Applications of walsh and related functions. With an introduction to sequence theory

Beauchamp, K. G.

doi:10.1016/0165-1684(85)90044-1

Cited by 101 publications

(112 citation statements)

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“…Walsh power spectra are based on the use of square waves instead of sine and cosine waves. The equivalent of a Fourier Transform can be generated using the Walsh transform (Beauchamp, 1975;Weedon, 1989). The BlackmanTukey method is based on the Fourier transform as part of the autocovariance sequence.…”

Section: De Vleeschouwer Et Al: Precessional and Half-precessionamentioning

confidence: 99%

Precessional and half-precessional climate forcing of Mid-Devonian monsoon-like dynamics

Vleeschouwer

Silva²,

Boulvain³

et al. 2012

Clim. Past

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Abstract.A Devonian magnetic susceptibility (MS) record obtained on limestones ranging from the Uppermost-Eifelian to the Lower-Givetian and located on the southern border of the Dinant Synclinorium in Belgium was selected for timeseries analysis. In these carbonate ramp and platform deposits, spectral analyses highlight persistent, high-frequency cycles in both the MS-signal and the microfacies curve, reflecting environmental and climate changes. These metrescale variations in the MS-signal are interpreted as changes in the flux of magnetic minerals towards the marine system, most likely controlled by monsoon rainfall-intensity. By combining chrono-and biostratigraphic information with theoretical knowledge of sedimentation rates in different depositional environments, these cycles are interpreted as astronomically driven and dominated by precession. It is hypothesized that during precession minima (longitude of the perihelion ≈270 • ) the trans-equatorial pressure gradient reaches a maximum and intensifies the northeasterly monsoonal circulation. The consequent increase in moisture transport towards the continent leads to enhanced precipitation and runoff, which, in turn, induces a higher flux of detrital material -including magnetic minerals responsible for the MS-signal -towards the marine system. Moreover, this unique high-resolution climate signal reveals halfprecessional cycles. These cycles suggest that during precession maxima (longitude of the perihelion ≈90 • ) southeasterly monsoonal circulation strengthened due to an increased inter-hemispheric pressure gradient.

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Section: De Vleeschouwer Et Al: Precessional and Half-precessionamentioning

confidence: 99%

Precessional and half-precessional climate forcing of Mid-Devonian monsoon-like dynamics

Vleeschouwer

Silva²,

Boulvain³

et al. 2012

Clim. Past

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“…Note that the raised parts of these functions are not all of equal length; for this reason, it is not accurate to speak of the frequency of a Walsh function. The term sequency is used instead, either to denote the number of zero crossings that the function makes in the course of a unit of time, or to denote the number of raised portions or lumps in the function in a unit of time (Beauchamp, 1984). In this paper, sequency will be used to refer to the number of raised portions.…”

Section: The Walsh Seriesmentioning

confidence: 99%

“…The Walsh spectrum does not in- performance of the two analyses is similar. The characteristics that are most relevant in choosing an analysis method are (1) the number of different values taken, relative to the number of time bins (75 different data values could be considered a small number if the data series were 1,000 points long), and (2) the number of plateaus, or runs of any single value, in the data set (Beauchamp, 1984). As the sequency or frequency of the periodicity increases, Walsh and Fourier transforms converge in how well they represent the data.…”

Section: Transformation Of a Summed Composite Rectangular Wavementioning

confidence: 99%

Analysis of periodic data using Walsh functions

Broadbent

Maksik

1992

Behavior Research Methods, Instruments, & Computers

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(Student Award Winning Paper for 1991) Data often contain periodic components plus random variability. Walsh analysis reveals periodicities by fitting rectangular functions to data. It is analogous to Fourier analysis, which represents data as sine and cosine functions. For many behavioral measures, Fourier transforms can produce spurious peaks in power spectra and fail to resolve separable components. Walsh analysis is superior for strongly discontinuous data. The strengths and weaknesses of each transform are discussed, and specific algorithms are given for the newer Walsh technique.There are many methods for analyzing periodicities in data. In the time domain, autocorrelations provide a convenient way to look for cyclic patterns. However, such time-domain analyses often fail to resolve multiple periodicities superimposed in a single data set, particularly if some frequency components have little amplitude relative to others. Frequency-domain analyses typically provide greater resolution and are therefore often a better approach than their time-based counterparts; but many frequency-based methods exist, and the choice of the best method for a given data set is not always straightforward. The purpose of this paper is to compare two methods, the familiar Fourier transform power spectrum and the less familiar Walsh transform power spectrum, and to demonstrate that for some types of data, the Walsh spectrum provides a much clearer and more detailed understanding of the data than does the Fourier spectrum. Algorithms for computing a Walsh spectrum will be presented and explained, and applications of the analysis method will be considered. Figure 1 shows a single trial from a rat pressing a lever for food on a 48-sec fixed-interval schedule in which food was delivered on a random half of the trials. Responses are shown as +Is, and time bins in which no responses occurred are shown as -Is. The trial illustrated is a typical example of responding following omission of food in this procedure; responses occurred in bursts throughout the interval. To determine whether these response bursts were periodically or randomly spaced, it is necessary to choose a method of periodic analysis that will provide an accurate representation of these data.This research was supported in part by grants from the National Institute of Mental Health (R01-MH44234) and the National Science Foundation (BNS 9110158). We would like to thank Russell M. Church for hishelp in exploring these analysis techniques and for aiding in the writing of this manuscipt. We would also like to thank Howard Kaplan for his comments on an earlier draft. Our thanks also to the anonymous reviewers for their comments and suggestions. Correspondence should be addressed

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“…(19) applies for the GDFT as well as the Hadamard or Walsh transform, cf. [27], but not for the discrete cosine transform (DCT) in general. A relation between DFT FBE and DCT FBE is established in [17] for gain factors with property of Eq.…”

Section: Prototype Filter Designmentioning

confidence: 99%