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The Multidimensional Quadratic Phase Fourier Transform: Theoretical Analysis and Applications

Sarga Varghese, Gita Rani Mahato, Manab Kundu

TL;DR

This work extends the quadratic-phase Fourier transform to multidimensional domains, defining the MQPFT and proving key theoretical results including a Plancherel identity and inversion theorems. It introduces three convolution-like operations in the MQPFT setting and derives corresponding convolution theorems, enabling analysis and design in the transform domain. A multidimensional Boas-type theorem is established, linking spectral localization to high-frequency behavior. Practical applications are demonstrated in solving integral equations and designing multiplicative filters, highlighting the MQPFT's potential for multidimensional signal processing and optical systems.

Abstract

The quadratic phase Fourier transform (QPFT) is a generalization of several well-known integral transforms, including the linear canonical transform (LCT), fractional Fourier transform (FrFT), and Fourier transform (FT). This paper introduces the multidimensional QPFT and investigates its theoretical properties, including Parseval's identity and inversion theorems. Generalized convolutions and correlation for multiple variables, extending the conventional convolution for single-variable functions, are proposed within the QPFT setting. Additionally, a Boas-type theorem for the multidimensional QPFT is established. As applications, multiplicative filter design and the solution of integral equations using the proposed convolution operation are explored.

The Multidimensional Quadratic Phase Fourier Transform: Theoretical Analysis and Applications

TL;DR

This work extends the quadratic-phase Fourier transform to multidimensional domains, defining the MQPFT and proving key theoretical results including a Plancherel identity and inversion theorems. It introduces three convolution-like operations in the MQPFT setting and derives corresponding convolution theorems, enabling analysis and design in the transform domain. A multidimensional Boas-type theorem is established, linking spectral localization to high-frequency behavior. Practical applications are demonstrated in solving integral equations and designing multiplicative filters, highlighting the MQPFT's potential for multidimensional signal processing and optical systems.

Abstract

The quadratic phase Fourier transform (QPFT) is a generalization of several well-known integral transforms, including the linear canonical transform (LCT), fractional Fourier transform (FrFT), and Fourier transform (FT). This paper introduces the multidimensional QPFT and investigates its theoretical properties, including Parseval's identity and inversion theorems. Generalized convolutions and correlation for multiple variables, extending the conventional convolution for single-variable functions, are proposed within the QPFT setting. Additionally, a Boas-type theorem for the multidimensional QPFT is established. As applications, multiplicative filter design and the solution of integral equations using the proposed convolution operation are explored.
Paper Structure (12 sections, 19 theorems, 139 equations)

This paper contains 12 sections, 19 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convolution and correlation
  4. Convolution: type 1
  5. Convolution: type 2
  6. Convolution: type 3
  7. Inversion theorem
  8. Boas theorem for multidimensional QPFT
  9. Potential Applications
  10. Solvability of Integral equations
  11. Designing of multiplicative filters in the multidimensional QPFT domain
  12. Conclusion

Key Result

Theorem 2.4

For functions $f,g \in L^2 (\mathbb{R}^N)$, the inner product of multidimensional QPFT of two functions $f$ and $g$ is same as the inner product of $f$ and $g$. i.e.,

Theorems & Definitions (44)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3
  • Definition 3.4
  • Theorem 3.5
  • ...and 34 more