Preface |
Acknowledgments |
Acronyms |
Signals and Their Mathematical Models / 1: |
Systems / 1.1: |
Signals / 1.2: |
Mathematical Models of Signals / 1.3: |
References |
Fourier Analysis / 2: |
Representations of Groups / 2.1: |
Complete Reducibility / 2.1.1: |
Fourier Transform on Finite Groups / 2.2: |
Properties of the Fourier Transform / 2.3: |
Matrix Interpretation of the Fourier Transform on Finite Non-Abelian Groups / 2.4: |
Complete reducibility |
Fast Fourier Transform on Finite Non-Abelian Groups / 2.5: |
Matrix Interpretation of the FFT / 3: |
Properties of the Fourier transform / 3.1: |
Matrix Interpretation of FFT on Finite Non-Abelian Groups |
Illustrative Examples / 3.2: |
Matrix interpretation of the Fourier transform on finite non-Abelian groups |
Complexity of the FFT / 3.3: |
Fast Fourier transform on finite non-Abelian groups / 3.3.1: |
Complexity of Calculations of the FFT |
Remarks on Programming Implememtation of FFT / 3.3.2: |
FFT Through Decision Diagrams / 3.4: |
Decision Diagrams / 3.4.1: |
Matrix interpretation of FFT on finite non-Abelian groups / 3.4.2: |
FFT on Finite Non-Abelian Groups Through DDs |
MMTDs for the Fourier Spectrum / 3.4.3: |
Illustrative examples |
Complexity of DDs Calculation Methods / 3.4.4: |
Optimization of Decision Diagrams / 4: |
Complexity of calculations of the FFT / 4.1: |
Reduction Possibilities in Decision Diagrams |
Group-Theoretic Interpretation of DD / 4.2: |
Remarks on programming implementation of FFT |
Fourier Decision Diagrams / 4.3: |
FFT through decision diagrams / 4.3.1: |
Fourier Decision Trees |
Decision diagrams / 4.3.2: |
Discussion of Different Decompositions / 4.4: |
FFT on finite non-Abelian groups through DDs / 4.4.1: |
Algorithm for Optimization of DDs |
Representation of Two-Variable Function Generator / 4.5: |
MTDDs for the Fourier spectrum |
Representation of Adders by Fourier DD / 4.6: |
Complexity of DDs calculation methods / 4.7: |
Representation of Multipliers by Fourier DD |
Complexity of NADD / 4.8: |
Fourier DDs with Preprocessing / 4.9: |
Matrix-valued Functions / 4.9.1: |
Fourier Transform for Matrix-Valued Functions / 4.9.2: |
Fourier Decision Trees with Preprocessing / 4.10: |
Group-theoretic Interpretation of DD |
Fourier Decision Diagrams with Preprocessing / 4.11: |
Construction of FNAPDD / 4.12: |
Algorithm for Construction of FNAPDD / 4.13: |
Fourier decision trees |
Algorithm for Representation / 4.13.1: |
Fourier decision diagrams / 4.14: |
Optimization of FNAPDD |
Functional Expressions on Quaternion Groups / 5: |
Fourier Expressions on Finite Dyadic Groups / 5.1: |
Algorithm for optimization of DDs |
Finite Dyadic Groups / 5.1.1: |
Representation of adders by Fourier DD / 5.2: |
Arithmetic Expressions / 5.3: |
Representation of multipliers by Fourier DD / 5.4: |
Arithmetic Expressions from Walsh Expansions |
Complexity of FNADD / 5.5: |
Arithmetic Expressions and Arithmetic-Haar Expressions / 5.5.1: |
Arithmetic-Haar Expressions and Kronecker Expressions / 5.5.2: |
Matrix-valued functions |
Different Polarity Polynomials Expressions / 5.6: |
Fourier transform for matrix-valued functions / 5.6.1: |
Fixed-Polarity Arithmetic-HaarExpressions / 5.6.2: |
Calculation of the Arithmetic-Haar Coefficients / 5.7: |
FFT-like Algorithm / 5.7.1: |
Calculation of Arithmetic-Haar Coefficients Through Decision Diagrams / 5.7.2: |
Gibbs Derivatives on Finite Groups / 6: |
Definition and Properties of Gibbs Derivatives on Finite Non-Abelian Groups / 6.1: |
Algorithm for representation |
Gibbs Anti-Derivative / 6.2: |
Partial Gibbs Derivatives / 6.3: |
Gibbs Differential Equations / 6.4: |
Matrix Interpretation of Gibbs Derivatives / 6.5: |
Fast Algorithms for Calculation of Gibbs Derivatives on Finite Groups / 6.6: |
Fourier expressions on finite dyadic groups / 6.6.1: |
Complexity of Calculation of Gibbs Derivatives |
Calculation of Gibbs Derivatives Through DDs / 6.7: |
Finite dyadic groups |
Calculation of Partial Gibbs Derivatives / 6.7.1: |
Fourier Expressions on Q[subscript 2] |
Linear Systems on Finite Non-Abelian Groups / 7: |
Linear Shift-Invariant Systems on Groups / 7.1: |
Linear Shift-Invariant Systems on Finite Non-Abelian Groups / 7.2: |
Arithmetic expressions from Walsh expansions |
Gibbs Derivatives and Linear Systems / 7.3: |
Arithmetic expressions on Q[subscript 2] / 7.3.1: |
Discussion |
Arithmetic expressions and arithmetic-Haar expressions / 8: |
Hilbert Transform on Finite Groups |
Some Results of Fourier Analysis on Finite Non-Abelian Groups / 8.1: |
Arithmetic-Haar expressions and Kronecker expressions |
Hilbert Transform on Finite Non-Abelian Groups / 8.2: |
Different Polarity Polynomial Expressions / 8.3: |
Hilbert Transform in Finite Fields |
Index |
Fixed-polarity Fourier expansions in C(Q[subscript 2]) |
Fixed-polarity arithmetic-Haar expressions |
Calculation of the arithmetic-Haar coefficients |
FFT-like algorithm |
Calculation of arithmetic-Haar coefficients through decision diagrams |
Definition and properties of Gibbs derivatives on finite non-Abelian groups |
Gibbs anti-derivative |
Partial Gibbs derivatives |
Gibbs differential equations |
Matrix interpretation of Gibbs derivatives |
Fast algorithms for calculation of Gibbs derivatives on finite groups |
Calculation of Gibbs derivatives through DDs |
Calculation of partial Gibbs derivatives |
Linear shift-invariant systems on groups |
Linear shift-invariant systems on finite non-Abelian groups |
Gibbs derivatives and linear systems |
Some results of Fourier analysis on finite non-Abelian groups |
Hilbert transform on finite non-Abelian groups |
Hilbert transform in finite fields |