利用参量结构解盲源分离算法研究
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摘要
盲源分离是在源信号与传输信道均未知的情况下,仅利用接收天线的观测数据估计源信号。在过去的二十多年里,盲源分离在诸多研究领域引起了极大关注,广泛应用于语音信号处理,无线通信,图像识别与增强,数据分析以及生物医学信号处理等领域。本文主要研究了利用参量结构解盲源分离新算法。综合起来,本文的主要工作可以概括如下:
1.在瞬时混合盲源分离情况下,重点关注了联合对角化问题,论述了基于正交联合对角化解盲源分离可辨识性定理以及基于非正交联合对角化解盲源分离可辨识性定理,为联合对角化算法提供了理论基础。随后,提出了一种基于多步分解(MSA)的正交联合对角化算法,该算法既可以像并行算法一样,一次性地恢复所有源信号,也可以根据一个参量指标排序,序贯地提取各个源信号。与经典的同样用于估计正交混迭矩阵,并恢复源信号的SOBI算法相比,所提算法在估计性能上具有优势。
2.在详细地分析了现存的、用于表征联合对角化近似程度的各代价函数优缺点的基础上,提出了一种基于F-范数代价函数的快速复数域非正交联合对角化算法(CVFFDIAG)。该算法采用乘性迭代机制,每步迭代求解一个严格对角占优更新矩阵,保证了联合对角化器可逆,避免了F-范数代价函数可能产生的平凡解;算法对代价函数的合理性近似,使得求解过程更加简单易操作;算法不需要预白化操作,不要求目标矩阵的正定性,且可以处理复数问题,因而具有极广的适用性;详细的计算复杂度分析表明了该算法计算复杂度低,且简单易执行;大量的仿真实验结果验证了算法收敛速度快且收敛后性能良好。正是由于具备这些优点,该算法几乎可以推广应用于联合对角化算法所能解决的任何问题,作为示例,本文还重点讨论了将算法用于阵列信号处理,解决波达方向估计和谐波恢复问题。
3.重点关注了卷积混合盲源分离中的联合块对角化问题,将提出的基于多步分解的正交联合对角化算法推广,并行地提出了一种基于多步分解的正交联合块对角化算法,解卷积混合盲源分离问题。
4.针对许多现存的联合块对角化算法需要预白化操作的缺点,提出了一种不需要预白化的非正交联合块内对角化算法(JBID),由于该算法不需要预白化操作,从而避免了白化引入的误差,而且使得混迭矩阵的块Toeplitz结构得以保留,该算法充分利用了混迭矩阵的块Toeplitz结构以及源信号相关矩阵组的块内对角化结构,实现了只用一步便解决了卷积混合盲源分离问题。
5.分析发现,当源信号为平稳信号时,卷积混合盲源分离问题中的参量具有更丰富的结构信息可以挖掘并利用,主要表现在,混迭矩阵具有块Toeplitz结构,源信号的连续延时相关矩阵具有块Toeplitz结构,且各块元素又具有对角化结构,并且,各个相关矩阵之间存在联系,具体表现为,它们之间有许多相同的非零元素,这些元素的位置有规律可循并可以作为预知的先验知识。为了充分利用这些结构特点,提出了一种基于块Toeplitz化和块内对角化算法(JBTBID)解卷积混合盲源分离问题。建立了三二次代价函数,并利用三迭代算法循环求取函数的最小点,由于需要充分考虑并利用这些结构信息,导致三迭代算法的三个子步的推导都非常复杂,本文给出了详细缜密的推导过程,并给出了算法严格的计算复杂度分析。JBTBID算法同样取消了多步分解算法中的预白化操作,并且也是只用一步便实现了卷积混合盲源分离,而且,在源信号近似满足平稳性假设的情况下,具有更好的分离性能,同时证明了算法是渐进收敛的。
6.本文还挖掘汇总了解卷积混合盲源分离中所涉及参量的结构特点,介绍了三种参量结构,其中,最典型的是:在JBTBID的假设条件下,如果采取不同的数据排列方式,则可以得到不同的参量结构特点,具体为:混迭矩阵是块内Toeplitz矩阵;所有的源信号延时相关矩阵都具有块对角结构;所有的源信号相邻延时相关矩阵都具有块内Toeplitz结构;各源信号相邻延时相关矩阵之间有许多共同的非零元素。很明显,在以后提出的新的解卷积混合盲源分离算法中,可以考虑利用这些参量结构以提高算法性能。
Blind source separation (BSS) aims to estimate the source signals, i.e. to demix the mixtures captured by a number of sensors, without knowledge about the actual sources or the mixing procedure. In the past two decades, BSS has received much attention in various fields, such as speech and audio processing, wireless communication, image recognition and enhancement, data analysis and biomedical signal processing, etc. This dissertation attempts to do some research on BSS algorithms based on the utilizations of parametric structures. The main contributions of this dissertation are summarized as follows:
1. The joint diagonalization structure in instantaneous BSS case is studied. The identifiability theorems of both the orthogonal joint diagonalization based BSS and non-orthogonal joint diagonalization based BSS are introduced, providing a theoretical basis for joint diagonalization based BSS algorithms. Based on these, a multi-step algorithm (MSA) for orthogonal joint diagonalization is proposed to solve BSS problem. This algorithm manages not only to recover all source signals simultaneously but also to extract source signals one by one. Compared with SOBI, which is one classic algorithm for orthogonal joint diagonalization, the MSA has better estimation performance.
