A new method of estimating wavelet with desired features from a given signal

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1 A new method of estimating wavelet with desired features from a given signal Anubha Gupta a '*, Shiv Dutt Joshi b, Surendra Prasad b adivision of Computer Engineering, Netaji Subhas Institute of Technology, Dwarka, Sector-III, Delhi , India Electrical Engineering Department, Indian Institute of Technology, Hauz Khas, Delhi, India Received 26 September 2003; received in revised form 23 July 2004 Abstract This paper proposes a new method of estimating both biorthogonal compactly supported as well as semi-orthogonal infinitely/compactly supported wavelet from a given signal. The method is based on maximizing projection of the given signal onto successive scaling subspace. This results in minimization of energy of signal in the wavelet subspace. The idea used to estimate analysis wavelet filter is similar to a sharpening filter used in image enhancement. First, a new method is proposed that helps in the design of 2-band FIR biorthogonal perfect reconstruction filter bank from a given signal. This leads to the design of biorthogonal compactly supported wavelet. It is also shown that a wavelet with desired support as well as desired number of vanishing moments can be designed with the proposed method. Next, a method is proposed to design semi-orthogonal wavelets that are usually infinitely supported wavelets. Here, corresponding to FIR analysis filters, the resulting synthesis filters are IIR filters that satisfy the property of perfect reconstruction. Keywords: Perfect reconstruction FIR/IIR biorthogonal filter bank; Multiresolution analysis; Matched wavelet 1. Introduction signal into a set of frequency channels of equal bandwidth on a logarithmic scale i.e., an analysis The discrete wavelet transform of a given signal of signal using constant Q-filters [14,15]. It is may be interpreted as a decomposition of the computed by expanding the signal into a family of functions, each of which is a dilation and translation of a function called mother wavelet cðtþ : Unlike Fourier methods, wavelet transforms do not have a unique basis. This is one of the reasons why wavelets are finding applications in diverse fields and is a topic of current research.

2 148 A. Gupta et al. / Signal Processing 85 (2005) Since the basis here is not unique, it is natural to seek a wavelet which is best in a particular context. Particularly in the context of signal/image compression, an issue of great research interest is to find a wavelet that can provide best representation for a given signal Relevant earlier work Daubechies proposed methods to find orthonormal [6] and biorthonormal [5] wavelet bases with compact support. The resulting wavelets were maximally regular. But both these techniques of designing wavelets were independent of the signals being analyzed. One of the earliest problems that were solved to find the best approximation of the given signal fðtþ with translates of a valid scaling function of finite fixed support N, dilated by a given factor M, at the proper scale J, was addressed by Tewfik et al. [24]. But here, the approximation at resolution J depends only on the scaling function fðtþ and not on the corresponding wavelets. Moreover, authors have minimized the upper bound of error norm rather than minimizing the actual L 2 distance between fðtþ and its approximation at scale J. Since the minimization of norm in time domain was complex it was carried out in the frequency domain by Gopinath et al. [8] assuming that the signal being analyzed is band-limited. The optimality was measured with respect to minimization of frequency domain Lp norm of the approximation error. Closed form expression for error norm was obtained with this constraint in the frequency domain but it resulted in complex equations that are difficult to solve. Similarly, an algorithm called Matching Pursuits is proposed by Mallat and Zhang [16] that decomposes any signal into a linear expansion of waveforms that are selected from a redundant dictionary of functions. But here the families of functions over which the signals are expanded are derived by translation and scaling of some fixed known function. Work has also been done to search for the best basis for signal enhancement in white Gaussian noise by Krim et al. [13]. This basis search is performed in families of orthonormal bases constructed with wavelet packets and is based on the criterion of minimal reconstruction error of the underlying signal. Chapa and Rao [3] have proposed an algorithm for designing wavelets matched to the given signal. Authors have proposed a solution to find a wavelet that looks like the desired signal for the case of orthonormal multiresolution analysis with bandlimited wavelets. Their method is however, suboptimal in the sense that it matches the amplitude spectrum and phase spectrum of the given signal and wavelet independently. Moreover, the technique is computationally expensive. Similarly, work has also been carried out to design signal adapted orthogonal or biorthogonal filter banks [25,34] but these methods again find solution of constrained minimization problems, in terms of coding gain criterion that lead to very complicated solutions. Design of applicationspecific wavelets have also been discussed in [1]. Aldroubi and Unser [2] discuss methods to find a matched wavelet by projecting the signal onto an existing wavelet basis, or transforming the wavelet basis under certain conditions such that the error norm between the desired signal and new wavelet basis is minimum. Again the design is constrained by initial choice of MRA Problem formulation and proposed method The problem of finding a wavelet matched to a given signal has been considered again in this paper and a new but simple method is proposed. Here, the signal is assumed to be deterministic. Let us say, we consider L2 error norm between the signals reconstructed at two successive scaling subspaces and minimize it. Or say, L 2 error norm between the signals reconstructed at initial scaling subspace and successive lower wavelet subspace is considered and maximized. This will project maximum energy of signal in scaling subspace and minimum in wavelet subspace. This implies that approximation on scaling subspace is closest to given signal and wavelet subspace indeed carries additional or finer information of signal. Thus, the method will naturally lead to a wavelet matched to a given signal. Motivated by the above discussion, the problem is formulated as follows: Find a closed form

