Discrete version of Richard´s theorem and applications to cascaded lattice realization of digital filter transfer matrices and functions

The well-known Richards\´ Theorem of the continuous-time filter theory is reformulated in the digital domain in a convenient manner, leading to a simple derivation of cascaded lattice digital filter structures, realizing lossless bounded transfer functions. The theorem is also extended to the matrix case, leading to a derivation of -input -output cascaded lattice filter structures with lossless building blocks, that realize an arbitrary digital Lossless Bounded Real (LBR) transfer matrix. Extensions to the synthesis of arbitrary, stable transfer matrices in the form of such cascaded lattices is also outlined. The derivation also places in evidence a means of testing the stability of an arbitrary transfer matrix of a discrete-time linear system.