Sampling Analysis For The Random Signals And Deterministic Signals In The Mixed Lebesgue Spaces

Sampling theories are the basis of modern communication theory and the most fundamental tools for signal processing.Sampling problem mainly consists of the following two aspects.Firstly,finding out the conditions which ensure the given discrete sampling set satisfies the sampling stability.Secondly,designing an efficient and fast reconstruction algorithm to realize the reconstruction of signals.Up to date,the findings of the sampling problems for signals are mainly focused on the different subspaces of the classical Lebesgue space,such as bandlimited space,shift invariant subspace,reproducing kernel subspace and so on.It is well known that the larger space could simulate the more signals and the corresponding sampling conclusions could be applied to more signal models.However,there also exist some signals which are composed by some independent variables with different properties in the real applications.In order to investigate the sampling problems for those signals,the mixed Lebesgue space,which could realize the separate integrability of each variable,is more suitable to model those signals than the classical Lebesgue space,which makes it compulsory for all of the variables to have the same properties.This paper aims to generalize the sampling theories for signals in the classical Lebesgue space to the mixed Lebesgue space,so that the relevant sampling theories could provide an essential academic support for dealing well with some more complex sampling problems.Thus,we investigate the sampling problems for signals in shift invariant subspace,multiply generated shift invariant subspace and reproducing kernel subspace of mixed Lebesgue space and weighted mixed Lebesgue space and obtain the following five sampling results.First,due to the limitation of the sampling devices,the obtained sampling value in reality is not the exact value of signal at each sampling point but is the local average value near the corresponding sampling location.Thus,we study the average sampling problems for signals in shift invariant subspace of mixed Lebesgue space.Firstly,we establish the sampling stability for two kinds of average sampling functionals.Secondly,using two kinds of iterative reconstruction algorithms with exponential convergence,we realize the recovering of signals.Finally,based on three forms of noises,we provide the corresponding error estimations respectively.Second,inspired by the properties of the moderated weight functions which could control the growth and decaying properties of signals,we consider making use of the weighted mixed Lebesgue space to simulate those signals,which are non-decaying or even grow at infinity.Similarly,based on two kinds of average sampling functionals we investigate the average sampling problems for signals in shift invariant subspace of weighted mixed Lebesgue space.In addition to establishing the corresponding sampling stability,we also use the iterative reconstruction algorithms with exponential convergence to realize the recovering of the corresponding signals.Different from the previous average sampling results for signals in shift invariant subspace of mixed Lebesgue space,the obtained average sampling results for signals in the corresponding subspace of weighted mixed Lebesgue space could be applied to more signal models.Third,the generator is usually required to belong to mixed Wiener amalgam space or weighted Wiener amalgam space when investigating the sampling problems for signals in shift invariant subspace of mixed Lebesgue space or weighted mixed Lebesgue space.However,this requirement which does not depend on the mixed norm index p,q greatly restricts the admissibility condition on the generator.For relaxing the corresponding restricted condition on the generator,we will investigate the average sampling problems for the non-decaying signals in shift invariant subspace of weighted mixed Lebesgue space based on the assumption that the generator belongs to the weighted hybrid-norm space with mixed form.Based on the given average sampling functional,we establish the corresponding sampling stability and realize the recovering of signals utilizing the corresponding iterative reconstruction algorithm.Furthermore,we also provide the estimations about the upper bound of the convergence rate and the error of the signal reconstruction which is caused by the noise.Fourth,many researches about the sampling problems for signals in mixed Lebesgue space and weighted mixed Lebesgue space are all obtained based on a pre-given sampling set.In fact,the random sampling problems for signals,which are based on the sampling set being constituted by the random sampling points satisfying a certain distribution,also attracts the researchers’ attention.Thus,we study the random convolution sampling stability problem for signals in multiply generated shift invariant subspace of weighted mixed Lebesgue space and prove that sampling stability for some subset of the defined multiply generated shift invariant subspace holds with high probability if the random sampling size is large enough.Fifth,due to the influence of various random noises,some random characteristics are inherent in physical signals.Compared with the deterministic signal models,the random process and random field are more suitable to model those signals.Thus,we investigate the average sampling problems for reproducing kernel homogeneous random fields in mixed Lebesgue space.First,we provide a uniformly convergence sampling result for the deterministic signals in reproducing kernel subspace of mixed Lebesgue space.Then,we prove that the quadratic sum of the corresponding reconstructed functionals is uniformly bounded based on the deterministic sampling result.Finally,we also provide a mean square convergence result for reproducing kernel homogeneous random fields in mixed Lebesgue space.