Research And Application Of The Fractional Convolution

As a generalization of the classical Fourier transform(FT), the fractional Fourier transform (FrFT) which adds a degree of freedom parameter-rotation angle and retains all the features of the classical Fourier and novel advantages moreover, is very propitious for processing of the non-stationary signals. It is widely used in scientific research and engineering in all aspects. The study of discrete fractional Fourier transform has formed a big climax, with the rapid development applications of the digital signal processing. Many scholars have derived different types of discrete fractional Fourier transform(DFrFT) and sampling theory in fractional Fourier transform domain. A novel algorithm based on fractional Fourier circular convolution theorem is proposed to deal with the fractional linear convolution. The main contributions and innovations of the dissertation are:1. The sampling theorem of the fractional Fourier transform and the hidden chirp periodicity of the discrete fractional Fourier transform is explained, the fractional circular convolution theorem which is the important foundation of the proposed algorithm. It is described in detail. The novel algorithm proposed in this paper is able to deal with the fractional linear convolution by making segment on the longer sequence.2. A novel algorithm based on fractional Fourier circular convolution theorem is proposed to deal with the fractional linear convolution, which is in accordance with the hidden periodicity of the discrete fractional Fourier transform. In the pre-processing, it applies the methods of overlap-save and overlap-add to make segment on the longer sequence. The algorithm is able to overcome the disadvantages of traditional fractional circular convolution theorem, which only can be used to calculate the convolution of two sequences with the similar length. The problem of setting the sequence length is discussed in detail. Simulation results show the effectiveness of this algorithm.3. The filtering algorithm of the linear frequency modulation (LFM) signal in the fractional Fourier domain is explored, it described relationship between the realization of the sweep filter and the filter in fractional Fourier domain. Based on the correctness of the algorithm, the application of fractional convolution algorithm on (LFM) signal filtering is proposed. The relevant principle is discussed. Analyses and simulation results also presented which show that the algorithm has a broader practical value.