INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS Int. J. Circ. ¹heor. Appl., 26, 539—546 (1998) LETTER TO THE EDITOR AN EFFICIENT DESIGN METHOD OF COSINE-MODULATED QMF BANKS SATISFYING PR PROPERTY YING TAN1* AND ZHENYA HE2 1Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei 230027, People+s Republic of China 2Department of Radio Engineering, Southeast University, Nanjing 210096, People+s Republic of China SUMMARY This paper deals with the design of cosine-modulated QMF banks satisfying the perfect-reconstruction property (PR). This problem has been formulated as a novel quadratic-constrained least-squares (QCLS) minimization problem in which all constrained matrices of the QCLS optimization problem are symmetric and positive definite. In order to efficiently solve this kind of QCLS optimization problems, we construct a cost function that is a convex function of our desired prototype filter coefficients. Therefore, a global minimizer of this problem can be easily obtained. Moreover, this design approach has the ability to obtain solutions that are a compromise between PR conditions and stopband performance. The results of two design examples are presented to support our derivations and analyses. ( 1998 John Wiley & Sons, Ltd. 1. INTRODUCTION Multirate signal processing and filter bank systems have found many applications in areas such as subband coders for speech signal and image coding during the past 20 years. Recently, the perfect reconstruction (PR) theory has been established.1~3 Among them, the PR cosine-modulated filter bank (PR, CMFB) has emerged as an attractive choice of filter bank with respect to implementation cost and design saving. The impulse responses of the analysis and synthesis filters are cosine-modulated versions of a prototype filter. It is shown in Reference 3 that the 2M polyphase components of the prototype filter can be grouped into M power-complementary pairs, where each pair is implemented as a two-channel lossless lattice filter bank. The lattice coefficients are optimized to minimize the stopband attenuation of the prototype filter, but this is a highly non-linear optimization problem with respect to lattice coefficients. Consequently, it is difficult to obtain the PR CMFB with high stopband attenuation. Recently, References 6 and 7 formulated the design problem as a QCLS problem, and obtained the PR CMFB with high stopband attenuation. In this paper, we at first derive another form of QCLS problem8 for designing the PR CMFB, where all constrained matrices are symmetric and positive definite. After that, we solve the QCLS optimization problem by penalty function method. Finally, the PR cosine-modulated QMF banks with high stopband attenuation can also be easily obtained as illustrated by two examples. What follows is a summary of the features of existing design methods and our proposed method. Drawbacks of existing PR CMFB design methods: (a) It is necessary to provide an appropriate initial approximate solution for the designed filter since it is very sensitive to the initial approximations. * Correspondence to: Y. Tan, Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei 230027, People’s Republic of China. Email: ytan@ipc.ustc.edn.cn CCC 0098—9886/98/050539—08$17.50 ( 1998 by John Wiley & Sons, Ltd. Received 26 July 1997 Revised 19 March 1998 540 LETTER TO THE EDITOR (b) All the methods follow an iterative procedure, so they share their calculational complexity. (c) The optimized solution may be a local minima. (d) A complex non-linear optimization problem should be converted into a linear optimization problem by linearizing the processing procedure. (e) Cost function / is a highly non-linear function with respect to the lattice coefficients. Consequently, PR filler banks, where the analysis filters have high stopband attenuation, are difficult to obtain. Advantages of our proposed method: (a) A global optimal solution can be reached as penalty factors approach infinity simultaneously. (b) No initialization approximation is needed for the method as it has global Lyapunov stability. (c) The precision of the solution can be controlled by appropriately choosing values of