Implementation of Operators via Filter Banks; Hardy Wavelets and Autocorrelation Shell

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1 Impementation of Operators via Fiter Banks; Hardy Waveets and Autocorreation She Gregory Beykin, Bruno Torresani To cite this version: Gregory Beykin, Bruno Torresani. Impementation of Operators via Fiter Banks; Hardy Waveets and Autocorreation She. Appied and Computationa Harmonic Anaysis, Esevier, 1995, 3, pp < /acha >. <ha-01313> HAL Id: ha Submitted on Nov 015 HAL is a muti-discipinary open access archive for the deposit and dissemination of scientific research documents, whether they are pubished or not. The documents may come from teaching and research institutions in France or abroad, or from pubic or private research centers. L archive ouverte puridiscipinaire HAL, est destinée au dépôt et à a diffusion de documents scientifiques de niveau recherche, pubiés ou non, émanant des étabissements d enseignement et de recherche français ou étrangers, des aboratoires pubics ou privés.

2 IMPLEMENTATION OF OPERATORS VIA FILTER BANKS; Hardy Waveets and Autocorreation She G. Beykin and B. Torrésani November 13, 1995 Abstract We consider impementation of operators via fiter banks in the framework of the Mutiresoution Anaysis. Our method is particuary efficient for convoution operators. Athough our method of appying operators to functions may be used with any waveet basis with a sufficient number of vanishing moments, we distinguish two particuar settings, namey, orthogona bases, and the autocorreation she. We appy our method to evauate the Hibert transform of signas and derive a fast agorithm capabe of achieving any given accuracy. We consider the case where the waveet is the autocorreation function of another waveet associated with an orthonorma basis and where our method provides a fast agorithm for the computation of the moduus and the phase of signas. Moreover, the resuting waveet may be viewed as being (approximatey, but with any given accuracy) in the Hardy space H (IR). I INTRODUCTION In this paper we introduce a method for design of digita fiters and consider their impementation and appication via fiter banks. The design of digita fiters is aways a trade-off between accuracy and efficiency. For a number of operators this trade-off obtained via traditiona fiter design techniques is not adequate, especiay if high precision is required. As exampes, consider the Hibert transform or operators of fractiona differentiation where an accurate traditiona impementation over a wide band necessariy impies a ong fiter. Signa processing is not the ony fied where fast and accurate impementation of such operators is of interest. In numerica anaysis Fast Mutipoe Method (FMM) [18, 9, 4] has been deveoped to address this probem. Athough this method proved to be efficient in numerica anaysis, it did not so far find its way into the signa processing. A possibe expanation for this may be the traditiona reiance on fitering operations in the signa processing community. In fact, mutiresoution techniques evoved in signa processing as subband coding techniques [8, 0]. The origina motivation for subband coding was essentiay optima representation and compression of signas. The introduction of the orthonorma bases of waveets[1, 16] and the concept of mutiresoution anaysis [13, 15] have ed to the deveopment of a broader concept of harmonic anaysis of signas where subband coding became a natura way of representing and anayzing signas. These notions aso migrated to numerica anaysis, where they Program in Appied Mathematics, University of Coorado, Bouder, Coorado 80304, USA CPT, CNRS-Luminy, Case 907, 1388 Marseie Cedex 09, France 1

3 were appied to the probem of efficient representation and appication of operators [3]. In particuar, it was shown in [3] how to use waveet bases to amost diagonaize certain casses of operators, for exampe, pseudodifferentia and Caderón-Zygmund operators. For signa processing appications the approach of [3] is aso of interest. Athough designed for numerica purposes, agorithms of [3] expicity use quadrature mirror fiters (QMFs) with the exact reproduction property and may be viewed as a ink between signa processing techniques and traditiona numerica computing. The method deveoped in this paper is different from that of [3] and may be aso easiy impemented both in software and hardware. It avoids the construction of non-standard forms which, athough quite efficient, are not as simpe as the fiter banks approach described in this paper. Briefy, we decompose a signa into different scaes (subbands) and impement operators as subband fiters. Let us, for exampe, consider a convoution operator T. The waveet ψ may be written as ˆψ(ξ) = m 1 (ξ/) ˆφ(ξ/), where ˆψ and ˆφ are the Fourier transform of the waveet and of the scaing function. The πperiodicsquare-integrabefunctionm 1 representsoneofthequadraturemirrorfiters(qmfs). Our approach is based on an observation that if the waveet ψ(x) is sufficienty we ocaized in the Fourier domain, one may write (Tψ)(ξ) m T (ξ/4) ˆφ(ξ/4), (1.1) where m T is a π-periodic function which is computed given the symbo of the operator T. The accuracy of the approximation in (1.1) is controed by the number of vanishing moments of the waveet ψ (it might be necessary to consider (1.1) on each scae separatey if the symbo of T is not homogeneous). As a resut, the operator T is impemented using fiters m T (may be different on different scaes), where m T pays a roe simiar to that of the fiter m 1 of the QMF pair. The major difference, which is aready visibe in equation (1.1), is that the m T fiter performs a scaing by a factor 4 instead of. This has some practica impications, which are discussed throughout the paper. The factor 4 may be repaced by a factor n,n, as a way to improve accuracy. In this way, the procedure for design of these subband fiters aows us to attain any desired accuracy. The approach of this paper (as that of [3]) may be traced back to Caderón-Zygmund and Littewood-Paey approach to harmonic anaysis of functions and operators which we appy here to design fiters given the symbo of an operator. Our method may be used with any waveets(associated with quadrature mirror fiters) which possess sufficient number of vanishing moments. In cases where the associated scaing function aso has vanishing moments (which impies that the corresponding coefficients are we approximated by sampes on fine scaes), our agorithm eads to a fast method for computing the Hibert transform (and, thus, moduus and phase) of signas. This is the case for the autocorreation waveets derived in [19] which we consider here in some detai. Athough our approach is quite genera, we concentrate on severa specific exampes, such as the Hibert transform, operators of differentiation and, more generay, convoution operators. In particuar, we consider the Hibert transform both as an exampe and an important specia case, because of its particuar status (it is one of the simpest and most popuar exampes of Caderón-Zygmund operators) and its reevance in signa processing. In our approach the Hibert transform (as we as a number of other operators) is competey expressed in terms of fiter banks, which makes it easy to hande for a wide variety of scientific communities. In addition to the Hibert transform, we construct derivative and integration operators incuding those of fractiona order.

