IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 7, JULY 2001 1433

On Factorization ofM -Channel ParaunitaryFilterbanks

Xiqi Gao, Truong Q. Nguyen, Senior Member, IEEE, and Gilbert Strang

Abstract—We systematically investigate the factorization ofcausal finite impulse response (FIR) paraunitary filterbanks withgiven filter length. Based on the singular value decompositionof the coefficient matrices of the polyphase representation, afundamental order-one factorization form is first proposed forgeneral paraunitary systems. Then, we develop a new struc-ture for the design and implementation of paraunitary systembased on the decomposition of Hermitian unitary matrices.Within this framework, the linear-phase filterbank and pairwisemirror-image symmetry filterbank are revisited. Their structuresare special cases of the proposed general structures. Comparedwith the existing structures, more efficient ones that only useapproximately half the number of free parameters are derived.The proposed structures are complete and minimal. Although thefactorization theory with or without constraints is discussed in theframework of -channel filterbanks, the results can be appliedto wavelets and multiwavelet systems and could serve as a generaltheory for paraunitary systems.

I. INTRODUCTION

I N THE research area of multirate filterbanks, wavelets, andmultiwavelets, the paraunitary concept plays a central role

[1]–[7]. The design and implementation of orthogonal systemsare based on multi-input and multi-output systems, of whichthe polyphase transfer matrices are paraunitary. A polyphasetransfer function matrix is paraunitary if, where the superscript stands for the conjugated transpose,

and is the identity matrix [1]. A complete factorization of theparaunitary matrix with or without constraints often provides anefficient structure for optimal design and fast implementation.

In this paper, we systematically investigate the factorizationof paraunitary matrices in the framework of -channel max-imally decimated FIR filterbanks. The results can be appliedto -band wavelets as well as multiwavelet systems. Fig. 1(a)shows a typical -channel maximally decimated filterbank,where and are the th analysis and synthesis sub-band filters. The polyphase implementation of the filterbank

Manuscript received March 10, 2000; revised March 7, 2001. The associateeditor coordinating the review of this paper and approving it for publication wasDr. Paulo J. S. G. Ferreira.

X. Gao was with the Department of Mathematics, Massachusetts Institute ofTechnology, Cambridge, MA 02139 USA, and with the Department of Electricaland Computer Engineering, Boston University, Boston, MA 02215 USA. He isnow with the Department of Radio Engineering, Southeast University, Nanjing,China (e-mail: xqgao@seu.edu.cn).

T. Q. Nguyen is with the Department of Electrical and Computer Engineering,Boston University, Boston, MA 02215 USA (e-mail: nguyent@engc.bu.edu).

G. Strang is with the Department of Mathematics, Massachusetts Institute ofTechnology, Cambridge, MA 02139 USA (e-mail: gs@math.mit.edu).

Publisher Item Identifier S 1053-587X(01)05354-5.

is illustrated in Fig. 1(b), where and are, respec-tively, type-I analysis polyphase matrix and type-II synthesispolyphase matrix [1], i.e.,

and

In a causal FIR orthogonal filterbank with filters of length, and are paraunitary matrices of order ,

and . We use “order” to denote thehighest power of in and . The order comes di-rectly from the filter lengths. Our objective is to develop com-plete and minimal factorization structures for paraunitary ma-tricesof given orderwith or without further constraints.

Many excellent works on this topic have been reported byother researchers [8]–[19]. For general paraunitary filterbanks,Vaidyanathanet al. propose a complete and minimal structurein [9] and [10], which shows that any paraunitary matrix ofMcMillan degree can be factorized into a product of de-gree-one paraunitary building blocks and an additional unitarymatrix. The McMillan degree of a multi-input and multi-outputcausal rational system is the minimum number of delay units(i.e., elements) required in implementation. With filtersof length , the McMillan degree of the polyphase matrixmay range from to . Conversely, parauni-tary matrices of different orders can have the same degree. Inpractice, we care more about the filter length than the degreeof the system. In this case, it is expected to have a factoriza-tion form for paraunitary matrices ofa given order. This is theinitial motivation of this work. With constraints such as linearphase and/or pairwise mirror-image symmetry on subband fil-ters, complete and minimal lattice structures of the paraunitarypolyphase matrices have been developed [12]–[18]. The con-strained paraunitary matrix of order can be expressedas the product of order-one paraunitary building blocksand an additional unitary matrix. These factorizations provideefficient structures for implementing linear-phase and mirror-image paraunitary filterbanks with length constraint. More re-cently, Rault and Guillemot prove that an paraunitarymatrix of order can be factorized into a -stageorder-one form if the ranks of the first and last coefficient ma-trices are not less than for even or for odd

1053–587X/01$10.00 ©2001 IEEE

1434 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 7, JULY 2001

Fig. 1. M -channel maximally decimated filterbank. (a) Basic structure. (b) Polyphase implementation.