penalty factors. In other words, the compromise between PR conditions and stopband attenuation can be set by the penalty factor. (d) The coefficients of the prototype filter can be optimized directly. (e) This method can be easily extended to the multidimensional multichannel QMF bank case the appropriately rearranging the coefficients of the prototype filter. The rest of this paper is organized as follows. In Section 2, we present the design problem formulation of paraunitary CMFB. In Section 3, we describe and analyse our method. In Section 4, two design examples are presented to illustrate the method. Finally, Section 5 concludes this paper. Notations. Bold-faced letters represent vectors and matrices. Superscript T denotes transposition, and the ‘tilde’ accent on a function is defined such that A3 (z)"A`(1/z*) where superscript astrisk denotes the conjugation of coefficients. I and J denote k]k identity and reversal matrices respectively. x x y denotes k k the largest integer less than or equal to x. 2. DESIGN OF PARAUNITARY COSINE-MODULATED QMF BANKS 2.1. Background for designing an M-channel maximally decimated QMF bank Figure 1 shows an M-channel maximally decimated QMF bank. QMF banks are used in a wide range of speech, image, and other applications, which involve the splitting of an input signal into subbands and, finally, the reconstruction of the original signal. There are two prevalent approaches to M-channel QMF filter design. They are the perfect reconstruction QMF banks and the pseudo-QMF banks. The latter have an efficient design procedure (only the prototype filter is designed), but the former achieves perfect reconstruction of the input i.e. without aliasing, magnitude, or phase distortion). In a cosine-modulated QMF bank, the transfer functions of the analysis and synthesis filters, denoted by H (z) and F (z) for k"0, 1,2, M!1, respectively, are obtained by modulating the transfer function P(z) of k k a linear-phase lowpass FIR filter prototype P with bandwidth n/(2M). So, the impulse responses of the Figure 1. M-channel maximally decimated filter bank ( 1998 John Wiley & Sons, Ltd. Int. J. Circ. ¹heor. Appl., 26, 539—546 (1998) 541 LETTER TO THE EDITOR analysis and synthesis filters H and F are given by k k A A A A B B B B n N!1 n! #h h (n)"2p(n) cos (2k#1) k k 2M 2 (1a) n N!1 n! !h f (n)"2p(n) cos (2k#1) k k 2M 2 (1b) where n and k are in the range of 0)n)N!1 and 0)k)M!1, respectively, and p(n) is the impulse response of the prototype filter P. From equation (1), the analysis and synthesis filters are related by f (n)"h (N!1!n) or F (z)"z~(N~1)HI (z) k k k k As shown in equation 2, in order to cancel significant aliasing terms and assume a relatively flat overall amplitude response, h must be chosen as k n for 0)k)M!1 h "(!1)k k 4 Here the lengths of H (z) and F (z) are the same, and are assumed to be multiples of 2M, i.e. N"2mM. k k It has been shown in Reference 3 that the PR conditions for M-channel cosine-modulated QMF bank when the prototype filter is linear-phase FIR can be written as 1 M GI (z)G (z)#GI (z)GI (z)" , 0)k) !1, even M M`k M`k k k 2M 2 M 1 GM k (z)Gk (z)#GI M`k (z)GM`k (z)" , 0)k) !1 odd M 2 2M 1 2GI (z)G (z)" (M~1)@2 (M~1)@2 2M G (2) where G (z) are the type I polyphase components of prototype filter P(z). Generally, for the case where the k prototype filter of cosine-modulated QMF bank has arbitrary-length,7 it turn out that the necessary and sufficient conditions for perfect reconstruction are the same as equation (2). Thus, for convenience, we only consider the case where the length of P(z) is an arbitrary multiple of 2M because the other cases are very similar. 