Towards quaternion quadratic‐phase Fourier transform

Abstract

The quadratic‐phase Fourier transform (QPFT) is a neoteric addition to the class of integral transforms and embodies a variety of signal processing tools like the Fourier, fractional Fourier, linear canonical, and special affine Fourier transform. In this paper, we generalize the quadratic‐phase Fourier transform to quaternion‐valued signals, known as the quaternion quadratic‐phase Fourier transform (Q‐QPFT). We initiate our investigation by studying the QPFT of 2D quaternionic signals, and later on, we introduce the Q‐QPFT of 2D quaternionic signals. Using the fundamental relationship between the Q‐QPFT and quaternion Fourier transform (QFT), we derive the inversion, Parseval's, and Plancherel's formulae associated with the Q‐QPFT. Some other properties including linearity, shift, and modulation of the Q‐QPFT are also studied. Finally, we formulate several classes of uncertainty principles (UPs) for the Q‐QPFT like Heisenberg‐type UP, logarithmic UP, Hardy's UP, Beurling's UP, and Donoho–Stark's UP. This study can be regarded as the first step in the applications of the Q‐QPFT in the real world.