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Symmetric Delay Factorization: Generalized Framework for ParaUnitary Filter Banks Patrick RAULT and Christine GUILLEMOT* C.C.E.T.T - 4, rue du Clos Courtel - 35 510 Cesson-Sevigne - FRANCE fax: 33 (0) 299 124 098 Email: [email protected] (*) INRIA, Institut National de Recherche en Informatique et en Automatique, at IRISA/INRIA Rennes, Campus de Beaulieu, 35042 Rennes Cedex, FRANCE, phone: 33 (0) 299 84 74 29 fax: 33 (0) 299 84 25 31 Email: Christine.Guillemo[email protected] Please address correspondence to Christine Guillemot.

March 23, 1998

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Abstract The Symmetric Delay Factorization (SDF) is introduced in [1] for synthesizing linear phase paraunitary lter banks and is applied successfully in [2] for designing Time-Varying Filter Banks (TVFB). This paper describes a minimal and complete generalized symmetric delay factorization valid for a larger class of paraunitary lter banks, and for an arbitrary (even and odd) number of channels. The approach presented here provides a unifying framework for linear phase paraunitary lter banks including linear phase Lapped Orthogonal Transforms (LOT) and for cosine-modulated lter banks, this for an arbitrary number of channels (odd or even). This approach opens new perspectives in the design of time-varying lter banks used for image and video compression, especially in the framework of region or object based coding. The generalized symmetric delay factorization relying on lattice structure representations leads also naturally to fast implementation algorithms.

I. Introduction

Notwithstanding a large number of standards for encoding audiovisual signals, compression remains a widely-sought capability especially at low bit rates (lower than 64 kbit/s), for audiovisual communications over voiceband or wireless networks. However, even if for most multimedia applications, compression remains a key issue, this is not the only one that has to be taken into account. Emerging applications in the area of interactive audiovisual services show a growing interest for interactivity, content-based capabilities, and for integration of information of dierent nature, namely of synthetic and natural type, capabilities not well supported by the existing standards. Content-based capabilities rely on audiovisual object representation and coding of the multimedia content, i.e. on the ability to encode nite support and arbitrary shape objects independently of the neighbouring regions. Hence, block transforms, such as the DCT, have been extended to 'shape-adaptive transforms'. The rst solutions introduced are based on the calculation of orthogonal basis vectors [3] and on the coding of rectangular blocks using frequency domain region-zeroing [4]. However these two transformations require high computational cost. The technique of design of shape-adaptive block transforms described in [3] has been applied to an Ni Nj DCT leading to a 'generalized' Shape-Adaptive DCT. The orthogonalization that must be performed for each region shape leads to a high computational complexity. A 'simpli ed' SA-DCT [5], relying on prede ned orthogonal sets of DCT basis functions turns out to provide good results. Submitted to IEEE Trans. on Signal Processing

March 23, 1998

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Similar developments have been done, in the domain of multirate lter banks, by rst considering signal extension techniques, in order to apply iterated wavelet structures to bounded support image regions [6]. Time-Varying Filter Banks (TVFB), considering in the lter design the nite size characteristics of the region, have then been developed [7]. These lter banks also called Time Bounded Filter Banks (TBFB), process a nite support region ( nite length segments), independently of neighboring regions. The transformed regions are thus autonomous objects that can be manipulated. Time-varying lter banks can also be used in the context of transform switching in order to adapt the transformation to local characteristics of a region or sub-region. For example, one may wish to vary the lters, from one region to another, the down/up sampling rates, the structures of decomposition or the number of bands. However, shifting from one arbitrary lter bank to another independently designed perfect reconstruction system normally results in a substantial amount of reconstruction distortion during the transition period. In order to preserve the exact reconstruction property in the transition area, time-varying transition lter banks are required. Methods of design of TVFBs and TBFBs relying on a Least Square Approach are developed in [7], [8]. However, the lter banks obtained do not usually achieve the perfect reconstruction [9]. Several solutions of perfect reconstruction two-channel time-varying lter banks, that can be implemented on a lattice structure, are described in [10], [11]. Another approach, based on the redesign of the analysis time-varying lter banks in the transition and the boundary areas has been developed in [12], [13], [14], [15] for the twochannel case, and later extended to M -channel lter banks in [16] and [17]. The design problem is turned into a matrix orthogonalization problem. However, the orthogonalization procedure does not allow to obtain boundary lters with appropriate frequency or coding gain characteristics. Further optimization of the lters requiring a high computational cost is needed. Dierent optimization strategies are proposed in [17], [18]. This method has been extended to non paraunitary lter banks at the expense of a growing computational cost [16]. Lattice structures for the M -channel linear-phase paraunitary lter bank is rst introduced in [1] and is proven to be complete and minimal for even-channel and for lter length March 23, 1998

Submitted to IEEE Trans. on Signal Processing

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multiple of the number of channels M . In other words, even-channel linear phase paraunitary lter banks can be synthesized by decomposing lter bank analysis and synthesis matrices into their polyphase components resulting in the so-called Polyphase Transfer Matrices (PTM). The PTM, being paraunitary, can be decomposed into a series of orthogonal matrices and delay stages [1]. This decomposition is called the Symmetric Delay Factorization (SDF). An alternate form of this factorization is developed in [23], and is later on extended to the case where the lter length is not restricted to be a multiple of the number of channels M [21] and to nonparaunitary lter banks [20]. A technique for designing time-varying lter banks, based on the so-called symmetric delay factorization of linear-phase paraunitary lter banks, is developed in [2]. In comparison with other structures of decomposition of the PTM [19], the SDF presents the advantage of decomposing the polyphase transfer matrix into only square orthonormal matrices, even at the boundary of nite length signals, simplifying signi cantly the design procedure of TVFB. The number of parameters to be optimized is signi cantly reduced and this is essential in the design of time-varying lter banks for which the number of parameters grows rapidly. The symmetric delay factorization provides also a lattice structure decomposition of the polyphase transfer matrix, leading directly to a fast implementation algorithm of the lter bank. However, the symmetric delay factorization technique, known so far, applies only to linear phase paraunitary lter banks. In addition, the approaches proposed lead to non square matrices at the boundary of nite length signals when the number of channels is odd [1]. This paper describes a new minimal and complete symmetric delay factorization of the polyphase transfer matrix valid for a large class of paraunitary lter banks. It provides a unifying framework for linear phase paraunitary lter banks including linear phase Lapped Orthogonal Transforms and for cosine-modulated lter banks, now widely used in speech, audio and image coding. A second key issue is that the approach does not make any assumption on the parity of the number of channels. It leads also for the case where the number of channels is odd to only square orthonormal matrices, even at the boundary of nite length signals. This generalized symmetric delay factorization approach opens new perspectives in the design of time-varying lter banks, and nds strong interest in the Submitted to IEEE Trans. on Signal Processing

March 23, 1998

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framework of image and video object-based coding. II. Notations

In terms of notation, the following conventions are adopted: In is a n n identity matrix. On is a n n null matrix. ()T stands for the transposition of matrices and vectors. It is assumed, without loss of generality, that analysis and synthesis lter banks are composed of respectively M lters hm and gm of same length L. III. Design by Symmetric Delay Factorization

Let E N (z) be the analysis lter bank polyphase transfer matrix. The matrix E N (z) can P be expressed as E N (z) = Nn=0 z?nPnN , where PN (n) are M M matrices. It has been shown in [19], that the polyphase transfer matrix can be expressed as a product of M M orthogonal matrices of z polynoms of degree 1

E N (z) = VN (z)VN ?1 (z):::V1 (z)V0 ;

(1)

where Vn(z) is given by

Vn(z) = n Wn1(z)WnT ;

(2)

and where 1 (z) is a matrix of size M M of the form

0 1 @IM ?1 O?1A : O

z

(3)

This factorization leads to an implementation of the lter bank under the form of a lattice [19], as shown in gure (1). However, this lattice, composed of cells with one delayed branch and M ? 1 non delayed branches is not symmetrical, leading to non square orthonormal matrices at the boundary of nit length signals. In order to overcome this problem, a new polyphase transfer matrix factorization framework, has been introduced in [1]. It is rst shown that the M -channel linear phase perfect reconstruction lter bank is composed of mu symmetric lters and of ml antisymmetric lters, where mu = ml = M=2 March 23, 1998

Submitted to IEEE Trans. on Signal Processing

6 M

M ?1

M ?1

B0

B2i

M ?1

M ?1 Z ?1

B2i+1

BN

Z ?1

M

Analysis lattice M

B0T

B2Ti M ?1

M ?1

Z ?1

B2Ti+1 M ?1

Z ?1

Synthesis lattice

BNT

M

M ?1

Fig. 1. M -Channel Lattice with one delayed channel.

if M is even, and where (mu; ml ) = ( M2?1 ; M2+1 ) or (mu; ml ) = ( M2+1 ; M2?1 ) if M is odd. A minimal factorization of the matrices Vn(z); n = 0; :::; N , is developed for a large class of linear phase paraunitary lter banks. This factorization is proved to be complete only for an even number of channels M , and can be expressed in this case by

Vn(z) = n Wn(z)WnT ; where

Wn = (z) = n =

0 1 p1(2) @IM=2 IM=2 A ; IM=2 ?IM=2 0 1 @ IM=2 ?O1M=2 A ; O z IM=2 0 M=2 1 @ Sn OM=2A ; OM=2

(4)

(5)

Tn

and where the matrices Sn and Tn are M=2 M=2 orthogonal matrices. This factorization leads to the lattice structured implementation shown in gure (2), where B2i and B2i+1 are orthogonal matrices. This polyphase transfer matrix factorization, also known as the symmetric delay Factorization (SDF) technique, has been applied in the design of time-varying lapped transform [2], [22], [23]. This leads, in the case where M is even, to a lattice implementation formed by only square orthonormal matrices, even at the boundary of nite length signals. Submitted to IEEE Trans. on Signal Processing

March 23, 1998

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However, this structure of decomposition applies only to linear phase lters, and its completeness is proved only for an even number of channels.. A minimal and complete generalized symmetric delay factorization, valid for a larger class of paraunitary lter banks, and for an arbitrary number of channels, is developed in the next section. The approach provides a unifying framework for linear phase paraunitary lter banks including linear phase Lapped Orthogonal Transforms (LOT) and for cosine-modulated lter banks, now widely used in image compression, and this for an arbitrary number of channels (odd or even). M

M

mu

mu

B0

Z ?1

B2i ml

ml

mu

mu

B0T ml

Z ?1

B2i+1

BN

M

ml

mu

Analysis Lattice ml

B2Ti ml

mu

ml

Z ?1

B2Ti+1 mu

mu Z ?1

Synthesis Lattice

BNT

M

ml

Fig. 2. M -Channel SDF Lattice.

