Spectral logic and its applications for the design of digital devices

Providing a complete and self-contained presentation of spectral logic and its applications, Spectral Logic and Its Applications for Design of Digital Devices gives readers a base for further study of abstract harmonic analysis over finite groups, which results in new methods and techniques to process signals modeled by functions on finite groups. The broad scope includes an introduction to harmonic analysis over finite groups, Walsh and Chrestenson transforms, decision diagrams and spectral techniques for their optimization, spectral analysis of Boolean functions, spectral synthesis and optimization of combinational and sequential devices, and spectral methods for testing computer hardware. INDICE: Preface. Acknowledgments. Acronyms. 1. Logic Functions. 1.1 Discrete Functions. 1.2 Tabular Representations of Discrete Functions. 1.3 Functional Expressions. 1.4 Decision Diagrams for Discrete Functions. 1.4.1 Decision trees. 1.4.2 Decision diagrams. 1.4.3 Decision diagrams for multiple-valued functions. 1.5 Spectral Representations of Logic Functions. 1.6 Fixed-polarity Reed-Muller Expressions of Logic Functions. 1.7 Kronecker Expressions of Logic Functions. 1.8 Circuit Implementation of Logic Functions. 2. Spectral Transformsfor Logic Functions. 2.1 Algebraic Structures for Spectral Transforms. 2.2 Fourier Series. 2.3 Bases for Systems of Boolean Functions. 2.3.1 Basis functions. 2.3.2 Walsh functions. 2.3.2.1 Ordering of Walsh functions. 2.3.2.2 Properties of Walsh functions. 2.3.2.3 Hardware implementations of Walsh functions. 2.3.3 Haar functions. 2.3.3.1 Ordering of Haar functions. 2.3.3.2 Properties ofHaar functions. 2.3.3.3 Hardware implementation of Haar functions. 2.3.3.4 Hardware implementation of the inverse Haar transform. 2.4 Walsh Related Transforms. 2.4.1 Arithmetic transform. 2.4.2 Arithmetic expressions from Walsh expansions. 2.5 Bases for Systems of Multiple-Valued Functions. 2.5.1 Vilenkin-Chrestenson functions and their properties. 2.5.2 Generalized Haar functions. 2.6 Properties of Discrete Walsh and Vilenkin-Chrestenson Transforms. 2.7 Autocorrelation and Cross-correlation Functions. 2.7.1 Definitions of autocorrelation and cross-correlation functions. 2.7.2 Relationships to the Walsh and Vilenkin-Chrestenson transforms, the Wiener-Khinchin theorem. 2.7.3 Properties of correlation functions. 2.7.4 Generalized autocorrelation functions. 2.8 Harmonic Analysis over an Arbitrary Finite Abelian Group. 2.8.1 Definition and properties of Fourier transform on finite Abelian groups. 2.8.2 Construction of group characters. 2.8.3 Fourier-Galois transforms. 2.9 Fourier Transform on Finite Non-Abelian Groups. 2.9.1 Representation of finite groups. 2.9.2 Fourier transform on finite non-Abelian groups. 3. Calculation of Spectral Transforms. 3.1 Calculation of Walsh Spectra. 3.1.1 Matrix interpretation of the Fast Walsh transform. 3.1.2 Decision diagram methods for calculation of spectral transforms. 3.1.3 Calculation of the Walsh spectrum through BDD. 3.2 Calculation of the Haar Spectrum. 3.2.1 FFT-like algorithms for the Haar transform. 3.2.2 Matrix interpretation of the Fast Haar transform. 3.2.3 Calculation of the Haar spectrum through BDD. 3.3 Calculation of the Vilenkin-Chrestenson Spectrum. 3.3.0.1 Matrix interpretation of the Fast Vilenkin-Chrestenson transform. 3.3.1 Calculation of the Vilenkin-Chrestenson transform through decision diagrams. 3.4 Calculation of the Generalized Haar Spectrum. 3.5 Calculation of Autocorrelation Functions. 3.5.1 Matrix notation for the Wiener-Khinchin theorem. 