## Mathematics of Multidimensional Fourier Transform AlgorithmsThe Fourier transform of large multidimensional data sets is an essen tial computation in many scientific and engineering fields, including seismology, X-ray crystallography, radar, sonar and medical imaging. Such fields require multidimensional arrays for complete and faithful modelling. Classically, a set of data is processed one dimension at a time, permitting control over the size of the computation and calling on well-established I-dimensional programs. The rapidly increasing availability of powerful computing chips, vector processors, multinode boards and parallel machines has provided new tools for carrying out multidimensional computations. Multidimensional processing offers a wider range of possible implementations as compared to I-dimensional the greater flexibility of movement in the data in processing, due to dexing set. This increased freedom, along with the massive size data sets typically found in multidimensional applications, places intensive demands on the communication aspects of the computation. The writ ing of code that takes into account all the algorithmic possibilities and matches these possibilities to the communication capabilities of the tar get architecture is an extremely time-consuming task. A major goal of this text is to provide a sufficiently abstra |

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### Contents

Multidimensional Tensor Product and FFT | 29 |

Finite Abelian Groups | 45 |

Bibliography | 61 |

Bibliography | 75 |

Lines 89 | 88 |

Bibliography | 112 |

Bibliography | 131 |

### Other editions - View all

Mathematics of Multidimensional Fourier Transform Algorithms Richard tolimieri,Myoung An,Chao Lu Limited preview - 2012 |

### Common terms and phrases

2-dimensional operation 2-dimensional planes A/B+ A1 and A2 algebra algorithm design architecture array automorphism B-periodic bº e B+ Chapter character group computational stages Cooley–Tukey FFT corresponding cyclic group dclock defined degree of parallelism Denote described direct product Discrete Fourier Transform distinct lines duality end do end Example Fast Fourier Transform fe L(A FFT algorithms finite abelian group finite field floating-point Fºf formula Fourier Trans Fourier Transform Algorithms function given GL(N global Good–Thomas gp(a granularity idempotents IEEE Trans implementation indexing set input integer isomorphism M-point mapping Mixed Radix multiply–add node processors number of lines output Parallel Comput permutation matrix primary factorization Proof reduced transform algorithm relatively prime RISC shift matrix Signal Processing skew-circulant SL(N STFSU stride permutations subblocks subgroup tensor product Tensor Product Factorization Theorem tion Tolimieri transpose twiddle factors vector operation write