A. Generalized Symmetric Delay Factorization

Let us consider an M -channel lter bank, composed of Finite Impulse Response lters of length L = M (N + 1), where N is an arbitrary integer. Let also

8 > :( M2?1 ; M2+1 )

if M is even, if M is odd.

(6)

The following theorem, providing a cascade form structure of the Polyphase Transfer Matrix of a paraunitary lter bank, is proved in dierent cases, where the paraunitary system is linear phase and also in the case where the system is a cosine-modulated lter bank. This cascade form structure represents a Symmetric Delay Factorization of the lter bank. March 23, 1998

Submitted to IEEE Trans. on Signal Processing

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Theorem 1: Let E N (z) be a FIR paraunitary matrix of polynoms of maximum degree N , representing the polyphase transfer matrix of a M -channel lter bank. The matrix P E N (z) can be expressed as E N (z) = Nn=0 z?n PnN , where PN (n) are M M matrices [19], [2]. If

rank(PNN ) ml and rank(P0N ) ml ;

(7)

then the matrix E N (z) can be decomposed as the product

E N (z) = VN (z)VN ?1 (z):::V1 (z)V0 ;

(8)

where, the matrices Vn(z), for n > 0, are given by

Vn(z) = Bno(z) Bn = Bn;1 Bn;2 o(z) =

0 @Imu

(9)

1 O A ; ?1

O z Iml

with Bn;1, matrix of size M mu , and Bn;2, matrix of size M ml , or equivalently where, the matrices Vn(z), for n > 0, are given by

Vn(z) = Bne(z) Bn = Bn;1 Bn;2 e(z) =

0 @Iml

(10)

1 O A ; ?1

O z Imu

with Bn;1, matrix of size M ml , and Bn;2, matrix of size M mu . In both cases, Bn is a unitary matrix and Bn;1 must be orthogonal to Bn;2. Note that in practice, the matrices Bi as explained in section (IV) are Givens plane rotations. Note also that in the case where the number of channels M is odd, mu being dierent from ml , a "Pseudo-Symmetric" Delay Factorization is obtained. Submitted to IEEE Trans. on Signal Processing

March 23, 1998

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B. Preliminary Results

In order to prove theorem 1, let us rst demonstrate the two following preliminary results. Theorem 2: Let E N (z) be a FIR paraunitary matrix of polynoms of maximum degree N , representing the Polyphase Transfer Matrix of a M -channel lter bank. The matrix P E N (z) verify E N (z) = Nn=0 z?nPnN , where PnN are M M matrices. If the lter bank is linear phase, then the matrices P0N and PNN verify

rank(PNN ) = rank(P0N ) M2 :

(11)

Let hk (z) be the kth lter of length L = M (N + 1) of a M -channel lter bank of Polyphase Transfer Matrix E N (z). The matrix E N (z) has elements in the eld of polynomials in z. The elements of the matrix E N (z), [E N (z)]k;l , are the z-transforms of the sequences [2]

Proof:

[eN (n)]k;l = hk (nM + l);

(12)

and [PnN ]k;l = [eN (n)]k;l. In [1], it is demonstrated that a linear phase paraunitary lter bank has mu symmetric lters and ml antisymmetric lters. The impulse response of the kth lter veri es hk (n) = sgn(k)hk (L ? 1 ? n), where sgn(k) = 1 if hk is symmetric, and sgn(k) = ?1 if hk is antisymmetric. As a consequence, the elements [eN (n)]k;l verify [eN (n)]k;l = sgn(k)[eN (N ? n)]k;M ?1?l:

(13)

So, the kth row of PNN is the time-reverse of the kth row of sgn(k)P0N , and

rank(PnN ) = rank(PNN?n):

(14)

Since the lter bank is paraunitary, the matrices P0N and PNN verify (P0N )T PNN = 0, and span two orthogonal spaces of maximum dimension M . Since P0N and PNN span two orthogonal spaces of the same rank, obviously they verify rank(P0N ) = rank(PNN ) M2 . March 23, 1998

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Hence, by taking a linear phase paraunitary lter bank such as

rank(P0N ) = ml or rank(PNN ) = ml ;

(15)

constraint which is veri ed for a large class of linear phase paraunitary lter banks useful from a practical standpoint, then this lter bank will satisfy theorem 1's assumption (equation 7). Theorem 3: Let E N (z), be a FIR paraunitary matrix of polynoms of maximum degree N , representing the Polyphase Transfer Matrix of a M -channel lter bank. The PTM P E N (z) veri es E N (z) = Nn=0 z?nPnN , where PnN are M M matrices. If the lter bank is a cosine-modulated lter bank, then the matrices PnN , n = 0; :::; N verify

8 > : ml

if n is even if n is odd:

(16)

Let us consider a paraunitary cosine-modulated lter bank obtained by modulation of a linear phase lowpass prototype h(n) of length L = (N + 1)M = 2pM . The impulse response of the kth lter can be expressed as [24], [25]

Proof:

hk (n) = ck;nh(n) ? where, ck;n = cos (2k + 1) (n ? L ? 1 ) ? ; 2 2M 4

(17)

Ce = fck;ng0kM ?1;0nM ?1; Co = fck;ng0kM ?1;M n2M ?1:

(18) (19)

ck;n = ck;n+2M ; ck;pM +n = ck;(p+1)M ?n; 8p 2 f0; : : : ; N g ck;(2p+1)M +ml = 0; if M is odd

(20)

Let Ce and Co the M M square matrices de ned by

The equations

Submitted to IEEE Trans. on Signal Processing

March 23, 1998

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expressing, in the modulation function ck;n, ml symmetry relations for n 2 f0; : : : ; M ? 1g, and ml symmetry relations for n 2 fM; : : : ; 2M ? 1g can be veri ed easily. The ranks of the matrices Ce and Co are thus rank(Ce) = mu and rank(Co) = ml . Since the matrices P2Ni and P2Ni+1 can be obtained from Ce and Co by the relations [2]

P2Ni = Ceg2i P2Ni+1 = Cog2i+1

(21) (22)

where gn is a vector of length M de ned by gn(l) = h(nM + l), then rank(P2Ni ) mu and rank(P2Ni+1 ) ml . From this result, let us show that cosine modulated lter banks satisfy theorem 1's assumptions (equation 7). Since the ith row of P0N is equal to the ith row of Ce multiplied by h(i), if the linear phase prototype lter is such as,

h(n) = 6 0 8n 2 f0; : : : ; mu ? 1; mu + 1; : : : ; M ? 1g h(mu) = 6 0 if M is even

(23)

ml rank(P0N ) mu ; rank(PNN ) = ml ;

(24)

then,

which ensures that theorem 1 assumptions are veri ed. C. Generalized Symmetric Delay Factorization: Proof of Theorem 1

Theorem 1 can be proved by using a degree reduction procedure. Let us consider E N (z), PTM matrix of degree N of the M -channel lter bank. The matrix E N (z) veri es P E N (z) = Nn=0 z?nPnN . Let us assume that the matrix E N (z) is such that theorem 1 assumptions (7) are satis ed. As shown in theorems 2 and 3, this is the case for linear phase paraunitary lter bank and for cosine-modulated lter banks respecting only respectively constraints (15) and (24). March 23, 1998

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Let the orthogonal matrix BN = (BN;1; BN;2) be such as, T PN = 0 BN; 1 N T PN = 0 BN; 2 0 T T BN;1 BN; 2 = BN;1 BN;2 = 0

(25)

The existence and the paraunitarity of the two matrices BN;1 and BN;2 can be proved from theorems 2 or 3 and from the respective constraints (15) and (24). The proof is provided in annex (VII). Let the matrix E N ?1(z) be

E N ?1(z) = VN (z?1 )T E N (z) = (BN N (z+1 ))T E N (z) = (BN N

N X

(z+1 ))T (

n=0

z?n PnN );

(26)

where VN (z?1 ) = BN N (z+1 ) is de ned as follows: if rank(PNN ) ml , then VN is de ned as in equation (10), and N (z) = e(z) if rank(P0N ) ml , then VN is de ned as in equation (9), and N (z ) = o (z ) otherwise, VN can be de ned as in equations (9) or (10).

