faster calculation of the percolation correlation length on spatial networks the divergence of the correlation length at criticality is an important phenomenon of percolation in two dimensional systems substantial speed ups to the calculation of the percolation threshold and component distribution have been achieved by utilizing disjoint sets but existing algorithms of this sort cannot measure the correlation length here we utilize the parallel axis theorem to track the correlation length as nodes are added to the system allowing us to utilize disjoint sets to measure for the entire percolation process with arbitrary precision in a single sweep this algorithm enables direct measurement of the correlation length in lattices as well as spatial network topologies and provides an important tool for understanding critical phenomena in spatial systems

model categories of diagram spectra working in the category mathcal t of based spaces we give the basic theory of diagram spaces and diagram spectra these are functors mathcal d longrightarrow mathcal t for a suitable small topological category mathcal d when mathcal d is symmetric monoidal there is a smash product that gives the category of mathcal d spaces a symmetric monoidal structure examples include begin enumerate item prespectra as defined classically item symmetric spectra as defined by jeff smith item orthogonal spectra a coordinate free analogue of symmetric spectra with symmetric groups replaced by orthogonal groups in the domain category item gamma spaces as defined by graeme segal item mathcal w spaces an analogue of gamma spaces with finite sets replaced by finite cw complexes in the domain category end enumerate we construct and compare model structures on these categories with the caveat that gamma spaces are always connective these categories and their simplicial analogues are quillen equivalent and their associated homotopy categories are equivalent to the classical stable homotopy category monoids in these categories are strict ring spectra often the subcategories of ring spectra module spectra over a ring spectrum and commutative ring spectra are also model categories when this holds the respective categories of ring and module spectra are quillen equivalent and thus have equivalent homotopy categories this allows interchangeable use of these categories in applications 2000 mathematics subject classification primary 55p42 secondary 18a25 18e30 55u35

seeg dipole source localization based on an empirical bayesian approach taking into account forward model uncertainties electromagnetic brain source localization consists in the inversion of a forward model based on a limited number of potential measurements a wide range of methods has been developed to regularize this severely ill posed problem and to reduce the solution space imposing spatial smoothness anatomical constraint or sparsity of the activated source map this last criteria based on physiological assumptions stating that in some particular events e g epileptic spikes evoked potential few focal area of the brain are simultaneously actives has gained more and more interest bayesian approaches have the ability to provide sparse solutions under adequate parametrization and bring a convenient framework for the introduction of priors in the form of probabilistic density functions however the quality of the forward model is rarely questioned while this parameter has undoubtedly a great influence on the solution its construction suffers from numerous approximation and uncertainties even when using realistic numerical models in addition it often encodes a coarse sampling of the continuous solution space due to the computational burden its inversion implies in this work we propose an empirical bayesian approach to take into account the uncertainties of the forward model by allowing constrained variations around a prior physical model in the particular context of seeg measurements we demonstrate on simulations that the method enhance the accuracy of the source time course estimation as well as the sparsity of the resulting source map results on real signals prove the applicability of the method in real contexts

sparsity cognizant total least squares for perturbed compressive sampling solving linear regression problems based on the total least squares tls criterion has well documented merits in various applications where perturbations appear both in the data vector as well as in the regression matrix however existing tls approaches do not account for sparsity possibly present in the unknown vector of regression coefficients on the other hand sparsity is the key attribute exploited by modern compressive sampling and variable selection approaches to linear regression which include noise in the data but do not account for perturbations in the regression matrix the present paper fills this gap by formulating and solving regularized tls optimization problems under sparsity constraints near optimum and reduced complexity suboptimum sparse s tls algorithms are developed to address the perturbed compressive sampling and the related dictionary learning challenge when there is a mismatch between the true and adopted bases over which the unknown vector is sparse the novel s tls schemes also allow for perturbations in the regression matrix of the least absolute selection and shrinkage selection operator lasso and endow tls approaches with ability to cope with sparse under determined errors in variables models interesting generalizations can further exploit prior knowledge on the perturbations to obtain novel weighted and structured s tls solvers analysis and simulations demonstrate the practical impact of s tls in calibrating the mismatch effects of contemporary grid based approaches to cognitive radio sensing and robust direction of arrival estimation using antenna arrays

