Compressed Sensing Of Analog Signals In Function Spaces Based On Linear Canonical Transform

The classical Fourier transform is a basic tool for signal and system analysis,which plays a vital role in stationary signal processing and time invariant system analysis.However,for non-stationary signals and time-varying systems,Fourier analysis could only get approximate results,which are not necessarily optimal.Based on the Fourier analysis,a series of new transforms emerged,which greatly enriched the theoretical system of classical signal processing.Among them,the linear canonical transform(LCT)as the generalized form of Fourier transform,provides a new perspective for signal analysis and processing.In recent years,it has attracted much attention in signal processsing,and has been widely used in filter design,radar system analysis,time-frequency analysis,encryption and communication.As a fundamental problem of signal processing,sampling theory is the basis for the practical application of linear canonical transform.However,most of the existing LCT sampling theories are inherited from the Shannon-Nyquist sampling frame,and it has the constraint of bottleneck in reducing the sampling rate.Inspired by the compressed sensing in the field of signal processing in recent years.There is a new method to acquire the information contained in signal,which makes full use of the sparse characteristics of signals,and can effectively overcome this bottleneck.In view of this,from the perspective of function space,combined with the sparse characteristics of signals under the linear canonical transform.This thesis constructs the compressed sensing theory of analog signals,and the main results can be summarized as follows:Firstly,the sparse multitone and sparse multiband signal models are established from the perspective of bandlimited function space in LCT domain.Based on this,a compressed sampling and reconstruction method of bandlimited analog signal in linear canonical transform domain is proposed,and theoretical results are verified by numerical analysis and simulations.Secondly,the uncertainty principle of signal sparse representation under the linear canonical transform is constructed.The minimal coherence and tightness of uncertainty relation are also be analyzed.Then,we disucussed the uniqueness of sparse representation in the joint dictionary,which consists of two orthonormal bases.It made a theoretical foundation for establishing the compressed sensing of analog signals in the general function spaces of linear canonical transform.Finally,starting from the general function space of linear canonical transform.Based on the sparse characteristic of signals in the function space and compressed sensing,we proposed a new method of compressed sampling and reconstruction for analog signal without constrianed condition of bandlimited in LCT domain.The concrete numerical analysis is constructed to further illustrate the mechanism of proposed method,and the theoretical analysis also be verified by simulations.