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TWO-CHANNEL 1D AND 20 BIORTHONORMAL FILTER BANKS WITH CAUSAL STABLE IIR AND LINEAR PHASE FIR FILTERS t

See-May Phoong and P. P. Vaidyanathan Department of Electrical Engineering, 116-81

California Institute of Technology Pasadena, CA 91125

Abstract. A new class of twechannel biorthogonal filter banks is derived. The framework covers two use- ful subclasses: (i) causal stable IIR filter banks; (ii) linear phase FIR filter banks. Perfect reconstruction is structurally preserved and the structural complexity is very low. Filter banks of high frequency selectivity can be achieved by simply designing a single transfer function. Furthermore zeros of arbitrary multiplicity at aliasing frequency can be easily imposed, for the purpose of generating wavelets with regularity prop erty. We also map the proposed 1D framework into 2D. The mapping preserves: (i) perfect reconstruction; (ii) stability in the IIR case; (iii) linear phase in the FIR case; (iv) zeros at aliasing frequency and (v) frequency characteristic of the filters.

1. Introduction Fig. l.l(a) shows a twechannel filter bank and Fig.

l . l (b) shows its polyphase form. A number of per- fect reconstruction (PR) or nearly PR systems have been reported before [1]-[4]. The earliest good designs for the IIR filter banks were such that the analysis bank was paraunitary and the polyphase components of Ho(z) and Hl(z) were allpass (see pp. 201 of [l]). Even though all the IIR filters are causal stable, the reconstructed signal suffers from phase distortion. IIR filter banks with the perfect reconstruction property typically have noncausal stable filters or causal unsta- ble filters [2], [3]. Recently the authors in [4] proposed a IIR PR technique providing causal stable solutions. However there is no known good design technique for causal stable IIR filter banks even though this is the- oretically feasible. Furthermore an approximately lin- ear phase response in passband, as well as regularity requirements (i.e., forcing sufficient number of zeros of Ho(z) at T ) make the design more complicated. In this paper we will show how to achieve all of this. The s e lution also gives rise to an FIR special case, which is identical to the one proposed independently in [5].

In the 2D case, it was shown in [6] that FIR PR filter bank can be obtained by transformation. In this paper, we will provide a different mapping and show how to transform the system into a 2D PR filter

t Work supported by ONR Grant N00014-93-1-0231, and funds from Tektronix, Inc.

bank which retains all of the above crucial properties. Remarks: Some derivations in this paper are omitted,

the readers are refered to [9] for the details. While this paper was under preparation, we noticed that some of the material here is similar to that in [7], [8].

2. A Framework for I D Biorthogonal Filter Banks 1.1, where a twechannel system is

shown. Let us consider the polyphase matrix E(z) with the following form:

Consider Fig.

where P(z) is any causal stable transfer function. To achieve perfect reconstruction, the synthesis polyphase matrix R(z) is chosen to be:

R(z) = z-3N+1E-1 ( z 1. (2-2) In this case, the stability and causality of R(z) is also guaranteed. With (2.1) and (2.2) we get the following expressions for the analysis and synthesis filters:

Hl(2) = - P ( Z 2 ) H 0 ( Z ) + z -4N+1, ( 2 . 3 ~ ) Fo(z) = - H ~ ( - z ) , F ~ ( z ) = Ho(-z) . (2.3b)

Eqn. (2.3b) ensures that {Fo, Fl} is a lowpass/high- pass pair if {Ho, HI} is a lowpass/highpass pair. F'rom (2.1), we have the implementation of the filter bank shown in Fig. 2.1. The structure is similar to a ladder network structure.

Design of p( z ) First notice that the filters {Ho(z) ,H1(z)} can be

made ideal lowpasslhighpass filters if p ( z ) has the following magnitude and phase responses:

Ip(ej2")1 = 1, v U , ( 2 . 4 ~ )

(2.4b) This ideal choice of p ( z ) requires infinite complexity. Therefore, we have to design p(z) to approximate the

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conditions in (2.4). However the approximation will not change the perfect reconstruction property because the analysis and synthesis polyphase matrices satisfy R(z)E(z) = 0 . 5 ~ - ~ ~ + ' 1 , regardless of the choice of p(z) . Fig. 2.1 shows that the frequency responses of all the analysis and synthesis filters depend on one single function p( z ) only. The frequency selectivity of all four filters depends on how well p ( z ) approximates conditions (2.4). This makes the design procedure simple. We will consider two simple but useful approximations which correspond to the following two Cas@:

A. Causal Stable IIR Biorthogonal Filter Banks Here P(z) in Fig. 2.1 is taken to be the causal stable

real allpass function

where = 1 and a k are real. With this choice of /3(.z), Eqn (2.4a) is met exactly. We design the phase response of the allpass filter so that (2.4b) is approximately satisfied. This leads to a causal stable IIR biorthogonal system.

