IEICE TRANS. FUNDAMENTALS, VOL.E90–A, NO.8 AUGUST 2007

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PAPER

Design of M-Channel Perfect Reconstruction Filter Banks with
IIR-FIR Hybrid Building Blocks
Shunsuke IWAMURA†a) , Student Member, Taizo SUZUKI†† , Member, Yuichi TANAKA† , Student Member,and Masaaki IKEHARA† , Member

SUMMARY
This paper discusses a new structure of M-channel IIR
perfect reconstruction filter banks. A novel building block defined as acascade connection of some IIR building blocks and FIR building blocks ispresented. An IIR building block is written by state space representation,
where we easily obtain a stable filter bank by setting eigenvalues of thestate transition matrix into the unit circle. Due to cascade connection ofbuilding blocks, we are able to design a system with a larger number offree parameters while keeping the stability. We introduce the condition
which obtains the new building block without increasing of the filter orderin spite of cascade connection. Additionally, by showing the simulation
results, we show that this implementation has a better stopband attenuation
than conventional methods.
key words: M-channel infinite impulse response perfect reconstruction
filter banks, state space representation, FIR-IIR hybrid building blocks

1.

Introduction

Recently, many researchers have been studying multirate
signal processing. One of the most eﬃcient techniques forprocessing wideband digital signals in communication systems and compressing audio, image and video signals iscalled filter bank (FB). A perfect reconstruction (PR) FB design involves its analysis and synthesis polyphase matrices
E(z) and R(z) [1]. These matrices consist of polynomialsof z which is a delay. There are two types of transfer function of FBs, called finite impulse response (FIR) and infinite
impulse response (IIR). In an FIR case, the denominator oftransfer function is 1, so we do not need to consider its stability. On the contrary, in an IIR case, the denominator involves delays. Thus, it is necessary to consider its stability.
Meanwhile, IIR FBs have superiority in terms of the orderof the transfer function. In other words, they can be realized in lower order (much fewer multipliers and adders) thanFIR 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 diﬃcultly to ensure 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 cirManuscript received August 7, 2006.
Manuscript revised January 27, 2007.
Final manuscript received April 25, 2007.
†
The authors are with the Department of Electronics andElectrical Engineering, Keio University, Yokohama-shi, 223-8522
Japan.
††
The author is with TOPPAN PRINTING CO., LTD., Tokyo,
101-0024 Japan.
a) E-mail: iwamura@tkhm.elec.keio.ac.jp
DOI: 10.1093/ietfec/e90–a.8.1636cle) and the synthesis FB is anti-causal stable (poles of R(z)are outside the unit circle) have already been introduced.
For example, [3] proposed using allpass filters instead ofdelays in lossless matrices to design IIR PRFB, approximately linear-phase filters using complex allpass sections is
proposed in [4], and orthogonal IIR 2-channel PRFBs aredesigned 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 and may only be acceptable for finite length signals such as images. Although some design
methods have been proposed for the causal and stable case[7], they mostly treated the 2-channel FB cases. Recently,
an M-channel real IIR FB [8] was presented, but the design method includes a complicated stabilization procedureof the synthesis filters.
Another problem of IIR FB design is optimization
problem. High order IIR FB design needs a lot of computation. It is also diﬃcult to optimize higher order FBs as
a building block. To solve this problem, these FBs can beobtained by cascading lower order building blocks.
Organization: In this paper, we introduce a class of IIR
causal stable PRFBs obtained by cascade connection of FIR
and IIR building blocks. Firstly, we present the conventional
method for IIR PRFB by using the state space representation [9] and the structure of FIR degree-1 building blocks in
Sect. 2. In Sect. 3, we show the condition which obtains the
new building block without increasing the order for the caseof order-1 and 2. If we have the building block of order-1
and 2, we can easily obtain higher order IIR FBs by cascade
connection of these building blocks. Frequency responsesof the proposed FBs are presented and compared with theconventional method in Sect. 4.
Notation: Boldface small and capital letters represent
vectors and matrices, respectively. det(), ad j() and trace()denote determinant, adjoint and trace of a matrix. Superscripts † indicates the conjugate transpose of matrices andvectors. A McMillan degree of a building block is the minimum number of delay units required to realize the building
block.
2.

