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

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ISBN: 9780849371783 | 0849371783
Cover: Hardcover
Copyright: 3/25/1998

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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.

1 Introduction

(6)

1.1 Outline

(1)

1.2 The Organization

(4)

PART I: COMPUTATIONAL NUMBER THEORY

(218)

Thoughts on Part I

(2)

2 Computational Number Theory

(24)

2.1 Groups, Rings, and Fields

(2)

2.2 Elements of Number Theory

(4)

2.3 Integer Rings and Fields

(2)

2.4 Chinese Remainder Theorem for Integers

(4)

2.5 Number Theory for Finite Integer Rings

(8)

2.5.1 Permutation

(1)

2.5.2 Euler's Totient Function

(4)

2.5.3 Order of an Element

(4)

Notes

(1)

Bibliography

(1)

Problems

(3)

3 Polynomial Algebra

(32)

3.1 Algebra of Polynomials Over a Field

(5)

3.2 Roots of a Polynomial

(1)

3.3 Polynomial Fields and Rings

(2)

3.4 Chinese Remainder Theorem for Polynomials

(4)

3.5 CRT-P in Matrix Form

(4)

3.6 Lagrange Interpolation

(1)

3.7 Polynomial Algebra Over GF(p)

(3)

3.8 Order of an Element: Continued

(6)

Notes

(1)

Bibliography

(1)

Problems

(3)

4 Theoretical Aspects of the Discrete Fourier Transform and Convolution

(24)

4.1 The Discrete Fourier Transform

(3)

4.2 Basic Formulation of Convolution

(3)

4.3 Bounds on the Multiplicative Complexity

(7)

4.3.1 Multiplicative Complexity of Bilinear Forms

(2)

4.3.2 Dual of a Bilinear Form

(5)

4.4 Basic Formulation of Convolution Algorithms

(5)

4.4.1 The Acyclic Convolution Algorithm

(2)

4.4.2 The Cyclic Convolution Algorithm

(2)

4.4.3 Converting Acyclic to a Cyclic Convolution

(1)

4.4.4 Cyclic Convolution Beyond DFT

(1)

4.5 Matrix Exchange Property

(2)

Notes

(1)

Bibliography

(1)

Problems

(2)

5 Cyclotomic Polynomial Factorization and Associated Fields

(34)

5.1 Cyclotomic Polynomial Factorization Over Complex and Real Numbers

(1)

5.2 Cyclotomic Polynomial Factorization Over Rational Numbers

(9)

5.3 Cyclotomic Fields and Cyclotomic Polynomial Factorization

102

(8)

5.3.1 Cyclotomic Polynomial Factorization Over Cyclotomic Fields

103

(4)

5.3.2 Complex Cyclotomic Polynomials

107

(1)

5.3.3 n: a Power of 2

108

(2)

5.4 Extension Fields of Cyclotomic Fields and Cyclotomic Polynomial Factorization

110

(3)

5.5 A Preview of Applications to Digital Signal Processing

113

(6)

5.5.1 Applications to Simplifying Complex-Valued Computations

113

(3)

5.5.2 Expressing an One-Dimensional Computation as a Two-Dimensional Computation

116

(3)

Notes

119

(1)

Bibliography

120

(1)

Problems

121

(4)

6 Cyclotomic Polynomial Factorization in Finite Fields

125

(26)

6.1 Cyclotomic Polynomial Factorization

125

(5)

6.2 Factorization of (u(n) - 1)) Over GF (p)

130

(6)

6.2.1 Returning Back to GF(p)

131

(5)

6.3 Primitive Polynomials Over GF(p)

136

(6)

6.4 Complex Finite Fields and Cyclotomic Polynomial Factorization

142

(6)

6.4.1 Factorization of (j(2) + 1) Over GF(p)

142

(2)

6.4.2 Special cases

144

(1)

6.4.3 Cyclotomic Factorization Over Complex Finite Fields

144

(4)

Notes

148

(1)

Bibliography

148

(1)

Problems

148

(3)

7 Finite Integer Rings: Polynomial Algebra and Cyclotomic Factorization

151

(74)

7.1 Polynomial Algebra Over a Ring

151

(4)

7.1.1 Roots of a Polynomial Over Z(M)

153

(2)

7.2 Lagrange Interpolation

155

(4)

7.2.1 A Special Case of Lagrange Interpolation

157

(2)

7.3 Number Theoretic Transforms

159

(6)

7.3.1 Mersenne Number Transforms

162

(1)

7.3.2 Fermat Number Transforms

163

(2)

