\section{Introduction}\label{sec:intro}

Future wireless networks are envisaged to support emerging applications such as auto-driving, virtual/augmented reality (AR/VR), smart home, unmanned aerial vehicles (UAVs), and factory automation \cite{rahman2020enabling}, which require ultra reliable and low latency sensing,  communication, and computation. This thus calls for a paradigm shift in wireless networks, from the conventional communications only design, to a new design with sensing-communication-computation integration. Towards this end, integrated sensing and communication (ISAC)\footnote{In the literature, the ISAC is also referred to as joint radar communications (JRC) \cite{liu2020joint}, joint communication and radar (JCR) \cite{kumari2019adaptive}, dual-functional radar communications (DFRC) \cite{hassanien2016signaling,liu2018toward}, radar-communications (RadCom) \cite{sturm2011waveform}, and joint communication and radar/radio sensing (JCAS) \cite{rahman2020enabling}.}  \cite{liu2021integrated, liu2020joint,kumari2019adaptive,sturm2009ofdm,hassanien2016signaling,sturm2011waveform,donnet2006combining,dokhanchi2019mmwave,liu2018toward,liu2018mu,Eldar2020joint,mccormick2017simultaneous_I,mccormick2017simultaneous_II,dokhanchi2021multicasting,xu2021rate,liyanaarachchi2021joint} has recently attracted growing research
interests in both academia and industry, in which the radar sensing capabilities are integrated into wireless networks (e.g., 6G cellular \cite{wild2021joint} and WiFi 7 \cite{noauthor_undated-pr,tan2020wi}) for a joint communications and sensing design.
On one hand, with such integration, ISAC can significantly enhance the spectrum utilization efficiency and cost efficiency by exploiting the dual use of radio signals and infrastructures for both radar sensing and wireless communication \cite{wild2021joint}.
On the other hand, due to the advancements of millimeter wave (mmWave)/terahertz (THz), wideband transmission (on the order of 1 GHz), and massive antennas in wireless communications, ISAC can reuse the communication signals to provide high sensing resolution (in both range and angular) and sensing accuracy (in detection and estimation) to meet the stringent sensing requirements of emerging applications. In the industry, Huawei and Nokia envisioned ISAC as one of the key technologies for 6G \cite{wild2021joint,tan2021integrated}, and IEEE 802.11 formed the WLAN Sensing Task Group IEEE 802.11bf in September 2020, with the objective of incorporating wireless sensing as a new feature for next-generation WiFi systems (e.g., Wi-Fi 7) \cite{noauthor_undated-pr,tan2020wi}.

While ISAC is a relatively new research topic that emerged recently in communications society,  the interplay between wireless communications and radar sensing has been extensively investigated in the literature, some examples of which include the communication and radar spectrum sharing (CRSS) \cite{zheng2019radar,labib2017coexistence,sodagari2012projection, mahal2017spectral, saruthirathanaworakun2012opportunistic,  rihan2018optimum,liu2018mimo,qian2018joint,li2016optimum, liu2018mu,liu2017robust, chiriyath2017radar} and the radar-centric communication integration \cite{li2019integrated,sahin2017filter,zhang2017modified,zheng2019radar,hassanien2016signaling,huang2020majorcom,wu2020waveform,wang2018dual,ma2018novel,hassanien2016dual,hassanien2016phase}. In particular, CRSS features the spectrum sharing between separate radar sensing and communication systems, for which a large amount of research efforts have been put on managing the inter-system interference via, e.g.,  interference null-space projection \cite{sodagari2012projection, mahal2017spectral}, opportunistic spectrum sharing \cite{saruthirathanaworakun2012opportunistic}, and transmit beamforming optimization \cite{rihan2018optimum,liu2018mimo,qian2018joint,li2016optimum,liu2018mu,liu2017robust}. However, CRSS generally has limited performance in sensing and communication due to the limited coordination and information exchange between them \cite{rahman2020enabling}.
In another line of research called radar-centric communication integration, the radar signals are revised to convey information, thus incorporating communication capabilities in conventional radar systems. For instance, the authors in \cite{li2019integrated,sahin2017filter,zhang2017modified} incorporated continuous phase modulation (CPM) into the conventional linear frequency modulation (LFM) radar waveform to
convey information, \cite{hassanien2016phase,hassanien2016dual} embedded the information bits via controlling the amplitude and phase of radar side-lobes and waveforms, and \cite{huang2020majorcom,wu2020waveform,wang2018dual,ma2018novel} modified the radar waveforms via index modulation to represent information by the indexes of antennas, frequencies, or codes instead of directly modifying the radar waveforms.
However, the radar-centric communication integration has limited achievable data rates as the radar signaling is highly suboptimal for data transmission in general.


Different from the CRSS and radar-centric communication integration, the ISAC shares the same wireless infrastructures for radar sensing and communication, and optimizes the transceivers design for both objectives, thus further enhancing both sensing and communication performance. For example, the authors in \cite{sturm2009ofdm,sturm2011waveform,donnet2006combining,dokhanchi2019mmwave} exploited the orthogonal frequency division multiplexing (OFDM) waveform for probing and sensing. In particular,
thanks to its great success in wireless communications \cite{goldsmith2005wireless, heath2018foundations} and radar sensing \cite{fishler2004mimo, li2007mimo}, the multi-antenna or multiple-input multiple-output (MIMO) technique has played a key role in ISAC by exploiting the spatial degrees of freedom (DoF) via transmit beamforming to perform multi-target sensing and multi-user communication simultaneously (see, e.g., \cite{mccormick2017simultaneous_I,mccormick2017simultaneous_II,liu2018toward,liu2018mu,Eldar2020joint,dokhanchi2021multicasting,xu2021rate,liyanaarachchi2021joint}). The benefit of MIMO may become even more considerable in 5G and 6G, due to the employment of massive MIMO \cite{larsson2014massive} and mmWave/THz \cite{heath2016overview}. For instance, the authors in \cite{mccormick2017simultaneous_I,mccormick2017simultaneous_II} considered that one transmitter needs to simultaneously communicate with one receiver and sense one target, in which two transmit beamformers are designed towards the communication receiver and sensing target, respectively. Then, \cite{liu2018toward,liu2018mu,Eldar2020joint,dokhanchi2021multicasting,xu2021rate,liyanaarachchi2021joint} considered the case with multiple receivers and multiple sensing targets. To be more specific, \cite{liu2018toward} optimized the beamformers to minimize the cross interference between radar sensing and communication at the base station (BS), while matching the desired transmit beampattern for sensing. \cite{liu2018mu,Eldar2020joint} further addressed the beampattern matching problem subject to the signal-to-interference-plus-noise ratio (SINR) constraints at communication receivers and the per-antenna transmit power constraints at the BS. Most recently, \cite{dokhanchi2021multicasting} studied the joint information multicasting and radar sensing, \cite{xu2021rate} considered the ISAC beamforming design under a rate-splitting setup, and \cite{liyanaarachchi2021joint} investigated the full-duplex hybrid beamforming for ISAC.

 How to design the transmit signals/waveforms and optimize the associated transmit beamforming is an essential issue to be dealt with in ISAC with multiple antennas. In \cite{liu2018mu}, the authors considered a conventional design by directly reusing the communication waveforms for sensing and solved the SINR-constrained beampattern matching problems sub-optimally by the techniques of semidefinite relaxation (SDR). Such conventional designs, however, may only have limited DoF in transmitted signals especially when the number of users is smaller than that of transmit antennas. To fully exploit the full DoF of the ISAC system, \cite{Eldar2020joint} proposed to enhance the sensing capability by sending dedicated radar signals combined with communication signals. 
 Although the radar signals are assumed to result in harmful co-channel interference towards the communication receivers, it was shown counterintuitively in \cite{Eldar2020joint} that the adding of dedicated radar signal is beneficial for SINR-constrained beampattern matching. Despite the progress in \cite{Eldar2020joint}, however, there still lacks an in-depth investigation on under what conditions or setups the dedicated radar signal is needed for optimizing the ISAC performance, by considering the co-channel interference from sensing signals. Moreover, since the radar signal is typically known prior to the transmission, it is possible that the communication receiver can apply successive interference cancellation (SIC) to first cancel the interference caused by the radar signal, thus facilitating the communication signals' decoding to further enhance the ISAC performance. By considering this new type of communication receivers with SIC capabilities, how to optimize the joint information and radar beamforming has not been investigated in the literature yet. Besides that, the optimal beamforming design in \cite{Eldar2020joint} has great computational complexity and although the proposed zero-forcing technique in \cite{Eldar2020joint} relieves the computational burden of the optimal design, it inevitably results in sub-optimal solution. It is still unknown whether there exist other better design approaches which enjoy lower computational complexity without necessarily incurring performance loss. The aforementioned points thus motivate the current work.



