矩阵束分解算法及在通信信号处理中的应用
摘要
矩阵束的三角化与对角化分解,如广义Schur分解(GSD)、广义特征值分解(GEVD)与广义奇异值分解(GSVD),是矩阵计算的重要内容,并在无线通信与信号处理等工程领域中具有广泛的应用。
无线与移动通信的空时接收信号处理问题,常可归结为相关矩阵束的分解。这类矩阵束中常含有阵元移位与时滞互相关矩阵等非厄米特非正定复矩阵。强相关或相干信号常会导致相关矩阵接近奇异,即具有坏条件数。近奇异非对称正定的矩阵束分解的有效计算方法的研究具有一定难度,但对于无线空时通信信号处理问题具有重要意义。在矩阵束分解算法及应用于无线通信的研究中,论文取得了下列创新成果:
①提出非对称矩阵束( A, B )的广义Schur分解(GSD)的交叉迭代算法。将该算法应用于改进强相关阵列信号波达方向估计的ESPRIT子空间方法的计算实现,提高了方向角度的估计精度和分辨概率。
提出的交叉迭代算法通过在矩阵A, B间交换各自QR和RQ分解的酉因子,并乘上原有三角因子,实现矩阵束( A, B )的循环迭代更新,并最终收敛至三角矩阵束。论文证明了该算法的收敛性同直接计算单个矩阵AB ?1Schur分解的QR迭代算法收敛性等价。然而,由于包含矩阵求逆与乘积运算,若直接计算AB ?1,其条件数可能很大(即使A, B都是良态的),在有限字长条件下,QR迭代精度无法保证。交叉迭代算法绕开了矩阵直接求逆与乘积运算,得到的结果具有和单个矩阵A, B条件数大致相当的精度。该算法的速度不受B秩亏损的影响。
与矩阵束GSD的QZ算法相比,交叉迭代算法中两个矩阵A, B的QR(或RQ)分解及乘积更新过程在计算上相互独立,仅需交换酉因子,具有环型并行计算结构;交叉迭代算法的一步迭代计算主要集中于QR(与RQ)分解及相反过程(酉旋转矩阵与上三角阵的乘积),硬件和软件的实现都易于模块化与并行化。
②建立了自适应GSD和GEVD的递推交叉迭代算法,并用于解决快速时变环境中未知CDMA系统自适应盲辨识问题。
在实际的自适应应用中,根据实时更新的输入数据递推生成输出。与批处理的QZ算法相比,基于QR(与RQ)分解的交叉迭代算法更适合自适应实现。自适应秩一更新矩阵束GSD的一步交叉迭代的计算代价为( )O n 2。论文提出了自适应GSD和GEVD的递推交叉迭代算法,实现对广义特征值与广义特征向量的自适应更新,即论文提出的广义子空间追踪(GSST)。与子空间追踪不同,广义子空间追踪不要求相关矩阵是厄米特阵,无需对广义特征向量的正交性约束,更适用于辨识一般非列正交矩阵,如CDMA系统多用户扩频序列矩阵。
借助天线阵列,可以在CDMA系统期望用户扩频波形未知(如截听、灾害等)情况下,实现对用户有效扩频波形的盲辨识和接收信号的检测。针对多径信道的快速时变性和计算的低复杂性要求,论文提出了时变环境中未知CDMA系统自适应盲辨识广义子空间追踪方法。分别基于递推交叉迭代和递推更新lanczos迭代两种算法进行相关矩阵束的自适应广义特征值分解,利用广义特征向量自适应辨识未知期望用户信号的有效扩频序列,并实现接收信号的盲多用户检测。仿真表明了GSST算法的有效性,并比较了递推交叉迭代和递推Lanczos迭代两种GSST实现形式的收敛性能。
③提出了典型相关解(CCD)精确计算的正切CCD算法。应用于相关噪声环境中的多径CDMA盲多用户检测,改善了接收信号的检测输出性能。
典型相关分析(CCA)通过分析两个随机变量之间的典型相关性,寻找其中的公共模式。它综合考虑两组输入数据的互相关性及每组数据的自相关性。典型相关分解(CCD)实现两路线性变换数据之间典型相关性的最大化,从而得到该数据相关部分(即信号)的最优估计。作为三个矩阵乘积的奇异值分解,CCD的计算易受到矩阵坏条件数和元素误差的影响。基于对数据矩阵Gram-Schmidt正交化预处理的已有算法亦存在误差累积效应,在矩阵规模较大时尤为明显。
在结合矩阵束广义奇异值分解与乘积奇异值分解精确计算的正切算法基础上,论文建立了精确计算典型相关分解的正切CCD算法。该算法无需正交化预处理,不存在误差累积效应,应用于改进具有多接收阵元的有色噪声环境中多径CDMA盲多用户检测的子空间方法,可准确辨识信号与噪声子空间。仿真表明多径信道与用户有效特征波形盲估计及盲多用户检测性能得到明显改善;相对于原有方法,基于正切CCD的子空间方法的信号检测结果对矩阵规模大小和条件数不敏感,在高负载系统条件下性能优势更为明显。
Matrix pen triangle and diagonal decomposition,such as generalized Schur decomposition (GSD), generalized eigenvalue decomposition (GEVD) and generalized singular values decomposition (GSVD), is a important subject on matrix computing and Matrix pen decomposition has important application to wireless communication and signal processing.
The problem on wireless and space-time receptive signal processing often can be come down to matrix pen decomposition. This kind of matrix pen often incudes senser shift corrlection matrix and time delay correlation matrix, they are not Hermitian and positive. Strong correlation or Coherent signal often cause correlative matrix is close to singular, means has a bad condition. It is difficult that investigates approximative singular and unsymmetry positive matrix pen decomposition, but it is important to wireless commiunication signal processing.