3 A. Gupta et al. / Signal Processing 85 (2005) expression for extracting the compactly/infinitely supported wavelet which maximizes L2 error norm between the signals reconstructed at initial scaling subspace and successive lower wavelet subspace. Here a closed form expression that is simple to compute has been obtained for the estimation of analysis wavelet filter. In this paper, two methods have been proposed to estimate both biorthogonal and semi-orthogonal wavelet from a given signal. First, based on the design of analysis wavelet filter, corresponding FIR analysis and synthesis filters are designed such that the resulting filter bank structure is a 2-band biorthogonal perfect reconstruction FIR filter bank. This results in the biorthogonal compactly supported wavelet matched to a given signal. Next, a method is discussed that helps us in the design of wavelet with desired support and desired number of vanishing moments for the case of compactly supported wavelet. In the second method, scaling and wavelet subspaces are assumed to be orthogonal and then analysis scaling filter is designed from the estimated analysis wavelet filter. Here, corresponding to FIR analysis filters, synthesis filters are usually IIR filters for PR filter bank. These synthesis filters can only be implemented as anticausal but stable filters. This results in the design of a semiorthogonal infinitely/compactly supported wavelet matched to a given signal. Here, it is to emphasize that wavelets and IIR filter banks have been discussed in the past but not with a motivation to find a matched wavelet. Herley and Vetterli [12] discussed IIR filter banks, but there the IIR filter bank is constructed without any consideration for the input signal. Design of IIR filter banks have been studied in [18,21] also, but still it is a less studied area as compared to FIR filter bank in the theory of wavelet transforms. The method as discussed above is applied on two real life clips: one music and the other a speech clip. This method of estimating wavelet from the signal is much simpler than any other method proposed so far in the literature. Outline of the paper : This paper is organized into eight sections. Section 2 contains a brief review of the theories of wavelet transform and multirate filter banks. Section 3 discusses the proposed method of estimating analysis wavelet filter from a given signal. Section 4 first discusses how to design PR biorthogonal FIR filter bank based on the analysis wavelet filter estimated in Section 3. A procedure to design a compactly supported wavelet with desired support and number of vanishing moments is described next. In Section 5, a method to design semi-orthogonal infinitely/compactly supported matched wavelet is discussed based on the analysis wavelet filter estimated in Section 3. Simulation results on some real life clips like music/speech clips are presented at the end of Sections 4 and 5. The method is extended to binary tree structure in Section 6. The proposed methods are applied to the problem of compression in Section 7 as an application of the concept of matched wavelet. In the end, conclusions are presented in Section Preliminaries The subject of multi-scale signal representation/ analysis has been studied by applied mathematicians for a number of years. The works of Mallat [14,15] and Daubechies [5,6] evoked the interest of signal processing community in the theory of wavelet transforms. These papers established the connection between wavelet transforms and the theory of multirate filter banks. Here, in this section, a succinct review of wavelet transforms and multirate filter banks is presented. For detailed discussion on filter bank structures and wavelets one may refer to [4,7,17,19,22,23,27,30,32] Review of wavelet transforms A wavelet system can be orthogonal or biorthogonal. Consider a 2-band wavelet system shown in Fig. 1. a o(n) d-i(n) Fig band biorthogonal filter bank. fi(n)