7.4 Monic Polynomial Factorization

165

(12)

7.4.1 A Systematic Procedure

172

(3)

7.4.2 Monic Polynomial Factorization Over Z(M)

175

(2)

7.5 Extension of CRT-P Over Finite Integer Rings

177

(7)

7.5.1 The CRT-PR Reconstruction

180

(2)

7.5.2 CRT-PR in Matrix Form

182

(2)

7.6 Polynomial Algebra and CRT-PR: The Complex Case

184

(8)

7.6.1 Computations Over Complex Integer Rings

185

(1)

7.6.2 Factorization of j(2) + 1 Over GF(p)

186

(1)

7.6.3 Factorization Over a Complex Integer Ring

187

(3)

7.6.4 CRT-PR Over Complex Integer Rings

190

(2)

7.7 Number Theoretic Transforms: The Complex Case

192

(3)

7.8 Pseudo Number Theoretic Transforms

195

(1)

7.9 Polynomial Algebra and Direct Sum Properties in Integer Polynomial Rings

196

(22)

7.9.1 Mathematical Preliminaries

197

(2)

7.9.2 Polynomial Rings and Fields

199

(7)

7.9.3 Monic Polynomial Factorization in Z(q, Q(Theta))

206

(7)

7.9.4 CRT-PR in Z(q, Q(Theta)) and CZ(q, Q(Theta))

213

(5)

Notes

218

(1)

Bibliography

218

(2)

Problems

220

(5)

PART II: CONVOLUTION ALGORITHMS

225

(200)

Thoughts on Part II

227

(6)

8 Fast Algorithms for Acyclic Convolution

233

(42)

8.1 CRT-P Based Fast Algorithms for One-Dimensional Acyclic Convolution

234

(5)

8.1.1 Improvements in the Basic Procedure

237

(2)

8.2 Casting the Algorithm in a Bilinear Form

239

(11)

8.3 Multidimensional Approaches to One-Dimensional Acyclic Convolution

250

(4)

8.4 Multidimensional Acyclic Convolution Algorithms

254

(4)

8.5 Nesting and Split Nesting Algorithms for Multidimensional Convolution

258

(4)

8.6 Acyclic Convolution Algorithms Over Finite Fields and Rings

262

(9)

8.6.1 Algorithms Over Finite Fields

262

(5)

8.6.2 Algorithms Over Finite Integer Rings

267

(4)

Notes

271

(1)

Bibliography

272

(1)

Problems

272

(3)

9 Fast One-Dimensional Cyclic Convolution Algorithms

275

(62)

9.1 Bilinear Forms and Cyclic Convolution

276

(7)

9.2 Cyclotomic Polynomials and Related Algorithms Over Re and C

283

(2)

9.3 Cyclotomic Polynomials and Related Algorithms Over Z

285

(18)

9.4 Other Considerations

303

(3)

9.5 Complex Cyclotomic Polynomials and Related Algorithms Over CZ

306

(7)

9.5.1 Conjugate Symmetry Property of Complex Cyclotomic Polynomials

311

(1)

9.5.2 Computational Complexity Comparisons

312

(1)

9.6 The Agarwal-Cooley Algorithm

313

(5)

9.6.1 Conversion from One Dimension to Higher Dimensions

313

(3)

9.6.2 Computing Higher-Dimensional Cyclic Convolutions

316

(2)

9.7 Cyclic Convolution Algorithms Over Finite Fields and Rings

318

(12)

Notes

330

(1)

Bibliography

330

(2)

Problems

332

(5)

10 Two-and Higher-Dimensional Cyclic Convolution Algorithms

337

(44)

10.1 Polynomial Formulation and an Algorithm

338

(2)

10.2 Improvements and Related Algorithms

340

(1)

10.2.1 Split Nesting Algorithms

340

(1)

10.3 Discrete Fourier Transform Based Algorithms

341

(1)

10.4 Algorithms Based on Extension Fields

342

(17)

10.5 Algorithms for Multidimensional Cyclic Convolution

359

(2)

10.6 Algorithms for Two-Dimensional Cyclic Convolution in Finite Integer Rings

361

(12)

10.6.1 Algorithms for Integer Polynomial Rings

366

(4)

10.6.2 Selecting Q(Theta) for Cyclic Convolution Algorithms

370

(3)

Notes

373

(1)

Bibliography

374

(3)

Problems

377

(4)

11 Validity of Fast Algorithms Over Different Number Systems

381

(20)

11.1 Introduction

381

(1)

11.2 Mathematical Preliminaries

382

(1)