B. Linear phase F I R Biorthogonal Filter Banks To satisfy the condition (2.4b) exactly, P(z ) can be

chosen as a Type 2 linear phase function [l] (filter with a symmetric impulse response of even length):

k= 1

where 'Uk satisfy the condition cr=l Wk = 0.5 so that V(Go) = 1. The coefficients v k are optimized such that the amplitude response of V(ejw) is as close to unity as possible so that (2.4a) is well-approximated. This leads to a linear phase biorthogonal system.

Some Properties of the Proposed Filter Banks In the following, we will list some of the useful

properties of the proposed biorthogonal filter banks without proofs. The readers are refered to [9] for the details.

1. 2.

3.

4.

5.

Perfect reconstruction (PR) is preserved structurally. In the IIR case, all the analysis and synthesis filters are causal and stable if the allpass function p ( z ) is. In the FIR case, the filters are exact linear-phase. In the IIR case, we can force the phase response of the filters to be nearly linear in the passband. Fo(z) and Ho(z) have the same set of zeros on the unit circle. Both the product Ho(z)Fo(z) and the filter Ho(z ) are halfband.

Without subband quantization, the ladder structure shown in Fig. 2.1 preserves PR even when all the coefficients and the intermediate results are rounded off (similar to [ll]) . The proposed filter bank has a very low complexity. If p(z) is taken as (2.5) (IIR) or (2.6) (FIR), the analysis (or synthesis) bank requires approximately N multiplications per input sample, assuming that (IIR case) the allpass function is implemented by using the one multiplier lattice [l].

3. Imposition of Multiple Zeros at 7r The relation between continuoustime wavelet and

discretetime perfect reconstruction filter bank is well known [lo]. It was shown that in order to achieve wavelet functions of high regularity, the lowpass filters Ho(z ) and Fo(z) should have a sufficient number of zeros at the aliasing frequency 7r. For the proposed filter bank, because of Property 4, Fo(z) is guaranteed to have the same number of zeros at 7r as Ho(z ) . Therefore by imposing sufficient number of zeros at n for Ho(z) , we ensure the regularity of both the analysis and the synthesis wavelets. We will discuss only the IIR case since the derivation for the FIR case is very similar. Causal Stable IIR Wavelet Bases

Since the denominator does not provide any zeros, we consider only the numerator of Ho(z). Except for a delay, the numerator of Ho(z) can be written as:

N

(3.1) PR(W) = a k cos(2k - 1/2)w. k=O

To obtain r zeros at z = -1. we set

Note that when i is even, PZ'(7r) is always equal to zero. This proves that PR(w) always has an odd number of zeros at w = 7r. Therefore, we can write r = 2r0 + 1. In this case, we obtain a set of ro linear constraints as follows:

N

The set of linear constraints in (3.3) can be incorpe rated in the optimization of filter coefficients.

Maximally flat IIR wavelets To obtain a maximally flat solution, i.e., maximum

possible number of zeros at 7r consistent with the

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constraint that Ho(z) = 0 . 5 ( ~ - ~ ~ + z-A(z2)), we set r g = N . We have

(3.4) where Xk = 1 - 4k. As the matrix is nonsingular, the solution for ak will always exist. Furthermore, it can be shown that ak has the closed form solution [9]:

O S k S N , (22 - 1)

(3.5) where (f) = 6. Note that although these filters have a numerator of degree 4N, they have only 2N + 1 zeros at z = -1 . This implies that some of the zeros are not at z = -1. Therefore these IIR maximally flat filters are different from the Butterworth halfband filters.

4. Quincunx Fifier Banks Obtained by Mapping In this section, the proposed 1D biorthogonal filter

banks is mapped into the 2D quincunx filter banks with perfect reconstruction. Here the decimation matrix and the coset vectors are respectively:

Given any 1D biorthogonal system with the polyphase matrix of the form in (2.1), we will use the following transformation on the polyphase components:

(i) First replace the 1D transfer function p ( z ) with the separable 2D transfer function P(zo)P(zl).

