Direct Design of Orthogonal Filter Banks and Wavelets by Sequential Convex Quadratic Programming

Transcription

1 Direct Design of Orthogonl Filter Bnks nd Wvelets by Sequentil Convex Qudrtic Progrmming W.-S. Lu Dept. of Electricl nd Computer Engineering University of Victori, Victori, Cnd September 8 1

2 Abstrct Two-chnnel conjugte qudrture (CQ) FIR filter bnks re mong the most populr building blocks for multirte systems s they offer precise perfect reconstruction property. Most design methods for CQ filters re indirect in tht hlfbnd filter with nonnegtivity constrint is designed, followed by spectrum fctoriztion. This tlk describes direct method tht does not require designing the hlfbnd filter nd its fctoriztion, nd integrtes lest-squre nd minimx designs with vnishing moment nd other requirements into single design frmework. Exmples re supplied to help exmine design performnce, efficiency, nd its bility of getting globl solutions.

3 Outline Introduction Erly nd recent work Constrined liner updtes nd convex QP formultion for lest-squres design of conjugte qudrture (CQ) filters Constrined liner updtes nd n second-order cone progrmming (SOCP) formultion for minimx (equiripple) design of CQ filters Experimentl results 3

4 1. Introduction Two-chnnel FIR filter bnk x(n) H (z) F (z) x(n) ^ H 1 (z) F 1 (z) 1 1 Xˆ ( z) = [ F( zh ) ( z) + F1( zh ) 1( z)] X( z) + [ F( zh ) ( z) + F1( zh ) 1( z)] X( z) Perfect reconstruction (PR) conditions F ( z) H ( z) + F( z) H ( z) = z l 1 1 F ( z) H ( z) + F( z) H ( z) = 1 1 4

5 A conjugte qudrture (CQ) filter bnk ssumes H ( z) = z H ( z ), F ( z) = z H ( z ), F( z) = z H ( z ) ( N 1) 1 ( N 1) 1 ( N 1) the nd PR condition is utomticlly stisfied (no lising) nd the 1 st PR condition becomes H ( z) H ( z ) + H ( z) H ( z ) = (PS) 1 1 which is clled the power symmetric (PS) condition becuse it implies 1 ( j( π θ) ) j( π + θ) ( ) H e + H e = for ny θ 5

6 . Erly nd Recent Work Representtive erly nd recent work include Smith nd Brnwell (1984) Mintzer (1985) Vidynthn nd Nguyen (1987) Rioul nd Duhmel (1994) Lwton nd Michelli (1997) Tuqn nd Vidynthn (1998) Dumitrescu nd Popee () Ty (5, 6) 6

7 The most common design technique: A hlf-bnd filter P(z) is zero-phse FIR filter stisfying P(z) + P( z) = If we define P z = H z H z, then the PS condition 1 ( ) ( ) ( ) H ( z) H ( z ) + H ( z) H ( z ) = 1 1 becomes P(z) + P( z) = So P(z) is hlf-bnd filer nd, it is nonnegtive everywhere: ( ) P e = H ( e ) H ( e ) = H ( e ) (P) jω jω jω jω 7

8 Design steps: () Design lowpss hlf-bnd FIR filter P(z) with j nonnegtivity property Pe ( ω ) (b) Perform spectrl decomposition P z H z H z 1 ( ) = ( ) ( ) Vnishing moment (VM): the number of VMs equls to the number of zeros of H t ω = π : dh ( e ) l jω N 1 j l n n l h l n dω ω= π n= = ( ) ( 1) =, for l =, 1,, L 1 8

9 3. Lest-Squres Design of CQ Filters Problem Formultion Let H ( z) h z N 1 n = n with N even, nd h = [h h 1 h N-1 ] T n= A direct pproch: minimizing lest squres type objective function subject to the PS constrint: jω minimize H ( e ) dω h π ω subject to: H ( z) H ( z ) + H ( z) H ( z ) = 1 1 9

10 The objective function is positive definite qudrtic form: π ω ( jω ) T H e dω = h Qh where Q is symmetric positive definite Toeplitz mtrix: Q 1 = toeplitz π ω sinω sin[( N 1) ω] N 1 1

11 The constrint is the PS condition: H ( z) H ( z ) + H ( z) H ( z ) = (PS) 1 1 which is equivlent to set of N/ second-order equlity constrints: N 1 m n n+ m δ n= h h = ( m) for m=,1,,( N )/ where δ ( m) = 1 for m= nd δ ( m) = for m. 11

12 The design problem now becomes polynomil optimiztion problem (POP): N 1 m n= minimize h n n+ m T 1/ hqh= Q h subject to: h h = δ ( m) for m ( N ) / The POP cn be modified to include VM requirement: minimize h N 1 m n= N 1 n= T 1/ hqh= Q h subject to: h h = δ ( m) for m ( N ) / n n+ m n l ( 1) nh = for l=,1,, L 1 n 1

13 The POP cn be further modified to reduce the filter s phse response nonlinerity minimize h N 1 m n= N 1 n= 1/ Q h + w IN I N h n n+ m 1 1 subject to: h h = δ ( m) for m ( N ) / n l ( 1) nh = for l=,1,, L 1 n 13

14 Fetures of these problems: All polynomils re of second-order. The objective function is convex These re nonconvex problems becuse of the N/ second-order equlity constrints (the PS conditions). The PS constrints: exmples 1. N = 4 constrints: h + h + h + h = h h + h h =

15 . N = 1 constrints: h + h + + h = 1 ( terms) 1 19 h h + h h + + h h = h h + h h + + h h = h h + h h + + h h = h h + h h + h h + h h = h h + h h = (18 terms) (16 terms) (14 terms) (4 terms) ( terms) 15