universal manifold embedding for geometrically deformed functions assume we have a set of observations for example images of different objects each undergoing a different geometric deformation yet all the deformations belong to the same family as a result of the action of these deformations the set of different observations on each object is generally a manifold in the ambient space of observations in this paper we show that in those cases where the set of deformations admits a finite dimensional representation there is a mapping from the space of observations to a low dimensional linear space the manifold corresponding to each object is mapped to a distinct linear subspace of euclidean space the dimension of the subspace is the same as that of the manifold this mapping which we call universal manifold embedding enables the estimation of geometric deformations using the classical linear theory the universal manifold embedding further enables the representation of the object classification and detection problems in a linear subspace matching framework the embedding of the space of observations depends on the deformation model and is independent of the specific observed object hence it is universal we study two cases of this embedding that of elastic deformations of 1 d signals and the case of affine deformations of n dimensional signals

compressed sensing with basis mismatch performance bounds and sparse based estimator compressed sensing cs is a promising emerging domain that outperforms the classical limit of the shannon sampling theory if the measurement vector can be approximated as the linear combination of few basis vectors extracted from a redundant dictionary matrix unfortunately in realistic scenario the knowledge of this basis or equivalently of the entire dictionary is often uncertain i e corrupted by a basis mismatch bm error the consequence of the bm problem is that the estimation accuracy in terms of bayesian mean square error bmse of popular sparse based estimators collapses even if the support is perfectly estimated and in the high signal to noise ratio snr regime this saturation effect considerably limits the effective viability of these estimation schemes in the first part of this work the bayesian cramer rao bound bcrb is derived for cs model with unstructured bm we show that the bcrb foresees the saturation effect of the estimation accuracy of standard sparse based estimators as for instance the omp cosamp or the bp in addition we provide an approximation of this bmse threshold in the second part and in the context of the structured bm model a new estimation scheme called bias correction estimator bice is proposed and its statistical properties are studied the bice acts as a post processing estimation layer for any sparse based estimator and mitigates considerably the bm degradation finally the bice i is a blind algorithm i e is unaware of the uncorrupted dictionary matrix ii is generic since it can be associated to any sparse based estimator iii is fast i e the additional computational cost remains low and iv has good statistical properties to illustrate our results and propositions the bice is applied in the challenging context of the compressive sampling of non bandlimited impulsive signals

a discretization free sparse and parametric approach for linear array signal processing direction of arrival doa estimation in array pro cessing using uniform sparse linear arrays is concerned in this pa per while sparse methods via approximate parameter discretiza tion have been popular in the past decade the discretization may cause problems e g modeling error and increased computations due to dense sampling in this paper an exact discretization free method named as sparse and parametric approach spa is proposed for uniform and sparse linear arrays spa carries out parameter estimation in the continuous range based on well established covariance fitting criteria and convex optimization it guarantees to produce a sparse parameter estimate without discretization required by existing sparse methods theoretical analysis shows that the spa parameter estimator is a large snapshot realization of the maximum likelihood estimator and is statistically consistent in the number of snapshots under uncorrelated sources other merits of spa include improved resolution applicability to arbitrary number of snapshots ro bustness to correlation of the sources and no requirement of user parameters numerical simulations are carried out to verify our analysis and demonstrate advantages of spa compared to existing methods

uwb sparse diffuse channels part i channel models and bayesian estimators in this two part paper the problem of channel estimation in ultra wide band uwb systems is investigated due to the large transmission bandwidth the channel has been traditionally modeled as sparse however some propagation phenomena e g scattering from rough surfaces and frequency distortion are better modeled by a diffuse channel herein a novel hybrid sparse diffuse hsd channel model is proposed tailored to the hsd model channel estimators are designed for different scenarios that vary in the amount of side information available at the receiver an expectation maximization algorithm to estimate the power delay profile of the diffuse component is also designed the proposed methods are compared to unstructured and purely sparse estimators the numerical results show that the hsd estimation schemes considerably improve the estimation accuracy and the bit error rate performance over conventional channel estimators in part ii the new channel estimators are evaluated with more realistic geometry based channel emulators the numerical results show that even when the channel is generated in this manner the new estimation strategies achieve high performance moreover a mean squared error analysis of the proposed estimators is performed in the high and low signal to noise ratio regimes thus quantifying in closed form the achievable performance gains

hz algebra spectra are differential graded algebras we show that the homotopy theory of differential graded algebras coincides with the homotopy theory of h algebra spectra namely we construct quillen equivalences between the quillen model categories of unbounded differential graded algebras and h algebra spectra we also construct quillen equivalences between the differential graded modules and module spectra over these algebras we use these equivalences in turn to produce algebraic models for rational stable model categories we show that basically any rational stable model category is quillen equivalent to modules over a differential graded algebra with many objects

