Designing good causal filters for perfect reconstruction (PR) filter banks is a challenging task due to the unusual nature of the design constraints. We present a new least squares (LS) design methodology for approximating PRFBs that avoids most of these difficult constraints. The designer first selects a set of subband analysis filters from an almost unrestricted class of rational filters. Then, given some desired reconstruction delay, this design procedure produces the causal and rational synthesis filters that result in the best least squares approximation to a PRFB. This technique is built on a multi-input multi-output (MIMO) system model for filter banks derived from the filter bank polyphase representation. Using this model, we frame the LS approximation problem for PRFBs as a causal LS equalization problem for MIMO systems. We derive the causal LS solution to this design problem and present an algorithm for computing this solution. The resulting algorithm includes a MIMO spectral factorization that accounts for most of the complexity and computational cost for this design technique. Finally, we consider some design examples and evaluate their performance.

Least squares approximation of perfect reconstruction filter banks

Examines the realization matrix R=(A b; c d) defined by a state variable model of a linear, shift invariant, discrete time, scalar system. Several properties concerning the eigenvalues and singular values are derived, which are used to obtain tests for the minimality of the state variable model. An inequality is derived between the spectral norm of R and the L/sub Omega /-norm of its frequency response. The realization matrices of principal axis realizations are characterized in terms of their eigenvalues and singular values. qr-factorizations and bounds for the spectral norms of such realization matrices are derived. >

http://ieeexplore.ieee.org/iel4/81/4396/00168929.pdf

Linear algebraic properties of the realization matrix with applications to principal axis realizations

The problem of implementing a specified state variable model using orthogonal computations is addressed. Three approaches, namely, realization embedding, inner product, and matrix factorization are proposed. Output roundoff error and magnitude perturbances are also studied via simulations. The results presented show that orthogonality of the computation can be achieved while simultaneously specifying the dynamical behavior of a system. The advantages of implementing optimal state variable structures, where the optimality is defined in the sense of time-averaged properties, with orthogonal computations which are optimal in terms of numerical conditioning remains an open issue. >

Orthogonal implementations of state variable models

Orthogonal digital filters are an extremely robust class of filters which possess a number of desirable properties. The literature contains two different definitions of orthogonal filters. We shall discuss the implications of these definitions arid present two design methods for the authors' definition of orthogonal digital filters.

https://ieeexplore.ieee.org/iel4/6081/16283/00754008.pdf

On The Design Of Orthogonal Digital Filters

The authors present a computational toolkit which operates on rational polynomial structures. This family of structures represent almost all sequences whose z-transforms can be written as the ratio of two finite order polynomials in z. The toolkit may be applied to a variety of practical filter design problems. Some of the algorithms developed include spectral factorization, separation into causal and anticausal subsequences, computation of greatest common denominator, and derivation of equivalent minimum-phase polynomials. The authors briefly develop each algorithm and then present condensed pseudo code versions of each tool. >

A computational toolkit for applied z-transforms

Internal projections are used to approximate static linear systems The results are used trs (i) interpret the Kalman experiment and the Ho-Kalman algoritom for identifying finite dimensional linear systems, (ii) derive low rank digital filters with minimum round-of noise variance, and (iii) obtain low rank dynamical systems which approximate finite or infinite rank dynamical systems The second-order modes of a static or dynamical system arise as the natural modes to edit in any rank reduction scheme These modes are invariant to internal structure, so questions of structure play no role in rank reduction until internal constraints are applied and internal noise is considered The second-order modes of a dynamical system with an internal low rank projection are shown to be bounded from above by the second-order modes of the dynamical system without the internal projection This fundamental theorem provides the basis for answering a variety of questions about approximation errors

http://ieeexplore.ieee.org/iel4/5502/14788/00671502.pdf

C.T. Mullis

Papers

On The Design Of Orthogonal Digital Filters

A computational toolkit for applied z-transforms

Internal Projections For Rank Reduction In Linear Dynamical Systems\sup */