By investigating matrix decomposition algorithm and its application in wireless communication, the following innovating results are achieved:
①The cross-iterative algorithm of matrix generalized Schur decomposition was built up. And it is applied to improve the ESPRIT subspace method on direction of arrival of array signal strong correlating. The estimating precision and probability of recognition is improved.
By exchanging the unitary matrix factor from QR and RQ factorization between A, B , and multipling triangle factor inhere, the author derive out the circulative iterative program of pen( A, B ) that converges at triangle matrix pen at last, called cross-iterative algorithm. And it is proved that the convergence of algorithm is equivalent to the one of QR iterative of Schur decomposition of AB ?1. Of course, it is unpractical that straight computing AB ?1. Due to product and inverse calculate, the matrix condition number is bad and iterative computing accuracy can not be promised in limited words length condition. Cross-iterative algorithm avoids computing inverse and product of A &B and the result has approximately same condition number with A, B . The speed is free from the deficient rank of B .
In Cross-iterative algorithm, The QR&RQ factorization is independent each other, so do the product of Q and R. So the algorithm is parallel annular structurally.
The computing of one step iterative of cross-iterative algorithm fastens on QRF (RQF) and product of Q&R, the inverse program of former. So, it is fit to carry out modularly.
②Recursive cross-iterative algorithm on adaptive GSD and GEVD is set up and is applied to blind identify to unknown CDMA system in fast time varing environment.
In actual application, the input data is updated real time. Contrast with batch method, such as QZ algorithm, cross-iterative algorithm is easy to carry out adaptively. Its one step need ( )O n 2 flops to compute GSD of rank one update matrix pen. The author gives a recursive cross-iterative algorithm to achieves adaptive GSD and GEVD,namely generalized subspace tracking. This method with Low-complexity provides an effective way to rapidly calculate the decomposition of matrix pencil update eigenvalue. Distinguish from subspace tracking, the generalized subspace tracking need not the correlation matrix is Hermitian and the orthonormal restriction to generalized eigenvector, so it is fit to identify the ordinary matrix without rows orthogonal each other, such as multiusers’extend sequence matrix of CDMA system.
