text:
\begin{document}

\maketitle

\begin{abstract}
 A fully-digital wideband joint communication-radar (JCR) with a multiple-input-multiple-output (MIMO) architecture at the millimeter-wave (mmWave) band will enable high joint communication and radar performance with enhanced design flexibility.  A quantized receiver with few-bit analog-to-digital converters (ADCs) will enable a practical JCR solution with reduced power consumption for futuristic portable devices and autonomous vehicles. In this paper, we present a joint communication-radar proof-of-concept platform, named JCR70, to evaluate and demonstrate the performance of these JCR systems using real channel measurements in the 71-76 GHz band. We develop this platform by extending a mmWave communication set-up with an additional full-duplex radar receiver and by capturing the MIMO JCR channel using a moving antenna on a sliding rail. To characterize the JCR performance of our developed tested, we conduct several indoor and outdoor experiments and apply traditional as well as advanced processing algorithms on the measured data.  Additionally, we compare the performance of our JCR70 platform with the INRAS Radarbook, which is a state-of-the-art automotive radar evaluation platform at 77 GHz. The results demonstrate that a quantized receiver with 2-4 bit ADCs generally performed quite close to the high-resolution ADC for a signal-to-noise ratio of up to 5 dB. Our JCR70 platform with a fully digital JCR waveform at 73 GHz and 2 GHz bandwidth achieved higher resolution capability than the Radarbook due to higher bandwidth and larger synthesized antenna aperture.

\end{abstract}

\IEEEpeerreviewmaketitle

\section{Introduction}

Millimeter-wave (mmWave) communication and radar are key technologies for many demanding applications, such as automated driving~\cite{ChoVaGon:Millimeter-Wave-Vehicular-Communication:16} and smart connected devices~\cite{lien2016soli}. MmWave radars enable high-resolution sensing with a wide field of view, while mmWave communications provide a high data rate.  The combination of these two technologies into a single joint communication-radar (JCR) enables hardware reuse and a common signaling waveform. This leads to significant benefits in cost, power consumption, latency, spectrum efficiency, and market penetrability.

In this paper, we present a fully-digital multiple-input-multiple-output (MIMO) JCR platform, named JCR70, at 71-76 GHz band. Due to the fully-digital MIMO functionality and a software-defined architecture, JCR70 provides enhanced communication and radar performance with increased waveform and beamforming design flexibility. Unfortunately, a naive design of a fully-digital MIMO systems at the mmWave band will result in high cost, hardware complexity, and power consumption due to large bandwidths and high dimensions. To mitigate these issues, we synthesize a fully-digital MIMO testbed by moving an antenna connected with an RF chain and a high-speed, high-resolution analog-to-digital converter (ADC) on a slider. Further improvements can be achieved by using a low quantized receiver, as proposed in our theoretical paper~\cite{KumMazMez:Low-Resolution-Sampling-for-Joint:18}. Therefore, we also perform a proof-of-concept evaluation for a fully-digital JCR with low- and medium- resolution ADCs by emulating quantization effect on the data collected from JCR70 testbed. This will enable a practical fully-digital MIMO JCR solution for futuristic radio systems.

Because of hardware limitations, mmWave JCR prototyping using communication testbeds have been difficult~\cite{AjoSreLoc:On-the-Feasibility-of-Using-IEEE:19}. Recently, \cite{Kim:Experimental-Demonstration-of-MmWave:19} investigated the applicability of the IEEE 802.11ad technology at 60 GHz for communications on a vehicular testbed using the Tensorcom 802.11ad module. In~\cite{AjoSreLoc:On-the-Feasibility-of-Using-IEEE:19}, the feasibility of IEEE 802.11ad-based radar~\cite{KumChoGon:IEEE-802.11ad-based-Radar::17,GroLopVen:Opportunistic-Radar-in-IEEE:18} was performed indoors in the range domain using a Dell laptop with IEEE 802.11ad functionality at 60 GHz. The IEEE 802.11ad testbeds developed in~\cite{Kim:Experimental-Demonstration-of-MmWave:19,AjoSreLoc:On-the-Feasibility-of-Using-IEEE:19}, however, used analog beamforming and was not fully programmable. Additionally, the strong atmospheric absorption at 60 GHz makes it difficult for future outdoor-to-indoor communications, when compared to 71-76  GHz~\cite{RapR.WMur:Millimeter-Wave-Wireless:14}. The existing mmWave testbeds with software-defined radio architecture and fully digital waveform generation/processing have demonstrated gigabits-per-second communication data rate at 71-76 GHz band~\cite{CudKovTho:Experimental-mm-wave-5G-cellular:14,GomSisRib:Will-COTS-RF-Front-Ends:18}. These mmWave communication prototypes, however, used analog processing in the angular domain and were not leveraged for simultaneous radar operations.

To realize a practical fully-digital radio system using low-resolution ADCs with a high sampling rate, there is some theoretical work on MIMO communications~\cite{LiuLuoXio:Low-Resolution-ADCs-for-Wireless:19,MezAntNos:Multiple-parameter-estimation:10,MoSchHea:Channel-Estimation-in-Broadband:18}. In~\cite{MezAntNos:Multiple-parameter-estimation:10}, an iterative channel estimation method using Expectation-Maximization (EM) was proposed for the ultra-high frequency band. To leverage the sparsity in the mmWave MIMO channels, approximate message passing-based channel estimation algorithms were proposed and numerically analyzed in~\cite{MoSchHea:Channel-Estimation-in-Broadband:18}. Prior work, though, did not consider the self-interference effect that occurs in a full-duplex radar operation~\cite{LiuLuoXio:Low-Resolution-ADCs-for-Wireless:19,MezAntNos:Multiple-parameter-estimation:10,MoSchHea:Channel-Estimation-in-Broadband:18}. There is limited work on low-resolution ADCs for a full-duplex radar that transmits and receives simultaneously. In~\cite{ZahNagMod:One-Bit-Compressive-Radar:19}, time-varying thresholds and $\ell_1$-norm minimization was proposed for a single-input-single-output radar with 1-bit ADC. To realize a fully-digital wideband mmWave MIMO JCR,~\cite{KumMazMez:Low-Resolution-Sampling-for-Joint:18,MazMezHea:Low-Resolution-Millimeter-Wave:18} proposed the use of high-speed, low-resolution ADCs receivers. The Cram\'er Rao bounds demonstrated that the radars with 1-bit ADC perform closely to the infinite-bit ADC for a single-target scenario at low signal-to-noise ratio (SNR)~\cite{KumMazMez:Low-Resolution-Sampling-for-Joint:18,MazMezHea:Low-Resolution-Millimeter-Wave:18}. A fully-digital mmWave MIMO JCR proof-of-concept platform with high-speed, low-resolution ADCs, however, is unavailable.

In this paper, we present a low-complexity mmWave proof-of-concept platform for a fully-digital MIMO JCR using high-, medium-, and low-resolution ADCs at 71-76 GHz band and 2 GHz bandwidth. To perform mmWave JCR characterization of the communication-centric JCR testbed, we conduct several experiments in a static indoor and outdoor setting with multiple scatters in the range and angle domain. The main contributions of the proposed research are summarized as follows:
\begin{itemize}
\item We present a fully-digital mmWave MIMO platform for demonstrating and evaluating the performance of a wideband JCR system. We develop this testbed by first extending the National Instruments (NI) mmWave platform for 5G communications~\cite{niTestbed,CudKovTho:Experimental-mm-wave-5G-cellular:14} to a single-input-single-output (SISO) JCR mmWave testbed with a full-duplex radar receiver. Then, we synthesize a single-input-single-output (SIMO) testbed by moving the transmit (TX) antenna element on a slider. To our knowledge, this will be the first software-defined radio prototype for wideband MIMO joint communication and radar with a fully digital waveform generation/processing at the mmWave band.

\item We conduct measurement campaigns to collect mmWave JCR data using trihedral corner reflectors for a single- and a two-target scenarios. We also perform experiments for extended target scenarios using a bike in the indoor setting and using a car in the outdoor setting.

\item To estimate the JCR channel from the collected data using our testbed, we apply traditional FFT-based linear processing as well as advanced Bernoulli Gaussian (BG)-Generalized Approximate Message Passing (GAMP) and Gaussian mixture (GM)-GAMP algorithms with sparsity constraints. Additionally, we use the EM technique to optimize the hyperparameters of the BG or GM prior. The GAMP processing provides an enhanced channel estimate with reduced sidelobes and noise at the cost of higher computational complexity than the traditional FFT-based technique.

\item We compare the radar channel estimates obtained using our fully-digital JCR70 testbed against that from a state-of-the-art automotive radar. We use the Radarbook by INRAS~\cite{inras}, which is a leading automotive radar evaluation platform for rapid prototyping at 77 GHz band with 1 GHz bandwidth. The Radarbook platform uses frequency-modulated continuous-wave (FMCW) waveforms, employs analog pre-processing in the time domain, and supports time-division multiplexing-MIMO with 12-bit ADCs. Our experimental results demonstrate that our JCR70 testbed estimates the radar channel with higher resolution than the Radarbook.

\item For evaluating the performance of low- and medium- resolution ADCs, we collect measurements at the full-duplex radar receiver using 12-bit ADCs and emulate the quantization effect on the collected data, such as 1-bit ADC can be emulated by just keeping the most significant bit. We mitigate the self-interference from the JCR transmitter to the radar receiver by using directive TX antenna as well as by separating the TX and receive (RX) antennas. To our knowledge, this is the first experimental evaluation of a wideband fully-digital mmWave MIMO JCR with high-speed, few-bit resolution ADCs.

\item We analyze the performance of emulated $b$-bit ADC data using the normalized mean square error (NMSE) metric for radar channel estimate in the single-target, two-target, and extended target scenarios. Additionally, we also compare the communication and radar performances of the joint system with normalized mean square error (NMSE) for both 1-bit ADC and with the infinite-resolution ADC for the proposed system model. Our experimental results show that low-resolution ADCs perform close to the high-resolution ADCs. The performance gap reduces with decreasing SNR and increasing channel sparsity.

\end{itemize}

This paper is a significant extension of our submitted conference papers~\cite{KumMazHea:A-MIMO-Joint-Communication-Radar:20,KumMezHea:A-Low-Resolution-ADC-Proof-of-Concept-Development:20}. In addition to the detailed exposition, we have included multiple point target and extended target results, outdoor testing, JCR performance using $b$-bit ADCs, and comparison with traditional radars to demonstrate and evaluate the performance of our testbed.

The rest of this paper is organized as follows. We formulate a system model for our proposed JCR system in Section~\ref{sec:SystemModel}. In Section~\ref{sec:HardwarePlatform}, we describe our developed fully-digital SIMO hardware testbed with 2 GHz bandwidth at 71-76 GHz band. Then, we outline the software platform for our testbed with traditional as well as advanced receive processing algorithms in Section~\ref{sec:SoftwareTestbed}. In Section~\ref{sec:Results}, we describe the experimental and numerical results. Finally, we conclude our work and provide direction for future work in Section~\ref{sec:Conclusion}.

\textbf{Notation:} We use the following notation throughout the paper: The notation $\mathcal{N}(\mu,\sigma^2)$ is used for a complex Gaussian random variable with mean $\mu$ and variance $\sigma^2$. The operator $\vert \vert \cdot \vert \vert$ represent the square norm of a vector. The notation $(\cdot)^\rmT$, $(\cdot)^*$, and $(\cdot)^\rmc$ stand for transpose, Hermitian transpose, and conjugate of a matrix/vector, while $(\cdot)^{-1}$ represent the inverse of a square full-rank matrix. Additionally, $\mathrm{vec}(\cdot)$ vectorizes a matrix to a long vector, while $\circ$ represent Khatri-Rao product of matrices. The $b$-bit quantization function $\mathcal{Q}_b(\cdot)$ is applied component-wise and separately to the real and imaginary parts.

\section{System model} \label{sec:SystemModel}

In this section, we present the system model and assumptions pertaining to the development of radar digital signal processing (DSP) receiver algorithms for our JCR70 testbed with a fully-digital communication-centric waveform, MIMO architecture, and the use of high-speed ADCs. Although we did not incorporate some hardware impairments in the system model, like phase noise, and power amplifier non-linearity, they will be taken into account in our experimental proof-of-concept evaluation. As a first step to investigate the feasibility of a mmWave JCR system with low-resolution ADCs, we assume a static indoor setting. The proof-of-concept evaluation for dynamic scenarios is a subject of future work.

\begin{figure}[!t]
\centering
\includegraphics[clip,width=\columnwidth]{Fig/Figure1}
 \vspace{-0.2cm}
 \caption{ A full-duplex joint communication-radar scenario, where first a source vehicle sends preamble to the communicating target vehicle, while simultaneously receiving its radar echoes in the presence of self-interference. Then, both vehicles start communicating data in a full-duplex mode. The JCR receivers use low-resolution ADCs per receive RF chain to reduce the hardware complexity.}
 \label{fig:systemModel}
 \vspace{-0.7cm}
\end{figure}

We consider a mmWave JCR system, where a full-duplex source transmits the JCR waveform at a carrier wavelength $\lambda$ with a signaling bandwidth $W$ to a destination receiver at a distance $d_\comm$, while simultaneously receiving echoes from the surrounding targets, as shown in Fig.~\ref{fig:systemModel}. The receiver employs an $M$ element uniform linear array (ULA) with a $b$-bit ADC per antenna with an inter-element spacing of $d_0 \leq \frac{\lambda}{2}$. Nonetheless, the proposed technique can be employed to other arrays by using $b$-bit ADC for each antenna. The TX and RX antennas are assumed to be closely separated by a distance of $d_\mI$ m to reduce the self-interference due to the full-duplex operation, while making sure that the observed range and direction of a target is the same. The single directional TX antenna is assumed to have a notch in its antenna pattern aligned with the end-fire direction, allowing further suppression of direct self-interference due to full-duplex operation. The residual direct signal leakage of the TX signal to the RX antennas is incorporated in the system model. The analysis in this paper can be extended to TX antenna array with $b$-bit digital-to-analog converters (DACs), while maintaining a separate radio-frequency chain per antenna.