This paper studies the transmit beamforming design in a downlink ISAC system, in which a single BS equipped with an uniform linear array (ULA) sends combined information-bearing and dedicated radar signals to perform downlink multiuser communication and radar targets sensing simultaneously.
Our main contributions are listed as follows.
\begin{itemize}
	\item First, to measure the sensing performance, we consider both the beampattern matching criterion and a novel reduced-complexity design criterion which maximizes the minimum beampattern gain at the sensing angles of interest specified by the desired beampattern. This minimum beampattern maximization design is shown to achieve desired sensing beampattern with lower computational complexity. 
	\item Second, different from \cite{Eldar2020joint} assuming that the communication receivers suffer from unavoidable interference from radar signals,   we consider two types of communication receivers, without and with the capability of canceling the interference from the \textit{a-priori} known dedicated radar signals, which are referred to as Type-I and Type-II receivers, respectively.
	\item Third, the formulated beampattern matching and minimum beampattern gain maximization problems according to two different design criteria with both Type-I and Type-II receivers are non-convex and thus difficult to be optimally solved in general. Fortunately, by applying the SDR technique \cite{luo2010semidefinite}, we extend the result in \cite{Eldar2020joint} and rigorously prove that the SDR is not only always tight for Type-I receivers, but also for Type-II receivers under beampattern matching criterion. For the newly proposed minimum beampattern gain maximization criterion, we also rigorously prove that the SDRs are tight for both Type-I and Type-II receivers and thus globally optimal solutions can be obtained. Under two design criteria, we additionally prove that the problem formulated without dedicated radar signal in the relaxed domain yields the same optimal objective function value as that with dedicated radar signal and Type-I receivers.
	\item Fourth, under both design criteria, we rigorously prove that with Type-I receivers, there is no need to add dedicated radar signals when the wireless communication channels are line-of-sight (LOS); while with Type-II receivers, the addition of dedicated radar signals could always yield better performance in terms of beampattern matching error or minimum beampattern gain at the interested angles. 
	Combining all the aforementioned points, we reveal fundamental relationships on the optimal beampattern matching errors or the optimal minimum beampattern gains under different scenarios with Type-I receivers, with Type-II receivers, and without dedicated radar signals. 
	\item Numerical results demonstrate that the newly proposed design criterion not only yields enhanced worst-case radar sensing performance at the interested angles, but also enjoys significantly lower computational complexity in terms of execution time than its beampattern matching counterpart. Moreover, it is shown that under both design criteria, the case with Type-II receivers leads to significantly enhanced radar sensing performance as compared to that with Type-I receivers. With the fundamental relationships addressed above, we can thus conclude that with Type-II receivers, dedicated radar signal is beneficial in enhancing the sensing and communication performance under all the considered setups thanks to the radar interference cancellation capabilities at the receivers. Furthermore, it is observed that under Type-I receivers with high SINR requirements, dedicated radar signal is not needed even when the wireless channels follow Rayleigh fading.
\end{itemize}




The remainder of this paper is organized as follows. Section \ref{sec:system1} presents the system model. Section \ref{Section-joint} and \ref{S-maxmin} study the transmit beamforming designs for SINR-constrained beampattern matching and minimum beampattern gain maximization problems, respectively. Section \ref{Section_results} provides numerical results to validate the performance of our proposed designs. Section \ref{Section_conclusion} concludes this paper.

{\it Notations:} Boldface letters refer to vectors (lower  case) or matrices (upper case). For a square matrix $\mv{S}$, ${\text{tr}}(\mv{S})$ and $\mv{S}^{-1}$ denote  its trace  and inverse, respectively, while $\mv{S}\succeq \mv{0}$, $\mv{S}\preceq \mv{0}$, $\mv{S}\prec \mv{0}$ and $\mv{S}\nsucceq \mv{0}$ mean that $\mv{S}$ is positive semidefinite, negative semidefinite, negative definite, and non-positive semidefinite, respectively. For an arbitrary-size matrix $\mv{M}$, ${\text{rank}}(\mv{M})$, $\mv{M}^\dagger$, $\mv{M}^H$, and $\mv{M}^T$ denote its rank, pseudoinverse, conjugate transpose, and transpose, respectively, and $\mv{M}_{ik}$ denotes the element in the $i$th row and $k$th column of $\mv{M}$. $\mv{I}$ and $\mv{0}$ denote an identity
matrix and an all-zero matrix, respectively, with appropriate
dimensions. The distribution of a circularly symmetric complex Gaussian (CSCG) random vector with mean vector $\mv{x}$ and covariance matrix $\mv{\Sigma}$ is denoted by $\mathcal{CN}(\mv{x,\Sigma})$; and $\sim$ stands for ``distributed as''. $\mathbb{C}^{x\times y}$ denotes the space of $x\times y$ complex matrices. $\mathbb{R}$ denotes the set of real numbers. ${\mathbb{E}}(\cdot)$ denotes the statistical expectation. $\|\mv{x}\|$ denotes the Euclidean norm of a complex vector $\mv{x}$. $|z|$ and $z^*$ denote the magnitude of a complex number $z$ and its complex conjugate, respectively.
For two real vectors $\mv{x}$ and $\mv{y}$, $\mv{x}\ge \mv{y}$ means that $\mv{x}$ is greater than or equal to $\mv{y}$ in a component-wise manner.


\section{System Model}\label{sec:system1}

We consider a downlink ISAC system as shown in Fig. \ref{fig:system_model}, in which a BS sends wireless signals to perform radar sensing towards potential targets and downlink communication with $K$ users simultaneously. The BS is equipped with a ULA with $N>1$ antennas, and each user is equipped with a single antenna.


\begin{figure}
	\centering

	\setlength{\abovecaptionskip}{-4mm}
	\setlength{\belowcaptionskip}{-4mm}
	\includegraphics[width=3.5in]{figures/Shared_deployment.png}
	\caption{Illustration of the considered downlink ISAC system. } \label{fig:system_model}
\end{figure}




In order to perform sensing and communication at the same time, we suppose that the BS uses transmit beamforming to send information-bearing signals $s_k$'s for the $K$ users, together with a dedicated radar signal $\mv{s}_0 \in \mathbb{C}^{N \times 1}$. Let $\mathcal{K} \triangleq \{1,...,K\}$ denote the set of communication users. The transmitted information signals $\{s_k\}_{k=1}^K$ are assumed to be independent random variables, with zero mean and unit variance, while the radar signal $\mv{s}_0$ has zero mean and covariance matrix $\mv{R}_d = \mathbb{E}[\mv{s}_0\mv{s}_0^H] \succeq \mv{0}$ and is independent from $\{s_k\}_{k=1}^K$.
Let $\mv{t}_k \in \mathbb{C}^{N \times 1}$ denote the transmit beamforming vector for user $k \in \mathcal{K}$. By combining $s_k$'s and $\mv{s}_0$, the transmit signal $\mv{x} \in \mathbb{C}^{N \times 1}$ by the BS is expressed as
\begin{equation}
	\bm{x} = \sum_{k=1}^K \bm{t}_k s_k + \bm{s}_0.
\end{equation}
Notice that different from the single-beam transmission for information-bearing signals, we consider the general multi-beam transmission for the dedicated radar signal $\mv{s}_0$ (i.e., $\mv{R}_d$ is with general rank). In particular, suppose that  $\text{rank}(\mv{R}_d) = N_s$. This then corresponds to a set of $N_s$ radar signal beams, which can be obtained via the eigenvalue decomposition (EVD) of $\mv{R}_d$.
In this case, the sum transmit power at the BS is given as $\mathbb{E}\left[\| \sum_{k=1}^{K} \mv{t}_k s_k + \mv{s}_0 \|^2\right]   = \sum_{k=1}^{K} \|\mv{t}_k\|^2 + \text{tr}(\mv{R}_d)$. Suppose that the maximum sum transmit power budget at the BS is $P_0$. Then we have the following power constraint 
\begin{align}\label{equ:sum_power_constr}
	\sum_{k=1}^{K} \|\mv{t}_k\|^2 + \text{tr}(\mv{R}_d) \leq P_0.
\end{align}

