Orthogonal Filter Research Articles

Efficient wavelet adaptation for hybrid wavelet–large margin classifiers

Hybrid wavelet–large margin classifiers have recently proven to solve difficult signal classification problems in cases where solely using a large margin classifier like, e.g., the Support Vector Machine may fail. In this paper, we evaluate several criteria rating feature sets obtained from various orthogonal filter banks for the classification by a Support Vector Machine. Appropriate criteria may then be used for adapting the wavelet filter with respect to the subsequent support vector classification. Our results show that criteria which are computationally more efficient than the radius-margin Support Vector Machine error bound are sufficient for our filter adaptation and, hence, feature selection. Further, we propose an adaptive search algorithm that, once the criterion is fixed, efficiently finds the optimal wavelet filter. As an interesting byproduct we prove a theorem which allows the computation of the radius of a set of vectors by a standard Support Vector Machine.

An orthogonal family of quincunx wavelets with continuously adjustable order

We present a new family of two-dimensional and three-dimensional orthogonal wavelets which uses quincunx sampling. The orthogonal refinement filters have a simple analytical expression in the Fourier domain as a function of the order lamda, which may be noninteger. We can also prove that they yield wavelet bases of L2(R2) for any lambda > 0. The wavelets are fractional in the sense that the approximation error at a given scale a decays like O(a(lamda)); they also essentially behave like fractional derivative operators. To make our construction practical, we propose a fast Fourier transform-based implementation that turns out to be surprisingly fast. In fact, our method is almost as efficient as the standard Mallat algorithm for separable wavelets.

Design of ODMA Digital Waveforms Using Non-Convex Optimization Methods

This paper presents a new approach for the design of a set of orthogonal digital filters which can be used as message-carrying waveforms in orthogonality division multiple access (ODMA) communication systems. The filter set design problem is formulated as a constrained L 2 space optimization problem. For bandwidth efficiency, all the digital filters in the set are constrained to have approximately the same desired spectral shapes in a prescribed passband; To minimize intersymbol interference and co-channel interference, all the digital filters in the set are constrained to have low values of auto-correlation at nonzero translates of multiple symbol interval and low values of cross-correlation at all translates of multiple symbol interval. Methods for solving the proposed non-convex optimization problem are outlined. Numerical results are presented to illustrate the usefulness of the proposed method.

Linear Boundary Extensions for Finite Length Signals and Paraunitary Two-Channel Filterbanks

In this paper, we introduce a novel and general matrix formulation of artificial linear boundary extension methods for removing border effects inherent to any paraunitary two-channel size-limited filterbank. This new characterization of the transformation operator allows us to prove that perfect reconstruction (PR) of finite signals can be ensured under some conditions without using extra subband coefficients; in other words, we characterize the signal extension methods that lead to nonexpansive transforms. The necessary and sufficient condition we find allows us to show that some traditional extension techniques that are being used in an expansive way, such as the polynomial extension, lead in fact to nonexpansive invertible transforms; moreover, we can also prove that in contradiction to previous literature, not every transformation matrix associated with a linear extension is invertible even if using prototype filters of the same length. Apart from these invertibility criteria, we propose the first algorithm for the design of all linear extensions and their associated biorthogonal boundary filters that lead to nonexpansive and invertible transforms. Analogously, we provide the first method for the design of all linear extensions that yield orthogonal transforms: We construct an infinite number of orthogonal extensions, apart from the commonly used periodic extension, and their associated orthogonal boundary filters. The final contribution of the paper is a new algorithm for the design of smooth orthogonal extensions, which keep the orthogonality property and overcome the main drawback of periodization, that is, the introduction of subband coefficients of great amplitude near the boundaries in the transform domain.

The Schur algorithm applied to the design of optical multi‐mirror structures

AbstractWe show how to use the classical Schur algorithm to design multi‐mirror optical interferometers (or filters). Our derivation is simple and straightforward, clearly revealing its connection to the previously known orthogonal digital filter structures. We also give a complete detailed description of an FFT‐based algorithm for the reciprocal polynomial approximation of an arbitrary curve (or spectrum). The Schur algorithm can, in turn, be applied to the obtained polynomial to get the desired reflection coefficients of the mirrors. Copyright © 2004 John Wiley & Sons, Ltd.

