Research On Fractional Convolution And Its Application

With the rapid development of signal processing,the diversity of signal forms,new signal processing tools such as integral transformation constantly emerge,the traditional Fourier transform can no longer meet the needs of signal processing,so further research on the nature and application of these integral transformations has important theoretical significance and practical application value.Therefore,in this paper,we develop a systematic thorough study on fractional positive cosine series and fractional Hartley transform.The main contributions of this paper can be stated as follows:1.Firstly,two novel discrete convolution for the fractional sine series and the fractional cosine series are proposed.Secondly,three general convolution operations and four convolution operations with weight function of fractional cosine series are given,and then the relevant properties of the convolution operations are studied.Then,we use the theory of convolution correlation to obtain the convolution theorem for fractional sine and cosine series,respectively.Finally,discuss the relations between fractional sine and cosine convolution operations,and the Young's inequality for the correlation of fractional cosine series are derived.2.The fractional Hartley transform is studied based on the Hartley transform.Firstly,the fractional Hartley transform is defined.Secondly,several new types of convolution operations for fractional Hartley transform are proposed.Finally,the corresponding convolution theorem is derived from the theory related to the convolution operation of the fractional Hartley transform.3.Based on the definition of fractional cosine series and the convolution operation and convolution theorem.Firstly,the application of fractional cosine series the in the convolution equation is proposed.Secondly,we study the problem of convolution operation in several kinds of convolution equation and the correlation theory of convolution theorem.