Multi-Antenna Joint Radar and Communications: Precoder Optimization and Weighted Sum-Rate vs Probing Power Tradeoff

In order to further exploit the potential of joint multi-antenna radar-communication (RadCom) system, we propose two transmission techniques respectively based on separated and shared antenna deployments. Both techniques are designed to maximize the weighted sum rate (WSR) and the probing power at target's location under average power constraints at the antennas such that the system can simultaneously communicate with downlink users and detect the target within the same frequency band. Based on a Weighted Minimized Mean Square Errors (WMMSE) method, the separated deployment transmission is designed via semidefinite programming (SDP) while the shared deployment problem is solved by majorization-minimization (MM) algorithm. Numerical results show that the shared deployment outperforms the separated deployment in radar beamforming. The tradeoffs between WSR and probing power at target are compared among both proposed transmissions and two practically simpler dual-function implementations i.e., time division and frequency division. Results show that although the separated deployment enables spectrum sharing, it experiences a performance loss compared with frequency division, while the shared deployment outperforms both and surpasses time division in certain conditions.

coordinated design of both existing MIMO communication systems and MIMO radar systems to achieve coexistence. Given the existing infrastructure, a coexistence approach manages interference between radar and communication as much as it can. However, for uncoordinated coexistence design, some important phenomena are not considered in the simplified scenarios [11], while for coordinated coexistence, governmental and military agencies might be unwilling to upgrade the existing deployment [12].

B. DUAL-FUNCTION SYSTEM DESIGN
Accounting for the possible drawbacks aforementioned, designing a dual-function system that makes the best use of the spectrum for both detecting and communicating might be a better alternative. Early studies [13], [14] consider singleantenna dual-function platforms without utilizing multiantenna processing. Then, based on the waveform diversity of MIMO radar and the concept of space-division multiple access (SDMA) in MIMO communication, [15], [16] embed the information stream into radar pulses via a multi-antenna platform, detecting targets at the mainlobe and transmitting information streams at sidelobe. The sidelobe level is modulated via amplitude shift keying (ASK), where different powers correspond to different communication symbols in [15]. Likewise, [16] also develops phase shift keying (PSK) in this system by representing the symbols as the different phases of the signals received at the angle of the sidelobe. One significant restriction of such a dual-function system is that the rate is limited by the Pulse Repetition Frequency (PRF), which is far from satisfactory for communication requirements. To overcome this problem, [17] proposes a joint multi-antenna radar-communication (RadCom) system defined as a dual-function platform simultaneously transmitting probing signals to radar targets and serving multiple downlink users. Both functions are realized within the same frequency band. Specifically, two antenna deployments are mentioned. Separated deployment splits the antennas into two groups respectively working as MIMO radar and BS, while the shared deployment only transmits communication streams and the precoders are designed to form a desired radar beampattern and meet the SINR requirements for communication users. However, only communication SINR is adopted as a metric, but a more representative communication performance metric such as rate is not considered in this work. In addition, the performance of joint RadCom has not been compared with practically simpler implementation using orthogonal resources in time or frequency to fulfill the dual function, which is an essential criterion to decide where joint RadCom is worth the efforts.

C. CONTRIBUTION
In this paper, we propose two multi-antenna RadCom transmission design techniques based on separated and shared antenna deployments respectively. Both techniques enable the platform to simultaneously communicate with downlink users and probe one target of interest within the same frequency band. Major contributions are summarized as follows.
1) We propose transmission techniques that maximize the weighted sum rate (WSR) of communication and the probing power at target's location for both separated and shared deployments. Since WSR is the most representative metric of a communications system, we consider WSR maximization instead of SINR constraint at each user in [17]. To the best of our knowledge, we are the first to consider WSR maximization in the system model with precoders. We also consider probing power maximization at the target's location rather than turning this metric into beampattern approximation problem in [17]. This makes our transmission design more direct for typical MIMO radar tracking and scanning mode, and enables a more clear tradeoff comparison. However, by adopting WSR and probing power at target, the transmission design problem becomes difficult. Specifically, maximizing WSR as a sum of logarithms and probing power as a quadratic form under power constraints makes the optimization problem highly non-convex and intractable. 2) We propose WMMSE-SDP and WMMSE-MM algorithms respectively to solve the two proposed transmission design problems. For separated deployment, we propose to reformulate the problem into semidefinite programming (SDP) based on Weighted Minimized Mean Square Errors (WMMSE) method. For the shared deployment, nonconvex per-antenna power constraint makes the design problem even more difficult. We propose a majorization-minimization (MM) iterative algorithm based on WMMSE to effectively solve the problem. 3) We compare the performance of the proposed transmission techniques with practically simpler time-division and frequency-division dual-function implementations.
In order to provide a well-rounded evaluation of our proposed techniques, we compare the tradeoffs of both separated and shared multi-antenna RadCom deployments with time-division and frequency-division dualfunction implementation which might also be practical options because of plain and easy realization. Separated deployment has an advantage of realizing spectrum sharing compared with frequency division, but is surpassed by the latter in tradeoff performance. In contrast, the shared deployment outperforms frequency division with a significant tradeoff gain, and exceeds time division in certain conditions.

