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<p>Reconstructing a function from its windowed linear canonical transform is a fundamental topic in both theory and applications. In this paper, we propose a new inversion formula for the windowed linear canonical transform via Riemann sums. We show that for certain window functions, the Riemann sums are well-defined on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^2(\mathbb R)\)</EquationSource> </InlineEquation> and converge to the function or signal to be reconstructed as the sampling density tends to zero. We also derive the inversion formula in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^p(\mathbb {R})\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(1&lt; p &lt; \infty \)</EquationSource> </InlineEquation>, by the Poisson summation formula. Moreover, we present numerical results not only for various window functions but also for different types of signals, including non-stationary signals, to validate the effectiveness of the proposed inversion formula.</p>

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Inversion of the windowed linear canonical transform and numerical results

  • 期刊论文

  • 发表日期：

    2026年1月8日

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  • Yaoyao Han,
  • Chengmeng Xu

摘要

Reconstructing a function from its windowed linear canonical transform is a fundamental topic in both theory and applications. In this paper, we propose a new inversion formula for the windowed linear canonical transform via Riemann sums. We show that for certain window functions, the Riemann sums are well-defined on \(L^2(\mathbb R)\) and converge to the function or signal to be reconstructed as the sampling density tends to zero. We also derive the inversion formula in \(L^p(\mathbb {R})\) , \(1< p < \infty \) , by the Poisson summation formula. Moreover, we present numerical results not only for various window functions but also for different types of signals, including non-stationary signals, to validate the effectiveness of the proposed inversion formula.