A Dual-Tree Rational-Dilation Complex Wavelet Transform

Transcription

1 1 A Dual-Tree Rational-Dilation Comlex Wavelet Transform İlker Bayram and Ivan W. Selesnick Abstract In this corresondence, we introduce a dual-tree rational-dilation comlex wavelet transform for oscillatory signal rocessing. Like the shorttime Fourier transform and the dyadic dual-tree comlex wavelet transform, the introduced transform emloys uadrature airs of time-freuency atoms which allow to work with the analytic signal. The introduced wavelet transform is a constant-q transform, a roerty lacked by the short-time Fourier transform, which in turn makes the introduced transform more suitable for models that deend on scale. Also, the freuency resolution can be as high as desired, a roerty lacked by the dyadic dual-tree comlex wavelet transform, which makes the introduced transform more suitable for rocessing oscillatory signals like seech, audio and various biomedical signals. Index Terms Rational-dilation wavelet transform, dual-tree comlex wavelet transform, short-time Fourier transform, analytic signal, instantaneous freuency estimation. a) I. INTRODUCTION Desite its usefulness for iecewise smooth signals, the dyadic wavelet transform is less effective for the rocessing of oscillatory signals like seech, audio, various biomedical signals) because of its oor freuency resolution. In this regard, the rational-dilation wavelet transform RADWT), which rovides a finer freuency analysis, is more suitable [1] see Fig. 1). However, even though the RADWT has better freuency resolution, it is not simle to carry out certain oerations with the RADWT, such as the Hilbert transform, enveloe detection, instantaneous freuency estimation, which are relevant for oscillatory-signal rocessing [2]. For such tasks, the short-time Fourier transform STFT) is more useful because its atoms are analytic, which in turn allow for the easy construction of analytic signals. However, the time-freuency structure of the STFT is uite different from that of a wavelet transform, it being a constant bandwidth transform rather than a constant-q transform 1. In this aer, we introduce a dual-tree rational-dilation wavelet transform DT-RADWT) that inherits the good freuency resolution and constant-q roerty of the RADWT 2 and whose atoms form uadrature airs see Fig. 1a,b). In brief, a DT-RADWT frame consists of the union of a RADWT frame and its Hilbert transform 3. The DT-RADWT introduced in this aer is realized by two filter banks FB) oerating in arallel on the inut as shown in Fig. 2. Essentially, we reuire that the i th channel of the two filter banks, say FB 1 and FB 2, to comute inner roducts of the inut xn) with shifts of a bandass filter h in) and H d {h in)} resectively. Here, H d { } denotes the discrete-time Hilbert transform [6]. For the İ. Bayram was with Biomedical Imaging Grou, EPFL, Switzerland. He is now with the Deartment of Electronics and Communication Engineering, Istanbul Technical University, Maslak, 34469, Istanbul, Turkey. I. W. Selesnick is with the Deartment of Electrical and Comuter Engineering, Polytechnic Institute of New York University, Brooklyn, NY, 11201, USA. ilker.bayram@itu.edu.tr, selesi@oly.edu 1 The Q-factor of a function is defined to be the ratio of its center freuency to its bandwidth i.e. f c/ f). A transform is said to be constant-q if it emloys atoms that have the same Q-factor. See [3] and Section IV for an argument of the utility of constant-q transforms in audio rocessing. 2 The introduced DT-RADWT is also µ-shift invariant like the RADWT see [4] for the definition of µ-shift invariance) thanks to the secific filter freuency resonses described in [1] which are also used here. 3 To be recise, the Hilbert relation holds only for the bandass functions. The situation for lowass functions is similar to that in dyadic DT-CWT see [5]. Fre. f f/α f/α 2 f/α 3 0.2π 0.1π 0 0.1π 0.2π β α β α 2 β α 3 β b) c) Time Fig. 1. a) A number of atoms from different scales of a RADWT thick lines). DT-RADWT emloys these atoms as well as their Hilbert transforms thin lines), b) The freuency resonses of a number of analytic discrete-time) wavelets, c) Samling of the