Amplitude and Phase in the MüLler-Lyer Illusion

Perception ◽  
10.1068/p2916 ◽  
2000 ◽  
Vol 29 (2) ◽  
pp. 201-209 ◽  
Author(s):  
Bernt C Skottun
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Inverse Fourier Transform ◽  
Low Frequency ◽  
Phase Spectrum ◽  
The Other ◽  
Frequency Filtering ◽  
Lyer Illusion ◽  
Amplitude Spectra ◽  
Low Pass

It has previously been claimed that the Müller-Lyer illusion is the result of low-pass spatial filtering. One way to understand this would be that the distribution of amplitudes is what generates this illusion. This possibility was investigated by computing the 2-D Fourier transforms of the two Müller-Lyer stimuli and extracting their phase and amplitude spectra. These spectra were combined to create hybrid spectra having the phase of one Müller-Lyer figure and the amplitudes of the other. Images were then created by computing the inverse Fourier transform of the hybrid spectra. Except in cases where the analysis was performed patchwise on very small patches, the figures generated with the phase spectrum of the stimuli having outward-pointing fins appear the longer. This was also the case when stimuli were generated with flat amplitude spectra. Because they show that the Müller-Lyer illusion does not depend on any particular distribution of amplitudes, these demonstrations do not support the theory that the Müller-Lyer illusion is the result of low-frequency filtering.

An analysis of the spectrum of a gravity anomaly over an inclined dike

Geophysics ◽  
1985 ◽  
Vol 50 (9) ◽  
pp. 1500-1501
Author(s):  
B. N. P. Agarwal ◽  
D. Sita Ramaiah
Keyword(s):  
Fourier Transform ◽  
Gravity Anomaly ◽  
Fourier Spectrum ◽  
Fourier Transforms ◽  
Weighted Average ◽  
Window Function ◽  
Average Spectrum ◽  
Amplitude Spectra ◽  
Distance Data ◽  
The Fourier Transform

Bhimasankaram et al. (1977) used Fourier spectrum analysis for a direct approach to the interpretation of gravity anomaly over a finite inclined dike. They derived several equations from the real and imaginary components and from the amplitude and phase spectra to relate various parameters of the dike. Because the width 2b of the dike (Figure 1) appears only in sin (ωb) term—ω being the angular frequency—they determined its value from the minima/zeroes of the amplitude spectra. The theoretical Fourier spectrum uses gravity field data over an infinite distance (length), whereas field observations are available only for a limited distance. Thus, a set of observational data is viewed as a product of infinite‐distance data with an appropriate window function. Usually, a rectangular window of appropriate distance (width) and of unit magnitude is chosen for this purpose. The Fourier transform of the finite‐distance and discrete data is thus represented by convolution operations between Fourier transforms of the infinite‐distance data, the window function, and the comb function. The combined effect gives a smooth, weighted average spectrum. Thus, the Fourier transform of actual observed data may differ substantially from theoretic data. The differences are apparent for low‐ and high‐frequency ranges. As a result, the minima of the amplitude spectra may change considerably, thereby rendering the estimate of the width of the dike unreliable from the roots of the equation sin (ωb) = 0.

Modifications of Fourier transforms and singular continuous measures

1980 ◽  
Vol 87 (3) ◽  
pp. 383-392
Author(s):  
Alan MacLean
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Asymptotic Behaviour ◽  
Fourier Transforms ◽  
Sufficient Conditions ◽  
The Other ◽  
Absolutely Continuous ◽  
Necessary And Sufficient ◽  
Continuous Measures ◽  
Absolutely Continuous Measures

It has long been known, after Wiener (e.g. see (11), vol. 1, p. 108, (5), (8), §5·6)) that a measure μ whose Fourier transform vanishes at infinity is continuous, and generally, that μ is continuous if and only if is small ‘on the average’. Baker (1) has pursued this theme and obtained concise necessary and sufficient conditions for the continuity of μ, again expressed in terms of the rate of decrease of . On the other hand, for continuous μ, Rudin (9) points out the difficulty in obtaining criteria based solely on the asymptotic behaviour of by which one may determine whether μ has a singular component. The object of this paper is to show further that any such criteria must be complicated indeed. We shall show that the absolutely continuous measures on T = [0, 2π) whose Fourier transforms are the most well-behaved (namely, those of the form (1/2π)f(x)dx, where f has an absolutely convergent Fourier series) are such that one may modify their transforms on ‘large’ subsets of Z so that they become the transforms of singular continuous measures. Moreover, the singular continuous measures in question may be chosen so that their Fourier transforms do not vanish at infinity.