In a $T$ second coherent processing interval, we consider a single carrier physical layer TX JCR waveform with $\delta$ fraction of preamble symbols and $1-\delta$ fraction of communication data symbols. We assume that the training sequences possess good correlation properties for quantized channel estimation and has a low peak-to-average power ratio~\cite{MoSchHea:Channel-Estimation-in-Broadband:18}. The mmWave WLAN standard~\cite{ieee2012wireless} with Golay complementary sequences or 5G communications~\cite{KabPat:Zadoff-Chu-Spreading-Sequence:20} using Zadoff-Chu (ZC) sequences can realize this JCR frame structure. Similar to~\cite{KumChoGon:IEEE-802.11ad-based-Radar::17}, we exploit the training sequences used in the preamble with good properties for radar sensing. The preamble is denoted by an $N$-element vector $\bt \in \mathbb{C}^{N \times 1}$. The complex baseband JCR signal model at the source transmitter with $\Ts$ symbol period is
\begin{equation} \label{eq:Tsignal}
s_{\txm}(t) = \sum_{n=-\infty}^{\infty} s[n] g_\txm(t-nT_s), \\
\end{equation}
where $s[n]$ is the transmitted symbol with $\e{\vert s[n]} \vert^2 = \Es$, and $g_\txm(t)$ is the unit energy pulse-shaping filter. The transmitted symbol $s[n]$ could either correspond to the data part or the training sequence of the JCR waveform.

We represent the mmWave communication channel using a geometric wideband channel model with $N_\mcom$ sparse clusters. Each $n^\thm$ cluster is further assumed to consist of $P_\mcom[n]$ rays/paths between the source transmitter and the destination receiver. Each $p^\thm$ ray in the $n^\thm$ cluster is characterized by its complex channel power $G_\mcom[n,p]$ that includes path loss and antenna gain, physical angle-of-arrival (AoA) relative to broadside $\phi_\mcom[n,p] \in [ -\pi/2 , \pi/2 ]$, spatial AoA $\theta_\mcom[n,p] = \frac{d_0}{\lambda} \sin(\phi_\mcom[n,p])$, and delay $\tau_\mcom[n,p]$. For the $M$-element RX ULA, the array response vector $\textbf{a}(\theta) \in \mathbb{C}^{M \times 1}$ is
\begin{equation}
\textbf{a}(\theta) = \frac{1}{\sqrt{M}} \left[ 1, e^{-j 2 \pi \theta }, \cdots , e^{-j (M-1) 2 \pi \theta} \right]^\mathrm{T}.
\end{equation}
Denoting $G_\mrx(\theta[n,p])$ as the RX antenna element gain, $u_\mcom$ as the path loss exponent for communication channel, $L_\mcom$ as the communication loss factor that includes various losses like cable loss, impedance mismatch, the channel power $G_\comm[k,p]$ using the free space reference distance path loss model is expressed as~\cite{RapMacSam:Wideband-Millimeter-Wave-Propagation:15}
\begin{equation}
G_\mcom[n,p] =  \frac{{ \lambda^2 G_\mrx(\theta[n,p]) }}{(4 \pi)^2 (d_\mcom[n,p])^{u_\mcom} L_\mcom },
\end{equation}
where $u_\mcom$ is close to 2 for the mmWave line-of-sight (LoS) communications in outdoor urban \cite{RapMacSam:Wideband-Millimeter-Wave-Propagation:15} and rural scenarios~\cite{ MacSunRap:Millimeter-Wave-Wireless:16}. Under this model, the communication channel $\bh_\mcom(\tau) \in \mathbb{C}^{M\times 1}$ corresponding to delay $\tau$ with $g_\mrx(t)$ as the RX pulse shaping function is expressed as
\begin{equation} \label{eq:commCh1}
\bh_\mcom(\tau) =   \sum_{n = 0}^{N_\mcom-1} \sum_{p = 0}^{P_\mcom[n]-1} \sqrt{G_\mcom[n,p]} \mathbf{a}(\theta_\mcom[n,p]) g_\mrx(\tau-\tau_\mcom[n,p]).
\end{equation}

The radar channel is assumed to consist of $N_\mr$ clusters, which includes a small-delay cluster of residual self-interference due to the simultaneous TX and RX operation in radars, along with the other $N_\mr-1$ clusters corresponding to the reflections from multiple surrounding objects. The maximum delay spread of the radar channel is assumed to be $\tau_{\mathrm{max}}$, which is generally much smaller than the communication delay spread. Each of the $p^\thm$ ray in the $n^\thm$ cluster is characterized by its complex channel power $G_\mrad[n,p]$ that includes path loss, antenna gain, and radar cross-section (RCS) $\sigma_{\mathrm{RCS}}[n,p]$, physical angle-of-arrival (AoA) $\phi_\mrad[n,p]$, spatial AoA $\theta_\mrad[n,p] = \frac{d_0}{\lambda} \sin(\phi_\mrad[n,p]) $, and delay $\tau_\mrad[n,p]$.  The radar targets in this paper are either single/multiple point reflectors or extended targets. The two-way radar channel power, $G_\mr[n,p]$, corresponding to the $p^{\mathrm{th}}$ path in the $n^\thm$ LoS cluster with $L_\mrad$ as the radar loss factor, which includes various losses like cable loss, is given as
\begin{equation}
G_\mr[n,p] = \frac {\lambda^2  G_\mrx(\theta[n,p]) \sigma_{\mathrm{RCS}}[n,p]}{64 \pi^3 (d_\mr[n,p])^{u_\mr} L_\mrad} .
\end{equation}
We will experimentally estimate the path loss exponent $u_\mr$ for mmWave radar channels at 73 GHz using our set-up. We choose the cluster with $n = 0$ to represent the residual self-interference effect. Under this model, the radar channel with residual self-interference $\bh_\mI(\tau)$ is given as
\begin{equation} \label{eq:radCh1}
\begin{aligned}
\bh_{\mr}(\tau) &= \sum_{n = 1}^{N_\mr-1} \sum_{p = 0}^{P_\mr[n]-1} \!\! \sqrt{G_\mr[n,p]} \mathbf{a}(\theta_\mr[n,p]) g_\mrx(\tau-\tau_\mr[n,p])  \\&+ \bh_\mI(\tau),
\end{aligned}
\end{equation}
where $\bh_\mI(\tau) \! = \! \sum_{p = 0}^{P_\mr[0]-1} \!\!  \sqrt{G_\mr[0,p]} \mathbf{a}(\theta_\mr[0,p]) g_\mrx(\tau-\tau_\mr[0,p])$.

While the physical radar channel vector in~\eqref{eq:radCh1} is accurate, it is difficult to estimate because of the non-linear dependence on the unknown parameters, such as channel powers, delays, and AoAs. Due to the finite waveform bandwidth, however, the radar channel model can be represented by a discretized channel vector $\bh_\rmd[k]$ by uniformly sampling the delay domain at the Nyquist rate $\Delta\tau = 1/W$ with the aid of Fourier series expansion. Denoting $K = \lceil W \tau_{\mathrm{max}} + 1 \rceil $ range bins of delay resolution  $\Delta\tau = 1/W$, the frequency domain representation $\tilde{\bh}_\mr(f)$ of $\bh_\mr(\tau)$ is given by~\cite{bajwa2010compressed}
\begin{equation} \label{eq:radCh2}
\tilde{\bh}_\mr(f)  =    \sum_{k = 0}^{K-1}  \bh_\rmd[k] e^{j2 \pi \frac{k} {W }f}.
\end{equation}
Similarly, due to the finite RX antenna aperture, the channel at the $k^\thm$ range bin can be represented by a discretized channel vector $\bx_k$ with $M$ discrete angle bins of spatial angle resolution $\Delta\theta = 1/M$, such that they are related using the Fourier series expansion in the angle domain, and is given by
\begin{equation} \label{eq:radCh3}
\bh_\rmd[k] =  \sum_{m=0}^{M-1} x_k[m] \mathbf{a} \left( \frac{m}{M} \right) = \bA_\rmM \bx_k,
\end{equation}
where $\mathbf{a} \left( m/M \right)$ is the $m^\thm$ column of the RX beamforming matrix $\bA_\rmM \in \mathbb{C}^{M \times M}$.

Using the discretized channel matrix $\bX_\mr = [ \bx_0 \text{ } \cdots \text{ } \bx_{K-1}]$ in the range-angle domain, we can express the unquantized RX signal model in a linear equation format. Therefore, the unquantized RX radar signal corresponding to the training sequence vector $\bt $ with $\bD \in \mathbb{C}^{K \times N}$ as the circulant-shift matrix of $\bt$, where the $k^\thm$ row of $\bD$ is obtained by circularly shifting $\bt^\mathrm{T}$ by $k$, and the additive Gaussian noise matrix $\bW_\mr \in \mathbb{C}^{M \times N}$ with zero mean and variance $\sigma_{w}^2$ is given as
\begin{equation} \label{eq:rady}
\bY_{\mr} =     \bA_\rmM \bX_\mr  \bD+ \bW_{\mr} .
\end{equation}
We define the radar SNR as $\e{\vert \vert \bz_\mr \vert \vert^2}/\e{\vert \vert \bw_\mr \vert \vert^2}$ with the unquantized noiseless signal vector $\bz_\mr \triangleq \mathrm{vec}(\bA_\rmM \bX_\mr  \bD)$ and the noise vector $\bw_\mr \triangleq \mathrm{vec}(\bW_\mr)$.

The quantized RX radar signal using $b$-bit ADC is $\bQ_\mr =\mathcal{Q}_b\Big(  \bY_{\mr}\Big)$. The quantization operation introduces distortion due to the granularity of the quantizer and due to the clipping effects. These distortions cause the quantized RX signal to be non-linearly related to the radar channel matrix $\bX_\mr$ unlike the unquantized case.
Denoting the discretized radar channel vector as $\bx_\mr \triangleq \mathrm{vec}(\bX_\mr)$, the quantized complex-baseband RX radar signal vector defined as  $\bq_\mr \triangleq \mathrm{vec}(\bQ_\mr)$ is represented as
\begin{equation}\label{eq:qRad}
\bq_\mr =  \mathcal{Q}_b\Big( (\bD^\rmT \kron \bA_\rmM ) \bx_\mr + \bw_\mr \Big).
\end{equation}
The unquantized RX communication signal $\bq_\mcom$ can similarly be expressed using the discretized communication channel vector $\bx_\mcom$  in the range-angle domain as in~\eqref{eq:qRad}. To estimate the discretized JCR channel matrix in the range-angle domain using the quantized received JCR signal, we can either use the traditional method with a correlation-based time-domain processing algorithm and a fast Fourier transform (FFT)-based angle-domain technique~\cite{KumChoGon:IEEE-802.11ad-based-Radar::17}, or using advanced GAMP algorithms. These algorithms are described in detail in Section~\ref{sec:SoftwareTestbed}.

For radar performance evaluation,  we consider 1-bit to 8-bit ADC and compare it with 12-bit ADC available in our JCR testbed. The main advantage of the low-resolution architecture is that it can be implemented with low power consumption and reduced the overall complexity of the circuit~\cite{SinPonMad:Multi-Gigabit-communication:-the-ADC-bottleneck1:09}.

\section{Experimental JCR70 hardware platform} \label{sec:HardwarePlatform}

\begin{figure}[!h]
\centering
\includegraphics[width=0.55\columnwidth]{Fig/Figure2}

 \caption{The mmWave joint communication and radar set-up with mono-static radar and bi-static communication in a SISO configuration. The radar and communication share a common fully-digital waveform to enable hardware/spectrum reuse. The radar is in a full-duplex mode and the interference between the radar TX and RX depends on the distance between them or the isolation provided by the objects between them.}
\label{fig:JCRTestbed}
\end{figure}

\begin{figure*}[!h]
\centering
\includegraphics[width=1.6\columnwidth]{Fig/Figure3}

\caption{The block diagram of a SISO joint communication and radar testbed. The PXIe numbers correspond to parts from NI.}
\label{fig:JCRBlock}
\vspace{-1.0em}
\end{figure*}
This section describes the hardware for our mmWave wideband JCR70 platform. Our JCR70 platform is developed for the use case where a source JCR transmitter sends a signal to a communication RX and uses the echoes from surrounding targets and clutter to derive target range and AoA estimates at the source radar receiver.  First, we developed a full-duplex SISO JCR set-up with one JCR transmitter, one communication receiver, and one radar receiver. Then, we extended this set-up for a SIMO mmWave JCR system by moving TX antenna on a slider using a stepper motor to collect RX signals with multiple TX-RX inter-spacing simultaneously for communication and radar receivers.