efficient and robust compressed sensing using optimized expander graphs expander graphs have been recently proposed to construct efficient compressed sensing algorithms in particular it has been shown that any n dimensional vector that is k sparse can be fully recovered using o klogn measurements and only o klogn simple recovery iterations in this paper we improve upon this result by considering expander graphs with expansion coefficient beyond 3 4 and show that with the same number of measurements only o k recovery iterations are required which is a significant improvement when n is large in fact full recovery can be accomplished by at most 2k very simple iterations the number of iterations can be reduced arbitrarily close to k and the recovery algorithm can be implemented very efficiently using a simple priority queue with total recovery time o nlog n k we also show that by tolerating a small penalty on the number of measurements and not on the number of recovery iterations one can use the efficient construction of a family of expander graphs to come up with explicit measurement matrices for this method we compare our result with other recently developed expander graph based methods and argue that it compares favorably both in terms of the number of required measurements and in terms of the time complexity and the simplicity of recovery finally we will show how our analysis extends to give a robust algorithm that finds the position and sign of the k significant elements of an almost k sparse signal and then using very simple optimization techniques finds a k sparse signal which is close to the best k term approximation of the original signal

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

fast monte carlo algorithm for site or bond percolation we describe in detail an efficient algorithm for studying site or bond percolation on any lattice the algorithm can measure an observable quantity in a percolation system for all values of the site or bond occupation probability from zero to one in an amount of time that scales linearly with the size of the system we demonstrate our algorithm by using it to investigate a number of issues in percolation theory including the position of the percolation transition for site percolation on the square lattice the stretched exponential behavior of spanning probabilities away from the critical point and the size of the giant component for site percolation on random graphs

replacing model categories with simplicial ones in this paper we show that model categories of a very broad class can be replaced up to quillen equivalence by simplicial model categories

complex valued signal processing the proper way to deal with impropriety complex valued signals occur in many areas of science and engineering and are thus of fundamental interest in the past it has often been assumed usually implicitly that complex random signals are proper or circular a proper complex random variable is uncorrelated with its complex conjugate and a circular complex random variable has a probability distribution that is invariant under rotation in the complex plane while these assumptions are convenient because they simplify computations there are many cases where proper and circular random signals are very poor models of the underlying physics when taking impropriety and noncircularity into account the right type of processing can provide significant performance gains there are two key ingredients in the statistical signal processing of complex valued data 1 utilizing the complete statistical characterization of complex valued random signals and 2 the optimization of real valued cost functions with respect to complex parameters in this overview article we review the necessary tools among which are widely linear transformations augmented statistical descriptions and wirtinger calculus we also present some selected recent developments in the field of complex valued signal processing addressing the topics of model selection filtering and source separation

multi scale multi lag channel estimation using low rank approximation for ofdm this paper considers the estimation of multi scale multi lag msml channels the msml channel model is a good representation for wideband communication channels such as underwater acoustic communication and radar this model is characterized by a limited number of paths each parameterized by a delay doppler scale and attenuation factor herein it is shown that an ofdm signal after passing through the msml channel exhibits a low rank representation this feature can be exploited to improve the channel estimation by characterizing the received signal it is shown that the msml channel estimation problem can be adapted to a structured spectral estimation problem the challenge is that the unknown frequencies are very close to each other due to the small values of doppler scales this feature can be employed to show that the data matrix is approximately low rank by exploiting structural features of the received signal the prony algorithm is modified to estimate the doppler scales close frequencies delays and channel gains two strategies using convex and no convex regularizers to remove noise from the corrupted signal are proposed these algorithms are iterative based on the alternating direction method of multipliers a bound on the reconstruction of the noiseless received signal provides guidance on the selection of the relaxation parameter in the optimizations the performance of the proposed estimation strategies are investigated via numerical simulations and it is shown that the proposed non convex method offers up to 7 db improvement in low snr and the convex method offers up to 5 db improvement in high snr over prior methods for the msml channel estimation

simplicial structures on model categories and functors we produce a highly structured way of associating a simplicial category to a model category which improves on work of dwyer and kan and answers a question of hovey we show that model categories satisfying a certain axiom are quillen equivalent to simplicial model categories a simplicial model category provides higher order structure such as composable mapping spaces and homotopy colimits we also show that certain homotopy invariant functors can be replaced by weakly equivalent simplicial or continuous functors this is used to show that if a simplicial model category structure exists on a model category then it is unique up to simplicial quillen equivalence

