Perturbation effects on filters having multiple 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 to avoid this sensitivity. This paper considers the case of filters implemented in direct form and having multiple poles. An expression for the sensitivity for such filters to parameter perturbation is developed. The usual concept of pole position sensitivity for simple poles is an infinitesimal measure which becomes very conservative for practical quantization errors. The multiple pole result can be interpreted for the case of simple poles to provide a working estimate for pole sensitivity