2. Detailed analysis on merits and drawbacks of existed cost functions for joint diagonalization is done. Thereafter, a fast algorithm, named CVFFDIAG (Complex-Valued Fast Frobenius DIAGonalization), for seeking the non-unitary approximate joint diagonalizer of a given set of complex-valued target matrices is proposed. The proposed algorithm adopts a multiplicative update to minimize the Frobenius-norm formulation of the approximate joint diagonalization problem. In each of multiplicative iterations, a strictly diagonally-dominant updated matrix is obtained. This scheme ensures the invertibility of the diagonalizer and thus guarantees the avoidance of the trivial solution to Frobenius-norm cost function. The special approximation of the cost function, the ingenious utilization of some structures and the skilful denotation of concerning variables result in the highly computational efficiency of the algorithm. Furthermore, the CVFFDIAG relaxes several constraints on the target matrices, e.g. unitarity and positive-definiteness assumptions or the real-valued or Hermitian assumption, and thus has more general utilizations. Detailed computational load analysis also shows the low computational complexity of the algorithm. Extensive numerical simulations are performed to illustrate the fast convergence and good performance of the CVFFDIAG. Due to these obvious merits, the CVFFDIAG could be used to solve many problems. We take the algorithm being used to estimate the DOA estimation and harmonica retrieval in array signal processing area as an example to show its various utilizations.
3. The joint block diagonalization problem of convolutive BSS is paid much attention. The multi-step algorithm for orthogonal joint diagonalization to solve the instantaneous BSS problem is parallel extended to orthogonal joint block diagonalization to solve the convolutive BSS problem.
4. An non-orthogonal joint block-inner diagonalization (JBID) algorithm, getting rid of the pre-whitening operation, is proposed. The discarding of pre-whitening operation ensures the avoidance of whitening errors and preserves the block Toeplitz structure of the mixing matrix. By fully considering the block-inner diagonalization structure of source signals correlation matrices at different time lags and the block Toeplitz structure of the mixing matrix, the JBID attains to recover all source signals in only one step.
5. After careful analysis, we discover that there are abundant structure traits contained in parameters of convolutive BSS when the source signals are assumed to be real and stationary. These traits which could be used in our new algorithm are as follows. The mixing matrix possesses Toeplitz structure. The correlation matrices of source signals at successive time lags are block Toeplitz and block-inner diagonalization. Furthermore, these correlation matrices have many common entries. And the coordinates of these entries could be known ahead as a prior knowledge. The main idea behind the proposed algorithm is to implement the joint block Toeplitzation and block-inner diagonalization (JBTBID) of a set of correlation matrices of the observed vector sequence such that the mixture matrix can be extracted. For this purpose, a novel tri-quadratic cost function is introduced. The important feature of this tri-quadratic contrast function enables to develop an efficient algebraic method based on triple iterations for searching the minimum point of the cost function, which is called the triply iterative algorithm (TIA). Since the corresponding variables are highly structured, the derivation procedures in three sub-steps of TIA appear complex and need to be dealt with skill. But we still clearly explain the derivation procedures. We also take a close look at the computational complexity of our algorithm. This algorithm also discards the pre-whitening operation, manages to recover all source signals in only one step, and possesses good performance. Moreover, the asymptotical convergence of JBTBID is shown.
6. We furthermore take a close look at the parametric structure traits of convolutive BSS and introduce three kinds of structures. We now introduce the representative one among them. Following all assumptions of JBTBID, when the received data are arrayed in different way, different parametric structures can be derived. Concretely speaking, the mixing matrix is block-inner Toeplitz. The correlation matrices of source signals at successive time lags possess block diagonalization and block-inner Toeplitz structures. Also, these correlation matrices have many common entries and the coordinates of these entries could be known ahead. It is obvious that, all these structure traits could be utilized when considering a new algorithm for the convolutive BSS problem.
