Transmit Beamforming for MIMO Dual Functional Radar-Communication With IQI

Multiple-input-multiple-output (MIMO) dual-func- tional radar communication (DFRC) provides a solution to the severe spectrum scarcity challenge. MIMO systems may be prone to the hardware impairments (HWI). Considering I/Q-imbalance (IQI), the most common type of HWI, we study the optimal transmit beamforming design for MIMO DFRC. Both radar-centric and communication-centric scenarios are investigated. For the radar-centric design, the radar beampattern and communciations user signal-to-interference-plus-noise ratio (SINR) with IQI are derived. Then, the radar beampattern is optimized by using the semidefinite relaxation (SDR) method. For the communication-centric design, the achievable rate with IQI is derived. Then, the transmit beamforming is optimized with the constraint on the MIMO radar receive signal-to-noise ratio (SNR). Moreover, the computational complexity is analyzed. Numerical results verify that IQI amplitude mismatch, IQI phase mismatch and the IQI mitigation error can significantly degrade the system's overall performance. Also, for the communication-centric scenario, IQI phase mismatch at the CUs is much more important than that at the BS.

Transmit Beamforming for MIMO Dual Functional Radar-Communication With IQI Junqiu Wang , Yunfei Chen , Senior Member, IEEE, and Li Chen Abstract-Multiple-input-multiple-output (MIMO) dual-functional radar communication (DFRC) provides a solution to the severe spectrum scarcity challenge.MIMO systems may be prone to the hardware impairments (HWI).Considering I/Q-imbalance (IQI), the most common type of HWI, we study the optimal transmit beamforming design for MIMO DFRC.Both radar-centric and communication-centric scenarios are investigated.For the radarcentric design, the radar beampattern and communciations user signal-to-interference-plus-noise ratio (SINR) with IQI are derived.Then, the radar beampattern is optimized by using the semidefinite relaxation (SDR) method.For the communication-centric design, the achievable rate with IQI is derived.Then, the transmit beamforming is optimized with the constraint on the MIMO radar receive signal-to-noise ratio (SNR).Moreover, the computational complexity is analyzed.Numerical results verify that IQI amplitude mismatch, IQI phase mismatch and the IQI mitigation error can significantly degrade the system's overall performance.Also, for the communication-centric scenario, IQI phase mismatch at the CUs is much more important than that at the BS.Index Terms-Achievable rate, beamforming, beampattern, dual-functional radar-communication, I/Q imbalance, zeroforcing.

