Recent results by Van Buskirk have broken the record set by Yavne in 1968 for the lowest exact count of real additions and multiplications to compute a power-of-two discrete Fourier transform (DFT). Here, we present a simple recursive modification of the split-radix algorithm that computes the DFT with asymptotically about 6% fewer operations than Yavne, matching the count achieved by Van Buskirk's program-generation framework. We also discuss the application of our algorithm to real-data and real-symmetric (discrete cosine) transforms, where we are again able to achieve lower arithmetic counts than previously published algorithms

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A Modified Split-Radix FFT With Fewer Arithmetic Operations

This manuscript describes a number of algorithms that can be used to quickly evaluate a polynomial over a collection of points and interpolate these evaluations back into a polynomial. Engineers define the “Fast Fourier Transform” as a method of solving the interpolation problem where the coefficient ring used to construct the polynomials has a special multiplicative structure. Mathematicians define the “Fast Fourier Transform” as a method of solving the evaluation problem. One purpose of the document is to provide a mathematical treatment of the topic of the “Fast Fourier Transform” that can also be understood by someone who has an understanding of the topic from the engineering perspective.
The manuscript will also introduce several new algorithms that solve the fast multipoint evaluation problem over certain finite fields and require fewer finite field operations than existing techniques. The document will also demonstrate that these new algorithms can be used to multiply polynomials with finite field coefficients with fewer operations than Schonhage's algorithm in most circumstances.
A third objective of this document is to provide a mathematical perspective of several algorithms which can be used to multiply polynomials of size which is not a power of two. Several improvements to these algorithms will also be discussed.
Finally, the document will describe several applications of the “Fast Fourier Transform” algorithms presented and will introduce improvements in several of these applications. In addition to polynomial multiplication, the applications of polynomial division with remainder, the greatest common divisor, decoding of Reed-Solomon error-correcting codes, and the computation of the coefficients of a discrete Fourier Series will be addressed.

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Fast fourier transform algorithms with applications

http://www-math.mit.edu/~stevenj/preprints/ShaoJo07-preprint.pdf

Type-II/III DCT/DST algorithms with reduced number of arithmetic operations

A mathematical perspective

http://www-math.mit.edu/~stevenj/preprints/ShaoJo07-dct4-preprint.pdf

Type-IV DCT, DST, and MDCT algorithms with reduced numbers of arithmetic operations

A new algorithm for the fast computation of the discrete Fourier transform is introduced. The algorithm, called the conjugate pair FFT (CPFFT), is used to compute a length-2m DFT. The number of multiplications and additions required by the CPFFT is less than that required by the SRFFT algorithm.

I. Kamar

Papers

Conjugate pair fast Fourier transform