In this paper, a new design algorithm is presented for a family of linear phase paraunitary filter banks with generalized filter length and symmetric polarity. A number of new constraints on the distributions of filter length and symmetry polarity among the channels are derived. In the algorithm, the lengths of the filters are gradually reduced through a cascade of lattice structures. The derivations for filter banks with even and odd number of channels are formulated in a unified form.published_or_final_versio

Chan, KP

Chen, L

Nguyen, TQ

Zheng, Y

HKU Scholars Hub

Title A new synthesis procedure for linear-phase paraunitary digitalfilter banksAuthor(s) Chen, L; Chan, KP; Nguyen, TQ; Zheng, YCitationThe 1997 IEEE International Symposium on Circuits andSystems, Hong Kong, China, 9-12 June 1997. In IEEEInternational Symposium on Circuits and Systems Proceedings,1997, v. 4, p. 2377-2380Issued Date 1997URL http://hdl.handle.net/10722/45573Rights©1997 IEEE. Personal use of this material is permitted. However,permission to reprint/republish this material for advertising orpromotional purposes or for creating new collective works forresale or redistribution to servers or lists, or to reuse anycopyrighted component of this work in other works must beobtained from the IEEE.1997 IEEE International Symposium on Circuits and Systems, June 9-12, 1997, Hong Kong A new synthesis procedure for linear-phase paraunitary digital filter banks L. Chen, I<. P. Chan T. Q. Nguyen Y. Zheng Department of Computer Science University of Hong Kong Department of Electrical Computer Engineering Lniversity of Wisconsin, Madison Institute of AI Sr PR Vniversity of Shantou, China ABSTRACT In this paper, a new design algorithm is presented for a family of linear phase paraunitary filter banks with generalized filter length and symmetric polarity. A number of new constraints on the distributions of filter length and symmetry polarity among the chan- nels are derived. In the algorithm. the lengths of the filters are gradually reduced through a cascade of lat- tice structures . The derivations for filter banks with even and odd number of channels are formulated in a unified form. I. Introduction Recently, there has been considerable interest among researchers in the design of ,Wchannel maxi- mally decimated filter banks. as shown in Fig. 1. The  theory of linear-phase ( L P )  filter banks and their ap- plications have been addressed by many authors. In two-channel case, linear phase (symmetry) property has been imposed on the traditional subband filter banks [l, 2 %  31. It was shown that  among all cases of two-channel FIR perfect-reconstruction QMF struc- tures which yield LP filters, only two cases yield good filters in the practical sense [4. 51. The  results are extended to the case of arbitrary number :VI of channels in [6, 71. In parallel to  the work on the general perfect-reconstruction filter banks, which are biorthogonal, researches have also been done for pa- raunitary systems. Lattice factorizations were devel- oped in [8, 101 for even M . In [ l l ,  12,  131, the results were extended to the case where ibI is odd. In the lattice factorizations of [8, 111, the filter length is restricted to N = kAbI or 2kibI for even and odd i W ,  and the step size in increasing the filter 0-7803-3583-X/97 $10.00 01997 IEEE length is at  least ‘LI. This is not convenient when the number of channels is large, e.g.. &I = 8 or 16. In the phase of filter design, an increase of filter length by a large LLI implies a much higher non-linear pa- rameter space to be searched and the optimization program tends to be trapped in local minima. In the phase of implementing filter banks, i.e., in the analy- sis/synthesis systems. a large increase in filter length by AbI gives much higher computational complexity. This leads to the design of general equal length filter banks and unequal length filter banks. In [14, the theory and structures are stud- ied for a large subclass of M-channel LP perfect- reconstruction FIR filter banks whose the analysis and synthesis filters have length Li = k i M + P ,  where 3 is an arbitrary integer, 0 5 p < ibI, and ki is a non-negative integer. The extension of filter length from AV = k M  to k i M  + i? gives more flexibility in fine-tuning filter length to meet a given filter spec- ification. e.g., stopband attenuation. The  unified ,b’ leads to this class of filter banks to have a simple characterization of symmetric property in polyphase representation (Eq. (2.1) in [8] and Eq. (3) in [14]). I t  is noted that tree-structured filter banks [2, 31 can be viewed as a kind of unequal length filter banks. In terms of lapped orthogonal transforms [9, 15, 161, the freedom in filter length provides the possibility of overlap which is a fraction of the number of chan- nels .