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• In existing multiplexed spectral imaging methods based on matrix operations (especially HTS), the reconstructed results are similar to miscalculations in the case of under-sampling. In contrast, our FCTS based on frequency domain calculation discards only the high-frequency components and has higher coding efficiency and physical interpretability. • Compared with existing snapshot systems (especially CASSI), FCTS has no limitation on the number of dispersion channels and can use high dispersion elements such as gratings according to practical needs. The experimental system in this paper uses an Amici prism with a spectral resolution of about 4 nm, which is tested to be lower than 1 nm if a 300-line grating is used, and has an unparalleled advantage in spectral resolution for snapshot systems. • Real systems and simulations show that the FCTS frequency domain calculation method is less affected by optical misalignment, diffraction and distortion in optical systems. It is more robust to optical distortions. • Because of the Fourier transform process of FCTS under-sampling we only take the low frequency part of the Fourier spectrum. So, there are some high frequency details missing in the reconstruction. We can also use the common super resolution reconstruction or image enhancement methods to complete the image details to improve the image quality. We consider using Fourier coded aperture transform to solve the acquisition and reconstruction problem of three-dimensional (3-D) hyperspectral imaging. In recent years, Coded aperture snapshot spectral imaging (CASSI) system has received increasing attention. However, reconstruction errors are unavoidable due to its ill-posed problems and the number of dispersion channels is limited. The Hadamard transform spectral (HTS) imaging method has no limitation on the number of dispersion channels, but it usually requires a long data acquisition time. By exploring the process of coded spectral imaging, in this paper, we propose Fourier coded aperture transform spectral imaging (FCTS). We compared the performance of HTS and FCTS by theoretical analysis and experiments. The results show that FCTS has better concentrated energy than HTS on natural hyperspectral data. Compared with the Hadamard transform method, the Fourier basis mode has better physical correlation with the natural image, and the reconstruction quality of FCTS is better than that of HTS in the case of under-sampling. [ABSTRACT FROM AUTHOR]