## Abstract

Most traditional digital signal processing and control algorithms are inherently ill-conditioned when applied in situations in which data are taken at sampling rates that are high relative to the dynamics of the underlying continuous-time processes being sampled. This signal-processing regime is of particular interest in applications such as digital feedback control and wideband communications, in which high relative sampling rates are often dictated by system stability or format considerations rather than by signal-processing needs. Considerable recent progress toward ameliorating such ill-conditioning problems has been made through the use of a divided-difference operator, rather than the conventional shift operator, to represent the dynamics of sampled data. This approach leads to a novel systems calculus that allows for a unification of continuous and discrete time formulations, enables a smooth transition from sampled-data algorithms to their continuous-time counterparts, and consequently enhances the numerical conditioning of algorithms in the high-speed regime. This means of signal representation and analysis shows considerable promise relative to traditional analysis based on shift operators in the emerging era of high-speed realtime processing, and thus it should be of interest to researchers and practitioners in a variety of areas. The purpose of this paper is to organize and survey this recent work, and to present it in a unified and accessible form. This paper begins with an introductory section in which this need for a new approach suitable for high-speed processing is motivated in the context of several applications in control and communications, and an historical perspective of the use of difference operators in numerical analysis is presented. Secondly, the general systems calculus, based on divided-difference operators, is introduced to unify the continuous-time and discrete-time systems theories. This calculus is then used as a framework to treat the three problems of system state estimation; system identification and time-series modeling; and control system design. Realization aspects of algorithms based on the difference operator representation, including such issues as coefficient rounding and implementation with standard hardware, are also discussed.

Original language | English (US) |
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Pages (from-to) | 240-259 |

Number of pages | 20 |

Journal | Proceedings of the IEEE |

Volume | 80 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1992 |

## All Science Journal Classification (ASJC) codes

- General Computer Science
- Electrical and Electronic Engineering