Frequency-Domain Design of Overcomplete. Rational-Dilation Wavelet Transforms

Transcription

1 Frequency-Domain Design of Overcomlete 1 Rational-Dilation Wavelet Transforms İlker Bayram and Ivan W. Selesnick Polytechnic Institute of New York University Brooklyn, NY 1121 ibayra1@students.oly.edu, selesi@oly.edu Tel: , Fax: Abstract The dyadic wavelet transform is an effective tool for rocessing iecewise smooth signals; however, its oor frequency resolution its low Q-factor limits its effectiveness for rocessing oscillatory signals like seech, EEG, and vibration measurements, etc. This aer develos a more flexible family of wavelet transforms for which the frequency resolution can be varied. The new wavelet transform can attain higher Q-factors desirable for rocessing oscillatory signals or the same low Q-factor of the dyadic wavelet transform. The new wavelet transform is modestly overcomlete and based on rational dilations. Like the dyadic wavelet transform, it is an easily invertible constant-q discrete transform imlemented using iterated filter banks and can likewise be associated with a wavelet frame for L 2R. The wavelet can be made to resemble a Gabor function and can hence have good concentration in the timefrequency lane. The construction of the new wavelet transform deends on the judicious use of both the transform s redundancy and the flexibility allowed by frequency-domain filter design. I. INTRODUCTION The dyadic wavelet transform WT is an easily-invertible constant-q transform that is very effective for the sarse reresentation of iecewise smooth signals, like a scan-line from a tyical hotograhic image [1]. This roerty has made the dyadic WT a oular tool for several signal and image rocessing alications coding, denoising, deblurring, sharening, etc. However, the dyadic WT is less effective for rocessing signals of more oscillatory nature, like seech, EEG, and hysical vibration measurements, etc. Such signals are quasi-eriodic over short-time intervals; and when analyzing/filtering such signals one generally needs better frequency resolution than that rovided by the dyadic WT. Indeed, the dyadic WT has a very low Q-factor 1 and oor frequency resolution, as illustrated in Fig. 1. Other transforms, like the short-time Fourier transform, cosine modulated filter banks, and wavelet ackets, are often used for oscillatory-tye signals instead of the dyadic WT. But these transforms are not constant-q. This aer develos a family of wavelet constant-q transforms attaining a range of Q-factors; in 1 The Q-factor of a band-ass filter is the ratio of its center frequency to its bandwidth.

2 2 6 DYADIC WAVELET TRANSFORM 4 2 π/4 π/2 3π/4 π ω a Hω 2 Hω 2... Gω 2 Gω 2 b Fig. 1. imlementation. The critically-samled dyadic discrete wavelet transform has a low Q-factor. a Frequency decomosition and b iterated filter bank articular, comared to the dyadic WT, the Q-factor can be made higher and the frequency resolution can be made finer. The roosed WT can also attain the same low Q-factor as the dyadic WT. For many signal rocessing alications, overcomlete wavelet transforms outerform critically-samled wavelet transforms. Overcomlete WTs or frames exand an N-oint signal to a set of M-coefficients with M > N [21]. Several overcomlete invertible WTs are available, like the undecimated WT UWT, the dual-tree comlex WT CWT, the double-density WT, and others [13], [18], [28], [29]. However, many of these WTs 2 attain overcomleteness by increasing only the temoral samling in some or all frequency bands. But, in order to better utilize the redundancy of an overcomlete WT it can be beneficial to also reduce the frequency sacing between adjacent frequency bands. Additionally, most of the existing overcomlete WTs achieve either a limited range of redundancy factors M/N or can be much more overcomlete than is necessary or ractical. The family of wavelet transforms develoed in this aer attain over-comleteness by increasing samling in both time and frequency. Moreover, the roosed family of WTs rovides a rich range of redundancy factors. Based on the engineering literature, most current discrete wavelet transform imlementations are based on FIR filters. Indeed, much of the develoment of wavelet-based signal rocessing algorithms grew from Daubechies construction of FIR-based orthonormal wavelet bases [15]. However, the FFT-based imlementation of filter banks and WTs offers greater design flexibility; and this flexibility has been exloited in the develoment of certain WTs so as to attain attributes not ossible with FIR filters [17], [23], [33]. Likewise, the family of WTs introduced 2 We exclude the continuous integral wavelet transform because we restrict our interest to fully-discrete transforms that are easily and stably inverted.

3 3 in this aer is based on a frequency-domain design the filters do not have rational transfer functions and our imlementation is based on the FFT. We note that although the filters are not FIR, the time-domain filter resonse decays raidly and the wavelets are well localized in time and frequency. The family of overcomlete wavelet transforms introduced in this aer is based on rational non-dyadic dilations. This WT is develoed for discrete-time data, is aroximately shift-invariant, and is easily invertible in fact, selfinverting in the sense of [3] the frame is a tight frame. By using a dilation factor close to one, one obtains a WT where the wavelet is gradually dilated from scale to scale. In addition, the introduced WT is flexible a range of Q-factors and redundancy factors can be attained. Most revious research on wavelet transforms with rational dilations consider only the critically-samled case [5] [7], [11], [22]. An excetion is our revious work [3] in which we develoed overcomlete rational-dilation filter banks and WTs with FIR filters. However, the family of rational-dilation WTs roosed here, designed in the frequency-domain and imlemented using the FFT, have several advantages over the WTs develoed in [3]. First, in [3] each frequency band or scale is reresented by a number of signals roduced by different band-ass filters; the signals need to be interlaced together to form the subband signal an inconvenience. Second, the construction in [3] involves olynomial matrix sectral factorization, which while being straightforward in rincile, is more involved than the frequency-domain aroach roosed here. Consequently, in [3] we were unable to roduce WTs with good frequency resolution high Q-factor, even though that was one of our motivations. Third, the aroach roosed in this aer has greater flexibility; in articular, the redundancy factor of the transform is not comletely determined by the dilation factor as it is in [3]. The close relationshi, between invertible wavelet transforms for discrete data and multiresolution anaylsis MRA on the real line, is well recognized and areciated. For the dyadic case, self-inverting FIR FBs are related to comactly-suorted tight wavelet frames on the real line. Unfortunately for rational-dilation WTs, no MRA with rational dilation factors can be constructed based on self-inverting FIR FBs [6]; that is, there is no unique well defined wavelet on the real line associated with discrete rational-dilation wavelet transforms imlemented using FIR filters. However, in this aer we will show that a tight rational-dilation wavelet frame on the real line can be constructed using the roosed bandlimited filters 3 using a single wavelet. This is an extension of the construction by Auscher [1] of an orthonormal MRA with q wavelets. A. Samling the Time-Frequency Plane The manner in which the analysis/synthesis functions cover the time-frequency T-F lane, as illustrated in Fig. 2, rovides a useful way to understand the relationshis among various wavelet transforms. For examle, the dyadic WT, realized using the filter bank illustrated in Fig. 1b, can be understood to samle the T-F lane with {A = 2, B = 2} in Fig. 2. In contrast, the critically-samled rational-dilation WT, realized using the iterated FB in Fig. 3a, sets {A = q/, B = q/q }. 3 For the filter bank in Fig. 4, we will show that erfect reconstruction PR cannot be achieved using filters with rational transfer functions.

