**Differentiation**

[Basic Properties of Derivatives] [Applications of Differentiation] [Derivative Spectroscopy] [Trace Analysis] [The Importance of Smoothing Derivatives] [Video Demonstrations] [Spreadsheets] [Differentiation in Matlab and Octave] [Interactive tools] [Have a question? Email me]

The symbolic differentiation of functions is a topic that is introduced in all elementary Calculus courses. The numerical differentiation of digitized signals is an application of this concept that has many uses in analytical signal processing. The first derivative of a signal is the rate of change of y with x, that is, dy/dx, which is interpreted as the

(for
1< j <n-1).

where X'_{j} and Y'_{j} are the
X and Y values of the j^{th} point of the derivative, n
= number of points in the signal, and X is the difference between the X values of
adjacent data points. A commonly used variation of this
algorithm computes the average slope between three adjacent
points:

(for 2
< j <n-1).

This is called a *central-difference*
method; its advantage is that it does not involve a shift in the
x-axis position of the derivative.

The *second derivative* is the derivative
of the derivative: it is a measure of the *curvature* of
the signal, that is, the rate of change of the slope of the
signal. It can be calculated by applying the first derivative
calculation twice in succession. The simplest algorithm for
direct computation of the second derivative in one step is

(for 2
< j <n-1).

Similarly, higher derivative orders can be
computed using the appropriate sequence of coefficients: for
example +1, -2, +2, -1 for the third derivative and +1, -4, +6,
-4, +1 for the 4^{th }derivative, although these
derivatives can also be computed simply by taking successive
lower order derivatives.

It's also possible to compute *gap-segment*
derivatives in which the x-axis interval between the points in
the above expressions is greater than one; for example, Y_{j-2}
and Y_{j+2}, or Y_{j-3} and Y_{j+3},
etc. It turns out that this is equivalent to applying a
moving-average (rectangular) smooth in addition to the
derivative.

The Savitzky-Golay smooth can also be used as a differentiation algorithm with the appropriate choice of input arguments; it combines differentiation and smoothing into one algorithm.

**Basic
Properties of Derivative Signals**

The figure on the left shows the
results of the successive differentiation of a
computer-generated signal (click to see the full-sized figure).
The signal in each of the four windows is the first derivative
of the one before it; that is, Window 2 is the first derivative
of Window 1, Window 3 is the first derivative of Window 2,
Window 3 is the *second* derivative of Window 1, and so
on. You can predict the shape of each signal by recalling that
the derivative is simply the slope of the original signal: where
a signal slopes up, its derivative is positive; where a signal
slopes down, its derivative is negative; and where a signal has
zero slope, its derivative is zero. (Matlab/Octave
code for this figure.)

The sigmoidal signal shown in Window 1 has an *inflection
point* (point where where the slope is maximum) at the
center of the x axis range. This corresponds to the *maximum*
in its first derivative (Window 2) and to the *zero-crossing*
(point where the signal crosses the x-axis going either from
positive to negative or *vice versa*) in the second
derivative in Window 3. This behavior can be useful for
precisely locating the inflection point in a sigmoid signal, by
computing the location of the zero-crossing in its second
derivative. Similarly, the location of the maximum
in a peak-type signal can be computed precisely by computing the
location of the zero-crossing in its first derivative.

Another
important property of the differentiation of peak-type signals
is the effect of the peak width on the amplitude of derivatives.
The figure on the left shows the results of the successive
differentiation of two computer-generated Gaussian bands (click
to see the full-sized figure). The two bands have the same
amplitude (peak height) but one of them is exactly twice the
width of the other. As you can see, the *wider* peak has
the *smaller* derivative amplitude, and the effect becomes
more noticeable at higher derivative orders. In general, it is
found that that the amplitude of the n^{th} derivative
of a peak is inversely proportional to the n^{th} power
of its width, for signals having the same shape and amplitude.
Thus differentiation in effect discriminates against wider peaks
and the higher the order of differentiation the greater the
discrimination. This behavior can be useful in quantitative
analytical applications for detecting peaks that are
superimposed on and obscured by stronger but broader background
peaks. (Matlab/Octave code for this
figure). The amplitude of a derivative of a peak also depends on
the *shape *of the peak and is directly proportional to
its peak *height*.

