Approximations to complementary error function by method of least squares

A few simple approximations are derived for erfc (x) by method of least squares (MLS). The detailed error profiles are presented. It is shown how these approximations are useful in extracting the inverse of erfc(x). Finally a simple approximation with overall relative root mean square error (RRMS) of less than one percent is presented.     

Comparison of the linear least squares and nonlinear least squares spheres

Two important, to scientists and engineers, sphere fitting procedures, namely the linear least squares (LLS) and the non-linear least squares (NLLS) methods and their general random-error analysis are described. The first-order random errors of the center coordinates and the radius of the fitted sphere using the above mentioned procedures are derived in detail under the assumption that the variance-covariance matrix exists for the random error vector. With the additional trivariate normal error distribution assumption, the Maximum Likelihood (ML) estimators and their standard deviations are also derived. The effectiveness of these procedures are studied through computer stimulation.     

最小二乘与加权最小二乘空域矩阵滤波器设计

空域矩阵滤波器是一种新的信号处理技术,通过一个滤波矩阵与接收到的阵列数据相乘,可实现保留通带目标信号,抑制阻带干扰的目的.本文主要研究了最小二乘和加权最小二乘两类的空域矩阵滤波器.给出了空域滤波器设计基本原理,通过最优化问题得出了最优解.最小二乘空域矩阵滤波器是加权系数为1的加权加权最小二乘空域矩阵滤波器的特列.由加权最小二乘迭代仿真结果可以看出,迭代次数的增加使滤波器阻带响应极大值逐渐变小,可实现恒定阻带抑制效果,设计效率较高.     

DOA estimation by least squares approach

A least squares estimation technique for finding the directions of arrival (DOAs) of signals corrupted by additive white noise using a uniform linear array is developed. Although the estimates are not as accurate as those of subspace methods, the computations involved are simpler, and prior knowledge of the number of sources is not required     

Knowledge bounded least squares method for the identification of fuzzy systems

This paper presents the knowledge bounded least squares method that uses both linguistic information (i.e., human knowledge and experience) and numerical data to identify fuzzy models. Based on the concept of fuzzy interval systems, the basic idea of this method is: first, to utilize all the available linguistic information to obtain a fuzzy interval system and then use the obtained fuzzy interval system to give the admissible model set (i.e., the set of all fuzzy models which are acceptable and reasonable from the point of view of linguistic information); second, to find a fuzzy model in the admissible fuzzy model set which best fits the available numerical data. It is shown that such a fuzzy model can be obtained by a quadratic programming approach. By comparing this method with the least squares method, it is proved that the fuzzy model obtained by the proposed method fits the real model better than the fuzzy model obtained by the least squares method.     

超级电容模型的递推增广最小二乘辨识

为了更好的描述超级电容的电气特性,需要建立超级电容模型。提出系统辨识方法进行建模,阐述了系统辨识的原理和递推增广最小二乘法算法,在此基础上利用MATLAB编写递推增广最小二乘法程序估计出超级电容的传递函数,通过仿真对辨识结果进行验证。可知,该超级电容模型是正确有效的。     

Maximum likelihood, least squares, and penalized least squares for PET

The EM algorithm is the basic approach used to maximize the log likelihood objective function for the reconstruction problem in positron emission tomography (PET). The EM algorithm is a scaled steepest ascent algorithm that elegantly handles the nonnegativity constraints of the problem. It is shown that the same scaled steepest descent algorithm can be applied to the least squares merit function, and that it can be accelerated using the conjugate gradient approach. The experiments suggest that one can cut the computation by about a factor of 3 by using this technique. The results are applied to various penalized least squares functions which might be used to produce a smoother image.     

A comparison of the least squares method and the Burg method for autoregressive spectral analysis

The performance of autoregressive (AR) spectral estimates based on two different methods for computing the AR coefficients are compared. They are the recursive method as stated by Burg, which minimizes the residual power with respect to only one coefficient, and the straightforward but computationally less efficient least squares method (LSM) which minimizes the residual power with respect to all the AR coefficients simultaneously. It is shown that when the input signal consists of two equal-leveled sinusoids in white noise, the LSM estimate is highly superior with respect to resolution, positional bias, and spurious peaks in the spectrum.     

Adaptive blind channel estimation by least squares smoothing

A least squares smoothing (LSS) approach is presented for the blind estimation of single-input multiple-output (SIMO) finite impulse response systems. By exploiting the isomorphic relation between the input and output subspaces, this geometrical approach identifies the channel from a specially formed least squares smoothing error of the channel output. LSS has the finite sample convergence property, i.e., in the absence of noise, the channel is estimated perfectly with only a finite number of data samples. Referred to as the adaptive least squares smoothing (A-LSS) algorithm, the adaptive implementation has a high convergence rate and low computation cost with no matrix operations. A-LSS is order recursive and is implemented in part using a lattice filter. It has the advantage that when the channel order varies, channel estimates can be obtained without structural change of the implementation. For uncorrelated input sequence, the proposed algorithm performs direct deconvolution as a by-product     