D. ORGANIZATION
The rest of the paper is organized as follows. The system models and metrics of both separated and shared deployments are illustrated in Section II. In Section III, optimization problems of transmission designs for both deployments are formulated. Algorithms for solving the optimization problems are subsequently presented in Section IV. Section V demonstrates the simulation results and analysis. Section VI concludes the paper.

II. SYSTEM MODEL AND METRICS
In this work, we adopt the separated and shared deployment models of [17], where either deployment works simultaneously as a BS serving downlink users and a collocated MIMO radar probing the target of interest. Both deployments are equipped with a total of N t antennas, serving K singleantenna users indexed as K = {1, . . . , K}. Typically, we assume that both deployments use a uniform linear array (ULA) in our system model. The total power budget for either deployment is P t . We assume that the RadCom system works in a tracking mode as a radar, where there is typically one target of interest at the azimuth angle of θ m [18]. Beamforming is thus expected.  The separated deployment splits the antennas into two groups, i.e., a group of N tr antennas only transmitting radar signals and the other group of N tc antennas only transmitting communication signals. Both the communication precoders and the radar signals are designed to fulfill the dual function. The schematic diagram is shown in Fig. 1(a).
The received signal at user-k can be expressed as where h k ∈ C Ntc×1 and f k ∈ C Ntr×1 are respectively the non-line-of-sight (NLOS) channel vectors from communication antennas and radar antennas to user-k. s j [l] and n j [l] ∼ CN (0, σ 2 n ) are the communication symbols and receiving noise of user-j at the time index l. Without loss of generality, we assume σ 2 n = 1. p j ∈ C Ntc×1 is the precoder for user-j, and r l ∈ C Ntr×1 is the lth snapshot of radar antennas. The covariance matrix of transmit radar signal is R x = 1 L r l r H l with L be the length of signal on fast-time axis.
We first introduce WSR as the communication metric. The SINR of decoding s k at user-k is (2) where P = [p 1 , . . . , p K ] is the precoder matrix of the separated deployment. Therefore, the achievable rate at userk in the separated deployment can be denoted as Denoting the rate weight of user-k as µ k , WSR of the separated deployment is k∈K µ k R S k (P, R x ). Then, we consider our dual-function system works as a MIMO radar where the target location is known or estimated and we aim at maximizing cumulated power of the probing signals at the target location θ m . Note that this is a simple and classic MIMO radar beamforming scenario illustrated in [19]. The probing power is where a(θ m ) = [1, e j2πδsin(θm) , . . . , e j2π(Nt−1)δsin(θm) ] T ∈ C Nt×1 is the transmit steering vector of ULA, and δ is the normalized distance (relative to wavelength) between adjacent array elements. For other array structures, the expression of a(θ m ) needs to be changed. C t ∈ C Nt×Nt is the overall transmit covariance matrix. Assuming the radar signals are statistically independent with communication signals, we have Thus (4) is reformulated as

B. SHARED DEPLOYMENT
For the shared deployment, N t antennas all transmit precoded communication streams only and fulfill the dual function.
The schematic diagram is shown in Fig. 1(b). In this deployment, only the precoders are designed. The received signal at user-k is whereȟ k ∈ C Nt×1 is the channel vector between the shared system and user-k. Differently, without the interference caused by radar signals, the SINR at user-k is whereP = [p 1 , . . . ,p K ] is the precoder matrix for the shared deployment. Thus, the rate at user-k is and WSR of the shared deployment is k∈K µ k R S k (P, R x ). VOLUME 4, 2016 In the shared deployment, probing power at the target location θ m is P J T (θ m ) = a H (θ m )PP H a(θ m ). Although we do not focus on the radar matched filter design in this work, the transmit signal of shared deployment coincides with the transmit model of one collocated MIMO radar in [18] where the matched filter settings for radar detection can be referred to as an option. It is also derived in [18] that maximizing output signal-to-noise-ratio (SNR) of the detector is equivalent to maximizing P J T (θ m ).