greedy adaptive linear compression in signal plus noise models in this paper we examine adaptive compression policies when the sequence of vector valued measurements to be compressed is noisy and the compressed variables are themselves noisy the optimization criterion is information gain in the case of sequential scalar compressions the unit norm compression vectors that greedily maximize per stage information gain are eigenvectors of an a priori error covariance matrix and the greedy policy selects them according to eigenvalues of a posterior covariance matrix these eigenvalues depend on all previous compressions and are computed recursively a water filling solution is given for the optimum compression policy that maximizes net information gain under a constraint on the average norm of compression vectors we provide sufficient conditions under which the greedy policy for maximizing stepwise information gain actually is optimal in the sense of maximizing the net information gain in the case of scalar compressions our examples and simulation results illustrate that the greedy policy can be quite close to optimal when the noise sequences are white

a uniqueness theorem for stable homotopy theory roughly speaking the stable homotopy category of algebraic topology is obtained from the homotopy category of topological spaces by inverting the suspension functor yielding a linear approximation to the homotopy category of spaces the isomorphism classes of objects in the stable homotopy category represent the generalized cohomology theories defined by the eilenberg steenrod axioms es45 without the dimension axiom which distinguishes ordinary from generalized cohomology theories the first construction of the full stable homotopy category was given by boardman b69 nowadays manymodels for the stable homotopy category are known most of which have the additional structure of a closed model category in the sense of quillen q67 in m83 h r margolis introduced a short list of axioms and conjectured that they characterize the stable homotopy category up to an equivalence of categories in theorem 3 2 we prove that these axioms do uniquely specify the stable homotopy category whenever there is some underlying quillen model category we also prove amore structured version theuniqueness theorembelow which states that the model category of spectra itself is uniquely determined by certain equivalent conditions up to so called quillen equivalence of

sensitivity to basis mismatch in compressed sensing the theory of compressed sensing suggests that successful inversion of an image of the physical world broadly defined to include speech signals radar sonar returns vibration records sensor array snapshot vectors 2 d images and so on for its source modes and amplitudes can be achieved at measurement dimensions far lower than what might be expected from the classical theories of spectrum or modal analysis provided that the image is sparse in an apriori known basis for imaging problems in spectrum analysis and passive and active radar sonar this basis is usually taken to be a dft basis however in reality no physical field is sparse in the dft basis or in any apriori known basis no matter how finely we grid the parameter space the sources may not lie in the center of the grid cells and consequently there is mismatch between the assumed and the actual bases for sparsity in this paper we study the sensitivity of compressed sensing to mismatch between the assumed and the actual sparsity bases we start by analyzing the effect of basis mismatch on the best k term approximation error which is central to providing exact sparse recovery guarantees we establish achievable bounds for the l1 error of the best k term approximation and show that these bounds grow linearly with the image or grid dimension and the mismatch level between the assumed and actual bases for sparsity we then derive bounds with similar growth behavior for the basis pursuit l1 recovery error indicating that the sparse recovery may suffer large errors in the presence of basis mismatch although we present our results in the context of basis pursuit our analysis applies to any sparse recovery principle that relies on the accuracy of best k term approximations for its performance guarantees we particularly highlight the problematic nature of basis mismatch in fourier imaging where spillage from off grid dft components turns a sparse representation into an incompressible one we substantiate our mathematical analysis by numerical examples that demonstrate a considerable performance degradation for image inversion from compressed sensing measurements in the presence of basis mismatch for problem sizes common to radar and sonar

esprit estimation of signal parameters via rotational invariance techniques an approach to the general problem of signal parameter estimation is described the algorithm differs from its predecessor in that a total least squares rather than a standard least squares criterion is used although discussed in the context of direction of arrival estimation esprit can be applied to a wide variety of problems including accurate detection and estimation of sinusoids in noise it exploits an underlying rotational invariance among signal subspaces induced by an array of sensors with a translational invariance structure the technique when applicable manifests significant performance and computational advantages over previous algorithms such as mem capon s mlm and music

