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Transmit Waveform Design for Symbol-level Precoding-based One-bit Dual-functional Radar-communication
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摘要: 在大规模多输入多输出(MIMO)通信和雷达系统中，采用单比特数模转换器(DAC)是一种降低发射系统硬件成本和功耗的有效方法。该文研究单比特量化下雷达通信一体化系统的发射波形设计，在给定通信服务质量约束下最小化雷达发射波束图的积分旁瓣主瓣比，通过提升发射波束的功率集中程度以获得良好的发射波束赋形性能。针对单比特量化导致发射波形仅具有低自由度可行域的问题，该文采用符号级预编码技术，基于有益干扰(CI)原理充分利用空域和时域自由度来辅助波形设计。由于所提出的波形设计问题具有非凸分式二次目标函数和大量的非凸离散约束，该文提出了一种基于丁克尔巴赫(Dinkelbach)变换和交替方向乘子法(ADMM)的算法来有效求解该NP-难问题。仿真结果表明，所设计的波形能够显著降低对DAC分辨率的需求，并在满足下行用户通信质量需求的条件下具有良好的雷达发射波束图性能。Abstract: This study explores the use of one-bit Digital-to-Analog Converters (DAC) to mitigate the challenges of high hardware costs and excessive power consumption in large-scale Multiple-Input Multiple-Output (MIMO) communication and radar systems. The present study focuses on the design of one-bit transmit waveforms for dual-functional radar and communication systems. Under preset communication Quality of Service (QoS) constraints, the objective was to minimize the integral sidelobe-to-mainlobe ratio of the radar transmit beampattern. This should help enhance the power concentration of the transmitted beampattern and improve the performance of the beampattern synthesis. To address the limited Degrees of Freedom (DoF) caused by one-bit quantization, this study employs symbol-level precoding technology and then fully utilizes the DoFs in spatial and temporal domains to assist waveform design based on the principle of Constructive Interference (CI). To address the nonconvex fractional quadratic objective function and the multiple nonconvex discrete constraints inherent in the proposed waveform design problem, this study introduces an algorithm that combines the Dinkelbach transform with the Alternating Direction Method of Multipliers (ADMM). This approach effectively tackles the NP-hard problem. The numerical results demonstrate that the designed waveform significantly reduces the required DAC resolution and achieves excellent radar beampattern performance while satisfying the QoS requirements of downlink multiuser communications.
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图 1 雷达通信一体化系统示意图
Figure 1. Illustration of Dual-Functional Radar-Communication (DFRC) system
图 4 目标函数值和Dinkelbach辅助变量$ \xi $随迭代次数的变化曲线
Figure 4. The variations of objective function value and Dinkelbach auxiliary variable $ \xi $ with the number of iterations
图 5 不同发射波形设计下的波形序列相位分布图
Figure 5. Comparison of the phase diagram of waveform sequence in different transmit waveform design
图 6 不同发射波形设计下的雷达发射波束图
Figure 6. Comparison of the radar transmit beampattern in different transmit waveform design
图 7 发射阵列规模N对发射波束图性能的影响
Figure 7. The influence of different antennas number N on the performance of the transmit beampattern
图 8 平均误码率与通信性能需求的关系图
Figure 8. Average symbol error rate versus communication requirements
图 9 雷达性能与通信性能需求的关系图(实斜线$ K = 2 $，虚斜线$ K = 3 $)
