Moving-average filter

To illustrate behaviors of different filters, consider a moving average filter that averages over 11 terms. Such a filter has the frequency response shown in Fig. 8. Note that this filter has a relatively low gain of 0.55 at the break-point frequency of 0.05 cycles per minute. So in the range of... 

An example is the relatively simple moving average filter. In case of a digitized signal, the values of a fixed (odd) number of data points (a window) are added and divided by the number of points. The result is a new value of the center point. Then the window shifts one point and the procedure, which can be considered as a convolution of the sipal with a rectangular pulse function, repeats. Of course, other functions like a triangle, an exponential and a Gaussian, can be used. 

Filter out any fast ripple of period Aw/2 in V(z), due to interference with internal reverberations in the lens (Fig. 8.5(b)). This may be achieved most simply by convolving with a rectangular function of length Aw/2. This is known as a moving average filter it is equivalent to a sine filter in the Fourier domain, but is computationally somewhat more efficient. Because of its period the ripple removed at this stage is sometimes called water ripple. 

Averaging is a least-squares process that reduces the effects of noise, if the noise is zero-mean and fairly random [10], and the moving-average filter removes high-frequency noise well. It is less successful at removing low-frequency noise, since these nonzero-mean variations are less likely to be affected by the averaging. It also... 

FIGURE 10.13 FIR signal processing with a simple moving-average filter. 

FIGURE 10.14 Time-domain processing of noisy data with a ten-point, moving-average filter. The moving-average-filtered data are indicated as a solid line. The true signal is shown as a dotted line. 

FIGURE 10.15 Transfer functions for FIR filters shown as a function of window size. Transfer functions for simple, moving-average filters with windows of 5 (...), 10 (-), and... 

Selection of points to use in a three point moving average filter... 

Drift in a sensor or a combination of bias change and noise inflation would affect both the means and variances of the residuals. To determine which one of these sensor faults has occurred, an additional test is performed. The data for the variable under study is filtered bj using a moving average filter in order to eliminate the effects of process/instrument noise. If the filtered data have a non-stationary mean (changing over time as a... 

Figure 3.1 Moving-average filter for computing the first and last aver-a filter width of 2m +1 = 3, that is, age. Original signal value, o fil-m = 1. Note that for the extreme tered signal value, points, no filtered data can be calculated, since they are needed for...

After applying the moving-average filter, the data contain less noise. In the case of structured data, the filter width has to be chosen such that the structure of the data, for example, of a peak, is not distorted. 

Figure 3.2 demonstrates the filtering of raw data by using a 5-point moving-average filter (curve 1). In this example, the filter width of 5 points leads already to the distortion of peaks. This effect is enhanced if the filter width is further increased as demonstrated here for an 11-point filter (Figure 3.2, curve 2). The appropriate choice of the filter width is discussed as follows. 

Very efficient smoothing of data is obtained with filters that weight the raw data differently. In the case of the moving-average filter, all the data were weighted by the same factor, that is, l/(2m -I-1) (cf. Eq. (3.1)). A better fit results if weights are used that... 

Figure 3.2 Filtering of a discrete analytical signal consisting of k data points with different filters (1) A 5-point moving-average filter, (2) an...

The filter coefficients Cj are tabulated in Table 3.1 for different filter widths. Figure 3.2, curve 3, demonstrates the effect of a Savitzky-Golay filter with a filter width of 5 points applied to the raw data. Compared to the 5-point moving-average filter, the obviously better fit can be seen. 

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