On the structure of efficient DFT algorithms

Abstract

Bilinear Form This paper examines the structure of the prime length DFT algorithms that are developed by Winograds approach. It is shown that these algorithms have considerable structure and that this can be exploited to develop a straight forward design procedure that does not use the Chinese remainder theorem. This structure also allows the design of real-data programs, and the improvement the data transfer properties of the PFA.

Automating the design of prime length FFT programs

“…This form of convolution is presented with many examples in [2]. Using the matrix exchange property explained in [2] and [4] this form can be written as…”

“…Now DFT{z} = RA'DAI. In [4] Johnson observes that by permuting the elements on the diagonal of D, the output can be permuted, so that the R matrix can be hidden in D , and DFT{z} = A'DAx. From the knowledge of this form, once A is found, D can be found numerically [4].…”

“…This paper automates the process Winograd used for constructing prime length FFTs [2,8].for N 5 7 and that Johnson and Burrus [4] extended to N < 19. This paper presents a program that will design any prime length F F T in principle.…”

“…This paper uses three methods to facilitate the construction of prime length FFT modules. First, the matrix ezchange property [2,4,51 is used so that the transpose of the reduction operator can be used rather than the more complicated CRT reconstruction operator. This is then combined with the numerical method [4] for obtaining the multiplication coefficients rather than the direct use of the CRT.…”

“…First, the matrix ezchange property [2,4,51 is used so that the transpose of the reduction operator can be used rather than the more complicated CRT reconstruction operator. This is then combined with the numerical method [4] for obtaining the multiplication coefficients rather than the direct use of the CRT. We also deviate from the Toom-Cook algorithm, because it 'This work has been supported in part by DARPA under NDSEG fellowship N00014-89J-3204 requires too many additions for the lengths in which we are interested.…”