A general family of multivariable digital lattice filters

Abstract

Lattice structures are developed for the realization of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> -input <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> -output discrete-time all-pass transfer matrices <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H(z)</tex> , given in the form of a right matrix-fraction description <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(MFD): H(z)=</tex> <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N(z)D^{-l}(z)</tex> . The procedure is based on the generation of a sequence of all pass matrices of successively decreasing order, by matrix LBR two-pair extraction. Two cases are distinguished: the first case is when none of the intermediate allpass matrices is degenerate. For this case, the resulting structures are in the form of a cascade of matrix two-pairs separated by vector delays, with each two-pair being a multi-input multi-output digital filter structure characterized by an orthogonal transfer matrix of dimension <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(m + p) \times ( m + p)</tex> . The structures are in general either completely controllable or completely observable, depending upon the location of the delay elements. The synthesis technique also leads to a procedure for obtaining the greatest common right divisor between the polynomial matrices involved in the MFD. The results are extended to the cascaded-lattice synthesis of arbitrary stable transfer matrices by an embedding process. The developments of this paper automatically place in evidence a procedure for testing the stability of a transfer matrix. A special case of the resulting structures when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p = m =1</tex> gives rise to the well-known Gray-Markel digital lattice structures, whereas another special case with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p = 2</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m = 1</tex> leads to certain recently reported orthogonal digital filters. The second case, where some of the intermediate allpass matrices are degenerate, is handled separately, leading to a modified form of cascaded-multivariable lattice structures.