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ADVANCES IN IMAGING AND ELECTRON PHYSICS VOLUME 125

EDITOR-IN-CHIEF

PETER W. HAWKES CEMES-CNRS Toulouse, France

ASSOCIATE EDITORS

BENJAMIN KAZAN Xerox Corporation Palo Alto Research Center Palo Alto, California

TOM MULVEY Department of Electronic Engineering and Applied Physics Aston University Birmingham, United Kingdom

Advances in

Imaging and Electron Physics Edited by

PETER W. HAWKES CEMES-CNRS Toulouse, France

VOLUME 125

Amsterdam Boston London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo

∞ This book is printed on acid-free paper. C 2002, Elsevier Science (USA). Copyright

All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the Publisher. The appearance of the code at the bottom of the first page of a chapter in this book indicates the Publisher’s consent that copies of the chapter may be made for personal or internal use of specific clients. This consent is given on the condition, however, that the copier pay the stated per copy fee through the Copyright Clearance Center, Inc. (www.copyright.com), for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Copy fees for pre-2002 chapters are as shown on the title pages. If no fee code appears on the title page, the copy fee is the same as for current chapters. 1076-5670/2002 $35.00 Explicit permission from Academic Press is not required to reproduce a maximum of two figures or tables from an Academic Press chapter in another scientific or research publication provided that the material has not been credited to another source and that full credit to the Academic Press chapter is given.

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1

CONTENTS

Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Future Contributions . . . . . . . . . . . . . . . . . . . . . .

ix xi xiii

An Algebraic Approach to Subband Signal Processing Marilena Barnabei and Laura B. Montefusco

I. II. III. IV. V. VI. VII. VIII. IX.

Introduction . . . . . . . . . . . An Overview of Recursive Matrices . Recursive Operators of Filter Theory More Algebraic Results . . . . . . Analysis and Synthesis Filter Banks . M-Channel Filter Bank Systems . . Transmultiplexers . . . . . . . . The M-Band Lifting . . . . . . . Conclusion . . . . . . . . . . . References . . . . . . . . . . .

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1 4 13 22 30 37 44 48 60 61

Background . . . . . . . . . . . . . . . . . . . . . Fundamentals . . . . . . . . . . . . . . . . . . . . ALCHEMI Results . . . . . . . . . . . . . . . . . . Predicting Sublattice Occupancies. . . . . . . . . . . . Competing (or Supplementary) Techniques . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . Current Challenges and Future Directions (a Personal View). References . . . . . . . . . . . . . . . . . . . . .

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63 64 77 100 102 106 106 107

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120 122 141 149

Determining the Locations of Chemical Species in Ordered Compounds: ALCHEMI I. P. Jones

I. II. III. IV. V. VI. VII.

Aspects of Mathematical Morphology K. Michielsen, H. De Raedt, and J. Th. M. De Hosson

I. II. III. IV.

Introduction . . . . . . . . Integral Geometry: Theory . . Integral Geometry in Practice. Illustrative Examples . . . .

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V. Computer Tomography Images of Metal Foams . VI. Summary . . . . . . . . . . . . . . . . . Appendix A: Algorithm . . . . . . . . . . . Appendix B: Programming Example (Fortran 90) Appendix C: Derivation of Eq. (36) . . . . . . Appendix D: Proof of Eq. (56) . . . . . . . . Appendix E: Proof of Eq. (57) . . . . . . . . References . . . . . . . . . . . . . . . .

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170 176 176 178 182 184 188 190

Introduction . . . . . . . . . . . . . . . . . . . . . . . History . . . . . . . . . . . . . . . . . . . . . . . . . Ultrafast Scanning Probe Microscopy . . . . . . . . . . . . Junction Mixing STM . . . . . . . . . . . . . . . . . . . Distance Modulated STM . . . . . . . . . . . . . . . . . Photo-Gated STM . . . . . . . . . . . . . . . . . . . . Ultrafast STM by Direct Optical Coupling to the Tunnel Junction Conclusions . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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195 196 199 205 216 218 225 228 228

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232 237 252 270 291 317 325 327 336 340 349

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . II. Electron Diffraction and Diffraction Contrast . . . . . . . . . . .

356 358

Ultrafast Scanning Tunneling Microscopy G. M. Steeves and M. R. Freeman

I. II. III. IV. V. VI. VII. VIII.

Low-Density Parity-Check Codes—A Statistical Physics Perspective Renato Vicente, David Saad, and Yoshiyuki Kabashima

I. II. III. IV. V. VI. VII.

Introduction . . . . . . . . . . . . . . . Coding and Statistical Physics . . . . . . . Sourlas Codes . . . . . . . . . . . . . . Gallager Codes . . . . . . . . . . . . . MacKay–Neal Codes . . . . . . . . . . . Cascading Codes . . . . . . . . . . . . . Conclusions and Perspectives. . . . . . . . Appendix A. Sourlas Codes: Technical Details Appendix B. Gallager Codes: Technical Details Appendix C. MN Codes: Technical Details . . References . . . . . . . . . . . . . . .

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Computer-Aided Crystallographic Analysis in the TEM Stefan Zaefferer

CONTENTS

vii

III. A Universal Procedure for Orientation Determination from Electron Diffraction Patterns . . . . . . . . . . . . . . . . . . . . . IV. Automation of Orientation Determination in TEM . . . . . . . . . V. Characterization of Grain Boundaries . . . . . . . . . . . . . . VI. Determination of Slip Systems . . . . . . . . . . . . . . . . . VII. Phase and Lattice Parameter Determination . . . . . . . . . . . . VIII. Calibration . . . . . . . . . . . . . . . . . . . . . . . . . IX. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

369 377 391 397 403 408 411 413

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CONTRIBUTORS

Numbers in parentheses indicate the pages on which the author’s contributions begin.

Marilena Barnabei (1), Department of Mathematics, University of Bologna, I-40127 Bologna, Italy M. R. Freeman (195), Department of Physics, University of Alberta, Edmonton, AB T6G 2J1, Canada J. Th. M. De Hosson (119), Department of Applied Physics, Materials Science Centre and Netherlands Institute for Metals Research, University of Groningen, NL-9747 AG Groningen, The Netherlands H. De Raedt (119), Institute for Theoretical Physics, Materials Science Center, University of Groningen, NL-9747 AG Groningen, The Netherlands I. P. Jones (63), Center for Electron Microscopy, University of Birmingham, Birmingham B15 2TT, United Kingdom Yoshiyuki Kabashima (231), Department of Computational Intelligence and Systems Science, Tokyo Institute of Technology, Yokohama 2268502, Japan K. Michielsen (119), Institute for Theoretical Physics, Materials Science Center, University of Groningen, NL-9747 AG Groningen, The Netherlands Laura B. Montefusco (1), Department of Mathematics, University of Bologna, I-40127 Bologna, Italy David Saad (231), Neural Computing Research Group, University of Aston, Birmingham B4 7ET, United Kingdom G. M. Steeves (195), California NanoSystems Institute, University of California, Santa Barbara, California 931006 Renato Vicente (231), Neural Computing Research Group, University of Aston, Birmingham B4 7ET, United Kingdom Stefan Zaefferer (355), Max Plank Institute for Iron Research, D-40237 Düsseldorf, Germany ix

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PREFACE

The contributions to this latest volume of the Advances extend over a wide range of themes: subband signal processing, electron microscopy and diffraction, mathematical morphology, fast scanning tunneling microscopy and coding theory. The volume opens with a presentation by M. Barnabei and L. B. Montefusco of the algebraic theory, largely initiated and developed by them, that makes it possible to study filter banks systematically from the time-domain viewpoint. For this, a knowledge of recursive matrix theory is needed and this is provided in the opening paragraphs. Since banded recursive matrices can be described by two Laurent polynomials, these are likewise defined and their properties presented in the introductory sections. The authors then examine the connection between banded recursive matrices and some of the linear filters encountered in filter theory. In the remainder of the chapter, these notions are used to investigate filter bank systems in great detail. In the second chapter, I. P. Jones describes how the locations of chemical sites in ordered compounds are determined by channeling-enhanced microanalysis, the technique that is now universally referred to as ALCHEMI. In this short and complete monograph on the subject, the author first explains how the technique works, considers the accuracy to be expected and investigates related questions. In the long section that follows, results obtained by ALCHEMI are tabulated for crystals of various kinds; these tables are an invaluable source of information and include the references in which the results were first described. I. P. Jones concludes with a personal view of current challenges and future directions, to which he will, I hope, return one day in these Advances. Mathematical morphology appears frequently in these pages but the aspect of this subject considered by K. Michielsen, H. de Raedt, and J. Th. M. de Hosson has not been discussed here before. They present an approach to morphological image analysis whereby the shape and connectivity patterns of the individual pixels of an image are described by additive image functionals. It is known that the number of such functionals is three in the case of two-dimensional images and four in the three-dimensional case. The authors present the basic mathematics of these ideas in full detail, from which the newcomer to this aspect of mathematical morphology can learn all the necessary background. They then turn to a number of applications, which demonstrate just how useful the theory is; the case of metal foams is explored in full. In the fourth chapter, by G. M. Steeves and M. R. Freeman, we return to microscopy: a form of scanning probe microscopy that allows extremely rapid xi

xii

PREFACE

changes to be followed. This is very useful for the study of electron transport in transistors or nanowires, for example, or in DNA or for the observation of transitions in magnetic materials. The authors describe the various stages through which the technique has passed on its way to the remarkable performances being achieved today. Many examples are included, from which potential users of these instruments can assess the suitability of ultra-fast STM to address their own problems. Digitized images are large, commonly redundant data-sets and they frequently have to be transmitted along noisy channels. Somewhat surprisingly, despite all the effort that has been put into concatenated and Turbo codes, it is the low-density parity-check codes that have reached the best performance and these are the subject of the chapter by R. Vicente, D. Saad and Y. Kabashima. After extensive introductory material, which will enable the reader to comprehend the present situation in coding theory, the authors examine in detail four of the low-density parity-check codes, which perform particularly impressively: the Sourlas, Gallagher, MacKay–Neal and cascading codes. This chapter is an invaluable introduction to modern thinking in coding methods. This brings us to the final chapter, in which S. Zaefferer discusses automation of the interpretation of electron diffraction patterns. The existence of programs for performing this task is transforming diffraction pattern analysis, which daunted all but the experts in the past. Here, the author covers eight topics in detail: the determination of crystal orientation from single-crystal diffraction patterns; automatic evaluation of spot and Kikuchi patterns; techniques for determining orientation based on dark-field images or Debye–Scherrer ring patterns; simulation of diffraction patterns; grain boundary characterization; determination of the Burgers vector, crystallographic line direction, slip plane and nature of dislocations; phase identification and lattice-parameter fitting; and calibration of microscope operating characteristics (camera length, accelerating voltage and rotation). All this is extensively illustrated. In conclusion, I thank all the contributors for the care they have brought to their chapters and their efforts to ensure that their material, however complicated, is accessible to newcomers to the subject. Peter Hawkes

FUTURE CONTRIBUTIONS

T. Aach Lapped transforms G. Abbate New developments in liquid-crystal-based photonic devices S. Ando Gradient operators and edge and corner detection A. Arnéodo, N. Decoster, P. Kestener and S. Roux (vol. 126) A wavelet-based method for multifractal image analysis C. Beeli (vol. 127) Structure and microscopy of quasicrystals I. Bloch Fuzzy distance measures in image processing G. Borgefors Distance transforms B. L. Breton, D. McMullan and K. C. A. Smith (Eds) Sir Charles Oatley and the scanning electron microscope A. Bretto Hypergraphs and their use in image modelling Y. Cho (vol. 127) Scanning nonlinear dielectric microscopy E. R. Davies (vol. 126) Mean, median and mode filters H. Delingette Surface reconstruction based on simplex meshes A. Diaspro (vol. 126) Two-photon excitation in microscopy R. G. Forbes Liquid metal ion sources

xiii

xiv


E. Förster and F. N. Chukhovsky X-ray optics A. Fox The critical-voltage effect L. Frank and I. Müllerová Scanning low-energy electron microscopy L. Godo & V. Torra Aggregation operators A. Hanbury Morphology on a circle P. W. Hawkes (vol. 127) Electron optics and electron microscopy: conference proceedings and abstracts as source material M. I. Herrera The development of electron microscopy in Spain J. S. Hesthaven (vol. 127) Higher-order accuracy computational methods for time-domain electromagnetics K. Ishizuka Contrast transfer and crystal images G. Kögel Positron microscopy N. Krueger The application of statistical and deterministic regularities in biological and artificial vision systems A. Lannes (vol. 126) Phase closure imaging B. Lahme Karhunen-Loève decomposition B. Lencová Modern developments in electron optical calculations M. A. O’Keefe Electron image simulation


N. Papamarkos and A. Kesidis The inverse Hough transform M. G. A. Paris and G. d’Ariano Quantum tomography T.-c. Poon (vol. 126) Scanning optical holography E. Rau Energy analysers for electron microscopes H. Rauch The wave-particle dualism D. de Ridder, R. P. W. Duin, M. Egmont-Petersen, L. J. van Vliet and P. W. Verbeek (vol. 126) Nonlinear image processing using artificial neural networks O. Scherzer Regularization techniques G. Schmahl X-ray microscopy S. Shirai CRT gun design methods T. Soma Focus-deflection systems and their applications I. Talmon Study of complex fluids by transmission electron microscopy M. Tonouchi Terahertz radiation imaging N. M. Towghi Ip norm optimal filters Y. Uchikawa Electron gun optics D. van Dyck Very high resolution electron microscopy K. Vaeth and G. Rajeswaran Organic light-emitting arrays

xv

xvi


C. D. Wright and E.W. Hill Magnetic force microscopy F. Yang and M. Paindavoine (vol. 126) Pre-filtering for pattern recognition using wavelet transforms and neural networks M. Yeadon (vol. 127) Instrumentation for surface studies

ADVANCES IN IMAGING AND ELECTRON PHYSICS, VOL. 125

An Algebraic Approach to Subband Signal Processing MARILENA BARNABEI AND LAURA B. MONTEFUSCO Department of Mathematics, University of Bologna, I-40127 Bologna, Italy

I. Introduction . . . . . . . . . . . . . . . . . . . . . . II. An Overview of Recursive Matrices . . . . . . . . . . . . . A. Basics . . . . . . . . . . . . . . . . . . . . . . . . B. Recursive Matrices . . . . . . . . . . . . . . . . . . C. Monomial Recursive Matrices . . . . . . . . . . . . . . D. Matrix Representation of Operations on Laurent Polynomials . III. Recursive Operators of Filter Theory . . . . . . . . . . . . A. Basic Linear Operators . . . . . . . . . . . . . . . . . B. Other Decimation Operators . . . . . . . . . . . . . . C. Interchanging and Combining Basic Operators . . . . . . . IV. More Algebraic Results . . . . . . . . . . . . . . . . . . A. Block Toeplitz Matrices . . . . . . . . . . . . . . . . B. Products of Hurwitz Matrices . . . . . . . . . . . . . . V. Analysis and Synthesis Filter Banks . . . . . . . . . . . . . A. Analysis and Synthesis Operators . . . . . . . . . . . . B. Matrix Description of Analysis Filter Banks . . . . . . . . C. Matrix Description of Synthesis Filter Banks . . . . . . . VI. M-Channel Filter Bank Systems . . . . . . . . . . . . . . A. Matrix Description . . . . . . . . . . . . . . . . . . B. Alias-Free Filter Banks . . . . . . . . . . . . . . . . C. Perfect Reconstruction Filter Banks . . . . . . . . . . . VII. Transmultiplexers . . . . . . . . . . . . . . . . . . . . A. Matrix Description . . . . . . . . . . . . . . . . . . B. Perfect Reconstruction Transmultiplexers . . . . . . . . . VIII. The M-Band Lifting . . . . . . . . . . . . . . . . . . . A. Algebraic Preliminaries . . . . . . . . . . . . . . . . B. Lifting and Dual Lifting . . . . . . . . . . . . . . . . C. The Algorithms . . . . . . . . . . . . . . . . . . . . D. Factorization into Lifting Steps . . . . . . . . . . . . . IX. Conclusion . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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I. Introduction In recent years, subband filter theory has been developed considerably, due to its increasing number of applications, mainly in telecommunications and signal processing. More specifically, this theory has been fruitfully exploited in 1 Copyright 2002, Elsevier Science (USA). All rights reserved. ISSN 1076-5670/02 $35.00

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BARNABEI AND MONTEFUSCO

such areas as speech and image compression, coding of high-resolution video and audio, and advanced television. Moreover, the connection with multiresolution signal analysis, used, for example, in pattern analysis and computer vision, indicates the role played by filter bank theory even outside the previous application fields. The performance of application systems based on filter banks is significantly affected both by the properties of the individual analysis and synthesis filters and by the structure and quality of the overall system. This implies that the design of such systems requires imposing global conditions—such as alias or cross-talk cancelation and perfect reconstruction—as well as individual conditions on the filters involved, as, for example, passband or stopband deviation, phase characteristics, and the computational efficiency of their implementation. Hence, the design of M-band filter bank systems represents a challenging problem, due to the great number of constraints involved. In addition, the difficulties grow as soon as the number of bands taken into consideration increases, and filters are required to meet a general set of prespecified properties. Most of the early theoretical developments and design procedures of Mband filter bank systems were based on the so-called transform domain formulation (see, e.g., Vaidyanathan, 1993; Vetterli and Kovacevic, 1995; Woods, 1991; and references therein). This approach gives an explicit description of the global properties of the system, hence aliasing, magnitude, and phase distorsions can be controlled in the design. Transform domain formulation uses two different, but equivalent, analysis methods, called alias-component formulation and polyphase formulation. The first one yields simple conditions for alias-free and perfect reconstruction properties, but it does not provide a mechanism for choosing the analysis filters in order for the synthesis filters to be acceptable. Polyphase formulation partially overcomes the above-mentioned drawbacks, and permits great simplification of the theoretical results, leading at the same time to computationally efficient implementations of filter banks. However, this formulation does not provide a mechanism for choosing individual filters with suitable properties, since these constraints are not given in the polyphase domain, but on the filters themselves. Time-domain formulation was recently introduced to deal with this problem (see, e.g., Nayebi, Barnwell, and Smith, and references therein). Indeed, it has two major advantages. Firstly, many different systems can be formulated in exactly the same way, resulting in the same design procedure. Secondly, it offers the flexibility to balance the tradeoff between competing system features, such as reconstruction error and filter quality. Nevertheless, this formulation cannot provide the simplicity and elegance of the polyphase approach, since it only works in the real domain. This is due to the fact that the essence of the time-domain formulation is a set of bi-infinite matrix equations relating output samples to input samples, and that the full potential of the structure of these

SUBBAND SIGNAL PROCESSING

3

matrices and their products has not as yet been exploited. The main contribution of the present work and of the research papers which preceded it (Barnabei and Montefusco, 1998; Barnabei et al., 1998, 2000) consists in presenting a theoretical basis that allows us to study filter banks from the time-domain point of view, while having a well-structured algebraic framework at our disposal. This allows us to explicitly characterize the bi-infinite matrices describing the action of analysis and synthesis filter banks, together with their combined action. Moreover, it provides the possibility of completely recovering all the results obtained from the polyphase approach, while still working in the real domain. In addition, the purely algebraic language of this approach allows us to give full mathematical justification to several computational rules widely used in practical applications. More precisely, the two cornerstones on which our presentation is based are the recursive matrix theory and the algebra of Laurent polynomials. A banded recursive matrix is a bi-infinite matrix that can be completely described by two Laurent polynomials, and the same happens for the product of recursive matrices. This allows us to switch from the context of banded bi-infinite matrices to the algebra of Laurent polynomials, taking advantage of its abundant properties. The recursive matrix machinery can then be fruitfully used to represent and easily handle some linear operators—acting on finite or infinite sequences—that are widely used in filter theory. In fact, such linear operators are associated with particular bi-infinite Toeplitz or Hurwitz matrices, which are special cases of recursive matrices. For example, the operation of convolution followed by decimation—a typical filtering operation—can be represented by means of a Hurwitz matrix, while the upsampling operation corresponds to the transpose of a Hurwitz matrix. Hence, the behavior of these linear operators can be studied through an analysis of the properties of the products of Hurwitz and Toeplitz matrices and their transposes. On the other hand, this matrix based approach allows us to recover the elegant and simple results of polyphase formulation by exploiting the possibility of describing a recursive matrix in terms of the generating function of its rows. The basic step in this context is the bijection between a Laurent polynomial and the M-tuple of its decimated polynomials, namely, a complete set of its subsampled versions. In Barnabei et al. (1998), we reinterpreted the analysis and synthesis filter banks in the previous algebraic setting, giving an explicit and computationally efficient description of their action. Moreover, we showed that the operator describing an analysis/synthesis system is associated with a block Toeplitz matrix, and that the alias cancelation condition is equivalent to requiring that such a matrix is also a “scalar” Toeplitz matrix. In this way, we managed to obtain a simple algebraic characterization for alias-free and perfect reconstruction analysis/synthesis systems, handling the 2- and M-band cases with equal

4


ease. A similar characterization can be given for transmultiplexing systems, as we show in the present paper. The algebraic tools we have developed not only allow us to control the global properties of filter bank systems but also enable us to impose constraints on the filters themselves, namely, on the coefficients of the corresponding Laurent polynomials. However, it is computationally cumbersome to impose both of these conditions directly. In Barnabei et al. (2000), we show how it is possible to reduce this complexity, by constructing in various consecutive steps (lifting steps) perfect reconstruction systems whose filters satisfy prespecified requirements. This construction makes use only of linear combinations of Laurent polynomials, therefore dramatically reducing the computational complexity of designing M-band systems with good filtering properties. The present paper is organized as follows. In Sections II and IV we give a detailed description of the algebraic notions and results that constitute the theoretical foundations of the present work. In Sections III and V we describe the action of the operators that are the building blocks of filter bank theory in recursive matrix notation, and we obtain equivalent results in terms of decimated polynomials, which correspond to the polyphase components. Sections VI and VII are devoted to the study of M-band filter banks and transmultiplexers. We give characterizations of alias and cross-talk cancelation conditions and perfect reconstruction systems, both in terms of M-decimated matrices and in terms of Hurwitz matrices. Finally, in Section VIII we present the extension to the M-band case of the lifting scheme, a powerful tool, yielding an easy custom-designed construction of biorthogonal filters with preassigned features. By fully exploiting the properties of the algebra of Laurent polynomials, we succeed in describing the building aspects of the lifting scheme, and its decomposition aspects, hence obtaining a factorization procedure that leads to faster analysis–synthesis algorithms.

