Asymptotic equivalence of spectral representations

This paper develops necessary and sufficient conditions under which for every sequence of matrices diagonal in an orthonormal basis there exists an asymptotically equivalent sequence in the class of matrices diagonal in some other basis . These conditions can be expressed in terms of a doubly stochastic matrix A whose entries Aijare the square magnitudes of the inner product between the ith basis vector of and the basis vector of . We show that a sufficient condition for asymptotic equivalence is and a necessary condition is . This paper also discusses the implications of these results to real-time signal processing whereby it is the prevailing practice to approximate the actual input correlation matrix by matrices which are diagonal in a computationally more manageable basis.