Multidimensional Nonlinear Schur Parametrization of NonGaussian Stochastic Signals Part Three: Low-Complexity Staircase Solution

In the first part l2r of this set of papers we stated the multidimensional nonlinear Schur parametrization problem for higher-order stochastic sequences. In the second part l1r we proposed the recursive solution to this problem, deriving the general multidimensional nonlinear Schur parametrization algorithm. The goal of this paper is to introduce a low-complexity solution to the nonlinear Schur parametrization problem, following from a multidimensional generalization of the two-indexed matrix extension problem. To obtain the solution, we derive a global multidimensional nonlinear Schur algorithm, then formulate and prove generalized staircase extension and interpolation theorems. The obtained results allow to achieve a considerable complexity reduction of the Schur parametrization algorithms for higher-order stochastic sequences as well as of orthogonal nonlinear approximate filters (ofb band-structure) of the Volterra–Wiener class.

A novel short-window algorithm of phasor computation for EHV line protection

A novel phasor computation algorithm is presented in this paper. The real and imaginary part filters of this algorithm have identical magnitude frequency characteristic (IMFC) at all the frequencies and orthogonal phase frequency characteristic (PFC) at fundamental frequency. Considering the conditions of the same magnitude frequency characteristic (MFC) and the orthogonal PFC, the two-sample algorithm is derived. Once two-sample orthogonal filters with IMFC are obtained, the algorithm design is just to select proper shapes and lengths of windows, which determine the filtering performance and response-time, respectively. This method could be applied to design filter in any window-length, especially in short-window. Therefore, this class of phasor algorithms is suitable for EHV transmission line protection where a short-window is crucial for fast response and better operation performance. The proposed algorithm has been applied to a series of EHV relay devices by China Xuchang Relay Group.

Decision-Feedback Multiuser Detection With Orthogonal Feedforward Transformation

Our proposed orthogonal decision feedback detector (ODFD) contains an orthogonal feedforward filter and a decision feedback structure. Compared with the decorrelating decision feedback detector (DDFD), our QR-factorization-based ODFD is computationally more efficient in dynamically updating the feedforward and feedback filter pairs. This feature is beneficial when the number of users in a system changes frequently. Using the Gaussian approximation, which was used to compute the approximate bit-error probabilities (BEPs) of multiusers for the successive interference canceller (SIC), we calculate the approximate BEPs of multiusers for the DDFD and our ODFD. The Gaussian approximation leads to good approximate BEPs for DDFD and to fine approximate BEPs for ODFD, but to poor approximate BEPs for SIC. The accuracy of these approximate BEPs is also justified by the results from Monte Carlo simulations.

MODELING AND PREDICTIVE CONTROL OF MIMO NONLINEAR SYSTEMS USING WIENER-LAGUERRE MODELS

In this work, a Weiner-type nonlinear black box model was developed for capturing dynamics of open loop stable MIMO nonlinear systems with deterministic inputs. The linear dynamic component of the model was parameterized using orthogonal Laguerre filters while the nonlinear state output map was constructed either using quadratic polynomial functions or artificial neural networks. The properties of the resulting model, such as open loop stability and steady-state behavior, are discussed in detail. The identified Weiner-Laguerre model was further used to formulate a nonlinear model predictive control (NMPC) scheme. The efficacy of the proposed modeling and control scheme was demonstrated using two benchmark control problems: (a) a simulation study involving control of a continuously operated fermenter at its optimum (singular) operating point and (b) experimental verification involving control of pH at the critical point of a neutralization process. It was observed that the proposed Weiner-Laguerre model is able to capture both the dynamic and steady-state characteristics of the continuous fermenter as well as the neutralization process reasonably accurately over wide operating ranges. The proposed NMPC scheme achieved a smooth transition from a suboptimal operating point to the optimum (singular) operating point of the fermenter without causing large variation in manipulated inputs. The proposed NMPC scheme was also found to be robust in the face of moderate perturbation in the unmeasured disturbances. In the case of experimental verification using the neutralization process, the proposed control scheme was found to achieve much faster transition to a set point close to the critical point when compared to a conventional gain-scheduled PID controller.

Pipelined CORDIC-Based State-Space Orthogonal Recursive Digital Filters Using Matrix Look-Ahead