time-freuency lane by a tyical wavelet frame. The dilation factor α, and the shift arameter β determine the lattice. For RADWT and DT- RADWT, α and β can be selected by changing the samling factors of the filter bank. derivation of the euivalent bandass filters h in), see Section IV in [1]. The difference between the introduced DT-RADWT and the STFT is the time-freuency TF) distribution of their atoms. The STFT samles the TF lane uniformly wheras the TF samling attern for the DT- RADWT is non-uniform along the freuency axis see Fig. 1c). For ease of notation, let us consider signals defined on the real line for now. Given a window function h ), STFT atoms are comrised of time shifts and modulates of this function, i.e., STFT Atoms : {h n x) exjk )} k,n Z. 1) A number of STFT atoms are shown in Fig. 3. We remark that, for a fixed window function h ), the number of oscillations in the STFT atoms

2 2 H H H... line), as well as the discrete-time functions which lay a more direct role in the comutations. G G G H1) G 1) H G H... Fig. 2. DT-RADWT consists of two filter banks with rational samling factors oerating in arallel on the inut. The system in the dashed box, which is referred to as the rational-dilation DWT RADWT), was discussed in [1] here, the only reuirement on,, s is that the FB have as many outut coefficients as inut coefficients, i.e. / + 1/s 1. G Previous Work The dual-tree comlex wavelet transform DT-CWT) which emloys a uadrature air of dyadic wavelet frames i.e. α = 2 in 2)) was introduced by Kingsbury [9] see also [5]). In comarison to DT-CWT, the DT-RADWT is based on rational samling factors α, β in 2 and Fig.2) and can therefore attain higher freuency resolution than the DT- CWT. An interesting aroach to devise a constant-q transform was formulated by Gambardella [10]. His idea is to generalize the STFT by using window functions that deend on the center freuency of interest. As a secial case, if one scales the window function with the center freuency, the resulting transform becomes, I f, t) = fx) h x t) ) e j x dx 3) Fig. 3. atoms. a) Real thick lines) and imaginary arts thin lines) of a few STFT where f ) is the inut signal and h ) is the window function. If we set ψx) = hx) e j x, this can be written as, I f, t) = e j t fx) ψ x t) ) dx. 4) deend on the freuency arameter k in 1). This imlies a non-uniform treatment of different scales how succesfully STFT can searate two freuencies f 0 and f 1 deends not only on their ratio f 1/f 0 but their values as well. On the other hand, the introduced DT-RADWT emloys time shifts and dilations of the real and imaginary art of an analytic wavelet function ψ ), i.e., { } DT-RADWT Atoms : α n/2 Re{ψα n kβ)} k,n Z { } α n/2 Im{ψα n kβ)} k,n Z where α and β can be adjusted by changing the samling factors of the FBs in Fig. 2. Because the freuency tiling is achieved by scaling the wavelet function, the number of oscillations in fact the shae) of the atoms remains the same see Fig. 1a). Likewise, these atoms all have the same Q-factor. The resulting time-freuency distribution with resect to these arameters is shown in Fig. 1c. The TF samling atterns of the STFT and the DT-RADWT are different in mainly two asects. First, the subbands are distributed logarithmically for the DT-RADWT, whereas they are uniform for the STFT. Second, the samling eriod of each subband is different for DT-RADWT to account for the subbanddeendent bandwidths. This way, the DT-RADWT attains a modest redundancy as well as a stable analysis/synthesis imlementation, which is related to its being a tight frame. We note however that the STFT has a faster imlementation than the DT-RADWT because it makes more efficient use of the FFT due to its uniform samling with resect to freuency). Contribution Our main contribution in this corresondence is the construction of an analytic rational dilation wavelet transform using the dual-tree framework. In articular, we rovide a generalization of the sufficient condition derived in [7] which was also shown to be necessary in [8]). The discussions are valid for the underlying wavelets defined on the real 2) Uto a hase term and scaling), 3) is euivalent to a continuous wavelet transform. This