\subsection{SISO JCR testbed}
The mmWave JCR testbed in a SISO set-up with 2~GHz bandwidth is shown in Fig.~\ref{fig:JCRTestbed}.  This set-up extends the mmWave communication testbed developed by NI~\cite{niTestbed} for JCR functionality in a full-duplex configuration. We developed the JCR testbed using two NI PXIe-1085 express chassis. One of the chassis acts as the source JCR that consists of a communication transmitter and radar receiver and the other chassis acts as the destination receiver for the communication receiver. Each chassis houses NI PXIe-8135 controller, NI PXIe 7902 FPGA for baseband TX/RX processing, NI PXIe 3610 DAC module, NI PXIe 3630 ADC module, and NI PXIe 3630 for intermediate frequency (IF) up-/down-conversion. The IF-local oscillator (LO) module is connected to mmWave TX/RX head(s) for up-/down-conversion to 71-76 GHz band and then these mmWave heads are connected to the horn antennas for over-the-air JCR transmission. The two chassis can be synchronized using Rubidium clock. The block diagram of this set-up is shown in Fig.~\ref{fig:JCRBlock} and the specifications for these modules are given in Table~\ref{tab:HardSpecs}.

\begin{table*}[h]
\caption{Hardware specifications for our JCR70 testbed}
\centering

        \begin{tabular}{l|ll}
        \hline
        \textbf{Hardware} & \textbf{Description/Specifications} & \textbf{Value(s)} \\\hline
            \hline
            NI PXIe-7902 Baseband & FPGA  & Virtex-7 485T \\\hline
           \multirow{3}{*}{NI PXIe 3610 DAC} & \multicolumn{1}{l}{Resolution} & \multicolumn{1}{l}{14 bit} \\\cline{2-3}
                                 & \multicolumn{1}{l}{Sampling rate} & \multicolumn{1}{l}{3.072GS/s} \\\cline{2-3}
                                 & \multicolumn{1}{l}{Bandwidth} & \multicolumn{1}{l}{2 GHz} \\\hline
             NI PXIe 3620 IF-LO & IF tuning range & 8.5 -13.5 GHz \\\hline
           \multirow{3}{*}{NI PXIe 3630 ADC} & \multicolumn{1}{l}{Resolution} & \multicolumn{1}{l}{12 bit} \\\cline{2-3}
                                 & \multicolumn{1}{l}{Sampling rate} & \multicolumn{1}{l}{3.072GS/s} \\\cline{2-3}
                                 & \multicolumn{1}{l}{Bandwidth} & \multicolumn{1}{l}{2 GHz} \\\hline
            \multirow{2}{*}{NI 3647 RX mmWave head} & \multicolumn{1}{l}{Frequency band} & \multicolumn{1}{l}{71-76 GHz} \\\cline{2-3}
                                 & \multicolumn{1}{l}{Quantities} & \multicolumn{1}{l}{2} \\\hline
              \multirow{2}{*}{NI 3657 TX mmWave head} & \multicolumn{1}{l}{Frequency band} & \multicolumn{1}{l}{71-76 GHz} \\\cline{2-3}
                                 & \multicolumn{1}{l}{Quantities} & \multicolumn{1}{l}{1} \\\hline
              \multirow{2}{*}{Horn antennas} & \multicolumn{1}{l}{Gains} & \multicolumn{1}{l}{23 dBi and 10 dBi} \\\cline{2-3}
                                 & \multicolumn{1}{l}{Beamwidth} & \multicolumn{1}{l}{10 and 50 degrees} \\\hline
\multirow{2}{*}{Trihedral corner reflectors antennas} & \multicolumn{1}{l}{Edge length} & \multicolumn{1}{l}{4.3 and 3.2 inches} \\\cline{2-3}
                                 & \multicolumn{1}{l}{Radar cross-section} & \multicolumn{1}{l}{5 and 8 dBsm at 73 GHz} \\\hline

  \multirow{2}{*}{ Sliding rail} & \multicolumn{1}{l}{Length} & \multicolumn{1}{l}{21 cm} \\\cline{2-3}
                                 & \multicolumn{1}{l}{Synthetic antenna inter-spacing} & \multicolumn{1}{l}{1.69 mm} \\\hline

        \end{tabular}

    \label{tab:HardSpecs}
\end{table*}

This test-bed can be used for both real-time JCR prototype mode and real-time JCR channel sounding mode. To characterize the signal processing performance of the wideband mmWave JCR, we explore the JCR channel sounding mode that is more flexible and easy to use as compared to the real-time JCR prototype. The JCR channel sounding mode transmits repetitions of 2048 length ZC sequences and acquires the raw signals at both radar and communication receivers simultaneously. We then further process this acquired RX signals in MATLAB for JCR performance evaluations as described in Section~\ref{sec:SoftwareTestbed}.

\subsection{SIMO JCR testbed}
 \begin{figure}[!h]
\centering
\includegraphics[width=0.7\columnwidth]{Fig/Figure4}
 \caption{The SIMO JCR70 hardware platform, where the sliding motor is used to synthesize multiple digital RF chains.}
\label{fig:TXSlider}
\end{figure}

We have also synthesized a SIMO testbed by moving TX antenna on a slider using a stepper motor to collect RX signals with multiple TX-RX inter-spacing for communication and radar receivers simultaneously, as shown in Fig.~\ref{fig:TXSlider}. The inter-distance between two TX locations is kept less than or equal to half of the carrier wavelength to avoid any grating lobes.  The number of locations of the TX on the slider dictates the aperture length of the synthetic antenna, its angular resolution,  and the far-field distance. Due to the time-domain channel reciprocity,  the channel obtained using multiple TX locations and a fixed RX location will equivalently represent the channel that could have been obtained using multiple RX antenna locations with a fixed TX antenna location.  Therefore, moving the JCR transmitter to several locations with fixed RX antennas for radar and communication, we equivalently obtain the SIMO channel for both radar and communication simultaneously. In this paper, we use our mmWave SIMO JCR testbed for static joint communication and radar testing. The proof-of-concept development for dynamic scenarios is a subject of future work.

We benchmark our proposed JCR system against a state-of-the-art automotive radar. We use the Radarbook by INRAS~\cite{inras}, which is a leading automotive radar evaluation platform for rapid prototyping at 77 GHz band with 1 GHz bandwidth using an Infineon chipset. We mount the Radarbook on top of the NI radar RX adapter module, as shown in Fig.~\ref{fig:TXSlider}. The Radarbook uses FMCW waveforms, includes 4 TX and 8 RX antennas, and supports time-division multiplexing (TDM)-MIMO with 12-bit ADCs and maximum sampling rate of 80 MSps. It employs analog pre-processing in the time-domain to perform deramping that reduces the ADC sampling requirement. In the Radarbook, the software support is provided for basic functionality to control front end using MATLAB along with the direct access to the raw deramped complex radar outputs. The accessibility of the raw radar data is a main advantage of using the Radarbook versus a commercial radar that only provides access to the final estimated radar parameters~\cite{delphi}.

In the Radarbook, a virtual ULA array with $M = 29$ virtual elements and 1.948~mm element spacing is synthesized by using the TDM-MIMO technique described in~\cite{inras}. After the deramping in the analog domain and the virtual ULA construction in the digital domain, the radar output can be mathematically expressed similar to \eqref{eq:qRad} with $\bD$ as a $K$-point FFT matrix. The RX processing techniques to estimate the radar channel using the raw deramped radar output from the 29 virtual antenna elements is described in Section~\ref{sec:SoftwareTestbed}.

 \section{Experimental JCR70 software platform} \label{sec:SoftwareTestbed}

In this section, we outline the software platform for our JCR70 testbed, as shown in Fig.~\ref{fig:SoftwareTestbed}. First, we will describe the real-time TX signal generation and offline RX emulations for $b$-bit ADCs in presence of the self-interference effects using our JCR70 testbed.  Then, we will describe the real-time and offline RX processing algorithms. The real-time algorithms are implemented in LabVIEW and LabVIEW FPGA, whereas the offline emulations and processing are implemented in MATLAB. Additionally, we will explain the RX processing techniques for the Radarbook.

\begin{figure*}[tbp]
\centering
\includegraphics[width=1.4\columnwidth]{Fig/Figure5}

 \caption{Software block diagram for our testbed.}
\label{fig:SoftwareTestbed}

\end{figure*}

\subsection{Transmit and receive signals}
Zadoff-Chu training sequences of length 2048 are used as the training sequence. Multiple ZC sequences are sent in each scan, several scans are carried out at each TX antenna location, and the transmitter is moved on the slider for multiple steps using a motor to synthesize our SIMO testbed. For each scan at a particular TX location, the transmitter sends several repetitions of the training sequences and then wait to RX echoes for a predefined time interval of around 1~second, which is large enough to avoid any target range ambiguity. The JCR70 testbed employs around 10~seconds wait time before moving to the next TX location.

To emulate the received signal using medium- and low-resolution ADCs on the collected data, JCR70 performs uniform mid-rise quantization. For a scalar complex-valued $y$, the received quantized signal $q = \mathcal{Q}_b(y)$ is defined as
\begin{equation} \label{eq:Quantb} \begin{split}
q = \mathrm{sign}(\mathrm{Re}(x)) \left( \mathrm{min} \left(  \Bigg\lceil \frac{\mathrm{Re}(x) }{\Delta_\mathrm{Re}} \Bigg\rceil , 2^{b-1} \right) - \frac{1}{2} \right)\Delta_\mathrm{Re} + \\ \rmj  \text{ } \mathrm{sign}(\mathrm{Im}(x)) \left( \mathrm{min} \left(  \Bigg\lceil \frac{\mathrm{Im}(x) }{\Delta_\mathrm{Im}} \Bigg\rceil , 2^{b-1} \right) - \frac{1}{2} \right)\Delta_\mathrm{Im} \end{split},
\end{equation}
where $\Delta_\mathrm{Re} = \left( \e{\vert \mathrm{Re}(x)\vert^2}\right)^{0.5} \Delta_b$ and $\Delta_\mathrm{Im} = \left( \e{\vert \mathrm{Im}(x)\vert^2} \right)^{0.5} \Delta_b$. The quantization stepsize $\Delta_b$ is chosen to minimize the quantization distortion mean square error assuming a Gaussian input. The values of $\Delta_b$ can be found in~\cite{Max:Quantizing-for-minimum-distortion:60,Jay:Digital-Coding-of-Waveforms:84}.

\subsection{Receive processing}\label{sec:RXProc}

The raw communication or radar signal $\bq_\mcom$ or $\bq_\mr$ in \eqref{eq:qRad} received from the real-time JCR sounding testbed is used for estimating the JCR channel $\bx_\mcom$ or $\bx_\mr$ in the range-angle domain. We benchmark the radar channel estimate obtained using JCR70 against that from the Radarbook. The radar output obtained from the JCR70 or the Radarbook is processed either by using a traditional FFT-based linear processing or by using an advanced non-linear GAMP technique.

