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Resolution enhancement
Figure 9 shows a spectrum that consists of several poorly-resolved (that is,
partly overlapping) bands.
Figure 9. A resolution enhancement algorithm has been applied to the
signal on the left to artificially improve the apparent resolution of the
peaks. In the resulting signal, right, the component bands are narrowed so that
the intensities and positions can be measured.
The extensive overlap of the bands makes the accurate measurement of their
intensities and positions impossible, even though the signal-to-noise ratio is
very good. Things would be easier if the bands were more completely resolved,
that is, if the bands were narrower. Here use can be made of resolution
enhancement algorithms to artificially improve the apparent resolution of
the peaks. One of the simplest such algorithims is based on the weighted sum
of the original signal and the negative of its second derivative.
where Rj is the resolution-enhanced signal, Y is the original signal, Y'' is
the second derivative of Y, and k is a user-selected weighting factor. It is
left to the user to select the weighting factor k which gives the best
trade-off between resolution enhancement, signal-to-noise degradation, and
baseline undershoot. The optimum choice depends upon the width, shape, and
digitization interval of the signal. The result of the application of this
algorithm is shown on the right. The component bands have been artificially
narrowed so that the intensities and positions can be measured. However, the
signal-to-noise ratio is degraded.
Here's how it works. The figure below shows, in Window 1, a computer-generated peak (with a
Lorentzian shape) in red, superimposed on the negative of its second derivative in green). (Click on the figure to see a full-size figure).
The second derivative is amplified (by multiplying it by an adjustable constant) so that the negative sides of the inverted second derivative (from approximately X = 0 to 100 and from X = 150 to 250) are a mirror image of the sides of the original peak over those regions. In this way, when the original peak is added to the inverted second derivative, the two signals will approximately cancel out in the two side regions but will reinforce each other in the central region (from X = 100 to 150). The result, shown in Window 2, is a substantial (about 50%) reduction in the width, and an increase in height, of the peak. This works
best with lorentzian-shaped peaks; with gaussian-shaped peaks, the resolution enhancement is less dramatic (only about 20%). An interesting property
of this procedure is that it does not change the total peak area (that is, the area under the peak) because the total area under the curve of the derivative of a peak-shaped signal is zero (the area under the negatives lobes cancels the area under the positive lobes).
SPECTRUM, the freeware signal-processing application that accompanies this tutorial, includes this resolution-enhancement algorithim, with adjustable weighting factor and derivative smoothing width.
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Last updated June 1, 2001. 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 to2@umail.umd.edu.