First, we consider the radar target sensing, where the transmit beampattern\footnote{We focus on the transmit beampattern design, as the proper design can result in enhanced performance in echo wave processing \cite{stoica2007probing}.} is the key performance metric that has been widely adopted in the literature for MIMO radar signal design \cite{stoica2007probing}. The transmit beampattern generally depicts the transmit signal power distribution with respect to the sensing angle $\theta$ ranging from $[-\frac{\pi}{2}, \frac{\pi}{2}]$. In our considered ISAC system with both radar and information signals, the resultant transmit beampattern gain is expressed as 
\begin{align}
	\mathcal{P}(\theta) = \mathbb{E}\left[ |\mv{a}^H(\theta)(\sum_{k=1}^{K} \mv{t}_k s_k + \mv{s}_0)|^2 \right] = \mv{a}^H(\theta) \left(\sum_{k=1}^{K} \mv{t}_k \mv{t}^H_k + \mv{R}_d \right)\mv{a}(\theta),
\end{align}
where
\begin{align} \label{equ:steering}
	\mv{a}(\theta) = [1,e^{j 2\pi \frac{d}{\lambda} \sin \theta},...,e^{j 2\pi \frac{d}{\lambda} (N-1) \sin\theta}]^T
\end{align}
denotes the steering vector at angle $\theta$, with $\lambda$ and $d$ denoting the carrier wavelength and the spacing between two adjacent antennas, respectively. In practice, the transmit beampattern is designed based on the radar target sensing requirements. For instance, if we need to perform the detection task without knowing the direction of potential targets, a uniformly distributed beampattern is desired. By contrast, if we roughly know the directions of the targets, e.g., for target tracking, we only need to maximize the beampattern gains towards these interested potential directions \cite{stoica2007probing}.

Next, we consider the information reception at communication users. By letting $\mv{h}_i$ denote the channel vector from the BS to each user $i \in \mathcal{K}$, the received signal at user $i$ is then given as
\begin{equation} \label{equ:Rx_signal}
y_i = \mv{h}_i^H \mv{x} + n_i = \mv{h}_i^H (\sum_{k=1}^{K} \mv{t}_k s_k + \mv{s}_0) + n_i,
\end{equation}
where $n_i \sim \mathcal{CN}(0,\sigma_i^2)$ denotes the additive white Gaussian noise (AWGN) at the receiver of user $i$. It is observed in (\ref{equ:Rx_signal}) that each user $i$ suffers from the interference induced by both the information signals $\{s_k\}_{k \neq i}$, as well as the dedicated radar signal $\mv{s}_0$. Notice that $\mv{s}_0$ is pre-determined sequences that can be {\it a priori} known by both the BS and the users. Therefore, we consider the following two different types of  communication users, namely Type-I and Type-II receivers, which do not have and do have the capability of cancelling the resultant interference by the radar signals $\mv{s}_0$, respectively.

\begin{itemize}
\item \emph{Type-I receivers} (e.g., legacy users) do not have the capability of canceling the interference of the radar signals before decoding its desirable information signal. In this case, the SINR at receiver $i \in \mathcal{K}$ is
\begin{align}
\gamma_{i}^{\text{(I)}} (\{\mv{t}_i\}, \mv{R}_d) =\frac{\left|\mv{h}_i^{H} \mv{t}_i\right|^{2}}{\sum\limits_{k\in\mathcal K, k\neq i}\left|\mv{h}_i^{H} \mv{t}_{k}\right|^{2} + \mv{h}_i^H \mv{R}_d \mv{h}_i +\sigma_i^2}, \forall i \in \mathcal{K}.
\end{align}
\item \emph{Type-II receivers} are dedicatedly designed for
the ISAC system to have the capability of perfectly canceling the interference caused by the dedicated sensing signal $\mv{s}_0$. In this case, the SINR at receiver $i \in \mathcal{K}$ is
\begin{align}
\gamma_{i}^{\text{(II)}} (\{\mv{t}_i\})=\frac{\left|\mv{h}_i^{H} \mv{t}_i\right|^{2}}{\sum\limits_{k\in\mathcal K, k\neq i}\left|\mv{h}_i^{H} \mv{t}_{k}\right|^{2} +\sigma_i^2}, \forall i \in \mathcal{K}.
\end{align}
\end{itemize}

For the purpose of illustration, Figs. 2(a) and 2(b) show the ISAC system designs with Type-I and Type-II receivers, respectively. For comparison, Fig. 2(c) shows the conventional design without $\mv{s}_0$ \cite{liu2018mu}. Note that our consideration with both Type-I and Type-II receivers is new as compared to \cite{Eldar2020joint}, in which only Type-I receivers are considered.

\begin{figure}
	\centering

	\setlength{\abovecaptionskip}{-4mm}
	\setlength{\belowcaptionskip}{-4mm}
	\includegraphics[width=4.5in]{figures/System_Setup.png}
	\caption{Illustration of different ISAC transceivers designs: (a) Type-I receivers with dedicated radar signals at the BS \cite{Eldar2020joint}; (b) Type-II receivers with dedicated radar signals at the BS; (c) the conventional design without dedicated radar signal at the BS \cite{liu2018mu}.} \label{fig:system_setup}

\end{figure}

 It is assumed that the BS perfectly knows all the channel vectors $\mv{h}_i$'s and the desirable beampattern or interested sensing
 directions, while each user perfectly knows its own channel vector. This assumption is made to characterize the fundamental performance upper bounds and gain essential design insights for practical scenarios when such information is not perfectly known. Under this setup, our objective is to design the joint sensing and communication beamforming to maximize the radar sensing performance while ensuring the communication quality of service (QoS) requirements at each user, as will be illustrated in the next two sections in more detail.


\section{SINR-Constrained Beampattern Matching}\label{Section-joint}

In this section, we jointly optimize the information beamforming vectors $\mv{t}_k, \forall k \in \mathcal{K}$,
and the dedicated radar signal covariance matrix $\mv{R}_d$, to minimize the matching error between the overall transmit beampattern and a given desired beampattern, subject to the sum power constraint at the BS and a set of minimum SINR constraints at communication users.

\subsection{Problem Formulation}

Let $\{\tilde{\mathcal{P}}\left(\theta_{m}\right)\}_{m=1}^M$ denote a pre-designed beampattern, which specifies the desired transmit power distribution at the $M$ angles $\{\theta_m\}_{m=1}^M$ in the space. For target detection tasks, $\{\tilde{\mathcal{P}}\left(\theta_{m}\right)\}_{m=1}^M$ can be uniformly distributed \cite{stoica2007probing}, while for target tracking, it can be non-zero \textit{constant} at interested angles with potential targets and zero elsewhere \cite{liu2018mu}. Let $\Gamma_{i} \ge 0$ denote the minimum SINR requirement at each receiver $i \in \mathcal{K}$ and define 
\begin{align}
f_0(\{\mv{t}_k\}, \bm{R}_d, \alpha) \triangleq \sum\limits_{m=1}^{M}\left|\alpha \tilde{\mathcal{P}}\left(\theta_{m}\right) - \mv{a}^H(\theta_m) (\sum\limits_{k=1}^{K} \mv{t}_k \mv{t}^H_k + \mv{R}_d) \mv{a}(\theta_m) \right|^{2}.
\end{align}
Our objective is to minimize the beampattern matching error, for which the problem is formulated as follows with Type-I and Type-II receivers, respectively.

\begin{subequations}
	\begin{align}
	(\text{P}1): &\min\limits_{\{\bm{t}_k\}, \bm{R}_d, \alpha} f_0(\{\mv{t}_k\}, \bm{R}_d, \alpha) \\
	&\text { s.t. } \gamma_{i}^{\text{(I)}}(\{\mv{t}_i\}, \mv{R}_d) \geq \Gamma_{i}, \quad \forall i \in \mathcal{K}\\
	\label{equ:SINR-I}
	&\quad \quad \sum_{k=1}^{K} ||\mv{t}_k||^2  + \text{tr}(\mv{R}_d) = P_0\\
	\label{equ:sum-P1}
	&\quad \quad \mv{R}_d \succeq 0.
	\end{align}
\end{subequations}
\begin{subequations}
	\begin{align}
	(\text{P}2): &\min\limits_{\{\bm{t}_k\}, \bm{R}_d, \alpha} f_0(\{\mv{t}_k\}, \bm{R}_d, \alpha) \\
	&\text { s.t. } \gamma_{i}^{\text{(II)}}(\{\mv{t}_i\}) \geq \Gamma_{i}, \quad \forall i \in \mathcal{K}\\ 
	\nonumber
	&\quad \quad (\text{\ref{equ:SINR-I}}) \text{ and } (\text{\ref{equ:sum-P1}}).




