Designs of Orthogonal Filter Banks and Orthogonal Cosine-Modulated - PowerPoint PPT Presentation

Designs of Orthogonal Filter Banks and Orthogonal Cosine-Modulated Filter Banks Jie Yan Department of Electrical and Computer Engineering University of Victoria April 16, 2010 1 / 45

OUTLINE INTRODUCTION 1 LS DESIGN OF ORTHOGONAL FILTER BANKS AND 2 WAVELETS MIMINAX DESIGN OF ORTHOGONAL FILTER BANKS AND 3 WAVELETS DESIGN OF ORTHOGONAL COSINE-MODULATED FILTER 4 BANKS CONCLUSIONS AND FUTURE RESEARCH 5 2 / 45

} 2 2 2 2 } 1. INTRODUCTION A two-channel conjugate quadrature (CQ) filter bank − z − ( N − 1 ) H 0 ( − z − 1 ) H 1 ( z ) = G 0 ( z ) = H 1 ( − z ) G 1 ( z ) = − H 0 ( − z ) where H 0 ( z ) = ∑ N − 1 n = 0 h n z − n 0 ( ) H 0 ( ) z G z 1 ( ) H z G z 1 ( ) Analysis Filter Bank Synthesis Filter Bank 3 / 45

Two-Channel Orthogonal Filter Banks Perfect reconstruction (PR) condition N − 1 − 2 m ∑ h n ⋅ h n + 2 m = 훿 m for m = 0 , 1 , ..., ( N − 2 ) / 2 n = 0 Vanishing moment (VM) requirement: A CQ filter has L vanishing moments if N − 1 ( − 1 ) n ⋅ n l ⋅ h n = 0 for l = 0 , 1 , ..., L − 1 ∑ n = 0 4 / 45

Two-Channel Orthogonal Filter Banks Cont’d A least squares (LS) design of CQ lowpass filter H 0 ( z ) having L VMs ∫ 휋 ∣ H 0 ( e j 휔 ) ∣ 2 d 휔 minimize 휔 a subject to: PR condition and VM requirement The LS problem above can be expressed as h T Qh minimize N − 1 − 2 m ∑ subject to: h n ⋅ h n + 2 m = 훿 m for m = 0 , 1 , ..., ( N − 2 ) / 2 n = 0 N − 1 ( − 1 ) n ⋅ n l ⋅ h n = 0 for l = 0 , 1 , ..., L − 1 ∑ n = 0 5 / 45

Two-Channel Orthogonal Filter Banks Cont’d A minimax design minimizes the maximum instantaneous power of H 0 ( z ) over its stopband ∣ H 0 ( e j 휔 ) ∣ minimize maximize 휔 a ≤ 휔 ≤ 휋 subject to: PR condition and VM requirement The minimax problem can be further cast as minimize 휂 subject to: ∥ T ( 휔 ) ⋅ h ∥ ≤ 휂 for 휔 ∈ Ω N − 1 − 2 m ∑ h n ⋅ h n + 2 m = 훿 m for m = 0 , 1 , ..., ( N − 2 ) / 2 n = 0 N − 1 ( − 1 ) n ⋅ n l ⋅ h n = 0 for l = 0 , 1 , ..., L − 1 ∑ n = 0 6 / 45

�������������������� } } ��������������������� Orthogonal Cosine-Modulated Filter Banks An orthogonal cosine-modulated (OCM) filter bank [ 휋 ( ) ( ) + ( − 1 ) k 휋 ] k + 1 n − D h k ( n ) = 2 h ( n ) cos M 2 2 4 [ 휋 ( ) ( ) − ( − 1 ) k 휋 ] k + 1 n − D f k ( n ) = 2 h ( n ) cos M 2 2 4 for 0 ≤ k ≤ M − 1 and 0 ≤ n ≤ N − 1 x ( n ) y ( n ) + H 0 ( z ) M M F 0 ( z ) H 1 ( z ) M M F 1 ( z ) . . . . . . H M -1 ( z ) M M M -1 ( z ) F 7 / 45

