Hermite-Gaussian-like eigenvectors of the discrete Fourier transform matrix based on the singular-value decomposition of its orthogonal projection matrices

A technique is proposed for generating initial orthonormal eigenvectors of the discrete Fourier transform matrix F by the singular-value decomposition of its orthogonal projection matrices on its eigenspaces and efficiently computable expressions for those matrices are derived. In order to generate Hermite-Gaussian-like orthonormal eigenvectors of F given the initial ones, a new method called the sequential orthogonal procrustes algorithm (SOPA) is presented based on the sequential generation of the columns of a unitary matrix rather than the batch evaluation of that matrix as in the OPA. It is proved that for any of the SOPA, the OPA, or the Gram-Schmidt algorithm (GSA) the output Hermite-Gaussian-like orthonormal eigenvectors are invariant under the change of the input initial orthonormal eigenvectors.     

经典规范正交子波的一种简单广义化方法及其应用

从最简单的Haar尺度函数入手，提出一种简单而又快捷的方法，将每一个经典的规范正交子波基进行拓展得到一类新的规范正交子波基。新子波类中的每一个子波均继承了原始子波的许多基本性质，比如规范正交性，正则阶，时、频局域化特性等，同时也得到某些性能的改善，文中重点探讨广义Haar子波、广义Shannon子波和Meyer子波、Daubechies子波等的简单广义化；最后讨论新子波系统的一个直接应用：实（序列）信号解析子波变换的快速算法问题。     

Broadband macromodelling of passive components using orthonormal vector fitting

Vector fitting is widely accepted as a robust macromodelling tool for efficient frequency domain identification of passive components. The orthonormal vector fitting technique is introduced, which improves the numerical stability of the method, by using orthonormal rational functions. This leads to better conditioned equations, reduces the numerical sensitivity to the choice of starting poles significantly, limits the number of required iterations, and reduces the overall computation time.     

光滑正交小波库的构造及在信号去噪中的应用

针对实际应用中小波的多样性要求，提出应用构造光滑正交小波库的方法来解决这个矛盾，光滑正交小波库是基于规则性、非冗余性、完备性进行构造。文中把构造好的小波库应用到信号去噪中，实验结果证明，在绝大多数情况下基于正交小波库的去噪方法性能优于任何一种单一的小波。     

用余弦调制PR－FIR方法构造正交小波基

离散小波变换将离散时间信号分解为一系列不同分辨率下的离散近似信号和离散细节，紧支的正交规范小波与完全重构正交镜象滤波器（ＰＲ－ＱＭＦ）相对应。本文在“二带”正交小波基的构造条件下，利用余弦调制完全重构滤波器组的方法，实现了正交小波基的构造，计算模拟表明该方法非常简单、有效。     

Time-invariant orthonormal wavelet representations

A simple construction of an orthonormal basis starting with a so-called mother wavelet, together with an efficient implementation gained the wavelet decomposition easy acceptance and generated a great research interest in its applications. An orthonormal basis may not, however, always be a suitable representation of a signal, particularly when time (or space) invariance is a required property. The conventional way around this problem is to use a redundant decomposition. We address the time-invariance problem for orthonormal wavelet transforms and propose an extension to wavelet packet decompositions. We show that it,is possible to achieve time invariance and preserve the orthonormality. We subsequently propose an efficient approach to obtain such a decomposition. We demonstrate the importance of our method by considering some application examples in signal reconstruction and time delay estimation     

Wavelets and recursive filter banks

It is shown that infinite impulse response (IIR) filters lead to more general wavelets of infinite support than finite impulse response (FIR) filters. A complete constructive method that yields all orthogonal two channel filter banks, where the filters have rational transfer functions, is given, and it is shown how these can be used to generate orthonormal wavelet bases. A family of orthonormal wavelets that have a maximum number of disappearing moments is shown to be generated by the halfband Butterworth filters. When there is an odd number of zeros at π it is shown that closed forms for the filters are available without need for factorization. A still larger class of orthonormal wavelet bases having the same moment properties and containing the Daubechies and Butterworth filters as the limiting cases is presented. It is shown that it is possible to have both linear phase and orthogonality in the infinite impulse response case, and a constructive method is given. It is also shown how compactly supported bases may be orthogonalized, and bases for the spline function spaces are constructed     