**Applications of
Differentiation**

A simple example of the application of differentiation of experimental signals is shown in Figure 5. This signal is typical of the type of signal recorded in amperometric titrations and some kinds of thermal analysis and kinetic experiments: a series of straight line segments of different slope. The objective is to determine how many segments there are, where the breaks between then fall, and the slopes of each segment. This is difficult to do from the raw data, because the slope differences are small and the resolution of the computer screen display is limiting. The task is much simpler if the first derivative (slope) of the signal is calculated (Figure 5, right). Each segment is now clearly seen as a separate step whose height (y-axis value) is the slope. The y-axis now takes on the units of dy/dx. Note that in this example the steps in the derivative signal are not completely flat, indicating that the line segments in the original signal were not perfectly straight. This is most likely due to random noise in the original signal. Although this noise was not particularly evident in the original signal, it is more noticeable in the derivative.

*Figure 5. The signal on the left seems to be a more-or-less
straight line, but its numerically calculated derivative
(dx/dy), plotted on the right, shows that the line actually
has several approximately straight-line segments with
distinctly different slopes and with well-defined breaks
between each segment.*

It is commonly observed that differentiation degrades signal-to-noise ratio, unless the differentiation algorithm includes smoothing that is carefully optimized for each application. Numerical algorithms for differentiation are as numerous as for smoothing and must be carefully chosen to control signal-to-noise degradation.

A classic use of second differentiation in
chemical analysis is in the location of endpoints in
potentiometric titration. In most titrations, the titration
curve has a sigmoidal shape and the endpoint is indicated by the
*inflection point*, the point where the slope is maximum
and the curvature is zero. The first derivative of the titration
curve will therefore exhibit a *maximum* at the inflection
point, and the second derivative will exhibit a *zero-crossing*
at that point. Maxima and zero crossings are usually much easier
to locate precisely than inflection points.

*Figure 6 The signal on the left is the pH titration curve
of a very weak acid with a strong base, with volume in mL on
the X-axis and pH on the Y-axis. The endpoint is the point
of greatest slope; this is also an inflection point, where
the curvature of the signal is zero. With a weak acid such
as this, it is difficult to locate this point precisely from
the original titration curve. The endpoint is much more
easily located in the second derivative, shown on
the right, as the the zero crossing. *

Figure 6 shows a pH titration curve of a very weak acid with a strong base, with volume in mL on the X-axis and pH on the Y-axis. The volumetric equivalence point (the "theoretical" endpoint) is 20 mL. The endpoint is the point of greatest slope; this is also an inflection point, where the curvature of the signal is zero. With a weak acid such as this, it is difficult to locate this point precisely from the original titration curve. The second derivative of the curve is shown in Window 2 on the right. The zero crossing of the second derivative corresponds to the endpoint and is much more precisely measurable. Note that in the second derivative plot, both the x-axis and the y-axis scales have been expanded to show the zero crossing point more clearly. The dotted lines show that the zero crossing falls at about 19.4 mL, close to the theoretical value of 20 mL.

Another common use of differentiation is in the detection of peaks in a signal. It's clear from the basic properties described in the previous section that the first derivative of a peak has a downward-going zero-crossing at the peak maximum, which can be used to locate the x-value of the peak. But the presence of random noise in real experimental signal will cause many false zero-crossings simply due to the noise. To avoid this problem, one popular technique smooths the first derivative of the signal first, before looking for downward-going zero-crossings, and then takes only those zero crossings whose slope exceeds a certain predetermined minimum (called the "slope threshold") at a point where the original signal amplitude exceeds a certain minimum (called the "amplitude threshold"). By carefully adjusting the smooth width, slope threshold, and amplitude threshold, it is possible to detect only the desired peaks and ignore peaks that are too small, too wide, or too narrow. Moreover, because smoothing can distort peak signals, reducing peak heights, and increasing peak widths, this technique determines the position, height, and width of each peak by least-squares curve-fitting of a segment of original unsmoothed signal 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.