4 Our approach aso aows us to consider the foowing reated probem. In signa processing it is often usefu to dea with waveets that beong to the compex Hardy space H (IR), i.e., waveets such that their Fourier transform is zero for negative frequencies. For instance, such waveets are considered to be more efficient for the identification of chirps (i.e. ampitude and frequency moduated components) in signas. In particuar, it is easy to identify the carrier frequency and remove it by shifting in the frequency domain if necessary. However, there does not exist any orthonorma mutiresoution anaysis of H (IR) where the associated waveet has such a property [1]. Nevertheess, as we show in this paper, it is possibe to keep the agorithmic structure of mutiresoution anaysis and use waveets that approximate the Hibert transform of a given rea-vaued function with any given (but finite) accuracy. The sum of the origina waveet and i times its approximate Hibert transform yieds a new waveet that is approximatey in H (IR). As a direct consequence, we obtain a fast agorithm for the computation of the Hibert transform, and a pyramida agorithm for discrete waveet transform with compex anaytic (or progressive) waveet [10]. This may be thought of as a starting point to carry on an anaysis simiar to that deveoped in [6,?, 1]. The first part of the paper is devoted to the representation of operators in terms of fiter banks. To iustrate our approach, we derive in Section II the function m T in (1.1) for the Hibert transform. We then present in Section III a genera approach of fiter banks impementation of convoution operators (in these two sections, we consider compacty supported orthogona waveets and obtain O(N) agorithms). We then turn to the particuar case of the autocorreation of Daubechies compacty supported waveets in Section IV. We again consider first the Hibert transform and then, in Section V, deveop approximations of other operators (e.g., operators of fractiona differentiation and integration) by our technique. In the second part of the paper we address the signa processing probems using autocorreation waveets. In Section VI we consider computing the Hibert transform as we as moduus and phase of signas and, in Section VII, address the probem of decomposition of band-pass signas into ampitude and frequency moduated components. The main toos are the same as in the first part of th paper, namey fiter banks impementation of Hibert transform. However, the agorithm we describe is O(N ogn) since we choose to use a transation-invariant version of waveet transform. Finay, Section VIII is devoted to concusions. 3

5 II THE HILBERT TRANSFORM As a way of introduction, et us consider our approach for impementing the Hibert transform, (Hf)(x) = 1 π p.v. f(s) ds. (.1) x s The genera case and some other operators wi be considered in the foowing sections. Let us start with the usua MRA (see Appendix). It is we known that the Hibert transform of the waveet ψ(x) is sti a waveet. Since we wi be intererested in computing coefficients Hf,ψ j k, et us consider the function H ψ given in the Fourier domain by (H ψ)(ξ) = isgn(ξ)ˆψ(ξ) ( ) = isgn(ξ)m ξ 1 ˆφ( ξ ), (.) where H denotes the adjoint of H. Let us consider the 4π-periodic function m (ξ) = i k sgn(ξ +4πk)m 1 (ξ +4πk)χ [ π,π] (ξ +4πk). (.3) Our main observation is as foows: athough isgn(ξ)m 1 (ξ) is not a periodic function, the product isgn(ξ)m 1 (ξ)ˆφ(ξ) may be we approximated by m (ξ)ˆφ(ξ), due to the fast decay of ˆφ(ξ). Proposition II.1 are given by 1. The Fourier coefficients b of the function m (ξ) = 1 b e iξ/ 1 g k b = π k k for odd 0 for even (.4). The Fourier coefficients b have the foowing asymptotics b 1 O(( 1) M 1 ) (.5) Notice that asymptotics (.5) coincides with the asymptotics expected from the genera approach of [3]. Proof: 1. Consider the Fourier coefficients of m (ξ) which yieds (.4). π b = i m 1 (ξ)sgn(ξ)e iξ/ dξ (.6) 4π π = i π [ m 1 (ξ)e iξ/ m 1 ( ξ)e iξ/] dξ (.7) 4π 0 = 1 π g k sin(k )ξdξ (.8) π k 0 4

6 . The decay of the coefficients b 1 is governed by the reguarity of m (ξ) at the origin, i.e. by the number of vanishing moments of the waveet. The asymptotics (.5) is obtained using Tayor series expansion of (.4) and taking into account the vanishing moments of the sequence {g }: which proves (.5). b 1 = 1 g k π k k 1 (.9) = g k ( ) k p (.10) π 1 1 k p=0 = g k ( ) k p (.11) π 1 1 k p=m O(( 1) M 1 ) (.1) In other words, the Hibert transform is essentiay a oca operator on functions which have sufficient number of vanishing moments. Remarks: 1. We notice that the sequence {b 1 } may be viewed as the Hibert transform of the sequence {g k }. Wenote that for sequences the singuar behaviour of the Hiberttransform at the origin is avoided by the 1 term in the denominator, and the sow decay at infinity is repaced by (.5).. A statement simiar to Proposition II.1 may be proved for waveets with rationa m 0 and m 1. Let m 1 (ξ) = P(eiξ ) Q(e iξ (.13) ) be a rationa π-periodic function, where both P and Q are poynomias. Set p(ξ) = P(e iξ ), q(ξ) = Q(e iξ ) and consider p (ξ) = i k sgn(ξ +4πk)p(ξ +4πk)χ [ π,π] (ξ +4πk), (.14) and the 4π-periodic function m (ξ) = p (ξ) q(ξ) (.15) Since p(ξ) carries a the vanishing moments of m 1 (ξ), the quaity of the approximation of isgn(ξ)m 1 (ξ)ˆφ(ξ) by m (ξ)ˆφ(ξ) is controed by that of the approximation of isgn(ξ)p(ξ)ˆφ(ξ) by p (ξ)ˆφ(ξ). Such approximation was discussed in Proposition II.1. Using Proposition II.1, we obtain from (.) the foowing approximation where m H is a π-periodic function, (H ψ)(ξ) m H (ξ/4) ˆφ(ξ/4), (.16) m H (ξ) = m (ξ)m 0 (ξ). (.17) 5