4 150 A. Gupta et al. / Signal Processing 85 (2005) Biorthogonal wavelet system relaxes the condition of orthogonality in the wavelet basis and gives more flexibility in the design process. It uses the concept of dual basis [4] where, the scaling filterf 0 and its dual h0 ; wavelet filterf1 and its dual h1 are required to satisfy the following conditions for perfect reconstruction of the 2-band filter bank (Fig. 1): :(-l)7o(m-n), (1) i = (-l) B Ao(M-n), (2) where M is any odd delay. The scaling function fðtþ and wavelet function cðtþ are governed by 2-scale relations for the 2- band wavelet system as e Z ; (3) a o(n) a 0 Z" 1 G o(4 Gi(z) (n) I» a.i(n) d.i(n) f t Fig channel filter bank.» E(z) R(z) 1 I 1 a v- fct f A a o (n) A a o (n) Fig band filter bank using polyphase component matrices. Here, f 0 and f1 are the corresponding low and high pass filters, respectively. Dual scaling function f0ðtþ and dual wavelet function c0 ð tþ are related to h 0 and h1 via similar equations. The wavelet function is said to have vanishing moments of degree N if = 0 for yt 0 This equation can be transferred to discrete moments of f1 where kth moment of f1 is defined as miik) Z : (4) Similarly, the synthesis filters can be written in terms of type-ii, 2-component polyphase form as (6) [F 0 (z) Fi(z)] = [z- 1 1] Rðz 2 Þ : (8) The 2-band filter bank structure of Fig. 2 can now be redrawn using polyphase matrices EðzÞ and RðzÞ as shown in Fig. 3. The condition of perfect reconstruction is ( 5 ) RðzÞEðzÞ = cz- m \. (9) If (9) is satisfied, then the output of filter bank is a o (n) ca 0 ðn - n 0 Þ (10) and the output is merely a delayed version of input signal. Requiring the moments of cðtþ to be zero is equivalent to putting the discrete moments of f1 to zero. 3. Estimation of analysis wavelet filter from a given signal 2.2. Review of fundamentals of filter bank theory Consider an analysis filter bank structure of the 2-band wavelet system in Fig. 4 to which the Consider a 2-channel filter bank shown in Fig. 2. sampled version of given continuous time signal The analysis filters can be written in terms of aðtþ is applied as input. type-i, 2-component polyphase form as below: Here, «o( w ) a ( n ) sampled version of input signal. This signal can be considered as the (7) coefficients of expansion over the basis formed 1 by integer translate of scaling function at scale

5 A. Gupta et al. / Signal Processing 85 (2005) a o(n) h o(-n) + 2 hiffl + 2 as below d_i(n) = Ai(0)ab(2n) + hi(l)a o (2n + 1) + h1ð2þa 0 ð2n þ 2Þ + h1ð3þa 0 ð2n þ 3Þ þ h1ð4þa 0 ð2n þ 4Þ : ð19þ Fig. 4. Analysis filter bank structure of 2-band wavelet system. j 0 in the scaling subspace V0 : That is, a(t) = - n). (11) Now, V0 can be written as V 0 = F_i0JF_i. (12) Thus, input signal can be further decomposed into a_i(«) and d-i(n). The outputs a_i(«) and d-i(n) from analysis filter bank structure as shown in Fig. 4 can be considered as the coefficients of expansion in the successive lower scaling subspace and wavelet subspace respectively. They are related by the following equations: fl_i(n) = 2^ Ao(-fc)oo(2«- k) k and (13) (14) = information in the lower wavelet subspace. Here, both h 0 and h1 are considered to be FIR filters of length N so as to design a compactly supported wavelet. The continuous time signal is reconstructed from a 0 ðnþ and d-i(n) ~ -k), (15) (16) The error between these two signals is defined as eðtþ = a(t) - a(t). (17) Corresponding error energy is defined as = Z e 2 ðtþdt : (18) Let us assume that the length of filter h1 is N 5 ; then d-i(n) can be written in terms of filter weights The signal <i_i(«) provides the detail or high pass information. Therefore, we would like to put it in the form such that it naturally looks like high pass signal. Now, if the center weight h1ð2þ of high pass filter h1 is set to unity, then, (19) can be rewritten as where = a 0 ð2n þ 2) - {-[ + h1ð1þa 0 ð2n þ 1Þ + h1ð3þa 0 ð2n þ 3Þ þ h1ð4þa 0 ð2n þ 4)]} = a 0 ð2n þ 2) - a o (2n þ 2Þ = e 0 ð2n þ 2Þ ; ð20aþ a o (2n þ 2) = - h1ð0þa 0 ð2nþþ 1ð1Þa 0 ð2n + h1ð4þa 0 ð2n þ 4)]. ð20bþ Discussion on Eq. (20): The interpretation of Eq. (20a) in the manner given below, in fact, is the central idea of the present work. This equation has been put in the above form so as to derive interesting interpretation for the same. This plays a key role in the estimation of matched wavelet. With center weight fixed to unity, from (20b), a(,(2n þ 2Þ is the smoother estimate of a0ð2n þ 2Þ from the past as well as future samples. Thus, d-i(n) is the error in estimating a 0 ð2n þ 2Þ from its neighborhood and hence represents additional/ finer information. This idea to estimate the analysis wavelet filter is similar to a sharpening filter used in image enhancement. Since, d-i(n) represents error signal between actual value of a0ð2n þ 2Þ and its estimated value cio(2n þ 2Þ ; we should minimize this error signal and hence resulting filter h1 should be a high pass filter. Expanding signal aðtþ and a(t) in terms of dual wavelet basis functions, we have e(t) = a(t) - a(t),

