Discrete-time system modelling in Lp with orthonormal basis functions

In this paper, model sets for linear time-invariant systems spanned by fixed pole orthonormal bases are investigated. The obtained model sets are shown to be complete in L_p(T) (1<p<∞), the Lebesque spaces of functions on the unit circle T, and in C(T), the space of periodic continuous functions on T. The L_p norm error bounds for estimating systems in L_p(T) by the partial sums of the Fourier series formed by the orthonormal functions are computed for the case 1<p<∞. Some inequalities on the mean growth of the Fourier series are also derived. These results have application in estimation and model reduction.     

具有模型和实际差异的非线性系统最优控制算法及其收敛性

针对模型和实际之间的差异,提出了一种基于时变线性二次型问题的动态系统优化和参数估计集成的算法该算法能逼近实际问题最优解,给出了该模型收敛的一个充分条件,分析了它的最优性,仿真例子说明了该算法的有效性和实用性。     

Continuous-Time Blind Channel Deconvolution Using Laguerre Shifts

The objective of this paper is to study the problem of continuous-time blind deconvolution of a pulse amplitude modulated signal propagated over an unknown channel and perturbed by additive noise. The main idea is to use so-called Laguerre filters to estimate a continuous-time model of the channel. Laguerre-filter-based models can be viewed as an extension of finite-impulse-response (FIR) models to the continuous-time case, and lead to compact and parsimonious linear-in-the-parameters models.? Given an estimate of the channel, different symbol estimation techniques are possible. Here, the shift property of Laguerre filters is used to derive a minimum mean square error estimator to recover the transmitted symbols. This is done in a way that closely resembles recent FIR-based schemes for the corresponding discrete-time case.? The advantage of this concept is that physical a priori information can be incorporated in the model structure, like the transmitter pulse shape. Date received: July 23, 1998. Date revised: August 26, 1999.     

基于径向基函数的图像修复技术

图像修复是指恢复图像中破损区域的颜色信息或者去除图像中的多余物体。本文提出了一种新的基于径向基函数的图像修复算法，由用户交互地指定需要修复的区域，算法自动地计算破损区域的轮廓并沿轮廓法向扩张，确定合适的径向基函数重构区域，将该区域内图像的颜色值看作规则采样点上的高度场，把二维图像修复问题转化为三维散乱点重建问题，利用径向基函数曲面重建的优势来修补破损的图像。实验表明，该算法能正确、稳定地处理各种破损区域。     

一种基于正交基神经网络算法的传感器误差补偿方法

为了提高传感器的误差补偿精度,提出了一种基于正交基神经网络算法的传感器误差补偿方法.研究了神经网络算法的收敛性,为学习率的选择提供了理论依据.为了验证算法的有效性,给出了传感器误差补偿实例.研究结果表明,基于正交基神经网络算法的传感器误差补偿方法具有高的误差补偿精度,因而是一种有效的误差补偿方法.     

Parallel Control for Continuous-Time Linear Systems: A Case Study

In this paper, a new parallel controller is developed for continuous-time linear systems. The main contribution of the method is to establish a new parallel control law, where both state and control are considered as the input. The structure of the parallel control is provided, and the relationship between the parallel control and traditional feedback controls is presented. Considering the situations that the systems are controllable and incompletely controllable, the properties of the parallel control law are analyzed. The parallel controller design algorithms are given under the conditions that the systems are controllable and incompletely controllable. Finally, numerical simulations are carried out to demonstrate the effectiveness and applicability of the present method.      

A Fast Display Method of Sky Colour Using Basis Functions

Computer graphics are being used for visual environmental assessment or architectural designs. Displaying the sky as a background is indispensable in generating photorealistic images for such applications. In this paper, we propose a fast display method of the sky colour by expressing the intensity distribution of the sky using basis functions, even if the sun position and/or the camera position are altered. In the proposed method, cosine functions are used as basis functions. The sun altitude is altered at certain intervals and the distributions of the sky colour for each sun altitude are precalculated and stored efficiently using basis functions. The colour of the sky in the view direction of an arbitrary sun position can be obtained from the stored distributions and displayed quickly. © 1997 by John Wiley & Sons, Ltd.     