[19]. The resulting structure has the symmetric delay prop-erty that has been found to be valuable in object-based audiovi-sual signal compression.

It is interesting that all the reported results on factorizing aspecial class of a paraunitary system take the -stageorder-one form. These special systems also include the two-channel filterbanks [8] and cosine-modulated paraunitary filter-banks [20]–[25]. Does a generalized factorization for parauni-tary system exist? It is reasonable to handle the factorizationproblem of causal FIR paraunitary systems with or without con-straints in one framework.

Section II shows that if the degree-one building blocks areseparated into groups in which the parameter vectorswithin each group are orthogonal, then the product is parauni-tary of order , and the filter length is . This leadsto the fundamental order-one factorization form. In this section,we also prove that the order-one factorization is complete; anycausal FIR paraunitary matrix of order is allowed. Ourapproach is based on the singular value decomposition (SVD) ofcoefficient matrices. Notice that the SVD has also been used inthe design of -channel linear-phase biorthogonal filterbanksin [26]. The difference is that it was used to represent invertiblematrices that appear in the proposed lattice structure, whereaswe use it to derive the structure of a given paraunitary filterbank.We develop a more efficient structure for the design and imple-mentation in Section III. The CS decomposition and spectral de-composition of Hermitian unitary matrices are valuable for pa-

raunitary systems. In this framework, we revisit the linear-phaseand mirror-image filterbanks in Section IV. More efficient struc-tures than those reported are obtained, which implies more effi-cient design and implementation of these classes of filterbanks.Section V discusses several design examples.

Notations:Bold-faced letters indicate vectors and matrices.The tilde operation on a matrix function is defined by

, where the superscript denotes the con-jugated transpose. For real coefficient systems,is replaced by

(the transpose). The rank of the matrixis . anddenote the integer floor and ceiling of. and are the

identity and reverse identity matrices, respectively.

II. DEGREE-ONE FACTORIZATION AND ORDER-ONE

FACTORIZATION

A. Brief Review of Degree-One Factorization

Consider an -channel causal FIR filterbank with filterlength . The polyphase matrix of the analysis bankcan be expressed as

(1)

where . The filterbank is paraunitary if and only ifis paraunitary, i.e.,

(2)

GAO et al.: ON FACTORIZATION OF -CHANNEL PARAUNITARY FILTERBANKS 1435

For , this implies particularly that the following equationholds:

(3)

which implies that must be singular since is nonzero.This property is the starting point for a degree-one factoriza-tion [1], [10]. In fact, (3) contains more information on and

, which will be used to complete the factorization.Suppose that the McMillan degree of is . Since

is paraunitary, the order of its determinant is. Since is sin-gular, there exists an -dimensional unit vector such that

. Using , we can define an degree-oneparaunitary matrix . Then,

is a causal FIR paraunitary system of degree .As mentioned above, the first coefficient matrix ofmust be singular if . Therefore, the degree reductionprocess can be repeated until a constant unitary matrix is ob-tained. Therefore, for any paraunitary of degree

, there exist vectors and a unitary matrixso that can be factorized into [1]

(4)

where . Conversely, with ar-bitrary unit-norm vectors , , and arbitrary unitarymatrix , the product in (4) is always a causal FIR pa-raunitary system of degree.

It is easy to see that produced by (4) is not necessarily oforder . If we want to design a paraunitary filterbank withgiven filter lengths, the parameters in (4) can not be arbitrary,and it is not reasonable to fix the McMillan degree of the systemin advance. For order paraunitary , its McMillandegree can range from to . Any orderparaunitary of degree can be produced by (4) with

. This fact was used to design paraunitary filterbankswith filters of given length in [10].

B. Order-One Factorization

We now derive the order-one factorization from the de-gree-one structure by constraining the unit norm vectors andthen prove completeness for fixed order . Separatethe set of unit vectors into disjointsubsets. The th group contains vectorsending with . Suppose that the vectors within each groupare orthonormal. In this case, . Let be thematrix with columns from group

; then, (4) can be rewritten as

(5)

where

(6)

Since the columns of are orthonormal, we can findvectors to complete an unitary matrix (of which

the right submatrix is ). Then

(7)

where

diag (8)

Substituting (7) into (5), we obtain

(9)

where . Since are

unitary matrices, so are . The structure is demonstrated inFig. 2(a), where .