7 The coefficients d j k = Hf,ψj k may then be computed as Hf,ψ j k j/ ˆf(ξ)e ikjξ m ( j 1 ξ)ˆφ( j 1 ξ)dξ (.18) = b 1 ˆf(ξ)e i(k 1/)j 1 ξˆφ( j 1 ξ)dξ (.19) = b 1 s j 1 k +1/, (.0) since the coefficients b +1 are rea. Notice that as a consequence of equation (.0), we now need haf-integer sampes of the coefficients s j 1. In view of equation (.16), this may be interpreted as foows. The shift by 1/ actuay amounts to a switch from V j to V j 1, as sugested by the foowing emma: ( ) Lemma II.1 Let f(x) V 0 or f(x) W 0, and set f1(x) = f x+ 1. Then f1(x) V 1. The proof of Lemma II.1 is very simpe. Observe that f(x) V 0 is equivaent to ˆf(ξ) = q(ξ)ˆφ(ξ), where q(ξ) is a π-periodic square-integrabe function, and that transation by 1/ is equivaent to mutipication by exp{iξ/} in the Fourier space, so that ˆf1 (ξ) = (x) V 1. The argument for W 0 is exp{iξ/}m 0 (ξ/)ˆφ(ξ/) which, in turn, impies that f1 the same. From Lemma II.1 foows a simpe agorithm to compute (.0). In order to obtain haf-integer sampes on a scaes except the finest scae j = 0, it is simpy sufficient to avoid subsamping at the first step of the agorithm 1. At scae j = 0, we need to use interpoation to obtain haf-integer sampes s 0 k+1/, using the assumption that f V 0. We have s 0 k+1/ = f,φ0 k+ 1 = s 0 φ 0,φ 0 k+ 1, (.1) where coefficients φ 0,φ0 k+ 1 are easiy obtained using autocorreation function of scaing function described in Appendix. Summarizing, we obtain the foowing O(N) scheme for computing the coefficients s j k of the Hibert transform of a function f(x) on subspace V j,j 0, assuming that projection of f(x) onto an approximation space, say V 0 is known. 1. Compute the coefficients s j k, j 1 and k Z/ using equation (9.9) in Appendix and the Hibert differences d j k = Hf,ψj k via the pyramida agorithm given in equation (.0) on a scaes.. Use the usua fiters for the reconstruction on a scaes: repace equation (9.10) by the foowing s j k = ( ) h k s j+1 +g k dj+1 (.) 1 Notice that in the particuar case of the Hibert transform, we woud then use the even (non-subsamped) scaing function coefficients to compute the difference coefficients d j k, and the odd ones for the Hibert transform coefficients d j k. 6

8 We note that the computationa cost is a factor of two compared with the usua waveet transform. Remark: Recenty Auscher [1] and, independenty, Lemarié [11] have shown (as a part of a more genera resut) that given a Mutiresoution Anaysis (MRA), it is possibe to associate another MRA with the Hibert transform of the associated waveet ψ(x). We notice that our construction is different in the sense that we never need to consider the scaing function associated with the new MRA. Aso, we aways derive approximate formuas since our goa is to deveop efficient approximations suitabe for numerica impementations. 7

9 III IMPLEMENTATION OF OPERATORS VIA FILTER BANKS Let us now turn to the more genera case of fiter bank impementation of inear operators. We show that the approach we deveoped for the Hibert transform may be generaized to convoution operators with non-osciatory kernes. We aso anayze the connection with the BCR approach [3] and express the fiter bank impementation as an approximation (with controed accuracy) to the non-standard and standard forms (NS-form and S-form) approaches of [3], where we take into account ony a few bocks of corresponding representations. III.1 Convoution operators Let us consider a more genera convoution operator with symbo a(ξ), and compute coefficients d j k of the projection of g on W j, d j k = We ook for the coefficients b j such that ĝ(ξ) = a(ξ)ˆf(ξ), (3.1) g(x)ψ j j/ k (x)dx = π ĝ(ξ) ˆψ( j ξ)e ikjξ dξ. (3.) d j k ν bj ν s j 1 ν+k, (3.3) where s j 1 ν = (j 1)/ ˆf(ξ) ˆφ( j 1 ξ)e iνj 1ξ dξ, (3.4) π and index ν is not necessariy an integer. Typicay, we wi set ν to be haf-integer and, if more precision is needed, we wi demonstrate that ν can be taken to be in N Z. Using (3.1), we write (3.) as d j k = j/ a(ξ) π ˆf(ξ)m 1 ( j 1 ξ) ˆφ( j 1 ξ)e ikjξ dξ, (3.5) and note that it is sufficient to find an approximation or a(ξ)m 1 ( j 1 ξ) ˆφ( j 1 ξ)e ikjξ 1/ ν a(ξ)m 1 ( j 1 ξ) 1/ ν bj ν kˆφ( j 1 ξ)e iνj 1ξ, (3.6) bj ν e iνj 1ξ. (3.7) over the essentia support of the function ˆφ( j 1 ξ). Let us repace j 1 ξ by ξ in (3.7) and approximate a( j+1 ξ)m 1 (ξ) by the foowing 4π-periodic function (i.e., restrict ν to haf integers, ν = n/) m j # (ξ) = k a( j+1 (ξ +4πk))m 1 (ξ +4πk)χ [ π,π] (ξ +4πk). (3.8) Due to M 1 vanishing derivatives of m 1 (ξ) at points πk, k Z, at these points the break in the function due to the periodization occurs ony in the higher derivatives. Since derivatives of m 1 (ξ) vanish at ξ = 0, we can consider symbos that have a singuarity at ξ = 0, e.g., the Hibert transform and fractiona derivatives. 8

10 The Fourier coefficients of m j # may be found by computing b j n = 4π π π a( j+1 ξ)m 1 (ξ)e inξ/ dξ, (3.9) since they are reated to the b j ν coefficients by a compex conjugation and a redefinition of the index ν, and we have m j # (ξ) = 1 b j ne inξ/. (3.10) The coefficients b j n have a fast asymptotic decay. Let us consider two cases, first where the symbo a(ξ) has at east M continuous derivatives (M is the number of vanishing moments of the basis) and, second, where a(ξ) has a singuarity at ξ = 0 but has at east M continuous derivatives esewhere. In the first case we simpy integrate (3.9) by parts M 1 times using and notice that the boundary terms vanish so that n ( ) i m d m n dξ me inξ/ = e inξ/, (3.11) b j n = O(n M+1 ). (3.1) In the second case we spit the integra into two over [ π,0] and [0,π], and then integrate by parts to obtain again (3.1). Summarizing the resuts of this section, we show that the action of a convoution operator T with symbo a(ξ) on a function f(x) may be obtained as foows: Tf(x) = j d j k ψj k (x) = j k b j ns j 1 k n/ ψj k (x) (3.13) where the coefficients b j n are given in (3.9). This impies that in order to evauate T in the waveet basis, we compute d j k = b j ns j 1 k n/ (3.14) n Again, we notice that haf-integer sampes of the coefficients s j 1 are needed, and refer to the discussion in Section II. The fiter B j = {b j n} defined in (3.9) may be used simiar to the fiter G in QMF pair. Namey, a signa f is decomposed using the fiter pair H and B j and then reconstructed with the usua QMF pair H and G to yied the desired resut. Notice that the fiters B j depend on the scae. If the symbo a(ξ) is homogeneous of degree m, then b j n = jm b 0 n. Remark: It is easy to see the simiarities with the decomposition into a biorthogona basis. We note, however, that there is a singe MRA in our approach. III. Time-dependent symbos A number of interesting questions arises if we consider a more genera cass of symbos of pseudodifferentia operators, k n T σ f(x) = 1 σ(ξ,x)ˆf(ξ)e iξx dξ, (3.15) π IR 9

