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Robust Design of Constant Modulus Sequence and Receiver Filter in the Presence of Signal-dependent Clutter
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摘要: 该文针对信号相关杂波环境下的运动目标检测问题，研究一种稳健的慢时间发射波形和接收滤波器设计方法。首先，基于杂波2阶统计特性不确定时的最坏SINR (the Worst-case SINR, W-SINR)，建立非凸恒模约束下高维的发射-接收联合优化模型；然后，提出一种基于序列迭代的优化算法(Iterative Sequential Optimization, ISO)。每步迭代中，该算法将一个高维优化问题转化为多个1维分式规划问题，并通过丁克尔巴赫(Dinkelbach)方法进行求解。最后，仿真实验证明，ISO具有对抗不确定杂波信息的能力，使系统具有更好的适应能力；此外，相比半正定松弛(Semi-Definite Relaxation, SDR)与随机化方法，提出的算法在W-SINR优化值与计算复杂度上均具有明显的优势。
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关键词:
- 稳健设计 /
- 信干噪比(SINR) /
- 恒模序列 /
- 迭代序列优化
Abstract: In this paper, we focus on the detection of a moving point-like target embedded in uncertain signal-dependent clutter and develop robust transmit-code and receive-filter designs in slow-time. First, based on the Worst-case Signal-to-Interference-plus-Noise Ratio (W-SINR) when the second-order clutter statistics are uncertain, we establish a high-dimensional transmit-receive optimization model that considers the constant modulus constraint with non-convexity. Next, we propose an Iterative Sequential Optimization (ISO) algorithm. At each iteration, it converts a high-dimensional optimization into multiple one-dimensional fractional programming problems that can be efficiently solved using Dinkelbach’s method. Finally, we use numerical examples to confirm that the ISO can resist the uncertain knowledge of signal-dependent clutter, which enables the radar system to adapt to complicated environments. Moreover, compared to Semi-Definite Relaxation (SDR)-related and randomization methods, the proposed algorithm is superior with respect to both optimized W-SINR and computational time. -
图 1 W-SINR随迭代次数变化曲线(SOA2-EC, ISO和SOA2-CMC)
Figure 1. W-SINR versus iteration number for SOA2-EC, ISO and SOA2-CMC
图 3 基于常规设计和稳健设计的W-SINR随归一化不确定度的变化曲线(ISO和SOA2-CMC)
Figure 3. W-SINR against the normalized uncertainty size for ISO and SOA2-CMC, associated with the nominal design and the robust design
表 1 稳健的发射-接收联合设计算法
Table 1. Algorithm for the robust transmit-receive design
输入： ${{{s}}_0},\xi ,\sigma _{\left( {k,l} \right)}^2,{f_{\left( {k,l} \right)}},{\varepsilon _{\left( {k,l} \right)}},{\delta _{\left( {k,l} \right)}},k = 1,2, ·\!·\!· ,N,l = 1,2, ·\!·\!· ,L$ 输出：P1的最优解 $\left( {{{{s}}_{{\rm{opt}}}},{{{w}}_{{\rm{opt}}}}} \right)$ (1) 由式(11)得到 ${{{M}}_{{\rm{opt}}\left( {k,l} \right)}}$然后根据式(13)计算 $ \widehat{ {Σ} _{\rm{c}}}\left( {{s}} \right)$； (2) 对于 $n = 0,i = 0$然后初始化序列 ${{{s}}^{\left( 0 \right)}} = {{{s}}_0}$； (3) 由式(17)计算 ${{{w}}^{\left( 0 \right)}}$，根据式(14)计算 ${\stackrel \frown {\rho}} _0{\rm{ = }}\mathord{\stackrel \frown {\rho}} \left( {{{{s}}^{\left( 0 \right)}},{{{w}}^{\left( 0 \right)}}} \right)$； (4) n=n+1； (5) 根据式(19)和式(22)分别计算矩阵 ${{Σ}_{\rm{t}}}\left( {{{{w}}^{\left( {n - 1} \right)}}} \right)$和 ${Θ}\left( {{{{w}}^{\left( {n - 1} \right)}}} \right)$； (6) i=i+1； (7) 根据式(26)和式(27)分别计算 ${a_{r,i}},{b_{r,i}}\left( {r = 0,1,2} \right)$； (8) 在可行域内随机产生初始码字 ${s_{i,0}}$； (9) 解式(28)并得到最优解 ${s_{{\rm{opti}}}}$； (10) 如果i=N，输出 ${{{s}}^{\left( n \right)}} = {\left[ {{s_{{\rm{opt1}}}}\;{s_{{\rm{opt}}2}}\; ·\!·\!· {\rm{ }}{s_{{\rm{opt}}N}}} \right]^{\rm{T}}}$，否则返回步骤(6)； (11) 由式(17)计算 ${{{w}}^{\left( n \right)}}$，根据式(14)计算 ${\mathord{\stackrel \frown {\rho}} _n}{\rm{ = }}\mathord{\stackrel \frown {\rho}} \left( {{{{s}}^{\left( n \right)}},{{{w}}^{\left( n \right)}}} \right)$； (12) 如果 $\left| {{{\mathord{\stackrel \frown {\rho}} }_n} - {{\mathord{\stackrel \frown {\rho}} }_{n - 1}}} \right| \le \xi $ ( $\xi $是一个用来控制收敛的自定义参数)， 输出 ${{{s}}_{{\rm{opt}}}}{\rm{ = }}{{{s}}^{\left( n \right)}},{{{w}}_{{\rm{opt}}}}{\rm{ = }}{{{w}}^{\left( n \right)}}$；否则返回步骤(4)，直到收敛。 
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表 2 收敛条件 $\left| {{{\mathord{\stackrel \frown {ρ} } }_{n}} - {{\mathord{\stackrel \frown{{ρ}} } }_{{n} - {1}}}} \right| \!\!\le\!\! {10^{ - 3}}$下3种算法的迭代次数和计算时间
Table 2. Iteration number and computation time (in seconds) of all the three algorithms with the exit condition $\left| {{{\mathord{\stackrel \frown{{ρ}}} }_n} - {{\mathord{\stackrel \frown{{ρ}} } }_{{n} - {1}}}} \right| \le {{10}^{ - 3}}$
算法 迭代次数 计算时间(s) SOA2-EC 4 0.005 ISO 73 0.220 SOA2-CMC 45 23.920
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