DESIGN OF M-CHANNEL PERFECT RECONSTRUCTION FILTERBANKSWITH IIR-FIR HYBRID BUILDING BLOCKSShunsuke Iwamura, Taizo Suzuki, Yuichi Tanaka, and Masaaki Ikehara
Department of Electronics and Electrical Engineering, Keio University
Yokohama, Kanagawa, 223-8522 Japanemail: iwamura@tkhm.elec.keio.ac.jp

ABSTRACTThis paper discusses a new structure of M-channel IIR perfect reconstruction filterbanks. A novel design implementation for the new building block defined as a product of anIIR building block and an FIR building block is presented.
The IIR building blocks are written by state space representation, in which we can easily obtain a stability of the filterbank by placing its eigen values inside the unit circle. Dueto cascading of building blocks, we can get more free parameters. We introduce the condition how we obtain the newbuilding blocks without increasing of the filter order. Additionally, by showing the simulation results, we show that our
proposed FBs have better stopband attenuation than a conventional method.
Index Terms—M-channel Infinite Impulse Response Perfect Reconstruction Filterbanks, State Space Representation,
FIR-IIR hybrid building blocks

Although some design methods have been proposed forthe causal and stable case [7], they mostly treated the 2channel FB cases. Recently, an M-channel real IIR FB [8]was presented, but the design method includes a complicatedstabilization procedure of the synthesis filters.
In this paper, we introduce a class of IIR causal stable PRFBs obtained by cascading of FIR and IIR building
blocks. Firstly, we present the conventional method for IIR
PRFB by using state space representation [9] and the structure of FIR degree-1 building blocks in section 2. In section
3, we show the condition how we obtain the new building
block without increasing the order for the case of order-1 and2. Frequency responses of the proposed FBs are presented
and compared with the conventional method in section 4.
Notations: Boldface small and capital letters represent
vectors and matrices, respectively. det(), ad j() and trace()denote determinant, adjoint and trace of a matrix.

1. INTRODUCTION
2. REVIEW
2.1 Polyphase Structure for Filterbanks
Fig. 1 shows a typical structure of an M-channel maximallydecimated FB, where Hk (z) and Fk (z) are the k-th (for k =0 · · · M − 1) analysis and synthesis filter, respectively. In Fig.
2, the analysis and synthesis filters are shown by using thepolyphase matrices E(z) and R(z) represented as follows:[H0 (z) H1 (z) · · · HM−1 (z)]T = E(zM )e(z)T[F0 (z) F1 (z) · · · FM−1 (z)] = e(z)R(zM )e(z) = [1 z−1 · · · z−(M−1) ].
It is clear that the condition for PR is R(z)E(z) = cIR) (PR condition [1]).

x(n)

H0(z)

M

H1(z)

M

HM-1(z)M

Processing

Recently, many researchers have been studying multirate signal processing. One of the most efficient techniques for processing wideband digital signals in communication systems
and compressing audio, image and video signals is called asfilter bank (FB). A perfect reconstruction (PR) FB design involves its analysis and synthesis polyphase matrices E(z) andR(z) [1]. These matrices consist of polynomials of z which iscalled delay. There are two types of transfer function (TF) ofFBs, called finite impulse response (FIR) and infinite impulseresponse (IIR). In an FIR case, the denominator of TF is 1,
so we do not need to consider its stability. On the contrary,
in an IIR case, the denominator involves delays. Thus, it isnecessary to consider its stability. Meanwhile, IIR FBs havesuperiority in terms of the order of the TF. In other word, they
can be realized in lower order (much fewer multipliers andadders) than FIR FBs to obtain the desired response specifications [2]. While there are many well-developed FIR FB design theories, there are few satisfactory implementations for
IIR FB design because of the problem of its stability. Several
design methods for the case of IIR PRFBs whose analysis
FB is causal stable (poles of E(z) are inside the unit circle)
and the synthesis FB is anti-causal stable (poles of R(z) areoutside the unit circle) have already been introduced. For example, [3] proposed using all pass filters instead of delays inlossless matrices to design IIR PRFB, approximately linearphase filters using complex all pass sections is proposed in[4], and orthogonal IIR 2-channel PRFBs are designed using
all pass filters in [5]. For such FBs, synthesis filtering needs
to be performed in anti-causal fashion by introducing time reversal [6], which increases the storage cost of the system andmay only be acceptable for image processing applications.

M

F0(z)

M

F1(z)

M

FM-1(z)Figure 1: An M-channel filterbank.

