VLSI architecture for the Winograd Fourier Transform algorithm

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

/pdf/an-algorithm-for-the-machine-calculation-of-complex-fourier-2mwjmk0q9u.pdf

An algorithm for the machine calculation of complex Fourier series

http://www.norbertwiener.umd.edu/Research/Duhamel_Vetterli_FFT_90.pdf

Fast fourier transforms: a tutorial review and a state of the art

Note: V. K. Madisetti, D. B. Williams, Eds. Reference LCAV-CHAPTER-2005-009 Record created on 2005-06-27, modified on 2017-05-12

Fast Fourier transforms: a tutorial review and a state of the art

New algorithms for computing the Discrete Fourier Transform of n points are described. For n in the range of a few tens to a few thousands these algorithms use substantially fewer multiplications than the best algorithm previously known, and about the same number of additions.

/pdf/on-computing-the-discrete-fourier-transform-18ubnywxin.pdf

VLSI architecture for the Winograd Fourier Transform algorithm

References

An algorithm for the machine calculation of complex Fourier series

Fast fourier transforms: a tutorial review and a state of the art

Fast Fourier transforms: a tutorial review and a state of the art

On computing the discrete Fourier transform

The Interaction Algorithm and Practical Fourier Analysis

Related Papers (5)

The Arithmetic Cube

An architecture for a VLSI FFT processor

Multiplierless Winograd and prime factor FFT implementation

A VLSI architecture for DFT

A new fast DCT algorithm and its systolic VLSI implementation