Orthogonal recursive or infinite impulse response (IIR) digital filters can achieve a sharp transition band, have good finite word-length behavior, and are used in many modern digital signal processing (DSP) applications such as mobile communications. However, Givens rotation or coordinate rotation digital computer (CORDIC)-based fine-grain pipelined true orthogonal recursive digital filters have yet to be developed. In this paper, a state-space approach-based novel algorithm for designing fine-grain pipelined true orthogonal recursive digital filters is proposed using the matrix look-ahead technique. The algorithm is developed for designing general multi-input/multi-output (MIMO) digital filters, whereas the single-input/single-output (SISO) filters are treated as special cases. The filter synthesis procedure contains five major steps and only involves applying orthogonal transformations that are known to be numerically very reliable and, therefore, is ideal for very large scale integration (VLSI) implementations. The proposed filter architectures are pipelined at fine-grain level and, thus, can be operated at arbitrarily high sample rates. The total complexity is MN(m+p)+(m+N)p+p(p-1)/2 Givens rotations for MIMO filters, where N, m, p, and M are the filter order, number of inputs, number of outputs, and pipelining level, respectively. For the SISO case, the complexity reduces to (2M+1)N+1 Givens rotations, which is linear with respect to the filter order and pipelining level. Furthermore, the filter realizations consist of only Givens rotations, which can be mapped onto CORDIC arithmetic-based processors. Different filter design and realization approaches are explored, and the resulting topologies are compared. As an application, a pipelined intermediate frequency filter for an American mobile telephone system is designed using the proposed approach. Finally, finite word-length simulations are carried out for various orthogonal topologies. It is shown both theoretically and numerically how the proposed orthogonal filter realizations achieve low sensitivity due to finite word-length truncations.

Multidimensional Nonlinear Schur Parametrization of NonGaussian Stochastic Signals, Part Two: Generalized Schur Algorithm

In the first part l1r of this paper we stated the multidimensional nonlinear Schur parametrization problem for higher-order (and non-Gaussian) stochastic sequences as a generalization of the linear prediction/innovations filter problem for second-order stochastic sequences. The subject of this paper is a multidimensional nonlinear generalization of the linear Schur parametrization procedure for second-order stochastic sequences. We state and solve the problem geometrically, in the space of generalized (multi-indexed) matrices. We derive multidimensional nonlinear Levinson and Schur recursions, resulting in a complete parametrization schema for higher-order nonstationary stochastic sequences and show its connection to solution the non-linear orthogonal approximate filter problem of the Volterra–Wiener class. In the third part l2r we propose a low-complexity nonlinear Schur parametrization procedure, following from a multidimensional nonlinear generalization of the staircase matrix extension problem.

Multidimensional Nonlinear Schur Parametrization of NonGaussian Stochastic Signals. Part One: Statement of the Problem

The subject of this three-part paper is the multidimensional nonlinear Schur parametrization problem for higher-order (and non-Gaussian) stochastic signals. In the first part of this paper we state the nonlinear orthogonal parametrization problem as a generalization of the linear prediction/innovations filter problem. A unified geometric approach to the problem is introduced in the following three isometrically isomorphic spaces: of random variables (of higher-order stochastic sequences), of multi-indexed matrices, and of multidimensional z-polynomials. The nonlinear orthogonal approximate filters of the Volterra–Wiener class are considered. In the second part l30r of this paper we derive a generalized multidimensional nonlinear Schur parametrization algorithm for higher-order nonstationary stochastic sequences. The third part l33r of the paper is devoted to the complexity reduction problem in the nonlinear case, where a low-complexity Schur parametrization procedure, following from a nonlinear generalization of the staircase matrix extension problem l13r, is proposed.

WAVELET-BASED MEDICAL IMAGE COMPRESSION USING EZW: OBJECTIVE AND SUBJECTIVE EVALUATIONS

Recently, the wavelet transform has emerged as a cutting edge technology within the field of image compression research. Wavelet methods involve overlapping transforms with varying-length basis functions. This overlapping nature of the transform alleviates blocking artifacts, while the multi-resolution character of the wavelet decomposition leads to superior energy compaction and perceptual quality of the decompressed image. Embedded zerotree wavelet (EZW) coder is the first algorithm to show the full power of wavelet-based image compression. The main purpose of this paper is to investigate the impact and quality of orthogonal wavelet filter in compressing medical image by using EZW. Meanwhile, we also look into the effect of the level of wavelet decomposition towards compression efficiency. The wavelet filters used are Haar and Daubechies. The compression simulations are done on three modalities of medical images. The objective (based on PSNR) and subjective (perceived image quality) results of these simulations are presented.

Fast fractal coding of subbands using a non-iterative block clustering

We propose a new image coding scheme which uses a fast non-iterative algorithm for block clustering in fractal coding of wavelet transform coefficients. The original image is first decomposed into subbands containing information in different spatial directions and different scales, using an orthogonal wavelet filter bank. Subbands are encoded using local iterated function systems (LIFS) with range and domain blocks presenting horizontal or vertical directionalities. Their sizes are adapted to the correlation lengths and the resolution of each subband. This hybrid compression scheme allows to use the natural classification obtained by using a subband decomposition and reduce the computational complexity by a 12:1 factor. In order to increase this speedup, we propose a fast non-iterative algorithm for block clustering which classifies the directional domain and range blocks and leads to an accelerated LIFS generation in each subband. The proposed approach reduces the computational complexity of the basic fractal compression algorithm by a 50:1 factor and gives better PSNR/bit rate results, especially for images presenting directionalities.

