Adaptive Sub-Linear Time Fourier Algorithms

Abstract

We present a new deterministic algorithm for the sparse Fourier transform problem, in which we seek to identify k N significant Fourier coefficients from a signal of bandwidth N . Previous deterministic algorithms exhibit quadratic runtime scaling, while our algorithm scales linearly with k in the average case. Underlying our algorithm are a few simple observations relating the Fourier coefficients of time-shifted samples to unshifted samples of the input function. This allows us to detect when aliasing between two or more frequencies has occurred, as well as to determine the value of unaliased frequencies. We show that empirically our algorithm is orders of magnitude faster than competing algorithms. 1 See [GST08] for a "user-friendly" description of the improved algorithm. 2 Specifically, the runtime is O(k 2 ·log N ·|S|), where S is the set of samples read by the algorithm. This set takes the form S = log N =1 A−B , where A has ε-discrepancy on rank 2 Bohr sets, B ε-approximates the uniform distribution on [0, 2 − 1] ∩ Z, and A − B is the difference set. Using constructions from [Kat89] one has |A| = O(ε −1 log 4 N ), |B | = O(ε −3 log 4 N ); setting ε = Θ(k −1 ) and noting that | A − B | = O ( |A − B |) and |A − B | = O(|A||B |)

SFFT based ISAR Imaging

“…Researchers put forward two improved algorithms later: SFFTv2 [1] and SFFTv3 [3], enabling the computational complexity of SFFT decreasing to O (k*logn). At the same time, transformed algorithms used in various fields are developed, such as PS (Phase Shifted) for continuous Fourier transform [4] and 2D-SFFT [5].…”

“…As a continuous Fourier transform algorithm, PS (Phase Shift) has a very similar principle with SFFTv3, but when seeking for spectra points that bigger than zero, it uses two different sampling operations instead of spectra permutations [4].…”

Deterministic sparse FFT algorithms

“…Its complexity is O(M 2 log 4 N ). Further deterministic approaches can be found in [1], [2] and [29]. The latter proposes an algorithm with O(M log M ) runtime, though it only works with the a priori condition that time-shifted samples are available.…”

“…However, reconstruction algorithms usually incorporate iterative schemes to solve corresponding minimization problems with higher computational costs than N (log N ). The relationship between both approaches is described in more detail in [29].…”

“…In contrast to our approach, Potts, Tasche and Volkmer use the procedure to split polynomials of large sparsity into several polynomials of lower sparsity which can then be recovered by applying Prony's method. The idea of shifted sampling has originally been proposed in [29].…”