By replacing equations (25,9,10) in equation (26), the noncausal term of E N ?1(z) is cancelled. From equation (25), it appears that the degree of E N ?1 (z) is lower or equal to N ? 1. Since E N (z) has degree N and VN has degree 1, E N ?1 (z) cannot have a degree smaller than N ? 1. So the degree of E N ?1 (z) is precisely N ? 1, and the factorization is minimal. Since the unitary transformation VN (z?1 ) = BN N (z+1 ) is orthogonal, the matrix E N ?1 (z) is also paraunitary and we can write

E N (z) = (BN N (z?1 ))E N ?1(z): Submitted to IEEE Trans. on Signal Processing

(27) March 23, 1998

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In addition, the orthogonality property of VN guarantees that

rank(P0N ?1) rank(P0N ); rank(PNN??11) rank(PNN ):

(28) (29)

Hence E N ?1(z) veri es theorem 1 assumptions (7). Repeating the degree reduction procedure N times, we obtain the complete decomposition (8) where V0 is a nonzero vector. Then the lter can be implemented by the lattice structure of gure (2). In the 2-channel case, we obtain the classical lattice structure, as in [19], and valid for all 2-channel paraunitary ler banks. IV. Design and Implementation of Time-Bounded Filter Banks

This factorization leads, for time-bounded signals, to a representation of the analysis and synthesis lter banks under the form of a lattice structure, as shown in gure (3), where the grey blocks are transient blocks (analog to the transient lters). Note that, if M is even, then ml = mu = M=2. In the case where M is odd, ml = M2?1 and mu = M2+1 , mu being dierent from ml , a "Pseudo-Symmetric" Delay Factorization is obtained. The interesting property is that the transient blocks are square orthogonal blocks, which simpli es the design procedure. The size of the square orthogonal blocks depends on their location in the matrix. For example, the blocks in the upper part of gure (3) (left edge of the lter bank) are of respective size l l, (l + ml ) (l + ml ), and (mu + ml ) (mu + ml ). These blocks, representing unitary orthogonal matrices, can be designed by a cascade of Givens plane rotations [19], as explained in annex (VII). This decomposition in terms of Givens rotations inherently guarantees the orthogonality and the perfect reconstruction properties of the time-bounded lter bank. The parameters of the Givens rotations Bi can then be optimized so that the lters hk (n) derived from the matrices Bi by equations (8) and (10) verify criteria such as the classical frequency selectivity or the maximum coding gain. The complete design procedure consists thus in rst decomposing a time unbounded lter bank by using the symmetric delay procedure presented above. It consists then in pruning unnecessary paths and blocks, used for the segments of signal outside the bounds. For each block in each stage, the procedure consists in checking its input branches and in removing the equivalent number of output branches in order to have a square orthogonal March 23, 1998

Submitted to IEEE Trans. on Signal Processing

14 l ml mu

B0

B1

B0

B1

B0

B1

B0

B1

B0

B2 B2 B2

ml l ml mu

B3 B3

r mu

r

Fig. 3. Time-Bounded Lattice.

matrix. These new orthogonal matrices provide the degrees of freedom in the transitory lter banks. The remaining blocks, those in white on gure (3), are the same as in the time-invariant SDF and are kept unchanged. All the degrees of freedom are then spanned in a search for the minimum of a given cost function, expressing for example the frequency selectivity or the coding gain. Any change of the plane Givens rotations (each block of the lattice) allows to span all perfect reconstruction solutions described by the structure. Degrees of Freedom:

The transient blocks (in grey) having mu + ml inputs are M M orthogonal transient matrices whereas the blocks with ml (resp. mu) inputs are ml ml (resp. mu mu) transient orthononal matrices. Let us consider the case where l = r. For each boundary, the number of matrices of transition is given by: stage 2i + 1 ! i + 1 matrices of size M M stage 2i ! i matrices of size M M and one matrix of size l l

(30) (31)

for i 2 f0; : : : ; K ? 1g and where 2K represents the number of stages Bi . Given that an otrthogonal matrix M M has M (M ? 1)=2 degrees of freedom corresponding to Givens Submitted to IEEE Trans. on Signal Processing

March 23, 1998

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plane rotations angles (see annex(VII)), the number of degrees of freedom for each signal boundary is given by

X

K ?1

X

K ?1

!

i + 1 M (M2 ? 1) + K l(l ?2 1) ; i=0 i=0 M (M ? 1) + K l(l ? 1) = K2 2 2

=

i+

(32) (33)

2 is the total number of degrees of freedom for a 1-D signal. Table (I) shows the number of degrees of freedom according to the number of bands and of lattice stages with l = r = M2 , which turn out to be, in this particular case, similar to the number of degrees of freedom obtained in [2] for linear-phase paraunitary lter banks. M 2 4 6 8 10 12 14 16 20 24 32 1 1 7 18 34 55 81 112 148 235 342 616 2 4 26 66 124 200 294 406 536 850 1236 2224 K 3 9 57 144 270 435 639 882 1164 1845 2682 4824 4 16 100 252 472 760 1116 1540 2032 3220 4680 8416 5 25 155 390 730 1175 1725 2380 3140 4975 7230 13000 TABLE I Number of degrees of freedom for each boundary.

Examples: Examples of frequency responses obtained for transient time varying lowpass lters of a 4-channel cosine modulated lter bank, in the case where l = r = M=2, are given in gure (4). The time-invariant lter bank has been obtained by cosine-modulation of a lowpass prototype lter of length L = 2 2 4, with the coecients given in table (II). Figure (4.a) gives the frequency response at the signal boundary of the rst low-pass lter. The length of the equivalent lter bank is 6 taps. Figure (4.b) shows the frequency response at the signal boundary of the second low-pass lter. The length of the equivalent lter bank is 12 taps and the third low-pass lter is the time-invariant 16 taps lter shown in gure (4.c). March 23, 1998

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h[n] -1.3908586668E-02 -6.1494240116E-03 8.0034052982E-03 3.2265417503E-02 6.8581843589E-02 1.0752945970E-01 1.3994836685E-01 1.5909753229E-01 1.5909753229E-01 1.3994836685E-01 1.0752945970E-01 6.8581843589E-02 3.2265417503E-02 8.0034052982E-03 -6.1494240116E-03 -1.3908586668E-02 TABLE II Prototype Filter Coefficients. 3

2

2 1.8

1.8 2.5

1.6 1.6 1.4 2 1.4

1.5

1.2

1.2

1 0.8

1 1

0.6 0.8 0.4 0.5

a.

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

b.

0.6

0.4 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

c.

0.2 0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Fig. 4. Frequency responses of the low frequency transient lters

V. Design and Implementation of Transition Filter Banks in Filter Bank Switching

Time-varying lter banks are also useful in a context of lter bank switching. Indeed, instantaneous switching between two analysis lter banks, Hodd and Hnew , illustrated in gure (5), can be obtained by juxtaposition of two bounded lattices. In order to eliminate the distortion in the transition area, a new time-varying synthesis section (Tn) has to be Submitted to IEEE Trans. on Signal Processing

March 23, 1998

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designed for each transition. An alternate approach to the lter bank switching problem consists in designing 'soft' transitions also in the analysis lter banks. For example, in gure (6), in order to avoid the transition with transient matrices of size ml ml and matrices of size m0u m0u , that would result from the two juxtaposed bounded lattices, as seen in the previous sections. we merge the two matrices ml ml and m0u m0u in order to obtain new transients blocks with matrices of size (ml + m0u) (ml + m0u ). The angles of the orthogonal transient blocks are then optimized as explained in the section above.

Hold

Hnew

Gold T1 T2 T3 Gnew

y(n) x(n)

Fig. 5. Instantaneous Filter bank Switching:

Hold and Hnew are respectively the rst and the second analysis lter banks; Gold and Gnew are the corresponding synthesis lter banks; and T1 : : : T3 are the transition synthesis lter banks. Note that Figure (5) represents the particular case where the parameters r and l0 of the two juxtaposed bounded lattices are equal to zero, or in other words where the lter bank switching is performed at the boundary of blocks B0 and B00 . This corresponds to the case where the length of the juxtaposed bounded signals are multiples of respectively (mu + ml ) and (m0u + m0l ). This is why there is no transient block in the rst stage of the lattice. However a similar lattice representation can be obtained when r 6= l0 6= 0. VI. Conclusion

A new minimal and complete symmetric delay factorization formalism valid for a large class of paraunitary lter banks including linear phase lter banks, cosine-modulated lter banks, all the 2-channel cases, has been described without assumption on the parity of March 23, 1998

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B0 B0

B2

B1

B0

B2

B1

B0 ml mu m0l m0u

B2

B1

B2

B1

B0 B00

B10

B0

0

B20

B0

1

B00

B20

B10

B0

0

B0

2

B3 B3 B3 ml mu m0l mu m0u m0l

B30 B30

Fig. 6. \Soft" switching lattice: switching from a lattice (blocks B , i = 0; :::; 3) of (m + m ) bands to a new lattice (blocks B , i = 0; :::; 3) of (m + m ) bands. i

0

0

u

i

u

l

0

l

the number of channels. This formalism opens new perspectives in the design of of timebounded and time-varying lter banks. The symmetric delay factorization turns out to be a very powerful tool for the design of Time-Varying lter banks useful for the processing of time bounded signals or in the transition phase when switching from one lter bank to another one to process two regions with dierent characteristics. The approach allows to use Givens rotations in the design procedure that inherently guarantee the perfect reconstruction, even in presence of quantization of the lattice coecients, and that provide fast implementation algorithms. This approach, by preserving the square shape of the dierent blocks of the matrices of transition to be designed at the border of a nite length signal, provides a simpli ed method for synthesizing and implementing time-varying lter banks. VII. Acknowledgements

The authors wish to thank Tom Kalker from Philips Research Labs in Eindhoven for helpful discussions. Submitted to IEEE Trans. on Signal Processing

March 23, 1998

19

APPENDIX: EXISTENCE AND PARAUNITARITY OF THE MATRICES BN;1 AND BN;2

Let L, length of the analysis lter bank, be of the form L = NM + R, where 0 R < M . A component Ek;m(z) of the polyphase transfer matrix can be expressed as: Ek;m(z ) =

N X n=0

z?n hk (nM + m);

(34)

where the lters hk are extended by zeros, if needed. The polyphase transfer matrix can thus be written as E(z ) =

N X n=0

z?nPnN ;

(35)

where Pn is a M N matrix given by

0 h0 (nM ) : : : h0 (nM + m) : : : h0 (nM + M ? 1) B ... ... ... B PnN = B @

hM ?1 (nM ) : : : hM ?1(nM + m) : : : hM ?1 (nM + M ? 1)

1 CC CA :

(36)

Let E~ (z) = ET(z?1) be the polyphase transfer matrix of the synthesis lter bank, The perfect reconstruction condition can be formulated as N N X X n=0 m=0

zn?m (PnN )T PmN = I:

(37)

This equation can be rewritten as N ?l N X X l l=0

z

n=0

(PnN )T Pn+lN = I;

(38)

and is veri ed if and only if

8 l 2 [0; : : : ; N ] )

N ?l X m=0

(PnN )T PnN+l = l I:

(39)

This implies in particular that (P0N )T PNN = 0. The existence and paraunitarity of the two matrices BN;1 and BN;2 can be proved as follows. March 23, 1998

Submitted to IEEE Trans. on Signal Processing

20

Let us assume that theorem 1 assumptions are verifed, hence rank(P0N ) ml and rank(PNN ) ml . From relation (39), the M M matrices PNN and P0N verify (PNN )T P0N = 0, and as mu + ml = M we can deduce rank(PNN ) mu , and rank(P0N ) mu . In the odd case, we cannot have rank(PNN ) = rank(P0N ) = mu (this would mean that PNN and P0N span a space of dimension M + 1). We have thus the two following alternatives:

rank(PNN ) mu and rank(P0N ) = ml rank(PNN ) = ml and rank(P0N ) mu

(40) (41)

Let us consider, without loss of generality, case (40), and let us prove the existence and paraunitarity of the matrices BN;1 and BN;2 . The matrix BN;1, being of size M ml , and rank(PNN ) mu , can be constructed with ml orthogonal vectors of the space spanned by P0N , orthogonal to the space spanned by PNN . In practice, the matrix is then orthogonalized by using a procedure of Gram-Schmidt. Similarly, the matrix BN;2 of size M mu orthogonal to P0N , can be constructed by taking ml orthogonal vectors of the space spanned by PNN (space of dimension ml and orthogonal to the space spanned by P0N ) and one vector orthogonal to these ml vectors and to P0N .