Review

2.1 Polyphase Structure for Filter Banks
Figure 1 shows a typical structure of an M-channel maxi-

c 2007 The Institute of Electronics, Information and Communication Engineers
Copyright IWAMURA et al.: DESIGN OF M-CHANNEL PERFECT RECONSTRUCTION FILTER BANKS

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mally decimated FB, where Hk (z) and Fk (z) are the k-th (fork = 0 · · · M − 1) analysis and synthesis filter, respectively. In
Fig. 2, the analysis and synthesis filters are shown by using
the polyphase matrices E(z) and R(z) represented as follows:[H0 (z) H1 (z) · · · H M−1 (z)]T = E(z M )e(z)T[F0 (z) F1 (z) · · · F M−1 (z)] = e(z)R(z M )e(z) = [1 z−1 · · · z−(M−1) ].

2.2 IIR Filter Banks Using State Space Structure
The PR condition for general M × M polyphase matrices
E(z) and R(z) are
ad j(E(z))R(z) = 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 the state spacestructure.
As it is known, any rational function E(z) can be expressed in the state space structure [9],E(z) = D + C (zI − A)−1 B
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
equationFig. 1

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

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). E (z) andR (z) are called building blocks, and D is called a last block.
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 D of E(z) uniquely, in this paper, we focus on only the
structure of 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 numberof parameters is M 2 + 2mM + m2 .
2.2.1 Stability
The design of IIR FBs using the state space representation,
all transfer functions have the same denominator. So theanalysis filter bank is rewritten as
E(z) =

B(z)
,
A(z)

where B(z) is same form as FIR polyphase analysis matrices
and A(z) is m-th order polynomial of z. As stated earlier, indesigning IIR FBs, we have to take the stability into account.
From system theory, it is well known that poles of E(z)are same as the eigenvalues of A in the analysis bank. Similarly, the poles of R(z) are same as the eigenvalues of A∗ .
For causal stable analysis and synthesis filters, poles of E(z)and R(z) have to be inside unit circle, which implies eigenvalues of A and A∗ have to be inside unit circle. So thedesign problem boils down to characterizing the set of matrices {A, B, C, D} such that the eigenvalues of A and A∗ areinside the unit circle. However it is very hard to keep the stability on both analysis and synthesis bank when m is large.

An M-channel filterbank.

(b)
The polyphase representation of FBs.

(4)

(5)

(a)

Fig. 2

(3)

where
(1)

It is clear that the condition for PR is R(z)E(z) = cI (c ∈R) (PR condition [1]).

−1E(z) = D(I + C(zI − A)−1 B) = DE (z)Fig. 3A realization of the conventional (a) E (z), (b) R (z).

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• Since the rank of V†m Um is γm , the McMillan degree ofWm (z) as in (7) is γm . When γm = 1 and M/2, the structure is called degree-1 and order-1 building block, respectively. In this paper, we focus on the order-1 structure, since more design parameters can be used than the
degree-1 structure.
• The structure in (6) completely spans all causal FIR
PRFBs having anticausal FIR inverses. The spanned
analysis 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 givenby

2.2.2 Controllability and Observability
In designing FBs using state space representation, we haveto impose the both conditions observability and controllability to obtain minimal system E(z) [13].
• Controllability condition
The m × Mk matrixC(A, B) = [B AB A2 B · · · Ak B]
must be of rank m; this condition is called controllability condition.

−1−1
R(z) = E−1
0 W1 (z) · · · WL (z)• Observability condition
The m × Mk matrix

Due to possibly nonzero oﬀ-diagonal elements of−1
V†m Um , the order of W−1m (z ) can be greater than one.
Thus the synthesis bank could have diﬀerent filter
lengths from M(L+1). In this paper, we set V†m Um = Iγ .
††−1Hence, W−1m (z ) = I − Um Vm + zUm Vm , which is anticausal and satisfies R(z)E(z) = I for PR.

