Shift-invariant Spaces Research Articles

Characterization of Shift-Invariant Spaces on a Class of Nilpotent Lie Groups with Applications

Given a simply connected nilpotent Lie group having unitary irreducible representations that are square-integrable modulo the center, we use operator-valued periodization to give a range-function type characterization of shift-invariant spaces of function on the group. We then give characterizations of frame and Riesz families for shift-invariant spaces.

Linear combinations of frame generators in systems of translates

A finitely generated shift invariant space V is a closed subspace of L2(Rd) that can be generated by the integer translates of a finite number of functions. A set of frame generators for V is a set of functions whose integer translates form a frame for V. In this note we give necessary and sufficient conditions in order that a minimal set of frame generators can be obtained by taking linear combinations of the given frame generators. Surprisingly the results are very different from the recently studied case when the property to be a frame is not required.

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).

Sampling theorems in function spaces for frames associated with linear canonical transform

The linear canonical transform (LCT) has proven to be a powerful tool in optics and signal processing. Most existing sampling theories of this transform were derived from the LCT band-limited signal viewpoint. However, in the real world, many analog signals encountered in practical engineering applications are non-bandlimited. The purpose of this paper is to derive sampling theorems of the LCT in function spaces for frames without band-limiting constraints. We extend the notion of shift-invariant spaces to the LCT domain and then derive a sampling theorem of the LCT for regular sampling in function spaces with frames. Further, the theorem is modified to the shift sampling in function spaces by using the Zak transform. Sampling and reconstructing signals associated with the LCT are also discussed in the case of Riesz bases. Moreover, some examples and applications of the derived theory are presented. The validity of the theoretical derivations is demonstrated via simulations.

ERRATUM: "A RANGE FUNCTION APPROACH TO SHIFT-INVARIANT SPACES ON LOCALLY COMPACT ABELIAN GROUPS"

This paper develops several aspects of shift-invariant spaces on locally compact abelian groups. For a second countable locally compact abelian group G we prove a useful Hilbert space isomorphism, introduce range functions and give a characterization of shift-invariant subspaces of L2(G) in terms of range functions. Utilizing these functions, we generalize characterizations of frames and Riesz bases generated by shifts of a countable set of generators from L2(ℝn) to L2(G).

Uncertainty principles in finitely generated shift-invariant spaces with additional invariance

We consider finitely generated shift-invariant spaces (SIS) with additional invariance in L2(Rd). We prove that if the generators and their translates form a frame, then they must satisfy some stringent restrictions on their behavior at infinity. Part of this work (non-trivially) generalizes recent results obtained in the special case of a principal shift-invariant spaces in L2(R) whose generator and its translates form a Riesz basis.

Gabor frames and totally positive functions

Let $g$ be a totally positive function of finite type. Then the Gabor set $\{e^{2\pi i \beta l t} g(t-\alpha k), k,l \in Z \}$ is a frame for $L^2(R)$, if and only if $\alpha \beta <1$. This result is a first positive contribution to a conjecture of I.\ Daubechies from 1990. So far the complete characterization of lattice parameters $\alpha, \beta $ that generate a frame has been known for only six window functions $g$. Our main result now provides an uncountable class of functions. As a byproduct of the proof method we derive new sampling theorems in shift-invariant spaces and obtain the correct Nyquist rate.

Sampling and reconstruction for shift-invariant stochastic processes

In this paper, combining stochastic processes with shift-invariant spaces, we introduce shift-invariant stochastic processes. It is a general case of the classical band-limited stochastic processes and a kind of non-band-limited stochastic processes. Two sampling theorems are obtained for the shift-invariant stochastic processes. The results for band-limited stochastic processes and shift-invariant spaces are generalized by our new results.

Reconstruction from non-uniform samples: A direct, variational approach in shift-invariant spaces

We propose a new approach for signal reconstruction from non-uniform samples, without any constraint on their locations. We look for a function that minimizes a classical regularized least-squares criterion, but with the additional constraint that the solution lies in a chosen linear shift-invariant space-typically, a spline space. In comparison with a pure variational treatment involving radial basis functions, our approach is resolution dependent; an important feature for many applications. Moreover, the solution can be computed exactly by a fast non-iterative algorithm, that exploits at best the particular structure of the problem.