In the traditional FFT-based linear processing, first the range processing is performed for each antenna location and then the FFT is applied in the angle domain for each range bin to estimate the channel in the range-angle domain. In our JCR70 testbed, the range processing involves matched filtering of the raw received data with the known training sequence for each antenna location~\cite{KumChoGon:IEEE-802.11ad-based-Radar::17}. In the Radarbook, the range processing involves applying FFT on the raw deramped output obtained from each virtual MIMO antenna element.

\begin{algorithm}[tbp]
\caption{EM-GM-GAMP algorithm}
\label{GAMP_alg}
\begin{algorithmic}[1]
\STATE \textbf{Input:}  Observation vector $\bq$, $\bB= \bD^\rmT \kron \bA_\rmM $
\STATE \textbf{Initialize:} $\bz^0=\boldsymbol{0}$,
$\bx^0=\boldsymbol{0}$, $\boldsymbol{\psi}^0={\mathrm{
const}}$, $\ell \leftarrow 0$   \\
\REPEAT
\STATE $\ell \! \leftarrow \!  \ell+1$
\STATE Output Step:\\

 $\bz^{\ell} \! \leftarrow \! g_\ell(-\bB\hat{\bx}^{\ell-1}  + \boldsymbol{\psi}^{\ell-1} \circ\bz^{\ell-1},\bq,\boldsymbol{\psi}^{\ell-1})$\\
$\boldsymbol{\xi}^{\ell} \! \leftarrow \! (\bB \circ \bB^*)^{\mathrm{T}}
g'_\ell ( -\bB\hat{\bx}^{\ell-1}  + \boldsymbol{\psi}^{\ell-1} \circ\bz^{\ell-1}, \bq, \boldsymbol{\psi}^{\ell-1}) $ \\
\STATE Input Step: \\
$\bx^{\ell}  \! \leftarrow \!  f_\ell(\bB^{\mathrm {H}}
\bz^\ell  + \boldsymbol{\xi}^\ell \circ
\hat{\bx}^{\ell-1},  \boldsymbol{\xi}^\ell)$ \\
$\boldsymbol{\psi}^\ell \! \leftarrow \!   (\bB \circ \bB^*)
f'_\ell(\bB^{\mathrm{H}} \bz^\ell  +
\boldsymbol{\xi}^\ell \circ \hat{\bx}^{\ell-1},  \boldsymbol{\xi}^\ell )$
\STATE update the parameters $\boldsymbol{\Omega}$ using EM algorithm
\UNTIL{the cost does not significantly decrease or a maximum iteration count has been reached}

\RETURN $\bx^\ell$.
\end{algorithmic}
\end{algorithm}

We also perform advanced sparse reconstruction using the high-performing non-linear GAMP technique in Algorithm~\ref{GAMP_alg} to estimate the mmWave channel in the range-angle domain from the received signal obtained using our JCR70 testbed or the Radarbook~\cite{Ran:Generalized-approximate-message:11}. To leverage sparsity in the mmWave channels, we assume the channel coefficients $x_i$ of the mmWave channel in the range-angle domain, $\bx_\mr$ or $\bx_\mcom$, are drawn from the BG or GM model with unknown parameters $\boldsymbol{\Omega}$, having marginal probability distribution function
\begin{equation}
p_X(x_i;\boldsymbol{\Omega}) = \eta_0\delta(x_i)+\sum_{i=0}^{V-1} \eta_i \mathcal{N}({x_i; \mu_i,\nu_i}),
\end{equation}
where $\eta_0$ is the probability of $x_i = 0$ thereby enforcing sparsity, $\delta(\cdot)$ is the Dirac distribution, and the unknown parameters $\boldsymbol{\Omega}= [\{ \eta_i, \mu_i,\nu_i\}_{i=0}^{V-1}, V]$ with $\eta_i$ as the weight, $\mu_i$ as the mean and $\nu_i$ as the variance of the Gaussian mixture with $V$ components. By definition, $\sum_{i=0}^{V-1} \eta_i = 1$. For the BG model, $V=1$ and we use $\eta$ to represent $\eta_1$ for simplicity with $\eta_0 = (1-\eta)$.

The GAMP algorithm generally performs channel estimation better than the traditional processing in sparse environments. The GAMP algorithm, however, has more computational complexity than the traditional FFT-based linear processing. Therefore, a two-stage processing with traditional processing on the collected data with the second stage of GAMP processing on a smaller part of the channel estimate to improve the resolution would improve the resolution without increasing the complexity much.

In Algorithm~\ref{GAMP_alg}, we provide the pseudo-code of GAMP algorithm \cite{Ran:Generalized-approximate-message:11} to estimate the channel estimate $\bx_\mr$ or $\bx_\mcom$ with noiseless received signal $\bz_\mr$ or $\bz_\mcom$ based on the quantized observation $\bq_\mr$ or $\bq_\mcom$ in~\eqref{eq:Quantb}. The recursive approach breaks apart the entire estimation problem into smaller scalar estimations described by the input denoising function $f_\ell(v,\xi^\ell)$ and the output (residual) function $g_\ell(-u,q,\psi^{\ell-1})$. The minimum mean square error (MMSE) denoiser function $f_\ell(v,\xi^\ell)$ for estimating $x$ using BG-GAMP is calculated based on the posterior mean obtained from the Bernoulli-Gaussian prior $x \sim BG\left(\eta,0, \nu \right)$ along with the noisy observation $v|x\sim \mathcal{N}\left( \xi^{\ell} x, \xi^\ell \right)$:
\begin{equation}
f_\ell(v,\xi^\ell)=\e{x |v }= \zeta \frac{\nu }{\xi^\ell \nu+1} v,
\end{equation}
where
\begin{equation}
\begin{aligned}
\zeta &=\frac{\eta \mathcal{N}\left(v/\xi^\ell ;0, \nu + 1/\xi^\ell\right)}{(1-\eta) \mathcal{N}\left(v/\xi^\ell;0, 1/\xi^\ell \right)+\eta \mathcal{N}\left(v/\xi^\ell;0,\nu+1/\xi^\ell \right)}.
\end{aligned}
\end{equation}
The output (residual) function (applied element-wise for each real/imaginary component)~\cite{MezNos:Efficient-reconstruction-of-sparse:12} is
\begin{equation}
\begin{aligned}
&  g_\ell(-u,q,\psi^{\ell-1})= - \frac{u}{\psi^{\ell-1}} +\\
& ~ \frac{ {\mathrm{exp}}\left(-\frac{( q^{\mathrm {lo}}-u)^2}{\sigma_w^2+\psi^{\ell-1}}\right) - {\mathrm {exp}}\left(-\frac{( q^{\mathrm{ up}}-u)^2}{\sigma_w^2+\psi^{\ell-1}}\right)  }{2 \sqrt{ \pi (\sigma_w^2 + \psi^{\ell-1}) } \left(\mathrm{erf} \left(\frac{q^{\mathrm {up}}-u}{\sqrt{\sigma_w^2+\psi^{\ell-1}}}\right) \!- \! \mathrm{erf}\left( \frac{q^{\mathrm{low}}-u}{\sqrt{\sigma_w^2+\psi^{\ell-1}}}  \right)   \right) } \! ,
\end{aligned}
\end{equation}
with $q^{\mathrm {lo}}$ and $q^{\mathrm {up}}$ being the lower and upper quantization boundary.
The scalar input functions $f_\ell$ and output function $g_\ell$ are applied element-wise to vectors in the GAMP algorithm. The messages exchanged between the input and output steps consist of the results of the individual scalar estimations as well as the curvature around these optima. This is crucial for faster convergence. The curvature message vectors $\boldsymbol{\xi}^{\ell}$ and $\boldsymbol{\psi}^{\ell}$ are obtained by means of the Wirtinger derivatives $g'_\ell$ and $f'_\ell$ in step (5) and (6) with respect to the first argument. Similar expressions for the GM-GAMP algorithm can be calculated, and is given in~\cite{VilSch:Expectation-Maximization-Gaussian-Mixture-Approximate:13}. Additionally, we can use the EM technique to optimize the hyperparameters of the BG or GM prior and the resulting algorithm is called EM-BG-GAMP and EM-GM-GAMP~\cite{VilSch:Expectation-Maximization-Gaussian-Mixture-Approximate:13}.

 \section{Experimental results} \label{sec:Results}

 In this section, we describe different experiments conducted using our mmWave wideband testbed for the proof-of-concept performance evaluation of joint communication and radar at 73 GHz. The training sequence used in our mmWave JCR testbed is ZC sequence of length $N = 2048$ and the symbol rate is 1.536 GHz.  First, we describe the hardware characterization for JCR that includes RF hardware calibration. Then, we evaluate the performance of our testbed for radar channel estimation using fully digital radar with infinite-bit and 1-bit ADC at RX. We conducted our experiments using corner reflectors for precise characterization and extended targets such as a bike and a car for JCR characterization in a more realistic setting. Finally, we present the results for the wideband joint communication and radar at mmWave band. We also compare our results using the state-of-the-art INRAS Radarbook that uses FMCW waveform at 77 GHz.

To evaluate the JCR channel estimation performance using b-bit ADCs, we use the NMSE metric that we define as
\begin{equation}~\label{eq:NMSE}
\mathrm{NMSE}(\hat{\bx}) \triangleq \e{\frac{\vert \vert \bar{\bx} - \hat{\bx} \vert \vert^2}{\vert \vert \bar{\bx} \vert \vert^2}},
\end{equation}
where $\hat{\bx}$ is the estimated channel using our emulated data for $b$-bit ADCs with $1 \leq b \leq 8$ and $\bar{\bx}$ is the channel estimate collected using our testbed at a higher SNR with 12-bit ADC and traditional FFT processing. The NMSE metric defined in \eqref{eq:NMSE} is not the traditional NMSE with $\bar{\bx}$ denoting as the true value, which is unknown. The NMSE metric in \eqref{eq:NMSE}, however,  provides relative performance evaluation of low- and medium-resolution ADCs as compared to the high-resolution ADCs.

\subsection{Radar: Single-target scenarios}

  \subsubsection{Path loss}

\begin{figure}[htb]
\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{ \includegraphics[clip,width=0.45\columnwidth]{Fig/Figure6a}}

  \centerline{(a) Experimental set-up}\medskip
\end{minipage}

\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{\includegraphics[clip,width=0.69\columnwidth]{Fig/Figure6b}}

  \centerline{(b) Path loss}\medskip
\end{minipage}
\vspace{-2em}
\caption{Radar testing using JCR70 platform and a 4.3 inch trihedral corner reflector mounted on a toy vehicle for path loss measurements at different distances. The received power decrease with the distance (in meters) closely follows a path loss of 2 in the far-field region. }
\label{fig:Pathloss}

\end{figure}
  A screenshot of the measurement campaign for indoor radar sounding using the mmWave communication channel sounder setup is shown in Fig.~\ref{fig:Pathloss}(a). Indoor data collection has been performed using this setup for different distances using corner reflectors mounted on a mobile remote-controlled 1/6 hummer car. The processed measurement results in Fig.~\ref{fig:Pathloss}(b) shows the received power of the 4.3 inch target corner reflector at different distances $d$ in meters.  With increasing distance, the amplitude of the target corner reflector decreases and the slope follows a path loss of 2 at higher distances, where it is considered in the far-field region.

\subsubsection{RF hardware frequency response calibration}
\begin{figure}[htb]
\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{ \includegraphics[clip,width=0.7\columnwidth]{Fig/Figure7a}}

  \centerline{(a) Calibration set-up}\medskip
\end{minipage}

\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{\includegraphics[clip,width=0.65\columnwidth]{Fig/Figure7b}}

  \centerline{(b) Corner reflector response}\medskip
\end{minipage}
\vspace{-2em}
\caption{RF hardware frequency response calibration set-up (left) and corner reflector response after and before RF hardware calibration. The delay bin resolution is the symbol period. To collect RF hardware frequency response for calibration, we used two STA-30-12-F2 attenuators of 30 dB each between the transmitter and receiver to avoid RX damage.}
\label{fig:RFCal}

\end{figure}

 To be able to estimate the radar or communication channel accurately, we first measure the RF hardware channel response using loop-back mode and then we need to remove the impact of the hardware, which is known as calibration. In the loop-back mode, we connect the mmWave head of the transmitter and the receiver back-to-back with a 60 dB attenuator in between to measure the hardware frequency response as shown in Fig.~\ref{fig:RFCal}(a). We assume that the major non-flatness in the frequency response is caused by the TX-RX RF chain up to the mmWave head. This setup, however, assumes negligible non-flatness due to the RF frequency response of the waveguide connectors and passive components, such as horn antennas, themselves have flat response. Then, we equalize the hardware frequency response using frequency-domain MMSE equalizer.

Fig.~\ref{fig:RFCal}(b) shows the estimated channel response of a trihedral corner reflector of 4.3 inch mounted on a toy car before the equalization. This channel response deviates from the ideal narrow thumbtack shape. Therefore, we applied the RF hardware equalization and also changed the base of the corner reflector with narrow wooden flat surface to achieve near ideal thumbtack shape of the channel response. We see that equalized channel response has narrower mainlobe width and smaller sidelobes as compared to the unequalized one.

\subsubsection{Estimated radar channels in the range-azimuth domain}

     \begin{figure}[!h]
\centering
\includegraphics[width=0.65\columnwidth]{Fig/Figure8}
 \caption{Experimental set-up to evaluate the radar performance of our testbed for a single-target scenario using a corner reflector in the indoor lab.}
\label{fig:SceneSingle}
\vspace{-1.2em}
\end{figure}

  \begin{figure}[tbp]
\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{ \includegraphics[clip,width=0.65\columnwidth]{Fig/Figure10a}}

  \centerline{(a) JCR70 traditional processing}\medskip
\end{minipage}

\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{\includegraphics[clip,width=0.65\columnwidth]{Fig/Figure10b}}

  \centerline{(b) Radarbook traditional processing}\medskip
\end{minipage}
\begin{minipage}[b]{0.49\linewidth}
  \centering
  \centerline{ \includegraphics[clip,width=\columnwidth]{Fig/Figure10c}}

  \centerline{(c) JCR70 GAMP}\medskip
\end{minipage}

\begin{minipage}[b]{0.49\linewidth}
  \centering
  \centerline{\includegraphics[clip,width=\columnwidth]{Fig/Figure10d}}

  \centerline{(d) Radarbook GAMP}\medskip
\end{minipage}

\caption{Estimated radar channels for a single-target scenario with traditional and advanced processing using our testbed (left) as well as the Radarbook (right). The channel estimates in (c/d) with EM-BG-GAMP have reduced sidelobes than (a/b) with traditional processing.}
\label{fig:FFT_GAMP_Single}
\vspace{-1.6em}
\end{figure}

\begin{figure}[htb]
\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{ \includegraphics[clip,width=0.65\columnwidth]{Fig/Figure11a}}

  \centerline{(a) NMSE versus ADC resolution}\medskip
\end{minipage}

\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{\includegraphics[clip,width=0.65\columnwidth]{Fig/Figure11b}}

  \centerline{(b) NMSE versus SNR}\medskip
\end{minipage}
 \vspace{-0.7cm}
\caption{Estimated radar NMSE for the single-target scenario versus ADC resolutions at SNR equal to -5 dB (left) and versus SNRs for 12-, 4-, 3-, 2-, as well as 1- bit  ADC resolutions (right). The radar channel estimate with 2-bit ADCs perform closely to the high-resolution ADCs at low and medium SNRs, whereas with 3-bit ADCs perform closely at all considered SNR values.}
\label{fig:Single_NADC}
\vspace{-1.2em}
\end{figure}

To evaluate the radar performance of our JCR testbed in the range-angle domain, we placed a corner reflector of 0.1~m edge length at 3.21~m and 0$^{\circ}$ with respect to our testbed, used a horn antenna with 10 dBi gain, and moved the TX with $M = 86$ steps. Additionally, we used the Radarbook to estimate the radar channel for comparison and placed it above our setup, in the indoor lab as shown in Fig.~\ref{fig:SceneSingle}.  For benchmarking purposes, we measured the target distance and angle using a laser device with mm-level accuracy. We also determined the SIMO antenna pattern of our developed JCR70 testbed using this set-up and compared it with the ideal antenna pattern. The resulting antenna pattern using our testbed was found to be close to the ideal pattern, and the detailed description on the JCR70 antenna pattern can be found in our paper~\cite{KumMazHea:A-MIMO-Joint-Communication-Radar:20}.