In spectroscopy, the differentiation of spectra
is a widely used technique, particularly in infra-red,
u.v.-visible absorption,
fluorescence,
and reflectance
spectrophotometry, referred to as *derivative
spectroscopy.* Derivative methods have been used in
analytical spectroscopy for three main purposes: (a) spectral
discrimination, as a qualitative fingerprinting technique to
accentuate small structural differences between nearly identical
spectra; (b) spectral resolution enhancement, as a technique for
increasing the apparent resolution of overlapping spectral bands
in order to more easily determine the number of bands and their
wavelengths; (c) quantitative analysis, as a technique for the
correction for irrelevant background absorption and as a way to
facilitate multicomponent analysis. (Because
differentiation is a linear technique, the amplitude of a
derivative is proportional to the amplitude of the original
signal, which allows quantitative analysis applications
employing any of the standard
calibration techniques). Most commercial
spectrophotometers now have built-in derivative capability. Some
instruments are designed to measure the spectral derivatives
optically, by means of dual
wavelength or wavelength
modulation designs.

Because of the fact that the amplitude of the n^{th}
derivative of a peak-shaped signal is inversely proportional to
the n^{th} power of the width of the peak,
differentiation may be employed as a general way to discriminate
against broad spectral features in favor of narrow components.
This is the basis for the application of differentiation as a
method of correction for background signals in quantitative
spectrophotometric analysis. Very often in the practical
applications of spectrophotometry to the analysis of complex
samples, the spectral bands of the analyte (i.e. the compound to
be measured) are superimposed on a broad, gradually curved
background. Background of this type can be reduced by
differentiation.

This is illustrated by the figure on the left, which shows a simulated UV spectrum (absorbance vs wavelength in nm), with the green curve representing the spectrum of the pure analyte and the red line representing the spectrum of a mixture containing the analyte plus other compounds that give rise to the large sloping background absorption. The first derivatives of these two signals are shown in the center; you can see that the difference between the pure analyte spectrum (green) and the mixture spectrum (red) is reduced. This effect is considerably enhanced in the second derivative, shown on the right. In this case the spectra of the pure analyte and of the mixture are almost identical. In order for this technique to work, it is necessary that the background absorption be broader (that is, have lower curvature) than the analyte spectral peak, but this turns out to be a rather common situation. Because of their greater discrimination against broad background, second (and sometimes even higher-order) derivatives are often used for such purposes. See DerivativeDemo.m for a Matlab/Octave demonstration of this application.

It is sometimes (mistakenly) said that
differentiation "increases the sensitivity" of analysis. You can
see how it would be tempting to say something like that by
inspecting the three figures above; it does seems that the
signal amplitude of the derivatives is greater (at least
graphically) than that of the original analyte signal. However,
it is not valid to compare the amplitudes of signals and their
derivatives because they have different units. The units of the
original spectrum are absorbance; the units of the first
derivative are absorbance per nm, and the the units of the
second derivative are absorbance per nm^{2}. You can't
compare absorbance to absorbance per nm any more than you can
compare miles to miles per hour. (It's meaningless, for
instance, to say that 30 miles per hour is greater than 20
miles.) You can, however, compare the *signal-to-background
ratio* and the *signal-to-noise ratio*. For example,
in the above example, it would be valid to say that the
signal-to-background ratio is increased in the derivatives.

Loosely speaking, the opposite of differentiation is integration, so if you are given a first derivative of a signal, you might expect to regenerate the original (zeroth derivative) by integration. However, there is a catch; the constant term in original signal (like a flat baseline) is completely lost in differentiation; integration can not restore it. So strictly speaking, differentiation represents a net loss of information, and therefore differentiation is used only in situations where the constant term in the original signal is not of interest.

It is also often said that "differentiation
increases the noise". That is true, but it is not the main
problem. In fact, computing the unsmoothed first derivative of a
set of random numbers
increases its standard deviation by only the square root of 2,
simply due to the usual propagation
of errors. But even the slightest degree of smoothing
applied to the derivative will reduce this standard deviation
greatly. More important is that the signal-to-noise ratio
of an *unsmoothed *derivative is almost always much lower
(poorer) than that of the original signal, but smoothing is *always
*used in any practical application to control this problem
(See "The Importance of Smoothing Derivatives) below.