O(C, A) = [CT (CA)T (CA2 )T · · · (CAk )T ]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.
2.3 FIR Degree-1 Building Block
A class of causal M-channel FIR biorthogonal (BO) FBs oforder-L are factorized into [10]E(z) = WL (z) · · · W1 (z)E0( j = 1, 2, · · · m)

(6)

where E0 is an M × M nonsingular matrix and is called thefirst block. Wm (z)(m = 1 . . . L) is an M × M first-order BO
building block given by
Wm (z) = I − Um V†m + z−1 Um V†m(7)

where ·† denotes conjugate transpose and the M ×γm parameter matrices Um and Vm satisfy⎡⎤⎢⎢⎢ 1 × · · · × ⎥⎥⎥⎢⎢⎢ 0 1 · · · × ⎥⎥⎥⎢⎥⎥⎥V†m Um = ⎢⎢⎢⎢⎢ . . .
. . ... ⎥⎥⎥⎥⎥⎢⎢⎢ .. ..
⎥⎦⎣0 0 · · · 1 γm ×γm
for some integer 1 ≤ γm ≤ M, where × indicates possibly
nonzero elements. This is a generalization of the paraunitary
factorization given in [11] where Um = Vm , and has beenused for a factorization of the biorthogonal lapped transform(BOLT) [12]. The properties of the matrix are described asfollows:(8)

3.

PRFBs with IIR-FIR Hybrid Building Blocks

In this section, we introduce a novel building block obtained by cascade connection of the conventional IIR building blocks and the FIR BO building blocks.
As mentioned above, it is very hard to design IIR FBwith higher order directly using (3) and (4) while keeping
the stability. When we design the general IIR filters withhigher order, the IIR transfer functions are implemented bycascade connection of first and second order transfer functions, which is easy to impose the stability. Similarly, we
consider the IIR FB with higher order is implemented bycascade connection of lower order IIR FBs as
E(z) =

K
k=1

E1(k) (z)NE2(n) (z)

(9)

n=1

where E1 (z) and E2 (z) are the polyphase matrices with firstand second order denominators, respectively. Therefore thetotal order of the denominator is K + 2M.
This paper considers only cases where the order of the
new building block is one or two.
3.1 Design of Order-1 Hybrid Building Block
We consider the condition for the case of order-1 (m = 1) inthis part. The analysis matrix E (z) can be rewritten byE (z) = I + c(z − λ)−1 b†
I − z−1 (cb† − λI)=
1 − λz−1(10)where b and c are M ×1 vectors. Since λ is pole, |λ| has to beless than 1. In order to increase the number of free parameters while keeping the order of the numerator, we connect

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Fig. 4 The structure of order-1 IIR-FIR hybrid building blocks:(a) analysis filter (b) synthesis filter.

the first degree FIR building block to an IIR building blockas
I − z−1 (cb† − λI)· (I − UV† + z−1 UV† )1 − λz−1
I − UV† + z−1 G1 − z−2 G2=
1 − λz−1(a)E1 (z) =Fig. 5 The structure of IIR-FIR hybrid building blocks: (a) analysis filter(b) synthesis filter.

(11)
det(zI − A) = z2 − ztrace(A) + det(A)where the size of U and V is M × 1 and(zI − A)−1 =G1 = UV† − cb† + λI + (cb† − λI)UV† ,G2 = (cb† − λI)UV† .

zI − det(A)Adet(zI − A)

(16)zCB − det(A)CA−1 B
det(zI − A)
I + z−1 G1 + z−2 G2=
1 − z−1 trace(A) + z−2 det(A)E (z) =I −(cb† − λI)U = 0.

where
(12)

Therefore, the proposed order-1 building blocks E1 (z) andR1 (z) are expressed as
I − UV† + z−1 (UV† + cb† − λI)
E1 (z) =1 − λz−1†
R1 (z) = [I − UV − cb† + zUV† ].