With the orthogonality assumption on the parameter vectors,we have shown that of the -stage degree-one structurecan be expressed as the product of order-one parauni-tary matrices. From (9), with arbitrary unitary matrices, theorder of cannot be higher than . Therefore, we canuse these structures to design filterbanks with constrained filterlength. In fact, the structure of (9) was used in the design ofparaunitary system [27]. This order-one form is more suitablethan the degree-one form when the filter length is constrained.In the rest of this subsection, we will show that any causal FIRparaunitary matrix of order can be factorized intothe form of (9).

Given a causal FIR paraunitary matrix , it can be ex-pressed as in (1), where . Thus, the ranks ofand satisfy

(10)

The singular value decompositions of and are

(11)

whereand square diagonal matrices containing the

and singular values ofand ;

and matrices;and matrices.

The column vectors of each are orthonormal. Substituting(11) into , we have

(12)

This means that the column vectors of are orthogonal tothose of . Similarly, the column vectors of are orthog-onal to those of . The paraunitary property also implies

, which leads to and. Since the columns of and are

orthogonal, there are orthonormal vec-tors orthogonal to these columns. We place these vectors intothe columns of a matrix and create the square unitary ma-trix

(13)

Then, and can be expressed as

(14)

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Fig. 2. Order-one structure for paraunitary systems. (a) Basic form. (b) More efficient form.

Substituting (14) into (1), we have

(15)

where diag with; is a causal FIR transfer function

matrix of order . Equation (15) shows that the orderof can be reduced to for . Since ,and are paraunitary, is also paraunitary. There-fore, the above order-one reduction process can be repeated untilthe remainder is a constant unitary matrix. Then, is factor-ized into (9). The above form also holds for system with realcoefficients. In summary, we have the following result.

Theorem 1: Let be an causal FIR transfer func-tion matrix with order not exceeding . It is paraunitary if

and only if there exist unitary matrices , , , andintegers , so that can be factorized as (9)with defined by (8). In the real-coefficient case, arereal and orthogonal.

Comments:

1) There are unitary matrices in this order-one fac-torization. Any unitary matrix can be completely factor-ized by free parameters [1], [28]. For the real-co-efficient case, rotation parameters are re-quired. Thus, the total numbers of parameters areand for the complex-coefficient and real-coefficient cases, respectively.

2) When , the set of free integers can be con-strained as

(16)

To see this, we can check in (15). From, the first rows of and the last

rows of are and , respec-

GAO et al.: ON FACTORIZATION OF -CHANNEL PARAUNITARY FILTERBANKS 1437

tively. Consequently, , and. If we choose in the first

reduction step and repeat in the subsequent step, then. In this way, the paraunitary matrix

can be factorized as in (9) with a set of integerssatis-fying (16). If we choose in the firststep and repeat in the subsequent steps, the integers are inthe following order:

(17)

3) For a two-channel filterbank with real coefficients andwith and , the set of integers mustbe , and are realorthogonal matrices. Any orthogonal matrix can becompletely factorized as

where is the rotation angle .Let and ,

, where

and , are diagonal matrices withdiagonal entries ; then, reduces to the followingtwo-channel lossless lattice structure [8]:

diag (18)

where diag .4) Although any with order and with

can be factorized as in (9), the order of produced bythis structure with arbitrary unitary matrices is notnecessarily equal to , i.e., can be the zeromatrix. However, the order cannot be higher than .This situation appears in other special factorization formsfor special classes of filterbanks [8], [13]–[18]. To ensurethat , one needs to impose additional constrainton and such that

diag (19)

In practice, it seems that there is no reason to do so. Ifcan take arbitrary integer values from 0 to, the wholespace of produced by (9) with arbitrary unitary ma-trices is composed of causal paraunitary matrices withorder not exceeding .

III. M ORE EFFICIENT STRUCTURE

Since the order-one factorization form in (9) is complete fororder , it can be used to design paraunitary filterbankswith the constrained filter lengths. However, as the number ofchannels and the filter length increase, the number of free pa-rameters is very large, and the optimal solution is difficult to

obtain. In this section, we present a more efficient structure. Toachieve this, we first investigate the Hermitian unitary matrix,which plays an important role in the development.

A. Hermitian Unitary Matrix

Let be Hermitian and unitary

(20)

If is real, it is symmetric and orthogonal. Any complex uni-tary matrix can be expressed by parameters, whereas anyorthogonal matrix can be represented by angleparameters. The Hermitian constraint reduces thesedegrees of freedom. The problem is how to express this specialclass of unitary matrix with fewer parameters. Similar to the co-sine-sine (CS) decomposition for a general unitary matrix [29],we have the following result on Hermitian unitary matrix.