11 where symbo σ(ξ,x) S (IR ). The operator T σ may be expressed as an integra operator of the form T σ f(x) = K(x, y)f(y)dy. (3.16) IR Here K(x,y) S (IR ) is the distribution kerne of T σ, given by [ K(x,y) = F1 ](x y,x) 1 σ = L(x y,x), (3.17) where F 1 denotes Fourier transform with respect to the first variabe. To deveop our approach, we need to specify further the symbo cass we are working with. We restrict to the cass of the so-caed Caderón-Zygmund kernes, i.e. kernes K(x, y) such that x α yk(x,y) β C α,β x y 1+α+β. Let f(x) L (IR), and et us compute the projection of T σ f onto W j, ( ) T σ f,ψ j k = j/ L(x y,x)f(y)ψ j x k dxdy (3.18) IR IR Let us focus on the integra with respect to x first. We write ( ) ( L(x y,x)f(y)ψ j x k dx = L x y,k j) ( ) ψ j x k dx+r(y;j,k) (3.19) where R(y; j, k) is some remainder. It foows from genera arguments invoving the vanishing moments of ψ(x) that ( R(y;j,k) = O M(j 1/)). From now on, we assume that the remainder may be negected, i.e., that we are at a sufficienty fine scae. Assuming that we may change the order of summation in (3.18), we arrive at an approximation T σ f,ψ j k 1 σ(ξ,k j )ˆf(ξ)e ikjξ m 1 ( π j 1 ξ) ˆφ( j 1 ξ)dξ. (3.0) Repeating considerations of Section III.1, we construct the 4π-periodic function Setting m σ (ξ;k,j) = n Zm 1 (ξ +4πn)σ( j+1 (ξ +4πn),k j )χ [ π,π] (ξ +4πn). (3.1) we obtain the Fourier coefficients of m σ (ξ;k,j) b j k, = n m σ (ξ;k,j) = 1 b j k, eiξ/, (3.) g n 1 4π π π σ( j+1 ξ,k j )e i(n /)ξ dξ. (3.3) It is cear that the coefficients b j k, have the expected asymptotic behavior as, b j k, = O ( L 1). (3.4) Finay, we obtain the foowing agorithm for computing the waveet coefficients in (3.18), T σ f,ψ j k = b j k, sj 1. (3.5) This expression is simiar to (3.14), except that the sum is no onger a convoution. Thus, stricty speaking, the agorithm in (3.5) is not a fiter bank, since fiter bank agorithms are usuay understood to consist of convoutions. 10

12 III.3 Connection with BCR approach It is reasonabe to expect that a subcass of Caderón-Zygmund operators (see e.g., vo. of [16]) may be impemented numericay via fiter banks. Let us consider the cass of symbos S 0 1,1, where σ S0 1,1 satisfies α ξ β xσ(ξ,x) C(α,β)(1+ ξ ) β α. (3.6) It was shown in [3] that in waveet bases operators of this cass may be represented by sparse matrices. A information is contained in the foowing set of coefficients α j k = Tψ j k,ψj β j k = Tφ j k,ψj γ j k = Tψ j k,φj, (3.7) which gives rise to the NS-form, an aternative to the S-form consisting of the eements Tψ j k,ψj k (see [3] for more detais). To expain the reation of the fiter bank approach to that using NS-form, et us consider waveets with good ocaization in the Fourier domain (e.g., Batte-Lemarié waveets), so that for a given precision we need to consider interaction between scaes which are immediate neighbors. In this case we may consider the simpified S-form where ony interaction between neighboring scaes is taken into account. Thus, for a given subspace W j, ony its mappings from subspaces W j+1, W j and W j 1 are significant, and these are subspaces of V j. In this approximation we then consider the mapping V j W j which is exacty the one considered in Tψ(ξ) = m T (ξ/4)ˆφ(ξ/4), (3.8) where m T (ξ) = m # (ξ)m 0 (ξ) (3.9) and m # is defined in (3.8) (we suppress the scae index j). If more accuracy is required, one may consider mappings between more scaes, e.g., W j+, W j+1, W j, W j 1 and W j, which amounts to considering the mapping V j 3 W j. This corresponds to an approximation where Tψ(ξ) = m T (ξ/8)ˆφ(ξ/8), (3.30) m T (ξ) = m # (4ξ)m 0 (ξ)m 0 (ξ) (3.31) and m # is defined simiar to m #, except that the 4π-periodization in (3.8) is repaced with the 8π-periodization. Let us then consider again a convoution operator T with symbo a(ξ). Foowing [3], we consider J P 0 TP 0 = P J TP J + [Q j TQ j +Q j TP j +P j TQ j ] (3.3) j=1 J = P J TP J + [Q j TP j 1 +P j TQ j ]. (3.33) j=1 11

