Sampling is one of the fundamental topics in the signal processing community. Theorems proposed under this topic form the bridge between the continuous-time signals and discrete-time signals. Several sampling theorems, which aid in the reconstruction of signals in the linear canonical transform (LCT) domain, have been proposed in the literature. However, two main practical issues associated with the sampling of the LCT still remain unresolved. The first one relates to the reconstruction of the original signal from nonuniform samples and the other issue relates to the fact that only a finite number of samples are available practically. Focusing on these issues, this paper seeks to address the above from the LCT point of view. First, we extend several previously developed theorems for signals band-limited in the Fourier domain to signals band-limited in the LCT domain, followed by the derivation of the reconstruction formulas for finite uniform or recurrent nonuniform sampling points associated with the LCT. Simulation results and the potential applications of the theorem are also proposed.

Nonuniform sampling occurs in many applications due to imperfect timebase or random events. Periodic nonuniform sampling is a special type of nonuniform sampling. The properties and applications of periodic nonuniform sampling signals in traditional Fourier domain have been extensively studied, but so far none of the research papers covering the spectral analysis and reconstruction of periodic nonuniformly sampled signals in fractional Fourier domain have been published. This correspondence is to explore the spectral properties of periodic nonuniformly sampled signals associated with the fractional Fourier transform. First, based on the uniform fractional Fourier transform kernel, the general spectral representation of periodic nonuniformly sampled signals has been derived. Second, detailed analysis of periodic nonuniformly sampled chirp signals in the fractional Fourier domain have been performed. The results can be used to estimate the chirp rate and the sampling offsets. Finally, a more simple relationship between the discrete fractional spectrum of periodic nonuniformly sampled signals and the continuous spectrum has been obtained. Based on this relationship, the original continuous spectrum can be reconstructed from periodic nonuniformly sampled signals in fractional Fourier domain. In addition, the simulations are carried out to verify the correctness of the results.

The linear canonical transform is shown to be one of the most powerful tools for nonstationary signal processing. Based on the properties of the linear canonical transform and the classical Wigner‐Ville transform, this paper investigates the Wigner‐Ville distribution in the linear canonical transform domain. Firstly, unlike the classical Wigner‐Ville transform, a new definition of Wigner‐Ville distribution associated with the linear canonical transform is given. Then, the main properties of the newly defined Wigner‐Ville transform are investigated in detail. Finally, the applications of the newly defined Wigner‐Ville transform in the linear‐frequency‐modulated signal detection are proposed, and the simulation results are also given to verify the derived theory.

Convolution and correlation operations are very important in signal processing community, as well as in sampling, filter design and applications. In this study, the authors derive the convolution and correlation theorems for the two‐dimensional linear canonical transform (2D LCT). Moreover, they utilise the convolution theorem to investigate the sampling theorem for the band limited signal in the 2D LCT domain. They also discuss multiplicative filter for the band limited signal in the 2D LCT domain which has much lower computational load than the method in the 2D LCT domain.

Firefly algorithm (FA) is one of the newly developed nature inspired optimisation algorithm, inspired by the flashing behaviour of fireflies that a firefly tends to be attracted towards other fireflies with higher brightness. Thus FA has two advantages: local attractions and automatic regrouping. Based on these good properties, a novel image watermarking method based on FA in discrete wavelet transform (DWT)‐QR transform domain is proposed in this study. Structural similarity index measure and bit error rate are used in the objective function to trade‐off invisibility and robustness. The experiment results show that the proposed image watermarking method not only meet the need of invisibility, but also has better or comparable robustness as compared with some related methods.