Least-squares fitting of data by polynomials

In the last two columns, we looked at one approach to approximation, interpolating a set of points by piecewise-cubic polynomials forming a cubic spline. We now look at another approach - one that involves fitting a curve to a set of data without restricting that curve to coincide with the data points. Our focus is on least-squares approximation, and, in particular, least-square fitting of polynomials to data     

Identification of a class of multivariable systems from impulse response data: Theory and computational algorithm

A theoretical and algorithmic framework is proposed for the identification of rational transfer function matrices of a class of discrete-time multivariable systems. The proposed technique obtains an optimal approximation from the given (possibly noisy) measuredimpulse response data. It is assumed that the measured impulse response data corresponds to a system with a strictly proper transfer function matrix. The impulse response fitting error criterion is theoretically decoupled into a purely linear problem for estimating the optimal numerators and a nonlinear problem for the optimal denominators. Based on the proposed theoretical basis, an efficient computational algorithm is developed and illustrated with several examples. Research supported by AFOSR-89-0291 and by WRDC/WPAFB grant F33615-88-C-3605.     

结合自重变形特点的天线测量数据粗差剔除与定权方法

针对传统天线测量数据处理中粗差剔除和定权不合理问题,提出了一种结合粗差剔除和一次范数最小法定权的天线反射面多仰角测量数据处理方法.首先,根据反射面自重变形的特点,通过构造统计量对粗差点予以剔除,提高了测量数据的可靠性;其次,顾及测量点的变形特点,引入了抗差估计中的一次范数最小法对不同变形点进行定权,解决了等权最小二乘拟合的"虚假面型"问题.经过某口径13m天线反射面测量数据的实例计算,对比其他方法,新方法拟合结果能够合理、准确地反映出反射面的实际变形偏差,更适用于天线反射面自重变形数据的处理.     

Neural Network-Based IIR All-Pass Filter Design

This paper presents a neural network-based Lyapunov energy function for the weighted least-squares design of IIR all-pass filters. In the proposed method, the error reflecting the difference between the desired phase response and the phase of the designed IIR all-pass filter is formulated as a Lyapunov error criterion. Based on the neural network architecture and suitable Hopfield parameters, the optimal filter coefficients can be obtained when convergence is achieved. Furthermore, a weight updating function is proposed to achieve accurate approximation of the equiripple response. The simulation results indicate that the proposed technique can achieve high performance in a parallel manner.     

Optimal identification of discrete-time systems from impulseresponse data

An optimal method (OM) for estimation of the parameters of rational transfer functions from prescribed impulse response data is presented. The multidimensional nonlinear fitting error minimization problem has been theoretically decoupled into two subproblems of reduced computational complexities. The proposed approach is applicable for identifying rational models with arbitrary numbers of poles and zeros. The nonlinear denominator subproblem possesses weighted-quadratic structure which is utilized to formulate an efficient iterative minimization algorithm. The optimal numerator is found noniteratively with a linear least-squares approach that utilizes the optimized denominator. Both the decoupled subcriteria of OM posses global optimality properties. The Steiglitz-McBride (1960, SM) method is also decoupled for arbitrary numbers of poles and zeros (DSM-G). It is demonstrated that the denominator subproblem of DSM-G is theoretically optimal. For another existing decoupled SM method (DSM-J), it has been shown that only the numerator is theoretically optimal     

Periodic quasi-orthogonal spline bases and applications toleast-squares curve fitting of digital images

Presents a new covariant basis, dubbed the quasi-orthogonal Q-spline basis, for the space of n-degree periodic uniform splines with k knots. This basis is obtained analogously to the B-spline basis by scaling and periodically translating a single spline function of bounded support. The construction hinges on an important theorem involving the asymptotic behavior (in the dimension) of the inverse of banded Toeplitz matrices. The authors show that the Gram matrix for this basis is nearly diagonal, hence, the name "quasi-orthogonal". The new basis is applied to the problem of approximating closed digital curves in 2D images by least-squares fitting. Since the new spline basis is almost orthogonal, the least-squares solution can be approximated by decimating a convolution between a resolution-dependent kernel and the given data. The approximating curve is expressed as a linear combination of the new spline functions and new "control points". Another convolution maps these control points to the classical B-spline control points. A generalization of the result has relevance to the solution of regularized fitting problems.     

Approximations to Error Functions

This paper addresses approximations to error functions and points out three representative approximations, each with its own merits. Cody's approximation is the most computationally intensive of the three, it is not overly so, and there is no arguing over its accuracy. The other two approximations are much simpler computationally, and they both yield accuracies that would be considered more than sufficient in most practical situations. Absolute relative error provides an effective measure of goodness, and, for approximations to the Q-function, it also places a loose bound on the absolute error in the approximation. Cody's approximation is an effective surrogate for the true error function; the values provided by that approximation match the actual values of the error function to within roughly the precision of double-precision floating point arithmetic.     

