On the adjustment of coefficients, poles, zeros and residues in adaptive and estimation systems

The sensitivity properties of a network transfer function H(s) given in three different forms are compared. The three forms of H(s) are (i) given by coefficients, (ii) by poles and zeros, and (iii) by poles and residues. The comparison was motivated by an analog adaptive filter application in high-speed data transmission over digital subscriber lines, where a matching function is adapted to the subscriber line within an active hybrid network. The matching problem is formulated in terms of sensitivities, whereby the Schoeffler gain deviation is used as the relevant measure. Lower bounds on the summed squares of coefficient, pole-zero, and pole-residue sensitivities are given in closed form for the sensitivity sums. It is found that the coefficient representation yields a smaller gain deviation and, consequently, better resolution, or accuracy, of H(s) than the pole-zero representation. Furthermore, in all cases in which H(s) possesses only real poles and zeros, the pole-residue representation gives the smallest Schoeffler measure in the entire frequency band. In the case of a dominant-zero effect, the roles may be reversed, i.e., the pole-residue representation results in the largest gain deviation, at least in the neighbourhood of the zero-frequency. The results presented are quite general, and the conclusions applicable well beyond the initial matching problem.