	\end{align}
\end{subequations}
Here, $\alpha$ denotes a scaling factor and $\mv{a}\left(\theta_{m}\right)$ denotes the steering vector at direction $\theta_{m}$ as in (\ref{equ:steering}).
In problems (P1)\footnote{(P1) is different from that in \cite{Eldar2020joint} in the following two aspects. First, we consider the beampattern matching error as the sole objective function so that a fair analysis and comparison could be conducted involving \cite{liu2018mu}, \cite{Eldar2020joint} and our design. Second, this paper considers the sum-power constraint at the BS versus the per-antenna power constraints in \cite{Eldar2020joint} since it has been widely adopted in wireless communication networks.} and (P2), we consider the equality power constraints in (\ref{equ:SINR-I}) in order for the BS to use up all the transmit power to maximize the radar performance for target tracking or detection \cite{li2007mimo}.
Notice that both problems (P1) and (P2) are non-convex and thus are difficult to be optimally solved in general. Fortunately, we will use the SDR technique \cite{luo2010semidefinite} to obtain the optimal solutions, as will be shown next.



\subsection{Proposed Solution via SDR}

In this subsection, we propose to use the SDR technique to solve problems (P1) and (P2). Towards this end, we introduce new auxilary varaibles  $\mv{T}_k = \mv{t}_k \mv{t}_k^H, \forall k \in \mathcal{K}$, where  $\mv{T}_k \succeq 0$ and $\text{rank}(\mv{T}_k) = 1$. In this case, define
\begin{align} \label{equ:f1}
f_1(\{\bm{T}_k\}, \bm{R}_d, \alpha) \triangleq \sum\limits_{m=1}^{M}\left|\alpha \tilde{\mathcal{P}}\left(\theta_{m}\right) - \mv{a}^H(\theta_m) (\sum\limits_{k=1}^{K} \mv{T}_k + \mv{R}_d) \mv{a}(\theta_m) \right|^{2}.
\end{align}
Problems (P1) and (P2) are equivalently reformulated as (P1.1) and (P2.1) in the following, respectively.
\begin{subequations}
	\begin{align}
	(\text{P}1.1): & \min\limits_{\{\bm{T}_k\}, \bm{R}_d, \alpha} f_1(\{\bm{T}_k\}, \bm{R}_d, \alpha)\\
	\text { s.t. } &\frac{\operatorname{tr}\left(\bm{h}_{i} \bm{h}_{i}^H \bm{T}_{i}\right)}{\Gamma_{i}}-\sum\limits_{k \in \mathcal{K} \atop k \neq i} \operatorname{tr}\left(\bm{h}_{i} \bm{h}_{i}^H \bm{T}_{k}\right)-\operatorname{tr}\left(\bm{h}_{i} \bm{h}_{i}^H \bm{R}_{d}\right)-\sigma_{i}^{2} \geq 0, \forall i \in \mathcal{K}\\
	\label{equ:T_power}
	\quad \quad & \sum_{k=1}^{K} \text{tr}(\mv{T}_k)  + \text{tr}(\mv{R}_d) = P_0\\
	\label{equ:rank}
	\quad \quad &\mv{R}_d \succeq 0, \mv{T}_k \succeq 0, \text{rank}(\mv{T}_k) = 1, \forall k \in \mathcal{K}.
	\end{align}
\end{subequations}
\begin{subequations}
	\begin{align}
	(\text{P}2.1): &\min\limits_{\{\bm{T}_k\}, \bm{R}_d, \alpha} f_1(\{\bm{T}_k\}, \bm{R}_d, \alpha) \\
	\text { s.t. } &\frac{\operatorname{tr}\left(\bm{h}_{i} \bm{h}_{i}^H \bm{T}_{i}\right)}{\Gamma_{i}}-\sum\limits_{k \in \mathcal{K} \atop k \neq i} \operatorname{tr}\left(\bm{h}_{i} \bm{h}_{i}^H \bm{T}_{k}\right)
	-\sigma_{i}^{2} \geq 0, \forall i \in \mathcal{K} \\ \nonumber
	\quad \quad & \text{(\ref{equ:T_power})} \text{ and } \text{(\ref{equ:rank})}.
	\end{align}
\end{subequations}

However, problems (P1.1) and (P2.1) are still non-convex due to the rank-one constraints on $\mv{T}_k$'s. To deal with this issue, we relax the rank-one constraints and accordingly get the SDRs of problems (P1.1) and (P2.1), denoted as (SDR1) and (SDR2), respectively. Notice that both (SDR1) and (SDR2) are convex and thus can be solved optimally by convex optimization solvers such as CVX \cite{grant2014cvx}.
Suppose that the ranks of the obtained communication covariance matrices $\mv{T}_k$'s in the optimal solutions to (SDR1) and (SDR2) are all equal to 1, then they are also optimal for (P1.1) and (P2.1), respectively. Otherwise, we should further perform the Gaussian randomization \cite{luo2010semidefinite} to construct rank-one solutions that are generally sub-optimal. Fortunately, the following propositions show that there always exists optimal rank-one solutions to (SDR1) and (SDR2), and thus the Gaussian randomization is not needed in general.


\begin{proposition}\label{T_1} \emph{
		There always exists a globally optimal solution to problem (SDR1), denoted as $\{\{\tilde{\mv{T}}_k\},\tilde{\mv{R}}_d,\tilde{\alpha}\}$, such that
	\begin{align}
		\nonumber
		\text{rank}(\tilde{\mv{T}}_k) = 1, \forall k \in \mathcal{K}.
	\end{align}}

	\begin{proof}
		Notice that problem (P1.1) has a similar structure as the joint information and radar beamforming problem in \cite{Eldar2020joint}, (30) and can be similarly proved as in \cite[Theorem 1]{Eldar2020joint},
		for which the details are omitted for brevity.
	\end{proof}
\end{proposition}

Besides the tightness of (SDR1), we extend the result to (SDR2) and rigorously prove the tightness of (SDR2) also holds, as shown in the following proposition.

\begin{proposition}\label{T_2} \emph{
		There always exists a globally optimal solution to problem (SDR2), denoted as $\{\{\bar{\mv{T}}_k\},\bar{\mv{R}}_d,\bar{\alpha}\}$, such that
		\begin{align}
			\nonumber
			\text{rank}(\bar{\mv{T}}_k) = 1, \forall k \in \mathcal{K}.
	\end{align}}

	\begin{proof}
		See Appendix \ref{Proof_theorem P2}.
	\end{proof}
\end{proposition}

\begin{remark} \label{remark:Joint} \emph{
		By comparing problems (P1.1) and (P2.1), it is observed that every feasible solution to problem (P1.1) is also feasible to problem (P2.1) but not vice versa. Therefore, it is evident that problem (P2.1) always yields an equal or smaller beampattern matching error than (P1.1) due to the enlarged feasible region.}
\end{remark}

\subsection{Relationship with the Case without Radar Signal}

Up to now, when solving the corresponding problems optimally, we have the following relationship in terms of objective function value:
\begin{align}
	f_1^*|_{(P1.1)} = f_1^*|_{(SDR1)} \geq f_1^*|_{(SDR2)} = f_1^*|_{(P2.1)},
\end{align}
where $f_1$ is the objective function defined in (\ref{equ:f1}) and $*$ denotes the optimality. However, it is still not clear whether the addition of dedicated radar signal will enhance the sensing performance under either Type-I or Type-II receivers given the same power budget and SINR requirements. To facilitate the analysis of this question, we introduce problem (P3) as the beampattern matching problem in the case without dedicated radar signals, which can be expressed similarly as for (P1) and (P2) by setting $\mv{R}_d = 0$. In this case, by introducing $\mv{T}_k = \mv{t}_k \mv{t}_k^H$, with $ \mv{T}_k \succeq \mv 0$ and $\text{rank}(\mv{T}_k) \le 1$, $ \forall k \in \mathcal{K}$, we can equivalently re-express problem (P3) as
\begin{align}
(\text{P}3.1): &\min\limits_{\{\bm{T}_k\}, \alpha} f_3(\{\bm{T}_k\}, \alpha) = \sum\limits_{m=1}^{M}\left|\alpha \tilde{\mathcal{P}}\left(\theta_{m}\right) - \mv{a}^H(\theta_m) (\sum\limits_{k=1}^{K} \mv{T}_k) \mv{a}(\theta_m) \right|^{2} \\
\nonumber
&\text { s.t. } \frac{\operatorname{tr}\left(\bm{h}_{i} \bm{h}_{i}^H \bm{T}_{i}\right)}{\Gamma_{i}}-\sum\limits_{k \neq i, k \in \mathcal{K}} \operatorname{tr}\left(\bm{h}_{i} \bm{h}_{i}^H \bm{T}_{k}\right)
-\sigma_{i}^{2} \geq 0, \forall i \in \mathcal{K} \\
\nonumber
& \quad \quad \sum_{k=1}^{K} \text{tr}(\mv{T}_k) = P_0\\
\nonumber
& \quad \quad \mv{T}_k \succeq 0, \text{rank}(\mv{T}_k) = 1, \forall k \in \mathcal{K}.
\end{align}
By removing the rank-one constraints on $\mv{T}_k$'s, we have the SDR of (P3.1) as (SDR3). Then we have the following proposition.







