A Stochastic Approach For The Range Evaluation

Digital Signal Processing (DSP) applications have experienced a very strong development in the last decades, benefiting from the major advances of the semiconductor industry. All practical DSP implementations use fixed-point arithmetic to reduce the area and power consumption and obtain a cost-effective hardware. As a consequence, a conversion from the floating-point description of the algorithm to a fixed-point implementation that adjusts every bit-width in the datapath must be realized. This is an optimization process that consists in finding the minimal fractional part (numerical accuracy evaluation) and integer part (range estimation) wordlengths that still satisfy the performance constraints. In this thesis a stochastic approach for the range evaluation is presented. Our goal is to obtain a complete representation of the variability that incorporates the probabilistic behaviour and not only the maximal and minimal bounds. A method based on the Karhunen-Loeve Expansion is developed at the beginning for the case of linear time-invariant systems. Furthermore, the Polynomial Chaos Expansion is introduced in order to treat non-linear operations. The methods are applied to the optimization of the integer part wordlength when a slight degradation of the performances is acceptable. The range doesn't cover anymore the entire theoretical interval of variation, instead the occurrence of overflows is authorized with a constraint regarding their probability of appearance. Signals that have high variations of their amplitude are approximated with tight intervals so that the implementation cost can be reduced.