With the help of antenna array, you can realize the detection of blind identification and receiving signal of effective spreading spectrum waveform of user on the unknown of spread spectrum waveform (such as interception, disaster, etc.) of multipath CDMA system desire user. Taking into account the fast time-varying characteristic of multipath channel and the requirements on low complexity of calculation, the author puts forward generalied subspace tracking method on unknown CDMA system blinds identification with antenna array. Based on the recursion cross-iterative algorithm or recursion Lanzocs iterative, we achieves self-adaptive generalied eigenvalue decomposition of correlation matrix pen, and identifies the unknown user’available extend sequence and achieve blind multiuser detection. Computer simulation shows the validity of GSST, and makes a comparison of the two adaptive subspace tracking methods between Lanzocs iterative algorithm and recursive cross-iterative algorithm proposed by the author. The results show that cross-iterative algorithm is superior to lanzocs method for signal processing in the convergence rate and numerical accuracy of steady output.
③Tangent CCD algorithm on accurate calculating canonical correlation decompoction is built up. It is applied to blind multiuser detection of multipath CDMA with correlation noise, and the output permance of detector of recepted signal is improved.
Canonical Correlation Analysis (CCA) is a statistical analysis method to examine the correlation between two random variables, by analyzing the correlation between two linear transformations of data, then searching for the public model. It considers the correlation between two sets of input data and the inter-relation of data from each group. Canonical correlation decomposition (CCD) is the singular value decomposition of typical correlation coefficient matrix, can achieve maximize relevance between the two linear transformation of data, and thereby obtain the optimal estimation of the relevant parts of the data (i.e., signal). CCD is the singular value decomposition of the product of three matrixes, so it is vulnerable in the impact of bad matrix condition number and element error. The Gram-Schmidt orthogonal preprocessing of data matrix will also be affected by the accumulated error, particularly evident when the matrix is large-scale.
Based on the tangent algorithm combines the accurate calculation of the generalized singular value decomposition and the product of singular value decomposition, the author puts forward a tangent algorithm to achieve precise calculation of canonical correlation decomposition. The CCD tangent algorithm need not Gram-schmidt orthogonal preprocessin and exist not accumulated error impact.
The CCD tangent algorithm is applied to improve subspace method on multipath CDMA blind multi-user detection with unknown related ambient noise and it estimates accurately of effective characteristic waveform in multipath channel. Simulation shows that the blind multi-user detector has been significant improvement in performance. Compared to existing algorithms, the results of signal detection are not sensitive to the scale of the matrix and condition number.