Multiresolution Expansion and Approximation Order of Generalized Tempered Distributions

Let be the generalized tempered distributions of -growth with restricted order , where the function grows faster than any linear functions as . We show the convergence of multiresolution expansions of in the test function space of . In addition, we show that the kernel of an integral operator provides approximation order in in the context of shift-invariant spaces.

Construction of Frames for Shift-Invariant Spaces

We construct a sequence{ϕi(·-j)∣j∈ℤ, i=1,…,r}which constitutes ap-frame for the weighted shift-invariant spaceVμp(Φ)={∑i=1r∑j∈ℤci(j)ϕi(·-j)∣{ci(j)}j∈ℤ∈ℓμp, i=1,…,r},p∈[1,∞], and generates a closed shift-invariant subspace ofLμp(ℝ). The first construction is obtained by choosing functionsϕi,i=1,…,r, with compactly supported Fourier transformsϕ^i,i=1,…,r. The second construction, with compactly supportedϕi,i=1,…,r,gives the Riesz basis.

Shift invariant spaces on compact groups

We study the theory of shift invariant spaces in L2(G), where G is a compact group. We define a range function and show that a relation between an H-invariant space and the range function is valid as in the case of abelian group setting. Here H is assumed to be a closed normal subgroup of G. We also obtain a decomposition for an H-invariant space in terms of principle H-invariant spaces whose generators give rise to “generalized Parseval frames” and use this result to study H-preserving operators.

Signal Reconstruction in Multi-Windows Spline-Spaces Using the Dual System

The letter presents a non-massive parallel procedure to compute the biorthogonal dual system used for signal reconstruction in the case of spline-type spaces with multiple generators. The basis of this algorithm are the properties of the projection operator and the invertibility of the Gramian in the case of a Riesz basis. We use a parallel approach in both time and frequency for the computation of the dual system obtained by translation and sampling of a finitely number of atoms. Since there are many applications in signal and image processing where the spline-type spaces (also known as shift-invariant spaces) play a central role, fast computing methods are needed, especially in the multi-windows case, where the computations are expensive from the execution time and from memory storage point of view. We test the implementation on car crash data.

Sampling and Reconstruction of Signals in Function Spaces Associated With the Linear Canonical Transform

The linear canonical transform (LCT) has been shown to be useful and powerful in signal processing, optics, etc. Many results of this transform are already known, including sampling theory. Most existing sampling theories of the LCT consider the class of bandlimited signals. However, in the real world, many analog signals arising in engineering applications are non-bandlimited. In this correspondence, we propose a sampling and reconstruction strategy for a class of function spaces associated with the LCT, which can provide a suitable and realistic model for real applications. First, we introduce definitions of semi- and fully-discrete convolutions for the LCT. Then, we derive necessary and sufficient conditions pertaining to the LCT, under which integer shifts of a chirp-modulated function generate a Riesz basis for the function spaces. By applying the results, we present a more comprehensive sampling theory for the LCT in the function spaces, and further, a sampling theorem which recovers a signal from its own samples in the function spaces is established. Moreover, some sampling theorems for shift-invariant spaces and some existing sampling theories for bandlimited signals associated with the Fourier transform (FT), the fractional FT, or the LCT are noted as special cases of the derived results. Finally, some potential applications of the derived theory are presented.