(1)(c ∈y(n)

x(n)

z-1I

BM

z-1

E(z)

Processingz-1
MCM

z-1
R(z)

(a)

z-1
M
y(n)

MAMz-1I

B-C

A*Figure 2: The polyphase representation of FBs.
(b)
2.2 IIR Filterbanks Using State Space Structure
The PR condition for general M × M polyphase matrices
E(z) and R(z) are
R(z) = E−1 (z) =ad j(E(z))
det(E(z))(2)

where obviously E(z) is nonsingular. For IIR FBs, causal
stable synthesis filters are obtained if det(E(z)) is minimum
phase with E(z) being causal. The problem of constraining
E(z) with minimum phase determinant is resolved by considering the minimal factorization of E(z) in state space structure.
As it is known, any rational function E(z) can be expressed in state space structure [9],where D, A, B′ and C′ are M × M, m × m, m × M and M × m
matrices respectively (m ≤ M). A is called the state transition matrix. Then to simplify, we can rewrite the above
equation(3)

And the synthesis polyphase matrix R(z) is given by
R(z) = (I − C(zI − A∗ )−1 B)D−1 = R′ (z)D−1
whereA∗ = A − BC.

• Controllability condition
The m × Mk matrix
C (A, B) = [B AB A2 B · · · Ak B]
must be of rank m; this condition is called controllabilitycondition.
• Observability condition
The m × Mk matrix
O(C, A) = [CT (CA)T (CA2 )T · · · (CAk )T ]E(z) = D + C′ (zI − A)−1 B′

E(z) = D(I + C(zI − A)−1 B) = DE′ (z)Figure 3: A realization of the conventional filterbanks(a)E′ (z) (b)R′ (z)

(4)
(5)

Since R(z)E(z) = I, this implies R′ (z)E′ (z) = I. It is sufficient to focus on factorization of E(z) and R(z). The structure of the building block is shown in Fig. 3 (a) and (b).
Because R(z) can be calculated by matrices A, B, C and Dof E(z) uniquely, in this paper, we focus on only the structureof the building block in E(z)• Free parameters
There is no restriction of every matrix A, B, C and D.
These are all nonsingular matrices. Thus, the number ofparameters is M 2 + 2mM + m2 .
2.3 Controllability and Observability
In designing FBs using state space representation, we have toimpose the both conditions observability and controllability
to obtain minimal system E(z)[10].

must be of rank m; this condition is called observability
condition.
To keep E(z) minimal, additional constraints have to be imposed in the present design. It can be seen that if m ≤ M,
full rank matrices B and C are enough to satisfy both minimality conditions discussed above, irrespective of the rankof A. So, in the present design methods we assume m ≤ M(the dimension of the matrix A never exceeds the number ofchannels M) and rank(B) = rank(C) = m.
2.4 FIR Degree-1 Building Block
A class of causal M-channel FIR biorthogonal (BO) FBs oforder-L are factorized into [11]E(z) = WL (z) · · · W1 (z)E0( j = 1, 2, · · · m)

(6)

where E0 is an M × M nonsingular matrix and which have anFIR, anticausal inverse. In the above equation, each Wm (z)is a first-order BO building block given by††Wm (z) = I − Um Vm+ z−1 Um Vm(7)

where ·† denotes conjugate transpose and the M × γm parameter matrices Um and Vm satisfy

1 × ··· ×
 0 1 ··· × †VmUm =  ... ... . . . ... 00···

1γm ×γm-as product of (7) and (9) as

λz-1G(z) =I-UV̐

I-UV̐-cbz-1z=

UV̐+cb-λI

UV̐

(a)

(b)

where the size of U and V is M × 1 andFigure 4: The structure of Order-1 IIR-FIR hybrid building
blocks:(a) analysis filter (b) synthesis filter.
for some integer 1 ≤ γm ≤ M, where × indicates possibly
nonzero elements. This is a generalization of the paraunitaryorder-one factorization given in [12] where Um = Vm , andhas been used for factoring the biorthogonal lapped transform (BOLT) [13].
† U is γ , the McMillan degree of• Since the rank of Vmmm
Wm (z) as in (7) is γm .
• The structure in (6) completely spans all causal FIR
PRFBs having anticausal FIR inverses. The spannedanalysis filters have filter lengths no greater than M(L +1), and the McMillan degree of E(z) ranges from L toML, where L is the order of the FB.
• The Type-II synthesis polyphase matrix R(z) is given by−1−1R(z) = E−10 W1 (z) · · · WL (z)(8)−1 (z−1 ) = I − U V + zU V , which is anwhere Wmm mm mticausal and satisfies R(z)E(z) = I for PR. Due to the† U , thepossibly nonzero off-diagonal elements of Vm
m−1−1order of Wm (z ) can be greater than one, and thus
the synthesis bank can have filter lengths different fromM(L + 1).