Intelligibility of Filtered Synthetic Sentences

1967 ◽  
Vol 10 (2) ◽  
pp. 289-298 ◽  
Author(s):  
Charles Speaks
Keyword(s):  
Low Frequency ◽  
Normal Hearing ◽  
Frequency Filtering ◽  
Frequency Range ◽  
Frequency Bands ◽  
Low Pass ◽  
High Pass

The effects of frequency filtering on intelligibility of synthetic sentences were studied on three normal-hearing listeners. Performance-intensity (P-I) functions were defined for several low-pass and high-pass frequency bands. The data were analyzed to determine the interactions of signal level and frequency range on performance. Intelligibility of synthetic sentences was found to be quite dependent upon low-frequency energy. The important frequency for identification of the materials was approximately 725 Hz. These results are compared with previous findings concerning the intelligibility of single words in quiet and in noise.

scholarly journals Inversion of Fourier Transforms by Means of Scale-Frequency Series

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Nassar H. S. Haidar
Keyword(s):  
Fourier Transform ◽  
Nonlinear Programming ◽  
Fourier Transforms ◽  
Inverse Fourier Transform ◽  
Double Series ◽  
Nonlinear Programming Problem ◽  
Continuous Scale ◽  
Free Parameters ◽  
The Fourier Transform ◽  
Dual Scale

We report on inversion of the Fourier transform when the frequency variable can be scaled in a variety of different ways that improve the resolution of certain parts of the frequency domain. The corresponding inverse Fourier transform is shown to exist in the form of two dual scale-frequency series. Upon discretization of the continuous scale factor, this Fourier transform series inverse becomes a certain nonharmonic double series, a discretized scale-frequency (DSF) series. The DSF series is also demonstrated, theoretically and practically, to be rate-optimizable with respect to its two free parameters, when it satisfies, as an entropy maximizer, a pertaining recursive nonlinear programming problem incorporating the entropy-based uncertainty principle.

Tempered distributions with spectral gaps

1989 ◽  
Vol 106 (1) ◽  
pp. 143-162 ◽  
Author(s):  
Jean-Pierre Gabardo
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
The Other ◽  
Convolution Operators ◽  
Spectral Gaps ◽  
Tempered Distributions ◽  
Tempered Distribution ◽  
Other Hand

AbstractA tempered distribution on ℝ whose Fourier transform is supported in an interval [−Ω,Ω], where Ω>0, can be characterized by the behaviour of its successive derivatives. On the other hand, a tempered distribution on ℝ whose Fourier transform vanishes in an interval (−Ω,Ω), where Ω>0, can be characterized by the behaviour of a particular sequence of successive antiderivatives. Similar considerations apply to general convolution operators acting on J′(ℝn) and yield characterizations for tempered distributions having their Fourier transforms supported in sets of the form or , where and Ω>0.

scholarly journals Analysis of putative exoplanetary signatures found in light curves of two sdBV stars observed by Kepler

2019 ◽  
Vol 627 ◽  
pp. A86 ◽  
Author(s):  
A. Blokesz ◽  
J. Krzesinski ◽  
L. Kedziora-Chudczer
Keyword(s):  
Fourier Transform ◽  
Light Curve ◽  
Fourier Transforms ◽  
Detection Threshold ◽  
Low Frequency ◽  
Light Curves ◽  
Comparison Star ◽  
Data Set ◽  
Nearby Star ◽  
Lower Amplitude

Context. We investigate the validity of the claim that invokes two extreme exoplanetary system candidates around the pulsating B-type subdwarfs KIC 10001893 and KIC 5807616 from the primary Kepler field. Aims. Our goal was to find characteristics and the source of weak signals that are observed in these subdwarf light curves. Methods. To achieve this, we analyzed short- and long-cadence Kepler data of the two stars by means of a Fourier transform and compared the results to Fourier transforms of simulated light curves to which we added exoplanetary signals. The long-cadence data of KIC 10001893 were extracted from CCD images of a nearby star, KIC 10001898, using a point spread function reduction technique. Results. It appears that the amplitudes of the Fourier transform signals that were found in the low-frequency region depend on the methods that are used to extract and prepare Kepler data. We demonstrate that using a comparison star for space telescope data can significantly reduce artifacts. Our simulations also show that a weak signal of constant amplitude and frequency, added to a stellar light curve, conserves its frequency in Fourier transform amplitude spectra to within 0.03 μHz. Conclusions. Based on our simulations, we conclude that the two low-frequency Fourier transform signals found in KIC 5807616 are likely the combined frequencies of the lower amplitude pulsating modes of the star. In the case of KIC 10001893, the signal amplitudes that are visible in the light curve depend on the data set and reduction methods. The strongest signal decreases significantly in amplitude when KIC 10001898 is used as a comparison star. Finally, we recommend that the signal detection threshold is increased to 5σ (or higher) for a Fourier transform analysis of Kepler data in low-frequency regions.