II. An Overview of Recursive Matrices A. Basics In this section we recall the basic definitions and properties of Laurent series which will be widely used in the rest of the paper. Let K be any field, and K Z the K -vector space of bi-infinite sequences in K , (ai ), with i ∈ Z. We associate to the sequence (ai )i∈Z its generating function, namely, the formal series α := ai t i i∈Z

where t is a formal variable.


5

For every integer i, the coefficient ai of t i in α will be also denoted by i|α. In the following, we give as examples the most commonly used sequences with their respective generating functions. 1. The generating function of the infinite sequence whose elements are all equal to zero is the zero-series ζ := 0t i . i∈Z

2. The generating function of the infinite sequence (δi,0 )i∈Z , where δ is the Kronecker symbol, is the one-series, namely, the series υ := δi,0 t i . i∈Z

3. The generating function of the sequence (δi,1 )i∈Z is the identity series t, and, in general, the generating function of the sequence (δi,k )i∈Z is the series t k . The correspondence between a sequence of K Z and its generating function is a bijection. Hence, we shall often identify bi-infinite sequences in K and formal series. Moreover, if α is a formal series, we shall denote by [α] the bi-infinite column matrix whose elements are the coefficients of α. A Laurent series is a formal series α := ai t i i∈Z

in which only a finite number of coefficients ai with negative index i are nonzero, namely, an index h exists such that i < h ⇒ ai = 0. The minimum index d such that ad = 0 is the degree deg(α) of the Laurent series α. The degree of the zero-series ζ is conventionally defined to be −∞. For example, the series υ is a Laurent series of degree zero, while the series t is a Laurent series of degree 1. We recall that thesum α + β andthe (convolution) product αβ of two Laurent series α := i≥h ai t i , β := i≥k bi t i are the Laurent series defined as follows: α+β := (ai + bi )t i , i

αβ : =

i≥h+k

ci t i ,

6


where, for every index i, ci :=

i−k

a j bi− j .

j=h

Note that deg(α + β) = min(deg(α), deg(β)), while deg(αβ) = deg(α) + deg(β). The set L+ of all Laurent series turns out to be a field with respect to the operations previously defined. In particular, the identity elements for sum and product are the series ζ and υ, respectively, and, for every nonzero series α, its reciprocal series α −1 exists, namely, a Laurent series such that αα −1 = υ = α −1 α. Let α := i ai t i , β := j b j t j be Laurent series such that β has positive degree; the composition α ◦ β is defined as the Laurent series: i j i α(β) = α ◦ β := bjt . ai β = ai i

i

j

Note that deg(α ◦ β) = deg(α) · deg(β). The identity element with respect to composition is the series t. It is easily seen that a Laurent series α admits compositional inverse α, ˜ namely, another Laurent series such that α ◦ α˜ = t = α˜ ◦ α, whenever deg(α) = 1. In Figure 1, we give some examples of sequences in R Z whose generating functions are Laurent series. Note that the zeroth entry is boldface. The duality map is defined as the map T : K Z → K Z,

T(ai ) = (a−i ).

The map T is an involution and turns out to be a vector space automorphism. If α is the generating function of the sequence (ai ), the generating function of

Figure 1. Examples of Laurent series.


7

the dual sequence T(ai ) will be denoted by the symbol α*. Hence: α := ai t i ⇐⇒ α ∗ := a−i t i = ai t −i . i

i

i

+

The maximum subspace P of L that is invariant under the action of the duality map is the subspace of Laurent series with only a finite number of nonzero coefficients. The elements of P are called Laurent polynomials. We remark that the duality map α → α ∗ , when restricted to the subspace of Laurent polynomials, can be represented as follows: T(α) = α ◦ t −1 . In fact, even if the composition of a Laurent series α with the series t −1 is not recovered under the general definition, it makes sense in the special case in which α is a Laurent polynomial, and once again yields a Laurent polynomial.

B. Recursive Matrices We will be concerned here with the K -vector space M of bi-infinite matrices over the field K , namely, matrices M : Z × Z → K . Let M = [m i j ] be a matrix in M. For every i ∈ Z, the generating functions of the ith row of M will be denoted by M(i): M(i) = mi j t j . j

We now come to the definition of a recursive matrix, which can be loosely described as a bi-infinite matrix that can be completely determined by two series, the first of which gives the recurrence rule, while the second provides the boundary conditions. The formal definition of a recursive matrix can be given in several equivalent terms, each one of which highlights a different feature pre sented by these matrices. More precisely, let α = j≥d a j t j , β = j≥h b j t j be nonzero Laurent series. The (α, β)-recursive matrix is the unique matrix M ∈ M defined by the following equivalent conditions: r r

M(i) = α i β for every integer i. For every integer j, we have:

* m0 j = b j , * for i > 0, the element m i, j can be recursively computed from the elements of the preceding row of M, namely, as m i−1, j−s , (1) m i, j = s≥d

8


* for i < 0, the element m i, j can be computed from the elements of the following row of M, namely, a¯ s m i+1, j−s , (2) m i, j = s≥−d

where the a¯ s are the coefficients of the series α −1 , the reciprocal series of α. r

M is the unique solution, in M, of the following linear problem: F M G = M;

A(0) = β

(3)

where F is the bi-infinite forward shift matrix, while G is the bi-infinite Toeplitz matrix with generating function α. The (α, β)-recursive matrix will be denoted by R(α, β) and the series α, β will be called the recurrence rule and boundary value of R(α, β), respectively. Example II.1 In Figure 2, we show the recursive matrix R(α, υ), where α = 1 + t. The zeroth row and column are boldface. Note that the southeast section of the matrix is simply the Pascal triangle. The main property of recursive matrices is the fact that—under suitable conditions—they can be multiplied, and the product is again a recursive matrix, whose recurrence rule and boundary value can be explicitly described. More precisely, we have:

Figure 2. The “Pascal” matrix R(1 + t, υ).

9


Theorem II.1 (Product Rule) Let α, β, γ , δ be nonzero Laurent series, and let γ have positive degree. Then, R(α, β) × R(γ , δ) = R(α ◦ γ , (β ◦ γ )δ). Proof. Set R(α, β) = P = ( pi j ), R(γ , δ) = Q = (qi j ), M = P × Q. For every integer h, we have M(h) = ph j q ji t i = ph j Q( j) = ph j γ j δ i

j

j

j

h

= (P(h) ◦ γ )δ = ((α β) ◦ γ )δ = (α ◦ γ )h (β ◦ γ )δ,

which gives the assertion. The Product Rule immediately yields an explicit description of the recursive inverse of a recursive matrix (if it exists). Proposition II.1 Let α, β be nonzero Laurent series, with deg(α) = 1. Then the recursive matrix R(α, β) is invertible and its inverse is the recursive matrix R(α, ˜ (β ◦ α) ˜ −1 ). In other words, every recursive matrix whose recurrence rule has degree 1 admits a recursive inverse.

C. Monomial Recursive Matrices A notable class of recursive matrices is the class of recursive matrices whose recurrence rule is a monomial, namely, a Laurent series of the kind k α = t k , where k is any nonzero integer. If M = R(t , β) is such a matrix, with β = i bi t i , then, according to formula (1), the element m i j of M is given by m i j = b j−ki . We point out that this class of recursive matrices recovers three well-known types of bi-infinite structured matrices, namely: r

r

r

A Toeplitz matrix with generating function β is the recursive matrix R(t, β) (see Fig. 3). A Hurwitz matrix of step k and generating function β is the recursive matrix R(t k , β) (see Fig. 4). A Hankel matrix with generating function β is the recursive matrix R(t −1 , β) (see Fig. 5).

10


Figure 3. The Toeplitz matrix R(t, β).

These three classes of matrices are of special interest since they reveal a particularly pleasant algebraic behavior, and they are a crucial instrument in the algebraic theory of signal processing. For example, by multiplying a general recursive matrix by a Toeplitz matrix we succeed in changing the boundary value without affecting its recurrence rule. Hence, every recursive matrix can be seen as the product of a recursive matrix whose boundary value is υ, and a Toeplitz matrix. In fact, by the Product Rule, we get: Proposition II.2 Let α, β be nonzero Laurent series. Then the recursive matrix R(α, β) admits the following factorization: R(α, β) = R(α, υ) × R(t, β). As we have already remarked, Toeplitz, Hurwitz, and Hankel recursive matrices are widely used in the applications, together with their products and powers. For this reason, we summarize some important results on these matrices, which follow immediately from the Product Rule. Corollary II.1 The product of two Toeplitz recursive matrices is again a Toeplitz matrix, whose generating function is the product of the two generating

Figure 4. The Hurwitz matrix R(t 2 , β).


11

Figure 5. The Hankel matrix R(t −1 , β).

functions: R(t, α) × R(t, β) = R(t, αβ). In particular, two Toeplitz recursive matrices always commute. Corollary II.2 For every positive integer p, the pth power of the Toeplitz recursive matrix R(t, α) is the Toeplitz matrix R(t, α p ). Corollary II.3 The recursive inverse of the Toeplitz recursive matrix R(t, β) is the Toeplitz matrix R(t, β −1 ). Corollary II.4 The products of Toeplitz and Hurwitz recursive matrices are Hurwitz matrices; more precisely: R(t, α) × R(t k , β) = R(t k , (α ◦ t k )β); R(t k , β) × R(t, α) = R(t k , αβ).

Corollary II.5 The product of two Hurwitz recursive matrices is again a Hurwitz matrix with a different step: R(t h , α) × R(t k , β) = R(t h·k , (α ◦ t k )β). Corollary II.6 For every positive integer p, the pth power of the Hurwitz recursive matrix R(t k , α) is a Hurwitz matrix with a different step: p

R(t k , α) p = R(t k , β) where β=α

p−1 i=1

α◦t

ki

.

To give further results concerning Hankel matrices, we now restrict ourselves to the banded case, namely, we consider monomial recursive matrices

12


whose boundary value is a Laurent polynomial. These matrices will play a fundamental role in the applications we are going to consider. For such matrices, the Product Rule can be extended to the case when the recurrence rule of the second matrix is a monomial of negative degree. For example, we get the following results for the products of Toeplitz and Hankel banded matrices: Corollary II.7 The products of banded Toeplitz and Hankel matrices are Hankel matrices; more precisely: R(t, α) × R(t −1 , β) = R(t −1 , α ∗ β);

R(t −1 , β) × R(t, α) = R(t −1 , αβ).

Corollary II.8 The product of two banded Hankel matrices is a Toeplitz matrix: R(t −1 , α) × R(t −1 , β) = R(t, α ∗ β). As a consequence, the pth power of a banded Hankel matrix is either a Toeplitz or a Hankel matrix, according to the parity of the integer p: Corollary II.9 For every positive integer h, the (2h)th power of the Hankel recursive matrix R(t −1 , α) is a Toeplitz matrix: R(t −1 , α)2h = R(t, α h α ∗h ), while the (2h + 1)th power of R(t −1 , α) is a Hankel matrix: R(t −1 , α)2h+1 = R(t −1 , α h+1 α ∗h ).

D. Matrix Representation of Operations on Laurent Polynomials It is worth remarking that it is possible to represent some operations on Laurent polynomials by means of suitable recursive matrices using the Product Rule. For example: 1. The product of two Laurent polynomials α, β can be seen as the row vector [αβ]T = [α]T × R(t, β) = [β]T × R(t, α).

(4)

or, equivalently, as the column vector [αβ] = R(t, β ∗ ) × [α] = R(t, α ∗ ) × [β]. In fact, since R(t, α) × R(t, β) = R(t, αβ),

(5)


13

we get (4) by equating the zeroth row on both sides. Identity (5) follows from the well-known fact that the transpose of the Toeplitz matrix R(t, α) is the Toeplitz matrix R(t, α ∗ ). 2. The composition of two Laurent polynomials α, β is the row vector: [α ◦ β]T = [α]T × R(β, υ)

(6)

or, equivalently, the column vector [α ◦ β] = R(β, υ)T × [α].

(7)

In fact, again using the Product Rule, we have: R(t, α) × R(β, υ) = R(β, α ◦ β), and (6) is obtained by equating the zeroth row on both sides. 3. The dual polynomial α ∗ of a Laurent polynomial α is the column vector: [α ∗ ] = [α ◦ t −1 ] = R(t −1 , υ) × [α].

(8)

The above identity is a consequence of identity (7) and of the symmetry of Hankel matrices.

III. Recursive Operators of Filter Theory In this section, we show the connection between banded recursive matrices and some linear operators widely used in filter theory. This connection emphasizes the algebraic properties of such operators, hence leading to a simple and general description, together with a computationally efficient implementation of their combined action.

A. Basic Linear Operators This subsection is devoted to the analysis of some linear operators on the subspace P of Laurent polynomials, frequently used in practical applications, whose associated matrices reveal a recursive structure. r

Duality As we have already remarked, the duality map T : P → P can be represented as follows: T(σ ) = σ ◦ t −1 ,

(9)

where σ is any Laurent polynomial. Using identity (8), we get: [T(σ )] = [σ ∗ ] = R(t −1 , υ) × [σ ],

(10)

14


Figure 6. The action of the counter-identity matrix R(t −1 , υ).

that is, the duality map T is a linear operator, whose associated matrix is the Hankel matrix R(t −1 , υ) (the counter-identity matrix) (see Fig. 6). r

Filters Let α be a fixed Laurent polynomial, and Fα be the finite-length (FIR) filter corresponding to α, namely, the linear operator on the vector space P such that, for every σ ∈ P , Fα (σ ) = α ∗ σ . The action of a filter is usually represented by the following scheme:

As we have already seen (identity (5)), the matrix associated with the filter Fα is the Toeplitz matrix R(t, α), namely, [Fα (σ )] = [α ∗ σ ] = R(t, α) × [σ ].

(11)

In particular, the filter Ft h corresponding to the monomial t h is called an h-step shift and multiplies a given Laurent polynomial by t −h . When h > 0, the shift Ft h is called a delay, while when h < 0, it is called an advance. r

k-upsampler Let k be a fixed integer, k ≥ 2. The k-upsampling operator (or k-expander) Uk , represented as

is defined as follows: if σ =

i si

k−1

is a Laurent polynomial, k−1

k−1

k−1

U (σ ) = (↑ k)σ := (.. 0......0 s−1 0......0 s0 0.....0 s1 0.....0 ...). k

This is equivalent to the composition of σ with the series t k , namely, Uk (σ ) = σ ◦ t k . Hence, by identity (7), we get [Uk (σ )] = [σ ◦ t k ] = R(t k , υ)T × [σ ].

(12)


15

Figure 7. The action of the matrix R(t 3 , υ)T associated with the 3-upsampling operator U3 .

In other words, Uk is a linear operator, whose associated matrix is the transposed Hurwitz matrix R(t k , υ)T (see Fig. 7). r

k-downsampler For every fixed integer k ≥ 2, the k-downsampling operator Dk0 , represented as

is defined as follows: if σ =

Dk0 (σ ) = σ0 :=

i si t

i

i

is a Laurent polynomial, ski t i = ki|σ t i . i

It is immediately clear that the matrix associated with Dk0 is the Hurwitz matrix R(t k , υ), namely, k (13) D0 (σ ) = R(t k , υ) × [σ ]. B. Other Decimation Operators The downsampling operator defined above associates to a given Laurent polynomial σ the polynomial σ0 that only partially maintains the features of the signal. In order to recover the whole information contained in σ it is necessary to associate to σ all its k-decimated polynomials, namely, the k polynomials

16


σ0 , σ1 , . . . , σk−1 , defined as follows: σ0 := Dk0 (σ ), σr := Dk0 (t −r σ ) for r = 1, 2, . . . , k − 1. In other words, for every r = 0, 1, . . . , k − 1, σr := ski+r t i = ki + r |σ t i . i

(14)

i

Extending the notation introduced for the downsampling operator, we denote by Drk the linear operator such that Drk (σ ) = σr for r = 0, 1, . . . , k − 1. These operators will be called the k-decimation operators. By definition, the r th k-decimation operator can be written as Drk = Dk0 Ft r

(15)

and represented by

Hence, the matrix associated with the decimation operator Drk is given by the product R(t k , υ) × R(t, t r ) that yields, using the Product Rule, the Hurwitz matrix R(t k , t r ) (see Fig. 8), namely: [Drk σ ] = [σr ] = R(t k , t r ) × [σ ].

(16)

Every k-decimated polynomial of σ represents an aliased version of the original input signal. However, by using all of this incomplete information, it is possible to recover exactly the original signal. In fact, the explicit description of every decimated polynomial in terms of the coefficients of σ given by identity (14) allows us to state the following reconstruction rule: σ = σ0 ◦ t k + · · · + t r (σr ◦ t k ) + · · · + t k−1 (σk−1 ◦ t k ),

(17)


17

Figure 8. R(t 3 , t) × [σ ] = [σ1 ].

represented by the following scheme:

We have seen that the Hurwitz matrix R(t k , t r ) is associated with the operator whenever 0 ≤ r ≤ k − 1. In general, we can describe the linear operator associated with a Hurwitz matrix of the kind R(t k , t n ), without any restriction on the exponent n. More precisely, using the Product Rule, we get:

Drk

Proposition III.1 Let k, n be fixed integers, with k ≥ 2, and let q, r be the unique integers such that 0 ≤ r < k and n = kq + r . The linear operator associated with the Hurwitz matrix R(t k , t n ) is the operator Drk followed by the shift Ft q . In other words, R(t k , t n ) = R(t, t q ) × R(t k , t r ).

(18)

In Figure 8 the action of the linear operator associated with the matrix R(t 3, t) is shown. Lastly, it is useful—in view of further applications—to describe the k-decimated series of the product of two given series.

18


Proposition III.2 Let k ≥ 2 be a fixed integer and α, β two Laurent series. The zeroth k-decimated series of the product αβ is Dk0 (αβ) = α0 β0 + t

k−1

αi βk−i ,

i=1

and, for 0 < r < k, the r th k-decimated series of αβ is Drk (αβ) =

r i=0

αi βr −i + t

k−1

i=r +1

αi βr +k−i .

where αi , βi denote the ith k-decimated series of α, β, respectively. Proof. By identity (17), we have αβ = (α0 ◦ t k + · · · + t k−1 (αk−1 ◦ t k ))(β0 ◦ t k + · · · + t k−1 (βk−1 ◦ t k )) =

k−1

i, j=0

(t i+ j ((αi β j ) ◦ t k )).

For i, j = 0, . . . , k − 1, we have

⎧ ⎨α0 β0 if i = j = 0 Dk0 (t i+ j ((αi β j ) ◦ t k )) = tαi β j if i + j = k ⎩0 otherwise

The first identity now follows by linearity. The second identity can be proved by similar arguments. Example III.1 Let α = a−1 t −1 + a0 + a1 t,

β = b−1 t −1 + b0 + b1 t.

The 3-decimated series of α and β are, respectively: α0 = a 0 ,

α1 = a 1 ,

β0 = b0 ,

β1 = b1 ,

α2 = a−1 t −1 ,

β2 = b−1 t −1 .

We have: αβ = a−1 b−1 t −2 + (a−1 b0 + a0 b−1 )t −1 + (a0 b0 + a−1 b1 + a1 b−1 ) + (a0 b1 + a1 b0 )t + a1 b1 t 2 ;

D30 (αβ) = a0 b0 + a−1 b1 + a1 b−1 = α0 β0 + tα1 β2 + tα2 β1 ;

D31 (αβ) = a−1 b−1 t −1 + a0 b1 + a1 b0 = α0 β1 + α1 β0 + tα2 β2 .


