Smoothing Fourier filter

The Fourier transforms were performed in the standard way. No smoothing nor filtering was employed. Subtraction of the data from the least squares fit removes the constant or linear term characterizing a Markovian process. Fourier transform of the differences from the linear fit suppresses the enhancement of both the power and amplitude spectra at low frequencies. 

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... 

The dula analysis was performed uccortling io relerence 109. To determine Ihe smooth pari of ihe background subiracictl spectrum, a modified smoothing spline algorithm was used. The EXAFS Xlk) functions were multiplied with k1 and Fourier filtered in the ranges 1.40- .50 A for the bromine F.XAI S spectra and 1.00- .40 A for the magnesium EXAFS spectra. 

Filtering and smoothing are related and are in fact complementary. Filtering is more complicated because it involves a forward and a backward Fourier transform. However, in the frequency domain the noise and signal frequencies are distinguished, allowing the design of a filter that is tailor-made for these frequency characteristics. 

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... 

Function approximation comes naturally with the Fourier transition. Since tiny details of a function in real space relate to high-frequency components in Fourier space, restricting to low-order components when transforming back to real space (low-pass filtering) effectively smoothes the function to any desirable degree. There are special function decomposition schemes, like spherical harmonics, which especially build on this ability [128]. 

Formally, the sum of random electromagnetic-field fluctuations in any set of bodies can be Fourier (frequency) decomposed into a sum of oscillatory modes extending through space. The "shaky step" in this derivation, already mentioned, is that we treat the modes extending over dissipative media as though they were pure sinusoidal oscillations. Implicitly this treatment filters all the fluctuations and dissipations to imagine pure oscillations only then does the derivation transform these oscillations into the smoothed, exponentially decaying disturbances of random fluctuation. 

This procedure is equivalent to the Savitsky-Golay method, algorithms for which have been included in computer software for scientific instruments such as the Fourier-transform infrared (FTIR) spectrometer. Alternatives to smoothing are weighted least-squares fitting or optimal (Weiner) filtering techniques. ... 

The smoothing operations discussed above have been presented in terms of the action of filters directly on the spectral data as recorded in the time domain. By converting the analytical spectrum to the frequency domain, the performance of these functions can be compared and a wide variety of other filters designed. Time-to-frequency conversion is accomplished using the Fourier transform. Its use was introduced earlier in this chapter in relation to sampling theory, and its application will be extended here. 

Figure 12 A spectrum (a) and its Fourier transform before (b) and after applying a 13-point quadratic smoothing filter (c)...

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