Contents

• Introduction
  • Objective and Overview
  • Previous Work
  
  
• Review of Fresnel Integrals
  • Definitions
  • Approximations
  • Other Properties
  
  
• Detailed Proof of Concave Down Parabolic Phase for a DC-Centered Increasing Frequency Chirp Signal
  • Definition of a Linear Frequency Chirp Function and its FFT
  • Approximation of the FFT Sums with Integrals
  • Manipulation of the Integrals into Fresnel Integrals
  • Application of Fresnel Integral Approximations
  • Hann Window / Infinite Limits Resulting Phase and Magnitude Approximation
  • Modifications to Proof for Decreasing Frequency Chirp
  • Generalizations of Proof to Nonzero Center Frequencies
  
  
• Model Inversion
  • Choosing the Relevant Frequency Range
  • Inversion of the Magnitude Expression
  • Inversion of the Phase Expression
  
  
• Examples of Estimating by Model Inversion
  • Linear Frequency Chirps
  • Other Monotonic Nonstationary Signals
  • Nonmonotonic Nonstationary Signals
  
  
• Summary
• Summaries of Other Implemented Algorithms for Future Publication
  • Same Sign Large Limits Approximation
  • Small Limits Approximation
  • Comparative Plots
  
  
• Yi-Wen Liu's Model
• Error Propagation Analysis
  • Large Limits Approximation Error Propagation in Magnitude Estimate
  • Large Limits Approximation Error Propagation in Phase Estimate
  
  
• Appendix: Non-Validity of the Infinite Fresnel Integral Limits Approximation for the Rectangle Window
  • Definition of Approximation
  • Test of the Infinite Limits Approximation for the Rectangle Window
  
  
• Bibliography
• About this document ...