Orthogonal Cosine-Modulated Filter Banks Cont’d An M -channel OCM filter bank is uniquely characterized by its prototype filter (PF) The design of the PF of an OCM filter bank can be formulated as ∫ 휋 ∣ H 0 ( e j 휔 ) ∣ 2 d 휔 minimize 휔 s subject to: PR condition As the PF has linear phase, h is symmetrical. The design problem can be reduced to h T ˆ e 2 (ˆ h ) = ˆ P ˆ minimize h h T ˆ a l , n (ˆ h ) = ˆ Q l , n ˆ subject to: h − c n = 0 for 0 ≤ n ≤ m − 1 and 0 ≤ l ≤ M / 2 − 1 where the design variables are reduced by half to ˆ h = [ h 0 h 1 ⋅ ⋅ ⋅ h N / 2 − 1 ] T . 8 / 45

Overview and Contribution of the Thesis Overview We have formulated three nonconvex optimization problems LS design of CQ filter banks Minimax design of CQ filter banks Design of OCM filter banks Contribution of the thesis Several improved local design methods for the three problems Several strategies proposed for potentially GLOBAL solutions of the three problems 9 / 45

Global Design Method at a Glance Multiple local solutions exist for a nonconvex problem Algorithms in finding a locally optimal solution are available Start the local design algorithm from a good initial point How do we secure such a good initial point? 10 / 45

2. LS DESIGN OF ORTHOGONAL FILTER BANKS AND WAVELETS A least squares (LS) design of a conjugate quadrature (CQ) filter of length- N with L vanishing moments (VMs) can be cast as h T Qh minimize N − 1 − 2 m ∑ subject to: h n ⋅ h n + 2 m = 훿 m for m = 0 , 1 , ..., ( N − 2 ) / 2 n = 0 N − 1 ( − 1 ) n ⋅ n l ⋅ h n = 0 for l = 0 , 1 , ..., L − 1 ∑ n = 0 11 / 45

Local LS Design of CQ Filter Banks An effective direct design method is recently proposed by W.-S. Lu and T. Hinamoto Based on the direct design technique, we develop two local methods Sequential convex-programming (SCP) method Sequential quadratic-programming (SQP) method Both methods produce improved local designs than the direct method 12 / 45

Local LS Design of CQ Filter Banks Cont’d Sequential Convex-Programming Method Suppose we are in the k th iteration to compute 휹 h so that h k + 1 = h k + 휹 h reduces the filter’s stopband energy and better satisfies the constraints, then h T k + 1 Qh k + 1 = 휹 T h Q 휹 h + 2 휹 T h Qh k + h T k Qh k N − 1 N − 1 ( − 1 ) n ⋅ n l ⋅ ( 휹 h ) n = − ( − 1 ) n ⋅ n l ⋅ ( h k ) n ∑ ∑ n = 0 n = 0 N − 1 − 2 m N − 1 − 2 m ∑ ∑ ( h k ) n ( 휹 h ) n + 2 m + ( h k ) n + 2 m ( 휹 h ) n n = 0 n = 0 N − 1 − 2 m ∑ ≈ 훿 m − ( h k ) n ( h k ) n + 2 m n = 0 13 / 45

Local LS Design of CQ Filter Banks Cont’d With h bounded to be small, the k th iteration assumes the form 휹 T h Q 휹 h + 휹 T minimize h g k subject to: A k 휹 h = − a k C 휹 h ≤ b By using SVD to remove the equality constraint, the problem is reduced to x T ˆ Qx + x T ˆ minimize g k Cx ≤ ˆ ˆ subject to: b We modify the problem to make it always feasible as x T ˆ Qx + x T ˆ minimize g k subject to: Fx ≤ a which is a convex QP problem. 14 / 45

Local LS Design of CQ Filter Banks Cont’d Sequential Quadratic-Programming Method The design problem is a general nonlinear optimization problem minimize f ( h ) subject to: a i ( h ) = 0 for i = 1 , 2 , ... , p By using the first-order necessary conditions of a local minimizer, the problem can be reduced to 1 2 휹 T h W k 휹 h + 휹 T minimize h g k A k 휹 h = − a k subject to: ∣∣ 휹 h ∣∣ is small 15 / 45

