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Joint Transmit-receive Beam Design of FDA-MIMO Dual-function Radar-communication Systems for Radio-frequency Stealth
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摘要: 雷达通信一体化(DFRC)系统的射频隐身性能是雷达隐身探测和通信隐蔽传输的关键。然而，传统基于相控阵和MIMO体制的波束形成方案不具备距离维辐射能量控制能力，导致一体化发射信号容易被敌方无源探测系统截获。针对此问题，该文提出一种面向射频隐身的频控阵-多输入多输出(FDA-MIMO)雷达通信一体化发射接收波束联合设计方法。首先，构建基于正交波形生成、频率分集调制和发射波束形成加权的FDA-MIMO一体化发射信号模型，通过匹配滤波和接收波束形成获得雷达等效发射波束图与通信传输信道的距离角度二维表达式。其次，以通信信息嵌入和通信可达速率为约束条件，以雷达目标处的等效发射波束图功率最小化和输出信干噪比最大化为双优化目标函数，建立面向射频隐身的FDA-MIMO雷达通信一体化发射接收波束联合优化模型。最后，提出基于加权均方误差最小化(WMMSE)和共享交替方向乘子法(C-ADMM)的联合优化算法，推导各变量的闭式表达式并结合凸优化算法，实现低复杂度求解。仿真结果表明，该文所提方法的雷达探测与通信传输在距离角度二维平面上均为“点对点”模式，具备良好的射频隐身能力，同时能够提供较高的杂波和干扰抑制性能以及较低的通信误码率。Abstract: Efficient Radio-Frequency (RF) stealth is crucial for Dual-Function Radar-Communication (DFRC) systems that detect radar stealth and con vert communication transmission. However, traditional beamforming schemes based on phased arrays and Multiple-Input Multiple-Output (MIMO) systems lack the ability to control the radiation energy in the range dimension, resulting in the facile interception of integrated transmission signals by enemy-owned passive detection systems. To address this issue, a joint transmit-receive beamforming design for Frequency Diversity Array MIMO (FDA-MIMO) DFRC systems is designed herein to achieve RF stealth. First, an integrated transmission signal model based on orthogonal waveform generation, frequency diversity modulation, and weighted transmission beamforming is constructed. The two-dimensional expression of the distance angle between the radar equivalent transmission beam pattern and the communication transmission channel is obtained through matched filtering and reception beamforming. Second, with communication information embedding and communication reachable rate as constraints, a joint optimization model for FDA-MIMO radar communication integrated transmission and reception beams for RF stealth is established. The model aims to simultaneously minimize the equivalent transmission beam power at the radar target and maximize the output signal-to-noise ratio. Finally, a joint optimization algorithm based on weighted mean square error minimization and the consensus alternating direction multiplier method is proposed. Closed form expressions for each variable are derived and combined with convex optimization algorithms to achieve low-complexity solutions. The simulation results show that radar detection and communication transmission using the proposed method form a “point-to-point” pattern on the two-dimensional plane of range and angle, exhibiting good RF stealth capability. Simultaneously, this method can provide high clutter and interference suppression performance as well as a low communication bit error rate.
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图 2 FDA-MIMO雷达通信一体化发射接收波束联合优化算法流程图
