Research On Real Fourier Transform Algorithm Based On 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.However,with the widespread application of signal processing,the simple Fourier transform has large errors in signal processing.To solve this problem,this thesis derives an algebraic signal processing algorithm to process and reconstruct ultrasonic signals by the theory of orthogonal polynomials.Combined with Laguerre polynomials,this thesis defines a new translation operation and establish a signal model and construct a real Fourier transform algorithm for the signal model.The main contents include the following points:Firstly,on the basis of algebraic signal processing,a new translation operation is defined.Combined with Laguerre polynomials,an infinite signal model is established.According to the boundary extension,the finite signal model is established and the real Fourier transform algorithm is derived.Under this model,the real Fourier transform matrix and the inverse transform matrix of the finite signal can be obtained.Secondly,using the real Fourier transform algorithm of the finite signal to collect and reconstruct the ultrasonic signal,a new reconstructed signal is obtained and the mean square error analysis of the experimental results is carried out.The error is 7.36%.The analysis results show that,compared with other algorithms,the algorithm proposed in this paper has a better effect on signal reconstruction.Two algorithms are used to detect the sine and cosine signals in order to analyze the advantages and disadvantages of the real Fourier transform algorithm constructed in this article and the traditional real Fourier transform algorithm in frequency acquisition.The experimental results show that within a certain range of noise intensity,both algorithms can obtain accurate frequency information of the signal.The real Fourier transform algorithm constructed in this paper and the traditional real Fourier transform algorithm will produce certain errors in the frequency detection of the signal when the noise intensity is large.However,the real Fourier transform algorithm constructed in this paper has a smaller detection error on the signal frequency.