Linear filters smoothing functions

In single-scale filtering, basis functions are of a fixed resolution and all basis functions have the same localization in the time-frequency domain. For example, frequency domain filtering relies on basis functions localized in frequency but global in time, as shown in Fig. 7b. Other popular filters, such as those based on a windowed Fourier transform, mean filtering, and exponential smoothing, are localized in both time and frequency, but their resolution is fixed, as shown in Fig. 7c. Single-scale filters are linear because the measured data or basis function coefficients are transformed as their linear sum over a time horizon. A finite time horizon results infinite impulse response (FIR) and an infinite time horizon creates infinite impulse response (HR) filters. A linear filter can be represented as... 

Smoothing functions can be introduced in a variety of ways, for example, as sums of coefficients or as a method for fitting local polynomials. In the signal analysis literature, primarily dominated by engineers, linear filters are often reported as a form... 

Historically, treatment of measurement noise has been addressed through two distinct avenues. For steady-state data and processes, Kuehn and Davidson (1961) presented the seminal paper describing the data reconciliation problem based on least squares optimization. For dynamic data and processes, Kalman filtering (Gelb, 1974) has been successfully used to recursively smooth measurement data and estimate parameters. Both techniques were developed for linear systems and weighted least squares objective functions. 

Conceptually, die simplest methods are lineal" filters whereby the resultant smoothed data are a linear function of die raw data. Normally this involves using the surrounding... 

Some people are confused by the difference between Fourier filters and linear smoothing and resolution functions. In fact, both methods are equivalent and are related... 

Problems, however, arise if the intervals between the knots are not narrow enough and the spline begins to oscillate (cf. Figure 3.13). Also, in comparison to polynomial filters, many more coefficients are to be estimated and stored, since in each interval, different coefficients apply. An additional disadvantage is valid for smoothing splines, where the parameter estimates are not expectation-true. The statistical properties of spline functions are, therefore, more difficult to describe than in the case of linear regression (cf. Section 6.1). ... 

The purpose of the high- and low-pass filters, shown in Figure 9.2, is to eliminate interference signals such as electrode half-cell potentials and preamplifier offset potentials and to reduce the noise amplitude by the limitation of the amplifier bandwidth. Because the biosignal should not be distorted or attenuated, higher order sharp-cutting linear-phase filters have to be used. Active Bessel filters are preferred filter types because of their smooth transfer function. Separation of biosignal and interference is in most cases incomplete because of the overlap of their spectra. 

IR spectra are given as products (< )/i(decadic linear absorption coefficient) as a function of wavenumber, time domain, i.e., each term of the correlation function C(t) is multiplied by a Gaussian function exp —0.5a(tlt, axf ), where is the length of the simulation, and a is usually chosen around a value of 10 for gas phase simulations. This convolution only has the purpose to remove the numerical noise arising from the finite length of the Fourier transform of (5). These calculations are performed with our home-made code. 

In an analytical evaluation of an equivalent load function, a full nonlinear analysis with a flexible target and a deformable missile should be carried out, with a strong emphasis on a sensitivity study of the results to the wide variety of assumptions which usually affect such approaches (e.g. non-linear material properties and simulation of erosion effects). After the simulation, a smoothing process should be applied to the result to filter out as far as possible the unavoidable spurious noise from the numerical integration attention should be paid not to exclude physical high frequency effects from the load function. 

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