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A perturbative analysis for noisy spectral estimation

Lexing Ying

Abstract

Spectral estimation is a fundamental task in signal processing. Recent algorithms in quantum phase estimation are concerned with the large noise, large frequency regime of the spectral estimation problem. The recent work in Ding-Epperly-Lin-Zhang shows that the ESPRIT algorithm exhibits superconvergence behavior for the spike locations in terms of the maximum frequency. This note provides a perturbative analysis to explain this behavior. It also extends the discussion to the case where the noise grows with the sampling frequency.

A perturbative analysis for noisy spectral estimation

Abstract

Spectral estimation is a fundamental task in signal processing. Recent algorithms in quantum phase estimation are concerned with the large noise, large frequency regime of the spectral estimation problem. The recent work in Ding-Epperly-Lin-Zhang shows that the ESPRIT algorithm exhibits superconvergence behavior for the spike locations in terms of the maximum frequency. This note provides a perturbative analysis to explain this behavior. It also extends the discussion to the case where the noise grows with the sampling frequency.
Paper Structure (3 sections, 20 equations, 3 figures)

This paper contains 3 sections, 20 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Perturbative analysis
  3. General noise

Figures (3)

  • Figure 1: $p=0$. The slopes of the log-log plots for $\{x_k\}$ and $\{w_k\}$ match the perturbative analysis.
  • Figure 2: $p=0.25$. The slopes of the log-log plots for $\{x_k\}$ and $\{w_k\}$ match the perturbative analysis.
  • Figure 3: $p=0.75$. The slopes of the log-log plots for $\{x_k\}$ and $\{w_k\}$ match the perturbative analysis.

Theorems & Definitions (3)

  • Example 1
  • Example 2
  • Example 3