_{2},\theta_{3}\\}=\\{-28^{\circ},-24^{\circ},44^{\circ}\\}$,
(b) four targets with azimuth angles of
$\\{\theta_{1},\theta_{2},\theta_{3},\theta_{4}\\}=\\{-20^{\circ},-2^{\circ},35^{\circ},53^{\circ}\\}$,
and five targets with azimuth angles of
$\\{\theta_{1},\theta_{2},\theta_{3},\theta_{4},\theta_{5}\\}=\\{-49^{\circ},-46^{\circ},-40^{\circ},-28^{\circ},-13^{\circ}\\}$,
under MIMO sparse array and fully completed array from mixed-precision data.

In our next experiments, we explore scenarios involving three, four, and five
targets, each exhibiting different azimuth angles, using the same
configuration as considered in Fig. 2 (c). It is important to emphasize that
we consider the same number of one-bit and $10$-bit entries based on this
configuration while trying to recover the rank-$3$,$4$, and $5$ Hankel
matrices $\mathbf{X}$. As can be observed in Figs. 3 (a)-(c), we can
successfully recover the azimuth angles from the mixed-quantized data.

##  IV Discussion

We integrated mixed-precision ADCs into an SLA synthesized by a collaborative
radar sensing paradigm employed in high-resolution automotive imaging radar
applications. By implementing a properly designed uniform dithering scheme, we
achieved precise target detection in azimuth while notably suppressing
spectrum sidelobes. Our empirical findings indicated that the arrangement of
multi-bit and one-bit ADCs has minimal impact on Hankel matrix completion and
azimuth detection performance, yielding nearly identical results. However, a
remaining question for future exploration is the theoretical investigation of
this observed indifference in Hankel matrix completion performance concerning
the mixed ADC arrangement.

##  Appendix A Proof of Theorem 1

Consider the set of real low-rank matrices $\bar{\mathcal{K}}_{r}$ as defined
in (11). Initially, we focus on quantized matrix completion given
$\mathcal{Q}_{{}_{\Delta}}\left(\mathcal{P}_{\Omega}\left(\mathbf{X}\right)\right)$
for any $\mathbf{X}\in\bar{\mathcal{K}}_{r}$. Subsequently, with slight
adjustments, we extend the proof to mixed quantized complex matrix completion.
Consider the following lemma for the scalar quantizer:

######  Lemma 2.

[16, Appendix A] For real values $a,b\in\mathbb{R}$ and
$\tau\sim\mathcal{U}_{[-\frac{\Delta}{2},\frac{\Delta}{2}]}$, we have

$\mathbb{E}_{\tau}\left\\{\left|\mathcal{Q}_{{}_{\Delta}}(a+\tau)-\mathcal{Q}_{{}_{\Delta}}(b+\tau)\right|\right\\}=\left|a-b\right|.$
(24)

According to the aforementioned lemma, for any pair
$\left(\mathbf{X},\mathbf{Y}\right)\in\bar{\mathcal{K}}_{r}$, the following
$\ell_{1}$-distance equation between the qunatized values holds true:

######  Lemma 3.

If we have the uniform sampling in the matrix completion problem, for a pair
$\left(\mathbf{X},\mathbf{Y}\right)\in\bar{\mathcal{K}}_{r}$ and dither values
follow $\mathcal{U}_{[-\frac{\Delta}{2},\frac{\Delta}{2}]}$, the following
relation is obtained:

$\mathbb{E}_{\tau,(i,j)}\left\\{\left\|\mathcal{Q}_{{}_{\Delta}}\left(\mathcal{P}_{\Omega}\left(\mathbf{X}\right)\right)-\mathcal{Q}_{{}_{\Delta}}\left(\mathcal{P}_{\Omega}\left(\mathbf{Y}\right)\right)\right\|_{1}\right\\}=\frac{m^{\prime}}{n_{1}n_{2}}\left\|\mathbf{X}-\mathbf{Y}\right\|_{1}.$
(25)

###### Proof:

The proof of Lemma 3, is straightforward by considering Lemma 2 for the
expectation over the dither values. Since we use the uniform sampling in the
matrix completion, the expectation over the indices $(i,j)\in\Omega$ is
readily given by

$\mathbb{E}_{(i,j)}\left\\{\sum_{(i,j)\in\Omega}\left|X_{i,j}-Y_{i,j}\right|\right\\}=\sum_{(i,j)\in\Omega}\sum_{(i,j)\in[n_{1}]\times[n_{2}]}\frac{1}{n_{1}n_{2}}\left|X_{i,j}-Y_{i,j}\right|=\frac{m^{\prime}}{n_{1}n_{2}}\left\|\mathbf{X}-\mathbf{Y}\right\|_{1}.$
(26)