Figure 2. Flowchart of joint optimization algorithm of transmit-receive beam for FDA-MIMO DFRC
图 4 每个通信用户的可达速率与迭代次数的关系
Figure 4. Sum-rate versus iteration number of each communication user
图 5 传统角度维发射波束图(式(17))
Figure 5. Conventional angle-dimensional transmitted beampattern (Eq. (17))
图 6 距离角度二维等效发射波束图(式(15))
Figure 6. Two-dimensional range-angle equivalent transmitted beampattern (Eq. (15))
图 7 距离角度二维等效发射波束图(目标剖面)
Figure 7. Two-dimensional range-angle equivalent transmitted beampattern (range profiles)
图 8 距离角度二维等效发射波束图(杂波角度维剖面)
Figure 8. Two-dimensional range-angle equivalent transmitted beampattern (angle profiles of clutters)
图 9 距离角度二维等效发射波束图(杂波距离维剖面)
Figure 9. Two-dimensional range-angle equivalent transmitted beampattern (range profiles of clutters)
图 10 雷达角度维接收波束图(式(16))
Figure 10. Radar angle-dimensional received beampattern (Eq. (16))
图 11 距离角度二维发射接收双程波束图(式(14))
Figure 11. Two-dimensional range-angle equivalent transmit-received beampattern (equation (14))
图 13 距离角度二维误码率(三维平面图)
Figure 13. Two-dimentional range-angle communication BER (three-dimensional plane)
图 14 距离角度二维误码率(二维剖面图)
Figure 14. Two-dimentional range-angle communication BER (two-dimensional profile)
1 改进稳健接收波束形成(IRRB)算法
1. Improved Robust Received Beamforming algorithm (IRRB)
输入：N, $ {N_{\rm r}} $, E, Q, J, $ J' $, $ \varsigma $, $ \nu $, $ {\rho _{1, 1}} $, $ {\rho _{1, 2}} $, $ {\rho _{1, 3}} $, $ {\rho _{1, 4}} $, $ {\zeta _j} $, $ {\chi _{j, j'}} $, $ \eta $, $ {\theta _t} $, $ {\theta _q} $, $ {\theta _j} $, $ {\varphi _j} $, $ {\stackrel \frown{\theta } _{j, j'}} $, $ {\gamma _t} $, $ {\gamma _q} $, $ {\gamma _j} $。 输出：雷达角度维接收波束形成器$ {{\boldsymbol{e}}^{{\text{opt}}}} $。 步骤1：初始化：$ {{\boldsymbol{e}}^{(0)}} $, $ \left\{ {\vartheta _{j, j'}^{(0)}} \right\} $, $ {h^{(0)}} $, $ \left\{ {\varpi _{j, j'}^{(0)}} \right\} $, $ \mu _t^{(0)} $, $ \left\{ {\mu _{j, j'}^{(0)}} \right\} $, $ \left\{ {\mu _{j, j'}^{'(0)}} \right\} $, $ \left\{ {\eta _{j, j'}^{(0)}} \right\} $。 步骤2：令$ l = {\text{1}} $。 步骤3：利用式(39a)更新$ {{\boldsymbol{e}}^{(l)}} $。 步骤4：利用式(39b)更新$ \left\{ {\vartheta _{j, j'}^{(l)}} \right\} $。 步骤5：利用式(39c)更新$ {h^{(l)}} $。 步骤6：利用式(39d)更新$ \left\{ {\varpi _{j, j'}^{(l)}} \right\} $。 步骤7：利用式(38e)、式(38f)、式(38g)和式(38h)更新$ \mu _t^{(l)} $, $ \left\{ {\mu _{j, j'}^{(l)}} \right\} $, $ \left\{ {\mu _{j, j'}^{'(l)}} \right\} $和$ \left\{ {\eta _{j, j'}^{(l)}} \right\} $。 步骤8：如果$ l \ge N_{{\text{IRRB}}}^{{\text{max}}} $或者$ {\varOmega }_{\text{IRRB}}^{\text{(}l\text{)}}\le {\epsilon}_{\text{IRRB}} $，则停止迭代，输出$ {{\boldsymbol{e}}^{{\text{opt}}}} = {{\boldsymbol{e}}^{{\text{(}}l{\text{)}}}} $。否则，令$ l = l + 1 $，跳转步骤3。其中，
$ \varOmega _{{\text{IRRB}}}^{{\text{(}}l{\text{)}}} = \left| {{{\boldsymbol{e}}^{{\text{(}}l{\text{)H}}}}\left( {{{\boldsymbol{R}}_q} + {{\boldsymbol{R}}_n}} \right){{\boldsymbol{e}}^{{\text{(}}l{\text{)}}}} + \dfrac{\varsigma }{2}{h^{{\text{(}}l{\text{)}}2}} + \dfrac{\nu }{2}{{\left\| {{{\boldsymbol{e}}^{{\text{(}}l{\text{)}}}}} \right\|}^2}} \right| $，$ N_{{\text{IRRB}}}^{{\text{max}}} $和$ {\epsilon}_{\text{IRRB}} $分别表示最大迭代次数和停止门限。
