Orthogonal Filter Research Articles

A width-recursive depth-first tree search approach for the design of discrete coefficient perfect reconstruction lattice filter bank

The lattice structure two-channel orthogonal filter bank structurally guarantees the perfect reconstruction (PR) property. Thus, it is eminently suitable for hardware realization even under severe coefficient quantization condition. Nevertheless, its frequency response is still adversely affected by coefficient quantization. In this paper, a novel recursive-in-width depth-first tree search technique is presented for the design of lattice structure PR orthogonal filter banks subject to discrete coefficient value constraint. A frequency-response deterioration measure is developed to serve as a branching criterion. At any node, the coefficient which will cause the largest deterioration in the frequency response of the filter when quantized is selected for branching. The improvement in the frequency response ripple magnitude achieved by our algorithm over that by simple rounding of coefficient values differs widely from example to example ranging from a fraction of a decibel to over 10 dB.

Correlation spectroscopy with diffractive grating synthetic spectra and orthogonal subspace projection filters

We evaluate the feasibility of implementing correlation spec- troscopy for remote sensing with a passive IR sensor that incorporates a diffractive grating on which a synthetic spectrum is imbedded. The spec- tral diffraction reflectances of diffractive gratings are measured and com- pared to the theoretical design spectra and to the library absorbance spectra. Five gratings are evaluated: sulfur hexafluoride (SF6), ((dimethoxyphosphinothioyl)thio)butanedioic acid, diethyl ester (malathion), diisopropyl-methylphosphonate (DIMP), methylphosphono- fluoridic acid,(1-methylethyl) ester (Sarin or GB), and O-ethyl-S-(2- diisopropylamino)ethyl methylphosphonothioate (VX). While the gratings were designed from liquid phase infrared spectra, the measurements and models provide a satisfactory proof of principle for the exploitation of synthetic spectra in correlation spectroscopy. The predicted noise equivalent concentration pathlength (NECL) for detecting GB and VX in a desert environment is computed for distances up to 5 km. The device was found to have high sensitivity in terms of NECL. The selectivity of the grating is found to be poor, but can be improved dramatically with the use of orthogonal subspace projection (OSP) filters that reduce the false alarm (improve selectivity) from misidentifying a GB as VX (and vice versa) at the ''cost'' of a slight increase of the NECL (reduce sensitivity). © 2003 Society of Photo-Optical Instrumentation Engineers. (DOI: 10.1117/1.1531638) Subject terms: correlation spectroscopy; infrared spectroscopy; orthogonal sub- space projection (OSP); vapor detection; false alarm; diffractive optics; synthetic spectra.

Orthogonal Rational Functions: A Transformation Analysis

Finite impulse response (FIR) models are among the most basic tools in control theory and signal processing and are routinely used in almost all fields of application. The connections to orthogonal polynomials are well known. However, infinite impulse response (IIR) models often provide much more compact descriptions and in many cases give improved performance. The objective of this paper is to present a simple framework for the derivation and analysis of orthogonal IIR transfer functions, which are directly related to orthogonal rational functions. Orthogonality simplifies approximation analysis and leads to improved numerical properties. The basic idea is to use a fractional transformation to map the problem to a new domain, where an FIR description is most appropriate. This FIR representation is then mapped back to the original domain to give an orthogonal IIR representation. It is then straightforward to extend many results for FIR models to IIR model structures with arbitrary stable poles; i.e., properties of orthogonal polynomials are easily generalized to orthogonal rational functions. Much of the theory to be presented is classical, e.g., Laguerre and Kautz functions, and we will make use of well-known results in orthogonal filter theory. However, our main contribution is to present a uniform and transparent theory which also covers more novel results that have mainly been presented in the signals, systems, and control literature in the last decade.

Construction of Two-Dimensional Compactly Supported Orthogonal Wavelets Filters with Linear Phase

In this paper, a large class of two-dimensional orthogonal wavelet filters, (lowpass and highpass), are presented in explicit expression. We also characterize the filters with linear phase in this case. Some examples are also given, including non-separable filters with linear phase.

