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Error Analysis of Digital Filters Using Theorem Proving

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Theorem Proving in Higher Order Logics (TPHOLs 2004)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 3223))

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Abstract

When a digital filter is realized with floating-point or fixed-point arithmetics, errors and constraints due to finite word length are unavoidable. In this paper, we show how these errors can be mechanically analysed using the HOL theorem prover. We first model the ideal real filter specification and the corresponding floating-point and fixed-point implementations as predicates in higher-order logic. We use valuation functions to find the real values of the floating-point and fixed-point filter outputs and define the error as the difference between these values and the corresponding output of the ideal real specification. Fundamental analysis lemmas have been established to derive expressions for the accumulation of roundoff error in parametric Lth-order digital filters, for each of the three canonical forms of realization: direct, parallel, and cascade. The HOL formalization and proofs are found to be in a good agreement with existing theoretical paper-and-pencil counterparts.

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Author information

Authors and Affiliations

  1. Dept. of Electrical & Computer Engineering, Concordia University, 1455 de Maisonneuve W, Montreal, Quebec, H3G 1M8, Canada

    Behzad Akbarpour & Sofiène Tahar

Authors
  1. Behzad Akbarpour
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  2. Sofiène Tahar
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Editor information

Editors and Affiliations

  1. School of Computing, University of Utah,  

    Konrad Slind

  2. Electrical and Computer Engineering Department, Utah State University, 4120 Old Main Hill, 84341, Logan, UT, USA

    Annette Bunker

  3. School of Computing, Univ. of Utah, 84112, Salt Lake City, UT, USA

    Ganesh Gopalakrishnan

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© 2004 Springer-Verlag Berlin Heidelberg

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Akbarpour, B., Tahar, S. (2004). Error Analysis of Digital Filters Using Theorem Proving. In: Slind, K., Bunker, A., Gopalakrishnan, G. (eds) Theorem Proving in Higher Order Logics. TPHOLs 2004. Lecture Notes in Computer Science, vol 3223. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30142-4_1

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  • DOI: https://doi.org/10.1007/978-3-540-30142-4_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-23017-5

  • Online ISBN: 978-3-540-30142-4

  • eBook Packages: Springer Book Archive

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