Composite Signalling for DFRC: Dedicated Probing Signal or Not?
aa r X i v : . [ c s . I T ] S e p Composite Signalling for DFRC:Dedicated Probing Signal or Not?
Li Chen, Fan Liu,
Member, IEEE,
Jun Liu,
Senior Member, IEEE, and ChristosMasouros,
Senior Member, IEEE
Abstract
Dual-functional radar-communication (DFRC) is a promising new solution to simultaneously probethe radar target and transmit information in wireless networks. In this paper, we study the joint optimiza-tion of transmit and receive beamforming for the DFRC system. Specifically, the signal to interferenceplus noise ratio (SINR) of the radar is maximized under the SINR constraints of the communication user(CU), which characterizes the optimal tradeoff between radar and communication. In addition to simplyusing the communication signal for target probing, we further consider to exploit dedicated probingsignals to enhance the radar sensing performance. We commence by studying the single-CU scenario,where a closed-form solution to the beamforming design problem is provided. It is then proved that adedicated radar probing signal is not needed. As a further step, we consider a more complicated multi-CU scenario, where the beamforming design is formulated as a non-convex quadratically constrainedquadratic programming. The optimal solutions are obtained by applying semidefinite relaxation withguaranteed rank-1 property. It is shown that under the multi-CU scenario, the dedicated probing signalshould be employed to improve the radar performance at the cost of implementing an additionalinterference cancellation at the CU. Finally, the numerical simulations are provided to verify theeffectiveness of the proposed algorithm.
Index Terms
Spectrum sharing, radar-communication, signal to interference and noise ratio, probing signal,communication signal, joint beamforming.
L. Chen and J. Liu are with the Department of Electronic Engineering and Information Science, University of Science andTechnology of China. (e-mail: { chenli87, junliu } @ustc.edu.cn).F. Liu and C. Masouros are with the Department of Electronic and Electrical Engineering, University College London, London,WC1E 7JE, UK (e-mail: [email protected], [email protected]). I. I
NTRODUCTION
Communication and radar spectrum sharing (CRSS) has recently drawn significant attentiondue to the scarcity of the commercial wireless spectrum. For instance, the millimeter wave(mmWave) band is occupied by variety of radars [1], and has also been assigned as a newlicensed band to the 5G network [2]. It is well-recognized that communication and radar signalshave some common features in their waveforms. Although their purposes are dramaticallydifferent, it is feasible to use one type of signal for the other types purpose. Nevertheless,the use of radar (communication) signals for communication (radar) functionalities, introducesa number of challenges [3]–[5]. To address these challenges, the research of dual-functionalradar-communication (DFRC) is well-underway [6]–[8].In general, the aim of the DFRC is to implement both communication and radar functionalitieson the same hardware platform. Based on information theory, the work in [9] unified the radar andcommunication performance metric and discussed the performance bounds of the DFRC system.Furthermore, the weighted sum of the estimation and communication rates was analyzed as theperformance metrics in the DFRC system [10]. By leveraging the simple time-division scheme,the radar and the communication signals can be transmitted within different time slots, whichavoids the mutual interference [11]. To exploit the favorable time-frequency decoupling propertyof the orthogonal frequency division multiplexing (OFDM) waveform, the OFDM communicationsignal was adopted for target detection, where the range and Doppler processing are independentwith each other [12]. From a signal processing perspective, the implementation of mmWaveDFRC systems was fully studied in [13]. The unimodular signal design was discussed in [14]for DFRC architecture, where the information of downlink communication was modulated viathe ambiguity function (AF) sidelobe nulling in the prescribed range-Doppler cells.Beamforming design is essential to improve the performance of the DFRC signal processingin the spatial domain, which has been widely studied in the literature. Aiming to realize the dualfunctionalities, the work of [15] designed a transmit beampattern for multiple input multipleoutput (MIMO) radar with the communication information being embedded into the sidelobesof the radar beampattern. Considering both the separated and the shared antenna deployments,a series of optimization-based transmit beamforming approaches for the DFRC system werestudied in [16], where the communication signal was exploited for target detection. By imposingthe constraints of the radar waveform similarity and the constant modulus, the interference of themultiple communication users (CUs) was suppressed to improve the communication performancein [17]. Based on IEEE 802.11ad wireless local area network (WLAN) protocol, a joint waveformfor automotive radar and a potential mmWave vehicular communication system were proposed in [18]. The work of [19] further studied the feasibility of an opportunistic radar, which exploitedthe probing signals transmitted during the sector level sweep of the IEEE 802.11ad beamformingtraining protocol. In order to reduce the hardware complexity and the associated costs incurredin the mmWave massive MIMO system, a hybrid analog-digital beamforming structure wasproposed for the DFRC transmission in [20].It is worth pointing out that all the aforementioned works on the DFRC beamforming de-sign focused on formulating a desired transmit beampattern without considering the receivebeamformimg. To the best of our knowledge, the joint optimization of the transmit and receivebeamforming under the communication constraints has never been studied for the DFRC systembefore, despite the fact that the joint optimization of transmit and receive beamforming for theMIMO radar system has been extensively investigated in the recent literature [21]–[27]. To detectan extended target, the joint optimization of waveforms and receiving filters in the MIMO radarwas considered in [21]. In order to guarantee the constant modulus and similarity properties of theradar waveforms, numerous approaches were provided to maximize the signal to interferenceplus noise (SINR) of the radar, e.g., the sequential optimization algorithms (SOAs) in [22],the successive quadratically constrained quadratic programming (QCQP) refinement method in[23], the block coordinate descent (BCD) framework in [24], and the general majorization-minimization (MM) framework in [25]. To detect multiple targets, the joint optimization ofwaveforms and receiving filters in the MIMO radar was further studied in [26]. The problem ofbeampattern synthesis with sidelobe control was studied in [27] using constant modulus weights.The jointly design of the transmit and receive beamforming was provided in [28] based on apriori information on the locations of target and interferences.All these works improved the receive SINR of the echo signal based on the prior informationof the target and clutter. In contrast, for the DFRC system, there exist both probing signaland communication signal, which are coupled together with each other for drastically differentpurposes. As a consequence, the known MIMO radar-only designs are inapplicable for the latter.In this paper, we study the joint optimization of transmit and receive beamforming for the DFRCsystem. 