A Globally Optimal Bilinear Programming Approach to the Design of Approximate Hilbert Pairs of Orthonormal Wavelet Bases

It is understood that the Hilbert transform pairs of orthonormal wavelet bases can only be realized approximately by the scaling filters of conjugate quadrature filter (CQF) banks. In this paper, the approximate FIR realization of the Hilbert transform pairs is formulated as an optimization problem...

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Bibliographic Details
Published in:IEEE transactions on signal processing Vol. 58; no. 1; pp. 233 - 241
Main Authors: Wang, Jiang, Zhang, Jian Qiu
Format: Journal Article
Language:English
Published: New York, NY IEEE 01.01.2010
Institute of Electrical and Electronics Engineers
Subjects:
Applied sciences
Bilinear programming
conjugate quadrature filter (CQF)
Constraint optimization
Delay
Detection, estimation, filtering, equalization, prediction
Discrete transforms
Discrete wavelet transforms
dual-tree complex wavelet transform (DTWT)
Exact sciences and technology
Filter bank
Finite impulse response filter
Frequency
Hilbert transform
Information, signal and communications theory
Miscellaneous
orthonormal wavelet bases
Quadratic programming
Signal and communications theory
Signal processing
Signal, noise
Telecommunications and information theory
Wavelet transforms
dual-tree complex wavelet transform (DTWT)
conjugate quadrature filter (CQF)
Branch and bound method
L1 approximation
Programming
Bilinear programming
Orthogonality
Hilbert transformation
Constrained optimization
Hilbert transform
Wavelet transformation
Optimal solution
Wavelet base
Signal processing
Filter bank
orthonormal wavelet bases
ISSN:1053-587X, 1941-0476
Online Access:Get full text
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Summary:It is understood that the Hilbert transform pairs of orthonormal wavelet bases can only be realized approximately by the scaling filters of conjugate quadrature filter (CQF) banks. In this paper, the approximate FIR realization of the Hilbert transform pairs is formulated as an optimization problem in the sense of the lp ( p =1, 2, or infinite) norm minimization on the approximate error of the magnitude and phase conditions of the scaling filters. The orthogonality and regularity conditions of the CQF bank pairs are taken as the constraints of such an optimization problem. Whereafter the branch and bound technique is employed to obtain the globally optimal solution of the resulting bilinear program optimization problem. Since the orthogonality and regularity conditions are explicitly taken as the constraints of our optimization problem, the attained solution is an approximate Hilbert transform pair satisfying these conditions exactly. Some orthogonal wavelet bases designed herein demonstrate that our design scheme is superior to those that have been reported in the literature. Moreover, the designed orthogonal wavelet bases show that minimizing the l 1 norm of the approximate error should be advocated for obtaining better approximated Hilbert pairs.
ISSN:1053-587X
1941-0476
DOI:10.1109/TSP.2009.2029725