13 The action of first term Q j TP j 1 on f(x) L (IR) (putting together P j and Q j is motivated by Lemma II.1) may be evauated with the same type of approximation as discussed before, Q j TP j 1 f(x) = k, s j 1 φ j 1,T ψ j k ψj k (x). (3.34) Using notation of the previous section, we have Tφ j 1,ψ j k = 1 ( π j 1/ e i(k )j 1ξ a(ξ)m 1 ( j 1 ξ) ˆφ ξ) j 1 dξ (3.35) 1 ( π j 1/ e i(k )j 1ξ m j # (j 1 ξ) ˆφ ξ) j 1 dξ (3.36) = b j 1 n e i(k n/)ξ ˆφ(ξ) dξ. (3.37) n π Finay, using the definition of coefficients a 1 in Appendix, we obtain for the coefficient of ψ j k (x) in (3.34), Tφ j 1,ψ j k sj 1 = b j (k ) sj ) a n 1 (b j ( k)+(n 1) +bj ( k) (n 1) s j 1.,n (3.38) Comparing this expression with equation (3.13) and interchanging the order of summation, we recognize here the same structure as that described in Section III.1. We may interpret the summation in (3.38) as interpoation to obtain the haf-integer transates coefficients s j 1 k. Coefficients b k are given in (3.9). Let us now turn to the P j TQ j term. We notice that in order to describe this term, it is sufficient consider ony Q j+1 TQ j, P j TQ j f Q j+1 TQ j f, due to considerations above. This term represents mapping from scae j to scae j +1, and Q j+1 TQ j f(x) = k, d j k Tψj k,ψj+1 ψ j+1 (x) = q j ψj+1 (x). (3.39) As before, we approximate the coefficient ω j k = Tψj k,ψj+1 ω j k = 1 π j+1/ 1 π j+1/ n as foows e i( k)jξ a(ξ)m 0 ( j 1 ξ)m 1 ( j ξ)m 1 ( j 1 ξ) ( ˆφ ξ) j 1 dξ ˆφ( e i(4 k)j 1ξ a(ξ)m j+1 # (j ξ)m 0 ( j 1 ξ)m 1 ( j 1 ξ) ξ) j 1 dξ b j+1 n c n+k 4, (3.40) where we have set for simpicity m 0 (ξ)m 1 (ξ) = 1 c n e inξ. (3.41) n Again, we obtain for the coefficients q j in (3.39) a fiter bank type reation, q j = k ω j k dj k. (3.4) 1

14 The resuts of this section may be summarized as foows. From equations (3.13), (3.38) and (3.4) we have d j k = n = b j ns j 1 k n/ b j (k ) sj k ω j 1 k dj 1 k, a n 1 (b j ( k)+(n 1) +bj ( k) (n 1),n ) s j 1 (3.43) which is a fiter bank type representation obtained using eements of the NS-form. The terms in (3.43) may be interpreted as foows: the first two terms of the r.h.s. of (3.43) may be viewed as representing the haf-integer sampes discussed in Section II via interpoation within V j 1 and the third term is an eement of W j 1 so that (3.43) represents the mapping V j W j. 13

15 IV HILBERT TRANSFORM OF AUTOCORRELATION WAVELETS In Section II we have described an approximation of the action of the Hibert transform on waveets. These approximate fiters have to be appied to the coefficients of the function on subspaces V j. Since the discrete Hibert transform is usuay defined directy on the sampes of the function, it is advantageous to require that the coefficients s j k are (at east approximatey) the vaues of the function. This requirement may be satisfied by considering interpoating scaing functions. Exampes of such scaing functions proposed in [19] are obtained as autocorreations of the usua compacty supported scaing functions. The properties of such autocorreation waveets and scaing functions are described in Appendix. We note that by using symmetric interpoating waveets in this section, we give up orthogonaity of the basis. Inviewoftheappicationsweconsiderfurtherinthepaper, wewieaborateonthecase of the so-caed dyadic waveet transform (which is redundant with respect to the transation variabe, see Appendix). As a resut we obtain an O(N ogn) agorithm for decomposition and computing the Hibert transform. Let us make cear that the redundancy may be avoided (yieding an O(N) agorithm) if we foow considerations of Section II. IV.1 Representation of the Hibert transform Let us consider m 1 (ξ) = isgn(ξ) m 1 (ξ) (4.1) and denote by m c 1 (ξ) its restriction to the interva [ π,π]. Let m (ξ) = m c 1(ξ +4πk), (4.) and consider its Fourier series, m (ξ) = i k= ( ) kξ b k sin, (4.3) k=1 where m is a 4π-periodic function. The adverse effect of the restriction to [ π,π] and of the 4π-periodization in (4.) is weakened by the fact that the probematic point in mutipying by isgn(ξ) is at the origin ξ = 0, where m 1 has a zero of order L. Therefore, the sequence b k in (4.3) has fast decay and, since ˆΦ(ξ) is concentrated around ξ = 0, we may expect the product m (ξ)ˆφ(ξ) to be a good approximation for sgn(ξ) m 1 (ξ) ˆΦ(ξ). Let Tj f = T j (Hf) denote the jth scae of dyadic waveet transform of the Hibert transform of f. We prove Theorem IV.1 Let Ψ be the autocorreation of the Daubechies compacty supported waveet with L/ vanishing moments. Then the coefficients of T j f = T j (Hf) may be approximated by those obtained from the foowing pyramida agorithm: W j f(n) = b k 1 [S j f(n+k j 1 j ) S j f(n k j 1 + j )], (4.4) k=1 where the sequence b k is given by 0, for k = m, [ L/ b k = 1 ] 1 1 a 1, for k = m 1, (m 1)π =1 1 4( 1 m 1 ) 14 (4.5)

16 and decays at infinity as b k 1 = O((k 1) L 1 ). (4.6) In addition, the exact and approximate Hibert transform coefficients satisfy T j f W j f K (1+π) αl f, (4.7) where α is the Höder reguarity of the scaing function φ(x). The theorem is proved beow, and the proof foows the ines of that of Proposition II.1. We detai it for competeness. IV.1.1 Computation of coefficients b k 1 and their behavior for arge k We have b k = 1 1 π i = 1 8π π π π sin π L/ 1 8π =1 isgn(ξ)sin ( kξ a 1 π ( ) kξ m 1 (ξ) dξ ) sgn(ξ)dξ π The computation of the integras in (4.8) yieds (4.5). If k is arge enough, k 1 > ( 1), then 1 series, namey, sin ( ) kξ cos(( 1)ξ)sgn(ξ)dξ. 1 4( 1 (4.8) k 1 ) may be repaced by its Tayor 1 ( 1 4( 1 = 1 ) p (4.9) k 1 ) k 1 p=0 AccordingtoLemmaIX.ofAppendix, thesequence{a 1 }hasl 1vanishingevenmoments, and we have 1 L/ b k 1 = π(k 1) (k 1) p a 1 p ( 1) p IV.1. Pyramida agorithm p=l L/ = 1 (k 1) L 1 π =1 = O((k 1) L 1 ). =1 a 1 [( 1)] L Let us consider the approximate waveet transform of Hf, 1 ( 1 k 1 ) W j f(n+w) = 1 ˆf(ξ)e iξ(n+w) m ( π j 1 ξ) Φ( j 1 ξ)dξ = b k 1 [S j 1 f(n+w +k j 1 j ) S j 1 f(n+w k j 1 + j )]. k=1 By setting w = 0, we arrive at (4.4). (4.10) (4.11) 15