(15)

−1

From (15) and (16), the E (z) in (3) is formulated as

So as not to increase the order of the building block, thecondition isThis condition is satisfied by
λ = b† c
.
U =
c

(b)

(13)
(14)

As shown in (13) and (14), the building blocks are IIR system for E1 (z) and FIR system for R1 (z). Figure 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, the restriction imposed for λ = b† c  0 and V† c = 1. 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 Hybrid Building Block
In this subsection, we introduce the new design for the caseof order-2 (m = 2) similar to order-1. In this case, A, B and
C are a 2 × 2, 2 × M and M × 2 matrix, respectively, and(zI − A)−1 can be rewritten as follows

G1 = CB − trace(A)I,G2 = det(A)(I − CA−1 B).
Then, we denote the proposed building block of the analysis
polyphase matrix as follows:E2 (z) =I + z−1 G1 + z−2 G2
1 − trace(A)z−1 + det(A)z−2· I − UV† + z−1 UV†

where the size of U and V is M × 2. To eliminate the termof z−3 , the constraints not to increase the order of numerator
can be expressed asdet(A)(I − CA−1 B)UV† = 0.
then it is rewritten by
A = BC
.
U = C

(17)

Hence, our proposed order-2 building blocks E2 (z) andR2 (z) are represented asG1 + z−1 G2 + z−2 G3
1 − trace(A)z−1 + det(A)z−2
R2 (z) = [G1 − CB + zUV† ]E2 (z) =

(18)
(19)

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

where

E (i)
stop =
R(i)pass =


Figure 5 shows the structure of analysis and synthesis IIRFIR 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 are the restrictions for A = BC
and V† C = I. Thus, we have M 2 + 6M − 3 parameters
totally.
3.3 Higher Order Hybrid Building BlockIn previous section, we show the novel structure of order-1
and 2 hybrid building blocks. In Table 1, the form of TFs
are shown. By cascading these building blocks, building
blocks of all the arbitrary forms can be designed. Moreover,filter length diﬀers with the analysis and synthesis filter ofthe proposed order-2 hybrid building block. Therefore, we
can easily obtain unequal length filter banks.
ResultIn this section, we present the design examples of proposed
IIR PRFBs. In this paper, we optimize the cost function Φ
to design FB which is calculated as weighted linear combination of following errors:
M−1
(i)
(i)
(i)
E (i)
pass + E stop + R pass + R stop ,

Φ=
i=0

E (i)
pass =(20)ω∈passiW p (ω)(1 − |Hi (ω)|)2 dω

Table 1 TFs form.
analysis filter synthesis filter

Filter orderprop. order-1

×+×z−1×+×z−1× + ×zprop. order-2

×+×z−1 +×z−2
×+×z−1 +×z−2

× + ×zconv. order-1

×+×z−1×+×z−1×+×z−1×+×z−1

Table 2k0
1
2
3

W s (ω)|Hi (ω)|2 dωG1 = I − UV†
G2 = {CB − trace(A)I}E0 + UV†
G3 = det(A)(I − CA−1 B) − {CB − trace(A)I}.

4.

ω∈stopiR(i)stop =

ω∈passiω∈stopiW p (ω)(1 − |Fi (ω)|)2 dω
W s (ω)|Fi (ω)|2 dω

where Hi (e jω ) and Fi (e jω ) (fori = 0, 1, · · · M − 1) are thefrequency responses of the ith filter, which can be obtainedby substituting designed FBs using the proposed building(i)
blocks into (1). E (i)pass and E stop are error energies in the passband and stopband of the analysis filterbank, respectively.
(i)
Similarly, R(i)pass and R stop are error energies of the synthesis
filterbank. W p and W s are positive weighting functions. We
optimize the parameters of B, C, V and D using fminunc.m
in Matlab 2006a such that the cost function in (20) is minimized.
Remark: Order-1 building blocks should be optimized
by taking the λ inside the unit circle in order to achieve stability. Due to the condition A = BC, the stability might belost. In order to impose the stability, the optimization should
be done while checking the nonsingularity of A which is
2 × 2 matrix.
4.1 Design Example
Figure 6 show the magnitude responses in the analysis andsynthesis banks of proposed 4-channel IIR PRFBs. The
order-1 FB in Fig. 6(a) and (b) is designed by the proposedorder-1 building block in Sect. 3.1. Figure 6(c) and (d) illustrate the frequency responses of FBs designed by the
proposed building block in Sect. 3.2. Since the inverse of
the proposed order-2 building block has order-1 structure asshown in Table 1, the order-2 IIR PRFB is unequal length
FB. As a comparison, the freqency responses of the conventional IIR PRFB are shown with dotted lines in Fig. 6(c) and(d). Higher order FBs can be obtained by cascading connection of the proposed order-1 and 2 building blocks. For
example, Fig. 6(e) and (f) are the freqency responses of theorder-3 IIR PRFB by cascading E1 (z) and E2 (z). The frequency responses of the proposed method have better stopband attenuations than the conventional method. Table 2
compares stopband attenuation of the FBs. Each value denotes the largest value of |Hm (z)| and |Fm (z)| in stopband
domain respectively. Clearly we can obtain better performance by the proposed method. Furthurmore, by using