6 152 A. Gupta et al. / Signal Processing 85 (2005) e(t) = Substituting the expression of eðtþ in (18) = Z e 2 ðtþdt -k) -k]\ Using the following relations: At. b'(t-p) = J2 y/2h o (k)<l>'(2t -2p- k), k and we get, M -mþ dt = ð2t - 2p - kþ ; ð21þ ð22þ (23a) (23b) (23c) (23d) (23e) E k forr = 0 ; 1 ;...,j- a 0 ð2k þ pþa 0 ð2k þ rþ = 0 ð25bþ Here,jth filter weight is kept constant to value 1. Discussion on (25): The proposed method leads to a closed form expression (25b) where the bracketed term looks like deterministic autocorrelation function of decimated input signal. These are a set of N 1 linear equations in filter weights that can be solved simultaneously without much computational complexity. The solution gives corresponding weights of dual wavelet filter h1 i.e. the analysis high pass filter. Second derivative of the error energy E is computed to verify that it indeed results in the maximization of error energy as crf = -2 Xða 0 ð2n þ rþþ 2 = -ve. dh\ (26) Here, this work has been carried out for the 2- band wavelet system independently of [8,24]. In [24] L 2 -norm approximation error has been minimized but no closed form expression has been found, rather an upper bound on error norm has been minimized. Similarly, in [8], L p -norm error has been minimized in frequency domain giving very complex equations. Here, a simple closed form expression has been obtained in terms of weights of the analysis wavelet filter h1 and input signal samples. m q k p xa o (2k þ qþa 0 ð2k þ ð24þ In order that maximum input signal energy moves to scaling subspace, the energy E in the difference signal eðtþ in (24) should be maximized [9,10]. Therefore, derivative of E with respect to all but one filter weights is equated to zero. The resulting equation is E E rþ = 0 ð25aþ 4. Estimation of biorthogonal compactly supported wavelet 4.1. Design of FIR perfect reconstruction biorthogonal filter bank Consider the analysis/synthesis filter bank structure of Fig. 1. The four filters h 0; h1 ; f 0; f 1 are related by (1) and (2) for the condition of perfect reconstruction. First from (1), the scaling filter f 0 is computed. All these filters are FIR filters. Since, the integer translates of fðtþ and cðtþ form the basis of V0 and W0 respectively in L 2 ;

7 A. Gupta et al. / Signal Processing 85 (2005) f 0 ð2m nþ andf1 ð2m nþ form the basis of 2 for integer values of m. Similarly, h0ðn 2mÞ and h1ðn 2mÞ form the dual basis of 2 for integer values of m. Therefore, and 8(m\ mi) 2m1Þh1ðn - 2m 2 Þ e Z 0 Z : ð27þ (28) To find h 0; (27) and (28) are required to be evaluated for only those values of m for which the vectors f 0 ðn 2mÞ and h1ðn 2mÞ overlap with h 0 ðnþ : After computing h 0; the filter f1 can be found using (2). Hence, all the four filters of the 2-band wavelet system can be designed to form a FIR biorthogonal perfect reconstruction filter bank Design of biorthogonal compactly supported wavelet with desired support and vanishing moments A compactly supported wavelet with desired support can be designed starting from any chosen order of analysis wavelet filter. Say, initially the analysis wavelet filter of order 3 is chosen. And it is desired to design wavelet filter of order 7. Then, two extra zeros can be padded before and after the actual filter h1 and the design procedure mentioned in Section 4.1 above can be repeated. This will result in the design of wavelet filter of order 7. To design a wavelet with desired number of vanishing moments, conditions of vanishing moments (6) can be imposed onf1 and transferred to filter h 0 as h 0 and f1 are related by (2). These equations can then be used along with Eqs. (27) and (28) to compute the solution for h 0 and hence f1 to estimate wavelet with desired number of vanishing moments. Algorithm I. The algorithm to estimate compactly supported matched wavelet from given signal with desired support and vanishing moments is as below: Step 1: Estimate analysis wavelet filter h1 of order N1 from the given input signal using N 1 linear equations of (25). Step 2: If it is desired to design the wavelet filter f1 of order N24N1 ; then append extra zeros before and after h1 such that its order is N2: Step 3: Use (1) to compute synthesis scaling filterf 0: Step 4: To design wavelet with K number of vanishing moments, conditions of vanishing moments (6) are imposed on f1 and transferred to filter h 0 as h 0 andf1 are related by (2). Step 5: Using (27), (28) and (6), compute analysis scaling filter h0 : Step 6: Use (2) to compute synthesis wavelet filterf1 : Step 7: Plot scaling function and wavelet function from the scaling filter and wavelet filter. Hence, biorthogonal compactly supported wavelet with desired support and vanishing moments is estimated from the given signal Simulation results The simulation results are appended for two cases. First, when the given signals have shapes identical to standard wavelets and second, when the given signals are real life signals like music and speech. Case 1: To validate the theory, that the proposed method indeed results in wavelet matched to signals, two biorthogonal wavelets are considered and are applied as input to 2-band filter bank structure. First signal is a biorthogonal scaling function of bior2.4 (as listed in Wavelet Toolbox of MATLAB software). When the proposed method of compactly supported matched wavelet is applied to this clip, it results in the corresponding wavelet of bior2.4. With the resulting 2-band biorthogonal PR FIR filter bank, the detail coefficients of subsequent lower wavelet subspace has all coefficients equal to zero and the approximation coefficients have the coarser version of the input signal. This signifies that the resulting scaling function has a shape identical to input signal resulting in a matched wavelet. Similarly, another biorthogonal signal bior3.3 (as listed in Wavelet Toolbox of MATLAB software)