Lemma 1: Let be an matrix. Partition it as

, where is with and

. is Hermitian and unitary if and only if it canbe expressed as

diag diag (21)

where and are and unitary matrices, re-

spectively; , for even and

for odd , where , , and are real diagonal ma-trices with entries , , and

If is real, and can be real orthogonal.Appendix A provides the proof for Lemma 1. Compared with

the CS decomposition for a general unitary matrix [29], the pro-posed form for Hermitian unitary matrix contains unconstrainedangles . More importantly, only two reduced-size unitary ma-trices of order and are required, plusadditional angle parameters. The total number of parameters is

for a complex Hermitian unitary matrixand for a real symmetric orthogonal matrix. TheHermitian constraint reduces the number of parameters nearlyby half. In fact, to factorize an orthogonal matrix byGivens rotations, sign parameters are required besidesplanar rotation parameters [1]. The rotation angles take arbitraryvalues. The factorization of an orthogonal matrix is ,where the matrix represents the planar rotation matrix, andthe sign parameters are in the diagonal matrix. From (21),the sign parameters of and can be moved into , whichonly affects the angles . Therefore, the sign parameters ofand are not necessary for a symmetric orthogonal matrix.This is an advantage of using the unconstrained angles in theproposed CS decomposition.

The special matrix is characterized by sym-

metric orthogonal matrices . The eigen-

1438 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 7, JULY 2001

values of any symmetric orthogonal matrix are 1 or. It is easyto show that

(22)

where

and

diagdiag

Therefore, can be expressed as

(23)

where is a real diagonal matrix with diagonal entries of abso-lute value 1

for even and

for odd (24)

where and are real diagonal matrices with entriesand . Conversely, with arbitrary and

arbitrary real diagonal matrix with diagonal entries of abso-lute value 1, produced by (23) is symmetric orthogonal, andtherefore, produced by (21) is Hermitian unitary. In summary,we have the following result.

Lemma 2: Let be an matrix, and let it be parti-tioned as in Lemma 1. is Hermitian and unitary if and only ifit can be expressed as in (21) with defined by (23). If isreal, the two unitary matrices are real orthogonal.

This lemma gives the spectral decomposition for a Hermi-tian unitary matrix. The transform matrix takes the special formdiag , where and are unitary, and is definedby (24).

B. More Efficient Structure for Paraunitary System

For in (9), we can find unitary matrices so that

Therefore, any paraunitary causal FIR of order canalso be expressed as the form in (5) with the order-one matrix

defined by (7).

(25)

where

(26)

.(27)

It is easy to see that can be expressed as

(28)

where diag . Substituting (28) into(26), we have

(29)

where

(30)

The matrix is Hermitian unitary matrix. From Lemma 2, wecan express as

diag diag (31)

whereand and uni-

tary matrices, respectively;real diagonal matrix with diagonal entries 1or ;real orthogonal matrix of the form

for even and

for odd (32)

where and are real diagonal matriceswith entries and

.

Substituting (31) into (29)

diag diag(33)

where is a diagonal matrix with

[

Substituting (33) into (25), we obtain

diag (34)

where

diag .In (34), , , and are unitary matrices. Conversely, witharbitrary unitary matrices , , and and arbitrary angles

and produced by (34), they are causal FIR paraunitarywith order equal to or less than . For the real-coefficientcase, , , and are real orthogonal. In summary, wehave proved the following theorem.

GAO et al.: ON FACTORIZATION OF -CHANNEL PARAUNITARY FILTERBANKS 1439

Theorem 2: Let be an causal FIR transfer func-tion matrix with order not exceeding . It is paraunitary ifand only if there exist unitary matrices ,

unitary matrices , an unitary ma-trix , and a set of angles so that it can be factorized as in(34) with defined by (32) and the diagonal matricesof diagonal entries 1 or . For real coefficients, all the unitarymatrices are real orthogonal.

Comments:

1) The above theorem shows that the causal FIR paraunitarysystem can be characterized by unitary matricesof size , unitary matrices ofsize , an unitary matrix, and

additional angles. The total numbers offree parameters is forthe complex case and

for the real case. Compared with the structure of(9), a significant reduction of parameters is achieved. Thisgives more efficient design for the paraunitary system.Fig. 2(b) shows the structure in (34). At each stage of thelattice structure, the unitary matrix ,

unitary matrix , andmatrices are to be implemented instead of thematrix in the structure of (9). This implies a signifi-cant saving of the implementation cost.

2) in (34) is generally different from in (9),although (34) is derived from (9). However, they containthe same number of delay terms. To verify, one only needsto check their determinants

(35)

The power of indicates the number of inand since they are diagonal matrices with diagonalentries 1 or .

3) For real coefficients, the unitary matrices are real orthog-onal. The sign parameters of and are not nec-essary. The rotation parameters are sufficient. For a two-channel system, and are scalars and can be set to1, which simplifies the polyphase matrix as

(36)

where

It has the same implementation complexity as in (18).

