I Introduction
Communication and radar spectrum sharing (CRSS) has recently drawn significant attention due to the scarcity of the commercial wireless spectrum. For instance, the millimeter wave (mmWave) band is occupied by variety of radars [7786130], and has also been assigned as a new licensed band to the 5G network [7000981]. It is wellrecognized that communication and radar signals have some common features in their waveforms. Although their purposes are dramatically different, it is feasible to use one type of signal for the other type’s purpose. Nevertheless, the use of radar (communication) signals for communication (radar) functionalities, introduces a number of challenges [8828023, 8352726, 8737000]. To address these challenges, the research of dualfunctional radarcommunication (DFRC) is wellunderway [8828016, 8642926, 9005192].
In general, the aim of the DFRC is to implement both communication and radar functionalities on the same hardware platform. Based on information theory, the work in [7279172]
unified the radar and communication performance metric and discussed the performance bounds of the DFRC system. Furthermore, the weighted sum of the estimation and communication rates was analyzed as the performance metrics in the DFRC system
[7855671]. By leveraging the simple timedivision scheme, the radar and the communication signals can be transmitted within different time slots, which avoids the mutual interference [5483108]. To exploit the favorable timefrequency decoupling property of the orthogonal frequency division multiplexing (OFDM) waveform, the OFDM communication signal was adopted for target detection, where the range and Doppler processing are independent with each other [5776640]. From a signal processing perspective, the implementation of mmWave DFRC systems was fully studied in [8828030]. The unimodular signal design was discussed in [9119137] for DFRC architecture, where the information of downlink communication was modulated via the ambiguity function (AF) sidelobe nulling in the prescribed rangeDoppler cells.Beamforming design is essential to improve the performance of the DFRC signal processing in the spatial domain, which has been widely studied in the literature. Aiming to realize the dual functionalities, the work of [7347464] designed a transmit beampattern for multiple input multiple output (MIMO) radar with the communication information being embedded into the sidelobes of the radar beampattern. Considering both the separated and the shared antenna deployments, a series of optimizationbased transmit beamforming approaches for the DFRC system were studied in [8288677], where the communication signal was exploited for target detection. By imposing the constraints of the radar waveform similarity and the constant modulus, the interference of the multiple communication users (CUs) was suppressed to improve the communication performance in [8386661]. Based on IEEE 802.11ad wireless local area network (WLAN) protocol, a joint waveform for automotive radar and a potential mmWave vehicular communication system were proposed in [8114253]. The work of [8309274] further studied the feasibility of an opportunistic radar, which exploited the probing signals transmitted during the sector level sweep of the IEEE 802.11ad beamforming training protocol. In order to reduce the hardware complexity and the associated costs incurred in the mmWave massive MIMO system, a hybrid analogdigital beamforming structure was proposed for the DFRC transmission in [8999605].
It is worth pointing out that all the aforementioned works on the DFRC beamforming design focused on formulating a desired transmit beampattern without considering the receive beamformimg. To the best of our knowledge, the joint optimization of the transmit and receive beamforming under the communication constraints has never been studied for the DFRC system before, despite the fact that the joint optimization of transmit and receive beamforming for the MIMO radar system has been extensively investigated in the recent literature [4840496, 6649991, 7450660, 7762192, 8239836, 9034082, 8826594]. To detect an extended target, the joint optimization of waveforms and receiving filters in the MIMO radar was considered in [4840496]. In order to guarantee the constant modulus and similarity properties of the radar waveforms, numerous approaches were provided to maximize the signal to interference plus noise (SINR) of the radar, e.g., the sequential optimization algorithms (SOAs) in [6649991], the successive quadratically constrained quadratic programming (QCQP) refinement method in [7450660], the block coordinate descent (BCD) framework in [7762192], and the general majorizationminimization (MM) framework in [8239836]. To detect multiple targets, the joint optimization of waveforms and receiving filters in the MIMO radar was further studied in [9034082]. The problem of beampattern synthesis with sidelobe control was studied in [8826594] using constant modulus weights. The jointly design of the transmit and receive beamforming was provided in [6656878] based on a priori information on the locations of target and interferences.
