## Smoothing

[Smoothing Algorithms]  [Savitzky-Golay]  [[Noise Reduction]  [End Effects]  [Examples]  [The problem with smoothing]  [Optimization]  [When should you smooth a signal?]  [When should you NOT smooth a signal?]  [Dealing with spikes]  [Video Demonstration]  [Spreadsheets]   [Matlab/Octave]   [Live Script]   [Interactive tools]  [Have a question? Email me]

In many experiments in 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 (and a slower step response to signal changes). As long as the true underlying signal is actually smooth, then the true signal will not be much distorted by smoothing, but the high frequency noise will be reduced. In terms of the frequency components of a signal, a smoothing operation acts as a low-pass filter, reducing the high-frequency components and passing the low-frequency components with little change. If the signal and the noise is measured over all frequencies, then the signal-to-noise ratio will be improved by smoothing, by an amount that depends on the frequency distribution of the noise.

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 and divided by the sum of the coefficients, which becomes 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 boxcar 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 Sj the jth point in the smoothed signal, Yj the jth 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 D, then the standard deviation of the noise remaining in the signal after the first pass of an unweighted sliding-average smooth will be approximately D over the square root of m (D/sqrt(m)), where m is the smooth width. Despite its simplicity, this smooth is actually optimum for the common problem of reducing white noise while keeping the sharpest step response. The response to a step change is in fact linear, so this filter has the advantage of responding completely with no residual effect within its response time, which is equal to the smooth width divided by the sampling rate. Smoothing can be performed either during data acquisition, by programming the digitizer to measure and average multiple readings and save only the average, or after data acquisition ("post-run"), by storing all the acquired data in memory and smoothing the stored data. The latter requires more memory but is more flexible.

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 (see the spreadsheet UnitGainSmooths.xls). In both of these cases, the integer in the denominator is the sum of the coefficients in the numerator, which results in a "unit-gain" smooth that has no effect on the signal where it is a straight line and which preserves the area under peaks.

It is often useful to apply a smoothing operation more than once, that is, to smooth an already smoothed signal, in order to build longer and more complicated smooths. For example, the 5-point triangular smooth above is equivalent to two passes of a 3-point rectangular smooth.
Three passes of a 3-point rectangular smooth result in a 7-point "pseudo-Gaussian" or haystack smooth, for which the coefficients are in the ratio 1:3:6:7:6:3:1. The general rule is that n passes of a w-width smooth results in a combined smooth width of n*w-n+1. For example, 3 passes of a 17-point smooth results in a 49-point smooth. These multi-pass smooths are more effective at reducing high-frequency noise in the signal than a rectangular smooth, but they exhibit slower step response.

In all these smooths, the width of the smooth
m is chosen to be an odd integer, so that the smooth coefficients are symmetrically balanced around the central point, which is important because it preserves the x-axis position of peaks and other features in the smoothed signal. (This is especially critical for analytical and spectroscopic applications because the peak positions are often 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.

The
Savitzky-Golay smooth is a very widely-used smoothing algorithm that is based on the least-squares fitting of polynomials to segments of the data. The algorithm is discussed in http://www.wire.tu-bs.de/OLDWEB/mameyer/cmr/savgol.pdf. Compared to the sliding-average smooths of the same width, the Savitzky-Golay smooth is less effective at reducing noise, but more effective at retaining the shape of the original signal.  It is capable of differentiation as well as smoothing.  The algorithm is more complex and the computational times may be greater than the smooth types discussed above, but with modern computers the difference is not significant. Code in various languages is widely available online. See SmoothingComparison.html.

The shape of any smoothing algorithm can be determined by applying that smooth to a
signal consisting of all zeros except for one point, called a delta function  or impulse, as demonstrated by the simple Matlab/Octave script DeltaTest.m. The result is called the impulse response function (Graphic).