IV. FACTORIZATION OF PARAUNITARY SYSTEM WITH

CONSTRAINTS

In this section, we consider two special classes of paraunitaryfilterbanks using the above factorization for general paraunitarysystems. In applications such as image coding, linear phase isimportant. Several researchers have reported results on latticestructure for linear-phase filterbanks [13], [15], [17]. We willrevisit it in the following subsection and derive an alternativestructure with fewer parameters. The other class of filterbankconsidered in this section is the pairwise mirror-image sym-metry filterbank, which was first investigated by Nguyen andVaidyanathan in [12]. With the mirror-image constraint on thesubband filters, the number of free parameters can be reducedsignificantly, whereas a good stopband attenuation can be pre-served. A factorization was proposed in [12], and its complete-ness was proved in [14]. For these special filterbanks, we willfirst obtain the same or equivalent results as the reported works,which correspond to (9), and then achieve more efficient struc-tures related to the form in (34). The filterbanks considered herehave real coefficients, and is assumed to be even for sim-plicity.

A. Linear Phase Filterbank

In a linear phase paraunitary filterbank, all filters are eithersymmetric or antisymmetric, i.e.,

(37)

where is 1, and for symmetry and antisymmetry, respec-tively. Since , (37) also implies thatsynthesis filters are symmetric or antisymmetric. Equation (37)holds if and only if the polyphase matrix satisfies

(38)

where diag . For perfect reconstruc-tion, the symmetry polarities of the filters can not be indepen-dent. As shown in [13], there are symmetric and an-tisymmetric filters for even . For simplicity, it is often as-sumed that the first filters are symmetric, i.e.,diag .

The symmetry condition (38) implies particularly that. Therefore, ,

and the SVD of can be derived from that of .and in (11) can be chosen as

and . Let ,where both and are matrices.Then, . From (12),we have . Therefore,

. This meansthat the column vectors of each are orthogonal. Thereexist matrices , sothat and .Therefore, we can choose the orthogonal matrix as

(39)

1440 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 7, JULY 2001

where . With this matrix, (14) and(15) hold, and we can set . Thus, can beexpressed as

diag

diag (40)

where

and . Substituting (40) into (38), wehave

(41)

This means that the causal FIR paraunitary matrix oforder also satisfies the linear phase condition. The aboveprocess is repeated on until the order is reduced to zero.The zero-order paraunitary matrix satisfying the linear phasecondition can be expressed as diag ,where and are orthogonal matrices,

and . Therefore, for the linear phase

paraunitary filterbank, the polyphase matrix can be factorizedas in (9) with the unitary matrices of the following form:

diagdiagdiag

(42)and

diag (43)

Conversely, with arbitrary orthogonal matrices , pro-duced by (9), (42), and (43) is a causal FIR paraunitary matrixand satisfies the linear phase condition (38). This result is thesame as that presented in [15] and equivalent to that in [13].The structure is shown in Fig. 3(a), where the scale factors areignored. It is interesting that this structure is similar to that of(34) with and replaced by and the initialstage of the special form. The structure of (34) is for generalparaunitary systems. With the linear phase constraint, furtherreduction of parameters is possible.

Substituting (42) into (27), and then (30), we get

diag

diag

(44)

(45)

where;

;special symmetric orthogonal matrix.

The CS decomposition and spectral decomposition ofarethe following.

diag diag

diag diag

diag (46)

Therefore can be rewritten as

diag

diag

diag (47)

where and

Conversely, with arbitrary orthogonal matrices and ,, which is produced by (47), is a causal FIR paraunitary

matrix and satisfies the linear phase condition (38). In summary,we have the following result.

Theorem 3: Let be an real causal FIR transferfunction matrix with order not exceeding . Then, it isparaunitary and satisfies the linear phase condition (38) if andonly if there exist orthogonal matrices and

so that it can be factorized as in (47).Comments:

1) In this new structure, only one orthog-onal matrix is required at each stage, except for the firstone. Therefore, a significant reduction of free parame-ters is achieved, and the implementation cost is reduced.Fig. 3(b) illustrates this new structure, where the scalefactor is not included.

2) The spectral decomposition of in (46) implies that werestrict . The sign parametersof are generally necessary. In other words, if we justconsider the planar rotation part of as in (34),should be allowed to be any diagonal matrix with diag-onal entries 1 or to keep the completeness of this fac-torization.

B. Pairwise Mirror-Image Symmetry Filterbank

For the filterbank with even , the mirror-image conditioncan be imposed by the following relation [12]:

(48)

For simplicity, we can reorder the filters so that (48) becomes

(49)

GAO et al.: ON FACTORIZATION OF -CHANNEL PARAUNITARY FILTERBANKS 1441

Fig. 3. Structure for paraunitary system with linear phase property. (a) Existing form. (b) New form.

