Exponential smoothing filter

When rearranged this gives the exponential smoothing filter in Eq. (36), where the filter factor, 0 < a < 1, is defined by Eq. (37). 

The exponential shape of the filter follows directly by elaborating eq. (40.14) for a few consecutive data points (see Table 40.4). From this table we can see that a smoothed data point at time i is the average of all data points measured before, weighted with an exponentially decaying weight X< ) with d the distance of that data point from the measurement to be smoothed. Such shapes are also found for electronic filters with a given time constant. The effect of exponential smoothing is visualized in the plot of the a , and values (Fig. 40.25) listed in... 

Table 40.3. As one can see, the filter introduces a slower response to stepwise changes of the signal, as if it were measured with an instrument with a large response time. Because fluctuations are smoothed, the standard deviation of the signal is decreased, in this example from 2.58 to 1.95. A Gaussian peak is broadened and becomes asymmetric by exponential smoothing (Fig. 40.26).

Input mapping methods can be divided into univariate, multivariate, and probabalistic methods. Univariate methods analyze the inputs by extracting the relationship between the measurements. These methods include various types of single-scale and multiscale filtering such as exponential smoothing, wavelet thresholding, and median filtering. Multivariate methods analyze... 

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... 

For an IIR filter, the parameter T in Eq. (9) tends to infinity. IIR filters can be represented as a function of previous filter outputs and often can be computed with fewer multiplications and reduced data storage requirements compared to a FIR filter. A popular example of an IIR filter is the exponentially weighted moving average (EWMA) or exponential smoothing, which is represented as... 

Fig. 3 Filter used in (a) mean filtering and (b) exponential smoothing.

Fig. 14 (a) Bumps signal with mean shift and white noise oj varianee 0.5. (h) OLMS filtering using Haar (MSE = 0.1635J, (e) mean filtering (MSE = 0.2530J, (d) exponential smoothing (MSE - 0.2237). 

All impactor and filter samples were analyzed for up to 45 elements by instrumental neutron activation analysis (INAA) as described by Heft ( ). Samples were irradiated simultaneously with standard flux monitors in the 3-MW Livermore pool reactor. The x-ray spectra of the radioactive species were taken with large-volume, high-resolution Ge(Li) spectrometer systems. The spectral data were transferred to a GDC 7600 computer and analyzed with the GAMANAL code (1 ), which incorporates a background-smoothing routine and fits the peaks with Gaussian and exponential functions. 

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. 

Fig. 0 shows a typical curve of the GST for the exponential filter. It is clear that the approximate spectral transform well represents the profile of the exponential filter at low energies, where the curve is also smooth and monotonous. 

As can be seen in Fig.3 the exponential filter causes a decrease in the peak height (and increase in the peak width due to the constant peak area) even when the noise level is 0 % and should not be used when smoothing ac polarograms. The rectangular filter is found to give the best result. This is obviously due to the fact that the shape of the ac... 

Analog filters are used to smooth noisy experimental data. For example, an exponential filter can be used to damp out high-frequency fluctuations due to electrical noise hence, it is called a low-pass filter. Its operation is described by a first-order transfer function, or, equivalently, a first-order differential equation. 

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