FIR数字滤波器的递推最小二乘设计算法

本文考虑对称系数及反对称系数的ＦＩＲ数字滤波器的设计问题,设计准则选为最小加权平方误差准则,并将这个设计问题看成一个线性系统的辨识问题,辨识系统参数所需的输入数据由一随机抽样法产生,辨识算法采用递推最小二乘法．按随机抽样法产生的数据具有很强的激励,保证了被辨识参数的收敛性,同时又自然地实现了最小加权平方误差准则．两个设计范例说明了本文提出的设计方法的有效性．     

Optimal Design of Nonlinear-Phase FIR Filters With Prescribed Phase Error

Constrained least-squares design and constrained Chebyshev design of one- and two-dimensional nonlinear-phase FIR filters with prescribed phase error are considered in this paper by a unified semi-infinite positive-definite quadratic programming approach. In order to obtain unique optimal solutions, we propose to impose constraints on the complex approximation error and the phase error. By introducing a sigmoid phase-error constraint bound function, the group-delay error can be greatly reduced. A Goldfarb–Idnani based algorithm is presented to solve the semi-infinite positive-definite quadratic program resulting from the constrained least-squares design problem, and then applied after some modifications to the constrained Chebyshev design problem, which is proved in this paper to be equivalent also to a semi-infinite positive-definite quadratic program. Through design examples, the proposed method is compared with several existing methods. Simulation results demonstrate the effectiveness and efficiency of the proposed method.      

A Note on the Multiplicity of Poles in the Vector Fitting Macromodeling Method

Vector fitting is a robust macromodeling tool for rational approximation of spectral data obtained by full-wave electromagnetic simulators or high-frequency measurements. The technique iteratively calculates a suitable set of poles, and solves the residues of the transfer function in a two-step procedure. If poles with a higher order multiplicity occur during the pole-identification step, numerical problems and inaccuracies can result, especially if the normal equations are solved. This problem is illustrated on an RLC filter, and a generalization of the basis functions is proposed to resolve the issue     

一种拟合曲线的新拟合算法

非线性测量中对测量结果进行曲线拟合通常采用最小均方误差的标准进行系统参数的辨识,该方法未考虑输入样本的随机性。基于此,提出了一种新的拟合算法:考虑输入样本的随机性所采取的一种加权的最小均方误差拟合方法,利用已知的输入样本的统计特性对权值进行估计,对算法概率为1的任意逼近性给出证明,并对区间半长作出估计。实验表明,该算法有较理想的逼近效果。     

P~+-N-N~+功率二极管零输入态反向电流峰值研究

从半导体器件物理角度出发,分析了P+-N-N+功率二极管零输入态下正向恒定电流IF与反向电流峰值IRM的解析关系。根据误差函数与初等函数的近似关系,推出简约解析关系式:(IRM/IF)=a+b×ln(IF+1)。在检测电路中引入反向平衡电流源,利用仿真软件Silvaco-Atlas获得瞬态仿真实验数据。通过Matlab软件对实验数据进行拟合分析,并对简约式适用范围和相关结论进行了验证。研究结果对P+-N-N+功率二极管的应用和相应电路系统的可靠性具有指导意义。     

A coupled efficient and systematic full-wave time-domain macromodeling and circuit Simulation method for signal integrity analysis of high-speed interconnects

This paper presents an accurate and systematic approach for analysis of the signal integrity of the high-speed interconnects, which couples the full-wave finite difference time domain (FDTD) method with scattering (S) parameter based macromodeling by using rational function approximation and the circuit simulator. Firstly, the full-wave FDTD method is applied to characterize the interconnect subsystems, which is dedicated to extract the S parameters of the subnetwork consisting of interconnects with fairly complex geometry. Once the frequency-domain discrete data of the S parameters of the interconnect subnetwork is constructed, the rational function approximation is carried out to establish the macromodel of the interconnect subnetwork by employing the vector fitting method, which provides a more robust and accurate solution for the overall problem. Finally, the analysis of the signal integrity of the hybrid circuit can be fulfilled by using the S parameters based macromodel synthesis and simulation program with integrated circuits emphasis (SPICE) circuit simulator. Numerical experiments demonstrate that the proposed approach is accurate and efficient to address the hybrid electromagnetic (interconnect part) and circuit problems, in which the electromagnetic field effects are fully considered and the strength of SPICE circuit simulator is also exploited.     

