1st Edition

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

672 Pages
by CRC Press

672 Pages
by Routledge

Also available as eBook on:

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.

Introduction
Outline
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
Introduction
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)
Algorithms
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
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 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
Index

### Biography

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