Research On Non-stationary Signal Sampling And Spectral Analysis Based On Linear Canonical Transform

Signal sampling and spectrum analysis are the main research contents of the modern signal processing.In the traditional signal processing system,the theory of sampling and spectrum analysis based on Fourier transform is a powerful tool for the analysis of stationary signals.With the continuous expansion of the application fields of signal processing,the non-stationary signal has become the main research object of modern signal processing.However,the theory of sampling and spectrum analysis based on Fourier transform is not suitable for non-stationary signal.Simultaneously,the linear canonical transform(LCT)as a new signal analysis tool,which has become an important tool for non-stationary signal analysis.Due to its three free parameters,the LCT is more flexibility and has unique advantages in the analysis of non-stationary signals.However,at present,the basic theories of sampling and spectral analysis of non-stationary signal in the LCT domain is not completed yet.Hence,it is important and practical useful to study the sampling and spectrum analysis theory for non-stationary signal based on the LCT.Based on the advantages of the LCT in non-stationary signal processing,this thesis focuses on the analysis of decimation and interpolation in the LCT domain,the analysis of periodic nonuniformly sampled signal in the LCT domain,the theory of signal sampling and spectrum analysis based on linear canonical Hilbert transform and the analysis of random signals in the LCT domain.The main contents of this thesis are as follows:(1)The analysis of decimation and interpolation in the LCT domain has been studied.Based on the definition of equivalent LCT,the definition of equivalent filter in the LCT domain have been given at first.Then,by applying the definition,the direct structure and implementation structure for decimator and interpolator in the LCT domain have been proposed.Subsequently,the polyphase networks for decimator and interpolator in the LCT domain have been studied based on the polyphase structure of the equivalent filter.Finally,the LCT differential filters have been constructed,and the perfect reconstruction expressions for the LCT differential filters have been presented,which can reconstruct the signal from the LCT differential sampling.The proposed results show that the theory of decimation and interpolation in the LCT domain canreduce the storage and computation.It also provides the support for the multirate theory of non-stationary signals.(2)The analysis of periodic nonuniformly sampled signal in the LCT domain have been studied.First,the sampling theorem and spectral analysis theory of periodic nonuniform sampling signals in LCT domain are introduced based on the periodic nonunifornly sampled model.Then,due to the offset linear canonical transform(OLCT)is a further supplement and improvement of the LCT,the relationship between the digital spectrum of the periodic nonuniformly sampled signal and the spectrum of the original signal is presented.Subsequently,the spectrum analysis of periodic nonuniformly sampled chirp signal of is proposed,and a method to reconstruct the digital spectrum from periodic nonuniformly sampled signals in OLCT domain has been studied.Finally,the simulation results show the advantage of the above results in the analysis of periodic nonuniformly sampled non-stationary signals.(3)The theory of signal sampling and spectrum analysis based on linear canonical Hilbert transform(LCHT)has been studied.First,the uncertainty principles for one and two dimensional LCHT are proposed based on the definition and properties of the LCHT.Then,according to the definition of the offset linear canonical Hilbert transform(OLCHT),the properties and the generalized Bedrosian theorem of the OLCHT have been studied.In addition,a single sideband communication system based on the OLCHT has been proposed,which can save the bandwidth and modulate the signal.Finally,the simulation results are given to verify the effective of the above results in the non-stationary signal sampling and spectral analysis.(4)The analysis of sampling theory and power spectral density function for random signals in the LCT domain have been studied.Firstly,several nonuniform sampling theorems of random signals in the LCT domain are presented based on the definition of the LCT power spectral density function and the LCT correlation function.Next,the definitions of the OLCT power spectral density function and correlation function are given,and the power spectral density relations for the OLCT filter are derived.Subsequently,the multichannel sampling theorem for random signals in the OLCT domain is derived.In addition,some sampling theorems for random signals associated with the OLCT are also presented by designing different OLCT filters.Finally,the simulation results are presented to show the advantage and effectiveness of the theorems.Based on the above theoretical analysis and simulation verification,the main research work and achievements of this thesis are summarized.Meanwhile,the further research work is also forecasted.