All these works improved the receive SINR of the echo signal based on the prior information of the target and clutter. In contrast, for the DFRC system, there exist both probing signal and communication signal, which are coupled together with each other for drastically different purposes. As a consequence, the known MIMO radaronly designs are inapplicable for the latter. In this paper, we study the joint optimization of transmit and receive beamforming for the DFRC system. Specifically, the SINR of the radar is maximized under the SINR constraints of the CUs. Depending on the component of the DFRC transmit signal, we consider both the nondedicated probing signal case and the dedicated probing signal case. For the nondedicated probing signal case, the DFRC transmit signal is only composed of the communication signal of the CUs. For the dedicated probing signal case, besides the communication signal, the dedicated probing signal is added to the DFRC transmit signal to improve the radar performance. Under the single CU scenario, closedform solutions of the optimized beamforming are provided for both cases. And it can be proved that there is no need to employ dedicated probing signal for the single CU scenario. On top of that, we consider a more complicated scenario with multiple CUs. For the nondedicated probing signal case, the beamforming design is formulated as a nonconvex quadratically constrained quadratic programming (QCQP), and we show that the globally optimal solution can be obtained by applying semidefinite relaxation (SDR) with rank1 property. For the dedicated probing signal case, the rank1 property after applying SDR can be also proved, and the corresponding optimal solution shows that the dedicated probing signal should be employed to improve the SINR of the radar receiver. The main contributions of this paper can be summarized as follows.

Transmitreceive DFRC beamforming: We provide the solutions of the joint transmitreceive beamforming optimization to maximize the SINR of the radar under the SINR constraints of the CUs. For the single CU scenario, the closedform solutions of the optimized beamforming are derived. For the multiple CUs scenario, the optimal solutions are obtained by applying SDR with rank1 property.

Dualfunctional performance tradoff: The optimal performance tradeoff between the radar and the communication is characterized in terms of SINR. Compared to the timesharing scheme between the radar and the communication, the joint optimization of transmit and receive beamforming yields a more favorable tradeoff performance.

Dedicated radar probing signal or not: Compared to the existing works only exploiting communication signals for radar functionality, we consider the use of a dedicated radar probing signal to improve the radar performance. For the single CU scenario, our analysis shows that there is no need of dedicated probing signal. For the multiCU scenario, on the other hand, it is beneficial to employ the dedicated probing signal.
The remainder of the paper is organized as follows. Section II presents the system model. Section III studies a simplified single CU scenario. The study is further extended to a more complicated multiple CUs scenario in Section IV. Simulation results are provided in Section V, followed by concluding remarks in Section VI.
Notation
: We use boldface lowercase letter to denote column vectors, and boldface uppercase letters to denote matrices. Superscripts
and stand for Hermitian transpose and transpose, respectively. and represent the trace operation and the rank operator, respectively. is the set of complexvalued matrices. means thatobeys a complex Gaussian distribution with mean
and covariance . denotes the statistical expectation. denotes the Euclidean norm of a complex vector .Ii System Model
As illustrated in Fig. 1, we consider a DFRC MIMO system, which simultaneously probes the radar target and transmits information to the CUs. To be specific, it is composed of a DFRC base station (BS) with transmit antennas and receive antennas, singleantenna CUs indexed by .
Suppose there is a target and signaldependent interference sources indexed by . The target is located at angle and the interference sources are located at angle . Given the transmit signal , the received signal of the radar receiver is
(1)  
where and are the complex amplitudes of the target and the th interference source, respectively, and with and being the spacing between adjacent antennas normalized by the wavelength, respectively, and is the additive white Gaussian noise (AWGN) with each element subjects to . The symbol index is omitted for simplicity. Then, the output of the radar receiver is
(2)  
where is the receive beamforming vector for SINR maximization.
Further, given the transmit signal , the received signal of the CU is
(3) 
where is the multiple input single output (MISO) channel vector between the DFRC BS and the CU , and is the AWGN of the CU .
In this paper, we consider two cases according to the component of the DFRC BS’s transmit signal .

Case 1 (Nondedicated probing signal): In this case, the transmit signal of the DFRC BS is only composed of the communication signals of the CUs. That is
(4) where is the communication signals of the CU and the radar functionality is realized by the sum of the CUs’ communication signal.

Case 2 (Dedicated probing signal): In this case, the transmit signal of the DFRC BS is composed of both the communication signal of the CUs and the dedicated probing signal. That is
(5) where is the dedicated probing signal to enhance the radar performance.
