Examining the Deconvolution Process

Mathematically speaking, deconvolution refers to the method or methods used to solve the convolution integral equation 

There are many reasons why deconvolution algorithms produce unsatisfactory results. In the deconvolution of actual spectral data, the presence of noise is usually the limiting factor. For the purpose of examining the deconvolution process, we begin with noiseless data, which, of course, can be realized only in a simulation process. When other aspects of deconvolution, such as errors in the system response function or errors in base-line removal, are examined, noiseless data are used. The presence of noise together with base-line or system transfer function errors will, of course, produce less valuable results. 

For all the tests detailed in this chapter, deconvolution was carried out for 100 iterations using a relaxation function of the form 

In solving Eq. (1) we follow an iterative process that has been refined in the last several years (Jansson, 1970 Jansson et a/., 1970 Willson, 1973 Pliva et al., 1980 Blass and Halsey, 1981). In this iterative process, the zeroth approximation to o, the ideal spectrum, is taken as i, the measured spectrum, and the iterative process can be represented as 

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