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2 FDA-MIMO雷达通信一体化发射接收波束联合优化算法
2. Joint optimization algorithm of transmit-receive beam for FDA-MIMO DFRC
输入：N, $ {N_{\rm t}} $, $ {N_{\rm r}} $, $ {N_{{\mathrm{c}},u}} $, $ {N_{{\text{path}}}} $, E, Q, J, U, $ \kappa $, $ {\rho _1} $, $ {\rho _2} $, $ {\rho _3} $, $ {\alpha _{u, l}} $, $ \eta $, $ \beta $, $ {\theta _t} $, $ {\theta _q} $, $ {\theta _j} $, $ {\theta _u} $, $ {R_u} $, $ \phi _{u, l}^{({\mathrm{t}})} $, $ \phi _{u, l}^{({\mathrm{r}})} $, $ {R_t} $, $ {R_q} $, $ {R_{u, l}} $, $ \tau $, $ {\varepsilon _u} $, $ {d_{u, n}} $, $ {b_{u, n}} $, $ {\gamma _t} $, $ {\gamma _q} $, $ {\gamma _j} $。 输出：一体化发射波束形成器$ {{\boldsymbol{T}}^{{\text{opt}}}} $，雷达接收波束形成器$ {{\boldsymbol{u}}^{{\text{opt}}}} $，通信接收波束形成器$ \left\{ {{\boldsymbol{W}}_u^{{\text{opt}}}} \right\} $。 步骤1：初始化：$ {{\boldsymbol{e}}^{(0)}} $, $ {{\boldsymbol{T}}^{(0)}} $, $ {{\boldsymbol{v}}^{(0)}} $, $ \left\{ {{\boldsymbol{q}}_{u, n}^{(0)}} \right\} $, $ \left\{ {L_{u, n}^{(0)}} \right\} $, $ \left\{ {Q_{u, n}^{(0)}} \right\} $, $ \left\{ {\lambda _{u, n}^{(1)(0)}} \right\} $, $ \left\{ {\lambda _{u, n}^{(2)(0)}} \right\} $, $ \left\{ {\lambda _{u, n}^{(3)(0)}} \right\} $。 步骤2：利用式(21)和式(22)计算通信信道$ {{\boldsymbol{H}}_u} $。 步骤3：令$ k = 1 $。 步骤4：利用式(29)计算$ {e_{u, n}} $和$ {w_{u, n}} = 1/{e_{u, n}} $，并利用式(30)更新$ \left\{ {{\boldsymbol{W}}_u^{(k)}} \right\} $。 步骤5：通过算法1更新$ {{\boldsymbol{e}}^{(k)}} $。 步骤6：通过式(46a)、式(46b)和式(46c)获得$ {{\boldsymbol{d}}^{(k)}} $，并根据$ {\boldsymbol{d}} = {\text{vec}}\left( {\boldsymbol{T}} \right) $更新$ {{\boldsymbol{T}}^{(k)}} $。 步骤7：利用式(53)更新$ {{\boldsymbol{v}}^{(k)}} $。 步骤8：利用CVX工具箱求解凸优化模型(55)更新$ \left\{ {{\boldsymbol{q}}_{u, n}^{(k)}} \right\} $。 步骤9：依据$ \sum\nolimits_{u = 1}^U {\sum\nolimits_{n = 1}^N {{{\left| {{L_{u, n}}\left( \partial \right) - {d_{u, n}}{b_{u, n}}} \right|}^2}} } = {\tau ^2} $，通过二分法获得$ {\partial ^{{\text{opt}}}} $，然后利用式(58)更新$ \left\{ {L_{u, n}^{(k)}} \right\} $。 步骤10：依据$ \sum\nolimits_{n = 1}^N {{\omega _{u, n}}{{\left| {{Q_{u, n}}\left( \ell \right) - 1} \right|}^2}} = {\upsilon _u} $，通过二分法获得$ {\ell ^{{\text{opt}}}} $，然后利用式(61)更新$ \left\{ {Q_{u, n}^{(k)}} \right\} $。 步骤11：利用式(62a)、式(62b)和式(62c)更新$ \left\{ {\lambda _{u, n}^{(1)(k)}} \right\} $, $ \left\{ {\lambda _{u, n}^{(2)(k)}} \right\} $和$ \left\{ {\lambda _{u, n}^{(3)(k)}} \right\} $。 步骤12：如果$ k \ge N_{{\text{C-ADMM}}}^{{\text{max}}} $，则停止迭代，输出$ {{\boldsymbol{T}}^{{\text{opt}}}} = {{\boldsymbol{T}}^{{\text{(}}k{\text{)}}}} $, $ {{\boldsymbol{u}}^{{\text{opt}}}} = {{\boldsymbol{e}}^{{\text{(}}k{\text{)}}}} \otimes {{\boldsymbol{v}}^{{\text{(}}k{\text{)}}}} $, $ {\boldsymbol{W}}_u^{{\text{opt}}} = {\boldsymbol{W}}_u^{{\text{(}}k{\text{)}}} $。否则，令$ k = k + 1 $，跳转步骤4。
其中，$ N_{{\text{C-ADMM}}}^{{\text{max}}} $表示最大迭代次数。
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