I. INTRODUCTION
I N THE upcoming sixth-generation (6G) era, there will be a massive number of connected devices, which will require much more spectrum resources for operation.These requirements will impose a serious challenge on spectrum scarcity to limit the overall throughput of the communications systems [1].To alleviate this problem, joint radar and communications systems (JRC) have been proposed, allowing a variety of radar systems to share their spectrum with wireless communications systems [2].In particular, various dual-functional systems have been proposed [3].
Due to the inevitable limitations of traditional single-antenna systems, multiple-input-multiple-output (MIMO) dual-func-tional radar and communication (DFRC) has been extensively studied.One major approach to achieve MIMO DFRC is to design the waveform of the transmitted signals.The closedform globally optimal waveform designs for both omnidirectional and directional cases were obtained, and a weighted optimization was solved by a branch-and-bound algorithm under constant waveform modulus constraint in [4].A new technique for DFRC system enabling both sidelobe control of the transmit beamforming and waveform diversity was proposed to deliver information to multiple communication directions outside the radar's mainlobe [5].The subcarrier power in the waveform design problem for an orthogonal frequency division multiplexing (OFDM) DFRC system was studied in [6].Considering the range sidelobe control, a novel waveform design was proposed for MIMO DFRC system [7].The authors in [8] investigated the joint waveform design and passive beamforming in reconfigurable intelligent surface (RIS) assisted DFRC system to mitigate the high multi-user interference caused by the limited degrees of freedom of the waveform design.
In addition to waveforms, beamforming is also an efficient approach to achieve MIMO DFRC.The MIMO DFRC beamforming designs were studied to simultaneously detect radar targets and communicate with downlink communication with imperfect channel state information by maximizing the radar output power subject to probabilistic outage signal-to-interference-and-noise ratio (SINR) constraints in [9].A novel consensus alternating direction method of multipliers approach was proposed to deploy hybrid beamforming (HBF) for OFDM-DFRC system [10].Another HBF design along with direction-of-arrival estimation was proposed in multi-carrier DFRC systems to minimize the mean squared error between the generated spatial spectrum and the reference one under constraints of constant waveform modulus, communication quality of service and power as well as orthogonality [11].For millimeter wave MIMO DFRC scenario, a novel HBF was studied to minimize the gap between the realized radar beam pattern and the objective, which was subject to constraints of the total DFRC transmission power and the SINR of CU [12].The work of [13] proposed a joint design of transmit beamforming and receive filters for a coordinated two-cell network by formulating the non-convex optimization problem of minimizing the transmit power at two base stations (BSs) under SINR constraints.The authors in [14] investigated a DFRC scheme combining degrees of freedom in frequency and space to deliver digital information via index modulation.Beamforming has also been found to improve the security performance for MIMO DFRC systems.For example, the joint transmit waveform and receive beamforming design was proposed to maximize the radar SINR under the constraints of security and power budget when the radar target might be a potential eavesdropper [15].The transmit beamforming was designed to maximize the sum secrecy rate for more CUs leveraged by non-orthogonal multiple access in [16].Taking account of the physical layer security, a novel beamforming scheme was proposed and indicated that the corresponding DFRC radar waveforms could be regarded as the traditional artificial noise that was exploited for improved degrees of freedom and for drowning out the eavesdropping channel [17].
Moreover, DFRC has been incorporated in other 6 G technologies.In vehicle to everything, DFRC technique was employed to investigate a radar-assisted predictive beamforming design [18].The authors in [19] proposed an innovative single-target-multibeam radar beam alignment approach to mitigate inter-radar interference in vehicle-to-everything systems using DFRC.A novel DFRC cooperative sensing unmanned aerial vehicle network was designed to enhance the cooperative sensing ability [20].For edge computing, [21] created an DFRC based integrated architecture of communication, sensing and mobile-edge computing to perform radar detection and computation offloading simultaneously at the user terminals.The joint optimization of the RIS passive phase-shift matrix and transmit beamforming was studied to enhance the radar performance where the target was within a crowded area [22].Joint DFRC waveform, passive beamforming and RIS phase shift matrix were optimized in the RIS-assisted DFRC system to minimize multi-user interference under the strict beampattern constraint [8].