∎

Herein, we want to obtain the probability of how the $\ell_{1}$-distance of
the quantized values concentrates around its mean by using Hoeffding’s
inequality for the bounded random variable:

######  Lemma 4.

[18, Theroem 2.2.5] Let $\left\\{X_{i}\right\\}^{n}_{i=1}$ be independent,
bounded random variables satisfying $X_{i}\in[a_{i},c_{i}]$, then for any
$t>0$ it holds that

$\operatorname{Pr}\left(\left|\frac{1}{n}\sum_{i=1}^{n}\left(X_{i}-\mathbb{E}\left\\{X_{i}\right\\}\right)\right|\geq
t\right)\leq
2e^{-\frac{2n^{2}t^{2}}{\sum_{i=1}^{n}\left(c_{i}-a_{i}\right)^{2}}}.$ (27)

Also, we have

$\left|\mathcal{Q}_{{}_{\Delta}}(X_{i,j})-\mathcal{Q}_{{}_{\Delta}}(Y_{i,j})\right|\leq
K\Delta,\quad(i,j)\in\Omega.$ (28)

Therefore, we can write the following concentration inequality:

$\operatorname{Pr}\left(\left|\frac{1}{m^{\prime}}\left\|\mathcal{Q}_{{}_{\Delta}}\left(\mathcal{P}_{\Omega}\left(\mathbf{X}\right)\right)-\mathcal{Q}_{{}_{\Delta}}\left(\mathcal{P}_{\Omega}\left(\mathbf{Y}\right)\right)\right\|_{1}-\frac{1}{n_{1}n_{2}}\left\|\mathbf{X}-\mathbf{Y}\right\|_{1}\right|\leq\varepsilon\right)\leq
2e^{-\frac{2\varepsilon^{2}m^{\prime}}{K^{2}\Delta^{2}}}.$ (29)

We take a finite covering of $\bar{\mathcal{K}}_{r}$ by a $\rho$-net
$\mathcal{G}_{\rho}\subset\bar{\mathcal{K}}_{r}$ (for $0<\rho\leq\varepsilon$)
and we extend the concentration of $\ell_{1}$-distance to all vectors of
$\mathcal{G}_{\rho}$ by using the following fact:

$\log\left(|\mathcal{G}_{\rho}|\right)=\mathcal{H}\left(\bar{\mathcal{K}}_{r},\rho\right).$
(30)

For structured sets, the Kolmogorov $\rho$-entropy is upper bounded as [16]

$\mathcal{H}\left(\bar{\mathcal{K}}_{r},\rho\right)\leq\gamma^{2}\left(\bar{\mathcal{K}}_{r}\right)\log\left(1+\frac{\|\bar{\mathcal{K}}_{r}\|}{\rho}\right),$
(31)

where the gaussian complexity is upper-bounded for $\bar{\mathcal{K}}_{r}$ as

$\gamma^{2}\left(\bar{\mathcal{K}}_{r}\right)\leq r\left(n_{1}+n_{2}\right).$
(32)

Consequently, we have

$\operatorname{Pr}\left(\sup_{\mathbf{X},\mathbf{Y}\in\bar{\mathcal{K}}_{r}}\left|\frac{1}{m^{\prime}}\left\|\mathcal{Q}_{{}_{\Delta}}\left(\mathcal{P}_{\Omega}\left(\mathbf{X}\right)\right)-\mathcal{Q}_{{}_{\Delta}}\left(\mathcal{P}_{\Omega}\left(\mathbf{Y}\right)\right)\right\|_{1}-\frac{1}{n_{1}n_{2}}\left\|\mathbf{X}-\mathbf{Y}\right\|_{1}\right|\leq\varepsilon\right)\leq
2\left|\mathcal{G}_{\rho}\right|^{2}e^{-\frac{2\varepsilon^{2}m^{\prime}}{K^{2}\Delta^{2}}},$
(33)

or equivalently,

$\operatorname{Pr}\left(\sup_{\mathbf{X},\mathbf{Y}\in\bar{\mathcal{K}}_{r}}\left|\frac{1}{m^{\prime}}\left\|\mathcal{Q}_{{}_{\Delta}}\left(\mathcal{P}_{\Omega}\left(\mathbf{X}\right)\right)-\mathcal{Q}_{{}_{\Delta}}\left(\mathcal{P}_{\Omega}\left(\mathbf{Y}\right)\right)\right\|_{1}-\frac{1}{n_{1}n_{2}}\left\|\mathbf{X}-\mathbf{Y}\right\|_{1}\right|\leq\varepsilon\right)\leq
2e^{2\mathcal{H}\left(\bar{\mathcal{K}}_{r},\rho\right)-\frac{2\varepsilon^{2}m^{\prime}}{K^{2}\Delta^{2}}},$
(34)

which leads to obtain the required number of samples as follows:

