Accuracy of spike-train Fourier reconstruction for colliding nodes

TL;DR

This paper analyzes the limits of Fourier-based methods for reconstructing signals composed of clustered delta functions, establishing bounds on reconstruction errors and demonstrating the effectiveness of a decimation algorithm.

Contribution

It provides theoretical bounds for the worst-case reconstruction error of clustered nodes and introduces a decimation algorithm that achieves near-optimal accuracy.

Findings

Lower bound on worst-case reconstruction error for clustered nodes

Decimation algorithm attains error close to the theoretical lower bound

Reconstruction accuracy depends on cluster size and frequency interval

Abstract

We consider Fourier reconstruction problem for signals F, which are linear combinations of shifted delta-functions. We assume the Fourier transform of F to be known on the frequency interval [-N,N], with an absolute error not exceeding e > 0. We give an absolute lower bound (which is valid with any reconstruction method) for the "worst case" reconstruction error of F in situations where the nodes (i.e. the positions of the shifted delta-functions in F) are known to form an l elements cluster of a size h << 1. Using "decimation" reconstruction algorithm we provide an upper bound for the reconstruction error, essentially of the same form as the lower one. Roughly, our main result states that for N*h of order of (2l-1)-st root of e the worst case reconstruction error of the cluster nodes is of the same order as h, and hence the inside configuration of the cluster nodes (in the worst case scenario) cannot be reconstructed at all. On the other hand, decimation algorithm reconstructs F with the accuracy of order of 2l-st root of e.