4 4 freq. f B f/a A B f/a 2 f/a 3 f/a 4 A 2 B A 3 B A 4 B time Fig. 2. The time-frequency samling lattice of a discrete wavelet transform imlemented via an iterated two-channel filter bank, as in Figs. 1 and 3. The arameters A and B are determined by the samling factor of the low-ass branch and the redundancy of the filter bank resectively. Overcomlete wavelet transforms samle the T-F lane more densely. For examle, the double-density WT [28] doubles the temoral density of the dyadic WT and maintains its frequency sacing, leading to {A = 2, B = 1} in Fig. 2. In this aer, we roose a family of WTs realized using the iterated FB in Fig. 3b, which sets {A = q/, B = s}. Our revious work on overcomlete rational-dilation wavelet transforms [3] sets {A = q/, B = 1} and is therefore less flexible than the WT roosed here s = 1 in [3]. In articular, the roosed WT can samle the T-F lane more coarsely than the WT of [3] and it can therefore be less redundant. Indeed, the roosed WT is redundant by /s q, whereas the WT of [3] is redundant by /q. The latter redundancy can be large even for modest rational samling factors. For examle, with = 7, q = 8, which is relevant for classification of audio signals [11], the WT of [3] is redundant by a factor of 7. Such a high redundancy might be higher than is desired or necessary. Using s = 3 reduces the redundancy to 7/3. In addition to affecting the redundancy, s, in the filter bank of Fig. 3b, also affects the Q-factor and the timebandwidth roduct. We will demonstrate that, for a given rational dilation factor, q/, there is a trade-off between the Q-factor and the time-bandwidth roduct. B. Prior Constant-Q Transforms Numerous methods have been roosed to obtain constant-q transforms with high Q-factors see [8] and the references therein. For examle, Oenheim et al. describe a method that mas a discrete sequence to another whose discrete-time Fourier transform DTFT is related to the former via a frequency waring, thereby allowing one to obtain an unequally saced frequency samling [27]. In this direction, Smith and Abel [31] use a bilinear ma as the waring function to aroximate the Bark frequency waring. Makur and Mitra discuss in [24] the wared DFT which samles the unit circle on oints that are not equally saced as required by the conventional DFT. In this regard, the constant-q transform introduced by Brown in [8] may be interreted even though Brown s

5 5 Hω q Hω q... Gω q Gω q a Hω q Hω q... Gω s Gω s b Fig. 3. Rational-dilation discrete wavelet transforms. a The critically-samled rational WT with = q, b The overcomlete rational WT investigated in this aer. The dilation factor is q/. work recedes [24] as a short-time Fourier transform that uses a sliding window with the wared DFT instead of the conventional DFT also see [9] for an imlementation of the inverse transform. A second aroach to obtaining constant-q transforms with variable Q-factor is to aroximate the T-F lattice with simler building blocks. For instance, Diniz et al. [16] rovide an efficient imlementation of a transform called the bounded-q fast filter bank, insired by the bounded-q transform introduced by Mont-Reynaud [26]. The bounded-q transform first slits the sectrum into geometrically distributed frequency bins and then slits each of these bins uniformly, achieving a iecewise linear frequency decomosition. In a similar vein, Karmakar et al. [2] roose a criterion to select otimal wavelet acket trees to aroximate the Bark scale. The rational-dilation wavelet transform roosed in this aer rovides an alternative, with attractive roerties, to the methods above that set out to achieve a constant-q analysis with a high Q-factor. First, it is a tight frame, hence the inverse is given by the transose of the forward transform and can be imlemented as efficiently, a roerty lacked by the first grou of transforms. Second, it is a true constant-q transform, in contrast to the methods in the second grou. Therefore, we believe that the introduced rational-dilation WT may be a useful addition to the inventory of transforms designed for the task of constant-q analysis. C. Notation and Conventions We denote discrete-time sequences by lower case letters as in an with n Z. The discrete-time Fourier transform DTFT of an is denoted by Aω and is given by Aω = n an ex jωn. Note that Aω is 2π eriodic in ω. Unless otherwise stated, we assume all discrete-time signals are real valued, so that secifying a DTFT on ω [, π] will determine the DTFT on ω [ π, π]. Therefore, we restrict our attention to ω [, π] so as to simlify some of the notation. The Fourier transform of a function ft, with t R, is given by ˆfω = ft e jωt dt.

6 6 Hω q q H ω + Gω s s G ω Fig. 4. Analysis and synthesis filter banks for the imlementation of the roosed rational-dilation wavelet transform. The dilation factor is q/, a rational number. D. Organization In Section II, we derive the erfect reconstruction conditions and roose a secific set of erfect reconstruction filters. In Sections III and IV, we show that for the roosed filters, the rate changer and related iterated filter banks can be written so as to simlify the analysis of their behavior. In Section V we rovide examles. In Section VI we examine the Q-factor and time-bandwidth roduct of the roosed WT. In Section VII, we resent another interretation of the rational rate changer and discuss the underlying wavelet frame for L 2 R. II. PERFECT RECONSTRUCTION CONDITIONS In this section we derive the erfect reconstruction conditions for the filter bank in Fig. 4. This FB is the basic element of the roosed wavelet transform. The arameters, q, and s are ositive integers satisfying 1 < q and /q + 1/s 1. We also assume and q are corime. We will show that PR cannot be attained using filters with rational transfer functions. Hence our filter bank imlementation will be based on the FFT. For this reason, we formulate the PR conditions in the frequency domain. Let us consider the system in Fig. 5. We have, Uω = Xaω, 1 V ω = 1 b [ F ω Uω + F ω + 2π b U ω + 2π b + + F ω + b 1 2π b U ω + b 1 2π b ] F ω, 2 and Y ω = 1 a [ ω ω V + V a a + 2π ω ] + + V a a + a 12π. 3 a Noting that ω U a + k 2π b + n2π a = X ω + a k 2π b, 4 we obtain b 1 Y ω = C k ω X ω + a k 2π b k= 5

