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It is known that every linear method which determines the Hilbert transform from the samples of the function diverges (weakly). This paper presents strong evidence that all such methods even diverge strongly. It is shown that the common approximation method derived from the conjugate Fej'er means diverges strongly, and that all reasonable approximation methods with a finite search horizon diverge strongly. Moreover, the paper discusses the relation between strong divergence and the existence of adaptive approximation methods.

Antenna arrays used in compressive sensing based DOA estimation algorithms are generated randomly to minimize mutual coherence. This scheme suffers from practical limitations. For an antenna array that is sufficiently random, some elements of the array would almost always fall very close to each other, which is not practicable. Rectangular arrays, although very uniform and practicable, suffer from poor performance when used in compressive sensing algorithms that assume spatial sparsity. Aperiodic arrays seem to offer a compromise solution. This paper demonstrates that it is possible to design aperiodic antenna array by using a simple disturbance optimization scheme, that can be applied to multiple aperiodic array geometries. The optimization scheme uses a few parameters to generate an aperiodic geometry. We will also see that the optimized aperiodic array has better performance than several other geometries studied in this paper and performs very close to random array configuration.

This paper considers the problem of recovering a one or two dimensional discrete signal which is approximately sparse in its gradient from an incomplete subset of its Fourier coefficients which have been corrupted with noise. The results show that in order to obtain a reconstruction which is robust to noise and stable to inexact gradient sparsity of order s with high probability, it suffices to draw O(s log N) of the available Fourier coefficients uniformly at random. However, if one draws O(s log N) samples in accordance to a particular distribution which concentrates on the low Fourier frequencies, then the stability bounds which can be guaranteed are optimal up to log factors. The final result of this paper shows that in the one dimensional case where the underlying signal is gradient sparse and its sparsity pattern satisfies a minimum separation condition, then to guarantee exact recovery with high probability, for some M < N, it suffices to draw O(s log M logs) samples uniformly at random from the Fourier coefficients whose frequencies are no greater than M.

Adaptive dictionary learning has become a hot-topic research field during the past decade. Though several algorithms have been proposed and achieved impressive results, they are all computationally intensive due to the lack of structure in their output dictionaries. In this paper we build upon our previous work and take a geometric approach to develop better, more efficient algorithms that can learn adaptive structured dictionaries. While inheriting many of the advantages in the previous construction, the new algorithm better utilizes the geometry of data and effectively removes translational invariances from the data, thus able to produce smaller, more robust dictionaries. We demonstrate the performance of the new algorithm on two data sets, and conclude the paper by a discussion of future work.

In the context of analytic functions on the open unit disk, a weighted composition operator is simply a composition operator followed by a multiplication operator. The class of weighted composition operators has an important place in the theory of Banach spaces of analytic functions; for instance, it includes all isometries on $H^p$ $(p\neq 2)$. Very recently it was shown that only weighted composition operators preserve the class of outer functions. The present paper considers a particular question motivated by applications: Which smallest possible sets of test functions can be used to identify an unknown weighted composition operator? This stems from a practical problem in signal processing, where one seeks to identify an unknown minimum phase preserving operator on $L^2(\real_+)$ using test signals. It is shown in the present paper that functions that determine weighted composition operators are directly linked to the classical normal family of schlicht functions. The main result is that a pair of functions $\{f,g\}$ distinguishes between any two weighted composition operators if and only if there exists a zero-free function $h$ and a schlicht function $\sigma$ such that $\spn\{f,g\}=\spn\{h\sigma,h\}$. This solves completely the underlying signal processing problem and brings to light an intriguing geometric object, the manifold of planes of the form $\spn\{h\sigma,h\}$. As an application of the main result, it is proven that there exist compactly supported pairs in $L^2(\real_+)$ that can be used to identify minimum phase preserving operators.