17 Remarks: 1. As in the orthogona case, we note that for j = 1 (and ony in this case, since we are now using the dyadic waveet transform) we need sampes of S 0 f for haf-integer n. Thus, an additiona interpoation procedure is required for the first step of the agorithm, j = 1. An aternative is to set w = j, which woud yied haf integer sampes of the Hibert transform coefficients. We wi come back to this point ater on.. In order to compute the waveet coefficients of f(x) with respect to the Hibert transform of Ψ(x), we just need to use (4.4), since f,hψ jk = Hf,Ψ jk. IV.1.3 Accuracy estimate Let us introduce the foowing function Θ(ξ) = m ( ξ ) ( ξ Φ ) Ĥ Ψ(ξ) (4.1) where m is the 4π-periodic function given in equation (4.). The function Θ is aso a waveet, and has, in fact, the same number of vanishing moments as Ψ. Considering f L (IR), we fix j and compute and T j f W j f Θ j ĤΨ j f, (4.13) Θ j (ξ) ĤΨ j(ξ) = m ( j 1 ξ) m 1 ( j 1 ξ) φ( j 1 ξ) 4. (4.14) Since we know that for some positive K [5] ˆφ(ξ) K[1+ ξ ] αl, (4.15) and that m m 1 = 0 inside the interva [ π,π], it foows that This competes the proof of Theorem (IV.1). IV. Numerica exampes Θ j ĤΨ j 4K 4 (1+π) 4αL. (4.16) The coefficients b k 1 are easy to compute numericay using equation (4.5). We present in Figure 1 a pot of the approximate fiter m (ξ) (up to a factor i) for the case of the autocorreation of Daubechies waveet with 5 vanishing moments. As expected, we observe the sign fip in [ π,0]. We aso provide tabes of the twenty top coefficients b k 1 in (4.5) for the autocorreation of Daubechies compacty supported waveets with L =, 4, 6,...1 (the numerica vaues of the a 1 coefficients are isted in []). The coefficients b k 1 have been computed using Mathematica TM. 16

18 n L = L = 4 L = Tabe 1: Coefficients b 1 in (4.5), for L = to 6. 17

19 n L = 8 L = 10 L = Tabe : Coefficients b 1 in (4.5), for L = 8 to 1. 18

20 Figure1: The4π-periodicfunctionm (ξ)in(4.),fortheautocorreationofdaubechies waveet with M = Figure : Error of the approximation ˆΘ(ξ) ˆΨ(ξ) for the autocorreation of Daubechies waveet with M = and M = 5. To check numericay the accuracy of our approach, we compare ˆΘ(ξ) and ˆΨ(ξ) for positive vaues of ξ. The difference is potted in Figure for L = and L = 5, respectivey. In both cases ony the twenty top coefficients b 1 have been considered for the evauation of m (ξ). Tabe3containsanumericaestimatefortheabsoutevaueoftheerrormax ξ [0,π] ˆΘ ˆΨ, computed for L = to L = 1. IV.3 Improving the accuracy Itiscearfromtheestimate(4.16)andexampesinFigurethattheaccuracymaybecontroed by increasing the number of vanishing moments of waveets. Aso, the accuracy estimate may be improved by considering the restriction of m 1 to a arger interva. Let m c 1 denote the restriction of m 1 to the interva [ n π, n π], and set m (ξ) = k= m c 1(ξ + n πk). (4.17) 19

21 L max ˆΘ ˆΨ Tabe 3: Error max ξ [0,π] ˆΘ ˆΨ as a function of the number of vanishing moments, for L = to 1. Here m is a n+1 π-periodic function. Let b k denote the Fourier coefficients of m, b k = n π 1 n+ sin( n kξ)sgn(ξ)dξ π n π L/ 1 π n+ a 1 sin( n kξ)cos(( 1)ξ)sgn(ξ)dξ. π =1 The same computation as before yieds where and b k 1 = m (ξ) = i π (4.18) b k 1 sin((k 1) n ξ), (4.19) k=1 [ 1 1 (k 1)π L/ =1 ] 1 a 1 1 n ( 1, (4.0) k 1 ) b k 1 = O((k 1) L 1 ) (4.1) for k 1 > n ( 1). Let us consider the function Θ defined by Θ(ξ) = m ( ξ ) Φ( ξ ). For f L (IR) and fixed j we have W j f(m) = 1 ˆf(ξ)e iξm π Θ( j ξ)dξ (4.) = b k 1 [S j 1 f(m+k j n j n 1 ) S j 1 f(m k j n + j n 1 )]. k=1 The accuracy estimate is derived essentiay as before. The ony change is that π has to be repaced by n π. We then obtain as a generaization of (4.13) and (4.16). T j f W j f 4K 4 (1+ n π) 4αL f, (4.3) 0

22 V OTHER EXAMPLES A particuary simpe impementation (simiar to that for the Hibert transform) is possibe for the convoution operators with non-osciatory kerne considered in []. The coefficients of fiters ( simiar to m ) might be scae dependent. If the operator is homogeneous of some degree, then fiters on different scaes wi differ ony by a scaing factor. This section is devoted to the study of the action of various operators of differentiation and integration inuding those of fractiona order. Throughout the section, we wi ony use the autocorreation waveets that we used for the Hibert transform, eaving to the reader the (straightforward) computations in the orthogona case, as described in Section II. V.1 Derivative operators The Hibert transform is homogeneous of degree zero and, therefore, the operator (and, thus, m ) is the same on a scaes. Since derivative operators are homogeneous, the fiter wi be the same on a scaes except for a scaing factor, d n ( d dx nψ jk = nj n ) Ψ dx n. (5.1) jk Thus, it is sufficient to evauate the derivative operator on the function ψ. Our anaysis is based on an approximation of ξ m 1 (ξ) by the 4π-periodization of its restriction to [ π,π]. Let us consider (for simpicity) the case n = 1, and the 4π-periodic function m 3 (ξ) = k= Since m 3 is an odd function, we have (ξ +4πk) m 1 (ξ +4πk) χ [ π,π] (ξ +4πk). (5.) m 3 (ξ) = ( ) kξ δ k sin k=1 An expicit computation of the Fourier coefficients δ k yieds δ k = [ ( 1)k 1 k [ ( 1)k 1 k L/ 1 ( ( 1) k ) ] 1 a 1 =1, 0 L/ 1 a 1 =1 1 ( ( 1) k ). (5.3) ] + 1 k a k/ if k = ( 0 1) otherwise. As in the case of the Hibert transform, the behavior of the Fourier coefficients is governed by the number of vanishing moments of the waveet. Using Lemma IX. and the Tayor series expansion of δ k for k > (L 1), we estimate δ k as (5.4) L/ δ k = ( 1) k k L 1 (( 1)) L a 1 = O(k L 1 ) =1 1 ( ( 1) k ) (5.5) For a given precision, the series (5.5) may be truncated. As an exampe, we provide in Tabes 4 and 5 coefficients δ k for the autocorreation of the Daubechies waveet with 4, 5 and 6 vanishing moments. The shape of fiter m 3 in the Fourier domain is shown in Figure 3. 1