BN;1 being in the space spanned by P0N , BN;1 veri es condition (25). Similarly, BN;2 being orthogonal to P0N , the condition (25) is established. One can deduce that BN;1 is orthogonal to BN;2, proving condition (26). APPENDIX: FACTORIZATION IN TERMS OF GIVENS ROTATIONS

This section reviews, for sake of completeness, the decomposition of a unitary matrix in terms of a cascade of Givens plane rotations. Let us consider a unitary matrix P 2 Rmm, i.e., verifying PTP = I. This matrix can be written under the form [26], [19] :

0 1 S 0 A P = @ 0

Submitted to IEEE Trans. on Signal Processing

(42) March 23, 1998

21

where STS = Im?1, = 1 and where = m?2 0, with i de ned as:

8 > ci;m if k = l = i or k = l = m ? 1 > > > ?si;m if k = m ? 1 and l = i > > > :0 otherwise

(43)

and, where ci;m = cos(i;m) and si;m = sin(i;m ). The matrix i can be rewritten as

0 BB BB B i = B BB BB B@

1 CC CC si;m C CC : 0 C C ... C CA

... 0

... 0

0 ci;m 0 0 0 ... 0 ?si;m 0 0 ci;m

(44)

By applying a reduction procedure, it can be shown easily that the unitary matrix P can be decomposed in terms of a cascade of plane rotations known as the Givens plane rotations. The principle of the decomposition consists in searching m?2 such that

0 1 BB .. ... ... C CC BB . CC : Tm?2P = B BB C B@ 0 CCA

(45)

This is equivalent to searching m?2;m verifying

P (m ? 2; m ? 1)cos(m?2;m) ? P (m ? 1; m ? 1)cos(m?2;m ) = 0:

(46)

Solutions to the above equation are given by

m ? 2; m ? 1) ; tan(m?2;m ) = PP ((m ? 1; m ? 1) March 23, 1998

(47)

Submitted to IEEE Trans. on Signal Processing

22

if P (m ? 1; m ? 1) 6= 0, and by m?2;m = 2 , otherwise. By iterating the reduction procedure for Tm?1Tm?2P up to T0 : : : Tm?1Tm?2P we obtain

0 1 S 0 A: T0 Tm?2P = @ 0

(48)

The transformation by the matrix P can then be represented as shown in gure (7).

S

0;m 1;m

Fig. 7. Givens Reduction of the Matrix P

The factorization procedure can be applied on the matrix S, leading to the structure shown in gure (8), where (i) corresponds to the matrix 2 R (i+1)(i+1) . The 'butter y' structure of the matrix is shown in gure (9), in which the basic unit represents the transformation

0 1 0 10 1 y cos ( ) sin ( ) @ 0A = @ A : @x0A ?sin() cos()

y1

0

(0)

x1

(49)

(1) (m?2)

m?1

Fig. 8. Factorization in terms of Givens Plane Rotations. Submitted to IEEE Trans. on Signal Processing

March 23, 1998

23

0

i+1

Fig. 9. Butter y Structure of the Transformation given by ( ). Each crossing section implements the transformation of relation (49). i

References [1] A. K. Soman, P. P. Vaidyanathan, and T. Q. Nguyen, \Linear phase paraunitary lter banks: Theory, factorizations and designs," IEEE Trans. on signal Processing, vol. 40, no. 12, pp. 3480{3496, December 1993. [2] R. L. Queiroz, On Lapped Transforms, Ph.D. thesis, University of Texas at Arlington, 1994. [3] M. Gilge, T. Engelhardt, and R. Mehlan, \Coding of arbitrarily-shaped image segments based on a generalized orthogonal transform," EURASIP journal on Signal Processing and Image Communication, vol. 1, pp. 153{ 180, 1989. [4] H. H. Chen, M. R. Civanlar, and B. G. Haskell, \A block transform coder for arbitrarily-shaped image segments," in Proceedings of the International Workshop on Coding for Very Low Bitrate Video, University of Essex, England, April 1994. [5] T. Sikora and B. Makai, \Shape-adaptive DCT for generic coding of video," in Contribution to the International Organisation for Standardization, ISO/IEC Coding of Moving Pictures and Associated Audio, July 1994. [6] H. J. Barnard, Image and Video Coding using a Wavelet Decomposition, Ph.D. thesis, Delft University, 1994. [7] K. Nayebi, T. P. Barnwell and M. J. T. Smith, \Analysis-synthesis systems with time-varying lter bank structures," in Proceedings of the IEEE Int. Conf. on Acoustics, Speech and Signal Processing, ICASSP, March 1992, vol. IV, pp. 617{620. [8] K. Nayebi, T. P. Barnwell and M. J. T. Smith, \Time-domain lter bank analysis: A new design theory," IEEE Trans. on Signal Processing, vol. 40, pp. 1412{1428, June 1992. [9] I. Sodagar, Analysis and Design of Time-Varying Filter Banks, Ph.D. thesis, Georgia Institute of technology, 1994. [10] J. L. Arrowood Jr. and M. J. T. Smith, \Exact reconstruction analysis/synthesis lter banks with time-varying lters," in ICASSP, Mineapolis, Minnesota, April 1993, IEEE, vol. III, pp. 233{236. [11] J. C. Pesquet and H. Krim, \Time-varying lter banks for the analysis of PC processs," in EUropean SIgnal Processing COnference, EUSIPCO, September 1994, vol. III. [12] C. Herley, Wavelets and lter Banks, Ph.D. thesis, Columbia University, 1993. [13] C. Herley, J. Kovacevic, K. Ramchandran and M. Vetterli, \Time-varying orthonormal tilings of the timefrequency plane," in IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, ICASSP, 1993, vol. III, pp. 205{208. [14] C. Herley and M. Vetterli, \Orthogonal time-varying lter banks and wavelets," in IEEE Int. Symposium on Circuits and Systems, ISCAS, 1993, vol. I, pp. 391{394. March 23, 1998

Submitted to IEEE Trans. on Signal Processing

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[15] C. Herley and M. Vetterli, \Orthogonal time-varying lter banks and wavelets," IEEE Trans. on Signal Processing, vol. 42, no. 10, pp. 2650{2663, October 1994. [16] C. Herley, \Boundary lters for nite-length signals and time-varying lter banks," IEEE Trans. on Circuits and Systems-II: Analog and digital signal processing, vol. 42, no. 2, pp. 102{114, February 1995. [17] T. Kalker, \On optimal boundary and transition lters in time-varying lter banks," in IEEE Int. Conf. on Image Processing, Lausanne, September 1996, vol. I, pp. 625{628. [18] A. Mertins, \Time-varying and support preservative lter banks: Design of optimal transition and boundary lters via SVD," in IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, ICASSP, Detroit, May 1995, pp. 1316{1319. [19] P.P. Vaidyanathan, Multirate systems and lter banks, Prentice Hall, 1993. [20] Takayuki Nagai, Takaaki Fuchie, and Masaaki Ikehara, \Design of linear phase M -channel perfect reconstruction FIR lter banks," IEEE Trans. on Signal Processing, vol. 45, no. 9, pp. 2380{2387, September 1997. [21] Trac Duy Tran and Truong Q. Nguyen, \On M -channel linear phase FIR lter banks and application in image compression," IEEE trans. on Signal Processing, vol. 45, no. 9, pp. 2175{2187, September 1997. [22] R. L. de Queiroz and K. R. Rao, \Optimal orthogonal boundary lter banks," in IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, ICASSP, 1995, vol. III, pp. 1296{1299. [23] T. Q. Nguyen R. L. de Queiroz and K.R. Rao, \The GenLOT: Generalized linear-phase lapped orthogonal transform," IEEE Trans. on Signal Processing, vol. 44, no. 3, pp. 497{507, March 1996. [24] Henrique S. Malvar, Signal Processing with Lapped Transforms, Artech House, Inc, 1992. [25] R. D. Koilpillai and P. P. Vaidyanathan, \Cosine-modulated FIR lter banks satisfying perfect reconstruction," IEEE Trans. on Signal Processing, vol. 40, no. 4, pp. 770{783, April 1992. [26] Gene H. Golub and Charles F. Van Loan, Matrix computations, The John Hopkins University Press, 1989. [27] H. Barnard, J. H. Weber, and J. Biemond, \A region-based discrete wavelet transform," in Proceedings of the EUropean SIgnal Processing COnference, September 1994, pp. 1234{1237. [28] T. P. Barnwell K. Nayebi and M. J. T. Smith, \Design of low delay FIR analysis-synthesis lter banks in the context of periodically time-varying systems," in Proc. Conf. on Information Sciences and Systems, 1991. [29] P. Rault and C. Guillemot, \Pseudo symmetric delay factorization: A theory for M-odd channel paraunitary lter banks," in Proceedings of the International Multidimensional Digital Signal Processing Workshop, IMDSP, Belize City, BELIZE, March 1996. [30] P. Rault and C. Guillemot, \Symmetric delay factorization: A generalized theory for paraunitary lter banks," in Proceedings of the EUropean SIgnal Processing COnference, EUSIPCO, 1996.

