Asymptotic equivalence of spectral representations
Jacob Pearl · IEEE Transactions on Acoustics Speech and Signal Processing · 1975
This paper develops necessary and sufficient conditions under which for every sequence of matrices diagonal in an orthonormal basis{u}there exists an asymptotically equivalent sequence in the class of matrices diagonal in some other basis{\upsilon}. These conditions can be expressed in terms of a doubly stochasticN \times Nmatrix A whose entries Aijare the square magnitudes of the inner product between the ith basis vector of{\upsilon}and thejthbasis vector of{\upsilon}. We show that a sufficient condition for asymptotic equivalence is\lim\min{N \rightarrow \infin}[1-\frac{1}{N}tr(A^{T}A)] = 0and a necessary condition is\lim\min{N\rightarrow\infin}[tr(A^{T}A)]^{-1} = 0. 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.