One of the widest uses of the derivative signal
processing technique in practical analytical work is in the
measurement of small amounts of substances in the presence of
large amounts of potentially interfering materials. In such
applications it is common that the analytical signals are weak,
noisy, and superimposed on large background signals. Measurement
precision is often degraded by sample-to-sample baseline shifts
due to non-specific broadband interfering absorption,
non-reproducible cuvette (sample cell) positioning, dirt or
fingerprints on the cuvette walls, imperfect cuvette
transmission matching, and solution turbidity. Baseline shifts
from these sources are usually either wavelength-independent
(light blockage caused by bubbles or large suspended particles)
or exhibit a weak wavelength dependence (small-particle
turbidity). Therefore it can be expected that differentiation
will in general help to discriminate relevant absorption from
these sources of baseline shift. An obvious benefit of the
suppression of broad background by differentiation is that *variations*
in the background amplitude from sample to sample are also
reduced. This can result in improved precision or measurement in
many instances, especially when the analyte signal is small
compared to the background and if there is a lot of uncontrolled
variability in the background. An example of the improved
ability to detect trace component in the presence of strong
background interference is shown in Figure 7.

*Figure 7. The spectrum on the left shows a
weak shoulder near the center due to a small concentration of
the substance that is to be measured (e.g. the active
ingredient in a pharmaceutical preparation). It is difficult
to measure the intensity of this peak because it is obscured
by the strong background caused by other substances in the
sample. The fourth derivative of this spectrum is
shown on the right. The background has been almost completely
suppressed and the analyte peak now stands out clearly,
facilitating measurement. *

The spectrum on the left shows a weak shoulder
near the center due to the analyte. The signal-to-noise ratio is
very good in this spectrum, but in spite of that the broad,
sloping background obscures the peak and makes quantitative
measurement very difficult. The fourth derivative of this
spectrum is shown on the right. The background has been almost
completely suppressed and the analyte peak now stands out
clearly, facilitating measurement. An even more dramatic case is
shown in Figure 8. This is essentially the same spectrum as in
Figure 7, except that the concentration of the analyte is lower.
The question is: is there a detectable amount of analyte in this
spectrum? This is quite impossible to say from the normal
spectrum, but inspection of the fourth derivative (right) shows
that the answer is *yes*. Some noise is clearly evident
here, but nevertheless the signal-to-noise ratio is sufficiently
good for a reasonable quantitative measurement.

*Figure 8. Similar to Figure 7, but in the case the peak is
so weak that it can not even be seen in the spectrum on the
left. The fourth derivative (right) shows that a peak is
still there, but much reduced in amplitude (note the smaller
y-axis scale).*

This use of signal differentiation has become widely used in quantitative spectroscopy, particularly for quality control in the pharmaceutical industry. In that application the analyte would typically be the active ingredient in a pharmaceutical preparation and the background interferences might arise from the presence of fillers, emulsifiers, flavoring or coloring agents, buffers, stabilizers, or other excipients. Of course, in trace analysis applications, care must be taken to optimize signal-to-noise ratio of the instrument as much as possible.

**The Importance of
Smoothing Derivatives**

For the successful application of
differentiation in quantitative analytical applications, it is
essential to use differentiation in combination with sufficient
smoothing, in order to optimize the signal-to-noise ratio.
This is illustrated in the figure on the left. (Matlab/Octave code for this figure.)
Window 1 shows a Gaussian band with a small amount of added
white noise. Windows 2, 3, and 4, show the first derivative of
that signal with increasing smooth widths. As you can see,
without sufficient smoothing, the signal-to-noise ratio of the
derivative can be substantially poorer than the original signal.
However, with adequate amounts of smoothing, the signal-to-noise
ratio of the smoothed derivative can be better than that of the
unsmoothed original. This effect is even more striking in the
second derivative, as shown on the right (Matlab/Octave
code for this figure). In this case, the signal-to-noise
ratio of the unsmoothed second derivative (Window 2) is so poor
you can not even see the signal visually. What is particularly
interesting about the noise in these derivative signals,
however, is their "color".
This noise is not *white; *rather, it is *blue* -
that is, it has much more power at high frequencies than white
noise. The consequence of this is that it is especially subject
to reduction by *smoothing*.