\.I. Moreover, it  has been shown tha t  this class of filter banks can be used in processing finite-length signal with the symmetric extension method [U, 181. In this paper. we further investigate factorizations of linear-phase paraunitary systems. In Section 11, several known results are reviewed and new con- straints are derived. Section I11 and IV show that the filter banks can be designed by a successive of length reductions through a cascade of lattice struc- tures which is minimal in terms of the number of delays used for implementation. The  algorithm can also be applied to filter banks where the filter length is an integral multiple of the number of channels, o r ,  filter banks with equal length. In Section V ,  a brief summary is given and a design example is included to verify the theory. 2377 11. Basic constraints and properties Definition 2.1 Consider an M-channel linear- phase filter bank where among the -VI filters, there are pk + Vk filters that have length k M  + 0, 0 5 k < AA7, where pk and vk stand that  the numbers of sym- metric and anti-symmetric filters, respectively. A matrix tc-stack is defined to  indicate distributions of symmetry polarity among the filters . (1) 0, l,..', k, ' " ;  P t )  110, Pi,..., P k ,  . . .  -UO, - V i ,  .", - U k ,  . . . ,  -VI< Proposition 2.1 In an M-channel LP perfect- reconstruction filter bank with filter lengths Li = k i M  + p, 0 5 k; 5 IC, let 3 = k , ,  6 = t d k = O ~ k r  ip = C k = , v k .  The following results have been shown [8, 141 1 . I f  114 E Z, and P E  Z,, CL E Z,, b = y ,  v = +; 2.If 114 E Z,  and p E z,, + E  z,, 6 = ?+I, ;o = +-I; 3 , I f n / 1 ~ ~ , a n d p ~ ~ , ,  + E Z , ,  b = y , V = q ;  where 2, and 2, denote the sets of even and odd integers, respectively. Proposition 2.1 shows the total sum of the filter length and the total numbers of symmetric and anti- symmetric filters. However, the permissible length and symmetry polarity distributions are not given. For example, for a given total number p~ + VK of filters that have the longest length K i M  + ,!j', the re- lahon for the number of symmetric filters p~ and the number of anti-symmetric filters VK has not been shown. In the following, new results will be presented and lead to the unification of the lattice structures of filter banks with even and odd number of channels. Partitioning of E K ( ~ )  : It is noted that the perfect-reconstruction property is preserved in spite of the interchange of rows of the polyphase matrix EK(z). With row-wise permutations, EK(z) can al- ways be partitioned as 7 f i  f i  4 . I f  M E Z,  and P E  z,, i E z,, 6 = y, v = 2; M-1 Efc(z) = ( i:lfi , ( 2 )  where E,(z) and E,(z) represent p~ symmetric and V,Y anti-symmetric filters with the longest length L; = A'M + /3; E d ( ? )  stands for the rest iVf - ( p ~  + YK) filters. For convenience, the subsequent discussions assume that  EK (2) has already been arranged in the form of (2). Let EK(:) be ex- pressed as EK(z) = e(0) + . . . + e(K - l ) ~ - ( ~ - l )  + e(K)z-K, e(K) # 0, and e(O), e(K - 1) and e(li) are partitioned as E d ( z )  e , , l ( O )  e ~ , , ( o )  e , , l ( K  - 1) * e , r ( K  - 1) , , l ( r c )  ( e a , i ( o )  e d , l ( e a , . ( o ) )  - d , r O  . (ea. i (K e d . l  (K - l )  1  e a , r ( K  - 1;) . G ~ , ~ ( K )  I) , (3) where the left hand side part consists of j3' columns; and the right hand side part consists of - ,8 columns. The lower part of .(IC) and ed,,(K - l),  which is to  the right ofed , , (K- l ) ,  are set to  0, as the corresponding filters are shorter than ICM 1-3. Also, since t,he maximum length of filters are K'bf + d ,  all the elements on the right hand side of e(1i) are set to  zeros. Proposition 2.2 Foir a LP paraunitary filter bank EK(:), the polyphase components es(0) and ea(0) as defined i n  ( 3 )  satisfy (4) ( 5 )  ~ z n k ( e ~ ( 0 ) )  = r u n k ( e , ( o ) ) ,  2 5 K Tank(e jq l (0 ) )  :: rank(e,,L(O)), 15  K .  