\begin{proposition}\label{T_30} \emph{
		Suppose that $\{\{\tilde{\mv{T}}_k\},\tilde{\mv{R}}_d,\tilde{\alpha}\}$ is an optimal solution to problem (SDR1). Then there always exist an optimal solution to (SDR3), denoted by $\{\{\hat{\mv{T}}_k\},\hat{\alpha}\}$ with
		\begin{align}
		&\hat{\bm{T}}_{k} = \tilde{\bm{T}}_{k} + \beta_k \tilde{\mv{R}}_d, \sum_{k=1}^K \beta_k = 1, \beta_k \geq 0,\\
		&\hat{\alpha} = \tilde{\alpha},
		\end{align}
		where
		\begin{align}
			\sum\limits_{k=1}^K \tilde{\bm{T}}_{k} + \tilde{\mv{R}}_d = \sum\limits_{k=1}^{K} \hat{\bm{T}}_{k}.
		\end{align}
		As a result, $\{\{\hat{\mv{T}}_k\}, \hat{\alpha}\}$ in (SDR3) achieves the same optimal beampatterns (and the same optimal value) as that achieved by $\{\{\tilde{\mv{T}}_k\}, \tilde{\mv{R}}_d, \tilde {\alpha}\}$ in (SDR1). i.e., $f_3^*|_{(SDR3)} = f_1^*|_{(SDR1)}$.}
	\begin{proof}	
	See Appendix \ref{Proof_theorem P3}.
	\end{proof}
\end{proposition}


Based on proposition 3 together with the relationship shown in (14), we reveal the following fundamental relationship in general, 
\begin{align} \label{equ:Fund_1}
	f_3^*|_{(P3.1)} \geq f_3^*|_{(SDR3)} = f_1^*|_{(P1.1)} = f_1^*|_{(SDR1)} \geq f_1^*|_{(SDR2)} = f_1^*|_{(P2.1)},
\end{align}
where the first inequality follows from the reduced feasible region due to the addition of rank-1 constraints. 
Therefore, problem (P1) with dedicated radar signals and Type-I receivers (or equivalently (SDR1)) can generally achieve better performance than (P3) without dedicated radar signals, in terms of lower beampattern matching errors and the performance achieved by Type-II receivers is even better than that with Type-I receivers. Nevertheless, under specific conditions (see Proposition \ref{T_LOS}), the SDR of (P3.1) is indeed tight, i.e., $f_3^*|_{(P3.1)} = f_3^*|_{(SDR3)}$. And thus (P1) and (P3) achieve the same optimal matching error. In other words, the dedicated radar signals are not needed in this special case with Type-I receivers.

\begin{proposition}\label{T_LOS} \emph{
	When the wireless channels from the BS to the communication users are LOS (i.e., $\mv{h}_i$ is given in the form of $\mv{h}_i = [1,e^{j \phi_i},...,e^{j (N-1)\phi_i}]^T$ with $\phi_i = 2 \pi \frac{d}{\lambda} \sin(\theta_i)$, where $\lambda$ and $d$ denotes the carrier wavelength and the spacing between two adjacent antennas, respectively), the SDR of problem (P3.1) (or equivalent (P3)) is tight.}

	\begin{proof}
		See Appendix \ref{Proof_theorem_LOS}.
	\end{proof}
\end{proposition}

According to (\ref{equ:Fund_1}), it is clear that if better performance is indeed observed with Type-I or Type-II receivers (as the case shown in Section \ref{Subsection-A}), we can conclude that dedicated radar signals are beneficial in enhancing the joint radar sensing and communication performance under Type-I or Type-II receivers since if $\bm{R}_d = 0$ at the optimal solution to (P1.1) or (P2.1), then the solution is also a feasible solution to (P3.1), which incurs a contradiction.



	





\section{SINR-Constrained Minimum Beampattern Gain Maximization}\label{S-maxmin}

In this section, we propose an alternative radar sensing design criterion termed minimum beampattern gain maximization, based on which our objective is to maximize the minimum beampattern gain illuminated at desirable directions of potential sensing targets, subject to SINR constraints for individual communication users. We will show that this design leads to a better sensing beampattern at lower computation complexity than the beampattern matching design in Section \ref{Section-joint}.

\subsection{Problem Formulation}

In this design, our objective is to maximize the minimum beampattern gain at a given set of interested angles $\Theta \triangleq \{\theta_1,\theta_2,...,\theta_Q\}$, where $Q$ denotes the number of quantized angles of interest. For instance, for target detection tasks without knowing any prior information about targets' potential locations, $\{\theta_q\}_{q=1}^Q$ may correspond to the uniformly quantized angles ranging from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$. For target tracking tasks, the interested angles are specified based on the targets' potential locations (similarly as the angles with non-zero entries in the pre-designed beampattern $\tilde{\mathcal{P}}(\theta_m)$ in Section \ref{Section-joint}). In this case, the SINR-constrained minimum beampattern gain maximization problems are formulated as (P4) and (P5) in the following, by considering Type-I and Type-II communication receivers, respectively.
\begin{subequations}
	\begin{align}
		(\text{P}4): &\max \limits_{t,\bm{R}_d,\{\bm{t}_k\}}  t \\
		& \text { s.t. } \mv{a}^{H}\left(\theta\right) \left(\mv{R}_d + \sum\limits_{k=1}^{K} \mv{t}_k \mv{t}_k^H \right) \mv{a} \left(\theta\right) \geq t, \forall \theta \in \Theta\\
		&\quad \quad \gamma_{i}^{\text{(I)}}(\{\mv{t}_i\}, \mv{R}_d) \geq \Gamma_{i}, \quad \forall i \in \mathcal{K}\\
			\label{equ:sum-P5}
		&\quad \quad \sum_{k=1}^{K} ||\mv{t}_k||^2  + \text{tr}(\mv{R}_d) \leq P_0\\
		\label{equ:psd-P5}
		&\quad \quad \mv{R}_d \succeq 0.
	\end{align}
\end{subequations}
\begin{subequations}
	\begin{align}
		(\text{P}5): &\max \limits_{t,\bm{R}_d,\{\bm{t}_k\}}  t \\
		&\text { s.t. } \mv{a}^{H}\left(\theta\right) \left(\mv{R}_d + \sum\limits_{k=1}^{K} \mv{t}_k \mv{t}_k^H \right) \mv{a} \left(\theta\right) \geq t, \forall \theta \in \Theta \\
		&\quad \quad \gamma_{i}^{\text{(II)}}(\{\mv{t}_i\}) \geq \Gamma_{i}, \quad \forall i \in \mathcal{K}\\
	
		\nonumber
		&\quad \quad \text{(\ref{equ:sum-P5})} \text{ and } \text{(\ref{equ:psd-P5})}.
	\end{align}
\end{subequations}
In (\ref{equ:sum-P5}), we consider the inequality power constraints, and it can be easily shown that the optimality of problems (P4) and (P5) is attained when the constraints are met with equality. Notice that problems (P4) and (P5) are both non-convex. In the following, we will adopt the SDR technique to obtain their optimal solutions.

