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Enhanced Fractional Fourier Transform (FRFT) scheme based on closed Newton-Cotes rules

A. H. Nzokem

TL;DR

The paper tackles the accuracy of the one-dimensional FRFT and its use in inverting Fourier and Laplace transforms. It introduces a composite FRFT scheme by weighting FRFT inputs with closed Newton-Cotes weights of order $QN$, showing that the FRFT of a $QN$-long weighted sequence can be decomposed into two successive FRFTs (on $N$- and $Q$-long blocks). The authors prove (and numerically verify) that these composite FRFTs are commutative and algebraically consistent, and they apply the method to variance-gamma and generalized tempered stable density inversions, where the composite FRFTs outperform the plain non-weighted FRFT and the Newton-Cotes integration method, with the latter sometimes closer in certain regimes. The approach provides a practical, efficient improvement for FRFT-based transforms and density inversions in quantitative finance and signal processing. Potential impact includes faster, more accurate transform inversions and improved numerical quadrature for FRFT-based methods.

Abstract

The paper improves the accuracy of the one-dimensional fractional Fourier transform (FRFT) by leveraging closed Newton-Cotes quadrature rules. Using the weights derived from the Composite Newton-Cotes rules of order QN, we demonstrate that the FRFT of a QN-long weighted sequence can be expressed as two composites of FRFTs. The first composite consists of an FRFT of a Q-long weighted sequence and an FRFT of an N-long sequence. Similarly, the second composite comprises an FRFT of an N-long weighted sequence and an FRFT of a Q-long sequence. Empirical results suggest that the composite FRFTs exhibit the commutative property and maintain consistency both algebraically and numerically. The proposed composite FRFT approach is applied to the inversion of Fourier and Laplace transforms, where it outperforms both the standard non-weighted FRFT and the Newton-Cotes integration method, though the improvement over the latter is less pronounced.

Enhanced Fractional Fourier Transform (FRFT) scheme based on closed Newton-Cotes rules

TL;DR

The paper tackles the accuracy of the one-dimensional FRFT and its use in inverting Fourier and Laplace transforms. It introduces a composite FRFT scheme by weighting FRFT inputs with closed Newton-Cotes weights of order , showing that the FRFT of a -long weighted sequence can be decomposed into two successive FRFTs (on - and -long blocks). The authors prove (and numerically verify) that these composite FRFTs are commutative and algebraically consistent, and they apply the method to variance-gamma and generalized tempered stable density inversions, where the composite FRFTs outperform the plain non-weighted FRFT and the Newton-Cotes integration method, with the latter sometimes closer in certain regimes. The approach provides a practical, efficient improvement for FRFT-based transforms and density inversions in quantitative finance and signal processing. Potential impact includes faster, more accurate transform inversions and improved numerical quadrature for FRFT-based methods.

Abstract

The paper improves the accuracy of the one-dimensional fractional Fourier transform (FRFT) by leveraging closed Newton-Cotes quadrature rules. Using the weights derived from the Composite Newton-Cotes rules of order QN, we demonstrate that the FRFT of a QN-long weighted sequence can be expressed as two composites of FRFTs. The first composite consists of an FRFT of a Q-long weighted sequence and an FRFT of an N-long sequence. Similarly, the second composite comprises an FRFT of an N-long weighted sequence and an FRFT of a Q-long sequence. Empirical results suggest that the composite FRFTs exhibit the commutative property and maintain consistency both algebraically and numerically. The proposed composite FRFT approach is applied to the inversion of Fourier and Laplace transforms, where it outperforms both the standard non-weighted FRFT and the Newton-Cotes integration method, though the improvement over the latter is less pronounced.
Paper Structure (13 sections, 43 equations, 5 figures, 3 tables)

This paper contains 13 sections, 43 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Composite Newton-Cotes Quadrature Formulas
  3. Composite Rules
  4. Weights Computation
  5. Fast Fractional Fourier Transform (FRFT) and Composite Newton-Cotes Quadrature rules
  6. Fast Fourier Transform and Fractional Fourier Transform
  7. FRFT of QN-long weighted sequence
  8. Composite of FRFTs: FRFT of Q-long weighted sequence FRFT of N-long sequence
  9. Composite of FRFTs: FRFT of N-long weighted sequence FRFT of Q-long sequence
  10. ILLUSTRATION EXAMPLES
  11. Variance-Gamma VG Distribution
  12. Generalized Tempered Stable (GTS) Distribution
  13. Conclusion

Figures (5)

  • Figure 1: VG* Probability density error ($\textbf{$\check{f}$ - $f$}$): $\textbf{N=5000}$$\textbf{b-a=100}$
  • Figure 2: VG* Probability density error ($\textbf{$\hat{f}$ - $f$}$): $\textbf{N=5000}$$\textbf{b-a=100}$
  • Figure 3: VG* Probability density error ($\textbf{$\tilde{f}_{NQ}(x_{k})$ - $\tilde{f}_{QN}(x_{k})$}$): $\textbf{N=5000}$$\textbf{b-a=100}$
  • Figure 4: Risk Neutral Probability Density function (PDF)
  • Figure 5: GTS* Probability density error