19

C. Interchanging and Combining Basic Operators The general results about recursive matrices stated in the previous sections allow us to examine the interplay between the basic operators of filter theory. First of all, we study the conditions for interchanging the duality operator T and the k-decimation operators Drk . These operators do not commute unless r = 0, as shown in the next proposition: Proposition III.3 Let k be a fixed integer, k ≥ 2. We have TDk0 = Dk0 T,

and, for every integer r, 0 < r < k, TDrk = Ft −1 Dkk−r T, that is, for every Laurent polynomial σ, (σ0 )∗ = (σ ∗ )0 ,

(19)

(σr )∗ = t(σ ∗ )k−r .

(20)

and, for r > 0,

Proof. The operator

TDk0

is associated with the matrix

R(t −1 , υ) × R(t k , υ) = R(t −k , υ),

and Dk0 T is associated with

R(t k , υ) × R(t −1 , υ), which gives the same recursive matrix. For r > 0, the operator TDrk is associated with the matrix R(t −1 υ) × R(t k , t r ) = R(t −k , t r ),

while Dkk−r T is associated with the matrix

R(t k , t k−r ) × R(t −1 , υ) = R(t −k , t r −k ). The assertion now follows by noting that R(t, t −1 ) × R(t −k , t r −k ) = R(t −k , t k t r −k ) = R(t −k , t r ) and recalling that R(t, t −1 ) is the matrix associated with the shift Ft −1 (see Eq. (11)). Example III.2 Set k = 3, and let

σ = · · · + s−2 t −2 + s−1 t −1 + s0 + s1 t + s2 t 2 + s3 t 3 + s4 t 4 + · · ·

20


be a Laurent polynomial; then D31 σ = σ1 = · · · + s−2 t −1 + s1 + s4 t + · · · , and TD31 σ = (σ1 )∗ = · · · + s4 t −1 + s1 + s−2 t + · · · . On the other hand, σ ∗ = · · · + s4 t −4 + s3 t −3 + s2 t −2 + s1 t −1 + s0 + s−1 t + s−2 t 2 + · · · and D32 Tσ = (σ ∗ )2 = · · · + s4 t −2 + s1 t −1 + s−2 + · · · ; hence, (σ1 )∗ = t(σ ∗ )2 .

For the sake of simplicity, from now on we will write σr∗ for (σr )∗ . The problem of interchanging the k-decimation operators Drk and the h-step shift can easily be handled by a systematic use of the Product Rule. We give below a simple example of this kind of result: Proposition III.4 Let k ≥ 2 be a fixed integer, and σ a Laurent polynomial. We have: Ft h Drk (σ ) = Dk0 Ft n (σ )

(21)

where n = hk + r . Proof. The matrix associated with the left-hand side operator is R(t, t h ) × R(t k , t r ) = R(t k , t hk+r ) = R(t k , υ) × R(t, t n ),

and the last one is precisely the matrix associated with the operator Dk0 Ft n . The preceding proposition can be illustrated by the following scheme:

We now consider two results largely used in signal manipulation, namely, the two Noble identities (Strang and Nguyen, 1996). The first Nobel identity gives a commutation rule for the product of a filter and a decimator, while the second one gives a similar rule for the product of a filter and an expander. These identities can be rewritten in terms of recursive matrices, so that their proof is an immediate consequence of the factorization property of recursive


21

matrices (Proposition II.2) and Corollary II.4. More precisely, the first Noble identity, Fα Dk0 = Dk0 Fα◦t k ,

(22)

represented by the scheme

corresponds to the equality: R(t, α) × R(t k , υ) = R(t k , α ◦ t k ) = R(t k , υ) × R(t, α ◦ t k ), while the second Noble identity, Uk Fα = Fα◦t k Uk ,

(23)

represented by

corresponds to the following equality: R(t k , υ)T × R(t, α) = (R(t k , α ∗ ◦ t k ))T = R(t, α ◦ t k ) × R(t k , υ)T . We notice that the present approach, based on the correspondence between linear operators and recursive matrices, allows us to state a further identity that generalizes the results of both (21) and (22), whose proof is straightforward: Proposition III.5 Let k ≥ 2 be a fixed integer, and α a Laurent polynomial. We have: Fα Drk = Drk Fα◦t k = Dk0 Ft r (α◦t k ) .

(24)

The matrix-based presentation of the basic operators provides an effective way to describe the building blocks of a k-channel maximally decimated filter bank system. In fact, these are linear operators—analysis and synthesis operators—that can be expressed as the composition of filters with decimators or expanders. Such operators can therefore be represented in terms of banded recursive matrices. In particular, the analysis operator consists of a filter followed by a k-decimator, according to the following scheme:

22


This yields the operator Dk0 Fγ , whose associated matrix is the Hurwitz matrix R(t k , υ) × R(t, γ ) = R(t k , γ ).

(25)

The synthesis operator is the combined action of a k-expander followed by a filter:

which yields the operator Fα∗ Uk . Once again using the Product Rule, the synthesis operator corresponds to the transposed Hurwitz matrix R(t k , α)T . In fact, R(t, α ∗ ) × R(t k , υ)T = R(t, α)T × R(t k , υ)T = (R(t k , υ) × R(t, α))T = R(t k , α)T .

(26)

In practical applications of filter theory, the analysis of an input signal σ is followed by a synthesis that produces the output signal. It is therefore important to be able to describe, in terms of recursive matrices, the action of the analysis–synthesis operator:

namely, the operator Fα∗ Uk Dk0 Fγ . Making use of the preceding considerations, it immediately follows that this operator corresponds to the product R(t k , α)T × R(t k , γ ).

(27)

IV. More Algebraic Results In order to perform a detailed study of filter bank systems we need some further results about monomial matrices. More precisely, we introduce the notion of block Toeplitz matrix and study its main features. Furthermore, we succeed in giving an explicit expression for the product of a banded Hurwitz matrix and


23

a transposed banded Hurwitz matrix and vice versa. These results will allow us to give a compact and efficient description of the action of the analysis and synthesis operators together with their combined action. A. Block Toeplitz Matrices The notion of recursive matrix can be extended to the case of matrices whose entries are square matrices of fixed order k, yielding the notion of block recursive matrix. The definition and the main properties of such matrices are formally almost the same as in the scalar case, even if in the block case the sboundary value is a block Laurent polynomial, that is, a polynomial = i=−h P (i) t i (i) whose coefficients are k × k square matrices P (i) = [ plm ]: ⎞ ⎛ (−h) (−h) p00 p01 ··· ⎟ −h ⎜ (−h) = ⎝p ··· ··· ⎠t 10 (−h) ··· ··· pk−1,k−1 ⎞ ⎞ ⎛ (s) (s) ⎛ (−h+1) (−h+1) p00 p01 ··· p00 p01 ··· ⎟ s ⎟ −h+1 ⎜ ⎜ · · · + ⎝ p (s) · · · + ⎝ p (−h+1) · · · ··· ⎠t . ··· ⎠t 10 10 (s) (−h+1) · · · · · · pk−1,k−1 ··· ··· pk−1,k−1

Sometimes it will be useful to see the block Laurent polynomial as a polynomial matrix, namely, a matrix whose elements are scalar Laurent polynomials: ⎛ ⎞ π00 π01 ··· ⎟ ⎜ = ⎝π10 · · · ··· ⎠, · · · · · · πk−1,k−1 where

−h πi j = pi(−h) + pi(−h+1) t −h+1 + · · · · · · + pi(s)j t s . j t j

A detailed study of block recursive matrices, together with some applications to multiwavelet function theory, can be found in Bacchelli (1999) and Bacchelli and Lazzaro (2001). In the present paper, for the sake of simplicity, we will restrict our discussion to the case of block Toeplitz s matrices. Given a block Laurent polynomial = i=−h P (i) t i , the block Toeplitz matrix with generating function is defined as the bi-infinite matrix A, whose elements Ai j are k × k square matrices, such that Ai j = P ( j−i) for every pair of indices i, j, or, equivalently, A is the block recursive matrix R(t, ). Every block matrix can clearly be seen also as a scalar matrix, and vice versa. However, in general, a block Toeplitz matrix does not maintain its recursive structure when viewed as a scalar matrix. Nevertheless, given a k × k block

24


Toeplitz matrix, it is possible to describe the scalar generating function of its nth (scalar) row as follows: Proposition IV.1 Let A be the k × k block Toeplitz whose generating s matrix function is the block Laurent polynomial = i=−h P (i) t i . For every integer n, let r, q be the unique integers such that 0 ≤ r < k and n = kq + r . Then the scalar generating function of the nth row of A is given by: k−1 j k kq A(n) = t t (πr j ◦ t ) . (28) j=0

Proof. Since n = kq + r , the elements of the nth scalar row of A can be found in the r th row of the qth block row, whose generating function is t q . Hence: k−1 k−1 s s (i) ik (i) j+ik kq j kq t A(n) = t pr j t . pr j t =t i=−h j=0

j=0

i=−h

The assertion now follows from the fact that s pr(i)j t ik = πr j ◦ t k . i=−h

The preceding result allows us to give the following necessary and sufficient condition for a block Toeplitz matrix to be also a scalar Toeplitz matrix: Theorem IV.1 Let A be a (k × k)-block Toeplitz matrix whose generating function is the polynomial matrix . The matrix A is a scalar Toeplitz matrix if and only if the polynomial matrix is a t-circulant matrix, namely, it has the form ⎡ ⎤ φ0 φ1 φ2 ... φk−1 φ0 φ1 ··· φk−2 ⎥ ⎢tφk−1 ⎢ ⎥ tφ tφ φ · · · φ ⎢ k−1 0 k−3 ⎥ . = ⎢ k−2 (29) .. ⎥ .. .. .. ⎣ ... ⎦ . . . . tφ1 tφ2 · · · tφk−1 φ0

If this is the case, the generating function of the scalar Toeplitz matrix A is given by φ = φ0 ◦ t k + tφ1 ◦ t k + · · · + tk−1 φ k−1 ◦ t k , that is, for i = 0, 1, . . . , k − 1, φi = Dik φ.


25

Proof. We point out that the scalar Toeplitz matrix R(t, φ) can always be seen as the (k × k)-block Toeplitz matrix R(t, ), whose block generating function is the polynomial matrix ⎛ ⎞ π00 π01 ··· ··· ⎠, = ⎝π10 · · · · · · · · · πk−1,k−1

where the jth element of each row of is the jth k-decimated polynomial of the generating function of the corresponding row of the scalar Toeplitz matrix, that is, for every j = 0, 1, . . . , k − 1, πi j = Dkj (t i φ) = Dkj Ft −i φ

i = 0, 1, · · · , k − 1,

or equivalently, in matrix notation, [πi j ] = R(t k , t j ) × R(t, t −i ) × [φ] = R(t k , t j−i ) × [φ]. For every pair of indices i, j, let q, r be the unique integers such that 0 ≤ r ≤ k − 1 and j − i = kq + r . Recalling that 0 ≤ i, j ≤ k − 1, we get ⎧ 0 ⇒r = j −i ⎪ ⎨ for i ≤ j . q= ⇒r =k + j −i ⎪ ⎩−1 for i > j

By using relation (18), we get k r for i ≤ j R(t , t ) × [φ] , [πi j ] = R(t, t −1 ) × R(t k , t r ) × [φ] for i > j namely, πi j =

Drk φ = φr t Drk φ = t φr

for i ≤ j . for i > j

This proves the t-circulant structure of the matrix . Conversely, let A be a (k × k)-block Toeplitz matrix whose generating function is the polynomial matrix of the form (29), namely, such that, for every i, j, its (i, j)th entry πi j satisfies φr where r = j − i, for i ≤ j πi j = . t φr where r = k + j − i, for i > j Again by relation (18), we get: πi j = Dkj (t i φ).

26


By replacing this expression in (28) and using (17), we get the assertion. The explicit expression of the generating function φ is an immediate consequence of the preceding considerations.

B. Products of Hurwitz Matrices As shown in the previous section, the key elements in filter bank computations are banded Toeplitz and Hurwitz matrices, their transposes, and their products. Hence we devote the present subsection to a detailed study of such matrices. First of all, for the sake of simplicity, we introduce a compact notation to represent the set of the decimated polynomials of a given Laurent polynomial. More precisely, if α is a Laurent polynomial, the k-decimated vector of α will be the column vector ⎡ ⎤ α0 ⎢ α1 ⎥ ⎥ ∆k (α) := ⎢ (30) ⎣ ... ⎦ , αk−1

whose entries are the k-decimated polynomials of α. Similarly, the starred k-decimated vector of α will be the column vector ⎡ ∗ ⎤ α0 ⎢ ∗ .. ⎥ k ∗ (31) ∆ (α) := ⎣ α1 . ⎦ . ∗ αk−1

In view of further applications, we introduce a similar notation to denote the set of the k-decimated polynomials of an ordered k-tuple of Laurent polynomials α := (α (0) , α (1) , . . . , α (k−1) ). The k-decimated matrix of α is defined as the following matrix: ⎤ ⎡ (0) α0(1) · · · α0(k−1) α0 ⎥ ⎢ ⎥ ⎢ (0) ⎢ α1 α1(1) · · · α1(k−1) ⎥ ⎥ ⎢ ⎥. ∆k (α) = ⎢ (32) ⎢ . .. .. .. ⎥ ⎥ ⎢ .. . . . ⎥ ⎢ ⎦ ⎣ (0) α(k−1)

(1) α(k−1)

(k−1) · · · α(k−1)

We now show that the product of a banded Hurwitz matrix, and the transpose of another banded Hurwitz matrix with the same step, turns out to be a Toeplitz matrix, whose generating function can be described explicitly.

27


Theorem IV.2 Let α, β be nonzero Laurent polynomials, and k a fixed integer, k ≥ 2. Set H = R(t k , α), K = R(t k , β). Then, the product H × K T is a Toeplitz matrix, whose generating function λ is given by ∗ λ := Dk0 (αβ ∗ ) = α0 β0∗ + · · · + αk−1 βk−1 .

(33)

In other words, λ = ∆k (α)T × ∆k (β)∗ .

(34)

Proof. out that, given two Laurent polynomials γ := n We point δ := i=−n di t i , the (canonical) scalar product γ · δ := (c−n , c−n+1 , . . . , cn ) · (d−n , d−n+1 , . . . , dn ) =

can be written as n

i=−n

n

i=−n

n

i=−n

ci t i ,

= ci di

ci di = 0|γ ∗ δ = 0γ δ ∗ .

Now, setting P = [ pi j ] = H × K T , we have

pi j = H (i) · K ( j) = 0|H (i)K ( j)∗

= 0|t ki αt −k j β ∗ = k( j − i)|αβ ∗ .

Hence, P(i) =

Setting j − i = q, we get P(i) =

j

pi j t j =

k( j − i)|αβ ∗ t j . j

kq|αβ ∗ t q+i = t i Dk0 (αβ ∗ ), q

which proves that P is a banded Toeplitz matrix with generating function λ = Dk0 (αβ ∗ ). The second expression for λ is an immediate consequence of Propositions III.2 and III.3. Our next aim is to show that the product of the transpose of a banded Hurwitz matrix and a banded Hurwitz matrix with the same step k is once again a Toeplitz matrix, although in this case it is a block Toeplitz matrix. In order to do this, we need to show that the transpose of the banded Hurwitz matrix R(t k , α) presents some kind of recursive structure. More precisely,

28


Figure 9. The transposed Hurwitz matrix R(t 3 , α)T .

we can describe its row-generating functions by means of the k-decimated polynomials of α as follows: Theorem IV.3 Let α be a Laurent polynomial, and k a fixed integer, k ≥ 2. Set H := R(t k , α)T . For every integer n, let r, q be the unique integers such that 0 ≤ r < k and n = kq + r . Then we get the following expression for the generating function of the nth row of H : H (n) = t q+1 (α ∗ )k−r = t q αr∗ . In particular, H (0) = α0∗ .

Proof. Set K = R(t k , α). By Proposition II.2 we have: K = R(t k , υ) × R(t, α).

This means that every column of K can be seen as the k-downsampled version of the corresponding column of R(t, α). Recalling that the generating function of the nth column of R(t, α) is t n α ∗ , by Propositions III.3 and III.4, we get H (n) = Dk0 (t n α ∗ ) = t q+1 (α ∗ )k−r = t q αr∗ . In other words, as shown in Figure 9, the transpose of the banded Hurwitz matrix R(t k , α) can be seen as the “wedge” of k Toeplitz matrices, whose ∗ . generating functions are the polynomials α0∗ , α1∗ , . . . , αk−1 These polynomials turn out to be crucial in describing the blocks of a product of the kind R(t k , α)T × R(t k , β). In fact, we have:


29

Proposition IV.2 Let α, β be nonzero Laurent polynomials, and k a fixed integer, k ≥ 2. Set H := R(t k , α), K := R(t k , β). For every integer n, let r, q be the unique integers such that 0 ≤ r < k and n = kq + r . Then (H T × K )(n) = t kq (αr∗ ◦ t k )β.

(35)

Proof. Denote by Tr the Toeplitz matrix with generating function αr∗ , namely, Tr := R(t, αr∗ ). By Theorem IV.3, we have H T (n) = t q αr∗ = Tr (q). Hence, (H T × K )(n) = (Tr × K )(q), that is, the nth row of the product H T × K equals the qth row of the matrix Tr × K , which turns out to be the Hurwitz matrix R(t k , (αr∗ ◦ t k )β), and this gives the assertion. Relation (35) shows that, given two integers n 1 , n 2 such that n 1 − n 2 is a multiple of k, namely, n 1 = kq1 + r,

n 2 = kq2 + r,

then, (H T × K )(n 1 ) = t kq1 (αr∗ ◦ t k )β,

(H T × K )(n 2 ) = t kq2 (αr∗ ◦ t k )β.

Loosely speaking, the generating functions of the corresponding rows of the matrix H T × K are the same polynomial (αr∗ ◦ t k )β multiplied by different shifts. This highlights a block Toeplitz structure for the matrix H T × K , as shown by the next result. Theorem IV.4 Let α, β be nonzero Laurent polynomials, and k a fixed integer, k ≥ 2. Set H := R(t k , α), K := R(t k , β). Then the product H T × K is a (k × k)-block Toeplitz matrix, whose generating function is the block Laurent polynomial = i Pi t i , where ⎡ ⎤ i(α0 )∗ β0 · · · i(α0 )∗ βk−1 ⎦. ··· ··· ··· Pi = ⎣ i(αk−1 )∗ β0 · · · i(αk−1 )∗ βk−1

Equivalently, can be represented as the polynomial matrix: = ∆k (α)∗ × ∆k (β)T .

Proof. The block Toeplitz structure of the matrix H T × K is a consequence of the preceding considerations concerning formula (35). To give an

30


explicit description of these blocks, we make use of identity (17). In fact, we have: (H T × K )(n) = t kq (αr∗ ◦ t k )(β0 ◦ t k + t(β1 ◦ t k ) + · · · + t k−1 (βk−1 ◦ t k )) = t kq ((αr∗ β0 ) ◦ t k + t(αr∗ β1 ) ◦ t k + · · · + t k−1 (αr∗ βk−1 ) ◦ t k ). V. Analysis and Synthesis Filter Banks From now on, we use the tools presented in the previous sections to review several classic results from the filter bank literature, and mathematically describe their main features. Main applications of filter banks are found in subband coding schemes and transmultiplexers. Both of these applications make use of two basic devices, analysis and synthesis banks, which consist of M analysis operators and M synthesis operators, respectively. In fact, in subband coding, an analysis bank splits the input signal into M downsampled subband signals, which are then quantized and coded individually. After decoding, the reconstructed signal is obtained by means of a synthesis filter bank. Dually, the transmultiplexing scheme starts from several individual signals and, using a synthesis filter bank, combines them into a single signal from which the individual components can be recovered using an analysis filter bank. Given the importance of analysis and synthesis systems, we devote the present section to a matrix description of the corresponding operators.

A. Analysis and Synthesis Operators We start by giving an explicit description of the action of a single analysis and synthesis operator, together with their combined action. r

Analysis The analysis operator Dk0 Fγ is associated with the Hurwitz matrix R(t k , γ ). Hence, the output σˆ of its action on the input signal σ coincides with the zeroth column of the matrix R(t k , γ ) × R(t, σ ∗ ) = R(t k , γ σ ∗ ). Recalling that if β is any Laurent polynomial, the generating function of the zeroth column of the Hurwitz matrix R(t k , β) is given by the Laurent polynomial β0∗ = Dk0 (β ∗ ) (Proposition IV.3), we get σˆ = Dk0 (γ σ ∗ )∗ = Dk0 (γ ∗ σ ).

31


Using Propositions III.2 and III.3, the latter polynomial can be written as σˆ = Dk0 (γ ∗ σ ) = namely, σˆ = ∆k (γ )∗T × ∆k (σ ) = [γ0∗

k−1 (γi )∗ σi , i=0

⎡

⎤ σ0 ⎢ σ1 ⎥ ∗ ⎥ ]×⎢ γ1∗ · · · γk−1 ⎣ ... ⎦ .

(36)

σk−1

This yields the computationally efficient and easily parallelizable description of the action of an FIR analysis operator depicted below:

r

Synthesis The synthesis operator Fα∗ Uk corresponds to the transposed Hurwitz matrix R(t k , α)T .

