Perfect Reconstruction Nonuniform Filter Banks with Linear

Phase

Takayuki Nagai, Takaaki Fuchie,* and Masaaki Ikehara

Faculty of Science and Technology, Keio University, Yokohama, Japan 223-8522

SUMMARY

A nonuniform filter bank divides the band nonuni-

formly according to the properties of the signal, which is

expected to be more effective than a uniform division filter

bank in applications such as subband coding. The structure

of the nonuniform filter bank, however, is complex. In

particular, no investigation has been made of the linear-

phase nonuniform filter bank. From such a viewpoint, this

paper considers a perfect reconstruction nonuniform filter

bank with linear phase, and derives the necessary and

sufficient conditions for a perfect reconstruction system as

well as for a linear-phase filter. Then, a design method is

presented based on the amplitude distortion in the fre-

quency domain and the elimination of aliasing. By the

proposed method, the nonuniform filter bank can be de-

signed directly without using the equivalent transformation

to the uniform filter bank. © 2000 Scripta Technica, Elec-

tron Comm Jpn Pt 3, 83(8): 103�114, 2000

Key words: Nonuniform filter bank; perfect recon-

struction; linear phase.

1. Introduction

In digital signal processing in recent years, there have

been many approaches where the signal is divided into

subbands and is then processed. Typical of those is the

subband coding of speech and image signals. The useful-

ness of subband division is shown through applications to

various problems such as spectral analysis, adaptive signal

processing, and speech processing [2].

There have been presented a large number of design

methods for uniform filter banks, where the frequency band

is uniformly divided with the same sampling rate for each

channel. As another approach, on the other hand, there are

studies of filter banks where the sampling rate is ideally set

in each channel and the frequency band is divided nonuni-

formly. This kind of system is called a nonuniform filter.

The wavelet transform is one such example [1]. In compari-

son to the uniform filter, the nonuniform filter gives ade-

quate subband responses according to the distribution of the

frequency components of the given input signal, which is

helpful in processing more effectively signals with a

nonuniform frequency distribution, as in the cases of speech

and images. In subband coding, for example, a more effec-

tive compression transmission will be realized for speech

and image information [3]. Another advantage will be to

realize signal analysis with high resolution, focusing on a

particular frequency.

In various applications, the perfect reconstruction

property (PR) is required for the filter bank. In addition, it

is required that each filter should have linear phase (LP) so

that the group delay is kept constant in each channel. This

is especially important in image processing.

Approaches to nonuniform filter banks include meth-

ods that combine uniform filter banks, namely, the method

based on tree structure [4] and the method to combine

adjacent channels [5]. Those, however, are indirect meth-

ods, and are only quasi-optimal in the sense of the circuit

© 2000 Scripta Technica

Electronics and Communications in Japan, Part 3, Vol. 83, No. 8, 2000Translated from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. J81-A, No. 6, June 1997, pp. 916�927

*Presently with Sony Corp.

103

scale and the filter responses, in addition to which some

band partitions are not possible. In contrast, there are pro-

posals for the direct construction of a nonuniform filter

bank. References 8 to 10 propose the design methods

through modulation. The methods are easy, but the con-

structed system is not a perfect reconstruction system.

As to the perfect reconstruction system, Ref. 6 pre-

sented a design method for the nonparaunitary filter bank.

Reference 7 presented a design method for the paraunitary

filter bank, where a nonuniform filter bank is transformed

into a uniform filter bank by using two equivalent transfor-

mations, and then the nonuniform filter bank is designed

through the lattice structure. A problem of the method based

on the lattice structure is that the design cannot be executed

in the case of an integer sampling factor. Thus, nonlinear

optimization is required in the design. It should also be

noted that the above two methods do not consider the

linearity of the phase. The authors previously proposed a

design method for a linear-phase nonuniform filter bank,

but only the nonparaunitary filter bank can be designed

[11].

This paper considers the direct construction linear-

phase perfect reconstruction nonuniform FIR filter bank.

The important issue in constructing the nonuniform filter

bank is the realizable division shape, that is, the frequency

response and the down-sampling rate of each filter. This

point is discussed in Ref. 7.

