Floating Point Arithmetic and Digital Filters

摘要

In this paper, the roundoff noise properties of floating point digital filtersare examined. To make the analysis tractable, a high level model to deal with the errors in the inner product operation is developed. This model establishes a broad connection between coefficient sensitivity and roundoff noise. Along with the model, an efficient procedure to keep track of the addition scheme used in the inner product, and to compute the statistics of the errors is introduced. A systematic procedure based on the model is then developed to derive general expressions for the roundoff noise of FIR, direct form IIR, and state-space filters. The expressions in the context of state space filters are explored in some detail. Optimality issues are considered, and it is shown that when double precision accumulation is used, the optimal filters are similar in nature to those derived in the context of fixed point arithmetic with the essential difference that they also do depend on the spectrum of the input signal. Optimality with respect to addition schemes, and second-order filters are also examined in some detail. Roundoff noise, floating point arithmetic, state space filters.