7 7 Xω a Uω F ω b b F ω V ω a Y ω Fig. 5. One channel of the rational filter bank in Fig. 4. where C k ω = 1 ab [ ω F a + k 2π b We can now write the outut of the FB in Fig. 4 as, q 1 Y ω = L k ω X ω + k 2π q k= F ω ω + F a a + k 2π b + 2π F ω a s 1 + k= a + 2π a ω +F a + k 2π b + a 12π a + F ω ] a + a 12π. 6 a M k ω X ω + k 2π, 7 s where L k ω = 1 1 ω H q + k 2π q + n2π H ω + n2π, 8 n= M k ω = 1 [ G ω + k 2π ] G ω. 9 s s Proosition 1: For the FB in Fig. 4, there exist filters with non-zero rational transfer functions achieving PR if and only if + 1 = q = s. Proof: See Aendix A. Notice that + 1 = q = s imlies the FB is orthonormal. Therefore, in order to obtain erfect reconstruction filters for the overcomlete case, we must use filters which do not have rational transfer functions. We remark that if, L k ω = for k = 1, 2,..., q 1, 1 M k ω = for k = 1, 2,..., s 1, 11 L ω + M ω = 1, 12 then PR is granted 4. A. Proosed erfect reconstruction filters In this section we roose a low-ass filter Hω and a high-ass filter Gω satisfying the erfect reconstruction conditions derived above. The two filters will have ideal ass-bands and ideal sto-bands; however, the filters will not be ideal brick-wall filters because of the resence of transition bands. In fact, the transition bands are imortant because a brick-wall filter will have a sinc-tye time-domain resonse with oor decay, which we wish to avoid. 4 It can be shown that these conditons are also necessary if q and s are corime.

8 8 If we let, Hω =, for ω [ π q, π ], 13 then it can be shown that, L k ω =, k = 1, 2,..., q 1 14 L ω = 1 ω 2 H for ω [, π]. 15 q Similarly, if then, Gω =, ω [, 1 1 ] π, 16 s M k ω =, k = 1, 2,..., s 1 17 M ω = 1 s G ω 2 for ω [, π]. 18 In other words, with the choices 13 and 16, the only condition for PR is, by 12, 1 ω 2 1 H + q s G ω 2 = 1 for ω [, π]. 19 Equations 13 and 19 imly that Gω = [ ] s for ω q π, π. 2 Similarly, 16 and 19 imly Hω = q for ω [, 1 1 π ]. 21 s Notice that 13 and 21 determine Hω on all ω excet [1 1/sπ/, π/q]. Likewise, 16 and 2 determine Gω on all ω excet [1 1/sπ, π/q]. These will be the transition bands of Hω and Gω. From 19, the transition band of Gω is comletely determined by the transition band of Hω. These constraints on the filters are deicted in Fig. 6a for reference. It is easy to check that, rovided q + 1 s > 1, the transition bandwidths in Fig. 6 are ositive. That is, rovided the FB is overcomlete, Hω and Gω really do have transition bands. The width of the transition band will be imortant because it will influence the decay of hn and gn and of the analysis/synthesis functions of the designed wavelet transform. Even though Fig. 6 defines transition bands for the filters Hω and Gω, with different widths, we remark that for the low-ass branch, the filter Hω acts not on the inut signal but on the u-samled inut signal. In Section IV, we rewrite the low-ass branch so as to clarify the filtering oeration. 22

9 9 q Hω Gω s s 1π π s 1π q s s q π ω a π 6 Hω Gω 2!/4!/3!/2 2!/3! " b = 2, q = 3, s = 2. Fig. 6. a Proerties of roosed erfect reconstruction filters for the filter bank in Fig. 4. b Proosed filters with dilation factor q/ = 3/2 and s = 2. B. Transition Bands The transition band of Hω should be carefully secified so that both hn and gn are reasonably well time-localized, esecially because the roosed frequency resonses Hω and Gω have ideal sto-bands and ass-bands. On its transition band, [1 1/sπ/, π/q], the frequency resonse Hω can be set arbitrarily, as long as Hω q. We define the transition function θω on ω [, π] with θω 1, and set Hω = ω a [ q θ for ω 1 1 π b s, π ], 23 q where a and b are such that ω a/b mas [, π] to the transition band of Hω. Secifically, a = 1 1 π s, b = 1 q 1 1 s 1. We also define the comlementary transition function θ c ω as θ c ω := 1 θ 2 ω Then, the roosed PR filters can be written as [ q ω, 1 1 π s, Hω = q θ ω a b ω [ 1 1 π s, π q, ω [ π q, π], ω [, 1 1 s π, Gω = s ω a θc b ω [ 1 1 s π, q π, [ s ω q π, π].

