Recursive Phase Estimation with Applications to Heartbeat Processing Öffentlichkeit

Abstract

In this thesis, we explore the estimation of nonlinear, time-varying phase. We first introduce how to conceptualize and estimate phase using global techniques from Fourier analysis and Functional Data Analysis, before going on to develop recursive formulations using a constrained Extended Kalman Filter approach. We then evaluate the performance of the global and recursive methods on a variety of simulated examples, and discuss respective limitations of each. In general, we find that the recursive methods can outperform traditional estimation techniques in high-noise settings, and may be better suited to tracking local phase variations. Finally, we illustrate how the recursive models may be applied in order to robustly process and analyze heartbeat signals.

Table of Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Defining Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Sinusoidal Phase Estimation . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Discrete Fourier Transform . . . . . . . . . . . . . . . . . 3

2.1.2 Short-Time Fourier Transform . . . . . . . . . . . . . . . 4

2.1.3 Hilbert Transform . . . . . . . . . . . . . . . . . . . . . . 4

2.2 General Phase Estimation . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Curve Registration . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Recursive Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.1 Bayesian Filter . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.2 Extended Kalman Filter . . . . . . . . . . . . . . . . . . . 9

3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1 Model 1: Raw Phase Model . . . . . . . . . . . . . . . . . . . . . 11

3.1.1 State Space Model . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Model 2: Phase Function Model . . . . . . . . . . . . . . . . . . 13

3.2.1 B-Spline Functions . . . . . . . . . . . . . . . . . . . . . . 13

3.2.2 State-Space Model . . . . . . . . . . . . . . . . . . . . . . 16

3.3 State Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4 Practical Considerations . . . . . . . . . . . . . . . . . . . . . . . 18

4 Numerical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.1 Sinusoidal Examples . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2 Recursive Curve Registration . . . . . . . . . . . . . . . . . . . . 23

4.3 Simulated ECG Signals . . . . . . . . . . . . . . . . . . . . . . . 26

5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.1 Conclusions and Limitations . . . . . . . . . . . . . . . . . . . . . 28

5.2 Theoretical Considerations . . . . . . . . . . . . . . . . . . . . . . 29

5.3 Further Applications . . . . . . . . . . . . . . . . . . . . . . . . . 30

6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Statistics Applied Mathematics
Stichwort	curve registration extended kalman filter functional data analysis fourier analysis phase estimation
Committee Chair / Thesis Advisor	Bree Ettinger, Emory University
Committee Members	Alessandro Veneziani, Emory University Gordon Ramsay, Emory University

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