Figs.~\ref{fig:FFT_GAMP_Single}(a) and (b) shows the normalized radar channels estimated with traditional processing algorithm using our testbed and the Radarbook, while Figs.~\ref{fig:FFT_GAMP_Single}(c) and (d) show the estimated radar channels using EM-BG-GAMP algorithm with reduced sidelobes and noise. We observe the full-duplex effect in the traditional radar images. We used the tap corresponding to the self-interference effect in the channel estimate obtained at the first antenna location as the zero range reference. We found the tap corresponding to the corner reflector to be at a constant distance with respect to the zero reference for all $M$ steps, while its magnitude varied within 1~dB.

From Fig.~\ref{fig:FFT_GAMP_Single}(a), we see that the corner reflector is at 3.223 m and 1$^{\circ}$, demonstrating high-resolution sensing capability of our testbed. We also observe wall reflection near 4.9 m and several multipath reflections around it. The wall reflection is stronger than the corner reflector because it has a larger radar cross-section and because the corner reflector was placed quite far from the boresight resulting in smaller antenna gain. From Figs.~\ref{fig:FFT_GAMP_Single}(c) and (d), we observe that the resolution of the single-target image using our testbed is much higher than the Radarbook due to higher bandwidth and the larger number of synthesized antennas.

    \begin{figure*}[!]
\begin{minipage}[b]{0.49\linewidth}
  \centering
  \centerline{ \includegraphics[clip,width=0.65\columnwidth]{Fig/Figure13a}}

  \centerline{(a) JCR70 traditional processing}\medskip
\end{minipage}

\begin{minipage}[b]{0.49\linewidth}
  \centering
  \centerline{\includegraphics[clip,width=0.65\columnwidth]{Fig/Figure13b}}

  \centerline{(b) Radarbook traditional processing}\medskip
\end{minipage}
\begin{minipage}[b]{0.49\linewidth}
  \centering
  \centerline{\includegraphics[clip,width=0.65\columnwidth]{Fig/Figure13c}}

  \centerline{(c) JCR70 GAMP processing}\medskip
\end{minipage}
\begin{minipage}[b]{0.49\linewidth}
  \centering
  \centerline{\includegraphics[clip,width=0.65\columnwidth]{Fig/Figure13d}}

  \centerline{(d) Radarbook GAMP processing}\medskip
\end{minipage}
\vspace{-0.2cm}
\caption{Estimated radar channels for the two-target scenario using corner reflectors of 4.3 inch and 3.2 inch edge length with traditional and advanced processing using our testbed (left) as well as the Radarbook (right). The channel estimates in (c/d) with EM-BG-GAMP have reduced sidelobes than (a/b) with traditional processing. The channel estimates in (a/c) using our JCR70 testbed have recovered the two-target channel response better than the channel estimates in (b/d) using the Radarbook.}
\label{fig:FFT_GAMP_Two}
\vspace{-1.1em}
\end{figure*}

Figs.~\ref{fig:Single_NADC}(a) and (b) show the estimated radar NMSE variation with different ADC resolutions and SNRs for the single-target experiment using emulations on the data collected from our testbed. Fig.~\ref{fig:Single_NADC}(a) shows the estimated NMSE variation with different ADC resolutions  at -5 dB, using the traditional algorithm, EM-BG-GAMP, and EM-GM-GAMP.  The estimated NMSE decreases marginally till 2 bit ADC. The gap between the estimated NMSEs of any two consecutive ADC resolution is highest between the 1-bit ADC and 2-bit ADC. Fig.~\ref{fig:Single_NADC}(b) depicts the estimated NMSE variation with different SNRs for 12-bit, 4-bit, 3-bit, 2-bit, and 1-bit ADCs using the traditional FFT-based algorithm and EM-GM-GAMP technique. The gap between high-resolution ADCs and low-resolution ADCs increases with SNR. The gap between the estimated NMSEs of traditional processing and sparse GAMP technique decreases with SNR. From Figs.~\ref{fig:Single_NADC}(a) and (b), we see that 2-bit ADCs perform very closely to the high-resolution ADCs at low and medium SNRs, whereas 3-bit ADCs perform very closely at all considered SNR values.

\subsection{Radar: Two-target scenarios}

  \begin{figure}[tbp]
\centering
\includegraphics[width=0.4\columnwidth]{Fig/Figure12}
 \caption{Experimental set-up to evaluate the radar performance of our testbed for a two-target scenario using two corner reflectors in the indoor lab.}
\label{fig:SceneTwo}
\vspace{-1.2em}
\end{figure}

\begin{figure}[tbp]
\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{ \includegraphics[clip,width=0.65\columnwidth]{Fig/Figure14a}}

  \centerline{(a) NMSE versus ADC resolution}\medskip
\end{minipage}

\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{\includegraphics[clip,width=0.65\columnwidth]{Fig/Figure14b}}

  \centerline{(b) NMSE versus SNR}\medskip
\end{minipage}
\vspace{-0.7cm}
\caption{Estimated radar NMSE for the two-target scenario for different ADC resolutions at SNR equal to -5 dB (left) and versus SNRs for 12-, 4-, 3-, 2-, as well as 1- bit  ADC resolutions (right).  The radar channel estimate with 2-bit ADCs perform closely to the high-resolution ADCs at low SNRs, whereas with 3-bit ADCs perform closely at all considered SNR values.}
\label{fig:Two_NADC}

\end{figure}

  \begin{figure*}[tbp]
\begin{minipage}[b]{0.49\linewidth}
  \centering
  \centerline{ \includegraphics[clip,width=0.65\columnwidth]{Fig/Figure16a}}

  \centerline{(a) JCR70 traditional processing}\medskip
\end{minipage}

\begin{minipage}[b]{0.49\linewidth}
  \centering
  \centerline{\includegraphics[clip,width=0.65\columnwidth]{Fig/Figure16b}}

  \centerline{(b) Radarbook traditional processing}\medskip
\end{minipage}
\hspace{0.05cm}\begin{minipage}[b]{0.49\linewidth}
  \centering
  \centerline{ \includegraphics[clip,width=0.65\columnwidth]{Fig/Figure16c}}

  \centerline{(c) JCR70 GAMP processing}\medskip
\end{minipage}

\begin{minipage}[b]{0.49\linewidth}
  \centering
  \centerline{\includegraphics[clip,width=0.65\columnwidth]{Fig/Figure16d}}

  \centerline{(d) Radarbook GAMP processing}\medskip
\end{minipage}
\vspace{-0.2cm}
\caption{Estimated radar channels for the indoor bike scenario with traditional and advanced processing using our testbed (left) as well as the Radarbook (right). The channel estimates in (c/d) with EM-BG-GAMP have reduced sidelobes than (a/b) with traditional processing. The channel estimates in (a/c) using our JCR70 testbed have resolved the bike much better than the channel estimates in (b/d) using the Radarbook.}
\label{fig:ExtBike}
\vspace{-1.0em}
\end{figure*}

We performed a two-target experiment with two corner reflectors of 4.3 inch and 3.2 inch edge length in the indoor lab using our fully-digital SIMO testbed and the Radarbook. The two targets are closely placed in range and angle domain. Figs.~\ref{fig:FFT_GAMP_Two}(a) and (b) shows the estimated radar channels with traditional processing algorithm using our testbed and the Radarbook, while  Figs.~\ref{fig:FFT_GAMP_Two}(c) and (d) show the estimated radar channels using EM-BG-GAMP algorithm with reduced sidelobes and noise. Fig.~\ref{fig:SceneTwo} shows the experimental set-up for the two-target scenario. We use 30 steps to emulate SIMO using our testbed. In Fig.~\ref{fig:FFT_GAMP_Two}(a), we observe two scattering centers corresponding to the two corner reflectors used, unlike the Radarbook due to higher bandwidth and number of synthesized antennas. The GAMP processed image in (c) has recovered the amplitudes of the two corner reflectors better than the one in (d).

Figs.~\ref{fig:Two_NADC}(a) and (b) show the estimated radar NMSE variation with different ADC resolutions and SNRs for the two-target experiment using emulations on the data collected from our testbed. Fig.~\ref{fig:Two_NADC}(a) shows the estimated NMSE variation with different ADC resolutions at -5 dB, using the traditional algorithm, EM-BG-GAMP, and EM-GM-GAMP. The estimated NMSE decreases marginally till 3 bit ADC. The gap between the estimated NMSEs of any two consecutive ADC resolution is highest between the 1-bit ADC and 2-bit ADC. The gap between traditional FFT-based processing and advanced GAMP algorithms is smaller as compared to single-target scenarios. Fig.~\ref{fig:Two_NADC}(b) depicts the estimated NMSE variation with different SNRs for 12-bit, 4-bit, 3-bit, 2-bit, and 1-bit ADCs using traditional FFT-based algorithm and EM-GM-GAMP technique. The gap between high-resolution ADCs and low-resolution ADCs increases with SNR. The gap between the estimated NMSEs of traditional processing and sparse GAMP technique decreases with SNR for all considered ADC resolutions except 1-bit ADC, where it first decreases and then increases with SNR. From Figs.~\ref{fig:Two_NADC}(a) and (b), we see that 2-bit ADCs perform very closely to the high-resolution ADCs at low SNRs, and 3-bit ADCs perform very closely at all considered SNR values.

\subsection{Radar: Extended target scenarios}

\begin{figure}[tbp]
\centering
\includegraphics[width=0.65\columnwidth]{Fig/Figure15}
 \caption{Experimental set-up to evaluate the radar performance of our testbed using a bike in the indoor lab.}
\label{fig:SceneBike}
\vspace{-0.3cm}
\end{figure}

We used a bike in the indoor lab to evaluate the radar performance in the extended target scenario using our fully-digital SIMO testbed and the Radarbook. Figs.~\ref{fig:ExtBike}(a) and (b) shows the estimated radar channels with traditional processing algorithm using our testbed and the Radarbook, while  Figs.~\ref{fig:ExtBike}(c) and (d) show the estimated radar channels using EM-BG-GAMP algorithm with reduced sidelobes and noise. Fig.~\ref{fig:SceneBike} shows the experimental set-up for the extended target scenario using a bike.  We use 50 steps to emulate SIMO using our testbed. In Fig.~\ref{fig:ExtBike}(c), we observe that multiple scattering centers corresponding to different parts of the bike. Due to higher bandwidth and number of synthesized antennas, the resolution of the bike image using our testbed is much higher than the INRAS Radarbook.

\begin{figure}[tbp]
\vspace{-0.2cm}
\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{ \includegraphics[clip,width=0.65\columnwidth]{Fig/Figure17a}}

  \centerline{(a) NMSE versus ADC resolution}\medskip
\end{minipage}
\hspace{0.05cm}

\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{\includegraphics[clip,width=0.65\columnwidth]{Fig/Figure17b}}

  \centerline{(b) NMSE versus SNR}\medskip
\end{minipage}
 \vspace{-0.7cm}
\caption{Estimated radar NMSE for the bike experiment for different ADC resolutions at SNR equal to -5 dB (left) and versus SNRs for 12-, 4-, 3-, 2-, as well as 1- bit  ADC resolutions (right). The radar channel estimate with 2-bit ADCs perform closely to the high-resolution ADCs at low SNR, whereas with 3-bit ADCs perform closely for the medium SNR, and with 4-bit ADCs perform closely at all considered SNR values.}
\label{fig:Ext_Bike_NADC}
\vspace{-1.0em}
\end{figure}

