Prime Factor FFT Algorithm Research Articles

An In-place In-order Prime Factor FFT Algorithm with Reconfigurable Structure

An In-place In-order Prime Factor FFT Algorithm with Reconfigurable Structure

Multiplierless Winograd and Prime Factor FFT Implementation

This letter describes fixed-point implementations of the Winograd and prime factor FFT algorithms in which multiplications are replaced by additions, subtractions and shifts. Methods are described for minimizing the number of additions and subtractions, while achieving a required level of accuracy. VLSI implementation of the resulting FFTs could achieve very high speed and/or power efficiency. The method can be used to provide any chosen accuracy; examples are presented for 12 to 20 bit accuracy.

A two-port envelope model for building heat transfer

A zone envelope thermal model is derived from first principles, based on the exact two-port solution to the heat conduction equation. A single set of matrices, one for each frequency of interest, is used to describe the thermal behaviour of the total envelope. The set of matrices must be computed once for a particular zone and cooling loads for various conditions, or the passive response, can then be obtained. The method is based on the same principles underlying the CTF method. However, the transfer function is calculated for the whole envelope by combining the two-port matrices of the individual walls, after simplification of the room interior heat transfer network. The conduction transfer coefficients of the individual walls forming the zone are not required. Except for the simplification of interior radiation, the method is completely exact and easy to implement. It is also numerically efficient. For load prediction, an implementation in the frequency domain, based on the prime factor FFT algorithm is convenient. For time domain simulation with active control, the envelope conduction coefficients are easily obtained by transforming to the Z-domain. This represents a modern solution of the envelope conduction problem, which is convenient to implement in modern object oriented computer programming languages.

Computing the inverse DFT with the in-place, in-order prime factor FFT algorithm

We present a method for computing the inverse discrete Fourier transform (IDFT) by the in-place, in-order prime factor FFT algorithm (PFA). This is achieved by modifying the input and the output index mapping equations. This approach does not result in any additional cost in terms of program length and computational time. >

New Shortlength DFTs for the Prime Factor Implementation on DSP-Architectures

We found that for the implementation of the prime factor FFT algorithm on a modern digital signal processor (DSP), e.g. the ADSP 2100, the shortlength DFT-modules exhibiting the minimum multiply configuration as developed by S. Winograd are not the optimum choice. New modules which yield an increase of up to 15% in execution speed and a 1 bit increase in accuracy are designed by a novel powerful method

New high-speed prime-factor algorithm for discrete Hartley transform

Fast algorithms for computing the DHT of short transform lengths (N = 2, 3, 4, 5, 7, 8, 9 and 16) are derived. A new prime-factor algorithm is also proposed to compute the long-length DHTs from the short-length DHT algorithms. The short-length algorithms (except for N = 8 and N = 16) are such that the even and the odd parts of the DHT components are obtained directly, without any additional computation. This feature of the short-length algorithms makes the proposed prime-factor DHT algorithm more attractive and efficient. It is found that the proposed algorithm is more efficient compared to the radix-2 FHT in terms of the computational requirements, as well as the execution time for transform lengths higher than 30. It is also observed that the number of operations required for the computation of DHT by the prime-factor FFT algorithm for real-valued data is the same as those of the proposed algorithm for certain transform lengths, e.g. N = 30, 60, 252 etc., which do not contain 8 or 16 as a cofactor. However, for all other transform lengths the proposed algorithm has a lower computational complexity. It is further observed that the proposed algorithm is faster than the prime-factor FFT algorithm for real-valued series.

A prime factor fast W transform algorithm

A method for converting any nesting DFT algorithm to the type-I discrete W transform (DWT-I) is introduced. A nesting algorithm that differs from either the Windograd Fourier transform algorithm (WFTA) or the prime factor FFT algorithm (PFA) is presented. New small-N DETs, which are suitable for this nesting algorithm, are developed based on using sparse matrix decomposition. The proposed algorithm is more efficient that either WFTA or PFA for large N, and it is more flexible for the choice of transform length, because 32 points are used. For 2D processing, the proposed algorithm is more efficient than the polynomial transform.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A general in-place and in-order prime factor FFT algorithm

Starting from an index mapping for one to multi-dimensions, a general in-place and in-order prime factor FFT algorithm is proposed in this paper. In comparing with existing prime factor FFT algorithms, this algorithm saves about half of the required storage capacity and possesses a higher efficiency. In addition, this algorithm can easily implement the DFT and IDFT in a single subroutine.

