**Differentiation**

[Basic Properties of Derivatives] [Applications of Differentiation] [Peak detection] [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 Gaussian peak 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. Different
peak shapes have different derivatives shapes: the Matlab/Octave
function DerivativeShapeDemo.m
demonstrates the first derivative forms of 16 different model
peak shapes (graphic). Any
smooth peak shape with a single maximum has sequential
derivatives that exhibit a series of alternating maxima and
minima, the total number of which is one more than the
derivative order. The
even-order derivatives have a maximum or a minimum at the peak
center, and the odd-order derivatives have a zero-crossing at the peak center (Graphic, Matlab/Octave code). You can
also see here that the numerical magnitude of the derivatives
(y-axis values) is much less than the original signal, because
derivatives are the *differences *between adjacent y
values, divided by the independent variable increment. (It's the
same reason the odometer in
your car usually displays a much larger number than the
speedometer; the speedometer is essentially the first derivative
of the odometer).

An
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*.

Although differentiation changes the shape of peak-type signals
drastically, a *periodic **signal *like a sine wave
signal behaves very differently. The derivative of a sine wave
of frequency *f* is a phase-shifted sine wave, or cosine
wave, of the *same frequency* and with an amplitude
that is proportional to* **f*, as can be demonstrated
in Wolfram
Alpha. For this reason, when a music or speech signal is
differentiated, the music or speech is still completely
recognizable, but the low frequencies are attenuated and the
high frequencies amplified, and as a result, it sounds "thin" or
"tinny". See this demonstration
for an example.

**Applications of
Differentiation**

A simple example of the application of differentiation of experimental signals is shown in the figure below. 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 (below 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.

*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.

*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 zero crossing. *

The figure above 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. If there is *no noise* in the signal, then any data
point that has lower values on both sides of it will be a peak
maximum. But there is always at least a little noise in real
experimental signals, and that 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 over a wide range of peak widths
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 can be extended to
measure 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 and the
effect of random noise in the signal is reduced by curve fitting
over multiple data points in the peak. This technique has been
implemented in Matlab/Octave
and in spreadsheets.

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 above, 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 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.

There are several ways to measure the amplitude
of a derivative spectrum for quantitative analysis: the absolute
value of the derivative at a specific wavelength, the value of a
specific feature (such as a maximum), or the difference between
a maximum and a minimum. Another widely-used technique is the
zero-crossing measurement - taking readings derivative amplitude
at the wavelength where an interfering peak crosses the zero on
the y (amplitude) axis. In all these cases, it's important to
measure the standards and the unknown samples in exactly the
same way. Also, because the amplitude of a derivative of a
peak depends strongly on its width, it's important to control
environmental factors that might change spectral peak width
subtlety, such as temperature.

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 this figure:

*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 below. This is essentially the same spectrum as in the
figure above, 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.

*Similar to the previous figure, 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.

**Derivatives**
and Noise: The Importance of Smoothing

It is 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 of the sum or difference between two numbers,
e.g. `std(deriv1(randn(size(1:10000))))`. But even a little bit of smoothing
applied to the derivative will reduce this standard deviation
greatly, e.g.`
std(fastsmooth(deriv1(randn(size(1:10000))),2,3))`. 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, mainly
because the numerical amplitude of the derivative is usually
much smaller (as you can see for yourself in all the examples
on this page). But smoothing is *always *used
in any practical application to control this problem; with
optimal smoothing, the signal-to-noise of a derivative can
actually be *greater *than the unsmoothed
original. 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. Differentiation does not actually
add noise to the signal; if there were no noise at all in the
original signal, then the derivatives would also have no noise.

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. An overbar conventionally signifies an averaging or smoothing 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 for Macintosh OS8, 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 smooths the data and computes the first derivative of Y (column B) with
respect to X (column A), then applies that smoothing
and differentiation process successively to compute the smoothed
second and third derivatives. The same smoothing coefficients
(in row 5, columns K through AA) are applied successively for
each stage of differentiation; you can enter any set of numbers
here (preferably symmetrical about the center number in column
S). You can type or paste your own data into column A and B (X
and Y), rows 8 to 263.

DerivativeSmoothingWithNoise.xlsx
is a related spreadsheet that demonstrates the dramatic effect
of smoothing on the signal-to-noise ratio of derivatives on a
noisy signal. It uses the same signal as DerivativeSmoothing.xls,
but adds simulated white noise to the Y data. You can control
the amount of added noise.

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. If the data are sparsely sampled, a more accurate value for the peak position (x-axis value at the zero crossing) can be obtained by interpolating between the point before and the point after the zero-crossing using the Matlab/Octave “interp1” function: interp1([d(j) d(j+1)],[x(j)
x(j+1)],0)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) magnitude of
the derivative seems to be 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 at 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 script “iSignalDeltaTest” demonstrates the frequency response of the smoothing and differentiation functions of iSignal by a applying them to a delta function. Change the smooth type, smooth width, and derivative order and see how the power spectrum changes. |

This page is also available in French, at http://www.teilestore.de/edu/?p=8215, courtesy of Anna Chekovsky.

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