Recovery of missing samples for oversampling in shift invariant spaces

It is well known that in the classical Shannon sampling theory on band-limited signals, any finitely many missing samples can be recovered when the signal is oversampled at a rate higher than the minimum Nyquist rate. In this work, we consider the problem of recovering missing samples from multi-channel oversampling in a general shift-invariant space. We find conditions under which any finite or infinite number of missing samples can be recovered when they are from a single or two channels.

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.

Optimal shift invariant spaces and their Parseval frame generators

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.

Common problem in signal processing is reconstruction of the missing signal samples. Missing samples can occur by intentionally omitting signal coefficients to reduce memory requirements, or to speed up the transmission process. Also, noisy signal coefficients can be considered as missing ones, since they have wrong values due to the noise. The reconstruction of these coefficients is demanding task, considered within the Compressive sensing area. Signal with large number of missing samples can be recovered, if certain conditions are satisfied. There is a number of algorithms used for signal reconstruction. In this paper we have analyzed the performance of iterative gradient-based algorithm for sparse signal reconstruction. The parameters influence on the optimal performances of this algorithm is tested. Two cases are observed: non-noisy and noisy signal case. The theory is proved on examples.

The aim of this paper is to study sampling and reconstruction in a shift invariant space with multiple generators. On contrary to the classical case of a shift invariant space with a single generator, it is shown that cannot be a stable set of sampling for where when Further the problems of perturbation of a stable set of sampling and local reconstruction method are discussed along with an illustration and implementation.

In this paper, we introduce the notion of Walsh shift-invariant space and present a unified approach to the study of shift-invariant systems to be frames in L2(ℝ+). We obtain a necessary condition and three sufficient conditions under which the Walsh shift-invariant systems constitute frames for L2(ℝ+). Furthermore, we discuss applications of our main results to obtain some known conclusions about the Gabor frames and wavelet frames on positive half line.

Erratum to: Periodic operators and elements. Shift operators. Shift invariant spaces

To reconstruct a function from its sampling value is not always exact, error may arise due to a lot of reasons, therefore error estimation is useful in reconstruction. For non-uniform sampling in shift invariant space, three kinds of errors of the reconstruction formula are discussed in this article. For every kind of error, we give an estimation. We find the accuracy of the reconstruction formula mainly depends on the decay property of the generator and the sampling function.

The structure of shift–modulation invariant spaces: The rational case

A method is proposed for reconstructing an Orthogonal Frequency Division Multiplexing (OFDM) signal that contains data gaps, with the aim to improve demodulation. The main objective is to use the method in a passive radar application with missing data samples and to improve target detection. The OFDM signal is assumed to comply with the Digital Video Broadcasting Terrestrial standard. The proposed recovery method is based on optimisation of a novel objective function, which consists of two parts. The first part is a function of the energy in the out‐of‐band frequencies, whereas the second, and novel part, uses the location of data carriers in the constellation diagram. The method is evaluated using both simulations and real data. The authors show that the proposed method significantly improves the OFDM signal in just a few iteration steps. The proposed method improved the condition number more than a factor ten thousand millions compared to using the least square method on the out‐of‐band frequencies only. The authors also decode the symbols with the Viterbi decoding algorithm and show how the required number of iterations with the proposed algorithm depends on the amount of missing samples and on the Signal‐to‐Noise Ratio in order to achieve a Bit Error Rate of less than one in one hundred thousand millions.

This paper shows the interest of an interpolation method based on parametric modeling to retrieve missing samples in ECG signals. This problem occurs more and more with the emergence of telemedicine applications. The different links (fixed access network (PSTN), mobile access network (GSM/GPRS and future UMTS) or satellite interfacing (DVB-RCS technology)) involved in e-health applications are liable to induce errors on the transmitted data. These errors/losses can occur anytime and anywhere (according to the channel availability, memory overflows, protocols, etc) during a transmission process. Therefore the recovering of missing samples for biomedical signals is of great interest. The method used in this paper is based on a left-sided and right-sided autoregressive model, i.e., the interpolation algorithm uses the samples before and after the sequence of missing samples. An objective measure is used to assess the method performance. Results show that this interpolation method represents a really suitable technique to ECG signal reconstruction in a possible corrupted transmission.

Extra invariance of shift-invariant spaces on LCA groups

We consider a framework for generalized sampling of graph signals that extends sampling results in shift-invariant (SI) spaces to the graph setting. We assume that the input signal lies in a periodic graph spectrum subspace, which generalizes the standard SI assumption to graph signals. Sampling is performed in the graph frequency domain by an arbitrary graph filter. We show that under a mild condition on the sampling filter, perfect recovery is possible using a correction filter that can be represented as a spectral graph filter whose response depends on the prior subspace spectrum and on the sampling filter. This filter parallels the correction filter in SI sampling in standard signal processing. Since the input space and the sampling filter are almost arbitrary, our framework allows perfect recovery of many classes of input signals from a variety of different sampling patterns using a simple correction filter. For example, our method enables perfect recovery of non-bandlimited graph signals from their bandlimited measurements.

It is the purpose of this paper to feature the link between the theory of minimal norm interpolation over lattices by elements fromSobolev algebras H s (S d )with what is known as the theory of spline-type (or principal shift invariant) spaces. As an extremely useful tool allowing us to establish various kinds of robustness results we will present the Wiener amalgam spacesW(B, l p with general (smooth) local components and globall p behavior. For this reason a summary of their most important properties, including convolution relations and the behavior under the Fourier transform, is presented. The discussion of projection and minimal norm interpolation operators is not restricted to the pure Hilbert space setting for which these concepts were developed originally. Among others we show LP-stability of the (for p = 2 orthogonal) projection from LP onto the corresponding spline-type spaces with lP-coefficients. As a main result (which can be formulated in several different concrete ways) we show that for s > d/2 the mapping f ↦ Qs, a(f), from f to the minimal norm interpolation in Hs over the lattice aℤd, a >, depends continuously on the input parameters (s, a). It also extends to certain fractional LP-Sobolev spaces consisting of continuous functions in LP. In this modified setting the outcome of this procedure depends continuously on (s, a) in the LP sense. Moreover, the mapping is robust against small jitter errors. Wiener amalgam spaces tum out to be very useful, both for a precise formulation and in the proofs of such results.