Fast numerical calculation of the offset linear canonical transform

The offset linear canonical transform (OLCT) is an important research topic in many fields, and it has a more universal and elastic performance due to its extra parameters. However, although much work has been done concerning the OLCT, its fast algorithms are rarely addressed. In this paper, an O(N log⁡N) fast OLCT (FOLCT) algorithm that can significantly reduce the amount of calculation and improve accuracy is proposed. First, the discrete form of the OLCT is provided, and several important properties of its kernel are advanced. Next, the FOLCT based on the fast Fourier transform (FT) is derived for its numerical implementation. Then, the numerical results indicate that the FOLCT is a serviceable tool for signal analysis; additionally, the FOLCT algorithm can be used for the FT, fractional FT, linear canonical transform, and other transforms. Finally, its application to the detection of linear frequency modulated signals and optical image encryption, which is a basic case in signal processing, is discussed. The FOLCT can be effectively applied for the fast numerical calculation of the OLCT with valid and accurate results.