无线与移动通信的空时接收信号处理问题,常可归结为相关矩阵束的分解。这类矩阵束中常含有阵元移位与时滞互相关矩阵等非厄米特非正定复矩阵。强相关或相干信号常会导致相关矩阵接近奇异,即具有坏条件数。近奇异非对称正定的矩阵束分解的有效计算方法的研究具有一定难度,但对于无线空时通信信号处理问题具有重要意义。在矩阵束分解算法及应用于无线通信的研究中,论文取得了下列创新成果:
①提出非对称矩阵束( A, B )的广义Schur分解(GSD)的交叉迭代算法。将该算法应用于改进强相关阵列信号波达方向估计的ESPRIT子空间方法的计算实现,提高了方向角度的估计精度和分辨概率。
提出的交叉迭代算法通过在矩阵A, B间交换各自QR和RQ分解的酉因子,并乘上原有三角因子,实现矩阵束( A, B )的循环迭代更新,并最终收敛至三角矩阵束。论文证明了该算法的收敛性同直接计算单个矩阵AB ?1Schur分解的QR迭代算法收敛性等价。然而,由于包含矩阵求逆与乘积运算,若直接计算AB ?1,其条件数可能很大(即使A, B都是良态的),在有限字长条件下,QR迭代精度无法保证。交叉迭代算法绕开了矩阵直接求逆与乘积运算,得到的结果具有和单个矩阵A, B条件数大致相当的精度。该算法的速度不受B秩亏损的影响。
与矩阵束GSD的QZ算法相比,交叉迭代算法中两个矩阵A, B的QR(或RQ)分解及乘积更新过程在计算上相互独立,仅需交换酉因子,具有环型并行计算结构;交叉迭代算法的一步迭代计算主要集中于QR(与RQ)分解及相反过程(酉旋转矩阵与上三角阵的乘积),硬件和软件的实现都易于模块化与并行化。
②建立了自适应GSD和GEVD的递推交叉迭代算法,并用于解决快速时变环境中未知CDMA系统自适应盲辨识问题。
在实际的自适应应用中,根据实时更新的输入数据递推生成输出。与批处理的QZ算法相比,基于QR(与RQ)分解的交叉迭代算法更适合自适应实现。自适应秩一更新矩阵束GSD的一步交叉迭代的计算代价为( )O n 2。论文提出了自适应GSD和GEVD的递推交叉迭代算法,实现对广义特征值与广义特征向量的自适应更新,即论文提出的广义子空间追踪(GSST)。与子空间追踪不同,广义子空间追踪不要求相关矩阵是厄米特阵,无需对广义特征向量的正交性约束,更适用于辨识一般非列正交矩阵,如CDMA系统多用户扩频序列矩阵。
借助天线阵列,可以在CDMA系统期望用户扩频波形未知(如截听、灾害等)情况下,实现对用户有效扩频波形的盲辨识和接收信号的检测。针对多径信道的快速时变性和计算的低复杂性要求,论文提出了时变环境中未知CDMA系统自适应盲辨识广义子空间追踪方法。分别基于递推交叉迭代和递推更新lanczos迭代两种算法进行相关矩阵束的自适应广义特征值分解,利用广义特征向量自适应辨识未知期望用户信号的有效扩频序列,并实现接收信号的盲多用户检测。仿真表明了GSST算法的有效性,并比较了递推交叉迭代和递推Lanczos迭代两种GSST实现形式的收敛性能。
③提出了典型相关解(CCD)精确计算的正切CCD算法。应用于相关噪声环境中的多径CDMA盲多用户检测,改善了接收信号的检测输出性能。
典型相关分析(CCA)通过分析两个随机变量之间的典型相关性,寻找其中的公共模式。它综合考虑两组输入数据的互相关性及每组数据的自相关性。典型相关分解(CCD)实现两路线性变换数据之间典型相关性的最大化,从而得到该数据相关部分(即信号)的最优估计。作为三个矩阵乘积的奇异值分解,CCD的计算易受到矩阵坏条件数和元素误差的影响。基于对数据矩阵Gram-Schmidt正交化预处理的已有算法亦存在误差累积效应,在矩阵规模较大时尤为明显。
在结合矩阵束广义奇异值分解与乘积奇异值分解精确计算的正切算法基础上,论文建立了精确计算典型相关分解的正切CCD算法。该算法无需正交化预处理,不存在误差累积效应,应用于改进具有多接收阵元的有色噪声环境中多径CDMA盲多用户检测的子空间方法,可准确辨识信号与噪声子空间。仿真表明多径信道与用户有效特征波形盲估计及盲多用户检测性能得到明显改善;相对于原有方法,基于正切CCD的子空间方法的信号检测结果对矩阵规模大小和条件数不敏感,在高负载系统条件下性能优势更为明显。
Matrix pen triangle and diagonal decomposition,such as generalized Schur decomposition (GSD), generalized eigenvalue decomposition (GEVD) and generalized singular values decomposition (GSVD), is a important subject on matrix computing and Matrix pen decomposition has important application to wireless communication and signal processing.
The problem on wireless and space-time receptive signal processing often can be come down to matrix pen decomposition. This kind of matrix pen often incudes senser shift corrlection matrix and time delay correlation matrix, they are not Hermitian and positive. Strong correlation or Coherent signal often cause correlative matrix is close to singular, means has a bad condition. It is difficult that investigates approximative singular and unsymmetry positive matrix pen decomposition, but it is important to wireless commiunication signal processing.
By investigating matrix decomposition algorithm and its application in wireless communication, the following innovating results are achieved:
①The cross-iterative algorithm of matrix generalized Schur decomposition was built up. And it is applied to improve the ESPRIT subspace method on direction of arrival of array signal strong correlating. The estimating precision and probability of recognition is improved.