\subsection{Proposed Solution via SDR}

Similarly as in Section \ref{Section-joint}, we define $\mv{T}_k = \mv{t}_k \mv{t}_k^H$ with $ \mv{T}_k \succeq \mv 0$ and $\text{rank}(\mv{T}_k) \le 1$, $ \forall k \in \mathcal{K}$. Accordingly, problems (P4) and (P5) can be reformulated as (P4.1) and (P5.1) as follows, respectively.
\begin{subequations}
\begin{align}
(\text{P}4.1): &\max \limits_{t,\bm{R}_d,\{\bm{T}_k\}} t  \\ 
&\text { s.t. } \mv{a}^{H}\left(\theta\right) \left(\mv{R}_d + \sum\limits_{k=1}^{K} \mv{T}_k\right) \mv{a} \left(\theta\right) \geq t, \forall \theta \in \Theta \\
&\quad \quad \frac{\operatorname{tr}\left(\bm{h}_{i} \bm{h}_{i}^H \bm{T}_{i}\right)}{\Gamma_{i}}-\sum\limits_{k \neq i, k \in \mathcal{K}} \operatorname{tr}\left(\bm{h}_{i} \bm{h}_{i}^H \bm{T}_{k}\right)
-\operatorname{tr}\left(\bm{h}_{i} \bm{h}_{i}^H \bm{R}_{d}\right)-\sigma_{i}^{2} \geq 0, \forall i \in \mathcal{K} \\
\label{equ:sum-P41}
&\quad \quad	\operatorname{tr}\left(\sum\limits_{k=1}^{K} \mv{T}_k + \mv{R}_d\right) \leq P_{0} \\
\label{equ:psd-P41}
&\quad \quad	\mv{R}_d \succeq 0, \mv{T}_k \succeq 0, \text {rank}\left(\mv{T}_k\right) = 1, \quad \forall k \in \mathcal{K}.
\end{align}
\end{subequations}
\begin{subequations}
\begin{align}
(\text{P}5.1): &\max \limits_{t,\bm{R}_d,\{\bm{T}_k\}} t  \\ 
&\text { s.t. } \mv{a}^{H}\left(\theta\right) \left(\mv{R}_d + \sum\limits_{k=1}^{K} \mv{T}_k\right) \mv{a} \left(\theta\right) \geq t, \forall \theta \in \Theta\\ 
&\quad \quad	\frac{\operatorname{tr}\left(\bm{h}_{i} \bm{h}_{i}^H \bm{T}_{i}\right)}{\Gamma_{i}}-\sum\limits_{k \neq i, k \in \mathcal{K}} \operatorname{tr}\left(\bm{h}_{i} \bm{h}_{i}^H \bm{T}_{k}\right)
-\sigma_{i}^{2} \geq 0, \forall i \in \mathcal{K}\\
&\quad \quad	\text{(\ref{equ:sum-P41})} \text{ and } \text{(\ref{equ:psd-P41})}.
\end{align}
\end{subequations}
\begin{remark} \label{remark_3} \emph{
		Similar as Remark \ref{remark:Joint}, it is evident that (P5)/(P5.1) with Type-II communication receivers always yields an equal or  greater minimum beampattern gains at interested angles than (P4)/(P4.1) with Type-I receivers, due to the enlarged feasible region.}
\end{remark}

By removing the rank-one constraints, we have the SDRs of (P4.1) and (P5.1) as (SDR4) and (SDR5), respectively, which are both separable semidefinite programming (SSDP) problems that can be solved efficiently by CVX. Then we can show that the two SDRs are indeed tight in the following proposition.

\begin{proposition}\label{T_56} \emph{
		There always exists globally optimal solutions to problems (SDR4) and (SDR5), which are denoted as $\{\{\tilde{\mv{T}}_k\},\tilde{\mv{R}}_d,\tilde{\alpha}\}$and $\{\{\bar{\mv{T}}_k\},\bar{\mv{R}}_d,\bar{\alpha}\}$, respectively, such that
		\begin{align}
		\nonumber
		\text{rank}(\tilde{\mv{T}}_k) = 1, \forall k \in \mathcal{K},\\
		\nonumber
		\text{rank}(\bar{\mv{T}}_k) = 1, \forall k \in \mathcal{K}.
		\end{align}}
	\begin{proof}
		Similar as the proof for Propositions \ref{T_1} and \ref{T_2}, suppose that $\{\{{\mv{T}}_k^*\},\mv{R}_d^*,\alpha^*\}$($\{\{\hat{\mv{T}}_k\},\hat{\mv{R}}_d,\hat{\alpha}\}$)  is one optimal solution to SDR4 (SDR5), which does not necessarily meet the rank-one constraints. As shown in Appendix \ref{Proof_theorem P2}, we can always construct an alternative optimal solution $\{\{\tilde{\mv{T}}_k\},\tilde{\mv{R}}_d,\tilde{\alpha}\}$ ($\{\{\bar{\mv{T}}_k\},\bar{\mv{R}}_d,\bar{\alpha}\}$)  meeting the rank-one constraints, such that the same minimum beampattern gains can be achieved by $\{\{\tilde{\mv{T}}_k\},\tilde{\mv{R}}_d,\tilde{\alpha}\}$ ($\{\{\bar{\mv{T}}_k\},\bar{\mv{R}}_d,\bar{\alpha}\}$). As the detailed proof procedure is similar as that in Appendix \ref{Proof_theorem P2}, the details are skipped for brevity.
	\end{proof}
\end{proposition}




Following Proposition \ref{T_56}, it is evident that we can always obtain rank-one optimal solutions to (P4.1) and (P5.1) by solving (SDR4) and (SDR5). As a result, the optimal solutions to (P4) and (P5) can be obtained accordingly.

Besides the SDR tightness shown in Proposition \ref{T_56}, we can also show the interesting relationship between (SDR4) and the case without dedicated radar signal. To facilitate the illustration, we introduce the minimum beampattern gain maximization problem without radar signals, denoted by (P6), which corresponds to problem (P4)/(P5) by setting $\mv{R}_d= 0$. Furthermore, we can reformulate (P6) as (P6.1) by introducing $\mv{T}_k = \mv{t}_k \mv{t}_k^H$ with $ \mv{T}_k \succeq \mv 0$ and $\text{rank}(\mv{T}_k) \le 1$, $ \forall k \in \mathcal{K}$ as follows.
\begin{align}
(\text{P}6.1): &\max \limits_{t,\{\bm{T}_k\}} t  \\ \nonumber
&\text { s.t. } \mv{a}^{H}\left(\theta\right) \left(\sum\limits_{k=1}^{K} \mv{T}_k\right) \mv{a} \left(\theta\right) \geq t, \forall \theta \in \Theta \\ \nonumber
&\quad \quad	\frac{\operatorname{tr}\left(\bm{h}_{i} \bm{h}_{i}^H \bm{T}_{i}\right)}{\Gamma_{i}}-\sum\limits_{k \neq i, k \in \mathcal{K}} \operatorname{tr}\left(\bm{h}_{i} \bm{h}_{i}^H \bm{T}_{k}\right)
-\sigma_{i}^{2} \geq 0, \forall i \in \mathcal{K}\\\nonumber
&\quad \quad	\operatorname{tr}\left(\sum\limits_{k=1}^{K} \mv{T}_k\right) \leq P_{0} \\\nonumber
&\quad \quad	\mv{T}_k \succeq 0, \text {rank}\left(\mv{T}_k\right) = 1, \quad \forall k \in \mathcal{K}.
\end{align}

By removing the rank-one constraints, the SDR of (P6.1)/(P6) is denoted as (SDR6). Then we have the following two propositions, for which the detailed proofs are similar to those for Propositions \ref{T_30} and \ref{T_LOS}, and thus are omitted for brevity.

\begin{proposition}\label{T_60} \emph{
		Suppose that $\{\{\tilde{\mv{T}}_k\},\tilde{\mv{R}}_d,\tilde{\alpha}\}$ is an optimal solution to problem (SDR4). Then there always exist an optimal solution to (SDR6), denoted by $\{\{\hat{\mv{T}}_k\},\hat{\alpha}\}$ with
		\begin{align}
			&\hat{\bm{T}}_{k} = \tilde{\bm{T}}_{k} + \beta_k \tilde{\mv{R}}_d, \sum_{k=1}^K \beta_k = 1, \beta_k \geq 0,\\
			&\hat{\alpha} = \tilde{\alpha},
		\end{align}
		where
		\begin{align}
			\sum\limits_{k=1}^K \tilde{\bm{T}}_{k} + \tilde{\mv{R}}_d = \sum\limits_{k=1}^{K} \hat{\bm{T}}_{k}.
		\end{align}
		As a result, $\{\{\hat{\mv{T}}_k\}, \hat{\alpha}\}$ in (SDR6) achieves the same optimal beampatterns (and the same optimal value) as that achieved by $\{\{\tilde{\mv{T}}_k\}, \tilde{\mv{R}}_d, \tilde {\alpha}\}$ in (SDR4), i.e., $t^*|_{(SDR6)} = t^*|_{(SDR4)}$.}
\end{proposition}
\begin{proposition}\label{T_LOS2} \emph{
		When the wireless channels from the BS to the communication users are LOS (i.e., $\mv{h}_i$ is given in the form of $\mv{h}_i = [1,e^{j \phi_i},...,e^{j (N-1)\phi_i}]^T$ with $\phi_i = 2 \pi \frac{d}{\lambda} \sin(\theta_i)$), the SDR of problem (P6.1) (or equivalent (P6)) is tight.}
\end{proposition}
Similar to that in Section \ref{Section-joint}, in general, we have the following relationship between problems formulated under different setups in terms of objective function value $t$, the minimum beampattern gain at $\Theta$,
\begin{align}
	t^*|_{(SDR5)} = t^*|_{(P5.1)} \geq t^*|_{(P4.1)} = t^*|_{(SDR4)} = t^*|_{(SDR6)} \geq t^*|_{(P6)}.
\end{align}
Specially, when the wireless channels are LOS, we have $t^*|_{(SDR6)} = t^*|_{(P6)}$ according to Proposition \ref{T_LOS2}.


