The wave packet transform associated with the linear canonical transform

Linear canonical transforms (LCTs) are a family of integral transforms with wide application in optical, acoustical, electromagnetic, and other wave propagation problems. In this paper, a new kind of wave packet transform (WPT) associated with the LCT is proposed, this new WPT (WPTL) is defined based on the ideal of the LCT and the WPT. Some properties and physical meaning of the WPTL are investigated. In particular, we show a version of the resolution of the identity of WPTL. Moreover, the relationship between the WPTL and the Wigner distribution (WD) is derived. At last, we introduce the concept of the fractional wavepacketgram, which is defined as the modulus square of the WPTL. It is proved that the fractional wavepacketgram is a member of the Cohen class time–frequency distribution where the kernel is a scale dependent ambiguity function.