Modeling and prediction of environmental data in space and time using Kalman filtering

The Kalman filter is used in this paper as a framework for space time data analysis. Using Kalman filtering it is possible to include physically based simulation models into the data analysis procedure. Attention is concentrated on the development of fast filter algorithms to make Kalman filtering feasible for high dimensional space time models. The ensemble Kalman filter and the reduced rank square root filter algorithm are briefly summarized. A new algorithm, the partially orthogonal ensemble Kalman filter is introduced too. We will illustrate the performance of the Kalman filter algorithms with a real life air pollution problem. Here ozone concentrations in a part of North West Europe are estimated and predicted.

Superresolution ISAR imaging using 2‐D autoregressive lattice filters

AbstractA new efficient algorithm to obtain high‐resolution ISAR images in the case of limited frequency band and small angular section is presented. The method is based on the two‐dimensional (2‐D) autoregressive modeling of 2‐D Cartesian frequency backscattered data using 2‐D orthogonal lattice filters. ISAR images obtained for an experimental target are included to validate this technique. © 2002 John Wiley & Sons, Inc. Microwave Opt Technol Lett 32: 81–85, 2002.

Realization of an orthogonal filter with hypercomplex coefficients

AbstractThis paper proposes an orthogonal digital filter to simultaneously realize a pair of complex‐coefficient digital filters with power complementarity by means of hypercomplex‐coefficient all‐pass digital filters. Hypercomplex filters can realize equivalent characteristics with only half as great a filter order as complex‐coefficient digital filters and only one‐quarter as great an order as real‐coefficient digital filters. A method of realization of the hypercomplex‐coefficient all‐pass transfer function is discussed as well as a method of determining a transfer function with power complementarity for a certain complex‐coefficient digital filter. Further, an example of the hypercomplex‐coefficient transfer function is used and the effectiveness of the proposed method is shown. © 2001 Scripta Technica, Electron Comm Jpn Pt 3, 85(3): 52–60, 2002

Algorithm for fringe pattern normalization

In this work we present a new algorithm for fringe pattern normalization, that is, background suppression and modulation normalization. Normalization is necessary for several fringe pattern processing techniques. For example, this is the case of the regularization and phase sampling methods. In general, background suppression can be accomplished by high-pass filtering, however if modulation is not constant or almost constant over the field of view, normalization is a difficult task. The solution proposed is based in the use of two orthogonal bandpass filters, from which a normalized irradiance distribution is obtained. We have applied the method to simulated as well as experimental data with good results.

Wavelet packet-based subband adaptive equalization

This paper proposes adaptive algorithms for channel equalization in the wavelet packet transform domain. Adaptation to highly correlated time-varying channels is achieved by the use of the orthogonal tree-structured time-varying filter bank associated to this transform. The filter bank structure, that is, the type of decomposition, is obtained, for a fixed number of bands, as the one that leads to a quasi-optimum convergence rate of the adaptive filters in the different subbands. Then, we propose an NLMS-type adaptive strategy for two possible implementations, a block algorithm, BWPKNLMS, and a non-block approach, WPKNLMS. A theoretical analysis of both schemes is also provided: we obtain the mean squared error after the adaptive algorithm has converged and we find the expression for the optimal solution, to finally show that faster convergence is achieved. Experimental results for the learning curves show the efficiency of the proposed schemes and illustrate the previously presented theoretical results.

On a Class of Finitely Supported Scaling Functions with Properties of Interpolation and Symmetry

In this paper, a class of finitely supported orthogonal filters with interpolation and symmetry is constructed.

A unified algebraic transformation approach for parallel recursive and adaptive filtering and SVD algorithms