This relation is equivalent to the following condition on thepolyphase matrix:

(50)

where

and

with diag . Equation (50) im-plies particularly that . Therefore, as inthe linear phase filterbank, , and theSVD of can be derived from that of . and in(11) can be chosen as and .Let an -dimensional vector be orthogonal to the columnvectors of both and , i.e., and .Using the fact , we have and

. Thus, the vector is also orthogonalto the column vectors of both and . It is easy to verifythat is also orthogonal to . Therefore, andcould be two column vectors of in . In this way, wecan choose the orthogonal matrix of the following form:

(51)

where , are matrices. Noticethat the left and right submatrices of areand , respectively. Using this matrix, (14) and (15) hold, andwe can set . Thus, can be expressed as

diag

(52)

It is easy to verify that the causal FIR paraunitary matrixof order also satisfies the pairwise mirror-image

symmetry condition (50) with replaced by . Wecan repeat the above process on until the order isreduced to zero, whereas the mirror-image property is

1442 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 7, JULY 2001

Fig. 4. Structure for paraunitary system with pairwise mirror image symmetry property. (a) Form equivalent to the existing structures. (b) New form.

preserved at each step. The zero order paraunitary matrixsatisfying the mirror-image condition can be expressed as

diag , where and

are matrices. Therefore, for the pairwisemirror-image symmetry paraunitary filterbank, the polyphasematrix can be factorized as in (9) with the orthogonal matricesas in (53), shown at the bottom of the page, and definedby (43). Conversely, with arbitrary orthogonal matricesof the form (53) and defined by (43), , which isproduced by (9), is a causal FIR paraunitary matrix and satisfiesthe pairwise mirror-image symmetry condition (50). This resultis equivalent to those in [12] and [14]. The structure is shown in

Fig. 4(a). The real matrix is orthogonal if and onlyif the complex matrix is unitary.Thus, can be expressed by parameters, whereasgeneral orthogonal matrix needs parametersbesides sign parameters. The reduction of the parameters issignificant. This structure corresponds to our basic order-onefactorization form. Further reduction of the parameters is stillpossible, which is related to the structure of (34).

Substituting (53) into (27) and then (30), we can see thattakes the same form as , i.e., we have (54), shown at thebottom of the page, and is of the following form:

(55)

diag(53)

GAO et al.: ON FACTORIZATION OF -CHANNEL PARAUNITARY FILTERBANKS 1443

where and are symmetric. In addition, is a spe-cial symmetric orthogonal matrix. From Appendix B, we canexpress as

diag diag

diag (56)

where is orthogonal, and is defined as in (32) with even. This implies that we can express as in (34) with the

matrices as

diag

diag

For easy reference, we give the expression of in the fol-lowing:

diag diag

diag (57)

Conversely, with arbitrary orthogonal matrices and of

the form and a set of angles , , which

is produced by (57), is a causal FIR paraunitary matrix and sat-isfies the pairwise mirror-image symmetry condition (50). Insummary, we have the following result.

Theorem 4: Let be an real causal FIR transferfunction matrix with order not exceeding . It is paraunitaryand satisfies the pairwise mirror-image symmetry condition (50)if and only if there exist orthogonal matrices , an orthogonal

matrix of the form , and a set of angles

so that it can be factorized as (57).Comments:

1) Fig. 4(b) illustrates the structure of (57). At each stageof this new structure except the initial one, only one free

orthogonal matrix is required besidesangles in and . Compared with the structure

of (34) for general paraunitary systems and the one forthis special class of filterbank as shown in Fig. 4(a), thisstructure needs far fewer parameters.

2) In the implementation aspect, the structure in Fig. 4(b)has the same complexity as that in Fig. 2(b), although itis more efficient than that in Fig. 4(a). This means thatthe implementation cost does not decrease for the pair-wise mirror-image symmetry. In the linear phase filter-

bank, the linear phase constraint leads to improvementsin both design and implementation cost.

V. DESIGN EXAMPLE

In this section, we present several design examples of real-co-efficient filterbanks with even based on the proposed struc-tures in (34), (47), and (57). In different applications, variousobjective functions can be constructed to optimize the filter co-efficients. The most common functions are the stopband attenu-ation and coding gain. Our examples presented in this paper arebased on minimizing the stopband energy and maximizing thecoding gain. The stopband energy criterion measures the sum ofthe filters’ energy outside the designated passbands:

(58)

where denotes the stopband of the filter , andis equal to for the general paraunitary filterbank and linearphase filterbank and for the even-channel mirror-imagefilterbank. The coding gain measures the energy compactionability of a filterbank for the input signal. Let be the vari-ance of the input signal , and let be the variance of the

th subband signal . The objective function is defined as[32]

(59)

The commonly used AR(1) process with autocorrelation coeffi-cient was used in the optimization.

