Underdetermined Wideband DOA Estimation Based on Distributed Compressive Sensing
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摘要: 为解决基于稀疏阵列的宽带欠定信号到达角(DOA)估计问题，该文提出基于分布式压缩感知(DCS)的宽带DOA估计算法。首先，对稀疏阵列宽带信号处理模型进行理论推导与分析，将宽带信号DOA估计建模成DCS问题；其次，利用经典DCS算法实现稀疏阵列上的宽带欠定信号DOA估计；最后，引入网格失配误差，建立包含网格失配参数的DCS模型，并进行迭代求解，实现对DOA和网格失配参数的联合估计。仿真结果表明，该算法能够实现宽带欠定信号DOA估计，较现有成果而言，在保证测向精度的同时，具备分辨率高、运算速度快的优点。Abstract: In order to realize underdetermined wideband Direction Of Arrival(DOA) estimation based on sparse array, an algorithm on account of Distributed Compressive Sensing(DCS) is proposed. Firstly, wideband signal processing model based on sparse array is deduced and the underdetermined wideband DOA estimation is formulated as a DCS problem. Then, the DCS-Simultaneous Orthogonal Matching Pursuit(DCS-SOMP) algorithm is utilized to solve this problem. Finally, the off-grid problem is considered and a joint DCS model containing off-grid parameters is established. Estimations of DOAs and off-grid parameters are achieved through iterative solution. Simulation results show that the proposed algorithm is effective and have advantages in resolution and computational complexity.
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表 1 DCS-SOMP算法
输入：虚拟阵列接收数据${{\text{z}}_h}$，过完备字典集${\text{Φ}_h}\left( \psi \right)$，信号个数$K$。 输出：重构信号${{\text{s}}_h}$，支撑基列标集合$\varOmega$。 初始化：迭代计数$i = 1,{\varOmega_0}=\varnothing ,{\hat {\text{s}}_h} = {\text{0}}$，残差初值${{\text{r}}_{h, 0}} = {{\text{z}}_h}$。 步骤 1 支撑基选择：
$ {g_i} = \mathop {\arg \max }\limits_{g \in \left\{ {1, 2, \cdots , G} \right\}} \sum\limits_{h = 1}^H {\frac{{\left| {\left\langle {{{\text{r}}_{h, i - 1}}, {{\text{φ}} _{h, g}}} \right\rangle } \right|}}{{{{\left\| {{{\text{φ}} _{h, g}}} \right\|}_2}}}} ,{\varOmega_i}={\varOmega_{i - 1}} \cup \left\{ {{g_i}} \right\}$；步骤 2 残差更新：${\hat{\text{ s}}_h}={{\text{Φ}} _{{\varOmega_i}}}^\dagger {{\text{z}}_h},{{\text{r}}_{h, i}} = {{\text{z}}_h} - {{\text{Φ}} _{{\varOmega_i}}}{\hat {\text{s}}_h}$； 步骤 3 条件判断：若$i < K$，则$i = i + 1$跳至步骤1，否则跳至步
骤4；步骤 4 结果结算：$\varOmega={\varOmega_i},{{\text{s}}_h}={{\text{Φ}} _\varOmega}^\dagger {{\text{z}}_h}$。 
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表 2 DCS-JSOMP算法
输入：虚拟阵列接收数据${{\text{z}}_h}$，过完备字典集${\text{Φ}_h}\left( \text{Ψ} \right)$，网格失配字典${\text{Γ}_h}\left( \text{Ψ} \right)$，信号个数$K$。 输出：重构信号${{\text{s}}_h}$，支撑基列标集合$\varOmega$，网格失配误差${\text{Δ}} $。 初始化：迭代计数$i = 1,{\varOmega_0}=\varnothing,{\hat{\text{ s}}_h} = {\text{0}},\hat{\text{β}}_h={\text{0}}$，残差${{\text{r}}_{h, 0}} = {{\text{z}}_h}$。
步骤 1 支撑基选择：${c_g} = \sum\limits_{h = 1}^H {\frac{{\left| {\left\langle {{{\text{r}}_{h, i - 1}}, {{\text{φ }}_{h, g}}} \right\rangle } \right|}}{{{{\left\| {{{\text{φ}} _{h, g}}} \right\|}_2}}}} ,{d_g} = \sum\limits_{h = 1}^H {\frac{{\left| {\left\langle {{{\text{r}}_{h, i - 1}}, {\text{γ}_{h, g}}} \right\rangle } \right|}}{{{{\left\| {{\text{γ}_{h, g}}} \right\|}_2}}}} ,{g_i} = \mathop {\arg \max }\limits_{g \in \left\{ {1, 2, \cdots , G} \right\}} \sqrt {{c_g}^2 + {d_g}^2} ,{\varOmega_i}={\varOmega_{i - 1}} \cup \left\{ {{g_i}} \right\}$；步骤 2 残差更新：${\hat{\text{ s}}_h} = {{\text{Φ}} _{{\varOmega_i}}}^\dagger \left( {{{\text{z}}_h} - {{\text{Γ}} _{{\varOmega_i}}}{\hat{\text{β}}_h}} \right),{\hat{\text{β}}_h} = {{\text{Γ}} _{{\varOmega_i}}}^\dagger \left( {{{\text{z}}_h} - {{\text{Φ}} _{{\varOmega_i}}}{{\hat {\text{s}}}_h}} \right),{{\text{r}}_{h, i}} = {{\text{z}}_h} - {{\text{Φ}} _{{\varOmega_i}}}{\hat{\text{ s}}_h} - {{\text{Γ}} _{{\varOmega_i}}}{\hat{\text{β}}_h}$； 步骤 3 条件判断：若$i < K$，则$i = i + 1$跳至步骤1，否则跳至步骤4；
步骤 4 结果结算：$\varOmega={\varOmega_i},{{\text{s}}_h} = {{\text{Φ}} _\varOmega}^\dagger \left( {{{\text{z}}_h} - {{\text{Γ}} _\varOmega}{\hat{\text{β}}_h}} \right),{\text{β}_h} = {{\text{Γ}} _\varOmega}^\dagger \left( {{{\text{z}}_h} - {{\text{Φ}} _\varOmega}{{\text{s}}_h}} \right),{\text{Δ}} =\frac{1}{H}\sum\limits_{h = 1}^H {\frac{{{{\text{β}} _h}}}{{{{\text{s}}_h}}}} $。
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表 3 5种算法单次蒙特卡洛实验用时(s)
算法 信噪比变化 频域快拍次数变化 DCS-SOMP 0.1747 0.1943 DCS-JSOMP 0.3439 0.3784 SS-MUSIC 0.5021 0.5207 WNNSBL 3.3751 3.0231 OGSLIM 0.6068 0.6678
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