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 non-dedicatedprobing signal case and the dedicated probing signal case. For the non-dedicated probing signalcase, 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 probingsignal is added to the DFRC transmit signal to improve the radar performance. Under the singleCU scenario, closed-form 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 singleCU scenario. On top of that, we consider a more complicated scenario with multiple CUs. Forthe non-dedicated probing signal case, the beamforming design is formulated as a non-convexquadratically constrained quadratic programming (QCQP), and we show that the globally optimalsolution can be obtained by applying semidefinite relaxation (SDR) with rank-1 property. For thededicated probing signal case, the rank-1 property after applying SDR can be also proved, and thecorresponding optimal solution shows that the dedicated probing signal should be employed toimprove the SINR of the radar receiver. The main contributions of this paper can be summarizedas follows. • Transmit-receive DFRC beamforming : We provide the solutions of the joint transmit-receive beamforming optimization to maximize the SINR of the radar under the SINRconstraints of the CUs. For the single CU scenario, the closed-form solutions of the opti-mized beamforming are derived. For the multiple CUs scenario, the optimal solutions areobtained by applying SDR with rank-1 property. • Dual-functional performance tradoff : The optimal performance tradeoff between the radarand the communication is characterized in terms of SINR. Compared to the time-sharingscheme between the radar and the communication, the joint optimization of transmit andreceive beamforming yields a more favorable tradeoff performance. • Dedicated radar probing signal or not : Compared to the existing works only exploitingcommunication signals for radar functionality, we consider the use of a dedicated radarprobing signal to improve the radar performance. For the single CU scenario, our analysisshows that there is no need of dedicated probing signal. For the multi-CU scenario, on theother 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 morecomplicated 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 uppercaseletters to denote matrices. Superscripts ( · ) H and ( · ) T stand for Hermitian transpose and transpose,respectively. tr ( · ) and rank ( · ) represent the trace operation and the rank operator, respectively. C m × n is the set of complex-valued m × n matrices. x ∼ CN ( a, b ) means that x obeys a complexGaussian distribution with mean a and covariance b . E ( · ) denotes the statistical expectation. k x k denotes the Euclidean norm of a complex vector x . TargetDFRC BS CU 1
Prob. signal
Com. signal (cid:266)(cid:266) (cid:266)(cid:266)
TargetClutter 1 Clutter I (cid:266)(cid:266) Echo signal h K h (cid:266) (cid:266) a a I a CU K
Figure 1. System model of a DFRC MIMO system
II. S
YSTEM M ODEL
As illustrated in Fig. 1, we consider a DFRC MIMO system, which simultaneously probesthe radar target and transmits information to the CUs. To be specific, it is composed of a DFRCbase station (BS) with N t transmit antennas and N r receive antennas, K single-antenna CUsindexed by k ∈ { , · · · , K } .Suppose there is a target and I signal-dependent interference sources indexed by i ∈ { , · · · , I } .The target is located at angle θ and the interference sources are located at angle θ i , i ∈{ , · · · , I } . Given the transmit signal x ∈ C N t × , the received signal of the radar receiver is y = α a r ( θ ) a Tt ( θ ) x + I X i =1 α i a r ( θ i ) a Tt ( θ i ) x + z = α A ( θ ) x + I X i =1 α i A ( θ i ) x + z , (1)where α and α i are the complex amplitudes of the target and the i -th interference source, respec-tively, a t ( θ ) = [1 , e − j π ∆ t sin θ , · · · , e − j π ( N t − t sin θ ] T and a r ( θ ) = [1 , e − j π ∆ r sin θ , · · · , e − j π ( N r − r sin θ ] T with ∆ t and ∆ r being the spacing between adjacent antennas normalized by the wavelength,respectively, and z ∈ C N r × is the additive white Gaussian noise (AWGN) with each elementsubjects to CN (0 , . The symbol index is omitted for simplicity. Then, the output of the radarreceiver is r = w H y = α w H A ( θ ) x + w H X i ∈I α i A ( θ i ) x + w H z , (2) where w ∈ C N r × is the receive beamforming vector for SINR maximization.Further, given the transmit signal x ∈ C N t × , the received signal of the CU k is y k = h Hk x + z k , (3)where h k ∈ C N t × is the multiple input single output (MISO) channel vector between the DFRCBS and the CU k , and z k ∼ CN (0 , is the AWGN of the CU k .In this paper, we consider two cases according to the component of the DFRC BS’s transmitsignal x . • Case 1 (Non-dedicated probing signal): In this case, the transmit signal of the DFRC BSis only composed of the communication signals of the CUs. That is x = K X k =1 x k , (4)where x k is the communication signals of the CU k and the radar functionality is realizedby the sum of the CUs’ communication signal. • Case 2 (Dedicated probing signal): In this case, the transmit signal of the DFRC BS iscomposed of both the communication signal of the CUs and the dedicated probing signal.That is x = K X k =1 x k + x , (5)where x 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 θ and the interference { θ i } 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, andthe dedicated probing signal x is pseudo-random and assumed to be known in prior to the CUs.III. S INGLE
CU S
CENARIO
In this section, we consider a simplified scenario with single CU in the network. The beam-forming design of the DFRC BS is discussed for the non-dedicated probing signal case, and theclosed-form solution is provided. Then, the dedicated probing signal case is studied, and it canbe proved that there is no need of dedicated probing signal with single CU.