In addition, we impose the following two assumptions in this paper. 1) For the radar function, the angles of the target and the interference are assumed to be known to the DFRC BS. 2) For the communication function, the channel is assumed to be known to the DFRC BS, and the dedicated probing signal is pseudorandom and assumed to be known in prior to the CUs.
Iii Single CU Scenario
In this section, we consider a simplified scenario with single CU in the network. The beamforming design of the DFRC BS is discussed for the nondedicated probing signal case, and the closedform solution is provided. Then, the dedicated probing signal case is studied, and it can be proved that there is no need of dedicated probing signal with single CU.
Iiia Nondedicated Probing Signal Case
For the nondedicated probing signal case, the transmit signal of the DFRC BS in (4) with single CU can be rewritten as
(6) 
where and are the beamforming vector and the information symbol of the CU, respectively.
According to the output in (2), the SINR of the radar receiver can be expressed as
(7)  
And the output SINR of the radar receiver depends on the choice of the receive beamforming vector . The design of can be expressed as
(8) 
which is equivalent to the wellknow minimum variance distortionless response (MVDR) problem, and its solution can be given by
[4840496](9) 
where
(10) 
(11) 
where
(12) 
And the average SINR of the radar receiver can be given by
(13) 
For the CU , the received signal in (3) can be rewritten as
(14) 
where is the MISO channel vector between the DFRC BS and the CU. And the average SINR of the CU can be calculated as
(15) 
Then, we consider the beamforming optimization problem that maximizes the SINR of the radar receiver and satisfies the SINR of the CU, i.e.,
(16) 
where is the threshold of the CU’s SINR, and is the transmit power constraint of the DFRC BS.
Because is a nonlinear function of the transmit beamforming vector , Problem (P1.1) is generally nonconvex. Thus, we adopt the sequential optimization to find the transmit beamforming vector in an iterative fashion. Specifically, at the th iteration, we first compute , where is obtained in the th iteration. Thus, Problem (P1.1) can be rewritten as
(17) 
And the closedform solution of Problem (P1.2) can be given by the following proposition.
Proposition 1.
(Optimal beamforming with single CU) For the nondedicated probing signal case, the optimal beamforming vector of the DFRC BS with single CU i.e., the optimal solution to Problem (P1.2), can be given by
(18) 
(19) 
(20) 
where
is the dominant eigenvector of
, , , denoting the projection of into the null space of , and can be expressed as .Proof.
The proof is given in Appendix A. ∎
IiiB Dedicated Probing Signal Case
For the dedicated probing signal case, the transmit signal of the DFRC BS in (5) with single CU can be rewritten as
(21) 
where and are the beamforming vector and the symbol of the dedicated probing signal, respectively. In addition, and are independent and identically distributed (i.i.d.).
According to the output in (2), the SINR of the radar receiver can be expressed as
(22)  
By solving an equivalent MVDR problem, the corresponding receive beamforming vector to maximize the output SINR can be given by
(23) 
where
(24) 
(25) 
where
(26) 
And the average SINR of the radar receiver can be given by
(27) 
For the CU, it has a priori information of probing signal. After probing signal interference cancelling, its received SINR can also be expressed as (15). The beamforming optimization problem that maximizes the SINR of the radar receiver and ensures the SINR requirement of the CU can be expressed as
(28) 
Similarly, the sequential optimization can be adopted to find the transmit beamforming vector and in an iterative fashion, where we first compute at the th iteration with and being obtained from the th iteration. Thus, the beamforming optimization problem can be given by
(29) 
And the closedform solution of Problem (P2.2) can be given by the following proposition.
Proposition 2.
(No need of dedicated probing signal with single CU) For the dedicated probing signal case, the optimal beamforming vector of the dedicated probing signal with single CU, i.e., the optimal solution to Problem (P2.2), can be given by . Thus, there is no need to design the dedicated probing signal with single CU, and the optimal beamforming vector of the communication signal is also given by (18).
Proof.
The proof is given in Appendix B. ∎
Finally, the beamforming optimization algorithm for the single CU scencario can be summarized as Algorithm 1, which repeatedly updates based on until the improvement of the radar receiver’s SINR becomes insignificant.
Iv Multiple CUs Scenario
In this section, we proceed to consider a more complicated scenario with multiple CUs in the network. For the nondedicated probing signal case, the beamforming design is formulated as a nonconvex QCQP, and the globally optimal solutions can be obtained by applying SDR with rank1 property. For the dedicated probing signal case, the rank1 property after applying SDR can also be proved, and the corresponding optimal solution shows that the dedicated probing signal should be employed to improve the SINR of the radar receiver.
