Abstract
In this paper we provide a summary of recent and new results on finite word length effects in recursive least squares adaptive algorithms. We define the numerical accuracy and numerical stability of adaptive recursive least squares algorithms and show that these two properties are related to each other, but are not equivalent. The numerical stability of adaptive recursive least squares algorithms is analyzed theoretically and the numerical accuracy with finite word length is investigated by computer simulation. It is shown that the conventional recursive least squares algorithm gives poor numerical accuracy when a short word length is used. A new form of a recursive least squares lattice algorithm is presented which is more robust to round-off errors compared to the conventional form. Optimum scaling of recursive least squares algorithms for fixedpoint implementation is also considered.
Analyse
On présente un résumé des nouveaux résultats concernant les effets de la précision finie dans les algorithmes adaptatifs des moindres carrés récursifs. Pour ces algorithmes, on définit la piécision numérique, la stabilité numérique et l’on montre que ces deux propriétés sont liées sans être équivalentes. La stabilité numérique est analysée théoriquement et la précision numérique sur des mots de longueur finie est examinée par simulation sur ordinateur. On montre que l’algorithme récursif classique des moindres carrés atteint une faible précision numérique lorsque des mots de longueur faible sont utilisés. On présente une nouvelle forme d’algorithme des moindres carrés en treillis qui se montre plus résistant aux erreurs d’arrondi que l’algorithme classique. On considère aussi une normalisation optimale des algorithmes des moindres carrés récursifs pour une implantation en virgule fixe.
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Ling, F., Manolakis, D. & Proakis, J.G. Finite word-length effects in recursive least squares algorithms with application to adaptive equalization. Ann. Telecommun. 41, 328–336 (1986). https://doi.org/10.1007/BF02998639
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DOI: https://doi.org/10.1007/BF02998639
Key words
- Adaptive algorithm
- Rounding error
- Least squares method
- Recursivity
- Accuracy
- Stability
- Numerical calculation
- Digital simulation
- Equalization
- Error analysis