Hardware implementations of fixed-point Atan2

International audience—The atan2 function computes the polar angle arctan(x/y) of a point given by its cartesian coordinates. It is widely used in digital signal processing to recover the phase of a signal. This article studies for this context the implementation of atan2 with fixed-point inputs and outputs. It compares the prevalent CORDIC shift-and-add algorithm to two multiplier-based techniques. The first one reduces the bivariate atan2 function to two functions of one variable: the reciprocal, and the arctangent. These two functions may be tabulated, or evaluated using bipartite or polynomial approximation methods. The second technique directly uses piecewise bivariate polynomial approximations, in degree 1 and degree 2. It requires larger tables but has the shortest latency. Each of these approaches requires a relevant argument reduction, which is also discussed. All the algorithms are described with the same accuracy target (faithful rounding) and implemented with similar care in an open-source library. Based on synthesis results on FPGAs, their relevance domains are discussed