Nesting strategies for prime factor FFT algorithms

Several nesting techniques for the prime factor FFT algorithm are examined. These include the full nesting strategy of Winograd's algorithm, which gives a very low multiplication count; a split nesting technique, which requires the same number of multiplications as Winograd's FFT, but fewer additions; and a partial nesting technique, which further reduces the addition count at the cost of some extra multiplications and requires the lowest total number of floating-point operations. Applications to large multidimensional transforms, as well as to 1-dimensional transforms, are discussed. On large scientific computers where the multiplications can be overlapped with the additions, the prime factor algorithm without nesting remains the fastest and is the easiest to implement.

A new set of minimum-add small- n rotated DFT modules

A self-sorting in-place prime factor FFT algorithm introduced in an earlier paper was based on a set of modules for performing small discrete Fourier transforms (DFTs) with an optional “rotation” of the results. The modules were designed to reduce the total number of additions in the FFT algorithm. In specializing this algorithm to the case of real or conjugate-symmetric input data, it was found necessary to redesign these modules to impose a particular structure. In some cases, the new modules proved to require fewer operations than the old. We describe the design procedure and compare the new operation counts with those for Winograd's DFT modules. The algorithms for the new modules are given in detail; these will be useful in coding FFT algorithms of either the “prime factor” or conventional variety.

Implementation of a prime factor FFT algorithm on CRAY-1

Recent developments in algorithm design have made the Fast Fourier Transform even faster. We described the implementation on the CRAY-1 of a prime factor FFT algorithm which adapts some of these developments to a vector-processing scientific computer. Comparative times are given for the new and old versions of the FFT algorithm, applied to the problem of performing multiple simultaneous complex transforms. It is shown that worthwhile gains are obtained in both speed and storage requiremennts. The new algorithm is also vectorizable in the more difficult cases of a single transform. Finally, we use timing measurements of the new routine to estimate the value on the CRAY-1 of Hockney's parameter n 1 2 , which characterizes a computer in terms of its apparent degree of parallelism.

A prime factor FFT algorithm implementation using a program generation technique

This correspondence presents details of a new implementation of the prime factor FFT algorithm (PFA) for computing the discrete Fourier transform (DFT). This implementation applies a program generation technique to the PFA algorithm and saves about 40 percent of the execution time of the conventional one.

Implementation of a self-sorting in-place prime factor FFT algorithm

A “prime factor” Fast Fourier Transform algorithm is described which is self-sorting and computes the transform in place. With a view to implementation on a Cray-1 or Cyber 205, the form of the algorithm is chosen to minimize the number of additions. With an appropriate choice of index mapping in the derivation, we obtain the unexpected result that the required indexing is actually simpler than that for a conventional FFT. The construction of the necessary “rotated” DFT modules is described, and comparisons are presented between the new algorithm and the conventional FFT in terms of operation counts and timings on an IBM 3081; on this machine, the new transform algorithm takes about 60% of the time for the conventional FFT. A Fortran routine for the new algorithm is outlined.

A prime factor FFT algorithm with real valued arithmetic

A prime factor FFT algorithm involving only real valued arithmetic is devised to compute the discrete Fourier transform of a real sequence. This letter extends an approach proposed by Bracewell.

Note a note on prime factor FFT algorithms

The author shows that prime factor FFT algorithms offer little improvement over conventional FFT algorithms on computers such as the Cray-1 and Cyber 205 where the multiplications can be performed in parallel with the additions. A very modest gain may be obtained by using Good's algorithm (1958) with conventional small-n transforms. Winograd's technique (1978), despite its impressive reduction in the number of multiplications, is likely to be slower than the conventional algorithm, particularly on the Cray-1, where memory transfers will dominate the computation. 13 references.

An in-order, partially in-place mixed radix FFT algorithm

This correspondence describes a modified version of the Burrus and Rothweiler prime factor FFT algorithm for the powers of prime length. It allows a much wider selection of transform sizes, and calculates the DFT in order. Speed measurements show that the resulting program can be faster than Singleton's algorithm for some specific transform length.

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

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.

Comments on "A prime factor FFT algorithm using high-speed convolution"

The purpose of this correspondence is to point out that in Table II of the above paper, the number of additions reported for the radix-2 FFT algorithm are highly erroneous.