8 154 A. Gupta et al. / Signal Processing 85 (2005) is chosen and similar results are obtained. Input signals and resulting wavelets along with actual wavelets for signal 'bior3.3' are plotted in Fig. 5. Resulting filter coefficients are appended in Table 1. Case 2: The proposed method is applied on an audio and a speech clip. The analysis/synthesis filters are designed from the signal itself being based on this theory. Resulting scaling functions and wavelets have shapes similar to Spline, Coiflet wavelet or other shapes and are plotted in Fig. 6. These wavelets are biorthogonal compactly supported wavelets. The resulting analysis and synthesis filter coefficients for wavelet are appended in Table Estimation of semi-orthogonal wavelet Design of perfect reconstruction filter bank and hence design of semi-orthogonal wavelet First, we consider dual wavelet subspace to be orthogonal to the dual scaling wavelet subspace. Now, filters h0 and h1 at the analysis end are A j / \ (a) Fig. 5. Simulation results on standard wavelet bior3.3. (a) Input signal, (b) approximation coefficients, (c) detail coefficients (in 10 ), (d) estimated dual scaling function, (e) actual dual scaling function, (f) estimated dual wavelet function, (g) actual dual wavelet function, (h) estimated scaling function, (i) actual scaling function, (j) estimated wavelet function, (k) actual wavelet function. (k) Table 1 Analysis and synthesis filters with PR property for biorthogonal compactly supported matched wavelet Sr. no. Input signal No. of samples Estimated filters bior2.4 bior h o (-n) = 00 : : : : ] hi-n) = [ : ] f o (n) = 0000 :5 1 : 0 0 : ] /;(«) = 00 : : : : : ] ho(-n) = 0 : : : : ] hi-n) = : : ] / 0 (n) = 00 0 : : 0 0 : : ] f1ðn = [ : : : ]

9 A. Gupta et al. / Signal Processing 85 (2005) / I I A1 A / V / (b),, V,,, R (c) ( Fig. 6. Estimated biorthogonal scaling functions and wavelet functions for two clip along with dual scaling and dual wavelet functions, (a) Clip I; Fig. (b)-(e) Estimated wavelet and scaling functions with no. of vanishing moments of wavelet (synthesis) filter equal to 2 (Table 2): (b) scaling function, (c) wavelet function, (d) dual scaling function, (e) dual wavelet function. (f) Clip II; Fig. (g)-(j) estimated wavelet and scaling functions with no. of vanishing moments of wavelet (synthesis) filter equal to 3 (Table 2): (g) scaling function, (h) wavelet function, (i) dual scaling function, (j) dual wavelet function. Table 2 Analysis and synthesis filters with PR property corresponding to biorthogonal wavelets estimated from different signals Clip no./no. of samples No. of vanishing moments of wavelet (synthesis) filter Coefficients of analysis and synthesis filters 1. Speech clip, 2714 samples, 0 sampling rate: Hz 2. Music clip, samples, 0 sampling rate: Hz h0 = [O - 0 : : : : ] hi = 00 : : 0 0 : : ] / 0 = 00 : : 0 0 : : ] /; = 00 : : : : ] ho = 00 : : : : ] h 1 [ : : 0 0 : : ] / 0 = : : 0 0 : : ] / 1= [ : : : : ] ho = 00 : : : : : ] h =[ : : : ] / 0 = : : : 0 0 : : ] / 1= [ : : : ] h0 = [ : : : : ] h1 = [ : : 0-0 : ] / 0 = : : 0 0 : ] f1 = [ : : : : ]