On the other hand, low cost hardware components are widely used, in particular in MIMO radar and MIMO communications systems, to reduce the overall cost of the system.Both communications and radar performance are largely affected by hardware impairment (HWI) since these HWI will cause phase and amplitude mismatch, raise noise floor or distort image signals [23].In-phase/quadrature imbalance (IQI) is one of the most common HWI types which has been extensively studied during the past decade.In an OFDM system with maximum ratio combining detection, the analytical outage probability of half-duplex amplify-and-forward relaying with IQI was derived [24].The authors in [25] derived the outage probability's exact and tight analytical lower bounds over independent and non-identically distributed Nakagami-m fading channels as well as the tractable upper and lower bounds on the ergodic capacity in the presence of IQI with arbitrary SNR [25].The work of [26] considered both IQI and additive distortion to derive the outage probabilities of both amplify-and-forward and decode-and-forward relaying schemes and analyze their performances.A low-complexity joint analog and digital self-interference cancellation approach for the full duplex transceiver with IQI was proposed [27].
Besides IQI, other HWI types, such as power amplifier nonlinearity, phase noise and carrier frequency offset may also occur.For instance, the compensation method of the power amplifier nonlinearity was proposed in large-scale multi-user MIMO downlink systems [28].The authors in [29] studied the effect of power amplifier non-linearity on the closed form expressions for achievable sum-rate.The work of [30] exploited location-specific channel gain and transmitter-specific phase noise which were two intrinsic physical-layer features for massive MIMO.Considering phase noise at both transmitter and receiver, the compensation scheme for such practical imperfections in high-mobility scenarios was proposed for the non-stationary and time-varying millimeter wave MIMO communication systems [31].Closed-form expressions of achievable rate were derived considering quasi-static radio frequency mismatch, channel estimation error and carrier frequency offset [32].The achievable downlink sum-rate of massive MIMO system was derived [33].
All the above works have provided helpful insights on DFRC or HWI.Nevertheless, to the best of our knowledge, none has considered HWI in DFRC designs.Motivated by the above observation, in this work, we study a beamforming design for the DFRC system that simultaneously detects the target as a MIMO radar and communicates with multiple communications users by considering the effect of IQI.We first formulate the radar-centric transmit beamforming problem to optimally design the radar beampattern while guaranteeing the minimum SINR of each communications user.The radar performance metric and communications performance metric are derived in the presence of IQI.Due to the non-convex constraint, semidefinite relaxation (SDR) approximation is used to solve the optimization problem.For the communication-centric transmit beamforming problem, the achievable rate with IQI is derived.To reduce the optimization complexity, a zero-forcing beamforming method is used.The MIMO radar receiving SNR is calculated as the performance constraint of the radar function.Then, the communicationcentric transmit beamforming problem is formulated and solved.In summary, the main contributions of our work are as follows: 1) We consider HWI in the DFRC beamforming architecture that has been ignored in the previous works.2) We propose a radar-centric beamforming design method which combines MIMO radar beampattern with communications user SINR and is solved by using the SDR approxmation method.3) We study a communication-centric beamforming optimization via zero-forcing beamforming method to maximize the communications rate with constraint on the MIMO radar receiving SNR. 4) We demonstrate that both IQI parameters and IQI mitigation have significant impact on the DFRC performance.Also, IQI phase mismatch at the CUs has more significant impact than IQI phase mismatch at the BS for the communication-centric problem when they are of asymmetric levels.The remainder of this article is organized as follows.Section II presents the system model with IQI.Sections III and Section IV study the radar-centric beamforming problem and communication-centric beamforming problem, respectively.Simulation results and discussion are provided in Section V. Finally, conclusions are made in Section VI.
Notations: The italic letter denotes a scalar and the lower case boldface letter represents a vector.CN (a, b, c) denotes a complex Gaussian random variable with mean a, variance b and pseudo variance c.E(•) represents the expectation operation.Superscripts () H and () T stand for Hermitian transpose and transpose, respectively.tr(), diag(), and rank() represent the trace operation, the vector formed by the diagonal elements and the rank operator, respectively.C m×n is the set of complexvalued m × n matrices.