It makes no difference whether the smooth
operation is applied before or after the differentiation. What
is important, however, is the nature of the smooth, its smooth
ratio (ratio of the smooth width to the width of the original
peak), and the number of times the signal is smoothed. The
optimum values of smooth ratio for derivative signals is
approximately 0.5 to 1.0. For a first derivative, two
applications of a simple rectangular smooth or one application
of a triangular smooth is adequate. For a second derivative,
three applications of a simple rectangular smooth or two
applications of a triangular smooth is adequate. The general
rule is: for the n^{th} derivative, use at least n+1
applications of a rectangular smooth. (The Matlab signal
processing program iSignal automatically
provides the desired type of smooth for each derivative order).
Smoothing derivatives results in a substantial attenuation of
the derivative amplitude; in the figure on the right above, the
amplitude of the most heavily smoothed derivative (in Window 4)
is much less than its less-smoothed version (Window 3). However,
this won't be a problem, as long as the standard (analytical)
curve is prepared using the exact same derivative, smoothing,
and measurement procedure as is applied to the unknown samples.
Because differentiation and smoothing are both linear
techniques, the amplitude of a smoothed derivative is
exactly proportional to the amplitude of the original signal,
which allows quantitative analysis applications employing any of
the standard calibration
techniques. As long as you apply the same
signal-processing techniques to the standards as well as to the
samples, everything works.

Because of the different kinds and degrees of smoothing that might be incorporated into the computation of digital differentiation of experimental signals, it's difficult to compare the results of different instruments and experiments unless the details of these computations are known. In commercial instruments and software packages, these details may well be hidden. However, if you can obtain both the original (zeroth derivative) signal, as well as the derivative and/or smoothed version from the same instrument or software package, then the technique of Fourier deconvolution, which will be discussed later, can be used to discover and duplicate the underlying hidden computations.

Interestingly, neglecting to smooth a derivative was ultimately responsible for the failure of the first spacecraft of NASA's Mariner program on July 22, 1962, which was reported in InfoWorld's "11 infamous software bugs". In his 1968 book "The Promise of Space", Arthur C. Clarke described the mission as "wrecked by the most expensive hyphen in history." The "hyphen" was actually superscript bar over a radius symbol, handwritten in a notebook. The overbar signifies a smoothing or averaging function, so the formula should have calculated the smoothed value of the time derivative of a radius. Without the smoothing function, even minor variations of speed would trigger the corrective boosters to kick in, causing the rocket's flight to become unstable.

The first 13-second, 1.5 MByte video (SmoothDerivative2.wmv ) demonstrates the huge signal-to-noise ratio improvements that are possible when smoothing derivative signals, in this case a 4th derivative.

The second video, 17-second, 1.1 MByte, (DerivativeBackground2.wmv ) demonstrates the measurement of a weak peak buried in a strong sloping background. At the beginning of this brief video, the amplitude (Amp) of the peak is varied between 0 and 0.14, but the background is so strong that the peak, located at x = 500, is hardly visible. Then the 4th derivative (Order=4) is computed and the scale expansion (Scale) is increased, with a smooth width (Smooth) of 88. Finally, the amplitude (Amp) of the peak is varied again over the same range, but now the changes in the signal are now quite noticeable and easily measured. (These demonstrations were created in Matlab 6.5. If you have access to that software, you may download a set of Matlab Interactive Derivative m-files (15 Kbytes), InteractiveDerivative.zip so that you can experiment with the variables at will and try out this technique on your own signals).

SPECTRUM, the freeware signal-processing application that accompanies this tutorial, includes first and second derivative functions, which can be applied successively to compute derivatives of any order.

**Differentiation
in Spreadsheets**

Differentiation operations such as described above can readily be performed in spreadsheets such as Excel or OpenOffice Calc. Both the derivative and the required smoothing operations can be performed by the shift-and-multiply method described in the section on smoothing. In principle, it is possible to combine any degree of differentiation and smoothing into one set of shift-and-multiply coefficients (as illustrated here), but it's more flexible and easier to adjust if you compute the derivatives and each stage of smoothing separately in successive columns. This is illustrated by DerivativeSmoothing.ods (for OpenOffice Calc) and DerivativeSmoothing.xls (for Excel), which computes the smoothed first derivative, then applies that process successively three times to compute the second and third derivatives. (Note that in order to compute a symmetrical first derivative, the coefficients in columns J to Q must be the negatives of the positive coefficients in columns S to Z).