form of the transform was studied by Youngberg and Boll [11], Petersen and Boll [12]. In [12] the authors also show how to samle this constant-q transform i.e. samling the arameters and t), deriving the attern in Fig. 1c. This articular constant-q transform was also utilized for time-scaling of audio [13], an alication which we will discuss in Section IV. Brown [3] roosed a similar constant- Q transform by utilizing channel-deendent window functions and nonuniform samling in the freuency domain also see [14]). Freuency-waring [15] also see [16] for a comrehensive tutorial and further references) is another interesting aroach to obtain nonuniform freuency analysis. Here, the idea is to transform the inut signal xn) to another signal gn) so that their DTFTs satisfy Xe j ) = Ge jθ) ) 5) where θ) is a one-to-one function on the unit circle. By aroriately selecting θ), FFT of gn) allows one to nonuniformly samle Xe j ) in the freuency domain. In general, the maing that transforms xn) to gn) is not orthonormal. Noting the relation to Laguerre seuences, Evangelista and Cavaliere [17] refilter xn) so as to ensure that this maing is orthonormal. Following this orthonormalized freuencywaring, they aly a dyadic DWT to the freuency-wared signal to obtain an orthonormal time-freuency analysis similar to that of a rational-dilation DWT. Organization In Section II, we discuss some basic oerations that can be carried out naturally with the DT-RADWT. Following this, Section III rovides a sufficient condition which ensures that the two filter banks emloy uadrature airs of discrete-time atoms. We will also show that the two filter banks are related to uadrature airs of wavelets which can be combined to form an analytic wavelet. In Section IV, we discuss an alication that reuires both the analysis and synthesis caabilities of DT-RADWT to demonstrate the otential of the introduced transform. Section V is the conclusion.

3 3 xn) AFB 1 SFB 1 + } + H d {xn) a) AFB 2 SFB 2 Fig. 5. Using DT-RADWT, the analytic signal transform of a signal xn) can be erformed by swaing the oututs of the analysis filter banks and taking the difference. This follows because if xn) = i c if i n) for a collection of atoms f i n), the Hilbert transform of xn) is i c ih{f i n)}. The analysis art in the figure comute c i s for f i, swaing in the synthesis stage relaces f i by H{f i }. In the figure, AFB and SFB stand for analysis filter bank and synthesis filter bank resectively. b) Fig. 4. Decomosition of a honocardiogram PCG) signal using the RADWT. a) PCG signal, b) Several channels of the RADWT of the PCG signal. The RADWT arameters are = 2, = 3, s = 2 with reference to Fig. 2.) a) II. BASIC OPERATIONS WITH THE DT-RADWT Before going into the details of the construction, we first briefly discuss some basic oerations for which the DT-RADWT is useful. Consider the honocardiogram PCG) signal xn) shown in Fig. 4a. RADWT gives us a decomosition of xn) as xn) = #Chn. c ik) h in k s i) 6) k Z where h in) is the euivalent filter emloyed by the i th stage of RADWT, s i is a channel-deendent shift arameter see [1] for details) and c ik) = x ), h i k s i) 7) We can isolate the searate channels x in) as x in) = k Z c ik) h in k s i). 8) A number of x in) s are shown in Fig. 4b for a articular RADWT with arameters = 2, = 3, s = 2 we use the the filters described in Section II.B of [1]). The Hilbert transform of xn) can be comuted by swaing the coefficients as illustrated in Fig. 5. The Hilbert transform of a single channel x in) can be similarly comuted by setting to zero the coefficients of the rest of the channels. A articular channel and its Hilbert transform is shown in Fig. 6a. Now that we have x in) and H d {x in)}, we can form the analytic signal for x in) via x A i n) = x in) + j H d {x in)}. 