compressive parameter estimation in awgn compressed sensing is by now well established as an effective tool for extracting sparsely distributed information where sparsity is a discrete concept referring to the number of dominant nonzero signal components in some basis for the signal space in this paper we establish a framework for estimation of continuous valued parameters based on compressive measurements on a signal corrupted by additive white gaussian noise awgn while standard compressed sensing based on naive discretization has been shown to suffer from performance loss due to basis mismatch we demonstrate that this is not an inherent property of compressive measurements our contributions are summarized as follows a we identify the isometries required to preserve fundamental estimation theoretic quantities such as the ziv zakai bound zzb and the cramer rao bound crb under such isometries compressive projections can be interpreted simply as a reduction in effective snr b we show that the threshold behavior of the zzb provides a criterion for determining the minimum number of measurements for accurate parameter estimation c we provide detailed computations of the number of measurements needed for the isometries in a to hold for the problem of frequency estimation in a mixture of sinusoids we show via simulations that the design criterion in b is accurate for estimating the frequency of a single sinusoid

reduce and boost recovering arbitrary sets of jointly sparse vectors the rapid developing area of compressed sensing suggests that a sparse vector lying in a high dimensional space can be accurately and efficiently recovered from only a small set of nonadaptive linear measurements under appropriate conditions on the measurement matrix the vector model has been extended both theoretically and practically to a finite set of sparse vectors sharing a common sparsity pattern in this paper we treat a broader framework in which the goal is to recover a possibly infinite set of jointly sparse vectors extending existing algorithms to this model is difficult due to the infinite structure of the sparse vector set instead we prove that the entire infinite set of sparse vectors can be recovered by solving a single reduced size finite dimensional problem corresponding to recovery of a finite set of sparse vectors we then show that the problem can be further reduced to the basic model of a single sparse vector by randomly combining the measurements our approach is exact for both countable and uncountable sets as it does not rely on discretization or heuristic techniques to efficiently find the single sparse vector produced by the last reduction step we suggest an empirical boosting strategy that improves the recovery ability of any given suboptimal method for recovering a sparse vector numerical experiments on random data demonstrate that when applied to infinite sets our strategy outperforms discretization techniques in terms of both run time and empirical recovery rate in the finite model our boosting algorithm has fast run time and much higher recovery rate than known popular methods

the alternating descent conditional gradient method for sparse inverse problems we propose a variant of the classical conditional gradient method for sparse inverse problems with differentiable observation models such models arise in many practical problems including superres

mismatch in the classification of linear subspaces sufficient conditions for reliable classification this paper considers the classification of linear subspaces with mismatched classifiers in particular we assume a model where one observes signals in the presence of isotropic gaussian noise and the distribution of the signals conditioned on a given class is gaussian with a zero mean and a low rank covariance matrix we also assume that the classifier knows only a mismatched version of the parameters of input distribution in lieu of the true parameters by constructing an asymptotic low noise expansion of an upper bound to the error probability of such a mismatched classifier we provide sufficient conditions for reliable classification in the low noise regime that are able to sharply predict the absence of a classification error floor such conditions are a function of the geometry of the true signal distribution the geometry of the mismatched signal distributions as well as the interplay between such geometries namely the principal angles and the overlap between the true and the mismatched signal subspaces numerical results demonstrate that our conditions for reliable classification can sharply predict the behavior of a mismatched classifier both with synthetic data and in a motion segmentation and a hand written digit classification applications

a multivariate generalization of prony s method abstract prony s method is a prototypical eigenvalue analysis based method for the reconstruction of a finitely supported complex measure on the unit circle from its moments up to a certain degree in this note we give a generalization of this method to the multivariate case and prove simple conditions under which the problem admits a unique solution provided the order of the moments is bounded from below by the number of points on which the measure is supported as well as by a small constant divided by the separation distance of these points stable reconstruction is guaranteed in its simplest form the reconstruction method consists of setting up a certain multilevel toeplitz matrix of the moments compute a basis of its kernel and compute by some method of choice the set of common roots of the multivariate polynomials whose coefficients are given in the second step all theoretical results are illustrated by numerical experiments

variational bayesian inference of line spectra we address the fundamental problem of line spectral estimation in a bayesian framework we target model order and parameter estimation via variational inference in a probabilistic model in which the frequencies are continuous valued i e not restricted to a grid and the coefficients are governed by a bernoulli gaussian prior model turning model order selection into binary sequence detection unlike earlier works which retain only point estimates of the frequencies we undertake a complete bayesian treatment by estimating the posterior probability density functions pdfs of the frequencies and computing expectations over them thus we additionally capture and operate with the uncertainty of the frequency estimates aiming to maximize the model evidence variational optimization provides analytic approximations of the posterior pdfs and also gives estimates of the additional parameters we propose an accurate representation of the frequency pdfs by mixtures of von mises pdfs which yields closed form expectations we define the algorithm valse in which the estimates of the pdfs and parameters are iteratively updated valse is a gridless convergent method does not require parameter tuning can easily include prior knowledge about the frequencies and provides approximate posterior pdfs based on which the uncertainty in line spectral estimation can be quantified simulation results show that accounting for the uncertainty of frequency estimates rather than computing just point estimates significantly improves the performance the performance of valse is superior to that of state of the art methods and closely approaches the cramer rao bound computed for the true model order