Noise reduction. Smoothing usually reduces the noise in a signal. If the noise is "white" (that is, evenly distributed over all frequencies) and its standard deviation is D, then the standard deviation of the noise remaining in the signal after one pass of a rectangular smooth will be approximately D/sqrt(m), where m is the smooth width. If a triangular smooth is used instead, the noise will be slightly less, about D*0.8/sqrt(m). Smoothing operations can be applied more than once: that is, a previously-smoothed signal can be smoothed again. In some cases this can be useful if there is a great deal of high-frequency noise in the signal. However, the noise reduction for white noise is less in each successive smooth. For example, three passes of a rectangular smooth reduces white noise by a factor of approximately D*0.7/sqrt(m), only a slight improvement over two passes.  For a spreadsheet demonstration, see VariableSmoothNoiseReduction.xlsx.

The frequency distribution of noise, designated by noise color, substantially effects the ability of smoothing to reduce noise. The Matlab/Octave function "NoiseColorTest.m" compares the effect of a 20-point boxcar (unweighted sliding average) smooth on the standard deviation of white, pink, red, and blue noise, all of which have an original unsmoothed standard deviation of 1.0. Because smoothing is a low-pass filter process, it effects low frequency (pink and red) noise less, and effects high-frequency (blue and violet) noise more, than it does white noise.

 Original unsmoothed noise 1 Smoothed white noise 0.1 Smoothed pink noise 0.55 Smoothed blue noise 0.01 Smoothed red (random walk) noise 0.98

Note that the computation of standard deviation is independent of the order of the data and thus of its frequency distribution; sorting a set of data does not change its standard deviation. The standard deviation of a sine wave is independent of its frequency. Smoothing, however, changes both the frequency distribution and standard deviation of a data set.

End effects and the lost points problem. In the equations above, 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 the usual way. 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 Matlb/Octave 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. The reduced noise allows the signal characteristics (peak position, height, width, area, etc.) to be measured more accurately by visual inspection.

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 smooth 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. Be aware that the smooth width can be expressed in two different ways: (a) as the number of data points or (b) as the x-axis interval (for spectroscopic data usually in nm or in frequency units). The two are simply related: the number of data points is simply the x-axis interval times the increment between adjacent x-axis values. The smooth ratio is the same in either case.

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 noticeably 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 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. In all cases, the total area under the peak remains unchanged. If the peak widths vary substantially, an adaptive smooth, which allows the smooth width to vary across the signal, may be used.

The problem with smoothing is that it is often less beneficial than you might think. It's important to point out that smoothing results such as illustrated in the figure above may be deceptively impressive because they employ a single sample of a noisy signal that is smoothed to different degrees. This causes the viewer 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 problem can visualized by recording a number of independent samples of a noisy signal consisting of a single peak, as illustrated in the two figures below. These figures show ten superimposed plots with the same peak but with independent white noise, each plotted with a different line color, unsmoothed on the left and smoothed on the right. Clearly, the noise reduction is substantial, but close inspection of the smoothed signals on the right clearly shows the variation in peak position, height, and width between the 10 samples caused by the low frequency noise remaining in the smoothed signals. Without the noise, each peak would have a peak height of 2, peak center at 500, and width of 150. Just because a signal looks smooth does not mean there is no noise. Low-frequency noise remaining in the signals after smoothing will still interfere with precise measurement of peak position, height, and width.

 x=1:1000; for n=1:10,  y(n,:)=2.*gaussian(x,500,150)+whitenoise(x); end plot(x,y) x=1:1000; for n=1:10,  y(n,:)=2.*gaussian(x,500,150)+whitenoise(x);  y(n,:)=fastsmooth(y(n,:),50,3); end plot(x,y)

(The generating scripts below each figure require that the functions gaussian.m, whitenoise.m, and fastsmooth.m be downloaded from http://tinyurl.com/cey8rwh.)

