Empirical Tests for Evaluation of Multirate Filter Bank Parameters

Empirical test s have been develop ed for evalu ating the numerical prop­ ert ies of multirate ,'Vi -ba nd filter banks repr esen ted as N x M matrices of filter coefficients . Each test re turns a numerically observed est imate of a 1 x Al vector parameter in which th e m t h eleme nt corresp onds to t he m t h filter band. These vector valued par am et ers can be read­ ily converte d to scalar valued p arameters for com parison of filter b ank performance 01' optimization of filter bank design . However , th ey ar e in­ t ended primarily for t he cha rac terizat ion and ver ification of filter banks. By cha racterizing t he numerical perform anc e of analyt ic 0 1' algorit hmic designs, these test s facilit ate the exp erimental verification of th eor etical specifica t ions . Tests ar e in troduced , defined , a nd demonst rated for Al -shift biorthog­ onality and orthogonality err ors, AI-b and reconstruction error and de­ lay, frequency dom ain selectivi ty, t ime frequ ency un certainty, t ime do­ main regul arity and mom ent s, and vanish ing mom en t numbers, These test s cons t it ute th e verifi cation com ponent of th e first stage of the hier ­ archical three stage fram ework (with filter bank coefficients , single-level convolut ions. and multi-Ievel tran sform s) for spec ificat ion and verifica­ tion of t he reproducibili ty of wavelet transform algorit hms . Fil ter banks t est ed as exam ples include d a vari ety of real and C OIn­ plex orthogon al , biorthogonal , and non or th ogonallv!-b and syste ms with Al :::: 2. Coefficients for t hese filter banks were eit he r generated by com­ pu tational algori thms 0 1' obtain ed from published t abl es. An alysis of t hese exa m ples from th e publish ed lit er ature revealed pr eviously unde­ te cte d err ors of three differ ent kinds which have been called transmis­ sion, implement ation, and int er pretation err ors . The detect ion of th ese mistakes dernonstrates t he importance of t he evaluat ion methodology in re vea ling pa st and prevent ing future discrep anc ies between observed and expected results, and thus, in ins ur ing th e validi ty and reproducibili ty of resul ts and conclusions bas ed on t hose results.