The problem which is left is the condition for the filter

length and the symmetry (even or odd symmetry) of each

filter, in order to satisfy the requirement for the perfect

reconstruction. This point has already been discussed in

detail for the case of the uniform filter bank, based on the

properties of the polyphase matrix [13]. The derived con-

dition, however, is not necessarily sufficient. Even if the

filter length and the symmetry are chosen so as to satisfy

the condition, the existence of the solution is not always

guaranteed. Still, the condition is useful in the sense that

the range of selection for the designer is sufficiently speci-

fied. Another point is that the solution usually exists, except

for special cases.

The condition of this kind, however, has not been

discussed at all for the case of the nonuniform filter bank,

and it seems important to investigate this point. This paper

is based on the equivalent transformation given in Ref. 7.

In other words, the nonuniform filter bank is transformed

into a filter bank where all filters have the same down-sam-

pling rate. The necessary conditions for the filter length and

the symmetry are derived from the polyphase matrix.

Then, this paper proposes a design method for the

linear-phase perfect reconstruction nonuniform filter bank.

Direct construction, without using the equivalent transfor-

mation to the uniform filter, is considered in this paper. As

the first step in this paper, the square error for the perfect

reconstruction condition is represented by the quadratic

form of the filter coefficient vectors of the decomposition

filter and the synthesis filter. The above quadratic form is

used as the evaluation function.

Since the evaluation function is represented by the

quadratic form of the filter coefficient vectors, both for the

decomposition filter and for the synthesis filter, one of the

filter coefficient vectors is fixed and the other can be deter-

mined so that the evaluation function is minimized. By

iterating alternately the above processes, the solution is

obtained for which the evaluation function is zero (i.e., the

solution that satisfies the perfect reconstruction condition).

In other words, the nonuniform filter bank with the linear

phase can be designed directly, without using the nonlinear

optimization.

This paper is composed as follows. The next section

discusses the perfect reconstruction condition for the

nonuniform filter bank, as well as the equivalent transfor-

mation to the uniform filter bank. Then, the necessary

condition is derived for the filter length and the symmetry,

in order to satisfy the perfect reconstruction condition.

Section 4 presents the design method. Lastly, a design

example is shown, demonstrating the effectiveness of the

proposed method.

2. Nonuniform Filter Bank

2.1. Perfect reconstruction condition

Figure 1 shows the nonuniform filter bank with frac-

tional sampling factor. It is assumed that pi and qi are prime

to each other, and the filter bank is of the maximum deci-

mation. When all pi �i 0, 1, . . . , M � 1� are 1, the filter

bank is called a nonuniform filter bank with integer sam-

pling factor. In the following in this paper, the sampling

factor is represented as [p0 / q0 . . . pM�1 /qM�1].

Let the decomposition filter and the synthesis filter

be represented by Hk�z� and Fk�z�, respectively. Then, the

following relation holds between the input signal X�z� and

the output signal X̂�z�:

Fig. 1. Nonuniform filter bank.

104

where Wi e�j2S / i.

The condition for eliminating the amplitude distor-

tion corresponds to the case of l = 1 in the above relation.

It is given as follows:

where r is a positive integer.

When l is other than zero, it corresponds to the case

of aliasing cancellation. The condition is written as follows:

(The derivation is shown in the Appendix.) In the above, Q

is the least common multiple of qi, and ri is given as

ri Q/ qi �i 0, 1, . . . , M � 1�. J is the minimum integer

for which �l � QJ� / �pi� is an integer �J t 0�. If such J does

not exist, it is set that J = 0. ¬al is the maximum integer not

greater than a. G�a� is a function that takes the value 1 for a

= 0, and the value 0 otherwise.

Equations (2) and (3) give the perfect reconstruction

condition in this structure. The filter bank that satisfies the

perfect reconstruction condition can largely be divided into

paraunitary and nonparaunitary filter banks. The parauni-

tary filter bank is a structure where the synthesis filter is

given by the time reversal of the decomposition filter. In

other words, there exists the following relation:

where Li is the filter length of Hi�z�. If the above condition

is not satisfied, the filter bank is a nonparaunitary filter

bank.

