Smoothing algorithms. Most smoothing algorithms are based on the "shift and multiply" technique, in which a group of adjacent points in the original data are multiplied point-by-point by a set of numbers (coefficients) that defines the smooth shape, the products are added up to become one point of smoothed data, then the set of coefficients is shifted one point down the original data and the process is repeated. 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):
The triangular smooth is like the rectangular smooth, above, except that it implements a weighted smoothing function. For a 5-point smooth (m = 5):
End effects and the lost points problem. Note in the equations above that the 3-point rectangular smooth is defined only for j = 2 to n-1. There is not enough data in the signal to define a complete 3-point smooth for the first point in the signal (j = 1) or for the last point (j = n) , because there are no data points before the first point or after the last point. (Similarly, a 5-point smooth is defined only for j = 3 to n-2, and therefore a smooth can not be calculated for the first two points or for the last two points). In general, for an m-width smooth, there will be (m-1)/2 points at the beginning of the signal and (m-1)/2 points at the end of the signal for which a complete m-width smooth can not be calculated. What to do? There are two approaches. One is to accept the loss of points and trim off those points or replace them with zeros in the smooth signal. (That's the approach taken in most of the figures in this paper). The other approach is to use progressively smaller smooths at the ends of the signal, for example to use 2, 3, 5, 7... point smooths for signal points 1, 2, 3,and 4..., and for points n, n-1, n-2, n-3..., respectively. The later approach may be preferable if the edges of the signal contain critical information, but it increases execution time. The fastsmooth function discussed below can utilize either of these two methods.
Examples of smoothing. A simple example of smoothing is shown in Figure 4. The left half of this signal is a noisy peak. The right half is the same peak after undergoing a triangular smoothing algorithm. The noise is greatly reduced while the peak itself is hardly changed. Smoothing increases the signal-to-noise ratio and allows the signal characteristics (peak position, height, width, area, etc.) to be measured more accurately, especially when computer-automated methods of locating and measuring peaks are being employed.
Figure 4. The left half of this signal is a noisy peak. The right half is the same peak after undergoing a smoothing algorithm. The noise is greatly reduced while the peak itself is hardly changed, making it easier to measure the peak position, height, and width directly by graphical or visual estimation, but it does not improve measurements made by least-squares methods (see below).
The larger the smooth width, the greater the noise reduction, but also the greater the possibility that the signal will be distorted by the smoothing operation. The optimum choice of smooth width depends upon the width and shape of the signal and the digitization interval. For peak-type signals, the critical factor is the smoothing ratio, the ratio between the smooth width m and the number of points in the half-width of the peak. In general, increasing the smoothing ratio improves the signal-to-noise ratio but causes a reduction in amplitude and in increase in the bandwidth of the peak.
The figures above show examples of the effect of
three different smooth widths on noisy Gaussian-shaped
peaks. In the figure on the left, the peak has a (true)
height of 2.0 and there are 80 points in the half-width of
the peak. The red line is the original unsmoothed peak. The
three superimposed green lines are the results of smoothing
this peak with a triangular smooth of width (from top to
bottom) 7, 25, and 51 points. Because the peak width is 80
points, the smooth ratios of these three smooths are
7/80 = 0.09, 25/80 = 0.31, and 51/80 = 0.64, respectively.
As the smooth width increases, the noise is progressively
reduced but the peak height also is reduced slightly. For
the largest smooth, the peak width is slightly increased. In
the figure on the right, the original peak (in red) has a
true height of 1.0 and a half-width of 33 points. (It is
also less noisy than the example on the left.) The three
superimposed green lines are the results of the same three
triangular smooths of width (from top to bottom) 7, 25, and
51 points. But because the peak width in this case is only
33 points, the smooth ratios of these three smooths
are larger - 0.21, 0.76, and 1.55, respectively. You can see
that the peak distortion effect (reduction of peak height
and increase in peak width) is greater for the narrower peak
because the smooth ratios are higher. Smooth ratios of
greater than 1.0 are seldom used because of excessive peak
distortion. Note that even in the worst case, the peak
positions are not effected (assuming that the original peaks
were symmetrical and not overlapped by other peaks). If
retaining the shape of the peak is more important than
optimizing the signal-to-noise ratio, the Savitzky-Golay has
the advantage over sliding-average smooths.
