Spectral Logic and Its Applications for the Design of Digital Devices

ACKNOWLEDGMENTS.

LIST OF FIGURES.

LIST OF TABLES.

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.

1.5 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 TRANSFORMS FOR LOGIC FUNCTIONS.

2.1 Algebraic Structures for Spectral Transforms.

2.2 Fourier Series.

2.3 Bases for Systems of Boolean Functions.

2.4 Walsh Related Transforms.

2.5 Bases for Systems of Multiple-Valued Functions.

2.6 Properties of DiscreteWalsh andVilenkin–Chrestenson Transforms.

2.7 Autocorrelation and Cross-Correlation Functions.

2.8 Harmonic Analysis over an Arbitrary Finite Abelian Group.

2.9 Fourier Transform on Finite Non–Abelian Groups.

3. CALCULATION OF SPECTRAL TRANSFORMS.

3.1 Calculation of Walsh Spectra.

3.2 Calculation of the Haar Spectrum.

3.3 Calculation of the Vilenkin–Chrestenson Spectrum.

3.4 Calculation of the Generalized Haar Spectrum.

3.5 Calculation of Autocorrelation Functions.

4. SPECTRAL METHODS IN OPTIMIZATION OF DECISION DIAGRAMS.

4.1 Reduction of Sizes of Decision Diagrams.

4.2 Construction of Linearly Transformed Binary Decision Diagrams.

4.3 Construction of Linearly Transformed Planar BDD.

4.4 Spectral Interpretation of Decision Diagrams.

5. ANALYSIS AND OPTIMIZATION OF LOGIC FUNCTIONS.

5.1 Spectral Analysis of Boolean Functions.

5.2 Analysis and Synthesis of Threshold Element Networks.

5.3 Complexity of Logic Functions.

5.4 Serial Decomposition of Systems of Switching Functions.

5.5 Parallel Decomposition of Systems of Switching Functions.

6. SPECTRAL METHODS IN SYNTHESIS OF LOGIC NETWORKS.

6.1 Spectral Methods of Synthesis of Combinatorial Devices.

6.2 Spectral Methods for Synthesis of Incompletely Specified Functions.

6.3 Spectral Methods of Synthesis of Multiple-Valued Functions.

6.4 Spectral Synthesis of Digital Functions and Sequences Generators.

7. SPECTRAL METHODS OF SYNTHESIS OF SEQUENTIAL MACHINES.

7.1 Realization of Finite Automata by Spectral Methods.

7.2 Assignment of States and Inputs for Completely Specified Automata.

7.3 State Assignment for Incompletely Specified Automata.

7.4 Some Special Cases of the Assignment Problem.

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.

8.5 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.

9.2 Spectral Methods for Synthesis of Reliable Digital Devices.

9.3 Correcting Capability of Sequential Machines.

9.4 Synthesis of Fault-Tolerant Automata with Self-Error Correction.

9.5 Comparison of Spectral and Classical Methods.

10. SPECTRAL METHODS FOR TESTING OF DIGITAL SYSTEMS.

10.1 Testing and Diagnosis by Verification of Walsh Coefficients.

10.2 Functional Testing, Error Detection, and Correction by Linear Checks.

10.3 Linear Checks for Processors.

10.4 Linear Checks for Error Detection in Polynomial Computations.

10.5 Construction of Optimal Linear Checks for Polynomial Computations.

10.6 Implementations and Error-Detecting Capabilities of Linear Checks.

10.7 Testing for Numerical Computations.

10.8 Optimal Inequality Checks and Error-Correcting Codes.

10.9 Error Detection in Computer Memories by Linear Checks.

10.10 Location of Errors in ROMs by Two Orthogonal Inequality Checks.

10.11 Detection and Location of Errors in Random-Access Memories.

11. EXAMPLES OF APPLICATIONS AND GENERALIZATIONS OF SPECTRAL METHODS ON LOGIC FUNCTIONS.

11.1 Transforms Designed for Particular Applications.

11.2 Wavelet Transforms.

11.3 Fibonacci Transforms.

11.4 Two-Dimensional Spectral Transforms.

11.5 Application of the Walsh Transform in Broadband Radio.

APPENDIX A.

REFERENCES.

INDEX.