Discrete Fourier transforms when the number of data samples is prime

Rader, Charles M.

doi:10.1109/proc.1968.6477

Cited by 457 publications

(211 citation statements)

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“…The FFT provided by the FPGA and digital signal processor (DSP) companies is highly optimized for their chip and requires a length that is a power of two. There are also algorithms that can perform the FFT when L is a prime number, less efficiently however [19], [20]. A state of the art of FFT algorithms can be found in [17].…”

Section: Fft Lengthmentioning

confidence: 99%

Acquisition of modern GNSS signals using a modified parallel code-phase search architecture

2014

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The acquisition of global navigation satellite system signals can be performed using a fast Fourier transform (FFT). The FFT-based acquisition performs a circular correlation, and is thus sensitive to potential transitions between consecutive periods of the code. Such transitions are not occurring often for the GPS L1 C/A signal because of the low data rate, but very likely for the new GNSS signals having a secondary code. The straightforward solution consists in using two periods of the incoming primary code and using zeropadding for the local code to perform the correlation. However, this solution increases the complexity, and is moreover not efficient since half of the points calculated are discarded. This has led us to research for a more efficient algorithm, which discards less points by calculating several sub-correlations.It is applied to the GPS L5, Galileo E5a, E5b and E1 signals. Considering the radix-2 FFT, the proposed algorithm is more efficient for the L5, E5a and E5b signals, and possibly for the E1 signal. The theoretical number of operations can be reduced by 21 %, the processing time measured on a * Corresponding author Email address: jerome.leclere@epfl.ch (Jérôme Leclère)Preprint submitted to Signal Processing July 2, 2013 software implementation is reduced by 39 %, and the memory resources are almost halved for an FPGA implementation.

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Section: Fft Lengthmentioning

confidence: 99%

Acquisition of modern GNSS signals using a modified parallel code-phase search architecture

2014

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“…To reduce the storage and complexity of cyclically shifted ZC sequences' discrete fourier transform, several effective work has been done. CHARLES M. RADER expressed a fast fourier transform (FFT) algorithm of prime length M as a fast convolution [6][7]. But the computation of three FFTs of highly composite length N>M is still required.…”

Section: Introductionmentioning

confidence: 99%

Random Access Preamble Generation with Low Storage for LTE Networks

Xin

et al. 2017

ITM Web Conf.

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Abstract. Long Term Evolution (LTE) random access preambles are generated from cyclically shifted Zadoff-Chu(ZC) sequences of prime length by using so-called Discrete Fourier Transformation-Spread-Orthogonal Frequency Division Multiplexing (DFT-S-OFDM) modulation. In this paper we propose a novel preamble generation to reduce the storage and complexity of cyclically shifted ZC sequences' Discrete Fourier Transformation (DFT). The proposed preamble generation allows the DFT of cyclically shifted ZC sequences to rely only on several constant factors and the elements of non-cyclically shifted ZC sequences which have already been generated. This preamble generation merely requires the storage of several constant factors, while avoids large storage and complex computation of complex exponential function.

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“…Rader [14] and independently S. Winograd had developed such algorithms in the early 1970s, but for the most part these algorithms were viewed as mathematical oddities. These algorithms suffered from several disadvantages which at the time precluded their widespread adoption.…”

Section: Introductionmentioning

confidence: 99%

Lesser Known FFT Algorithms

Tolimieri¹,

An²

2001

Twentieth Century Harmonic Analysis — A Celebration

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ABSTRACT. The introduction of the Cooley-Tukey Fast Fourier transform (C-T FFT) algorithm in 1968 was a critical step in advancing the widespread use of digital computers in scientific and technological applications. Initial efforts focused on realizing the potential of the immense reduction in arithmetic complexity afforded by the FFT for computing the finite Fourier transform and convolution. On existing serial butterfly architectures, this limited implementations of the FFT to transform sizes a power of two.The original C-T FFT algorithms and its many extensions rested mainly on additive structures of the data indexing set and divide-and-conquer strategies for reducing complexity. The relative cost of multiplications as compared with additions on these early machines motivated the study of new algorithms for reducing multiplications usually at the cost of increasing additions. These new algorithms were based on multiplicative structures of the data indexing set. They resulted in greater flexibility as compared to earlier efforts by allowing for fast small prime size computations which could be embedded in a wide collection of large transform sizes.During the 1980s, the increasing importance of RISC and parallel computers removed some of the initial motivation for these multiplicative FFTs. Many RISC architectures featured a hardwired multiply and accumulate which permitted multiplications to be nested in additions. The goal was to arrange the computations so that most multiplications were followed by an addition. Parallel architectures placed the major algorithmic burden on controlling the data flow. The exotic data flow of the multiplicative FFTs were often incompatible or at least required an immense coding effort for efficient parallel computation. Only the need for flexible transform sizes justified continued efforts.The recent importance of FPGAs and reconfigurable hardware renews the need to reduce multiplication counts. In this talk, we will review some of the work on multiplicative approaches introduced mainly at IBM in the 1970s and 1980s and place these results within the realm of harmonic analysis.

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Discrete Fourier transforms when the number of data samples is prime

Cited by 457 publications

References 2 publications

Acquisition of modern GNSS signals using a modified parallel code-phase search architecture

Acquisition of modern GNSS signals using a modified parallel code-phase search architecture

Random Access Preamble Generation with Low Storage for LTE Networks

Lesser Known FFT Algorithms

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