Classical and Nonclassical Optical Diffusion

1990 ◽  
Vol 195 ◽  
Author(s):  
J. M. Drake ◽  
A. Z. Genack
Keyword(s):  
Fourier Transform ◽  
Diffusion Coefficient ◽  
Correlation Function ◽  
Optical Transmission ◽  
Optical Field ◽  
Photon Absorption ◽  
Absorption Time ◽  
Transmission Data ◽  
The Fourier Transform ◽  
Field Correlation

ABSTRACTWe present a brief summary of our recent work on classical and nonclassical optical diffusion in disorder metal oxides. Results on the optical transmission through a slab of titania spheres is presented and discussed in some detail illustrating the experimental method for obtaining the optical diffusion coefficient and photon absorption time from the transmission data. We conclude that the optical field-field correlation function with frequency is the Fourier transform of the time of flight distribution.

scholarly journals On the Importance of Statistical Homogeneity to the Scaling of Rain

2019 ◽  
Vol 36 (6) ◽  
pp. 1063-1078 ◽  
Author(s):  
A. R. Jameson
Keyword(s):  
Fourier Transform ◽  
Correlation Function ◽  
Power Spectrum ◽  
Wide Sense ◽  
Time Data ◽  
Statistical Homogeneity ◽  
Statistical Heterogeneity ◽  
Cross Correlations ◽  
The Fourier Transform ◽  
Universal Scaling Function

AbstractScaling studies of rainfall are important for the conversion of observations and numerical model outputs among all the various scales. Two common approaches for determining scaling relations are the Fourier transform of observations and the Fourier transform of a correlation function using the Wiener–Khintchine (WK) theorem. In both methods, the observations must be wide-sense statistically stationary (WSS) in time or wide-sense statistically spatially homogeneous (WSSH) in space so that the correlation function and power spectrum form a Fourier transform pair. The focus here is on developing an explicit understanding for the requirement. Statistically heterogeneous (either in space or time) data can produce serious scaling errors. This work shows that the effects of statistical heterogeneity appear as contributions from cross correlations among all of the distinct contributing rainfall components using either method so that the correlation function and its FFT do not form a transform pair. Moreover, the transform then also depends upon the time and location of the observations so that the “observed” power spectrum no longer represents a “universal” scaling function beyond the observations. An index of statistical heterogeneity (IXH) defined in previous work provides a way of determining whether or not a set of rain data may be considered to be WSS or WSSH. The greater IXH exceeds the null, the more likely the derived power spectrum should not be used for general scaling. Numerical simulations and some observations are used to demonstrate all of these findings.

Detection of Large-Scale Noisy Multi-Periodic Patterns with Discrete Double Fourier Transform. II. Study of Correlations Between Patterns

2020 ◽  
pp. 2150003
Author(s):  
V. R. Chechetkin ◽  
V. V. Lobzin
Keyword(s):  
Fourier Transform ◽  
Correlation Function ◽  
Geomagnetic Activity ◽  
Large Scale ◽  
Statistical Significance ◽  
Periodic Patterns ◽  
Pattern Correlation ◽  
Double Fourier Transform ◽  
The Fourier Transform ◽  
Combined Technique

The discrete double Fourier transform (DDFT) was developed to search for large-scale multi-periodic patterns in the presence of noise and is based on detection of the equidistant series of harmonics generated by the periodic patterns in the discrete Fourier transform (DFT) spectra. As DDFT retains all generic features of the Fourier transform, the corresponding pattern correlation function (PCF) related to DDFT can be introduced similarly to the data correlation function (DCF) related to DFT on the basis of the Wiener–Khinchin relationship. Peaks in PCF indicate the number of periodic patterns in a dataset under analysis and have direct correspondence with the counterpart peaks in the DFT spectrum. The close correspondence between positions of the peaks in the PCF and DFT spectra strongly enhances statistical significance of detected periodicities. Similar PCFs can also be defined for the cepstrum transform. The combined DFT–DCF and DDFT–PCF technique was applied to the detection of cycles in geomagnetic activity using disturbance storm-time (Dst) index. In addition to the known 27-day, semiannual and 11-year cycles of geomagnetic activity, we have also found the annual cycle of activity. The results were compared with those obtained by the cepstrum transform. A multiple cross-check makes the combined technique much more efficient and robust in comparison with the detection based on a unique particular method.

On-line electron-optical correlation computing in the CTEM

1978 ◽  
Vol 36 (1) ◽  
pp. 88-89 ◽  
Author(s):  
R. Guckenberger ◽  
W. Hoppe
Keyword(s):  
Fourier Transform ◽  
Correlation Function ◽  
Transfer Function ◽  
Inverse Fourier Transform ◽  
Main Peak ◽  
Optical Correlation ◽  
Auto Correlation ◽  
Auto Correlation Function ◽  
On Line ◽  
The Fourier Transform

Light diffractograms of electron micrographs are frequently used to study the transfer function of the microscope. In order to utilize diffractograms for control operations in the microscope, several attempts have been undertaken to obtain on-line diffractograms /1 - 3/. Alternatively correlation functions (CF) may be used /4-8/. In this paper we describe an electron-optical device for the computation of such CF and its on-line operation in a microscope.The auto-correlation function (ACF) is the inverse Fourier transform of the squared modulus of the Fourier transform (diffractogram) of an image. Therefore it also contains the transfer function. It is its zero peak (main peak) which is of particular interest. In noisy images the main ACF-peak of the noise contributes in an unwanted way to the main ACF-peak of the image. This can be avoided if the ACF will be computed of two images which are identical except for noise /9/ (noise-reduced ACF= NRACF).

