A General Family of Multivariable Digital Lattice Filters

Absfruct-Lattice structures are developed for the realization of M -input p-output discrete-time all-pass transfer matrices H(z), given in the form of a right matrix-fraction description (MFD): H(r) = N(z) D’ (z). 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 (m + p) X ( M + p). 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 p = m = 1 gives rise to the well-known Gray-Markel digital lattice structures, whereas another special case with p = 2 and M = 1 leads to certain recently reported orthogonal digital fitlers. 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.