Coefficient sensitivity analysis for block‐state realization of state‐space digital filters

This paper presents an analysis of the frequency characteristics of the coefficient sensitivity in the block‐state realization. The expression for the coefficient sensitivity is derived first, and the following properties are shown. In contrast to the usual state‐space realization, the coefficient sensitivity in the block‐state realization has the periodicity with period (2π/L), where L is the block length. There is also a symmetry for the response in a period. Then the following property is shown in the coefficient quantization in the block‐state realization. The peaks of the coefficient sensitivity, which usually appear only at the end of the passband, appear many times iteratively in other frequency ranges, resulting in a particular coefficient quantization error. This point must be considered in the design of the filter by block‐state. A measure for the coefficient sensitivity is defined anew to evaluate the coefficient sensitivity over the whole frequency range. It is shown that this measure of the coefficient sensitivity is an extension of the ordinary state‐space realization. The expression of the measure is given by the state covariance matrix of the subfilters for the elements of transfer function in the block‐state realization, i.e., the L‐input/L‐output system, together with the noise matrix. Finally, a computation example is shown.