Empirical Tests for the Evaluation of Multirate Filter Bank Parameters

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 themth element corresponds to themth 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 forM -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 expected results, and thus, in insuring the validity and reproducibility of results and conclusions based on those results.