Figs.~\ref{fig:Ext_Bike_NADC}(a) and (b) show the estimated radar NMSE variation with different ADC resolutions and SNRs for the bike experiment using emulations on the data collected from our testbed. Fig.~\ref{fig:Ext_Bike_NADC}(a) shows the estimated NMSE variation with different ADC resolutions at -5 dB, using the traditional algorithm, EM-BG-GAMP, and EM-GM-GAMP.  The estimated NMSE decreases marginally till 4 bit ADC. The gap between the estimated NMSEs of any two consecutive ADC resolution is highest between the 1-bit ADC and 2-bit ADC. The gap between traditional processing and advanced GAMP algorithms is smaller as compared to single- and two-target scenarios. Fig.~\ref{fig:Ext_Bike_NADC}(b) depicts the estimated NMSE variation with different SNRs for 12-bit, 4-bit, 3-bit, 2-bit, and 1-bit ADCs using traditional FFT-based algorithm and EM-GM-GAMP technique. The gap between high-resolution ADCs and low-resolution ADCs increases with SNR. The gap between the estimated NMSEs of traditional processing and sparse GAMP technique for high-resolution ADCs decreases with SNR, whereas it first decreases and then increases with SNR for the low-resolution ADCs. From Figs.~\ref{fig:Ext_Bike_NADC}(a) and (b), we see that 2-bit ADCs perform very closely to the high-resolution ADCs at low SNR, 3-bit ADCs perform very closely for the medium SNR, and 4-bit ADCs perform very closely at all considered SNR values.

  \begin{figure}[tbp]
\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{ \includegraphics[clip,width=0.65\columnwidth]{Fig/Figure18a}}

\end{minipage}

 \vspace{-0.7cm}
\caption{Outdoor JCR measurement scenario using a car on the top of a parking garage. The cart with the JCR70 transmitter and radar receiver is kept in front of the car, whereas the cart with the JCR70 communication receiver is kept behind the car. }
\label{fig:SceneOutdoorJCR}

\end{figure}

 \begin{figure}[tbp]
\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{ \includegraphics[clip,width=0.6\columnwidth]{Fig/Figure19a}}

  \centerline{(a) Traditional processing}\medskip
\end{minipage}

\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{\includegraphics[clip,width=0.6\columnwidth]{Fig/Figure19b}}

  \centerline{(b) GAMP processing}\medskip
\end{minipage}
 \vspace{-0.7cm}
\caption{Estimated radar channels for the outdoor car scenario with traditional and advanced processing using our testbed (left) and the Radarbook (right). The channel estimates in (a) have reduced sidelobes than (b).  Due to the high-resolution of our JCR70 testbed, the channel estimate in (b) shows several scattering centers corresponding to the car.}
\label{fig:FFT_GAMP_Car}
\vspace{-1.6em}
\end{figure}

Additionally, we also conduct outdoor joint communication and radar experiments with 50 steps using Subaru Crosstrek as the vehicle target on Speedway parking garage in UT Austin, as shown in Fig.~\ref{fig:SceneOutdoorJCR}. From Fig.~\ref{fig:SceneOutdoorJCR} and Fig.~\ref{fig:FFT_GAMP_Car}(b), we see that the JCR transmitter and the communication receiver were separated by 7.51~m. The vehicle target was placed in between the JCR transmitter and the communication receiver. The directivity of the horn antenna reduced the reflections from the railings and poles around our set-up. Fig.~\ref{fig:FFT_GAMP_Car}(a) shows the estimated radar channel around the car reflections using the traditional FFT-based algorithm with an interpolation of factor 4, while Fig.~\ref{fig:FFT_GAMP_Car}(b) shows the estimated radar channel using the advanced EM-BG-GAMP technique. We see that Fig.~\ref{fig:FFT_GAMP_Car}(b) has reduced sidelobes and noise as compared to Fig.~\ref{fig:FFT_GAMP_Car}(a). The range-spread of the car is much wider in range than that of the bike in Fig.~\ref{fig:ExtBike}. Due to the high-resolution of our testbed, we can also see multiple scattering centers corresponding to the car which makes the car radar image look quite different than the bike.

\subsection{Joint communication-radar}

\begin{figure}[tbp]
\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{ \includegraphics[clip,width=0.65\columnwidth]{Fig/Figure20a}}

  \centerline{(a) Radar}\medskip
\end{minipage}

\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{\includegraphics[clip,width=0.65\columnwidth]{Fig/Figure20b}}

  \centerline{(b) Communication}\medskip
\end{minipage}
\vspace{-0.7cm}
\caption{Estimated radar channel (left) and communication channel (right) using our JCR70 testbed. In the radar channel estimate, the full-duplex effect is observed around 0~m along with reflections from surrounding objects, such as the communication receiver at 1.75~m. In the communication channel, the LoS path between the communication TX and RX is observed at 1.75 m.}
\label{fig:FFT_JCR}
\vspace{-0.7em}
\end{figure}

The performance of our fully-digital SIMO wideband testbed is also evaluated for the simultaneous communication and radar modes at 73 GHz. We conducted JCR experiments with 15 steps in the same indoor lab as in \figref{fig:SceneSingle}.

 \begin{figure}[tbb]
\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{ \includegraphics[clip,width=0.65\columnwidth]{Fig/Figure21a}}

  \centerline{(a) Radar}\medskip
\end{minipage}

\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{\includegraphics[clip,width=0.65\columnwidth]{Fig/Figure21b}}

  \centerline{(b) Communication}\medskip
\end{minipage}
\vspace{-0.7cm}
\caption{Estimated NMSEs for radar (left) and communication (right) with different ADC resolutions at SNR of -15 dB. The JCR estimate with 3-bit ADCs perform closely to the high-resolution ADCs case.}
\label{fig:JCR_NADC_bit}
\vspace{-0.7em}
\end{figure}

Fig.~\ref{fig:FFT_JCR}(a) shows the estimated indoor radar channel in the range-azimuth domain using the traditional processing, while Fig.~\ref{fig:FFT_JCR}(b) shows the estimated communication channel in the outdoor setting. In the estimated radar channel image, we observe the full-duplex effect unlike the estimated communication channel image. The delay-spread in the radar channel is found to be higher than the communication channel because of the long-distance targets. In Fig.~\ref{fig:FFT_JCR}(a), we observe that the direct path corresponding to the communication receiver is at 1.75~m and -8.4 degrees. The wall reflection is more spread as compared to the communication receiver around 4.98~m because of its spatial extent and strong multipath effect due to the large radar cross-section. We also see the reflection of the metallic chassis at 3.613~m. The angular resolution and dynamic range of the JCR image in Fig.~\ref{fig:FFT_JCR} is worse than the earlier traditional FFT-processed radar images, such as in Fig.~\ref{fig:FFT_GAMP_Car}, due to the lower number of antenna steps used in this JCR experiment. A detailed analysis of this JCR setting with high-resolution ADCs and the BG-GAMP processing algorithm can be found in our conference paper~\cite{KumMazHea:A-MIMO-Joint-Communication-Radar:20}.

In Figs.~\ref{fig:JCR_NADC_bit}(a) and (b), we also compare the JCR performance using the NMSE metric for different ADC resolutions at an SNR of -15 dB for the indoor setting. Due to the lower radar SNR resulting from the two-way radar channel as compared to the one-way communication channel, we see that radar NMSE is worse than the communication NMSE. We also see that the gap between the traditional FFT-based method and the sparse GAMP method is lower in radar than the communication. This could be due to the smaller delay spread and sparser channel in the LoS communication channel as compared to the indoor radar channel. From Figs.~\ref{fig:JCR_NADC_bit}(a) and (b), we see that ADC resolution of 3 bit performs very closely to the high-resolution ADC case.

\section{Conclusion and discussion} \label{sec:Conclusion}

In this paper, we developed a low-complexity proof-of-concept platform for a fully-digital joint communication-radar sounding testbed with SIMO functionality and different ADC resolutions at 73 GHz carrier frequency with 2 GHz bandwidth. For the precise radar characterization of our JCR70 measurement platform, we conducted experiments for the single-target, two-target scenarios using trihedral corner reflectors. We also conducted experiments for extended target scenarios using a bike indoors and a car outdoors for demonstrating and evaluating the performance of our testbed. We applied both traditional FFT-based and advanced GAMP processing algorithms for JCR channel estimations.

The results in this paper demonstrate the high-resolution capability with a wide field of view of our low-complexity wideband a fully-digital joint communication-radar testbed. The GAMP-based processing provided enhanced radar and communication channel estimates with reduced sidelobes and noise as compared to the traditional processing.  Our JCR70 platform with a fully-digital JCR waveform achieved higher resolution capability in the range-angle domain than the state-of-the-art automotive radar. A quantized receiver of 2-4 bit ADCs performed very closely to the high-resolution ADCs. The quantized receiver with 1-bit ADC performed closely to high-resolution ADCs at low SNR. The performance gap, however, grows with increasing SNR and decreasing channel sparsity. The normalized mean square errors for radar channel estimates were found higher than the communication channel estimates due to the two-way radar channel with colocated TX-RX instead of the one-way communication channel with widely separated TX-RX.

The insights in this paper can be taken into account for designing a JCR waveform and developing receive processing for radar and communication with improved performance. The next step is to extend the mmWave MIMO JCR proof-of-concept development for a dynamic scenario, to demonstrate its capability for next-general high-performance automotive and drone applications.

\bibliographystyle{IEEEtran}

\bibliography{IEEEabrv,refProp1}

\end{document}


TITLE: On a coupled fixed point
QUESTION [2 upvotes]: Let $X$ be a Banach space.
An element $(x^*,y^*)X\times X$ is called a coupled fixed point of a mapping
$T : X ×X \rightarrow  X$ if $$T(x^*,y^*) = x^*$$ and $$T(y^*,x^*) = y^*$$.
While studying this type of fixed point, this question has come to mind: why we don't transform this problem into a classical problem of fixed points, by putting $F(x,y)=(T(x,y),T(y,x))$, so we will have $$F(x^*,y^*)=(x^*,y^*)$$
So, why add a whole fixed point theory - coupled fixed point - , whereas it can be considered as a classic fixed point: $F(x,y)=(x,y)$

REPLY [3 votes]: Perhaps there is some nice structure in the study of the coupled fixed points?
I'll use optimization as an example: Sure, we could say that all unconstrained optimization can be written as minimizing a single function $g$. However, often times we can exploit the structure of  $g$ to get nicer methods for solving the problem. For instance, if $g$ is the sum of two functions, then we can activate each summand independently (via methods such as the forward-backward algorithm). This yields a lighter computational load while still solving the problem. In this way, we have used the underlying structure of $g$ which may not have been visible if we just used the standard theory.
EDIT:
Continuing the example, suppose that $g=f+h$ is the sum of two convex lower-semicontinuous functions where $f$ has a Lipschitzian gradient and $h$ might not be differentiable. We also suppose that the proximity operator $\textrm{prox}_h$ can be computed. The proximity operator is a minimization tool that is often similar in cost to a gradient step, but it requires fewer assumptions on $h$. Now, if we want to minimize $g$, then in general $g$ will not be differentiable. Standard methods for just the problem of "minimize $g$" would dictate that we need either $\nabla g$ (which may not exist) or the ability to compute $\textrm{prox}_g$ (which is only rarely computable given only $\nabla f$ and $\textrm{prox}_{h}$).
Nonetheless, thanks to the structure, we can solve this problem using only activations of $\textrm{prox}_h$ and $\nabla f$ using the Forward-backward algorithm. Sweeping some details under the rug,  the Forward-Backward algorithm allows us to solve the minimization task by iterating two steps: at every iteration $n\in\mathbb{N}$,

First, we perform a gradient step $y_n=x_n-\gamma\nabla f(x_n)$ (where $\gamma$ is a parameter).

Second, compute the next iterate by performing the proximal step $x_{n+1}=\textrm{prox}_h y_n$.

In addition to making this problem tractable, notice that we do not need to compute both $\nabla f$ and $\textrm{prox}_h$ at the same time. This allows us to save some storage space.


TITLE: Integral representation of Dilogarithm
QUESTION [3 upvotes]: I have a question regarding how to obtain a certain integral representation of the dilogarithm, namely:
$$\DeclareMathOperator{\li}{Li_2}\li(x) = -\int_{0}^{1}\frac{x\ln t}{1-tx}\mathrm dt\quad ,|x|<1\qquad (a)$$
I know that the dilogarithm is commonly defined as
$$\li(x) = -\int_{0}^{x}\frac{\ln(1-t)}{t}\mathrm dt \overset{t\to xt}{=} -\int_{0}^{1}\frac{\ln(1-xt)}{t}\mathrm dt\qquad(b)$$
But how do we go from $(b)$ to $(a)$? My main issue is reconciling the limits of integration after substitution. For example,
$$\li(x) = -\int_{0}^{x}\frac{\ln(1-t)}{t}\mathrm dt \stackrel{t\to 1-p}{=} -\int_{1}^{1-x}\frac{\ln p}{1-p}\mathrm dp\stackrel{p\to xy}{=}-\int_{\frac{1}{x}}^{\frac{1}{x}-1}\frac{x\ln y}{1-xy}\mathrm dy\qquad (c)$$
I checked $(c)$ and discovered that the integral is undefined...I can't figure out why that is the case. The integrands in $(a)$ and $(c)$ are the same, but the limits differ. Are the limits I calculated from my substitutions incorrect? Also, why are the limits in $(a)$ correct?