Local LS Design of CQ Filter Banks Cont’d where p ∑ ∇ 2 ( 흀 k ) i ∇ 2 = h f ( h k ) − h a i ( h k ) W k (13a) i = 1 ] T [ = ∇ h a 1 ( h k ) ∇ h a 2 ( h k ) ⋅ ⋅ ⋅ ∇ h a p ( h k ) A k (13b) g k = ∇ h f ( h k ) (13c) [ a 1 ( h k ) a p ( h k ) ] T a k = a 2 ( h k ) ⋅ ⋅ ⋅ (13d) By removing the equality constraint using the SVD or QR decomposition, the problem assumes the form of a QP problem. Once the minimizer 휹 ∗ h is found, the next iterate is set to h k + 1 = h k + 휹 ∗ h , 흀 k + 1 = ( A k A T k ) − 1 A k ( W k 휹 ∗ h + g k ) 16 / 45

Global LS Design of Low-Order CQ Filter Banks The LS design problem is a polynomial optimization problem (POP) Two recent breakthroughs in solving POPs Global solutions of POPs are made available by Lasserre’s method Sparse SDP relaxation is proposed for global solutions of POPs of relatively larger scales MATLAB toolbox SparsePOP and GloptiPoly can be used to find global solutions of POPs, but only for POPs of limited sizes 17 / 45

Global LS Design of Low-Order CQ Filter Banks Cont’d Example: Design a globally optimal LS CQ filter with N = 6 , L = 2 and 휔 a = 0 . 56 휋 MATLAB toolbox GloptiPoly and SparsePOP are utilized to produce the globally optimal solution ⎡ 0 . 33268098788629 ⎤ 0 . 80689591454849 ⎢ ⎥ ⎢ ⎥ 0 . 45986215652386 h ( 6 , 2 ) ⎢ ⎥ = ⎢ ⎥ LS − 0 . 13501431772967 ⎢ ⎥ ⎢ ⎥ − 0 . 08543638600240 ⎣ ⎦ 0 . 03522516035714 However, GloptiPoly and SparsePOP fail to work as long as the filter length N is greater than or equal to 18 18 / 45

Global LS Design of High-Order CQ Filter Banks A common pattern shared among globally optimal low-order impulse responses. 0.8 N = 6, L = 2 N = 8, L = 2 0.7 N = 10, L = 2 0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 0 0.2 0.4 0.6 0.8 1 19 / 45

Global LS Design of High-Order CQ Filter Banks Cont’d h 6 : Globally optimal impulse response when N = 6 h zp 8 : Impulse response generated by zero-padding h 6 h 8 : Globally optimal impulse response when N = 8 0.8 h 6 (N=6, L=2) zp 0.7 h 8 h 8 (N=8, L=2) 0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 0 0.2 0.4 0.6 0.8 1 Generate initial point by zero-padding! 20 / 45

Global LS Design of High-Order CQ Filter Banks Cont’d Global design strategy in brief: Design a globally optimal CQ filter of short length, say 4, using 1 e.g. GloptiPoly Generating an impulse response for higher order design by 2 zero-padding Apply the SCP or SQP method with the zero-padded impulse 3 response as the initial point to obtain the optimal impulse response of higher order Follow this concept in an iterative way, until desired filter length 4 is reached 21 / 45

Global LS Design of High-Order CQ Filter Banks Cont’d The designs obtained are quite likely to be globally optimal because: Zero-padded initial point sufficiently close to the global minimizer. 1 The local design methods are known to converge to a nearby 2 minimizer. 22 / 45

Design Examples Potentially globally optimal design of an LS CQ filter with N = 96 , L = 3 and 휔 = 0 . 56 휋 0 −20 −40 −60 −80 −100 −120 0 0.2 0.4 0.6 0.8 1 Normalized frequency 23 / 45