Walsh Spectral Estimates with Applications to the Classification of EEG Signals

A basis for the processing of EEG signals using the discrete, orthogonal set of Walsh functions is presented. The Walsh power spectrum is examined from the point of view of its statistical properties, especially as it relates to spectral resolution. Features, selected from the spectrum of sleep EEG data are compared to corresponding Fourier features. Each feature set is used to classify the data using a minimum-distance clustering algorithm. The results show that the Walsh spectral features classify the data in much the same way as the Fourier spectral features. This provides sufficient justification for usage ofWalsh spectral features in place of Fourier spectral features, enabling one to take advantage of the vast computational superiority of the fast Walsh transform over the fast Fourier transform.     

A generalized concept of frequency and some applications

The system of sine and cosine functions has been distinguished historically in communications. Whenever the term frequency is used, reference is made implicitly to these functions; hence the generally used theory of communication is based on the system of sine and cosine functions. In recent years other complete systems of orthogonal functions have been used for theoretical investigations as well as for equipment design. Analogs to Fourier series, Fourier transform, frequency, power spectra, and amplitude, phase, and frequency modulation exist for many systems of orthogonal functions. This implies that theories of communication can be worked out on the basis of these systems. Most of these theories are of academic interest only. However, for the complete system of the orthogonal Walsh functions, the implementation of circuits by modem semiconductor techniques appears to be competitive in a number of applications with the implementation of circuits for the system of sine and cosine functions.     

Comparison of Walsh and Fourier spectroscopy of geomagneticreversals and nonsinusoidal palaeoclimatic time series

Higher resolving capabilities and theoretical appropriateness of Walsh spectral techniques as compared to Fourier spectral analyses are presented for synthetic and nonsinusoidal geotime series. Theoretical developments of Walsh transform techniques and a comparative study of Walsh and Fourier spectral estimates are presented. The Walsh spectral technique is applied specifically to two actual time series data of geomagnetic reversals in binary telegraphic wave form and nonsinusoidal palaeomagnetic and palaeoclimate time series. Walsh spectra reveal periodicities in Milankovitch frequency bands and provide exceptionally well-resolved spectral lines. The possible physical significance of these orbital periodicities is discussed. A comparative example of autocorrelation analysis in the real time domain and dyadic time domain is also presented using a telegraphic signal model of actual geomagnetic reversal time series. and the result is briefly discussed. The computational efficiency of the Walsh function could be exploited further for many other binary and nonsinusoidal geophysical/geological time series     

Walsh-to-Fourier Spectral Conversion for Periodic Waves

Relations are developed for the determination of the Fourier spectra of frequency-limited periodic waves from truncated Walsh spectra. The matrix conversion process is simplest if the highest-order Walsh coefficient in the spectru to be converted is 2n, where n is an integer. For such cases, compensation for truncation consists of a diagonal matrix that premultiplies the Walsh to Fourier conversion matrix and the elements of which are [(sinx)/x]-2 terms. Element values range between unity and less than ?2/4. The same compensation matrix is used for determning the Walsh spectra. of sequency-limited waves from 2n Fourier expansion terms. Examples are included which demonstrate the spectral conversion processes, Walsh to Fourier and Fourier to Walsh.     

Average Walsh Power Spectrum for Periodic Signals

A circular shift-invariant Walsh power spectrum for deterministic periodic sequences is defined. For a sequence with period N, the power spectrum is the average of the Walsh power spectra of all N possible distinct circular shifts. The Average Walsh Power Spectrum (AWPS) consists of (N/2) + 1 coefficients, each representing a distinct sequency. A fast transformation from the arithmetic autocorrelation function of a periodic sequence to its AWPS is presented.     

Some Comments Concerning Walsh Functions

The use of explicit forms for Walsh functions removes much of the confusion surrounding these interesting functions and permits simple proofs of their properties. Thus, for example, their period is far easier to determine than Alexandridis found, but their Fourier spectra are more complex than Schreiber's approximation suggests. For wave-form analysis, certain Walsh functions--the regular symmetric square waves--are more useful than are the others.     

