[Smoothing
Algorithms] [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]
[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 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):

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

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.

In all these smooths, the width of the smooth

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 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, 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 are greater than the smooth types discussed above, but with modern computers the difference is not significant and 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

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

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, 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 |

0.98 |

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 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. 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.

**
**

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) |

It should be clear that smoothing can seldom *
completely *eliminate noise, because most noise is
spread out over a wide 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 noise or single-point spikes) is there
hope of anything close to complete noise elimination.

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 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 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, 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 the 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 on the left 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 preferred.

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 may result.

For a more detailed 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:

(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 emphasizelong-termbehavior overshort-term, or

(b)if the signal will be subsequently analyzed by a method that would be degraded by the presence of too much high-frequency noise in the signal, for example if the heights of peaks are to be determinedvisually or graphicallyor 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 theunsmootheddata, 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 theunsmootheddata; 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 tolocate peaksbut it should not be used tomeasure peaks.

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. 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 propagation-of-error calculations and the bootstrap method.

**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. 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*. See http://en.wikipedia.org/wiki/Median_filter.
The killspikes.m function uses a
different approach; it locates and eliminates the spikes by
"patches over them" using linear interpolation from the
signal before and after. Unlike conventional smooths, these
functions can be profitably applied *prior *to
least-squares fitting functions. (On the other hand, if it's
the *spikes *that are actually the signal of
interest, and other components of the signal are interfering
with their measurement,
see CaseStudies.html#G).

**An alternative to smoothing **to reduce noise in the
set of ten unsmoothed signals
used above is 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 is enough
to judge 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.

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 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, 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 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) and column E does 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 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 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). In this spreadsheet, the smooth is applied three times in succession, resulting in an effective smooth width of 49 points applied to column G.

Compared to Matlab/Octave, spreadsheets are much slower, less flexible, and less easily automated. For example, in these spreadsheets, to change the signal or the number of points in the signal, or to change the smooth width or type, you have to modify the spreadsheet in several places, whereas to do the same using the Matlab/Octave "fastsmooth" function (below), you need only change the input arguments of a single line of code. And combining several different techniques into one spreadsheet is more complicated than writing a Matlab/Octave script that does the same thing.

SegmentedSmooth.m,
illustrated on the right, is
a segmented multiple-width data
smoothing function, based on the fastsmooth
algorithm, which can be useful if the widths of the peaks or
the noise level varies substantially across 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
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 examples.
DemoSegmentedSmooth.m
demonstrates the operation with different signals consisting
of noisy variable-width peaks
that get progressively wider, like the figure on the right.

Diederick
has published a Savitzky-Golay
smooth function in Matlab, which you can download from the Matlab
File Exchange. It's included in the iSignal function.
Greg
Pittam has published a modification of the fastsmooth
function that tolerates NaNs (Not a Number) in the data file
(nanfastsmooth(Y,w,type,tol))
and a version for smoothing angle data (nanfastsmoothAngle(Y,w,type,tol)).

SmoothWidthTest.m is a simple 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

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).

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

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).

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. (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)**

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, a median filter, and a condense 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.
Click to view larger figures.

**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:

`>> load ``NoisySignal`

` >> 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 February, 2017. This page is part of "

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