It is known, that the Shuffle/Exchange-Network (S/E) is well suited to perform the parallel Fast Fourier Transform (FFT). In this paper we show, that it is optimal for this purpose in a quite general sense:

Shuffle/exchange is the natural interconnection scheme for the parallel fast fourier transform

An efficient method for the calculation of the interactions of a 2' factorial ex- periment was introduced by Yates and is widely known by his name. The generaliza- tion to 3' was given by Box et al. (1). Good (2) generalized these methods and gave elegant algorithms for which one class of applications is the calculation of Fourier series. In their full generality, Good's methods are applicable to certain problems in which one must multiply an N-vector by an N X N matrix which can be factored into m sparse matrices, where m is proportional to log N. This results inma procedure requiring a number of operations proportional to N log N rather than N2. These methods are applied here to the calculation of complex Fourier series. They are useful in situations where the number of data points is, or can be chosen to be, a highly composite number. The algorithm is here derived and presented in a rather different form. Attention is given to the choice of N. It is also shown how special advantage can be obtained in the use of a binary computer with N = 2' and how the entire calculation can be performed within the array of N data storage locations used for the given Fourier coefficients. Consider the problem of calculating the complex Fourier series N-1 (1) X(j) = EA(k)-Wjk, j = 0 1, * ,N- 1, k=0

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An algorithm for the machine calculation of complex Fourier series