$m^{\prime}\gtrsim\varepsilon^{-2}r\left(n_{1}+n_{2}\right)\log\left(1+\frac{\|\bar{\mathcal{K}}_{r}\|}{\rho}\right).$
(35)

By considering the concentration inequality (34), with probability exceeding
$1-e^{-\frac{\varepsilon^{2}m^{\prime}}{K^{2}\Delta^{2}}}$ we can write

$\frac{1}{n_{1}n_{2}}\left\|\mathbf{X}-\mathbf{Y}\right\|_{1}-\varepsilon\leq\frac{1}{m^{\prime}}\left\|\mathcal{Q}_{{}_{\Delta}}\left(\mathcal{P}_{\Omega}\left(\mathbf{X}\right)\right)-\mathcal{Q}_{{}_{\Delta}}\left(\mathcal{P}_{\Omega}\left(\mathbf{Y}\right)\right)\right\|_{1}\leq\frac{1}{n_{1}n_{2}}\left\|\mathbf{X}-\mathbf{Y}\right\|_{1}+\varepsilon,$
(36)

or equivalently,

$\left\|\mathbf{X}-\mathbf{Y}\right\|_{1}\leq\frac{n_{1}n_{2}}{m^{\prime}}\left\|\mathcal{Q}_{{}_{\Delta}}\left(\mathcal{P}_{\Omega}\left(\mathbf{X}\right)\right)-\mathcal{Q}_{{}_{\Delta}}\left(\mathcal{P}_{\Omega}\left(\mathbf{Y}\right)\right)\right\|_{1}+n_{1}n_{2}\varepsilon.$
(37)

Now, we want to extend the proof for the pair
$(\mathbf{X},\mathbf{Y})\in\mathcal{K}_{r}$ while considering the mixed
quantization scheme. According to Lemma 3, for the distance
$D_{{}_{\Delta_{1},\Delta_{2}}}^{r}$ defined in (16), we can write

$\displaystyle\mathbb{E}_{\tau,(i,j)}\left\\{\left\|\mathcal{Q}_{{}_{\Delta_{1}}}\left(\operatorname{Re}\left(\mathcal{P}_{\Omega_{1}}\left(\mathbf{X}\right)\right)\right)-\mathcal{Q}_{{}_{\Delta_{1}}}\left(\operatorname{Re}\left(\mathcal{P}_{\Omega_{1}}\left(\mathbf{Y}\right)\right)\right)\right\|_{1}\right\\}$
$\displaystyle=\frac{m^{\prime}_{1}}{n_{1}n_{2}}\left\|\operatorname{Re}\left(\mathbf{X}-\mathbf{Y}\right)\right\|_{1},$
(38)
$\displaystyle\mathbb{E}_{\tau,(i,j)}\left\\{\left\|\mathcal{Q}_{{}_{\Delta_{2}}}\left(\operatorname{Re}\left(\mathcal{P}_{\Omega_{2}}\left(\mathbf{X}\right)\right)\right)-\mathcal{Q}_{{}_{\Delta_{2}}}\left(\operatorname{Re}\left(\mathcal{P}_{\Omega_{2}}\left(\mathbf{Y}\right)\right)\right)\right\|_{1}\right\\}$
$\displaystyle=\frac{m^{\prime}_{2}}{n_{1}n_{2}}\left\|\operatorname{Re}\left(\mathbf{X}-\mathbf{Y}\right)\right\|_{1},$

with $\alpha=2\Delta_{1}$. Similarly for $D_{{}_{\Delta_{1},\Delta_{2}}}^{i}$
defined in (17), we can write

