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Perturbation effects on filters having localised poles

It is well known that the actual poles implemented in digital filters are increasingly sensitive to decreasing pole separation and that practical filters are implemented usually as a cascade of second order sections or the dual form to avoid this sensitivity. This paper considers the case of filters implemented in direct form and having localised poles. Expressions that describe pole position sensitivity for simple isolated poles and for multiple poles have previously been developed. This paper addresses the problem of pole sensitivity due to pole interaction when one or more poles are placed in "close proximity" to other poles.