# Estimating the Frequency of a Clustered Signal

## TL;DR

This paper develops new methods for accurately estimating the central frequency of clustered signals with Fourier spectrum in a narrow band, improving bounds on recovery accuracy and establishing fundamental limits.

## Contribution

It introduces generic conditions for frequency estimation, improves bounds for $k$-Fourier-sparse signals, and provides a new ratio bound with independent applications.

## Key findings

- Achieves frequency recovery within $ ilde{O}(k^3)$ error bound.
- Improves previous bounds from $O( ilde{O}(k^5)^{1.5})$ to $ ilde{O}(k^3)$.
- Establishes a lower bound of $ ilde{O}(k^2)$ for frequency estimation accuracy.

## Abstract

We consider the problem of locating a signal whose frequencies are "off grid" and clustered in a narrow band. Given noisy sample access to a function $g(t)$ with Fourier spectrum in a narrow range $[f_0 - \Delta, f_0 + \Delta]$, how accurately is it possible to identify $f_0$? We present generic conditions on $g$ that allow for efficient, accurate estimates of the frequency. We then show bounds on these conditions for $k$-Fourier-sparse signals that imply recovery of $f_0$ to within $\Delta + \tilde{O}(k^3)$ from samples on $[-1, 1]$. This improves upon the best previous bound of $O\big( \Delta + \tilde{O}(k^5) \big)^{1.5}$. We also show that no algorithm can do better than $\Delta + \tilde{O}(k^2)$. In the process we provide a new $\tilde{O}(k^3)$ bound on the ratio between the maximum and average value of continuous $k$-Fourier-sparse signals, which has independent application.