Witten´s complex of spectral deformation and Morse theory

A sound-morphing method (Osaka 1995) and a speaker adaptation scheme (Shiraki 1995) both use essentially spectral deformation. One of the basic problems of these methods is how to deform speech spectra gradually while maintaining intelligible speech as a specific individual. In this note, we consider the problem to be the study of differentiable manifolds (spectra) and differentiable maps. Then, naturally, two manifolds are considered equivalent if they are diffeomorphic: there exists a differentiable map from one to the other with a differentiable inverse. So the problem turns out to be how to construct numerical and algebraic invariants determined by the diffeomorphism class of the manifold. We use Morse theory (Floer 1989 and Fukaya 1995) in order to extract topological invariants of each given speech spectrum from their critical points, i.e., our focus is attention to the peaks and troughs of the speech-spectrum which are important characteristics of the perceptual property of speech