Computational Frameworks for the Fast Fourier Transform

"This finely crafted work fills a gap in the library of books on the fast Fourier Transform (FFT). It provides a complete and elegant mathematical formulation of the family of algorithms that compute FFTs. It is written for students and professionals who already have a working knowledge of computational linear algebra. It leads the student directly into the current major fields of FFT research ...This work is an excellent contribution to the modern FFT literature.' J. R. Hubbard, Computing Reviews 'This comprehensive book contains many examples, problems, notes and hints to the literature. It will be very useful for students as well as researchers who are interested in the application of the FFT.' M. Tasche (Rostock), Zentralblatt fur Mathematik "The fast Fourier transform (FFT) is one of the truly great computational developments. As the author emphasizes in the preface, the central feature of the FFT is essentially the factorization of the discrete Fourier transform (DFT) matrix. This point of view will not only unify the FFT and make it more understandable to outsiders, but also give key aspects for advanced scientific computing, e.g. vectorization or parallelization... This is only a brief glimpse of an excellent survey on the FFT, which will be valuable to all who wish to use it.' S. Hitotumatu, Mathematical Reviews ' ... the FFT has found application in almost every field of modern science, including fields as diverse as astronomy, acoustics, image processing, fluid dynamics, petroleum exploration, medicine and quantum physics, among many others. It is not an exercise in hyperbole to say that the world as we know it would be different without the FFT.' David H. Bailey, SIAM Review "It is generally accepted that 'Life as we know it would be very different without the FFT.' The author adds to this 'Life as we know it would be considerably different if, from the 1965 Cooley-Tukey paper onwards, the FFT community had made systematic and heavy use of the matrix-vector notation.' This book contains a very readable and up-to-date presentation of FFT techniques, their theory and application. Together with many explicit computational algorithms, the extensive annotated list of references add greatly to the scientific value of this reference text.' Short Book Reviews

• Chapter 1: The Radix-2 Frameworks. Matrix Notation and AlgorithmsThe FFT IdeaThe Cooley-Tukey Radix-2 FactorizationWeight and Butterfly ComputationsBit Reversal and TranspositionThe Cooley-Tukey FrameworkThe Stockham Autosort FrameworksThe Pease FrameworkDecimation in Frequency and Inverse FFTsChapter 2: General Radix Frameworks. The General Radix IdeasIndex Reversal and TranspositionMixed-Radix FactorizationsRadix-4 and Radix-8 FrameworksThe Split-Radix FrameworksChapter 3: High Performance Frameworks. The Multiple DFT ProblemMatrix TranspositionThe Large Single-Vector FFT ProblemMultidimensional FFT ProblemDistributed-Memory FFTsShared-Memory FFTsChapter 4: Selected Topics. Prime Factor FFTsConvolutionFFTs of Real DataFast Trigonometric TransformsFast Poisson SolversBibliographyIndex.