Fourier spectrum extrapolation and enhancement using supportconstraints

The use of support constraints for improving the quality of Fourier spectra estimates is discussed. It is shown that superresolution is an additive phenomenon that is a function of the correlation scale induced by the support constraint, and it is independent of the bandwidth of the measured Fourier spectrum. It is also shown for power spectra that support constraints, due to the enforced correlation of power spectra, reduce the variance of measured power spectra. These theoretical results are validated via computer simulation in the area of speckle interferometry, with very good agreement shown between theory and simulation     

Walsh Versus Fourier Estimators of the EEG Power Spectrum

Walsh-based discrete algorithms offer a fast, simple alternative to Fourier methods for estimating EEG power spectra, but their performance has been criticized. Previous comparisons of Fourier and Walsh estimators are reexamined, and they are compared experimentally, on simulated EEG. The evidence suggests that they should perform comparably for EEG monitoring.     

Walsh Orthogonal Functions in Geometrical Feature Extraction

Walsh functions are used in designinq a feature extraction algorithm. The ?axis-symmetry? property of the Walsh functions is used to decompose geometrical patterns. An axissymmetry (a.s.)-histogram is obtained from the Walsh spectrum of a pattern by adding the squares of the spectrm coefficients that correspond to a given a.s.-number ? and plotting these against ?. Since Walsh transformation is not positionally invariant, the sequency spectrum does not specify the pattern uniquely. This disadvantage is overcome by performing a normalization on the input pattern through Fourier transformation. The a.s.-histogram is obtained from the Walsh spectrum coefficients of the Fourier-normalized rather than the original pattern. Such histogram contains implicit information about symmetries, periodicities, and discontinuities present in a figure. It is shown that a.s.-histograms result in great dimensionality reduction in the feature space, which leads to a computationally simpler classification task, and that patterns which differ only in translations or 90° rotation have equal a.s.-histograms.     

海杂波AR谱多重分形特性及微弱目标检测方法

该文研究了海杂波功率谱的多重分形特性。为了克服频谱傅里叶分析的缺点,用现代谱估计的方法来计算海杂波的功率谱。AR模型是一个线性预测模型,它通过序列的自相关函数矩阵来估计功率谱,并且具有更精确的频谱分辨率。该文主要分析基于AR谱估计的海杂波功率谱的多重分形特性,以及在微弱目标检测中的应用。首先,以分数布朗运动(FBM)模型为例,证明其功率谱具有多重分形特性。其次,根据X波段雷达的实测海杂波数据,通过多重去趋势分析法(MF-DFA)验证了海杂波AR谱的多重分形特性。最后,分析了海杂波AR谱的广义Hurst指数以及影响参数,并提出一种基于局部AR谱广义Hurst指数的目标检测方法。实验结果表明,该种检测方法具有海杂波背景下微弱目标检测的能力。与现有的分形检测方法和传统的CFAR检测方法对比,该算法在低信杂比情况下具有较好的检测性能。     

Ternary Walsh Transform and Its Operations for Completely and Incompletely Specified Boolean Functions

The new ternary Walsh transform is considered in this paper. Such a ternary Walsh transform can be used in a similar manner as the standard Walsh transform for binary logic functions as shown here. It is based on the Kronecker product as well as the Galois field and new ternary operations. The same hardware implementation can be used for both forward and inverse ternary Walsh transforms based on its fast algorithms and properties. The ternary Walsh transform is suitable for processing both completely and incompletely specified Boolean functions. Its properties for the decomposition and symmetry detections of the Boolean functions are shown.     

客观测定胶片分辨率的实验研究

提出了用Fourier功率谱法评价胶片分辨率值的理论判据.这是一种用光学信息处理原理对胶片分辨率图案的Fourier谱进行分析处理的客观评价方法.用胶片分辨率图案的Fourier变换频谱的最外次极点作为客观评价胶片分辨率的尺度,并找出了胶片分辨率与其Fourier变换频谱的最外次极点有近似线性关系.     

Sinusoids versus Walsh functions

In most of the applications contemplated for Walsh functions these binary waveforms would replace the more usual sinusoids, as the fast-Walsh-transform algorithm appears to make them very attractive for many kinds of signal processing. This paper begins with a brief review of the characteristics of Walsh functions and of their applications. Some old and some new interrelations are presented between sinusoids and Walsh functions, but the principal aim of the paper is to investigate the truncation and roundoff errors associated with the use of Fourier and of Walsh series. By employing simplifying approximations it is found that, for long samples of smooth signals, far more terms are required in the Walsh-series representation and greater accuracy is required of their coefficients for a given rms total error. Even for discontinuous signals the Walsh series may require substantially more terms, thus counterbalancing the computational advantage of the fast Walsh transform. This relative inefficiency of the Walsh-series representation of long waveforms may explain why it has not proven particularly effective in applications.     

