Signal Reconstruction And Discrete Algorithm In Fractional Domain

Nonstationary signal processing based on the conventional Fourier transform hascertain limitations. As the potent tool to analyze the nonstationary signals, the fractionalFourier transform (FRFT) and the linear canonical transform (LCT) have and attractedmore attentions in signal processing community and become an important issue recently.The sampling, reconstruction and discretization of the FRFT and the LCT are the basis andkey point for practical applications in modern signal processing. Therefore, this paper focuson the research of signal reconstruction and fractional periodic matrices. For the uniformlysampled signals, the reconstruction of a finite discrete-time signal from the partialinformation of its discrete LCT and some known samples under some restrictions ispresented. Its advantage is that the number of known samples can be less than half of thetotal number of the finite discrete-time signal when the sampling rate is larger than theNyquist rate. For the nonuniformly sampled signals, the fractional spectrum of two types ofperiodic nonuniformly sampled signal are given, then reconstruction of uniformly sampledsignals from nonuniform short samples in fractional Fourier domain is proposed. Itsadvantage is that some high-frequency contents are prevented in perfect reconstruction.Then based on the linear summation of fractional matrices, we can generate variousfractional versions by the order partition in the circle and establish the double random phaseencoding used by the new fractional matrices to multi-order forms, which can besignificantly improved data security. The main content of this paper includes:Firstly, the basic theory of uniformly sampling in fractional domain is reviewed,including the definition and main properties of FRFT and LCT and their correspondingdiscrete algorithm, the fractional spectrum of uniformly sampled signals.Secondly, signal reconstruction from partial information of discrete linear canonicaltransform is presented. A finite discrete-time signal can be reconstructed from its discreteLCT magnitude or phase and some known samples under some restrictions.Thirdly, the fractional spectrum representation of periodic nonuniformly sampledsignals is researched. Then the reconstruction of uniformly sampled sequence from randomnonuniformly sampled sequence in the fractional Fourier domain is proposed.Finally, the fractionalization approach for periodic matrix is investigated. Then the double random phase encoding using2D fractional kernel matrix generation by periodicmatrix sequence is presented.