III. PROBLEM FORMULATION
The transmission design problem for the separated deployment can be expressed as where P is the precoders of communication streams, R x is the transmit covariance matrix of radar signals, P c and P r are the transmit power budgets of radar and communication sub-arrays respectively. The first counterpart of the objective function (9a) represents WSR while the rest parts denote probing power at the target. Both metrics are maximized via regularization with a parameter ρ. Although communication and radar signals are separately transmitted by two sub-systems, they show mutual effects on each other when operating simultaneously, i.e., radar signals cause interference to communication users and communication signals are supposed to help probe the target. Constraint (9b) is the uniform elemental power constraint in radar implementation [19], and (9c) is the total power constraint in communication implementation. (9d) restricts R x to be semi-definite. However, the dual function can also be fulfilled by the shared deployment, of which the transmission design problem can be formulated as Likewise, WSR and probing power maximization are combined in the objective function via regularization. There is also an elemental power constraint for all antennas. The total power budget constraint for communication is omitted because it is certainly satisfied when the elemental power restriction is met. It needs pointing out that the maximum probing power design approach used in (9d) and (10) might have drawbacks when extended to the scenario of multiple targets according to [19].

IV. WMMSE-BASED SOLVING ALGORITHMS
It is clear that both separated and shared transmission design problems (9) and (10) are non-convex because of the intractable form of WSR and maximizing a quadratic power function in objective functions. However, this problem can be reformulated using the WMMSE approach and solved through the WMMSE-based Alternating Optimization (WMMSE-AO) algorithm following [20].

A. WMMSE-SDP ALGORITHM FOR SEPARATED TRANSMISSION
We decode s k at user-k via an equalizer g k , and get the estimationŝ k of s k asŝ k = g k y S k . Subsequently, the Mean Square Errors (MSE) of estimation, defined as ε k Optimum equalizers are obtained by letting ∂ε k ∂g k = 0, which are also the MMSE equalizers given by Minimized MSEs (MMSEs) based on g MMSE k are given by Hence, by comparing (13) with (2), we rewrite SINRs of decoding the intended streams at user-k as γ S By allocating a positive weight w k to user-k's rate, we define the augmented WMSEs as ξ k w k ε k − log 2 (w k ). After optimizing over the equalizers and weights, the Rate-WMMSE relationships are where the optimum equalizers and the optimum weights are resulting from meeting the first order optimality conditions. Using the rate-WMMSE relationships, we can then reformulate (9) as min P,Rx,w,g where w = [w 1 , w 1 , . . . , w K ] is the vector of all MSE weights. g = [g 1 , g 2 , . . . , g K ] is the vector of all equalizers. It is worth noting that the second term −a H 2 (θ m ) PP H a 2 (θ m ) in (16a) is non-convex. To make it convex, we first reformulate this part as where we denote Z(θ m ) = N tc I − a 2 (θ m )a H 2 (θ m ). For the definition of steering vector in (4), it is clear that a 2 (θ m )a H 2 (θ m ) is a rank-1 matrix with the eigenvalue of a 2 (θ m ) 2 = N tc . Therefore, Z(θ m ) is semidefinite. By omitting the constant part, we have that minimiz- Note that when {w, g} are fixed, (18) is a semidefinite programming (SDP) convex problem that can be efficiently solved by CVX toolbox, and optimum {w * , g * } can be updated following (15). Therefore, we here use the WMMSEbased AO algorithm with details in [20] to solve the problem, which is summarized in Algorithm 1. After having the optimum R x , the radar snapshots can be further obtained using algorithms in [21].