23 n L = 8 L = 10 L = Tabe 4: Approximate coefficients δ k in (5.5) for the derivative of the autocorreation waveets with 4, 5 and 6 vanishing moments.

24 n L = 8 L = 10 L = Tabe 5: Approximate coefficients δ k in (5.5) for the derivative of the autocorreation waveets with 4, 5 and 6 vanishing moments (continued) Figure 3: Approximate fiter m 3 (ξ) in (5.) for the derivative of the autocorreation of the Daubechies waveet with 6 vanishing moments 3

25 Figure 4: Approximate fiter m 4 (ξ) and the symbo ξ for the second derivative of the autocorreation of the Daubechies waveet with 5 vanishing moments. V. Second order derivative For the second order derivative we have to consider the Fourier expansion of m 4 (ξ), the 4πperiodization of ξ m 1 (ξ) χ [ π,π] (ξ) (5.6) Since m 4 (ξ) is even, we have m 4 (ξ) = where 0 = π 3 + a 1 1 k = 8( 1)k k +( 1) k a 1 0 = 8( 1)k k +a 0 1 ( +( 1) k a 1 ( ) kξ k cos, (5.7) k=0 ( ) 1 ( 1 k + 1 ) ( 1+ k ( ) ) 4( 1) k k + π 3 1 ( 1 k ) + 1 ( 1+ k ) ) if k = ( 0 1) esewhere. (5.8) The derivation of (5.8) is straightforward and is simiar to that of (5.5). The graph of m 4 (ξ) is shown in Figure 4, together with that of ξ. V.3 Derivative of the Hibert transform The symbo of the composition of the Hibert transform and differentiation is σ(ξ) = i ξ and this operator appears prominenty in the inversion of the Radon transform on the pane, e.g. in X-ray tomography. Setting m 5 (ξ) = σ(ξ) m 1 (ξ) χ [ π,π] (ξ) = i ( ) kξ µ k cos, (5.9) 0 4

26 Figure 5: Approximate fiter im 5 (ξ) in (5.10) and the symbo ξ for autocorreation of the Daubechies waveet with 5 vanishing moments. the evauation of the Fourier coefficients yieds µ 0 = π µ 4k = 0 µ (k 1) = π a k 1 µ k 1 = 1 ( ) ( 1) +( k 1 a π 1 ( k 1 ) ) ( 1) ( k 1. ) (5.10) The graph of im 5 (ξ) is shown in Figure 5, together with that of ξ. V.4 Fractiona derivatives Contrary to what the name suggests, it is better to view fractiona derivatives as integra operators since they are non-oca in a way simiar to the Hibert transform. This non-oca behavior is manifested in the Fourier domain by the action of the operator that breaks the function at ξ = 0. However, in the waveet representations such operators are approximatey oca if appied to functions which do not have projections on the coarsest subspace. In particuar, band-imited signas are an exampe of such cass of functions. The projection on the coarsest subspace has to be treated separatey (if necessary) and requires very few operations since the function is represented by a sma number of sampes. Using the foowing definition of fractiona derivatives, ( α xf)(x) = + (x y) α 1 + Γ( α) f(y)dy, (5.11) where α 1,... (if α < 0, then (5.11) defines fractiona anti-derivatives), we find its representation in the Fourier domain as a(ξ) = e iαπ/ ξ α + +e iαπ/ ξ α, (5.1) where ξ α + = ξ α for ξ > 0 and is zero otherwise, and ξ α = ξ α for ξ < 0 and is zero otherwise. Since α xψ jk = αj ( α xψ) jk, (5.13) 5

27 Figure 6: Moduus of the approximate fiter m 6 (ξ) in (5.14) for the derivative of order 1/ of the Hibert transform of autocorreation of Daubechies waveet with 6 vanishing moments (dashed ine: the symbo ξ 1/ ). it is sufficient to evauate the operator on the function Ψ. We have, as before, m 6 (ξ) = k= The Fourier coefficients of m 6 (ξ) are given by Setting we obtain a(ξ +4πk) m 1 (ξ +4πk) χ [ π,π] (ξ +4πk). (5.14) γ = 1 π a(ξ) m 1 (ξ) e iξ/ dξ = 1 π 4π π π Re m 1 (ξ) e iαπ/ ξ α e iξ/ dξ. (5.15) 0 u α k = 1 π ξ α cos 4π 0 ( kξ +απ ) dξ, (5.16) γ = u α 1 ) a k 1 (u α +(k 1) +uα (k 1). (5.17) k Again, the decay of the γ coefficients is governed by the reguarity of m 6. Since the π-periodic function m 1 (ξ) vanishes at ξ = 0 and ξ = π together with its derivatives of order up to L 1, one directy obtains the asymptotics of γ, γ = O( L 1 ) (5.18) As an exampe, we dispay in Figure 6 the derivative of order α = 1/ using the autocorreation of Daubechies waveets with 6 vanishing moments. We note that m 6 (ξ) is compex-vaued. V.5 Integration operators Let us now consider integration operator with symbo σ(ξ) = 1 iξ. (5.19) The same procedure as before yieds m 7 (ξ) = 1 iξ m 1(ξ) = k λ k sin ( ) kξ. (5.0) 6

28 Figure 7: Approximate fiter im 7 (ξ) in (5.0) and the symbo 1/ξ for the primitive of the autocorreation of the Daubechies waveet with 5 vanishing moments. For the coefficients λ k, we obtain λ k = i ( Si(kπ) 1 ) a 1 [Si((k +( 1))π) Si((k ( 1))π)], (5.1) π where Si(x) = x 0 sin(y) dy. y The graph of im 7 (ξ) is shown in Figure 7, together with that of 1/ξ. 7

