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Joint Waveform Optimization and Antenna Position Selection for MIMO Radar Beam Scanning
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摘要: 为实现集中式多输入多输出(MIMO)雷达波束扫描，本文在峰值平均功率比(PAPR)、能量以及布尔(天线位置选择)约束下，基于min-max波束图匹配准则，首次提出MIMO雷达天线位置和多组探测波形(一组波形对应一个独立的波束图)的联合优方法。由于非凸PAPR约束、布尔约束以及min-max目标函数的非凸非光滑性导致了优化问题成为典型的大规模NP-难问题。为求解该NP-难优化问题，该文首先利用Lawson算法将min-max问题转化为迭代加权最小二乘(ILS)问题，然后根据上界函数最小化(MM)准则简化ILS优化问题，最后用交替方向乘子法(ADMM)求解简化后的上界优化问题。数值仿真结果检验了所提算法的有效性。
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关键词:
- MIMO雷达发射波束图 /
- 峰值平均功率比(PAPR) /
- 稀疏阵列天线 /
- 天线选择 /
- Lawson算法 /
- 上界函数最小化(MM) /
- 交替方向乘子法(ADMM)
Abstract: In this study, under the Peak-to-Average Power Ratio (PAPR), energy, and binary (for antenna position selection) constraints, we proposed an antenna position selection and beam scanning method for colocated Multiple-Input Multiple-Output (MIMO) radar system using the min-max beampattern amplitude matching criterion. In our design, antenna positions and a set of probing waveforms were jointly determined to match a set of beampattern masks, and hence realize the beam scan. The resultant problem was large-scale, nonconvex, nonsmooth, and typical nondeterministic hard, because of the PAPR and nonconvex binary constraints, and the max and modulus operations in the objective function. To address these issues, we first transformed the min-max problem into the Iterative weighted Least Squares (ILS) problem using the Lawson algorithm, replaced the nonsmooth nonconvex objective function with the convex majorization function, and finally applied the alternating direction method of multipliers to solve the majorized ILS problem. Finally, several numerical examples were given to show the effectiveness of the proposed algorithms. -
图 3 仿真样例2(
$L = 20$ ,$\hat L = 12$ )Figure 3. Simulation example 2 (
$L = 20$ ,$\hat L = 12$ )
图 4 仿真样例2(
$L = 17$ ,$\hat L = 12$ )Figure 4. Simulation example 2 (
$L = 17$ ,$\hat L = 12$ )
图 5 仿真样例2(
$L = 20$ ,$\hat L = 15$ )Figure 5. Simulation example 2 (
$L = 20$ ,$\hat L = 15$ )表 1 基于波形优化和天线位置选择的MIMO雷达波束扫描算法
Table 1. Algorithm for joint waveform optimization and antenna position selection for MIMO radar beam scanning
输入参数：I, ${\boldsymbol{\varTheta }}_m^i$, ${\boldsymbol{\varTheta }}_s^i$, ${\boldsymbol{\varTheta }}_n^i$, ${\boldsymbol{\varTheta }}_t^i$, ${{\boldsymbol{W}}^i},\forall i$, ${\zeta _l},\forall l$, $\gamma $, $\hat L$, L, $\rho $, ${\delta _{{\rm{out}}} } = {10^{ - 8} }$, ${\delta _{{\rm{inner}}} } = {10^{ - 8} }$,
$\{ \{ {{\boldsymbol{X}}^i}(0),{{\boldsymbol{Z}}^i}(0),{{\boldsymbol{\varPsi }}^i}(0)\} _{i = 1}^I,{\boldsymbol{\alpha }}(0),{\boldsymbol{\bar s}}(0)\} $, $\tilde R_{m,q}^i(0) = W_{m,q}^i\left[ {{{\boldsymbol{W}}^i}; \cdots ;{{\boldsymbol{W}}^I}} \right]$, $\forall i,\forall m,\forall q$, ${K_{{\rm{max}}} }$, $k = 0$。1 WHILE {$|{\rm{obj}}(k) - {\rm{obj}}(k - 1)| > {\sigma _{{\rm{out}}} }$ & $k \le {K_{ {\rm{max} } } }$}(Lawson迭代) 2 $t = 0$;
3 WHILE $\left\{ {\dfrac{1}{I}\displaystyle\sum\limits_{i = 1}^I {\left\| { { {\boldsymbol{Z} }^i}(t) - { {\boldsymbol{Z} }^i}(t - 1)} \right\|_{\rm F}^2} > {\sigma _{{\rm{inner}}} } } \right\}$(M-ADMM迭代)4 $ \bullet $ 更新${\boldsymbol{d}}_m^i(t + 1)$； 5 $ \bullet $ 顺序更新$\{ {\boldsymbol{\bar s}}(t + 1),{\boldsymbol{\alpha }}(t + 1),\{ {{\boldsymbol{X}}^i}(t + 1)\} _{i = 1}^I\} $； 6 $ \bullet $ 更新$\{ {{\boldsymbol{Z}}^i}(t + 1)\} _{i = 1}^I$； 7 $ \bullet $ 更新$\{ {{\boldsymbol{\varPsi }}^i}(t + 1)\} _{i = 1}^I$； 8 $t: = t + 1$; 9 ENDWHILE(M-ADMM迭代结束) 10 ${{\boldsymbol{Z}}^i}(k + 1) = {{\boldsymbol{Z}}^i}(t - 1),\forall i,{\beta ^i}(k + 1) = {\bar h^i}({{\boldsymbol{Z}}^i}(k + 1)),\forall i$; 11 更新Lawson权值$w_m^i(k)$; 12 ${\rm{obj}}(k) = {\max _{m,i} }\{ G_m^i({ {\boldsymbol{Z} }^i}(k + 1),{\beta ^i}(k + 1))\}$; 13 $k: = k + 1$; 14 ENDWHILE(Lawson迭代结束) 输出：探测波形${{\boldsymbol{Z}}^i}( \star ) = {{\boldsymbol{Z}}^i}(t),i = 1,2,\cdots,I$，天线位置${\boldsymbol{\bar s}}( \star ) = {\boldsymbol{\bar s}}(t)$。 
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