Nonuniform fast linear canonical transform based on low rank approximation

Yannan Sun; Jing Liu; Yannan Sun; Jing Liu

doi:10.3934/math.20251253

AIMS Mathematics

2025, Volume 10, Issue 12: 28470-28487. doi: 10.3934/math.20251253

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Research article

Nonuniform fast linear canonical transform based on low rank approximation

Yannan Sun ^,,
Jing Liu

School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013, Jiangsu, China

Received: 20 September 2025 Revised: 25 October 2025 Accepted: 11 November 2025 Published: 03 December 2025
MSC : 11F20, 11M20

The investigations of the discrete and fast linear canonical transform (LCT) are becoming one of the hottest research topics in modern signal processing and optics. Among them, the fast calculation of LCT for nonuniform data is one of the key problems. In this paper, two novel fast algorithms based on low-rank approximation are presented. First, we propose two methods for approximate nonuniform time-domain sampling with uniform sampling. Second, we utilize a low rank matrix to approximate the nonuniform LCT kernel, combined with the exponential function and Taylor series. Then, the fast algorithms for nonuniform sampling in the time domain are developed, which cost $ K $ FFTs. Finally, we extend the fast algorithm to nonuniform LCT in the frequency and transform domains. The effectiveness of the proposed algorithm is verified by simulations.
- linear canonical transform,
- fast Fourier transform,
- Taylor series,
- approximation theory,
- nonuniform sampling
Citation: Yannan Sun, Jing Liu. Nonuniform fast linear canonical transform based on low rank approximation[J]. AIMS Mathematics, 2025, 10(12): 28470-28487. doi: 10.3934/math.20251253

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Abstract

The investigations of the discrete and fast linear canonical transform (LCT) are becoming one of the hottest research topics in modern signal processing and optics. Among them, the fast calculation of LCT for nonuniform data is one of the key problems. In this paper, two novel fast algorithms based on low-rank approximation are presented. First, we propose two methods for approximate nonuniform time-domain sampling with uniform sampling. Second, we utilize a low rank matrix to approximate the nonuniform LCT kernel, combined with the exponential function and Taylor series. Then, the fast algorithms for nonuniform sampling in the time domain are developed, which cost $ K $ FFTs. Finally, we extend the fast algorithm to nonuniform LCT in the frequency and transform domains. The effectiveness of the proposed algorithm is verified by simulations.

References

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