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

CRC Press, Mar 25, 1998 - Technology & Engineering - 672 pages
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.

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### Contents

 Introduction 1 FAST FOURIER TRANSFORM FFT 2 Thoughts on Part I 9 Polynomial Algebra 35 Theoretical Aspects of the Discrete Fourier Transform 67 Cyclotomic Polynomial Factorization and Associated Fields 91 Cyclotomic Polynomial Factorization in Finite Fields 125 Polynomial Algebra and Cyclotomic 151
 Fault Tolerance for Integer Sequences 401 Thoughts on Part III 427 OneDimensional Data Sequences 433 Multidimensional Data Sequences 493 Thoughts on Part IV 525 A Number Theoretic Approach to Fast Algorithms 531 Properties of Polynomial Transforms Over Zp for V q q 552 On Fast Algorithms for OneDimensional Digital Signal 561

 Thoughts on Part II 227 Fast Algorithms for Acyclic Convolution 233 Fast OneDimensional Cyclic Convolution Algorithms 275 Two and HigherDimensional Cyclic Convolution Algorithms 337 Validity of Fast Algorithms Over Different Number Systems 381
 Cyclotomic Polynomial Factorization in Finite Integer 585 Error Control Techniques for Data Sequences Defined 605 APPENDIX A Small Length Acyclic Convolution Algorithms 619 Index 633 Copyright