Another example of a derivative application is the spreadsheet SecondDerivativeXY2.xlsx, which demonstrates locating and measuring changes in the second derivative (a measure of curvature or acceleration) of a time-changing signal. This spreadsheet shows the apparent increase in noise caused by differentiation and the extent to which the noise can be reduced by smoothing (in this case by two passes of a 5-point triangular smooth). The smoothed second derivative shows a large peak the point at which the acceleration changes (at x=30) and plateaus on either side showing the magnitude of the acceleration before and after the change (y=2 and 4, respectively).

**Differentiation in Matlab
and Octave**

Differentiation functions such as described above
can easily be created in Matlab or Octave. Some
simple derivative functions for equally-spaced time series
data: deriv, a first derivative
using the 2-point central-difference method, deriv1, an unsmoothed first
derivative using adjacent differences, deriv2, a second derivative using
the 3-point central-difference method, a third derivative
deriv3 using a 4-point formula, and
deriv4, a 4th derivative using a
5-point formula. Each of these is a simple Matlab function
of the form d=deriv(y);
the input argument is a signal vector "y", and the
differentiated signal is returned as the vector "d". For data that are
not
equally-spaced on the independent variable (x) axis, there
are versions of the first and second derivative
functions, derivxy and secderivxy, that take two input
arguments (x,y), where x
and y are vectors
containing the independent and dependent variables. Click
on these links to inspect the code, or right-click to
download for use within Matlab. The following code shows how to use the first derivative to find all the maxima in a smooth x,y data set by locating the points of zero-crossing, that is, the points at which the first derivative "d" (computed by derivxy.m) passes from positive to negative. In this example, the “sign” function is a built-in function that returns 1 if the element is greater than zero, 0 if it equals zero and -1 if it is less than zero. The routine prints out the value of x and y at each zero-crossing: d=derivxy(x,y);for j=1:length(x)-1 if sign(d(j))>sign(d(j+1)) disp([x(j) y(j)]) endendIf the data are noisy, many false zero crossings will be reported; smoothing the data will reduce that. ProcessSignal.m, a Matlab/Octave command-line function that performs smoothing and differentiation on the time-series data set x,y (column or row vectors). 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 (shown above) is an interactive function for Matlab that performs differentiation and smoothing for time-series signals, up to the 5 ^{th} derivative, automatically including
the required type of smoothing. Simple keystrokes allow
you to adjust the smoothing parameters (smooth type,
width, and ends treatment) while observing the effect on
your signal dynamically. In the example shown above, a
series of three peaks ay x=100, 250, and 400, with heights
in the ratio 1:2:3, are buried in a strong curved
background; the smoothed second and fourth derivatives are
computed to suppress that background. View the code here or download the ZIP file with sample data for
testing. (Version 2 of iSignal, November 2011, computes
derivatives with respect to the x-axis vector, correcting
for non-uniform x-axis intervals). Unfortunately,
iSignal does not currently work in Octave.The older version of Interactive Derivative works only in Matlab 6.5. It's a collection of functions and scripts for interactive differentiation with sliders that allow you to adjust the derivative order, smooth width, and scale expansion continuously while observing the effect on your signal dynamically. Requires Matlab 6.5; will not work with newer versions. Click here to download the ZIP file "InteractiveDerivative.zip" that also includes supporting functions, self-contained demos. Run InteractiveDerivativeTest to see how it works. Also includes DerivativeDemo.m, which demonstrates the application of differentiation to the detection of peaks superimposed on a strong, variable background. Generates a signal peak, adds random noise and a variable background, then differentiates and smooths it, and measures the signal range and signal-to-noise ratio (SNR). Interactive sliders allow you to control all the parameters. This was used to create the video demonstration DerivativeBackground2.wmv. Note: you can right-click on any of the m-file links above and select Save Link As... to download them to your computer for use within Matlab. |

Last updated January 2015. This page is maintained by Prof. Tom O'Haver , Professor Emeritus, Department of Chemistry and Biochemistry, The University of Maryland at College Park. Comments, suggestions and questions should be directed to Prof. O'Haver at toh@umd.edu.

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