Submitted to IEEE Trans. on Signal Processing

March 23, 1998

Symmetric Delay Factorization: Generalized Framework for ParaUnitary Filter Banks Patrick RAULT and Christine GUILLEMOT* C.C.E.T.T - 4, rue du Clos Courtel - 35 510 Cesson-Sevigne - FRANCE fax: 33 (0) 299 124 098 Email: [email protected] (*) INRIA, Institut National de Recherche en Informatique et en Automatique, at IRISA/INRIA Rennes, Campus de Beaulieu, 35042 Rennes Cedex, FRANCE, phone: 33 (0) 299 84 74 29 fax: 33 (0) 299 84 25 31 Email: Christine.Guillemo[email protected] Please address correspondence to Christine Guillemot.

March 23, 1998

2

Abstract The Symmetric Delay Factorization (SDF) is introduced in [1] for synthesizing linear phase paraunitary lter banks and is applied successfully in [2] for designing Time-Varying Filter Banks (TVFB). This paper describes a minimal and complete generalized symmetric delay factorization valid for a larger class of paraunitary lter banks, and for an arbitrary (even and odd) number of channels. The approach presented here provides a unifying framework for linear phase paraunitary lter banks including linear phase Lapped Orthogonal Transforms (LOT) and for cosine-modulated lter banks, this for an arbitrary number of channels (odd or even). This approach opens new perspectives in the design of time-varying lter banks used for image and video compression, especially in the framework of region or object based coding. The generalized symmetric delay factorization relying on lattice structure representations leads also naturally to fast implementation algorithms.

I. Introduction

Notwithstanding a large number of standards for encoding audiovisual signals, compression remains a widely-sought capability especially at low bit rates (lower than 64 kbit/s), for audiovisual communications over voiceband or wireless networks. However, even if for most multimedia applications, compression remains a key issue, this is not the only one that has to be taken into account. Emerging applications in the area of interactive audiovisual services show a growing interest for interactivity, content-based capabilities, and for integration of information of dierent nature, namely of synthetic and natural type, capabilities not well supported by the existing standards. Content-based capabilities rely on audiovisual object representation and coding of the multimedia content, i.e. on the ability to encode nite support and arbitrary shape objects independently of the neighbouring regions. Hence, block transforms, such as the DCT, have been extended to 'shape-adaptive transforms'. The rst solutions introduced are based on the calculation of orthogonal basis vectors [3] and on the coding of rectangular blocks using frequency domain region-zeroing [4]. However these two transformations require high computational cost. The technique of design of shape-adaptive block transforms described in [3] has been applied to an Ni Nj DCT leading to a 'generalized' Shape-Adaptive DCT. The orthogonalization that must be performed for each region shape leads to a high computational complexity. A 'simpli ed' SA-DCT [5], relying on prede ned orthogonal sets of DCT basis functions turns out to provide good results. Submitted to IEEE Trans. on Signal Processing

March 23, 1998

3

Similar developments have been done, in the domain of multirate lter banks, by rst considering signal extension techniques, in order to apply iterated wavelet structures to bounded support image regions [6]. Time-Varying Filter Banks (TVFB), considering in the lter design the nite size characteristics of the region, have then been developed [7]. These lter banks also called Time Bounded Filter Banks (TBFB), process a nite support region ( nite length segments), independently of neighboring regions. The transformed regions are thus autonomous objects that can be manipulated. Time-varying lter banks can also be used in the context of transform switching in order to adapt the transformation to local characteristics of a region or sub-region. For example, one may wish to vary the lters, from one region to another, the down/up sampling rates, the structures of decomposition or the number of bands. However, shifting from one arbitrary lter bank to another independently designed perfect reconstruction system normally results in a substantial amount of reconstruction distortion during the transition period. In order to preserve the exact reconstruction property in the transition area, time-varying transition lter banks are required. Methods of design of TVFBs and TBFBs relying on a Least Square Approach are developed in [7], [8]. However, the lter banks obtained do not usually achieve the perfect reconstruction [9]. Several solutions of perfect reconstruction two-channel time-varying lter banks, that can be implemented on a lattice structure, are described in [10], [11]. Another approach, based on the redesign of the analysis time-varying lter banks in the transition and the boundary areas has been developed in [12], [13], [14], [15] for the twochannel case, and later extended to M -channel lter banks in [16] and [17]. The design problem is turned into a matrix orthogonalization problem. However, the orthogonalization procedure does not allow to obtain boundary lters with appropriate frequency or coding gain characteristics. Further optimization of the lters requiring a high computational cost is needed. Dierent optimization strategies are proposed in [17], [18]. This method has been extended to non paraunitary lter banks at the expense of a growing computational cost [16]. Lattice structures for the M -channel linear-phase paraunitary lter bank is rst introduced in [1] and is proven to be complete and minimal for even-channel and for lter length March 23, 1998

Submitted to IEEE Trans. on Signal Processing

4

multiple of the number of channels M . In other words, even-channel linear phase paraunitary lter banks can be synthesized by decomposing lter bank analysis and synthesis matrices into their polyphase components resulting in the so-called Polyphase Transfer Matrices (PTM). The PTM, being paraunitary, can be decomposed into a series of orthogonal matrices and delay stages [1]. This decomposition is called the Symmetric Delay Factorization (SDF). An alternate form of this factorization is developed in [23], and is later on extended to the case where the lter length is not restricted to be a multiple of the number of channels M [21] and to nonparaunitary lter banks [20]. A technique for designing time-varying lter banks, based on the so-called symmetric delay factorization of linear-phase paraunitary lter banks, is developed in [2]. In comparison with other structures of decomposition of the PTM [19], the SDF presents the advantage of decomposing the polyphase transfer matrix into only square orthonormal matrices, even at the boundary of nite length signals, simplifying signi cantly the design procedure of TVFB. The number of parameters to be optimized is signi cantly reduced and this is essential in the design of time-varying lter banks for which the number of parameters grows rapidly. The symmetric delay factorization provides also a lattice structure decomposition of the polyphase transfer matrix, leading directly to a fast implementation algorithm of the lter bank. However, the symmetric delay factorization technique, known so far, applies only to linear phase paraunitary lter banks. In addition, the approaches proposed lead to non square matrices at the boundary of nite length signals when the number of channels is odd [1]. This paper describes a new minimal and complete symmetric delay factorization of the polyphase transfer matrix valid for a large class of paraunitary lter banks. It provides a unifying framework for linear phase paraunitary lter banks including linear phase Lapped Orthogonal Transforms and for cosine-modulated lter banks, now widely used in speech, audio and image coding. A second key issue is that the approach does not make any assumption on the parity of the number of channels. It leads also for the case where the number of channels is odd to only square orthonormal matrices, even at the boundary of nite length signals. This generalized symmetric delay factorization approach opens new perspectives in the design of time-varying lter banks, and nds strong interest in the Submitted to IEEE Trans. on Signal Processing

March 23, 1998

5

framework of image and video object-based coding. II. Notations

In terms of notation, the following conventions are adopted: In is a n n identity matrix. On is a n n null matrix. ()T stands for the transposition of matrices and vectors. It is assumed, without loss of generality, that analysis and synthesis lter banks are composed of respectively M lters hm and gm of same length L. III. Design by Symmetric Delay Factorization

Let E N (z) be the analysis lter bank polyphase transfer matrix. The matrix E N (z) can P be expressed as E N (z) = Nn=0 z?nPnN , where PN (n) are M M matrices. It has been shown in [19], that the polyphase transfer matrix can be expressed as a product of M M orthogonal matrices of z polynoms of degree 1

E N (z) = VN (z)VN ?1 (z):::V1 (z)V0 ;

(1)

where Vn(z) is given by

Vn(z) = n Wn1(z)WnT ;

(2)

and where 1 (z) is a matrix of size M M of the form

0 1 @IM ?1 O?1A : O

z

(3)

This factorization leads to an implementation of the lter bank under the form of a lattice [19], as shown in gure (1). However, this lattice, composed of cells with one delayed branch and M ? 1 non delayed branches is not symmetrical, leading to non square orthonormal matrices at the boundary of nit length signals. In order to overcome this problem, a new polyphase transfer matrix factorization framework, has been introduced in [1]. It is rst shown that the M -channel linear phase perfect reconstruction lter bank is composed of mu symmetric lters and of ml antisymmetric lters, where mu = ml = M=2 March 23, 1998

Submitted to IEEE Trans. on Signal Processing

6 M

M ?1

M ?1

B0

B2i

M ?1

M ?1 Z ?1

B2i+1

BN

Z ?1

M

Analysis lattice M

B0T

B2Ti M ?1

M ?1

Z ?1

B2Ti+1 M ?1

Z ?1

Synthesis lattice

BNT

M

M ?1

Fig. 1. M -Channel Lattice with one delayed channel.

if M is even, and where (mu; ml ) = ( M2?1 ; M2+1 ) or (mu; ml ) = ( M2+1 ; M2?1 ) if M is odd. A minimal factorization of the matrices Vn(z); n = 0; :::; N , is developed for a large class of linear phase paraunitary lter banks. This factorization is proved to be complete only for an even number of channels M , and can be expressed in this case by

Vn(z) = n Wn(z)WnT ; where

Wn = (z) = n =

0 1 p1(2) @IM=2 IM=2 A ; IM=2 ?IM=2 0 1 @ IM=2 ?O1M=2 A ; O z IM=2 0 M=2 1 @ Sn OM=2A ; OM=2

(4)

(5)

Tn

and where the matrices Sn and Tn are M=2 M=2 orthogonal matrices. This factorization leads to the lattice structured implementation shown in gure (2), where B2i and B2i+1 are orthogonal matrices. This polyphase transfer matrix factorization, also known as the symmetric delay Factorization (SDF) technique, has been applied in the design of time-varying lapped transform [2], [22], [23]. This leads, in the case where M is even, to a lattice implementation formed by only square orthonormal matrices, even at the boundary of nite length signals. Submitted to IEEE Trans. on Signal Processing

March 23, 1998

7

However, this structure of decomposition applies only to linear phase lters, and its completeness is proved only for an even number of channels.. A minimal and complete generalized symmetric delay factorization, valid for a larger class of paraunitary lter banks, and for an arbitrary number of channels, is developed in the next section. The approach provides a unifying framework for linear phase paraunitary lter banks including linear phase Lapped Orthogonal Transforms (LOT) and for cosine-modulated lter banks, now widely used in image compression, and this for an arbitrary number of channels (odd or even). M

M

mu

mu

B0

Z ?1

B2i ml

ml

mu

mu

B0T ml

Z ?1

B2i+1

BN

M

ml

mu

Analysis Lattice ml

B2Ti ml

mu

ml

Z ?1

B2Ti+1 mu

mu Z ?1

Synthesis Lattice

BNT

M

ml

Fig. 2. M -Channel SDF Lattice.