2.2. Equivalent transformation to uniform

filter bank

Reference 7 presented a method that transforms a

nonuniform filter bank to a uniform filter bank, where all

subchannels have the same down-sampling rate. Although

this transformation is not needed in the proposed design

method, it is briefly described in the following, since the

transformation is needed in deriving the necessary condi-

tion for the linear-phase property in the next section. For

the details, see Ref. 7.

The branch, where the up-sampling by p and the

down-sampling by q is applied, can be represented by p

filters with down-sampling by q and the inverse polyphase

transform (IPT) with size p. In other words, when p and q

in the sampling factor p /q are prime to each other, the

following transformation can be applied.

[Transform 1]

When the sampling factor is p /q, the branch with the

filtering by H�z� is represented by the parallel structure of

p filters, where

and

E0�z�, . . . , Ep�1 above represent the polyphase components

when H�z� is polyphase-decomposed by p.

In the above process, if the relation

q0 . . . qM�1 Q applies to all denominators of the sam-

pling factor, this transform provides 6k 0M�1pk Q branches

with down-sampling by Q. Consequently, no processing is

required. When [1/4 3/4], for example, transform 1 gives

the 4-decomposition filter bank. When, on the other hand,

not all qk are equal, Transform 1 gives pk branches with the

down-sampling by qk for each branch k. In order to trans-

form this system to a system with Q branches with the

down-sampling by Q, each branch is represented by

Q/ qk rk branches with the down-sampling by Q, plus the

inverse polyphase transform of size rk. In other words, the

following transformation is applied.

[Transform 2]

The branch with the sampling factor 1/q and filtering

by H�z� is transformed into Q/ q branches, in each of which

By the above two transforms, M-decomposition

nonuniform filter bank can be transformed into the filter

bank with Q branches and the synthesis part. Consider the

nonuniform filter bank, where each branch has the sampling

factor pk /qk. As the first step, each branch Hk�z� is decom-

posed by transform 1 into pk branches as follows (Trans-

form 1 in Fig. 2).

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(1)

105

Then, by transform 2, Hk,vk

g �z� is decomposed into

rk Q/qk branches as follows (Transform 2 in Fig. 2):

In order to provide Hk,vkrk � u

k

gg �z� with the serial num-

bered suffix, the expression is rewritten as follows:

where

In the above, k is the branch number in Fig. 1. vk is the

branch number in regard to the pk branches resulting from

transform 1 of the k-th branch. uk is the branch number in

regard to the rk branches resulting from the further transfor-

mation by transform 2.

By the above procedure, each channel is decomposed

into Q branches with Higgg�z�. Thus, the perfect reconstruc-

tion condition for the nonuniform filter bank is reduced to

the perfect reconstruction condition for the Q-channel filter

bank. It should be noted that the IPT block satisfies the

perfect reconstruction condition (the matrix in the

polyphase representation is composed only of delay) [11].

As is well known, the perfect reconstruction condi-

tion for the Q-channel filter bank is that the matrix of the

polyphase representation is composed only of delay [1]. In

the next section, the linear-phase condition is imposed on

the nonuniform filter, and the properties of the filters

Higgg�z�, obtained by the transformation to the uniform filter

bank, are examined. It is intended to analyze the properties

of the obtained Q u Q polyphase matrix, and to derive the

necessary condition for the filter length and the symmetry

of each filter.

3. Necessary Condition for Symmetry and

Filter Length

For the individual filter to have the linear phase, there

must be a certain kind of symmetry in the filter coefficients.

The symmetry, however, is not completely arbitrary, in

order to satisfy both the perfect reconstruction condition

and the linear-phase property. The symmetry must be se-

lected within a certain constraint. Similarly, there exists a

constraint on the filter length. The analysis of this condition

is important, since the solution satisfying the perfect recon-

struction condition is not obtained unless the solution is

selected to satisfy the above conditions. This section derives

the necessary condition for the symmetry of the decompo-

sition filter and the filter length for the linear-phase perfect

reconstruction nonuniform FIR filter bank.

Let Hk�z�, k 0, . . . , M � 1 be the linear-phase de-

composition filter, and the filter length be pk�Nk � 1� � 1

(only this case is considered in the following). When Q is

given, let 0 d ik d Q � 1, and mk be an integer. Then,

Nk � 1 is represented uniquely as mkQ � ik.