It should be clear that smoothing can seldom completely eliminate noise, because most noise is spread out over a range of frequencies, and smoothing simply reduces the noise in part of its frequency range. Only for some very specific types of noise (e.g. discrete frequency sine-wave noise or single-point spikes) is there hope of anything close to complete noise elimination. Smoothing does make the signal smoother and it does reduce the standard deviation of the noise, but whether or not that makes for a better measurement or not depends on the situation. And don't assume that just because a little smoothing is good that more will necessarily be better. Smoothing is like alcohol; sometimes you really need it - but you should never overdo it.

The figure on the right below is another example signal that illustrates some of these principles. 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 the same 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 peak (from x=0 to 100) and 1.0 for the second peak (from x=100 to 200). 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 least-squares curve fitting method to be covered later, the fit of the first peak is more stable with the noise and the measured parameters of that 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). You can download this data file, "udx", in TXT format or in Matlab MAT format.

Optimization of smoothing. As smooth width increases, the smoothing ratio increases, noise is reduced quickly at first, then more slowly, and the peak height is also reduced, slowly at first, then more quickly. The noise reduction depends on the smooth width, the smooth type (e.g. rectangular, triangular, etc), and the noise color, but the peak height reduction also depends on the peak width. The result is that the signal-to-noise (defined as the ratio of the peak height of the standard deviation of the noise) increases quickly at first, then reaches a maximum. This is illustrated in the animation at the top of this page for a Gaussian peak with white noise (produced by this Matlab/Octave script). The maximum improvement in the signal-to-noise ratio depends on the number of points in the peak: the more points in the peak, the greater smooth widths can be employed and the greater the noise reduction. This figure also illustrates that most of the noise reduction is due to high frequency components of the noise, whereas much of the low frequency noise remains in the signal even as it is smoothed.

Which is the best smooth ratio?  It depends on the purpose of the peak measurement. If the ultimate objective of the measurement is to measure the peak height or width, then smooth ratios below 0.2 should be used and the Savitzky-Golay smooth is preferred. But if the objective of the measurement is to measure the peak position (x-axis value of the peak), larger smooth ratios can be employed if desired, because smoothing has little effect on the peak position (unless peak is asymmetrical or the increase in peak width is so much that it causes adjacent peaks to overlap). If the peak is actually formed of two underlying peaks that overlap so much that they appear to be one peak, then curve fitting is the only way to measure the parameters of the underlying peaks. Unfortunately, the optimum signal-to-noise ratio corresponds to a smooth ratio that significantly distorts the peak, which is why curve fitting the unsmoothed data is often the preferred method for measuring peaks position, height, and width. Peak area is not changed by smoothing, unless it changes your estimate of the beginning and the ending of the peak.

In quantitative chemical analysis applications based on calibration by standard samples, the peak height reduction caused by smoothing is not so important. 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, as shown in the figure on the left (for a simple sliding-average rectangular smooth). 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 unknown.) 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 will result.

For a more detailed comparison of all four smoothing types considered above, see SmoothingComparison.html.

When should you smooth a signal?  There are four reasons to smooth a signal:

(a) for cosmetic reasons, to prepare a nicer-looking or more dramatic graphic of a signal for visual inspection or publications, especially in order to emphasize long-term behavior over short-term, or
(b) If the signal contains mostly high-frequency ("blue") noise, which can look bad but has less effect on the low-frequency signal components (e.g. the positions, heights, widths, and areas of peaks) than white noise.
(c) if the signal will be subsequently analyzed by a method that would be degraded by the presence of  too much noise in the signal, for example if the heights of peaks are to be determined visually or graphically or by using the MAX function, of the the widths of peaks is measured by the halfwidth function, or if the location of maxima, minima, or inflection points in the signal is to be determined  automatically by detecting zero-crossings in derivatives of the signal. Optimization of the amount and type of smoothing is important in these cases (see Differentiation.html#Smoothing). But generally, if a computer is available to make quantitative measurements, it's better to use least-squares methods on the unsmoothed data, rather than graphical estimates on smoothed data. If a commercial instrument has the option to smooth the data for you, it's best to disable the smoothing and record and save the unsmoothed data; you can always smooth it yourself later for visual presentation and it will be better to use the unsmoothed data for an least-squares fitting or other processing that you may want to do later. Smoothing can be used to locate peaks but it should not be used to measure peaks.
(d). The formal limit of detection and limit of quantification of an analytical method
(reference 89) may be improved by smoothing or averaging, depending on the detailed method of signal measurement, as described in the previous section and demonstrated by the Matlab/Octave script SNRdemo.m.