2.2. Quadratic-constraint formulation of PR-CM QMF bank design A convenient and simple filter design method is directly optimizing the impulse response p(n) of prototype filter P(z). For this purpose, we must rewrite the PR conditions in equation (2) as some quadratic-constraint form in p(n). According to the polyphase representation of filter bank8, kth polyphase filter of Type I can be expressed as m~1 G (z)" + g (n)z~n k k n/0 (3) The PR conditions for even M case in equation (2) can be expressed as 1 " d gTS g #gT S g M`k r M`k M r k r k ( 1998 John Wiley & Sons, Ltd. (4) Int. J. Circ. ¹heor. Appl., 26, 539—546 (1998) 542 LETTER TO THE EDITOR where g "[g (0), g (1),2, g (m!1)]T is the impulse response vector of the kth polyphase filter G (z), k k k k k g "[g (0), g (1), , g (m!1)]T is the coefficient vector of (M#k)-th polyphase filter G (z), M`k M`k M`k 2 M`k M`k 0 I 0 0 1, r"0 m~r # and d " S" is the unit-pulse. r r 0 0 I 0 0, rO0 m~r m]m m]m Let gT"pTT , where p"[p(0), p(1),2, p(2mM!1)]T which is an (2mM)-dimensional coefficient vector k k of prototype filter P(z) and T ’s (i, j)-entry k 1, i"2jM#k [T ] " , k i,j 0, otherwise C D C D G G and T is 2mM]m matrix. k Substituting the related equations of above into equation (4), we can obtain 1 pTA p" d k,r M r (5) S TT is a group of symmetrical matrices since S is a symmetrical matrix for where A "T S TT#T r M`k r M`k k,r k r k each r(r"0,2, m!1) by its definition. In addition, A is also a group of sparse matrices with each column k,r or now only containing two 1 s in our current problem. By checking equation (5), we have known that equation (5) implies a normalized constraint for the prototype filter, pTp"l. We combined the above two constraints and then obtained pTp"1, a pT[aA #I] p" dr#1 k,r M (6) where a is a constant which is used to guarantee matrices B "aA #I to be diagonal entry dominant k,r k,r when we choose 2mM 2mM a(max + [Ak,r]i,j!min + [Ak,r]i,j . i/l i/l iOj For our problem, each column or row of A has only two 1 s. Consequently, if we choose a)0.5, B must k,r k,r be diagonal entry dominant matrices. On the other hand, we have BT "B by checking equation (6). So, k,r k,r B is positive definite by matrix theory. k,r By using the linear phase property of the prototype filter, g can be rewritten as gT"pT [I J]T , where 1@2 k k k p "[p(0), p(1),2, p(mM!1)]T. In terms of p , the above equation can be re-expressed as 1@2 1@2 pT C p "1 1@2 k,r 1@2 (7) where CD 1 I a C " [I J]B and q " d #1O0 k,r q k,r J r M r r Obviously, there exist CT "C , that is to say C is a group of symmetric and positive definite matrices. k,r k,r k,r Note that k goes from 0 to (M/2)!1 and r is in the range of 0 to m!1, therefore there are mM/2 constrained conditions in equation (7). In a similar way, PR conditions for the odd M case in equation (2) are given by pT C p "1, 1@2 k,r 1@2 p "1 pT C 1@2 ((M~1)@2),r 1@2 (8) where C ((M~1)@2),r ( 1998 John Wiley & Sons, Ltd. CD 1 I a " [I J] T S TT , q" d #1, (M~1)@2 (M~1)@2 r r q J 2M r r Int. J. Circ. ¹heor. Appl., 26, 539—546 (1998) 543 LETTER TO THE EDITOR M M and, r is in the same range as equation (7) and k goes from 0 to x y !1, so there are m x y constrained 2 2 conditions in equation (8). In summary, the PR conditions in equation (2) are rewritten as quadratic-constraint conditions on p as 1@2 in equations (7) and (8) with normalized condition in which all the constrained matrices are symmetric and positive definite. Since the system in this case is a perfect reconstruction, according to PR theory in equation 2 it is sufficient to only minimize the stopband energy of the prototype filter under the mM/2 quadratic-constrained conditions and normalized condition. The cost of the stopband energy /": n D P (e+u) D 2 du can be (n@2M)`e 0 further simplified as /"pTQp, where Q is a real, symmetric, and positive-definite matrix with (i, j)th entry P n cos(iu!N/2)cos( ju!N/2) du (9) (n@2M)`e In concise form, the optimization problem of prototype filter design for PR cosine-modulated QMF bank can be restated as [Q] "4 i,j min p pT Qp 1@2 1@2 1@2 s.t. pT C p "1, pT p "1/2 1@2 1@2 1@2 k,n 1@2 (10) where index k and n are chosen to have the same value ranges as that in equation (7) or equation (8). 