The impulse response of the linear-phase filter

Hk�z� has either even or odd symmetry, with the following

relation:

where

Using the polyphase element Ek,i�z�, the filter Hk�z� isdefined as

(12)

(9)

(10)

(11)

Fig. 2. Transform 1, 2.

(13)

106

Applying Eq. (12) to Eq. (14),

Then, the following relation applies:

Letting

and rewriting i in Eq. (16) using tvkc

.

By the reasoning in Lemma 1 (Appendix), the above

relation is written as

The second equation of the above can be modified as

By Lemma 2 (Appendix),

Letting the filter after applying transform 1 be Hk,vk

g , the

following conditions are necessary for the linear phase, as

is seen from Eqs. (8), (19), and (21):

As a simple example, consider the case of [1/4 3/4].

The branch with 1/4 is not transformed by transform 1, and

it is obvious that Eq. (22) holds as is. The branch with 3/4

is transformed into the following three branches by trans-

form 1:

In the above, E1,i�z� is the i-th component in the polyphase

decomposition of H1�z� by 3.

The filter length of H1�z� is 3�N1 � 1� � 1. When

H1�z� is polyphase-decomposed by 3, E1,0�z� is a filter with

length N1 and is always symmetrical by itself. In other

words, Eq. (22) applies. E1,1�z�, E1,2�z� are filters of length

N1 � 1, and are in relation of time reversal to each other.

Considering the delays z and z2, it is seen that Eq. (23)

applies.

Let the filter after applying transform 2 be Hk,rkvk � u

k

gg .

When vk 0, uk 0, it follows from Eq. (22) that

When vk 0, 1 d uk d rk � 1, it follows from Eq. (22) that

Consequently,

When 1 d vk d pk � 1, 0 d uk d rk � 1, it follows from

Eq. (23) that

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(14)

107

Using rkqk Q, the above expression is modified as

Consequently,

Thus, Eqs. (24), (26), and (29) represent the symme-

try for the linear phase after transforms 1 and 2 are applied.

Consider, as another simple example, the case of [1/2

1/4 1/4]. The branch with 1/4 is not transformed by trans-

forms 1 and 2, and it is obvious that Eq. (24) directly

applies. The branch with 1/2 is transformed by transform 2

into the following two branches:

H0,0gg �z� is H0�z� itself, and Eq. (24) applies. As to H0,1

gg �z�, itis seen that Eq. (26) applies, considering the delay z2.

Consider then the polyphase representations of the

above. By representing Eq. (24) by polyphase and is trans-

formed considering that Nk � 1 mkQ � ik,

where Ek,i,lgg �z� is the l-th component when Hk,i

gg �z� is

polyphase-decomposed by Q.

Applying the variable transformation to the above

expression,

Comparing the respective terms on both sides of the above

equation, the condition for the polyphase element required

for the decomposed filter to have the linear phase, is given

as follows:

As in Ref. 13, consider the case of i0 i1 . . . iM�1 I.

Then, Eq. (32) is written as

Similarly, modifying the expressions for Eqs. (26)

and (29), they are represented as

As in the case of Eq. (10), let

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

108

Then, the expressions are written as

In other words, for the polyphase matrix Eccc to have the

above form is the condition for each filter to have the linear

phase.

The above expressions can be rewritten as

where L1�z�, L2�z�, L3�z� are the diagonal matrices with

as the elements. L4 is the following matrix:

where L4,k is the pkrk u pkrk matrix given by

P is the following permutation matrix:

GI is the I u I inverse unit matrix.

For the perfect reconstruction system, it is necessary

that detEccc�z� bz�K (K is a nonzero integer). Forming the

determinants of both sides of Eq. (39),

By comparing both sides of the above equation, the follow-

ing two conditions are derived:

(40)

(36)

(37)

(38)

(39)

(41)

(42)

(43)

(44)

(46)

(45)

109

Equation (45) represents the constraint for the filter

length of the decomposition filter. Equation (46) represents

the constraint concerning the number of filters with coeffi-

cients with even symmetry, and the number of filters with

coefficients with odd symmetry. In other words, in order to

design the system with a linear phase satisfying the perfect

reconstruction condition, the filter length and the symmetry

of each filter must be determined so that Eqs. (45) and (46)

are satisfied.Those conditions agree with the case of uni-

form decomposition �pk rk 1, Q M�, that is, with the

result in Ref. 13. It should be noted that those conditions

are the necessary condition, and the existence of the solu-

tion is not always guaranteed. They, however, can well be

a guide for the designer in selecting the filter length and the

symmetry.

