Wiener spectrum

Models are constructed which suggest that these optical measurements can be used to determine the effective particle size distribution parameters, mean diameter and sigma. Assumptions include multilayer particle deposit, the lognormal distribution of the diameters of the spherical, opaque particles, and no sorting of size classes during particle deposition. The optical measurement include edge trace analysis to derive the contrast transfer function, and density fluctuation measurements to derive the Wiener spectrum. Algorithms to perform these derivations are outlined. 

The corresponding functions which do contain information over a range of dimensions are the contrast transfer function, CTF, and the Wiener spectrum, or noise content of the deposit as a function of spatial frequency (8). These functions also have been found to contain information which correlates to the particle size distribution, as will now be discussed. 

The statistical fluctuation or noise level of a toner image is also by these postulates dependent on the particle size distribution. The function which relates the statistical fluctuations to spatial frequency is the Wiener spectrum, which is the Fourier transform of the optical density autocorrelation function. In terms of toner images, it is a measure of the dimensional extent over which the presence or absence of a particular toner particle will contribute to density. The density fluctuations can be measured as a function of position, normally with a slit aperture. This is schematically represented in Figure 6 where the left-hand sketch is related to large particles and the right-hand one to small particles. The density data can be used to calculate noise power or Wiener spectrum (8). Formally, the Wiener spectrum is ... 

The dimension of W(v) is (distance)T, e.g. ym2. For comparison with particle parameters, the square root of the Wiener spectrum values where used. This gives a function of linear distance dimension and is analogous to use of an RMS value such as granularity. Figure 7 shows examples of these experimental Wiener spectra, plotted as the square root of W(v) vs. log (v), for toners fractions 18 and parent toner. Note that the bimodal parent toner yields a spectrum with an inflection point, another supportive observation for the correlation of image parameters and toner PSD. 

A theoretical model to relate the Wiener spectrum to the toner deposit parameters is difficult to construct because the mathematical difficulties of dealing with projections of transforms of probability distributions quickly "hide" any simple relationships. Models have been constructed however for a crowded monolayer photographic emulsion (11), and for multilayers of emulsion (12). Although the analysis was done for one-dimensional geometry, extension to two dimensions was outlined. A different approach will be used here, which relies on the linearity property of the Fourier transform, and assumes that the location of the toner particles is independent of neighbors. 

For a first approximation to a model for the Wiener spectrum constructed from the toner particle distribution data, consider a fraction of a deposit consisting of n-j toner particles of circular projection area and diameter d-j. The expected circular aperture Wiener spectrum for a deposit of these particles is (10,2),... 

Two estimates of how well the particle size distribution shows through the computed Wiener spectrum were made. First, the extrapolated value of (0) was determined. This should be proportional to the granularity of the deposit (2, 6), and according to the relation of equation (3), should in turn correlate with dm. Figure 9 shows this comparison, yielding a correlation coefficient of 0.985. Secondly, the spatial frequency at which (v) falls to the extrapolated maximum /w(0) should be correlated with the average particle size of those contributing to the spectrum. 

Figure 8. Wiener spectrum for toner Sample 1. Key --------------, calculated from...

The correlation analyses of toner particle size and size distribution parameters and image quality characteristics of toner deposits as measured by the spectral dependence of contrast transfer function and noise show high coefficients of correlation Specifically the Wiener spectrum data appear to yield the weight geometric mean and standard deviation of the toner population in this study. Therefore the Wiener spectrum may be another analytical tool in characterizing particle populations. It must be pointed out that the analysis reported here is mainly empirical. Further work is needed to refine the models and to examine the limits of applicability of these tests. Factors such as particle clumping, non-uniform depositions and optical limitations are specific areas for examination. 

It is a well known fact, called the Wiener-Khintchine Theorem [gardi85], that the correlation function and power spectrum are Fourier Transforms of one another ... 

As a further application of the Wiener-Khinchine theorem, we shall now calculate the power density spectrum of the shot noise process. The autocorrelation function for such a process is given by Campbell s theorem, Eq. (3-262), repeated below... 

