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 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.     

A state space approach to the design of globally optimal FIR energycompaction filters

We introduce a new approach for the least squared optimization of a weighted FIR filter of arbitrary order N under the constraint that its magnitude squared response be Nyquist(M). Although the new formulation is general enough to cover a wide variety of applications, the focus of the paper is on optimal energy compaction filters. The optimization of such filters has received considerable attention in the past due to the fact that they are the main building blocks in the design of principal component filter banks (PCFBs). The newly proposed method finds the optimum product filter F_opt(z)=H_opt(Z)H_opt (z^-1) corresponding to the compaction filter H_opt (z). By expressing F(z) in the form D(z)+D(z^-1), we show that the compaction problem can be completely parameterized in terms of the state-space realization of the causal function D(z). For a given input power spectrum, the resulting filter F_opt(z) is guaranteed to be a global optimum solution due to the convexity of the new formulation. The new algorithm is universal in the sense that it works for any M, arbitrary filter length N, and any given input power spectrum. Furthermore, additional linear constraints such as wavelets regularity constraints can be incorporated into the design problem. Finally, obtaining H_opt(z) from F_opt(z) does not require an additional spectral factorization step. The minimum-phase spectral factor H_min(z) can be obtained automatically by relating the state space realization of D_opt(z) to that of H _opt(z)     

Design of Time–Frequency Localized Filter Banks: Transforming Non-convex Problem into Convex Via Semidefinite Relaxation Technique

We present a method for designing optimal biorthogonal wavelet filter banks (FBs). Joint time–frequency localization of the filters has been chosen as the optimality criterion. The design of filter banks has been cast as a constrained optimization problem. We design the filter either with the objective of minimizing its frequency spread (variance) subject to the constraint of prescribed time spread or with the objective of minimizing the time spread subject to the fixed frequency spread. The optimization problems considered are inherently non-convex quadratic constrained optimization problems. The non-convex optimization problems have been transformed into convex semidefinite programs (SDPs) employing the semidefinite relaxation technique. The regularity constraints have also been incorporated along with perfect reconstruction constraints in the optimization problem. In certain cases, the relaxed SDPs are found to be tight. The zero duality gap leads to the global optimal solutions. The design examples demonstrate that reasonably smooth wavelets can be designed from the proposed filter banks. The optimal filter banks have been compared with popular filter banks such as Cohen–Daubechies–Feauveau biorthogonal wavelet FBs, time–frequency optimized half-band pair FBs and maximally flat half-band pair FBs. The performance of optimal filter banks has been found better in terms of joint time–frequency localization.     

Minimax design of two-channel nonuniform-division FIR filter banks

The paper deals with the minimax design of two-channel nonuniform-division filter (NDF) banks. Based on a linearisation scheme, the design problem is formulated as an optimisation problem with linear constraints. The authors present a method to design a two-channel NDF bank using a modified dual-affine scaling variant of Karmarkar's (1984) algorithm. This method provides the optimal results that the linear-phase FIR analysis and synthesis filters have equiripple stopband response and the resulting NDF bank also shows equiripple reconstruction error behaviour. The effectiveness of the proposed design technique is demonstrated by several simulation examples     

Symmetric Orthogonal Complex-Valued Filter Bank Design by Semidefinite Programming

A new design method for complex-valued two-channel finite impulse response (FIR) filter banks with both orthogonality and symmetry properties is developed. Based on a novel linear matrix inequality (LMI) characterization of trigonometric curves, the optimal design of perfect-reconstruction filter banks is cast into a semidefinite programming (SDP) problem. The dimension of the resulting SDP problem is further reduced by exploiting convex duality. Consequently, the globally optimal solution can be found for any practical filter length and desired regularity order.     

Two-dimensional FIR compaction filter design

The design of signal-adapted multirate filter banks has been an area of research interest. The authors present the design of a 2-D finite impulse response (FIR) compaction filter followed by a 2-D FIR filter bank that packs the maximum energy of the input process into a few subbands. The energy compaction property of the 2-D compaction filter is extremely good for higher filter orders and converges to the ideal optimal solution as the order tends to infinity. The design procedure is very straightforward and involves a 2-D spectral factorisation     

General analysis of two-band QMF banks

The two-channel perfect-reconstruction quadrature-mirror-filter banks (PR QMF banks) are analyzed in detail by assuming arbitrary analysis and synthesis filters. Solutions where the filters are FIR or IIR correspond to the fact that a certain function is monomial or nonmonomial, respectively. For the monomial case, the design problem is formulated as a nonlinear constrained optimization problem. The formulation is quite robust and is able to design various two-channel filter banks such as orthogonal and biorthogonal, arbitrary delay, linear-phase filter banks, to name a few. Same formulation is used for causal and stable PR IIR filter bank solutions     

