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Digital Signal Processing Algorithms

Number Theory, Convolution, Fast Fourier Transforms, and Applications

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## Book Description

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.

## Table of Contents

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

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

Index

## Reviews

"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