$\displaystyle\mathbb{E}_{\tau,(i,j)}\left\\{\left\|\mathcal{Q}_{{}_{\Delta_{1}}}\left(\operatorname{Im}\left(\mathcal{P}_{\Omega_{1}}\left(\mathbf{X}\right)\right)\right)-\mathcal{Q}_{{}_{\Delta_{1}}}\left(\operatorname{Im}\left(\mathcal{P}_{\Omega_{1}}\left(\mathbf{Y}\right)\right)\right)\right\|_{1}\right\\}$
$\displaystyle=\frac{m^{\prime}_{1}}{n_{1}n_{2}}\left\|\operatorname{Im}\left(\mathbf{X}-\mathbf{Y}\right)\right\|_{1},$
(39)
$\displaystyle\mathbb{E}_{\tau,(i,j)}\left\\{\left\|\mathcal{Q}_{{}_{\Delta_{2}}}\left(\operatorname{Im}\left(\mathcal{P}_{\Omega_{2}}\left(\mathbf{X}\right)\right)\right)-\mathcal{Q}_{{}_{\Delta_{2}}}\left(\operatorname{Im}\left(\mathcal{P}_{\Omega_{2}}\left(\mathbf{Y}\right)\right)\right)\right\|_{1}\right\\}$
$\displaystyle=\frac{m^{\prime}_{2}}{n_{1}n_{2}}\left\|\operatorname{Im}\left(\mathbf{X}-\mathbf{Y}\right)\right\|_{1}.$

Consequently, for $D_{{}_{\Delta_{1},\Delta_{2}}}^{r}$, the following
concentration inequality is obtained:

$\operatorname{Pr}\left(\sup_{\mathbf{X},\mathbf{Y}\in\mathcal{K}_{r}}\left|D^{r}_{{}_{\Delta_{1},\Delta_{2}}}\left(\mathbf{X},\mathbf{Y}\right)-\frac{1}{n_{1}n_{2}}\left\|\operatorname{Re}\left(\mathbf{X}-\mathbf{Y}\right)\right\|_{1}\right|\leq(\varepsilon_{1}+\varepsilon_{2})\right)\leq
2\max\left(e^{-\frac{\varepsilon^{2}_{1}m^{\prime}_{1}}{\Delta^{2}_{1}}},e^{-\frac{\varepsilon^{2}_{2}m^{\prime}_{2}}{K^{2}\Delta^{2}_{2}}}\right),$
(40)

where $\varepsilon_{1}$ is associated with the concentration inequality of
one-bit data and $\varepsilon_{2}$ is related to the multi-bit data. Similarly
for $D_{{}_{\Delta_{1},\Delta_{2}}}^{i}$, we have

$\operatorname{Pr}\left(\sup_{\mathbf{X},\mathbf{Y}\in\mathcal{K}_{r}}\left|D^{i}_{{}_{\Delta_{1},\Delta_{2}}}\left(\mathbf{X},\mathbf{Y}\right)-\frac{1}{n_{1}n_{2}}\left\|\operatorname{Im}\left(\mathbf{X}-\mathbf{Y}\right)\right\|_{1}\right|\leq(\varepsilon_{1}+\varepsilon_{2})\right)\leq
2\max\left(e^{-\frac{\varepsilon^{2}_{1}m^{\prime}_{1}}{\Delta^{2}_{1}}},e^{-\frac{\varepsilon^{2}_{2}m^{\prime}_{2}}{K^{2}\Delta^{2}_{2}}}\right).$
(41)

In the consistency scenario, we have

$\displaystyle\mathcal{Q}_{{}_{\Delta_{1}}}\left(\operatorname{Re}\left(\mathcal{P}_{\Omega_{1}}\left(\mathbf{X}\right)\right)\right)$
$\displaystyle=\mathcal{Q}_{{}_{\Delta_{1}}}\left(\operatorname{Re}\left(\mathcal{P}_{\Omega_{1}}\left(\mathbf{Y}\right)\right)\right),$
(42)
$\displaystyle\mathcal{Q}_{{}_{\Delta_{2}}}\left(\operatorname{Re}\left(\mathcal{P}_{\Omega_{2}}\left(\mathbf{X}\right)\right)\right)$
$\displaystyle=\mathcal{Q}_{{}_{\Delta_{2}}}\left(\operatorname{Re}\left(\mathcal{P}_{\Omega_{2}}\left(\mathbf{Y}\right)\right)\right).$

A similar result is true for the imaginary part. Combining the consistency
result with (40) and (41) leads to

$\left\|\operatorname{Re}\left(\mathbf{X}-\mathbf{Y}\right)\right\|_{1}+\left\|\operatorname{Im}\left(\mathbf{X}-\mathbf{Y}\right)\right\|_{1}\leq
2n_{1}n_{2}(\varepsilon_{1}+\varepsilon_{2}),$ (43)

where the triangle inequality implies

$\|\mathbf{X}-\mathbf{Y}\|_{\mathrm{F}}\leq
2n_{1}n_{2}(\varepsilon_{1}+\varepsilon_{2}).$ (44)

Note that the factor of $4$ in the probability arises from the fact that both
real and imaginary parts have the same probability. According to the union
bound, the overall probability is twice as large. Setting $\rho\leq
2n_{1}n_{2}(\varepsilon_{1}+\varepsilon_{2})$ completes the proof.

