Explicit Formulas for Orthogonal IIR Wavelets - Semantic Scholar

Explicit Formulas for Orthogonal IIR Wavelets Ivan W. Selesnick (new address)

Electrical Engineering Polytechnic University 6 Metrotech Center Brooklyn New York 11201 Tel. (718) 260-3416 [email protected]

August 10, 1997 EDICS: SP 2.4.4, Wavelets and Filter Banks

Abstract Explicit solutions are given for the rational function P (z ) for two classes of IIR orthogonal 2-band wavelet bases, for which the scaling lter is maximally at. P (z ) denotes the rational transfer function H (z )H (1=z ) where H (z ) is the (lowpass) scaling lter. The rst is the class of solutions that are intermediate, between the Daubechies and the Butterworth wavelets. It is found that the Daubechies, the Butterworth, and the intermediate solutions are uni ed by a single formula. The second is the class of scaling lters realizable as a parallel sum of two allpass lters (a particular case of which yields the class of symmetric IIR orthogonal wavelet bases). For this class, an explicit solution is provided by the solution to an older problem in group delay approximation by digital allpole lters.

 This

work has been supported by NSF and Nortel.

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1 Introduction The results of this paper supplement the paper by Herley and Vetterli [6] in which orthogonal IIR wavelets are examined. (See also pages 139-141, 276-278 of [28].) This paper describes explicit formulas for P (z ) for the design of orthogonal 2-band IIR wavelets, with and without symmetry, for which the scaling lter is maximally at. P (z ) denotes the rational transfer function H (z )H (1=z ) where H (z ) is the (lowpass) scaling lter. Necessarily, the associated orthogonal IIR lter banks are noncausal, which makes them substantially more dicult to use for some real-time applications. However, some of the applications for which wavelet analysis has proven most successful, compression and denoising being perhaps the two most notable examples, are often performed in `batch-mode' on nite length data sets. In such cases, the noncausality of the lter is less problematic. Several authors have discussed and/or advocated the use of IIR lter banks for certain applications [1, 2, 8, 10, 13, 22, 23].

2 Preliminaries Let the rational function H (z ) denote the transfer function of the real-valued scaling lter of an orthogonal 2-band wavelet basis. As such, H (z ) is also the lowpass lter of a 2-channel orthogonal lter bank [4]. The central equation, that H (z ) must satisfy, is

H (z )H (1=z ) + H (?z )H (?1=z ) = 1:

(1)

When H (z ) is a polynomial solution to (1), the corresponding wavelet and scaling functions are compactly supported, and the lter bank is FIR. On the other hand, rational solutions to (1) produce in nitely supported wavelets, and describe IIR lter banks that are necessarily noncausal. It is convenient to de ne the function P (z )

P (z ) = H (z )H (1=z ):

(2)

P (z ) + P (?z ) = 1

(3)

The solutions P (z ) of

are sought, where P (z ) permits a factorization as in (2). This paper considers maximally at solutions H (z ). Those are solutions for which H (z ) possesses the maximum number of zeros at z = ?1 possible, given the degrees of the numerator and denominator of H (z ). Requiring H (z ) be maximally at is an attractive strategy for several reasons which have been described elsewhere [4]. However, it is noted here, that requiring maximal atness makes possible a straight forward 1

design technique and produces useful wavelet bases. From this requirement, it follows that P (z ) has the form ?1 N N (4) P (z ) = (1 + z ) F(1(+z )z ) FN (z ) D where FN (z ) and FD (z ) are `numerator' and `denominator' polynomials, as noted in [6]. This paper provides closed form solutions for P (z ) given the degrees of FN (z ) and FD (z ). Two special cases are well known. Certainly the most important solution, described by Daubechies, has FD (z ) = 1. Another solution is the Butterworth wavelet [26], with FN (z ) = 1. The intermediate solutions described by Herley and Vetterli have both FN (z ) and FD (z ) of positive degree [6]. However, an explicit formula for P (z ) for these intermediate solutions has not been provided. Both the Daubechies and the Butterworth solutions P (z ) are `halfband' instances of digital lters, known before the elegant theory of wavelet analysis was developed. In Section 3 we introduce an explicit formula for the intermediate solutions P (z ), by drawing from previous work on the design of generalized digital Butterworth lters. The formula provided in Section 3 specializes to each of the Daubechies and the Butterworth solutions. In Section 4 we considers another class of orthogonal scaling lters: maximally at scaling lters realizable as a parallel sum of allpass lters, a special case of which yields symmetric wavelets. Herley and Vetterli described IIR orthogonal 2-channel lter banks with symmetric lters and give examples of maximally at solutions H (z ), where H (z ) is composed of allpass lters. This paper supplements the description given in [6] by noting that an explicit formula for H (z ) in this case comes directly from the solution to an older problem in group delay approximation for digital allpole lters.