Consider, as an example, the case of [1/4 3/4]. In this

case, M = 2 and Q = 4. Letting I = 3, Eq. (39) is written as

Then, Eqs. (45) and (46) take the forms

Equation (48) indicates that m0 and m1 should both be either

even or odd. Equation (49) indicates that J0 r1, J1 c1,

that is, H0 and H1 should have different kinds of symmetry.

4. Design of Perfect Reconstruction

Nonuniform Filter Bank

This section proposes a new design method for the

linear-phase perfect reconstruction nonuniform filter bank.

As the first step, the square error for the perfect reconstruc-

tion condition is formulated as a quadratic form of the filter

coefficient vectors. Then, the design of the perfect recon-

struction nonuniform filter bank is considered by minimiz-

ing the square error for the perfect reconstruction condition

as the evaluation function. It is not necessary to apply the

nonlinear optimization. The evaluation function is mini-

mized by iteratively solving the linear equation.

4.1. Formulation of perfect reconstruction

condition by filter coefficient vector

In the following, for simplicity, only the nonuniform

filter with the integer sampling factor is considered. The

reasoning is similar in the case of the fractional sampling

factor. In the proposed method, the square error for the

perfect reconstruction condition is minimized as the evalu-

ation function as described above.

The evaluation function ) is defined as

In the above, Hk �ejZ�, Fk

�ejZ� represent the zero-

phase responses of the decomposition and the synthesis

filters, respectively. Since each filter has the linear phase,

those are rewritten as follows using the filter coefficient

vectors:

where hk and fk are the coefficient vectors of the k-th

decomposition and the synthesis filters, respectively.

ci�Z�, si�Z� are the triangular function vectors, which are

determined by the respective filter length and the kind of

symmetry. D and E are the weight assigned to the amplitude

distortion elimination condition and the aliasing canceling

condition, respectively.

Defining the filter coefficient vectors as

The first term of Eq. (50) is written as follows, using the

matrix representation

(47)

(48)

(49)

(50)

(51)

(52)

(53)

110

where

Equation (53) can further be written as

where

Similarly for the term )b concerning the aliasing

cancellation, the expression is written as

where

Finally, the evaluation function ) )a � )b is given

by

where

4.2. Design algorithm for nonparaunitary

nonuniform filter bank

As discussed in the previous section, the square error

for the perfect reconstruction condition can be formulated

as a quadratic form of the filter coefficient vectors. Then, it

is possible, as in Ref. 12, to derive the solution that makes

the evaluation function eventually almost zero, by alter-

nately solving the equations for the decomposition and the

synthesis filters. It is shown in Ref. 12 for this process that

the evaluation function monotonically decreases and the

filter bank with a satisfactory response can be designed,

even though the evaluation function does not include the

evaluation of the frequency characteristics. It should be

noted that the evaluation function never tends to zero unless

the symmetry or the filter length is selected so as to satisfy

the condition of Section 3.

The design algorithm for the nonparaunitary case is

as follows.

(1) Based on the existing method for filter design,

the decomposition filter is designed so that the desired

frequency response is realized.

(2) The coefficient vector h of the decomposition

filter is fixed, and the coefficient vector f of the synthesis

filter is determined by Eq. (56).

(3) The coefficient vector f of the synthesis filter is

fixed as the value obtained by the above step, and the

coefficient vector h of the decomposition filter is similarly

determined.

(4) If the value of the evaluation function ) is

sufficiently small �) d 10�10�, the procedure ends. If not,

the algorithm is continued (i.e., the procedure goes back to

step 2).

4.3. Design algorithm for paraunitary

nonuniform filter bank

The major difference between the paraunitary and

nonparaunitary filter banks is in the choice of the synthesis

filter. In the case of the paraunitary filter bank, the synthesis

filter is given as the time reversal of the decomposition

filter. Consequently, the design is completed by the above

method.