11.3 Chinese Remainder Theorem Over Finite Integer Rings

383

(3)

11.4 Interrelationships Among Algorithms Over Different Number Systems

386

(9)

11.4.1 Analysis of Cyclic Convolution Algorithms

388

(3)

11.4.2 Analysis of Acyclic Convolution Algorithms

391

(4)

11.5 Analysis of Two-Dimensional Cyclic Convolution Algorithms

395

(2)

Notes

397

(1)

Bibliography

398

(1)

Problems

399

(2)

12 Fault Tolerance for Integer Sequences

401

(24)

12.1 A Framework for Fault Tolerance

401

(3)

12.1.1 Fault Detection and Correction

404

(1)

12.2 Mathematical Structure of C Over Z(M)

404

(5)

12.2.1 Decoding Algorithms Over Z(M)

407

(2)

12.3 Coding Techniques Over Z(q)

409

(8)

12.3.1 Decoding Algorithms Over Z(q)

412

(3)

12.3.2 A Fast Algorithm for Decoding Over Z(q)

415

(2)

12.3.3 Some General Remarks

417

(1)

12.4 Examples and SFC-DFD Codes

417

(4)

Notes

421

(1)

Bibliography

422

(1)

Problems

423

(2)

PART III: FAST FOURIER TRANSFORM (FFT) ALGORITHMS

425

(98)

Thoughts on Part III

427

(6)

13 Fast Fourier Transform: One-Dimensional Data Sequences

433

(60)

13.1 The DFT: Definitions and Properties

433

(6)

13.1.1 Properties of DFT

436

(3)

13.1.2 Bluestein's Algorithm

439

(1)

13.2 Rader's FFT Algorithm, n = p, p an Odd Prime

439

(4)

13.3 Rader's FFT Algorithm, n = p(c), p an Odd Prime

443

(4)

13.4 Cooley-Tukey FFT Algorithm, n = a . b

447

(4)

13.5 FFT Algorithms for n a Power of 2

451

(23)

13.5.1 Radix-2 FFT

451

(13)

13.5.2 Radix-4 FFT

464

(5)

13.5.3 Split Radix FFT

469

(2)

13.5.4 Rader-Brenner Algorithm

471

(2)

13.5.5 Brunn's Algorithm

473

(1)

13.6 The Prime Factor FFT n = a . b, (a, b) = 1

474

(7)

13.6.1 Matrix Formulation of Prime Factor FFT

478

(3)

13.7 The Winograd FFT Algorithm

481

(3)

Notes

484

(1)

Bibliography

485

(2)

Problems

487

(6)

14 Fast Fourier Transform: Multidimensional Data Sequences

493

(30)

14.1 The Multidimensional DFT: Definition and Properties

494

(3)

14.2 FFT for n = p, p an Odd Prime

497

(2)

14.3 Multidimensional FFT Algorithms for n a Power of 2

499

(2)

14.3.1 Two-Dimensional Radix-2 FFT

499

(2)

14.4 Matrix Formulation of Multidimensional DFT and Related Algorithms

501

(4)

14.4.1 Prime Factor Algorithm in a Multidimensional Setting

503

(1)

14.4.2 Nesting and Other Algorithms

504

(1)

14.5 Polynomial Version of Rader's Algorithm

505

(2)

14.6 Polynomial Transform Based FFT Algorithms

507

(10)

14.6.1 n(1) = n(2) = ... = n(d) = n = p, p an Odd Prime

507

(4)

14.6.2 n = p(c), p an Odd Prime

511

(3)

14.6.3 Polynomial Transform Based FFT Algorithms for n = 2(c)

514

(3)

Notes

517

(1)

Bibliography

518

(1)

Problems

519

(4)

PART IV: RECENT RESULTS ON ALGORITHMS IN FINITE INTEGER RINGS

523

(96)

Thoughts on Part IV

525

(6)

Paper one: A Number Theoretic Approach to Fast Algorithms for Two-Dimensional Digital Signal Processing in Finite Integer Rings

531

(30)

Paper two: On Fast Algorithms for One-Dimensional Digital Signal Processing in Finite Integer and Complex Integer Rings

561

(24)

Paper three: Cyclotomic Polynomial Factorization in Finite Integer Rings with Applications to Digital Signal Processing

585

(20)

Paper four: Error Control Techniques for Data Sequences Defined in Finite Integer Rings

605

(14)

APPENDIX A Small Length Acyclic Convolution Algorithms

619

(4)

APPENDIX B Classification of Cyclotomic Polynomials

623

(10)

Index

633

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

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