Coherence function

It is the self-coherence function that is measured in Fourier transform spectroscopy. Writing the measured on-axis intensity at the output of the Michelson interferometer as... 

This paper discusses the impact of wind action on natural-draft cooling towers. The structure of the wind load may be divided into a static, a quasistatic, and a resonant part. The effect of surface roughness of the shell and of wind profile on the static load is discussed. The quasistatic load may be described by the variance of the pressure fluctuations and their circumferential and meridional correlations. The high-frequency end of the pressure spectra and of the coherence functions are used for the analysis of the resonant response. It is shown that the resonant response is small even for very high towers, however, it increases linearly with wind velocity. Equivalent static loads may be defined using appropriate gust-response factors. These loads produce an approximation of the behavior of the structure and in general are accurate. 11 refs, cited. 

The second method, the coherence function, was developed by Bell Northern Research (Canada). It uses the coherent (signal) and non-coherent (noise) powers to derive a quality measure [CCITT86sgl2con46,1986],... 

For interferometric detection, the correlation of the electric fields at the various pairs of detectors is calculated. It can be shown that the mutual coherence function of the electric fields at detector placements rx = (xl7 yx) and r2 = (x2, y2) can be written as [65]... 

The form of the coherence function of a near-field imaging array approaches the far-field planar array limit when R0 -> 00 because at very large radii of curvature, the surface of the spherical array now becomes flat. When Ra —r 00, exp(/<5) = exp[ikz (cos2 — cosazimuthal angles are virtually zero. As Z0 00, exp(/<5) =... 

It has been shown that instead of Kramers-Kronig transforms, another method involving a coherence function could be used to validate the data. The coherence function, y, is defined as... 

The power spectra may be directly obtained using dynamic signal analyzers that measure signals as a function of time and perform the fast Fourier transform. The coherence function takes values betweaen 0 and 1 and characterizes statistical validity of the frequency response measurements ... 

When using an interferometer one is measuring coherence functions (Haniff 2007). In this section the focus is on the spatial coherence function, which is the case associated with measuring the electric field from a source at two locations but at the same time. This is equivalent to the Young s two slit experiment. 

The output of an interferometer are the so-called complex visibilities, also called spatial coherence function. When observing an object with an interferometer, the light intensity / is the result of the superposition of electromagnetic waves coming from the apertures. Considering an instrument with two apertures P and P2, the corresponding electric fields at the apertures are Ei (t) and 2 (t), respectively. At the recombination point Q, the intensity of the recombined signal is... 

The Van-Cittert-Zernike theorem describes the relation between the complex visibility of an object and its brightness distribution on the plane of the sky. It states that for sources in the far field the normalised value of the spatial coherence function (or complex visibility) is equal to the Fourier transform of the normalised sky brightness distribution. If a source with a brightness distribution 7(0), where 0 = 6y)... 

Going back to Eq.2.34, the complex visibility or spatial coherence function was defined as the mutual coherence function when r = 0. According to the van-Cittert-Zernike theorem, the normalised spatial coherence funetion is the Fourier transform of the normalised sky brightness distribution (Eq.2.35). The temporal coherence function is defined as the mutual coherence function for b = 0, and according to the Weiner-Khinchin theorem, the normalised value of the temporal coherence function is equal to the Eourier transform of the normalised spectral energy distribution of the source, this is... 

Considering that/r = vh, where 5 is the spectroscopical optical path difference due to the movable mirror (as in Eq. 2.9), the measurement of the mutual coherence function by a Multi-Fourier Transform Interferometer of the waves obtained by two apertures for a given baseline and measuring a specific source intensity I 9, n) is... 

To recover the information from the source, a two step process needs to be performed. First, to obtain the spectroscopy-resolved source image one has to perform a Fourier integration of the obtained in 5 to obtain first the spectrally resolved mutual coherence function, P(u, v, v), which in interferometer notation is... 

Statistics of the Estimates. The mean and the variance of the sample estimates of the coherence function ate derived in Ref. 4, and I only reproduce the final results here. In general, the cross-spectral density is evaluated by doing the Fourier transform of a windowed (using a lag window) sequence of cross-correlation estimates. The choice of the smoothing window therefore determines the variance in the estimates of cross-spectral density (numerator in the expression of coherence). The variance of the smoothed coherence estimator is given by ... 

Z. Bohdanecky, P. Lansky, and T. Radii, An integral measure of the coherence function between pairs of EEG recordings, Electroencephalogr. Clin. Neurophysiol. 54 587-590 (1982). 

By transforming the signals to the frequency domain and calculating their coherence functions, it is therefore possible to find a quantitative relationship between the waveforms of signals with similar source mechanisms. This method is intended allow for rapid, systematic classification of the AE signals to recognize similarities and differences in signal pattern. 

Application of the coherence function instead of the classical Kramers-Kronig transforms has also been proposed in the literature [37, 605] but is rarely used in practice. 

The coherence function y ( ) can be calculated to determine the validity of a transfer function measurement if the extent of extraneous input and nonlinearity is... 

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