We consider the problem of recovering a complete (i.e., square and invertible) matrix $\mathbf A_0$, from $\mathbf Y \in \mathbb R^{n \times p}$ with $\mathbf Y = \mathbf A_0 \mathbf X_0$, provided $\mathbf X_0$ is sufficiently sparse. This recovery problem is central to the theoretical understanding of dictionary learning, which seeks a sparse representation for a collection of input signals, and finds numerous applications in modern signal processing and machine learning. We give the first efficient algorithm that provably recovers $\mathbf A_0$ when $\mathbf X_0$ has $O(n)$ nonzeros per column, under suitable probability model for $\mathbf X_0$. In contrast, prior results based on efficient algorithms provide recovery guarantees when $\mathbf X_0$ has only $O(n^{1-\delta})$ nonzeros per column for any constant $\delta \in (0, 1)$. Our algorithmic pipeline centers around solving a certain nonconvex optimization problem with a spherical constraint, and hence is naturally phrased in the language of manifold optimization. To show this apparently hard problem is tractable, we first provide a geometric characterization of the high-dimensional objective landscape, which shows that with high probability there are no "spurious" local minima. This particular geometric structure allows us to design a Riemannian trust region algorithm over the sphere that provably converges to one local minimizer with an arbitrary initialization, despite the presence of saddle points. The geometric approach we develop here may also shed light on other problems arising from nonconvex recovery of structured signals., Comment: 104 pages, 5 figures. Due to length constraint of publication, this long paper are subsequently divided into two papers (arXiv:1511.03607 and arXiv:1511.04777). Further updates will be made only to the two papers

In 2004 Casazza, Kutyniok and Lammers introduced the R-duals of sequences in a general Hilbert space. The purpose was to obtain a general version of the duality principle in Gabor analysis. It was shown that the R-duals cover the duality principle for tight Gabor frames and Gabor Riesz bases. In this paper we discuss the relationship between the R-duals and a variant, called R-duals of type III, introduced in 2014. In contrast to the original R-duals, it is known that the R-duals of type III generalize the duality principle for all Gabor frames, but we believe that a smaller and more convenient class will work as well. The purpose of the paper is to give a focussed presentation of the R-duals of type I and III that can trigger the research on this.

Compressed sensing (CS) is one of the great successes of computational mathematics in the past decade. There are a collection of tools which aim to mathematically describe compressed sensing when the sampling pattern is taken in a random or deterministic way. Unfortunately, there are many practical applications where the well studied concepts of uniform recovery and the Restricted Isometry Property (RIP) can be shown to be insufficient explanations for the success of compressed sensing. This occurs both when the sampling pattern is taken using a deterministic or a non-deterministic method. We shall study this phenomenon and explain why the RIP is absent, and then propose an adaptation which we term ‘the RIP in levels’ which aims to solve the issues surrounding the RIP. The paper ends by conjecturing that the RIP in levels could provide a collection of results for deterministic sampling patterns.

Parameter estimation for exponential sums is a classical problem in signal processing. Recently, a new concept for estimating parameters of bivariate exponential sums has been proposed. The resulting method relies on parameter estimations for univariate exponential sums along several lines in the plane. These (univariate) parameter estimations are being used to first compute the projections of the unknown bivariate frequency vectors onto these lines, before they are combined to obtain estimations for the sought frequency vectors of the bivariate exponential sum. In this paper, we address theoretical questions concerning this new concept, namely (a) how many lines are needed for exact reconstruction, and (b) how to recover linear combinations of shifted positive definite functions.

Many relevant applications of signal processing rely on the separation of sources from a mixture of signals without a prior knowledge about the mixing process. Given a mixture of signals f = ∑ i f i , the task of signal separation is to estimate the components f i by using specific assumptions on their time-frequency behaviour or statistical characteristics. Time-frequency data is often very high-dimensional which affects the performance of signal separation methods quite significantly. Therefore, the embedding dimension of the time-frequency representation of f should be reduced prior to the application of a decomposition strategy, such as independent component analysis (ICA) or non-negative matrix factorization (NNMF). In other words, a suitable dimensionality reduction method should be applied, before the data is decomposed and then back-projected. But the choice of the dimensionality reduction method requires particular care, especially in combination with ICA and NNMF, since non-negative input data are required. In this paper, we introduce a generic concept for the construction of suitable non-negative dimensionality reduction methods. Furthermore, we discuss the two different decomposition strategies NNMF and ICA for single channel signal separation in combination with non-negative principal component analysis (NNPCA), where our main interest is in acoustic signals with transitory components.

