Some properties of the Givens' reduction applied to linear prediction analysis of speech.

Abstract

Described here are some physical interpretations for the computational process of the Givens' reduction, an efficient algorithm for obtaining the least-squares solution of a set of over-determined linear equations. These physical interpretations are brought into view by applying the Givens' reduction to the linear predictive analysis of speech, and would not have suggested themselves if the Givens' reduction were considered only in connection with the solution of an ordinary set of over-determined linear equations. The attractive advantages found in the computational process of the Givens' reduction are the following:(1) The forward and the backward prediction errors for each data sample are automatically obtained in the working matrix of the augmented Givens' reduction.(2) The direct time-update recursion for the K-parameter, the reflection coefficients in linear prediction analysis, is implicitly performed sample by sample.(3) An index representing the rank of the covariance matrix of the input data sequence is available in the working vector. These advantages come out from the physical interpretation of the corresponding parts of the working areas for the Givens' reduction. The behavior of these values in the working areas are empirically confirmed by actual analysis of speech.