The theory and design of arbitrary-length cosine-modulated filterbanks and wavelets, satisfying perfect reconstruction

It is well known that FIR filter banks that satisfy the perfect-reconstruction (PR) property can be obtained by cosine modulation of a linear-phase prototype filter of length N=2mM, where M is the number of channels. In this paper, we present a PR cosine-modulated filter bank where the length of the prototype filter is arbitrary. The design is formulated as a quadratic-constrained least-squares optimization problem, where the optimized parameters are the prototype filter coefficients. Additional regularity conditions are imposed on the filter bank to obtain the cosine-modulated orthonormal bases of compactly supported wavelets. Design examples are given     

Optimization of filter banks using cyclostationary spectralanalysis

This paper proposes a design method of optimal biorthogonal FIR filter banks that minimize the time-averaged mean squared error (TAMSE) when the high-frequency subband signal is dropped. To study filter banks from a statistical point of view, cyclostationary spectral analysis is used since the output of the filter bank for a wide-sense stationary input is cyclostationary. First, the cyclic spectral density of the output signal is derived, and an expression for the TAMSE is presented. Then, optimal filter banks are given by minimizing the TAMSE with respect to the coefficients of the filters under the biorthogonality condition. By imposing the additional constraints on the coefficients, the optimal biorthogonal linear phase filter bank can be obtained     

Analysis, design, and implementation of two-channel linear-phasefilter banks: a new approach

The problem of designing two-channel perfect-reconstruction FIR filter banks with linear-phase analysis and synthesis filters is revisited. Based on a new algebraic formulation, all the possible factorized forms for this two-band filter bank are derived. We thus obtain complete and canonical solutions for the filter banks, composed of odd-order symmetric and antisymmetric filters (type-A systems) and for those built with symmetric even order filters (type-B systems). A strong characteristic of these new cascade structures, which, until now, had not been identified, is related to a defectivity property. Taking this into account is the key issue to cover all the FIR solutions and to design cascade structures being robust to the quantization of their parameters. Design examples are provided that illustrate our method     

Multicriterion optimized QMF Bank design

The quadrature mirror filter (QMF) bank with multicriterion constraints such as minimal aliasing and/or minimal error coding is among the most important problems in filterbank design, for solving which linear algebra-based methods are still heuristic and do not always work, especially for large filter length. It is shown in this paper that this problem can be reduced either to convex linear matrix inequality (LMI) optimization (when filters are of nonlinear phase) or to semi-infinite linear (SIP) programming (when filters are of linear phase), which can be very efficiently solved either by the standard LMI solvers or our previously developed SIP solver. The proposed computationally tractable optimization formulations are confirmed by several simulations.     

A direct approach to H2 optimal deconvolution ofperiodic digital channels

This paper studies the H₂ optimal deconvolution problem for periodic finite impulse response (FIR) and infinite impulse response (IIR) channels. It shows that the H₂ norm of a periodic filter can be directly quantified in terms of periodic system matrices and linear matrix inequalities (LMIs) without resorting to the commonly used lifting technique. The optimal signal reconstruction problem is then formulated as an optimization problem subject to a set of matrix inequality constraints. Under this framework, the optimization of both the FIR and IIR periodic deconvolution filters can be made convex, solved using the interior point method, and computed by using the Matlab LMI Toolbox. The robust deconvolution problem for periodic FIR and IIR channels with polytopic uncertainties are further formulated and solved, also by convex optimization and the LMIs. Compared with the lifting approach to the design of periodic filters, the proposed approach is simpler yet more powerful in dealing with multiobjective deconvolution problems and channel uncertainties, especially for IIR deconvolution filter design. The obtained solutions are applied to the design of an optimal filterbank yielding satisfactory performance     

Theory of rate-distortion-optimal, constrainedfilterbanks-application to IIR and FIR biorthogonal designs

We design filterbanks that are best matched to input signal statistics in M-channel subband coders, using a rate-distortion criterion. Previous research has shown that unconstrained-length, paraunitary filterbanks optimized under various energy compaction criteria are principal-component filterbanks that satisfy two fundamental properties: total decorrelation and spectral majorization. In this paper, we first demonstrate that the two properties above are not specific to the paraunitary case but are satisfied for a much broader class of design constraints. Our results apply to a broad class of rate-distortion criteria, including the conventional coding gain criterion as a special case. A consequence of these properties is that optimal perfect-reconstruction (PR) filterbanks take the form of the cascade of principal-component filterbanks and a bank of pre- and post-conditioning filters. The proof uses variational techniques and is applicable to a variety of constrained design problems. In the second part of this paper, we apply the theory above to practical filterbank design problems. We give analytical expressions for optimal IIR biorthogonal filterbanks; our analysis validates a conjecture by several researchers. We then derive the asymptotic limit of optimal FIR biorthogonal filterbanks as filter length tends to infinity. The performance loss due to FIR constraints is quantified theoretically and experimentally. The optimal filters are quite different from traditional filters. Finally, a sensitivity analysis is presented     