quantized spectral compressed sensing cramer rao bounds and recovery algorithms efficient estimation of wideband spectrum is of great importance for applications such as cognitive radio recently sub nyquist sampling schemes based on compressed sensing have been proposed to greatly reduce the sampling rate however the important issue of quantization has not been fully addressed particularly for high resolution spectrum and parameter estimation in this paper we aim to recover spectrally sparse signals and the corresponding parameters such as frequency and amplitudes from heavy quantizations of their noisy complex valued random linear measurements e g only the quadrant information we first characterize the cramer rao bound under gaussian noise which highlights the trade off between sample complexity and bit depth under different signal to noise ratios for a fixed budget of bits next we propose a new algorithm based on atomic norm soft thresholding for signal recovery which is equivalent to proximal mapping of properly designed surrogate signals with respect to the atomic norm that motivates spectral sparsity the proposed algorithm can be applied to both the single measurement vector case as well as the multiple measurement vector case it is shown that under the gaussian measurement model the spectral signals can be reconstructed accurately with high probability as soon as the number of quantized measurements exceeds the order of k log n where k is the level of spectral sparsity and n is the signal dimension finally numerical simulations are provided to validate the proposed approaches

backbone and elastic backbone of percolation clusters obtained by the new method of burning we introduce a method of obtaining information about percolation clusters in which the clusters are burned in different ways this method yields the backbone and the elastic backbone their number of loops shortest path and other quantities it works in any dimension and can be applied also to other clusters we obtain accurate estimates for the backbone exponents in two and three dimensions

esprit a subspace rotation approach to estimation of parameters of cisoids in noise the application of a subspace invariance approach esprit to the estimation of parameters frequencies and powers of cisoids in noise is described esprit exploits an underlying rotational invariance of signal subspaces spanned by two temporally displaced data sets the new approach has several advantages including improved resolution over pisarenko s technique for harmonic retrieval

music for multidimensional spectral estimation stability and super resolution this paper presents a performance analysis of the multiple signal classification music algorithm applied on d dimensional single snapshot spectral estimation while s true frequencies are located on the continuum of a bounded domain inspired by the matrix pencil form we construct a d fold hankel matrix from the measurements and exploit its vandermonde decomposition in the noiseless case music amounts to identifying a noise subspace evaluating a noise space correlation function and localizing frequencies by searching the s smallest local minima of the noise space correlation function in the noiseless case 2s d measurements guarantee an exact reconstruction by music as the noise space correlation function vanishes exactly at true frequencies when noise exists we provide an explicit estimate on the perturbation of the noise space correlation function in terms of noise level dimension d the minimum separation among frequencies the maximum and minimum amplitudes while frequencies are separated by 2 rayleigh length rl at each direction as a by product the maximum and minimum non zero singular values of the multidimensional vandermonde matrix whose nodes are on the unit sphere are estimated under a gap condition of the nodes under the 2 rl separation condition if noise is i i d gaussian we show that perturbation of the noise space correlation function decays like sqrt log bf n bf n as the sample size bf n increases when the separation among frequencies drops below 2 rl our numerical experiments show that the noise tolerance of music obeys a power law with the minimum separation of frequencies

sparse total least squares analysis and greedy algorithms linear systems where both the matrix and the right hand side are affected by noise or uncertainties are typically solved via a total least squares tls approach we study the sparse tls problem with solutions having s nonzero elements we present a set of conditions under which the sparse least squares ls and tls problems have the same support the conditions are expressed in terms of the restricted isometry constants and the minimum nonzero principal angle between subspaces made of s columns of the matrix we also study several greedy algorithms for solving the sparse tls problem inspired by our theoretical results we infer that an effective approach is to compute the support of the ls solution then solve the standard tls problem restrained to that support indeed doing so with orthogonal matching pursuit omp gives consistently good results another greedy strategy is to extend the support with the column that minimizes the smallest angle with the subspace sas corresponding to the already chosen support which is a combined ls tls heuristic we show via simulations that sas gives the best results for small problem sizes and high and medium signal to noise ratio snr while omp is best or nearly best in almost all cases