\subsection{Complexity Analysis} \label{Section_complexity}

In this subsection, we analyze the computational complexity for solving problem (SDR1) /(SDR2) versus that for (SDR4)/(SDR5) in order to show the benefit of the newly proposed minimum beampattern gain maximization design. Notice that problems (SDR1) and (SDR2) are both quadratic semidefinite programs (QSDPs). Given a solution accuracy $\epsilon$, the worst-case complexity to solve the QSDP with the primal-dual interior-point method is $\mathcal{O}(K^{6.5}N^{6.5}\log(1/\epsilon))$ \cite{Eldar2020joint}. By contrast, problems (SDR4) and (SDR5) are SSDPs. It follows from \cite{luo2010semidefinite} that given a solution accuracy $\epsilon$, the worst-case complexity to solve the SSDP with the primal-dual interior-point method is $\mathcal{O}( \max \{KN+N+1,Q+K+1\}^4 (KN+N+1)^{1/2} \log(1/\epsilon)) \approx \mathcal{O}( K^{4.5} N^{4.5} \log(1/\epsilon)) $, which is significantly less than that for solving (SDR1) and (SDR2). This shows the benefit of the proposed minimum beampattern gain maximization criterion over its beampattern matching counterpart, in terms of the computational complexity.


\section{Numerical Results} \label{Section_results}

In this section, we provide numerical results to validate the performance of our proposed transmit beamforming designs under beampattern matching and minimum beampattern gain maximization criteria together with both Type-I and Type-II receivers. We begin the analysis by first focusing on the first design criterion in Section \ref{Subsection-A}, comparing our design with the two-stage design approach in \cite{liu2018mu}. In Section \ref{Subsection-B}, we compare the performance of two types of receivers in the second design criterion. Finally, the overall comparison between the performance with both type of receivers under two design criteria is given in Section \ref{Subsection-C}. The impact of number of users on the performance and the computational complexity reduction brought by the second design criteria are also studied in Section \ref{Subsection-C}. 

Throughout the whole section, we will adopt the following settings: half-wavelength spacing is assumed at the uniform linear array. The power constraints  of all the approaches considered in simulation are sum-power constraints with power budget $P_0 = 0.1$ Watt (W) or 20 dBm. The received signal at each communication user is corrupted with an additive white Gaussian noise of equal variance $\sigma^2 = -70$ dBm. Without loss of generality, we set $\Gamma_i = \Gamma, \forall i \in \mathcal{K}$. A pathloss of 80 dB is assumed between the BS and each communication receiver. The desired beampattern is composed of five main beams, whose direction set $\Theta_d=\{-\frac{\pi}{3}, -\frac{\pi}{6}, 0, \frac{\pi}{6} ,\frac{\pi}{3}\}$. The width of each main beam $\delta = \frac{\pi}{18}$ and the grids are obtained by specifying $M=101$ discrete sensing angles ranging from $[-\frac{\pi}{2}:\frac{\pi}{100}:-\frac{\pi}{2}]$. In conclusion, the normalized $\tilde{\mathcal{P}}(\theta)$ is defined as 


\begin{align}
	\tilde{\mathcal{P}}(\theta)=\left\{\begin{array}{l}
		1, \theta_d-\frac{\delta}{2} \leq \theta \leq \theta_d+\frac{\delta}{2}, \theta_d \in \Theta_d \\
		0, \text { otherwise }
	\end{array}\right.
\end{align}


\subsection{SINR-Constrained Beampattern Matching} \label{Subsection-A}

This subsection considers the beampattern matching design. Figs. \ref{fig:A-I-Rich-a} and \ref{fig:A-I-Rich-b} show the beampattern matching error versus the SINR threshold $\Gamma$ in the case with Rayleigh fading channels and LOS channels, respectively, in which we set $N = 8$ and $K = 5$ and the results are obtained by averaging over 50 random channel realizations. Besides the proposed joint beamforming design with Type-I and Type-II receivers via solving problems (P1)/(SDR1), and (P2)/(SDR2), we also consider the performance upper bound with radar sensing only (i.e., beampattern matching in (P1)/(P2) with the SINR constraints omitted) and the benchmark design without radar signal via solving (SDR3) (with $\mv{R}_d = 0$). In addition, we further consider the two-stage design approach without radar signals in \cite{liu2018mu} for comparison, where in the first stage, the optimal radar transmit beamforming without SINR constraints is obtained to minimize the beampattern matching error, while in the second stage, the information beamforming vectors are optimized via SDR to minimize its difference with the obtained optimal radar transmit beamforming subject to SINR constraints\footnote{Notice that in \cite{liu2018mu}, Gaussian randomization is employed to obtain a feasible but sub-optimal rank-one information beamforming solution. Here, we directly adopt the high-rank solution before the Gaussian randomization for the purpose of comparison only, which generally achieves better performance or a lower average matching error than that in \cite{liu2018mu}.}.

\begin{figure}[htb]
	\centering

	\setlength{\abovecaptionskip}{-4mm}
	\setlength{\belowcaptionskip}{-4mm}
	\setlength{\abovecaptionskip}{+3pt}
	\subfigure[The case with Rayleigh fading channels]{ \label{fig:A-I-Rich-a}
		\includegraphics[width=3.1in]{figures/Joint_SINR_avg_error_Rich_2-eps-converted-to.pdf}}
	\subfigure[The case with LOS channels]{ \label{fig:A-I-Rich-b}
		\includegraphics[width=3.1in]{figures/Joint_SINR_avg_error_LOS_2-eps-converted-to.pdf}}
	\caption{Average beampattern matching error versus the SINR threshold $\Gamma$. }
	\label{fig:A-I-Rich}
\end{figure}

 In Fig. \ref{fig:A-I-Rich}, it is observed that the proposed joint design with Type-II receivers achieves much lower matching error than the other designs under both channel conditions, especially when the SINR threshold $\Gamma$ becomes high. As mentioned at the end of Section \ref{Section-joint}, this validates the necessity of adding dedicated radar signals under Type-II receivers. It is also observed that the performance achieved by the proposed design with Type-I receiver (via solving (P1)/(SDR1)) is exactly same as that by (SDR3), which is consistent with Proposition \ref{T_30}. Besides, under Rayleigh fading channels, it is observed in the simulations that the solution to (SDR1) satisfies that $\mv{R}_d = 0$ and $\text{rank}(\mv{T}_i) = 1, \forall i \in \mathcal{K}$ when $\Gamma \geq 6 $dB and thus there is no need to add dedicated radar signal in this scenario.
 This is similar to the scenario with LOS channels, where it follows from Propositions \ref{T_30} and \ref{T_LOS} that there is no need to add dedicated radar signal over the whole SINR range with Type-I receivers. Furthermore, it is observed that our proposed joint designs with both Type-I and Type-II receivers achieve much smaller average beampattern matching error than the two-stage design without radar signals over the whole SINR range.

\begin{figure}[htb]
	\centering

	\setlength{\abovecaptionskip}{-4mm}
	\setlength{\belowcaptionskip}{-4mm}
	\includegraphics[width=3.8in]{figures/Tx_beampattern_joint_Rich-eps-converted-to.pdf}
	\caption{Obtained transmit beampattern via the beampattern matching under Rayleigh fading channels with $\Gamma$ = 20dB. The black dotted line specifies the desired transmit beampattern $\tilde{\mathcal{P}}(\theta)$.}
	 \label{fig:A-II-Tx-Beam}
\end{figure}

Fig. \ref{fig:A-II-Tx-Beam} shows the obtained transmit sensing beampattern under $\Gamma$ = 20 dB by considering Rayleigh fading channels. It is observed that our proposed designs with both Type-I and Type-II receivers allocate more transmit power into desired directions specified by the desired transmit beampattern, while the two-stage separate design leads to more undesired power leakage into uninterested anlges. Furthermore, the proposed joint design with Type-II receivers is observed to outperform that with Type-I receiver, with higher power in interested regions and lower leakage in uninterested angles. The benchmark scheme without $\mv{R}_d$ (via solving SDR3) is observed to have exactly the same beampattern as the joint design with Type-I receivers, which is consistent with Proposition \ref{T_30}.


