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

Abstract

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.