Two-channel IIR QMF banks with approximately linear-phase analysisand synthesis filters

Perfect linear-phase two-channel QMF banks require the use of finite impulse response (FIR) analysis and synthesis filters. Although they are less expensive and yield superior stopband characteristics, perfect linear phase cannot be achieved with stable infinite impulse response (IIR) filters. Thus, IIR designs usually incorporate a postprocessing equalizer that is optimized to reduce the phase distortion of the entire filter bank. However, the analysis and synthesis filters of such an IIR filter bank are not linear phase. In this paper, a computationally simple method to obtain IIR analysis and synthesis filters that possess negligible phase distortion is presented. The method is based on first applying the balanced reduction procedure to obtain nearly allpass IIR polyphase components and then approximating these with perfect allpass IIR polyphase components. The resulting IIR designs already have only negligible phase distortion. However, if required, further improvement may be achieved through optimization of the filter parameters. For this purpose, a suitable objective function is presented. Bounds for the magnitude and phase errors of the designs are also derived. Design examples indicate that the derived IIR filter banks are more efficient in terms of computational complexity than the FIR prototypes and perfect reconstruction FIR filter banks. Although the PR FIR filter banks when implemented with the one-multiplier lattice structure and IIR filter banks are comparable in terms of computational complexity, the former is very sensitive to coefficient quantization effects     

A new approach to the design of critically sampled M-channeluniform-band perfect-reconstruction linear-phase FIR filter banks

A new approach is presented for the design of uniform-band M-channel perfect-reconstruction (PR) FIR filter banks employing linear-phase analysis and synthesis filters. The technique designs on the impulse responses of the analysis filters directly. The design problem is formulated as an optimization program. The filter bank's PR feature can either be implicitly enforced through a set of mathematical relationships among the analysis filters' coefficients or through a set of constraints in the optimization program. The former approach results in a filter bank whose PR feature's dependency on hardware and software is eliminated or, at least, minimized. The synthesis filters are then obtained by a set of relationships that describe each synthesis filter as a function of the analysis filters. The criterion for optimality is “least-squares,” where the square of the difference between the ideal and actual frequency responses is integrated over the appropriate frequency bands for all M analysis filters and minimized     

Optimal design of two-channel nonuniform-division FIR filter bankswith -1, 0, and +1 coefficients

This paper deals with the optimal design of two-channel nonuniform-division filter (NDF) banks whose linear-phase FIR analysis and synthesis filters have coefficients constrained to -1, 0, and +1 only. Utilizing an approximation scheme and a weighted least squares algorithm, we present a method to design a two-channel NDF bank with continuous coefficients under each of two design criteria, namely, least-squares reconstruction error and stopband response for analysis filters and equiripple reconstruction error and least-squares stopband response for analysis filters. It is shown that the optimal filter coefficients can be obtained by solving only linear equations. In conjunction with the proposed filter structure, a method is then presented to obtain the desired design result with filter coefficients constrained to -1, 0, and +1 only. The effectiveness of the proposed design technique is demonstrated by several simulation examples     

Design of Oversampled Interleaved DFT Modulated Filter Bank Using 2Block Gauss-Seidel Method

In this paper, we present several new properties of the recently introduced interleaved DFT modulated filter bank and an efficient algorithm for designing the filter bank. The periodicity and symmetry properties of the overall transfer function and aliasing transfer functions are stated. Then the design of the filter bank is formulated into a constrained optimization problem that jointly minimizes the overall distortion and aliasing distortion subject to fixed bounds on the stopband energy, transition-band energy, and passband flatness of the prototype filters. The constrained optimization problem is solved by the 2block Gauss-Seidel method, which alternatively optimizes the analysis PF pair and the synthesis PF pair. Since the overall distortion and aliasing distortion are jointly minimized, the proposed algorithm can lead to filter banks with small reconstruction error, even when the filter banks behave with a low redundancy ratio and short PFs. The convergence of the proposed algorithm is proved. Numerical examples and comparisons with the existing method are included to demonstrate the performance of the proposed algorithm.     

Theory and design of optimum FIR compaction filters

The problem of optimum FIR energy compaction filter design for a given number of channels M and a filter order N is considered. The special cases where N相似文献   

Convergence results on the design of QMF banks

This correspondence considers the design of quadrature mirror filter (QMF) banks whose analysis and synthesis filters are FIR and have linear phase. The design criterion is of least-squares type. An iterative computation algorithm is proposed, and its convergence is proven that offers new insight to the design of QMF banks and relates it to a more general nonlinear optimization problem