Synthesis of spectrum shaping digital filters of recursive design

Brophy, F.; Salazar, Andres C.

doi:10.1109/tcs.1975.1084023

Cited by 17 publications

(1 citation statement)

References 16 publications

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“…Over the years, the applications that have benefited most from minimax rational approximations come from recursive filter design in signal processing [13,23] and the representation of special functions [18,19]. Apart from such practical motivations, we believe it worthwhile to pursue robust numerical methods for computing these approximations because of their fundamental importance to approximation theory.…”

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confidence: 99%

Rational Minimax Approximation via Adaptive Barycentric Representations

Filip¹,

Nakatsukasa²,

Trefethen³

et al. 2018

SIAM J. Sci. Comput.

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Computing rational minimax approximations can be very challenging when there are singularities on or near the interval of approximation -precisely the case where rational functions outperform polynomials by a landslide. We show that far more robust algorithms than previously available can be developed by making use of rational barycentric representations whose support points are chosen in an adaptive fashion as the approximant is computed. Three variants of this barycentric strategy are all shown to be powerful: (1) a classical Remez algorithm, (2) a "AAA-Lawson" method of iteratively reweighted least-squares, and (3) a differential correction algorithm. Our preferred combination, implemented in the Chebfun MINIMAX code, is to use (2) in an initial phase and then switch to (1) for generically quadratic convergence. By such methods we can calculate approximations up to type (80, 80) of |x| on [−1, 1] in standard 16-digit floating point arithmetic, a problem for which Varga, Ruttan, and Carpenter required 200-digit extended precision.Key words. barycentric formula, rational minimax approximation, Remez algorithm, differential correction algorithm, AAA algorithm, Lawson algorithm AMS subject classifications. 41A20, 65D151. Introduction. The problem we are interested in is that of approximating functions f ∈ C([a, b]) using type (m, n) rational approximations with real coefficients, in the L ∞ setting. The set of feasible approximations is(1.1)Given f and prescribed nonnegative integers m, n, the goal is to computewhere · ∞ denotes the infinity norm over [a, b], i.e., f − r ∞ = max x∈ [a,b] |f (x) − r(x)|. The minimizer of (1.2) is known to exist and to be unique [58, Ch. 24]. Let the minimax (or best) approximation be written r * = p/q ∈ R m,n , where p and q have no common factors. The number d = min {m − deg p, m − deg q} is called the defect of r * . It is known that there exists a so-called alternant (or reference) set consisting of ordered nodes a x 0 < x 1 < · · · < x m+n+1−d b, where f − r * takes

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confidence: 99%

Rational Minimax Approximation via Adaptive Barycentric Representations

Filip¹,

Nakatsukasa²,

Trefethen³

et al. 2018

SIAM J. Sci. Comput.

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Two Design Techniques for Digital Phase Networks

Brophy¹,

Salazar²

1975

Bell System Technical Journal

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Two computer‐aided algorithms for the design of all‐pass digital filters are presented. The first technique is based on a linear programming approach to solving the approximation problem posed by the minimax design of an all‐pass digital filter. A new iterative algorithm with stability constraints is offered for direct form design. The second technique implements a gradient search for those quadratic factors of an all‐pass transfer function that lead to a locally optimal approximation (in the least‐squares sense) of a desired phase function. New initial guess procedures and the parameterization of linear‐phase offset enhance the least‐squares design procedure. Examples illustrating the result of both procedures are included.

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Arithmetic in the Design of Linear Time-Invariant Filters

de Dinechin,

Kumm

2023

Application-Specific Arithmetic

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Synthesis of spectrum shaping digital filters of recursive design

Cited by 17 publications

References 16 publications

Rational Minimax Approximation via Adaptive Barycentric Representations

Rational Minimax Approximation via Adaptive Barycentric Representations

Two Design Techniques for Digital Phase Networks

Arithmetic in the Design of Linear Time-Invariant Filters

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