3 Intermediate Daubechies-Butterworth Wavelets In [6] Herley and Vetterli describe a class of rational solutions to (1) that are intermediate | between the compactly supported orthogonal wavelets of Daubechies [4] and the orthogonal IIR Butterworth wavelets, in the sense that each of FN (x) and FD (x) are of positive degree. It is explained that the intermediate solutions can be obtained by solving a linear system of equations, which is described in [6]. It turns out, however, that the generalized digital Butterworth lter provides an explicit formula for P (z ). The version of the generalized Butterworth lter described in this paper is characterized by three parameters L, M , and N . N denotes the number of zeros of H (z ) at z = ?1. L denotes the number of zeros that shape the passband, and M denotes the number of poles (away from the origin). For example, in Figure 1, L = 3, M = 2, N = 6. The position of the cut-o frequency and the roll-o rate is controlled by the triplet (L; M; N ). Generalized Butterworth lters are appealing because sometimes, a lter having diering numerator and denominator degrees can achieve an improved trade-o between performance and implementation complexity [9, 17]. 2

Like Daubechies [4] and Herrmann [7], the generalization of the digital Butterworth lter in [19, 20] uses the transformation x = 12 (1 ? cos !) with z = ej! . With this change of variables, the sought rational function P (x) has a lowpass behavior over the interval [0; 1]. The maximally at behavior requires that P (x) has a zero of order N at x = 1 (this corresponds to the stopband ! = ). The at behavior at x = 0, requires that P (x) ? 1 has a high order zero at x = 0. The maximal order of that zero is L + M + 1. From [19], the function P (x) uniquely de ned by these conditions is given by N (5) P (x) = T (1f(1??x)x)NS (Sx()x)g M where   L  X N ? M + k ? 1 L + M ? k xk (6) S (x) = k M k=0

and TM denotes polynomial truncation (discarding all terms beyond the M th power). It is assumed that M is even, M  0, L  0, and N > M . ?
?
For negative values in the binomial coecient, the convention [16], n+kk?1 = (?1)k ?kn for k  0 is used. The rst M +1 coecients of the numerator of P (x) ? 1 are zero because those coecients are common to both the numerator and denominator of P (x). The later coecients of the numerator of P (x) ? 1 are simply the coecients of (1 ? x)N S (x); those are zero by design [19]. Figure 1 illustrates the maximally at scaling lter H (z ) obtained using (5) with L = 3, M = 2, N = 6. It should be noted that P (x) is not halfband (does not satisfy (3)), unless N = L + M + 1. Only then is the resulting generalized Butterworth lters a valid scaling lter of an orthogonal wavelet lter bank. To obtain H (z ) from P (z ), it must be possible to spectrally factor P (z ) as in (2). This is indeed possible because P (x) is positive over the interval (0; 1). However, the z -domain transfer function H (z ) is most p readily obtained by mapping the poles and zeros of P (x) via z = 1 ? 2x  (1 ? 2x) ? 1. That is the standard mapping, used for example in [7]. The formula (5) specializes to the well-known formula for maximally at symmetric FIR lters, rst given in [7]. For M = 0, N = L + 1, the halfband instance of that lter, (the Daubechies polynomial) is retrieved. For L = 0, N = M + 1 a classical halfband Butterworth lter is retrieved. Some unanswered questions remain. Note that for the polynomial (FIR) solution, it has been shown [21] p that (i ) The zeros of H (z ), other than those at z = ?1, lie near j1 + z j = 2. (ii ) The slope of jH (!)j2 p p is about N= at ! = =2. (iii ) The transition from 1 to 0 has a width of about 2 2=N . What are the analogous results for the rational (IIR) solution?