10 156 A. Gupta et al. / Signal Processing 85 (2005) related as hi(n) = (-!)" h o (M-n), (29) where M is odd delay. Thus, h 0 is designed using (29). From analysis filters h 0 and h1 ; polyphase decomposition matrix EðzÞ is formed as in (7). Objective: The aim is to find synthesis filters such that resulting filter bank is a perfect reconstruction filter bank. Therefore, RðzÞ should be formed such that RðzÞ and EðzÞ satisfy (9) for the design of perfect reconstruction filter bank. First, matrix EðzÞ is decomposed and is written in Smith-McMillan form as E(z) = U(z)/L(z)V(z), (30) where, UðzÞ and VðzÞ are unimodular matrices and LðzÞ is the diagonal matrix. Next, RðzÞ is formed as such that R(z)E(z) = det(/t(z))i. )V-\z) (31) (32) All pole part of I=detðLðzÞÞ is absorbed in the filters corresponding to RðzÞ : Thus, corresponding to FIR analysis filters, synthesis filters are usually IIR filters that result in the design of perfect reconstruction filter bank. The resulting structure of PR filter bank is drawn as in Fig. 7. These IIR filters on synthesis end are implemented as anti-causal but stable filters. Reference material on the role of anticausal and stable filters can be found in [28,29]. The resulting wavelet corresponding to high pass synthesis filter is hence semi-orthogonal infinitely/compactly supported matched wavelet. a o(n) H n(z) + 2 Hi(z) a.i(n d-i(n) * 2 F 0(z)/det(A(z 2 )) Fig band IIR perfect reconstruction filter bank. a o(n) Algorithm II. The algorithm to estimate semiorthogonal infinitely/compactly supported wavelet from a given signal is as below: Step 1: Estimate analysis wavelet filter h1 of order N1 from given input signal using N 1 linear equations of (25). Step 2: Use (29) to compute analysis scaling filter h 0: Step 3: Form polyphase decomposition matrix EðzÞ from analysis filters h 0 and h1 : Carry out Smith-McMillan form decomposition of EðzÞ as in (30). Find RðzÞ and then using RðzÞ and detðlðzþþ ; compute synthesis filters and hence design the structure of perfect reconstruction filter bank as shown in Fig. 7. Step 4: Plot scaling function and wavelet function from the synthesis scaling filter and wavelet filter. The resulting wavelet corresponding to high pass synthesis filter is usually infinitely supported. However, a subclass of these wavelets have finite support when detðlðzþþ is a monomial and results in compactly supported wavelet. Hence, a semi-orthogonal infinitely/compactly supported wavelet is estimated from given signal Simulation results The simulations results are appended for two cases. First, when the given signals have shapes identical to standard wavelets and second, when the given signals are music and speech signals. Case 1: To validate the theory, that the proposed method indeed results in wavelet matched to signals, two signals are applied as input to 2-band filter bank structure. The first signal is a square wave. When the proposed method of semi-orthogonal matched wavelet is applied to this clip, it results in the corresponding Haar wavelet which should be the result for the matched wavelet. With the resulting 2-band PR filter bank, the detail coefficients of subsequent lower wavelet subspace has all coefficients equal to zero and the approximation coefficients have the coarser version of the input signal. This signifies that the resulting scaling function should have shape identical to input signal resulting in matched wavelet. The second signal is the triangular wave signal. Again the resulting scaling function and

11 A. Gupta et al. / Signal Processing 85 (2005) wavelet are having shapes of triangular function. As above, the detail coefficients of subsequent lower wavelet subspace have all coefficients equal to zero and the approximation coefficients have the coarser version of the input signal. This method differs from the method discussed in Section 4 by the fact that here the scaling subspace and wavelet subspace are orthogonal to each other. Input signals and resulting wavelets along with actual wavelets for the signals considered are plotted in Fig. 8. The resulting filter coefficients are appended in Table 3. Case 2: Again, the proposed method is applied on two clips: one music clip and the other a speech clip. The analysis/synthesis filters designed from the signal itself based on this theory are tabulated in Table 4. The semi-orthogonal infinitely/compactly supported matched wavelet and corresponding scaling clips in Fig. 9. function are plotted for all the 6. Wavelets and binary tree structured filter banks Consider a tree structured analysis filter bank as shown in Fig. 10. The 2-band wavelet structure method can be extended to a tree structured filter bank wherein the decimation ratio in each branch is a power of two. Here, first corresponding to input signal, matched wavelet and corresponding 2-band filter bank structure is found. Then, the low pass filtered signal a_i(«) is used again to find out the matched wavelet and corresponding structure consisting of filters h10 and h11 : Hence, every low pass filtered signal can be decomposed (j) (k) (m) (n) \ \ Vh \ 1 \/ \/ V Fig. 8. Simulation results of semi-orthogonal matched wavelet with square wave input and triangular wave input. Clip I; Fig. (a)-(g): (a) Input square wave signal, (b) approximation coefficients, (c) detail coefficients, (d) estimated scaling function, (e) estimated wavelet function, (f) estimated dual scaling function, (g) estimated dual wavelet function. Clip II; Fig. (h)-(n): (h) Input triangular wave signal, (i) approximation coefficients, (j) detail coefficients, (k) estimated scaling function, (l) estimated wavelet function, (m) estimated dual scaling function, (n) estimated dual wavelet function.