A. Generalized Symmetric Delay Factorization

Let us consider an M -channel lter bank, composed of Finite Impulse Response lters of length L = M (N + 1), where N is an arbitrary integer. Let also

8 > :( M2?1 ; M2+1 )

if M is even, if M is odd.

(6)

The following theorem, providing a cascade form structure of the Polyphase Transfer Matrix of a paraunitary lter bank, is proved in dierent cases, where the paraunitary system is linear phase and also in the case where the system is a cosine-modulated lter bank. This cascade form structure represents a Symmetric Delay Factorization of the lter bank. March 23, 1998

Submitted to IEEE Trans. on Signal Processing

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Theorem 1: Let E N (z) be a FIR paraunitary matrix of polynoms of maximum degree N , representing the polyphase transfer matrix of a M -channel lter bank. The matrix P E N (z) can be expressed as E N (z) = Nn=0 z?n PnN , where PN (n) are M M matrices [19], [2]. If

rank(PNN ) ml and rank(P0N ) ml ;

(7)

then the matrix E N (z) can be decomposed as the product

E N (z) = VN (z)VN ?1 (z):::V1 (z)V0 ;

(8)

where, the matrices Vn(z), for n > 0, are given by

Vn(z) = Bno(z) Bn = Bn;1 Bn;2 o(z) =

0 @Imu

(9)

1 O A ; ?1

O z Iml

with Bn;1, matrix of size M mu , and Bn;2, matrix of size M ml , or equivalently where, the matrices Vn(z), for n > 0, are given by

Vn(z) = Bne(z) Bn = Bn;1 Bn;2 e(z) =

0 @Iml

(10)

1 O A ; ?1

O z Imu

with Bn;1, matrix of size M ml , and Bn;2, matrix of size M mu . In both cases, Bn is a unitary matrix and Bn;1 must be orthogonal to Bn;2. Note that in practice, the matrices Bi as explained in section (IV) are Givens plane rotations. Note also that in the case where the number of channels M is odd, mu being dierent from ml , a "Pseudo-Symmetric" Delay Factorization is obtained. Submitted to IEEE Trans. on Signal Processing

March 23, 1998

9

B. Preliminary Results

In order to prove theorem 1, let us rst demonstrate the two following preliminary results. Theorem 2: Let E N (z) be a FIR paraunitary matrix of polynoms of maximum degree N , representing the Polyphase Transfer Matrix of a M -channel lter bank. The matrix P E N (z) verify E N (z) = Nn=0 z?nPnN , where PnN are M M matrices. If the lter bank is linear phase, then the matrices P0N and PNN verify

rank(PNN ) = rank(P0N ) M2 :

(11)

Let hk (z) be the kth lter of length L = M (N + 1) of a M -channel lter bank of Polyphase Transfer Matrix E N (z). The matrix E N (z) has elements in the eld of polynomials in z. The elements of the matrix E N (z), [E N (z)]k;l , are the z-transforms of the sequences [2]

Proof:

[eN (n)]k;l = hk (nM + l);

(12)

and [PnN ]k;l = [eN (n)]k;l. In [1], it is demonstrated that a linear phase paraunitary lter bank has mu symmetric lters and ml antisymmetric lters. The impulse response of the kth lter veri es hk (n) = sgn(k)hk (L ? 1 ? n), where sgn(k) = 1 if hk is symmetric, and sgn(k) = ?1 if hk is antisymmetric. As a consequence, the elements [eN (n)]k;l verify [eN (n)]k;l = sgn(k)[eN (N ? n)]k;M ?1?l:

(13)

So, the kth row of PNN is the time-reverse of the kth row of sgn(k)P0N , and

rank(PnN ) = rank(PNN?n):

(14)

Since the lter bank is paraunitary, the matrices P0N and PNN verify (P0N )T PNN = 0, and span two orthogonal spaces of maximum dimension M . Since P0N and PNN span two orthogonal spaces of the same rank, obviously they verify rank(P0N ) = rank(PNN ) M2 . March 23, 1998

Submitted to IEEE Trans. on Signal Processing

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Hence, by taking a linear phase paraunitary lter bank such as

rank(P0N ) = ml or rank(PNN ) = ml ;

(15)

constraint which is veri ed for a large class of linear phase paraunitary lter banks useful from a practical standpoint, then this lter bank will satisfy theorem 1's assumption (equation 7). Theorem 3: Let E N (z), be a FIR paraunitary matrix of polynoms of maximum degree N , representing the Polyphase Transfer Matrix of a M -channel lter bank. The PTM P E N (z) veri es E N (z) = Nn=0 z?nPnN , where PnN are M M matrices. If the lter bank is a cosine-modulated lter bank, then the matrices PnN , n = 0; :::; N verify

8 > : ml

if n is even if n is odd:

(16)

Let us consider a paraunitary cosine-modulated lter bank obtained by modulation of a linear phase lowpass prototype h(n) of length L = (N + 1)M = 2pM . The impulse response of the kth lter can be expressed as [24], [25]

Proof:

hk (n) = ck;nh(n) ? where, ck;n = cos (2k + 1) (n ? L ? 1 ) ? ; 2 2M 4

(17)

Ce = fck;ng0kM ?1;0nM ?1; Co = fck;ng0kM ?1;M n2M ?1:

(18) (19)

ck;n = ck;n+2M ; ck;pM +n = ck;(p+1)M ?n; 8p 2 f0; : : : ; N g ck;(2p+1)M +ml = 0; if M is odd

(20)

Let Ce and Co the M M square matrices de ned by

The equations

Submitted to IEEE Trans. on Signal Processing

March 23, 1998

11

expressing, in the modulation function ck;n, ml symmetry relations for n 2 f0; : : : ; M ? 1g, and ml symmetry relations for n 2 fM; : : : ; 2M ? 1g can be veri ed easily. The ranks of the matrices Ce and Co are thus rank(Ce) = mu and rank(Co) = ml . Since the matrices P2Ni and P2Ni+1 can be obtained from Ce and Co by the relations [2]

P2Ni = Ceg2i P2Ni+1 = Cog2i+1

(21) (22)

where gn is a vector of length M de ned by gn(l) = h(nM + l), then rank(P2Ni ) mu and rank(P2Ni+1 ) ml . From this result, let us show that cosine modulated lter banks satisfy theorem 1's assumptions (equation 7). Since the ith row of P0N is equal to the ith row of Ce multiplied by h(i), if the linear phase prototype lter is such as,

h(n) = 6 0 8n 2 f0; : : : ; mu ? 1; mu + 1; : : : ; M ? 1g h(mu) = 6 0 if M is even

(23)

ml rank(P0N ) mu ; rank(PNN ) = ml ;

(24)

then,

which ensures that theorem 1 assumptions are veri ed. C. Generalized Symmetric Delay Factorization: Proof of Theorem 1

Theorem 1 can be proved by using a degree reduction procedure. Let us consider E N (z), PTM matrix of degree N of the M -channel lter bank. The matrix E N (z) veri es P E N (z) = Nn=0 z?nPnN . Let us assume that the matrix E N (z) is such that theorem 1 assumptions (7) are satis ed. As shown in theorems 2 and 3, this is the case for linear phase paraunitary lter bank and for cosine-modulated lter banks respecting only respectively constraints (15) and (24). March 23, 1998

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Let the orthogonal matrix BN = (BN;1; BN;2) be such as, T PN = 0 BN; 1 N T PN = 0 BN; 2 0 T T BN;1 BN; 2 = BN;1 BN;2 = 0

(25)

The existence and the paraunitarity of the two matrices BN;1 and BN;2 can be proved from theorems 2 or 3 and from the respective constraints (15) and (24). The proof is provided in annex (VII). Let the matrix E N ?1(z) be

E N ?1(z) = VN (z?1 )T E N (z) = (BN N (z+1 ))T E N (z) = (BN N

N X

(z+1 ))T (

n=0

z?n PnN );

(26)

where VN (z?1 ) = BN N (z+1 ) is de ned as follows: if rank(PNN ) ml , then VN is de ned as in equation (10), and N (z) = e(z) if rank(P0N ) ml , then VN is de ned as in equation (9), and N (z ) = o (z ) otherwise, VN can be de ned as in equations (9) or (10).

