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Fast Tensor-based Three-dimensional Sparse Bayesian Learning Space-Time Adaptive Processing Method
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摘要: 当机载雷达处于非正侧视工作模式时，非平稳杂波会对运动目标检测造成严重干扰。传统三维空时自适应处理(3D-STAP)方法通过构造俯仰-方位-多普勒三维自适应滤波器，可有效抑制非平稳杂波，然而巨大的系统自由度导致其在非均匀杂波环境下训练样本严重不足。虽然稀疏恢复(SR)技术可有效改善样本需求，但庞大的运算开销又使得该技术难以应用于实际。针对上述问题，该文结合机载雷达回3阶张量结构提出一种新的快速三维稀疏贝叶斯学习STAP方法，通过采用运算开销更低的张量处理将大规模矩阵求解拆分为多个小规模矩阵计算，从而大幅降低运算复杂度。详尽的数值实验验证了所提张量基SR-STAP方法可在维持SR-STAP小样本处理性能不变的基础上，将运行时间直接降低数个量级，因此是一种更适用于实际工程的SR-STAP处理方式。Abstract: When airborne radar is applied to the non-side-looking mode, moving target detection performance considerably degrades because of the nonstationary clutter. Conventional three-dimensional (3D) Space-Time Adaptive Processing (STAP) can effectively eliminate the nonstationary clutter via adaptively constructing an elevation-azimuth-Doppler 3D filter. However, large system degrees of freedom lead to a shortage of training samples in a heterogeneous environment. Although introducing the Sparse Recovery (SR) technology substantially reduces the sample requirement, the practical application of this technology is limited by computational complexities. To solve the above problems, this paper proposes a fast 3D sparse Bayesian learning STAP, based on the third-order tensor structure of echo data. In the proposed method, large-scale matrix calculation is decomposed into small-scale matrix calculation using a low-complexity tensor-based operation, thus considerably reducing the computational load. Exhaustive numerical experiments verify that the proposed method directly reduces the computational load by several orders of magnitude compared with that of the existing SR-STAP algorithms, while maintaining the SR-STAP performance. Therefore, the tensor-based method is a superior processing method than the vector-based method in engineering.
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图 5 不同SR-STAP方法的SCNR损失结果
Figure 5. The SCNR loss results of different SR-STAP methodss
表 1 TMSBL算法
Table 1. TMSBL algorithm
输入：方位字典${{\boldsymbol{S}}_{\rm{a}}}$，俯仰字典${{\boldsymbol{S}}_{\rm{e}}}$，多普勒字典${{\boldsymbol{S}}_{\rm{d}}}$，训练样本集合${\boldsymbol{X}}$。 输出：稀疏系数${\boldsymbol{\varXi}}$ 初始化：过完备字典${\boldsymbol{S}} = {{\boldsymbol{S}}_{\rm{e}}} \otimes {{\boldsymbol{S}}_{\rm{a}}} \otimes {{\boldsymbol{S}}_{\rm{d}}}$，稀疏控制系数
${{\boldsymbol{\gamma}} _0} = {{\boldsymbol{e}}_{ {N_{\rm{a} } }{M_{\rm{e} } }{K_{\rm{d} } } } }$，均值${Y_0} = 1$，噪声方差$\sigma _0^2$，最大迭代
次数${i_{\max} }$，收敛阈值$\mu $。1：对于 $i = 1:{i_{\max} }$执行 2: 对于$j = 1:{N_{\rm{a}}}{M_{\rm{e}}}{K_{\rm{d}}}$执行 3: 计算方位索引${ {\rm{loc} }_{\rm{a}}}$，俯仰索引${{\rm{loc}}_{\rm{e}}}$和多普勒索引${{\rm{loc}}_{\rm{d}}}$ 4: ${\mathcal{T}_{:,:,:,j} } = $
${\mathcal{F}_{N,M,K} }\left\{ { {\mathcal{V}_{NK} }\left\{ { {\gamma _j}{\boldsymbol{S} }_{:,{ {\rm{loc} }_{\rm{d} } } }^{\rm{d} }{ {\left( { {\boldsymbol{S} }_{:,{ {\rm{loc} }_{\rm{a} } } }^{\rm{a} } } \right)}^{\rm{T} } } } \right\}{ {\left( { {\boldsymbol{S} }_{:,{ {\rm{loc} }_{\rm{e} } } }^{\rm{e} } } \right)}^{\rm{T} } } } \right\}$5: 结束循环 6: ${\boldsymbol{C} } = {\mathcal{M}_{NMK,NMK} }\left\{ {\mathcal{T}{ \times _2}{\boldsymbol{S} }_{\rm{d} }^{\rm{H} }{ \times _3}{\boldsymbol{S} }_{\rm{a} }^{\rm{H} }{ \times _4}{\boldsymbol{S} }_{\rm{e} }^{\rm{H} } } \right\} + {\sigma ^2}{\boldsymbol{I}}$ 7: ${\boldsymbol{Y}} = {\mathcal{M}_{ {N_{\rm{a}}}{M_{\rm{e}}}{K_{\rm{d}}},NMK} } $
