Root Projection of One-Sided Time Series

Area of Science:

• Signal Processing
• Numerical Analysis
• Time Series Analysis

Background:

• Accurate determination of high-degree polynomial roots, especially from Z transforms of time series with limited dynamic range, has been a significant challenge.
• Existing methods struggle with polynomials derived from time series data where coefficient dynamic range is typically below 100 dB.

Purpose of the Study:

• Introduce a novel root projection (RP) technique for accurately solving high-degree polynomials.
• Develop a Gram-Schmidt method for implementing RP on large-dimension vectors.
• Demonstrate the utility of RP in modifying time series Z transforms for enhanced signal processing.

Main Methods:

• The study presents the root projection (RP) technique.
• A Gram-Schmidt method is employed for implementing RP on high-dimensional vectors.
• RP utilizes Z transform roots to create a weighted least squares modification of the time series.

Main Results:

• The root projection technique enables accurate determination of high-degree polynomial roots.
• The modified time series exhibits an adjusted root distribution in its Z transform.
• Applications in filtering and deconvolution are demonstrated, including noise reduction and Prony's method generalization.

Conclusions:

• Root projection offers a robust method for solving complex polynomial root-finding problems in time series analysis.
• The technique provides a powerful tool for signal processing tasks such as filtering and deconvolution.
• Boundary root projection shows promise for effective front-end noise reduction.