Iva Nondedicated Probing Signal Case
For the nondedicated probing signal case, the transmit signal of the DFRC BS in (4) with multiple CUs can be given by
(30) 
where and are the beamforming vector and the i.i.d. information symbol of the CU , respectively.
According to the output in (2), the SINR of the radar receiver can be written as
(31)  
By solving an equivalent MVDR problem, the corresponding receive beamforming vector to maximize the output SINR can be given by
(32) 
where
(34) 
where
(35) 
And the average SINR of the radar receiver can be given by
(36)  
The received signal of the CU in (3) can be rewritten as
(37) 
and the average SINR of the CU can be calculated as
(38) 
Again, we consider the beamforming optimization problem that maximizes the SINR of the radar receiver by imposing the individual SINR constraints of the CUs, i.e.,
(39) 
Note that Problem (P3.1) is also nonconvex. Thus, we adopt the sequential optimization to find the transmit beamforming vector in an iterative way. Specifically, at the th iteration, we first compute , where is obtained in the th iteration. As a result, Problem (P3.1) can be rewritten as
(40) 
While Problem (P3.2) is still a nonconvex QCQP, and we can solve it via SDR with , i.e,
(41) 
where .
Although the rank1 constraints have been removed for the convexity of the problem, the optimal solution can be proved to have the rank1 property by the following proposition.
Proposition 3.
(Rank1 property of Problem (P3.3)) For the nondedicated probing signal case with multiple CUs, there is always a solution to Problem (P3.3) satisfying that . Thus, the optimal beamforming vector for the CU , i.e., the optimal solution to Problem (P3.2) can be given by with .
Proof.
The proof is given in Appendix C. ∎
Above all, the beamforming optimization algorithm for the multiCU scenario and the nondedicated probing signal case can be provided as Algorithm 2. It repeatedly updates given until convergence.
IvB Dedicated Probing Signal Case
For the dedicated probing signal case, the transmit signal of the DFRC BS in (5) with multiple CUs can be given by
(42) 
where and are the beamforming vector and the symbol of the dedicated probing signal, respectively. And and are i.i.d..
According to the output in (2), the SINR of the radar receiver can be expressed as
(43)  
By solving an equivalent MVDR problem, the corresponding receive beamforming vector to maximize the output SINR can be given by
(44) 
where
(46) 
where
(47) 
And the average SINR of the radar receiver can be given by
(48)  
For the CU , it has a priori information on the probing signal. After the probing signal interference cancelling, its received SINR can also be expressed as (38). Thus, the beamforming optimization problem that maximizes the SINR of the radar receiver and satisfies the SINR constraints of the CUs can be formulated as
(49) 
Let us then employ the sequential optimization to find the transmit beamforming vector and in an iterative fashion, where we first compute at the th iteration with and being obtained from the th iteration. Thus, the beamforming optimization problem can be again formulated as
(50) 
which can be solved using the SDR by letting and , i.e,
(51) 
Although the rank1 constraints have been removed for the convexity of the problem, the optimal solution can be guaranteed to have rank1 property by the following proposition. Furthermore, the dedicated probing signal should be employed according to the following proposition.
Proposition 4.
(Rank1 property of Problem (P4.3)) For the nondedicated probing signal case with multiple CUs, there is always a solution to Problem (P4.3) satisfying that and . Thus, the optimal beamforming vector for the CU and the dedicated probing signal, i.e., the optimal solution to Problem (P4.2) can be given by and with and . Specifically, , where , is the dominant eigenvector of , and .
Proof.
The proof is given in Appendix D. ∎
Remark 1.
(Dedicated probing signal is employed or not) It can be observed that any feasible solution to Problem (4.2) is also feasible for Problem (3.2) with , and vice versa. If , a higher SINR of radar receiver can be achievable, which will also be verified by the simulation results in the next section. Therefore, it is beneficial to employ the dedicated probing signal for multiple CUs scenario. The benefit is achieved at the cost of implementing an additional interference cancellation with a priori known probing signals by all CUs.
Finally, we can also use the solution and to update and , and it is repeated until the improvement of the radar receiver’s SINR becomes insignificant as illustrated in Algorithm 3.
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