Research On Signal Compressive Sensing Based On Fractional Fourier Transform

Compressive sensing is a novel theory of sampling and compression,which samples signal directly at lower rates under the different condition from Nyquist theorem.And then compressive sensing recovers the original signal from the sampled values by reconstruction algorithm.Owing to the sampling theorem is restricted by its large bandwidth and high sampling rate requirements in dealing with linear frequency modulation signal.As well as the existing sparse representation algorithms cannot sparse such signal well.In this paper,by exploiting the character of fractional Fourier transform is especially suitable for processing such signal,we propose to transform linear frequency modulation of highly dense in time domain into fraction Fourier domain to form a sparse representation.We also design the sparse representation dictionary and measurement matrix,and analyze the signal recover algorithm.Therefrom,we systematically study the compressive sensing scheme and its feasibility of processing linear frequency modulation;the main work is as follows:Firstly,the orthogonal dictionary of centered discrete fractional Fourier transform based on singular value decomposition is proposed.Compared with combined dictionary,frame dictionary and learning dictionary with very high computational complexity,the dictionary is established by orthogonal transform basis,which has the advantages of simple structure and comparatively faster achievement.Moreover,compared with the simple orthogonal dictionary constructed by conventional Fourier transform or wavelet transform,this paper presents the orthogonal dictionary has higher flexibility of fractional order and better sparse effect.Secondly,in order to make the measurement matrix to save almost all of the useful information of original signal in the sampling process,and to recover the original signal from the measured signal with high probability,we construct the measurement matrix based on chaotic system with strong stability in this paper.Finally,a new algorithm of variable number matching pursuit is proposed.In this algorithm,the number of atoms selected at each iteration is variable,which overcomes the shortcoming of the number of atoms in the traditional matching pursuit algorithm is too small or too much and the number of atoms selected at each iteration is fixed.The experimental results show that the improved reconstruction algorithm not only greatly reduces the reconstruction time,but also improves the quality of signal reconstruction.