The multi-channel sampling theorem has flourished as one of the nicest alternatives to the classical Shanon's sampling theorem which relies on non-uniform or multi-channel data acquisition. Although, a primary attempt to study the multi-channel sampling theorem for the multi-dimensional linear canonical transform (LCT) was made by D. Wei and Y. Li (2014) [16], however, it lacks the inherent multi-dimensional nature in the sense that the associated kernel is a product of N-copies of the usual one dimensional kernel, restricting the degrees of freedom only to three. The goal of this paper is to fill this gap by studying the lattice-based multi-channel sampling theorem for the multi-dimensional LCT with $N(2N+1)$ degrees of freedom. The idea is accomplished in two steps: firstly, an elegant convolution structure is presented for constructing the multiplicative filters; secondly, the sampling lattices are constructed via general separable and non-separable matrices which often provide more flexibility and better performance in the context of multi-dimensional signal analysis. In the sequel, the Shanon's sampling theorem for the multi-dimensional LCT is deduced as a particular case of the proposed multi-channel sampling theorem. Finally, some potential applications including signal reconstruction and image super-resolution are presented to validate the obtained results.

多通道采样定理已经成为经典香农采样定理的最佳替代方案之一，该定理依赖于非均匀或多通道数据采集。尽管 D. Wei 和 Y. Li (2014) [16] 首次尝试研究多维线性正则变换 (LCT) 的多通道采样定理，但是，它缺乏固有的多维从某种意义上说，相关内核是通常的一维内核的N副本的乘积，将自由度限制为三个。本文的目标是通过研究多维 LCT 的基于格的多通道采样定理来填补这一空白$N(2N+1)$自由程度。这个想法分两步完成：首先，提出了一个优雅的卷积结构来构造乘法滤波器；其次，采样格是通过一般的可分离和不可分离矩阵构建的，这通常在多维信号分析的背景下提供更大的灵活性和更好的性能。在后续中，多维 LCT 的香农采样定理被推导出为所提出的多通道采样定理的一个特例。最后，提出了一些潜在的应用，包括信号重建和图像超分辨率，以验证所获得的结果。