10 1 Furthermore, we roose for the transition function, θω, that the Daubechies filter [15] with two vanishing moments be used to be recise, the DTFT magnitude thereof, restricted to [, π] and normalized so that θω 1. Secifically, we set θω = cosω 2 cosω for ω [, π]. 26 We make this choice for θω because then the transition bands of Hω and Gω will have the same behavior, namely for θω in 26 we have θ c ω = θπ ω. Figure 6b illustrates Hω and Gω with this transition function for = 2, q = 3, and s = 2. Using for θω a Daubechies filter having a higher number of vanishing moments will increase the differentiability of Hω. However, if the number of vanishing moments is large, then the frequency resonse Hω will be closer to a brick wall and the time-domain resonse hn will be closer to a sinc function which has oor slow decay. Because we wish that the analysis/synthesis functions of the constructed wavelet transform have good timefrequency localization roerties, we use a Daubechies filter with a small number of vanishing moments. Note that the roosed construction need not use a Daubechies filter any set of erfect reconstruction filters could be used in the construction. FFT filter bank imlementation. Our imlementation of the roosed rational-dilation filter bank for finite-length inut signals is based on the FFT. Secifically, our imlementation consists of using the FFT to imlement circular convolution for low-ass and high-ass filtering. Our imlementation rovides erfect reconstruction for signals of any length. For erfect reconstruction of the wavelet transform, the length of the signal at each level should be a multile of q and of s; that is, the signal length should be a multile of the least common multile of q and s, denoted lcmq, s. This condition can be understood as follows: If a discrete-time eriodic signal with eriod N is down-samled by q, the resulting signal will be eriodic with eriod N/q only if N is divisible by q. Because the two-channel analysis filter bank contains down-samlers by both q and s, the inut signal should be divisible by both q and s. Otherwise, circular convolution filtering does not readily lead to the erfect reconstruction roerty. In case the signal length is not a multile of lcmq, s, we zero-ad the signal to the next multile of lcmq, s. We erform the necessary zero-adding at each level of the wavelet transform. As a consequence of the zeroadding, each subband signal may be a few samles longer than the minimum; however, the transform is already overcomlete so the additional few samles caused by zero-adding at each level will usually be negligible. As a second consequence of the zero-adding, the length of the reconstructed signal may be a few samles longer than the original signal but these additional signal values will be equal to zero in the absence of wavelet-domain rocessing. Instead of zero-adding at each stage of the wavelet transform we could zero-ad the original signal. However, zero-adding the original signal instead of level-by-level would require zero-adding to the next multile of lcmq J, s where J is the number of levels. This can lead to much more zero-adding than level-by-level zeroadding. Avoiding that length extension is esecially imortant for the high Q-factor case because a high Q-factor

11 11 2 Hω 3 a 2 H2ω Rω 3 b Hω 2 Rω 3 c Fig. 7. Provided that Hω is bandlimited to π/3, and Hω and Rω are defined as in 28 and 29, the three systems are equivalent. transform will generally have more levels large J and q will be greater in comarison to a low Q-factor transform leading to excessive zero-adding. Our imlementation avoids this. On the other had, our imlementation can aggravate the artificial discontinuity introduced by eriodic boundary extensions. We assume the signal is sufficiently longer than the wavelet at each level so that eriodic boundary artifacts are minor. Although the described FFT-imlementation may be comutationally subotimal for some lengths, it does rovide erfect reconstruction without requiring the signal be of a secial length. We have used this imlementation for seech and EEG signals longer than 1, samles. III. REWRITING THE FILTER BANK In order to make more exlicit the filtering oeration the low-ass channel erforms on the inut signal, in this section we rewrite the filter bank in Fig. 4 so that the filtering recedes the u-samler. In general, the order of the u-samler and the low-ass filter Hω in Fig. 4 can not be exchanged. However, for the roosed low-ass filter Hω in Section II-A, the exchange of the u-samler and low-ass filter is ossible because the roosed low-ass filter is aroriately bandlimited. Rewriting the filter bank in Fig. 4 so that the filter acts directly on the inut signal instead of on the u-samled inut signal clarifies the behavior of the filter bank. For, when a -fold u-samling recedes the filter Hω as in Fig. 4, the inut signal is convolved not with the imulse resonse hn, but with each of the down-samled sub-sequences hqn + k for k q 1. Therefore, each sub-sequence should be assessed its frequency resonse, frequency-resolution, time-domain ringing, time-frequency localization, etc. The band-limited roerty of the roosed low-ass filter Hω not only simlifies the erfect reconstruction equations, it also facilitates the assessment of the designed filter bank. Consider the rate changer in Fig. 7a. For clarity, we set = 2 and q = 3. Suose that Hω is bandlimited to π/3, Hω = for ω [π/3, π], 27

12 12 Hω π 2π 3 π π 3 3 2π 3 π Hω π 2π 3 π π 3 3 2π 3 π H2ω π 2π 3 π π 3 3 2π 3 π Rω π 2π 3 π π 3 3 2π 3 π Fig. 8. The frequency resonse Hω may be written as H2ω Rω, if it is bandlimited to π/3. as illustrated in Fig. 8. If we define Hω as a 2π-eriodic function, Hω := 1 ω 2 H for ω [, π], 28 2 and Rω as an ideal low-ass filter, 2 ω [, π/3], Rω := ω π/3, π]. then we can write, as illustrated in Fig. 8, Hω = H2ω Rω, 29 3 as deicted in Fig. 7b. Using noble identities, this system is equivalent to the one shown in Fig. 7c. Hence we can interret the system in Fig. 7a as filtering followed by an ideal rate changer, 5 rovided Hω is bandlimited to π/3. This filter bank identity is valid for general and q, rovided Hω is bandlimited to π/q. Figure 9 illustrates the general FB identity where Hω and R,q ω are defined as Hω = 1 ω H for ω [, π], 31 ω [, π/q], R,q ω = ω π/q, π] By an ideal rate changer we mean a -fold u-samler, followed by an ideal low-ass filter, followed by a q-fold down-samler. The cut-off frequency of the low-ass filter should be min{π/, π/q}. See for examle Eqn in [25].