29 VI THE HILBERT TRANSFORM OF SIGNALS We now turn to signa processing probems. The purpose of this Section is to iustrate one of the appications of our method for computing the Hibert transform of a signa. As we sha see, it is interesting to work in a context in which the scaing function has vanishing moments, since the sampes of the signa may then be identified (within a certain accuracy) with the coefficients of its projection onto some V j space. For this reason we sha use autocorreation waveets (other choices such as high order Batte-Lemarié waveets or Coifets, whose scaing function aso possess vanishing moments, woud do the job as we). The autocorreation waveets aso offer an advantage of a trivia reconstruction formua (simpe summation of waveet coefficients over scaes, see equation (9.31) in Appendix, the price to pay being an O(N og(n)) compexity). VI.1 Band-pass signas Let f C r (IR), r > L, and assume aso that Hf C r (IR), with r > L. Then, according to (9.31), we have the foowing waveet decompositions (we refer to (9.8) and (9.9) in Appendix for the description of our notation), f(k) = S j0 f(k)+o( j0(l 1) ) (6.1) J = S J f(k)+ T j f(k)+o( j0(l 1) ) (6.) j=j 0 +1 [Hf](k) = S j0 [Hf](k)+O( j0(l 1) ) (6.3) J = S J [Hf](k)+ T j [Hf](k)+O( j0(l 1) ) (6.4) But the Hibert transform is anti sef-adjoint, so that for j 0 arge enough we have j=j 0 +1 T j [Hf](k) = Hf,ψ jk = f,[hψ] jk, (6.5) [Hf](k) S J [Hf](k)+ J j=j 0 +1 where W j f is defined in (4.4). We then obtain the foowing W j f(k), (6.6) Theorem VI.1 Let f C r (IR) be such that Hf C r (IR), with r,r > L. Then [Hf](n) = S J [Hf](n)+ W j f(n)+o((1+π) αl ). (6.7) Therefore, as ong as for a sufficienty sparse scae J the ow-pass component S J f(k) of a signa f(k) can be negected, the agorithm in (6.7) provides a good approximation of the Hibert transform of f. Remark: The above is an O(N ogn) agorithm, because we used a redundant (without subsamping) version of waveet decomposition agorithm. The same agorithm with subsamping requires O(N) operations, as shown in the first part of the paper. An aternative woud be to use spine waveets and the associated Lagrange interpoation to obtain the connection between approximation coefficients and sampes, or to use the more genera agorithms deveoped in [7]. 8

30 Signa Ampitude Time Figure 8: An exampe of sound /one two/ samped at 8kHz. VI. Exampes Speech signa (or at east voiced speech) is an exampe of signas that may be modeed as superpositions of ampitude and frequency moduated components (see e.g. [14, 1]). Athough waveet decompositions do not seem optima for appying the anaysis we have in mind to speech, et us consider it as an iustration. In Figure 8 we show a haf a second exampe of sound /one two/ samped at 8kHz. In Figures 9 and 10 we show the band-pass component of the signa j T jf(n) and the corresponding approximate Hibert transform j W jf(n), respectivey. Figure 11 is a zoom of Figures 9 and 10 in which the rea and imaginary parts of the reconstructed anaytic signa, Z f (n) = j (T j f(n)+iw j f(n)), (6.8) are represented. Finay, in Figure 1 we shows the squared moduus Z f (n), i.e. the square of the instantaneous ampitude of the signa (in the sense of Vie []). Notice that the instantaneous ampitude sti has a ot of osciations characteristic of the presence of many additive components in the signa within the considered frequency band. 9

31 Reconstructed signa: Rea part Ampitude Time Figure 9: Rea part of the reconstructed signa in (6.8). Reconstructed signa: Imaginary part Ampitude Time Figure 10: Imaginary part of the reconstructed signa in (6.8). 30

32 500 Rea (dashed) and Imaginary parts 50 Ampitude Time Figure 11: Zoom of rea and imaginary parts of the reconstructed signa in (6.8). Moduus 600 Ampitude Time Figure 1: Moduus of the reconstructed signa in (6.8). 31

33 VII ON THE REPRESENTATION OF SIGNALS BY LOCAL PHASES AND AMPLITUDES It is we-known that an arbitrary continuous-time signa may be represented (e.g. using the method described in the previous Section) in terms of its oca phase or its phase derivative, the instantaneous frequency, and oca ampitude (foowing the pioneering work of J. Vie []). More precisey, writing Z f (x) = f(x)+i[hf](x), (7.9) we obtain an anaytic function (the so-caed anaytic signa) which may be associated with the so-caed canonica pair, A f (x) = Z f (x) instantaneous ampitude, ω f (x) = argz f (x) instantaneous phase. (7.10) The instantaneous frequency is then defined as ν f (x) = 1 π ω f(x). (7.11) The purpose of such representation is to obtain the oca phase and ampitude in the hope that they are much ess osciatory that the origina signa (and then more easiy compressibe if the target appication is compression). Moreover, in such a case, the instantaneous ampitude and frequency are often intimatey connected with physica quantities. In genera, however, the instantaneous frequency and ampitude may be as compicated as the signa itsef. This is particuary cear in the exampe shown in the previous section (see Figure 1), where the goba ampitude of the considered speech signa has fast osciations. Anexpanationofthisfactisasfoows. Inthespeechsigna, agivenphonememayoften be modeed as a superposition of short chirps, each having its own instantaneous frequency. It is then cear that there is no natura way of assigning a unique instantaneous frequency to such a phoneme, since the instantaneous frequency osciates fast due to the interferences between chirps. An adequate description of the speech signa thus has to take into account this muticomponent character of the signa. It is natura to expect that by spitting the frequency band, the ampitudes of the subbands wi have sower osciations, so that the representation of the subbands in terms of oca phase and ampitude becomes usefu. Moreover, if the considered subband contains one and ony one of the chirps of the phoneme, standard approximations (see for instance [6,?]) show that anaytic signa provides a good approximation of the behavior of the component. It turns out that in the discrete case such representation is quite easy to obtain from our approximate Hibert transform agorithm. Indeed, the main aspect of our approach is to derive approximate expressions for the Hibert transform of the waveet Ψ(x), together with a fast agorithm for the computation of the corresponding coefficients. As a by-product, our method yieds a decomposition of band-pass signas as f(n) = Re j Z j f(n), (7.1) where Z j f(n) = T j f(n)+iw j f(n) (7.13) 3