Signal Approximation via the Gopher Fast Fourier Transform

We consider the problem of quickly estimating the best k-term Fourier representation for a given frequency-sparse band-limited signal (i.e., function) f : [0,2π] → C. In essence, this requires the identification of k of the largest magnitude frequencies of f̂ ∈CN , and the estimation their Fourier coefficients. Randomized sublinear-time Monte Carlo algorithms, which have a small probability of failing to output accurate answers for each input signal, have been developed for solving this problem [1, 2]. These methods were implemented as the Ann Arbor Fast Fourier Transform (AAFFT) and empirically evaluated in [3]. In this paper we present and evaluate the first implementation, called the Gopher Fast Fourier Transform (GFFT), of the more recently developed sparse Fourier transform techniques from [4]. Our experiments indicate that different variants of GFFT generally outperform AAFFT with respect to runtime and sample usage.