Ebook
Fourier Analysis on Finite Groups with Applications in Signal Processing and System DesignISBN: 9780471745426
230 pages
August 2005, WileyIEEE Press

The majority of publications in spectral techniques consider Fourier transform on Abelian groups. However, nonAbelian groups provide notable advantages in efficient implementations of spectral methods.
Fourier Analysis on Finite Groups with Applications in Signal Processing and System Design examines aspects of Fourier analysis on finite nonAbelian groups and discusses different methods used to determine compact representations for discrete functions providing for their efficient realizations and related applications. Switching functions are included as an example of discrete functions in engineering practice. Additionally, consideration is given to the polynomial expressions and decision diagrams defined in terms of Fourier transform on finite nonAbelian groups.
A solid foundation of this complex topic is provided by beginning with a review of signals and their mathematical models and Fourier analysis. Next, the book examines recent achievements and discoveries in:
 Matrix interpretation of the fast Fourier transform
 Optimization of decision diagrams
 Functional expressions on quaternion groups
 Gibbs derivatives on finite groups
 Linear systems on finite nonAbelian groups
 Hilbert transform on finite groups
Among the highlights is an indepth coverage of applications of abstract harmonic analysis on finite nonAbelian groups in compact representations of discrete functions and related tasks in signal processing and system design, including logic design. All chapters are selfcontained, each with a list of references to facilitate the development of specialized courses or selfstudy.
With nearly 100 illustrative figures and fifty tables, this is an excellent textbook for graduatelevel students and researchers in signal processing, logic design, and system theoryas well as the more general topics of computer science and applied mathematics.
Acknowledgments.
Acronyms.
1 Signals and Their Mathematical Models.
1.1 Systems.
1.2 Signals.
1.3 Mathematical Models of Signals.
References.
2 Fourier Analysis.
2.1 Representations of Groups.
2.1.1 Complete Reducibility.
2.2 Fourier Transform on Finite Groups.
2.3 Properties of the Fourier Transform.
2.4 Matrix Interpretation of the Fourier Transform on Finite NonAbelian Groups.
2.5 Fast Fourier Transform on Finite NonAbelian Groups.
References.
3 Matrix Interpretation of the FFT.
3.1 Matrix Interpretation of FFT on Finite NonAbelian Groups.
3.2 Illustrative Examples.
3.3 Complexity of the FFT.
3.3.1 Complexity of Calculations of the FFT.
3.3.2 Remarks on Programming Implememtation of FFT.
3.4 FFT Through Decision Diagrams.
3.4.1 Decision Diagrams.
3.4.2 FFT on Finite NonAbelian Groups Through DDs.
3.4.3 MMTDs for the Fourier Spectrum.
3.4.4 Complexity of DDs Calculation Methods.
References.
4 Optimization of Decision Diagrams.
4.1 Reduction Possibilities in Decision Diagrams.
4.2 GroupTheoretic Interpretation of DD.
4.3 Fourier Decision Diagrams.
4.3.1 Fourier Decision Trees.
4.3.2 Fourier Decision Diagrams.
4.4 Discussion of Different Decompositions.
4.4.1 Algorithm for Optimization of DDs.
4.5 Representation of TwoVariable Function Generator.
4.6 Representation of Adders by Fourier DD.
4.7 Representation of Multipliers by Fourier DD.
4.8 Complexity of NADD.
4.9 Fourier DDs with Preprocessing.
4.9.1 Matrixvalued Functions.
4.9.2 Fourier Transform for MatrixValued Functions.
4.10 Fourier Decision Trees with Preprocessing.
4.11 Fourier Decision Diagrams with Preprocessing.
4.12 Construction of FNAPDD.
4.13 Algorithm for Construction of FNAPDD.
4.13.1 Algorithm for Representation.
4.14 Optimization of FNAPDD.
References.
5 Functional Expressions on Quaternion Groups.
5.1 Fourier Expressions on Finite Dyadic Groups.
5.1.1 Finite Dyadic Groups.
5.2 Fourier Expressions on Q_{2}.
5.3 Arithmetic Expressions.
5.4 Arithmetic Expressions from Walsh Expansions.
5.5 Arithmetic Expressions on Q_{2}.
5.5.1 Arithmetic Expressions and ArithmeticHaar Expressions.
5.5.2 ArithmeticHaar Expressions and Kronecker Expressions.
5.6 Different Polarity Polynomials Expressions.
5.6.1 FixedPolarity Fourier Expressions in C(Q_{2}).
5.6.2 FixedPolarity ArithmeticHaar Expressions.
5.7 Calculation of the ArithmeticHaar Coefficients.
5.7.1 FFTlike Algorithm.
5.7.2 Calculation of ArithmeticHaar Coefficients Through Decision Diagrams.
References.
6 Gibbs Derivatives on Finite Groups.
6.1 Definition and Properties of Gibbs Derivatives on Finite NonAbelian Groups.
6.2 Gibbs AntiDerivative.
6.3 Partial Gibbs Derivatives.
6.4 Gibbs Differential Equations.
6.5 Matrix Interpretation of Gibbs Derivatives.
6.6 Fast Algorithms for Calculation of Gibbs Derivatives on Finite Groups.
6.6.1 Complexity of Calculation of Gibbs Derivatives.
6.7 Calculation of Gibbs Derivatives Through DDs.
6.7.1 Calculation of Partial Gibbs Derivatives.
References.
7 Linear Systems on Finite NonAbelian Groups.
7.1 Linear ShiftInvariant Systems on Groups.
7.2 Linear ShiftInvariant Systems on Finite NonAbelian Groups.
7.3 Gibbs Derivatives and Linear Systems.
7.3.1 Discussion.
References.
8 Hilbert Transform on Finite Groups.
8.1 Some Results of Fourier Analysis on Finite NonAbelian Groups.
8.2 Hilbert Transform on Finite NonAbelian Groups.
8.3 Hilbert Transform in Finite Fields.
References.
Index.
CLAUDIO MORAGA, PhD, is Professor, Department of Computer Science, Dortmund University, Germany.
JAAKKO T. ASTOLA, PhD, is Professor, Institute of Signal Processing, Tampere University of Technology, Finland.