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Fast Fourier Transforms by C. Sidney Burrus, Matteo Frigo, Steven G. Johnson, - HTML preview
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Fast Fourier Transforms
Table of Contents
Chapter
1
.
Preface: Fast Fourier Transforms
1.1
.
References
Chapter
2
.
Introduction: Fast Fourier Transforms
2.1
.
References
Chapter
3
.
Multidimensional Index Mapping
3.1
.
The Index Map
Case 1.
Case 2.
Type-One Index Map:
Type-Two Index Map:
In-Place Calculation of the DFT and Scrambling
Efficiencies Resulting from Index Mapping with the DFT
The FFT as a Recursive Evaluation of the DFT
References
Chapter
4
.
Polynomial Description of Signals
4.1
.
Polynomial Reduction and the Chinese Remainder Theorem
The DFT as a Polynomial Evaluation
References
Chapter
5
.
The DFT as Convolution or Filtering
5.1
.
Rader's Conversion of the DFT into Convolution
The Chirp Z-Transform (or Bluestein's Algorithm)
Goertzel's Algorithm (or A Better DFT Algorithm)
The First-Order Goertzel Algorithm
The Second-Order Goertzel Algorithm
Analysis of Arithmetic Complexity and Timings
Conclusions
The Quick Fourier Transform (QFT)
References
Chapter
6
.
Factoring the Signal Processing Operators
6.1
.
The FFT from Factoring the DFT Operator
Algebraic Theory of Signal Processing Algorithms
References
Chapter
7
.
Winograd's Short DFT Algorithms
7.1
.
The Bilinear Structure
Winograd's Complexity Theorems
The Automatic Generation of Winograd's Short DFTs
Introduction
Matrix Description
Programming the Design Procedure
The Reduction Stage
The Polynomial Product Stage
Multiple Factors
Discussion and Conclusion
References
Chapter
8
.
DFT and FFT: An Algebraic View
8.1
.
Polynomial Algebras and the DFT
Polynomial Algebra
Chinese Remainder Theorem (CRT)
Polynomial Transforms
DFT as a Polynomial Transform
Algebraic Derivation of the Cooley-Tukey FFT
Matrix Notation
Radix-2 FFT
General-radix FFT
Discussion and Further Reading
Algebraic Derivation of Transform Algorithms
Algebraic Signal Processing Theory
References
Chapter
9
.
The Cooley-Tukey Fast Fourier Transform Algorithm
9.1
.
Modifications to the Basic Cooley-Tukey FFT
The Split-Radix FFT Algorithm
Evaluation of the Cooley-Tukey FFT Algorithms
The Quick Fourier Transform, An FFT based on Symmetries
Input and Output Symmetries
Further Reductions if the Length is Even
Arithmetic Complexity and Timings
Conclusions
References
Chapter
10
.
The Prime Factor and Winograd Fourier Transform Algorithms
10.1
.
The Prime Factor Algorithm
The Winograd Fourier Transform Algorithm
Modifications of the PFA and WFTA Type Algorithms
Evaluation of the PFA and WFTA
References
Chapter
11
.
Implementing FFTs in Practice
11.1
.
Introduction
Review of the Cooley-Tukey FFT
Goals and Background of the FFTW Project
FFTs and the Memory Hierarchy
Understanding FFTs with an ideal cache
Cache-obliviousness in practice
Memory strategies in FFTW
Adaptive Composition of FFT Algorithms
The problem to be solved
DFT problem examples
The space of plans in FFTW
Rank-0 plans
Rank-1 plans
Direct plans
Cooley-Tukey plans
Plans for higher vector ranks
Indirect plans
Plans for prime sizes
Discussion
The FFTW planner
Generating Small FFT Kernels
SIMD instructions
Numerical Accuracy in FFTs
Concluding Remarks
Acknowledgements
References
Chapter
12
.
Algorithms for Data with Restrictions
12.1
.
Algorithms for Real Data
Special Algorithms for input Data that is mostly Zero, for Calculating only a few Outputs, or where the Sampling is not Uniform
Algorithms for Approximate DFTs
References
Chapter
13
.
Convolution Algorithms
13.1
.
Fast Convolution by the FFT
Fast Convolution by Overlap-Add
Fast Convolution by Overlap-Save
Block Processing, a Generalization of Overlap Methods
Introduction
Block Signal Processing
Block Convolution
Block Recursion
Block State Formulation
Block Implementations of Digital Filters
Multidimensional Formulation
Periodically Time-Varying Discrete-Time Systems
Multirate Filters, Filter Banks, and Wavelets
Distributed Arithmetic
Direct Fast Convolution and Rectangular Transforms
Number Theoretic Transforms for Convolution
Results from Number Theory
Number Theoretic Transforms
References
Chapter
14
.
Comments: Fast Fourier Transforms
14.1
.
Other work and Results
References
Chapter
15
.
Conclusions: Fast Fourier Transforms
15.1
.
References
Chapter
16
.
Appendix 1: FFT Flowgraphs
16.1
.
Signal Flow Graphs of Cooley-Tukey FFTs
Chapter
17
.
Appendix 2: Operation Counts for General Length FFT
17.1
.
Figures
Chapter
18
.
Appendix 3: FFT Computer Programs
18.1
.
Goertzel Algorithm
Second Order Goertzel Algorithm
Second Order Goertzel Algorithm 2
Basic QFT Algorithm
Basic Radix-2 FFT Algorithm
Basic DIT Radix-2 FFT Algorithm
DIF Radix-2 FFT Algorithm
Basic DIF Radix-4 FFT Algorithm
Basic DIF Radix-4 FFT Algorithm
Basic DIF Split Radix FFT Algorithm
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