In this paper we derive the Fisher information matrix and the Cramer-Rao lower bound for the non-additive white Gaussian noise model yk = |{x, f k ) + μk|2, 1 ≤ k ≤ m, where {f 1 , · · ·, f m } is a spanning set for Cn and (μ 1 , …, μ m ) are i.i.d. realizations of the Gaussian complex process CN(0, ρ2). We obtain closed form expressions that include quadrature integration of elementary functions.

It is well-known that there exist bandlimited signals for which certain sampling series are divergent. One possible way of circumventing the divergence is to adapt the sampling series to the signals. In this paper we study adaptivity in the number of summands that are used in each approximation step, and whether this kind of adaptive signal processing can improve the convergence behavior of the sampling series. We approach the problem by considering approximation processes in general Banach spaces and show that adaptivity reduces the set of signals with divergence from a residual set to a meager or empty set. Due to the non-linearity of the adaptive approximation process, this study cannot be done by using the Banach-Steinhaus theory. We present examples from sampling based signal processing, where recently strong divergence, which is connected to the effectiveness of adaptive signal processing, has been observed.

An extension of the bivariate cubic spline on the uniform grid is derived in this paper to have different tensions in different square cells of the grid. The resulting function can be interpreted also as a bivariate extension of the univariate spline in piecewise constant tension which was applied to adaptive interpolation of digital images for their magnification and rotation. The bivariate function will hopefully make it possible to magnify and rotate images better and even to deform images into any shapes. A locally supported basis, which is crucial for the practical use of the bivariate functions, has not been constructed at the moment and its construction is left for the next step of study.

Linear sampling of sparse vectors via sensing matrices has been a much investigated problem in the past decade. The nonlinear sampling methods, such as quadratic forms are also studied marginally to include undesired effects in data acquisition devices (e.g., Taylor series expansion up to two terms). In this paper, we introduce customized nonlinear sampling techniques that provide possibility of sparse signal recovery. The main advantage of the nonlinear method over the conventional linear schemes is the reduction in the number of required samples to 2k for recovery of k-sparse signals. We also introduce a low-complexity reconstruction method similar to the annihilating filter in the sampling of signals with finite rate of innovation (FRI). The disadvantage of this nonlinear sampler is its sensitivity to additive noise; thus, it is suitable in scenarios dealing with noiseless data such as network packets, where the data is either noiseless or it is erased. We show that by increasing the number of samples and applying denoising techniques, one can improve the performance. We further introduce a modified version of the proposed method which has strong links with spectral estimation methods and exhibits a more stable performance under noise and numerical errors. Simulation results confirm that this method is much faster than l 1 -norm minimization routines, widely used in linear compressed sensing and thus much less complex.

In Compressed Sensing (CS), the matrices that satisfy the Restricted Isometry Property (RIP) play an important role. But it is known that the RIP properties of a matrix Φ and its ‘weighted matrix’ GΦ (G being a non-singular matrix) vary drastically in terms of RIP constant. In this paper, we consider the opposite question: Given a matrix Φ, can we find a non-singular matrix G such that GΦ has compliance with RIP? We show that, under some conditions, a class of non-singular matrices (G) exists such that GΦ has RIP-compliance with better RIP constant. We also provide a relationship between the Unique Representation Property (URP) and Restricted Isometry Property (RIP), and a direct relationship between RIP and sparsest solution of a linear system of equations.

A tight frame in Rn is a redundant system which has a reconstruction formula similar to that of an orthonormal basis. For a unit-norm frame F = {f i }k i=1 , a scaling is a vector c = (c(l),…, c(k)) e Rk≥0 such that {c(i)f i }k i=1 is a tight frame in Rn. If a scaling c exists, we say that F is a scalable frame. A scaling c is a minimal scaling if {fi : c{i) > 0} has no proper scalable subframes. In this paper, we present the uniqueness of the orthogonal partitioning property of any set of minimal scalings and provide a construction of scalable frames by extending the standard orthonormal basis of Rn.

