Effect of the smoothing matrix

The smoothing matrix can now be viewed as a filter which will have different effects on the two components. We attack the analysis of its effects by doing a further factorisation, exploiting the fact that the z transform can be regarded as a representation of the smoothing matrix as a polynomial in the unit subdiagonal. Factorisation of a circulant matrix is exactly equivalent to factorisation of the generating function as a polynomial. 

This factorisation is made unique by the convention that every factor must have rows which sum to unity. Also, because the mask is palindromic, we use the form of the symbol divided by such a power of yfz as will make the coefficients of positive and negative powers of yfz equal. Thus a = (1 + z)/2 /z. 

Each such factor has the effect of scaling the signal and artifact components by factors determined as follows. 

Three consecutive original signal components after sampling are 

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