Moving average filters calculation

Starting in the time domain we arrive at more or less the same conclusions. A popular filter is the moving average. It calculates the average over a small window on the data, a window that is slid over the data. When the Xj are the data points and yj is the output of the moving average, the calculation for a window of 4 points would be ... 

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...

The output of the simple moving average filter is the average of the M -F 1 most recent values of x(n). Intuitively, this corresponds to a smoothed version of the input, but its operation is more appropriately described by calculating the frequency response of the filter. First, however, the z-domain representation of the filter is introduced in analogy to the s- (or Laplace-) domain representation of analog filters. The z transform of a causal discrete-time signal x(n) is defined by... 

The magnitude and phase response of the simple moving average filter, with M = 7, are calculated from j (g 27r/) shown in Fig. 8.95. The filter is seen clearly to act as a crude low-pass, smoothing filter with a linear-phase response. The sampHng frequency periodicity in the magnitude and phase response is a property of discrete-time systems. The hnear-phase response is due to the term in... 

This process of moving the filter function through the raw data and calculating weighted averages is called convolution, and the digitally filtered data d, d2, 3,. .. are called the... 

The calculation of moving average and Savitsky-Golay filters is illustrated in Table 3.4. 

The convolution theorem states diat /, g and h are Fourier transforms of F, G and H. Hence linear filters as applied directly to spectroscopic data have their equivalence as Fourier filters in die time domain in other words, convolution in one domain is equivalent to multiplication in die other domain. Which approach is best depends largely on computational complexity and convenience. For example, bodi moving averages and exponential Fourier filters are easy to apply, and so are simple approaches, one applied direct to die frequency spectrum and die other to die raw time series. Convoluting a spectrum widi die Fourier transform of an exponential decay is a difficult procedure and so die choice of domain is made according to how easy the calculations are. 

The filter is then moved to the next position and the weighted average is again calculated (Fig. 3.24, bottom). The value for point 8 is... 

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