Linear canonical transforms (LCTs) are a family of integral transforms with wide applications in optical, acoustical, electromagnetic, and other wave propagation problems. In this paper, we propose the random discrete linear canonical transform (RDLCT) by randomizing the kernel transform matrix of the discrete linear canonical transform (DLCT). The RDLCT inherits excellent mathematical properties from the DLCT along with some fantastic features of its own. It has a greater degree of randomness because of the randomization in terms of both eigenvectors and eigenvalues. Numerical simulations demonstrate that the RDLCT has an important feature that the magnitude and phase of its output are both random. As an important application of the RDLCT, it can be used for image encryption. The simulation results demonstrate that the proposed encryption method is a security-enhanced image encryption scheme.

The affine Fourier transform (AFT) plays an important role in many fields of optics and signal processing. The Gabor transform (GT) is a kind of linear time–frequency representation (TFR). Compared with many bilinear TFRs, the GT does not have the cross‐term problem. In this study, the authors propose a time–frequency analysis method based on the AFT and GT. First, they obtain an affine relation between the AFT and the modified GT (MGT). Since the MGT is closely related to the AFT, they can use it as an assistant tool for signal processing in the AFT domain. Moreover, many useful relations between the AFT and the MGT are derived, such as recovery relation, projection relation and power integration relation. Then, they demonstrate that the AFT also has the affine relation with other TFRs, such as the Gabor–Wigner transform and the general class of quadratic distribution. Last, using the new time–frequency analysis method associated with the AFT and MGT, they present the filter design for multiple component chirp signal separation. Moreover, the simulation results illustrate the effectiveness of the proposed method.