Identifiability and Well-Posedness of Shaping-Filter Parameterizations: A Global Analysis Approach

Abstract

In this paper, we study the well-posedness of the problems of determining shaping filters from combinations of finite windows of cepstral coefficients, covariance lags, or Markov parameters. For example, we determine whether there exists a shaping filter with a prescribed window of Markov parameters and a prescribed window of covariance lags. We show that several such problems are well-posed in the sense of Hadamard; that is, one can prove existence, uniqueness (identifiability), and continuous dependence of the model on the measurements. Our starting point is the global analysis of linear systems, where one studies an entire class of systems or models as a whole, and where one views measurements, such as covariance lags and cepstral coefficients or Markov parameters, from data as functions on the entire class. This enables one to pose such problems in a way that tools from calculus, optimization, geometry, and modern nonlinear analysis can be used to give a rigorous answer to such problems in an algorithm-independent fashion. In this language, we prove that a window of cepstral coefficients and a window of covariance coefficients yield a bona fide coordinate system on the space of shaping filters, thereby establishing existence, uniqueness, and smooth dependence of the model parameters on the measurements from data.