Spread function shift variant

We have returned, for the time being, to the general case of the shift-variant spread function. Solving for on, we obtain... 

The matrix notation serves to stress that the technique is applicable to shift-variant spread functions, that is, where sjk slm for j — k = l — m. Many of the deconvolution methods described here with the convolution notation may thus be generalized. In the convolution notation, the present method may be expressed by the equation... 

Under these conditions, the spread function is termed shift invariant and s is called a Toeplitz matrix. When Eq. (17) does not hold, the spread function is called shift variant, and the spreading can no longer be described by simple convolution. 

Shift-Variant (-Invariant) Convolution Convolutions in which the same spreading function is applied to every time element or data. This is the case in linear chromatography where the spreading caused by each plate on the passing distribution is the same. In nonlinear chromatography, the effect of each plate on the profile depends on the concentration in that plate. The convolution is said to be shift-variant. As a consequence the rules of variance addition do not apply. 

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