3. PRFBS WITH IIR-FIR HYBRID BUILDINGBLOCKSIn this section, we introduce a novel building block obtainedby product of the conventional IIR building block and theFIR BO building block. This paper considers the cases where
the number of degrees of the new building block is one ortwo.
3.1 Design of Order-1 PRFBsE′ (z) = I + c(z − λ )−1 b=

G1= UV† − cb + λ I + (cb − λ I)UV† ,G2= (cb − λ I)UV† .

So as not to increase the order of the building block, the condition is(cb − λ I)U = 0.
Then it is rewritten by{λ = bc.
U =c(10)

Polyphase matrices E(z) and R(z) using proposed building
blocks are expressed as
I − UV† + z−1 (UV† + cb† − λ I)
(11)
1 − λ z−1
R(z) = [I − UV† − cb + zUV† ]D−1 .
(12)E(z) = DAs shown in (11) and (12), the systems of FB are IIR system
for E(z) and FIR system for R(z). Fig. 4 (a) and (b) show
the structures of Order-1 IIR-FIR Hybrid building blocks.
• Free parameters
In the conventional method, there are (M + 1)2 parameters for the case of Order-1. In our method, a restriction
imposed for λ which is same as matrix A in (3). There
are no restrictions of matrices D, b and c. Thus, the total
number of free parameters is M 2 + 3m − 1.
3.2 Design of Order-2 PRFBs
In this subsection, we introduce the new design for the caseof order-2 (m = 2) similar to order-1. We denote the proposedbuilding block of analysis polyphase matrix as follows:
G(z) = [I + C(zI − A)−1 B][I − UV† + z−1 UV† ]where the size of U and V is M × 2. Then G(z) can berewritten as
G(z) =I + z−1 G1 + z−2 G2[I−UV† +z−1 UV† ]1 − trace(A)z−1 + det(A)z−2

whereWe consider the condition for the case of order-1 (m = 1) inthis part. Analysis matrix E′ (z) can be rewritten by

I − z−1 (cb − λ I)
1 − λ z−1

I − z−1 (cb − λ I)· (I − UV† + z−1 UV† )1 − λ z−1
I − UV† + z−1 G1 − z−2 G2
1 − λ z−1

G1

= CB − trace(A)I,G2= det(A)(I − CA−1 B).

The constraints can be expressed as(9)where b and c are a row and a column vector respectively.
Since λ is pole, we obtain the stable filterbank if λ is placed
inside unit circle. Our proposed building block is representeddet(A)(I − CA−1 B)UV† = 0.
then it is rewritten by{A = BC.
U= C(13)det(A)z-1-trace(A)z-1

E'0-CBz

E'0z-1

UV

E'1(b)

z-1

(a)

(b)

Figure 6: Frequency responses of 4-channel order-2 prop. FB:(a) analysis bank (b) synthesis bank.

E'2(a)
Figure 5: The structure of IIR-FIR hybrid building blocks:(a) analysis filter (b) synthesis filter.

Hence, our proposed E(z) and R(z) are represented as
E(z) = D

E′0 + z−1 E′1 + z−2 E′2
1 − trace(A)z−1 + det(A)z−2R(z) = [E′0 − CB + zUV† ]D−1(14)(15)

(a)

(b)

whereFigure 7: Frequency responses of 4-channel order-2 conv. FB[9]: (a) analysis bank (b) synthesis bank.

E′0 = I − UV†
E′1 = {CB − trace(A)I}E0 + UV†
E′2 = det(A)(I − CA−1 B) − {CB − trace(A)I}.
As it is mentioned above, filterbanks of the proposed method
have both IIR and FIR filters. By cascading these new building blocks with order-1 and 2, we can obtain a higher orderfilterbank. Fig. 5 shows the structure of analysis and synthesis IIR-FIR hybrid building blocks.
• Free parameters
In the conventional method, the number of free parameters is M 2 + 4M + 4 for the case of Order-2. In proposed
method, there is some restrictions same as the case ofOrder-1. Thus, we have M 2 +6M −3 parameters totall1y.
4. RESULTS
In this section, we present the design examples of proposed
IIR PRFBs. In this paper, we optimize the cost function Φ todesign FB which is calculated as the weighted linear combination of these filters as follows:)
M−1 ((i)(i)Φ = ∑ E pass + Estop ,i=0(i)
E pass =(i)Estop =∫

5. CONCLUSION
In this paper, a new design approach of M-channel IIR
PRFBs based on the IIR-FIR hybrid building blocks are presented. we impose the restriction in the building blocks tokeep their order despite cascading of building blocks. Since
our method has more free parameters than the conventional
one, the proposed FBs have better stopband attenuation. Asa feature, our IIR PRFBs have IIR analysis filters and FIR
synthesis filters and the order of the synthesis filters are consistently.
REFERENCES

Wp (ω )(1 − |Hi (ω )|) d ω2

∫In Fig. 6, we show the frequency responses of 4-channel
order-2 proposed IIR PRFBs. These frequency responses
have better stopband attenuation than the conventional FBin Fig.7. Also Table 1 compares stopband attenuations of
the filterbanks. It is obvious that the proposed FBs have better performance than the conventional one in the same order. Also Fig. 8 shows frequency responses of the 8-channel
order-2 proposed IIR PRFB.

