Hybrid Dealiased Convolutions

Abstract

Efficient algorithms for computing linear convolutions based on the fast Fourier transform are developed. A hybrid approach is described that combines the conventional practice of explicit dealiasing (explicitly padding the input data with zeros) and implicit dealiasing (mathematically accounting for these zero values). The new approach generalizes implicit dealiasing to arbitrary padding ratios and includes explicit dealiasing as a special case. Unlike existing implementations of implicit dealiasing, hybrid dealiasing tailors its subtransform sizes to the convolution geometry. Hybrid dealiasing also extends implicit dealiasing to efficiently compute convolutions with real-valued inputs. Multidimensional convolutions are implemented with hybrid dealiasing by decomposing them into lower-dimensional convolutions. Convolutions of complex-valued, Hermitian symmetric, and real- valued inputs of equal length are illustrated with pseudocode and implemented in the open-source FFTW++ library. Hybrid dealiasing is shown to match or outperform explicit dealiasing in one dimension and greatly outperform explicit dealiasing in two and three dimensions.