## References

  * [1] S. Sun, A. P. Petropulu, and H. V. Poor, “MIMO radar for advanced driver-assistance systems and autonomous driving: Advantages and challenges,” _IEEE Signal Processing Magazine_ , vol. 37, no. 4, pp. 98–117, 2020.
  * [2] S. Sun and C. Li, “1 + 1 is greater than 2: Collaborative automotive radar imaging exploiting spatial diversity,” in _2023 20th European Radar Conference (EuRAD)_ , 2023, pp. 323–326.
  * [3] J. A. Nanzer, S. R. Mghabghab, S. M. Ellison, and A. Schlegel, “Distributed phased arrays: Challenges and recent advances,” _IEEE Transactions on Microwave Theory and Techniques_ , vol. 69, no. 11, pp. 4893–4907, 2021.
  * [4] B. K. Chalise, D. M. Wong, M. G. Amin, A. F. Martone, and B. H. Kirk, “Detection, mode selection, and parameter estimation in distributed radar networks: Algorithms and implementation challenges,” _IEEE Aerospace and Electronic Systems Magazine_ , vol. 37, no. 11, pp. 4–22, 2022.
  * [5] G. Zhang, T. Huang, Y. Liu, X. Wang, and Y. C. Eldar, “Direction finding in partly calibrated arrays exploiting the whole array aperture,” _IEEE Transactions on Aerospace and Electronic Systems_ , vol. 59, no. 5, pp. 5978–5992, 2023.
  * [6] C. Kong, A. Mezghani, C. Zhong, A. Swindlehurst, and Z. Zhang, “Multipair massive MIMO relaying systems with one-bit ADC s and DAC s,” _IEEE Transactions on Signal Processing_ , vol. 66, no. 11, pp. 2984–2997, 2018.
  * [7] J. Mo and R. Heath, “High SNR capacity of millimeter wave MIMO systems with one-bit quantization,” in _2014 Information Theory and Applications Workshop (ITA)_. IEEE, 2014, pp. 1–5.
  * [8] N. Liang and W. Zhang, “Mixed-ADC massive MIMO,” _IEEE Journal on Selected Areas in Communications_ , vol. 34, no. 4, pp. 983–997, 2016.
  * [9] X. Zhang, Y. Cheng, H. So, and J. Li, “A novel mixed-ADC architecture for DoA estimation,” _IEEE Signal Processing Letters_ , 2024.
  * [10] C. Mollén, J. Choi, E. Larsson, and R. Heath, “Achievable uplink rates for massive MIMO with coarse quantization,” in _2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2017, pp. 6488–6492.
  * [11] A. Eamaz, F. Yeganegi, Y. Hu, S. Sun, and M. Soltanalian, “Automotive radar sensing with sparse linear arrays using one-bit Hankel matrix completion,” _Accepted to present at the 2024 IEEE Radar Conference (RadarConf24)_ , 2024\.
  * [12] A. Eamaz, K. Mishra, F. Yeganegi, and M. Soltanalian, “UNO: Unlimited sampling meets one-bit quantization,” _IEEE Transactions on Signal Processing_ , vol. 72, pp. 997–1014, 2024.
  * [13] M. Gottinger, M. Hoffmann, M. Christmann, M. Schütz, F. Kirsch, P. Gulden, and M. Vossiek, “Coherent automotive radar networks: The next generation of radar-based imaging and mapping,” _IEEE Journal of Microwaves_ , vol. 1, no. 1, pp. 149–163, 2021.
  * [14] S. Sun and Y. Zhang, “4D automotive radar sensing for autonomous vehicles: A sparsity-oriented approach,” _IEEE Journal of Selected Topics in Signal Processing_ , vol. 15, no. 4, pp. 879–891, 2021.
  * [15] T. Cai and W. Zhou, “A max-norm constrained minimization approach to 1-bit matrix completion.” _J. Mach. Learn. Res._ , vol. 14, no. 1, pp. 3619–3647, 2013.
  * [16] L. Jacques, “Small width, low distortions: quantized random embeddings of low-complexity sets,” _IEEE Transactions on information theory_ , vol. 63, no. 9, pp. 5477–5495, 2017.
  * [17] A. Eamaz, F. Yeganegi, D. Needell, and M. Soltanalian, “Harnessing the power of sample abundance: Theoretical guarantees and algorithms for accelerated one-bit sensing,” _arXiv preprint arXiv:2308.00695_ , 2023.
  * [18] R. Vershynin, _High-dimensional probability: An introduction with applications in data science_. Cambridge university press, 2018, vol. 47.