Error Analyzing Method for Rotation Inertial Navigation System Based on Fourier Transform

2013 ◽  
Vol 336-338 ◽  
pp. 982-987 ◽  
Author(s):  
Jian Hua Cheng ◽  
Bing Yu Wang ◽  
Dai Dai Chen
Keyword(s):  
Fourier Transform ◽  
Navigation System ◽  
Inertial Navigation System ◽  
Inertial Navigation ◽  
Low Frequency ◽  
Pass Filter ◽  
Low Pass Filter ◽  
Low Pass ◽  
Frequency Components ◽  
Navigation Errors

According with the problem that the error propagative characteristics of rotation inertial navigation system can not be quantitatively analyzed, a new error analyzing method is proposed. As the inertial navigation system can be equivalent to a low pass filter, the method converted the complex gyro drifts into signal which can be quantitatively analyzed after modulating through Fourier transform, thus the low frequency components of the error can be extracted. The relationship between navigation errors and gyro drifts is built by using the conventional error equations of strapdown inertial navigation system. The comparative analyses and computer simulation of conventional, uniaxial unidirectional and uniaxial reciprocating system prove the correctness and feasibility of this method. This method provides an effective reference for other same types of control system error analyses.

RADIATION FROM A FINITE CYLINDRICAL EXPLOSIVE SOURCE

Geophysics ◽  
1977 ◽  
Vol 42 (7) ◽  
pp. 1384-1393 ◽  
Author(s):  
Anas M. Abo‐Zena
Keyword(s):  
Fourier Transform ◽  
Laplace Transform ◽  
Fourier Transforms ◽  
Inverse Fourier Transform ◽  
Applied Pressure ◽  
Inverse Laplace Transform ◽  
Laplace And Fourier Transforms ◽  
Explosion Pressure ◽  
Explosive Source ◽  
The Laplace Transform

For an elastic material with an infinite circular cylindrical hole, the exact solution due to a pressure on a finite length of the cylinder is obtained as a function of the Laplace transform parameter on time and Fourier transform parameter on the z-coordinate (the axis of the cylinder). The applied pressure is a function of the time and the position z. Numerical inversion of the Laplace and Fourier transforms are required to determine the field quantities in the time and space parameters. In the far field, the inverse Fourier transform can be obtained by an asymptotic expansion. It remains to obtain the inverse Laplace transform numerically. We have found that for cylinders whose radius is small compared with the smallest wavelength of interest, an analytical solution can be obtained. Graphical results for the cases of instantaneous explosion and progression of the detonation with constant velocity are given. In both cases an exponential decay of the explosion pressure is assumed.

scholarly journals On the L1-convergence of Fourier transforms

1996 ◽  
Vol 60 (3) ◽  
pp. 405-420
Author(s):  
Dᾰng Vũ Giang ◽  
Ferenc Móricz
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Absolutely Continuous ◽  
Inversion Formulas ◽  
Tauberian Conditions ◽  
Tauberian Type ◽  
Necessary And Sufficient

AbstractWe study cosine and sine Fourier transforms defined by F(t):= (2/π) and (t):= (2/π), where f is L1-integrable over[0, ∞]. We also assume than F are locally absolutely continuous over [0, ∞). In particular, this is the case if both f(x) and xf(x) are (L1-integrable over [0, ∞). Motivated by the inversion formulas, we consider the partial integras Sν (f, x):= and ν(f, x):= , the modified partial integrals uν (f, x):= sν(f, x) - F(ν)(sin νx)/x and ũν(f, x):= ν(f, x) + (ν) (cos νx)/x, where ν > 0. We give necessary and sufficient conditions for(L1 [0, ∞)-convergence of uν (f) and ũν (f) as well as for the L1 [0, X]-convergence of sν (f) and ν(f) to f as ν← ∞, where 0 < X < ∞ is fixed. On the other hand, in certain cases we conclude that sν(f) and ν(f) cannot belong to (L1 [0,∞). Conequently, it makes no sense to speak of their (L1 [0, ∞)-convergence as ν ← ∞.As an intermediate tool, we use the Cesàro means of Fourier transforms. Then we prove Tauberian type results and apply Sidon type inequalities in order to obtain Tauberian conditions of Hardy-Karamata kind.We extend these results to the complex Fourier transform defined by G(t):= , where g is L1- integrable over (−∞, ∞).

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