Recursive solution to the multichannel filtering problem
Recursive solution to the multichannel filtering problem
复制标题
多通道滤波问题的递归求解
DOI：
10.1029/jz070i008p01885
复制
发表时间：
1965
影响因子：
--
通讯作者：
E. Robinson
中科院分区：
文献类型：
--
作者：
R. Wiggins;E. Robinson
由传感器阵列(地震仪、水听器、天线、气象站)组成的系统在多个通道上接收地球物理数据。这种多通道数据可以通过多通道最小二乘(Wiener)数字滤波器来处理；该滤波器是通过要求每个通道的期望输出和实际输出之间的均方误差之和最小化来确定的。这一要求导致了一组联立的线性方程，称为法方程，其中每个元素本身都是一个方阵。数字滤波器的系数由这些法方程的解给出。给出了法方程数值解的精确递推算法。在该递归过程中，利用了法方程的矩阵是自相关矩阵的事实，因此是Toeplitz矩阵的形式，即沿任何对角线具有相等元素的Toeplitz矩阵的形式。在递归过程中产生的一组辅助系数是矩阵值多项式的系数，这些多项式是单位圆上正交的经典(标量值)多项式的对应物。这组辅助系数还可用于对应于移位正常方程右侧的时间索引的侧向递归。这种侧向递归在应用中是有价值的，因为它代表了一种相对便宜的确定结合了多通道输入信号和多通道输出信号之间的最佳时延的滤波器的方法。对于具有m个(矩阵值)系数的滤波器，在所提出的递归过程中，求解法方程所需的机器时间与m~2成正比，而对于解联立方程组的传统技术，则为m~3。使用递归过程的另一个优点是它只需要与m成比例的计算机存储空间，而不是像在传统技术的情况下那样与m2成比例。
Systems made up of arrays of sensors (seismographs, hydrophones, antennas, meteorological stations) receive geophysical data on a number of channels. Such multichannel data can be processed by means of a multichannel least-squares (Wiener) digital filter; the filter is determined by requiring that the sum of the mean-square errors between the desired outputs and the actual outputs for each channel be minimized. This requirement leads to a set of simultaneous linear equations, called the normal equations, where each element is itself a square matrix. The coefficients of the digital filter are given by the solution of these normal equations. An exact recursive procedure is derived for the numerical solution of the normal equations. In this recursive procedure advantage is taken of the fact that the matrix of the normal equations is an autocorrelation matrix, and hence is in the form of a Toeplitz matrix, namely one with equal elements along any diagonal. A set of auxiliary coefficients that are generated in the recursive procedure are the coefficients of matrix-valued polynomials which are the counterparts of the classical (scalar-valued) polynomials orthogonal on the unit circle. This set of auxiliary coefficients may also be used for a sideward recursion that corresponds to shifting the time index of the right-hand side of the normal equations. This sideward recursion is valuable in applications because it represents a relatively inexpensive way of determining the filter that incorporates the optimum time delay between the multichannel input signal and the multichannel output signal. The machine time required to solve the normal equations for a filter with m (matrix-valued) coefficients is proportional to m2 for the proposed recursive procedure, as compared with m3 for conventional techniques for the solution of simultaneous equations. Another advantage of using the recursive procedure is that it only requires computer storage space proportional to m, rather than to m2 as in the case of the conventional techniques.