1st Edition

Digital Signal Processing Algorithms Number Theory, Convolution, Fast Fourier Transforms, and Applications

By Hari Krishna Copyright 2000

    Digital Signal Processing Algorithms describes computational number theory and its applications to deriving fast algorithms for digital signal processing. It demonstrates the importance of computational number theory in the design of digital signal processing algorithms and clearly describes the nature and structure of the algorithms themselves. The book has two primary focuses: first, it establishes the properties of discrete-time sequence indices and their corresponding fast algorithms; and second, it investigates the properties of the discrete-time sequences and the corresponding fast algorithms for processing these sequences.
    Digital Signal Processing Algorithms examines three of the most common computational tasks that occur in digital signal processing; namely, cyclic convolution, acyclic convolution, and discrete Fourier transformation. The application of number theory to deriving fast and efficient algorithms for these three and related computationally intensive tasks is clearly discussed and illustrated with examples.
    Its comprehensive coverage of digital signal processing, computer arithmetic, and coding theory makes Digital Signal Processing Algorithms an excellent reference for practicing engineers. The authors' intent to demystify the abstract nature of number theory and the related algebra is evident throughout the text, providing clear and precise coverage of the quickly evolving field of digital signal processing.

    The Organization
    PART I: Computational Number Theory
    Computational Number Theory
    Groups, Rings, and Fields
    Elements of Number Theory
    Integer Rings and Fields
    Chinese Remainder Theorem for Integers
    Number Theory for Finite Integer Rings
    Polynomial Algebra
    Algebra of Polynomials over a Field
    Roots of a Polynomial
    Polynomial Fields and Rings
    The Chinese Remainder Theorem for Polynomials
    CRT-P in Matrix Form
    Lagrange Interpolation
    Polynomial Algebra over GF(p)
    Order of an Element
    Theoretical Aspects of Discrete Fourier Transform and Convolution
    The Discrete Fourier Transform
    Basic Formulation of Convolution
    Bounds on the Multiplicative Complexity
    Basic Formulation of Convolution Algorithms
    Matrix Exchange Property
    Cyclotomic Polynomial Factorization and Associated Fields
    Cyclotomic Polynomial Factorization over Complex and Real Numbers
    Cyclotomic Polynomial Factorization over Rational Numbers
    Cyclotomic Fields and Cyclotomic Polynomial Factorizations
    Extension Fields of Cyclotomic Fields and Cyclotomic Polynomial Factorization
    A Preview of Applications to Digital Signal Processing
    Cyclotomic Polynomial Factorization in Finite Fields
    Cyclotomic Polynomial Factorization
    Factorization of (un - 1) over GF (p)
    Primitive Polynomials over GF (p)
    Complex Finite Fields and Cyclotomic Polynomial Factorization
    Finite Integer Rings: Polynomial Algebra and Cyclotomic Factorization
    Polynomial Algebra over a Ring
    Lagrange Interpolation
    Number Theoretic Transforms
    Monic Polynomial Factorization
    Extension of CRT-P over Finite Integer Rings
    Polynomial Algebra and CRT-PR: The Complex Case
    Number Theoretic Transforms: The Complex Case
    Pseudo Number Theoretic Transforms
    Polynomial Algebra and Direct Sum Properties in Integer Polynomial Rings
    PART II: Convolution Algorithms
    Thoughts on Part II
    Fast Algorithms for Acyclic Convolution
    CRT-P Based Fast Algorithms for One-Dimensional Acyclic Convolution
    Casting the Algorithm in Bilinear Formulation
    Multidimensional Approaches to One-Dimensional Acyclic Convolution
    Multidimensional Acyclic Convolution Algorithms
    Nesting and Split Nesting Algorithms for Multidimensional Convolution
    Acyclic Convolution Algorithms over Finite Fields and Rings
    Fast One-Dimensional Cyclic Convolution Algorithms
    Bilinear Forms and Cyclic Convolution
    Cyclotomic Polynomials and Related Algorithms over Re and C
    Cyclotomic Polynomials and Related Algorithms over Z
    Other Considerations
    Complex Cyclotomic Polynomials and Related Algorithms over CZ
    The Agarwal-Cooley Algorithm
    Cyclic Convolution Algorithms over Finite Fields and Rings
    Two- and Higher Dimensional Cyclic Convolution Algorithms
    Polynomial Formulation and an Algorithm
    Improvements and Related Algorithms
    Discrete Fourier Transform Based Algorithms
    Algorithms Based on Extension Fields
    Algorithms for Multidimensional Cyclic Convolution
    Algorithms for Two-Dimensional Cyclic Convolution in Finite Integer Rings
    Validity of Fast Algorithms over Different Number Systems
    Mathematical Preliminaries
    Chinese Remainder Theorem over Finite Integer Rings
    Interrelationships among Algorithms over Different Number Systems
    Analysis of Two-Dimensional Cyclic Convolution Algorithms
    Fault Tolerance for Integer Sequences
    A Framework for Fault Tolerance
    Mathematical Structure of C over Z(M)
    Coding Techniques over Z(q)
    Examples and SFC-DFD Codes
    PART III: Fast Fourier Transform (FFT)
    Thoughts on Part III
    Fast Fourier Transform: One-Dimensional Data Sequences
    The DFT: Definitions and Properties
    Rader's FFT Algorithm, n=p, p an Odd Prime
    Rader's FFT Algorithm, n=pc, p an Odd Prime
    Cooley-Tukey FFT Algorithm, n=a . b
    FFT Algorithms for n a Power of 2
    The Prime Factor FFT n=a . b, (a,b) =1
    The Winograd FFT Algorithm
    Fast Fourier Transform: Multidimensional Data Sequences
    The Multidimensional DFT: Definition and Properties
    FFT for n=p, p an Odd Prime
    Multidimensional FFT Algorithms for n a Power of 2
    Matrix Formulation of Multidimensional DFT and Related Algorithms
    Polynomial Version of Rader's Algorithm
    Polynomial Transform Based FFT Algorithms
    PART IV: Recent Results on Algorithms in Finite Integer Rings
    Thoughts on Part IV
    Paper One: A Number Theoretic Approach to Fast Algorithms for Two-Dimensional Digital Signal Processing in Finite Integer Rings
    Paper Two: On Fast Algorithms for One-Dimensional Digital Signal Processing in Finite Integer and Complex Integer Rings
    Paper Three: Cyclotomic Polynomial Factorization in Finite Integer Rings with Applications to Digital Signal Processing
    Paper Four: Error Control Techniques for Data Sequences Defined in Finite Integer Rings
    A. Small Length Acyclic Convolution Algorithms
    B. Classification of Cyclotomic Polynomials


    Hari Krishna

    "It is hard to criticise this book. It does what it does well…this is an unusual and useful book, which definitely ought to find its niche…"
    --Mark Sandler, Applied Signal Processing, Vol. 5, No. 4