Corollary 2.1 For a.n ;VI-channel LP paraunitary filter bank with filter lengths L; = k ; h + y ,  0 5 9 5 - I., 0 5 k; 5 I<. The filters with the longest length L; = Ii-bf + ,O can not be all symmetric or all anti-symmetric. Proposition 2.3 Let El(:) be a FIR LP parau- nitary filter bank whose K-stack is ( ,,:: Then, ,,; ) - y o .  -U1 0, if A4 - p is even, 1, if M - /3 is odd. ( 6 )  0. if ,b' is even, pJ-vo .= = 1, if /3 is odd, i.e., the numbers of 5,ymmetric and anti-symmetric filters are balanced. 111. Lattice factorization Based on Proposition 2.2,  we show the "order re- duction" process to l i  = 1. The  next section will present a general initialization of the lattice struc- ture. Proposition 3.1 Let EK(z),  2 5 I< be a FIR LP paraunitary filter bank whose rc-stack is 0 ,  1 ,  2 , ' . '  , h - - 2 ,  A - - 1 ,  PO, P I ?  P 2 ,  " '  - U o ,  - V i ,  -U?, ' " ,  - U j , - - 2 ,  - U i < - i ,  --Vi< ( 7) Then it cain always be factored into a left hand side factor ZK(::) and a LP paraunitary filter bank E K - ~ ( z )  with the order I< reduced by 1, EK(z) = Z,y(z)E~- , (z ) ,  where the 6-stack of E ~ - l ( z )  > P i c - 2 ,  P X - 1  , ( 8 )  , - d .  r, -I U;, P i ,  P ; ,  ... - I / ; ,  -.U;, --U;, ... 0 ,  1, 2 ,  " '  k = A - - l  Pk = P k f C ,  = Uk f c ,  k Z K - 1  4 = P k f P k t 1  - < ,  k = I i - 2  Vi = U k f Y k + i  - < ,  k =  K - 2  @ k  = P k ,  O < k < K - 3  , U; = "k O < k < K - 3  is related to that of EK(z) in (7) as (9) where c = mnk(e , (O) ) .  Proof: i) Partztzonzng of lE,(z) and E,(z): The  proof is achieved through an order reduct ion process and is divided into three steps. Let E,(=)  be expressed as E, ( 2 )  = es(0) + e,( 1)z-1 + . . . + eS(J<)L-K, e,(hr) # 0. Denote the rank of e,(O) as 2378 E ~ ( z )  consists of c symmetric filters and all the rows of the polyphase components e/,(O) are linearly in- dependent, i.e., mnk(e i (0 ) )  = c ;  E Y ( ~ )  represents the rest K - c symmetric filters with e’,’(O) set to press E,(=) as ( :![:; ) = VvKE, (z ) .  Hence, 0 throug E linear combinations. Likewise, we can ex- where & I  = d i a g ( U p K , V ~ ~ , I , ~ ~ - P K - v K ) .  T and QS exchanges E:( z )  and E’, (2). ii) Reductzon of t h e  order of E:(:) and E’,(z) by A4 : Proposition 2.2 shows that  r u n k ( e a ( 0 ) )  = mnk(e,(O)) = F. Pairwisely combining the c sym- metric and c anti-symmetric rows in E: ( z )  and E’, ( z )  yields FK ( z )  satisfying the pairwise time-reversed property 1 Fl;(z) = T ( r )  (E:{:;) , T ( c )  = - ( J q  ) . (13) fi J, -I, It can be shown that  F K ( ~ )  can be factored into the “reduced form” FK- 1 ( z )  Fi,-(z) = A~,- (z)FI<-~(z) ,  Alii(.) = A(=)W.t< (14) r‘ =-11, ) ; U, and V, are two orthonormal ma- trices. Moreover, FK-~( : )  with pairwise time- reversed property can be expressed a s  FI<-l(z) = T ( c )  ( z!:[fi ) ,  where E’,’(z) and E:’(:) are LP filters with symmetric and anti-symmetric property, respectively. Hence, where A’(z) = T(c )A(z )T(c ) .  It  yields the first and second equations in (9). iii) Reductzon of the order ofElg(z) and E:(:) by 2 M  : As shown in ( l l ) ,  the first iz.l taps of the filters in E:(:) are set to zero. Due to the symmetry of E:(z), the last (highest order) Df taps are also zeros. As a result, E’,’(z) can be shifted to the left by z ,  resulting in a reduction in length of 2 M ,  E:(=) = =-‘E&”’(:). Similarly, E:(z) can be shifted to the left by z to lower order filters E:“(:), namely, E:(:) = z-’E:’’(z). It gives the third and fourth equations in (9).  