Sampling Problems In Quasi Shift-invariant Spaces

With the development of digital acquisition and reconstruction technology of signals,the research on sampling problems has been widely used in communications,radar and other electronic systems.In recent years,Shannon’s sampling theorem has been gradually generalized,from the band-limited space to shift-invariant spaces,and then to the more general quasi shift-invariant spaces,and sampling methods have also been generalized from point-by-point sampling to average sampling,adaptive sampling and random sampling.In view of this,this paper conducts in-depth research on adaptive sampling,random sampling of quasi shift-invariant signals and average sampling of quasi shift-invariant random signals.The main results are as follows:1.Based on the time encoding machines,the adaptive sampling and reconstruction of signals in quasi shift-invariant space are studied,and the reconstruction algorithm of quasi shift-invariant signals is mainly studied.First,when the generator and output sequence meet certain attenuation conditions,an accurate reconstruction algorithm with exponential convergence is established based on the two adaptive sampling schemes of Crossing and Integrate-and-Fire.Second,an IF sampler that can generate finite adaptive average samples is discussed,and an approximate reconstruction algorithm for convergence speed linearly dependent sampler threshold parameters is established.2.His research focuses on the average sampling and reconstruction of quasi shift-invariant stochastic processes.Combining stochastic processes in quasi shift-invariant spaces,the quasi shift-invariant stochastic processes are introduced.First,for the quasi shift-invariant stochastic process,two average sampling theorems are established in the sense of mean square convergence.Second,the truncation errors of the corresponding two average sampling schemes are estimated.3.The main research focuses on the random sampling stability of multi-window quasi shift-invariant signals.Under the condition that the generator satisfies certain decay property,the sampling stability of the energy concentration signal in the quasi shift-invariant space is established.In addition,with the help of the relative separation of the translation set and the spectral analysis of the positive-semidefinite Hilbert-Schmidt localization operator,it is proved that when there are enough sampling points,the sampling stability of the energy concentration signal is established with high probability.The sampling model and reconstruction method in quasi shift-invariant spaces studied in this paper cover adaptive sampling,average sampling,random sampling,and their application in electronic systems,which have certain practical reference value for further research on sampling problems in quasi shift-invariant spaces.