Figure 9. Radar performance versus communication requirements (solid lines represents the $ K = 2 $ scenario, and dashed lines represents the $ K = 3 $ scenario)
1 梯度投影算法
1. Gradient projection algorithm
输入：$ \xi ,{\boldsymbol{e}},{{\boldsymbol{\varOmega}} _{{\text{s}},{\text{R}}}},{{\boldsymbol{\varOmega }}_{{\text{m}},{\text{R}}}},{\boldsymbol{w}},{\boldsymbol{H}},\alpha ,\rho ,{\kappa _1},{\kappa _2},{\tau _0},{\delta _1},{\delta _2} $。 输出：$ {{\boldsymbol{x}}_{\text{R}}} $。 初始化：$ {\boldsymbol{x}}_{\text{R}}^{(0)} \in \mathcal{I} $，设置$ r = 0 $； 第1步：根据式(24)计算梯度$ \nabla f({\boldsymbol{x}}_{\text{R}}^{(r)}) $，令$ r = r + 1 $，$ \tau = {\tau _0} $； 第2步：计算$ {\boldsymbol{\tilde x}}_{\text{R}}^{(r)} = {\boldsymbol{x}}_{\text{R}}^{(r - 1)} - \tau \nabla f({\boldsymbol{x}}_{\text{R}}^{(r - 1)}) $； 第3步：若$ {\boldsymbol{\tilde x}}_{\text{R}}^{(r)} \in \mathcal{I} $，则令$ {\boldsymbol{x}}_{\text{R}}^{(r)} = {\boldsymbol{\tilde x}}_{\text{R}}^{(r)} $，跳至第6步，否则令$ {{\boldsymbol{v}}^{(0)}} = {\boldsymbol{\tilde x}}_{\text{R}}^{(r)} $, $ i = 0 $； 第4步：令$ i = i + 1 $，根据式(25)计算$ {{\boldsymbol{\tilde v}}^{(i)}} = {\mathcal{P}_\mathcal{B}}({{\boldsymbol{v}}^{(i - 1)}}) $，根据式(26)计算$ {{\boldsymbol{v}}^{(i)}} = {\mathcal{P}_\mathcal{H}}({{\boldsymbol{\tilde v}}^{(i)}}) $； 第5步：若$ \left\| {{{\boldsymbol{v}}^{(i)}} - {{\boldsymbol{v}}^{(i - 1)}}} \right\| < {\delta _2} $，令$ {\boldsymbol{x}}_{\text{R}}^{(r)} = {{\boldsymbol{v}}^{(i)}} $，进行下一步，否则返回第4步； 第6步：若$ f({\boldsymbol{x}}_{\text{R}}^{(r)}) < f({\boldsymbol{x}}_{\text{R}}^{(r - 1)}) + {\kappa _1}\nabla f{({\boldsymbol{x}}_{\text{R}}^{(r - 1)})^{\rm T}}({\boldsymbol{x}}_{\text{R}}^{(r)} - {\boldsymbol{x}}_{\text{R}}^{(r - 1)}) $，进行下一步，否则令$ \tau = {\kappa _2}\tau $，返回第2步； 第7步：若$ {\boldsymbol{x}}_{\text{R}}^{(r)} - {\boldsymbol{x}}_{\text{R}}^{(r - 1)} < {\delta _1} $，输出$ {{\boldsymbol{x}}_{\text{R}}} $，否则返回第1步。 
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2 Dinkelbach-ADMM算法
2. Dinkelbach-ADMM algorithm
输入：$ {\varTheta _{\text{s}}},{\varTheta _{\text{m}}},{{\boldsymbol{h}}_k},{\varPsi _k},{s_k}[l],\varPhi ,{P_{{\text{tot}}}},\rho ,\nu ,{\varepsilon _1},{\varepsilon _2} $。 输出：x。 初始化：$ \{ {{\boldsymbol{e}}^{(0)}},{\boldsymbol{x}}_{\text{R}}^{(0)},{{\boldsymbol{w}}^{(0)}}\} $，设置$ t = 0 $； 第1步：根据式(28)构造$ {{\boldsymbol{x}}^{(0)}} $，根据式(12)更新$ {\xi ^{(0)}} $； 第2步：令$ t = t + 1 $, $ k = 0 $； 第3步：令$ k = k + 1 $，根据式(21)更新$ {{\boldsymbol{e}}^{(k)}} $，根据算法1更新$ {\boldsymbol{x}}_{\text{R}}^{(k)} $，根据式(17)更新$ {{\boldsymbol{w}}^{(k)}} $； 第4步：若$ {\left\| {{{\boldsymbol{e}}^{(k)}} - {\boldsymbol{x}}_{\text{R}}^{(k)}} \right\|_2} < {\varepsilon _1} $或$ \rho {\left\| {{\boldsymbol{x}}_{\text{R}}^{(k)} - {\boldsymbol{x}}_{\text{R}}^{(k - 1)}} \right\|_2} < {\varepsilon _2} $，根据式(27)和式(28)更新$ {{\boldsymbol{x}}^{(t)}} $，根据式(12)更新$ {\xi ^{\left( t \right)}} $，进行下一步，否则返
回第3步；第5步：若$ \left| {{\xi ^{(t)}} - {\xi ^{(t - 1)}}} \right| < \nu $，输出x，否则返回第2步。
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