12 158 A. Gupta et al. / Signal Processing 85 (2005) Table 3 Analysis and synthesis filters with PR property for semi-orthogonal infinitely/compactly supported matched wavelet Sr. no. Input signal No. of samples Estimated filters Square wave signal Triangular wave signal H 0 ðzþ = 1 þ l.oz- 1 H l (z)=\.q-z- 1 F 0 ðzþ = 0 : 5 þ 0.5Z- 1 F1ðzÞ = 0 : 5-0.5Z- 1 H 0 ðzþ = 0.5Z- 1 þ l.oz- 2 þ 0.5z- 3 Fo(z) 2 Fi(z) = 1 þ 6z- 2 þ z- 4 _2 þ 4z-' þ 2z- 2 1 þ 6z- 2 þ z- 4 Table 4 Analysis and synthesis filters with PR property corresponding to semi-orthogonal wavelets estimated from different signals Clip no./no. of samples 1. Speech clip, 2714 samples, sampling rate: Hz 2. Music clip, samples, sampling rate: Hz Coefficients of analysis and synthesis filters H 0 ðzþ = Z- 1 þ z- 2-3 þ Z - 4 þ z- H1ðzÞ = 0 : Z- 1 þ l.oz Z - 3 þ Z - 4 :, Z- 1 þ Z - 2 þ z- 3 þ 11 : 22566lz" 4 þ z- 5 1 þ 18z- 2 þ 40z- 4 þ 18z- 6 þ z- 8 3 : Z- 1 þ Z Z - 3 þ Z - 4 Fi(z) 3 1 þ 18z- 2 þ 40z- 4 þ 18z- + z- H 0 ðzþ = Z- 1 þ l.oz- 2 þ Z - 3 H1ðz = þ l.oz z Z z- 2 þ Z - 3 Fo(z) 1 1 þ 4.7z- 2 þ z þ Z Z - Fi(z) = 1 þ 4.7z- 2 þ z _^ y h V y 0 03 I) (a) (c) (d) Fig. 9. Semi-orthogonal infinitely supported wavelets and scaling functions for two real life clips. Clip I; speech clip (a)-(b): (a) scaling function, (b) wavelet function. Clip II; music clip (c)-(d): (c) scaling function, (d) wavelet function.

13 A. Gupta et al. / Signal Processing 85 (2005) a.3(n) d.3(n) and 5=3 [26]. References for reading material on compression and comparison of filters can be found in [20,26,31,33]. Here, results are compiled with bits per sample and peak signal-to-noise ratio, PSNR. PSNR is defined as PSNR= 10 log1 max n jaðnþj 2 (33) a o (n) Fig. 10. Tree structured analysis filter bank. d i(n) h 3(-n) + 2 h 2(-n) + 4 d. 2(n) t? f 3(n)! 4 Here, N is equal to total number of samples in the input signal aðnþ and eðnþ is the error between input signal sample and reconstructed signal sample. To see compression, the process of uniform quantization is applied on different subband signals. Separate quantization step is chosen for each sub-band based on the variance of signal of these sub-bands. If an overall bit rate of R bits/ sample is required, then best bit allocation for subbands [26] is d. 3(n) Fig band perfect reconstruction filter bank structure. further to find a 2-band matched wavelet filter bank structure. Here, iteration is done over three branches to form a 4-band structure. The resulting scaling function and wavelets can then be determined using two scale relations. Tree structured filter banks have also been discussed in [23]. The various filters on the analysis and synthesis end can then be combined resulting in the overall filter bank structure as shown in Fig Applications The concept of matched wavelet as proposed in this paper is tested on the application of compression. Here, four different clips are used and results of compression of both compactly supported and infinitely supported matched wavelets are compared with standard wavelets i.e. biorthogonal 9=7 G\G 2 for A:= 1 ; 2 ; (34) where a\ is the variance of signal in scaling subband and a\ is the variance of signal in wavelet sub-band. R1 is the number of bits required for scaling sub-band and R 2 is the number of bits required per sample for wavelet sub-band. Based on the number of bits per sample required for each sub-band signal, number of quantization steps are chosen and uniform quantization is carried out for these sub-band signals. After quantization, Huffman coding is employed and required bits per sample with PSNR are computed and tabulated for different clips. Two signals used in the earlier sections are again considered and results of compression are compiled in Table 5. It is observed that matched wavelet discussed here gives better results for compression as compared to standard wavelets biorthogonal 9=7 and 5=3 : Discussion on results obtained: On comparing matched wavelet with the standard wavelets Bi 9=7 and Bi 5=3 ; it is observed that good results of compression are obtained for the matched wavelets for both biorthogonal compactly supported wavelet as well as semi-orthogonal infinitely/ compactly supported wavelet.