By exchanging the unitary matrix factor from QR and RQ factorization between A, B , and multipling triangle factor inhere, the author derive out the circulative iterative program of pen( A, B ) that converges at triangle matrix pen at last, called cross-iterative algorithm. And it is proved that the convergence of algorithm is equivalent to the one of QR iterative of Schur decomposition of AB ?1. Of course, it is unpractical that straight computing AB ?1. Due to product and inverse calculate, the matrix condition number is bad and iterative computing accuracy can not be promised in limited words length condition. Cross-iterative algorithm avoids computing inverse and product of A &B and the result has approximately same condition number with A, B . The speed is free from the deficient rank of B .
In Cross-iterative algorithm, The QR&RQ factorization is independent each other, so do the product of Q and R. So the algorithm is parallel annular structurally.
The computing of one step iterative of cross-iterative algorithm fastens on QRF (RQF) and product of Q&R, the inverse program of former. So, it is fit to carry out modularly.
②Recursive cross-iterative algorithm on adaptive GSD and GEVD is set up and is applied to blind identify to unknown CDMA system in fast time varing environment.
In actual application, the input data is updated real time. Contrast with batch method, such as QZ algorithm, cross-iterative algorithm is easy to carry out adaptively. Its one step need ( )O n 2 flops to compute GSD of rank one update matrix pen. The author gives a recursive cross-iterative algorithm to achieves adaptive GSD and GEVD,namely generalized subspace tracking. This method with Low-complexity provides an effective way to rapidly calculate the decomposition of matrix pencil update eigenvalue. Distinguish from subspace tracking, the generalized subspace tracking need not the correlation matrix is Hermitian and the orthonormal restriction to generalized eigenvector, so it is fit to identify the ordinary matrix without rows orthogonal each other, such as multiusers’extend sequence matrix of CDMA system.
With the help of antenna array, you can realize the detection of blind identification and receiving signal of effective spreading spectrum waveform of user on the unknown of spread spectrum waveform (such as interception, disaster, etc.) of multipath CDMA system desire user. Taking into account the fast time-varying characteristic of multipath channel and the requirements on low complexity of calculation, the author puts forward generalied subspace tracking method on unknown CDMA system blinds identification with antenna array. Based on the recursion cross-iterative algorithm or recursion Lanzocs iterative, we achieves self-adaptive generalied eigenvalue decomposition of correlation matrix pen, and identifies the unknown user’available extend sequence and achieve blind multiuser detection. Computer simulation shows the validity of GSST, and makes a comparison of the two adaptive subspace tracking methods between Lanzocs iterative algorithm and recursive cross-iterative algorithm proposed by the author. The results show that cross-iterative algorithm is superior to lanzocs method for signal processing in the convergence rate and numerical accuracy of steady output.
③Tangent CCD algorithm on accurate calculating canonical correlation decompoction is built up. It is applied to blind multiuser detection of multipath CDMA with correlation noise, and the output permance of detector of recepted signal is improved.
Canonical Correlation Analysis (CCA) is a statistical analysis method to examine the correlation between two random variables, by analyzing the correlation between two linear transformations of data, then searching for the public model. It considers the correlation between two sets of input data and the inter-relation of data from each group. Canonical correlation decomposition (CCD) is the singular value decomposition of typical correlation coefficient matrix, can achieve maximize relevance between the two linear transformation of data, and thereby obtain the optimal estimation of the relevant parts of the data (i.e., signal). CCD is the singular value decomposition of the product of three matrixes, so it is vulnerable in the impact of bad matrix condition number and element error. The Gram-Schmidt orthogonal preprocessing of data matrix will also be affected by the accumulated error, particularly evident when the matrix is large-scale.
Based on the tangent algorithm combines the accurate calculation of the generalized singular value decomposition and the product of singular value decomposition, the author puts forward a tangent algorithm to achieve precise calculation of canonical correlation decomposition. The CCD tangent algorithm need not Gram-schmidt orthogonal preprocessin and exist not accumulated error impact.
The CCD tangent algorithm is applied to improve subspace method on multipath CDMA blind multi-user detection with unknown related ambient noise and it estimates accurately of effective characteristic waveform in multipath channel. Simulation shows that the blind multi-user detector has been significant improvement in performance. Compared to existing algorithms, the results of signal detection are not sensitive to the scale of the matrix and condition number.
引文
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