Research Of The Fourier Transform Algorithm In Algebraic Signal Processing

In recent decades,algebraic signal processing has been a new and linear method of signal processing.In 1996,Paul A.Fuhrmann presented a polynomial representation of linear algebra and definitions of algebra and module,which provided the theoretical basis for the future study of algebraic signal processing.In 2008,Markus Puschel and Jose M.F.Moura proposed a theoretical framework of algebraic signal processing,established a one-dimensional time signal model and a one-dimensional space signal model.they also established related concepts of filters,convolution,spectral decomposition,Fourier transform and frequency response in algebraic signal processing.In algebraic signal processing,various shift operators can be defined.The corresponding signal models based on these operators are established and different Fourier transforms are obtained.In this paper,two new shift operators are defined and then two signal models are established,The main research work is as follows(1)A new shift operator is defined,and then a one-dimensional infinite signal model and a one-dimensional finite signal model based on Hermite polynomials are established,the Fourier transform algorithm of the model is also obtained.By using this algorithm a simulation experiment of the data compression for the electrocardiogram signals of recording 100 in the MIT-BIH database is given.The Percent Root mean square Different(PRD)of the reconstructed signal compared to the original signal is smaller.The experimental results show that the Fourier transform algorithm of the signal model based on Hermite polynomials is effective for the compression of ECG signals.(2)Another shift operator is defined,and then a one-dimensional infinite signal model and a one-dimensional finite signal model based on Legendre polynomials are established,the Fourier transform algorithm of the models is also derived.By using this algorithm the frequency information of the cosine signals and sinusoidal signals containing noise is extracted.The experimental results show that when the appropriate threshold is selected,the frequency information of the cosine signals and sinusoidal signals containing noise can be accurately extracted by using the Fourier transform algorithm which is derived from the algebraic signal model based on Legendre polynomials.