Modified correlation theorem for the linear canonical transform with representation transformation in quantum mechanics

As generalization of the fractional Fourier transform, the linear canonical transform (LCT) has been used in several areas, including optics and signal processing. Many properties of the LCT are currently known including correlation theorem, but however, these do not generalize very nicely to the classical result for the FT. This paper presents a modified correlation theorem in the LCT domain with representation transformation in quantum mechanics. The derivation is direct and concise since the authors make full use of the Dirac’s representation theory. The proposed theorem is compared with the existing correlation theorem in the literature for the LCT and found to be better and befitting proposition. As an application, the proposed correlation theorem is used to determine the power spectral density of frequency-modulated (FM) signal, which shows that the same FM signal can be transmitted with less bandwidth requirement.