Stable modelling of a continuous covariance function with application to continuous speech

We consider the modelling of a covariance function δ(t) + k(t) where t is continuous time. For instance, it is of interest to have a method for computing a continuous model of the oral cavity. To achieve this one uses a sampling of the speech signal which is high compared to the Nyquist frequency and one hopes to be able to distill from it a time continuous reflection function. The problem here, however, is one of numerical stability. By increasing the sampling and using classical methods one will lose control on accuracy. In order to stabilize the procedure we have investigated how the celebrated Schur method [1],[2],[3] may be extended to the time-continuous case. It turns out that a novel equation arises which expresses the causality of the modeling filter and which, when used on our problem, achieves a high degree of numerical stability. Incidentally we show also how the method is related to classical inverse scattering problems for lossless transmission lines. In fig. 1-4 we show a number of covariance testfunctions, their corresponding continuous-time reflection functions and their errors. As a conclusion we may state that we arrive at an optimal behaviour of the algorithm, even in the presence of noise, as is illustrated in fig. 5. In conclusion we discuss the relation of our technique with methods presented in estimation theory [4],[5], in electromagnetism [6] and in embedding theory [7].