Sparse Linear Canonical Transform Theory And Its Application

Most of the actual signals are nonlinear and non-stationary signals.For example,most of the signals produced in complex industrial processes and radar observation signals have non-stationary characteristics.Therefore,such signals become the main research object of modern signal processing.The theory of non-stationary signal processing has rapidly attracted the attention of many scholars.In recent years,it has developed rapidly and become one of the researching hot-spots in the field of signal and information processing.At present,many engineering problems can be transformed into problems that the non-stationary signals are processed in the transform domain.Therefore,as an orthogonal transform based on Chirp basis function decomposition,the discrete linear canonical transform(DLCT)is a kind of signal processing tool that is very suitable for non-stationary signal analysis.However,the three free parameters of the transform also lead to higher computational complexity.In the case of large amount of data,the computing disadvantage is more prominent,especially in the modern signal processing era where machine learning and the new technology of big data prevail,and an efficient fast algorithm is more needed to promote the wide application of DLCT.On the other hand,for these non-stationary signals,sometimes we want to know not only the frequency information,but also the change with time,so a single analysis in the time domain or the frequency domain fails to get good results.In recent years,the LCT,as a generalized form of Fourier transform(FT),has been introduced into the field of signal processing.Because it is a global transform,it cannot reveal the time-varying relationship of the frequency of non-stationary signals just like FT.With the emergence of short-time linear canonical transform(STLCT),a new possibility is provided for time-frequency analysis of non-stationary signals.In-depth research on the theory and application of this transform is of great significance for the application of this time-frequency analysis tool to practical engineering in the future.In this paper,we mainly study the fast algorithm of sparse discrete linear canonical transform(SDLCT).Then the definition of STLCT is taken as the starting point to build a theoretical framework including its properties and the inverse transform as well as the discretization method.On this basis,combining with the idea of sparse Fourier transform,the shorttime sparse linear canonical transform(ST-SLCT)is obtained,which further reduces the computational complexity of STLCT.The specific research work is as follows:The first part aims to propose SDLCT.Most practical signals in nature exhibit sparsity in linear canonical domain.By exploiting this property,SDLCT is proposed based on Pei's algorithm to substantially reduce the computational complexity when dealing with large data sets that are sparse in the linear canonical domain.A numerical example is given to realize the permutation operation of the algorithm,and the adjacent Fourier coefficients are separated well.Then,on the basis of the proposed algorithm,its applications in pulse compression of LFM signal and moving target detection of SAR are studied,and the pulse compression algorithm of sparse linear canonical matched filter and moving target detection algorithm of SDLCT are proposed.Thus,the sparsity of signal in the linear canonical domain is utilized to concentrate the signal energy and improve the compression performance and the detection probability.In the second part,the study of the property and inverse transform of STLCT are presented in this paper.We introduce the definition of STLCT,which is obtained by extending the physical meaning of the traditional short-time Fourier transform(STFT)to the linear canonical domain,and can be interpreted as a set of the linear canonical domain filters.Firstly,based on this definition,some basic properties and inverse transform of the SDLCT are given.Secondly,we derive the uncertainty principle and convolution theorem of STLCT.Finally,the time-canonical-frequency of SDLCT are analyzed by using the time-frequency window analysis idea of STFT,and the discrete form as well as filter interpretation are given to help us to study the transform as deeply as possible.At the same time,we study practical application of STLCT in time-frequency analysis of chirp signals.The effectiveness of the transform is proved by simulation.In the third part,ST-SLCT algorithm is proposed.Firstly,the original input signal is multiplied by the linear frequency modulation function to obtain the input signal in the short-time sparse Fourier transform stage.Secondly,the Gaussian window function is used to intercept the input signal.In order to reduce the amount of data,a two-dimensional complex function set is generated and a short-time sparse Fourier transform stage is formed by permutation operation,down-sampling as well as location and estimation loops of the intercepted local signals.Finally,another chirp function is used to modulate the result into the short-time linear canonical domain.On the basis of the proposed algorithm,the relationship between the algorithm and the parameters of LFM signal is derived,and the simulation results with or without noise are given.In order to further demonstrate the effectiveness of the algorithm,we also analyze the resolution of the variable frequency signal,BPSK and QFM.