A. Non-dedicated Probing Signal Case
For the non-dedicated probing signal case, the transmit signal of the DFRC BS in (4) withsingle CU can be rewritten as x = u s, (6)where u ∈ C N t × and s ∈ CN (0 , are the beamforming vector and the information symbol ofthe CU, respectively.According to the output in (2), the SINR of the radar receiver can be expressed as γ ( I ) R = (cid:12)(cid:12) α w H A ( θ ) x (cid:12)(cid:12) E "(cid:12)(cid:12)(cid:12)(cid:12) w H I P i =1 α i A ( θ i ) x (cid:12)(cid:12)(cid:12)(cid:12) + w H w = | α | (cid:12)(cid:12) w H A ( θ ) x (cid:12)(cid:12) w H (cid:20) I P i =1 | α i | A ( θ i ) uu H A H ( θ i ) + I (cid:21) w . (7)And the output SINR of the radar receiver depends on the choice of the receive beamformingvector w . The design of w can be expressed as max w (cid:12)(cid:12) w H A ( θ ) x (cid:12)(cid:12) w H (cid:20) I P i =1 | α i | A ( θ i ) uu H A H ( θ i ) + I (cid:21) w , (8)which is equivalent to the well-know minimum variance distortionless response (MVDR) prob-lem, and its solution can be given by [21] w ∗ = Σ ( u ) − A ( θ ) xx H A H ( θ ) Σ ( u ) − A ( θ ) x , (9)where Σ ( u ) = " I X i =1 | α i | A ( θ i ) uu H A H ( θ i ) + I . (10)Substituting (9) into (7), the SINR of the radar receiver can be calculated as γ ( I ) R = x H Φ ( u ) x , (11)where Φ ( u ) = | α | A H ( θ ) Σ ( u ) − A ( θ ) . (12)And the average SINR of the radar receiver can be given by ¯ γ ( I ) R = E (cid:2) x H Φ ( u ) x (cid:3) = u H Φ ( u ) u . (13)For the CU k , the received signal in (3) can be rewritten as y k = h H u s + z k , (14)where h ∈ C N t × is the MISO channel vector between the DFRC BS and the CU. And theaverage SINR of the CU can be calculated as ¯ γ C = (cid:12)(cid:12) h H u (cid:12)(cid:12) , (15)Then, we consider the beamforming optimization problem that maximizes the SINR of the radarreceiver and satisfies the SINR of the CU, i.e., (P1 .
1) max u ¯ γ ( I ) R = u H Φ ( u ) u s . t . ¯ γ C = (cid:12)(cid:12) h H u (cid:12)(cid:12) Γ u H u ≤ P , (16)where Γ is the threshold of the CUs SINR, and P is the transmit power constraint of the DFRCBS.Because ¯ γ ( I ) R is a nonlinear function of the transmit beamforming vector u , Problem (P1.1)is generally non-convex. Thus, we adopt the sequential optimization to find the transmit beam-forming vector u in an iterative fashion. Specifically, at the m -th iteration, we first compute Φ = Φ (cid:2) u ( m − (cid:3) , where u ( m − is obtained in the ( m − -th iteration. Thus, Problem (P1.1)can be rewritten as (P1 .
2) max u ¯ γ ( I ) R = u H Φ u s . t . ¯ γ C = (cid:12)(cid:12) h H u (cid:12)(cid:12) ≥ Γ u H u ≤ P . (17)And the closed-form solution of Problem (P1.2) can be given by the following proposition. Proposition 1. (Optimal beamforming with single CU) For the non-dedicated probing signalcase, the optimal beamforming vector of the DFRC BS with single CU i.e., the optimal solutionto Problem (P1.2), can be given by u ∗ = √ P ˆg , Γ ≤ P (cid:12)(cid:12) h H ˆg (cid:12)(cid:12) (cid:16) α ˆh + β ˆg ⊥ (cid:17) , P k h k ≥ Γ > P (cid:12)(cid:12) h H ˆg (cid:12)(cid:12) , (18) α = s Γ k h k α g | α g | , (19) β = s P − Γ k h k β g | β g | , (20)where g is the dominant eigenvector of Φ , ˆg = g / k g k , ˆh = h / k h k , g ⊥ = g − ( ˆh H g ) ˆh denoting the projection of g into the null space of ˆh , ˆg ⊥ = g ⊥ / k g ⊥ k and g can be expressedas g = α g ˆh + β g ˆg ⊥ . Proof.
The proof is given in Appendix A.
B. Dedicated Probing Signal Case
For the dedicated probing signal case, the transmit signal of the DFRC BS in (5) with singleCU can be rewritten as x = u s + v s , (21)where v ∈ C N t × and s ∼ CN (0 , are the beamforming vector and the symbol of thededicated probing signal, respectively. In addition, s and s are independent and identicallydistributed (i.i.d.).According to the output in (2), the SINR of the radar receiver can be expressed as γ ( II ) R = (cid:12)(cid:12) α w H A ( θ ) x (cid:12)(cid:12) E "(cid:12)(cid:12)(cid:12)(cid:12) w H I P i =1 α i A ( θ i ) x (cid:12)(cid:12)(cid:12)(cid:12) + w H w = | α | (cid:12)(cid:12) w H A ( θ ) x (cid:12)(cid:12) w H (cid:20) I P i =1 | α i | A ( θ i ) ( uu H + vv H ) A H ( θ i ) + I (cid:21) w . (22)By solving an equivalent MVDR problem, the corresponding receive beamforming vector tomaximize the output SINR can be given by w ∗ = Σ ( u , v ) − A ( θ ) xx H A H ( θ ) Σ ( u , v ) − A ( θ ) x , (23)where Σ ( u , v ) = " I X i =1 | α i | A ( θ i ) (cid:0) uu H + vv H (cid:1) A H ( θ i ) + I . (24)Substituting (23) into (22), the SINR of the radar receiver can be calculated as γ ( II ) R = x H Φ ( u , v ) x , (25)where Φ ( u , v ) = | α | A H ( θ ) Σ ( u , v ) − A ( θ ) . (26)And the average SINR of the radar receiver can be given by ¯ γ ( II ) R ( u , v ) = E (cid:2) x H Φ ( u , v ) x (cid:3) = u H Φ ( u , v ) u + v H Φ ( u , v ) v . (27)For the CU, it has a priori information of probing signal. After probing signal interferencecancelling, its received SINR ¯ γ C ( u ) can also be expressed as (15). The beamforming optimiza-tion problem that maximizes the SINR of the radar receiver and ensures the SINR requirementof the CU can be expressed as (P2 .