We consider the problem of spatiotemporal sampling in an evolutionary process xn = Anx where an unknown operator A driving an unknown initial state x is to be recovered from a combined set of coarse spatial samples {χ| Ωο , x(1)| Ωι ,· · ·, x(N)| ΩN }. In this paper, we will study the case of infinite dimensional spatially invariant evolutionary process, where the unknown initial signals x are modeled as l2(Z) and A is an unknown spatial convolution operator given by a filter α e l1 (Z) so that Ax = a ∗ x. We show that {x|Ω m , x(1)|Ω m , ···, x(N)|Ω m :N≥2m −, Ω m = mZ} contains enough information to recover the Fourier spectrum of a typical low pass filter a, if x is from a dense subset of l2 (Z). The idea is based on a nonlinear, generalized Prony method similar to [2]. We provide an algorithm for the case when a and x are both compactly supported. Finally, We perform the accuracy analysis based on the spectral properties of the operator A and the initial state x and verify them in several numerical experiments.

In this paper we present an improvement of architecture of the Asynchronous Sigma-Delta time encoding machine, which makes it suitable for sampling the wideband sparse analog signals. Since the values of the samples are encoded into sampling instants, this is a signal-dependent sampling scheme. By superseding the commonly used multitone model of the input signal with multiband stochastic model, the randomness of the sampling instants is obtained. This allows for the reconstruction of signal using compressive sampling methods without additional randomization hardware. The simulation results validate the presented approach.

Bilinear inverse problems (BIPs), the resolution of two vectors given their image under a bilinear mapping, arise in many applications, such as blind deconvolution and dictionary learning. However, there are few results on the uniqueness of solutions to BIPs. For example, blind gain and phase calibration (BGPC) is a structured bilinear inverse problem, which arises in inverse rendering in computational relighting, in blind gain and phase calibration in sensor array processing, in multichannel blind deconvolution (MBD), etc. It is interesting to study the uniqueness of solutions to such problems. In this paper, we define identifiability of a bilinear inverse problem up to a group of transformations. We derive conditions under which the solutions can be uniquely determined up to the transformation group. Then we apply these results to BGPC and give sufficient conditions for unique recovery under subspace or joint sparsity constraints. For BGPC with joint sparsity constraints, we develop a procedure to determine the relevant transformation groups. We also give necessary conditions in the form of tight lower bounds on sample complexities, and demonstrate the tightness by numerical experiments.

In this paper, a new class of deterministic sparse matrices derived from Quasi-Cyclic (QC) low-density parity-check (LDPC) codes is presented for compressed sensing (CS). In contrast to random and other deterministic matrices, the proposed matrices are generated based on circulant permutation matrices, which require less memory for storage and low computational cost for sensing. Its size is also quite flexible compared with other existing fixed-sizes deterministic matrices. Furthermore, both the coherence and null space property of proposed matrices are investigated, specially, the upper bounds of signal sparsity k is given for exactly recovering. Finally, we carry out many numerical simulations and show that our sparse matrices outperform Gaussian random matrices under some scenes.

Operator sampling considers the question of when operators of a given class can be distinguished by their action on a single probing signal. The fundamental result in this theory shows that the answer depends on the area of the support S of the so-called spreading function of the operator (i.e., the symplectic Fourier transform of its Kohn-Nirenberg symbol). |S| 1 it is impossible. In the critical case when |S| = 1, the picture is less clear. In this paper we characterize when regular operator sampling (that is, when the probing signal is a periodically-weighted delta train) is possible when S is a parallelogram of area 1.

