Optimization Problems over Positive Pseudopolynomial Matrices
SIAM Journal on Matrix Analysis and Applications
Hilbert transform pairs of orthogonal wavelet bases: necessary and sufficient conditions
IEEE Transactions on Signal Processing
A Higher Density Discrete Wavelet Transform
IEEE Transactions on Signal Processing
The design of approximate Hilbert transform pairs of wavelet bases
IEEE Transactions on Signal Processing
On the parameterization of positive real sequences and MA parameterestimation
IEEE Transactions on Signal Processing
Image analysis using a dual-tree M-band wavelet transform
IEEE Transactions on Image Processing
Symmetric self-Hilbertian filters via extended zero-pinning
Signal Processing
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We propose a parameterization of the finite-impulse-response (FIR) filters that generate the higher density discrete wavelet transform (HD-DWT). The parameterization has the form of a linear system whose variables are the coefficients of two positive trigonometric polynomials. The parameterization allows the easy separation of independent variables and can be used to transform the optimization of HD-DWT into semidefinite programming (SDP) form. Using the parameterization, we are able to optimize filters whose length is greater than the minimal one imposed by regularity constraints. We also propose two optimization methods for the dual-tree HD-DWT. The first generalizes the design based on allpass approximation of the half sample delay. The second is a brute force optimization based on generating values for the independent variables by using successive sections through the admissible set. As optimization criteria, we use an analyticity measure in the latter method and stopband energy in the former ones. For each type of design, we present examples and compare them with previous work.