Structured total least squares methods in signal processing
Structured total least squares methods in signal processing
复制标题
DOI：
10.5281/zenodo.35986
复制
发表时间：
1996
期刊：
影响因子：
--
通讯作者：
P. Lemmerling;S. Huffel;B. Moor
中科院分区：
文献类型：
--
作者：
P. Lemmerling;S. Huffel;B. Moor
在许多信号处理应用中，人们必须求解一个超定的线性方程组\(Ax\approx b\)，同时使\(A\)和\(b\)的误差最小化。总体最小二乘法（TLS）计算校正量\(\Delta A\)和\(\Delta b\)，使得\((A + \Delta A)x = b + \Delta b\)且\(\left\|\left[\Delta A\ \Delta b\right]\right\|_F\)最小。当\([A\ b]\)的不同元素上的噪声是独立同分布的零均值且等方差的高斯噪声时，所得的参数向量\(x\)是一个最大似然（ML）估计。在许多应用中，由于\([\Delta A\ \Delta b]\)中存在的结构，这些最后的条件不成立。在这种情况下，TLS不会产生参数向量\(x\)的最大似然估计，因为奇异值分解（SVD，它是获得TLS解的标准方法）不保持结构。因此，近年来已经开发了几种结构化总体最小二乘法：约束总体最小二乘法（CTLS）[1][2]、结构化总体最小二乘法（STLS）[3]以及结构化总体最小范数法（STLN）[8][7]。与普通的TLS不同，这些方法通过将\([A\ b]\)的结构施加于\([\Delta A\ \Delta b]\)，从而得到参数向量\(x\)的最大似然估计。
In many signal processing applications, one has to solve an overdetermined system of linear equations Ax ≈ b, while minimizing the errors on A and b. The Total Least Squares (TLS) method calculates corrections ΔA and Δb such that (A + ΔA)x = b + Δb and ||[ΔA Δb]||F is minimal. The resulting parameter vector x is ä Maximum Likelihood (ML) estimate when the noise on the different entries of [A b] is i.i.d. Gaussian noise with zero mean and equal variance. In many applications, these last conditions do not hold because of the structure present in [ΔA Δb]. Under those circumstances, the TLS will not yield a ML estimate of the parameter vector x since the SVD (which is the standard way to obtain the TLS solution) is not structure preserving. Therefore, several structured Total Least Squares methods have been developed in recent years: Constrained Total Least Squares (CTLS) method [1][2], the Structured Total Least Squares (STLS) method [3] and the Structured Total Least Norm (STLN) method [8] [7]. As opposed to the ordinary TLS these methods yield a ML estimate of the parameter vector x, by imposing the structure of [A b] to [ΔA Δb].