13 13 Hω q a Ideal rate changer Hω R,q ω q b Fig. 9. If Hω is bandlimited to π/q, then the two systems are equivalent. The frequency resonses, Hω and R,qω, are defined in 31 and 32. Hω R,q ω q q R,q ω H ω + Gω s s G ω Fig. 1. An equivalent filter bank to the one in Fig. 4 rovided Hω is bandlimited to π/q. Because the filter Hω roosed in Section II-B is bandlimited to π/q, we can aly the identity to the rationaldilation filter bank in Fig. 4. We obtain the FB in Fig. 1. The filter Hω is bandlimited to π/q. The system following Hω is an ideal rate changer. Rewriting the filter bank in Fig. 4 as in Fig. 1 reveals the role of the low-ass filter in the system. Notice that, because the low-ass filter Hω in Fig. 1 band-limits the inut signal to π/q, the ideal rate changer causes no aliasing. This is in contrast to a critically-samled filter bank. It is the filter Hω that acts directly on the inut signal. Thus it is the transition band of Hω and not that of Hω, that lays a direct art in determining the analysis/synthesis functions of the imlemented DWT. In fact, if we relace Hω with Hω in Fig. 6, then the transition bands coincide, as illustrated in Fig. 11. Note also, that for fixed and q, reducing s will widen the transition band, which in turn will lead to a faster decay of the time-domain resonse hn. The influence of s will be illustrated by examles in Section V. IV. ITERATED FILTER BANKS In Section III, we rewrote the filter bank in Fig. 4 as the filter bank in Fig. 1 so that the low-ass filter acts directly on the inut signal instead of on an u-samled version thereof. In this section, we will similarly rewrite the iterated filter bank. First, we consider the iterated rate changer with a generic low-ass filter. Second, we assume the filters are bandlimited like the filters roosed in Section II-B. Consider the rate changer iterated for j stages, as illustrated in Fig. 12a. It can be shown using noble identities [32] that this system is equivalent to the one in Fig. 12b where H j ω = j 1 k= H q j 1 k k ω. 33 In using this formula to obtain H j ω, one must kee in mind that Hω is eriodic by 2π.

14 14 q Hω Gω s s 1π s q π ω π = 2, q = 3, s = 2 3/2 Hω Gω 2!/2 2!/3! " Fig. 11. s = 2. a Proerties of the filters Hω and Gω. The transition bands are the same. b Proosed filters with dilation factor 3/2 and j stages Hω q Hω q... Hω q a j H j ω q j b H j ω j R j,q j ω q j c Fig. 12. The systems in a and b are equivalent. If Hω is bandlimited to π/q then all three systems are equivalent. To further simlify the system in Fig. 12b, we wish to rewrite it as in Fig. 7c; namely, we wish to exchange the order of the u-samler and the low-ass filter. Provided Hω is bandlimited to π/q we have such a result in the following roosition. The roosition is exected because Hω is bandlimited to the Nyquist rate for the rate changer. Proosition 2: Suose Hω is bandlimited to π/q. Then, H j ω as defined in 33 is bandlimited to π/q j. More recisely, we have j 1 k= H j ω = H q j 1 k k ω ω [, π/q j ], 34 ω π/q j, π].

15 15 Proof: See Aendix B. We remark that 34 is not the same as 33. In general, 33 does not ensure that H j ω is bandlimited, whereas 34 does. Defining H j ω = 1 ω j H j j for ω [, π], 35 R j,q j ω = j ω [, π/q j ], 36 ω π/q j, π] and using the filter bank identity in Fig. 9, we have that all three systems in Fig. 12 are equivalent. The filter H j ω is bandlimited to /q j π. We now consider a band-ass channel of the iterated filter bank, as illustrated in Fig. 13a. In this case, if we define G j+1 ω = H j ω Gq j ω, 37 then the two systems in Fig. 13a and 13b are equivalent using, again, the noble identities. In addition, if Hω is bandlimited to π/q, then by Pro. 2, H j ω is bandlimited to π/q j, and in turn, G j+1 ω is also bandlimited to π/q j. Therefore, H j ω Gq j ω ω [, π/q j ], G j+1 ω = ω π/q j, π]. Using the filter bank identity illustrated in Fig. 9, we define Ḡj+1ω as Ḡ j+1 ω = 1 ω j G j+1 j for ω [, π], 39 and we rewrite the system in Fig. 13b as Fig. 13c. All three systems in Fig. 13 are equivalent. The filter Ḡj+1ω in Fig. 13c acts directly on the inut signal instead of an u-samled version thereof. The filtered signal is then rocessed with an ideal rate changer, and subsequently down-samled by s. The wavelet. For the rational-dilation wavelet transform, we define the wavelet in the frequency-domain as jω ˆψω = lim j j/2ḡj. 4 q q We will show in Section VII that secific shifts and dilations of the wavelet ψt yield a tight frame for L 2 R. Redundancy factor. The redundancy of the overcomlete rational-dilation wavelet transform is found as follows. We first define the redundancy factor of the iterated FB with j stages, Red j, q, s, as the number of analysis FB coefficients er inut samle, which can be shown to be Red j, q, s = 1 1 /q j+1 j s 1 /q q The redundancy of the wavelet transform iterated FB is given by Red, q, s = lim j Red j, q, s = 1 s /q. 42

16 16 j H j ω q j Gω s a j G j+1 ω q j s b Ḡ j+1 ω j R j,q j ω q j s c Fig. 13. The systems in a and b are equivalent. If Hω is bandlimited to π/q then all three systems are equivalent. V. EXAMPLES This section resents four examles to illustrate the frequency resonses of the iterated filter bank, the wavelet, and the effect of the arameters, q, and s thereon. Each examle is based on the iterated filter bank in Fig. 4 and the filters roosed in II-B. The transition function θω is given by 26. Examle 1: In this examle, we set = 2, q = 3, s = 2. The dilation factor is 3/2 and the redundancy is 3/2. Iterating the FB yields the frequency resonses Ḡjω shown in Fig. 14. Note that each frequency resonse is exactly constant over art of its ass-band i.e. has a flat to. The wavelet ψt and its Fourier transform ˆψω are also shown in the figure. The wavelet somewhat resembles a sinc wavelet; however, it decays more raidly. Examle 2: We set = 2, q = 3, s = 1. The dilation factor is 3/2 and the redundancy is 3. Iterating the FB yields the frequency resonses shown in Fig. 15. Comared with Examle 1, the frequency resonses are less frequency selective. In addition, the band-ass filters are not exactly constant over any art of the ass-band excet for the high-ass filter at stage 1, which is constant for 3π/4 ω π. The wavelet ψt and its Fourier transform ˆψω are also shown in the figure. Comared with Examle 1, the wavelet has fewer oscillations and much less ringing. In fact, the wavelet closely resembles the Mexican hat function the second derivative of the Gaussian function. This examle mirrors the examle in [3] with 2 vanishing moments; in that examle the redundancy was also 3 and the analysis/synthesis functions also resembled the Mexican hat function. For the FIR solutions in [3] there is no uniquely defined wavelet, so we refer instead to the discrete-time analysis/synthesis functions. Examle 3: We set = 7, q = 8, s = 5. The dilation factor is 8/ and the redundancy is 8/5 = 1.6. The iterated frequency resonses, the wavelet ψt and its Fourier transform ˆψω are shown in Fig. 16. Comared with Examles 1 and 2, the Q-factor is higher; there are more band-ass filters covering the same frequency range. Like Examle 1, each frequency resonse is exactly constant over art of its ass-band. The wavelet has more oscillations than the wavelets of Examles 1 and 2; accordingly, this wavelet has a higher Q-factor. Also, note that this wavelet has some ringing; that is, there are several oscillations resent outside the rimary oscillatory interval.