结合密钥和随机标准正交基的音频伪装方案

传统基于Tangram的音频伪装方法所采用的变换模型为仿射变换模型,变换精度低且不满足基本的正交关系,从而无法保证秘密音频与公开音频之间的拟合精度,同时当分段变换音频为恒值序列时,需添加随机扰动以保证变换后音频的恢复质量,由此会降低信道传输音频的听觉质量。针对此问题,提出一种结合密钥和随机标准正交基的音频伪装方法。首先对秘密音频和公开音频分段,利用密钥构造随机标准正交基;其次通过秘密音频小段在随机标准正交基上的投影来对秘密音频小段进行表达,从中选取包括均值幅值较大的前k个投影系数,并记录对应的索引位置;再次通过EMD-q密写方法嵌入到对应的公开音频小段中形成信道公开传输音频;最后通过信道公开传输音频提取的变换参数结合密钥重构秘密音频。实验表明,所提方法可充分利用随机标准正交基重构不同精度的秘密音频,且随着选取的幅值系数增多,恢复的秘密音频质量也越来越好,同时所述策略严格依赖于密钥,只有掌握正确密钥的用户才能进行高精度的重构。     

Exponentially Accurate Approximations to Piece-Wise Smooth Periodic Functions

A family of simple, periodic basis functions with built-in discontinuities are introduced, and their properties are analyzed and discussed. Some of their potential usefulness is illustrated in conjunction with the Fourier series representation of functions with discontinuities. In particular, it is demonstrated how they can be used to construct a sequence of approximations which converges exponentially in the maximum norm to a piece-wise smooth function. The theory is illustrated with several examples and the results are discussed in the context of other sequences of functions which can be used to approximate discontinuous functions.     

Orthonormal basis functions for memory predistorter in oversampled systems

Volterra model or memory polynomial model are commonly used to describe the nonlinearity with memory effects for power amplifier (PA) modeling as well as digital predistorter designs. Different monomial terms of the Volterra model or memory polynomial model are highly correlated, which become a challenge during the fixed‐point implementation of the coefficients estimation as the data matrix is ill‐conditioned. Previous works derived orthogonal basis functions to eliminate the correlation among different monomial terms. Conversely, models of the PAs or the digital predistorters work in the oversampled domain to capture the adjacent band and/or out of band emissions. The correlation among data samples, which was neglected in previous works, can also be a major issue to the numerical instability in the coefficients estimation. In this article, we propose a set of new orthonormal basis functions to eliminate the correlation among different monomial terms as well as the correlation among data samples. As the proposed orthonormal basis functions can be predetermined and implemented with look up tables, the fixed‐point implementation is feasible and online computational complexity is greatly reduced. Simulation and experimental results show that the proposed orthonormal basis functions outperform the conventional ones in terms of condition number reduction as well as spectral regrowth suppression. © 2014 Wiley Periodicals, Inc. Int J RF and Microwave CAE 25:202–212, 2015.     

Continuous‐time stable and unstable system modelling with orthonormal basis functions

In this paper, model sets for linear time‐invariant continuous‐time systems which are spanned by fixed‐pole orthonormal bases are investigated. The obtained model sets are shown to be complete in the Lebesque spaces L_p (1<p<∞) and in C, the space of complex‐valued functions that are continuous on the extended imaginary axis. The L_p norm error bounds for estimating systems in L_p by the partial sums of the Fourier series formed by the orthonormal functions are computed for the case 1<p<∞. Some inequalities on the 𝓁_p means of the Fourier coefficients are also derived. These results have application in estimation and model reduction of stable and unstable continuous‐time linear time‐invariant systems. A numerical example illustrates the use of the basis functions for the approximation of unstable infinite‐dimensional dynamics. Copyright © 2000 John Wiley & Sons, Ltd.     

Large Scattered Data Fitting Based on Radial Basis Functions

Solving large radial basis function （RBF） interpolation problem with non-customized methods is computationally expensive and the matrices that occur are typically badly conditioned. In order to avoid these difficulties, we present a fitting based on radial basis functions satisfying side conditions by least squares, although compared with interpolation the method loses some accuracy, it reduces the computational cost largely. Since the fitting accuracy and the non-singularity of coefficient matrix in normal equation are relevant to the uniformity of chosen centers of the fitted RBE we present a choice method of uniform centers. Numerical results confirm the fitting efficiency.     

基于多项式基函数的泛函网络构造方法与逼近理论

用构造性方法证明:对于给定的r阶多项式函数,可以具体地构造出一个三层泛函网络,以任意精度逼近该多项式,所构造的网络的中问神经元个数仅与多项式基函数的阶数r有关,并能用r表达.该文所得结果对于基于多项式基函数的泛函网络逼近任意函数类的网络具体构造和逼近具有理论指导意义.     