Care must be used in the design of algorithms that employ smoothing. For example, in a popular technique for peak finding and measurement discussed later, 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, rather than simply taking the maximum of the smoothed data. That way, 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 as least-squares 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 the algebraic propagation-of-error calculations and by the bootstrap method. Even a visual estimate of the quality of the signal is compromised by smoothing, which makes the signal look better than it really is.

Dealing with spikes and outliers. Sometimes signals are contaminated with very tall, narrow "spikes" or "outliers" occurring at random intervals and with random amplitudes, but with widths of only one or a few points. For example, optical spectroscopy using photomultiplier tube detectors is subject to spikes caused by "cosmic rays" from outer space passing through the front window of the tube, creating a pulse of Cherenkov radiation. Spikes not only look ugly, but they also upset 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. See http://en.wikipedia.org/wiki/Median_filter. For a practical example, see Appendix C. A different approach is used by the killspikes.m function; it locates and eliminates the spikes by "patching over them" using linear interpolation from the signal points before and after the spike. Unlike conventional smooths, these functions can be profitably applied prior to least-squares fitting functions. (On the other hand, if the spikes themselves are actually the signal of interest, and the other components of the signal are interfering with their measurement, see CaseStudies.html#G).

An alternative to smoothing
to reduce noise in repeatable signals, such as the set of ten unsmoothed signals above, is simply to compute their average, called ensemble averaging, which can be performed in this case very simply by the Matlab/Octave code plot(x,mean(y)); the result shows a reduction in white noise by about sqrt(10)=3.2. This improves the signal-to-noise ratio enough to see that there is a single peak with Gaussian shape, which can then be measured by curve fitting (covered in a later section) using the Matlab/Octave code peakfit([x;mean(y)],0,0,1), with the result showing excellent agreement with the position (500), height (2), and width (150) of the Gaussian peak created in the third line of the generating script (above left). A huge advantage of ensemble averaging is that the noise at all frequencies is reduced, not just the high-frequency noise as in smoothing. This is a big advantage if the signal or the baseline drifts.

Condensing oversampled signals. Sometimes signals are recorded more densely (that is, with smaller x-axis intervals) than really necessary to capture all the important 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 better by replacing groups of adjacent points by their averages. The later approach has the advantage of using rather than discarding 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, but with no change in frequency distribution of the noise). The Matlab/Octave script testcondense.m demonstrates the effect of boxcar averaging using the condense.m function to reduce noise without changing the noise color. Shows that the boxcar reduces the measured noise, removing the high frequency components but has little effect on the the peak parameters. Least-squares curve fitting on the condensed data is faster and results in a lower fitting error, but no more accurate measurement of peak parameters.

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).

SPECTRUM, the freeware Macintosh signal-processing application, includes rectangular and triangular smoothing functions for any number of points.

Spreadsheets. Smoothing can be done in spreadsheets using the "shift and multiply" technique described above. In the spreadsheets smoothing.ods and smoothing.xls (screen image) the set of multiplying coefficients is contained in the formulas that calculate the values of each cell of the smoothed data in columns C and E. Column C performs a 7-point rectangular smooth (1 1 1 1 1 1 1). Column E performs a 7-point triangular smooth (1 2 3 4 3 2 1), applied to the data in column A. You can type in (or Copy and Paste) any data you like into column A, and you can extend the spreadsheet to longer columns of data by dragging the last row of columns A, C, and E down as needed. But to change the smooth width, you would have to change the equations in columns C or E and copy the changes down the entire column. It's common practice to divide the results by the sum of the coefficients so that the net gain is unity and the area under the curve of the smoothed signal is preserved.  The spreadsheets UnitGainSmooths.xls and UnitGainSmooths.ods  (screen image) contain a collection of unit-gain convolution coefficients for rectangular, triangular, and Gaussian smooths of width 3 to 29 in both vertical (column) and horizontal (row) format. You can Copy and Paste these into your own spreadsheets.