17 17 4 P = 2, Q = 3, S = 2, DILATION = 1.5, REDUNDANCY = 1.5 P = 2, Q = 3, S = 1, DILATION = 1.5, REDUNDANCY = ω / π.5.5 WAVELET t FOURIER TRANSFORM OF WAVELET 3/4 3/2 2 ω / π ω / π.4.2 WAVELET t FOURIER TRANSFORM OF WAVELET 3/2 2 ω / π Fig. 14. Examle 1: The iterated frequency resonses, the wavelet ψt, and its Fourier transform ˆψω. Fig. 15. function. Examle 2: The wavelet resembles the Mexican hat P = 7, Q = 8, S = 5, DILATION = 1.14, REDUNDANCY = 1.6 P = 7, Q = 8, S = 3, DILATION = 1.14, REDUNDANCY = ω / π WAVELET t 2 1 FOURIER TRANSFORM OF WAVELET 32/35 8/7 2 ω / π ω / π t WAVELET FOURIER TRANSFORM OF WAVELET 16/21 8/7 2 ω / π Fig. 16. Examles 1 and 2. Examle 3: The wavelet has a higher Q-factor than Fig. 17. Examle 4: The wavelet has a higher Q-factor than Examles 1 and 2, and less ringing than Examle 3.

18 18 Examle 4: We set = 7, q = 8, s = 3. The dilation factor is 8/ and the redundancy is 8/ The iterated frequency resonses, the wavelet ψt and its Fourier transform ˆψω are shown in Fig. 17. Like Examle 2, the band-ass filters do not have flat tos; their shae is more Gaussian. This is because the smaller value of s, comared with Examle 3, leads to Hω and Gω having wider transitions bands. The wavelet does not have the ringing behavior resent in Examle 3. The wavelet resembles a cosine function multilied by a Gaussian function; that is, a Gabor function. The aroximate Gaussian shae is resent in both the time-domain and frequency-domain. Accordingly, the wavelet in this examle is better localized in the time-frequency lane the Gabor function being otimally localized than Examle 3. Comared to Examles 1 and 2, the Q-factor is higher the wavelet has more oscillations than the wavelets of Examles 1 and 2. Note that these roerties good time-frequency localization, resemblance of the wavelet to a Gabor function, absence of ringing, and high Q-factor are attained with a modest factor of redundancy less than 3. Note that for both dilations factors, q/ = 3/2 and q/ = 8/7, increasing the redundancy decreasing s reduced the ringing resent in the wavelet and reduced the Q-factor. Finding the best arameters, q, and s so at so obtain aroximately a desired Q-factor and desired absence of ringing, may require insecting the wavelet generated by several different sets of arameter. A. The Quality-Factor VI. QUALITY-FACTOR AND TIME-BANDWIDTH PRODUCT One motivation for develoing a rational-dilation wavelet transform is the ability to achieve a high Q-factor fullydiscrete self-inverting wavelet transform. Examle 4 in Section V illustrates that for suitably chosen arameters, q, and s the roosed transform can achieve high Q-factors with good time-frequency localization. In this section, we examine the Q-factor as a function of the arameters. To find the Q-factor of the roosed wavelet transform we find the Q-factor of the band-ass filters Ḡjω. We will use the filters Hω and Gω roosed in Section II-B and the formulas develoed in Section IV. For Hω and Gω, bandlimited as in 13 and 16, it can be shown using 34, 38, and 39 that the frequency suort of Ḡjω is [ j 1 su{ḡjω} = In addition, if 1 1 s > q 2, q j ] π, j 1 s q j 1 π, j > then Ḡjω will be exactly constant over art of its ass-band it will have a flat to as in Examles 1 and 3. When 44 is satisfied i.e. when Ḡjω has a flat to, then the two transition bands of Ḡjω are [ j 2 q j j 1 ] [ j 1 π, s q j 1 π and q j j ] π, s q j π for j > 1. If the transition function θω is chosen such that θω 2 is half-band for examle 26 then the assband edges, defined by the half-ower frequencies, are the midoints of the transition bands 45. The resonant 44 45

19 19 frequency of Ḡjω, denoted RF j and defined as the geometric mean of the left and right ass-band edges, is given by RF j = j 2 1 q j s + π. 46 q q Using 45, the bandwidth of Ḡjω is given by BW j = j 2 1 q j s + 1 π. 47 q q The Q-factor, given by the ratio of the resonant frequency to the bandwidth, is Q j = RF j 1 =, j > BW j q 1 /q Since Q j is a constant, indeendent of the stage index j, the iterated FB is a constant-q transform excluding the first stage. The Q-factor formula 48 is valid only when 44 is satisfied, ie. when Ḡjω has a flat to, as in Examles 1 and 3. When the band-ass filters do not have flat tos, it is difficult to find a formula for the Q-factor and we calculate the Q-factor numerically. Note that the formulas 42 and 44 imly that Ḡ j ω will never have a flat to when the redundancy is greater than two. If one wishes that the wavelet be well localized in time-frequency and relatively free of ringing, then one should select the arameters, q and s so that the redundancy is great than two in order to avoid the flat-to behaviour. Figure 18 illustrates the Q-factor for numerous sets, q, s. In the figure, each set of connected oints corresonds to a single, q air and reresents the Q-factor as a function of redundancy 42. The arameters, q, s used to roduce Examles 1-4 in Section V are indicated in the figure. The figure indicates that increasing the redundancy by reducing s has the effect of reducing the Q-factor, a behaviour reflected in Examles 1-4. B. Time-Frequency Localization For the roosed wavelet transform, Section VI-A examined the frequency resolution as a function of the arameters, q, and s, by comuting the Q-factor for numerous sets, q, s. We can also examine the wavelets time-frequency localization as a function of the arameters by comuting the time-bandwidth roduct [14]. Figure 19 illustrates the time-bandwidth roduct, σ t σ ω, for numerous sets, q, s. The frequency variance σω 2 is comuted using only the ositive ω-axis because the wavelet ψt, being a real-valued band-ass function, has a two-sided sectrum. We comute the time-bandwidth roduct numerically. As Fig. 19 illustrates, as we increase the redundancy decrease s the time-bandwidth decreases and aroaches its otimal value of 1/2. Note σ t σ ω 1/2. The examles in Section V reflect the behavior illustrated in Fig. 19. Examles 1 and 2 have the same dilation factor of 1.5, with Examle 2 having the higher redundancy lower s. Accordingly, Examle 2 has the lower Q-factor and the smaller time-bandwidth roduct; in fact, its time-bandwidth roduct is ractically otimal.

