[Basic Properties of Derivative Signals] [Applications of Differentiation] [Derivative Spectroscopy] [Trace Analysis] [The Importance of Smoothing Derivatives] [Video Demonstrations] [Spreadsheets] [Differentiation in Matlab and Octave] [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 *slope* of
the tangent to
the signal at each point. Assuming that the x-interval between
adjacent points is constant, the simplest algorithm for computing
a first derivative is:

(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.

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. 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.)

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.

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.

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 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. 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.

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. 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) DerivativeDemo.m (shown above) is a self-contained Matlab/Octave demo function that uses ProcessSignal.m and plotfit.m to demonstrate an application of differentiation to the quantitative analysis of a peak buried in an unstable background (e.g. as in various forms of spectroscopy). The object is to derive a measure of peak amplitude that varies linearly with the actual peak amplitude and is minimally effected by the background and the noise. To run it, just type DerivativeDemo at the command prompt. You can change several of the internal variables (e.g. Noise, BackgroundAmplitude) to make the problem harder or easier. Note that, despite the fact that the magnitude of the derivative is numerically smaller than the original signal (because it has different units), the signal-to-noise ratio of the derivative is better and is much less effected by the background instability. (Execution time: 0.065 seconds in Matlab; 2.2 seconds in Octave).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 December, 2013. This page is maintained by Prof. Tom O'Haver , 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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