In this paper, a unified algebraic transformation approach is presented for designing parallel recursive and adaptive digital filters and singular value decomposition (SVD) algorithms. The approach is based on the explorations of some algebraic properties of the target algorithms' representations. Several typical modern digital signal processing examples are presented to illustrate the applications of the technique. They include the cascaded orthogonal recursive digital filter, the Givens rotation-based adaptive inverse QR algorithm for channel equalization, and the QR decomposition-based SVD algorithms. All three examples exhibit similar throughput constraints. There exist long feedback loops in the algorithms' signal flow graph representation, and the critical path is proportional to the size of the problem. Applying the proposed algebraic transformation techniques, parallel architectures are obtained for all three examples. For cascade orthogonal recursive filter, retiming transformation and orthogonal matrix decompositions (or pseudo-commutativity) are applied to obtain parallel filter architectures with critical path of five Givens rotations. For adaptive inverse QR algorithm, the commutativity and associativity of the matrix multiplications are applied to obtain parallel architectures with critical path of either four Givens rotations or three Givens rotations plus two multiply-add operations, whichever turns out to be larger. For SVD algorithms, retiming and associativity of the matrix multiplications are applied to derive parallel architectures with critical path of eight Givens rotations. The critical paths of all parallel architectures are independent of the problem size as compared with being proportional to the problem size in the original sequential algorithms. Parallelism is achieved at the expense of slight increase (or the same for the SVD case) in the algorithms' computational complexity.

Adaptive 1-D and 2-D Filtering with Internally Passive Structures

Summary Internally passive filters such as wave digital filters (WDF) and orthogonal filters, which have desirable properties, are developed as filter structures for adaptive filters. The equations for updating the multiplier coefficients as well as for computing the gradients are derived for these filters. The stability monitoring during the adaptation process for these filters is simple to implement. Simulation results are included to demonstrate the capability of these filters in system identification and noise cancellation. Moreover, motivated by applications of multidimensional adaptive filters in video compression and image enhancement, the proposed adaptive wave digital filters have been extended to two dimensions and applied as adaptive fan filters. Computer simulations are also conducted to show their potential applications to two-dimensional system identification and image enhancement.

A tutorial on modern lossy wavelet image compression: foundations of JPEG 2000

One of the purposes of this article is to give a general audience sufficient background into the details and techniques of wavelet coding to better understand the JPEG 2000 standard. The focus is on the fundamental principles of wavelet coding and not the actual standard itself. Some of the confusing design choices made in wavelet coders are explained. There are two types of filter choices: orthogonal and biorthogonal. Orthogonal filters have the property that there are energy or norm preserving. Nevertheless, modern wavelet coders use biorthogonal filters which do not preserve energy. Reasons for these specific design choices are explained. Another purpose of this article is to compare and contrast "early" wavelet coding with "modern" wavelet coding. This article compares the techniques of the modern wavelet coders to the subband coding techniques so that the reader can appreciate how different modern wavelet coding is from early wavelet coding. It discusses basic properties of the wavelet transform which are pertinent to image compression. It builds on the background material in generic transform coding given, shows that boundary effects motivate the use of biorthogonal wavelets, and introduces the symmetric wavelet transform. Subband coding or "early" wavelet coding method is discussed followed by an explanation of the EZW coding algorithm. Other modern wavelet coders that extend the ideas found in the EZW algorithm are also described.

On the Construction and Frequency Localization of Finite Orthogonal Quadrature Filters

We introduce a new method to construct finite orthogonal quadrature filters using convolution kernels and show that every filter with value 1 at the origin can be obtained from an even nonnegative kernel. We apply the method to estimate the optimal frequency localization of finite filters. The frequency localization γp of a finite filter m0 is given by the distance in Lp-norm between |m0|2 and the Shannon low-pass filter. For each N>0 there is a filter mN0 of length 2N minimizing the value of γp. We prove that for such a minimizing sequence we have γpp(mN0)=O(1/N), 1⩽p⩽2, and this estimate is optimal. We construct several new families of both MRA and non-MRA filters with optimal asymptotic frequency localization.

Uniform Analytic Construction of Wavelet Analysis Filters Based on Sine and Cosine Trigonometric Functions

Based on sine and cosine functions, the compactly supported orthogonal wavelet filter coefficients with arbitrary length are constructed for the first time. When N = 2k−1 and N = 2k, the unified analytic constructions of orthogonal wavelet filters are put forward, respectively. The famous Daubechies filter and some other well-known wavelet filters are tested by the proposed novel method which is very useful for wavelet theory research and many application areas such as pattern recognition.

