Research On Two-dimensional Fourier Transform Algorithm In Algebraic Signal Processing

In recently,the discrete Fourier transform used in digital signal processing can be represented by a decomposition matrix of canonical modules under the theoretical framework of algebra.This feature gives us a deeper understanding of the discrete Fourier transform,and uses algebraic methods to derive most of the known algorithms,and the problem of finding algorithms for different transforms has always been a hot topic in the field of signal processing.And with the gradual deepening of research on algebraic signal processing,algebraic signal processing is not limited to the study of one-dimensional signals,and multi-dimensional signals,as one of the important expressions of information,are also very important in the field of signal processing.Under the premise of one-dimensional signal processing,this article come up with the concept of two-dimensional signal model.The main research contents are summarized as follows:(1)Summarized the classic one-dimensional signal model and the Fourier transform in the model,and made a systematic discussion of the concepts in the model.(2)Based on the one-dimensional Fourier transform,a new two-dimensional translation operator is defined,and a two-dimensional(separable)signal model is constructed.For finitelength spatial signals,a finite signal model is established by constructing different boundary conditions,and the two-dimensional Fourier transform in the model is derived.Finally,through simulation experiments,it is verified that the deduced transform algorithm has a better image energy concentration effect than the classic one-dimensional Fourier transform.(3)Based on the one-dimensional cosine transform,a new two-dimensional translation operator is defined,a new two-dimensional infinite signal model and a two-dimensional finite signal model are established,and the two-dimensional cosine transform in the model is derived.The transformation algorithm is used to process the noisy image,and it is verified through experimental comparison that the denoising effect of the two-dimensional cosine transform on the image is better than that of the classic Fourier transform.