Showing papers on "Cyclotomic fast Fourier transform published in 2018"

A nonuniform fast Fourier transform based on low rank approximation

Abstract: By viewing the nonuniform discrete Fourier transform (NUDFT) as a perturbed version of a uniform discrete Fourier transform, we propose a fast and quasi-optimal algorithm for computing the NUDFT based on the fast Fourier transform (FFT). Our key observation is that an NUDFT and DFT matrix divided entry by entry is often well approximated by a low rank matrix, allowing us to express a NUDFT matrix as a sum of diagonally scaled DFT matrices. Our algorithm is simple to implement, automatically adapts to any working precision, and is competitive with state-of-the-art algorithms. In the fully uniform case, our algorithm is essentially the FFT. We also describe quasi-optimal algorithms for the inverse NUDFT and two-dimensional NUDFTs.

Fast Fourier Transforms for Nonequispaced Data

Abstract: In this chapter, we describe fast algorithms for the computation of the DFT for d-variate nonequispaced data, since in a variety of applications the restriction to equispaced data is a serious drawback. These algorithms are called nonequispaced fast Fourier transforms and abbreviated by NFFT.

Fast inverse nonlinear Fourier transform.

Abstract: This paper considers the non-Hermitian Zakharov-Shabat scattering problem which forms the basis for defining the $\mathrm{SU}(2)$-nonlinear Fourier transform (NFT). The theoretical underpinnings of this generalization of the conventional Fourier transform is quite well established in the Ablowitz-Kaup-Newell-Segur formalism; however, efficient numerical algorithms that could be employed in practical applications are still unavailable. In this paper, we present two fast inverse NFT algorithms with $O(KN+N{log}^{2}N)$ complexity and a convergence rate of $O({N}^{\ensuremath{-}2})$, where $N$ is the number of samples of the signal and $K$ is the number of eigenvalues. These algorithms are realized using a new fast layer-peeling (LP) scheme $[O(N{log}^{2}N)]$ together with a new fast Darboux transformation (FDT) algorithm $[O(KN+N{log}^{2}N)]$ previously developed by V. Vaibhav [Phys. Rev. E 96, 063302 (2017)]. The proposed fast inverse NFT algorithm proceeds in two steps: The first step involves computing the radiative part of the potential using the fast LP scheme for which the input is synthesized under the assumption that the radiative potential is nonlinearly bandlimited, i.e., the continuous spectrum has a compact support. The second step involves addition of bound states using the FDT algorithm. Finally, the performance of these algorithms is demonstrated through exhaustive numerical tests.

R-FFAST: A Robust Sub-Linear Time Algorithm for Computing a Sparse DFT

Abstract: The fast Fourier transform is the most efficiently known way to compute the discrete Fourier transform (DFT) of an arbitrary $ n$ -length signal, and has a computational complexity of $O( n\log n)$ . If the DFT $ \vec {X}$ of the signal $ \vec {x}$ has only $k$ non-zero coefficients (where $ k ), can we do better? We addressed this question and presented a novel fast Fourier aliasing-based sparse transform (FFAST) algorithm that cleverly induces sparse-graph alias codes in the DFT domain, via a Chinese-remainder-theorem-guided sub-sampling operation in the time-domain. The induced sparse-graph alias codes are then exploited to devise a fast and iterative onion-peeling style decoder that computes $ k$ -sparse DFT of an $ n$ -length signal using only $O( k)$ time-domain samples and $O( k\log k)$ computations. In this paper, we generalize the FFAST framework by Pawar and Ramchandran to the noisy setting where the time-domain samples are corrupted by white Gaussian noise. We show that the noise-robust R-FFAST algorithm computes a $ k$ -sparse DFT of an $ n$ -length signal using $O( k\log ^{3} n)$ noise-corrupted time-domain samples in $O( k\log ^{4} n)$ complexity, i.e., sub-linear sample and time complexity . In Section IX , we provide extensive simulation results validating the empirical performance of the R-FFAST algorithm, e.g., we show that the R-FFAST algorithm computes a 50-sparse DFT of an ≈ 10 million length signal using only 4800 noisy samples with an effective signal-to-noise ratio of 5 dB. We also provide comparison of the run-time performance of several existing sparse Fourier transform implementations with that of the R-FFAST and show that it is almost 20 times faster, for comparable settings, than the state-of-the-art algorithm, while simultaneously providing better support recovery guarantees. While our theoretical results are for signals with a uniformly random support of the non-zero DFT coefficients and additive white Gaussian noise, we provide simulation results, which demonstrate that the R-FFAST algorithm performs well even for signals like magnetic resonance images, that have an approximately sparse Fourier spectrum with a non-uniform support for the dominant DFT coefficients.