scholarly journals Strain Sensitivity Enhancement of Broadband Ultrasonic Signals in Plates Using Spectral Phase Filtering

2021 ◽  
Vol 11 (6) ◽  
pp. 2582
Author(s):  
Lucas M. Martinho ◽  
Alan C. Kubrusly ◽  
Nicolás Pérez ◽  
Jean Pierre von der Weid
Keyword(s):  
Fourier Transform ◽  
Guided Waves ◽  
Strain Level ◽  
Energy Concentration ◽  
Short Time Fourier Transform ◽  
Time Frequency ◽  
Frequency Representation ◽  
The Fourier Transform ◽  
Short Time ◽  
Sensitivity Gain

The focused signal obtained by the time-reversal or the cross-correlation techniques of ultrasonic guided waves in plates changes when the medium is subject to strain, which can be used to monitor the medium strain level. In this paper, the sensitivity to strain of cross-correlated signals is enhanced by a post-processing filtering procedure aiming to preserve only strain-sensitive spectrum components. Two different strategies were adopted, based on the phase of either the Fourier transform or the short-time Fourier transform. Both use prior knowledge of the system impulse response at some strain level. The technique was evaluated in an aluminum plate, effectively providing up to twice higher sensitivity to strain. The sensitivity increase depends on a phase threshold parameter used in the filtering process. Its performance was assessed based on the sensitivity gain, the loss of energy concentration capability, and the value of the foreknown strain. Signals synthesized with the time–frequency representation, through the short-time Fourier transform, provided a better tradeoff between sensitivity gain and loss of energy concentration.

Determination of starch crystallinity with the Fourier-transform terahertz spectrometer

2021 ◽  
Vol 262 ◽  
pp. 117928
Author(s):  
Shusaku Nakajima ◽  
Shuhei Horiuchi ◽  
Akifumi Ikehata ◽  
Yuichi Ogawa
Keyword(s):  
Fourier Transform ◽  
The Fourier Transform ◽  
Starch Crystallinity

Translation uniqueness of phase retrieval and magnitude retrieval of band-limited signals

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Lung-Hui Chen
Keyword(s):  
Fourier Transform ◽  
Entire Function ◽  
Convex Hull ◽  
Scattering Theory ◽  
Phase Retrieval ◽  
Indicator Function ◽  
Band Limited ◽  
The Fourier Transform ◽  
Complex Valued ◽  
Spectral Invariant

Abstract In this paper, we discuss how to partially determine the Fourier transform F ⁢ ( z ) = ∫ - 1 1 f ⁢ ( t ) ⁢ e i ⁢ z ⁢ t ⁢ 𝑑 t , z ∈ ℂ , F(z)=\int_{-1}^{1}f(t)e^{izt}\,dt,\quad z\in\mathbb{C}, given the data | F ⁢ ( z ) | {\lvert F(z)\rvert} or arg ⁡ F ⁢ ( z ) {\arg F(z)} for z ∈ ℝ {z\in\mathbb{R}} . Initially, we assume [ - 1 , 1 ] {[-1,1]} to be the convex hull of the support of the signal f. We start with reviewing the computation of the indicator function and indicator diagram of a finite-typed complex-valued entire function, and then connect to the spectral invariant of F ⁢ ( z ) {F(z)} . Then we focus to derive the unimodular part of the entire function up to certain non-uniqueness. We elaborate on the translation of the signal including the non-uniqueness associates of the Fourier transform. We show that the phase retrieval and magnitude retrieval are conjugate problems in the scattering theory of waves.

scholarly journals Convolutors on $${\mathcal S}_{\omega }({\mathbb R}^N)$$

2021 ◽  
Vol 115 (4) ◽  
Author(s):  
Angela A. Albanese ◽  
Claudio Mele
Keyword(s):  
Fourier Transform ◽  
Dual Space ◽  
Strong Operator ◽  
Dual Spaces ◽  
Structure Theorems ◽  
The Fourier Transform ◽  
Decreasing Functions

AbstractIn this paper we continue the study of the spaces $${\mathcal O}_{M,\omega }({\mathbb R}^N)$$ O M , ω ( R N ) and $${\mathcal O}_{C,\omega }({\mathbb R}^N)$$ O C , ω ( R N ) undertaken in Albanese and Mele (J Pseudo-Differ Oper Appl, 2021). We determine new representations of such spaces and we give some structure theorems for their dual spaces. Furthermore, we show that $${\mathcal O}'_{C,\omega }({\mathbb R}^N)$$ O C , ω ′ ( R N ) is the space of convolutors of the space $${\mathcal S}_\omega ({\mathbb R}^N)$$ S ω ( R N ) of the $$\omega $$ ω -ultradifferentiable rapidly decreasing functions of Beurling type (in the sense of Braun, Meise and Taylor) and of its dual space $${\mathcal S}'_\omega ({\mathbb R}^N)$$ S ω ′ ( R N ) . We also establish that the Fourier transform is an isomorphism from $${\mathcal O}'_{C,\omega }({\mathbb R}^N)$$ O C , ω ′ ( R N ) onto $${\mathcal O}_{M,\omega }({\mathbb R}^N)$$ O M , ω ( R N ) . In particular, we prove that this isomorphism is topological when the former space is endowed with the strong operator lc-topology induced by $${\mathcal L}_b({\mathcal S}_\omega ({\mathbb R}^N))$$ L b ( S ω ( R N ) ) and the last space is endowed with its natural lc-topology.

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