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
m
-input
p
-output cascaded lattice filter structures with lossless building blocks, that realize an arbitrary
p \times m
digital Lossless Bounded Real (LBR) transfer matrix. Extensions to the synthesis of arbitrary, stable
p \times m
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
p \times m
transfer matrix of a discrete-time linear system.