Sampling In Shift-invariant Spaces Research Articles

Reducing and invariant subspaces under two commuting shift operators

In this article, we characterize reducing and invariant subspaces of the space of square integrable functions defined in the unit circle and having values in some Hardy space with multiplicity. We consider subspaces that reduce the bilateral shift and at the same time are invariant under the unilateral shift acting locally. We also study subspaces that reduce both operators. The conditions obtained are of the type of the ones in Helson and Beurling-Lax-Halmos theorems on characterizations of the invariance for the bilateral and unilateral shift. The motivations for our study were inspired by recent results on Dynamical Sampling in shift-invariant spaces.

Injectivity of Gabor phase retrieval from lattice measurements

We establish novel uniqueness results for the Gabor phase retrieval problem: if $\mathcal{G} : L^2(\mathbb{R}) \to L^2(\mathbb{R}^2)$ denotes the Gabor transform then every $f \in L^4[-\tfrac{c}{2},\tfrac{c}{2}]$ is determined up to a global phase by the values $|\mathcal{G}f(x,\omega)|$ where $(x,\omega)$ are points on the lattice $b^{-1}\mathbb{Z} \times (2c)^{-1}\mathbb{Z}$ and $b>0$ is an arbitrary positive constant. This for the first time shows that compactly-supported, complex-valued functions can be uniquely reconstructed from lattice samples of their spectrogram. Moreover, by making use of recent developments related to sampling in shift-invariant spaces by Gr\"ochenig, Romero and St\"ockler, we prove analogous uniqueness results for functions in shift-invariant spaces with Gaussian generator. Generalizations to nonuniform sampling are also presented. Finally, we compare our results to the situation where the considered signals are assumed to be real-valued.

Non-ideal sampling in shift-invariant spaces associated with quadratic-phase Fourier transforms

Non-ideal sampling has nourished as one of the most attractive alternatives to classical sampling, which relies on shift-invariant spaces. The present study focuses on investigating the non-ideal sampling in shift-invariant spaces associated with the quadratic-phase Fourier transforms. The primary aim is to formulate novel convolution structures in quadratic-phase Fourier domains and invoke such structures to develop the generalized shift-invariant spaces. Moreover, we present the non-ideal sampling procedure via generalised shift-invariant spaces in the quadratic-phase Fourier domains by employing the proposed generalised convolutions.

Compressed Sampling in Shift-Invariant Spaces Associated With FrFT

Compressed Sampling in Shift-Invariant Spaces Associated With FrFT

Recovery of Compressed Sampling in Shift-Invariant Spaces Base on L1 Norm

Compressed sampling in shift-invariant spaces (SI) is an effective method for sampling of sparse signals. But, reconstruction of compressed sampling may be unstable. In the paper, the possibility of stable reconstruction under a sufficient sparsity is proven. Further, we consider the situation where the minimal L1 norm is used to recover sparse signals from the noisy data. The result shows that they are stable. Finally, we show that the minimal L1 norm through the simulation, and explain the applicability of our algorithm to sampling systems.

Phaseless Sampling and Reconstruction of Real-Valued Signals in Shift-Invariant Spaces

Sampling in shift-invariant spaces is a realistic model for signals with smooth spectrum. In this paper, we consider phaseless sampling and reconstruction of real-valued signals in a shift-invariant space from their magnitude measurements on the whole Euclidean space and from their phaseless samples taken on a discrete set with finite sampling density. We introduce an undirected graph to a signal and use connectivity of the graph to characterize whether the signal can be determined, up to a sign, from its magnitude measurements on the whole Euclidean space. Under the local complement property assumption on a shift-invariant space, we find a discrete set with finite sampling density such that signals in the shift-invariant space, that are determined from their magnitude measurements on the whole Euclidean space, can be reconstructed in a stable way from their phaseless samples taken on that discrete set. In this paper, we also propose a reconstruction algorithm which provides a suboptimal approximation to the original signal when its noisy phaseless samples are available only. Finally, numerical simulations are performed to demonstrate the robust reconstruction of box spline signals from their noisy phaseless samples.

Sampling theorems for shift-invariant spaces, Gabor frames, and totally positive functions

We study nonuniform sampling in shift-invariant spaces and the construction of Gabor frames with respect to the class of totally positive functions whose Fourier transform factors as hat{g}(xi )= prod _{j=1}^n (1+2pi idelta _jxi )^{-1} , e^{-c xi ^2} for delta _1,ldots ,delta _nin mathbb {R}, c >0 (in which case g is called totally positive of Gaussian type). In analogy to Beurling’s sampling theorem for the Paley–Wiener space of entire functions, we prove that every separated set with lower Beurling density >1 is a sampling set for the shift-invariant space generated by such a g. In view of the known necessary density conditions, this result is optimal and validates the heuristic reasonings in the engineering literature. Using a subtle connection between sampling in shift-invariant spaces and the theory of Gabor frames, we show that the set of phase-space shifts of g with respect to a rectangular lattice alpha mathbb {Z}times beta mathbb {Z} forms a frame, if and only if alpha beta <1. This solves an open problem going back to Daubechies in 1990 for the class of totally positive functions of Gaussian type. The proof strategy involves the connection between sampling in shift-invariant spaces and Gabor frames, a new characterization of sampling sets “without inequalities” in the style of Beurling, new properties of totally positive functions, and the interplay between zero sets of functions in a shift-invariant space and functions in the Bargmann–Fock space.