II. SYSTEM MODEL
The DFRC BS has N t transmit antennas to serve K single-antenna communication users (CUs) indexed by k ∈ {1, . . ., K}.The DFRC BS also operates as a MIMO radar so that it has N r receive antennas for receiving the return radar signal.Thus, the BS waveform transmission and echo reception are not time divisioned but instead simultaneously on two sets of antenna.Consequently, no full-duplex mode is needed.For convenience, we let BS are the dual-functional waveforms where the element c k of c is simultaneously intended for the k-th CU as the communications symbols and for radar probing.The dual-functional waveforms c is precoded by the transmit beamforming matrix is the precoder for c k .The goal of our work is to design B with the following assumptions: 1) each c k has zero-mean and they are uncorrelated with each other with c ∼ CN (0, is the flat Rayleigh fading instantaneous downlink channel matrix where h k ∈ C N ×1 is the physical channel vector between the BS antennas and the k-th CU, and H remains unchanged during one transmission; 3) the transmitter has knowledge of H obtained by exploiting wireless channel reciprocity via uplink channel estimation in a time-division duplex mode [34], [35].

A. Receiced Signal At the CUs
After precoding, the C N ×1 signal transmitted by the BS is given as For the i-th antenna at the BS, the IQI coefficient is denoted as where g T,i and φ T,i are the amplitude mismatch and phase mismatch for BS antenna i, respectively.By denoting the BS IQI coefficient in Let the symbols received at the CUs be where n [n 1 , . . ., n K ] T is the additive white Gaussian noise (AWGN) vector with n ∼ CN (0, I K , 0).At the receivers for the CUs, there also exists IQI as where g R,k and φ R,k are the amplitude mismatch and phase mismatch at the k-th CU, respectively.Similarly, using the diagonal matrix K 1 ∈ C K×1 and K 2 ∈ C K×1 , the distorted received signal at the CU is Substituting ( 1), ( 2) and ( 3) into ( 4), one has For simplicity, we denote being the effective channel vector between the BS antennas and the k-th CU.Similarly, we denote 5) is simplified as where z = K 1 n + K 2 n * and z is improper Gaussian due to the introduction of n * term.

B. MIMO Radar Receiving Signal
Meanwhile, the IQI distorted transmitted signal x T at the BS is also used for radar detection.Given the transmit signal x T in (2), the echo signal received by another set of antennas at the radar receiver is where α 0 is the complex amplitude proportional to the radar cross section (RCS) of the target [36], [37], θ denotes the target direction, a t (θ) = a r (θ) [1, e j2πΔ sin θ , . . ., e j2π(N −1)Δ sin θ ] T are the steering vectors of the transmit antenna array and the receive antenna array, respectively, with Δ being the spacing between adjacent antennas normalized by the wavelength and n 0 is the AWGN with CN (0, I N , 0).Using linearly independent waveforms that yield linearly independent radar return signals reflected from different targets, data-dependent array algorithms, such as Capon, amplitude and phase estimation (APES) and the combined method of Capon and APES (CAPES), can be employed to estimate θ and α 0 [36].
Similarly, the radar receiver has IQI, whose coefficient diagonal matrix K r1 has the i-th diagonal element as where g r,i and φ r,i are the receiving amplitude mismatch and phase mismatch at the i-th radar receiving antenna, respectively.Accordingly, K r2 = I − K * r1 .Thus, y 0 becomes The final output of the radar receiver is where w ∈ C N ×1 is the radar receive beamforming vector designed to achieve the maximum output radar SNR.To further expand ( 9) with ( 8) and ( 7), one has Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply. where and Our purpose is to optimally design the transmit precoding B and the radar receive beamforming vector w.

III. RADAR-CENTRIC BEAMFORMING
The MIMO communications system exploits the spatial diversity using an array of transceive antennas and rich multipath channels to increase capacity and enhance communications performance.It requires sampling, quantization, symbol mapping, space/time encoding, RF up-conversion, matched-filtering and the corresponding inverse operations at the receiver [38].The MIMO radar system exploits the additional spatial degrees of freedom to provide more flexible resource management, improved parameter identifiability and much better angular and range resolution [39].It focuses on radar processing, such as matched-filtering, beamforming, Doppler detection, range detection and peak detection [2].They have different design purposes.The proposed MIMO DFRC system uses an integrated dual-functional waveform to simultaneously communicate with multiple downlink users and detect radar targets for the tradeoff between the radar performance and the communications performance.In this section, we will consider the radar-centric beamforming design problem.The performance metrics of MIMO radar and multiuser MIMO communication with IQI will be derived in Sections III-A and III-B, respectively.

A. MIMO Radar Beampattern With IQI
For MIMO radar, its desired beamforming shall synthesize the transmit beam towards the target.Therefore, the radar beampattern is adopted as the main radar performance metric in this work to optimize the transmit beamforming [3], [40] and [41].To calculate the transmit beampattern (transmit power) at a given angular direction θ, one first defines the radar transmit waveform covariance matrix as The transmit beampattern for the DFRC system is [49, (10)] Also, from [49, (11)], the radar cross correlation pattern is where θ 1 and θ 2 represent different values of θ.The radar transmit waveform covariance matrix R determines both the transmit beam pattern and cross correlation pattern in ( 14) and (15).Thus, optimally designing the covariance matrix R is critical.In this sense, the design of beampattern is equivalent to designing the covariance matrix of the probing signals [3], [40].
To generate a beampattern with a desired 3 dB main-beam width, the radar-centric beampattern problem proposed is where θ 0 is the location of the main-beam, (θ 2 − θ 1 ) determines the 3 dB mainlobe beam width, Ω denotes the sidelobe region and P t is the total transmit power.This is the conventional convex optimization problem studied in [41] and will be used as a benchmark radar beampattern.

