On the diagonal approximation of the auto-correlation function with the wavelet basis which is optimal with respect to the relative entropy

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 {/spl phi//sub i/} is used as the basis, an optimal diagonal approximation of the covariance matrix /spl Gamma/ is not the diagonal form /spl Sigma//sub h/(/spl phi/~/sub h//sup /spl tau///spl Gamma//spl phi//sub h/)/spl phi//sub h//spl phi/~/sub h//sup /spl tau// using the so-called 'wavelet spectrum' but /spl Sigma//sub h/(/spl phi/~/sub h//sup /spl tau///spl Gamma//sup -1//spl phi//sub h/)/sup -1//spl phi//sub h//spl phi/~/sub h//sup /spl tau//. Further, several examples are given where Haar wavelets are used.