Vikram M. Gadre

An uncertainty principle for real signals in the fractional Fourier transform domain

The fractional Fourier transform (FrFT) can be thought of as a generalization of the Fourier transform to rotate a signal representation by an arbitrary angle /spl alpha/ in the time-frequency plane. A lower bound on the uncertainty product of signal representations in two FrFT domains for real signals is obtained, and it is shown that a Gaussian signal achieves the lower bound. The effect of shifting and scaling the signal on the uncertainty relation is discussed. An example is given in which the uncertainty relation for a real signal is obtained, and it is shown that this relation matches with that given by the uncertainty relation derived.

Time–frequency localized three-band biorthogonal wavelet filter bank using semidefinite relaxation and nonlinear least squares with epileptic seizure EEG signal classification

Abstract In this paper, we design time–frequency localized three-band biorthogonal linear phase wavelet filter bank for epileptic seizure electroencephalograph (EEG) signal classification. Time–frequency localized analysis and synthesis low-pass filters (LPF) are designed using convex semidefinite programming (SDP) by transforming a nonconvex problem into a convex SDP using semidefinite relaxation technique. Three-band parameterized lattice biorthogonal linear phase perfect reconstruction filter bank (BOLPPRFB) is chosen and nonlinear least squares algorithm is used to determine its parameters values that generate the designed analysis and synthesis LPF such that the band-pass and high-pass filters are also well localized in time and frequency domain. The designed analysis and synthesis three-band wavelet filter banks are compared with the standard two-band filter banks like Daubechies maximally regular filter banks, Cohen–Daubechies–Feauveau (CDF) biorthogonal filter banks and orthogonal time–frequency localized filter banks. Kruskal–Wallis statistical test is employed to measure the statistical significance of the subband features obtained from the various two and three-band filter banks for epileptic seizure EEG signal classification. The results show that the designed three-band analysis and synthesis filter banks both outperform two-band filter banks in the classification of seizure and seizure-free EEG signals. The designed three-band filter banks and multi-layer perceptron neural network (MLPNN) are further used together to implement a signal classifier that provides classification accuracy better than the recently reported results for epileptic seizure EEG signal classification.

An Eigenfilter-Based Approach to the Design of Time-Frequency Localization Optimized Two-Channel Linear Phase Biorthogonal Filter Banks

We present a novel eigenfilter-based approach to the design of time-frequency optimized, linear-phase, biorthogonal FIR filter banks. We first design a linear-phase, low-pass analysis filter, followed by a complementary linear-phase, low-pass synthesis filter. The optimality criterion used is uncertainty-based time-frequency localization, where the objective function is a convex combination of time variance and frequency variance of the respective filters. The objective function to be minimized is formulated in a convex-quadratic form and the perfect reconstruction (PR) and vanishing moment (VM) conditions are imposed in the eigen design of filters as a set of linear equality constraints. The PR and VM conditions are expressed in the time domain matrix formulation, so that these can directly be incorporated into the eigenfilter design. Using the Rayleigh principle, the optimal filter is obtained as an eigenvector corresponding to the minimum eigenvalue of the real symmetric positive-definite matrix associated with the optimization criterion. Thus, our formulation reduces the design problem of time-frequency optimal filter banks to an eigenfilter-based problem. Furthermore, the filter banks designed in this manner are found to be regular and are valid candidates for wavelet filter banks, allowing for the construction of linear phase wavelets. We present a few examples to show that the smooth wavelets can be constructed using the proposed method.

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.

Eigenfilter Approach to the Design of One-Dimensional and Multidimensional Two-Channel Linear-Phase FIR Perfect Reconstruction Filter Banks

We present an eigenfilter-based approach for the design of two-channel linear-phase FIR perfect-reconstruction (PR) filter banks. This approach can be used to design 1-D two-channel filter banks, as well as multidimensional nonseparable two-channel filter banks. Our method consists of first designing the low-pass analysis filter. Given the low-pass analysis filter, the PR conditions can be expressed as a set of linear constraints on the complementary-synthesis low-pass filter. We design the complementary-synthesis filter by using the eigenfilter design method with linear constraints. We show that, by an appropriate choice of the length of the filters, we can ensure the existence of a solution to the constrained eigenfilter design problem for the complementary-synthesis filter. Thus, our approach gives an eigenfilter-based method of designing the complementary filter, given a ldquopredesignedrdquo analysis filter, with the filter lengths satisfying certain conditions. We present several design examples to demonstrate the effectiveness of the method.

