Generalised modal realisation as a practical and efficient tool for FWL implementation

Abstract

Finite word length (FWL) effects have been a critical issue in digital filter implementation for almost four decades. Although some optimisations may be attempted to get an optimal realisation with regards to a particular effect, for instance the parametric sensitivity or the round-off noise gain, the purpose of this article is to propose an effective one, i.e. taking into account all the aspects. Based on the specialised implicit form, a new effective and sparse structure, named ρ -modal realisation, is proposed. This realisation meets simultaneously accuracy (low sensitivity, round-off noise gain and overflow risk), few and flexible computational efforts with a good readability (thanks to sparsity) and simplicity (no tricky optimisation is required to obtain it) as well. Two numerical examples are included to illustrate the ρ -modal realisation's interest.

Keywords:

• coefficient sensitivity
• dynamic scaling
• FWL implementation
• implicit framework
• modal realisation
• pole sensitivity
• round-off noise gain

Notes

Notes

1. Exactly-implemented parameters mentioned here are those that are not modified by the process of quantisation (those completely represented by a number of bits lower that the word length used for implementation).

2. ρDFIIt is evaluated by the methods proposed in Li and Zhao (2004). It is under the strict L ₂-scaling constraints, without L ₂-scaling on intermediate variables.

3. From a practical viewpoint, these measures are programmed in the toolbox developed by the authors and freely available at http://fwrtoolbox.gforge.inria.fr.