By replacing equations (25,9,10) in equation (26), the noncausal term of E N ?1(z) is cancelled. From equation (25), it appears that the degree of E N ?1 (z) is lower or equal to N ? 1. Since E N (z) has degree N and VN has degree 1, E N ?1 (z) cannot have a degree smaller than N ? 1. So the degree of E N ?1 (z) is precisely N ? 1, and the factorization is minimal. Since the unitary transformation VN (z?1 ) = BN N (z+1 ) is orthogonal, the matrix E N ?1 (z) is also paraunitary and we can write

E N (z) = (BN N (z?1 ))E N ?1(z): Submitted to IEEE Trans. on Signal Processing

(27) March 23, 1998

13

In addition, the orthogonality property of VN guarantees that

rank(P0N ?1) rank(P0N ); rank(PNN??11) rank(PNN ):

(28) (29)

Hence E N ?1(z) veri es theorem 1 assumptions (7). Repeating the degree reduction procedure N times, we obtain the complete decomposition (8) where V0 is a nonzero vector. Then the lter can be implemented by the lattice structure of gure (2). In the 2-channel case, we obtain the classical lattice structure, as in [19], and valid for all 2-channel paraunitary ler banks. IV. Design and Implementation of Time-Bounded Filter Banks

This factorization leads, for time-bounded signals, to a representation of the analysis and synthesis lter banks under the form of a lattice structure, as shown in gure (3), where the grey blocks are transient blocks (analog to the transient lters). Note that, if M is even, then ml = mu = M=2. In the case where M is odd, ml = M2?1 and mu = M2+1 , mu being dierent from ml , a "Pseudo-Symmetric" Delay Factorization is obtained. The interesting property is that the transient blocks are square orthogonal blocks, which simpli es the design procedure. The size of the square orthogonal blocks depends on their location in the matrix. For example, the blocks in the upper part of gure (3) (left edge of the lter bank) are of respective size l l, (l + ml ) (l + ml ), and (mu + ml ) (mu + ml ). These blocks, representing unitary orthogonal matrices, can be designed by a cascade of Givens plane rotations [19], as explained in annex (VII). This decomposition in terms of Givens rotations inherently guarantees the orthogonality and the perfect reconstruction properties of the time-bounded lter bank. The parameters of the Givens rotations Bi can then be optimized so that the lters hk (n) derived from the matrices Bi by equations (8) and (10) verify criteria such as the classical frequency selectivity or the maximum coding gain. The complete design procedure consists thus in rst decomposing a time unbounded lter bank by using the symmetric delay procedure presented above. It consists then in pruning unnecessary paths and blocks, used for the segments of signal outside the bounds. For each block in each stage, the procedure consists in checking its input branches and in removing the equivalent number of output branches in order to have a square orthogonal March 23, 1998

Submitted to IEEE Trans. on Signal Processing

14 l ml mu

B0

B1

B0

B1

B0

B1

B0

B1

B0

B2 B2 B2

ml l ml mu

B3 B3

r mu

r

Fig. 3. Time-Bounded Lattice.

matrix. These new orthogonal matrices provide the degrees of freedom in the transitory lter banks. The remaining blocks, those in white on gure (3), are the same as in the time-invariant SDF and are kept unchanged. All the degrees of freedom are then spanned in a search for the minimum of a given cost function, expressing for example the frequency selectivity or the coding gain. Any change of the plane Givens rotations (each block of the lattice) allows to span all perfect reconstruction solutions described by the structure. Degrees of Freedom:

The transient blocks (in grey) having mu + ml inputs are M M orthogonal transient matrices whereas the blocks with ml (resp. mu) inputs are ml ml (resp. mu mu) transient orthononal matrices. Let us consider the case where l = r. For each boundary, the number of matrices of transition is given by: stage 2i + 1 ! i + 1 matrices of size M M stage 2i ! i matrices of size M M and one matrix of size l l

(30) (31)

for i 2 f0; : : : ; K ? 1g and where 2K represents the number of stages Bi . Given that an otrthogonal matrix M M has M (M ? 1)=2 degrees of freedom corresponding to Givens Submitted to IEEE Trans. on Signal Processing

March 23, 1998

15

plane rotations angles (see annex(VII)), the number of degrees of freedom for each signal boundary is given by

X

K ?1

X

K ?1

!

i + 1 M (M2 ? 1) + K l(l ?2 1) ; i=0 i=0 M (M ? 1) + K l(l ? 1) = K2 2 2

=

i+

(32) (33)

2 is the total number of degrees of freedom for a 1-D signal. Table (I) shows the number of degrees of freedom according to the number of bands and of lattice stages with l = r = M2 , which turn out to be, in this particular case, similar to the number of degrees of freedom obtained in [2] for linear-phase paraunitary lter banks. M 2 4 6 8 10 12 14 16 20 24 32 1 1 7 18 34 55 81 112 148 235 342 616 2 4 26 66 124 200 294 406 536 850 1236 2224 K 3 9 57 144 270 435 639 882 1164 1845 2682 4824 4 16 100 252 472 760 1116 1540 2032 3220 4680 8416 5 25 155 390 730 1175 1725 2380 3140 4975 7230 13000 TABLE I Number of degrees of freedom for each boundary.

Examples: Examples of frequency responses obtained for transient time varying lowpass lters of a 4-channel cosine modulated lter bank, in the case where l = r = M=2, are given in gure (4). The time-invariant lter bank has been obtained by cosine-modulation of a lowpass prototype lter of length L = 2 2 4, with the coecients given in table (II). Figure (4.a) gives the frequency response at the signal boundary of the rst low-pass lter. The length of the equivalent lter bank is 6 taps. Figure (4.b) shows the frequency response at the signal boundary of the second low-pass lter. The length of the equivalent lter bank is 12 taps and the third low-pass lter is the time-invariant 16 taps lter shown in gure (4.c). March 23, 1998

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h[n] -1.3908586668E-02 -6.1494240116E-03 8.0034052982E-03 3.2265417503E-02 6.8581843589E-02 1.0752945970E-01 1.3994836685E-01 1.5909753229E-01 1.5909753229E-01 1.3994836685E-01 1.0752945970E-01 6.8581843589E-02 3.2265417503E-02 8.0034052982E-03 -6.1494240116E-03 -1.3908586668E-02 TABLE II Prototype Filter Coefficients. 3

2

2 1.8

1.8 2.5

1.6 1.6 1.4 2 1.4

1.5

1.2

1.2

1 0.8

1 1

0.6 0.8 0.4 0.5

a.

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

b.

0.6

0.4 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

c.

0.2 0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Fig. 4. Frequency responses of the low frequency transient lters

V. Design and Implementation of Transition Filter Banks in Filter Bank Switching

Time-varying lter banks are also useful in a context of lter bank switching. Indeed, instantaneous switching between two analysis lter banks, Hodd and Hnew , illustrated in gure (5), can be obtained by juxtaposition of two bounded lattices. In order to eliminate the distortion in the transition area, a new time-varying synthesis section (Tn) has to be Submitted to IEEE Trans. on Signal Processing

March 23, 1998

17

designed for each transition. An alternate approach to the lter bank switching problem consists in designing 'soft' transitions also in the analysis lter banks. For example, in gure (6), in order to avoid the transition with transient matrices of size ml ml and matrices of size m0u m0u , that would result from the two juxtaposed bounded lattices, as seen in the previous sections. we merge the two matrices ml ml and m0u m0u in order to obtain new transients blocks with matrices of size (ml + m0u) (ml + m0u ). The angles of the orthogonal transient blocks are then optimized as explained in the section above.

Hold

Hnew

Gold T1 T2 T3 Gnew

y(n) x(n)

Fig. 5. Instantaneous Filter bank Switching:

Hold and Hnew are respectively the rst and the second analysis lter banks; Gold and Gnew are the corresponding synthesis lter banks; and T1 : : : T3 are the transition synthesis lter banks. Note that Figure (5) represents the particular case where the parameters r and l0 of the two juxtaposed bounded lattices are equal to zero, or in other words where the lter bank switching is performed at the boundary of blocks B0 and B00 . This corresponds to the case where the length of the juxtaposed bounded signals are multiples of respectively (mu + ml ) and (m0u + m0l ). This is why there is no transient block in the rst stage of the lattice. However a similar lattice representation can be obtained when r 6= l0 6= 0. VI. Conclusion

A new minimal and complete symmetric delay factorization formalism valid for a large class of paraunitary lter banks including linear phase lter banks, cosine-modulated lter banks, all the 2-channel cases, has been described without assumption on the parity of March 23, 1998

Submitted to IEEE Trans. on Signal Processing

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B0 B0

B2

B1

B0

B2

B1

B0 ml mu m0l m0u

B2

B1

B2

B1

B0 B00

B10

B0

0

B20

B0

1

B00

B20

B10

B0

0

B0

2

B3 B3 B3 ml mu m0l mu m0u m0l

B30 B30

Fig. 6. \Soft" switching lattice: switching from a lattice (blocks B , i = 0; :::; 3) of (m + m ) bands to a new lattice (blocks B , i = 0; :::; 3) of (m + m ) bands. i

0

0

u

i

u

l

0

l

the number of channels. This formalism opens new perspectives in the design of of timebounded and time-varying lter banks. The symmetric delay factorization turns out to be a very powerful tool for the design of Time-Varying lter banks useful for the processing of time bounded signals or in the transition phase when switching from one lter bank to another one to process two regions with dierent characteristics. The approach allows to use Givens rotations in the design procedure that inherently guarantee the perfect reconstruction, even in presence of quantization of the lattice coecients, and that provide fast implementation algorithms. This approach, by preserving the square shape of the dierent blocks of the matrices of transition to be designed at the border of a nite length signal, provides a simpli ed method for synthesizing and implementing time-varying lter banks. VII. Acknowledgements

The authors wish to thank Tom Kalker from Philips Research Labs in Eindhoven for helpful discussions. Submitted to IEEE Trans. on Signal Processing

March 23, 1998

19

APPENDIX: EXISTENCE AND PARAUNITARITY OF THE MATRICES BN;1 AND BN;2

Let L, length of the analysis lter bank, be of the form L = NM + R, where 0 R < M . A component Ek;m(z) of the polyphase transfer matrix can be expressed as: Ek;m(z ) =

N X n=0

z?n hk (nM + m);

(34)

where the lters hk are extended by zeros, if needed. The polyphase transfer matrix can thus be written as E(z ) =

N X n=0

z?nPnN ;

(35)

where Pn is a M N matrix given by

0 h0 (nM ) : : : h0 (nM + m) : : : h0 (nM + M ? 1) B ... ... ... B PnN = B @

hM ?1 (nM ) : : : hM ?1(nM + m) : : : hM ?1 (nM + M ? 1)

1 CC CA :

(36)

Let E~ (z) = ET(z?1) be the polyphase transfer matrix of the synthesis lter bank, The perfect reconstruction condition can be formulated as N N X X n=0 m=0

zn?m (PnN )T PmN = I:

(37)

This equation can be rewritten as N ?l N X X l l=0

z

n=0

(PnN )T Pn+lN = I;

(38)

and is veri ed if and only if

8 l 2 [0; : : : ; N ] )

N ?l X m=0

(PnN )T PnN+l = l I:

(39)

This implies in particular that (P0N )T PNN = 0. The existence and paraunitarity of the two matrices BN;1 and BN;2 can be proved as follows. March 23, 1998

Submitted to IEEE Trans. on Signal Processing

20

Let us assume that theorem 1 assumptions are verifed, hence rank(P0N ) ml and rank(PNN ) ml . From relation (39), the M M matrices PNN and P0N verify (PNN )T P0N = 0, and as mu + ml = M we can deduce rank(PNN ) mu , and rank(P0N ) mu . In the odd case, we cannot have rank(PNN ) = rank(P0N ) = mu (this would mean that PNN and P0N span a space of dimension M + 1). We have thus the two following alternatives:

rank(PNN ) mu and rank(P0N ) = ml rank(PNN ) = ml and rank(P0N ) mu

(40) (41)

Let us consider, without loss of generality, case (40), and let us prove the existence and paraunitarity of the matrices BN;1 and BN;2 . The matrix BN;1, being of size M ml , and rank(PNN ) mu , can be constructed with ml orthogonal vectors of the space spanned by P0N , orthogonal to the space spanned by PNN . In practice, the matrix is then orthogonalized by using a procedure of Gram-Schmidt. Similarly, the matrix BN;2 of size M mu orthogonal to P0N , can be constructed by taking ml orthogonal vectors of the space spanned by PNN (space of dimension ml and orthogonal to the space spanned by P0N ) and one vector orthogonal to these ml vectors and to P0N .

