8. CONVOLATION

Convolation is an operation performed on two signals which involves multiplying one signal by a delayed or shifted version of another signal, integrating or averaging the product, and repeating the process for different delays. Convolution is a useful process because it accurately describes some effects that occur widely in scientific measurements, such as the influence of a low-pass filter on an electrical signal or of the spectral bandpass of a spectrometer on the shape of a spectrum. 

Figure 11. Convolution is used here to determine how the atomic line spectrum in Window 1 (top left) will appear when scanned with a spectrometer whose slit function (spectral resolution) is described by the Gaussian function in Window 2 (top right). The Gaussian function has already been rotated so that its maximum falls at x=0. The resulting convoluted spectrum (bottom center) shows that the two lines near x=110 and 120 will not be resolved but the line at x=40 will be partly resolved.

9. DECONVULATION

Deconfulation is the converse of confulation in the sense that division is the converse of multiplication^*. In fact, the deconvolution of one signal from another is usually performed by dividing the two signals in the Fourier domain^**. The practical significance of deconvolution is that it can be used as an artificial (i.e. computational) way to reverse the result of a convolution occurring in the physical domain, for example, to reverse the signal distortion effect of an electrical filter or of the finite resolution of a spectrometer. Two examples of the application of deconvolution are shown in Figures 12 and 13. 

10. FOURIER FILTER

The Fourier filter is a type of filtering function that is based on manipulation of specific frequency components of a signal. It works by taking the Fourier transform of the signal, then attenuating or amplifying specific frequencies, and finally inverse transforming the result. The example shown here is a simple low-pass, sharp cut-off filter, which simply cuts off all frequencies above a user-specified limit. The assumption is made here that the frequency components of the signal fall predominantly at low frequencies and those of the noise fall predominantly at high frequencies. The user tries to find a cut-off frequency that will allow most of the noise to be eliminated while not distorting the signal significantly. An example of the application of the Fourier filter is given in Figure 14.

REFERENCES

1.     Douglas A. Skoog, Principles of Instrumental Analysis, Third Edition, Saunders, Philadelphia, 1984. Pages 73-76.

2.     Gary D. Christian and James E. O'Reilly, Instrumental Analysis, Second Edition, Allyn and Bacon, Boston, 1986. Pages 846-851.

3.     Howard V. Malmstadt, Christie G. Enke, and Gary Horlick, Electronic Measurements for Scientists, W. A. Benjamin, Menlo Park, 1974. Pages 816-870.

4.     Stephen C. Gates and Jordan Becker, Laboratory Automation using the IBM PC, Prentice Hall, Englewood Cliffs, NJ, 1989.

5.     Muhammad A. Sharaf, Deborah L Illman, and Bruce R. Kowalski, Chemometrics, John Wiley and Sons, New York, 1986.

6.     Peter D. Wentzell and Christopher D. Brown, Signal Processing in Analytical Chemistry, in Encyclopedia of Analytical Chemistry, R.A. Meyers (Ed.), p. 9764–9800, John Wiley & Sons Ltd, Chichester, 2000 (http://myweb.dal.ca/pdwentze /paper/c2..pdf).

7.     Constantinos E. Efstathiou, Educational Applets in Analytical Chemistry, Signal Processing, and Chemometrics. (http://www.chem.uoa.gr/Applets /Applet Index2. htm).

8.      A. Felinger, Data Analysis and Signal Processing in Chromatography, Elsevier Scice (19 May 1998).