A Nonrecursive Equation for the Fourier Transform of a Walsh Function

Convolution of a discrete Walsh function with a rectangular pulse simplifies the derivation of an expression for the Fourier transform of a Walsh function. The nonrecursive transform equation that is developed is a function of the bits of the Gray code number for the order of the Walsh function.     

A comparison between two methods for determining the tracking endurance of organic materials

An experimental investigation of the correlation between two accelerated tests (the Tracking Endurance Wheel and the IEC Publication 112) for determining the tracking endurance of organic materials is reported. It is shown that the qualitative classification of materials in both methods in consistent.     

A comparison of algorithms for detection of spikes in the electroencephalogram

Identification of the short transient waveform, called a spike, in the cortical electroencephalogram (EEG) plays an important role during diagnosis of neurological disorders such as epilepsy. It has been suggested that artificial neural networks (ANN) can be employed for spike detection in the EEG, if suitable features are provided as input to an ANN. In this paper, we explore the performance of neural network-based classifiers using features selected by algorithms suggested by four previous investigators. Of these, three algorithms model the spike by mathematical parameters and use them as features for classification while the fourth algorithm uses raw EEG to train the classifier. The objective of this paper is to examine if there is any inherent advantage to any particular set of features, subject to the condition that the same data are used for all feature selection algorithms. Our results suggest that artificial neural networks trained with features selected using any one of the above three algorithms as well as raw EEG directly fed to the ANN will yield similar results.     

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.     

Spectral Decomposition of EEG Intervals Using Walsh and Fourier Transforms

This paper presents a comparison of the use of features derived from Walsh and Fourier transforms for classification of short segments of EEG data. The result was that features obtained from Fourier transforms produced consistently better classification results than Walsh function-derived features.     

A system for tracking changes in the mid-latency evoked potentialduring anesthesia

Describes a method to measure changes in the mid-latency auditory evoked potential (MLAEP) during anesthesia. It is claimed that the position of the Nb-trough of the MLAEP indicates the level of consciousness. The component shows graded changes corresponding to the dose of anesthetic and it exhibits stable reproducible properties between different subjects. The authors propose a system that reduces the disturbances in an averaged MLAEP using fewer realizations than in the standard averaging procedure. The resulting smoothing error is reduced if the number of stimulus is decreased. Unfortunately, the variance of the waveform estimate is, thereby, increased. An improved method must be used in order to estimate the Nb-trough within a prescribed time interval of one minute. The procedure is based on inherent properties of the MLAEP and the noise. A simulation and examples of the performance on real data recorded during surgery are shown     

A Note of the Relation between Reed-Muller Expansions and Walsh Transforms

In this note we will show that there is a direct relation between the Reed-Muller expansions and orthogonal expansions for Boolean functions which, from the mathematical point of view, rest upon completely different bases.     

加窗离散傅里叶变换性能分析和比对

离散傅里叶变换（DFT）是数字信号处理的常用方法,但通过加窗对信号进行时域截短会使频谱分析出现误差。不同窗函数对信号频谱产生的影响不同,对5种常用窗函数的特性进行了比较和适用性分析,通过仿真实验证明了针对不同信号特点和处理要求,应选择合适的窗函数,以减小频谱泄漏和栅栏效应对信号分析的影响。     

The use of fuzzy integrals and bispectral analysis of the electroencephalogram to predict movement under anesthesia

The objective of this study was to design and evaluate a methodology for estimating the depth of anesthesia in a canine model that integrates electroencephalogram (EEG)-derived autoregressive (AR) parameters, hemodynamic parameters, and the alveolar anesthetic concentration. Using a parameters, and the alveolar anesthetic concentration. Using a parametric approach, two separate AR models of order ten were derived for the EEG, one from the third-order cumulant sequence and the other from the autocorrelation lags of the EEG. Since the anesthetic dose versus depth of anesthesia curve is highly nonlinear, a neural network (NN) was chosen as the basic estimator and a multiple NN approach was conceived which took hemodynamic parameters, EEG derived parameters, and anesthetic concentration as input feature vectors. Since the estimation of the depth of anesthesia involves cognitive as well as statistical uncertainties, a fuzzy integral was used to integrate the individual estimates of the various networks and to arrive at the final estimate of the depth of anesthesia. Data from 11 experiments were used to train the NN's which were then tested on nine other experiments. The fuzzy integral of the individual NN estimates (when tested on 43 feature vectors from seven of the nine test experiments) classified 40 (93%) of them correctly, offering a substantial improvement over the individual NN estimates.     