Generating High-Accuracy Simulation Models Using Problem-Tailored Orthogonal Polynomials Basis

The problem of computing high-accuracy simulation models for systems described by tabulated frequency data is of paramount importance in the modeling arena. Standard algorithms for this task involve generating rational function approximations to the data. However, for complicated data sets, high-order approximations are required. Unfortunately, numerical conditioning problems arise when attempting to fit high-order rational approximations to the data, effectively limiting the accuracy of the models that can be generated. While robust fitting schemes based on orthogonal polynomial exist, they usually pose strict constraints on the data points, which are either hard or even impossible to guarantee. Furthermore, the approximation must still be translated such that it can directly be used inside a simulator. In this paper, we present an algorithm for robustly generating such a model using only the data given. The model is supported on a problem-tailored orthogonal polynomial basis. We also present a method for directly generating a state-space model associated with a rational function described in terms of such polynomials, effectively making the model amenable for simulation. An extension to the MIMO case is described and it is shown that the method is easily included with existing passivity enforcing procedures. Finally, we demonstrate the proposed technique by constructing approximations to several real-world data sets     

Cubic-spline interpolation: part 2

The authors discuss an approach to approximation based on interpolating a set of points. The one interpolator that they concentrated on, the cubic spline, is relatively straightforward to implement and can be made to be computationally efficient. They also point out that another approach to approximation involves fitting a curve to a set of data points without forcing that curve to pass through any of those data points. This approach is especially useful when the data available is recognized to contain an error component     

Autoregressive model fitting with noisy data by Akaike's information criterion (Corresp.)

Davisson [131, [141 has considered the problem of determining the "order" of the signal from noisy data. Although interesting theoretically, his result is difficult to use in practice. In this correspondence, we exploit one well-known fact concerning autoregressive (AR) signals plus white noise, and using Akaike's information criterion [15], [17], we have developed one efficient procedure for determining the order of the AR signal from noisy data. The procedure is illustrated numerically using both artificially generated and real data. The connection between the preceding problem and the classical statistical problem of factor analysis is discussed.     

Throughput Optimization for Wireless LANs in the Presence of Packet Error Rate Constraints

In [1], we optimized single-user throughput by selecting the transmitted bit rate and payload size as a function of channel conditions. However, the approach did not consider a packet error rate (PER) constraint, and the payload size obtained could yield excessively high packet error rates. We propose and solve the optimization problem of maximizing throughput by varying the PHY layer data rate and the payload size subject to a packet error rate constraint. The resulting SNR thresholds for adapting the PHY data rate and the corresponding payload sizes are drastically different than those obtained without the PER constraint.     

Mathematical methods for the design of color scanning filters

The problem of the design of color scanning filters is addressed in this paper. The problem is posed within the framework of the vector space approach to color systems. The measure of the goodness of a set of color scanning filters presented in earlier work is used as an optimization criterion to design color scanning filters modeled in terms of known, smooth, nonnegative functions. The best filters are then trimmed using the gradient of the mean square DeltaE(ab) error to obtain filters with a lower value of perceptual error. The results obtained demonstrate the utility of the method.     

A weighted least-squares method for the design of stable 1-D and2-D IIR digital filters

We present a new approach to the least-squares design of stable infinite impulse response (IIR) digital filters. The design is accomplished by using an iterative scheme in which the denominator polynomial obtained from the preceding iteration is treated as a part of the weighting function, and each iteration is carried out by solving a standard quadratic programming problem that yields a stable rational function. When the iteration converges, a stable and truly least-squares solution is obtained. The method is then extended to address the least-squares design of stable IIR two-dimensional (2-D) filters. Examples are included to illustrate the proposed design techniques     

Data record-based criteria for the selection of an auxiliary vector estimator of the MMSE/MVDR filter

When the auxiliary vector (AV) filter generation algorithm utilizes sample average estimated input data statistics, it provides a sequence of estimates of the ideal minimum mean-square error or minimum-variance distortionless-response filter for the given signal processing/receiver design application. Evidently, early nonasymptotic elements of the sequence offer favorable bias/variance balance characteristics and outperform in mean-square filter estimation error the unbiased sample matrix inversion (SMI) estimator as well as the (constraint) least-mean square, recursive least-squares, "multistage nested Wiener filter", and diagonally-loaded SMI filter estimators. Selecting the most successful (in some appropriate sense) AV filter estimator in the sequence for a given data record is a critical problem that has not been addressed so far. We deal exactly with this problem and we propose two data-driven selection criteria. The first criterion minimizes the cross-validated sample average variance of the AV filter output and can be applied to general filter estimation problems; the second criterion maximizes the estimated J-divergence of the AV filter output conditional distributions and is tailored to binary phase-shift-keying-type detection problems.