Quaternion Wigner–Ville distribution associated with the linear canonical transforms

• The QLCT, a generalization of the classical 2D Fourier transform, has applications in color image and signal processing.
• This paper presents new notions for the quaternion Wigner distribution (QLWD) and the corresponding ambiguity function in the QLCT setting.
• Connection with the corresponding analytic signal are investigated.
• Examples with bandpass analytic signal illustrate the features of the proposed distributions.
• A novel algorithm for the detection of quaternion-valued linear frequency-modulated signal is presented by the proposed QLWD.

The quaternion linear canonical transform (QLCT), a generalization of the classical 2D Fourier transform, has gained much popularity in recent years because of its applications in many areas, including color image and signal processing. There are relationship between Wigner distribution and ambiguity function. But, these relations are only suitable for complex-valued signals, and have not been investigated in quaternion linear canonical transforms. The purpose of this paper is to propose an equivalent relationship for the quaternion Wigner distribution and quaternion ambiguity function in the QLCT setting. First, we propose the 2D quaternion Wigner distribution (QLWD) and quaternion ambiguity function associated with the QLCT. Next, the relationship between these two novel concepts are derived. Moreover, the connection with the corresponding analytic signal are investigated. Examples with bandpass analytic signal illustrate the features of the proposed distributions. Finally a novel algorithm for the detection of quaternion-valued linear frequency-modulated signal is presented by using the proposed QLWD.