threshold behavior of the maximum likelihood estimator of frequency we present accurate simplifications of the rife and boorstyn 1974 performance equations for the maximum likelihood estimator of frequency the simplicity of the result allows quick calculation of the onset of threshold as a function of sample size and signal to noise ratio snr the accuracy of the new expression is demonstrated via simulation

sparse semi parametric estimation of harmonic chirp signals in this paper we present a method for estimating the parameters detailing an unknown number of linear possibly harmonically related chirp signals using an iterative sparse reconstruction framework the proposed method is initiated by a re weighted group sparsity approach followed by an iterative relaxation based refining step to allow for high resolution estimates numerical simulations illustrate the achievable performance offering a notable improvement as compared to other recent approaches the resulting estimates are found to be statistically efficient achieving the corresponding cramer rao lower bound

direction of arrival estimation using co prime arrays a super resolution viewpoint we consider the problem of direction of arrival doa estimation using a recently proposed structure of nonuniform linear arrays referred to as co prime arrays by exploiting the second order statistical information of the received signals co prime arrays exhibit o mn degrees of freedom with only m n sensors a sparsity based recovery algorithm is proposed to fully utilize these degrees of freedom the suggested method is based on the developing theory of super resolution which considers a continuous range of possible sources instead of discretizing this range onto a grid with this approach off grid effects inherent in traditional sparse recovery can be neglected thus improving the accuracy of doa estimation we show that in the noiseless case it is theoretically possible to detect up to mn 2 sources with only 2m n sensors the noise statistics of co prime arrays are also analyzed to demonstrate the robustness of the proposed optimization scheme a source number detection method is presented based on the spectrum reconstructed from the sparse method by extensive numerical examples we show the superiority of the suggested algorithm in terms of doa estimation accuracy degrees of freedom and resolution ability over previous techniques such as music with spatial smoothing and discrete sparse recovery

frequency rate estimation at high snr the problem of estimating the frequency rate of change of complex valued frequency modulated signals from noisy observations is considered the performance of four related estimators is studied both analytically and by means of simulations and their relationship to the estimators proposed by djuric and kay 1990 and lang and musicus 1989 is established

robust recovery of stream of pulses using convex optimization abstract this paper considers the problem of recovering the delays and amplitudes of a weighted superposition of pulses this problem is motivated by a variety of applications such as ultrasound and radar we show that for univariate and bivariate stream of pulses one can recover the delays and weights to any desired accuracy by solving a tractable convex optimization problem provided that a pulse dependent separation condition is satisfied the main result of this paper states that the recovery is robust to additive noise or model mismatch

gridless line spectrum estimation and low rank toeplitz matrix compression using structured samplers a regularization free approach this paper considers the problem of compressively sampling wide sense stationary random vectors with a low rank toeplitz covariance matrix certain families of structured deterministic samplers are shown to efficiently compress a high dimensional toeplitz matrix of size n times n producing a compressed sketch of size o sqrt r times o sqrt r the reconstruction problem can be cast as that of line spectrum estimation whereby in absence of noise toeplitz matrices of any size n can be exactly recovered from compressive sketches of size o sqrt r times o sqrt r no matter how large n is in the presence of noise and finite data the line spectrum estimation algorithm is combined with a novel denoising technique that only exploits a positive semidefinite psd toeplitz constraint to denoise the compressed sketch using a simple least squares minimization framework a major advantage of the algorithm is that it does not require any regularization parameter it also enjoys lower computational complexity owing to its ability to predict the unobserved entries of the low rank toeplitz matrix explicit bounds on the reconstruction error are established and it is shown that the psd constraint on the denoiser is sufficient to ensure stable reconstruction from a sketch of size o sqrt r times o sqrt r extensive simulations demonstrate that the proposed algorithm provides better performance over random samplers and algorithms that use nuclear norm based regularizers

cramer rao bounds for deterministic modal analysis the accuracy with which deterministic modes can be identified from a finite record of noisy data is determined by computing the cramer rao bound on the error covariance matrix of any unbiased estimator of mode parameters the bound is computed for many of the standard parametric descriptions of a mode including autoregressive and moving average parameters poles and residues and poles and zeros asymptotic frequency domain versions of the cramer rao bound bring insight into the role played by poles and zeros application of the bound to second and fourth order systems illustrates the influence of mode locations on the ability to identify them application of the bound to the estimation of an energy spectrum illuminates the accuracy of estimators that presume to resolve spectral peaks