16 Constrined Liner Updtes Suppose we re in the kth itertion of the lgorithm nd we wnt to updte filter coefficients from h (k) to chieve two things: to h (k+1) = h (k) + d to reduce the filter s stopbnd energy hqh T to better stisfy constrints N 1 m n= h h = δ ( m), m N/ 1 n n+ m The stopbnd energy t h (k+1) is equl to The constrints t h (k+1) becomes 1/ ( k ) Q ( d + h ) 16

17 h h + h d + d h + d d = δ ( m) ( k) ( k) ( k) ( k) n n+ m n n+ m n n+ m n n+ m n n n n Imposing constrints on the smllness of increment vector d: di β for i = 1,,, N then the nd -order constrints cn be linerized: h h + h d + d h + d d ( k) ( k) ( k) ( k) n n+ m n n+ m n n+ m n n+ m n n n n ( k ) known term, denoted by s ( m) = δ ( m) for m=,1,,( N )/ very smll, drop ( k) ( k) ( k) ( k) hn hn+ m + hn dn+ m + dnhn+ m n n n liner updtes 17

18 This leds to set of (N )/ liner equtions: n h d + d h = δ ( m) s ( m) u ( m) ( k) ( k) ( k) ( k) n n+ m n n+ m n which cn in mtrix-vector nottion be expressed s C d = u ( k) ( k) The smllness constrint on d is given by d β for 1 i n Ad b i The liner constrint on VMs is given by N 1 n= ( 1) n ( d + h ) = for l L 1 Dd = v n l ( k) ( k) n n 18

19 A Qudrtic Progrmming (QP) Formultion Summrizing the nlysis, the solution strtegy is to itertively updte the filter coefficients from h (k) to h (k+1) = h (k) + d (k) where d (k) is obtined by solving the QP problem 1/ ( k ) minimize Q ( d + h ) d subject to: Ad b ( k) ( k) C u d = ( k ) D v 19

20 If the phse-response nonlinerity is lso concern, we updte h (k) to h (k+1) = h (k) + d (k) by solving the QP problem 1/ ( k) ( k) minimize Q ( d + h ) + w IN I ( ) 1 N d + h 1 d subject to: Ad b ( k) ( k) C u d = ( k ) D v

21 Problem Formultion 4. Minimx Design of CQ Filters The formultion in this cse is chnged to N 1 m n= N 1 n= jω minimize mximize H ( e ) h subject to: h h = δ ( m) for m ( N ) / n n+ m n l ( 1) nh = for l=,1,, L 1 n ω ω π 1

22 Constrined Liner Updtes Like in the lest squres design, the constrined liner updte gives jω minimize mximize H ( e ) d ω ω π subject to: Ad b ( k) ( k) C u d = ( k ) D v

23 Deling with the objective function, we write N 1 jω = n jnω = T ω T n= H ( e ) h e h c( ) jh s( ω) T [ ] [ ] c( ω) = 1 cosω cos( N 1) ω, s( ω) = sinω sin( N 1) ω hence T jω ( T ) ( T ) c( ω) H( e ) = h c( ω) + h s( ω) = h F( ω) h T s( ω) T H (, ) ( ) ( ) ( ) e h + d = F ω h + d = F ω d + g jω ( k) ( k) ( k) ( k) ( k) ( k) 3

24 This converts the minimx problem into minimize η ( k) ( k) subject to: F( ωi) d + g η for { ωi} [ ω, π] Ad b ( k) ( k) C u d = ( k ) D v which is n SOCP problem. 4

25 5. Experimentl Results We performed totl of 18 LS designs with (filter length, stopbnd edge) being (N = 3, ω.58π = ), (N = 64, ω =.57π ), nd (N = 96, ω =.56π ), nd the number of vnishing moments L from, 1,, to 5. The ssocited QP were solved by the Optimiztion Toolbox provided by MthWorks Inc. The design performnce ws evluted in terms of stopbnd energy nd stisfction of the PS conditions. 5

26 Tble 1: Lest squres with N = 3, ω =.58π L Energy in Stopbnd Lrgest Eqution Error

27 LS design with N = 3, ω =.58π, nd L = Normlized frequency 7

28 Tble : Lest squres with N = 64, ω =.57π L Energy in Stopbnd Lrgest Eqution Error

29 LS design with N = 64, ω =.57π, nd L = Normlized frequency 9

30 Tble 3: Lest squres with N = 96, ω =.56π L Energy in Stopbnd Lrgest Eqution Error

31 LS design with N = 96, ω =.56π, nd L = Normlized frequency 31

32 We lso performed totl of 18 minimx designs with (filter length, stopbnd edge) being (N = 3, ω ω =.58π ), (N = 64, ω =.57π ), nd (N = 96, =.56π ), nd the number of vnishing moments L from, 1,, to 5. The ssocited SOCP problems were solved by SeDuMi 1.1R ( freewre mintined by the Advnced Optimiztion Lbortory t McMster). The design performnce ws evluted in terms of instntneous stopbnd energy nd stisfction of the PS conditions. 3

33 Tble 4: Minimx with N = 3, ω =.58π L Instntneous Energy in Stopbnd Lrgest Eqution Error

34 Minimx design with N = 3, ω =.58π, nd L = Normlized frequency 34

35 Tble 5: Minimx with N = 64, ω =.57π L Instntneous Energy in Stopbnd Lrgest Eqution Error

36 Minimx design with N = 64, ω =.57π, nd L = Normlized frequency 36

37 Tble 6: Minimx with N = 96, ω =.56π L Instntneous Energy in Stopbnd Lrgest Eqution Error

38 Minimx design with N = 96, ω =.56π, nd L = Normlized frequency 38