Multichannel ARMA modeling by least squares circular latticefiltering

The paper makes an attempt to develop least squares lattice algorithms for the ARMA modeling of a linear, slowly time-varying, multichannel system employing scalar computations only. Using an equivalent scalar, periodic ARMA model and a circular delay operator, the signal set for each channel is defined in terms of circularly delayed input and output vectors corresponding to that channel. The orthogonal projection of each current output vector on the subspace spanned by the corresponding signal set is then computed in a manner that allows independent AR and MA order recursions. The resulting lattice algorithm can be implemented in a parallel architecture employing one processor per channel with the data flowing amongst them in a circular manner. The evaluation of the ARMA parameters from the lattice coefficients follows the usual step-up algorithmic approach but requires, in addition, the circulation of certain variables across the processors since the signal sets become linearly dependent beyond certain stages. The proposed algorithm can also be used to estimate a process from two correlated, multichannel processes adaptively allowing the filter orders for both the processes to be chosen independently of each other. This feature is further exploited for ARMA modeling a given multichannel time series with unknown, white input     

Total least mean squares algorithm

Widrow (1971) proposed the least mean squares (LMS) algorithm, which has been extensively applied in adaptive signal processing and adaptive control. The LMS algorithm is based on the minimum mean squares error. On the basis of the total least mean squares error or the minimum Raleigh quotient, we propose the total least mean squares (TLMS) algorithm. The paper gives the statistical analysis for this algorithm, studies the global asymptotic convergence of this algorithm by an equivalent energy function, and evaluates the performances of this algorithm via computer simulations     

Noise-constrained least mean squares algorithm

We consider the design of an adaptive algorithm for finite impulse response channel estimation, which incorporates partial knowledge of the channel, specifically, the additive noise variance. Although the noise variance is not required for the offline Wiener solution, there are potential benefits (and limitations) for the learning behavior of an adaptive solution. In our approach, a Robbins-Monro algorithm is used to minimize the conventional mean square error criterion subject to a noise variance constraint and a penalty term necessary to guarantee uniqueness of the combined weight/multiplier solution. The resulting noise-constrained LMS (NCLMS) algorithm is a type of variable step-size LMS algorithm where the step-size rule arises naturally from the constraints. A convergence and performance analysis is carried out, and extensive simulations are conducted that compare NCLMS with several adaptive algorithms. This work also provides an appropriate framework for the derivation and analysis of other adaptive algorithms that incorporate partial knowledge of the channel     

基于最小二乘互相关算法的图像定位匹配研究

提出了一种基于最小二乘互相关的图像定位匹配算法。该算法将图像互相关信息和最小二乘法结合实现图像定位匹配,匹配精度可以达到亚像素甚至1/100像素级,同时利用金字塔分层来提高定位匹配速度;通过本算法在印刷品质量自动化检测系统中的应用,验证了该算法的高精度与高速度特性。1     

Convergence of polynomial least squares estimators

The rate of convergence of polynomial least squares estimators is derived as a function of its order. Some interesting properties of derivative matrices are exhibited.     

A robust recursive least squares algorithm

A new algorithm is developed, which guarantees the normalized bias in the weight vector due to persistent and bounded data perturbations to be bounded. Robustness analysis for this algorithm has been presented. An approximate recursive implementation is also proposed. It is termed as the robust recursive least squares (RRLS) algorithm since it resembles the RLS algorithm in its structure and is robust with respect to persistent bounded data perturbation. Simulation results are presented to illustrate the efficacy of the RRLS algorithm     

基于相关函数的递推最小二乘算法及其在回波消除中的应用

本文给出一种新的类似于RLS（recursive least squares)算法的递推最小二乘算法，该算法直接对输入信号的相关函数进行处理而不是对输入信号本身进行处理，理论分析表明了该算法的收敛性。该算法应用于回波消除问题中，克服了常规自适应滤波算法在出现双方对讲的情况下需停止调节自适应滤波器系数这一不足。计算机模拟仿真表明该算法在双方对讲的情况下有良好的收敛性能。     

Multiband least squares FIR filter design

This paper gives two methods for the least squares approximation design of FIR digital filters. They both allow multiple pass and stop bands and arbitrary transition bands, all with explicit control of band edges. The first method is as simple to use as a window method. This design method builds up an optimal multiband frequency response by sequentially adding and subtracting optimal lowpass filters with transition bands. However, it assumes the ideal response has spline transition functions and does not allow error weighting. The second method allows different weights in the multiple passbands, stopbands and transition bands but it requires numerical solution of simultaneous equations which may be ill-conditioned. These methods compliment the Parks-McClellan algorithm which minimizes the Chebyshev error     

基于最小二乘的航天器天文导航定姿算法研究

星敏感器是目前精度最高的姿态测量仪器之一,具有自主性和稳定性,是航天器主要的定姿设备.研究采用最小二乘法对星敏感器的测量信息进行处理得到航天器姿态信息,通过数字仿真证明姿态精度可以达到角秒级,为某工程型号应用提供理论依据.     

摄像机标定中总体最小二乘法抗噪性的研究

为了降低摄像机标定中图形加性噪声给标定精度带来的不良影响,提出了基于总体最小二乘法的摄像机标定方法。由于总体最小二乘法具有消除或降低噪声的功能,本文将其用于求解单应性矩阵,既提高了单应性矩阵的精度,又为摄像机内外参数和畸变系数的精确测量提供了理论依据。在此基础上借助OpenCV函数库获取图形中角点高精度坐标的功能,在Visual C＋＋环境下实现了对摄像机的标定。数值实验和实际标定实验均表明,提出的标定方法具有更高的精度和抗噪声能力。