What’s the Frequency, Kenneth?: Sublinear Fourier Sampling Off the Grid

Abstract

Abstract. We design a sublinear Fourier sampling algorithm for a case of sparse off-grid frequency recovery. These are signals with the form f (t) = P k j=1 aje iω j t +ν, t ∈ Z Z; i.e., exponential polynomials with a noise term. The frequencies {ωj} satisfy ωj ∈ [η, 2π − η] and min i =j |ωi − ωj| ≥ η for some η > 0. We design a sublinear time randomized algorithm, which takes O(k log k log(1/η)(log k + log( a 1/ ν 1)) samples of f (t) and runs in time proportional to number of samples, recovering {ωj} and {aj} such that, with probability Ω(1), the approximation error satisfies |ω j − ωj| ≤ η/k and |aj − a j | ≤ ν 1/k for all j with |aj| ≥ ν 1/k.

Rapidly computing sparse Legendre expansions via sparse Fourier transforms

“…Over the last several years SFTs have been improved significantly in both theory and practice. Recent work includes better implementations [25,31,43], improvements in runtime complexity bounds (both upper and lower) [24,31,28], adaptation of the methods to the recovery of superpositions of sinusoids with non-integer frequencies [8], and improvements in theoretical error guarantees [1,27,43,28]. In particular, entirely deterministic SFTs exist [27,28] that are guaranteed to always return a near-optimal sparse trigonometric polynomial, y s :…”

Compressive Parameter Estimation for Sparse Translation-Invariant Signals Using Polar Interpolation

“…While revised formulations have been proposed that can deal with noisy measurements at the expense of additional computation by using total variation norms [18] or more elaborate matrix factorizations [19], the restriction to uniform low-rate (or low-count) sampling acquisition schemes appears to remain necessary. Additional efforts based on group testing [20] require randomized sampling schemes.…”

A note on the high-dimensional sparse Fourier transform in the continuous setting*

“…The sample complexity and runtime complexity of the SFT are mainly affected by the sparsity, and less affected by the bandwidth. The SFT originated from the work on the Hadamard transform [2] and has received continuous attention from applied mathematics [3][4][5][6][7][8], signal processing [9][10][11][12][13][14], and theoretical computer science communities [15][16][17][18][19][20] over the last two decades. Most of the relevant works deals with the discrete case where the frequencies are on the grid.…”

“…It is natural for people to consider the case that the frequencies are in a continuous region. This has led researchers to establish the sparse Fourier transform in the one-dimensional continuous setting [18,24,25].…”