Caputo–FRFT Coupling Induced by Compatible Fractional Convolution: Representation and Discrete Realization
- 1Yantai Research Institute, Harbin Engineering University, Yantai 264000, China
Abstract
Caputo fractional derivatives describe causal power-law memory, whereas the fractional Fourier transform represents continuous rotations in the time–frequency plane. This study derives an Abel–Gaussian-regularized coupling for endpoint-compatible causal signals. Unlike a direct composition of the two operators, the coupling is induced by embedding the causal Caputo convolution into a chirp-conjugated fractional convolution compatible with the adopted transform phase convention. Evaluation of the two transform branches shows that their quadratic phases cancel the additional convolution phase. On a compactly supported smooth core, removal of the regularization yields a rotation–memory–rotation factorization in which the fractional-memory multiplier acts at the ordinary Fourier node and the outer rotation maps the weighted spectrum to the prescribed fractional domain. At the discrete level, modulo-four reduction of the transform order is kept separate from decomposition of the derivative order, while a one-sided Grünwald–Letnikov approximation preserves causal history accumulation. Numerical experiments confirm fixed-spectral rotation identities and forward–inverse reconstruction at double-precision roundoff, approximately first-order convergence to an analytic Caputo reference, and decreasing low-frequency discrepancies under grid refinement. Further tests distinguish the effects of the sampling interval, observation length, fractional order, and nonzero endpoint traces. Exact boundary-term retention preserves the finite record, whereas compatible tapering produces a modified surrogate. For the tested dense implementations, the spectral implementation is faster than the direct full-history Grünwald–Letnikov implementation, while the shared dense operators dominate the storage requirements.
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BibTeX
@article{sun2026caputofrft,
title = {{Caputo–FRFT Coupling Induced by Compatible Fractional Convolution: Representation and Discrete Realization}},
author = {Xiao Sun and Yi Fan and Yanyan Wang and Haitao Zhu},
journal = {Fractal and Fractional},
year = {2026},
month = sep,
doi = {10.3390/fractalfract10100684},
url = {https://www.mdpi.com/2504-3110/10/10/684},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Caputo–FRFT Coupling Induced by Compatible Fractional Convolution: Representation and Discrete Realization
AU - Xiao Sun
AU - Yi Fan
AU - Yanyan Wang
AU - Haitao Zhu
PY - 2026
DA - 2026/09/01
JO - Fractal and Fractional
DO - 10.3390/fractalfract10100684
UR - https://www.mdpi.com/2504-3110/10/10/684
AB - Caputo fractional derivatives describe causal power-law memory, whereas the fractional Fourier transform represents continuous rotations in the time–frequency plane. This study derives an Abel–Gaussian-regularized coupling for endpoint-compatible causal signals. Unlike a direct composition of the two operators, the coupling is induced by embedding the causal Caputo convolution into a chirp-conjugated fractional convolution compatible with the adopted transform phase convention. Evaluation of the two transform branches shows that their quadratic phases cancel the additional convolution phase. On a compactly supported smooth core, removal of the regularization yields a rotation–memory–rotation factorization in which the fractional-memory multiplier acts at the ordinary Fourier node and the outer rotation maps the weighted spectrum to the prescribed fractional domain. At the discrete level, modulo-four reduction of the transform order is kept separate from decomposition of the derivative order, while a one-sided Grünwald–Letnikov approximation preserves causal history accumulation. Numerical experiments confirm fixed-spectral rotation identities and forward–inverse reconstruction at double-precision roundoff, approximately first-order convergence to an analytic Caputo reference, and decreasing low-frequency discrepancies under grid refinement. Further tests distinguish the effects of the sampling interval, observation length, fractional order, and nonzero endpoint traces. Exact boundary-term retention preserves the finite record, whereas compatible tapering produces a modified surrogate. For the tested dense implementations, the spectral implementation is faster than the direct full-history Grünwald–Letnikov implementation, while the shared dense operators dominate the storage requirements.
SN - 2504-3110
ER -