20 2 QUALITY FACTOR q / 1 / 9 19 / 17 9 / 8 17 / 15 8 / 7 15 / 13 7 / 6 13 / 11 6 / 5 11 / 9 5 / 4 9 / 7 4 / 3 7 / 5 3 / 2 5 / 3 2 / 1 s = 5 s = 4 s = 3 s = 2 EXAMPLE 1: = 2, q = 3, s = 2 EXAMPLE 2: = 2, q = 3, s = 1 EXAMPLE 3: = 7, q = 8, s = 5 EXAMPLE 4: = 7, q = 8, s = REDUNDANCY s = 1 TIME BANDWIDTH PRODUCT q / 1 / 9 9 / 8 8 / 7 7 / 6 6 / 5 5 / 4 4 / 3 3 / 2 2 / 1 q / = DILATION FACTOR s = 5 s = 4 s = 3 s = 2 EXAMPLE 1: = 2, q = 3, s = 2 EXAMPLE 2: = 2, q = 3, s = 1 EXAMPLE 3: = 7, q = 8, s = 5 EXAMPLE 4: = 7, q = 8, s = REDUNDANCY s = 1 Fig. 18. The quality-factor of the roosed wavelet transform for various arameter sets, q, s. Given a and q, the Q- factor decreases as the redundancy increases. Examles 1-4 are highlighted in the figure. Fig. 19. The time-bandwidth roduct of the wavelet ψt for various arameter sets, q, s. Given and q, the time-bandwidth roduct decreases as the redundancy increases. The dashed line at.5 is the lower bound for the time-bandwidth roduct. Examles 1-4 are highlighted in the figure. VII. REINTERPRETING THE RATE CHANGER In this section, we rovide another interretation for the system comrised of an u-samler, filter and a downsamler. This is slightly more general than the interretation in Section III and allows us to define fractional down-samlers, leading to an understanding of rational FBs arallel to integer down-samled FBs. This in turn roves useful for deriving the underlying wavelet frame for L 2 R and understanding its relation with the introduced rational DWT, similar to the relation between the integer-dilation wavelet frames and integer down-samled DWTs. In Fig.2a, the outut signal values are the inner roducts of the inut signal values with the filter s imulse resonse values. Secifically, we can write, y 1 n = x 1 k, f 1 qn k. 49 Notice that f 1 n + q for q = 1, 2,..., 1 give the olyhase comonents of f 1 n. We thus see that the outut of the system in Fig. 2a is obtained by samling the convolution of the inut with not the filter, but its olyhase comonents. In general, it is not easy to determine the relationshi among the olyhase comonents of the filter. However, the articular choice 13 leads to a certain relation that in turn will allow us to interret the low-ass analysis channel of the rational FB as a filtering followed by a fractional down-samling oeration, which we will describe now. Once again, to gain insight, let us start with the simle system shown in Fig. 21a. Defining the olyhase comonents of F z via, F z = F z 2 + z 3 F 1 z 2, 5

21 21 x 1 n F 1 ω q y 1 n a x 2 n F 2 ω r y 2 n b Fig. 2. The role of the filter in the first system in a is more imlicit comared to the one in b. 2 F ω 3 a F ω F 1 ω 3 2 b e jω F ω e j3ω/2 c 3 2 e jω F ω 3 2 d Fig. 21. The analysis channel. F z and F 1 z are olyhase comonents of F z. Setting F ω = F ω/2, the systems in a, b, c are equivalent. We define the fractional down-samler by 3/2 to be the system following F ω. This imlies that all the systems are equivalent. it can be shown that this system is equivalent to the one shown in Fig. 21b. Now, if we choose F ω such that F ω = for ω [π/2, π], and define F ω = 1/2 F ω/2 for ω [, π], then it turns out that F ω = F ω and F 1 ω = e j3ω/2 F ω. This establishes the equivalence of Fig. 21b and 21c. Now consider the system following F ω in Fig. 21c. Suose we aly this system to an inut xn, and call the outut yn. For integer n, the uer channel sets y2n = x3n. Interreting e j3ω/2 as an advance oerator by 3/2, we can likewise say that the lower channel sets y2n + 1 = x3n + 3/2. Combining these, we get, formally, yn = x3n/2. This observation motivates us to define a fractional down-samling oeration by 3/2. Using this definition, we thus say that the system in Fig. 21a is equivalent to the one in Fig. 21d. via For general samling factors, q, we have, Proosition 3: For, q corime, suose that the olyhase comonents of Hz, namely H k z, are defined Then, 1 Hz = z qk H k z. k= H k ω = ejkqω/ if and only if ω H for ω [, π] Hω = for ω [π/, π]. 53

22 22 q := q + e j q ω q + e jω e j2 q ω q + e jω π π q.. b e j 1 q ω q e jω a Fig. 22. Two equivalent definitions for a fractional down-samler by q/. Hω q Hω// q Fig. 23. The two systems are equal if and only if 53 holds. Proof: See Aendix 3. By Pro. 3, a generalization of the fractional down-samler for arbitrary corime airs, q can be made as shown in Fig. 22a. Once again interreting e j kqω/ as a fractional advance oerator by kq/, we see that, for an inut xn, this system sets the outut yn = xqn/. An equivalent system to the fractional down-samler is given in Fig 22b. This equivalency follows from Pro. 3. By the definition in Fig. 22a and Pro. 3, we see that the two systems in Fig. 23 are equivalent. We can now rewrite the FB in Fig. 4. Noting that Hω by 13 satisfies 53, we define Hω as in 31. Using this, we conclude that the FBs in Fig. 4 and 24 are equivalent. In Fig. 24 the filtering is alied directly to the inut signal, not to the u-samled inut signal. Fig. 24 shows more exlicitly the discrete-time sequences with which the inut signal is convolved namely the fractional time-shifts of hn and ḡn. The definition of the fractional down-samler makes rovides a clearer understanding of the sectral analysis rovided by the iterated rational FBs as well. In fact, by Proosition 2 and 39 the iterated FB with n stages is Hω q q H ω + Gω s s G ω Fig. 24. This FB is equivalent to the FB in Fig. 4.

