Windowed Octonion Quadratic Phase Fourier Transform: Sharp Inequalities, Uncertainty Principles, and Examples in Signal Processing

Abstract

In this paper, we define the Windowed Octonion Quadratic Phase Fourier Transform (WOQPFT) and derive its inversion formula, including its essential properties, such as linearity, anti-linearity, parity, scaling, modulation, shifting, and joint time-frequency shifting, as well as its link to Octonion Quadratic Phase Fourier Transform (OQPFT). Additionally, we derive the Riemann-Lebesgue lemma using this transform. Following the present analysis, we formulated Sharp Pitt’s and Sharp Hausdorff-Young’s inequalities. Further, Logarithmic, Heisenberg’s, and Donoho-Stark’s uncertainty principles are also formulated. The practical application of WOQPFT and the five elementary examples of signal theory are discussed, and their particular cases are analyzed through graphical visualization, including interpretation.

Keywords

• Windowed octonion quadratic phase Fourier transform
• sharp Pitt’s inequality
• logarithmic uncertainty principle
• sharp Hausdorff-young’s inequality
• Heisenberg’s uncertainty principle
• Donoho-Stark’s uncertainty principle