Cubic Boolean functions with highest resiliency

We classify those cubic m-variable Boolean functions which are (m-4)-resilient. We prove that there are four types of such functions, depending on the structure of the support of their Walsh spectra. We are able to determine, for each type, the Walsh spectrum and, then, the nonlinearity of the corresponding functions. We also give the dimension of their linear space. This dimension equals m-k where k=3 for the first type, k=4 for the second type, k=5 for the third type, and 5/spl les/k/spl les/9 for the fourth type.     

Order and parameter identification in continuous linear systems via Walsh functions

This letter presents a method, via Walsh functions, for simultaneous identification of the order and parameters of a single-input-single-output linear lumped continuous dynamic system based on input-output data from an arbitrary but active record. Walsh spectra are impressively immune to zero-mean additive noise to some extent, and so are the identification algorithms employing them.     

Discrete Generalized Fresnel Functions and Transforms in an Arbitrary Discrete Basis

The idea of generalized Fresnel functions, which traces back to expressing a discrete transform as a linear convolution, is developed in this paper. The generalized discrete Fresnel functions and the generalized discrete Fresnel transforms for an arbitrary basis are considered. This problem is studied using a general algebraic approach to signal processing in an arbitrary basis. The generalized Fresnel functions for the discrete Fourier transform (DFT) are found, and it is shown that DFT of even order has two generalized Fresnel functions, while DFT of odd order has a single generalized Fresnel function. The generalized Fresnel functions for the conjunctive and Walsh transforms and the generalized Fresnel transforms induced by these functions are considered. It is shown that the generalized Fresnel transforms induced by the Walsh basis and the corresponding generalized Fresnel functions are unitary and that the generalized Fresnel transforms induced by the conjunctive basis and the corresponding generalized Fresnel functions consist of powers of the golden ratio. It is also shown that the Fresnel transforms induced by the generalized Fresnel functions for the Walsh and conjunctive transforms have fast algorithms.     

Variances of spectral parameters with a Gaussian shape (Corresp.)

In the paper [1] by the late M. J. Levin, maximum likelihood estimates and limits on their variances are derived for the parameters of the power spectrum of a zero-mean stationary Gaussian process. In this note we rederive the variances of these estimates in a somewhat different manner and give numerical results for power spectra with a Gaussian shape.     

多输出Plateaued函数的密码学性质

该文对多输出Plateaued函数的一些密码学性质进行了研究,以多输出函数的特征函数为工具,建立了多输出Plateaued函数的差分转移概率与其Walsh谱及阶数之间的关系。给出了多输出Plateaued函数的Walsh谱值在一定条件下的分布情形,指出多输出Plateaued函数的在其输出分量函数的任意非零线性组合函数均为非平衡函数时,其输入变量个数、输出变量个数与其阶数之间的关系满足。     

两类Plateaued函数的Walsh谱值分布

Based on the properties of trace functions and quadratic forms, this paper presents value distributions of Walsh spectrum of the Plateaued functions of the form Tr(R(x)) with n=3r or 4r variables, where r > 1 is an odd integer. Our results can be used to determine the numbers of non-zero Walsh spectrum values and the nonlinearities of these functions, and estimate their resiliency orders. Especially, the value distributions can be used to deduce the tight lower bounds of the second order nonlinearity of two classes of Boolean functions. It is demonstrated that our bounds are better than the previously obtained bounds.     

Short-time spectral and autocorrelation analysis in the Walsh domain

A short-time dyadic autocorrelation function (dacf) and a short-time Walsh energy spectrum of the first kind are defined in the Walsh-Fourier domain. The "natural" choice of the short-time functions does not lead to a Walsh-Fourier transform pair (dyadic Wiener-Khintchine theorem), and thus a second kind of short-time dacf and short-time Walsh energy spectrum are defined as the Walsh-Fourier transforms of the first kind. This leads to a meaningful and convenient Walsh transform pair between the first short-time Walsh energy spectrum and the second short-time dacf. The measurement procedures for both kinds of functions are discussed, and the mean values of these short-time functions are shown to be related to the corresponding long-time functions.