This paper provides a random complex measurement matrix and an algorithm for complex phase retrieval of sparse or approximately sparse signals from the noisy magnitudes of the measurements obtained with this matrix. We compute explicit error bounds for the recovery which depend on the noise-to-signal ratio, the sparsity s, the number of measured quantitites m, and the dimension of the signal N. This requires m to be of the order of s ln(N/s). In comparison with sparse recovery from complex linear measurements, our phase retrieval algorithm requires six times the number of measured quantities.

Recently progress has been made in compressive sensing by replacing simplistic sparsity models with more powerful denoisers. In this paper, we develop a framework to predict the performance of denoising-based signal recovery algorithms based on a new deterministic state evolution formalism for approximate message passing. We compare our deterministic state evolution against its more classical Bayesian counterpart. We demonstrate that, while the two state evolutions are very similar, the deterministic framework is far more flexible. We apply the deterministic state evolution to explore the optimality of denoising-based approximate message passing (D-AMP). We prove that, while D-AMP is suboptimal for certain classes of signals, no algorithm can uniformly outperform it.

The performance guarantees in recovery of a sparse vector in a compressed sensing scenario, besides the reconstruction technique, depends on the choice of the sensing matrix. The so-called restricted isometry property (RIP) is one of the well-used tools to determine and compare the performance of various sensing matrices. It is a standard result that random (Gaussian) matrices satisfy RIP with high probability. However, the design of deterministic matrices that satisfy RIP has been a great challenge for many years now. The common design technique is through the coherence value (maximum modulus correlation between the columns). In this paper, based on the Paley graphs, we introduce deterministic matrices of size q+1/2 × q with q a prime power, such that the corresponding Gram matrix is real-valued. We show that the coherence of these matrices are less than twice the Welch bound, which is a lower bound valid for general matrices. It should be mentioned that the introduced matrix differs from the equiangular tight frame (ETF) of size q-1/2 × q arising from the Paley difference set.

Visual inference over a transmission channel is increasingly becoming an important problem in a variety of applications. In such applications, low latency and bit-rate consumption are often critical performance metrics, making data compression necessary. In this paper, we examine feature compression for support vector machine (SVM)-based inference using quantized randomized embeddings. We demonstrate that embedding the features is equivalent to using the SVM kernel trick with a mapping to a lower dimensional space. Furthermore, we show that universal embeddings — a recently proposed quantized embedding design — approximate a radial basis function (RBF) kernel, commonly used for kernel-based inference. Our experimental results demonstrate that quantized embeddings achieve 50% rate reduction, while maintaining the same inference performance. Moreover, universal embeddings achieve a further reduction in bit-rate over conventional quantized embedding methods, validating the theoretical predictions.

Pairwise comparison matrices arise in numerous applications including collaborative filtering, elections, economic exchanges, etc. In this paper, we propose a new low-rank model for pairwise comparison matrices that accommodates non-transitive pairwise comparisons. Based on this model, we consider the regime where one has limited observations of a pairwise comparison matrix and wants to reconstruct the whole matrix from these observations using matrix completion. To do this, we adopt a recently developed alternating minimization algorithm to this particular matrix completion problem and derive a theoretical guarantee for its performance. Numerical simulations using synthetic data support our proposed approach.

We consider the class of continuous-time autoregressive (CAR) processes driven by (possibly non-Gaussian) Lévy white noises. When the excitation is an impulsive noise, also known as compound Poisson noise, the associated CAR process is a random non-uniform exponential spline. Therefore, Poisson-type processes are relatively easy to understand in the sense that they have a finite rate of innovation. We show in this paper that any CAR process is the limit in distribution of a sequence of CAR processes driven by impulsive noises. Hence, we provide a new interpretation of general CAR processes as limits of random exponential splines. We illustrate our result with simulations.

In this paper we discuss a system of dynamical sampling, i.e. sampling a signal x that evolves in time under the action of an evolution operator A. We examine the timespace sampling that allows for reconstruction of x. Here we describe the possible reconstruction systems when the system also contains an unknown constant forcing term σ. We give conditions under which both x and σ can be reconstructed from the spacio-temporal set of sampling.