REPLY [1 votes]: But how do we go from (b) to (a)? - just integrate by parts © $$(b)=-\ln t\ln(1-xt)\Big|_{t=0}^{t=1}-\int_0^1\frac{x\ln t}{1-xt}\,dt=(a).$$


TITLE: The Large Hadron Collider produce material residues?
QUESTION [1 upvotes]: In the LHC, particles are accelerated until they collide, releasing their energy in the form of many other particles.
My question is this:  What happens to all those new particles and to the old particles that aren't destroyed in collision?
Do they dissolve in the nether, are they absorbed by the container, or are they simply so insignificant that they are left inside?

REPLY [2 votes]: The collisions produce a shower of decay chain particles that impinge on the calorimeters (sensors that detect particle energy) surrounding the experiments. Particles in the beam that are not scattered every which-way by collisions can be diverted to a beam dump. Beam dumps are large targets, several meters long, that are designed to absorb the astonishing kinetic energy of the beam.
I recall during the CMS (Compact Muon Solenoid, one of the LHC experiments) talk broadcast today that the CMS calorimeter pixels have been noticeably degrading over time due to the high energy particles hitting them. This is probably because the incident particles are of sufficient energy to introduce crystallographic defects into the lead tungstate crystals used in the calorimeters.


TITLE: A characterization of pure sheaf
QUESTION [5 upvotes]: I have started reading the book "The Geometry of moduli of sheaves" by Huybrechts and Lehn.
This is a statement in this book at page no.3 the last line.
"$E$ is pure if and only if all associated points of $E$ have the same dimension."
Definition for associated points of a sheaf is as follows:
$Ass(E) = \{x \in X \vert m_x \in AssE_x \}$.
Point has always dimension 0 ,So my question is, what does this mean by saying that associated points have same dimension?
Does this mean that local rings $O_x$ has the same dimension for all $x$ associated point of $E$ ?
Can anyone suggest me a good reference which supports me while studying this book?

REPLY [4 votes]: I found a P.hd. thesis by Alain Leytem entitled with "Torsion and purity on non-integral schemes and singular sheaves in the fine Simpson moduli spaces of one-dimensional sheaves on the projective plane ".The above asked question is answered there and give a good exposition to the relation between torsion sheave and purity of coherent sheaves over Noetherian scheme. Link is given below:
http://orbilu.uni.lu/handle/10993/23380


TITLE: Probability density during equilibration
QUESTION [0 upvotes]: An ensemble of states $x$ are initially drawn from a non-equilibrium probability density $P_0(x)$.
Over time they will evolve towards the equilibrium distribution $P_\mathrm{eq}(x)$.
What I want to know is, at least approximately, the probability distribution as a function of time $t$, i.e. $P(x,t)$.
Intuitively, I would expect the decay from $P_0$ to $P_\mathrm{eq}$ to be well-described as exponential
$$ P(x,t) = \exp(-t/\tau)P_0(x) + [1-\exp(-t/\tau)]P_{\mathrm{eq}}(x) $$
This would be consistent with my experience with autocorrelation and diffusion equations.
But is it generally correct? And is there a theoretical basis for it? I'm particularly in search of something citable that addresses this question.

REPLY [2 votes]: First of all, (equilibrium) thermodynamics does not tell us anything about the dynamical behavior of the system.
One could say that this question is "outside the scope" of the framework of thermodynamics.
So while it is quite universally true that any system will approach an equilibrium state when it is in contact with a thermal reservoir, the way that the equilibrium state is approached is non-universal: it depends on the concrete system we are talking about.
In your question, you do not mention any concrete system, or what "x" actually is.
I am going to show you one example of a system, where your expectation is wrong, and another example, where your expectation is correct.

Consider the overdamped diffusion of a particle in a 1D Harmonic potential
$$ V(x) = kx^2 . $$
The equation of motion for its probability density $p_t(x)$ at time $t$ is then given by the Smoluchowski equation
$$ \partial_t p_t(x) = \frac{1}{m\gamma} \left( \partial_x kx + k_BT \partial_x^2 \right) p_t(x) , $$
where $m$ is the particle mass, $\gamma$ the friction constant, $k_B$ Boltzmann's constant and $T$ the temperature.
(See for example [H. Risken, The Fokker-Planck Equation, Springer 1996].)
Let us assume that the probability distribution is initially not centered, e.g.
$$ p_0(x) = \frac{1}{\sqrt \pi} \mathrm e^{-(x-10)^2} . $$
We don't even need to solve the equations of motion to see that the correct solution is obviously not $p_t = \mathrm e^{-t/\tau} p_0 + (1 - \mathrm e^{-t/\tau}) p_{\text{eq}}$.
As that would mean that the particles "teleport" from their initial position around $x=10$ to their final position around $x=0$.
The correct solution will instead have the particles diffuse slowly from the initial to the final position, which means that in an intermediate position, the probability density will first go up and then down again.
Consider the thermalization of a qubit with Hamiltonian
$$ H = \frac{\hbar\omega}{2} \sigma_z $$
(where $\sigma_z$ is the Pauli matrix).
The state of the qubit is given by a density matrix $\rho$. (I am taking the freedom here to let your index "x" describe a discrete set of state variables.)
Under a standard weak coupling master equation, the components
$$ \rho(t) = \frac 1 2 \begin{pmatrix} 1 + p_z(t) & p_x(t) - \mathrm i p_y(t) \\ p_x(t) + \mathrm i p_y(t) & 1 - p_z(t) \end{pmatrix} $$
have the following time evolution (assuming $\omega$ is small):
\begin{align}
p_x(t) &= \mathrm e^{-\Gamma_2 t} p_x(0) \\
p_y(t) &= \mathrm e^{-\Gamma_2 t} p_y(0) \\
p_z(t) &= \mathrm e^{-\Gamma_1 t} p_z(0) + (1 - \mathrm e^{-\Gamma_1 t}) p_{z,\text{eq}}
\end{align}
where $\Gamma_1$ and $\Gamma_2$ are some rates.
(Details: [H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press 2002].)


TITLE: Why Is The Category Of Sets A Grothendieck Topos?
QUESTION [0 upvotes]: Why Is The Category Of Sets A Grothendieck Topos? How can just object and relation lead to a Grothendieck Topos which {!} 'suggests the possibility of synthesis of algebraic geometry, topology, and arithmetic' [ReS, The vision]?

REPLY [1 votes]: Your question is wholly unclear.
However, based on your comment on your previous posting of this question, it seems you're wholly misunderstanding the comment.
That Set is a sheaf category is supposed to be obvious. This fact alone doesn't "suggest the possibility...".
Instead, what the comment is presumably referring to is the whole collection of facts:

Set is a topos
Every (sober) topological space can be identified with a topos
Every scheme can be identified with a topos with extra structure (a locally ringed topos)

So the point is, by studying toposes, you are learning about all of these subjects simultaneously. Conversely, each of these different subjects has practices that can be adapted to the study of toposes.
(disclaimer: this is based on my knowledge of the subjects, not on my knowledge of Grothendieck's writings specifically)


TITLE: Homeomorphism of Quotient Space
QUESTION [5 upvotes]: I am relatively new to topology and thought one should be able to prove the following:
Let $X = A \cup B$ be a topological space and $A,B \subset X$ (Maybe we need them to be closed subspaces?). Then $X/A \cong B/(A \cap B)$.
This at least seems pretty intuitively, but i do not really know how to prove that. A hint/solution would be amazing!
I tried something along the lines of:
Since $X = A \cup B$, the identity on X yields a homeomorphism $X \cong A \cup B$. This induces a homeomorphism $X/A \cong (A \cup B)/A$. Now I would like the union to respect quotients, such that $(A \cup B)/A \cong A/A \cup B/(A \cap B)$. Now the left space is a singleton and i kind of get what I want.

REPLY [4 votes]: Let us first go back to the definition. If $A$ is a subset of $X$, let $\sim_A$ be the equivalence relation on $X$ defined by
$$
  x \sim_A y \iff \text{$x =y$ or ($x\in A$  and $y \in A$)}
$$
This is well-defined, even if $A$ is the empty set. Now, the quotient space $X/A$ is by definition $X/{\sim_A}$. Let $0$ denote the the equivalence class $[A]$ of $\sim_A$. Then, as a set,
$$
  X/{\sim_A} = \begin{cases}
  X & \text{if $A$ is empty}\\
  (X - A) \cup \{0\} & \text{if $A$ is nonempty}
\end{cases}
$$
The open subsets of $X/{\sim_A}$ are the subsets of $X-A$ open in $X$ and the subsets $U \cup \{0\}$ such that $U \cup A$ is open in $X$.
Back to your question. Since $X = A \cup B$, one gets $X - A = B - (A \cap B)$. Thus, as sets, $X/A$ and $B/(A \cap B)$ are both equal to $B$ if $A \cap B$ is empty and to $(B - (A \cap B)) \cup \{0\}$ otherwise. They are also isomorphic as topological spaces: the open sets are in both cases the sets $U$ or $U \cup \{0\}$ such that $U \cup (A\cap B)$ is open in $B$.


TITLE: Describing the kernel of this surjection
QUESTION [0 upvotes]: Let $k$ be a field. For a finite dimensional $k$-vector space $V$, write $k[V]$ for the symmetric algebra, which is noncanonically isomorphic to some polynomial ring in finitely many variables.
Take a short exact sequence of $k$-vector spaces $0 \rightarrow U \rightarrow V \rightarrow W \rightarrow 0$. This induces a surjection $f : k[V] \rightarrow k[W]$. There is a map $U \otimes_k k[V] \rightarrow k[V]$ sending $u \otimes f$ to $u \otimes f$. My question is, what is the kernel of this map, and is the image of this map equal to the kernel of $f$?
Note that $U \otimes k[V] \rightarrow V \otimes k[V]$ is an injection. On each graded piece, $V \otimes S_n (V) \rightarrow S_{n+1} (V)$ is not an injection; we must symmetrize. So the kernel of $U \otimes S_n (V) \rightarrow S_{n+1} (V)$ should consist of relations induced by permuting terms.
Take an element of the kernel of $S_{n+1} (V) \rightarrow S_{n+1}(W)$. For now, let it just be a monomial $v_1 \otimes \cdots \otimes v_n$, so that the monomial $f(v_1) \otimes \cdots \otimes f(v_n)$ is $0$ in $W$. We should be able to say that at least one of the entries in this term is in the kernel of $f$, right?

REPLY [1 votes]: Let's pick bases so everything is very concrete in terms of polynomials.  We may pick a basis for $U$ and extend it to a basis for $V$.  Let us write our basis for $U$ as $x_1,\dots,x_n$ and the basis for $V$ as $x_{n+1},\dots,x_{n+m}$ (though no assertion is being made that these bases are finite; this is just convenient notation).  We may then consider $x_{n+1},\dots,x_{n+m}$ as a basis for $W$.  We have $k[V]=k[x_1,\dots,x_{n+m}]$ and $k[W]=k[x_{n+1},\dots,x_{n+m}]$ and $f:k[V]\to k[W]$ is the map sending $x_1,\dots,x_n$ to $0$.  Your map $g:U\otimes k[V]\to k[V]$ is just the multiplication map, when you consider an element of $U$ as a homogeneous linear polynomial in $x_1,\dots,x_{n}$.
In particular, this makes it obvious that the image of $g$ and the kernel of $f$ are both the ideal $(x_1,\dots,x_n)\subseteq k[V]$, so yes, they are equal.
The kernel of $g$ can be described reasonably explicitly.  Note that $U\otimes k[V]$ can naturally be considered as a $k[V]$-module and $g$ is $k[V]$-linear, so $\ker g$ is a $k[V]$-submodule of $U\otimes k[V]$.  As a $k[V]$-module, $\ker g$ is generated by the elements $x_i\otimes x_j-x_j\otimes x_i$ for all $1\leq i<j\leq n$.  Much more broadly, the exact sequence with $f$ and $g$ which you are considering has a natural extension to a free resolution of $k[W]$ as a $k[V]$-module known as the Koszul complex:
$$\dots\to \bigwedge^3 U\otimes k[V]\to \bigwedge^2 U\otimes k[V]\stackrel{h}\to \bigwedge^1 U\otimes k[V]\stackrel{g}\to \bigwedge^0 U\otimes k[V]\stackrel{f}\to k[W]\to 0.$$  Here $\bigwedge^0 U\cong k$ so $\bigwedge^0 U\otimes k[V]$ can be naturally identified with $k[V]$, and $\bigwedge^1 U\cong U$ so $\bigwedge^1 U\otimes k[V]$ can naturally be identified with $U\otimes k[V]$.  The map labelled $h$ then sends $(x_i\wedge x_j)\otimes 1$ to $x_i\otimes x_j-x_j\otimes x_i$ and so its image (the kernel of $g$) is exactly the submodule of $U\otimes k[V]$ generated by these elements, as described above.  More generally, the map $d_r:\bigwedge^{r} U\otimes k[V]\to \bigwedge^{r-1} U \otimes k[V]$ in this sequence is given by $$d_r((x_{i_1}\wedge\dots \wedge x_{i_r})\otimes 1)=\sum_{j=1}^r (-1)^j (x_{i_1}\wedge\dots\wedge \hat{x}_{i_j}\wedge\dots\wedge x_{i_r})\otimes x_{i_j}$$ where the hat indicates $x_{i_j}$ is omitted.  You can find more information on this resolution (and how to prove it is exact) at the link above, or in many textbooks on commutative algebra.


