Direct Design of Orthogonal Filter Banks and Wavelets

Transcription

1 Direct Design of Orthogonal Filter Banks and Wavelets W.-S. Lu T. Hinamoto Dept. of Electrical & Computer Engineering Graduate School of Engineering University of Victoria Hiroshima University Victoria, Canada Hiroshima, Japan 1

2 Outline Introduction Early and recent work Constrained linear updates and a convex QP formulation for least-squares design of conjugate quadrature (CQ) filters Constrained linear updates and an second-order cone programming formulation for minimax design of CQ filters Experimental results

3 Two-channel FIR filter bank 1. Introduction 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 3

4 A conjugate quadrature (CQ) filter bank assumes 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 automatically satisfied (no aliasing) and the 1 st PR condition becomes H ( z) H ( z ) + H ( z) H ( z ) = (PS) 1 1 which is called the power symmetric (PS) condition because it implies 1 ( j( π θ) ) j( π + θ) ( ) H e + H e = for any θ 4

5 . Early and Recent Work Representative early and recent work include Smith and Barnwell (1984) Mintzer (1985) Vaidyanathan and Nguyen (1987) Rioul and Duhamel (1994) Lawton and Michelli (1997) Tuqan and Vaidyanathan (1998) Dumitrescu and Popeea () Tay (5, 6) 5

6 The most common design technique: A half-band filter P(z) is a zero-phase FIR filter satisfying P(z) + P( z) = Let 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 a half-band filer and it is nonnegative: ( ) P e = H ( e ) H ( e ) = H ( e ) (P) jω jω jω jω 6

7 Design steps: (a) Design a lowpass half-band FIR filter P(z) with j nonnegativity property Pe ( ω ) (b) Perform a spectral decomposition P z H z H z 1 ( ) = ( ) ( ) Vanishing moment (VM): the number of VMs equals to the number of zeros of H at 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 7

8 Problem Formulation Let 3. Least-Squares Design of CQ Filters H ( z) h z N 1 n = n with N even, and h = [h h 1 h N-1 ] T n= A direct approach: minimizing a least squares type objective function subject to the PS constraint: jω minimize H ( e ) dω h π ω a subject to: H ( z) H ( z ) + H ( z) H ( z ) = 1 1 8

9 The objective function is a positive definite quadratic form: π ω a ( jω ) T H e dω = h Qh with Q a symmetric positive definite Toeplitz matrix: Q 1 = toeplitz π ωa sinωa sin[( N 1) ωa] N 1 The constraint is the PS condition: H ( z) H ( z ) + H ( z) H ( z ) = (PS) 1 1 that is equivalent to N/ second-order equality constraints: 9

10 N 1 m n n+ m δ n= h h = ( m) for m=,1,,( N )/ with δ ( m) = 1 for m= and δ ( m) = for m. The design problem now becomes a polynomial optimization 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 ) / 1

11 The POP can 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 Features of these problems: All polynomials are of second-order. The objective function is convex Nonconvex problems because of the N/ 11

12 second-order equality constraints (the PS conditions). Examples of the PS constraints 1. N = 4 constraints: h + h + h + h = h h + h h = 1 3. N = 1 constraints: h + h + + h = 1 ( terms) 1 19 h h + h1 h3 + + h17 h19 h h4 + h1 h5 + + h15 h19 = = (18 terms) (16 terms) h h + h h + + h h = (14 terms) h h + h h + h h + h h = (4 terms) h h + h h = ( terms)

13 Constrained Linear Updates In the kth iteration of the algorithm we update filter coefficients from h (k) things: to h (k+1) = h (k) + d to achieve two to reduce the filter s stopband energy hqh T to better approximate constraints N 1 m n= h h = δ ( m), m N/ 1 n n+ m The stopband energy at h (k+1) is equal to 1/ ( k ) Q ( d + h ) The constraints at h (k+1) becomes 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 13

14 Imposing constraints on the smallness of increment vector d: di β for i = 1,,, N the nd -order constraints can be linearized: 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 ) a known term, denoted by s ( m) = δ ( m) for m=,1,,( N )/ very small, drop ( k) ( k) ( k) ( k) hn hn+ m + hn dn+ m + dnhn+ m n n n linear updates 14

15 This leads to a set of (N )/ linear equations: n h d + d h = δ ( m) s ( m) u ( m) ( k) ( k) ( k) ( k) n n+ m n n+ m n which can be expressed as C d = u ( k) ( k) The smallness constraint on d is given by d β for 1 i n Ad b i The linear constraint on VMs is given by N 1 n= ( 1) n ( d + h ) = for l L 1 Dd = v n l ( k) ( k) n n 15

16 A Quadratic Programming (QP) Formulation Summarizing, the solution strategy is to iteratively update the filter coefficients from h (k) to h (k+1) = h (k) + d (k) with d (k) obtained by solving the QP problem 1/ ( k ) minimize Q ( d + h ) d subject to: Ad b ( k) ( k) C u d = ( k ) D v 16

17 Problem Formulation 4. Minimax Design of CQ Filters The formulation in this case is changed to N 1 m n= N 1 n= jω minimize maximize H ( e ) h subject to: h h = δ ( m) for m ( N ) / n n+ m a n l ( 1) nh = for l=,1,, L 1 n ω ω π 17

18 Constrained Linear Updates Like in the least squares design, the constrained linear update gives jω minimize maximize H ( e ) d ω ω π a subject to: Ad b ( k) ( k) C u d = ( k ) D v 18

19 Dealing 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 T( ω) h T s( ω) T H (, ) ( ) ( ) ( ) e h + d = T ω h + d = T ω d + g jω ( k) ( k) ( k) ( k) ( k) ( k) 19

20 This converts the minimax problem into minimize η T ω d + g η ω ω π ( k) ( k) subject to: ( i) for { i} [ a, ] Ad b ( k) ( k) C u d = ( k ) D v which is an SOCP problem.

21 5. Experimental Results 5.1 Comparison with designs by Smith-Barnwell s method Filter H (z) Largest Eq. Error N = 8 N = 16 N = 3 H (z) of [] Refined H (z) < 1-15 H (z) of [] Refined H (z) < 1-15 H (z) of [] Refined H (z) <

22 1 N = 3, L =, ω a = Normalized frequency

23 5. LS and minimax designs with N = 96 and L =, 1,, 5 Least squares with N = 96, ω =.56π L Energy in Stopband Largest Equation Error < < < < < a 3

24 Minimax with N = 96, ω =.56π a L Instantaneous Energy in Stopband Largest Equation Error <

25 Minimax design with N = 96, L = 3, ω =.56π : a Normalized frequency 5