3.5.2 Wiener-Khinchin theorem over decision diagrams. 3.5.3 In-place calculation of autocorrelation coefficients by decision diagrams. 4. Spectral Methods in Optimization of Decision Diagrams. 4.1 Reduction of Sizes of Decision Diagrams. 4.1.1 K-procedure for reduction of sizes of decision diagrams. 4.1.2 Properties of the K-procedure 168. 4.2 Construction of Linearly Transformed Binary Decision Diagrams. 4.2.1 Procedure for construction of Linearly Transformed Binary DecisionDiagrams. 4.2.2 Modified K-procedure. 4.2.3 Computing autocorrelation by symbolic manipulations. 4.2.4 Experimental results on the complexity of Linearly Transformed Binary Decision Diagrams. 4.3 Construction of Linearly Transformed Planar BDD. 4.3.1 Planar decision diagrams. 4.3.2 Construction of planar LT-BDD by Walsh coefficients. 4.3.3 Upper bounds on the number of nodes in planar BDDs. 4.3.4 Experimental results for complexity of planar LT-BDDs. 4.4 SpectralInterpretation of Decision Diagrams. 4.4.1 Haar Spectral Transform Decision Diagrams. 4.4.2 Haar transform related decision diagrams. 5. Analysis and Optimization of Logic Functions. 5.1 Spectral Analysis of Boolean Functions. 5.1.1 Linear functions. 5.1.2 Self-dual and anti-self-dual functions. 5.1.3 Partially self-dual and partially anti-self-dual functions. 5.1.4 Quadratic forms, functions with ºat autocorrelation. 5.2 Analysis and Synthesis of Threshold Element Networks. 5.2.1 Threshold elements. 5.2.2 Identification of single threshold functions. 5.3 Complexity of Logic Functions. 5.3.1 Definition of complexityof systems of switching functions. 5.3.2 Complexity and the number of pairs of neighboring minterms. 5.3.3 Complexity criteria for multiple-valued functions. 5.4 Serial Decomposition of Systems of Switching Functions. 5.4.1 Spectral methods and complexity. 5.4.2 Linearization relative to the number of essential variables. 5.4.3 Linearization relative to the entropy based complexity criteria. 5.4.4 Linearization relative to the numbers of neighboring pairs of minterms. 5.4.5 Classification of switching functions by linearization. 5.4.6 Linearization of multiple-valued functions relative to the number of essential variables. 5.4.7 Linearization for multiple-valued functions relative to the entropy based complexity criteria. 5.5 Parallel Decomposition of Systems of Switching Functions. 5.5.1 Polynomial approximation of completely spedified functions. 5.5.2 Additive approximation procedure. 5.5.3 Complexity analysis of polynomial approximations. 5.5.4 Approximation methods for multiple-valued functions. 5.5.5 Estimation on the numbers of non-zero coefficients. 6. Spectral Methods in Synthesis of Logic Networks. 6.1 Spectral Methods of Synthesis of Combinatorial Devices. 6.1.1 Spectral representations of systems of logic functions. 6.1.2 Spectral methods for design of combinatorial devices. 6.1.3 Asymptotically optimal implementation of systems of linear functions. 6.1.4 Walsh and Vilenkin-Chrestenson bases for design of combinatorial networks. 6.1.5 Linear transforms of variables in Haar expressions. 6.1.6 Synthesis with Haar functions. 6.1.6.1 Minimization of the number of non-zero Haar coefficients. 6.1.6.2 Determination of optimal linear transform of variables. 6.1.6.3 Eñciency of the linearization method. 6.2 Spectral Methods for Synthesis of Incompletely Spedified Functions. 6.2.1 Synthesis of incompletely spedified switching functions. 6.2.2 Synthesis of incompletely spedified functions by Haar expressions. 6.3 Spectral Methods of Synthesis of Multiple-Valued Functions. 6.3.1 Multiple-valued functions. 6.3.2 Network implementations of multiple-valued functions. 