The following point, however, should be noted. In

order to provide the time reversal relation between the

decomposition and synthesis filters, the average of hk and

fk is used in each iteration. In other words, instead of

determining the decomposition filter hk in step 3, the aver-

(54)

(55)

(56)

111

age of the decomposition filter and the synthesis filter is

determined as

which is used as the decomposition filter in the next itera-

tion.

5. Design Example

In this study, a 6-channel paraunitary nonuniform

filter bank with the integer sampling factor is designed. The

sampling factor is set as [1/8 1/8 1/8 1/8 1/4 1/4]. The filter

length is set as 12 for each fil ter. Then,

Q 8, M 6, pk 1, rk [1 1 1 1 2 2], mk 1, I 3. Cal-

culating the right-hand side of Eq. (45), the result is 2 (i.e.,

even), and the condition for the filter length described in

Section 3 is satisfied. Since detP detL4 1, it follows

from Eq. (46) that J0J1J2J3J42J5

2 1.

In this study, [S A S A S A] is selected as one of the

cases where the above condition is satisfied. Figure 3 shows

the frequency response of the designed decomposition fil-

ter. Figure 4 shows the error for the perfect reconstruction

as a function of the number of iterations. It is seen that the

error monotonically decreases, down to a sufficiently small

value.

6. Conclusions

This paper has considered the linear-phase perfect

reconstruction nonuniform FIR filter bank, and presented

the theory and the design procedure. As the first step, the

necessary condition for the filter length and the symmetry

of the nonuniform filter bank to satisfy the linear phase and

the perfect reconstruction is derived. The condition pro-

vides a useful guide in the design.

Then, a method is shown that directly designs the

nonuniform filter bank. The proposed method is based on

the magnitude distortion elimination and the aliasing can-

cellation in the frequency domain. The nonuniform filter

bank is directly designed without applying the equivalent

transformation to the uniform filter bank. As another point,

the nonlinear optimization is not required in the design, and

the solution satisfying the perfect reconstruction condition

can be derived by iteratively solving the linear equation.

A problem for the future is the application to image

coding and other problems.

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APPENDIX

Derivation of Eq. (3)

Let Wqi

pil WQ

rip

il. Then, the quantity inside the braces

on the right-hand side of Eq. (1) is written as

When l z 0, lc takes the values at the interval of ri in the

range ri d lc d Q � ri. In order to make lc 1 . . . Q � 1,

G�¬lc / ril � lcri

� is multiplied with the above expression. Then,

in order to align the indexes for the value of WQp

ilc, it is set

that {pilc mod Q} lcc. Then, one can write

Substituting the above expression into lc, and rewriting lsas l, Eq. (3) is obtained.

[Lemma 1]

Let tv {qv mod p}. Then,

where 1 d v d p � 1.

[Proof] There hold

Dividing qv by p, let the quotient be R1. Then,

Dividing q�p � v� by p, let the quotient be R2. Then,

By Eq. (A.2),

Consequently, by Eq. (A.5),

It follows from Eqs. (A.5) and (A.6) that

It follows from Eqs. (A.5) and (A.7) that

"

[Lemma 2]

If dv satisfies dv ¬qv /pl, then

[Proof] There hold

It follows from Eq. (A.10) that d�p�v� is written as

By Lemma 1, t�p�v� � tv p. Consequently,

"

(A.1)

(A.2)

(A.3)

(A.4)

(A.5)

(A.6)

(A.7)

(A.8)

(A.9)

(A.10)

(A.11)

(A.12)

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AUTHORS (from left to right)

Takayuki Nagai (member) graduated from the Department of Electrical Engineering of Keio University in 1993 and

completed his doctoral program in 1997. He holds a D.Eng. degree. He is presently a researcher at Keio University, where he

is engaged in research on digital signal processing.

Takaaki Fuchie graduated from the Department of Electrical Engineering of Keio University in 1995 and completed his

master�s program in 1997. He is presently with Sony Corp. In graduate school, he engaged in research on digital signal

processing.

Masaaki Ikehara (member) graduated from the Department of Electrical Engineering of Keio University in 1984 and

completed his doctoral program in 1989. He holds a D.Eng. degree. He then joined Nagasaki University as a lecturer, a post he

has occupied at Keio University since 1992. He is engaged in research on circuit theory and digital signal processing.

114