In the design, we factorize the orthogonal matrices with theGivens rotation. The complete parameter sets for the generalparaunitary filterbank and linear-phase filterbank contain signparameters, whereas the mirror-image filterbank only involvesangle parameters. For the general paraunitary filterbank,signparameters are required to express each in (34). For thelinear phase filterbank, the sign parameters of thein (47) areneeded to ensure the structure’s completeness. For simplicity,we fixed the sign parameters in the design. All the in(34) were set to be diag , and the sign param-eters of the in (47) were set to be 1. For the general pa-raunitary filterbank, this setting of the sign parameters assuresthat the designed filterbank has the symmetric delay propertyas discussed in [19]. To speed up the design procedure, the gra-dients of the objective functions can be calculated with explicitexpressions. In the design, we can use a constrained filterbankto initialize a general paraunitary system since the factoriza-tion form of the former is a special one of the later. In addi-tion, a high-order system can be initialized by a lower one. In

diag k=0(54)

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TABLE ISTOPBAND ENERGY OF FOUR-CHANNEL

FILTER BANKS

TABLE IICODING GAIN OF FOUR-CHANNEL FILTER BANKS

our design, these two methods were used. Tables I and II listthe results of four-channel paraunitary systems with differentorders. The stopbands were set to be

. From the tables, it canbe seen that both the stopband energy and coding gain of thepairwise mirror-image symmetry system are very close to thegeneral one, except in the zero-order case. The coding gain ofthe linear phase system is comparable with the others, whereasthe performance of the stopband attenuation is worse. The mag-nitude responses of four-channel filterbanks with areshown in Fig. 5.

VI. CONCLUSION

In this paper, we presented a general factorization theory for-channel causal FIR paraunitary filterbanks with constrained

filter length. Based on the singular value decomposition of thecoefficient matrices, any paraunitary matrix of order ,which corresponds to general paraunitary filterbanks with filtersof length , can be factorized into order-one parauni-tary building blocks in addition to the initial unitary matrix. Fur-thermore, the CS decomposition and spectral decomposition ofHermitian unitary matrices are investigated and applied to de-velop more efficient structures for paraunitary systems. Underthe theoretical framework for the general paraunitary systems,the linear phase filterbank and pairwise mirror-image symmetryfilterbank are revisited. It has been shown that the existing fac-torization structures of the two special paraunitary systems fallinto the proposed general structures. Moreover, we obtain moreefficient structures for these special paraunitary systems.

Fig. 5. Magnitude responses of four-channel paraunitary filterbanks withfilter lengthL = 32. The horizontal and vertical axes represent frequency andmagnitude response, respectively. Top: General paraunitary filterbank. Middle:Linear phase filterbank. Bottom: Mirror-image filterbank.

Although we have not mentioned the minimality of the struc-tures, it is easy to verify that all the structures as shown inFig. 2(a) through Fig. 4(b) are minimal. Moreover, the factoriza-tion theory for the paraunitary matrix with or without constraintsis not restricted to the filterbank. It can be applied to waveletsand multiwavelet systems and serve as a general theory for pa-raunitary systems.

APPENDIX APROOF OFLEMMA 1

Proof: It is easy to check that with any unitary, andangles, the matrix produced by (21) satisfies and

. We only need to prove the converse.

GAO et al.: ON FACTORIZATION OF -CHANNEL PARAUNITARY FILTERBANKS 1445

The Hermitian property of implies ,, and . The unitary property

of implies , , and, . Since is Hermitian, there

exists an unitary matrix and a real diagonal matrixso that

(60)

Substituting into the unitary condition on , we have. Since is a non-neg-

ative definite matrix, the diagonal entries of the diagonalmatrix are non-negative. Therefore, we can ex-press as diag . Let

diag ; then, ,and . From [30], we know thatthere exists an matrix so that and

. When is even, is a square unitary matrix.When is odd, there exists a vectorso that is a unitary

matrix. Letevenodd

; then

(61)

where the zero vector does not appear abovefor even .Without loss of generality, we assume that diag ,where the diagonal entries of are nonzero. In this case,diag , where contains all the diagonal entries ofabsolute value 1 of . Using (60), (61), and the relation

, we have

diag diag

(62)

where . The left side of

(62) is unitary, and so is the right side. The unitary property im-plies , , , and

. Therefore, we have ,, and . Now that is Hermitian

unitary, it can be expressed as , whereis unitary, and is a real diagonal matrix with diagonal en-tries of absolute value 1. By replacing with diag ,(62) becomes

(63)

Therefore, can be expressed as (21). For real, the Hermitianmatrices , , and are real symmetric, and all theunitary matrices in the above derivation can be real orthogonal.This completes the proof.

