4. Smoothing

In many experiments in physical science, the true signal amplitudes (y-axis values) change rather smoothly as a function of the x-axis values, whereas many kinds of noise are seen as rapid, random changes in amplitude from point to point within the signal. In the latter situation it may be useful in some cases to attempt to reduce the noise by a process called smoothing. In smoothing, the data points of a signal are modified so that individual points that are higher than the immediately adjacent points (presumably because of noise) are reduced, and points that are lower than the adjacent points are increased. This naturally leads to a smoother signal. As long as the true underlying signal is actually smooth, then the true signal will not be much distorted by smoothing, but the noise will be reduced.

Ø Smoothing algorithms.

The simplest smoothing algorithm is the rectangular or unweighted sliding-average smooth; it simply replaces each point in the signal with the average of m adjacent points, where m is a positive integer called the smooth width. For example, for a 3-point smooth (m = 3):

for j = 2 to n-1, where S_j the j^th point in the smoothed signal, Y_j the j^th point in the original signal, and n is the total number of points in the signal. Similar smooth operations can be constructed for any desired smooth width, m. Usually m is an odd number. If the noise in the data is "white noise" (that is, evenly distributed over all frequencies) and its standard deviation is s, then the standard deviation of the noise remaining in the signal after the first pass of an unweighted sliding-average smooth will be approximately s over the square root of m (s/sqrt(m)), where m is the smooth width.

The triangular smooth is like the rectangular smooth, above, except that it implements a weighted smoothing function. For a 5-point smooth (m = 5):

for j = 3 to n-2, and similarly for other smooth widths. This is equivalent to two passes of a 3-point rectangular smooth. This smooth is more effective at reducing high-frequency noise in the signal than the simpler rectangular smooth. Note that again in this case, the width of the smooth m is an odd integer and the smooth coefficients are symmetrically balanced around the central point, which is important point because it preserves the x-axis position of peaks and other features in the signal. (This is especially critical for analytical and spectroscopic applications because the peak positions are sometimes important measurement objectives).

Note that we are assuming here that the x-axis intervals of the signal is uniform, that is, that the difference between the x-axis values of adjacent points is the same throughout the signal.  This is also assumed in many of the other signal-processing techniques described in this essay, and it is a very common (but not necessary) characteristic of signals that are acquired by automated and computerized equipment.