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

### Common terms and phrases

acyclic convolution algorithm algorithm for cyclic algorithms for computing bilinear form chapter Chinese remainder theorem coefficients computation X(u computational complexity computationally efficient algorithms congruence convolution of length corresponding CRT-I CRT-P reconstruction cyclic convolution algorithm cyclotomic factorization cyclotomic field cyclotomic polynomials CZ(M CZ(p CZ(q CZ(r decoding defined mod deg(X(u denoted Derive computationally efficient described digital signal processing dimensional cyclic convolution direct sum Discrete Fourier Transform element of order Example expressed Fermat FFT algorithm finite fields finite integer rings GF(pm GF(q given IEEE Transactions irreducible polynomial Lemma matrix mod P(u mod u2 modulo polynomial monic factors monic polynomial multidimensional multiplicative complexity MULTs NTTs number systems Number Theory obtained one-dimensional cyclic convolution polynomial factorization polynomial of degree polynomial products polynomial rings polynomial transform polynomial X(u polynomials defined primitive element primitive polynomial properties recursive relatively prime requires root of unity Step two-dimensional cyclic convolution unique valid vector