1) max u ¯ γ (II) R = u H Φ ( u , v ) u + v H Φ ( u , v ) v s . t . ¯ γ C = (cid:12)(cid:12) h H u (cid:12)(cid:12) ≥ Γ u H u + v H v ≤ P , (28)Similarly, the sequential optimization can be adopted to find the transmit beamforming vector u and v in an iterative fashion, where we first compute Φ = Φ (cid:2) u ( m − v ( m − (cid:3) at the m -th iteration with u ( m − and v ( m − being obtained from the ( m − -th iteration. Thus, thebeamforming optimization problem can be given by (P2 .
2) max u ¯ γ ( II ) R = u H Φ u + v H Φ v s . t . ¯ γ C = (cid:12)(cid:12) h H u (cid:12)(cid:12) ≥ Γ u H u + v H v ≤ P , (29) Algorithm 1
Beamforming design of DFRC BS for the single CU scenario.Initialize { θ , θ , · · · , θ I } and { α , α , · · · , α I } .Initialize u (0) = [1 , · · · , T (cid:14) √ P N t .Initialize the convergence threshold ∆ , and m = 0 . repeat Set m = m + 1 .Calculate Φ = Φ (cid:2) u ( m − (cid:3) and ¯ γ ( I ) R (cid:2) u ( m − (cid:3) according to (12) and (13), respectively.Calculate u ( m ) according to (18) in Proposition 1.Calculate ¯ γ ( I ) R (cid:2) u ( m ) (cid:3) according to (13). until (cid:12)(cid:12)(cid:12) ¯ γ ( I ) R (cid:2) u ( m ] (cid:1) − ¯ γ ( I ) R (cid:2) u ( m − (cid:3)(cid:12)(cid:12)(cid:12) ≤ ∆ .And the closed-form 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 probingsignal 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 v ∗ = . Thus, there is no need todesign the dedicated probing signal with single CU, and the optimal beamforming vector of thecommunication signal u ∗ 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 summa-rized as Algorithm 1, which repeatedly updates u ( m − based on u ( m ) until the improvement ofthe radar receivers SINR becomes insignificant.IV. M ULTIPLE CU S S CENARIO
In this section, we proceed to consider a more complicated scenario with multiple CUs in thenetwork. For the non-dedicated probing signal case, the beamforming design is formulated as anon-convex QCQP, and the globally optimal solutions can be obtained by applying SDR withrank-1 property. For the dedicated probing signal case, the rank-1 property after applying SDRcan also be proved, and the corresponding optimal solution shows that the dedicated probingsignal should be employed to improve the SINR of the radar receiver.
A. Non-dedicated Probing Signal Case
For the non-dedicated probing signal case, the transmit signal of the DFRC BS in (4) withmultiple CUs can be given by x = K X k =1 u k s k , (30)where u k ∈ C N t × and s k ∈ CN (0 , are the beamforming vector and the i.i.d. informationsymbol of the CU k , respectively.According to the output in (2), the SINR of the radar receiver can be written as γ ( I ) R = (cid:12)(cid:12) α w H A ( θ ) x (cid:12)(cid:12) E "(cid:12)(cid:12)(cid:12)(cid:12) w H I P i =1 α i A ( θ i ) x (cid:12)(cid:12)(cid:12)(cid:12) + w H w = | α | (cid:12)(cid:12) w H A ( θ ) x (cid:12)(cid:12) w H (cid:20) I P i =1 | α i | A ( θ i ) (cid:18) K P k =1 u k u Hk (cid:19) A H ( θ i ) + I (cid:21) w . (31)By solving an equivalent MVDR problem, the corresponding receive beamforming vector tomaximize the output SINR can be given by w ∗ = Σ ( { u k } ) − A ( θ ) xx H A H ( θ ) Σ ( { u k } ) − A ( θ ) x , (32)where Σ ( { u k } ) = " I X i =1 | α i | A ( θ i ) K X k =1 u k u Hk ! A H ( θ i ) + I . (33)Substituting (32) into (31), the SINR of the radar receiver can be calculated as γ ( I ) R = x H Φ ( { u k } ) x , (34)where Φ ( { u k } ) = | α | A H ( θ ) Σ ( { u k } ) − A ( θ ) . (35)And the average SINR of the radar receiver can be given by ¯ γ ( I ) R = E (cid:2) x H Φ ( { u k } ) x (cid:3) = K X k =1 u Hk Φ ( { u k } ) u k (36)The received signal of the CU k in (3) can be rewritten as y k = h Hk u k s k + X j = k h Hk u j s j + z k , (37)and the average SINR of the CU k can be calculated as ¯ γ C,k ( { u k } ) = (cid:12)(cid:12) h Hk u k (cid:12)(cid:12) P j = k (cid:12)(cid:12) h Hk u j (cid:12)(cid:12) . (38)Again, we consider the beamforming optimization problem that maximizes the SINR of theradar receiver by imposing the individual SINR constraints of the CUs, i.e., (P3 .
1) max { u k } ¯ γ (I) R = K P k =1 u Hk Φ ( { u k } ) u k s . t . ¯ γ C,k = (cid:12)(cid:12) h Hk u k (cid:12)(cid:12) P j = k (cid:12)(cid:12) h Hk u j (cid:12)(cid:12) ≥ Γ k , ∀ k K P k =1 u Hk u k ≤ P . (39)Note that Problem (P3.1) is also non-convex. Thus, we adopt the sequential optimization to findthe transmit beamforming vector u in an iterative way. Specifically, at the m -th iteration, wefirst compute Φ = Φ [ { u ( m − k } ] , where { u ( m − k } is obtained in the ( m − -th iteration. Asa result, Problem (P3.1) can be rewritten as (P3 .