9) x A i n) can be used to comute the enveloe as env in) = x A i n) see the dashed line in Fig. 6a). We can also estimate the instantaneous freuency of x in) by tracking the hase of x A i n). For the articular filters that we use in DT-RADWT, it can be shown that x A i n) is obtained by LTI filtering xn) with a bandass filter f A i n). Since we know the center freuency of f A i n), we can write x A i n) as, x A i n) = env in) exj i n + φn))) 10) where i denotes the center freuency of f A i n) and φn) is the cumulative hase due to the deviation of the instantaneous freuency of x A i n) from i. From this exression, we take the instantaneous freuency as φ n) + i where φ n) is the rincial value of φn + 1) φn). We remark that x A i n + 1) x A i n) could have been used to directly obtain the instantaneous freuency, but this can lead to hase-wraing roblems. Taking the rincial value of φn + 1) φn) does not b) Fig. 6. a) A articular channel of the PCG in Fig. 4a thick line), its Hilbert transform thin line) and the enveloe signal dashed line), b) instantaneous freuency of the signal in a). guarantee to avoid hase-wraing roblems but it is less likely to encounter such roblems. The instantaneous freuency obtained through φ n) + i is shown in Fig. 6b. In brief, we think that the DT-RADWT will be useful in situations where one needs to erform time-freuency oerations on logarithmically distributed freuency subbands. III. DUAL-TREE RADWT : A SUFFICIENT PHASE CONDITION The RADWT consists of an iterated FB as shown in Fig. 2 see the system inside the dashed rectangle). For the i th bandass channel, the analysis FB essentially comutes the inner roducts of the inut with shifts of a discrete-time bandass function h in). We will construct another FB, with the same samling factors, such that h in), the function whose shifts form the i th bandass channel of this new iterated FB, satisfies h in) = H d {h in)}. We remark that both of the iterated FBs are associated with uniue wavelet functions ψx), ψx) [1] this is not the case in general see [18] for a detailed discussion) similarly as in dyadic wavelet frames. We will also show that ψx) = H{ψx)}, where H{ } denotes the Hilbert transform for L 2R). In the rest of this section, we rovide sufficient hase conditions on the filters for realizing a DT-RADWT. For this, let us first recall the sufficient) erfect reconstruction conditions for the RADWT for a more detailed treatment, we refer to [1]). Perfect Reconstruction Conditions for RADWT The RADWT is self-inverting or forms a tight frame) if the analysis and synthesis FBs shown in Fig. 7 have the erfect reconstruction PR) roerty. A set of sufficient conditions on the filters H) and G) ensuring PR can be derived 4 as notice that H), G) are 2π-eriodic 4 For general samling factors, FIR filters cannot yield a RADWT that is also a tight frame [1].

4 4 H) G) H ) s G ) Fig. 7. Rational-dilation wavelet transform is a tight frame rovided this filter bank has the erfect reconstruction roerty. s 1)π H) π s 1)π G) 0 s s π Fig. 8. The erfect reconstruction conditions given in 11) imly certain restrictions on the freuency resonses of H) and G). In addition to a band limit, there is a constant freuency resonse band. functions of ), [ ] π H) = 0, for, π, 11a) [ G) = 0, for 0, 1 1 ) ] π, 11b) s ) 1 2 H + 1 s G) 2 = 1 for [ π, π]. 11c) These conditions imly a transition band as well as a constant-resonse band shown uto a ossible hase factor) for H) and G) as deicted in Fig. 8. Phase Condition for the Second FB The RADWT essentially comutes the inner roducts of the inut with a set of discrete-time functions {h i,k } i I,k Z indexed by subband i and osition k. We will construct an additional FB using filters H, G such that the resulting set of functions { h i,k } i I,k Z are the discretetime Hilbert transforms of { h i,k } i I,k Z, save for the functions in the lowass subband. To derive a sufficient condition, we consider the system shown in Fig. 9, that usamles, filters, downsamles the inut and then comutes the inner roduct of the outut with a discrete-time function un). The system in Fig. 9 mas the inut discrete-time function to a number. Since it is also linear, it can be regarded as a correlator with a discrete-time function yn). Here, the correlator with un) may be regarded as a device that mas the inut signal to a articular samle of one of the bandass channels. In other words, we think of un) as one of h i,k0 n) described above i.e. the discrete-time function at subband i, osition k 0). Given this, yn) will be eual to h i+1,k1 n) i.e. the