The spreadsheets MultipleSmoothing.xls and MultipleSmoothing.ods (screen image) demonstrate a more flexible method in which the coefficients are contained in a group of 17 adjacent cells (in row 5, columns I through Y), making it easier to change the smooth shape and width (up to a maximum of 17) just by changing those 17 cells. (To make a smaller smooth, just insert zeros for the unused coefficients; in this example, a 7-point triangular smooth is defined in columns N - T and the rest of the coefficients are zeros). In this spreadsheet, the smooth is applied three times in succession in columns C, E, and G, resulting in an effective maximum smooth width of  n*w-n+1 = 49 points applied to column G.

The example in the graphic on the right shows smoothing applied to a step change occurring at x=111; without smoothing (blue line) the step is almost invisible. For example, the smoothed signal could be used to trigger an alarm whenever it exceeds a value of .2, warning that a step has occurred, whereas the raw unsmoothed signal would be completely unsuitable for that purpose.

Another set of spreadsheets that uses this same AVERAGE(INDIRECT()) technique is SegmentedSmoothTemplate.xlsx, a segmented multiple-width data smoothing spreadsheet template that can apply individually specified different smooth widths to different regions of the signal. This is especially useful if the widths of the peaks or the noise level varies substantially across the signal. In this version there are 20 segments. SegmentedSmoothExample.xlsx is an example with built-in data (graphic); note that the plot is conveniently lined up with the columns containing the smooth widths for each segment. A related sheet GradientSmoothTemplate.xlsx and GradientSmoothExample2.xlsx (graphic) performs a linearly increasing (or decreasing) smooth width across the entire signal, given only the start and end values, automatically generating as many segments and different smooth widths as are necessary. (It also enforces the restriction, in column C, that each smooth width must be an odd number, to prevent an x-axis shift in the smoothed data).

Smoothing in Matlab and Octave. The "mean" function, in both Matlab and Python, implements a single sliding average smooth. The custom function fastsmooth implements shift and multiply type smooths using a faster recursive algorithm. (Click on this link to inspect the code, or right-click to download for use within Matlab). This is a Matlab function of the form s=fastsmooth(a,w, type, edge). The argument "a" is the input signal vector; "w" is the smooth width (a positive integer); "type" determines the smooth type: type=1 gives a rectangular (sliding-average or boxcar) smooth; type=2 gives a triangular smooth, equivalent to two passes of a sliding average; type=3 gives a pseudo-Gaussian or "p-spline" smooth, equivalent to three passes of a sliding average; these shapes are compared in the figure on the left.  (See SmoothingComparison.html for a comparison of these smoothing modes). The argument "edge" controls how the "edges" of the signal (the first w/2 points and the last w/2 points) are handled. If edge=0, the edges are zero. (In this mode the elapsed time is independent of the smooth width. This gives the fastest execution time). If edge=1, the edges are smoothed with progressively smaller smooths the closer to the end. (In this mode the execution time increases with increasing smooth widths). The smoothed signal is returned as the vector "s". (You can leave off the last two input arguments: fastsmooth(Y,w,type) smooths with edge=0 and fastsmooth(Y,w) smooths with type=1 and edge=0). Compared to convolution-based smooth algorithms, fastsmooth uses a simple recursive algorithm that typically gives faster execution times for very large smooth widths; it can smooth a 1,000,000 point signal with a 1,000 point sliding average in less than 0.1 second. Here's a simple example of fastsmooth demonstrating the effect on white noise (graphic).

x=1:100;
y=randn(size(x));
plot(x,y,x,fastsmooth(y,5,3,1),'r')
xlabel('Blue: white noise.    Red: smoothed white noise.')