B. Multiuser MIMO Communication SINR
For multiuser MIMO communication, the precoder is designed to guarantee the minimum receiving SINR at each CU.In multiuser transmit beamforming, the precoder should be designed to guarantee a certain level of SINR at the users.Here, it is assumed that the transmitter has knowledge of the instantaneous downlink channel H.This knowledge can be obtained, for example, by exploiting wireless channel reciprocity when operating in time-division duplex mode, i.e., the downlink channel is obtained via uplink channel estimation.Fairness SINR, which is the lowest SINR among all communciations downlinks, is used as the performance metric for multiple CUs.It is required to be higher than a given threshold to guarantee a minimal level of communication quality of service at each user, i.e., For the k-th CU, from (6) one has where h T 1k B k c k is the desired signal part, 1 is the overall interference made of the multi-user interference in the first part and the IQI interference in the second part [42], and 2 is the noise.The interference power is Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Using assumptions on c i , it can be obtained as Similarly, we have and The SINR γ k at the k-th user can be calculated from ( 20), ( 21) and (22) as

C. Problem Formation and Solution
The goal of radar-centric DFRC beamforming is to optimize the radar beam pattern with constraints on the transmit power and communication quality of service.Therefore, we establish the radar-centric optimization problem to minimize the loss function on radar beam pattern defined in (16), with the per-antenna power constraint and the fairness SINR constraint (17) for each downlink user as From (25), one can rewrite (17) as Algorithm 1: Radar-Centric Optimization via SDR.1: Remove the constraint 'rank(R k ) = 1' in (27g) of the initial problem (27) to obtain the SDR convex problem (28).2: Solve (28) and obtain the solutions of ( 28 Thus, (24) becomes However, the optimization problem ( 27) is non-convex because of the rank-one constraints in (27g).To make it convex, these constraints can be dropped, leading to the following semidefinite relaxation (SDR) problem as an approximation to (27) as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Each part of ( 28) is either linear or semidefinite so (28) is a convex problem which can be solved by using the MATLAB CVX tools [43], [44].Denote the solutions to the approximated optimization problem in (28) as R 0 , R 1 , . . ., R K (they do not have analytical expressions.).If these solutions to (28) are exactly rank-one, they are also the optimal solutions to the original non-convex problem in (27).However, if they are not exactly rank-one and the SDR problem in (28) is not tight, following the method used in [49, (32) and (33)], the solutions to the original non-convex problem in ( 27) can be calculated and approximated by using the solutions to (28) as the following and From ( 29) and ( 30), one has and where these new solutions used by the solutions to the SDR problem in (28) satisfy the rank-one constraint in (27g).These steps are summarized in Algorithm 1.Since it is a special case of the quadratic semidefinite programming problem, its computational complexity is O(K 6.5 N 6.5 log(1/ )) as the worst where is the solution accuracy [38], [39].If R0 in (31) equals to R 0 , then R0 , . . ., RK immediately satisfy all the rest constraints of (27) and they are also the optimal solutions to (27).
Their effectiveness depends on how close R0 in ( 31) is to R 0 .Therefore, they are not guaranteed to be the optimal solutions to (27) but simulation results later show that they offer good performances.

IV. COMMUNICATION-CENTRIC BEAMFORMING
In this section, we will focus on the communication-centric design to optimize the communications performance while guaranteeing the minimal requirement on the radar performance.

