Empirical Tests for Evaluation of Multirate Filter Bank Parameters

Abstract

Empirical tests have been developed for evaluating the numerical properties of multirate M -band filter banks represented as N × M matrices of filter coefficients. Each test returns a numerically observed estimate of a 1 × M vector parameter in which the m th element corresponds to the m th filter band. These vector valued parameters can be readily converted to scalar valued parameters for comparison of filter bank performance or optimization of filter bank design. However, they are intended primarily for the characterization and verification of filter banks. By characterizing the numerical performance of analytic or algorithmic designs, these tests facilitate the experimental verification of theoretical specifications.Tests are introduced, defined, and demonstrated for M -shift biorthogonality and orthogonality errors, M -band reconstruction error and delay, frequency domain selectivity, time frequency uncertainty, time domain regularity and moments, and vanishing moment numbers. These tests constitute the verification component of the first stage of the hierarchical three stage framework (with filter bank coefficients, single-level convolutions, and multi-level transforms) for specification and verification of the reproducibility of wavelet transform algorithms.Filter banks tested as examples included a variety of real and complex orthogonal, biorthogonal, and nonorthogonal M -band systems with M ≥ 2. Coefficients for these filter banks were either generated by computational algorithms or obtained from published tables. Analysis of these examples from the published literature revealed previously undetected errors of three different kinds which have been called transmission, implementation, and interpretation errors. The detection of these mistakes demonstrates the importance of the evaluation methodology in revealing past and preventing future discrepancies between observed and 1 2 expected results, and thus, in insuring the validity and reproducibility of results and conclusions based on those results.

Constraint-selected and search-optimized families of Daubechies wavelet filters computable by spectral factorization

“…For binomial space methods [16] to generate Daubechies wavelets, ξ represents the set of binary codes that specifies z for F (z). In both cases, λ represents a criterion obtained from an individual property or a weighted combination of properties computed from z and/or f (such as the filter's time-domain regularity [14], phase nonlinearity [16], et c.) that characterizes F (z).…”

“…Numerical estimates of defined filter characterization parameters λ are used as selection criteria for all other families subjected to optimization in combinatorial searches of the root sets. These criteria [14] include the phase nonlinearity pnl(A), time-domain regularity tdr(A), frequency-domain selectivity fds(A), and time-frequency uncertainty tfu(A). Most of the orthogonal families are defined by pnl(A) selecting for varying degrees of asymmetry or symmetry.…”

“…DCOMN family is the union of the even-indexed DCOMA and odd-indexed DCOLS families. Complete details for the algorithms to compute each of the various selection criteria can be found elsewhere [12,14].…”

“…All filters of all families were demonstrated to meet or surpass requirements for orthogonality, biorthogonality, and reconstruction when tested [14] in 2-band wavelet filter banks. In general, reconstruction errors ranged from "perfect" at O(10 −16 ) to "near-perfect" at O(10 −8 ) as K ranged from K = 1 to K = 24 for both orthogonal and biorthogonal classes.…”

Least and most disjoint root sets for Daubechies wavelets

“…For binomial space methods [6] to generate Daubechies wavelets, ξ represents the set of binary codes that specifies z for F (z). In both cases, λ represents a criterion obtained from an individual property or a weighted combination of properties computed from z and/or f (such as the filter's time-domain regularity [12] and phase nonlinearity [6], et c.) that characterizes F (z).…”