The symbolic integration of functions and the calculation of definite integrals
are topics that are introduced in elementary Calculus courses. The numerical
integration of digitized signals finds application in analytical signal
processing mainly as a method for measuring the areas under the curves of
peak-type signals. Peak area measurements are very important in
chromatography. Quantitation in chromatography is customarily carried out on
the basis of peak area rather than peak height measurement. The reason for
this is that peak area is less sensitive to the influence of peak broadening
(dispersion) mechanisms. These broadening effects, which arise from many
sources, cause chromatographic peaks to become shorter, broader, and more
unsymmetrical, but have little effect on the total area under the peak. The
peak area remains proportional to the total quantity of substance passing into
the detector. Therefore peak area measurements are often found to be more
reliable than peak height measurement.
The simple numeric integration of a
digital, e.g. by Simpson's rule, will convert a series of peaks into a series
of steps, the height of each of which is proportional to the area under that
peak. This is a commonly used method in proton NMR spectroscopy, where the
area under each peak or multiplet is proportional to the number of equivalent
hydrogens responsible for that peak. But this works well only if the peaks are
well separated from each other (e.g. well resolved).
In chromatographic
analysis one often has the problem of measuring the the area under the curve of
the peaks when they are not well resolved or are superimposed on a background.
For example, Figure 15 shows a series of four computer-synthesized Gaussian
peaks that all have the same height, width, and area, but the separation
between the peaks on the right is insufficient to achieve complete resolution.
The classical way to handle this problem is to draw two vertical lines from the
left and right bounds of the peak down to the x-axis and then to measure the
total area bounded by the signal curve, the x-axis (y=0 line), and the two
vertical lines. This is often called the the perpendicular drop method, and it
is an easy task for a computer, although very tedious to do by hand. The idea
is illustrated for the second peak from the left in Figure 15. The left and
right bounds of the peak are usually taken as the valleys (minima) between the
peaks. Using this method it is possible to estimate the area of the second
peak in this example to an accuracy of about 0.3% and the second and third
peaks to an accuracy of better than 4%, despite the poor resolution.
Figure 15. Peak area measurement for overlapping peaks, using the
perpendicular drop method.
SPECTRUM, the freeware signal-processing
application that accompanies this tutorial, includes an integration
function, as well as peak area measurement by perpendicular drop or
tangent skim methods, with mouse-controlled setting of start and stop
points. Peak area measurememt using Matlab. iSignalis
a downloadable user-defined Matlab function that performs various signal processing
functions described in this tutorial, including measurememt of peak
area using Simpson's Rule and the perpendicular drop method. Click to view or right-click > Save link as...here, or you can download theZIP filewith sample data for testing.
For highly overlapped peaks of Gaussian or Lorentzian shapes, the most
accurate measurements can be made with iterative least-squares peak
fitting, for example using the Interactive Peak Fitter (http://terpconnect.umd.edu/~toh/spectrum/InteractivePeakFitter.htm).
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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