As in the previous case, the output σˆ is the zeroth column of the matrix R(t k , α)T × R(t, σ ∗ ) = R(t k , α)T × R(t, σ )T = R(t k , (σ ◦ t k )α)T , that is, σˆ = (σ ◦ t k )α. Recalling that α=

k−1 i=0

t i (αi ◦ t k ),

we get the following algebraic expression for the output of a synthesis operator: σˆ =

k−1 i=0

t i ((σ αi ) ◦ t k )

32


that can be schematized as follows:

Bearing in mind the bijection between a Laurent polynomial and its kdecimated vector, the action of the synthesis operator can be also described as follows: ∆k (σˆ ) = ∆k (α) σ,

(37)

that is, ⎤ ⎤ ⎡ σˆ 0 α0 σ ⎢ σˆ 1 ⎥ ⎢ α1 σ ⎥ ⎢ . ⎥ = ⎢ . ⎥. ⎣ .. ⎦ ⎣ .. ⎦ σˆ k−1 αk−1 σ ⎡

This last expression turns out to be particularly efficient for practical applications, both due to its reduced computational complexity and because of its inherent parallel structure. r

Analysis–synthesis By identities (36) and (37), the output σˆ of the analysis–synthesis operator Fα∗ Uk Dk0 Fγ is the unique polynomial whose k-decimated vector is given by ∆k (σˆ ) = ∆k (α) × ∆k (γ )∗T × ∆k (σ ), namely, ⎤ ⎤ ⎡ σˆ 0 α0 ⎢ σˆ 1 ⎥ ⎢ α1 ⎥ ⎢ . ⎥ = ⎢ . ⎥ × [γ ∗ 0 ⎣ .. ⎦ ⎣ .. ⎦ αk−1 σˆ k−1 ⎡

⎡

⎤ σ0 ⎢ σ1 ⎥ ∗ ⎥ γ1∗ · · · γk−1 ]×⎢ ⎣ ... ⎦ . σk−1

(38)

33


On the other hand, the analysis–synthesis operator is associated with the matrix R(t k , α)T × R(t k , γ ). By Theorem IV.4, this matrix is a (k × k)block Toeplitz matrix, whose generating function is the polynomial matrix: = ∆k (α)∗ × ∆k (γ )T . The above highlights the following general result: Proposition V.1 Let Q : P → P be a linear operator associated with the block Toeplitz matrix R(t, Ω), where Ω is the square polynomial matrix ⎡

ω0,0 ⎢ ω1,0 Ω=⎢ ⎣ ...

ωk−1,0

··· ··· .. .

ω0,1 ω1,1 .. . ω1,k−1

⎤ ω0,k−1 ω1,k−1 ⎥ ⎥ ωi, j ∈ P . .. ⎦ .

· · · ωk−1,k−1

Then the image σˆ = Qσ of the Laurent polynomial σ under the operator Q can be described in terms of its k-decimated polynomials as follows: ∆k (σˆ ) = Ω∗ × ∆k (σ ), where ⎡

∗ ω0,0

⎢ ω∗ ⎢ Ω∗ = ⎢ 1,0 ⎣ ... ∗ ωk−1,0

∗ ω0,1 ∗ ω1,1 .. . ∗ ω1,k−1

···

··· .. .

∗ ω0,k−1 ∗ ω1,k−1 .. .

∗ · · · ωk−1,k−1

⎤

⎥ ⎥ ⎥. ⎦

B. Matrix Description of Analysis Filter Banks In a maximally decimated M-channel analysis bank the input signal is divided into M filtered versions of equal bandwith, which are then maximally decimated, namely, subsampled by the same integer M, according to the scheme shown below, where, for i = 0, 1, . . . , M − 1, we have denoted by

34


Fγ (i) (γ (i) ∈ P ) the analysis filters:

Hence, the size M analysis operator, which associates the M-tuple (σ (0) , σ , . . . , σ (M−1) ) of its filtered and decimated versions to the input signal σ , is uniquely represented by the M-tuple of Laurent polynomials (1)

γ = γ (0) , γ (1) , . . . , γ (M−1) , and will be denoted by the symbol Cγ , namely, Cγ (σ ) = D0M Fγ (0) (σ ), D0M Fγ (1) (σ ), . . . , D0M Fγ (M−1) (σ ) . The M-tuple γ will be called the analysis filter vector. Its zeroth component γ (0) usually denotes a low-pass filter, while the components γ (1) , γ (2) , . . . , γ (M−1) denote high-pass filters. Recalling that, by (25), each operator D0M Fγ (i) is associated with the banded Hurwitz matrix Ci := R(t M , γ (i) ), the size M analysis operator Cγ can be associated with the matrix column vector ⎡

⎤ C0 ⎢ C1 ⎥ ⎢ . ⎥. ⎣ .. ⎦ C M−1

35


Its action can therefore be represented as: ⎤ ⎤ ⎡ [σ (0) ] C0 ⎢ [σ (1) ] ⎥ ⎢ C ⎥ ⎢ ⎥ ⎢ 1 ⎥ .. ⎢ ⎥ = ⎢ .. ⎥ × [σ ]. ⎣ ⎦ ⎣ . ⎦ . (M−1) C M−1 [σ ] ⎡

(39)

Making use of formula (36), we immediately get a computationally more efficient description of the action of an analysis filter bank, as follows: ⎡ (0) ∗ γ0 ⎢ (1) ⎥ ⎢ ⎢ σ ⎥ ⎢ (1) ∗ ⎢ ⎥ ⎢ γ0 ⎢ ⎥=⎢ ⎢ .. ⎥ ⎢ ⎢ . ⎥ ⎢ ··· ⎣ ⎦ ⎣ (M−1) ∗ (M−1) σ γ0 ⎡

σ (0)

⎤

(0) ∗ γ1 (1) ∗ γ1 ···

(M−1) ∗ γ1

···

(0) ∗ γ M−1 (1) ∗ γ M−1

⎤

⎡

σ0

⎤

⎥ ⎢ ⎥ ⎥ ⎢ σ1 ⎥ ⎥ ⎢ ⎥ ⎥×⎢ ⎥ ⎥ ⎢ . ⎥, ··· . . . ⎥ ⎢ .. ⎥ ⎦ ⎣ ⎦ (M−1) ∗ σ M−1 · · · γ M−1 ···

or, equivalently, ⎡

⎤ σ (0) ⎢ σ (1) ⎥ ⎢ .. ⎥ = ∆ M (γ)∗T × ∆ M (σ ), ⎣ . ⎦ σ (M−1)

(40)

where ∆ M (γ)∗ is the starred M-decimated matrix of γ.

C. Matrix Description of Synthesis Filter Banks A maximally upsampled M-channel synthesis bank upsamples by M the M input signals and filters them, before summing them to create a single output signal. This is represented in the scheme below, where, for i = 0, 1, . . . ,

36


M − 1, we have denoted by Fα(i)∗ (α (i) ∈ P ) the synthesis filters:

Hence, the size M synthesis operator that associates the output σˆ to the M input signals (σ (0) , σ (1) , . . . , σ (M−1) ), is uniquely represented by the M-tuple of Laurent polynomials α = α (0) , α (1) , . . . , α (M−1) , and will be denoted by the symbol Aα , namely,

M−1 Aα σ (0) , σ (1) , . . . , σ (M−1) = Fα(i)∗ U M σ (i) . i=0

The M-tuple α will be called the synthesis filter vector. As in the analysis case, the component α (0) is a low-pass filter, and the components α (1) , α (2) , . . . , α (M−1) are high-pass filters. Recalling that, by (26), each operator Fα(i)∗ U M is associated with the transpose of the banded Hurwitz matrix Ai := R(t M , α (i) ), the size M synthesis operator Aα can be associated with the matrix row vector T A0 A1T · · · A TM−1 . Its action can therefore be represented as:

[σˆ ] =

A0T

⎡

⎢ ⎢ A1T · · · A TM−1 × ⎢ ⎣

[σ (0) ] [σ (1) ] .. . [σ (M−1) ]

⎤

⎥ ⎥ ⎥. ⎦

(41)

A description of the M-decimated vector of the output signal σˆ , can be obtained

37


by exploiting formula (37): ⎡ ⎤ ⎡ (0) σˆ 0 α0 ⎥ ⎢ (0) ⎢ ⎢ σˆ 1 ⎥ ⎢ α1 ⎢ ⎥ ⎢ ⎢ . ⎥=⎢ . ⎢ .. ⎥ ⎢ .. ⎣ ⎦ ⎣ (0) σˆ M−1 α(M−1)

or, equivalently,

α0(1) α1(1) .. . (1) α(M−1)

· · · α0(M−1)

⎤

⎡

σ (0)

⎥ ⎢ · · · α1(M−1) ⎥ ⎢ σ (1) ⎥ ⎢ ×⎢ . .. .. ⎥ ⎢ . . ⎥ . ⎦ ⎣ .

(M−1) · · · α(M−1)

σ (M−1)

⎤ σ (0) ⎢ σ (1) ⎥ . ⎥ ∆ M (σˆ ) = ∆ M (α) × ⎢ ⎣ .. ⎦ . σ (M−1) ⎡

⎤

⎥ ⎥ ⎥ ⎥, ⎥ ⎦

(42)

VI. M-Channel Filter Bank Systems In the present section, we take advantage of the preceding considerations to give an algebraic description of a maximally decimated M-channel filter bank FIR system. This kind of system accepts as input a signal, hence, a Laurent polynomial, and produces as output another Laurent polynomial. It can therefore be interpreted as a linear operator on the vector space P of Laurent polynomials. The theoretical results stated above allow us to give an explicit description of this operator by means of its associated matrix, which turns out to be recursive. By studying the algebraic features of this matrix we can recover the conditions for the alias cancelation and perfect reconstruction properties of the analysis–synthesis filter bank system. Such a system is usually described in the literature using the polyphase decomposition of the filters. In the present algebraic setting, this corresponds to considering the decimated vectors of the Laurent polynomials representing the analysis and synthesis filters. Hence, the polyphase approach can be recovered using the description in terms of generating functions of the recursive matrices involved, and recalling the bijection between a Laurent polynomial and its decimated vector.

A. Matrix Description A maximally decimated M-channel filter bank consists of a size M analyisis operator Cγ , followed by a size M synthesis operator Aα , according to the

38


following scheme:

It is therefore represented by the linear operator Lα,γ = Aα Cγ . Hence, an M-channel filter bank is uniquely identified by the two M-tuples of Laurent polynomials γ = γ (0) , γ (1) , . . . , γ (M−1) , α = α (0) , α (1) . . . , α (M−1) ,

namely, the analysis and synthesis filter vectors, and represented by the two M-tuples of Hurwitz matrices (C0 , C1 , . . . , C M−1 ), where, for i = 0, 1, . . . , M − 1, Ci = R t M , γ (i) ,

(A0 , A1 , . . . , A M−1 ), Ai = R t M , α (i) .

Making use of formulas (39) and (41), it is easily seen that the action of the linear operator Lα,γ can be represented as ⎤ ⎡ C0 ⎢ C1 ⎥ ⎥ [σˆ ] = A0T A1T · · · A TM−1 × ⎢ ⎣ ... ⎦ × [σ ]. C M−1 Hence, by (27), its associated matrix is

A0T × C0 + A1T × C1 + · · · + A TM−1 × C M−1 .

(43)

39


By Theorem IV.4, every summand in (43) is a block Toeplitz matrix, hence, so is the matrix associated with the linear operator Lα,γ . We have therefore the following result, whose proof is a straightforward consequence of the same theorem: Theorem VI.1 Let Ai = R(t M , α (i) ), Ci = R(t M , γ (i) ), i = 0, 1, . . . , M − 1, be banded Hurwitz matrices. The matrix A0T × C0 + A1T × C1 + · · · + A TM−1 × C M−1 is an (M × M)-block Toeplitz matrix, whose generating function is the polynomial matrix given by ⎡ (0) ∗ α0 ⎢ ∗ ⎢ α (0) ⎢ = ⎢ 1. ⎢ .. ⎣ (0) ∗ α M−1

(1) ∗ α0 (1) ∗ α1 .. . (1) ∗ α M−1

∗ ⎤ ⎡ (0) γ0 γ1(0) · · · α0(M−1) ∗ ⎥ ⎢ ⎢ (1) γ1(1) · · · α1(M−1) ⎥ ⎥ ⎢ γ0 ⎥×⎢ . .. .. .. ⎥ ⎢ .. . . . ⎦ ⎣ (M−1) ∗ · · · α M−1 γ0(M−1) γ1(M−1)

⎤

(0) · · · γ M−1

(1) · · · γ M−1 .. .. . . (M−1) · · · γ M−1

⎥ ⎥ ⎥ ⎥, ⎥ ⎦

or, equivalently, = ∆ M (α)∗ × ∆ M (γ)T . The preceding result represents one of the most interesting achievements of the present algebraic approach, as it gives a description of a filter bank system both in terms of the associated matrix (time-domain formulation) and of generating function (polyphase formulation). In fact, using Proposition V.1, we immediately get the following description of the action of Lα,γ in terms of M-decimated vectors: ∆ M (σˆ ) = ∗ × ∆ M (σ ) = ∆ M (α) × ∆ M (γ)∗T × ∆ M (σ ).

(44)

The matrix ∗ is usually called the transmission matrix or transfer function matrix in the filter bank literature, and (44) is the well-known polyphase characterization of the action of an M-band filter bank.

40


B. Alias-Free Filter Banks The output σˆ of an M-channel filter bank is clearly better, the closer it is to the corresponding input signal σ . The best case is obviously when the output is a shifted and possibly scaled version of the input: in this case, the filter bank is said to have the perfect reconstruction property. This situation will be discussed in the next subsection. In this subsection we will consider the most important case of approximate reconstruction, namely, the case in which σˆ can be obtained from σ by multiplying it by a fixed Laurent polynomial. In the filter bank language this is called alias-free reconstruction, and the analysis–synthesis system is said to possess the alias-free property. Since such a system corresponds to a linear operator, we restate in algebraic language the definition of alias-free property for a linear operator on P . A linear operator Q : P → P will be called alias-free if and only if there exists a nonzero Laurent polynomial τ such that σˆ = τ σ for every Laurent polynomial σ . Equivalently, Q is alias-free if and only if it is a filter, namely, its associated matrix is a banded Toeplitz matrix. In this case, the Laurent polynomial τ will be said to be the distortion function of the operator Q. We note that, when Q is the linear operator describing the combined action of the analysis and synthesis operators of an M-channel filter bank system, the preceding definition gives the usual notion of alias-free system, as given, for example, in Vaidyanathan (1993). As we have already seen, the matrix associated with the linear operator describing the action of an M-channel filter bank system is a block Toeplitz matrix. Hence, the alias cancelation condition is equivalent to requiring that this matrix is also a scalar Toeplitz matrix, whose generating function is the distortion function of the system. The connection between block and scalar Toeplitz matrices has been analyzed in detail in Section IV.A, hence, using Theorem IV.1, we are able to characterize alias-free systems and give an explicit expression of their distortion function. Theorem VI.2 Let Lα,γ be the linear operator describing the action of an M-channel filter bank relative to the two M-tuples γ = γ (0) , γ (1) , . . . , γ (M−1) ,

α = α (0) , α (1) . . . , α (M−1) ,


41

The operator Lα,γ is alias-free if and only if the matrix = ∆ M (α)∗ × ∆ M (γ)T ⎡ (0) ∗ (1) ∗ ∗⎤ ⎡ (0) (0) ⎤ γ0 γ1(0) · · · γ M−1 · · · α0(M−1) α0 α0 ⎥ ⎢ ∗ (1) ∗ ∗⎥ ⎢ (1) ⎥ ⎢ (1) ⎢ α (0) γ1(1) · · · γ M−1 · · · α1(M−1) ⎥ α1 ⎥ ⎢ γ0 ⎥ ⎢ 1 =⎢ . ⎥×⎢ . .. .. .. ⎥ .. .. .. . ⎥ ⎢ ⎢ .. . ⎥ . . . . . ⎣ ⎦ ⎣ . ⎦ (0) ∗ (1) ∗ (M−1) ∗ (M−1) (M−1) (M−1) α M−1 · · · α M−1 α M−1 γ1 · · · γ M−1 γ0

is t-circulant. In this case, the distortion function of Lα,γ is the Laurent polynomial τ=

M−1 i=0

∗ α0(i) ◦ t M γ (i) .

Proof. The first statement is an immediate consequence of Theorems IV.1 and VI.1. We point out that, in the alias-free case, by Theorem VI.1, the generating function of the scalar Toeplitz matrix is given by φ=

M−1 j=0

tj

M−1 i=0

(i) ∗ M (i) M α0 (◦t ) γ j ◦ t ,

or, equivalently, φ=

M−1 i=0

(i) (i) ∗ M−1 (i) ∗ M M−1 α0 ◦ t M γ (i) . α0 (◦t ) t j γj ◦ tM = j=0

i=0

Hence, the distortion function is τ = φ∗ =

M−1 i=0

(i) M (i) ∗ α0 ◦ t (γ ) .

An interesting question in filter bank design is whether, given an analysis filter vector, it is possible to choose an appropriate synthesis filter vector in order to cancel aliasing. The answer to this problem is given by the next result (see Vetterli and Kovacevic, 1995). Proposition VI.1 Given an analysis filter vector γ = γ (0) , γ (1) , . . . , γ (M−1) ,

42


alias-free reconstruction is possible if and only if the determinant of the M-decimated matrix ∆ M (γ) is different from the zero polynomial, namely, the matrix ∆ M (γ) has full rank. Obviously, the dual problem of finding an alias-canceling analysis filter vector when a synthesis filter vector is given has a similar solution.

C. Perfect Reconstruction Filter Banks We now come to the characterization of perfect reconstruction systems. Let Lα,γ be the linear operator describing the action of an M-channel filter bank system relative to the two M-tuples γ = γ (0) , γ (1) , . . . , γ (M−1) , α = α (0) , α (1) . . . , α (M−1) .

The operator Lα,γ is said to have the perfect reconstruction property whenever a nonzero constant c ∈ K and an integer h exist such that Lα,γ (σ ) = c t −h σ for every Laurent polynomial σ . This is equivalent to the fact that Lα,γ is alias-free and its associated matrix is the scalar Toeplitz matrix R(t, c t h ). Theorem VI.2 immediately yields the following algebraic characterization of perfect reconstruction systems: Theorem VI.3 Let Lα,γ be the linear operator describing the action of an M-channel filter bank system relative to the two M-tuples γ = γ (0) , γ (1) , . . . , γ (M−1) , α = α (0) , α (1) . . . , α (M−1) .

The operator Lα,γ has the perfect reconstruction property if and only if both of the following conditions are satisfied: 1. the matrix = ∆ M (α)∗ × ∆ M (γ)T is t-circulant; 2. the following identity holds: M−1 i=0

(i) ∗ M (i) α0 ◦ t γ = c t h ,


43

that is, setting h = q M + r, with 0 ≤ r < M, M−1 (i) ∗ (i) 0 for j = r α0 γ j = c t q for j = r. i=0

In other words, the perfect reconstruction condition is equivalent to requiring that the matrix above is t-circulant and the elements of its first row are all zero polynomials, except one, which must be a monomial. If this is the case, we can always suitably shift and scale the filters involved in such a way that the matrix is simply the identity matrix. So, without loss of generality, we can restate the perfect reconstruction condition as follows: Proposition VI.2 The operator Lα,γ describing the action of an M-channel filter bank system has the perfect reconstruction property if and only if the following condition is satisfied: ∆ M (α)∗ × ∆ M (γ)T = I.

(45)

Due to the fact that we are working with FIR filters, namely, the filter vectors α and γ are M-tuples of Laurent polynomials, it follows from the above condition (45) that the determinants of both matrices ∆ M (α)∗ and ∆ M (γ) are nonzero monomials. As a consequence, the two matrices are the inverse one of the other, hence, condition (45) is equivalent to the following: ∆ M (γ)T × ∆ M (α)∗ = I.

(46)

The perfect reconstruction condition (46) can also be stated in terms of Hurwitz matrices: Proposition VI.3 The M-channel filter bank described by the two filter vectors γ, α corresponds to a perfect reconstruction operator if and only if the Hurwitz matrices Ai = R t M , α (i) , Ci = R t M , γ (i) , i = 0, 1, . . . , M − 1 satisfy the following conditions:

Ci × A Tj = δi, j I,

i, j = 0, 1, . . . , M − 1,

or, in matrix notation, ⎤ ⎡ ⎤ ⎡ C0 I 0 ··· 0 ⎢ C1 ⎥ T T ⎢0 I ··· 0 ⎥ ⎢ . ⎥ × A A · · · AT 0 1 M−1 = ⎣· · · · · · · · · · · ·⎦ . ⎣ .. ⎦ 0 0 ··· I C M−1

(47)

44


Proof. It is sufficient to remark that condition (46) can be equivalently written as T ∗ ∆ M γ (i) × ∆ M α ( j) = δi j ,

i, j = 0, 1 . . . , M − 1.

