Root Projection of One-Sided Time Series

Until recently it has been impossible to accurately determine the roots of polynomials of high degree, even for polynomials derived from the Z transform of time series where the dynamic range of the coefficients is generally less than 100 dB. In a companion paper, two new programs for solving such polynomials were discussed and applied to signature analysis of one-sided time series [1], We present here another technique, that of root projection (RP), together with a Gram-Schmidt method for implementing it on vectors of large dimension. This technique utilizes the roots of the Z transform of a one-sided time series to construct a weighted least squares modification of the time series whose Z transform has an appropriately modified root distribution. Such a modification can be employed in a manner which is very useful for filtering and deconvolution applications [2]. Examples given here include the use of boundary root projection for front end noise reduction and a generalization of Prony’s method.