1.在瞬时混合盲源分离情况下,重点关注了联合对角化问题,论述了基于正交联合对角化解盲源分离可辨识性定理以及基于非正交联合对角化解盲源分离可辨识性定理,为联合对角化算法提供了理论基础。随后,提出了一种基于多步分解(MSA)的正交联合对角化算法,该算法既可以像并行算法一样,一次性地恢复所有源信号,也可以根据一个参量指标排序,序贯地提取各个源信号。与经典的同样用于估计正交混迭矩阵,并恢复源信号的SOBI算法相比,所提算法在估计性能上具有优势。
2.在详细地分析了现存的、用于表征联合对角化近似程度的各代价函数优缺点的基础上,提出了一种基于F-范数代价函数的快速复数域非正交联合对角化算法(CVFFDIAG)。该算法采用乘性迭代机制,每步迭代求解一个严格对角占优更新矩阵,保证了联合对角化器可逆,避免了F-范数代价函数可能产生的平凡解;算法对代价函数的合理性近似,使得求解过程更加简单易操作;算法不需要预白化操作,不要求目标矩阵的正定性,且可以处理复数问题,因而具有极广的适用性;详细的计算复杂度分析表明了该算法计算复杂度低,且简单易执行;大量的仿真实验结果验证了算法收敛速度快且收敛后性能良好。正是由于具备这些优点,该算法几乎可以推广应用于联合对角化算法所能解决的任何问题,作为示例,本文还重点讨论了将算法用于阵列信号处理,解决波达方向估计和谐波恢复问题。
3.重点关注了卷积混合盲源分离中的联合块对角化问题,将提出的基于多步分解的正交联合对角化算法推广,并行地提出了一种基于多步分解的正交联合块对角化算法,解卷积混合盲源分离问题。
4.针对许多现存的联合块对角化算法需要预白化操作的缺点,提出了一种不需要预白化的非正交联合块内对角化算法(JBID),由于该算法不需要预白化操作,从而避免了白化引入的误差,而且使得混迭矩阵的块Toeplitz结构得以保留,该算法充分利用了混迭矩阵的块Toeplitz结构以及源信号相关矩阵组的块内对角化结构,实现了只用一步便解决了卷积混合盲源分离问题。
5.分析发现,当源信号为平稳信号时,卷积混合盲源分离问题中的参量具有更丰富的结构信息可以挖掘并利用,主要表现在,混迭矩阵具有块Toeplitz结构,源信号的连续延时相关矩阵具有块Toeplitz结构,且各块元素又具有对角化结构,并且,各个相关矩阵之间存在联系,具体表现为,它们之间有许多相同的非零元素,这些元素的位置有规律可循并可以作为预知的先验知识。为了充分利用这些结构特点,提出了一种基于块Toeplitz化和块内对角化算法(JBTBID)解卷积混合盲源分离问题。建立了三二次代价函数,并利用三迭代算法循环求取函数的最小点,由于需要充分考虑并利用这些结构信息,导致三迭代算法的三个子步的推导都非常复杂,本文给出了详细缜密的推导过程,并给出了算法严格的计算复杂度分析。JBTBID算法同样取消了多步分解算法中的预白化操作,并且也是只用一步便实现了卷积混合盲源分离,而且,在源信号近似满足平稳性假设的情况下,具有更好的分离性能,同时证明了算法是渐进收敛的。
6.本文还挖掘汇总了解卷积混合盲源分离中所涉及参量的结构特点,介绍了三种参量结构,其中,最典型的是:在JBTBID的假设条件下,如果采取不同的数据排列方式,则可以得到不同的参量结构特点,具体为:混迭矩阵是块内Toeplitz矩阵;所有的源信号延时相关矩阵都具有块对角结构;所有的源信号相邻延时相关矩阵都具有块内Toeplitz结构;各源信号相邻延时相关矩阵之间有许多共同的非零元素。很明显,在以后提出的新的解卷积混合盲源分离算法中,可以考虑利用这些参量结构以提高算法性能。
Blind source separation (BSS) aims to estimate the source signals, i.e. to demix the mixtures captured by a number of sensors, without knowledge about the actual sources or the mixing procedure. In the past two decades, BSS has received much attention in various fields, such as speech and audio processing, wireless communication, image recognition and enhancement, data analysis and biomedical signal processing, etc. This dissertation attempts to do some research on BSS algorithms based on the utilizations of parametric structures. The main contributions of this dissertation are summarized as follows:
1. The joint diagonalization structure in instantaneous BSS case is studied. The identifiability theorems of both the orthogonal joint diagonalization based BSS and non-orthogonal joint diagonalization based BSS are introduced, providing a theoretical basis for joint diagonalization based BSS algorithms. Based on these, a multi-step algorithm (MSA) for orthogonal joint diagonalization is proposed to solve BSS problem. This algorithm manages not only to recover all source signals simultaneously but also to extract source signals one by one. Compared with SOBI, which is one classic algorithm for orthogonal joint diagonalization, the MSA has better estimation performance.