An effective wavelet matrix transform approach for efficientsolutions of electromagnetic integral equations

An effective numerical method based on wavelet matrix transforms for efficient solution of electromagnetic (EM) integral equations is proposed. Using the wavelet matrix transform produces highly sparse moment matrices which can be solved efficiently. A fast construction method for various orthonormal or nonorthonormal wavelet basis matrices is also given. It has been found that using nonsimilarity wavelet matrix transforms such as nonsimilarity nonorthonormal cardinal spline wavelet (NSNCSW) transform, one can obtain a much higher compression rate and much better accuracy of the approximate solutions than using similarity wavelet transforms such as Daubechies' (1992) orthonormal wavelet (DOW) transform. Numerical examples are given to show the validity and effectiveness of the method     

SVD-updating using orthonormal μ-rotations

In this paper the implementation of the SVD-updating algorithm using orthonormal μ-rotations is presented. An orthonormal μ-rotation is a rotation by an angle of a given set of μ-rotation angles (e.g., the angles Φ_i = arctan2^-i) which are choosen such that the rotation can be implemented by a small amount of shift-add operations. A version of the SVD-updating algorithm is used where all computations are entirely based on the evaluation and application of orthonormal rotations. Therefore, in this form the SVD-updating algorithm is amenable to an implementation using orthonormal μ-rotations, i.e., each rotation executed in the SVD-updating algorithm will be approximated by orthonormal μ-rotations. For all the approximations the same accuracy is used, i.e., onlyr?w (w: wordlength) orthonormal μ-rotations are used to approximate the exact rotation. The rotation evaluation can also be performed by the execution of μ-rotations such that the complete SVD-updating algorithm can be expressed in terms of orthonormal μ-rotations. Simulations show the efficiency of the SVD-updating algorithm based on orthonormal μ-rotations.     

On orthonormal wavelets and paraunitary filter banks

The known result that a binary-tree-structured filter bank with the same paraunitary polyphase matrix on all levels generates an orthonormal basis is generalized to binary trees having different paraunitary matrices on each level. A converse result that every orthonormal wavelet basis can be generated by a tree-structured filter bank having paraunitary polyphase matrices is then proved. The concept of orthonormal bases is extended to generalized (nonbinary) tree structures, and it is seen that a close relationship exists between orthonormality and paraunitariness. It is proved that a generalized tree structure with paraunitary polyphase matrices produces an orthonormal basis. Since not all phases can be generated by tree-structured filter banks, it is proved that if an orthonormal basis can be generated using a tree structure, it can be generated specifically by a paraunitary tree     

Orthonormal (Fourier and Walsh) models of time-varying evokedpotentials in neurological injury

The hypothesis that injury-related changes in evoked potential (EP) signals can be modeled by orthonormal basis functions is tested. Two models of time-varying EP signals are evaluated: the Fourier series model (FSM) and the Walsh function model (WFM). The Fourier and Walsh coefficients are estimated with the aid of an adaptive least-mean-squares (LMS) technique. Results from computer simulations illustrate how selection of model order and of the adaptation rate of the estimator affect the signal-to-noise ratio (SNR). The FSM results in a somewhat higher steady-state SNR than does the WFM; however, the WFM is less computationally complex than is the FSM. These two orthonormal functions are used to evaluate transient response to hypoxic hypoxia in anesthetized cats. Trends of the first five frequencies (Fourier) and sequencies (Walsh) show that the lower frequencies and sequencies may be sensitive indicators of hypoxic neurological injury     

Two-Mode Waveguide for Equal Mode Velocities: Correction (Letters)

The T-septum waveguide was analyzed by Elliott using the orthonormal block method. The numerical results did not compare favorably with experimental measurements and it was suggested that the disparity was related primarily to the assumption of zero-thickness membranes for the septum. Later, Silvester analyzed the T-septum waveguide using a finite-element method and found very good agreement with the measured points, yet the septum thickness was again assumed to be infinitesimal. This letter is being written to dispel the implication that the orthonormal block method of analysis of the T-septum waveguide suffers for lack of accuracy. The universal curves as shown by Elliott will be presented here in corrected form along with experimental results further corroborating both Elliott's and Silvester's work.     

