6. Resolution enhancement

Figure 9 shows a spectrum on the left that consists of several poorly-resolved (that is, partly overlapping) bands. 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. 

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.

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 R_j 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 flatness. 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).