IOSS Criterion for the Absence of Limit Cycles in Interfered Digital Filters Employing Saturation Overflow Arithmetic
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IOSS Criterion for the Absence of Limit Cycles in Interfered Digital Filters Employing Saturation Overflow Arithmetic Choon Ki Ahn
Received: 21 February 2012 / Revised: 6 November 2012 / Published online: 27 November 2012 © Springer Science+Business Media New York 2012
Abstract In this paper, a new criterion for the absence of limit cycles in fixed-point state-space digital filters with saturation overflow arithmetic and external interference is proposed via an input/output-to-state stability (IOSS) approach. The criterion ensures not only the asymptotic stability, but also the reduction of the effect of the external interference. This criterion is represented by linear matrix inequality (LMI) and, hence, is computationally tractable. Via a numerical example, we demonstrate the effectiveness of the proposed IOSS criterion. Keywords Input/output-to-state stability (IOSS) · Asymptotic stability · Digital filters · Finite wordlength effects · Linear matrix inequality (LMI)
1 Introduction When we design a digital filter using fixed-point arithmetic on a digital computer or on special-purpose digital hardware, nonlinearities such as quantization and overflow due to finite wordlength are inevitable. If the total number of quantization steps is large, then the effects of these nonlinearities can be regarded as decoupled and can be investigated separately. The stability properties of digital filters employing saturation arithmetic have attracted the attention of several researchers [8, 10–12, 15–18]. When we implement a high-order digital filter in hardware, it usually breaks down into several biquad filters before implementation. Then, there exist interferences between these biquad filters. These interferences may result in malfunction as well as destruction in the last [1–5, 14, 21]. However, most existing criteria for the stability of digital filters are only available under specific conditions, while in unfavorable C.K. Ahn () School of Electrical Engineering, Korea University, 145, Anam-ro, Seongbuk-gu, Seoul, 136-701, Korea e-mail: [email protected]
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Circuits Syst Signal Process (2013) 32:1433–1441
environments with external interferences or parametric uncertainties, these existing criteria are not applicable. Recently, Ahn proposed some new stability criteria for interfered digital filters [1–5]. The input/output-to-state stability (IOSS) concept was introduced in [7, 13, 19]. The IOSS approach is a nice method to deal with the stability of dynamical systems. The IOSS property means that no matter what the initial state is, if the input and the observed output are small, then eventually also the state of the system will become small. This approach is an attractive tool for the stability analysis of digital filters because we can obtain general conclusions on stability using only input–output characteristics. At this point, the following natural question arises: can we obtain an IOSS criterion for digital filters with external interference? This paper answers this interesting question. To the best of the
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