(48)

The assertion now follows from Theorem IV.2. A pair (α, γ) satisfying one of the above equivalent conditions (45), (46), or (47) will be called a pair of dual (or biorthogonal) filter vectors. The simplest pair of dual filter vectors is the “trivial” pair (δ, δ), where δ = (1, t, t 2 , . . . , t M−1 ). As in the alias-free case, a crucial problem in FIR filter bank design is finding conditions under which, given an analysis filter bank, there exists a corresponding synthesis bank yielding a perfect reconstruction system. As we have already remarked, the answer to this problem can be given in algebraic language as follows: Proposition VI.4 Given an analysis filter vector γ = γ (0) , γ (1) , . . . , γ (M−1) ,

perfect reconstruction is possible if and only the determinant of the Mdecimated matrix ∆ M (γ) is a nonzero monomial. If this is the case, the dual M-tuple α of γ is uniquely determined by condition (45). Obviously, if (α, γ) is a pair of dual filter vectors, the same holds for the pair (γ, α).

VII. Transmultiplexers In the previous section, we considered the problem of decomposing a given signal into components, from which the original signal can be recovered. We will now consider the dual problem of starting from many input signals and combining them into a single output, from which the original signals can be recovered. This problem has some important applications, for example, in digital telephone networks, where several users share a common channel to transmit information. An efficient solution to this problem can be given in terms of synthesis and analysis filter banks, so all the algebraic tools previously developed can also be used in this dual context.

45


A. Matrix Description A device synthesizing a single signal from M input signals, followed by the inverse operation of recovering the initial M inputs, is called an Mtransmultiplexer. It can be algebraically seen as a size M synthesis operator Aα , followed by a size M analyisis operator Cγ according to the following scheme:

It is therefore represented by the linear operator Tα,γ = Cγ Aα . As in the subband coding case, an M-transmultiplexer is also uniquely identified by the two M-tuples of Laurent polynomials γ = γ (0) , γ (1) , . . . , γ (M−1) , α = α (0) , α (1) . . . , α (M−1) ,

namely, the synthesis and analysis filter vectors, and represented by the two M-tuples of Hurwitz matrices (A0 , A1 , . . . , A M−1 ),

where, for i = 0, 1, . . . , M − 1, Ai = R t M , α (i) ,

(C0 , C1 , . . . , C M−1 ), Ci = R t M , γ (i) .

Making use of formulas (39) and (41), the action of the linear operator Tα,γ can be represented as ⎤ ⎤ ⎡ ⎤ ⎡ ⎡ [σˆ (0) ] [σ (0) ] C0 ⎢ [σˆ (1) ] ⎥ ⎢ C1 ⎥ T T ⎢ [σ (1) ] ⎥ ⎥. ⎢ ⎥ = ⎢ . ⎥ × A A · · · AT ⎢ (49) .. . 0 1 M−1 × ⎣ ⎦ ⎣ ⎦ ⎣ .. ⎦ .. . [σˆ (M−1) ]

C M−1

[σ (M−1) ]

46


Its associated matrix is therefore ⎡ ⎤ C0 × A0T C0 × A1T · · · C0 × A TM−1 ⎥ ⎢ ⎢ C1 × A0T C1 × A1T · · · C1 × A TM−1 ⎥ ⎥. ⎢ ⎣ ⎦ ··· ··· ··· T T T C M−1 × A0 C M−1 × A1 · · · C M−1 × A M−1

(50)

By Theorem IV.2, every entry Ci × A Tj in (50) is a Toeplitz matrix with generating function T ∗ λi, j = ∆ M γ (i) × ∆ M α ( j) .

Hence, the matrix associated with the linear operator Tα,γ is a Toeplitz block matrix. The above matrix represents a description of the transmultiplexing system from an algebraic point of view, that corresponds to the so-called time-domain formulation. The analog of the polyphase formulation is obtained equivalently either by translating formula (49) in terms of generating functions, or by exploiting the description of synthesis and analysis filter banks given in (42) and (40), respectively. The first approach yields the following expression for the ith output signal: σˆ (i) =

M−1 j=0

λi,∗ j σ ( j) =

M−1 j=0

T ∗ ∆ M γ (i) × ∆ M α ( j) σ ( j) ,

(51)

while the second point of view gives the result stated below.

Proposition VII.1 Let α, γ be the synthesis and analysis filter vectors identifying an M-transmultiplexer. The action of the corresponding linear operator Tα,γ = Cγ Aα can be described as follows: ⎡ (0) ⎤ ⎡ (0) ⎤ σ σˆ (1) ⎥ ⎢ ⎢ σˆ (1) ⎥ ⎢ .. ⎥ = ∆ M (γ)∗T × ∆ M (α) × ⎢ σ .. ⎥ . ⎣ . ⎦ ⎣ . ⎦ (M−1) σ (M−1) σˆ B. Perfect Reconstruction Transmultiplexers The above matrix description of transmultiplexers allows us to simply characterize filter vectors that yield cross-talk-free and perfect reconstruction transmultiplexers. These two kinds of systems are the analogs of alias-free and perfect reconstruction filter banks.


47

In particular, an M-transmultiplexer (and, hence, the corresponding operator) is said to be free from cross-talk when every output signal σˆ (i) only depends on the corresponding input σ (i) , namely, nonzero Laurent polynomials τ (0) , τ (1) . . . , τ (M−1) exist such that, for every i = 0, 1, . . . , M − 1, σˆ (i) = τ (i) σ (i) . This last condition is clearly satisfied whenever the Toeplitz block matrix (50) associated with the linear operator Tα,γ is block diagonal. This happens if and only if Ci × A Tj = 0 for i = j,

(52)

namely, T ∗ λi, j := ∆ M γ (i) × ∆ M α ( j) = 0 for i = j.

If this is the case, for every i = 0, 1, . . . , M − 1, ∗ σ (i) . σˆ (i) = λi,i

Similarly, by using the synthesis–analysis filter bank description, the crosstalk-free condition can be stated as follows: Proposition VII.2 The M-transmultiplexer identified by the synthesis and analysis filter vectors α, γ is free from cross-talk if and only if the matrix ∆ M (γ)∗T × ∆ M (α) is a diagonal matrix. The question whether, given a synthesis filter vector, it is possible to choose an appropriate analysis filter vector in order to cancel cross-talk, can be asked here as well. The answer to this problem is similar to the alias-free case. In fact, easy considerations lead to the following result: Proposition VII.3 Given a synthesis filter vector α = (α (0) , α (1) , . . . , α (M−1) ), cross-talk elimination is possible if and only if the determinant of the Mdecimated matrix ∆ M (α) is different from the zero polynomial, namely, the matrix ∆ M (α) has full rank. We remark that, even if the condition stated in the preceding proposition is the same as the condition given in Proposition VI.1, the analogy holds only

48


in the case when a filter vector (analyisis or synthesis) is given, and we are looking for a suitable filter vector allowing for alias or cross-talk cancelation. On the contrary, when a complete system is given, namely, both filter vectors are fixed, the alias-free and cross-talk-free conditions are not in general equivalent. The situation is completely different if we consider perfect reconstruction systems. In fact, the operator Tα,γ describing the action of an M-transmultiplexer is said to have the perfect reconstruction property when every output signal σˆ (i) coincides with the corresponding input σ (i) , namely, for every i = 0, 1, . . . , M − 1, σˆ (i) = σ (i) . The matrix description of the action of Tα,γ allows us to state the perfect reconstruction property as follows: Proposition VII.4 The operator Tα,γ describing the action of an M-transmultiplexer has the perfect reconstruction property if and only if one of the following equivalent conditions is satisfied: (53) Ci × A Tj = δi, j I, i, j = 0, 1, . . . , M − 1, M (i) M (i) where Ai = R t , α , Ci = R t , γ , i = 0, 1, . . . , M − 1; or ∆ M (γ)T × ∆ M (α)∗ = I.

(54)

The two equivalent conditions above are exactly the same as those given in Propositions VI.2 and VI.3, showing that, in the perfect reconstruction case, transmultiplexers and analysis–synthesis systems are dual. In fact, we have: Proposition VII.5 A perfect reconstruction M-channel filter bank system is equivalent to a perfect reconstruction M-transmultiplexer. This implies that Proposition VI.4, originally stated for an analysis–synthesis system, remains valid in the transmultiplexing case as well.

VIII. The M-Band Lifting So far, we have mostly been interested in studying conditions under which two assigned filter vectors yield a perfect reconstruction system. However, perfect reconstruction is only one of the requirements needed in practical applications, since in most cases the filters themselves have to satisfy some properties, such


49

as frequency selectivity, phase characteristics, and computational efficiency. All this together makes the design of an M-band perfect reconstruction system a complex problem, due to the great number of parameters involved, and the difficulties grow as soon as the number of bands taken into consideration increases. In the two-band case, a powerful tool, yielding an easy custom-design construction of biorthogonal filters, is the well-known lifting scheme (Sweldens, 1996, 1997), which on the one hand can be seen as a simple tool for the construction of biorthogonal filters with preassigned features (building property), and on the other hand yields a factorization procedure that leads to faster analysis–synthesis algorithms (decomposition property). The aim of this section is to exploit the algebraic tools previously developed in order to carefully analyze both aspects of the lifting scheme in the M-band setting and to show and characterize its capabilities.

A. Algebraic Preliminaries The lifting scheme can be described as a procedure that starts from a simple perfect reconstruction system and builds new biorthogonal pairs of filter vectors, so that the new filter coefficients satisfy prespecified requirements. To give a mathematical description of this procedure, it is necessary, given a dual pair (α, γ) of biorthogonal filter vectors, to characterize in algebraic form all the dual pairs (α, ¯ γ) ¯ that share the same zeroth component α (0) of α (lowpass filter) and the same (M − 1)-tuple of components (γ (1) , γ (2) , . . . , γ (M−1) ) of γ (high-pass filters). To do this, we recall that a pair of filter vectors (α, γ) is a dual (or biorthogonal) pair whenever it satisfies identity (46), or, equivalently, the following identity: ∆ M (γ)∗T × ∆ M (α) = I

(55)

and we give two preliminary results, that represent the theoretical support of the lifting scheme. Theorem VIII.1 Let (α, γ) be a dual pair of filter vectors. For any fixed index i = 1, 2, . . . , M − 1, every Laurent polynomial γˆ (i) such that ∗T ∆ M γˆ (i) × ∆ M α ( j) = δi j

j = 1, 2, . . . , M − 1

(56)

50


is given by γˆ (i) = γ (i) + μ(i) ◦ t M γ (0)

as μ(i) ranges over the set of all Laurent polynomials. Proof. Conditions (56) yield a system of M-1 linear equations in the M (i) unknowns (γˆ0(i) )∗ , (γˆ1(i) )∗ , . . . , (γˆ M−1 )∗ , namely, ⎧ (1) (i) ∗ (i) ∗ =0 + · · · + α (1) ˆ M−1 α0 γˆ0 ⎪ M−1 γ ⎪ ⎪ ⎪ . . . ⎪ ⎨ (i) ∗ ∗ (57) α0(i) γˆ0(i) + · · · + α (i) =1 ˆ M−1 M−1 γ ⎪ ⎪ . . . ⎪ ⎪ ⎪ (i) ∗ ⎩ (M−1) (i) ∗ + · · · + α (M−1) ˆ M−1 = 0 γˆ0 α0 M−1 γ

The M-tuple ∆ M (γ (i) )∗ is a solution of the system (57), while ∆ M (γ (0) )∗ is a solution of the associated homogeneous linear system. Hence, in the quotient field L+ of Laurent series, the general solution of (57) is given by ∗ ∗ χ = ∆ M γ (i) + τ ∆ M γ (0) , (58)

where τ is a Laurent series. In other terms, the components of the vector χ are given by: ⎡ (i)∗ ⎡ ⎤ ⎤ χ0 γ0 + τ γ0(0)∗ ⎢ χ ⎥ ⎢ (i)∗ ⎥ ⎢ 1 ⎥ ⎢ γ + τ γ (0)∗ ⎥ 1 1 ⎥. ⎥ ⎢ ⎢ ⎥ ⎢ .. ⎥ = χ = ⎢ .. ⎦ ⎣ ⎣ . ⎦ . χ M−1

(i)∗ (0)∗ γ M−1 + τ γ M−1

We now need to characterize the polynomial solutions of system (57). We have assumed that the filter vector γ admits a dual vector α, which implies that det∆ M (γ)∗ is a nonzero monomial. By using the Laplace expansion formula for the determinant, we find that the g.c.d. of the M-tuple ((γ0(0) )∗ , (γ1(0) )∗ , . . . , (0) ∗ ) ) is a monomial. This implies that the solution χ given by (57) is a (γ M−1 polynomial vector if and only if τ is a Laurent polynomial. At this point we note that, given any polynomial solution χ of (57), its dual polynomial vector is ⎡ (i) ⎤ γ0 + τ ∗ γ0(0) ⎢ (i) ⎥ ⎢ γ + τ ∗ γ (0) ⎥ ⎥ 1 1 χ∗ = ⎢ ⎢ ⎥ .. ⎣ ⎦ . (0) (i) + τ ∗ γ M−1 γ M−1


51

and hence, by the reconstruction rule (17), the Laurent polynomial whose M-decimated vector coincides with χ∗ is ∗ ◦ tM) γˆ (i) = χ0∗ ◦ t M + t(χ1∗ ◦ t M ) + · · · + t M−1 (χ M−1 = γ0(i) ◦ t M + (τ ∗ ◦ t M ) γ0(0) ◦ t M + t γ1(i) ◦ t M + (τ ∗ ◦ t M ) γ1(0) ◦ t M (i) (0) + t M−1 γ M−1 ◦ t M + (τ ∗ ◦ t M ) γ M−1 ◦ tM

= γ (i) + (τ ∗ ◦ t M )γ (0)

the thesis now follows by setting μ(i) = τ ∗ . Theorem VIII.2 Given a biorthogonal pair of filter vectors (α, γ), every Laurent polynomial γˆ (0) such that ∗T × ∆ M α (0) = 1 ∆ M γˆ (0)

is given by

(59)

γˆ (0) = γ (0) + λ(1) ◦ t M γ (1) + · · · + λ(M−1) ◦ t M γ (M−1) ,

as λ(1) , λ(2) , . . . , λ(M−1) range over the set of all Laurent polynomials. Proof. Condition (59) yields a linear equation in the M unknowns (γˆ0(i) )∗ , namely,

(i) )∗ , (γˆ1(i) )∗ , . . . , (γˆ M−1

∗ ∗ (0) ∗ α0(0) γˆ0(0) + α1(0) γˆ1(0) + · · · + α (0) ˆ M−1 = 1. M−1 γ

(60)

The M-tuple ∆ M (γ (0) )∗ is a solution of equation (60), while the M-tuples ∆ M (γ (1) )∗ , ∆ M (γ (2) )∗ , . . . , ∆ M (γ (M−1) )∗ are M − 1 linearly independent solutions of the associated homogeneous equation. Hence, the general solution of (60) is given by the vector ∗ ∗ ∗ (61) χ = ∆ M γ (0) + λ(1) ∆ M γ (1) + · · · + λ(M−1) ∆ M γ (M−1) ,

where λ(1) , λ(2) , . . . , λ(M−1) belong to the field of Laurent series. We now prove that the solution χ is a polynomial vector if and only if every λ(i) is a Laurent polynomial. Indeed, suppose that χ is a polynomial solution of Eq. (60). Without loss of generality, we can write every λ(i) as the ratio of two Laurent polynomials, namely, λ(i) = τ (i) /ρ, with τ (i) , ρ ∈ P , i = 1, 2, . . . , M − 1. Then, setting χ′ := ρ(χ − ∆ M γ (0)∗ ), the (M-1)-tuple (τ (1) , . . . , τ (M−1) ) is a polynomial solution of the system of M linear equations in the M − 1

52


unknowns φ (1) , . . . , φ (M−1) : ⎡ (1) ∗ (2) ∗ γ0 ... γ0 (2) ∗ ⎢ (1) ∗ ⎢ γ1 γ1 ... ⎢ ⎢. . . ... ... ⎣ (1) ∗ (2) ∗ ... γ M−1 γ M−1

γ0(M−1)

∗ ⎤

⎡

φ (1)

⎤

⎡

χ0′

⎤

∗ ⎥ ⎢ (2) ⎥ ⎢ ′ ⎥ ⎢ φ ⎥ ⎢ χ1 ⎥ γ1(M−1) ⎥ ⎥×⎢ ⎥ ⎢ ⎥ ⎥ ⎢ .. ⎥ = ⎢ .. ⎥ , ... ⎦ ⎣ . ⎦ ⎣ . ⎦ (M−1) ∗ ′ χ M−1 φ (M−1) γ M−1

(62)

where we have denoted by χi′ the ith component of the polynomial vector χ′ . It is known (Heger, 1858, p. 111) that a linear system whose coefficients belong to a Euclidean domain E admits a solution in E whenever its matrix of coefficients has the same rank k, say, as the augmented matrix, and the g.c.d. of the nonzero minors of order k of the augmented matrix divides the g.c.d. of the minors of the same order of the matrix of coefficients. In our case, the M minors of order M − 1 of the matrix of coefficients are the cofactors of the elements of the zeroth row of the matrix ∆ M (γ)∗ , and therefore their g.c.d. is a monomial. On the other hand, the g.c.d. of the minors of order M − 1 of the augmented matrix is divisible by ρ, and this implies that ρ is a monomial. Theorems VIII.1 and VIII.2 can be equivalently formulated in terms of Hurwitz matrices. In fact, Propositions II.2 and VI.3 allow us to state the following results: Theorem VIII.3 Let (α, γ) be a pair of dual filter vectors. Set Ai = R t M , α (i) , Ci = R t M , γ (i) , i = 0, 1, . . . , M − 1.

For any fixed index i = 1, 2, . . . , M − 1, every Hurwitz matrix Cˆ i = R(t M , γˆ (i) ) such that Cˆ i × A Tj = δi j I, j = 1, 2, . . . , M − 1 is given by Cˆ i = Ci + Ti × C0 ,

as Ti = R t, μ(i) ranges over the set of banded Toeplitz matrices.

(63)

Theorem VIII.4 Let (α, γ) be a pair of dual filter vectors. Set Ai = R t M , α (i) , Ci = R t M , γ (i) , i = 0, 1, . . . , M − 1. Every Hurwitz matrix Cˆ 0 = R t M , γˆ (0) such that Cˆ 0 × A0T = I

is given by Cˆ 0 = C0 +

M−1 j=1

Q j × Cj,

(64)


53

as Q j = R t, λ( j) , j = 1, 2, . . . , M − 1

range over the set of all banded Toeplitz matrices.

B. Lifting and Dual Lifting The preceding results allow us to extend the lifting scheme to the M-band case, giving a characterization of the algebraic form of all biorthogonal pairs of finite filter vectors that can be obtained from a given pair, either by modifying the analysis low-pass filter and consequently updating the synthesis high-pass filters (lifting), or by modifying the analysis high-pass filters and consequently updating the synthesis low-pass filter (dual lifting). This characterization only makes use of linear combinations of Laurent polynomials, therefore dramatically reducing the computational complexity of designing M-band systems with good filtering properties. Theorem VIII.5 (Lifting Scheme) Let (α, γ) be a dual pair of filter vectors, with α = α (0) , α (1) , . . . , α (M−1) , γ = γ (0) , γ (1) , . . . , γ (M−1) .

Every dual pair (α, γ) of the form α = α (0) , αˆ (1) , . . . , αˆ (M−1) , can be obtained by choosing

γˆ (0) = γ (0) + αˆ (i) = α (i) −

λ(i)

∗

γ = γˆ (0) , γ (1) , . . . , γ (M−1) . M−1 j=1

( j) M ( j) λ ◦t γ ,

◦ t M α (0) ,

i = 1, 2, . . . , M − 1,

(65)

(66) (67)

where λ(1) , λ(2) , . . . , λ(M−1) are arbitrary Laurent polynomials. Proof. Set Ai = R(t M , α (i) ), Ci = R(t M , γ (i) ), i = 0, 1, . . . , M − 1, Aˆ i = R(t M , αˆ (i) ), i = 1, 2, . . . , M − 1, Cˆ 0 = R(t M , γˆ (0) ). By Proposition VI.3, the condition of (α, γ) being a dual pair is equivalent to the following three relations involving the associated Hurwitz matrices: i) Cˆ 0 × A0T = I ,

ii) C j × Aˆ iT = δi j I,

iii) Cˆ 0 × Aˆ iT = 0,

i, j = 1, 2, . . . , M − 1, and i = 1, 2, . . . , M − 1.

54


Making use of the symmetry of the duality relation, Theorems VIII.3 and VIII.4 imply that conditions i) and ii) are satisfied if and only if Cˆ 0 = C0 +

M−1 j=1

Qj × Cj

and Aˆ i = Ai + Ti × A0 ,

i = 1, 2, . . . , M − 1,

where Q j = R(t, λ( j) ), Ti = R(t, μ(i) ) are arbitrary banded Toeplitz matrices. We now get, for every i = 1, 2, . . . , M − 1: M−1 Cˆ 0 × Aˆ iT = C0 + Q j × C j × AiT + A0T × TiT j=1

= C0 × +

AiT

M−1 j=1

+ C0 × A0T × TiT

Q j × C j × AiT +

M−1 j=1

Q j × C j × A0T × TiT

∗ = 0 + TiT + Q i + 0 = R t, μ(i) + R t, λ(i) .

