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Formalisation en Coq des algorithmes de filtre numérique calculés en précision finie

Abstract : Digital filters have numerous applications, from telecommunications to aerospace. To be used in practice, a filter needs to be implemented using finite precision (floating- or fixed-point arithmetic). Resulting rounding errors may become especially problematic in embedded systems: tight time, space, and energy constraints mean that we often need to cut into the precision of computations, in order to improve their efficiency. Moreover, digital filter programs are strongly iterative: rounding errors may propagate and accumulate through many successive iterations. As some of the application domains are critical, I study rounding errors in digital filter algorithms using formal methods to provide stronger guaranties. More specifically, I use Coq, a proof assistant that ensures the correctness of this numerical behavior analysis. I aim at providing certified error bounds over the difference between outputs from an implemented filter (computed using finite precision) and from the original model filter (theoretically defined with exact operations). Another goal is to guarantee that no catastrophic behavior (such as unexpected overflows) will occur. Using Coq, I define linear time-invariant (LTI) digital filters in time domain. I formalize a universal form called SIF: any LTI filter algorithm may be expressed as a SIF while retaining its numerical behavior. I then prove the error filters theorem and the Worst-Case Peak Gain theorem. These two theorems allow us to analyze the numerical behavior of the filter described by a given SIF. This analysis also involves the sum-of-products algorithm used during the computation of the filter. Therefore, I formalize several sum-of-products algorithms, that offer various trade-offs between output precision and computation speed. This includes a new algorithm whose output is correctly rounded-to-nearest. I also formalize modular overflows, and prove that one of the previous sum-of-products algorithms remains correct even when such overflows are taken into account.
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https://theses.hal.science/tel-03202580
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Submitted on : Tuesday, April 20, 2021 - 9:18:08 AM
Last modification on : Friday, August 5, 2022 - 2:58:08 PM
Long-term archiving on: : Wednesday, July 21, 2021 - 6:13:47 PM

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  • HAL Id : tel-03202580, version 1

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Diane Gallois-Wong. Formalisation en Coq des algorithmes de filtre numérique calculés en précision finie. Traitement du signal et de l'image [eess.SP]. Université Paris-Saclay, 2021. Français. ⟨NNT : 2021UPASG016⟩. ⟨tel-03202580⟩

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