Optimal duration-bandwidth localized antisymmetric biorthogonal wavelet filters

We present a design of a new class of compactly supported antisymmetric biorthogonal wavelet filter banks which have the analysis as well as the synthesis filters of even-length. Here, the analysis and the synthesis filters are designed to have minimum joint duration-bandwidth localization (JDBL). The design of filters has been formulated as a direct time-domain linearly constrained eigenvalue problem that does not involve any parametrization and iterations. The optimal analysis and synthesis filters have been obtained as the eigenvectors of the positive definite matrices. The closed form analytic expression for the objective function has been presented. The perfect reconstruction and regularity conditions have been incorporated in the design by employing time-domain matrix characterization. The method can control duration and bandwidth localizations of the analysis and synthesis filters, independently. A few design examples have been presented and compared with previous works. The performance of the optimal filter banks designed by employing the proposed method has been evaluated in image coding and signal denoising applications. HighlightsA design of a new class of even-length antisymmetric biorthogonal filer banks is proposed.The optimality criterion chosen is joint duration-bandwidth localization (JDBL) of the filters.The method presents a direct time-domain approach that does not involve any integrations, iterations and parameterization.The closed form expressions for the objective function and constraints are provided.The method provides globally optimal filters that are obtained as eigenvectors of a positive definite matrix.The performance of the designed filter banks has been evaluated in image compression and signal denoising applications.

Function learning using wavelet neural networks

A new architecture based on wavelets and neural networks is proposed and implemented for learning a class of functions. The performance of such networks is analyzed for function learning. These functions belong to a common class but possess minor variations. The scheme developed makes use of wavelet neural network. It is useful to have a small dimensional network that can approximate a wide class of functions. The network has two levels of freedom. By this the network not only selects the parameters of the basis wavelets but also provides a variation in the choice.

Time-frequency localization optimized biorthogonal wavelets

The time-frequency product of any function in L2 (R) is bounded by the uncertainty principle. This paper presents a method to design linear phase biorthogonal wavelets with the time-frequency localization as the optimality criterion, improving on the previous designs on the same theme. The design philosophy is to optimize the time-frequency product of the wavelet, after fixing the number of vanishing moments of the analysis and synthesis lowpass filters of the corresponding filter bank. The regularity of the discrete filters has also been evaluated.

Design of Time---Frequency Optimal Three-Band Wavelet Filter Banks with Unit Sobolev Regularity Using Frequency Domain Sampling

In this paper, we design three-band time–frequency-localized orthogonal wavelet filter banks having single vanishing moment. We propose new expressions to compute mean and variances in time and frequency from the samples of the Fourier transform of the asymmetric band-pass compactly supported wavelet functions. We determine discrete-time filter of length eight that generates the time–frequency optimal time-limited scaling and wavelet functions using cascade algorithm. Time–frequency product (TFP) of a function is defined as the product of its time variance and frequency variance. The TFP of the designed functions is close to 0.25 with unit Sobolev regularity. Three-band filter banks are designed by minimizing a weighted combination of TFPs of wavelets and scaling functions. Interestingly, empirical results show that time–frequency optimal, filter banks of length nine, designed with the proposed methodology, have unit Sobolev regularity, which is maximum achievable with single vanishing moment. Design examples for length six and length nine filter banks are given to demonstrate the effectiveness of the proposed design methodology.

Vector multirate filtering and matrix filter banks

Multichannel or multi-input multi-output multirate filtering is discussed. Vector multirate systems are defined, and the issues of aliasing cancellation and perfect reconstruction are addressed in the vector context. The conditions for a vector linear periodically time-variant system to become time-invariant are derived. The idea of doubly complementary filters is generalized to the vector context, and one method for constructing doubly complementary filters is outlined. The construction of a vector perfect reconstruction system is briefly described.<<ETX>>