On the Role of Biorthonormality in Representation of Random Processes

Abstract-The representation of a random process by a set of uncorrelated random variables is examined. The main result indicates that a basis decorrelates a random process if and only if it satisfies an integral equation similar to the type satisfied by the Karhunen-Loeve expansion, but by relaxing the requirement of orthogonality of the representation functions. Index Terms- Karhunen-Lohe expansion, biorthonormal bases, decorrelation of random processes. Theorem I: Given a random process { z (t), t E T} which satisfies the conditions listed above, a basis {dt(t)} of &(T) decorrelates z(t) if and only if it satisfies the following equation:.I; R(s,t)?L,(t)dt = nzdz(s), vi (3) s, 4 z (t)?L, (4 dt = &, , VZ,.i. where {&(t)} is the biorthonormal basis of {qA(t)}, i.e. Prooj? Since {dz(t)} is a basis of Lz(T), the following equations hold z(t) = 1.i.m. Ca,dz(t) 2 I