\subsection{SINR-Constrained Minimum Beampattern Gain Maximization} \label{Subsection-B}

In this subsection, we consider the designs with minimum beampattern gain maximization. Instead of considering the specified $M$ point angles in Section \ref{Subsection-A},  here we only need to choose a subset of them (with positive desired beampattern gains) as the angles of interest $\Theta = \{ \theta_1, \theta_2, ..., \theta_Q\}$. In particular, in the simulation, $\Theta$ is composed of the angles with non-zero desired beampattern gains in Fig. \ref{fig:A-II-Tx-Beam}.

\begin{figure}[htb]
	\centering

	\setlength{\abovecaptionskip}{-4mm}
	\setlength{\belowcaptionskip}{-4mm}
	\includegraphics[width=3.8in]{figures/Tx_beampattern_max_min_Rich-eps-converted-to.pdf}
	\caption{Obtained transmit beampattern via the minimum beampattern gain maximization by considering Rayleigh fading channels with $\Gamma$ = 20dB. The angles of interest $\Theta$ are equivalently specified by the black dotted line.} \label{fig:B-II-Rich-Beam}
\end{figure}

Fig. \ref{fig:B-II-Rich-Beam} shows the obtained transmit beampattern via the minimum beampattern gain maximization. It is observed that compared with Type-I receivers, the transmit beampattern with Type-II receivers has higher beampattern gains at all angles of interest and lower undesired power leakage at those uninterested angles, thus showing the benefit of dedicated radar signal when the radar interference cancellation is implemented. Furthermore, the transmit beampattern achieved by (SDR6) without $\mv{R}_d$ is observed to be the same as that with Type-I receivers (via (P4)/(SDR4)). This is consistent with Proposition \ref{T_60}.


\subsection{Joint Comparision} \label{Subsection-C}

In this section, we will compare the performance of all the proposed design with both types of receivers under two design criteria.

\begin{figure}[htb]
	\centering
	\setlength{\abovecaptionskip}{-4mm}
	\setlength{\belowcaptionskip}{-4mm}
	\setlength{\abovecaptionskip}{+3pt}
	\subfigure[The case with Rayleigh fading channels]{ \label{fig:B-I-Rich-a}
		\includegraphics[width=3.1in]{figures/Beampattern_gain_SINR_all_Rich_2-eps-converted-to.pdf}}
	\subfigure[The case with LOS channels]{ \label{fig:B-I-Rich-b}
		\includegraphics[width=3.1in]{figures/Beampattern_gain_SINR_all_LOS-eps-converted-to.pdf}}
	\caption{The minimum beampattern gain versus SINR threshold $\Gamma$. }

	\label{fig:B-I-Rich}
\end{figure}


Figs. \ref{fig:B-I-Rich-a} and \ref{fig:B-I-Rich-b} show the obtained minimum beampattern gain in $\Theta$ versus the SINR threshold $\Gamma$ by considering the Rayleigh fading and LOS channels, respectively, where  $N = 8$ and $K = 5$. The results are obtained by averaging over 50 random channel realizations. It is observed that the minimum beampattern gain maximization designs achieve much higher worst-case beampattern gains at angles of interest than the beampattern matching designs. In particular, under both design criteria, the case with Type-II receivers is observed to have much better performance than other schemes, especially when the SINR threshold becomes high (e.g., $\Gamma \geq 10 $dB), thanks to its capability of cancelling the interference caused by radar signals.
Furthermore, in Fig. \ref{fig:B-I-Rich-a} and Fig. \ref{fig:B-I-Rich-b}, it is observed that similar as the case under SINR-constrained beampattern matching criterion, the performance achieved via solving (P4)/(SDR4) is exactly same as that by (SDR6), which is consistent with Proposition \ref{T_60}. Besides, under Rayleigh fading channels, it is observed in the simulations that the solution to (SDR4) always satisfies that $\mv{R}_d = 0$ and $\text{rank}(\mv{T}_i) = 1, \forall i \in \mathcal{K}$ when $\Gamma \geq 3$ dB and thus there is no need to add dedicated radar signals in this case. This is similar to the scenario with Type-I receivers under LOS channels, where it follows from Propositions \ref{T_60} and \ref{T_LOS2} that there is no need to add dedicated radar signal over the whole SINR range. 






\begin{figure}[htb]
	\centering

	\setlength{\abovecaptionskip}{-4mm}
	\setlength{\belowcaptionskip}{-4mm}
	\includegraphics[width=3.8in]{figures/Tx_beampattern_all_Rich-eps-converted-to.pdf}
	\caption{Obtained transmit beampattern by the two different design criteria, by considering the Rayleigh fading channels with $\Gamma$ = 20dB. The black dotted line specifies the desired transmit beampattern $\tilde{\mathcal{P}}(\theta)$.} \label{fig:B-III-Rich}
\end{figure}
Fig. \ref{fig:B-III-Rich} shows the obtained transmit beampattern by the beampattern matching and minimum beampattern gain maximization designs under Type-II receivers. It is observed that both designs allocate more power at angles of interest while reducing the leakage to other angles, as compared with the two-stage separate design approach. Furthermore, it is observed that the minimum beampattern gain maximization achieves more balanced beampattern gains over the angles of interest than the beampattern matching design.
\begin{figure}[htb]
	\centering

	\setlength{\abovecaptionskip}{-4mm}
	\setlength{\belowcaptionskip}{-4mm}
	\setlength{\abovecaptionskip}{+3pt}
	\subfigure[The case with Rayleigh fading channels]{ \label{fig:B-IV-Rich-a}
		\includegraphics[width=3.1in]{figures/Beampattern_gain_Users_all_Rich-eps-converted-to.pdf}}
	\subfigure[The case with LOS channels]{ \label{fig:B-IV-Rich-b}
		\includegraphics[width=3.1in]{figures/Beampattern_gain_Users_all_LOS-eps-converted-to.pdf}}
	\caption{Minimum beampattern gain at angles of interest $\Theta$ obtained by five different methods with $\Gamma$ = 20dB.} \label{fig:B-IV-Rich}
\end{figure}

Fig. \ref{fig:B-IV-Rich} shows the minimum beampattern gain at $\Theta$ versus the number of communication users $K$, where we set the SINR threshold as $\Gamma = 20$ dB and the number of antennas as $N=8$. The results in Figs. \ref{fig:B-IV-Rich-a} and \ref{fig:B-IV-Rich-b} are obtained by averaging over 50 Rayleigh and LOS channel realizations, respectively. As number of users increases, the minimum beampattern gain at $\Theta$ are observed to decrease. The proposed minimum beampattern gain maximization design with Type-II receivers is observed to outperform all other approaches under both Rayleigh fading and LOS channels.

Finally, Fig. \ref{fig:C-III-Rich} compares the computational complexity in terms of execution time between the minimum beampattern gain maximization (by solving (SDR4)) versus the beampattern matching (by solving (SDR1)), which is obtained by averaging over 50 random realizations. It is observed that as the number of antennas $N$ increases, the execution time of solving (SDR1) grows much faster than that of solving (SDR4), indicating the benefit of the minimum beampattern gain maximization in terms of the implementation complexity. This is consistent with the analysis in Section \ref{Section_complexity}.
\begin{figure}[htb]
	\centering

	\setlength{\abovecaptionskip}{-4mm}
	\setlength{\belowcaptionskip}{-4mm}
	\includegraphics[width=3.6in]{figures/Complexity_two_criteria-eps-converted-to.pdf}
	\caption{The execution time versus the number of antennas $N$ at the BS.} \label{fig:C-III-Rich}
\end{figure}

\section{Conclusion} \label{Section_conclusion}

This paper considered a downlink ISAC system, in which one BS equipped with ULA jointly designs the information and radar transmit beamforming to perform downlink multiuser communication and target sensing simultaneously. We studied two design criteria - the beampattern matching and the newly proposed low complexity one aiming to maximize the minimum beampattern gain. By considering the addition of dedicated radar signals, we introduced two types of communication receivers (Type-I and Type-II) without and with the capability of radar interference cancellation. Though non-convex in general, we optimally solved the two design problems under both types of receivers, by applying the technique of SDR.  Under both design criteria, it is revealed that equipping the ISAC system with dedicated radar signal in general enhances the performance under both types of receivers, and Type-II receivers yield better performance compared with Type-I receivers. Furthermore, when the wireless channels are LOS, dedicated radar signal is not needed with Type-I receivers. It was shown that the minimum beampattern gain maximization with Type-II receivers achieves the best joint sensing and communication performance with significantly lower computational complexity. 