14 160 A. Gupta et al. / Signal Processing 85 (2005) Table 5 Results of compression on different clips Type of wavelet Bits/sample Input signal 1: music clip, no. of samples: Bi9/ Bi 5/ Estimated biorthogonal wavelet 1.55 Estimated semi-orthogonal wavelet 1.55 Input signal 2: speech clip, no. of samples: 2713 Bi9/ Bi 5/ Estimated biorthogonal wavelet 2.34 Estimated semi-orthogonal wavelet 2.35 PSNR Here, simple coding method is used in compression to show the concept of matched wavelet. However, efficient coding methods can be employed to use this method on the application of compression of signals. 8. Conclusions In this paper, a new method of estimating matched wavelet has been proposed. The key idea lies in the estimation of analysis wavelet filter from a given signal. The idea to estimate this filter is similar to a sharpening filter used in image enhancement. Based on this approach, first analysis wavelet filter is estimated. Once analysis wavelet filter is estimated, the issue is how to design analysis scaling filter. There is no method to find scaling filter uniquely from wavelet filter. MRA is unique corresponding to scaling function but corresponding to wavelet function we can have many scaling functions and hence MRA. Therefore, after estimating analysis wavelet filter, the problem is then viewed from the signal processing aspect and hence issue of perfect reconstruction is talked about. Here, it is to emphasize that we can have many alternate designs, like biorthogonal, semi-orthogonal and orthogonal wavelet design. First, we focused on PR biorthogonal FIR filter bank that leads to design of biorthogonal compactly supported matched wavelet. Next, we assumed wavelet subspace to be orthogonal to scaling subspace and hence discussed the design of semi-orthogonal wavelet that are usually infinitely supported wavelets, whereas design of orthogonal wavelets was presented by the authors in [11]. The proposed method is applied on two real life clips: one music and one speech clip. Here, it is observed that the resulting scaling functions look like splines, Coiflet or Daubechies scaling function or takes some other shape. Use of concept of matched wavelet is shown in the context of signal compression. It is observed that matched wavelet discussed here gives better results for compression as compared to standard wavelets biorthogonal 9=7 and 5=3 : The method can be further extended to form a matched binary tree-like structure resulting in filter bank with decimation ratio of powers of two. References [1] P. Abry, A. Aldroubi, Designing multiresolution analysistype wavelets and their fast algorithms, J. Fourier Anal. Appl. 2 (2) (1995) [2] A. Aldroubi, M. Unser, Families of multiresolution and wavelet spaces with optimal properties, Numer. Func. Anal. 14 (5-6) (1993) [3] J.O. Chapa, R.M. Rao, Algorithm for designing wavelets to match a specified signal, IEEE Trans. Signal Process. 48 (12) (2000) [4] T. Chen, P.P. Vaidyanathan, Vector space framework for unification of one- and multidimensional filter bank theory, IEEE Trans. Signal Process. 42 (8) (1994) [5] A. Cohen, I. Daubechies, J.C. Feauveau, Biorthogonal bases of compactly supported wavelets, Commun. Pur. Appl. Math. 45 (1992) [6] I. Daubechies, Orthonormal bases of compactly supported wavelets, Commun. Pur. Appl. Math. 41 (11) (1988) [7] R.A. Gopinath, C.S. Burrus, An Introduction to Wavelets and Wavelet Transform, Prentice-Hall, Englewood Cliffs, NJ, [8] R.A. Gopinath, J.E. Odegard, C.S. Burrus, Optimal wavelet representation of signals and wavelet sampling theorem, IEEE Trans. Circuits Syst. II 41 (4) (1994) [9] A. Gupta, S.D. Joshi, S. Prasad, On a new approach for estimating wavelet matched to signal, in: Proceedings of Eighth National Conference on Communications, IIT Bombay, 2002, pp

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