Shift-invariant and sampling spaces associated with the special affine Fourier transform

The Special Affine Fourier Transformation or the SAFT generalizes a number of well known unitary transformations as well as signal processing and optics related mathematical operations. Shift-invariant spaces also play an important role in sampling theory, multiresolution analysis, and many other areas of signal and image processing. Shannon's sampling theorem, which is at the heart of modern digital communications, is a special case of sampling in shift-invariant spaces. Furthermore, it is well known that the Poisson summation formula is equivalent to the sampling theorem and that the Zak transform is closely connected to the sampling theorem and the Poisson summation formula. These results have been known to hold in the Fourier transform domain for decades and were recently shown to hold in the Fractional Fourier transform domain by A. Bhandari and A. Zayed. The main goal of this article is to show that these results also hold true in the SAFT domain. We provide a short, self-contained proof of Shannon's theorem for functions bandlimited in the SAFT domain and then show that sampling in the SAFT domain is equivalent to orthogonal projection of functions onto a subspace of bandlimited basis associated with the SAFT domain. This interpretation of sampling leads to least-squares optimal sampling theorem. Furthermore, we show that this approximation procedure is linked with convolution and semi-discrete convolution operators that are associated with the SAFT domain. We conclude the article with an application of fractional delay filtering of SAFT bandlimited functions.

Periodic Non-uniform Sampling for Sparse Signals in Shift-invariant Spaces

Periodic Non-uniform sampling in shift-invariant spaces is an effective method for lowrate sampling of continuous time sparse signals, but, recovery of sampled time sparse signals must be unstable in general. In the paper, the possibility of stable recovery under a combination of sufficient sparsity is proposed. The sampling and reconstruction of sparse signals were transformed into matrix and vector operations by using minimum L1 normal. Necessary condition of reconstruction is analyzed. Finally, we are taking multiband sinusoidal signals as an example to prove that the method can achieve the sampling and reconstruction of sparse signals from recovery successful rate and integer of system.

Direct Radio Frequency Sampling System on Software-defined Radio

A traditional assumption underlying most data converters is that the signal should be sampled at a rate exceeding twice the highest frequency. In this paper, we employ a method for low-rate sampling of multi- band signals via applying periodic nonuniform sampling in shift-invariant spaces generated by m kernels with period T. So, the sampling and reconstruction of signals were transformed into matrix and vector operations, the generalized inverse can be use to find the answer and an interpolator is used to insure that complete reconstruction will be achieved. Finally, we validate the method in MATLAB; the conclusion of simulation shows the frame-work presented here is feasible.

Sampling in shift-invariant spaces of functions with polynomial growth

Under suitable conditions, a function f(t) in a principal shift-invariant space can be recovered from its uniform samples { f(n)} n∈ℤ with a simple sampling formula. Provided that the generator ϕ has compact support, we consider the sampling problem in the bigger space V ϕ ≔ {∑a n ϕ(t − n) : a n ∈ ℂℤ}. In this space, there exist infinite functions with the same samples y n = f(n). We show that polynomial growth conditions give the uniqueness: if y n has polynomial growth, there is a unique function of polynomial growth f ∈ V ϕ, satisfying f(n) = y n , n ∈ ℤ. This function is given by the known sampling formula. The same result is proved also when we consider average samples y n = f ∗ h(n).

A New Method to Sampling of Multiple Bandpass Signals

A traditional assumption underlying most data converters is that the signal should be sampled at a rate exceeding twice the highest frequency. In this paper, we employ a method for low-rate sampling of multi- band signals via applying periodic nonuniform sampling in shift-invariant spaces generated by m kernels with period T. To avoid the non-ideal filter in traditional method, we use reconstruction functions to recovery the sampled signal completely. Finally, we validate the method in MATLAB; the conclusion of simulation shows the frame-work presented here is feasible.

A matrix pencil approach to the existence of compactly supported reconstruction functions in average sampling

The aim of this work is to solve a question raised for average sampling in shift-invariant spaces by using the well-known matrix pencil theory. In many common situations in sampling theory, the available data are samples of some convolution operator acting on the function itself: this leads to the problem of average sampling, also known as generalized sampling. In this paper we deal with the existence of a sampling formula involving these samples and having reconstruction functions with compact support. Thus, low computational complexity is involved and truncation errors are avoided. In practice, it is accomplished by means of a FIR filter bank. An answer is given in the light of the generalized sampling theory by using the oversampling technique: more samples than strictly necessary are used. The original problem reduces to finding a polynomial left inverse of a polynomial matrix intimately related to the sampling problem which, for a suitable choice of the sampling period, becomes a matrix pencil. This matrix pencil approach allows us to obtain a practical method for computing the compactly supported reconstruction functions for the important case where the oversampling rate is minimum. Moreover, the optimality of the obtained solution is established.