On Convergence of Volterra Series Expansion of a Class of Nonlinear Systems

A fundamental issue in conducting the analysis and design of a nonlinear system via Volterra series theory is how to ensure the excitation magnitude and/or model parameters will be in the appropriate range such that the nonlinear system has a convergent Volterra series expansion. To this aim, parametric convergence bounds of Volterra series expansion of nonlinear systems described by a NARX model, which can reveal under what excitation magnitude or within what parameter range a given NARX system is able to have a convergent Volterra series expansion subject to any given input signal, are investigated systematically in this paper. The existing bound results often are given as a function of the maximum input magnitude, which could be suitable for single‐tone harmonic inputs but very conservative for complicated inputs (e.g. multi‐tone or arbitrary inputs). In this study, the output response of nonlinear systems is expressed in a closed form, which is not only determined by the input magnitude but also related to the input energy or waveform. These new techniques result in more accurate bound criteria, which are not only functions of model parameters and the maximum input magnitude but also consider a factor reflecting the overall input energy or wave form. This is significant to practical applications, since the same nonlinear system could exhibit chaotic behavior subject to a simple single‐tone input but might not with respect to other different input signals (e.g. multi‐tone inputs) of the same input magnitude. The results provide useful guidance for the application of Volterra series‐based theory and methods from an engineering point of view. The Duffing equation is used as a benchmark example to show the effectiveness of the results.     

基于子域基函数的任意频率选择表面数值分析

频率选择表面的电磁散射特性问题是飞行器隐身技术中的重要研究内容.采用了谱域法和子域屋顶基函数相结合的方法对任意单元形状的周期结构进行分析.首先将单元划分成若干个小网格,每个网格上的电流密度用屋顶基函数表征,然后通过电流密度和散射场的关系以及边界条件,得到导体面上的感应电流,从而分析结构的反射和传输特性.同时给出了贴片型、环形和孔径型的仿真算例,分析了单元上的电流分布规律,并与仿真的结果进行比较分析,验证了方法的有效性和通用性.     

DP 基关于L2 内积的对偶函数及其应用

给出DP 基关于L2 内积对偶函数的显示表示公式。首先介绍了计算机辅助几何设 计中多项式对偶函数的一般理论;然后根据两组多项式基的转换公式,给出了相应对偶函数的 转换公式。随后利用此转换公式和Bernstein 基对偶函数的表示公式,推导出DP 基关于L2 内积 对偶函数的显示表示。讨论了对偶函数在计算机辅助几何设计中的应用。     

Building Adaptive Basis Functions with a Continuous Self-OrganizingMap

This paper introduces CSOM, a continuous version of the Self-Organizing Map(SOM). The CSOM network generates maps similar to those created with theoriginal SOM algorithm but, due to the continuous nature of the mapping,CSOM outperforms the SOM on function approximation tasks. CSOM integratesself-organization and smooth prediction into a single process. This is adeparture from previous work that required two training phases, one toself-organize a map using the SOM algorithm, and another to learn a smoothapproximation of a function. System performance is illustrated with threeexamples.     

连续时间时变参数系统的指数稳定性和Lp指数稳定性

本文给出具有慢时变参数的连续时间线性齐次系统指数稳定的一个一般性条件，并给出具有随机时变参数的连续时间线性齐次系统Ｌｐ指数稳定的一个充分条件。这些结果可用来分析带有未建模动态，确定性噪声或随机噪声及慢时变参数的自适应控制系统的稳定性问题。     

基于径向基函数与B样条的散乱数据拟合方法

针对散乱数据的曲面拟合问题,提出一种径向基函数与B样条插值结合使用的曲面拟合方法.通过分片径向基函数插值,三维散乱点,再从分片插值曲面上获取预先设定好的有序网格点的值,最后利用张量积B样条插值有序网格点,从而得到拟合曲面.该方法较好地解决散乱数据插值和拟合的计算不稳定性问题,最后给出算法实例.     

Asymptotically optimal orthonormal basis functions for LPV system identification

A global model structure is developed for parametrization and identification of a general class of Linear Parameter-Varying (LPV) systems. By using a fixed orthonormal basis function (OBF) structure, a linearly parametrized model structure follows for which the coefficients are dependent on a scheduling signal. An optimal set of OBFs for this model structure is selected on the basis of local linear dynamic properties of the LPV system (system poles) that occur for different constant scheduling signals. The selected OBF set guarantees in an asymptotic sense the least worst-case modeling error for any local model of the LPV system. Through the fusion of the Kolmogorov n-width theory and Fuzzy c-Means clustering, an approach is developed to solve the OBF-selection problem for discrete-time LPV systems, based on the clustering of observed sample system poles.