Ws (ω )|Hi (ω )|2 d ω

where i ∈ {0, 1] and H(e jω ) is an impulse response of the ith(i)(i)
filter. E pass and Estop are called passband energy and stopband energy respectively. Wp and Ws are positive weightingfunction.

[1] P. P. Vaidyanathan “Multirate systems and filter banks,”Englewood Cliffs, NJ: Prentice Hall, 1992.
[2] Hoang Duong Tuan, Tran Thai Son, Pierre Apkarian andTruong Q. Nguyen “Low-order IIR filter bank design,IEEE Transactions on Circuits and Systems-I: RegularPapers, vol. 52, no. 8, p.p. 1673–1683 Aug. 2005
[3] Z. Doğanata and P.P. Vaidyanathan, “Minimal structures
for the implementation of digital rational lossless sys-Table 1: Comparison of the stopband attenuations of 4-channel fiterbanks

k0123conv. FB in [9](order-2)
Hk (z)Fk (z)-18.5 dB -16.3 dB-14.0 dB -13.9 dB-16.0 dB -13.4 dB-18.1 dB -17.9 dB

prop. FB(order-1)
Hk (z)Fk (z)-17.1 dB -17.2 dB-12.8 dB -12.6 dB-13.7 dB -13.8 dB-18.3 dB -18.2 dB

prop. FB
(order-2)
Hk (z)Fk (z)-22.5 dB -25.6 dB-23.5 dB -24.6 dB-27.0 dB -27.0 dB-23.8 dB -20.6 dB

turally Regular Biorthogonal Filter Banks,” IEEE Transactions on Signal Processing, vol. 54, no. 2, Feb. 2006.
[12] X. Gao, T. Q. Nguyen, and G.Strang, “On factorization of M-channel paraunitary filterbanks,” IEEE Transactions on Signal Processing, vol. 49, no. 7, pp. 1433–1446, Jul. 2001.
[13] P. P. Vaidyanathan and T. Chen, “Role of anticausal inverses in multirate filter-banks Part II: The FIR case, factorizations, and biorthogonal lapped transforms,” IEEETransactions on Signal Processing, vol. 43, no. 5, pp.
1103–1115, May 1995.
(a)

(b)

Figure 8: Frequency responses of 8-channel order-2 prop. FB:(a) analysis bank (b) synthesis banktems,” IEEE Transactions on Acoustics, Speech and Signal Processing, vol. 38, no. 12, pp. 2058–2074, Dec.
1990.
[4] F. Argenti, V. Cappellini, A. Sciorpes and A. N. Venetsanopolous, “Design of IIR linear-phase QMF banks
based on complex allpass sectioins,” IEEE Transactions
on Signal Processing, vol. 44, no. 5, pp. 1262–1267,
May 1996.
[5] X. Zhang and T. Yoshikawa, “Design of orthonormal IIR
wavelet filter banks using allpass filters,” Signal Processing, vol.78, no.1, pp. 91–100, 1999
[6] M. J. T. Smith and S. L. Eddins, “Analysis/Synthesistechniques for subband image coding,” IEEE Transactions on Acoustics, Speech and Signal Processing, vol.
38, no. 8, pp. 1446–1456, Aug. 1990.
[7] M. J. T. Smith, “IIR analysis/synthesis systems in subband image coding,” Boston, MA: Kluwer, 1991.
[8] S. Basu., C. H. Chiang, and H.M. Choi, ”Wavelets andperfect-reconstruction subband coding with causal stable
IIR filters,” IEEE Transactions on Circuits and SystemsII, vol. 42, pp. 24–38, Jan. 1998
[9] Viswanadha Reddy, P. and Makur, A. “Design of M
channel IIR causal stable perfect reconstruction filterbanks using similarity transformation,” IEEE International Conderence on Image Processing International
Conference, vol. 1, pp. 1045–1048 Sept. 2003[10] I. Gohberg, L. Rodman and P.Lancaster, “Invariant
Subspaces of Matrices with Applications,” New York.
John Wiley, 1986.
[11] Ying-Yui Chen , Soontorn Oraintara and Kevin Amaratunga “Theory and Factorization for a Class of Struc-