A. MIMO Communications Achievable Rate With IQI
Different from the previous section, the communicationcentric beamforming problem maximizes the achievable rate of the MU-MIMO communications given constraints on the MIMO radar beampattern and the transmit power budget.We will first derive the achievable rate of the communications system with IQI.Since z is an improper Gaussian, we have its pseudo-covariance matrix as and the covariance matrix as Similarly, for y one has where Cy is not a zero matrix so that y is also an improper Gaussian.
To calculate the system achievable rate, define the complex augmented random vector z of z as z = [ z T z H ] T .The entropy of z is [45] where C z is defined as From [46], one has Thus, (37) becomes The achievable rate can be obtained as [47]

B. MIMO Radar Optimal SNR of Radar Receive Signal
Now we focus on the MIMO radar performance which maintains the necessary quality of the radar receive signal.Consider M 1 x + M 2 x * in (10) as the received radar information part and (10) as the noise part.From (10), the desired w is required to maximize the SNR of the MIMO radar as This is a typical Reighley quotient problem, which is equivalent to Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply. where For a Reighlay quotient, one has Thus, the corresponding optimal SNR of the MIMO radar is calculated as

C. Problem Formation and Solution
Therefore, the communication-centric optimization problem can be formulated as where τ is the threshold on the MIMO radar SNR.
The problem in ( 47) is too complex to solve.Thus, we propose an alternative zero-forcing based beamforming scheme to reduce the complexity [48].From (6), one sees both multi-user interference and IQI interference in the received signal.The key idea is to design the transmit beamforming to simultaneously eliminate the multi-user interference and IQI interference between different CUs.Thus, we have the additional constraints as 48), one has Algorithm 2: Communication-Centric Optimization via ZF.1: Carry out SVD for the effective channel matrix as in ( 48)-(55).2: Recompute the achievable sum rate as (57).3: Establish the convex optimization problem as (59).4: Solve (59) via the algorithm in [38].5: Obtain Bk for each k by (55).
Therefore, B k is in the null space of Ĥk which can be expressed as where (a) is from the singular value decomposition (SVD), ) forms an orthogonal basis consisting of N − rank( Ĥk ) right singular vectors for the null space of Ĥk and [ where are unitary matrices, and Then, the block diagonalization transmit beamforming can be used as where Using the above equations, one has and Therefore, (36) becomes and (42) becomes where the detailed procedure is given in Appendix A. For BB H , one has Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Therefore, ( 47) is equivalent to the following concave problem (the detailed procedure is given in Appendix B) as The computational complexity of SVD for the effective channel matrix of the k-th CU is O(N 3 ).The problem in (59) can be solved using the algorithm in [37] with dual-variable bisection search of max(L 1 , L 2 ), where L 1 and L 2 are the number of iterations for the transmit power constraint and the MIMO radar receive SCNR constraint, respectively [37].Therefore, the total computational complexity is K max(L 1 , L 2 )O(N 3 ).By tuning r, one can obtain the solution to (59) at different levels of radar-communications tradeoff.

V. NUMERICAL RESULTS AND DISCUSSION
In this section, the performances of the proposed DFRC beamforming designs, i.e. the SDR radar-centric beamforming and zero-forcing communciation-centric beamforming, are evaluated via Monte Carlo simulation.These simulation results provide validation for the efficiency of the proposed beamforming approaches.In all experiments, the following settings are used unless specified otherwise.Both the DFRC BS and the MIMO radar receiver are equipped with uniform linear arrays (ULAs) with the same number of elements and half wavelength spacing between adjacent antennas.The total transmit SNR budget is set as P t = 30 dB.The multi-user communications channels are assumed to obey Rayleigh fading, so the elements of the channel matrices H are i.i.d.standard complex Gaussian random variables CN (0, 1, 0).The AWGN at each user and MIMO radar receiving antenna also has a variance of σ 2 = 1.For the ideal MIMO radar beampattern, the relevant parameters are set as main beam In the simulation, the number of communications users K and the number of antennas for DFRC BS N change to test their impact on the performance of the proposed joint beamforming approach.The amplitude mismatch g T,i , g R,i and g r,i are uniformly generated from the common interval g : [g l , g u ] while the phase mismatch φ T,i , φ R,i and φ r,i are uniformly generated from the common interval φ : [φ l , φ u ] (in this case BS and CUs have the symmetric IQI level).We tune the values of g l , g u , φ l and φ u to study the effect of IQI.Both problems involved in (28) and ( 59) are solved by using the MATLAB CVX toolbox.All the following simulation results are obtained by averaging over 1000 Monte Carlo runs.