$\cdot\left\{ {\mathcal{D} \odot \left( { {\mathcal{C}^{ - 1} }{ \times _1}{\boldsymbol{S} }_{\rm{d} }^{\rm{H} }{ \times _2}{\boldsymbol{S} }_{\rm{a} }^{\rm{H} }{ \times _3}{\boldsymbol{S} }_{\rm{e} }^{\rm{H} } } \right)} \right\}{\boldsymbol{X} }$8: 对于$j = 1:{N_{\rm{a}}}{M_{\rm{e}}}{K_{\rm{d}}}$执行 9: ${\varSigma _{j,j} } = {\gamma _j} - { {\boldsymbol{Q} }_{j,:} }{ {\boldsymbol{T} }_{:,j} }$ 10: ${\gamma _{j + 1} } = \dfrac{1}{L}\left\| { { {\boldsymbol{Y} }_{j,:} } } \right\|_2^2 + {\varSigma _{j,j} }$ 11: ${D_j} = \dfrac{ { {\varSigma _{j,j} } } }{ { {\gamma _j} } }$ 12: 结束循环
13: ${\sigma ^2} = \dfrac{1}{ {NMKL} }\left\| {{\boldsymbol{X}} - {\boldsymbol{SY}}} \right\|_{\rm{F} }^2$
$+ \dfrac{ {\sigma _{i - 1}^2} }{ {NMK} }\mathop \sum \limits_{j = 1}^{ {N_{\rm{a} } }{M_{\rm{e} } }{K_{\rm{d} } } } \left( {1 - {D_j} } \right) $14: 如果 $\left\| {{\boldsymbol{Y}} - {{\boldsymbol{Y}}_{i - 1} } } \right\|_{\rm{F} }^2/\left\| {\boldsymbol{Y}} \right\|_{\rm{F} }^2 \le \mu$ 跳出循环 15：结束循环 
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表 2 计算复杂度
Table 2. Computational complexity
方法 复乘法次数 计算复杂度 OMP $\left( {NMK{N_{\rm a}}{M_{\rm e}}{K_{\rm d}} + r_{\rm s}^3 + NMKr_{\rm s}^2 + 2NMK{r_{\rm s}}} \right)L{K_{\rm OMP}}$ $O\left( {NMK{N_{\rm a}}{M_{\rm e}}{K_{\rm d}}L{K_{\rm OMP}}} \right)$ MIAA $\left( {2{N_{\rm a}}{M_{\rm e}}{K_{\rm d}}{{\left( {NMK} \right)}^2} + {{\left( {NMK} \right)}^3} + \left( {L + 1} \right){N_{\rm a}}{M_{\rm e}}{K_{\rm d}}NMK + L{N_{\rm a}}{M_{\rm e}}{K_{\rm d}}} \right){K_{\rm MIAA}}$ $O\left( {{N_{\rm a}}{M_{\rm e}}{K_{\rm d}}{{\left( {NMK} \right)}^2}{K_{\rm MIAA}}} \right)$ MFOCUSS $\left( {NMK{N_{\rm a}}{M_{\rm e}}{K_{\rm d}}L + {{\left( {NMK} \right)}^3} + 2{{\left( {NMK} \right)}^2}{N_{\rm a}}{M_{\rm e}}{K_{\rm d}} + NMK{{\left( {{N_{\rm a}}{M_{\rm e}}{K_{\rm d}}} \right)}^2}} \right){K_{\rm MFOC}}$ $O\left( {NMK{{\left( {{N_{\rm a}}{M_{\rm e}}{K_{\rm d}}} \right)}^2}{K_{\rm MFOC}}} \right)$ MSBL $ \begin{gathered} \left( {{{\left( {{N_{\rm a}}{M_{\rm e}}{K_{\rm d}}} \right)}^3} + 4NMK{{\left( {{N_{\rm a}}{M_{\rm e}}{K_{\rm d}}} \right)}^2} + \left( {3{{\left( {NMK} \right)}^2} + {L^2} + 2NMKL} \right){N_{\rm a}}{M_{\rm e}}{K_{\rm d}}} \right. \hfill \\ \left. { + 4{{\left( {NMK} \right)}^3}/3 + {{\left( {NMK} \right)}^2} + NMKL} \right){K_{\rm MSBL}} \hfill \\ \end{gathered} $ $O\left( {{{\left( {{N_{\rm a}}{M_{\rm e}}{K_{\rm d}}} \right)}^3}{K_{\rm MSBL}}} \right)$ MFCSBL $\left( { {N_{\rm a} }{M_{\rm e} }{K_{\rm d} }\left( {5{ {\left( {NMK} \right)}^2} + 2L{{NMK} } + 4{{NMK} } + 3} \right) + { {\left( {NMK} \right)}^3} + L{{NMK} } + L} \right){K_{ {\rm{MFCSBL} } } }$ $O\left( { {N_{\rm a} }{M_{\rm e} }{K_{\rm d} }{ {\left( {NMK} \right)}^2}{K_{\rm MFCSBL} } } \right)$ TMSBL $ \begin{gathered} \left( {{N_{\rm a}}{M_{\rm e}}{K_{\rm d}}(3NMK + NM + N + {L^2} + 2NMKL + NM{K^2} + NK{M^2})} \right. + 2{(NMK)^3}/3 \hfill \\ \left. { + {{(NMK)}^2}(1 + {K_{\rm d}} + {M_{\rm e}}) + NMK({N_{\rm a}}{M_{\rm e}}KN + NM{N_{\rm a}}{K_{\rm d}} + L)} \right){K_{\rm TMSBL}} \hfill \\ \end{gathered} $ $O\left( {{N_{\rm a}}{M_{\rm e}}{K_{\rm d}}NM{K^2}{K_{\rm TMSBL}}} \right)$
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表 3 雷达系统参数
Table 3. Radar system parameters
参数 符号 数值 载机速度 ${v_{\rm{p}}}$ 150 m/s 载机高度 $H$ 8000 m 行/列阵元数 $M/N$ 6/8 脉冲数 $K$ 8 雷达波长 $\lambda $ 0.1 m 脉冲重复频率 ${f_{\rm{r}}}$ 8100 Hz 阵元间距 $d$ 0.05 m 主波束指向 $\theta /\varphi $ –90°/0° 杂噪比 – 60 dB
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