4 Wavelet Filters from Allpass Sums It is widely appreciated that the only polynomial solution to (1) that produces a real-valued orthogonal basis of symmetric wavelets, is the Haar solution. To obtain a 2-band wavelet basis of symmetric real-valued 3

functions with more regularity than the Haar solution, without giving up orthogonality, requires the basis functions and lters have in nite support. To this end, Herley and Vetterli gave the following form for H (z )   (7) H (z ) = 12 A(z 2 ) + z ?1 A(z ?2 ) where A(z ) is an allpass lter of degree N , and N is even. This is a parallel sum of allpass lters, a structure that has been well studied. Such lters can be implemented with low complexity structures that are robust to nite precision eects [15]. Noting that A(z )=A(z ?1 ) = A2 (z ), it is straightforward to verify that H (z ) in (7) satis es (1). Note that H (z ) has half-sample symmetry1. Because the stable noncausal impulse response of H (z ) is symmetric, H (z ) produces symmetric wavelets. An example is provided in [6]. Some general properties of symmetric IIR lter banks and wavelet bases are also discussed in [3]. Given the form (7), an allpass lter A(z ) is sought so that H (z ) is maximally at. An explicit solution exists and is supplied by the solution to a group delay approximation problem. To introduce this connection, note that the magnitude response jH (ej! )j of (7) is not aected by multiplication of H (z ) by an allpass function. In particular, H (z ) can be divided by the allpass function A(z ?2 ) without aecting the number of zeros of H (z ) at z = ?1. Therefore, we can consider instead of (7), the sum 1 A2 (z 2 ) + z ?1  : (8) 2 It is clear that if A2(z 2 )  ?z ?1 at z = ?1, then the desired lowpass behavior of H (z ) is obtained. It follows that the group delay of the allpass function A2 (z 2 ) should be 1 at ! = . In turn, it follows that the group delay of A(z 2 ) should be 1=2 at ! = , and so the group delay of A(z ) should be 1=4 at ! = 0. Writing the allpass function A(z ) as ?N A(z ) = z DD(z(1) =z ) (9) where D(z ) is of degree N , we nally nd that the group delay of the digital allpole lter 1=D(z ) should approximate2  = 1=8 ? N=2 at ! = 0. To obtain the maximally at H (z ), given the form (7), the maximum number of derivatives of the group delay of 1=D(z ) should be made to vanish at ! = 0. For this solution, H (z ) possesses 2N + 1 zeros at z = ?1. The solution is given by the digital allpole lter [5, 25], the group delay of which is maximally at at ! = 0: 1 1 (10) D(z ) = PNn=0 anz ?n where   an = (?1)n Nn (2 +(2N )n+ 1) : (11) n Half-sample symmetry is symmetry of the form h(n) = h(no ? n) where no is an odd integer. Note that if the group delay of 1=D(z) is  , then the group delay of A(z) is 2 + N . Therefore, if the group delay of A(z) is to be X , then the group delay of 1=D(z) is to be (X ? N )=2. 1

2

4

The value of the group delay at ! = 0 is  . The pochhammer symbol (x)n denotes the rising factorial: (x)n = (x)
(x +1)
(x +2)


(x + n ? 1). Interestingly (11) is also useful for the construction of biorthogonal wavelet bases where both the analysis and synthesis IIR lters are stable causal [12]. The solution (11) is generalized in [18]. The group delay of the solution (11) has a at characteristic at z = 1 only. Achieving a at delay characteristic at both z = 1 and z = ?1, where the degrees of atness at z = 1 and z = ?1 are not necessarily the same, is considered in [18]. An explicit solution to that problem is given, and is used for the design of lowpass lters realizable as a parallel sum of two allpass lters.

