球谐域变分稀疏贝叶斯学习框架下的声源定位理论和方法研究

The existed acoustic source localization methods in the spherical harmonics domain have low spatial resolution and accuracy. Moreover, the performance of these methods is affected by multipath coherence and noise. It limits the development and application of sound field reproduction and spatial rendering. This project studies the acoustic pressure information acquired by spherical microphone arrays. The real-valued array model is constructed in the spherical harmonics domain. Based on the model, acoustic source localization theory is developed within the framework of variational sparse Bayesian learning. This research provides the theoretical foundation and method support for fast and accurate localization estimation in three-dimensional space. First, the complex-valued array model is built and the frequency and location information are decoupled in the spherical harmonics domain. A unitary matrix is designed to transfer the complex-valued steering matrix into the corresponding real-valued one. The joint restricted isometry property is found for multiple dictionaries and the real-valued tensor slice sparse model is constructed in the spherical harmonics domain. Secondly, the projection criterion is developed based on the l1 norm to find a robust low-dimensional signal subspace. In the subspace, the signal has jointly sparse property which is adopted to construct the hierarchical prior distribution model. Variational Bayesian approach is used to obtain the approximate posteriors of these sparse signals. The dictionary atoms can be automatically selected to reduce the computational complexity during the Bayesian iterative learning. Thirdly, the necessary and sufficient condition is deduced to get the accurate reconstruction of the sparse signals. The Cramer-Rao lower bound (CRLB) is developed to evaluate the performance of the location estimation and find the relation between the mean squared errors and the sparse signals. The spatial resolution can be found based on the CRLB.

球谐域传统声源定位方法存在分辨率低、受信号多径相干及噪声影响较大等问题，严重制约着声场还原及方位感重现的发展和应用。本项目以球阵列获取的三维空间声压信息为研究对象，构建球谐域实值阵列信号模型，发展变分稀疏贝叶斯学习框架下的声源定位理论, 为混响声场环境下快速准确的声源方位估计提供理论基础和方法支撑。主要包括:(1)建立球谐域频率和方位解耦合的复值阵列信号模型，设计酉变换矩阵，研究多字典联合约束等距特性，发展球谐域实值张量切片稀疏表示模型;(2)探索基于l1范数的投影准则，研究低维信号子空间的联合稀疏特性，构建多稀疏信号的等级先验分布模型，发展变分贝叶斯方法寻找其近似后验分布，探索字典原子的自动选择机制，减少变分贝叶斯估计方法的计算复杂度;(3)研究准确重构稀疏信号的充要条件，推导方位估计性能的克拉美罗下界，揭示均方误差性能和稀疏信号的关系，探索基于克拉美罗下界的方位分辨极限。

针对球阵列获取的信号源波达方向估计问题，提出了球谐域的稀疏信号模型及定位与跟踪算法，主要包括：1) 利用球傅里叶变换将空域阵列信号模型转化到球谐域，根据勒让德函数关于球谐自由度的对称性，构建酉矩阵将复值模型转化为实值模型，对空域角度进行采样，构建实球谐域的稀疏表示模型，进而考虑离散化带来声源偏离网格的情形，研究多字典联合约束等距特性，进一步改进为偏离网格的实值稀疏模型。2) 运用稀疏贝叶斯学习的方法推导信号的后验分布，重构稀疏信号。在稀疏重构的过程中，采用期望最大化算法估计角度偏差，进而估计信号源的波达方向。为了进一步降低算法的复杂度，发展基于投影基选择的变分稀疏贝叶斯学习方法，采用正交投影方法从匹配矩阵中选择基原子，并利用最小二乘准则估计角度偏差，发展了一种新的角度估计方法。3) 将变分稀疏贝叶斯学习与卡尔曼滤波结合来估计移动信号源的位置。运用变分稀疏贝叶斯学习方法求得稀疏信号的状态方差及噪声方差，并使用卡尔曼滤波来估计信号的当前位置，通过两步相互迭代联合估计信号的移动位置。4) 针对实时定位问题，开展了旋转不变算法的研究， 发展了新的球谐函数递归关系，将俯仰角和方位角解耦合求解，提出了球谐域分步旋转不变方法，解决了传统方法在俯仰角估计上的局限性。并将算法转换至实值域，提出了球谐域实值分步算法。由于实值方法在可估计声源的数目上存在局限性，进一步提出了球谐域半实值分步算法。利用酉变换矩阵的性质，建立实值信号子空间与复值导向矩阵的关系，从而可以利用复球谐函数的递推关系，既能降低计算量，还能提高DOA估计的性能。5) 构造球阵列波达方向估计的Fisher信息矩阵，根据未知向量中可能包含的确定性和随机性两种成分，分别推导了条件模型和非条件模型下的克拉美罗界，为角度估计误差提供了下界。推导估计算法的理论均方误差，揭示均方误差与信号的关系，为性能提升提供有力依据。