Hence, condition iii) holds if and only if ∗ −R t, μ(i) = R t, λ(i) ,

namely,

∗ μ(i) = − λ(i) .

The above result shows how, starting from a given perfect reconstruction M-channel filter bank, it is possible to modify the analysis low-pass filter, and consequently update the synthesis high-pass filters, while still maintaining the perfect reconstruction property. The polynomials λ(i) in Eqs. (66) and (67) above represent the degrees of freedom that are left after imposing the biorthogonality conditions, and hence they give us full control over all dual pairs (α, γ) of the form (65). In practical situations, especially in the context of image compression, it is sometimes more useful to modify the analysis high-pass filters by adding some desirable properties, such as vanishing moments, prefixed shape, etc., and consequently updating the low-pass synthesis filter. This second construction, known as the dual lifting scheme, is simply obtained by applying the lifting scheme to the pair (γ, α), where α now denotes the analysis filter vector, and γ the synthesis filter vector:

55


Theorem VIII.6 (Dual Lifting Scheme) Let (α, γ) be a dual pair of filter vectors, with γ = γ (0) , γ (1) , . . . , γ (M−1) . α = α (0) , α (1) , . . . , α (M−1) ,

Every dual pair (α, γ) of the form α = αˆ (0) , α (1) , . . . , α (M−1) ,

γ = γ (0) , γˆ (1) , . . . , γˆ (M−1)

can be obtained by choosing γˆ (i) = γ (i) + λ(i) ◦ t M γ (0) , i = 1, 2, . . . , M − 1, αˆ (0) = α (0) −

M−1 j=1

(68)

( j)∗ M ( j) λ ◦t α ,

(69)

where λ(1) , λ(2) , . . . , λ(M−1) are arbitrary Laurent polynomials. To fully exploit the capabilities offered by the algebraic framework introduced in the previous subsection, we now give an explicit formulation of the lifting scheme in terms of Hurwitz matrices. Theorem VIII.7 Let (C0 , C1 , . . . , C M−1 ), (A0 , A1 , . . . , A M−1 ) be two Mtuples of banded Hurwitz matrices corresponding to a perfect reconstruction M-channel filter bank, namely, Ci × A Tj = δi, j I, i, j = 0, 1, . . . , M − 1. Every pair of M-tuples of banded Hurwitz matrices of the form (A0 , Aˆ 1 , . . . , Aˆ M−1 )

(Cˆ 0 , C1 , . . . , C M−1 ), satisfying ⎡

Cˆ 0 C1 .. .

⎤

⎡

I

0

...

0

⎤

⎢ ⎥ ⎢ I ... 0 ⎥ ⎢0 ⎥ ⎢ ⎥ ⎥ × A0T Aˆ 1T · · · Aˆ TM−1 = ⎢ ⎢ ⎥ ⎦ ⎣. . . . . . . . . . . .⎦ ⎣ C M−1 0 0 ... I

can be obtained by choosing

Cˆ 0 = C0 + and

M−1 i=1

Q i × Ci

Aˆ j = A j − Q Tj × A0 , j = 1, 2, . . . , M − 1, where Q 1 , Q 2 , . . . , Q M−1 are arbitrary banded Toeplitz matrices.

(70)

(71)

56


C. The Algorithms The matrix formulation of lifting suggests a computationally efficient strategy for the realization of the lifted filters. In fact, it is not necessary to explicitly evaluate the new filters given by formulas (66) and (67), since, using (70) and (71), we find that the output of the new filters can be computed starting from the output of the old system, simply by knowing the Toeplitz matrices Q i . More precisely, we can sketch the following optimized analysis–synthesis algorithm. Analysis: M−1 (0) σ Q i × Ci × [σ ] = Cˆ 0 × [σ ] = C0 × [σ ] + i=1

M−1 = σ (0) + Q i × σ (i) ,

(72)

i=1

where

Synthesis:

(i) σ = Ci × [σ ] i = 1, 2, . . . , M − 1.

M−1 M−1 Aˆ iT × σ (i) = A0T × σ (0) + AiT × σ (i) , [σ ] = A0T × σ (0) + i=1

i=1

where we have used (72) to get

(0) (0) M−1 Q i × σ (i) . − = σ σ i=1

We refer to Lazzaro (1999) for a detailed analysis of the computational aspects of this algorithm.

D. Factorization into Lifting Steps The lifting scheme represents a powerful tool for M-band biorthogonal filter construction, particularly due to the fact that it can be iterated. In fact, after adding suitable properties to the analysis low-pass filter, one can use the dual lifting scheme to add other properties to the analysis high-pass filters, thereby obtaining filters with any desired properties after a finite number of lifting steps.


57

The flexibility of the lifting scheme prompts us to ask whether an M-channel biorthogonal filter bank can be obtained using lifting. In the two-band case, it has been shown (Daubechies and Sweldens, 1998) that every dual pair of finite filters can be obtained with a finite number of lifting steps. In the following, we show that a similar result also holds in the M-channel case, even if this case needs some additional work, due to the higher complexity of handling M × M polynomial matrices. To prove this result, we must make some preliminary considerations. First of all, recalling the bijective correspondence between a filter vector and its M-decimated matrix and using the reconstruction rule given in (17), we remark that the lifting steps (66) and (67) can be rewritten in matrix notation as follows: ∆ M (γ) = ∆ M (γ) × Λ M (λ),

∆ M (α) = ∆ M (α) × Λ M (−λ∗ )T , where λ is the vector whose components are the M − 1 polynomials λ(1) , λ(2) , . . . , λ(M−1) , and Λ M (λ) is the matrix ⎤ ⎡ 1 0 ··· 0 (1) 1. · ·. · 0. ⎥ ⎢ λ Λ M (λ) = ⎣ .. .. ⎦ , .. .. . (M−1) λ 0 ··· 1 which will be called the lifting matrix relative to λ. Consequently, ⎡ 1 ⎢0. ∗ T M Λ (−λ ) = ⎣ . . 0

−λ(1)∗ 1. .. 0

⎤ · · · −λ(M−1)∗ · ·. · 0. ⎥ ∗ −1 T M ⎦ = (Λ (λ ) ) . .. .. ··· 1

The dual lifting step (68), (69) can be similarly represented as ∆ M (γ) = ∆ M (γ) × Λ M (λ)T ,

∆ M (α) = ∆ M (α) × Λ M (−λ∗ ). Moreover, we define the notion of partial lifting matrix as follows: if N < M, let µ = (μ(1) , μ(2) , . . . , μ(N −1) ) be an (N -1)-tuple of Laurent polynomials. The N-order partial lifting matrix relative to µ will be the M × M matrix ! I 0 M−N N (µ) = ΛM . 0 Λ N (µ)

58


We are now able to prove that every dual pair of filter vectors can be obtained with a finite number of lifting and dual lifting steps, alternated with permutations, starting from a suitably scaled trivial pair. Theorem VIII.8 Let (α, γ) be a dual pair of filter vectors. Laurent polynoj mial vectors λk , k = 0, 1, . . . , M − 2, µh , j = 0, 1, . . . , n h , h = 2, 3, . . . , M exist, with (M − k − 1) and (h − 1) components, respectively, such that ∆ M (α) = D

M−2

Λ M−k M (λk )

M

(73)

Ωh ,

h=2

k=0

where D is a diagonal matrix whose determinant is a nonzero monomial, and, for every h = 2, 3, . . . , M, Ωh =

nh j=0

j T ΛhM µh P (h) j ,

P (h) j being suitable permutation matrices acting on the last h columns, and n h an integer bounded by the maximum degree of the nonzero elements of the hth row of ∆ M (α). Moreover, T

∆ M (γ)∗ =

2

h=M

Ω−1 h

0

−1 Λ M−k M (−λk )D .

(74)

k=M−2

Proof. The proof is essentially based on the fact that any square matrix Σ with Laurent polynomial elements whose determinant is a nonzero monomial can be diagonalized by means of right elementary operations, as stated in Gantmacher (1959, Ch.VI, §2). We recall here the main steps of the proof, to highlight the connections between the elementary operations involved and the matricial formulation of lifting and dual lifting steps. The diagonalization process consists of two phases, the first of which reduces the matrix Σ to a lower triangular form by right multiplication by a finite number of permutation matrices and by matrices, with polynomial entries, of the form ⎤ ⎡ 1 0 0 ··· 0 ⎥ ⎢0 1 0 ··· 0 0 ⎥ ⎢· · · · · · · · · · · · · · · ⎢ ⎥ ⎢0 ⎥ 0 · · · 0 1 ⎢ ⎥. ⎢ 1 ξ 1 ξ2 · · · ξ h ⎥ ⎢ ⎥ ⎢ 0 1 0 ··· 0 ⎥ ⎣ ··· ··· ··· ··· ··· ⎦ 0 0 0

···

0

1

59


Note that these can be seen as dual partial lifting matrices. The second phase reduces the lower triangular matrix previously obtained to a diagonal matrix D by right multiplication by a finite number of matrices of the form ⎤ ⎡ 1 0 0 ··· 0 ⎥ ⎢0 1 0 ··· 0 0 ⎥ ⎢· · · · · · · · · · · · · · · ⎥ ⎢ ⎥ ⎢0 0 ··· 0 1 ⎥, ⎢ ⎢ 1 0 0 ··· 0 ⎥ ⎥ ⎢ ⎢ ψ1 1 0 ··· 0 ⎥ ⎣ ··· ··· ··· ··· ··· ⎦ 0 ψk 0

···

0

1

which correspond to partial lifting steps. We point out that the elementary operations involved in the whole process do not change the determinant of the original matrix, up to a sign. Therefore, the diagonal elements of the matrix D are nonzero monomials. Formula (73) can now be easily deduced from the above factorization result, while (74) is an immediate consequence of (73) and (45). We remark that the proof of Theorem VIII.8 is based upon a factorization theorem which greatly differs from the well-known Smith factorization result, since it makes use only of right multiplication by elementary matrices. In fact, in this context, these are the only operations which can be seen as lifting steps. Example VIII.1 We give now an example of the previous factorization theorem, by considering the factorization of the 4-decimated matrix ∆4 (α) relative to the 4-band filters, with two vanishing moments, constructed in Montefusco and Lazzaro (1997), namely, α (0) = .269 + .394t + .519t 2 + .644t 3 + .230t 4 + .105t 5 − .019t 6 − .144t 7 ,

α (1) = −.098 − .072t − .206t 2 − .180t 3 + .751t 4 + .343t 5 − .064t 6 − .472t 7 , α (2) = .500 − .500t − .500t 2 + .500t 3 ,

α (3) = .217 − .677t + .657t 2 − .237t 3 + .053t 4 + .024t 5 − .004t 6 − .033t 7 , where, for the sake of simplicity, we have written only the first three digits. We have ⎡ ⎤ .269 + .230t −.098 + .751t .500 .217 + .053t ⎢.394 + .105t −.072 + .343t −.500 −.677 + .024t ⎥ ⎥, ∆4 (α) = ⎢ ⎣.519 − .019t −.206 − .064t −.500 .657 − .004t ⎦ .644 − .144t −.180 − .472t .500 −.237 − .033t

60


and the algorithm described in the proof of Theorem VIII.8 yields ∆4 (α) := DΛ4 (λ0 )Λ3 (λ1 )Λ2 (λ2 )Ω2 Ω3 Ω4 , where ⎡ ⎤ .5 0 0 0 0 0 ⎥ ⎢0 .7169 D := ⎣ ⎦, 0 0 1.448 0 0 0 0 −1.9255t ⎡

1 ⎢0 Λ (λ1 ) := ⎣ 0

⎡

1 0 −1 ⎢ −.697t 1 Λ4 (λ0 ) := ⎢ ⎣−.345t −1 0 −.259t −1 0 ⎤ ⎡ 0 1 0 0 0⎥ 0 ⎢0 1 2 , Λ (λ2 ) := ⎣ 0 0 1 0⎦ 0 0 .102t −1 1 ⎤ ⎡ 0 0 1 0 0 0 0 ⎥ ⎢0 1 0 × 1 −.064 − .456t ⎦ ⎣0 0 0 0 1 0 0 1

0 0 1 0 .609t −1 1 0 −.207t −1 0 ⎡ 1 0 0 T (2) ⎢0 1 2 Ω2 = Λ µ2 P0 = ⎣ 0 0 0 0 T T Ω3 = Λ3 µ03 P0(3) Λ3 µ13 ⎤ ⎡ ⎤ ⎡ ⎡ 1 0 0 0 1 0 0 1 0 0 0 ⎢0 1 −.238 + 1.527t −.625⎥ ⎢0 0 1 0⎥ ⎢0 1 .306 × × =⎣ 0 0 1 0 ⎦ ⎣0 1 0 0⎦ ⎣0 0 1 0 0 0 0 0 0 1 0 0 0 1 T Ω4 = Λ4 µ04 P0(4) ⎡ ⎤ ⎡ 1 −.196 + 1.502t +.539 + .460t +.4343 + .106t 0 0 1 0 0 ⎢0 ⎥ ⎢0 1 =⎣ ⎦ × ⎣1 0 0 0 1 0 0 0 0 1 0 0 3

0 0 1 0

⎤ 0 0⎥ ⎥, 0⎦

1 ⎤ 0 0⎥ , 0⎦ 1 ⎤ 0 0⎥ , 1⎦ 0

⎤ 0 .071⎥ 0 ⎦ 1 1 0 0 0

⎤ 0 0⎥ . 0⎦ 1

It is easy to check that the above factorization reduces the computational complexity of the analysis–synthesis algorithm.

IX. Conclusion We have shown how the recursive matrix machinery can be fruitfully used to represent and easily handle the linear operators describing the action of an M-band filter bank. The present work gives a new outlook to this topic,


61

stating the theory in time-domain language, while still maintaining the main advantages of the polyphase approach. Hence, it permits simple and transparent proofs of the theoretical results, leading, at the same time, to computationally efficient implementations of basic operations of filter theory. In conclusion, the present paper can be seen as an improvement of the existing results on this subject, and it provides a simple tool for the study and the construction of M-band systems. Acknowledgments This research was supported by MIUR, Cofin2000 and R.F.O. projects, and C.N.R. Grant No. 99.1707.

References Akansu, A. N., and Smith, M. J. T., eds. (1996). Subband and Wavelet Transforms. Boston: Kluwer Academic. Bacchelli, S. (1999). Block Toeplitz and Hurwitz matrices: A recursive approach. Adv. Appl. Math. 23, 199–210. Bacchelli, S., and Lazzaro, D. (2001). Some practical applications of block recursive matrices. Comput. Math. Appl. 41, 1183–1198. Barnabei, M., Brini, A., and Nicoletti, G. (1982). Recursive matrices and umbral calculus. J. Algebra 75, 546–573. Barnabei, M., Guerrini, C., and Montefusco, L. B. (1998). Some algebraic aspects of signal processing. Linear Algebra Appl. 284(1–3), 3–17. Barnabei, M., Guerrini, C., and Montefusco, L. B. (2000). An algebraic framework for biorthogonal M-band Filters, in Recent Trends in Numerical Analysis, edited by D. Trigiante. Nova Science Publ., pp. 17–34. Barnabei, M., and Montefusco, L. B. (1998). Recursive properties of Toeplitz and Hurwitz matrices. Linear Algebra Appl. 274, 367–388. Daubechies, I., and Sweldens, W. (1998). Factoring wavelet transforms into lifting steps. J. Fourier Anal. Appl. 4(3), 247–269. Gantmacher, F. R. (1959). The Theory of Matrices. New York: Chelsea. ¨ Heger, I. (1858). Uber die Auflösung eines Systemes von mehreren unbest immtem Gleichungen des erten Grades in ganzen Zahlen. Denkschriften der Königlichen Akademie der Wissenschaften (Wien), Mathematisch-naturwissenshaftliche Klasse. 14(2), 1–122. Lazzaro, D. (1999). Biorthogonal M-band filter construction using the lifting scheme. Num. Algorithm (22), 53–72. Montefusco, L. B., and Lazzaro, D. (1997). Discrete orthogonal transform and M-band wavelets for image compression, in Surface Fitting and Multiresolution Methods, edited by L. Schumacker, A. Le Mehaute, and C. Rabut. pp. 261–270. Nayebi, K., Barnwell, T. P., and Smith, M. J. T. (1987). Time domain conditions for exact reconstruction in analysis/synthesis systems based on maximally decimated filter banks. Proc. Southeastern Symposium on System Theory, pp. 498–503. Strang, G., and Nguyen, T. (1996). Wavelets and Filter Banks. Wellesley-Cambridge Press.

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Suter, B. W. (1998). Multirate and Wavelet Signal Processing. Boston: Academic Press. Sweldens, W. (1996). The lifting scheme: A custom-designed, construction of biorthogonal wavelets. Appl. Comput. Harmon. Anal. 3(2), 186–200. Sweldens, W. (1997). The lifting scheme: A construction of second generation wavelets. Siam J. Math. Anal. 29(2), 511–546. Tolhuizen, L. M. G., Hollmann, H. D., and Kalker, A. C. M. (1995). On the realizability of bi-orthogonal M-dimensional 2-band filter banks. IEEE Trans. Signal Processing 43(3) 640– 648. Vaidyanathan, P. P. (1993). Multirate Systems and Filter Banks. Englewood Cliffs, NJ: Prentice Hall Signal Processing Series. Vaidyanathan, P. P., and Mitra, S. K. (March 1988). Polyphase networks, block digital filtering, LPTV systems, and alias-free QMF banks: A unified approach based on pseudocirculats. IEEE Trans. Acoust. Speech Signal Processing ASSP-36, 381–391. Vetterli, M., and Kovacevic, J. (1995). Wavelets and Subband Coding. Englewood Cliffs, NJ: Prentice Hall Signal Processing Series. Woods, J. W. (1991). Subband Image Coding. Boston: Kluwer Academic.

ADVANCES IN IMAGING AND ELECTRON PHYSICS, VOL. 125

Determining the Locations of Chemical Species in Ordered Compounds: ALCHEMI I. P. JONES Center for Electron Microscopy, School of Engineering, University of Birmingham, Birmingham B15 2TT, United Kingdom

I. Background . . . . . . . . . . . . . . . . . . . . . . . . . . II. Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . A. What Is ALCHEMI? . . . . . . . . . . . . . . . . . . . . . B. The Two Basic ALCHEMI Analyses . . . . . . . . . . . . . . 1. Dilute Solutions . . . . . . . . . . . . . . . . . . . . . . 2. Concentrated and Less Strongly Ordered Solutions . . . . . . . 3. A Different Way of Doing Things . . . . . . . . . . . . . . C. The Accuracy of ALCHEMI . . . . . . . . . . . . . . . . . . D. Delocalization and EELS ALCHEMI . . . . . . . . . . . . . . E. Optimizing ALCHEMI . . . . . . . . . . . . . . . . . . . . III. ALCHEMI Results . . . . . . . . . . . . . . . . . . . . . . . A. Minerals . . . . . . . . . . . . . . . . . . . . . . . . . . B. Intermetallics . . . . . . . . . . . . . . . . . . . . . . . . C. Functional Materials . . . . . . . . . . . . . . . . . . . . . D. Quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . E. Investigations Involving a Priori Calculations of Electron Distributions F. EELS ALCHEMI . . . . . . . . . . . . . . . . . . . . . . G. Concentrated Solutions . . . . . . . . . . . . . . . . . . . . IV. Predicting Sublattice Occupancies . . . . . . . . . . . . . . . . . V. Competing (or Supplementary) Techniques . . . . . . . . . . . . . VI. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. Current Challenges and Future Directions (a Personal View) . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

63 64 64 66 66 67 71 73 75 76 77 77 81 93 93 93 97 97 100 102 106 106 107

I. Background The term ALCHEMI was coined by Spence and Taftø in 1983. It stands for Atom Location by Channeling Enhanced Microanalysis. The channeling of electrons between and along the atom planes is exploited via the electrons’ inelastic interactions with the atoms. These give rise to energy losses in the beam and excite x-rays, both of which enable the identification of the chemical natures of the atoms at various positions in the structure. All diffracted waves, whether associated with photons or with particles with rest mass, adopt an inhomogeneous distribution in a regular crystalline or quasicrystalline structure. 63 Copyright 2002, Elsevier Science (USA). All rights reserved. ISSN 1076-5670/02 $35.00

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I. P. JONES

For heavy particles (e.g., neutrons), this may be understood in terms of particle channeling between the atoms. For x-rays and electrons, the most convenient description is in terms of dynamic diffraction theories. The distribution of radiation or particles in the solid is sensitive to its angle of incidence, particularly around channeling orientations, which, for massy particles, are parallel to the channels between atoms and, for wavelike radiation, are close to Bragg positions. The likelihood of inelastic interactions, including the generation of x-rays, therefore varies from point to point in the solid. In a chemically ordered crystal or quasicrystal, this opens up the possibility of identifying separately the chemical nature of the atoms at each location. Such a technique has been employed using x-rays [the “Borrmann effect” (Borrmann, 1941; Batterman, 1969] and also using ion beams (Morgan, 1973). (See Spence and Taftø, 1982, 1983, and Spence, 1992), for reviews and for further information.) In the electron context, it had for some time prior to 1983 been realized that the channeling of the beam electrons in a transmission electron microscope (TEM), which gives rise via (mainly) thermal diffuse scattering to anomalous absorption and the characteristic bend contours of TEM (Hashimoto et al., 1962), would also give rise to anomalous generation of x-rays (Duncumb, 1962). Cherns et al. (1973) showed, among other things, that the x-ray emission oscillated with specimen thickness, demonstrating practically that the generated x-ray intensity depended on the whole electron wavefunction (including Bloch wave interference) rather than on the individual Bloch waves, mirroring an earlier discussion with respect to electron absorption. Taftø (1979) showed that such channeling in an ordered compound Ax By could give rise to a change in the ratio of characteristic x-ray signals IA :IB . He was clearly aware of the potential of this effect for identifying the chemical nature of the atoms at various positions in the unit cell. The crucial subsequent contribution of Taftø (1982), Taftø and Spence (1982a,b), and Spence and Taftø (1982, 1983) was to show how much firm quantitative chemical information of this type could be deduced from such an experiment without the need for complicated Bloch wave calculations. Earlier reviews of ALCHEMI have been published by Otten (1983), Krishnan (1988), Spence (1992), Buseck and Self (1992), and Horita (1998).

