Root Projections Research Articles

Deconvolution of Acoustic Emission and Other Causal Time Series.

A new technique, root projection (RP), is given for quantitative deconvolution of causal time series in the presence of moderate amounts of noise. Deconvolution is treated as a well-conditioned but underdetermined problem and a priori information is employed to obtain comparable noise reduction to that achieved by singular value decomposition (SVD) techniques while providing more accurate frequency information about the inverse. Two detailed examples arc given. The first gives noise analysis for alternate methods for deconvolution with a Gaussian kernel. The second example presents a model acoustic emission transducer calibration problem with typical noisy and incomplete output data. This example is treated by the use of a robust cross-cutting algorithm combining both the RP and SVD methods.

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.