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Application de la théorie des nombres à la conception optimale et à l'implémentation de très faible complexité des filtres numériques

Abstract : The main objective of our study is to develop fast algorithms for an optimal design and an implementation with low complexity of digital filters. The optimization criterion is the mean squared error at the filter output. Thus, we have studied and developed new algorithms for synthesis of finite impulse response (FIR) filters related to the two techniques of block filtering, overlap-save (OLS) and overlap-add (OLA). These two filtering techniques consist in processing the signal by blocks and use the fast Fourier transform (FFT) to reduce the complexity of the convolution calculation. Our algorithms, based on the matrix model development of the OLA and OLS structures, use the linear algebra properties, especially those of circulant matrices. To further reduce the complexity and the distortion, we have looked further into the mathematical foundations of the Fermat Number Transform (FNT). This transform is a particular case of the Number Theoretic Transforms (NTT) defined in the Galois field. Compared to the FFT, the FNT allows a calculation without rounding error and a large reduction of the number of multiplications necessary to carry out the convolution product. To highlight this transform, we have proposed and studied a new design of OLS and OLA filtering using the FNT. We have developed a low complexity algorithm for the optimal synthesis of filters using the properties of circulant matrices that we have developed in the Galois field. The simulation results of the block filtering with fixed-point implementation have shown that the use of the FNT instead of the FFT reduces the complexity and the filtering errors, as well as the cost of optimal filter synthesis.
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https://tel.archives-ouvertes.fr/tel-00490369
Contributor : Stéphane Azou <>
Submitted on : Tuesday, June 8, 2010 - 12:24:32 PM
Last modification on : Wednesday, September 16, 2020 - 9:56:36 AM
Long-term archiving on: : Friday, September 17, 2010 - 1:20:48 PM

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  • HAL Id : tel-00490369, version 1

Citation

Ali Daher. Application de la théorie des nombres à la conception optimale et à l'implémentation de très faible complexité des filtres numériques. Traitement du signal et de l'image [eess.SP]. Université de Bretagne occidentale - Brest, 2009. Français. ⟨tel-00490369⟩

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