Discrete Fourier Transform Matrix Research Articles

Recursive generation of the DIF-FFT algorithm for 1-D DFT

It is shown by using the Walsh-Hadamard matrix and a conversion matrix, a DIF-FFT algorithm can be developed for the radix-2 discrete Fourier transform (DFT). The recursive generation of the DFT matrix leads to its sparse matrix factorization. The flowgraph based on these sparse matrices is identical to that of the DFT-FFT algorithm. >

A new discrete Fourier transform algorithm using butterfly structure fast convolution

This paper proposes a new approach to computing the discrete Fourier transform (DFT) with the power of 2 length using the butterfly structure number theoretic transform (NTT). An algorithm breaking down the DFT matrix into circular matrices with the power of 2 size is newly introduced. The fast circular convolution, which is implemented by the NTT based on the butterfly structure, can provide significant reductions in the number of computations, as well as a simple and regular structure, The proposed algorithm can be successively implemented following a simple flowchart using the reduced size submatrices. Multiplicative complexity is reduced to about 21 percent of that by the classical FFT algorithm, preserving almost the same number of additions.

The fast Hartley transform

A fast algorithm has been worked out for performing the Discrete Hartley Transform (DHT) of a data sequence of N elements in a time proportional to Nlog <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> N. The Fast Hartley Transform (FHT) is as fast as or faster than the Fast Fourier Transform (FFT) and serves for all the uses such as spectral analysis, digital processing, and convolution to which the FFT is at present applied. A new timing diagram (stripe diagram) is presented to illustrate the overall dependence of running time on the subroutines composing one implementation; this mode of presentation supplements the simple counting of multiplies and adds. One may view the Fast Hartley procedure as a sequence of matrix operations on the data and thus as constituting a new factorization of the DFT matrix operator; this factorization is presented. The FHT computes convolutions and power spectra distinctly faster than the FFT.

Redundancy techniques and fast algorithms for a special large linear system

In a recently developed approach to the optical inverse scattering problem, the need arose to solve very large systems of linear equations with a nonsparse matrix. The entries in the matrix are determined by the specifications for the data collection pattern. The matrix is not a Discrete Fourier Transform matrix, and it not anenable to FFT methods. In this paper we show how a combination of the techniques of “redundancies” and “Kronecker decompositions” may be used with a specification for the data collection to produce a fast, accurate algorithm which solves the linear system. This algorithm has been implemented on a sequential computer, but parallel computation is clearly feasible.