A. Radar-Centric Transmit Beamforming
First, the proposed SDR transmit beamforming approach in (28) is studeied using the MIMO radar transmit beampatterns defined in (14).IQI parameters are set as g l = 0.95, g u = 1, φ l = 0 • and φ u = 5 • .Communications SINR threshold Γ is chosen as P t min{K,10} .The transmit beampatterns for N = 20 are depicted in Fig. 1 with K = 0, 2 or 4. When K = 0, it represents the radar-only beam pattern without any DFRC multi-users.Its beampattern has the overall best shape [41].Such desired shape requires that the beam power in the mainlobe be extremely high and the beam power in the sidelobe be as low as possible, so that most transmit signal power is concentrated in the mainlobe to detect the target better.When K = 2, it is shown that the beam power of the mainlobe is smaller than that of 'Radar-Only' while the beam power of the sidelobe is much greater than that of 'Radar-Only'.From this perspective, adding two communications users degrades the beampattern performance of the MIMO radar-only system.When K = 4, the degradation is larger than that for K = 2.This phenomenon discloses the tradeoff that one cannot simultaneously achieve both optimal MIMO radar performance and optimal multiuser communications performance.This is the reason for choosing radar-centric or communication-centric architectures.'RadarCommOpt' is the beampattern from (28) while 'RadarCommApp' is the beampattern from the approximation solution in (30) and (31).When K = 2, there is a slight mismatch between 'RadarCommApp' Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.and 'RadarCommOpt'.However, when K = 4, the mismatch becomes quite smaller, which indicates the effectiveness of the proposed SDR beamforming solution in (30) and (31).This observation further validates the SDR beamforming approximation, when K is large enough.
In Fig. 2, we fix N = 20, K = 4 and φ l = φ u = 0 • to explore amplitude mismatch's impact on DFRC system.Four intervals of [g l , g u ] are provided where a lower g l and g u stands for a more severe amplitude mismatch.The beampatterns of different amplitude mismatch levels are explicitly separated in Fig. 2. The greater amplitude mismatch is, the worse the beampattern will be.Therefore, amplitude mismatch has a degrading effect on the system performance.In Fig. 3, phase mismatch's impact on DFRC system is illustrated by fixing N = 20, K = 4 and g l = g u = 1.Similarly, it also has a degrading effect on the beampattern performance, though it is not as significant as that of the amplitude mismatch.

B. Communication-Centric Transmit Beamforming
In Fig. 4, the IQI parameters are specified as g l = 0.95, g u = 1, φ l = 0 • and φ u = 5 • .Y axis is the communciations achievable rate C in (57) and X axis is the ratio threshold r of the DFRC radar SNR.Hence, we tune r to realize the tradeoff between communications performance and radar performance, since a bigger r reflects more consideration on the radar side.For all curves, C gradually decreases as r increases.Adding more communication users or more BS antennas will greatly increase the overall C. When N = 10, C decreases faster as K increases.However, increasing K does not significantly speed up C's decrease when N = 20, which means that it is useful to equip much more DFRC BS antennas to enhance the overall DFRC system performance.Fixing N = 20, K = 4 and φ l = φ u = 0 • , we demonstrate amplitude mismatch's impact on the communication-centric DFRC system in Fig. 5 by using the same intervals of [g l , g u ] as Fig. 2. Each curve is explicitly separated and the achievable rate decreases as the amplitude mismatch gets more severe.In Fig. 6, phase mismatch is also found to have a degrading effect on the communications achievable rate performance.Fig. 2, Fig. 3, Fig. 5 and Fig. 6 show that IQI is of critical importance to be considered and compensated for improving future MIMO DFRC system performance.