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     

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     

A comparison between techniques for global surface interpolation inshaped reflector analysis

Usual geometrical optics based syntheses of shaped reflectors provide numerically defined surfaces which must be interpolated for subsequent evaluation of the antenna secondary field to check against design specifications. The implementation of three techniques capable of performing global interpolations of numerically defined (3D) surfaces, namely, the quintic pseudosplines, polynomial Fourier series, and Jacobi polynomial-sinusoidal expansions, is explored. The relative superior stability of the first is demonstrated for the synthesis of shaped reflectors presenting critical behavior of their principal curvatures     

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.     

A comparison of the Hartley, Cas-Cas, Fourier, and discrete cosinetransforms for image coding

The coding method used for the comparison utilizes a marginal bit-allocation scheme and Lloyd-Max quantizer. Gamma, Laplacian, and Gaussian models are compared for the distribution of the transform coefficients. The discrete cosine transform outperforms the other three transforms based on the mean-square quantization error     

A novel view of Fourier analysis

A different view of a familiar subject often provides insight that leads to innovations and new discoveries. A novel view of the Fourier transform, Fourier series, and discrete Fourier transform is presented in this letter.     

Analytical comparison between time- and frequency-domain techniques for phase-noise analysis

In the literature, different techniques have been presented for the phase-noise analysis of free-running oscillator circuits. In order to give some insight into the relationships existing between them, an analytical comparison is carried out in this paper between three different approaches. Two of them are time-domain approaches, based on Floquet's theory and the impulse sensitivity function, respectively, and the third one is the carrier modulation approach, in frequency domain. The application of Floquet's theory enables the calculation of periodic sensitivity functions to the noise perturbations. Here, the possibility to determine these functions through harmonic balance is demonstrated. This allows applying the whole stochastic characterization of phase noise, obtained from time-domain analysis, to circuits simulated through harmonic balance. For illustration, calculations in a cubic-nonlinearity oscillator are presented.     

A comparison of the time involved in computing fast Hartley and fast Fourier transforms

It is shown that the DFT of a real sequence, formed via the Fast Hartley Transform, can be computed at most only 2 times faster than by using a complex Fast Fourier Transform. However, more sophisticated FFT algorithms exist which give the same speedup factor. A simple FHT subroutine is presented to illustrate the similarity of the FHT and FFT butterflies in their simplest forms.     

A quantitative analysis of SNR in the short-time Fourier transformdomain for multicomponent signals

A quantitative analysis is given for the signal-to-noise ratio (SNR) in the short-time Fourier transform domain for multicomponent signals in additive white noise. It is shown that the SNR is increased on the order of O(N/K), where K is the number of components of a signal, N/T is the sampling rate, and T is the window size. The SNR increase rate is optimal for given K. For this result, the SNR definition is generalized, which is suitable for signals not only in the time domain but also in other domains. This theory is illustrated by one numerical example     

A Fourier analysis for a fast simulation algorithm [for powerconvertors]

A powerful discrete modeling approach to the simulation of a switching converter that has appeared in power electronics literature over the last several years is briefly reviewed, and some desirable traits for its matrix exponential subroutine are discussed. The key result is a systematic and generic Fourier analysis that operates on the steady-state solution as provided by a discrete model. The Fourier analysis algorithm was tested on a phase-controlled parallel-loaded resonant converter which is useful as a fixed-frequency low-distortion AC source. Three verifications of the algorithm were made: a phasor analysis, an alternate closed-form analysis directed specifically to the problem at hand, and an analysis using experimental data. The performance of the algorithm was good for the case considered