Total generalized variation (TGV) is a generalization of total variation (TV). This method has gained more and more attention in image processing due to its capability of reducing staircase effects. As the existence of high order regularity, TGV tends to blur edges, especially when noise is excessive. In this paper, we propose an iterative weighted total generalized variation (WTGV) model to reconstruct images with sharp edges and details from compressive sensing data. The weight is iteratively updated using the latest reconstruction solution. The splitting variables and alternating direction method of multipliers (ADMM) are employed to solve the proposed model. To demonstrate the effectiveness of the proposed method, we present some numerical simulations using partial Fourier measurement for natural and MR images. Numerical results show that the proposed method can avoid staircase effects and keep fine details at the same time.

Exploiting Compressive Sensing (CS) in MIMO radars, we can remove the need of the high rate A/D converters and send much less samples to the fusion center. In distributed MIMO radars, the received signal can be modeled as a block sparse signal in a basis. Thus, block CS methods can be used instead of classical CS ones to achieve more accurate target parameter estimation. In this paper a new method of energy allocation to the transmitters is proposed to improve the performance of the block CS-based distributed MIMO radars. This method is based on the minimization of an upper bound of the sensing matrix block-coherence. Simulation results show a significant increase in the accuracy of multiple targets parameter estimation.

An equiangular tight frame (ETF) is a set of unit vectors whose coherence achieves the Welch bound, and so is as incoherent as possible. ETFs arise in numerous applications, including compressed sensing. They also seem to be rare: despite over a decade of active research by the community, only a few construction methods have been discovered. One known method constructs ETFs from combinatorial designs known as balanced incomplete block designs. In this short paper, we provide an updated, more explicit perspective of that construction, laying the groundwork for upcoming results about such frames.

In this paper we present a new way to obtain a bound on the number of measurements sampled from certain distributions that guarantee uniform stable and robust recovery of low-rank matrices. The recovery guarantees are characterized by a stable and robust version of the null space property and verifying this condition can be reduced to the problem of obtaining a lower bound for a quantity of the form inf x∊T ||Ax||2. Gordon's escape through a mesh theorem provides such a bound with explicit constants for Gaussian measurements. Mendelson's small ball method allows to cover the significantly more general case of measurements generated by independent identically distributed random variables with finite fourth moment.

Computerized tomography allows us to reconstruct a bivariate function from its Radon samples. The reconstruction is based on the filtered back projection (FBP) formula, which gives an analytical inversion of the Radon transform. The FBP formula, however, is numerically unstable. Therefore, suitable low-pass filters of finite bandwidth are employed to make the reconstruction by FBP less sensitive to noise. The objective of this paper is to analyse the reconstruction error occurring due to the use of a low-pass filter. To this end, we prove L2 error estimates on Sobolev spaces of fractional order. The obtained error estimates are affine-linear with respect to the distance between the filter's window function and the constant function 1 in the L∞-norm. Our theoretical results are supported by numerical simulations, where in particular the predicted affine-linear behaviour of the error is observed.

In this paper we give an account of recent developments in the duality theory of Gabor frames. We prove the Wexler-Raz biorthogonality relations and the duality principle for co-compact Gabor systems on second countable, locally compact abelian groups G. Our presentation does not rely on the existence of uniform lattices in G.

On the real line, it is well-known that (the decay of) the one-dimensional continuous wavelet transform can be used to characterize the regularity of a function or distribution, e.g. in the sense of Holder regularity, but also in the sense of characterizing the wave front set. In higher dimensions — especially in dimension two — this ability to resolve the wave front set has become a kind of benchmark property for anisotropic wavelet systems like curvelets and shearlets. Summarizing a recent paper of the authors, this note describes a novel approach which allows to prove that a given wavelet transform is able to resolve the wave front set of arbitrary tempered distributions. More precisely, we consider wavelet transforms of the form W ψ u(x, h) = 〈u|T x D h ψ〉, where the wavelet ψ is dilated by elements h e H of a certain dilation group H ≤ GL (Rd). We provide readily verifiable, geometric conditions on the dilation group H which guarantee that (χ,ξ) is a regular directed point of u iff the wavelet transform W ψ u has rapid decay on a certain set depending on (χ,ξ). Roughly speaking, smoothness of u near x in direction ξ is equivalent to rapid decay of wavelet coefficients W ψ u (y, h) for y near x if D h ψ is a small scale wavelet oriented in a direction near ξ. Special cases of our results include that of the shearlet group in dimension two (even with scaling types other than parabolic scaling) and also in higher dimensions, a result which was (to our knowledge) not known before. We also briefly describe a generalization where the group wavelet transform is replaced by a discrete wavelet transform arising from a discrete covering/partition of unity of (a subset of) the frequency space Rd.

