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<p>The fractional Fourier transform (FrFT) is a tool for analyzing non-stationary signals, whose integral operator is commonly evaluated using a sampling-type discrete FrFT (DFrFT). Such an evaluation involves complex convolutions, which can be implemented using the Fermat number transform (FNT) to reduce the overall arithmetic complexity of the computation process. In this paper, we explain how such a possibility can be exploited by expressing the DFrFT as a circular convolution over the ring of integers and a finite field. In this context, we introduce two novel number-theoretic transforms: the local-input FNT (LiFNT) and the local-output inverse FNT (LoIFNT), which further decrease the number of additions required by the aforementioned strategy. Using such transforms, we propose an algorithm for calculating partial points of an <i>N</i>-point DFrFT based on the 2D convolution scheme; the method saves at least 4<i>N</i> multiplications compared to one that employs non-local transforms. Finally, we present numerical simulations, including applications in radar echo modeling, to verify the effectiveness of the proposed algorithms.</p>

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Digital Computation of the Fractional Fourier Transform Based on Modulo-\(2^b+1\) Convolutions

  • 期刊论文

  • 发表日期：

    2026年6月29日

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  • Eulogio Gutierrez-Huampo,
  • José R. de Oliveira Neto,
  • Juliano B. Lima

摘要

The fractional Fourier transform (FrFT) is a tool for analyzing non-stationary signals, whose integral operator is commonly evaluated using a sampling-type discrete FrFT (DFrFT). Such an evaluation involves complex convolutions, which can be implemented using the Fermat number transform (FNT) to reduce the overall arithmetic complexity of the computation process. In this paper, we explain how such a possibility can be exploited by expressing the DFrFT as a circular convolution over the ring of integers and a finite field. In this context, we introduce two novel number-theoretic transforms: the local-input FNT (LiFNT) and the local-output inverse FNT (LoIFNT), which further decrease the number of additions required by the aforementioned strategy. Using such transforms, we propose an algorithm for calculating partial points of an N-point DFrFT based on the 2D convolution scheme; the method saves at least 4N multiplications compared to one that employs non-local transforms. Finally, we present numerical simulations, including applications in radar echo modeling, to verify the effectiveness of the proposed algorithms.