Distributed Field Reconstruction in Wireless Sensor Networks Based on Hybrid Shift-Invariant Spaces

We use the theory and algorithms developed for so-called shift-invariant spaces to develop a novel distributed architecture for sampling and reconstructing time-varying non-bandlimited physical fields in wireless sensor networks. We introduce hybrid shift-invariant spaces that generalize conventional shift-invariant spaces and can adapt to local smoothness properties of the field. Using shift-invariant spaces with compactly supported generator functions allows us to split the global reconstruction into several smaller local problems that can be solved independently. Capitalizing on the sparsity of the matrices involved in the reconstruction, we propose direct and iterative reconstruction algorithms whose complexity per time slot scales only linearly with the number of sensor nodes. We furthermore analyze the impact of sensor localization errors on the mean square error of the reconstructed field. Numerical simulations illustrate that the proposed field reconstruction scheme performs close to bandlimited reconstruction and is less sensitive to sensor location errors while providing a significant reduction in computational complexity.

Random sampling in shift invariant spaces

The set of sampling in a shift invariant space plays an important role in signal processing and has many applications. This paper addresses the problem when some randomly chosen samples X={xj:j∈J} form a set of sampling in a shift invariant space. That is, when the inequality of the form cp‖f‖Lp(Rd)p≤∑xj∈X|f(xj)|p≤Cp‖f‖Lp(Rd)p holds uniformly for all functions f in a shift invariant space, where cp and Cp are positive constants (1≤p≤∞). We prove that with overwhelming probability, the above sampling inequality holds for certain compact subsets of the shift invariant space when the sampling size is sufficiently large.

Shift-Invariant and Sampling Spaces Associated With the Fractional Fourier Transform Domain

Shift-invariant spaces play an important role in sampling theory, multiresolution analysis, and many other areas of signal and image processing. A special class of the shift-invariant spaces is the class of sampling spaces in which functions are determined by their values on a discrete set of points. One of the vital tools used in the study of sampling spaces is the Zak transform. The Zak transform is also related to the Poisson summation formula and a common thread between all these notions is the Fourier transform. In this paper, we extend some of these notions to the fractional Fourier transform (FrFT) domain. First, we introduce two definitions of the discrete fractional Fourier transform and two semi-discrete fractional convolutions associated with them. We employ these definitions to derive necessary and sufficient conditions pertaining to FrFT domain, under which integer shifts of a function form an orthogonal basis or a Riesz basis for a shift-invariant space. We also introduce the fractional Zak transform and derive two different versions of the Poisson summation formula for the FrFT. These extensions are used to obtain new results concerning sampling spaces, to derive the reproducing-kernel for the spaces of fractional band-limited signals, and to obtain a new simple proof of the sampling theorem for signals in that space. Finally, we present an application of our shift-invariant signal model which is linked with the problem of fractional delay filtering.

Local sampling set conditions in weighted shift-invariant signal spaces

How to represent a continuous signal in terms of a discrete sequence is a fundamental problem in sampling theory. Most of the known results concern global sampling in shift-invariant signal spaces. But in fact, the local reconstruction from local samples is one of the most desirable properties for many applications in signal processing, e.g. for implementing real-time reconstruction numerically. However, the local reconstruction problem has not been given much attention. In this article, we find conditions on a finite sampling set X such that at least in principle a continuous signal on a finite interval is uniquely and stably determined by their sampling value on the finite sampling set X in shift-invariant signal spaces.

Recovery of missing samples for oversampling in shift invariant spaces

It is well known that any finite missing samples of a band-limited signal can be recovered from the remaining known samples when the signal is oversampled at a rate higher than the minimum Nyquist rate. We consider a similar problem for oversampled signals in shift invariant spaces with compactly supported continuous Riesz generators. Using a multi-channel oversampling, we find conditions under which finitely many missing samples can be recovered.

MULTI-CHANNEL SAMPLING ON SHIFT-INVARIANT SPACES WITH FRAME GENERATORS

Let φ be a continuous function in L2(ℝ) such that the sequence {φ(t - n)}n∈ℤ is a frame sequence in L2(ℝ) and assume that the shift-invariant space V(φ) generated by φ has a multi-banded spectrum σ(V). The main aim in this paper is to derive a multi-channel sampling theory for the shift-invariant space V(φ). By using a type of Fourier duality between the spaces V(φ) and L2[0, 2π] we find necessary and sufficient conditions allowing us to obtain stable multi-channel sampling expansions in V(φ).