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GPS-based Optimal FIR Filtering of Clock Models
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Authors: Yuriy Shmaliy (Guanajuato University, FIMEE, Mexico) 
Book Description:
This book addresses novel results in the field of optimal finite impulse response (FIR) estimation and steering of the local clock time errors using the Global Positioning System (GPS) timing signals. The studies are motivated by permanently increased demands for accuracy of the local timescales in different areas of applications of wire and wireless digital systems. The main limitations of accuracy here are the GPS time uncertainty caused by different satellites in a view and the sawtooth noise induced by the commercially available GPS timing receivers owing to the principle of the one pulse per second (1PPS) signal formation. Due to the GPS time uncertainty, flicker components of the clock noise, and non Gaussian sawtooth noise, the standard Kalman algorithms may become unstable and noisy, even when the sawtooth correction is applied. We show that an efficient way of providing stable and accurate filtering, smoothing, prediction, and steering of the local clock errors is to use the optimal FIR structures, which are inherently bounded input/bounded output (BIBO) stable and more robust against temporary uncertainties and round-off errors. Moreover, unbiased polynomial FIR solutions having strong engineering features become actually optimal by large averaging horizons typically used in timekeeping. Such solutions are found and investigated in detail theoretically and for real measurements. Based upon, it is stated that optimal (unbiased) FIR estimators are likely the best candidates to use in the modern filtering, prediction, and synchronizations algorithms intended for the estimation and steering of local clocks.

Table of Contents:
Preface

1. Introduction

1.1 GPS-based Clock Estimation and Steering pp.1-2
1.2 Main Sources of Time Errors. pp.3-3
1.2.1 GPS Time Error pp.3-7
1.2.2 Sawtooth Noise pp.8-11
1.2.3 Clock Noise pp.12-17
1.3 Discrete-Time Clock Model in State Space pp.18-19
1.3.1 Three State Clock Model pp.20-20
1.4 Methods for Optimal Filtering of Clock Errors pp.21-21
1.4.1 Simple Averaging pp.22-23
1.4.2 Weighted Average pp.24-24
1.4.3 Regression Estimates pp.25-26
1.4.4 Weighting Function Approach pp.27-27
1.4.5 Linear Scalar Filtering (Bayesian Approach) pp.27-29
1.4.6 Kalman Filter Algorithm pp.30-35
1.5 Resume pp.36-36

2. FIR Filtering of Discrete-Time Clock Models pp.37-37
2.1 State Space Clock Model on a Finite Horizon pp.37-38
2.1.1 Transformation of the Clock Model pp.38-40
2.2 Optimal FIR Filtering of the TIE pp.41-43
2.2.1 Covariance Function Matrices pp.44-45
2.2.2 Errors with Large Averaging Horizons pp.46-47
2.2.3 Optimal FIR Filter Gain for Large Averaging Horizons pp.47-48
2.2.4 Estimate Errors pp.48-50
2.3 Unbiased FIR Filtering of the TIE p.51-52
2.3.1 Polynomial FIR Filter Gain pp.52-53
2.3.2 Unique Polynomial FIR Filter Gains pp.54-55
2.4 Optimum Horizons and Sampling Intervals pp.56-57
2.4.1 General Relations pp.58-59
2.4.2 Optimal Nopt and opt for Low l and K pp.60-61
6 Contents
2.4.3 Verifications for the First Clock State pp.61-62
2.4.4 Verifications for Higher Clock States pp.63-65
2.5 FIR Filtering Algorithms pp.66-67
2.5.1 Basic Algorithm pp.68-69
2.5.2 Thinning Algorithm pp.70-71
2.6 Resume pp.72-73

3 FIR Prediction of Clock Instabilities pp.73-73
3.1 Prediction of Discrete Time State Space Models pp.73-74
3.1.1 Predictive Clock Model on a Finite Horizon pp.74-75
3.2 Optimal Predictive FIR Filtering of the TIE pp.76-77
3.2.1 Large Averaging Horizons pp.78-78
3.2.2 Errors in the Predictive Estimate pp.78-79
3.3 Unbiased Predictive FIR Filtering pp.80-81
3.3.1 Polynomial Gains pp.81-82
3.3.2 Properties of the p-step Dependent Gain pp.82-82
3.3.3 A p-step Ramp Gain pp.82-83
3.3.4 A p-step Quadratic Gain pp.84-85
3.3.5 Generalizations pp.86-86
3.4 Prediction Algorithms pp.86-87
3.4.1 Structures of Prediction Algorithms pp.87-88
3.4.2 Potentials of FIR Prediction pp.88-88
3.5 Resume pp.88-90

4 FIR Smoothing pp.91-91
4.1 Smoothing of Discrete Time State Space Models pp.91-92
4.1.1 State Space Clock Model for Smoothing on a Finite
Horizon pp.92-93
4.2 Optimal Smoothing FIR Filtering of Time Errors pp.94-95
4.2.1 Large Averaging Horizons, N 1 pp.95-95
4.3 Unbiased Smoothing FIR Filtering pp.95-96
4.3.1 A p-lag Ramp Gain pp.96-97
4.3.2 A p-lag Quadratic Gain pp.98-100
4.3.3 A p-lag Cubic Gain pp.101-102
4.3.4 Generalizations pp.103-104
4.4 Resume pp.105-106

5. Applications
5.1 Clock Identification on Finite Horizons
5.2 Unbiased FIR Filters vs. Optimal Filters
5.2.1 Ramp Optimal FIR Filter
5.2.2 Two State Kalman Filter
5.3 Filtering of Noisy Clock States
5.3.1 Time Interval Error
5.3.2 Fractional Frequency Onset

A. Clock Noise Characterization
A.1 Representation in the Frequency Domain
A.2 Representation in the Time Domain
A.2.1 Standard Estimates of Time and Frequency Random
Errors

B. Review of Discrete-time Unbiased FIR Estimators
B.1 Gains for Unbiased FIR Filtering
B.1.1 Ramp Gain .
B.1.2 Quadratic Gain
B.1.3 Cubic Gain
B.2 Gains for Unbiased FIR Predictors and Smoothers
B.2.1 Several Special Cases

References
Index

   Series:
      Electrical Engineering Developments
   Binding: Hardcover
   Pub. Date: 2009, 4th quarter
   Pages: 7 x 10, 200pp.
   ISBN: 978-1-60741-835-1
   Status: AV
  
Status Code Description
AN Announcing
FM Formatting
PP Page Proofs
FP Final Production
EP Editorial Production
PR At Prepress
AP At Press
AV Available
  
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GPS-based Optimal FIR Filtering of Clock Models