II. Fundamentals A. What Is ALCHEMI? When a TEM specimen is oriented near the Bragg position for a given reflection, the beam electrons are channeled (Hirsch et al., 1965; see Fig. 1a). When the specimen is negative of the Bragg position (negative refers to the

CHEMICAL SPECIES LOCATIONS IN ORDERED COMPOUNDS

65

Figure 1. (a) Bloch wave channeling concentrates the beam electrons on the atom planes when s is −ve and between the planes when s is +ve. (b) In an ordered crystal the electrons are channeled along the heavier atoms (here the unshaded circles) and the apparent chemical composition changes. (Reprinted with permission from Jones, I. P., 1996. How ordered are intermetallics?, in Towards the Millennium. London: Institute of Materials, pp. 267–280.)

deviation parameter “s”; negative “s” corresponds to tilting the crystal toward the symmetry orientation), the electrons are channeled along the atom planes. When the specimen is tilted positive of the Bragg position, the electrons are channeled between the atom planes. This gives rise to the familiar black bend contour, the anomalous absorption depending largely on thermal diffuse scattering. The characteristic x-ray production, which contributes a small amount

66

I. P. JONES

to electron absorption, follows a similar (but not identical) distribution, since both inelastic processes occur close to the atom nuclei. When an ordered specimen is oriented close to a fundamental reflection Bragg position, a similar scenario ensues, and the enhanced x-ray emission reflects the overall composition of the reflecting planes (i.e., that of the crystal as a whole). When an ordered specimen is oriented close to a superlattice Bragg position (a superlattice reflection is one which would disappear if order turned to disorder), the electrons are channeled more down one type of atom position than another. If each type of atom position is considered to be the origin of a sublattice, then tilting the crystal about a superlattice Bragg position favors one sublattice or another (see Fig. 1(b)). In an ALCHEMI experiment, the x-ray or EEL (electron energy loss) spectrum is recorded at kinematic (i.e., not strongly diffracting) and superlattice dynamic orientations, and these spectra are used to infer information concerning the chemical compositions of the two or more sublattices, or, to put it another way, the various atom locations in the crystal. The dynamic orientation may be systematic or axial. There are two ways of analyzing the data. One is appropriate for dilute solutions and one for concentrated solutions. In the former type of analysis [which was the original analysis given by Taftø (1982), Spence and Taftø (1982, 1983), and Taftø and Spence (1982a,b)], the solute atoms whose position is being defined by the experiment are assumed not to perturb the host lattice concentrations or locations. The total amount of solute (i.e., the overall chemical composition of the compound) is not required to be known. In the concentrated solution analysis, the composition is required to be known, and all types of atoms enter the analysis on an equal footing. The dilute solution analysis is a limiting case of the concentrated solution analysis. B. The Two Basic ALCHEMI Analyses 1. Dilute Solutions This is the original analysis given by Taftø (1982), Taftø and Spence (1982a,b), and Spence and Taftø (1982, 1983) and subsequently restated and tidied up by Bentley (1986), Goo (1986), and Otten and Buseck (1987). If A and B occupy separate sublattices and a small amount of C is added, how does C partition between the A and B sublattices? Note that we are not asking to know how much C there is, nor what is the exact chemical composition of the compound. In this case, as Taftø and Spence outlined, the A and B sublattices act as internal calibrators of the dynamic channeling. Imagine that the element C partitions such that a fraction p of it occupies the A sublattice and thus a fraction (1 − p)


67

occupies the B sublattice. Furthermore, imagine that the numbers of A, B and C x-rays from the kinematic analysis are NA , NB , and NC and that those from the channeling experiment are N′A , N′B , and N′C . Then if RA = N′A /NA etc., the channeling electron beam current along the A sublattice relative to that in the kinematic condition is RA and for the B sublattice is RB . Then N′c = R C = pRA + (1 − p)RB Nc and p=

RC − RB RA − R B

And there is no more to it than that. The beauty of this is that it is an easy experiment, and does not involve any complicated (and uncertain) dynamic electron diffraction calculations. With a splendid and magisterial disregard for detail, Spence and Taftø did not notice until the proof stage of their J. Mic. (1983) paper that their equations are overdefined. This (in fact trivial) blemish was subsequently removed (independently) by Goo (1986), Bentley (1986), and Otten and Buseck (1987). The simplicity and robustness of this basic ALCHEMI technique explains why it has been so much applied. A list of analyses is given later in this review. Everything written above applies equally well whether a systematic row orientation or a zone axis orientation is used. The former is usually called planar and the latter axial ALCHEMI. Krishnan and Thomas (1984) have examined the extension of this approach to more complicated situations involving several solute species. Their paper is a nice illustration of the advantages of the geometric approach introduced below. More complicated situations (geometrically, chemically . . ..) are perhaps better tackled initially on their own merits. 2. Concentrated and Less Strongly Ordered Solutions Here we do not assume that A occupies one sublattice and that B occupies the other. The state of order is represented by an ordering tie line, or OTL (Matsumura et al., 1991; Hou and Fraser, 1997) (at least for two sublattices) (see Fig. 2). The compositions of each sublattice, SL1 and SL2, describe the state of order of the compound. A kinematic measurement returns the overall composition, which is known and enables the k factors (Cliff and Lorimer, 1975) to be determined. In the simplest case of two equally inhabited sublattices, the overall composition is midway between SL1 and SL2. The dynamic

68

I. P. JONES

Figure 2. SL1 and SL2 are the chemical compositions of two equally populated sublattices and the overall composition is midway between them. In an ALCHEMI experiment the apparent chemical composition moves toward SL1 or SL2.

measurement of chemical concentration will move toward SL1 or SL2,∗ depending upon which type of plane is favored by the particular channeling conditions obtaining (Jones, 1996; Hou et al., 1996). Thus the ALCHEMI experiment defines the slope of the OTL. Because the channeling is never perfect, it cannot define the length of the OTL and thus the absolute state of order of the compound. To do this requires a further, independent, piece of information. Four possibilities are (i) Assume some geometric constraint—e.g., that the tie line intersects (ends at) one side of the triangle. (ii) Measure a superlattice extinction distance. (iii) Calculate the extent of the channeling. (iv) Constrain the beam using a different method. The simplest example of (i) might be thought to be where the tie line intersects two sides of the triangle. This is equivalent to the situation analyzed by Method I, but in fact the OTL analysis as described here (Method II) is not applicable because if the composition of the compound were known exactly, then the state of order would be known also. It is also worth noting that on entropic ∗ Exactly which way the apparent composition moves is quite informative, since tilting negative of the Bragg position will favor the sublattice with the higher structure factor (usually the heavier one). A particularly elegant example of this may be found in Taftø and Gjønnes (1988).


69

Figure 3. The chemical composition of one sublattice is assumed. This, along with a knowledge of the overall composition, allows the composition of the other sublattice to be fixed by an ALCHEMI experiment.

grounds this situation, strictly speaking, is forbidden, although it is a useful approximation. There are several examples in the literature of analyses which assume that the tie line is anchored at one side of the triangle. These include those of Otten (1983), Shindo et al. (1986, 1988), Walls (1992), Anderson and Bentley (1994), Horita et al. (1995), and Hao et al. (2000a). These presentations are all algebraic, but add little to the simple geometric construction of Figure 3. Approach (ii), where the supplementary, independent piece of information is a superlattice extinction distance, was introduced and pioneered by Matsumura and colleagues in an investigation of CuAuPd alloys (Matsumura et al., 1991, 1998; Kuwano et al., 1996; and Morimura et al., 1997). In their original experiments, the 001 superlattice extinction distance was measured via the intersecting Kikuchi line method of Gjønnes and Høier (1971). They subsequently applied the method to other compounds and structures (see Horita et al., 1995). This approach does not seem to have been taken up by other researchers, although a combination of approaches (ii) and (iii) (see also below) has obvious attractions. Approach (iii), where the absolute length of the OTL is determined by an iterated Bloch Wave simulation, was introduced by Hou and Fraser (1997) and subsequently used by Sarosi et al. (see Jones, 2001). A conventional Bloch wave calculation predicts the electron intensity at every depth through the foil. This is averaged and then folded with the x-ray generation profile for the atoms. This may be a delta function, or may include delocalization

Figure 4. Real-space crystallography. (a) A scan along undoped BSCCO unit cell establishes the periodicity of the X-ray signal. (b) Static analyses from the HAADF image establish the rough position of the dopant Dy atoms in the unit cell. (From Shang et al., 1999. Phil. Mag. Lett. 79, 741–745, by permission of Taylor and Francis Ltd, http://www.tandf.co.uk/journals)


71

(see Section II.D). The absorbed electrons are assumed to contribute the average composition over their remaining trajectories. Because the real and imaginary extinction distances depend upon the answer (the ordering state of the crystal), the calculation must be iterated until the correct experimental x-ray signals are predicted. This occurs quite rapidly (2 or 3 iterations). In ALCHEMI, the dynamic scattering of the electron beam effectively produces a series of small probes at each atom column, repeated over the area of the beam. It is also possible to perform this experiment in real space (considering ALCHEMI as a reciprocal space method) by condensing down the electrons to form a single probe which can be scanned along the unit cell (see Fig. 4; Shang et al., 1999). This approach (iv) really falls outside the remit of this review (ALCHEMI). For this type of analysis (Method II, concentrated solutions), it is worth noting that for three components and three sublattices the OTL must be replaced by an ordering tie triangle, and for four sublattices, a quadrilateral. For four components, we must use a tetrahedron, with the possibility of ordering tie lines, triangles, quadrilaterals (not necessarily planar), again. The ordering state for m sublattices and n elements is described by an m × n matrix, p. Assuming that the overall composition is known, p has (m − 1)(n − 1) degrees of freedom (or parameters to be determined) (Jones and Pratt, 1983). For example, for 3 elements/components and three sublattices: Element → Sublattice ↓

P11 P21 P31

P12 P22 P32

P13 P23 P33

the unshaded elements are independent. 3. A Different Way of Doing Things An approach rather different from the previous two is to measure the variation in x-ray signal either across a range of systematic orientations (Zaluzec and Smith, 2001; Anderson, 2001) or around a zone axis (Josefsson et al., 1994; Rossouw et al., 1996a,b, 1997, 2000; Bastow and Rossouw, 1998; Saitoh et al., 2000; and Rossouw, 2001). These x-ray channeling patterns are called ICPs by Rossouw and colleagues (Incoherent Channeling Patterns) and their measurement derives from original work by Bielicki (1983) and early development by Christenson and Eades (1989). Similarity of detail gives a qualitative clue as to where an element resides (see Fig. 5). A recent attempt to quantify the axial ICP technique (in the sense of extracting sublattice occupancies) was unsuccessful

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I. P. JONES

a

b

Figure 5. (a) 111 ICPs for TiAl containing 1% Ta. The Ta ICP is similar to the Ti ICP and dissimilar to the Al ICP. Therefore, Ta is on the Ti sublattice. (b) Calculated versions of (a). (Courtesy Chris Rossouw.)


73

(Rossouw et al., 2001). There appears to be no report of quantification of the planar version, either. Where the technique has proved particularly helpful is in placing solutes where there is no unequivocal reference lattice. When there is a sparsely inhabited interstitial lattice, which does not sufficiently contribute its own diffracted beams (e.g., Rossouw and Miller, 1999a,b) or a situation where different sublattices do not have detectably different chemical compositions (e.g., Rossouw et al., 2001), it is necessary to compute the ALCHEMI response and compare the result with experiment. The ICP approach (whether axial or systematic) provides a convenient framework for this. In the ALCHEMI results section, I have included a table of examples of this type of investigation (see Table 5, pp. 98–99). Although ICPs do not seem to have been applied to bulk specimens (for this purpose), there seems no reason why they should not be.

C. The Accuracy of ALCHEMI It is clearly desirable to have some idea of how accurate ALCHEMI measurements are. The approach universally adopted (either explicitly or implicitly) is to plot the ALCHEMI results in some form against the strength of channeling, on the principle that the stronger the channeling the more significant is the measurement and the more the weight which should be given to it. Standard regression analysis returns estimates of the errors involved. Thus, in terms of the OTL analysis described above, a straight line is fitted through the experimental points and the further away from the fulcrum (the overall concentration) the experimental measurements lie (i.e., the stronger the channeling), the more effect they have on the OTL. The extension to more components and sublattices is trivial. The best known and most commonly applied approach is that introduced by Rossouw and colleagues and named by them “statistical ALCHEMI” (Rossouw et al., 1989, 1996a). Using nomenclature similar to that in the 1989 paper, and taking the simplest situation where element C is added to ordered compound AB: IA (1 − pxC ) NA = kA IA = channeling intensity on A sublattice. kA is an x-ray generation/ proportionality factor IB (1 − (1 − p)xC ) kB IA xC p + IB xC (1 − p) NC = kC

NB =

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I. P. JONES

Eliminating IA and IB ,

# " (1 − p)kB NB xC pkA NA + NC = kC 1 − xC p 1 − xC (1 − p)

Writing

NC = αA NA + αB NB

(this defines the OTL)

(1)

then αA = Thus

xC pkA kC 1 − xC p xC =

and

αA αA +

kA kC

αB =

+

xC (1 − p)kB kC 1 − xC (1 − p) αB

αB +

kB kC

and p=

αA αB $ % =1− $ % kA kB xC αA + xC αB + kC kC

(2)

Best fit values of αA and αB are determined from the experimental data, and one of these is then used to calculate p. In this formulation, xC is determined from a combination of channeling chemical analyses, but it could just as easily be determined from a kinematic analysis. Note also that kA /kC and kB /kC require to be known and so the analysis in this form is subject to the logical quibble referred to above, that if, for example, xA were measured exactly, p would be known automatically. In the experimental implementation of Rossouw et al., the electron beam is scanned around a zone axis under computer control for an extended period. It is the weight of data thus accrued which makes the statistical ALCHEMI results reported apparently the most precise, currently. (Whether they are the most accurate remains to be seen.) The advantage of using axial rather than systematic ALCHEMI is that the channeling can be stronger, giving more significant results more quickly. The disadvantage is that delocalization (see Section II.D) corrections appear to be larger (also see below). Statistical ALCHEMI is said (by its authors) to be less affected by delocalization than other approaches, although there appears to be no justification of, nor evidence for, this assertion. In fact, in later versions of their method, Rossouw and colleagues include an additional constant in the fitting Eq. (1) which was said to account for delocalization and some of the deficiencies of their EDX software. Certainly, delocalization will, as this implies, move the OTL (Walls, 1992; Fraser, Hou,


75

and Amancherla, private communication). Including a constant like this would be expected to improve the precision of the fit, although whether it improves the accuracy is not obvious (see also Lábár et al., 2000). Other authors who have used straight line fits of ALCHEMI results against strength of channeling include Anderson (1997), Lábár et al., 2000; Hao et al., 2000a; and Jiang, to be published). Otten (1983) and Hao, Yang et al. (1999), among others, have analyzed error transmission in Method I of ALCHEMI.

D. Delocalization and EELS ALCHEMI It is a fundamental assumption of ALCHEMI that the original ionization events all take place at the nuclei of the atoms, and thus that atoms on the same crystal plane see the same electron beam intensity through the specimen. In fact, the probability of ionization goes through a maximum at a distance from the nucleus which increases as the energy required decreases. In classical terms, this is the impact parameter. This effect is called delocalization. Also leading to delocalization of the ionization event from the crystal lattice point is thermal vibration of the atoms. In an ordered crystal, the different atoms may vibrate with different amplitudes. Pre-ALCHEMI, Bourdillon et al. (1981) measured delocalization effects in copper and copper–platinum, using a selection of K, L, and M x-ray peaks. They also estimated impact parameters using time dependent perturbation theory, giving the formula b=

hv 1.24 2π E

(3)

(v, speed of incoming electrons; E, energy loss.) Subsequent to the inception of ALCHEMI, Pennycook (1988) reestimated b as $ %" $ % $ %# hv E 16E 1/2 b= ln ln (4) 2πEt Et Et (v, speed of incoming electrons; E their energy; Et threshold for ionization) for x-ray excitation and " $ %# hv 4E −1/2 ln 2π E for EELS, and advocated the use of simple correction factors which depend on energy. Other approximate treatments are due to Ma and Gjønnes (1992), Walls (1992), Anderson and Bentley (1994), and Horita (1996). (See Horita et al., (1993), for a review and a description of a more experimental approach.)

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I. P. JONES

More exact and lengthy treatments have been reported by Rossouw and Maslen (1987), von Hugo et al. (1988), Nüchter and Sigle (1995), and by Allen and Rossouw (1993), Allen et al. (1994), Josefsson et al. (1994), Oxley and Allen (1998, 1999, 2000), and Oxley et al. (1999), the latter series including impressive agreement between theory and experiment for the variation of the oxygen K peak with orientation for a spinel, as well as for other examples. These calculations are not trivial. Furthermore, simple corrections as advocated by Pennycook and others appear not to be very accurate (Oxley and Allen, 2000) and have not found general acceptance. What is generally accepted, however, is that delocalization corrections are much less important for systematic ALCHEMI than for zone axis ALCHEMI (Rossouw and Maslen, 1987; Spence et al., 1988; Qian et al., 1991; Munroe and Baker, 1992; Horita et al., 1993; Hao et al., 2000a; Lábár, 1999; and Rossouw et al., 2001)∗ and decreases as beam voltage decreases (Munroe and Baker, 1992; Ma and Gjønnes, 1992). A reasonable strategy would therefore seem to be (i) to seek to avoid delocalization effects by using, where possible, x-ray peaks of similar energies (e.g., Tian et al., 1992), (ii) if this is not possible, to consider using planar, rather than axial, ALCHEMI [not always feasible—e.g., Otten and Buseck (1987)]. (iii) Finally, if delocalization must be taken into account, it seems sensible to perform a full calculation, perhaps as part of a larger calculation— for example, the Bloch wave calculation approach (iii) to Method II (see Section II.B). The application of EELS to ALCHEMI has been inhibited by the problems of delocalization [which can be worse than for x-ray generation (Qian et al. (1992)] and of more difficult quantification procedures as compared with EDX. Thus, EELS has not made much of a contribution to standard ALCHEMI analyses. Where it does show considerable promise, however, is in identifying the oxidation states of particular ions at specific crystallographic sites. Strangely, this aspect seems to have been neglected ever since the early work of Taftø and Krivanek (1982) on Fe++ and Fe+++ distributions in a chromite spinel (see also Taftø, 1984; Self and Buseck, 1983; and Krishnan, 1989).

E. Optimizing ALCHEMI The statistical aspects of ALCHEMI have been discussed in Section II.C, and the question of planar vs axial ALCHEMI in Section II.D. It is always ∗ Presumably the comment to the opposite effect by Reviere et al. (1993) is a typographical error, especially since it is contradicted earlier in the same paper.


77

beneficial to cool the specimen (Spence et al., 1986; Rossouw and Maslen, 1987). Aside from reducing any differences in thermal vibration amplitudes between different elements, more significantly, reducing the temperature reduces the rate at which electrons are dechanneled into diffuse waves which only contribute a uniform background. In addition, like so much microanalysis, the accuracy of ALCHEMI is often limited by the build-up of contamination, which is slowed down by cooling the specimen. Following an earlier paper by Nakata et al. (1991c), Jiang, Hou et al. (1999) have analyzed planar ALCHEMI and have shown, theoretically and experimentally, that (i) the best channeling orientation is the symmetry orientation (this is because the –ve s effects of both ± g sum) (ii) the optimum beam voltage corresponds to a level of dynamic interaction (Jones, 1976) ws = 21/2 + ξg /ξ2g . (iii) Under these conditions, the maximum value of channeling (i.e., the proportion of the OTL which can be visited) is about 0.3–0.5, although this value can be much reduced by absorption. (iv) The optimum foil thickness is ξg /4 to ξg /3 (at the optimum level of dynamic interaction). (v) Beam convergence may safely be increased to approximately the Bragg angle. (vi) Channeling and ALCHEMI experiments disappear at the critical voltage. Beam voltage [(ii) and (vi) above], but not in the context of delocalization, has been investigated and discussed previously by Thomas et al. (1985), Krishnan et al. (1986), and Fujimoto et al. (1994), and there appears to be some disagreement about the effect of a critical voltage on ALCHEMI experiments, particularly for zone axis orientations (see also Horita et al., 1997). Secondary defects (e.g., antiphase domain boundaries (Jiang, Rong et al., 1999) predictably degrade ALCHEMI measurements as their density increases. III. ALCHEMI Results The results of ALCHEMI experiments reported in the literature are presented in the form of a series of tables with commentaries. The classification into tables is not mutually exclusive: some compounds appear in more than one table.