APPENDIX BDECOMPOSITION OFSPECIAL SYMMETRIC ORTHOGONAL

MATRICES IN SECTION IV-B

Lemma 3: Let be a real even-order matrix of the form

, where and are symmetric. is

orthogonal if and only if it can be expressed as

diag diag (64)

where is orthogonal, and and are diagonal matrices withentries and .

Proof: It is easy to verify that with arbitrary orthogonaland angles , the matrix produced by (64) is an orthogonal

matrix of the form , where and are

symmetric. In the following, we prove the converse.The orthogonality of implies that . Since

and are symmetric, . For these com-muting matrices, there exists an orthogonal matrixso that[31]

(65)

where and are diagonal matrices containing the eigen-values of and . The above equation means that we canexpress as in (64). The orthogonality of also implies that

. Substituting (65) into it, we have. Therefore, we can expressand by angles as stated

in the lemma. This completes the proof, and we have an easycorollary.

Corollary 1: Let be a real even-order matrix of the form

, where and are symmetric. is

orthogonal if and only if it can be expressed as

diag diag

diag (66)

where is orthogonal, and and are diagonal with entriesand .

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Xiqi Gao received the Ph.D. degree in electrical en-gineering from Southeast University, Nanjing, China,in 1997.

He joined the Department of Radio Engineering,Southeast University, in April 1992. From September1999 to August 2000, he was a visiting scholar atMassachusetts Institute of Technology, Cambridge,and Boston University, Boston, MA. His currentresearch interests include wavelets and filterbanks,image and video processing, and space-time signalprocessing for mobile communication.

Dr. Gao received the first- and second-class prizes of the Science and Tech-nology Progress Awards of the State Education Ministry of China in 1998

Troung Q. Nguyen (SM’95) received the B.S., M.S., and Ph.D degrees inelectrical engineering from the California Institute of Technology, Pasadena,in 1985, 1986, and 1989, respectively.

He was with the Lincoln Laboratory, Massachusetts Institute of Technology(MIT), Cambridge, from June 1989 to July 1994 as Member of Technical Staff.During the academic year 1993–1994, he was a visiting lecturer at MIT and anadjunct professor at Northeastern University, Boston, MA. From August 1994to July 1998, he was with the the Electrical and Computer Engineering Depart-ment, University of Wisconsin, Madison. He is now a Full Professor at BostonUniversity. His research interests are in the theory of wavelets and filterbanksand applications in image and video compression, telecommunications, bioin-formatics, medical imaging and enhancement, and analog/digital conversion. Heis the coauthor (with Prof. G. Strang) of a popular textbookWavelets and Fil-terbanks(Wellesley, MA: Wellesley-Cambridge, 1997) and the author of sev-eral Matlab-based toolboxes on image compression, electrocardiogram com-pression, and filterbank design. He also holds a patent on an efficient designmethod for wavelets and filterbanks and several patents on wavelet applications,including compression and signal analysis.

Prof. Nguyen received the IEEE TRANSACTIONS ON SIGNAL PROCESSING

Paper Award (in the Image and Multidimensional Processing area) for the paperhe co-wrote wth Prof. P. P. Vaidyanathan on linear-phase perfect-reconstructionfilterbanks in 1992. He received the NSF Career Award in 1995 and is currentlythe Series Editor (for Digital Signal Processing) for Academic Press. He servedas an Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING

from 1994 to 1996 and of the IEEE TRANSACTIONS ONCIRCUITS AND SYSTEMS

from 1996 to 1997. He received the Technology Award from Boston Universityfor his invention on Integer DCT (with Y. J. Chen and S. Oraintara). HeCo-Organizes (with Prof. G. Strang at MIT) several workshops on waveletsand also served on the DSP Technical Committee of the IEEE CAS society.

Gilbert Strang was an undergraduate at the Massachusetts Institute ofTechnology (MIT), Cambridge, and a Rhodes Scholar at Balliol College,Oxford, U.K. He received the Ph.D. degree from the University of California,Los Angeles. Since then, he has been a Professor of mathematics at MIT. Hehas published a monograph with G. Fix entitled “An Analysis of the FiniteElement Method,” and six textbooks:Introduction to Linear Algebra(1993,1998);Linear Algebra and Its Applications(1976, 1980, 1988);Introduction toApplied Mathematics(1986);Calculus(1991);Wavelets and Filter Banks[withT. Nguyen (1996)];Linear Algebra, Geodesy, and GPS[with K. Borre (1997)].

Dr. Strang served as President of SIAM from 1999 to 2000. He is an HonoraryFellow of Balliol College and has been a Sloan Fellow and a Fairchild Scholar.He is a Fellow of the American Academy of Arts and Sciences.