It's important to point out that smoothing results
such as illustrated in the figures above may be deceptively optimistic
because they employ a single
sample of a noisy signal that is smoothed to
different degrees. Smoothing is essentially a type of
low-pass filtering that reduces the high-frequency
components of a signal while retaining the low-frequency
components. This causes the viewer to overestimate the
quality of a smoothed noisy signal, because one tends to underestimate the
contribution of low-frequency noise, which is hard
to estimate visually because there are so few low-frequency
cycles in the signal record. This error can be
remedied by taking a large number of independent samples of
noisy signal. The same sort of error occurs when least-squares methods methods
are used to measure the parameters such as the slope,
intercept, height, position, and width of noisy
The figure on
the right is another example signal that illustrates some of
these principles. You can download the data file "udx"
in TXT format or in Matlab MAT format. The signal consists of two
Gaussian peaks, one located at x=50 and the second at x=150.
Both peaks have a peak height of 1.0 and a peak half-width
of 10, and a normally-distributed random white noise with a
standard deviation of 0.1 has been added to the entire
signal. The x-axis sampling interval, however, is different
for the two peaks; it's 0.1 for the first peaks and 1.0 for
the second peak. This means that the first peak is
characterized by ten times more points that the second peak.
It may look like
the first peak is noisier than the second, but that's just
an illusion; the signal-to-noise ratio for both peaks is 10.
The second peak looks less noisy only because there are
fewer noise samples there and we tend to underestimate the
dispersion of small samples. The result of this is that when
the signal is smoothed, the second peak is much more likely
to be distorted by the smooth (it becomes shorter and wider)
than the first peak. The first peak can tolerate a much
wider smooth width, resulting in a greater degree of noise
reduction. (Similarly, if both peaks are measured with the peakfit method, the
results on the first peak will be about 3 times more
accurate than the second peak, because there are 10 times
more data points in that peak, and the measurement precision
improves roughly with the square root of the number of data
points if the noise is white).
Optimization of smoothing. Which is the best smooth ratio? It depends on the purpose of the peak measurement. If the objective of the measurement is to measure the true peak height and width, then smooth ratios below 0.2 should be used. (In the example on the left above, the original peak (red line) has a peak height greater than the true value 2.0 because of the noise, whereas the smoothed peak with a smooth ratio of 0.09 has a peak height that is much closer to the correct value). Measuring the height of noisy peaks is much better done by curve fitting the unsmoothed data rather than by taking the maximum of the smoothed data (see CurveFittingC.html#Smoothing). But if the objective of the measurement is to measure the peak position (x-axis value of the peak), much larger smooth ratios can be employed if desired, because smoothing has no effect at all on the peak position (unless the increase in peak width is so much that it causes adjacent peaks to overlap).
In quantitative analysis applications, the peak height reduction caused by smoothing is not so important, because in most cases calibration is based on the signals of standard solutions. If the same signal processing operations are applied to the samples and to the standards, the peak height reduction of the standard signals will be exactly the same as that of the sample signals and the effect will cancel out exactly. In such cases smooth widths from 0.5 to 1.0 can be used if necessary to further improve the signal-to-noise ratio. In practical analytical chemistry, absolute peak height measurements are seldom required; calibration against standard solutions is the rule. (Remember: the objective of quantitative analysis is not to measure a signal but rather to measure the concentration of the analyte.) It is very important, however, to apply exactly the same signal processing steps to the standard signals as to the sample signals, otherwise a large systematic error may result.
For a comparison of all four smoothing types
considered above, see SmoothingComparison.html.
When should you smooth a signal? There are two reasons to smooth a signal: (1) for cosmetic reasons, to prepare a nicer-looking graphic of a signal for visual inspection or publication, and (2) if the signal will be subsequently processed by an algorithm that would be adversely effected by the presence of too much high-frequency noise in the signal, for example if the heights of peaks are to be determined graphically or by using the MAX function, or if the location of maxima, minima, or inflection points in the signal is to be automatically determined by detecting zero-crossings in derivatives of the signal. Optimization of the amount and type of smoothing is very important in these cases (see Differentiation.html#Smoothing).
Care must be used in the design of algorithms that employ smoothing. For example, in a popular technique for peak finding and measurement, peaks are located by detecting downward zero-crossings in the smoothed first derivative, but the position, height, and width of each peak is determined by least-squares curve-fitting of a segment of original unsmoothed data in the vicinity of the zero-crossing. Thus, even if heavy smoothing is necessary to provide reliable discrimination against noise peaks, the peak parameters extracted by curve fitting are not distorted by the smoothing.