2) max { u k } ¯ γ (I) R = K P k =1 u Hk Φ u k s . t . ¯ γ C,k = (cid:12)(cid:12) h Hk u k (cid:12)(cid:12) P j = k (cid:12)(cid:12) h Hk u j (cid:12)(cid:12) ≥ Γ k , ∀ k K P k =1 u Hk u k ≤ P (40)While Problem (P3.2) is still a non-convex QCQP, and we can solve it via SDR with U k = u k u Hk ,i.e, (P3 .
3) max { U k } ¯ γ ( I ) R = K P k =1 tr ( Φ U k )s . t . ¯ γ C,k = tr ( H k U k )Γ k − P j = k tr ( H k U j ) ≥ , ∀ k K P k =1 tr ( U k ) ≤ P , U k (cid:23) , ∀ k , (41)where H k = h k h Hk . Algorithm 2
Beamforming design of DFRC BS for multiple CUs scenario without dedicatedprobing signal.Initialize { θ , θ , · · · , θ I } and { α , α , · · · , α I } .Initialize n u (0) k o = [1 , · · · , T . √ KP N t .Initialize the convergence threshold ∆ , and m = 0 . repeat Set m = m + 1 .Calculate Φ = Φ hn u ( m − k oi and ¯ γ ( I ) R hn u ( m − k oi according to (35) and (36), respec-tively.Optimize { U ∗ k } according to Problem (P3.3).Calculate { u ∗ k } satisfying u ∗ k ( u ∗ k ) H = U ∗ k based on the Proposition 3.Calculate ¯ γ ( I ) R hn u ( m ) k oi according to (36). until (cid:12)(cid:12)(cid:12) ¯ γ ( I ) R hn u ( m ) k oi − ¯ γ ( I ) R hn u ( m − k oi(cid:12)(cid:12)(cid:12) ≤ ∆ .Although the rank-1 constraints have been removed for the convexity of the problem, theoptimal solution can be proved to have the rank-1 property by the following proposition. Proposition 3. (Rank-1 property of Problem (P3.3)) For the non-dedicated probing signal casewith multiple CUs, there is always a solution to Problem (P3.3) satisfying that rank ( U ∗ k ) = 1 , ∀ k .Thus, the optimal beamforming vector for the CU k , i.e., the optimal solution to Problem (P3.2)can be given by u ∗ k with u ∗ k u ∗ Hk = U ∗ k . Proof.
The proof is given in Appendix C.Above all, the beamforming optimization algorithm for the multi-CU scenario and the non-dedicated probing signal case can be provided as Algorithm 2. It repeatedly updates { u ( m ) k } given { u ( m − k } until convergence. B. Dedicated Probing Signal Case
For the dedicated probing signal case, the transmit signal of the DFRC BS in (5) with multipleCUs can be given by x = K X k =1 u k s k + v s , (42)where v ∈ C N t × and s ∼ CN (0 , are the beamforming vector and the symbol of the dedicatedprobing signal, respectively. And s k , ∀ k and s are i.i.d.. According to the output in (2), the SINR of the radar receiver can be expressed as γ ( II ) R = (cid:12)(cid:12) α w H A ( θ ) x (cid:12)(cid:12) E "(cid:12)(cid:12)(cid:12)(cid:12) w H I P i =1 α i A ( θ i ) x (cid:12)(cid:12)(cid:12)(cid:12) + w H w = | α | (cid:12)(cid:12) w H A ( θ ) x (cid:12)(cid:12) w H (cid:20) I P i =1 | α i | A ( θ i ) (cid:18) K P k =1 u k u Hk + vv H (cid:19) A H ( θ i )+ I (cid:21) w (43)By solving an equivalent MVDR problem, the corresponding receive beamforming vector tomaximize the output SINR can be given by w ∗ = Σ ( { u k } , v ) − A ( θ ) xx H A H ( θ ) Σ ( { u k } , v ) − A ( θ ) x , (44)where Σ ( { u k } , v )= I X i =1 | α i | A ( θ i ) K X k =1 u k u Hk + vv H ! A H ( θ i )+ I . (45)Substituting (44) into (43), the SINR of the radar receiver can be calculated as γ ( II ) R = x H Φ ( { u k } , v ) x , (46)where Φ ( { u k } , v ) = | α | A H ( θ ) Σ ( { u k } , v ) − A ( θ ) . (47)And the average SINR of the radar receiver can be given by ¯ γ ( II ) R = E (cid:2) x H Φ ( { u k } , v ) x (cid:3) = K X k =1 u Hk Φ ( { u k } , v ) u k + v H Φ ( { u k } , v ) v (48)For the CU k , it has a priori information on the probing signal. After the probing signalinterference cancelling, its received SINR ¯ γ C,k ( { u k } ) can also be expressed as (38). Thus, thebeamforming optimization problem that maximizes the SINR of the radar receiver and satisfiesthe SINR constraints of the CUs can be formulated as (P4 .
1) max { u k } , v ¯ γ (II) R = K P k =1 u Hk Φ u k + v H Φ v s . t . ¯ γ C,k = (cid:12)(cid:12) h Hk u k (cid:12)(cid:12) P j = k (cid:12)(cid:12) h Hk u j (cid:12)(cid:12) ≥ Γ k , ∀ k K P k =1 u Hk u k + v H v ≤ P (49)Let us then employ the sequential optimization to find the transmit beamforming vector { u k } and v in an iterative fashion, where we first compute Φ = Φ [ { u ( m − k } , v ( m − ] at the m -th iteration with { u ( m − k } and v ( m − being obtained from the ( m − -th iteration. Thus, thebeamforming optimization problem can be again formulated as (P4 .
2) max { u k } , v ¯ γ (II) R = K P k =1 u Hk Φ u k + v H Φ v s . t . ¯ γ C,k = (cid:12)(cid:12) h Hk u k (cid:12)(cid:12) P j = k (cid:12)(cid:12) h Hk u j (cid:12)(cid:12) ≥ Γ k , ∀ k K P k =1 u Hk u k + v H v ≤ P , (50)which can be solved using the SDR by letting U k = u k u Hk and V k = v k v Hk , i.e, (P4 .
