A Robust Sparse Fourier Transform in the Continuous Setting

TL;DR

This paper introduces a robust algorithm for computing sparse Fourier transforms in the continuous domain, effectively handling noise and improving accuracy over previous methods with similar sample complexity.

Contribution

It presents the first robust continuous sparse Fourier transform algorithm with linear sample complexity in sparsity and logarithmic in noise, surpassing prior noise tolerance limitations.

Findings

Achieves near-optimal noise tolerance in continuous sparse Fourier transform

Provides precise frequency recovery under noisy conditions

Reduces sample complexity compared to previous methods

Abstract

In recent years, a number of works have studied methods for computing the Fourier transform in sublinear time if the output is sparse. Most of these have focused on the discrete setting, even though in many applications the input signal is continuous and naive discretization significantly worsens the sparsity level. We present an algorithm for robustly computing sparse Fourier transforms in the continuous setting. Let $x(t) = x^*(t) + g(t)$, where $x^*$ has a $k$-sparse Fourier transform and $g$ is an arbitrary noise term. Given sample access to $x(t)$ for some duration $T$, we show how to find a $k$-Fourier-sparse reconstruction $x'(t)$ with $$\frac{1}{T}\int_0^T |x'(t) - x(t) |^2 \mathrm{d} t \lesssim \frac{1}{T}\int_0^T | g(t)|^2 \mathrm{d}t.$$ The sample complexity is linear in $k$ and logarithmic in the signal-to-noise ratio and the frequency resolution. Previous results with similar sample complexities could not tolerate an infinitesimal amount of i.i.d. Gaussian noise, and even algorithms with higher sample complexities increased the noise by a polynomial factor. We also give new results for how precisely the individual frequencies of $x^*$ can be recovered.