B. WMMSE-MM ALGORITHM FOR SHARED TRANSMISSION
For the shared transmission design problem, we first follow the same path in Section IV.A and reformulate (10) with WMMSE method. To simplify, we omit the repetitive parts and directly give the reformulated problem as where The optimum equalizers and weights are respectively We can see that (19) is non-convex because of the quadratic equality constraint, which also makes it difficult to solve. In the following part, we propose an MM-based iterative algorithm to solve this non-convex problem.
At first, to reformulate the problem into a more explicit form, we define p v = vec(P) and D p,k = 0 Nt×(k−1)Nt I Nt 0 Nt×(K−k)Nt , k ∈ K. (22) Then, the objective function in (19) can be rewritten as Afterwards, (19) is equivalent to Nt . According to the MM framework [22], we then construct the majorization function of f (p v ). We first recall Lemma 1 from [23] that is Lemma 1: Let L, M be the n × n Hermitian matrices and M L. For any point

VOLUME 4, 2016
According to Lemma 1, we chose M = λ max (Q)I where λ max (Q) means the largest eigenvalue of Q, and have Here the equality is achieved at p v = p t v . By omitting the constant items in (26), we can subsequently construct the majorization function of f (p v ) as where q = k∈K µ kwkǧk D H p,kȟ k . Then, (25) can be solved by iterating However, (28) can be further investigated to find a closedform solution. First, we denotê Then, the optimization problem (28) is equivalent to In order to show the essence more clearly, we further denotẽ q j = [q j ,q Nt+j , . . . ,q (K−1)Nt+j ] T , j = 1, 2, . . . , N t , (31) whereq i and [p v ] j respectively denote the ith entry ofq and the jth entry of p v . We further define the real form as Then (30) can be reformulated as Following Cauchy-Schwartz inequality, we have where the last equality follows from the constraint p r j 2 2 = P t /N t . For the condition of the equality, it is obvious that the optimal solutionp r j andq r j should be colinear, i.e., Equivalently, we havẽ for j = 1 to N t do 9:q j = [q j ,q Nt+j , . . . ,q (K−1)Nt+j ] T ; ] v usingp * j by inverse operation of (32); 13: t + +; 14: until p

V. NUMERICAL RESULTS
In this section, we provide numerical results to validate the performance of both separated and shared transmission for joint multi-antenna RadCom system, and further reveal the advantages of shared transmission.
We set that the platform adopts a ULA where N t = 16 with half-wavelength spacing, and serves K = 4 downlink users. We assume the total transmit power budget is P t = 20dBm and the noise power at each user is 0dBm. Target location is set to be θ m =0°. The channel vectors of users are generated obeying the i.i.d. complex Gaussian distribution. In the separated deployment, radar and communication subsystems fairly share the available resources, i.e., N tc = N tr = 1 2 N t and P c = P r = 1 2 P t . Although radar power budget is normally higher than communication, an even power allocation here is more representative for comparison. Moreover, to ensure fair comparison between the separated and shared transmission, we set up the same channel environment in the simulations, i.e.,ȟ k = [f k ; h k ].

A. TRANSMIT BEAMPATTERN COMPARISON
In order to evaluate the performance of transmit beamforming at the target, we first demonstrate the beampattern obtained by the separated and shared transmission.  We can see in Fig. 2(a) that when WSR=2.4bps/Hz, shared transmission can nearly achieve the same beampattern as the MIMO radar equipped with the same number of antennas, showing a 3dB probing power gain at target's location over separated transmission. Fig. 2(b) displays that when WSR increases by 2.9bps/Hz, shared transmission experiences a 1.45dB loss of probing power at target. However, it is clear that shared transmission still keeps a 4.88dB gain over separated transmission, which is even larger than the gain in Fig. 2(a). Fig. 2 also reveals that there is a tradeoff between maximizing probing power at target and maximizing WSR. Fig. 3 shows the average transmit power at each antenna in both separated and shared transmissions corresponding to the two scenarios in Fig. 2. Recall that the first eight antennas in the separated deployment are intended for the radar function, it is clear that the per-antenna power constraints in both shared and separated transmissions are met successfully, which proves that our algorithms handle the power constraints effectively for both transmissions.