3) max { U k } , V ¯ γ ( II ) R = K P k =1 tr ( Φ U k ) +tr ( Φ V )s . t ¯ γ C,k = tr ( H k U k )Γ k − P j = k tr ( H k U j ) ≥ , ∀ k K P k =1 tr ( U k ) + tr ( V ) ≤ P U k (cid:23) , ∀ k, V (cid:23) . (51)Although the rank-1 constraints have been removed for the convexity of the problem, theoptimal solution can be guaranteed to have rank-1 property by the following proposition. Further-more, the dedicated probing signal should be employed according to the following proposition. Proposition 4. (Rank-1 property of Problem (P4.3)) For the non-dedicated probing signal casewith multiple CUs, there is always a solution to Problem (P4.3) satisfying that rank ( U ∗ k ) = 1 , ∀ k and rank ( V ∗ ) ≤ . Thus, the optimal beamforming vector for the CU k and the dedicatedprobing signal, i.e., the optimal solution to Problem (P4.2) can be given by u ∗ k and v ∗ with u ∗ k u ∗ Hk = U ∗ k and v ∗ v ∗ H = V ∗ . Specifically, v ∗ = √ τ P ˆg , where ˆg = g / k g k , g is thedominant eigenvector of Φ , and ≤ τ ≤ . Algorithm 3
Beamforming design of DFRC BS for multiple CUs scenario with dedicated probingsignal.Initialize { θ , θ , · · · , θ I } and { α , α , · · · , α I } .Initialize n u (0) k o = v (0) = [1 , · · · , T .p ( K + 1) P N t . Initialize the convergence threshold ∆ , and m = 0 . repeat Set m = m + 1 .Calculate Φ = Φ hn u ( m − k o , v ( m − i according to (47) and ¯ γ (II) R hn u ( m − k o , v ( m − i according to (48).Optimize { U ∗ k } and V ∗ according to Problem (P4.3);Calculate { u ∗ k } and v ∗ satisfying u ∗ k ( u ∗ k ) H = U ∗ k and v ∗ v ∗ H = V ∗ based on the Proposition4.Calculate ¯ γ (II) R hn u ( m ) k o , v ( m ) i according to (48). until (cid:12)(cid:12)(cid:12) ¯ γ (II) R hn u ( m ) k o , v ( m ) i − ¯ γ (II) R hn u ( m − k o , v ( m − i(cid:12)(cid:12)(cid:12) ≤ ∆ . Proof.
The proof is given in Appendix D.
Remark 1. (Dedicated probing signal is employed or not) It can be observed that any feasiblesolution to Problem (4.2) is also feasible for Problem (3.2) with v ∗ = 0 , and vice versa. If v ∗ = 0 , a higher SINR of radar receiver can be achievable, which will also be verified bythe simulation results in the next section. Therefore, it is beneficial to employ the dedicatedprobing signal for multiple CUs scenario. The benefit is achieved at the cost of implementingan additional interference cancellation with a priori known probing signals by all CUs.Finally, we can also use the solution { u ( m − k } and v ( m − to update { u ( m ) k } and v ( m ) , and it isrepeated until the improvement of the radar receivers SINR becomes insignificant as illustratedin Algorithm 3. V. S IMULATION R ESULTS
In this section, we evaluate the performance of our proposed design via numerical simulations.We assume that both the DFRC BS and the radar receiver are equipped with uniform linear arrays(ULAs) with the same number of elements. The interval between adjacent antennas of the DFRCBS and the radar receiver is half-wavelength. The transmit power constraint of DFRC BS is setas P = 20 dBm. A target is located at the spatial angle θ = 0 ◦ with power | α | = 10 dB,
15 20 25 30
SINR of CU (dB) S I NR o f r ada r ( d B ) Optimal designTime sharing
Figure 2. Tradeoff between the SINR constraint of CU and the SINR of radar, N t = N r = 8 , h = [0.21 - 0.02i,-0.56 +0.65i,0.57- 0.23i,-0.93 + 0.47i,-0.19 + 1.35i,-0.19 + 0.11i,1.05 - 0.21i,1.02 - 0.35i]. and four fixed interferences are located at the spatial angels θ = − ◦ , θ = − ◦ , θ = 30 ◦ , θ = 60 ◦ , respectively. The power of each interference is | α i | = 30 dB, ∀ i . The channel vectorof each CU is randomly generated from i.i.d. Rayleigh fading. A. Single CU Scenario
In Fig. 2, the tradeoff between the SINR constraint of CU and the SINR of radar is shownfor the single CU scenario by varying the threshold of the CU’s SINR Γ . It is easy to identifytwo boundary points of the tradeoff, i.e., “Radar benchmark” and “Communication benchmark”.When the constraint of the CU’s SINR is inactive, the radar’s SINR is around dB and the CU’sSINR is around dB. When the constraint of the CU’s SINR is active, the maximum feasibleconstraint of the CU’s SINR is around dB, and the corresponding radar’s SINR is around dB. The optimal tradeoff between the SINR of CU and the SINR of radar is characterized by thesolid line. It can be observed that the SINR of the radar decreases with the increase of the CU’sSINR constraint, when the CU’s SINR constraint is above dB. Also, the tradeoff between theSINR of CU and the SINR of radar can be achieved by time sharing, which is shown by thedotted line. However, the optimal beamforming design for the DFRC BS yields better tradeoffperformance.In Fig. 3, the optimized beampatterns with different constraints of the CU’s SINR are illus-trated. The nulls are clearly placed at the locations of interferences, i.e., θ = − ◦ , θ = − ◦ , θ = 30 ◦ , θ = 60 ◦ , and the target is located at θ = 0 ◦ . When the SINR constraint of the CUis dB, it is inactive, and the beampattern is actually the optimal beampattern to maximize -80 -60 -40 -20 0 20 40 60 80 θ (deg) -140-120-100-80-60-40-20020 B ea m pa tt e r n ( d B i ) Γ =15dB Γ =25dB Figure 3. Optimized beampatterns with different constraints of the CU’s SINR, N t = N r = 8 , h = [0 . − . i, − .
