This paper describes the “fractional Fourier transform,” which admits computation by an algorithm that has complexity proportional to the fast Fourier transform algorithm. Whereas the discrete Fourier transform (DFT) is based on integral roots of unity $e^{{{ - 2\pi i} / n}} $, the fractional Fourier transform is based on fractional roots of unity $e^{ - 2\pi i\alpha } $ where $\alpha $ is arbitrary. The fractional Fourier transform and the corresponding fast algorithm are useful for such applications as computing DFTs of sequences with prime lengths, computing DFTs of sparse sequences, analyzing sequences with noninteger periodicities, performing high-resolution trigonometric interpolation, detecting lines in noisy images, and detecting signals with linearly drifting frequencies. In many cases, the resulting algorithms are faster by arbitrarily large factors than conventional techniques.

The fractional Fourier transform and applications

Two recently developed ideas, the conversion of a discrete Fourier transform (DFT) to convolution and the implementation of short convolutions with a minimum of multiplications, are combined to give efficient algorithms for long transforms Three transform algorithms are compared in terms of the number of multiplications and additions Timing for a prime factor fast Fourier transform (FFT) algorithm using high-speed convolution, which was programmed for an IBM 370 and an 8080 microprocessor, is presented

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A prime factor FFT algorithm using high-speed convolution

In some signal processing applications, it is desirable to build very high performance fast Fourier transform (FFT) processors. To meet the performance requirements, these processors are typically highly pipelined. Until the advent of VLSI, it was not possible to build a single chip which could be used to construct pipeline FFT processors of a reasonable size. However, VLSI implementations have constraints which differ from those of discrete implementations, requiring another look at some of the typical FFT'algorithms in the light of these constraints.

Pipeline and Parallel-Pipeline FFT Processors for VLSI Implementations

Numerical Fourier and Bessel transforms in logarithmic variables

This paper presents a Fortran program that calculates the discrete Fourier transform using a prime factor algorithm. A very simple indexing scheme is employed that results in a flexible, modular algorithm that efficiently calculates the DFT in-place. A modification of this algorithm gives the output both in-place and in-order at a slight cost in flexibility. A comparison shows it to be faster than both the Cooley-Tukey algorithm and the Winograd nested algorithm.

An in-place, in-order prime factor FFT algorithm

It is shown how the Chinese Remainder Theorem (CRT) can be used to convert a one-dimensional cyclic convolution to a multi-dimensional convolution which is cyclic in all dimensions. Then, special algorithms are developed which, compute the relatively short convolutions in each of the dimensions. The original suggestion for this procedure was made in order to extend the lengths of the convolutions which one can compute with number-theoretic transforms. However, it is shown that the method can be more efficient, for some data sequence lengths, than the fast Fourier transform (FFT) algorithm. Some of the short convolutions are computed by methods in an earlier paper by Agarwal and Burrus. Recent work of Winograd, consisting of theorems giving the minimum possible numbers of multiplications and methods for achieving them, are applied to these short convolutions.

J. Cooley

Papers

New algorithms for digital convolution

New algorithms for digital convolution