4.1 More General Allpass Sums It is straightforward to show that, in fact, any sum   (12) H (z ) = 21 A1 (z 2 ) + z ?1 A2 (z 2 ) : where A1(z ) and A2(z ) are allpass lters, satis es (1). Again, the maximally at group delay lter 1=D(z ) described above provides an explicit solution to this problem. For example, in [29], the linear system of equations that must be solved to obtain the maximally at solution, when A2(z ) is a pure delay z ?d , is described, but no explicit solution was given. In this case, when H (z ) = (A(z 2 ) + z ?(2d+1) )=2, the allpass lter A(z ) is given by (9), (10), (11), but now with  = d=2+1=4 ? N=2. For N = 3, d = 2, Figure 2 illustrates the scaling lter H (z ) and the scaling function. It is interesting to note that frequency selective lters that are realizable as a parallel sum of an allpass lter and a pure delay, have approximately linear-phase in the passband. It should also be noted that the classical Butterworth lter can be realized as a parallel sum of allpass lters, bringing one back to Section 3. Indeed, each of four the classical lowpass (Butterworth, Chebyshev I and II, elliptic) transfer functions can be realized as such [27]. In fact, the design of orthogonal IIR lter banks with elliptic lters, realized as allpass sums, has been described in [11]. It should be emphasized that the allpass sum scaling lters described in this paper are maximally at with respect to that structure. Accordingly, it should be noted that the maximally at intermediate DaubechiesButterworth scaling lters of Section 3 are not generally realizable as allpass sums.

5 Conclusion In conclusion, it is found that the maximally at orthogonal scaling lter associated with FIR, IIR, symmetric IIR, and approximately linear-phase IIR orthogonal wavelet bases, are obtained as instances of maximally

at digital lters. While this was well known for the Daubechies FIR and Butterworth IIR solutions, the recognition that it is also true for (i ) the intermediate Daubechies-Butterworth, (ii ) the symmetric IIR solutions described in [6] and (iii ) the approximately linear-phase IIR solution in [29], makes available new explicit formulas for P (z ). 5

In case (i ), a generalization of the classical digital Butterworth lter, that includes the well known maximally at symmetric FIR lter and Butterworth lter as special cases, provides the solution. In cases (ii ) and (iii ), the digital allpole lter, for which the group delay is maximally at at DC, provides the solution. Further work includes extending the results here to the M -band case, as has been done in [24] for the FIR scaling lters.

References [1] R. Ansari. IIR lter banks and wavelets. In A. N. Akansu and M. J. T. Smith, editors, Subband and Wavelet Transforms: Design and Applications, chapter 4, pages 113{148. Kluwer, 1996. [2] F. Argenti, G. Benelli, and A. Sciorpes. IIR implementation of wavelet decomposition for digital signal analysis. Electron. Lett., 28(5):513{515, Feb 27 1992. [3] S. Basu and H.-M. Choi. Linear phase IIR wavelets and perfect reconstruction subband coding. In Proc. IEEE Int. Conf. Systems, Man and Cybernetics, volume 4, pages 507{512, Le Touquet, October 17-20 1993. [4] I. Daubechies. Ten Lectures On Wavelets. SIAM, 1992. [5] A. Fettweis. A simple design of maximally at delay digital lters. IEEE Trans. Audio Electroacoust., 20:112{114, June 1971. [6] C. Herley and M. Vetterli. Wavelets and recursive lter banks. IEEE Trans. on Signal Processing, 41(8):2536{2556, August 1993. [7] O. Herrmann. On the approximation problem in nonrecursive digital lter design. IEEE Trans. on Circuit Theory, 18(3):411{413, May 1971. Also in [14]. [8] J. H. Husoy and R. A. Ramstad. Applications of an ecient parallel IIR lter bank to image subband coding. Signal Processing, 20(4):279{292, 1990. [9] L. B. Jackson. An improved Martinez/Parks algorithm for IIR design with unequal numbers of poles and zeros. IEEE Trans. on Signal Processing, 42(5):1234{1238, May 1994. [10] C. W. Kim, R. Ansari, and A. E. Cetin. A class of linear-phase regular biorthogonal wavelets. In Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing (ICASSP), volume 4, pages 673{676, San Francisco, March 23-26 1992. [11] H. Ochi, U. Iyer, and M. Nayeri. A design method of orthonormal wavelet bases based on IIR lters. IEICE Trans. Fundamentals, E77-A(8):1410, August 1994. Abstract only, original in Japansese. [12] S.-M. Phoong, C. W. Kim, P. P. Vaidyanathan, and R. Ansari. A new class of two-channel biorthogonal lter banks and wavelet bases. IEEE Trans. on Signal Processing, 43(3):649{665, March 1995. 6