A. Minerals I put Table 1 first because this is where ALCHEMI started. All of the analyses (except those involving a priori calculations of the Bloch wave distribution) are of the “dilute” variety, which may reflect the fact that the chemical bonding

TABLE 1 ALCHEMI Results: Minerals Host compound

Addition

Amphibole

Result

Reference

78

Is not possible to analyze via ALCHEMI Mn on Mg sites, Fe slight preference for Mg Consistent with how formula is written after delocalization correction

Otten (1987)

(X + Z) sites: Mg, Si, Ca, Fe Y sites: Al

Balboni et al. (1994)

Answer implicit in how formula is down written and to some extent in the chemical analysis 1. X3 Y2 Z3 O12 X dodecahedral 2+ Y octahedral 3+ Z tetrahedral 4+ 2. Straight line fit vs. channeling—see Section II.C 3. Calculations here are supportive, rather than essential. 4. Need axial rather than planar ALCHEMI to separate all three types of sites.

Dolomite

Mn, Fe

Garnet (pyrope) Mg3 Al2 Si3 O12 (Mg2.33 Fe0.53 Ca0.33 Mn0.01 ) (Al1.74 Ti0.04 Cr0.04 )Si3 O8 Garnet (pyrope) Mg3 Al2 Si3 O12 (Mg2.883 Fe0.074 Ca0.029 ) Al1.997 Si3 O12 Garnet: Almandine Fe3 Al2 Si3 O12

Fe, Mn, Ca Ti, Cr

Ca, Mn

Ca, Mn on Fe (dodecahedral) site

Lábár (1999)

Garnet: Grossular Ca3 Al2 Si3 O12 Ilmenite

Fe

Fe on Al (octahedral) site

Lábár et al. (2000, 2001)

Mg, Al, Mn, Cr Cr

Mg and Mn on Fe sites, Al on Ti sites, Cr slight preference for Ti 0, 0.25, 0 site

McCormick and Smyth (1987) Rossouw and Miller (1999a,b)

Mullite

Comment

McCormick and Smyth (1987) 1. O8 or O12 ? 2. Mn undetectable 3. Planar ALCHEMI too weak

See Table V

Olivine (Mg0.9 Fe0.1 Ni0.004 Mn0.002 )2 SiO4 Olivine

Ni, Fe, Mn, Cr

% Occupancy of M1 sites: Fe 50, Mn 0, Ni 100

Taftø and Spence (1982a)

% Occupancy of M1 sites: Fe 45, Mn 13, Ni 83, Cr∼64. Doesn’t change much with annealing.

McCormick et al. (1987)

Good agreement with x-ray results

Smyth and Taftø (1982)

At this stage ALCHEMI was called CHEXE! I will leave it for you to judge whether it would have caught on quite so quickly without a name change. There must be a lesson here somewhere.

All Al on M1+T. Most Fe and Mg on M1. 71% of Al on T1 site

McCormick (1986)

020 planar ALCHEMI. Results imply most of the vacancies are on M2. Only one reference lattice, but Al:Si known.

U, Th, Sr, Zr, Mo on Ca sublattice &2% Fe on Ca 28% Fe on Ti Sr and U → Ca Zr → Ti Irradiation results in loss of order

Taftø, Clarke, and Spence (1983) Rossouw et al. (1988b) Zaluzec and Smith (2001)

75% Fe on tetrahedral; Mn on tetrahedral; Ti on octahedral

Taftø, Clarke, and Spence (1983)

Olivine (forsterite) % on M1 of un-heat treated 6 days/300◦ C 48 h/600◦ C 45 h/900◦ C 24 h/1000◦ C

79

Omphacite (nonstoichiometric pyroxene) Orthoclase (K0.90 Na0.08 ) (Al0.99 Fe0.01 Si3.01 )O8 Perovskite CaTiO3 Perovskite CaTiO3

U, Th, Sr, Zr, Mo, Fe Sr, Zr, U

Perovskite CaTiO3

Ulvo-spinel

Fe, Ti, Mn

Fe 50 47 50 50 50

Ni 97 87 83 83 80

Mn Ca 1 0 15 0 15 0 15 0 15 0

Taftø and Buseck (1983)

X-ray profiles across systematic rows give sensitive detection of disordering. Various details in irradiated profiles referred to but not interpreted. Equivalent to high resolution 1-D ICP. (See Calculated table.)

(continues)

TABLE 1—Continued Host compound Spinels: ZnCr0.4 Fe1.6 O4 MgAl2 O4 TiFe2 O4 Spinel (Cr0.37 Fe0.23 Al0.23 Mg0.17 )3 O4

Addition

Ti, V

Spinel (Cr0.37 Fe0.23 Al0.23 Mg0.17 )3 O4 Spinel Cr2 MnO4

Fe

Spinel MgO(Al2 O3 )n , n=1 and 2.4

Spinel MgAl2 O4 Spinel MgAl2 O4

Spinel ZnAl2 O4 3 K-feldspars: sanidine low microcline orthoclase Zirconolite CaZrTi2 O7

3% Fe, 0.3% Mn

Yb

Result

Reference

Cr, Fe octahedral, Zn tetrahedral “normal” spinel “inverse” spinel Cr, Al, Ti and V octahedral. Mg and 3/4 of Fe are tetrahedral Fe2+ on tetrahedral sites, Fe3+ on octahedral sites (Cr1.79 Fe0.21 )(Mn0.9 Fe0.1 )O4

Taftø and Liliental (1982)

Qualitative “pre-ALCHEMI” ALCHEMI

Taftø (1982)

Quantitative “pre-ALCHEMI” ALCHEMI

Taftø and Krivanek (1982)

“pre-ALCHEMI” EELS ALCHEMI Original “statistical” ALCHEMI paper. Corrected for oxygen delocalization by full calculation

Unirradiated n =1 90% Al3+ on O sites, 60% Mg2+ on T sites n = 2.4 27% Mg2+ and 20% Al3+ on T sites Irradiated Disordered—stoichiometric more so than nonstoichiometric Normal spinel (Al octahedral, Mg tetrahedral) If MgAl2 O4 = Mg1−x Alx [Al2−x Mgx ]O4 , x measured at 0.17 62% of Fe and all Mn on Zn (tetrahedral) sites 2t1 : 0.52 (random, as expected) 0.92 0.67 20% of Yb on Ca sublattice, rest on Zr sublattice

Rossouw et al. (1989) Soeda et al. (2000)

Comment

Qian et al. (1992)

Used both x-rays and EELS

Anderson (2001)

Similar method to that of Zaluzec and Smith (2001) (see above). (See Table V.)

Taftø and Spence (1982b) McLaren and FitzGerald (1982) Turner et al. (1991)

Al:Si ordering. 2t1 = fraction of Al on T1 sites. Use fact that Al:Si = 1:3 (see Taftø and Buseck (1983)) Assume Yb not on Ti sites.


81

in minerals is of the covalent or ionic type and that this is less tolerant of partial (dis)ordering than the metallic bond. Earth science applications of ALCHEMI were reviewed in 1988 by Smyth and McCormick.

B. Intermetallics Tables 2A–2D have been organized in terms of host intermetallic and, secondarily, date of publication. I have generally used upper case for composition (at% throughout) except where a well-defined stoichiometric compound is involved, in which case I have used subscripts, but always I have made clarity of presentation the controlling factor. The most notable feature is how good the overall agreement is between different authors, which should give us confidence in the results. The host lattice composition has an important influence on occupancy by a third element: several authors have followed the very useful policy of examining a series of compounds such as A50−x B50 Cx , A50−x/2 B50−x/2 Cx , and A50 B50−x Cx , in which case it is the middle member of the series which reveals most clearly element C’s propensities. In terms of a Bragg–Williams interpretation (see Section 4), the occupancies reflect a competition between A–B, A–C, and B–C bond enthalpies. Only when the three bond enthalpies are comparable, in the sense that HA−C /HA−B ∼ HB−C /HA−B (Ochiai et al., 1984; Chiba et al., 1991; and Jiang, 1999) will there be any noticeable effect of A–B stoichiometry on C occupancy. When HA−C /HA−B ≫ HB−C /HA−B , for example, C always substitutes for B and vice versa. Currently the most satisfactory way of comparing and interpreting ALCHEMI results from different, perhaps complicated, alloys of this type is to derive bond enthalpies by matching with Bragg–Williams calculations. There is an obvious connection between sublattice occupancies and the underlying phase diagram, primarily because both depend on the (pairwise aspect of the) bonding. Ochiai et al. (1984) correlated the directions of solubility lobes on the ternary isothermal cross-sections with sublattice occupancy, and there has been general subsequent confirmation of this (see Fig. 6; but see Wu et al., 1989, and Kao et al., 1994). Ultimately (if the solubility lobes illustrated in Fig. 6 become lines), the phase diagram can virtually dictate the ALCHEMI result [see, for example, Chu et al., 1998 (Nb33 Cr42 V25 ) and Takasugi et al., 1990 (Ni3 (Si,Ti)]. This is equally—perhaps more—true of other classes of materials; for example, see Table 1, Balboni et al. (1994). Yang and Hao (1999, 2000) and Yang et al. (2000) have extended this correlation, via bond enthalpies, to the prediction of phase boundaries. This is not always an immediately obvious connection (Rossouw et al., 1996b).

TABLE 2A ALCHEMI Results: Intermetallics: B2 Compounds Host compound

Addition

Nb75Al15Ti10 Nb60Al15Ti25 Nb45Al15Ti40 Nb79Ti10Al11

82

Fe50Al40Ni10 Ni50Al40Fe10

Nb51Ti34Al15 Nb57Ti28Al15 Nb68Ti17Al15 Nb50Ti25Al25 Nb36Ti24Al40 Nb35V40Al25 Nb65V20Al15 NiAl 3, 5, 10% V

V

Result

Reference

Al occupies opposite sublattice from Nb and Ti.

Hou and Fraser (1997)

Al tends to adopt opposite sublattice from Nb and Ti (see Leonard and Vasudevan (2000) above). Al adopts opposite sublattice from Ni and Fe. Ni occupies Fe sublattice in first alloy but Fe occupies Al sublattice in second alloy. Al tends to adopt opposite sublattice from Nb and Ti (i.e., Al-Nb and Al-Ti bonds stronger than Nb-Ti).

Amancherla et al. (2000)

Al all on one sublattice (assumed), Nb and V fill up remaining space impartially. V on Al sublattice

Rong et al. (2002)

Leonard and Vasudevan (2000)

Darolia et al. (1989)

Comment Introduced Bloch wave calculation of OTL length. See also Table 6. OTL framework, but only slopes reported. Includes data from Hou and Fraser (1997). Implies stronger Ni-Al than Fe-Al bonding. Good agreement with Bragg–Williams calculations. See Table 6. OTL framework, but no independent measurement, so ordering tendency (i.e., slope) only. See Table 6.

How much V is in solution? Consistent with phase diagram (note Heusler phase Ni2 AlV).

NiAl: Ni50Al47V3 Ni51Al47.5Mn1.5 Ni51.3Al43.5Cr5.2 NiAl: Ni50Al49Cr1 NiAl: Ni50Al40Fe10 Ni40Al50Fe10 Ni47Al51Fe2 Ni49.75Al49.75Fe0.5 NiAl: Ni50Al48Cu2 Ni48Al50Cu2

83

Ni50Al47-xTi3 Cux, x=1,3,6 Ni50-xAl47Ti3 Cux,x=1,3 NiAl: Ni50-xAl50Fex Ni50-x/2Al50-x/2Fex Ni50Al50-xFex x = 0.25, 2, 5 and 10 FeAl: Fe66Al28Cr6 Ni50Al30Fe20

V, Mn, Cr

V, Mn, Cr all on Al sublattice

Munroe and Baker (1990a, 1992)

Cr Fe

Cr on Al sublattice % Fe on Ni sublattice: 25 88 94 55

Field et al. (1991) Anderson et al. (1995)

Cu All Cu on Al 80% Cu on Ni

Bastow and Rossouw (1998)

Ti inhabits Al sites Cu is indifferent

Wilson and Howe (1999)

Fe

Fe has slight preference for Ni sublattice (e.g., 60 : 40 in x/2 alloys)

Anderson et al. (1999)

Cr

% Occupancy of Fe sublattice 29 30

Munroe and Baker (1990b)

Fe has slight preference for Ni sublattice

NMR gave precise result, but needed calibrating by ALCHEMI. ICPs play supporting role.

Compare results with APFIM, EXAFS, magnetic susceptibility and NMR. See also Intermetallics: other structures table. Assumption of perfect Ni and Al ordering clearly incorrect: need to iterate to improve accuracy (see also Shindo et al. (1988)). (continues )

TABLE 2A—Continued Host compound FeAl: (Fe60 Al40 )95 Cr5 (Fe60 Al40 )90 Cr10 FeAl: Fe49Al50X1

84

FeAl: Fe50Al45X5, X=Ti, V, Cr, Mn, Co, Ni, Cu Fe52Al45Ti3 NiTi Ni50Ti50-xXx Ni50-x/2Ti50-x/2Xx Ni50-xTi50Xx 0<x≤3

Addition

Result

X = Cu, Ni, Co, Mn, Cr, V, Ti Ti, V, Cr, Mn, Co, Ni, Cu

All Cr in solution (up to 6%) on Al sites % X on Fe: Cu Ni Co Mn Cr V T: 96 98 100 100 52 5 93 % X on Fe: Ti V Cr Mn Co Ni Cu 15 19 26 74 99 99 100

Cr

Cr, Mn, Fe, Co, Cu Co, Fe, Pd on Ni. Sc on Ti. Others indifferently distributed, depanding on composition

Ti61.5Al24.5Nb14 Ti65Al24Nb11

(Ti65Al35)90.8V3Fe 6.2 Ti46V30Cr14Al10

(Ti48Al2)(Ti14Nb12Al24) 55% of Nb on Ti sites

V, Fe

V, Fe prefer Al sublattice Cr on Ti sublattice Al on V sublattice

Reference Munroe and Baker (1990c) Kong and Munroe (1994) Anderson (1997)

Nakata, et al. (1991a,b). See also Shimizu and Tadaki (1992) and Tadaki, Nakata, and Shimizu (1995) (reviews) Banerjee et al. (1987) Qian et al. (1991)

Inkson et al. (1993) Li et al. (1998)

Comment

Host element mixing allowed. Planar ALCHEMI: delocalization unimportant. Aging caused significant changes. Good agreement with Bragg–Williams.

B2 structure quenched in. Axial and planar gave same answer. Delocalization more effect for axial.

TABLE 2B ALCHEMI Results: Intermetallics: L12 Compounds Host compound

Addition

Al62Ti27Cr10.5X0.5

Al74.2Ti19Ni6.8

85

Al3 Ti: Al62.8Ti23.8Cu13.4 Al65.0Ti23.1Fe11.9 Al65.1Ti23.0Ni11.9 Ni3 Al: Ni73Al21Fe6 Ni70Al24Co6 Ni76Al21Hf3 Ni3 Al: Ni70Al25Co Ni75Al20Cr5 Ni70Al25Fe5 Ni72.5Al22.5Fe5 Ni75Al20Fe5 Ni3 (Co, Cr) Ni3 Al: Ni75Al20Mn5/600◦ C Ni75Al20Mn5/900◦ C Ni75Al16Mn9/600◦ C Ni3 Al: Ni76Al22Hf2B0.24

Hf Zr W Ni

Result % X occupancy of Al sublattices: 0.0 0.0 10.4 Ni prefers Al sublattice

Cu, Fe, Ni

Reference

Comment

Rossouw et al. (2000)

ICPs play supporting role

Munroe and Baker (1990a, 1991) Ma and Gjønnes (1992)

Implied by formula

Cu, Fe, Ni all on Al Fe, Co, Hf

Fe, Cr, Co

Al

% X on Al: 50 16 100 % Occupancy on Ni sublattice 94 6 70 41 23 Al prefers (Co, Cr) sublattice.

Mn

% Mn on Ni sites 2 25 26

Hf

66% Hf on Al sublattice

Bentley (1986, 1989a)

Axial ALCHEMI. Correction of Spence–Taftø formula.

Shindo et al. (1988)

Ni-Co bonds not exceptionally strong. Al-Cr bonds not exceptionally strong. Fe has moderate preference for Al sublattice: Fe-Ni bonds stronger than Fe-Al Ni3 (Co, Cr) part of two phase mixture in sprayed coating

Fox and Tatlock (1989) Shindo et al. (1990)

Munroe and Baker (1992)

(continues)

TABLE 2B—Continued Host compound Ni3 Al: Ni75Al23Pd2 Ni74Al24Pd2 Ni73Al25Pd2

Addition Pd

Ni3 Al: Ni78Al17Ta5 Ni75Al21Ta4 Ni74Al22Ta4

86

Ni3 Al: Ni75Al17Ti8

Ni3 Al Ni3 Al: Ni76Al21Hf3 Ni3 Al: Ni75.3Al24Zr0.7 Ni73.8Al25.5Zr0.7 Ni3 Al: Ni75Al20Mn5/ 600◦ C Ni75Al20Mn5/900◦ C Ni75Al16Mn9/600◦ C

Result

Reference

% Pd on Ni sites: 100 75 92

Chiba et al. (1991)

All Ta on Al sublattice

Tian et al. (1992) Horita et al. (1997)

Horita et al. (1993) Ti

% Ti on Al sites = 100

2% Re Hf

65% Re on Al sites

Miyazaki et al. (1994) Anderson and Bentley (1995)

78% Hf on Al sites Gu et al. (1997) Zr

Zr on Al sites

Mn

% Mn on Ni sites 2 25 26

Shindo et al. (1990)

Comment Shindo et al. (1986, 1988) analysis. The second and third results make a strange combination. Quoted errors are high. Implies strong Ni-Ta bonding. Consistent with phase diagram. Axial ALCHEMI—minimize delocalization corrections by using TaMα and AlK (similar energies). [100] zone axis gives weak channeling. Axial ALCHEMI. Delocalization corrections necessary. Empirical approach based on disordered alloy of similar composition. Good discussion of delocalization. Axial ALCHEMI with delocalization correction

TABLE 2C ALCHEMI Results: Intermetallics: L10 Compounds Host compound

Addition

Result

Reference

87

TiAl: Ti43Al55Nb2 TiAl: Ti48Al47Nb5 Ti43Al52Nb5

Nb Nb

Nb on Ti sublattice Nb “follows” Ti

Shindo et al. (1986) Konitzer et al. (1986)

TiAl: Ti50Al45Ga5 Ti46.5Al48.5Ga5 TiAl: Ti50Al48V2 Ti44Al54V2 TiAl: Ti50Al48Cr2 Ti44Al54Cr2 TiAl: Ti46Al54Nb2 Ti45.5Al53Mn1.5 Ti47.4Al50V2.6 Ti46.9Al51Cr2.1 TiAl: Ti50Al48Zr2

Ga

All Ga on Al sublattice

Ren et al. (1991)

V

Slightly more V on Ti sublattice V on Ti sublattice Cr mainly on Al sublattice Cr on both sublattices % X on Ti site: 89 3 98 43 All Zr on Ti

Huang and Hall (1991a)

Cr Nb, Mn, V, Cr

Zr

Comment

Authors conscious Ti and Al may not be perfectly ordered

Huang and Hall (1991b)

V small preference for Ti sublattice Cr prefers Al sublattice

Mohandas and Beaven (1991)

Axial and planar ALCHEMI gave same result

Chen et al. (1992) (continues)

TABLE 2C—Continued Host compound

Addition

TiAl: Ti49.5Al49.5Mn1 Ti48Al50Mn2 Ti48al48Mn4 Ti47.5Al47.5Mn5 Ti48Al49Cr3 2% Zr alloy unspecified (but see Chen et al. (1992) (previous line of table)) Ti49Al49Zr1Cr1

88

TiAl TiAl: Ti51Al41Mo8 TiAl: (TiAl)95 Mn5 TiAl: Ti50Al47Cr3 Ti48Al48Cr4 Ti47Al50Cr3 Ti50Al49Ni1 Ti49Al49Ni2 Ti49Al50Ni1 Ti50Al49Zr1 Ti49Al49Zr2 Ti49Al50Zr1

Result

Reference

Comment

Reviere et al. (1993) Mn occupies two sublattices equally

31% of Cr occupies Ti sublattice 96% Zr occupies Ti sublattice

V, Fe (

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