When should you NOT smooth a
signal? One common situation where you should not
smooth signals is prior to statistical procedures such
curve fitting, because: (a) smoothing will not
significantly improve the accuracy of parameter measurement
by least-squares measurements between separate independent
signal samples; (b) all smoothing algorithms are
at least slightly "lossy", entailing at least some
change in signal shape and amplitude, (c) it is harder to
evaluate the fit by inspecting the residuals if the data are
smoothed, because smoothed
noise may be mistaken for an actual signal, and (d)
smoothing the signal will seriously underestimate the
parameters errors predicted by propagation-of-error
calculations and the bootstrap method.
Smoothing can be used to locate peaks
but it should not be used to measure peaks.
Dealing with spikes. Sometimes signals are contaminated with very tall, narrow “spikes” occurring at random intervals and with random amplitudes, but with widths of only one or a few points. It not only looks ugly, but it also upsets the assumptions of least-squares computations because it is not normally-distributed random noise. This type of interference is difficult to eliminate using the above smoothing methods without distorting the signal. However, a “median” filter, which replaces each point in the signal with the median (rather than the average) of m adjacent points, can completely eliminate narrow spikes with little change in the signal, if the width of the spikes is only one or a few points and equal to or less than m. It can be applied prior to least-squares functions. See http://en.wikipedia.org/wiki/Median_filter.
Condensing oversampled signals. Sometimes signals are recorded more densely (that is, with smaller x-axis intervals) than really necessary to capture all the features of the signal. This results in larger-than-necessary data sizes, which slows down signal processing procedures and may tax storage capacity. To correct this, oversampled signals can be reduced in size either by eliminating data points (say, dropping every other point or every third point) or by replacing groups of adjacent points by their averages. The later approach has the advantage of using rather than discarding extraneous data points, and it acts like smoothing to provide some measure of noise reduction. (If the noise in the original signal is white, and the signal is condensed by averaging every n points, the noise is reduced in the condensed signal by the square root of n, with no change in frequency distribution of the noise).
Video Demonstration. This 18-second, 3 MByte video (Smooth3.wmv) demonstrates the effect of triangular smoothing on a single Gaussian peak with a peak height of 1.0 and peak width of 200. The initial white noise amplitude is 0.3, giving an initial signal-to-noise ratio of about 3.3. An attempt to measure the peak amplitude and peak width of the noisy signal, shown at the bottom of the video, are initially seriously inaccurate because of the noise. As the smooth width is increased, however, the signal-to-noise ratio improves and the accuracy of the measurements of peak amplitude and peak width are improved. However, above a smooth width of about 40 (smooth ratio 0.2), the smoothing causes the peak to be shorter than 1.0 and wider than 200, even though the signal-to-noise ratio continues to improve as the smooth width is increased. (This demonstration was created in Matlab 6.5.
Diederick has published a Savitzky-Golay smooth function in Matlab, which you can download from the Matlab File Exchange.
Here's a simple experiment in Matlab or Octave that creates a Gaussian peak, smooths it, compares the smoothed and unsmoothed version, then uses the peakfit.m function (version 3.4 or later) to show that smoothing reduces the peak height (from 1 to 0.786) and increases the peak width (from 1.66 to 2.12), but has little effect on the total peak area (a mere 0.2% change). Smoothing is useful if the signal is contaminated by non-normal noise such as sharp spikes or if the peak height, position, or width are measured by simple methods, but there is no need to smooth the data if the noise is white and the peak parameters are measured by least-squares methods, because the results obtained on the unsmoothed data will be more accurate (see CurveFittingC.html#Smoothing).>> x=[0:.1:10]';
The Matlab/Octave user-defined function medianfilter.m, medianfilter(y,w), performs a median-based filter operation that replaces each value of y with the median of w adjacent points (which must be a positive integer).
ProcessSignal, a Matlab/Octave command-line function that performs smoothing and differentiation on the time-series data set x,y (column or row vectors). It can employ all the types of smoothing described above. Type "help ProcessSignal". Returns the processed signal as a vector that has the same shape as x, regardless of the shape of y. The syntax is Processed=ProcessSignal(x, y, DerivativeMode, w, type, ends, Sharpen, factor1, factor2, SlewRate, MedianWidth)
iSignal is an interactive function for Matlab that performs smoothing for time-series signals using all the algorithms discussed above, including the Savitzky-Golay smooth, with keystrokes that allow you to adjust the smoothing parameters continuously while observing the effect on your signal instantly. Version 2.2 also includes a median filter and a condense function. Other functions include differentiation, peak sharpening, and least-squares peak measurement. View the code here or download the ZIP file with sample data for testing.
iSignal for Matlab. Click to view larger figures.
you can right-click on any of the m-file links on
this site and select Save
to download them to your computer for use within Matlab.
Unfortunately, iSignal does not currently work in Octave.