[13] L. L. Presti and G. Olmo. A realizable paraunitary perfect reconstruction QMF bank based on IIR lters. Signal Processing, 49(2):133{143, March 1996. [14] L. R. Rabiner and C. M. Rader, editors. Digital Signal Processing. IEEE Press, 1972. [15] M. Renfors and T. Saramaki. A class of approximately linear phase digital lters composed of allpass sub lters. In Proc. IEEE Int. Symp. Circuits and Systems (ISCAS), volume 2, pages 678{681, San Jose, May 5-7 1986. [16] J. Riordan. Combinatorial Identities. John-Wiley and Sons, 1968. [17] T. Saramaki. Design of optimum wideband recursive digital lters. In Proc. IEEE Int. Symp. Circuits and Systems (ISCAS), pages 503{506, 1982. [18] I. W. Selesnick. Maximally at lowpass lters realizable as allpass sums. IEEE Trans. on Circuits and Systems II, submitted December 1996. [19] I. W. Selesnick and C. S. Burrus. Generalized digital Butterworth lter design. IEEE Trans. on Signal Processing, submitted, September 1995. [20] I. W. Selesnick and C. S. Burrus. Generalized digital Butterworth lter design. In Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing (ICASSP), volume 3, pages 1367{1370, Atlanta, May 7-10 1996. [21] J. Shen and G. Strang. The zeros of the Daubechies polynomials. Proc. Amer. Math. Soc., 1996. [22] M. J. T. Smith. IIR analysis/synthesis systems. In J. W. Woods, editor, Subband Image Coding, pages 101{142. Kluwer Academic Publishers, 1991. [23] M. J. T. Smith and S. L. Eddins. Analysis/synthesis techniques for subband image coding. IEEE Trans. on Acoust., Speech, Signal Proc., 38(8):1446{1456, August 1990. [24] P. Steen, P. Heller, R. A. Gopinath, and C. S. Burrus. Theory of regular M -band wavelet bases. IEEE Trans. on Signal Processing, 41(12):3497{3511, December 1993. [25] J. P. Thiran. Recursive digital lters with maximally at group delay. IEEE Trans. on Circuit Theory, 18(6):659{664, November 1971. [26] T. E. Tuncer and G. V. H. Sundri. Orthonormal wavelet representation using Butterworth lters. In Proc. SPIE, Adaptive and Learning Systems, pages 122{128, Orlando, April 20-21 1992. [27] P. P. Vaidyanathan, S. K. Mitra, and Y. Neuvo. A new approach to the realization of low-sensitivity IIR digital lters. IEEE Trans. on Acoust., Speech, Signal Proc., 34(2):350{361, April 1986. [28] M. Vetterli and J. Kovacevic. Wavelets and Subband Coding. Prentice Hall, 1995. [29] H. Yasuoka and M. Ikehara. A recursive orthogonal wavelet device. Electronics and Communications in Japan, Part 3, 77(11):91{101, 1994.

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Pole−zero plot of H(z) 1

0.5

6

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−0.5

−1 −1

−0.5

0

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Freq. response square magnitude 1

0.8

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Scaling function 1

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Figure 1: Intermediate Daubechies-Butterworth scaling lter. L = 3, M = 2, N = 6.

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Pole−zero plot of H(z)

1.5 1 0.5 7

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Freq. response square magnitude 1

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Figure 2: Maximally at scaling lter realizable as the sum of an allpass lter and a pure delay. H (z ) =   1 A(z 2 ) + z ?5 where the degree of A(z ) is 3. 2

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