BN;1 being in the space spanned by P0N , BN;1 veri es condition (25). Similarly, BN;2 being orthogonal to P0N , the condition (25) is established. One can deduce that BN;1 is orthogonal to BN;2, proving condition (26). APPENDIX: FACTORIZATION IN TERMS OF GIVENS ROTATIONS

This section reviews, for sake of completeness, the decomposition of a unitary matrix in terms of a cascade of Givens plane rotations. Let us consider a unitary matrix P 2 Rmm, i.e., verifying PTP = I. This matrix can be written under the form [26], [19] :

0 1 S 0 A P = @ 0

Submitted to IEEE Trans. on Signal Processing

(42) March 23, 1998

21

where STS = Im?1, = 1 and where = m?2 0, with i de ned as:

8 > ci;m if k = l = i or k = l = m ? 1 > > > ?si;m if k = m ? 1 and l = i > > > :0 otherwise

(43)

and, where ci;m = cos(i;m) and si;m = sin(i;m ). The matrix i can be rewritten as

0 BB BB B i = B BB BB B@

1 CC CC si;m C CC : 0 C C ... C CA

... 0

... 0

0 ci;m 0 0 0 ... 0 ?si;m 0 0 ci;m

(44)

By applying a reduction procedure, it can be shown easily that the unitary matrix P can be decomposed in terms of a cascade of plane rotations known as the Givens plane rotations. The principle of the decomposition consists in searching m?2 such that

0 1 BB .. ... ... C CC BB . CC : Tm?2P = B BB C B@ 0 CCA

(45)

This is equivalent to searching m?2;m verifying

P (m ? 2; m ? 1)cos(m?2;m) ? P (m ? 1; m ? 1)cos(m?2;m ) = 0:

(46)

Solutions to the above equation are given by

m ? 2; m ? 1) ; tan(m?2;m ) = PP ((m ? 1; m ? 1) March 23, 1998

(47)

Submitted to IEEE Trans. on Signal Processing

22

if P (m ? 1; m ? 1) 6= 0, and by m?2;m = 2 , otherwise. By iterating the reduction procedure for Tm?1Tm?2P up to T0 : : : Tm?1Tm?2P we obtain

0 1 S 0 A: T0 Tm?2P = @ 0

(48)

The transformation by the matrix P can then be represented as shown in gure (7).

S

0;m 1;m

Fig. 7. Givens Reduction of the Matrix P

The factorization procedure can be applied on the matrix S, leading to the structure shown in gure (8), where (i) corresponds to the matrix 2 R (i+1)(i+1) . The 'butter y' structure of the matrix is shown in gure (9), in which the basic unit represents the transformation

0 1 0 10 1 y cos ( ) sin ( ) @ 0A = @ A : @x0A ?sin() cos()

y1

0

(0)

x1

(49)

(1) (m?2)

m?1

Fig. 8. Factorization in terms of Givens Plane Rotations. Submitted to IEEE Trans. on Signal Processing

March 23, 1998

23

0

i+1

Fig. 9. Butter y Structure of the Transformation given by ( ). Each crossing section implements the transformation of relation (49). i

References [1] A. K. Soman, P. P. Vaidyanathan, and T. Q. Nguyen, \Linear phase paraunitary lter banks: Theory, factorizations and designs," IEEE Trans. on signal Processing, vol. 40, no. 12, pp. 3480{3496, December 1993. [2] R. L. Queiroz, On Lapped Transforms, Ph.D. thesis, University of Texas at Arlington, 1994. [3] M. Gilge, T. Engelhardt, and R. Mehlan, \Coding of arbitrarily-shaped image segments based on a generalized orthogonal transform," EURASIP journal on Signal Processing and Image Communication, vol. 1, pp. 153{ 180, 1989. [4] H. H. Chen, M. R. Civanlar, and B. G. Haskell, \A block transform coder for arbitrarily-shaped image segments," in Proceedings of the International Workshop on Coding for Very Low Bitrate Video, University of Essex, England, April 1994. [5] T. Sikora and B. Makai, \Shape-adaptive DCT for generic coding of video," in Contribution to the International Organisation for Standardization, ISO/IEC Coding of Moving Pictures and Associated Audio, July 1994. [6] H. J. Barnard, Image and Video Coding using a Wavelet Decomposition, Ph.D. thesis, Delft University, 1994. [7] K. Nayebi, T. P. Barnwell and M. J. T. Smith, \Analysis-synthesis systems with time-varying lter bank structures," in Proceedings of the IEEE Int. Conf. on Acoustics, Speech and Signal Processing, ICASSP, March 1992, vol. IV, pp. 617{620. [8] K. Nayebi, T. P. Barnwell and M. J. T. Smith, \Time-domain lter bank analysis: A new design theory," IEEE Trans. on Signal Processing, vol. 40, pp. 1412{1428, June 1992. [9] I. Sodagar, Analysis and Design of Time-Varying Filter Banks, Ph.D. thesis, Georgia Institute of technology, 1994. [10] J. L. Arrowood Jr. and M. J. T. Smith, \Exact reconstruction analysis/synthesis lter banks with time-varying lters," in ICASSP, Mineapolis, Minnesota, April 1993, IEEE, vol. III, pp. 233{236. [11] J. C. Pesquet and H. Krim, \Time-varying lter banks for the analysis of PC processs," in EUropean SIgnal Processing COnference, EUSIPCO, September 1994, vol. III. [12] C. Herley, Wavelets and lter Banks, Ph.D. thesis, Columbia University, 1993. [13] C. Herley, J. Kovacevic, K. Ramchandran and M. Vetterli, \Time-varying orthonormal tilings of the timefrequency plane," in IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, ICASSP, 1993, vol. III, pp. 205{208. [14] C. Herley and M. Vetterli, \Orthogonal time-varying lter banks and wavelets," in IEEE Int. Symposium on Circuits and Systems, ISCAS, 1993, vol. I, pp. 391{394. March 23, 1998

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[15] C. Herley and M. Vetterli, \Orthogonal time-varying lter banks and wavelets," IEEE Trans. on Signal Processing, vol. 42, no. 10, pp. 2650{2663, October 1994. [16] C. Herley, \Boundary lters for nite-length signals and time-varying lter banks," IEEE Trans. on Circuits and Systems-II: Analog and digital signal processing, vol. 42, no. 2, pp. 102{114, February 1995. [17] T. Kalker, \On optimal boundary and transition lters in time-varying lter banks," in IEEE Int. Conf. on Image Processing, Lausanne, September 1996, vol. I, pp. 625{628. [18] A. Mertins, \Time-varying and support preservative lter banks: Design of optimal transition and boundary lters via SVD," in IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, ICASSP, Detroit, May 1995, pp. 1316{1319. [19] P.P. Vaidyanathan, Multirate systems and lter banks, Prentice Hall, 1993. [20] Takayuki Nagai, Takaaki Fuchie, and Masaaki Ikehara, \Design of linear phase M -channel perfect reconstruction FIR lter banks," IEEE Trans. on Signal Processing, vol. 45, no. 9, pp. 2380{2387, September 1997. [21] Trac Duy Tran and Truong Q. Nguyen, \On M -channel linear phase FIR lter banks and application in image compression," IEEE trans. on Signal Processing, vol. 45, no. 9, pp. 2175{2187, September 1997. [22] R. L. de Queiroz and K. R. Rao, \Optimal orthogonal boundary lter banks," in IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, ICASSP, 1995, vol. III, pp. 1296{1299. [23] T. Q. Nguyen R. L. de Queiroz and K.R. Rao, \The GenLOT: Generalized linear-phase lapped orthogonal transform," IEEE Trans. on Signal Processing, vol. 44, no. 3, pp. 497{507, March 1996. [24] Henrique S. Malvar, Signal Processing with Lapped Transforms, Artech House, Inc, 1992. [25] R. D. Koilpillai and P. P. Vaidyanathan, \Cosine-modulated FIR lter banks satisfying perfect reconstruction," IEEE Trans. on Signal Processing, vol. 40, no. 4, pp. 770{783, April 1992. [26] Gene H. Golub and Charles F. Van Loan, Matrix computations, The John Hopkins University Press, 1989. [27] H. Barnard, J. H. Weber, and J. Biemond, \A region-based discrete wavelet transform," in Proceedings of the EUropean SIgnal Processing COnference, September 1994, pp. 1234{1237. [28] T. P. Barnwell K. Nayebi and M. J. T. Smith, \Design of low delay FIR analysis-synthesis lter banks in the context of periodically time-varying systems," in Proc. Conf. on Information Sciences and Systems, 1991. [29] P. Rault and C. Guillemot, \Pseudo symmetric delay factorization: A theory for M-odd channel paraunitary lter banks," in Proceedings of the International Multidimensional Digital Signal Processing Workshop, IMDSP, Belize City, BELIZE, March 1996. [30] P. Rault and C. Guillemot, \Symmetric delay factorization: A generalized theory for paraunitary lter banks," in Proceedings of the EUropean SIgnal Processing COnference, EUSIPCO, 1996.

Submitted to IEEE Trans. on Signal Processing

March 23, 1998