2. Detailed analysis on merits and drawbacks of existed cost functions for joint diagonalization is done. Thereafter, a fast algorithm, named CVFFDIAG (Complex-Valued Fast Frobenius DIAGonalization), for seeking the non-unitary approximate joint diagonalizer of a given set of complex-valued target matrices is proposed. The proposed algorithm adopts a multiplicative update to minimize the Frobenius-norm formulation of the approximate joint diagonalization problem. In each of multiplicative iterations, a strictly diagonally-dominant updated matrix is obtained. This scheme ensures the invertibility of the diagonalizer and thus guarantees the avoidance of the trivial solution to Frobenius-norm cost function. The special approximation of the cost function, the ingenious utilization of some structures and the skilful denotation of concerning variables result in the highly computational efficiency of the algorithm. Furthermore, the CVFFDIAG relaxes several constraints on the target matrices, e.g. unitarity and positive-definiteness assumptions or the real-valued or Hermitian assumption, and thus has more general utilizations. Detailed computational load analysis also shows the low computational complexity of the algorithm. Extensive numerical simulations are performed to illustrate the fast convergence and good performance of the CVFFDIAG. Due to these obvious merits, the CVFFDIAG could be used to solve many problems. We take the algorithm being used to estimate the DOA estimation and harmonica retrieval in array signal processing area as an example to show its various utilizations.
3. The joint block diagonalization problem of convolutive BSS is paid much attention. The multi-step algorithm for orthogonal joint diagonalization to solve the instantaneous BSS problem is parallel extended to orthogonal joint block diagonalization to solve the convolutive BSS problem.
4. An non-orthogonal joint block-inner diagonalization (JBID) algorithm, getting rid of the pre-whitening operation, is proposed. The discarding of pre-whitening operation ensures the avoidance of whitening errors and preserves the block Toeplitz structure of the mixing matrix. By fully considering the block-inner diagonalization structure of source signals correlation matrices at different time lags and the block Toeplitz structure of the mixing matrix, the JBID attains to recover all source signals in only one step.
5. After careful analysis, we discover that there are abundant structure traits contained in parameters of convolutive BSS when the source signals are assumed to be real and stationary. These traits which could be used in our new algorithm are as follows. The mixing matrix possesses Toeplitz structure. The correlation matrices of source signals at successive time lags are block Toeplitz and block-inner diagonalization. Furthermore, these correlation matrices have many common entries. And the coordinates of these entries could be known ahead as a prior knowledge. The main idea behind the proposed algorithm is to implement the joint block Toeplitzation and block-inner diagonalization (JBTBID) of a set of correlation matrices of the observed vector sequence such that the mixture matrix can be extracted. For this purpose, a novel tri-quadratic cost function is introduced. The important feature of this tri-quadratic contrast function enables to develop an efficient algebraic method based on triple iterations for searching the minimum point of the cost function, which is called the triply iterative algorithm (TIA). Since the corresponding variables are highly structured, the derivation procedures in three sub-steps of TIA appear complex and need to be dealt with skill. But we still clearly explain the derivation procedures. We also take a close look at the computational complexity of our algorithm. This algorithm also discards the pre-whitening operation, manages to recover all source signals in only one step, and possesses good performance. Moreover, the asymptotical convergence of JBTBID is shown.
6. We furthermore take a close look at the parametric structure traits of convolutive BSS and introduce three kinds of structures. We now introduce the representative one among them. Following all assumptions of JBTBID, when the received data are arrayed in different way, different parametric structures can be derived. Concretely speaking, the mixing matrix is block-inner Toeplitz. The correlation matrices of source signals at successive time lags possess block diagonalization and block-inner Toeplitz structures. Also, these correlation matrices have many common entries and the coordinates of these entries could be known ahead. It is obvious that, all these structure traits could be utilized when considering a new algorithm for the convolutive BSS problem.
引文
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