6.3.3Completion of multiple-valued functions. 6.3.4 Complexity of linear multiple-valued networks. 6.3.5 Minimization of numbers of non-zero coefficients in thegeneralized Haar spectrum for multiple-valued functions. 6.4 Spectral Synthesis of Digital Functions and Sequences Generators. 6.4.1 Function generators. 6.4.2 Design criteria for digital function generators. 6.4.3 Hardware complexity of digital function generators. 6.4.4 Bounds for the number of coefficients in Walsh expansions of analytical functions. 6.4.5 Implementation of switchingfunctions represented by Haar series. 6.4.6 Spectral methods for synthesis ofsequence generators. 7. Spectral Methods of Synthesis of Sequential Machines.7.1 Realization of Finite Automata by Spectral Methods. 7.1.1 Finite structural automata. 7.1.2 Spectral implementation of excitation functions. 7.2 Assignments of States and Inputs for Completely Spedified Automata. 7.2.1 Optimization of the assignments for implementation of the combinational part by using the Haar basis. 7.2.2 Minimization of the number of highest order non-zero coefficients. 7.2.3 Minimization of the number of lowest order non-zero coefficients. 7.3 State Assignment for Incompletely Spedified Automata. 7.3.1 Minimization of higher order non-zero coefficients in representation of incompletely spedified automata. 7.3.2 Minimization of lower order non-zero coefficients in spectral representation of incompletely spedified automata. 7.4 Some Special Cases of the Assignment Problem. 7.4.1 Preliminary remarks. 7.4.2 Autonomous automata. 7.4.3 Assignment problem for automata with fixed encoding of inputs or internal states. 8. Hardware Implementation of Spectral Methods. 8.1 Spectral Methods of Synthesis with ROM. 8.2 Serial Implementation of Spectral Methods. 8.3 Sequential Haar Networks. 8.4 Complexity of Serial Realization by Haar Series. 8.4.1 Optimization of sequential spectral networks. 8.5 Parallel Realization of Spectral Methods of Synthesis. 8.6 Complexity of Parallel Realization. 8.7 Realization by Expansions over Finite Fields. 9. Spectral Methods of Analysis and Synthesis of Reliable Devices. 9.1 Spectral Methods for Analysis of Error Correcting Capabilities. 9.1.1 Errors in combinatorial devices. 9.1.2 Analysis of error correcting capabilities. 9.1.3 Correction of arithmetic errors. 9.2 Spectral Methods for Synthesis of Reliable Digital Devices. 9.2.1 Reliable systems for transmission and logic processing. 9.2.2 Correction of single errors. 9.2.3 Correction of burst errors. 9.2.4 Correction of errors with differentcosts. 9.2.5 Correction of multiple errors. 9.3 Correcting Capability of Sequential Machines. 9.3.1 Error models for finite automata. 9.3.2 Computing an expected number of corrected errors. 9.3.2.1 Simplified calculation of characteristic functions. 9.3.2.2 Calculation of two-dimensional autocorrelation functions. 9.3.3 Error-correcting capabilities of linear automata. 9.3.4 Error-correcting capability of group automata. 9.3.5 Error-correcting capabilities of counting automata. 9.4 Synthesis of Fault-Tolerant Automata with Self-Error-Correction. 9.4.1 Fault-tolerant devices. 9.4.2 Spectral implementation of fault-tolerant automata. 9.4.3 Realization of sequential networks with self-error-correction. 9.5 Comparison of Spectral and Classical Methods. 10. Spectral Methodsfor Testing of Digital Systems. 10.1 Testing and Diagnosis by Verification ofWalsh Coefficien

• ISBN: 978-0-471-73188-7
• Editorial: John Wiley & Sons
• Encuadernacion: Cartoné
• Páginas: 640
• Fecha Publicación: 13/06/2008
• Nº Volúmenes: 1
• Idioma: Inglés