SegmentedSmooth.m, illustrated on the right, is a segmented version of fastsmooth, which applies different smooth widths to different segments of the signal. The syntax is the same as fastsmooth.m, except that the second input argument "smoothwidths" can be a vector: SmoothY = SegmentedSmooth (Y, smoothwidths, type, ends).  The function divides Y into a number of equal-length regions defined by the length of the vector 'smoothwidths', then smooths each region with a smooth of type 'type' and width defined by the elements of vector 'smoothwidths'. In the graphic example in the figure on the right, smoothwidths=[31 52 91], which divides up the signal into three equal regions and smooths the first region with smoothwidth 31, the second with smoothwidth 51, and the last with smoothwidth 91.  Any number of smooth widths and sequence of smooth widths can be used. Type "help SegmentedSmooth" for other examples. The demonstration script DemoSegmentedSmooth.m shows the operation with signal (DemoSegmentedSmooth.mat) consisting of noisy variable-width peaks that get progressively wider, like the figure on the right. If the peak widths increase or decrease regularly across the signal, you can calculate the smoothwidths vector by giving only the number of segments ("NumSegments") , the first value, "startw", and the last value, "endw", like so:

wstep=(endw-startw)/NumSegments;

smoothwidths=startw:wstep:endw;

SmoothWidthTest.m is a demonstration script that uses the fastsmooth function to demonstrate the effect of smoothing on peak height, noise, and signal-to-noise ratio of a peak. You can change the peak shape in line 7, the smooth type in line 8, and the noise in line 9. A typical result for a Gaussian peak with white noise smoothed with a pseudo-Gaussian smooth is shown on the left. Here, as it is for most peak shapes, the optimal signal-to-noise ratio occurs at a smooth ratio of about 0.8. However, that optimum corresponds to a
significant reduction in the peak height, which could be a problem. A smooth width about half the width of the original unsmoothed peak produces less distortion of the peak but still achieves a reasonable noise reduction. SmoothVsCurvefit.m is a similar script, but is also compares curve fitting as an alternative method to measure the peak height without smoothing.

This effect is explored more completely by the text below, which shows an experiment in Matlab or Octave that creates a Gaussian peak, smooths it, compares the smoothed and unsmoothed version, then uses the max(), halfwidth(), and trapz() functions to print out the peak height, halfwidth, and area. (max and trapz are both built-in functions in Matlab and Octave, but you have to download halfwidth.m. To learn more about these functions, type "help" followed by the function name).

x=[0:.1:10]';
y=exp(-(x-5).^2);
plot(x,y)
ysmoothed=fastsmooth(y,11,3,1);
plot(x,y,x,ysmoothed,'r')
disp([max(y) halfwidth(x,y,5)  trapz(x,y)])
disp([max(ysmoothed) halfwidth(x,ysmoothed,5)  trapz(x,ysmoothed)])

1       1.6662       1.7725
0.78442       2.1327       1.7725

These results show that smoothing reduces the peak height (from 1 to 0.784) and increases the peak width (from 1.66 to 2.13), but has no effect on the peak area, as long as you measure the total area under the broadened peak.

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 least-squares results obtained on the unsmoothed data will be more accurate (see CurveFittingC.html#Smoothing).

The Matlab/Octave user-defined function condense.m
, condense(y,n), returns a condensed version of y in which each group of n points is replaced by its average, reducing the length of y by the factor n. This operation is also called "bunching".  (For x,y data sets, use this function on both independent variable x and dependent variable y so that the features of y will appear at the same x values).