discrete-time function at subband i + 1, osition k 1). We will first investigate how un) and yn) are related. Next, we will relace un) and H) with their counterarts in the second FB, denoted by ũn), H). This new system will allow us to comare the functions in the second FB, namely h i,k0 that is, ũn)) and h i+1,k1 that is, ỹn)) with the functions in the first FB, namely h i,k0 and h i+1,k1. In articular, our intent is to show that, if h i,k0 and h i,k0 are related by a discrete-hilbert transform, then h i+1,k1 and h i+1,k1 are also related by a discrete-hilbert transform, thereby imlying the Hilbert relation between the two FBs, by an induction argument. Let us return to the system in Fig. 9. We remark that yn) is imlicitly determined by,, H) and un). We now take a look at this relation. π + s H), un), yn) Fig. 9. Usamling, filtering, downsamling followed by correlator. Under 11)a, this is euivalent to comuting the inner roduct of the inut with yn), whose DTFT, Y ), is given in 14). We take H to be bandlimited to π/ and also <. In this setting, the outut of this system for a given inut signal xn) is, 1 π ) ) 1 2π X H U ) d. 12) π Using the bandlimitedness of H the exression is not correct otherwise), we can change variables to write this as, 1 π ) ) 1 2π X) H U d. 13) π Therefore, we see that Y ) = 1 ) H U ) for [ π, π]. 14) Now consider another system, that utilizes a different filter H and a different function ũn). Suose further that where Ũ) = U) e jθ) for [ π, π], 15) Θ) = sign) π 2 + a for [ π, π]. 16) We will try to find H) such that Ỹ ) = 1 H /) Ũ/), satisfies Ỹ ) = Y ) e jθ) for [ π, π]. 17) This will imly, by induction, the Hilbert relation we are after we will make this claim clearer below. In an attemt to satisfy 17), take with H) = H) e jτ) for [ π, π], 18) τ) = b for [ π, π]. 19) Noting that Ỹ ) = Y ) = 0 for [π/, π], we have Ỹ ) = 1 ) ) H Ũ for [ π, π] = Y )e j [Θ/) τ/)] for [ π, π]. 20) Therefore Ỹ ) = Y ) e jθ) holds if i.e., if Θ/) τ/) = Θ), 21) sign) π 2 + a b = sign) π 2 + a. 22) This is euivalent to ) b = a. 23) Therefore, considering the FB structure in Fig. 2, if we set the filters in the second FB as G) = G) e jθ), H) = H) e jτ), where Θ), τ) are defined by 16), 19) and ) b = a, we see, by an induction argument, that the bandass functions of the iterated FB constructed by H), G), are discrete-time Hilbert transforms of the bandass functions of the iterated FB constructed by H), G) uto a shift by 1/a. We can remove this shift by delaying the lowass filter

5 5 in the first stage of the second FB by /a, i.e. by relacing H) by see the second FB in Fig. 2) H 1) ) := H)e j /a = H) e j1/b+/a). 24) If we further ask that 1/a + 1/b) = /2), so as to distribute the shifts of the scaling function evenly see [5] for an argument), we find that recall a = )b), a = 2, b = 2. 25) Given H, G, this secifies the hase terms of H and G comletely. We remark however that it is the relation 23) that will yield Hilbert airs of wavelets which we discuss in the following subsection. In summary, we have shown the following roosition. Proosition 1. For the two FBs in Fig. 2, with samling factors,, s, suose H) and G) satisfy the conditions in 11). Let If Θ) = sign) π H) = H) e j ) /2, G) = G) e j Θ), H 1) ) = H) e j /2, for [ π, π]. 26) 27a) 27b) 27c) then rovided it is not the highass or lowass subband, the i th subband of the two FBs comute inner roducts of the inut with discrete time seuences that are related by the discrete-time Hilbert transform. Hilbert Pairs of Wavelets In [1], we defined the wavelets via the infinite roduct formula as, ) ˆψ) = G ) ) k 1 H. 28) We also recall that the wavelets are bandlimited to π/. Now as in the revious section, if we set G) = G) e j Θ) 29) H) = H) e j τ), 30) then for π/. ) ˆ ψ) = G { [ ) = ˆψ) ex j Θ Now using 1 H τ ) ) k ) )]} k 31). 32) Θ) = sign) π 2 + a, 33) τ) = b along with the sufficient condition )b = a, we obtain, ) ) ) k Θ + τ = sign) π 2 + a b 1 34) 35) ) k 36) = sign) π 2. 