A parametric approach to the realization of second-order digital filter sections

A general procedure is presented for exploring all real statespace realizations of second-order discrete-time transfer functions with complex-conjugate poles. The procedure is based on the sequential selection of two complex scalar parameters-an eigenvector parameter that determines the natural modes of the system, and a scaling parameter that determines the coupling of the natural modes to the input and output. For the realization of a digital filter, these parameters can be chosen to provide minimal computational complexity and low sensitivity to the undesirable finite word effects of fixed-point arithmetic. It is shown that it is possible to construct many state-space realizations that require the same number of multiplications as the canonical direct form, but which can have low sensitivity to roundoff error, low sensitivity to coefficient quantization, and immunity to overflow limit cycles. The realization procedure is illustrated by a numerical example.