Évaluation analytique de la précision des systèmes en virgule fixe

Romuald Rocher 1
1 CAIRN - Energy Efficient Computing ArchItectures with Embedded Reconfigurable Resources
Inria Rennes – Bretagne Atlantique , IRISA-D3 - ARCHITECTURE
Abstract : Digital signal processing applications are specified in floating-point in order to prevent problems due to computing precision. However, in order to satisfy cost constraints, application implementation in embedded systems requires fixed point arithmetic using. Thus, the application defined in floating point arithmetic must be converted into a fixed-point specification. To reduce applications time-to-market, tools to automate floating-point to fixed-point conversion are needed. In these outils, an important stage corresponds to precision evaluation of fixed-point specification. Indeed, fixed-point arithmetic generates noise sources due to the elimination of some bits while a format change. These noises propagates through the system and modifie computing precision. Computing precision damages must be contained in order to assure algorithm integrity and application performances. Application precision can be evaluated by fixed-point simulations but they require high computing time. Simulations approaches to optimise a fixed-point specification lead to very important optimisation time. Thus, the aim of this thesis is to propose a new approach to evaluate automatically and analytically a fixed-point system precision. The precision is determined through the Signal to Quantization Noise Ratio (SQNR) of the considered application. Methods have been proposed for linear and time invariant (LTI) systems as non-LTI and non recursive systems. Thus, the aim of this thesis is to propose a method for automatic evaluation of all types of fixed-point systems, and especially, non-LTI systems with a recursion as adaptive filters. First, precision evaluation models for adaptives filters are proposed. Then, a general model for all types of systems is presented. The first aspect of our work concerns analytical models development for precision evaluation of particular applications such as adaptive filters. Indeed, these non-LTI systems can not be treated by classical technics. For these applications, existing models are only valid for a convergent rounding quantization law. Proposed models by our approach take into account all quantization laws. An analytical expression of the noise power on the system output is proosed for the different adaptive algorithms. These models have been integrated into an IP (Intellectual Properties) generator allowing to generate an software or hardware component which is arithmetically optimised. The secund aspect of our work corresponds to the definition of a general approach for precision analytical evaluation valid for all systems including arithmetic operations. This method is based on an approach using matrix which simplifies transform algorithms treatment (FFT,DCT). For recursive systems, the recurrence is unrolled. The complexity of our approach is determined and a linear prediction model has been developped in order to reduce this complexity. This model accelerates recurrence unrolling. The model has been implemented in the Matlab tool and is integrated to the automatic tool of floating-point to fixed-point conversion. This approach allows to optimise operators width in an optimisation process of application cost (power consumption, architecture area). These two approaches are evaluated and compared in terms of precision and computing time for different applications such as, the Least Mean Square (LMS) or the Affine Projection Algorithms (APA). The two methods leads to very realistic noise power estimations. Model execution times have been evaluated on Matlab. The linear prediction approach reduce dramatically the noise power computing time. While an optimisation process of operators width, the execution times have been measured and compared to those obtained by an approach using fixed-point simulations. Our approach reduces computing time compared to simulation approaches after only some iterations. These results show the interest of our methodology in order to reduce fixed-point systems development time.
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Romuald Rocher. Évaluation analytique de la précision des systèmes en virgule fixe. Traitement du signal et de l'image [eess.SP]. Université Rennes 1, 2006. Français. ⟨tel-00609822⟩

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