Permutation Based Design of Orthogonal Block Transformsand Filter Banks

In this work, a new efficient design technique for orthogonal block transforms, lapped orthogonal transforms and 4-channel perfect reconstruction subband filter banks is developed. The technique consists of permutation and sign change operations on a reference vector. This approach can be thought of as a generalization of the Hadamard transform in the sense that the reference vector {\mbox{ h }}_{0} (which will be a prototype low-pass filter also forming one of the basis functions of the transform) will in general have components that are not identically 1's. The design technique, a constructive method based on Hadamard arrays, provides a convenient means to explore new transforms. The merit of our method is that the number of unknowns and equality constraints are both reduced significantly which render the design procedure much more feasible while guaranteeing at the same time linear phase.

Characterisation of symmetric∕antisymmetric orthonormal length-4 multi-filters

It is shown that any symmetric/antisymmetric orthonormal multi-filter of length-4 with multiplicity 2 can be derived from an orthogonal scalar wavelet filter of length-4 or length-8, if the refinement mask of the multi-filter at the origin is diagonal with entries 1 and 0.

Constraint-selected and search-optimized families of Daubechies wavelet filters computable by spectral factorization

A unifying algorithm has been developed to systematize the collection of compact Daubechies wavelets computable by spectral factorization of a symmetric positive polynomial. This collection comprises all classes of real and complex orthogonal and biorthogonal wavelet filters with maximal flatness for their minimal length. The main algorithm incorporates spectral factorization of the Daubechies product filter into analysis and synthesis filters. The spectral factors are found for search-optimized families by examining a desired criterion over combinatorial subsets of roots indexed by binary codes, and for constraint-selected families by imposing sufficient constraints on the roots without any optimizing search for an extremal property. Daubechies wavelet filter families have been systematized to include those constraint-selected by the principle of separably disjoint roots, and those search-optimized for time-domain regularity, frequency-domain selectivity, time-frequency uncertainty, and phase nonlinearity. The latter criterion permits construction of the least and most asymmetric and least and most symmetric real and complex orthogonal filters. Biorthogonal symmetric spline and balanced-length filters with linear phase are also computable by these methods. This systematized collection has been developed in the context of a general framework enabling evaluation of the equivalence of constraint-selected and search-optimized families with respect to the filter coefficients and roots and their characteristics. Some of the constraint-selected families have been demonstrated to be equivalent to some of the search-optimized families, thereby obviating the necessity for any search in their computation.

2D FIR Wiener filter realisation using orthogonallattice structures

A new realisation algorithm for 2D FIR Wiener filters is proposed. The reduced-order 2D orthogonal lattice filter structure is used as the principal component as in the 1D case. A numerical example is included to verify the formulation.

Pipelined CORDIC-based cascade orthogonal IIR digital filters

CORDIC-based cascade orthogonal infinite-impulse response (IIR) digital filters possess desirable properties for VLSI implementations such as local connection, regularity, absence of limit cycle and overflow oscillations, and good finite word-length behavior. However, the achievable sample rate of these filters is limited, since these structures cannot be pipelined at finer levels (such as bit or multi-bit level) due to the presence of feedback loops. In this paper, we present a novel approach to design pipelined CORDIC-based cascade orthogonal IIR digital filters using the transfer function approach. We first present a systematic way to synthesize cascade orthogonal IIR digital filters using scalar lossless inverse scattering theory, and realize the filter transfer function as a cascade inter-connection of orthogonal sections where each section implements one real zero or a pair of complex conjugate zeroes of the transfer function. In this way, the filter achieves low sensitivity over the entire filter spectrum. Novel pipelining techniques for both coarse-grain and fine-grain pipelining of these filters are then proposed. In coarse-grain pipelining, we present a novel method based on retiming and orthogonal matrix decomposition techniques which can increase the maximum filter sample rate to O(1) level which is independent of the filter order. In fine-grain pipelining, we present a novel method based on constraint filter design and polyphase decomposition techniques which could increase the maximum filter sample rate to any desired level. The proposed architecture for coarse-grain pipelining consists of only Givens rotations, and the one for fine-grain pipelining consists of only Givens rotations and a few additions. Both architectures can be realized using CORDIC arithmetic-based processors. Finally, finite word-length simulations are carried out to compare the performance of different topologies.