用方形区域内的标准正交多项式重构波前

提供了一种方形区域上归一化Zernike正交基的生成方法。它采用线性无关组Gram-Schimdt正交组构造方法,根据线性代数内积、欧氏空间及其正交性和范数的相关概念,对标准Zernike多项式进行正交处理,得到了一组新的正交多项式Z-square多项式。采用该正交基实现了方形区域内波前模式的拟合,它不仅可由Z-square模式的集合直接对波前进行表示,而且也可以通过线性反变换,将Z-square多项式表示成标准的Zernike模式的线性组合,使被分解的波前模式与像差之间有明确的对应关系。实验表明,它不仅可以对透镜设计中的波前像差函数进行有效的拟合,而且也能对Hartmann-Shack波前传感器测试得到的实际相位数据进行拟合。     

Extended RLS lattice adaptive filters

This paper solves the problem of developing exact fast weighted RLS lattice adaptive filters for input signals induced by general orthonormal filter models. The resulting algorithm can be viewed as a counterpart of the extended fast fixed-order RLS adaptive filters previously derived.     

Circuit diagnosis and Thevenin equivalents

Roytman and Swamy have used the properties of orthonormal excitations to establish one method of circuit diagnosis. In this letter the method is extended to determine the driving-point or Thevenin equivalent impedance for each node pair.     

Deconvolution by thresholding in mirror wavelet bases

The deconvolution of signals is studied with thresholding estimators that decompose signals in an orthonormal basis and threshold the resulting coefficients. A general criterion is established to choose the orthonormal basis in order to minimize the estimation risk. Wavelet bases are highly sub-optimal to restore signals and images blurred by a low-pass filter whose transfer function vanishes at high frequencies. A new orthonormal basis called mirror wavelet basis is constructed to minimize the risk for such deconvolutions. An application to the restoration of satellite images is shown.     

A tensor decomposition approach to data compression and approximation of ND systems

The method of Proper Orthogonal Decompositions (POD) is a data-based method that is suitable for the reduction of large-scale distributed systems. In this paper we propose a generalization of the POD method so as to take the ND nature of a distributed model into account. This results in a novel procedure for model reduction of systems with multiple independent variables. Data in multiple independent variables is associated with the mathematical structure of a tensor. We show how orthonormal decompositions of this tensor can be used to derive suitable projection spaces. These projection spaces prove useful for determining reduced order models by performing Galerkin projections on equation residuals. We demonstrate how prior knowledge about the structure of the model reduction problem can be used to improve the quality of approximations. The tensor decomposition techniques are demonstrated on an application in data compression. The proposed model reduction procedure is illustrated on a heat diffusion problem.     

Simplified method for the construction of an orthonormal base forCPFSK signals

The authors present a simplified method to derive a 2M dimensional orthonormal base for M-ary CPFSK signals based on the well-known Gram-Schmidt procedure. On the basis of this method, the simulated bit error performance of 4, 8 and 16-ary CPFSK modulation in additive white Gaussian noise channels is presented     

Optimal free parameters in orthonormal approximations

We consider orthonormal expansions where the basis functions are governed by some free parameters. If the basis functions adhere to a certain differential or difference equation, then an expression can be given for a specific enforced convergence rate criterion as well as an upper bound for the quadratic truncation error. This expression is a function of the free parameters and some simple signal measurements. Restrictions on the differential or difference equation that make this possible are given. Minimization of either the upper bound or the enforced convergence criterion as a function of the free parameters yields the same optimal parameters, which are of a simple form. This method is applied to several continuous-time and discrete-time orthonormal expansions that are all related to classical orthogonal polynomials     

Irregular Segmented Region Compression Coding Based on Pulse Coupled Neural Network

An irregular segmented region coding algorithm based on pulse coupled neural network（PCNN） is presented. PCNN has the property of pulse-coupled and changeable threshold, through which these adjacent pixels with approximate gray values can be activated simultaneously. One can draw a conclusion that PCNN has the advantage of realizing the regional segmentation, and the details of original image can be achieved by the parameter adjustment of segmented images, and at the same time, the trivial segmented regions can be avoided. For the better approximation of irregular segmented regions, the Gram-Schmidt method, by which a group of orthonormal basis functions is constructed from a group of linear independent initial base functions, is adopted. Because of the orthonormal reconstructing method, the quality of reconstructed image can be greatly improved and the progressive image transmission will also be possible.