## Abstract

The Cauchy index of a rational transfer function evaluated over an interval of the real line has proved a useful tool in various linear systems applications.

In this paper, we define the unit circle Cauchy index and give some computational methods of it. This provides a systematic view of many problems and results on unit circle positivity, polynomial root distribution, and stability. Results on positivity of polynomials in r and 2-' which are real on lz/ = 1 are relevant in checking discrete positive realness and the stability of two-dimensional digital filters, while results on the zero distribution of a polynomial relative to the boundary of the unit circle are of relevance in studying the stability of discrete-time systems.

In this paper, we define the unit circle Cauchy index and give some computational methods of it. This provides a systematic view of many problems and results on unit circle positivity, polynomial root distribution, and stability. Results on positivity of polynomials in r and 2-' which are real on lz/ = 1 are relevant in checking discrete positive realness and the stability of two-dimensional digital filters, while results on the zero distribution of a polynomial relative to the boundary of the unit circle are of relevance in studying the stability of discrete-time systems.

Original language | English |
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Pages (from-to) | 803–818 |

Journal | SIAM Journal of Applied Math |

Volume | 44 |

Issue number | 4 |

Publication status | Published - Aug 1983 |