Easy generation of small-Ndiscrete Fourier transform algorithms

Stasinski, Ryszard

doi:10.1049/ip-g-1.1986.0022

Cited by 9 publications

(8 citation statements)

References 2 publications

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“…and a n are cyclic group elements modulo N , the group size is N − 1. Rader algorithms exist also when N is a power of a prime number [21]. Their structure is more complicated, hence, a good idea is to add such DFTs to the structure of the developed here algorithms later, when Winograds improvements to the Rader algorithm are implemented, section 3.…”

Section: Nestingmentioning

confidence: 99%

Fast Discrete Fourier Transform algorithms requiring less than 0(NlogN) multiplications

Stasinski¹

2023

Preprint

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In the paper it is shown that there exist infinite classes of fast DFT algorithms having multiplicative complexity lower than O(N log N ), i.e. smaller than their arithmetical complexity. The derivation starts with nesting of Discrete Fourier Transform (DFT) of size N = q1 • q2 • . . . qr, where qi are powers of prime numbers: DFT is mapped into multidimensional one, Rader convolutions of qi-point DFTs extracted, and combined into multidimensional convolutions processing data in parallel. Crucial to further optimization is the observation that multiplicative complexity of such algorithm is upper bounded by 0(N log Mmax), where Mmax is the size of the greatest structure containing multiplications. Then the size of the structures is diminished: Firstly, computation of a circular convolution can be done as in Rader-Winograd algorithms. Secondly, multidimensional convolutions can be computed using polynomial transforms. It is shown that careful choice of qi values leads to important reduction of Mmax value: Multiplicative complexity of the new DFT algorithms is O(N log c log N ) for c ≤ 1, while for more addition-orietnted ones it is O(N log 1/m N ), m is a natural number denoting class of qi values. Smaller values of c, 1/m are obtained for algorithms requiring more additions, part of algorithms for c = 1, m = 2 have arithmetical complexity smaller than that for the radix-2 FFT for any comparable DFT size, and even lower than that of split-radix FFT for N ≤ 65520. The approach can be used for finding theoretical lower limit on the DFT multiplicative complexity.

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Section: Nestingmentioning

confidence: 99%

Fast Discrete Fourier Transform algorithms requiring less than 0(NlogN) multiplications

Stasinski¹

2023

Preprint

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“…y -x 0 t x l t X 2 (4) For fulfilling the premse 4, y should be hvided by 3, but because of implementation reasons it is usually avided by 4. It is reasonable to use such small-N DFT modules, for whxh t l u s effect does not exists.…”

mentioning

confidence: 99%

“…It is reasonable to use such small-N DFT modules, for whxh t l u s effect does not exists. The mdules consists of [4]: polynomal reciuction/reconstruction algorithms, operations linked with computation of DFT sample X ( 0 ) , and &stribution of data sample x(O), and polynomal product (PP) algorithms. In an WFTA a part of PPs can be replaced by forward and inverse (reduced) polynomal transforms (FTs) [3], [4].…”

mentioning

confidence: 99%

“…The mdules consists of [4]: polynomal reciuction/reconstruction algorithms, operations linked with computation of DFT sample X ( 0 ) , and &stribution of data sample x(O), and polynomal product (PP) algorithms. In an WFTA a part of PPs can be replaced by forward and inverse (reduced) polynomal transforms (FTs) [3], [4]. The only reduction/reconstruction algorithms and FTs of interest are due to circular convolutions for sizes being powers of 2, [3].…”

mentioning

confidence: 99%

“…For Fenrat prime numbers we must correct operations due to computation of X(0), and dxtribution of x(0) samples [4]. Fig.…”

mentioning

confidence: 99%

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Minimization of rounding errors in WFTA programs

Stasinski

Lukasik

ICASSP-88., International Conference on Acoustics, Speech, and Signal Processing

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In the paper two ideas limed with bgh-precision computation of W A S using fixed-point arithmetic are analysed Firstly, a use of the best, optimzed small-N DFT modules is considered m e sizes of these modules appear to be equal to F e m t prim numbers, and powers of 2. Secondly, the computation of W A S with a full accuracy of intermediate results is stuhed ?he properties of the received algorithms seem to be comparable to those of FFTs in the f o m r case, W l e sirmlar to those of D€T computed dmectly, or using an RNS in the latter one. IN'IRO~ION

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Computation of prime factor DFT and DHT/DCCT algorithms using cyclic and skew-cyclic bit-serial semisystolic IC convolvers

Gudvangen

Holt

1990

IEE Proc. F Radar Signal Process. UK

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Easy generation of small-Ndiscrete Fourier transform algorithms

Cited by 9 publications

References 2 publications

Fast Discrete Fourier Transform algorithms requiring less than 0(NlogN) multiplications

Fast Discrete Fourier Transform algorithms requiring less than 0(NlogN) multiplications

Minimization of rounding errors in WFTA programs

Computation of prime factor DFT and DHT/DCCT algorithms using cyclic and skew-cyclic bit-serial semisystolic IC convolvers

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