Fast Algorithms and Structures P.. Vetterli Introduction •A Historical Perspective•Motivation or: why dividing is also conquering•FFTs with Twiddle Factors •FFTs Based on Costless Mono-
Trang 1Fast Algorithms
and Structures
P Duhamel
´
Ecole Nationale Sup ´erieure des T ´el ´ecommunications (ENST)
7 Fast Fourier Transforms: A Tutorial Review and a State of the Art P Duhamel and M.
Vetterli
Introduction •A Historical Perspective•Motivation (or: why dividing is also conquering)•FFTs
with Twiddle Factors •FFTs Based on Costless Mono- to Multidimensional Mapping•State of
the Art •Structural Considerations•Particular Cases and Related Transforms•Multidimensional
Transforms •Implementation Issues•Conclusion
8 Fast Convolution and Filtering Ivan W Selesnick and C Sidney Burrus
Introduction •Overlap-Add and Overlap-Save Methods for Fast Convolution•Block Convolution
•Short and Medium Length Convolution•Multirate Methods for Running Convolution•
Convo-lution in Subbands •Distributed Arithmetic•Fast Convolution by Number Theoretic Transforms
•Polynomial-Based Methods•Special Low-Multiply Filter Structures
9 Complexity Theory of Transforms in Signal Processing Ephraim Feig
Introduction •One-Dimensional DFTs•Multidimensional DFTs•One-Dimensional DCTs•
Mul-tidimensional DCTs •Nonstandard Models and Problems
10 Fast Matrix Computations Andrew E Yagle
Introduction •Divide-and-Conquer Fast Matrix Multiplication•Wavelet-Based Matrix
Sparsifi-cation
promi-nence primarily through the discovery of efficient algorithms for computing various trans-forms (mainly the Fourier transtrans-forms) in the 1970s In addition to fast Fourier transtrans-forms (FFTs), discrete cosine transforms (DCTs) have also gained importance owing to their performance being very close to the statistically optimum Karhunen Loeve transform
Transforms, convolutions, and matrix-vector operations form the basic tools utilized by the signal processing community, and this section reviews and presents the state of art in these areas of increasing importance
The chapter by Duhamel and Vetterli, “Fast Fourier Transforms: A Tutorial Review and a State of the Art”, presents a thorough discussion of this important transform Selesnick and Burrus present
c 1999 by CRC Press LLC
Trang 2an excellent survey of filtering and convolution techniques in the chapter “Fast Convolution and Filtering”
One approach to understanding the time and space complexities of signal processing algorithms
is through the use of quantitative complexity theory, and Feig’s “Complexity Theory of Transforms
in Signal Processing” applies quantitative measures to the computation of transforms Finally, Yagle presents a comprehensive discussion of matrix computations in signal processing in “Fast Matrix Computations”
c 1999 by CRC Press LLC