(ii) Fkplace all the remaining 1D delay z-l with the 2D delay zO1 z l l . This results in nonseparable analysis and synthesis

filters as we will see. Under this transformation, the analysis polyphase matrix E(z) of the 2D system can be written as:

p(zo)p(zl) ) ( -P(z ,2P(zd 2 ;) ( (yN (Z0z1)-2N+1 1

By using the noble identities 111, we can write the analysis and synthesis filters as:

It can be shown that all the properties listed in Sec. 2 are preserved under the transformation. Furthermore if the 1D transfer function p ( z ) approximates the conditions in (2.4) well, then we can show that Ho(z ) and Hl(z) are respectively diamond and diamond- complement filters.

5. Numerical Examples We will provide examples for the causal stable IIR

case only. For the linear phase FIR case, see [9]. Example I. 1D IIR filter banks: In this example, the

transfer function @(z) is taken to be an allpass function A(z) with order N = 3. The filter bank has very low complexity: To implement the analysis (or synthesis) bank, we need only 3 multiplications per input sample! By using the eigenfilter approach, we optimize the coefficients ak such that maximum attenuation in the stopband of Ho(z) is achieved. The coefficients are obtained as a1 = 0.473, a2 = -0.094, and a3 = 0.025. The passband edge wp = 0 . 4 ~ and the stopband egde U, = 0 . 6 ~ . The stopband attenuations 6,(Ho) M 41.9 dB and 6,(H1) = 32 dB. The magnitude responses of all four filters are shown in Fig. 5.1.

Example 2. 2 0 IIR filter banks: In this example, we transform the 1D filter bank in Example 1 into the 2D case by using the mapping in Section 4. Since N = 3, the allpass function A(z ) needs only 3 multiplications. Since the complexity of the 2D analysis (or synthesis) bank is equal to twice that of A ( z ) , we need only 6 multiplications per input pixel to implement the analysis (synthesis) bank. The responses of &(a) and Hl(z) are shown in Fig. 5.2(a) and (b) respectively. The stopband attenuation 6,(Ho) % 42 dl3 and S,(H1) M 32 dB.

References P. P. Vaidyanathan, Multirate systems and filter banks, Englewood Cliffs, NJ: Prentice Hall, 1993. T. A. Ramstad, IIR filter bank for subband coding of images, Proc. I E E E ISCAS, pp. 827-830, ESPOO, Finland, 1988. C. Herley, and M. Vetterli, Wavelets and recursive filter banks, I E E E P a n s . o n Signal Proc., vol. 41, no. 8, pp. 253656, Aug. 1993. S. Basu, C.-H. Chiang, and H. M. Choi, Wavelets and perfect reconstruction subband coding with cau- sal stable IIR filters, preprint. H. Kiya, M. Yae, and M. Iwahashi, A linear phase two-channel filter bank allowing perfect reconstruc- tion, Proc. I E E E ISCAS, pp. 951-954, San Diego, 1992. D. B. H. Tay, and N. G. Kingsbury, Flexible de- sign of multidimensional perfect reconstruction FIR 2-band filters using transformations of variables, I E E E B a n s . on IP, vol. 2, pp. 466480, Oct. 1993. C. W. Kim, and R. Ansari, FIR/IIR exact recon- struction filter banks with applications to subband

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coding of images," N dwest CAS Symp., Monterey, CA. Mav 1991.

[8] C. W. Kim, and R. Ansari, " Subband decomposition procedure for quincunx sampling grid," Prvc. SPIE Visual Comm. and Image Proc., Boston, Ma., Nov. 1991.

[9] S.-M. Phoong, and P. P. Vaidyanathan, "A new class of twcxhannel biorthogonal filter banks and wavelet bases," Techniml Report, Caltech, Pasadena CA, Oct. 1993.

[lo] I. Daubechies, "Orthonormal bases of compactly supported wavelets," Commun. Pure Appl. Math., vol. 41, pp. 909-996, Yov. 1988.

[ll] A. M. Bruekers, and Ed W. M. van den Enden, "New networks for mrfect inversion and Derfect reconstruc- tion," IEEE jour. Selected Areas in Comm., pp. 130- 137, Jan. 1992.

(a) t I

2 -1 ai Fig. 1.1 (a) A 2 channel filter bank system.

(b) The polyphase representation.

-1 2

-2N+1

P

Fig. 2.1 A framework for a new dass of biorthogonal filter bank. (a) Analysis (b) Synthesis bank.

R(z)

- .......... - - - --

0.0 0.1 0 2 0.3 0.4 0.5 " a l b e d F " y

Fig. 5.1 An 1 D IIR biorthogonal filter bank.

, . ".-I -

no n1 FO F1

Fig. 5.2 An 20 IIR bimhogonal filter bank (a) Ho(z) and (b) Hi(z).