Search

Many applications of finite impulse response (FIR) digital filters impose strict format constraints on the filter coefficients. Such requirements increase the complexity of determining optimal designs for the problem at hand. We introduce a fast and efficient method, based on the computation of good nodes for polynomial interpolation and Euclidean lattice basis reduction. Experiments show that it returns quasi-optimal finite wordlength FIR filters; compared to previous approaches it also scales remarkably well (length 125 filters are treated in $<$ 9 s). It also proves useful for accelerating the determination of optimal finite wordlength FIR filters. [ABSTRACT FROM AUTHOR]

The 2008 revision of the IEEE-754 standard, which governs floating-point arithmetic, recommends that a certain set of elementary functions should be correctly rounded. Successful attempts for solving the Table Maker's Dilemma in binary64 made it possible to design CRlibm, a library which offers correctly rounded evaluation in binary64 of some functions of the usual libm. It evaluates functions using a two step strategy, which relies on a folklore heuristic that is well spread in the community of mathematical functions designers. Under this heuristic, one can compute the distribution of the lengths of runs of zeros/ones after the rounding bit of the value of the function at a given floating-point number. The goal of this paper is to change, whenever possible, this heuristic into a rigorous statement. The underlying mathematical problem amounts to counting integer points in the neighborhood of a curve, which we tackle using so-called exponential sums techniques, a tool from analytic number theory. [ABSTRACT FROM AUTHOR]