Window smoothing

DLC coatings are already in production in several areas (optical and IR windows) and appear particularly well-suited for abrasion and wear applications due to their high hardness and low coefficient of friction. They have an extremely smooth surface and can be deposited with little restriction of geometry and size (as opposed to CVD diamond). These are important advantages and DLC coatings will compete actively with existing hard coatings, such as titanium carbide, titanium nitride, and other thin film... 

In the accumulation process explained in the previous section, data points collected during several scans and measured at corresponding time values, are added. One could also consider to accumulate the values of a number of data points in a small segment or window in the same scan. This is the principle of smoothing, which is explained in more detail below. 

Fig. 40.22. Distortion (hJhn) of a Gaussian peak for various window sizes (indicated within parentheses). (a) Moving average, (b) Polynomial smoothing.

Another unwanted effect of smoothing is the alteration of the frequency characteristics of the noise. This calls for caution. Because low frequencies present in the noise are not removed, the improvement of the signal-to-noise ratio may be limited. This is illustrated in Fig. 40.23 where one can see that after smoothing with a 25 point window low-frequency noise is left. In Section 40.5.3 filtering... 

Fig. 40.23. Polynomial smoothing (noise = N(0,3%)) 5-point 17-point 25-point smoothing window and the noise left after smoothing.

Fig. 40.24. Polynomial smoothing window of 7 data points fitted with polynomials of degrees 0,1,2, 3 and 4.

Fig. 40.30. Smoothed second-derivative (window 7 data points, second-order) according to Savitzky-Golay.

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... 

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