23 23 H n ω qn n qn n H n ω Ḡ n ω qn 1 s n 1 qn 1 s n 1 Ḡ nω xn.. + ˆxn Ḡ 1 ω s s Ḡ 1ω Fig. 25. The iterated FB can be interreted as a constant-q transform. equivalent to the system in Fig. 25 where H n ω and Ḡnω are defined by 35 and 39. A. Underlying Wavelet Frame Using the low-ass filter Hω in 31 we define the scaling function via the infinite roduct as ˆφω = q H k ω = q q H ˆφ q ω q ω. 54 k=1 It follows by Pro. 2 that φt is bandlimited to [ π, π]. The scaling function satisfies the scaling equation q q φt = hn φ t n, 55 n obtained by taking the inverse Fourier transform of 54. Based on the scaling function, we define the wavelet function as ψt = q n q gn φ t n. 56 Equivalently, ˆψω = q G q ω ˆφ q ω 57 = q G q ω q H k ω. q 58 k=2 Equivalence of this definition to the definition in 4 follows readily. We also see from 4 that ψt is bandlimited to [1 1/sqπ/, qπ/] see Figs These definitions lead to, Proosition 4: For ft L 2 R, suose we are given q q xk = ft, φ t + k. 59 Then,

24 24 i y 1 k = ft, φt + k is obtained by alying the low-ass analysis system in Fig. 24 to xk. ii y 2 k = ft, ψ t + sq k is obtained by alying the high-ass analysis system in Fig. 24 to xk. Proof: Part ii follows readily from the definition of the wavelet function. For the roof of art i, see Aendix 4. Using this result we also have Proosition 5: The set { q k/2 q kt s ψ + q }k,n Z n is a tight frame for L 2 R. Proof: See Aendix E. These roositions suggest that the rational-dilation DWT is a fast algorithm for comuting the coefficients of a rational-dilation wavelet frame exansion of a finite energy function also see [2] for the critically-samled case. Besides redundancy, this wavelet frame differs from Auscher s orthonormal MRA with a rational dilation factor in the number of wavelets required. With the removal of the u-samler in the high-ass channel, the introduced wavelet frame is constructed by a single wavelet function, whereas the orthonormal MRA required q wavelets to san L 2 R also see [19] where non-refinable frame sequences are constructed, which yield an FB scheme with rational samling factors. VIII. CONCLUSION This aer shows that the develoment of self-inverting overcomlete wavelet transforms based on rational dilations can be conveniently accomlished using the frequency-domain design of over-samled filter banks. Even though the design and imlementation is erformed in the frequency domain and the discrete-time analysis/synthesis functions are not of comact suort, they decay raidly and are well localized in the time-frequency lane. As shown in the aer, the roosed rational-dilation wavelet transform can attain a range of redundancies and Q-factors making the transform more flexible than the dyadic wavelet transform. In articular, the roosed wavelet transform can have a substantially higher Q-factor than the dyadic wavelet transform; equivalently, it can attain imroved frequency resolution. It is therefore exected that the roosed wavelet transform will rovide an efficient reresentation of a broader class of signals: including short-time eriodic/oscillatory signals like EEG, seech, and other signals arising from hysical vibration henomena. Using the roosed wavelet transform with a low Q-factor yields wavelets similar to the Mexican hat wavelet, a dense samling of the time-frequency lane, and an exactly invertible aroximation to the continuous wavelet transform. Additionally, in contrast to other rational-dilation discrete wavelet transforms, the roosed discrete wavelet transform corresonds to a wavelet frame for the real line based on a single wavelet, ψt, defined through an infinite roduct formula. We have used the roosed transform to decomose an EEG signal into a sum of rhythmic and transient comonents; one goal of this alication is to imrove the accuracy of the hase locking value between two

25 25 EEG channels. The transform may also be useful for seech enhancement and for the restoration of clied seech. For these alications it is tyical to use the short-time Fourier transform STFT; however, the frequency resolution of a constant-q transform unlike the STFT scales with frequency. A constant-q filter bank is a uniform filter bank on a log scale which is a natural frequency sacing for analyzing and rocessing sounds and other oscillatory signals. For examle, the human ear has a near constant-q frequency resolution roerty above 5 Hz. APPENDIX A PROOF OF PROPOSITION 1 Notice that + 1 = q = s imlies that the FB is critically samled. There exist orthonormal FIR FBs with such samling factors. Now suose + 1 = q = s is not valid. Then, we need the FB to have at least as many outut samles as the inut samles. This requires that, /q + 1/s > 1 imlying s < q. Suose there exist filters with rational transfer functions achieving PR. To obtain a contradiction, we will rovide an inut that cannot be recovered. Let D = min2π/q, 2π/s 2π/q and define the set B = [2π/q, 2π/q + D. Now if, 1 for ω [, D, Xω = for ω [D, 2π]. then there exists a unique k {1, 2,..., q 1} s.t. X ω + n 2π = δ n,k for ω B, n =, 1,...,, q q Also, X ω + n 2π = for ω B, n =, 1,..., s s Hence we have, by 7, Y ω = L k ω for ω B. Since L k ω is the DTFT of a filter with a non-zero rational transfer function, it cannot be zero on a finite measure. Thus, Y ω on a subset of B with non-zero measure. Since Xω = on B, the FB is not PR. 6 We begin with a lemma. Lemma 1: Let N, M be corime and define the sets, N 1 [ k S 1 = N a 2N, k N + a ] M 1, S 2 = 2N k= Also set c = min a/2n, b/2m. Then, S 1 S2 = [ c, c] if and only if am + bn < 2. APPENDIX B PROOF OF PROPOSITION 2 k= [ k M b 2M, k M + b ]. 63 2M 64