Substituting the results in Step (ii)  and iii) gives the lac toriza- tion EK(:) = Q1Q’A’(z)EK-1(:) where A’(z) = d i a g ( A ’ ( z ) ,  z- i IPK- .q ,  : ‘IvK-<, Ilvf-pK-uK) and - (16) IV. Initialization Proposition 4.1 Let El(:) be a FIR LP parauni- tary matrix whose r;-stacl; is ( @:) . Then it can always be factored into a right hand side fac- tor e(:) and an initial matrix f20 El(=) =Go@(=),  (17) 0 ,  --uo, - V I  where & = v ,>. U, V and W are or- thonormal matrices having centro-symmetric prop- erty [8, 111. ( Proof:  Let El(:) be expressed as El(.) = e(o)+  e(I)z-*, where e(0) and e(1) be partitioned as in (3) es , i (o )  e s , r ( O )  ) , ( :s,L(l) e5.1(o) ea , , (0 )  e5,1(1) . (18) Following a similar procedure of derivation of (12) and employing (51, e(0) and e(1) can be factored as ) ( e d , l ( o )  0 e:,i(O) e : , r (o )  e: , lO)  0 [ y o )  s‘::“ e:,#) j, [ i l 1 ( 1 )  i j (19) e d , l  ( O )  where rank(e’,, ,(O)) = c’. Proposition 2.1, 2.2 and 2.3 show that the numbers of symmetric and anti-symmetric filters in E:(=) and E’,(:) are equal; E/, (2) and Eh(:) consist of balanced number of sym- metric and anti-symmetric filters, i.e., the number of symmetric filters is equal to  or greater than the number of anti-symmetric filters by 1; E d ( z )  itself consists of balanced number of symmetric and anti- symmetric filters. Pairwisely combining these sym- metric and anti-symmetric rows to  pairwise time- reversed rows, e(0) and e( 1) become eL,r(1) 0 ) ? ( ; ; ) . (20) We focus on the left hand side B elements in eL,l(0), e;,/(O) and e;,,(l). The centro-symmetric rows in e&,,(O) form a basis for a po + vo dimensional space. With a centro-symmetric matrix performing linear combinations of columns of e&,,(O), we can turn the central p0 + Y O  columns of e;,,(O) into a full-ranked centro-symmetric matrix e’&(O) while a t  the same time automatically set the polyphase components to  the left. right and above it to zeros. In particular, e(0)  and e( 1) become c,,(o) 0 e:,;(o) e:,,(o) 0 e&r(o) 0 0 ( g ‘  2379 e:,; (1) 0 e”’ 0 c , 4 ( 1 )  0 8 ) . (22) 0 0 0 The columns of e:,lI(O), e:,’2(0), ey,i(l) and eL,4(l) satisfy the pairwise time-reversed pro erty. Similar to the order reduction process in (147, we can cas- cade Al(z) = WlA(i) to the right of e(0) and e(1), turning e(0) into 0 and e(1) into 0. The difference between Al(z) = W , A ( z )  and A(;) in (14) lies in that  A(z) is placed to  the right of W1. Furthermore, permuting eL,r(0) and e’,,r(0) gives e(0)  = 8 ) . (24) U 0 el;(o) Note that  the row in e:,?(O) are orthonormal. Simi- lar to the deduction of ( 2 l ) ,  with a centro-symmetric matrix performing linear combinations of columns of e;,?-(O), we can turn the central hf - (po + YO) - 2 ~ ’  columns of eL,r(0) into a full-ranked centro- symmetric matrix e:,?(O) while a t  the same time au- tomatically set the polyphase components to  the left, rigjht and above it to zeros, i.e., e:,’F(o) 0 e:,y(o) e: ,m 0 With column-wise permutations, e”(i’(0) and e:,’‘(O) can be merged into a centro-symmetric matrix e,. e( 0) becomes 8 ) . (25) 0 o e&’(o) where ec(0), ey,r(0) and e:(O) are centro-symmetric. V. Conclusion and design example We have presented a new algorithm for designing a family of general linear-phase paraunitary. The lattice factorization is obtained through a succession of .‘order reduction” processes, compromising planar rotations which force more than half of the rows to zeros, and delay chains which shift the shorten filters to  lower orders. A 9-channel filter bank with length {X?, 32, 32, 32, 32, 32, 23, 23, 23) and alternative symmetric and anti-symmetric property is shown in Fig 2.  References [li J. D. Johnston, “A filter family designed for use in quadraturemirror filter banks,” in Proc. IEEE Int. Conf. on ASSP,  Apr. 1980, pp. 291-294. [21 J. Woods and S .  O’Neil, “Subband coding of images,” IEEE Trans. on ASSP,  pp. 1278-1288,1986. [31 S. Mallat, “A theory for multiresolution signal decom- position: The wavelet representation,” IEEE Trans. on PAMI, pp. 674-693, 1989. “Two-channel perfect-reconstruction FIR QMF structures which yield linear-phase analysis and synthesis filters,” IEEE Trans. on ASSP,  pp. 676-690,1989. [41 T. Q. Nguyen and P. P. Vaidyanathan, 0 -10 -20 -30 Figure 2: A 9-channel LP filter banks [SI M. Vetterli, “Filter banks allowing perfect reconstruc- tion,” Signal Processing, pp. 219-244, Apr. 1986. [6] T. Q. Nguyen and P. P. 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A new synthesis procedure for linear-phase paraunitary digital filter banks

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A new synthesis procedure for linear-phase paraunitary digital filter banks

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