3. PENALTY FUNCTION METHOD FOR QCLS PROBLEM A cost function for our QCLS minimization problem can be constructed as l J(X, b)"XTQX# + b p (X) (11) i i i/1 where b is a positive constant, p (X) is a penalty function satisfying the following conditions: (a) it is i i a continuous and differentiable function of the unknown variable X, (b) p (X)*0, ∀X, (c) p (X)"0 iff i i XTA X"1 such that i p (X)"(XTA X!1)2 and b '0 i i i According to penalty function (PS) theory,11 the minimizer of equation (11) is the exact solution of the original QCLS optimization problem in equation (10) when penalty factor b approaches infinity. Let Mbk, i i i"1, 2,2, l, k"1, 2,2 be l sequences simultaneously tending to infinity such that for each k, bk*0 and i bk`1*bk . Further, let Xk be the minimizer of J. Then, we have the following limit theorem. i i ¹heorem 1. ¸imit point of the sequence MXkN is a solution of equation (11). That is, a maximum objective function value exactly satisfying constraints can be arrived at the limit point of the sequence MXkN. So, the problem in hand has been transformed to an unconstrained non-linear optimization problem. The solution of the unconstrained non-linear optimization problem of equation (11) is just the solution of the constrained non-linear optimization problem of equation (10) as b approaches i positive infinity. In the following analysis, we assume that b is a fixed positive value. i Firstly, we define the feasible solution space as #"MX D QX"+ l (X)A X, where l (X)" i i i i 2b (1!XTA X)N which consists of all the equilibrium points of J. i i ¸emma 1. For ∀X , X 3# and X OX , then XTA X "0, i"1, 2,2, l. 1 2 1 2 1 i 2 ( 1998 John Wiley & Sons, Ltd. Int. J. Circ. ¹heor. Appl., 26, 539—546 (1998) 544 LETTER TO THE EDITOR The proof of Lemma 1 can be found in Reference 8. Clearly, the statement of Lemma 1 means every two different vectors in feasible region are mutually orthogonal with respect to each constrained matrix A . i ¹heorem 2. XM is a stable global minimizer of J in equation (11) if and only if XM satisfies: E XM E "max E X E , i"1,2, l, where, E X E stands for the weighted norm of X with matrix M, and # is Ai X|# Ai M the feasible solution space. For the proof of Theorem 2 refer to equation 8, the following four corollaries bring out the significant features of Theorem 2. Corollary 1. For a given set of b , every local minimizer of J is also a global minimizer. i Since the Hessian matrix H(XM ) at every local minimizer XM of J satisfies l XM @H(XM )XM " + 8b E XM E 4 *0 Ai i i/1 therefore the result of Corollary 1 follows directly from Theorem 2 by recognizing that the Hessian matrix H(XM ) at point XM is at least positive semidefinite when XM is a local minimizer of J. Corollary 2. ¹he minimizer of J is unique (except for the sign). Corollary 3. ¹he equilibrium points except the stable minimizer correspond to saddle points of J. This follows from the fact that H(XM ) is indefinite at the equilibrium point XM if it is a vector not satisfying the condition of Theorem 2. Corollary 4. ¹he minimum energy value of J is J "+ b (1!E XM E4A ) when a global minimizer arrived. i .*/ i i In the next section, we will present two examples to show the efficiency and flexibility of our method, in which the penalty functions are chosen in the form p (x)"(XTA X!1)2 and all the penalty parameters in i i equation (11) are chosen to have the same values as in our examples. 4. ILLUSTRATIVE EXAMPLES In order to verify our analysis and derivations in the preceding sections, we designed an 8-channel PR CMFB as an example using the proposed method. Let M"8 and m"3; the filter length is N"48. For this case, there are 13 quadratic-constraints and 24 unknown variables. In our PS method, we choose the penalty factor b "1)0e#03 and give random initial values. Table I gives the filter coefficients of the designed i prototype filter by our PS method. The magnitude responses of the designed prototype filter and analysis bank H (z) are shown in Figures 2(a) and 2(b). In Figure 2(a), the prototype designed by conventional k TC — LAT method in Reference 3 is also shown by dashed-line for comparison. It can be seen that the stopband attenuation of the designed analysis filter is about !40)32 dB, which is about 3 dB hiher than those designed using conventional approaches.3 Here the designed filter is only approximate PR CMFB since the penalty factors have finite positive values. The maximum error of the constraints is less than 10~6 which is very small and can be neglected in some areas. As the second example, we design a 16-channel PR CMFB. We choose larger penalty factors b"1)0e#06. The