Gaussian zero crossing

Within the framework of scale-space filtering, inflexion points of F(t) appear as extrema in dF(t)/dt and zero crossings in d2F(t)/dt2. Thus, filtering a signal by the Laplacian (second derivative) of a Gaussian will generate the inflexion points at various scales (Marr and Hildreth, 1980). In the same spirit, if the wavelet is chosen to be the first derivative of a scaling function, i.e., ip(t) = d(t)/ dt, then from Eqs. (5a) and (11) we... 

For systems which are suflBciently dilute so that the positions ry and r/ of different particles are uncorrelated but not so dilute that the scattered field may become non-Gaussian, the cross-terms in // have zero averages, and... 

The curves intersect the line y = 1 whenever any of the following conditions are met n = 0, x = 0, or acos (x) = 0. This accounts for the tip of the bell-shaped curves in Gaussian Mountain Range. The bells are centered at the origin at x = 0. Zero crossings satisfy n/2 = nx acos(x). 

The first application of differentiation to spectroscopy was, of course, the deconvolution of superposed peaks [41-43], which are frequently found in spectral investigations. From then on, many scientists analyzed peak overlapping by summation of synthetic signals, mostly of the Gaussian type [12,14,15, 44-48]. Derivatives, zero crossing, and extrema of different distribution functions such as Gauss, Lorentz, Student, T3, and others, are not difficult to estimate unless there are superpositions of two or more bands. For pure Gaussian functions, see Table 2-3. 

Table 2-3. Extrema and zero crossing for Gaussian functions and their derivatives (d to d" ) after [46].

In most electron-diffraction studies both the electron beam and the molecular beam are assumed to have vanishing cross-sections. This approximation is probably satisfactory for the electron beam. The effect of a non-zero cross-section for the molecular beam varies with the density distribution and with the nozzle-to-plate distance. - Rundgren has found that for a Gaussian density distribution with standard deviation, am, the observed values for will be too large by approximately... 

The two-network method has been carefully examined. All the previous two-network results were obtained in simple extension for which the Gaussian composite network theory was found to be inadequate. Results obtained on composite networks of 1,2-polybutadiene for three different types of strain, namely equibiaxial extension, pure shear, and simple extension, are discussed in the present paper. The Gaussian composite network elastic free energy relation is found to be adequate in equibiaxial extension and possibly pure shear. Extrapolation to zero strain gives the same result for all three types of strain The contribution from chain entangling at elastic equilibrium is found to be approximately equal to the pseudo-equilibrium rubber plateau modulus and about three times larger than the contribution from chemical cross-links. 

Fig. 9. Magnetic circular dichroism selection rules. (A) Transitions between Zeeman-split y (or S) = i paramagnetic ground (G) and excited (E) states. Right circularly polarized (RCP) and left circularly polarized (LCP) transitions are indicated. (B) MCD spectra associated with the energy scheme in (A). The dashed bands correspond to individual left and right polarized transitions, whereas the solid line spectra give the dichroism. The C-term spectrum corresponds to a Gaussian band shape, which peaks at the absorption maximum, whereas an A-term reflects its first derivative which crosses zero at the absorption maximum.

The set of measurements of the concentrations of the species in S,- is obtained as a function of the externally controlled concentrations of the species Ii and I2 at each of the selected time points. Figure 7.2 is a plot of the time series for each of the species in this system. One time point is taken every 10 s for 3,600 s. The effects of using a much smaller set of observations are discussed later. The first two plots are the time series for the two externally controlled inputs. The concentrations of Ii and I2 at each time point are chosen from a truncated Gaussian (normal) distribution centered at 30 concentration units with a standard deviation of 30 units. The distribution is truncated at zero concentration. The choice of Gaussian noise guarantees that in the long time limit the entire state-space of the two inputs is sampled and that there are no autocorrelations or cross-correlations between the input species. Thus all concentration correlations arise from the reaction mechanism. The bottom five times series are the responses of the species S3 to S7 to the concentration variations of the inputs. 

Figure 9.15. Typical trajectories of a Gaussian stochastic process x(t) with zero mean and Gaussian (a) or exponential (i>) correlation function. Circles are crossing points of x = 0. Trajectories were generated by regular sampling in the frequency domain, (c) corresponds to the Debye relaxation spectrum with a cutoff frequency. Reorganization energy of the discarded part of the spectrum is 7% of the total. The sampling pattern was the same as in (b).

The simple cross-correlation estimator is used extensively in the form of a matched filter implementation to detect a finite number of known signals (in other words, simultaneous acquisition of multiple chaimels of known signals). When these deterministic signals are embedded in white Gaussian noise, the matched filter (obtained from cross-correlation estimate at zero lag, k = 0, between the known signal sequence and the observed noisy signal sequence) gives the optimum detection performance (in the Bayes sense ). 

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