Comparison of stopband attenuations of 4-channel filterbanks.

conv. 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 dBprop. FB
(order-1)
Hk (z)Fk (z)−21.8 dB
−23.4 dB
−16.1 dB
−16.2 dB
−20.4 dB
−19.5 dB
−19.8 dB
−19.6 dBprop. 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 dBprop. FB
(order-3)
Hk (z)Fk (z)−29.3 dB
−28.3 dB
−27.4 dB
−29.3 dB
−28.6 dB
−28.6 dB
−28.3 dB
−30.8 dB

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(a)

(b)

(c)

(d)

(e)
(f)
Fig. 6 Frequency responses of 4-channel order-1 prop. FB: (a)(b) analysis and synthesis bank. 4channel order-2 prop. FB: (c)(d) analysis and synthesis bank (dotted line: order-2 conv. FB). 4-channel
order-3 prop. FB: (e)(f) analysis and synthesis bank. 8-channel order-2 prop. FB: (g)(h) analysis andsynthesis bank.

same structure of proposed building blocks, we can easily
expand the number of channels. For example, Fig. 6(g) and(h) show frequency responses of 8-channel order-2 proposed
IIR PRFB.

5.

Conclusion

In this paper, new design approaches of M-channel IIR
PRFBs based on the IIR-FIR hybrid building blocks are presented. We impose the condition on the building blocks with

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(g)

(h)
Fig. 6state space representation to keep the order despite cascading of building blocks. Since the new method has more freeparameters than conventional one, we obtain better stopband
attenuations. As a feature, our IIR PRFBs have IIR analysis
filters and FIR synthesis filters and the order of the synthesis
filters are consistently one.

Continued.

pp.1103–1115, May 1995.
[13] I. Gohberg, L. Rodman, and P. Lancaster, Invariant Subspaces of Matrices with Applications, John Wiley, New York, 1986.

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Shunsuke Iwamura
received the B.E. and
M.E. degree in electrical engineering from Keio
University, Yokohama, Japan in 2005, 2007, respectively. His research interests are a designof IIR perfect reconstruction filter banks and efficient lifting factorizations of filter banks forlossy/lossless image compression.

Taizo Suzuki
received the B.E. and M.E.
degrees from the Department of Electrical Engineering, Keio University in 2004 and 2006, respectively. Since 2006, he has been engaged in
Toppan Printing Co., LTD. His research interests are in the area of multirate signal processing, filter design problem and image coding.

Yuichi Tanaka
received the B.E. and M.E.
degrees in electrical engineering from Keio University, Yokohama, Japan, in 2003 and 2005,
respectively. He is currently a Ph.D. candidate at Keio University, Yokohama, Japan, under the supervision of Prof. Masaaki Ikehara.
His research interests are in the field of multirate digital signal processing particulary various
and eﬀective filter banks design and digital image compression.

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Masaaki Ikehara
received the B.E., M.E., and Dr.Eng. degrees in electrical engineering from Keio University, Yokohama, Japan, in 1984, 1986,
and 1989, respectively. He was appointed Lecturer at Nagasaki University,
Nagasaki, Japan, from 1989 to 1992. In 1992, he joined the Faculty ofEngineering, Keio University. From 1996 to 1998, he was a Visiting Researcher at the University of Wisconsin, Madison, and Boston University,
Boston, MA. He is currently an Professor with the Department of Electronics and Electrical Engineering, Keio University. His research interests arein the areas of multirate signal processing, wavelet image coding, and filter
design problems.