This is the autocorrelation and by the Wiener-Khintchine theorem the power spectrum of the disturbance is given by its Fourier transform,... 

Wiener-Khintchine theorem). The right-hand side of this equation is often called the power spectrum. It is given by the autocorrelation function, Eq. 2.55. The Fourier transform of the autocorrelation function is related to the spectral moments,... 

In general, the energy spectrum is calculated by using the auto-correlation function Rt, (r) based on Wiener-Khintchine s theorem as follows ... 

There are several ways of detecting peaks in such noisy signals. The Wiener-Hopf filter minimizes the expectation value of the noise power spectrum and may be used to optimally smooth the original noisy profile [19]. An alternative approach described by Hindeleh and Johnson employs knowledge of the peak shape. It synthesizes a simulated diffraction profile from peaks of known width and shape, for all possible peak amplitudes and positions, and selects that combination of peaks that minimizes the mean square error between the synthesized and measured profiles [20], This procedure is illustrated... 

Once we evaluate (t) we can obtain the power spectrum of the randomly modulated harmonic oscillator using the Wiener-Khintchine theorem (7.76)... 

Godsil, C.D. and Gutman, I. (1999) Wiener index, graph spectrum, line graph. Acta Chim. Hung. -Mod. Chem., 136, 503-510. 

Gutman, 1. (2003a) Hyper-Wiener index and Laplacian spectrum. J. Serb. Chem. Soc., 68, 949-952. 

Gutman, I., Vidovic, D. and Furtula, B. (2003b) Chemical applications of the Lapladan spectrum. VII. Studies of the Wiener and Kirchhoff indices. Indian J. Chem., 42, 1272—1278. 

Equation (41a) means that the function B( r) is equivalent to the volume integral of the density matrix y(ri, ri) under the condition of r = r - r, and Eq. (41b) means that B(r) is the autocorrelation function of the position wave function (r). The latter is an application of the Wiener-Khintchin theorem (Jennison, 1961 Bracewell, 1965 Champeney, 1973), which states that the Fourier transform of the power spectrum is equal to the autocorrelation function of a function. Equation (41c) implies not only that B(r) is simply the overlap integral of a wave function with itself separated by the distance r (Thulstrup, 1976 Weyrich et al., 1979), but also that the momentum density p(p) and the overlap integral S(r) are a pair of the Fourier transform. The one-dimensional distribution along the z axis, B(0, 0, z), for example, satisfies... 

Qualitative justification for the highly random nature of the power spectrum in a chaotic system is readily given. That is, IT[co, x(0)] is a function of a single trajectory. Since a single trajectory is known to depend sensitively on initial conditions, any such function is expected to show such extreme sensitivity as well. Nonetheless, it is necessary to reconcile this result with standard textbook statements28 of the Wiener-Khinchin theorem,29 which equate the power spectrum and spectral density S(co, M) ... 

According to the Wiener-Khinchine theorem, the power spectrum of a fluctuating quantity is given by the (time) Fourier transform of the autocorrelation function. So for the bending mode one derives from (7.4), for... 

TJie Fourier transform of the first-order correlation function G r) represents the normalized frequency spectrum of the incident light-wave intensity I (o>) (Wiener-Khintchine theorem) [930, 935]. 

The spectral density function of the fluctuation can be calculated from the autocorrelation function by the Wiener-Khintchine relation (Wiener, 1930 Khintchine, 1934). The original formulation of the theorem refers to stationary stochastic processes for a possible generalisation see, for example, Lampard, 1954. The relationship connects the autocorrelation function to the spectrum ... 

These properties give a peculiar look to G as shown in Fig. 5.5. Physically, this seems to reflect the fact that the oscillators whose natural frequencies are close to cuo are pulled perfectly into the central frequency to form a sharp peak, while this results in a remarkable population decrease around this peak. This is the very feature that caught N. Wiener s attention (1965), in connection with the spectrum of the a rhythm of the human brain waves. 

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