LEADER 07779nam 22007334a 450 001 9911018808903321 005 20200520144314.0 010 $a9786610277933 010 $a9781280277931 010 $a1280277939 010 $a9780471745433 010 $a047174543X 010 $a9781601193766 010 $a1601193769 010 $a9780471745426 010 $a0471745421 024 7 $a10.1002/047174543X 035 $a(CKB)1000000000355340 035 $a(EBL)233622 035 $a(SSID)ssj0000072005 035 $a(PQKBManifestationID)11110061 035 $a(PQKBTitleCode)TC0000072005 035 $a(PQKBWorkID)10090527 035 $a(PQKB)10590128 035 $a(MiAaPQ)EBC233622 035 $a(CaBNVSL)mat05237943 035 $a(IDAMS)0b00006481095e37 035 $a(IEEE)5237943 035 $a(OCoLC)173691873 035 $a(Perlego)2760345 035 $a(EXLCZ)991000000000355340 100 $a20050823d2005 uy 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 10$aFourier analysis on finite groups with applications in signal processing and system design /$fRadomir S. Stankovic, Claudio Moraga, Jaakko Astola 210 $aPiscataway, NJ $cIEEE Press ;$aHoboken, N.J. $cWiley-Interscience$dc2005 215 $a1 online resource (262 p.) 300 $aDescription based upon print version of record. 311 08$a9780471694632 311 08$a0471694630 320 $aIncludes bibliographical references and index. 327 $aPreface -- 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 Non-Abelian Groups -- 2.5 Fast Fourier Transform on Finite Non-Abelian Groups -- References -- 3 Matrix Interpretation of the FFT -- 3.1 Matrix Interpretation of FFT on Finite Non-Abelian 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 Non-Abelian 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 Group-Theoretic 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 Two-Variable 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 Matrix-valued Functions -- 4.9.2 Fourier Transform for Matrix-Valued 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 Q2. 327 $a5.3 Arithmetic Expressions -- 5.4 Arithmetic Expressions from Walsh Expansions -- 5.5 Arithmetic Expressions on Q2 -- 5.5.1 Arithmetic Expressions and Arithmetic-Haar Expressions -- 5.5.2 Arithmetic-Haar Expressions and Kronecker Expressions -- 5.6 Different Polarity Polynomials Expressions -- 5.6.1 Fixed-Polarity Fourier Expressions in C(Q2) -- 5.6.2 Fixed-Polarity Arithmetic-HaarExpressions -- 5.7 Calculation of the Arithmetic-Haar Coefficients -- 5.7.1 FFT-like Algorithm -- 5.7.2 Calculation of Arithmetic-Haar Coefficients Through Decision Diagrams -- References -- 6 Gibbs Derivatives on Finite Groups -- 6.1 Definition and Properties of Gibbs Derivatives on Finite Non-Abelian Groups -- 6.2 Gibbs Anti-Derivative -- 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 Non-Abelian Groups -- 7.1 Linear Shift-Invariant Systems on Groups -- 7.2 Linear Shift-Invariant Systems on Finite Non-Abelian 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 Non-Abelian Groups -- 8.2 Hilbert Transform on Finite Non-Abelian Groups -- 8.3 Hilbert Transform in Finite Fields -- References -- Index. 330 $aDiscover applications of Fourier analysis on finite non-Abelian groups The majority of publications in spectral techniques consider Fourier transform on Abelian groups. However, non-Abelian 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 non-Abelian 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 non-Abelian 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 non-Abelian groups. Hilbert transform on finite groups Among the highlights is an in-depth coverage of applications of abstract harmonic analysis on finite non-Abelian groups in compact representations of discrete functions and related tasks in signal processing and system design, including logic design. All chapters are self-contained, each with a list of references to facilitate the development of specialized courses or self-study. With nearly 100 illustrative figures and fifty tables, this is an excellent textbook for graduate-level students and researchers in signal processing, logic design, and system theory-as well as the more general topics of computer science and applied mathematics. 606 $aSignal processing$xMathematics 606 $aFourier analysis 606 $aNon-Abelian groups 615 0$aSignal processing$xMathematics. 615 0$aFourier analysis. 615 0$aNon-Abelian groups. 676 $a621.382/2 700 $aStankovic$b Radomir S$01731465 701 $aMoraga$b Claudio$0845321 701 $aAstola$b Jaakko$01837676 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9911018808903321 996 $aFourier analysis on finite groups with applications in signal processing and system design$94422724 997 $aUNINA