Optimal diagonal approximation of the covariance function with respect to the relative entropy using the wavelet basis
Fuminori Sakaguchi · Electronics and Communications in Japan (Part III Fundamental Electronic Science) · 1993
Abstract If the covariance function of a random signal can be written in a diagonal form via the wavelet basis, this random signal can be regarded as a superposition of the wavelets which arise randomly. However, it is known that, in general, such an expression is not possible. In this paper, in place of a perfect diagonalization, an optimal approximate diagonalization in the sense of the relative entropy is investigated theoretically. Especially, it is shown that when a set of wavelets forming complete orthonormal sets (expressed in a vector form as {ϕi} is used as the basis, an optimal diagonal approximation of the covariance matrix Γ is not the diagonal form\documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_k {\left({\bar \phi _k ^T \Gamma \phi _k} \right)} \phi _k \bar \phi _k ^T $\end{document} using the so‐called “wavelet spectrum” but\documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_k {\left({\bar \phi _k ^T \Gamma ^{- 1} \phi _k} \right)^{- 1}} \phi _k \bar \phi _k ^T $\end{document} Further, several examples are given where Haar wavelets are used.