37) Recalling ˆψ) = ˆ ψ) = 0 for > π/, this imlies that ψ ) is the Hilbert transform of ψ ). We remark that for the bandass discrete-time functions, the atoms from the first FB were the Hilbert transforms of the atoms from the second FB. This difference stems from the definition of the wavelets the wavelets are defined using the filter freuency resonses wheras the discrete-time atoms come from the time-reverse of the filter imulse resonses which involve the conjugates of the filter freuency resonses. IV. AN APPLICATION : TIME-SCALING AUDIO DT-RADWT is not a tool restricted to erform analysis or synthesis only it can be useful for doing both, thanks to its tight frame roerty. Therefore it can be used for rocessing signals which tyically reuire an analysis as well as a synthesis scheme. In the following, we will briefly resent an alication, namely, time-scaling of audio signals, that makes use of both the analysis and the synthesis arts of the transform. Consider a multicomonent model for an audio signal given by ft) = K A it) cosϕ it)). 38) In rincile, this signal can be scaled in time by a constant α without varying the instantaneous freuency content) [19], using f αt) = K A iα t) cosα 1 ϕ iα t)). 39) Lacking a reresentation as in 38), we have to address the roblem of defining/searating the K comonents, in order for the scaling formula 39) to work. One articular method is to emloy the hase vocoder or STFT see [20]). In that case, one obtains an exansion of fx) as, ft) = I b it) ex j i t + ϕ ) it)), 40) where i runs over the subbands of STFT with a total of I subbands), and i denotes the center freuency of the i th subband. In this exansion, one can regard each subband signal as a single comonent signal, with known center freuency. Therefore, one can comute the deviations from the center freuency by monitoring the change in the hase term. The time-scaled version of fx) is then obtained as, f αt) = I b iα t) ex j i t + t 0 ϕ iα y) dy )). 41) Phase-vocoder based time-scaling method is very fast and erforms uite well even for values of α far from unity. However, one artifact is that the time-scaled audio outut sounds hasy the outut is erceived as if the source is far away from the microhone for a discussion of this effect and some remedies see [21] and the references therein). The only reuirement in the formulation 41) based on the hase vocoder is that the subbands be narrow enough in freuency, in order to be able to calculate the instantaneous freuency using the hase information. Therefore this aroach is suitable also for DT-RADWT with a high enough Q-factor. We can use the DT-RADWT in lace of the STFT to obtain a time-scaled version of the inut. One motivation for relacing the STFT with a constant-q transform like the DT-RADWT is the constant-q behavior of human audio ercetion for freuencies above 500 Hz [22]. A articularly relevant model would therefore be a constant-q filter bank on this freuency range, which we can aroximate with an analytic wavelet

6 6 Freuency Freuency Original Time Time-Scaled by 3/2 Time Fig. 10. The time-freuency distributions of the original and the time-scaled signals, using the DT-RADWT, are similar as exected. Ideally, we would like the two scalograms to be the same uto a scaling along the time axis. transform 5. We also remark that for this articular method, an analytic transform like the DT-RADWT is more suitable than the RADWT since it facilitates comutation of the instantaneous freuency. The idea of using an analytic constant-q transform for audio time-scaling was in fact mentioned by Youngberg in [13]. However the imlementation in [13] reuires aroximations of continuous-time functions. On the other hand, the DT-RADWT is fully discrete and rovides exact reconstruction. An examle is shown in Fig. 10. The scalograms are seen to be similar, as exected from a successful time-scaling algorithm. An interesting and ositive side effect when we use the DT-RADWT for which we do not have an exlanation) is the lack of hasiness that is heard clearly for the same time-scaling factor when the hase vocoder is used. However, for seech, in some of our exeriments some distortion occurs with the DT-RADWT, which is not resent when the hase-vocoder is utilized. 