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

An orthogonal approximation for the 8-point discrete cosine transform (DCT) is introduced. The proposed transformation matrix contains only zeros and ones; multiplications and bit-shift operations are absent. Close spectral behavior relative to the DCT was adopted as design criterion. The proposed algorithm is superior to the signed discrete cosine transform. It could also outperform state-of-the-art algorithms in low and high image compression scenarios, exhibiting at the same time a comparable computational complexity.

A DCT Approximation for Image Compression

Video processing systems such as HEVC requiring low energy consumption needed for the multimedia market has lead to extensive development in fast algorithms for the efficient approximation of 2-D DCT transforms The DCT is employed in a multitude of compression standards due to its remarkable energy compaction properties Multiplier-free approximate DCT transforms have been proposed that offer superior compression performance at very low circuit complexity Such approximations can be realized in digital VLSI hardware using additions and subtractions only, leading to significant reductions in chip area and power consumption compared to conventional DCTs and integer transforms In this paper, we introduce a novel 8-point DCT approximation that requires only 14 addition operations and no multiplications The proposed transform possesses low computational complexity and is compared to state-of-the-art DCT approximations in terms of both algorithm complexity and peak signal-to-noise ratio The proposed DCT approximation is a candidate for reconfigurable video standards such as HEVC The proposed transform and several other DCT approximations are mapped to systolic-array digital architectures and physically realized as digital prototype circuits using FPGA technology and mapped to 45 nm CMOS technology

Improved 8-Point Approximate DCT for Image and Video Compression Requiring Only 14 Additions

Video processing systems such as HEVC requiring low energy consumption needed for the multimedia market has lead to extensive development in fast algorithms for the efficient approximation of 2-D DCT transforms. The DCT is employed in a multitude of compression standards due to its remarkable energy compaction properties. Multiplier-free approximate DCT transforms have been proposed that offer superior compression performance at very low circuit complexity. Such approximations can be realized in digital VLSI hardware using additions and subtractions only, leading to significant reductions in chip area and power consumption compared to conventional DCTs and integer transforms. In this paper, we introduce a novel 8-point DCT approximation that requires only 14 addition operations and no multiplications. The proposed transform possesses low computational complexity and is compared to state-of-the-art DCT approximations in terms of both algorithm complexity and peak signal-to-noise ratio. The proposed DCT approximation is a candidate for reconfigurable video standards such as HEVC. The proposed transform and several other DCT approximations are mapped to systolic-array digital architectures and physically realized as digital prototype circuits using FPGA technology and mapped to 45 nm CMOS technology.

Improved 8-point Approximate DCT for Image and Video Compression Requiring Only 14 Additions

In this paper, a new radix-2/8 fast Fourier transform (FFT) algorithm is proposed for computing the discrete Fourier transform of an arbitrary length N=q/spl times/2/sup m/, where q is an odd integer. It reduces substantially the operations such as data transfer, address generation, and twiddle factor evaluation or access to the lookup table, which contribute significantly to the execution time of FFT algorithms. It is shown that the arithmetic complexity (multiplications+additions) of the proposed algorithm is, in most cases, the same as that of the existing split-radix FFT algorithm. The basic idea behind the proposed algorithm is the use of a mixture of radix-2 and radix-8 index maps. The algorithm is expressed in a simple matrix form, thereby facilitating an easy implementation of the algorithm, and allowing for an extension to the multidimensional case. For the structural complexity, the important properties of the Cooley-Tukey approach such as the use of the butterfly scheme and in-place computation are preserved by the proposed algorithm.

A new radix-2/8 FFT algorithm for length-q/spl times/2/sup m/ DFTs

In this paper, we propose an efficient 8 times 8 multiplication-free transform operator for image compression by appropriately introducing some zeros in the 8 times 8 signed DCT matrix We also develop fast algorithms for the computation of the proposed transform as well as for its inverse Compared to the existing 8 times 8 approximated DCT matrices, it is shown that savings of 125% in the number of additions can easily be achieved using the proposed transform operator without noticeable degradations in the reconstructed images We also present simulations on some standard test images to show the efficiency of the proposed transform in image compression

A multiplication-free transform for image compression

In this paper, a general class of split-radix fast Fourier transform (FFT) algorithms for computing the length-2m DFT is proposed by introducing a new recursive approach coupled with an efficient method for combining the twiddle factors. This enables the development of higher split-radix FFT algorithms from lower split-radix FFT algorithms without any increase in the arithmetic complexity. Specifically, an arbitrary radix-2/2s FFT algorithm for any value of s, 4les sles m, is proposed and its arithmetic complexity analyzed. It is shown that the number of arithmetic operations (multiplications plus additions) required by the proposed radix-2/2s FFT algorithm is independent of s and is (2m-3)2m+1+8 regardless of whether a complex multiplication is carried out using four multiplications and two additions or three multiplications and three additions. This paper thus provides a variety of choices and ways for computing the length-2m DFT with the same arithmetic complexity.

A General Class of Split-Radix FFT Algorithms for the Computation of the DFT of Length- $2^{m}$

In this paper, improved algorithms for radix-4 and radix-8 FFT are presented. This is achieved by re-indexing a subset of the output samples resulting from the conventional decompositions in the radix-4 and radix-8 FFT algorithms. These modified radix-4 and radix-8 algorithms provide savings of more than 33% and 42% respectively in the number of twiddle factor evaluations or accesses to the lookup table compared to the corresponding conventional FFT algorithms without imposing any additional complexity.

Improved radix-4 and radix-8 FFT algorithms

A radix-2/16 decimation-in-frequency (DIF) fast Fourier transform (FFT) algorithm and its higher radix version, namely radix-4/16 DIF FFT algorithm, are proposed by suitably mixing the radix-2, radix-4 and radix-16 index maps, and combing some of the twiddle factors. It is shown that the proposed algorithms and the existing radix-2/4 and radix-2/8 FFT algorithms require exactly the same number of arithmetic operations (multiplications+additions). Moreover, by using techniques similar to these, it can be shown that all the possible split-radix FFT algorithms of the type radix-2/sup r//2/sup rs/ for computing a 2/sup m/-point DFT require exactly the same number of arithmetic operations.

S. Bouguezel

Papers

A new radix-2/8 FFT algorithm for length-q/spl times/2/sup m/ DFTs

A multiplication-free transform for image compression

A General Class of Split-Radix FFT Algorithms for the Computation of the DFT of Length- $2^{m}$

Improved radix-4 and radix-8 FFT algorithms

Arithmetic complexity of the split-radix FFT algorithms