C. Further Impact of the IQI
In the above, IQI is perfectly known.However, in many realistic scenarios, it is difficult to obtain an accurate value of IQI.Alternatively, IQI mitigation methods could be utilized.For each diagonal element of the IQI matrices K 1 , K r1 and G 1 , its estimation error is assumed to be complex Gaussian with CN (0, σ 2 e , 0), where σ 2 e is the error variance.Here, σ e is set as 0.1, 0.05 or 0.01 to study the impact of imperfect IQI mitigation on both radar-centric DFRC beamforming and communication-centric DFRC beamforming.Other relevant parameters are N = 20, K = 4, g l = 0.95, g u = 1, φ l = 0 • and φ u = 5 • .In Fig. 7, one finds that the beampattern is compromised as σ e increases.Similarly, in Fig. 8 the achievable rate decreases when the error variance increases.These results show the importance of achieving accurate and efficient IQI mitigation.Next, the case when the BS and CUs have asymmetric IQI levels is studied.The amplitude mismatches at the BS g T,i and g r,i are uniformly generated from the interval g b : [g bl , g bu ] while the amplitude mismatch at the CU g R,i is uniformly generated from the interval  and φ c .From Fig. 10, one sees that the overall achievable rate improves as φ c gets better while φ b gets worse.This interesting result indicates that the phase mismatch at the CU side has much more significant impact than that at the BS side.

D. System Analysis
Radar-centric and communications-centric are two different perspectives of optimization for DFRC.One can also carry out the joint optimization by constructing the weighted sum of the radar performance and the communications performance.This might provide more insights to improve the overall performance of DFRC.Also, new transmit beamforming for the emerging ultra-massive MIMO communications is interesting.For the CUs, this work can be extended to the multi-antenna case.Also, for multiple radar targets with high mobility, the DFRC system must consider the Doppler effect, return signal time delay and clutter interference.Finally, other HWI types can be investigated.

VI. CONCLUSION
In this work, we have extended the conventional DFRC beamforming designs to a more realistic scenario considering the existence of IQI.In order to achieve the optimal DFRC performance, both radar-centric and communication-centric optimizations have been proposed.For the radar-centric formulation, the beampattern design under constraints of mainlobe beam power, sidelobe beam power and each communications user's minimum SINR has been proposed.SDR has been employed to obtain a good approximation.For the communication-centric beamforming design, zero-forcing based approach has been proposed to greatly reduce the computation complexity.IQI amplitude mismatch, IQI phase mismatch and IQI mitigation error have been shown to have significant effect on the overall performance.For the communication-centric scenario, IQI phase mismatch at the CUs has more significant impact than IQI phase mismatch at the BS.Therefore, future massive MIMO DFRC system shall consider mitigating IQI and improve the IQI estimation accuracy.

B. Formation of the Communication-Centric Problem
If (57) and (58) are obtained by the zero-forcing method, the problem in (47) Since both the left side of (61b) and the right side of (61d) are linear with of p 1 , . . ., p K , one has the following ancillary Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
concave optimization problem: whose maximum objective value is denoted as Λ m .Therefore, (61b) can be rewritten as where r is the ratio threshold of DFRC radar SNR in [0, 1].For (61a), denoting log 2 (h Therefore, f k (p k ) is a concave function so (61a) is also concave.
Then, one has the equivalent communication-centric problem as (59).
g c : [g cl , g cu ].Also, the phase mismatches at the BS φ T,i and φ r,i are uniformly generated from the interval φ b : [φ bl , φ bu ] while the phase mismatch at the CU φ R,i is uniformly generated from the interval φ c : [φ cl , φ cu ].For the radar-centric scenario, we set φ b ( • ) = φ c ( • ) = [0, 0] and fix g c = [0.95,1] to explore the effect of aymmetric level of g b .Fig. 9 shows that the amplitude mismatch still has a significant impact on the radar beampattern performance.For the communication-centric scenario, we set g b = g c = [1, 1] to explore the effect of aymmetric level of φ b

2 |C y | 2 I−2g 2 R,k +4g 2 R
and (56), one finds that the multiplications of C y , Cy , C z and Cz are commutative since all of them are diagonal matrices of the same order.Therefore, one hasC = log 2 |C y K − C −1 y Cy CH y C −T y |C z | 2 I K − C −1 z Cz CH z ,k sin 2 φ R,k 4 becomes a power allocation problem: 4 k p 2 k + (1 + g 2 R,k )h 2 k p k + g 2 R,k cos 2 φ R,k ) as f k (p k ), one has