A spatially distributed system for signal sampling and reconstruction consists of huge amounts of small sensing devices with limited computing and telecommunication capabilities. In this paper, we discuss stability of such a sampling/reconstruction system and develop a distributed algorithm for fast reconstruction of high-dimensional signals.

We study the problem of recovering an unknown vector x e Rn from measurements of the form y i = |aT i x|2 (for i = 1,…, m), where the vectors a i e Rn are chosen independently at random, with each coordinate a ij e R being chosen independently from a fixed sub-Gaussian distribution D. However, without making additional assumptions on the random variables a ij — for example on the behavior of their small ball probabilities — it may happen some vectors x cannot be uniquely recovered. We show that for any sub-Gaussian distribution V, with no additional assumptions, it is still possible to recover most vectors x. More precisely, one can recover those vectors x that are not too peaky in the sense that at most a constant fraction of their mass is concentrated on any one coordinate. The recovery guarantees in this paper are for the PhaseLift algorithm, a tractable convex program based on a matrix formulation of the problem. We prove uniform recovery of all not too peaky vectors from m = 0(n) measurements, in the presence of noise. This extends previous work on PhaseLift by Candes and Li [8].

New spatial sampling geometries such as nested and coprime arrays have recently been shown to be capable of localizing O(M2) sources using only M sensors. However, these results are based on the assumption that the sampling locations are exactly known and they obey the specific geometry exactly. In contrast, this paper considers an array with perturbed sensor locations, and studies how such perturbation affects source localization using the co-array of a nested or coprime array. An iterative algorithm is proposed in order to jointly estimate the directions of arrival along with the perturbations. The directions are recovered by solving a sparse representation problem in each iteration. Identifiability issues are addressed by deriving Cramer Rao lower bound for this problem. Numerical simulations reveal successful performance of our algorithm in recovering of the source directions in the presence of sensor location errors.1

We consider the traditional compressed sensing problem of recovering a sparse solution from undersampled data. We are in particular interested in the case where the measurements arise from a partial circulant matrix. This is motivated by practical physical setups that are usually implemented through convolutions. We derive a new optimization problem that stems from the traditional l 1 minimization under constraints, with the added information that the matrix is taken by selecting rows from a circulant matrix. With this added knowledge it is possible to simulate the full matrix and full measurement vector on which the optimization acts. Moreover, as circulant matrices are well-studied it is known that using Fourier transform allows for fast computations. This paper describes the motivations, formulations, and preliminary results of this novel algorithm, which shows promising results.

In this paper we are concerned with the reconstruction of a class of measures on the square from the sampling of its Fourier coefficients on some sparse set of points. We show that the exact reconstruction of a weighted Dirac sum measure is still possible when one knows a finite number of non-adaptive linear measurements of the spectrum. Surprisingly, these measurements are defined on a model set, i.e quasicrystal.

The main aim of this paper is to study some boundedness inequalities of certain semi-discrete operators. These operators allow to unify some inequalities for both the Shannon sampling operators and Kantorovich-type operators.

Let Ω ⊂ RN be a bounded measurable set and let Λ ⊂ RN be a lattice. Assume that Ω tiles RN multiply at level K when translated by Λ. Then this paper derives conditions on sets of sampling functions such that they form a frame for the Paley-Wiener space PW Ω of functions bandlimited to Ω. Moreover, the canonical dual frame is derived, which allows signal recovery for all signals in PW Ω from a multichannel sampling system samples.