B. TRADEOFF COMPARISON
By varying the regularization parameter ρ in (9) and (10), we obtain the tradeoff between WSR and probing power at target for both the shared and the separated transmissions in Fig. 4 via Monte Carlo experiments.
To give a well-rounded comparison, we also provide in Fig. 4 two simple implementations that also achieve the dual function by orthogonalizing the resources in time (i.e. timedivision) or frequency (i.e. frequency-division).
Specifically, N t -antenna frequency-division implementation means that a N t -antenna system simultaneously transmits precoded communication streams and radar probing signals respectively with P t /2 power budget but within different frequency bands. There is thus no interference between the radar and communication functions because of the frequency orthogonality. To be fair, we assume the communication precoders are optimized via SDMA based on multi-user linear precoding (MU-LP) in [24], which maximizes WSR as well. N t -antenna time-division implementation means a system that spends a fraction α of time working as a N t -antenna BS with MU-LP and 1 − α of time working as a N tantenna MIMO radar with a power budget of P t on the same frequency band. Since radar and communication functions are realized orthogonally, WSR and probing power of both frequency-division and time-division implementations can be independently obtained by using classic methods in [24] and [19] respectively.
In Fig. 4, we can see that separated transmission experiences a considerable performance loss as the cost of realizing spectrum sharing. To be specific, separated transmission reaches the same achievable probing power as frequencydivision implementation, but sees an approximately 1bps/Hz WSR loss because it only uses half the number of antennas to transmit communication streams compared with frequencydivision. Also, separated transmission shows a dual-function tradeoff because of the interference imposed by radar signals on communication users, which is a cost of sharing the same band compared with frequency division. Therefore, although separated transmission meets the RadCom requirement, it seems not to be a wise choice as the resources could be used more efficiently to improve the overall performance.
In contrast, it is obvious that the shared transmission shows advantages compared with all dual-function implementations. First, it outperforms separated transmission with a maximum WSR gain of about 2bps/Hz, which results from that the former adopts twice the number of antennas and twice the power budget to transmit communication streams. We can also see that the shared transmission achieves at least 3dB gain of probing power at target given the same WSR. Second, it is clear that shared transmission surpasses frequency-division implementation with a maximum 3dB probing power gain or around 1bps/Hz WSR gain, with an additional advantage of realizing spectrum sharing. Third, as for time division, it needs pointing out that α varies depending on practical scenarios where the radar tracking and BS communication task are arranged based on specific demands. Therefore, for the convenience of comparison, we only provide a key baseline with α = 0.51. For larger α, time division outperforms shared transmission, but shared transmission still has the advantage of being able to fulfill the dual function simultaneously.

VI. CONCLUSION
To conclude, we propose two transmission design techniques that maximize WSR and probing power at target for both separated and shared RadCom deployments. We propose WMMSE-SDP and WMMSE-MM algorithms to solve the non-convex and intractable transmission design problems respectively. Numerical results show that our proposed algorithms are effective, and that shared deployment outperforms separated deployment given the same antenna number and power budget. Compared with practically simpler dualfunction implementations based on time/frequency division, both separated and shared transmissions have an advantage of being able to operate the dual function simultaneously within the same frequency band. Separated transmission is less efficient in exploiting the resource and experiences a considerable performance loss compared with frequency division. In contrast, shared transmission outperforms frequency division with a maximum 3dB probing power gain or 1bps/Hz WSR gain. However, in some conditions, shared transmission is surpassed by time-division implementation but still exceeds in the capability of operating the dual function simultaneously. He has authored two books, 190 peer-reviewed international research papers, and 150 standards contributions, and is the inventor of 80 issued or pending patents among which 15 have been adopted in the specifications of 4G standards and are used by billions of devices worldwide. His research area is communication theory and signal processing for wireless networks. He has been a TPC member, a symposium chair, or a TPC chair of many symposia on communication theory, signal processing for communication and wireless communication for several leading international IEEE conferences. He was an Elected Member of the IEEE Signal Processing Society SPCOM Technical Committee. He served as an Editor for the IEEE TRANS-ACTIONS ON COMMUNICATIONS, the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, and the IEEE TRANSACTIONS ON SIGNAL PROCESSING. He has also been a (lead) guest editor for special issues of the EURASIP Journal on Wireless Communications and Networking, IEEE ACCESS, the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS and the IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING. He was an Editor for the 3GPP LTE-Advanced Standard Technical Report on CoMP.
JIANYUN ZHANG received the B.S. degree in radar signal processing from the Electronic Engineering Institute, Hefei, China, in 1984, and the M.S. and Ph.D. degrees in digital signal processing from Xidian University, Xi'an, China, in 1989 and 1994, respectively. From 1995 to 2001, he was an Associate Professor with the Electronic Engineering Institute, where he became a Professor from 2001 to 2016. Since 2016, he has been a Professor with the College of Electronic Engineering, National University of Defense Technology. His research interests include high-speed digital signal processing, estimation theory, array signal processing, radar signal processing, and radar system theory. VOLUME 4, 2016