The Chirp-Fourier transform is one of the most important tools of the modern signal processing. It has been widely used in the fields of ultrasound imaging, parameter estimation, and so on. The key to its application lies in the sampling and fast algorithms. In practical applications, nonuniform sampling can be caused by sampling equipment and other reasons. For the nonuniform sampling, we utilized function approximation and interpolation theory to construct different approximation forms of Chirp-Fourier transform kernel function, and proposed three fast nonuniform Chirp-Fourier transform algorithms. By analyzing the approximation error and the computational complexity of these algorithms, the effectiveness of the proposed algorithms was proved.

The investigations of the discrete and fast linear canonical transform (LCT) are becoming one of the hottest research topics in modern signal processing and optics. Among them, the fast calculation of LCT for nonuniform data is one of the key problems. In this paper, two novel fast algorithms based on low-rank approximation are presented. First, we propose two methods for approximate nonuniform time-domain sampling with uniform sampling. Second, we utilize a low rank matrix to approximate the nonuniform LCT kernel, combined with the exponential function and Taylor series. Then, the fast algorithms for nonuniform sampling in the time domain are developed, which cost $K$ FFTs. Finally, we extend the fast algorithm to nonuniform LCT in the frequency and transform domains. The effectiveness of the proposed algorithm is verified by simulations.