6 V. CONCLUSION We roosed a dual-tree RADWT that extends the RADWT, similarly as the DT-CWT extends the dyadic DWT. The DT-RADWT allows us to choose the scaling factor of the wavelet frame and the Q-factors of the atoms. This in turn leads to a finer freuency resolution than the dyadic wavelet frames and facilitates the rocessing of signals which are known to ossess uasi-eriodic comonents. Therefore, using DT- RADWT, one can erform tasks that are tyically out of the scoe of DT-CWT. Also, unlike the STFT, it is a constant-q transform which can be a desired roerty for audio-rocessing and/or, signal rocessing based on scales. Here, we mainly focused on the construction and discussed an alication to demonstrate the otential of DT-RADWT. In fact, one disadvantage T 3T/2 of DT-RADWT, in comarison to an FFT imlementation of STFT, is the comutation time reuired. We hoe to address this issue in the near future. REFERENCES [1] İ. Bayram and I. W. Selesnick, Freuency-domain design of overcomlete rational dilation wavelet transforms, IEEE Trans. Signal Processing, vol. 57, no. 8, , Aug [2] L. Cohen, Time-Freuency Analysis. Prentice Hall, [3] J. C. Brown, Calculation of a constant Q sectral transform, J. Acoust. Soc. Amer., vol. 89, no. 1, , Jan [4] R. Yu, A new shift-invariance of discrete-time systems and its alication to discrete wavelet transform analysis, IEEE Trans. Signal Processing, vol. 57, no. 7, , Jul [5] I. W. Selesnick, R. G. Baraniuk, and N. G. Kingsbury, The dual-tree comlex wavelet transform - A coherent framework for multiscale signal and image rocessing, IEEE Signal Processing Magazine, vol. 22, no. 6, , Nov [6] A. V. Oenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing. Prentice-Hall, [7] I. W. Selesnick, Hilbert transform airs of wavelet bases, IEEE Signal Processing Letters, vol. 8, no. 6, , Jun [8] R. Yu and H. Özkaramanlı, Hilbert transform airs of orthogonal wavelet bases: Necessary and sufficient conditions, IEEE Trans. Signal Processing, vol. 53, no. 12, , Dec [9] N. G. Kingsbury, Comlex wavelets for shift invariant analysis and filtering of signals, J. of Al. and Com. Harm. Analysis, vol. 10, no. 3, , May [10] G. Gambardella, A contribution to the theory of short-time sectral analysis with nonuniform bandwidth filters, IEEE Trans. Circuit Theory, vol. 18, no. 4, , Jul [11] J. E. Youngberg and S. F. Boll, Constant-Q signal analysis and synthesis, in Proc. IEEE Int. Conf. on Acoustics, Seech and Signal Proc. ICASSP), [12] T. L. Petersen and S. F. Boll, Critical band analysis-synthesis, IEEE Trans. Acoust., Seech, and Signal Proc., vol. 31, no. 3, , Jun [13] J. E. Youngberg, Rate/Pitch modification using the constant-q transform, in Proc. IEEE Int. Conf. on Acoustics, Seech and Signal Proc. ICASSP), [14] J. C. Brown and M. S. Puckette, An efficient algorithm for the calculation of a constant Q transform, J. Acoust. Soc. Amer., vol. 92, no. 5, , Nov [15] A. V. Oenheim, D. H. Johnson, and K. Steiglitz, Comutation of sectra with uneual resolution using the fast Fourier transform, Proc. IEEE, vol. 59, no. 2, , Feb [16] A. Harma, M. Karjalainen, L. Savioja, V. Välimäki, U. K. Laine, and J. Huoaniemi, Freuency-wared signal rocessing for audio alications, J. Audio Eng. Soc., vol. 48, no. 11, , [17] G. Evangelista and S. Cavaliere, Freuency-wared filter banks and wavelet transforms: A discrete-time aroach via Laguerre exansion, IEEE Trans. Signal Processing, vol. 46, no. 10, , Oct [18] P. Auscher, Wavelet bases for L 2 R) with rational dilation factor, in Wavelets and Their Alications, M. B. R. et al., Ed. Jones and Barlett, Boston, [19] S. Mallat, A Wavelet Tour of Signal Processing. Academic Press, [20] M. Dolson, The hase vocoder: A tutorial, Comuter Music Journal, vol. 10, no. 4, , [21] J. Laroche and M. Dolson, Imroved hase vocoder time-scale modification of audio, IEEE Trans. Signal Processing, vol. 7, no. 3, , May [22] H. Fastl and E. Zwicker, Psychoacoustics: Facts and Models, Third ed. Sringer, The uality factor of these filters do not determine the dilation factor. It is in fact suggested that, for modelling the eriheral auditory system, one should consider the overla in freuency to be as high as ossible, leading to a dilation factor close to unity see [22], Matlab code for time-scaling using the DT-RADWT along with some examles is available at htt://web.itu.edu.tr/ibayram/dtradwt/