. Cependant, certains résultats présentés ici s'´ etendent aisémentaisémentà des cas plus généraux, et il est donc possible que la démarche entreprise dans cette thèse, qui donne les outils de base

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. Dans-le-cas, u L est pair, il existe un complément bi-orthogonal

L. 'algorithme-qui-le-détermine, En effet, puisque la longueur des filtres (symétriques) est paire, ces filtres admettent nécessairement chacun un zérò a z = ?1 On peut doncécriredoncécrire G(z) = (1 + z ?1 )F (z) et G ? (z) = (1 + z ?1 )F ? (z) En observant (5.4) on remarque qu'alors le couple formé des filtres de longueur impaire, (1 + z ?1 ) 2 F (z) et F ? (z) est solution, o` u F ? (z) se déterminè a partir de (1 + z ?1 ) 2 F (z) par l'algorithme CB 0 . Il suffit donc, appelé CB 1 dans la suite, se déduit immédiatement du précédent

C. K. Cette-idée-se-généralise-immédiatementimmédiatementà-l-'algorithme, partir d'un filtre passe-bas G(z) de longueur L ayant K zéroszérosà z = ?1, de déterminer un complément biorthogonal G ? (z) ayantégalementayantégalement K zéroszérosà z = ?1. Pour ce faire, il suffit d'ajouter K tels zéroszérosà G(z), d'appliquer CB 0, ) est alors L ? = L + 2(K ? 1). Imposer une forte contrainte de platitude augmente donc la dissymétrie des longueurs

. Cette-méthode-pourrait-se-généraliser-au-cas, on imposerait un nombre différent de zéros zérosà z = ?1 dans les deux filtres. On a préféré cependant utiliser le même K, afin de simplifier leprobì eme et de s'approcher du cas orthogonal en

C. Appliquer-l-'algorithme and G. , z) (cf. § 5.2.2) pour obtenir son complément birothogonal G ? (z)

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A. C. Calcul, . De-filtres:-bo?ites-`-abo?bo?itesbo?ites-`-bo?ites-`-a, . Outils, . Matlab, and . Magnitude, H) plots magnitude (frequency) response of FIR filter described by vector H. By default, a linear scale is assumed, MAGNITUDE, issue.1

A. Attenuation-(-h and B. , gives stop-band attenuation, in dB, of half-band FIR filter with transition band (0.25 ? B/2, 0.25 + B/2), where B is the normalized transition bandwidth) assumes H corresponds to a square magnitude, =ATTENUATION(. . . ) also gives the tolerance ? in the stop band

P. Phasevar-phase and Z. Or, This is a measure of phase distortion : If it is close to zero, then H is close to being linear phase. If it is zero, then the filter is linear phase CAUTION : This measure is generally spoiled by zeros in the stop band's magnitude response, causing diracs to appear in the group delay. However, phase transitions by ? should not be a problem. [var1,var2]=PHASEVAR(H) gives group delay variations in the first (0, ?/2) and second (?/2, ?) half band, respectively. This measure is more significant for half-band filters (var1 for low-pass filters, var2 for half-band filters) and avoids the problem of Diracs

M. See and . Zeroes, Zeroes on the unit circle are not selectable : They are retained with twice less multiplicity.) ZP can be obtained from design routines like REMEZWAV, etc. The resulting filter is normalized such that NORM(H)=1, but this may be inappropriate for e.g. MAGNITUDE (gain problem). (Note that checking the result with ZEROES will add computational round-off errors

P. Phasesort, describe filters H n of the same magnitude response (as given by FACTORALL), returns this matrix, sorted from the closest to the farthest from linear phase. A filter is closer to linear phase if its group delay deviation in the pass-band, as given by PHASEVAR, is smaller, HSORTED,INDEX]=PHASESORT(H) also gives the corresponding indexes of filters n = 0, 1, . . . (=phase codes as described in FACTORALL). [HSORTED,INDEX,VAR]=PHASESORT(H) also gives the group delay variations

R. H=remezz-(-l and K. , gives optimum low-pass filter described by vector H, of length L, with K zeroes at the Nyquist frequency, normalized transition bandwidth B, normalized frequency offset ? = (? p +? s )/2?0.25, and (optional) weight coefficient C = ? 1 /? 2 (pass-band attenuation is greater as C is larger). [H,? 1 ]=REMEZZ(. . . ) also gives the maximum deviation ? 1 in the passband (? 2 = ? 1 /C, attenuation is ?20 log 10,0) uses global variable glob as initial guess of alternations and set glob for next call (to minimize the number of iterations)

D. Deriv, the nth derivative of the scaling function (father wavelet) associated to low-pass FIR filter described by vector H. DERIV(H) sets n to 1. DERIV(H,G,n) plots the nth derivative of the (mother) wavelet associated to low-pass and high-pass filters described by vectors H and G, respectively. More generally, DERIV(H,G,n) plots the nth derivative of the limit function of an iterative " subdivision " procedure whose initial sequence is G, DERIV(H,n) sets G = H. By default, the number of iterations is the maximum permissible on this computer

R. Reg, Hölder Regularity of filter described by vector H, as estimated by a sharp upper bound. (The regularity may be negative.) Removal of zeroes in H at half the sampling frequency is done assuming remainder values < 10 ?3, REG(H,Z) forces the number Z of such zeros

B. Biorth-k-=-k+1, If L =length(H) is even, L ? =length(H ? )=length(H) If L =length(H) is odd