56 +0 . i, . − . i, − .
93 + 0 . i, − .
19 + 1 . i, − .
19 + 0 . i, . − . i, . − . i ] . the radar’s SINR. When the SINR constraint of the CU increases to dB, the performance ofbeampattern becomes worse from the radar’s viewpoint. Particularly, both the peak to sideloberatio (PSLR) and the main beam power both decrease with the increase of the SINR constraintof the CU. Furthermore, the optimized beampatterns with different numbers of the transmit andreceive antennas are shown in Fig. 4. The SINR constraint of the CU is fixed as dB. Whenthe number of the DFRC BS antennas increases, the performance of beampattern becomes betterfrom the radar’s viewpoint. In particular, the PSLR increases with the number of the DFRC BSantennas, and the main beam width decreases with the number of the DFRC BS antennas. B. Multiple CUs Scenario
In Fig. 5, we evaluate the convergence performance of the proposed algorithm for differentnumbers of CUs. The SINR of the radar versus the number of iterations is provided. The SINR ofthe radar converges to a fixed value with only 2 iterations. And the converged SINR performancedecreases with the increase of the number of the CUs. Furthermore, the same converged SINRof the radar for dedicated probing signal case and non-dedicated probing signal case is observed,when the number of CU is 1. And the converged SINR of the radar improves with employingdedicated probing signal for the multiple CUs scenario.The average performance of the tradeoff between the SINR constraints of CUs and the averageSINR of radar is evaluated in Fig. 6 through Monte Carlo simulations. The SINR of radarexponentially decreases with the increase of the CUs’ SINR constraint. And the average SINR ofradar becomes worse when the number of the CU increases. That is because the feasible region -80 -60 -40 -20 0 20 40 60 80 θ (deg) -160-140-120-100-80-60-40-20020 B ea m pa tt e r n ( d B i ) N t = N r =6 N t = N r =8 N t = N r =10 Figure 4. Optimized beampatterns with different numbers of the DFRC BS antennas, N t = N r , h = [1 . − . i, − . − . i, . − . i, − .
32 + 0 . i, .
21 + 0 . i, − . − . i ] , h = [ h , . − . i, .
90 + 0 . i ], h = [ h , − . − . i, . − . i ], Γ = 20 dB.
Iteration index S I NR o f r ada r ( d B ) K=1-Non-dedicatedK=1-DedicatedK=2-Non-dedicatedK=2-DedicatedK=3-Non-dedicatedK=3-Dedicated
Figure 5. Convergence performance of the proposed algorithm for different numbers of CU, N t = N r = 6 , Γ = 20 dB, h = [ − . − . i, − . − . i, . − . i, . − . i, .
21 + 0 . i, − .
19 + 0 . i ] , h = [ − .
69 + 0 . i, − .
38 +0 . i, − .
32 + 1 . i, . − . i, − .
17 + 0 . i, − . − . i ] , h = [0 . − . i, − . − . i, . − . i, − . − . i, .
06 + 0 . i, − . − . i ] of the radar beamforming optimization becomes smaller when either the CUs’ SINR constraintor the number of the CUs increase. Furthermore, it can be observed that the average SINR ofradar is the same for the dedicated probing signal case and non-dedicated probing signal case,when the number of CU is 1. That verifies Proposition 2. And, it can also observed that theaverage SINR of the radar improves with employing dedicated probing signal for multiple CUsscenario. In the case that the CUs’ SINR constraints are loose, the improvement is not obvious
15 16 17 18 19 20 21 22 23 24 25
SINR of CU (dB) S I NR o f r ada r ( d B ) K=1-Non-dedicatedK=1-DedicatedK=2-Non-dedicatedK=2-DedicatedK=3-Non-dedicatedK=3-Dedicated
Figure 6. Tradeoff between the SINR constraints of CUs and the average SINR of radar, N t = N r = 8 . because the CUs’ SINR constraints are inactive. When the CUs’ SINRs are constrained to belarge, the improvement becomes obvious. For example, when the number of CU is 2 and CUs’SINR constraint is 10 dB, it improves about 1 dB.In Fig. 7, the average SINR of radar versus the numbers of CUs is illustrated with differentnumber of transmit and receive antennas through Monte Carlo simulations. The averageSINR of radar decreases with the increase of the number of CUs, and it increases with theincrease of the number of transmit and receive antennas. When the number of transmit andreceive antennas is large, the decrease in the average SINR of radar with the number of CUs isnot obvious. It can also be observed that the average SINR of radar improves by employing thededicated probing signal. And the improvement increases with the number of CUs.VI. C
ONCLUSION
In this paper, we have proposed the joint optimization of transmit and receive beamformingfor the DFRC system. The optimal tradeoff of SINR between radar and communication hasbeen characterized through maximizing the SINR of the radar under the SINR constraints ofthe CUs. For the single CU scenario, we have given the closed-form solution of the optimizedbeamforming, and it has been proved that there is no need of dedicated probing signals. For themultiple CUs scenario, the beamforming design has been formulated as a non-convex QCQP.We have obtained the optimal solutions by applying SDR with rand-1 property, and it has beenproved that the dedicated probing signal should be employed to improve the SINR of the radar.Numerical results have been provided to show that our algorithm is effective. The number of CUs S I NR o f r ada r ( d B ) Non-dedicated probing signalDedicated probing signal
Nt=10Nt=15 Nt=20Nt=5
Figure 7. Average SINR of radar with different number of CUs and antennas, N t = N r , Γ = 25 dB.