magnitude responses of the designed prototype and analysis bank are shown in Figures 3(a) and 3(b), respectively. The maximum error of quadratic constrained conditions is less than 10~9, which can be ignored in many practical application fields. ( 1998 John Wiley & Sons, Ltd. Int. J. Circ. ¹heor. Appl., 26, 539—546 (1998) 545 LETTER TO THE EDITOR Table I. The coefficients of prototype (N"48) of 8-channel PR CMFB by our PS method for b"1.e#03 n p (n) 1@2 n p (n) 1@2 1 2 3 4 5 6 7 8 9 10 11 12 !3)138047627341292e-003 !3)749265236720407e-003 !4)798317517895776e-003 !7)568400992714401e-003 !1)203752077867772e-002 !1)524549626657524e-002 !1)673603975199828e-002 !1)697064663406244e-002 !1)550869354619807e-002 !1)192078528085052e-002 !5)987753050649529e-003 3)222310033767170e-003 13 14 15 16 17 18 19 20 21 22 23 24 1)688249086906735e-002 3)391606968769700e-002 5)340043348450080e-002 7)495679737521221e-002 9)796611627184632e-002 1)216234343259736e-001 1)447886766314165e-001 1)664158030709835e-001 1)852527850735704e-001 2)002476303839510e-001 2)107604016518628e-001 2)162036408160893e-001 Figure 2. The magnitude responses of the designed prototype (a) and analysis filter bank (b) for 8-channel PR CM-QMF bank Figure 3. The magnitude responses of the designed prototype (a) and analysis filter bank (b) for 16-channel PR CM-QMF bank 5. CONCLUSIONS AND REMARKS In this paper, a prototype design of PR CMFB is formulated as a new kind of QCLS optimization problem in which all constrained matrices are symmetric and positive definite. To solve the QCLS minimization problem, we use a penalty function approach, which has been verified to have Lyapunov global stability, to design PR cosine-modulated QMF banks. Design examples show the effectiveness of our method. ( 1998 John Wiley & Sons, Ltd. Int. J. Circ. ¹heor. Appl., 26, 539—546 (1998) 546 LETTER TO THE EDITOR ACKNOWLEDGEMENTS The authors would like to thank Dr X. Q. Gao, University of Southeast, for many helpful discussions. The authors also thank the anonymous reviewers for their constructive comments on the manuscript. This work was supported in part by the Climbing Programme—National Key Project for Fundamental Research in China, Grant NSC92097 and partly by National Natural Foundation of China. REFERENCES 1. T. Karp and N. J. Fliege, ‘MDFT filter banks with perfect reconstruction’, Proc. IEEE Int. Symp. on Circuits and Systems (ISCAS), IEEE, New York, 1995, pp. 744—747. 2. P. P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice-Hall, Englewood Cliffs, NJ, 1993. 3. R. D. Koilpillai and P. P. Vaidyanathan, ‘Cosine-modulated FIR filter banks satisfying perfect reconstruction’, IEEE ¹rans. Signal Processing, SP-40(3), 770—783 (1992). 4. T. A. Ramstad and J. P. Tanem, ‘Cosine-modulated analysis—synthesis filter bank with critical sampling and perfect reconstruction’, Proc. IEEE ICASSP, 1991, pp. 1789—1792. 5. C. Y. Maa and M. Shanblatt, ‘Linear and quadratic programming neural network analysis’, IEEE ¹rans. Neural Network, NN-3(6), 580—594 (1992). 6. T. Q. Nguyen, ‘Digital filter bank design quadratic-constrained formulation’, IEEE ¹rans. Signal processing, SP-43(9), 2103—2108 (1995). 7. T. Q. Nguyen and R. D. Koilpillai, ‘The theory and design of arbitrary-length cosine-modulated filter banks and wavelets, satisfying perfect reconstruction’, IEEE ¹rans. Signal Processing, SP-44(3), 473—483 (1996). 8. Y. Tan, X. Q. Gao and Z. Y. He, ‘Design of perfect reconstruction cosine-modulated QME banks’, Proc. IEEE Int. Symp. on Circuits and Systems (ISCAS), HK, IEEE, New York, 1997, pp. 2441—2445. 9. H. Xu, W. Lu and A. Antoniou, ‘Efficient iterative design method for cosine-modulated QMF banks’, IEEE ¹rans. Signal Processing, SP-44(7), 1657—1672 (1996). 10. Y. Tan and Z. Y. He, ‘Linear phase paraunitary cosine modulated filter bank design formula’, Proc. 3rd IEEE Int. Conf. on Electronics, Circuit and Systems (ICECS), Greece, 1996, pp. 898—901. 11. P. M. Pardalos and J. B. Rosen, Constrained Global Optimization: Algorithms and Applications, Springer Berlin, 1987. ( 1998 John Wiley & Sons, Ltd. Int. J. Circ. ¹heor. Appl., 26, 539—546 (1998)

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