Reconstruction of compressed samples in shift-invariant space base on matrix factorization

According to the model of compressed sampling in shift-invariant space, the over-complete dictionary is usually a function matrix, which increases the complexity of reconstructing samples. In order to reduce the complexity of reconstructing samples, this paper proposes a reconstruction method based on matrix factorization. The method transforms over-complete dictionary into the multiplication of a reversible function matrix and a constant matrix using least squares error, and converts the samples reconstruction into the computing of constant matrix, which reduces the complexity of samples reconstruction. Meanwhile, the feasibility and convergence of method is analyzed in theory. Finally, the convergence and reconstruction error of the method is validated by simulation; and it is shown that the method is effective.

Generalized average sampling in shift invariant spaces

Average sampling is motivated by realistic needs, e.g., physical limitation of acquisition devices or characteristics of sampling procedure. As an extension of the average sampling, we study generalized average sampling in shift invariant spaces with frame generators, in which averages are taken from suitable channeled version of signals. Illustrative examples are given.

Regular and Irregular Sampling Theorem for Multiwavelet Subspaces

We study the sampling theorem for frames in multiwavelet subspaces. Firstly, a sufficient condition under which the regular sampling theorem holds is established. Then, notice that irregular sampling is also useful in practice; we consider the general cases of the irregular sampling and establish a general irregular sampling theorem for multiwavelet subspaces. Finally, using this generalized irregular sampling theorem, we obtain an estimate for the perturbations of regular sampling in shift-invariant spaces.

Multivariate generalized sampling in shift-invariant spaces and its approximation properties

Nowadays the topic of sampling in a shift-invariant space is having a significant impact: it avoids most of the problems associated with classical Shannon's theory. Under appropriate hypotheses, any multivariate function in a shift-invariant space can be recovered from its samples at Zd. However, in many common situations the available data are samples of some convolution operators acting on the function itself: this leads to the problem of multivariate generalized sampling in shift-invariant spaces. This extra information on the functions in the shift-invariant space will allow to sample in an appropriate sub-lattice of Zd. In this paper an L2(Rd) theory involving the frame theory is exhibited. Sampling formulas which are frame expansions for the shift-invariant space are obtained. In the case of overcomplete frame formulas, the search of reconstruction functions with prescribed good properties is allowed. Finally, approximation schemes using these generalized sampling formulas are included.

Local reconstruction for sampling in shift-invariant spaces

The local reconstruction from samples is one of most desirable properties for many applications in signal processing, but it has not been given as much attention. In this paper, we will consider the local reconstruction problem for signals in a shift-invariant space. In particular, we consider finding sampling sets X such that signals in a shift-invariant space can be locally reconstructed from their samples on X. For a locally finite-dimensional shift-invariant space V we show that signals in V can be locally reconstructed from its samples on any sampling set with sufficiently large density. For a shift-invariant space V(ϕ 1, ..., ϕ N ) generated by finitely many compactly supported functions ϕ 1, ..., ϕ N , we characterize all periodic nonuniform sampling sets X such that signals in that shift-invariant space V(ϕ 1, ..., ϕ N ) can be locally reconstructed from the samples taken from X. For a refinable shift-invariant space V(ϕ) generated by a compactly supported refinable function ϕ, we prove that for almost all $(x_0, x_1)\in [0,1]^2$ , any signal in V(ϕ) can be locally reconstructed from its samples from $\{x_0, x_1\}+{\mathbb Z}$ with oversampling rate 2. The proofs of our results on the local sampling and reconstruction in the refinable shift-invariant space V(ϕ) depend heavily on the linear independent shifts of a refinable function on measurable sets with positive Lebesgue measure and the almost ripplet property for a refinable function, which are new and interesting by themselves.

High order Parzen windows and randomized sampling

In this paper high order Parzen windows stated by means of basic window functions are studied for understanding some algorithms in learning theory and randomized sampling in multivariate approximation. Learning rates are derived for the least-square regression and density estimation on bounded domains under some decay conditions on the marginal distributions near the boundary. These rates can be almost optimal when the marginal distributions decay fast and the order of the Parzen windows is large enough. For randomized sampling in shift-invariant spaces, we consider the situation when the sampling points are neither i.i.d. nor regular, but are noised from regular grids by probability density functions. The approximation orders are estimated by means of the regularity of the approximated function and the density function and the order of the Parzen windows.

IRREGULAR SAMPLING IN SHIFT INVARIANT SPACES OF HIGHER DIMENSIONS

We consider irregular sampling in shift invariant spaces V of higher dimensions. The problem that we address is: find ε so that given perturbations (λk) satisfying sup |λk| &lt; ε, we can reconstruct an arbitrary function f of V as a Riesz basis expansions from its irregular sample values f(k + λk). A framework for dealing with this problem is outlined and in which one can explicitly calculate sufficient limits ε for the reconstruction. We show how it works in two concrete situations.