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). killspikes.m is a threshold-based filter for eliminating narrow spike artifacts. The syntax is fy= killspikes(x, y, threshold, width). Each time it finds a positive or negative jump in the data between y(n) and y(n+1) that exceeds "threshold", it replaces the next "width" points of data with a linearly interpolated segment spanning x(n) to x(n+width+1), See killspikesdemo. Type "help killspikes" at the command prompt.

ProcessSignal is 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)

Real-time smoothing in Matlab is discussed in Appendix Y.

Live script for smoothing
Here is a Matlab Live Script (download link: DataSmoothing.mlx) for performing several types of smoothing applied to experimental data stored on disk. It can perform:

Clicking the "Open data file" button in line 1 opens a file browser, allowing you to navigate to your data file (in .csv or .xlsx format; the script assumes that your x,y data are in the first two columns). All the variables and settings appear in the Matlab workspace as usual; the finished smoothed data are in the vector "sy".

The script has several interactive controls. The startpc and endpc sliders in lines 7 and 8 allow you to select which portion of the data range to process, from 0% to 100% of the total range of the data file.

The SmoothType drop-down menu in line 13 selects the smoothing algorithm; each has one or more controls specific to that type in lines 16 to 30. The first choice is the recursive sliding average (fastsmooth.m) algorithm explained above. The smooth width and number of passes are controlled by the sliders in lines 16 and 17. Each The other controls are explained in the accompanying comment lines (in green). Fourier filtering, Savitsky-Golay and wavelet denoising are topics that will be explained in later sections.

The PlotBeforeAndAfter checkbox in line 3 gives you the option of plotting the original signal (in black) along with the processed signal (in red).

The FrequencySpectra checkbox in line 4 allows you to show the frequency spectrum of the original and/or processed signals (see HarmonicAnalysis.html). Note: to view the graphic plots to the right of the code, as shown above, right-click on the empty space on the right and select "Disable synchronous scrolling".

Smoothing in iSignal

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, segmented smooth, a median filter, and a condense (bunching) function, with keystrokes that allow you to adjust the smoothing parameters continuously while observing the effect on your signal instantly, making it easy to observe how different types and amounts of smoothing effect noise and signal, such as the height, width, and areas of peaks. Other functions include differentiation, peak sharpening, interpolation, least-squares peak measurement, and a frequency spectrum mode that shows how smoothing and other functions can change the frequency spectrum of your signals. The simple script "iSignalDeltaTest" demonstrates the frequency response of iSignal's smoothing functions by applying them to a single-point spike, allowing you to change the smooth type and the smooth width to see how the the frequency response changes. View the code here or download the ZIP file with sample data for testing.

iSignal for Matlab, showing the smoothing of a signal (upper panel) that is contaminated by a strong parasitic oscillation. The frequency spectrum is shown in the lower panel. As the smooth width increases, the oscillation is suppressed and the signal peaks come into view. Too much smoothing, however, broadens and shortens the peaks. Click to view larger figure.

You try it: Here's an example of a very noisy signal with lots of high-frequency (blue) noise totally obscuring a perfectly good peak in the center at x=150, height=1e-4; SNR=90.  First, download NoisySignal into the Matlab path, then execute these statements:

>> isignal(x,y);

Use the A and Z keys to increase and decrease the smooth width, and the S key to cycle through the available smooth types. Hint: use the Gaussian smooth and keep increasing the smooth width until the peak shows.

Note: you can right-click on any of the m-file links on this site and select Save Link As... to download them to your computer for use within Matlab. Unfortunately, iSignal does not currently work in Octave.

An earlier version of his page is available in French, at http://www.besteonderdelen.nl/blog/?p=4169, courtesy of Natalie Harmann and Anna Chekovsky.

Last updated April, 2023. This page is part of "A Pragmatic Introduction to Signal Processing", created and maintained by Prof. Tom O'Haver , Department of Chemistry and Biochemistry, The University of Maryland at College Park. Comments, suggestions, bug reports, and questions should be directed to Prof. O'Haver at toh@umd.edu.
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