ON VECTOR KARHUNEN-LOEVE TRANSFORMS AND OPTIMAL VECTOR TRANSFORMS

In this letter, we first prove that the vector Karhunen-Loeve (VKL) tr ansform for any finite many vector-valued signal, x(1),...,x(L), exist s. The VKL transform is equivalent to the scalar KL transform for the scalar-valued signal X = (x(1)(T),...,x(L)(T))(T). Based on VKL transf orms, we provide a necessary and sufficient condition for the existenc e of the optimal vector transforms (unitary). With the condition, one can see that the optimal unitary vector transforms do not exist in mos t cases, and therefore needs to use suboptimal unitary vector transfor ms. We then prove that the optimal nonunitary vector transform for x(1 ),...,x(L) exists when all eigenvalues of the correlation matrix of th e signal X are nonzero. We formulate the optimal vector transforms via the VKL transforms.