Orthogonal and projected orthogonal matched filter detection

This paper considers the generic problem of detecting in the presence of additive noise, which one from a set of known signals has been received. In place of the classical matched filter (MF) receiver we propose a modified receiver. When the transmitted signals are linearly independent this receiver is referred to as an orthogonal matched filter (OMF) receiver, and when the transmitted signals are linearly dependent it is referred to as a projected orthogonal matched filter (POMF) receiver. Two equivalent representations of the receiver are developed with different implications in terms of implementation. In the first, the demodulator consists of a MF demodulator followed by an optimal whitening transformation on a space formed by the transmitted signals, that optimally decorrelates the MF outputs prior to detection. In the second, the demodulator consists of a bank of correlators with correlating signals that are projections of a set of orthogonal signals, and are closest in a least-squares sense to the transmitted signals. We provide simulation results that suggest that in certain cases the OMF and POMF receivers can significantly increase the probability of correct detection over the MF receiver in non-Gaussian noise with only a minor impact on performance in Gaussian noise.

Parameterization of 3-channel non-separable 2-D wavelets and filters

We propose a complete parameterization presentation of the 3-channel bivariate non-separable orthogonal FIR filter, and describe the sufficient condition of generating continuous wavelet bases. Given the results above, a non-separable, compactly supported, orthogonal, continuous parameterized bivariate wavelet bank is set up here.

Signal extension for orthogonal filter banks and its application to image coding

AbstractIn this paper, a signal extension method is proposed for realization of nonextension convolution in a two‐channel orthogonal FIR filter bank. Methods of performing nonextension convolution include the symmetric extension method for a linear phase filter bank and the periodic extension method applicable to an arbitrary filter bank. However, the periodic extension method is not appropriate for image compression because it becomes discontinuous at the signal edge. In this paper, for application to an orthogonal filter bank, the input signal of the decomposed filter bank is extended in such a way that the input signal in the synthesized filter bank has symmetry similar to that in the symmetric extension method. In this way, nonextension convolution is realized. In this case, it is shown that there are degrees of freedom in the extension signals determined by the input signal and the coefficients of the decomposed filter bank. The extended signal is calculated by singularity decomposition in such a way that the extended signal becomes smooth. Finally, the proposed method is applied to the coding of static images and the effectiveness of the method is demonstrated. © 2003 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 87(3): 41–48, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.10125

Framing pyramids

Burt and Adelson (1983) introduced the Laplacian pyramid (LP) as a multiresolution representation for images. We study the LP using the frame theory, and this reveals that the usual reconstruction is suboptimal. We show that the LP with orthogonal filters is a tight frame, and thus, the optimal linear reconstruction using the dual frame operator has a simple structure that is symmetric with the forward transform. In more general cases, we propose an efficient filterbank (FB) for the reconstruction of the LP using projection that leads to a proved improvement over the usual method in the presence of noise. Setting up the LP as an oversampled FB, we offer a complete parameterization of all synthesis FBs that provide perfect reconstruction for the LP. Finally, we consider the situation where the LP scheme is iterated and derive the continuous-domain frames associated with the LP.

On spectral analysis using models with pre-specified zeros

The fundamental theory of Lindquist and co-workers on the rational covariance extension problem provides a very elegant framework for ARMA spectral estimation. Here the choice of zeros is completely arbitrary, and can be used to tune the estimator. An alternative approach to ARMA model estimation with pre-specified zeros is to use a prediction error method based on generalizing autoregressive (AR) modeling using orthogonal rational filters . Here the motivation is to reduce the number of parameters needed to obtain useful approximate models of stochastic processes by suitable choice of zeros, without increasing the computational complexity.The objective of this contribution is to discuss similarities and differences between these two approaches to spectral estimation.

Adaptive and Intelligent Systems for Generator Monitoring and Protection Purposes

New adaptive scheme for signal parameter measurement for generator monitoring and protection is presented. The algorithm is based on the orthogonal components of currents and voltages obtained with adaptive orthogonal filters. The filter data window length and coefficients are adapted according to coarsely estimated signal frequency. As a result, the measurements can be performed with high accuracy in wide frequency band conditions, i.e. also for generator start-up, etc. The measured signals are further used for training of artificial neural networks with the aim to design a robust and effective classifier of generator operation mode. The main attention is paid to the phenomena accompanying pole slipping and out-of-step conditions.