Future research may focus on beamforming designs with further practical constraints, e.g.constant modulus constraints. Also, our work is based on the prior information of both radar andcommunication. Thus, the robust design will be considered with the imperfect prior informationof the radar and the imperfect CSI of the communication.A
PPENDIX AP ROOF OF P ROPOSITION Γ ≤ (cid:12)(cid:12) h H ˆg (cid:12)(cid:12) , the SINR constraint of the CU is inactive. Thus, Problem (P1.2) reduces tothe MIMO radar beamforming optimization problem without communication constraints, i.e., max u ¯ γ ( I ) R = u H Φ u s . t . u H u ≤ P , (52)and the corresponding optimal solution is u ∗ = √ P ˆg .If P | h | ≥ Γ > P (cid:12)(cid:12) h H ˆg (cid:12)(cid:12) , the SINR constraint is active. The optimal beamforming vectorshould lie in the pace jointly spanned by ˆh and the projection of g into the null space of ˆh , i.e., ˆg ⊥ , where g H ⊥ ˆh = . The transmit power of u ∗ allocated in the direction of ˆh should satisfy theSINR constraint with the coefficient α in (19), and the left power is allocated in the directionof ˆg ⊥ with the coefficient β in (20), which improves the SINR of the radar receiver withoutinfluencing the SINR of the CU.If Γ > P | h | , the SINR constraint cannot be satisfied even with maximum ratio transmissionto the CU. A PPENDIX BP ROOF OF P ROPOSITION τ P with ≤ τ ≤ , Problem (P2.2)can be decomposed into the following two problems, i.e., (P2 .
3) max v v H Φ v s . t . v H v ≤ τ P (53)and (P2 .
4) max u u H Φ u s . t . ¯ γ C = (cid:12)(cid:12) h H u (cid:12)(cid:12) ≥ Γ u H u ≤ (1 − τ ) P . (54)For Problem (P2.3), the optimal beamforming vector of the probing signal can be calculatedas v ∗ = p τ P ˆg , (55)where ˆg = g / k g k and g is the dominant eigenvector of Φ . For Problem (P2.4), assuming P = τ P , the optimal beamforming vector of the probing signal can be calculated according toProposition 1, i.e., u ∗ = p (1 − τ ) P ˆg , Γ ≤ (1 − τ ) P (cid:12)(cid:12) h H ˆg (cid:12)(cid:12) (cid:16) α ′ ˆh + β ′ ˆg ⊥ (cid:17) , (1 − τ ) P | h | ≥ ΓΓ > (1 − τ ) P (cid:12)(cid:12) h H ˆg (cid:12)(cid:12) , (56) α ′ = s Γ k h k α g | α g | , (57) β ′ = s (1 − τ ) P − Γ k h k β g | β g | , (58)where g is the dominant eigenvector of Φ , ˆg = g / k g k , ˆh = h / k h k , g ⊥ = g − ( ˆh H g ) ˆh denoting the projection of g into the null space of ˆh , ˆg ⊥ = g ⊥ / k g ⊥ k and g can be expressedas g = α g ˆh + β g ˆg ⊥ .According to (55), one has u ∗ + v ∗ = √ P ˆg , Γ ≤ (1 − τ ) P (cid:12)(cid:12) h H ˆg (cid:12)(cid:12) (cid:16) α ′ ˆh + β ′ ˆg ⊥ (cid:17) , (1 − τ ) P | h | ≥ ΓΓ > (1 − τ ) P (cid:12)(cid:12) h H ˆg (cid:12)(cid:12) , (59)Thus, u ∗ + v ∗ can achieve the best performance when τ = 0 . That is the optimal beamformingvector of the probing signal v ∗ = for Problem (P2.2), which completes the proof.A PPENDIX CP ROOF OF P ROPOSITION min { X l } L P l =1 tr ( B l X l )s . t . L P l =1 tr ( C ml X l ) ⊲ m b m , m = 1 , · · · , M X l ≻ , l = 1 , · · · , L , (60)where { B l } , { C ml } are all Hermitian matrices (not necessarily positive semidefinite), b m ∈R , ∀ m , ⊲ m ∈ {≥ , ≤ , = } , ∀ m , and { X l } are all Hermitian matrices.Suppose the above SDP is feasible and bounded, where the optimal value is attained. Then,according to [29], it always has an optimal solution { X ∗ l } such that L X l =1 [rank ( X ∗ l )] ≤ M. (61)Based on the above result, it can be proved that there is always a solution to Problem (P3.3)satisfying that K X k =1 [rank ( U ∗ k )] ≤ K + 1 . (62)Meanwhile, due to the SINR constraints of each CU, one has U ∗ k = 0 , ∀ k and then rank ( U ∗ k ) ≥ , ∀ k . Thus, rank ( U ∗ k ) = 1 , ∀ k according to (62), which completes the proof.A PPENDIX DP ROOF OF P ROPOSITION K X k =1 [rank ( U ∗ k )] + [rank ( V ∗ )] ≤ K + 1 . (63)Meanwhile, due to the SINR constraints of each CU, U ∗ k = 0 , ∀ k , and then rank ( U ∗ k ) ≥ , ∀ k .Thus, one has rank ( U ∗ k ) = 1 , ∀ k and rank ( V ∗ ) ≤ according to (63).Next, we prove the necessity of employing the dedicated probing signal. The Lagrangianfunction of Problem (P4.3) is L = µP − K X k =1 λ k + K X k =1 tr ( D k U k ) + tr ( EV ) , (64)where D k = Φ + H k Γ k − X j = k λ j tr ( H j ) − µ I , (65)and E = Φ − µ I , (66)with λ k ≥ , ∀ k and µ ≥ being the dual variables associated with the SINR constraint of theCU k and the transmit power constraint, respectively. And the dual problem is min { λ k } ,µ µP − K P k =1 λ k s . t . D k ≺ , ∀ k, E ≺ (67)Note that Problem (P4.3) is convex. Thus, strong duality holds and the KKT conditions arenecessary and sufficient for any optimal solution to Problem (P4.3). Due to the fact that E = Φ − µ I (cid:22) . We have that µ ∗ ≥ κ , where κ is the dominant eigenvalue of Φ . If µ ∗ = κ , wehave that v ∗ = √ τ P ˆg , with < τ ≤ . If µ ∗ > κ , we have that v ∗ = . It completes the proof.R EFERENCES [1] J. Choi, V. Va, N. Gonzalez-Prelcic, R. Daniels, C. R. Bhat, and R. W. Heath, “Millimeter-wave vehicular communicationto support massive automotive sensing,”
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