TITLE: Let $\{x_n\}$ be a sequence of positive real numbers such that $\text{lim} \big(\frac{x_{n+1}}{x_n}\big) = L>1$. Then $\{x_n\}$ diverges.
QUESTION [1 upvotes]: Let $\{x_n\}$ be a sequence of positive real numbers  such that $\text{lim} \big(\frac{x_{n+1}}{x_n}\big) = L>1$. Then $\{x_n\}$ diverges.
My attempt:
Since $L> 1$ write $L = 2k +1$ for $k>0$. Choose $\epsilon = k$. Then there exists a natural number $N$ such that $n \geq n$ implies $ L - k < \frac{x_{n+1}}{x_n} < L + k$. Therefore, $\frac{x_{n+1}}{x_n} > 1 + k$, i.e. $x_{n+1} > x_n (1+k)$ for all $n \geq N$.
Now we see that $$x_{N+m} > x_{N+m - 1} (1+k) > x_{N+m - 2} (1+k)^2 > \cdots> x_N (1+k)^m$$ for all positive integer $m$.
By Bernoulli's inequality we have $x_{N+m} > x_N (1+ mk)$ for all positive integer $k$.
Now I want to know how to conclude that $\{x_n\}$ diverges. I know that one way for this is to show $\{x_n\}$ is  not bounded, which is by definition for each $M>0$ there exist a natural number $N$ such that $n \geq N$ implies $|x_n| > M$.
Please help me.

REPLY [1 votes]: $N$ is fixed and so is $k\gt 0$.
You have got $x_{N+m}\gt x_N(1+k)^m\gt x_N(1+mk)$ for all positive integers $m$.
Note that RHS $\to \infty$ as $m\to \infty$ (i.e., RHS can be made arbitrarily large by choosing $m$ large enough).
It follows that $x_{N+m}$ is arbitrarily large and hence the sequence $\{x_n\}$ is not bounded.


TITLE: How to calculate modulo negative number?
QUESTION [0 upvotes]: Why 134 mod -26 is equal 22?
I found an explanation, which says: 134 = (-26 * -6) - 22
However, 134 = (-26*-5)+4, so why 134 mod -26 is not equal 4
if 134=(26*5)+4  => 134 mod 26=4
All i could find, is about a situation, if i take a modulo of negative number (-134 mod 26), but that is not what i want.

REPLY [1 votes]: The first sentence in the post is wrong.
$-136$ in not congruent to $22$ modulo $-26$. It is congruent to $-22$.
In the second sentence you had an error (now fixed). You mean $-26 \times -5$.
$4$ and $-22$ are the same answer modulo $-26$ since they differ by a multiple of $-26$.
Arithmetic modulo $m$ is essentially the same as arithmetic modulo $-m$.
Sometimes you want to require an answer that lies between $0$ and the positive value $|m|-1$, sometimes any equivalent answer will do.
If you are working with or writing a computer program to do modular arithmetic the program will make a choice - usually the representative between $0$ and $|m|-1$.


TITLE: Prove that $n\mid\phi(a^n-1)$ in "Topics in Algebra 2nd Edition" by I. N. Herstein. Any natural solution that uses $\operatorname{Aut}(G)$?
QUESTION [1 upvotes]: I am reading "Topics in Algebra 2nd Edition" by I. N. Herstein.
The following problem is Problem 16 on p.71 in Herstein's book.

Let $\phi(n)$ be the Euler $\phi$-function. If $a>1$ is an integer, prove that $n\mid\phi(a^n-1)$

This problem is for Section 2.8 "Automorphisms".
So, I think the author's solution for this problem uses $\operatorname{Aut}(G)$ for some group $G$.
But the natural solution below doesn't use $\operatorname{Aut}(G)$ for some group $G$.
https://math.stackexchange.com/a/398227/384082
Please tell me a natural solution that uses $\operatorname{Aut}(G)$ for some group $G$.

REPLY [3 votes]: Note that $\phi(a^n - 1)$ measures the number of automorphisms of $\mathbb{Z}/(a^n - 1)\mathbb{Z}.$ There is a subgroup of order $n$ in this group: if $\phi$ is the automorphism sending $1$ to $a,$ then $\phi$ generates a subgroup of order $n.$ The statement follows from Lagrange's Theorem.


TITLE: Generators of a certain ideal
QUESTION [4 upvotes]: In view of Mariano Suárez-Alvarez's answer I see how badly phrased my question was, and decided to rewrite it. The drawback is that some comments of Martin Brandenburg are now incomprehensible, but I thought it would suffice to say here that Martin made some legitimate constructive criticisms to the original wording of the question. By the way, thank you also to Vladimir Dotsenko for his comments.
Let $K$ be a commutative ring, and let $X_1,\dots,X_n$ be indeterminates. Here $n$ is an integer $\ge3$. For $1\le i < j\le n$ put
$$
x_{ij}:=\frac{1}{X_i-X_j}
$$
and let $Y_{ij}$ be an indeterminate. Let $I$ be the kernel of the $K$-algebra morphism
$$
\varepsilon:K[(Y_{ij})]\to K[(x_{ij})],\quad Y_{ij}\mapsto x_{ij}.
$$

Is $I$ finitely generated? If it is, can one give an explicit finite set of generators?

Note that the identity
$$
\frac{1}{a-b}\ \frac{1}{a-c}+\frac{1}{b-a}\ \frac{1}{b-c}+\frac{1}{c-a}\ \frac{1}{c-b}=0.
$$
shows that $I$ is nonzero.
(I put the homological algebra tag because the ultimate goal is to know whether there is a functorial free resolution of $K[(x_{ij})]$, viewed as a $K[(Y_{ij})]$-module, and, if it exists, what can be said about it.)
The question had been posted before on Mathematics Stack Exchange (link).

REPLY [4 votes]: A polynomial $f\in K[\underline Y]$ is in the kernel of your map iff $f$ is zero in the quotient $$\frac{k[X,Y]}{\bigl((X_i-X_j)Y_{i,j}-1:1\leq i<j\leq n\bigr)}.$$In other words, your kernel is the intersection of the ideal in the denominator with the ring $k[Y]$, $$\ker\varepsilon=k[Y]\cap\bigl((X_i-X_j)Y_{i,j}-1:1\leq i<j\leq n\bigr).$$This intersection is generated by the elements of a Groebner basis which only contain $Y$s, assuming you are using a monomial order which eliminates the $X$s; this is explained in the book by Cox, Little and O'Shea, for example.
Doing small examples shows that

$(\star)$ the intersection is generated by all polynomials of the form $$Y_{i,j} Y_{i,k}+ Y_{j,k}Y_{j,i}+Y_{k,i}Y_{k,j}$$ with $i$, $j$ and $k$ distinct. (I am identifying $Y_{i,j}$ with $-Y_{j,i}$ here when $i\neq j$)

Ordering the variables as in $$X_1,X_2,X_3,X_4,Y_{1,2},Y_{1,3},Y_{1,4},Y_{2,3},Y_{2,4},Y_{3,4}$$ for $n=4$ we find the Groebner basis
$$\begin{array}{l}
 Y_{2,3} Y_{2,4}+Y_{3,4} Y_{2,4}-Y_{2,3} Y_{3,4} \\\\
 Y_{1,3} Y_{1,4}+Y_{3,4} Y_{1,4}-Y_{1,3} Y_{3,4} \\\\
 Y_{1,2} Y_{1,4}+Y_{2,4} Y_{1,4}-Y_{1,2} Y_{2,4} \\\\
 Y_{1,2} Y_{1,3}+Y_{2,3} Y_{1,3}-Y_{1,2} Y_{2,3} \\\\
 X_3 Y_{3,4}-X_4 Y_{3,4}-1 \\\\
 X_2 Y_{2,4}-X_4 Y_{2,4}-1 \\\\
 X_2 Y_{2,3}-X_3 Y_{2,3}-1 \\\\
 X_1 Y_{1,4}-X_4 Y_{1,4}-1 \\\\
 X_1 Y_{1,3}-X_3 Y_{1,3}-1 \\\\
 X_1 Y_{1,2}-X_2 Y_{1,2}-1
\end{array}$$
The same pattern is seen for all $n$. It is very easy to see that all these polynomials are in $((X_i-X_j)Y_{i,j}-1:1\leq i<j\leq n)$, and it should not be difficult to show that they are a Groebner basis in general. I expect checking that the above claim $(\star)$ can actually be proved without much pain.


TITLE: A question about a limit involving nested radicals
QUESTION [5 upvotes]: $$\lim\limits_{n\rightarrow \infty}1＋\sqrt{2+\sqrt[3]{3＋…\sqrt[n]{n}}}$$
Any hint will be appreciated

REPLY [0 votes]: The limit exist (since the sequence is increasing and bounded), but it has probably no close form.
By matlab,
function [f]=fxy(x,y)

if x==1
    f=y^(1/y);
else
    f=(y+fxy(x-1,y+1))^(1/y);
end

and
format long

for n=1:1:20
    f(n)=fxy(n,1);
end

plot(f,'o')

give the graph :

and we have
$$\lim_{n\to\infty}f(n,1)\approx2.911639216245824$$
read more


TITLE: maximum of multivariable convex function
QUESTION [0 upvotes]: There is a statement that says that every real convex function of one variable on compact convex set attains maximum on its boundary. Is there analogous result for real convex function of more then one variable?

REPLY [0 votes]: Suppose $f : U \to \Bbb R, U \subseteq \Bbb R^n$ is convex, and $U$ is compact and convex.
Let $p$ be any interior point of $U$ and $\ell$ be any line passing through $p$. Because $U$ is compact and convex, $\ell \cap U$ is a line segment, a compact convex set, and the endpoints are on the boundary of $U$. Further, the restriction $f|_\ell$ of $f$ to $\ell$ is convex, so by the result you've quoted, the maximum of $f|_\ell$ is at one of the endpoints $b$. Therefore $f(b) \ge f(p)$.
That is, for every interior point $p$ of $U$, there is at least one boundary point $b$ with $f(b) \ge f(p)$. Since $U$ is compact, $f$ must have a maximum, which by this result must be taken on by a point on the boundary.


TITLE: Bundle over $\mathbb{C}^{n}\setminus{0}$
QUESTION [5 upvotes]: Let $\pi:\mathbb{C}^{n}\setminus{0}\rightarrow\mathbb{CP}^{n-1}, n\geq 3$ be the projection from affine space without the origin to the projective space. If we pull back the tangent bundle of $\mathbb{CP}^{n-1}$ we would get a nontrivial bundle over $\mathbb{C}^{n}\setminus{0}$. Now my question would be: what is $H^{1}(\mathbb{C}^{n}\setminus{0},\pi^{*}\mathcal{T}_{\mathbb{CP}^{n-1}})$?

REPLY [9 votes]: $\newcommand{tot}{\mathbb{C}^n\setminus 0}\newcommand{tan}{\mathcal{T}_{\mathbb{P}^{n-1}}}$
Since morphism $\pi$ is affine, for any quasicoherent sheaf $\mathcal{F}$ on $\mathbb{C}^n\setminus 0$ its higher direct images $R^{>0}\pi_*\mathcal{F}$ vanish, so from Leray spectral sequence we get $H^i(\tot,
\mathcal{F})=H^i(\mathbb{P}^{n-1},\pi_*\mathcal{F})$. Applying this to $\pi^*\mathcal{T}_{\mathbb{P}^{n-1}}$ we get $$H^1(\tot,\pi^*\tan)=H^1(\mathbb{P}^{n-1},\pi_*\pi^*\tan)=H^1(\mathbb{P}^{n-1},\tan\otimes\pi_*\mathcal{O}_{\tot})$$ by projection formula.
Note that $\pi$ is the projection from the total space of $
\mathcal{O}_{\mathbb{P}^{n-1}}(-1)$ minus zero section.(the following is corrected thanks to Jason Starr). The direct image of the structure sheaf of the total space itself is $Sym(\mathcal{O}(-1))=\bigoplus\limits_{n\geq 0}S^n\mathcal{O}(-1)=\bigoplus\limits_{n\geq 0}\mathcal{O}(-n)$. So, if we throw away the zero section, sections of structure sheaf are allowed to have a pole along the zero section, so  $\pi_*\mathcal{O}_{\tot}=\bigoplus\limits_{n\in\mathbb{Z}}\mathcal{O}(-n)$. So, the problem is reduced to computing $H^1(\mathbb{P}^{n-1},
\tan\otimes \mathcal{O}(-k))$. From Euler exact sequence we get a long exact seqeunce
$\dots\to H^1(\mathbb{P}^{n-1},\mathcal{O}(1-k)^{\oplus n})\to H^1(\mathbb{P}^{n-1},\tan\otimes \mathcal{O}(-k))\to H^2(\mathbb{P}^{n-1},\mathcal{O}(-k))\to \dots$
Both left and right groups are zero, because line bundles on $\mathbb{P}^{n-1}$ can have nonzero cohoomology only in degrees $0,n-1>2$ so $H^1(\tot,\pi^*\tan)=0$