1 Introduction.- 1.1 Introductory Remarks.- 1.2 Notations.- 1.3 The Structure of the Book.- 2 Elements of Number Theory and Polynomial Algebra.- 2.1 Elementary Number Theory.- 2.1.1 Divisibility of Integers.- 2.1.2 Congruences and Residues.- 2.1.3 Primitive Roots.- 2.1.4 Quadratic Residues.- 2.1.5 Mersenne and Fermat Numbers.- 2.2 Polynomial Algebra.- 2.2.1 Groups.- 2.2.2 Rings and Fields.- 2.2.3 Residue Polynomials.- 2.2.4 Convolution and Polynomial Product Algorithms in Polynomial Algebra.- 3 Fast Convolution Algorithms.- 3.1 Digital Filtering Using Cyclic Convolutions.- 3.1.1 Overlap-Add Algorithm.- 3.1.2 Overlap-Save Algorithm.- 3.2 Computation of Short Convolutions and Polynomial Products.- 3.2.1 Computation of Short Convolutions by the Chinese Remainder Theorem.- 3.2.2 Multiplications Modulo Cyclotomic Polynomials.- 3.2.3 Matrix Exchange Algorithm.- 3.3 Computation of Large Convolutions by Nesting of Small Convolutions.- 3.3.1 The Agarwal-Cooley Algorithm.- 3.3.2 The Split Nesting Algorithm.- 3.3.3 Complex Convolutions.- 3.3.4 Optimum Block Length for Digital Filters.- 3.4 Digital Filtering by Multidimensional Techniques.- 3.5 Computation of Convolutions by Recursive Nesting of Polynomials.- 3.6 Distributed Arithmetic.- 3.7 Short Convolution and Polynomial Product Algorithms.- 3.7.1 Short Circular Convolution Algorithms.- 3.7.2 Short Polynomial Product Algorithms.- 3.7.3 Short Aperiodic Convolution Algorithms.- 4 The Fast Fourier Transform.- 4.1 The Discrete Fourier Transform.- 4.1.1 Properties of the DFT.- 4.1.2 DFTs of Real Sequences.- 4.1.3 DFTs of Odd and Even Sequences.- 4.2 The Fast Fourier Transform Algorithm.- 4.2.1 The Radix-2 FFT Algorithm.- 4.2.2 The Radix-4 FFT Algorithm.- 4.2.3 Implementation of FFT Algorithms.- 4.2.4 Quantization Effects in the FFT.- 4.3 The Rader-Brenner FFT.- 4.4 Multidimensional FFTs.- 4.5 The Bruun Algorithm.- 4.6 FFT Computation of Convolutions.- 5 Linear Filtering Computation of Discrete Fourier Transforms.- 5.1 The Chirp z-Transform Algorithm.- 5.1.1 Real Time Computation of Convolutions and DFTs Using the Chirp z-Transform.- 5.1.2 Recursive Computation of the Chirp z-Transform.- 5.1.3 Factorizations in the Chirp Filter.- 5.2 Rader's Algorithm.- 5.2.1 Composite Algorithms.- 5.2.2 Polynomial Formulation of Rader's Algorithm.- 5.2.3 Short DFT Algorithms.- 5.3 The Prime Factor FFT.- 5.3.1 Multidimensional Mapping of One-Dimensional DFTs.- 5.3.2 The Prime Factor Algorithm.- 5.3.3 The Split Prime Factor Algorithm.- 5.4 The Winograd Fourier Transform Algorithm (WFTA).- 5.4.1 Derivation of the Algorithm.- 5.4.2 Hybrid Algorithms.- 5.4.3 Split Nesting Algorithms.- 5.4.4 Multidimensional DFTs.- 5.4.5 Programming and Quantization Noise Issues.- 5.5 Short DFT Algorithms.- 5.5.1 2-Point DFT.- 5.5.2 3-Point DFT.- 5.5.3 4-Point DFT.- 5.5.4 5-Point DFT.- 5.5.5 7-Point DFT.- 5.5.6 8-Point DFT.- 5.5.7 9-Point DFT.- 5.5.8 16-Point DFT.- 6 Polynomial Transforms.- 6.1 Introduction to Polynomial Transforms.- 6.2 General Definition of Polynomial Transforms.- 6.2.1 Polynomial Transforms with Roots in a Field of Polynomials.- 6.2.2 Polynomial Transforms with Composite Roots.- 6.3 Computation of Polynomial Transforms and Reductions.- 6.4 Two-Dimensional Filtering Using Polynomial Transforms.- 6.4.1 Two-Dimensional Convolutions Evaluated by Polynomial Transforms and Polynomial Product Algorithms.- 6.4.2 Example of a Two-Dimensional Convolution Computed by Polynomial Transforms.- 6.4.3 Nesting Algorithms.- 6.4.4 Comparison with Conventional Convolution Algorithms.- 6.5 Polynomial Transforms Defined in Modified Rings.- 6.6 Complex Convolutions.- 6.7 Multidimensional Polynomial Transforms.- 7 Computation of Discrete Fourier Transforms by Polynomial Transforms.- 7.1 Computation of Multidimensional DFTs by Polynomial Transforms.- 7.1.1 The Reduced DFT Algorithm.- 7.1.2 General Definition of the Algorithm.- 7.1.3 Multidimensional DFTs.- 7.1.4 Nesting and Prime Factor Algorithms.- 7.1.5 DFT Computation Using Polynomial Transforms Defined in Modified Rings of Polynomials.- 7.2 DFTs Evaluated by Multidimensional Correlations and Polynomial Transforms.- 7.2.1 Derivation of the Algorithm.- 7.2.2 Combination of the Two Polynomial Transform Methods.- 7.3 Comparison with the Conventional FFT.- 7.4 Odd DFT Algorithms.- 7.4.1 Reduced DFT Algorithm. N = 4.- 7.4.2 Reduced DFT Algorithm. N = 8.- 7.4.3 Reduced DFT Algorithm. N = 9.- 7.4.4 Reduced DFT Algorithm. N = 16.- 8 Number Theoretic Transforms.- 8.1 Definition of the Number Theoretic Transforms.- 8.1.1 General Properties of NTTs.- 8.2 Mersenne Transforms.- 8.2.1 Definition of Mersenne Transforms.- 8.2.2 Arithmetic Modulo Mersenne Numbers.- 8.2.3 Illustrative Example.- 8.3 Fermat Number Transforms.- 8.3.1 Definition of Fermat Number Transforms.- 8.3.2 Arithmetic Modulo Fermat Numbers.- 8.3.3 Computation of Complex Convolutions by FNTs.- 8.4 Word Length and Transform Length Limitations.- 8.5 Pseudo Transforms.- 8.5.1 Pseudo Mersenne Transforms.- 8.5.2 Pseudo Fermat Number Transforms.- 8.6 Complex NTTs.- 8.7 Comparison with the FFT.- Appendix A Relationship Between DFT and Conyolution Polynomial Transform Algorithms.- A.1 Computation of Multidimensional DFT's by the Inverse Polynomial Transform Algorithm.- A.1.1 The Inverse Polynomial Transform Algorithm.- A.1.2 Complex Polynomial Transform Algorithms.- A.1.3 Round-off Error Analysis.- A.2 Computation of Multidimensional Convolutions by a Combination of the Direct and Inverse Polynomial Transform Methods.- A.2.1 Computation of Convolutions by DFT Polynomial Transform Algorithms.- A.2.2 Convolution Algorithms Based on Polynomial Transforms and Permutations.- A.3 Computation of Multidimensional Discrete Cosine Transforms by Polynomial Transforms.- A.3.1 Computation of Direct Multidimensional DCT's.- A.3.2 Computation of Inverse Multidimensional DCT's.- Appendix B Short Polynomial Product Algorithms.- Problems.- References.

Fast Fourier transform and convolution algorithms

In this extended abstract, we present several new layouts for the shuffle-exchange graph, including one which requires only 0(n2/log2n) area. The optimal layout is described and analyzed in section 3. The analysis is heavily dependent on several combinatorial results which we state in section 2 and prove in the appendix. The other layouts are described in section 4. Although these layouts are not asymptotically optimal (most require 0(n2/log3/2n) area), the theory behind their development is interesting and may eventually lead to good practical layouts as well as other asymptotically optimal layouts.

Shuffle/exchange is the natural interconnection scheme for the parallel fast fourier transform

References

An algorithm for the machine calculation of complex Fourier series

Fast Fourier transform and convolution algorithms

New layouts for the shuffle-exchange graph(Extended Abstract)

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