a fast and accurate reconstruction algorithm for compressed sensing of complex sinusoids the standard compressed sensing cs theory reconstructs a signal by recovering a sparse representation of the signal over a pre specified dictionary for cs of complex sinusoids this dictionary is usually set to be a dft matrix corresponding to a uniform frequency grid however such a setting can make conventional cs reconstruction methods degrade considerably since component frequencies of practical signals do not necessarily align with the specified grid to deal with this problem we apply a linear approximation to the true unknown dictionary and establish a more accurate model for sparse approximation of practical complex sinusoids based on this model signal reconstruction is reformulated as a problem that recovers two sparse coefficient vectors over two known dictionaries under the constraint that the vectors share the same support to solve such a problem we develop a fast iterative algorithm under a variational bayesian inference framework results of extensive numerical experiments demonstrate that the algorithm can achieve cs of complex sinusoids with low computational cost as well as high reconstruction accuracy

algebras and modules in monoidal model categories in recent years the theory of structured ring spectra formerly known as a infty and e infty ring spectra has been simplified by the discovery of categories of spectra with strictly associative and commutative smash products now a ring spectrum can simply be defined as a monoid with respect to the smash product in one of these new categories of spectra in this paper we provide a general method for constructing model category structures for categories of ring algebra and module spectra this provides the necessary input for obtaining model categories of symmetric ring spectra functors with smash product gamma rings and diagram ring spectra algebraic examples to which our methods apply include the stable module category over the group algebra of a finite group and unbounded chain complexes over a differential graded algebra 1991 mathematics subject classification primary 55u35 secondary 18d10

singular value decomposition and improved frequency estimation using linear prediction linear prediction based lp methods for fitting multiple sinusoid signal models to observed data such as the forward backward fblp method of nuttall 5 and ulrych and clayton 6 are very ill conditioned the locations of estimated spectral peaks can be greatly affected by a small amount of noise because of the appearance of outliers lp estimation of frequencies can be greatly improved at low snr by singular value decomposition svd of the lp data matrix the improved performance at low snr is also better than that obtained by using the eigenvector corresponding to the minimum eigenvalue of the correlation matrix as is done in pisarenko s method and its variants

on the stable resolution limit of total variation regularization for spike deconvolution the stability of spike deconvolution which aims at recovering point sources from their convolution with a point spread function psf is known to be related to the separation between those sources when the observations are noisy it is critical to ensure support stability where the deconvolution does not lead to spurious or oppositely missing estimates of the point sources in this paper we study the resolution limit of stably recovering the support of two closely located point sources using the beurling lasso estimator which is a convex optimization approach based on total variation regularization we establish a sufficient separation criterion between the sources depending only on the psf above which the beurling lasso estimator is guaranteed to return a stable estimate of the point sources with the same number of estimated elements as that of the ground truth our result highlights the impact of psf on the resolution limit in the noisy setting which was not evident in previous studies of the noiseless setting towards the end we show that the same resolution limit applies to resolving two close located sources in conjunction of other well separated sources

compressive music revisiting the link between compressive sensing and array signal processing the multiple measurement vector mmv problem addresses the identification of unknown input vectors that share common sparse support even though mmv problems have been traditionally addressed within the context of sensor array signal processing the recent trend is to apply compressive sensing cs due to its capability to estimate sparse support even with an insufficient number of snapshots in which case classical array signal processing fails however cs guarantees the accurate recovery in a probabilistic manner which often shows inferior performance in the regime where the traditional array signal processing approaches succeed the apparent dichotomy between the probabilistic cs and deterministic sensor array signal processing has not been fully understood the main contribution of the present article is a unified approach that revisits the link between cs and array signal processing first unveiled in the mid 1990s by feng and bresler the new algorithm which we call compressive music identifies the parts of support using cs after which the remaining supports are estimated using a novel generalized music criterion using a large system mmv model we show that our compressive music requires a smaller number of sensor elements for accurate support recovery than the existing cs methods and that it can approach the optimal bound with finite number of snapshots even in cases where the signals are linearly dependent

a fast and accurate single frequency estimator a novel frequency estimator for a single complex sinusoid in complex white gaussian noise is proposed the estimator is more computationally efficient that the optimal maximum likelihood estimator yet attains equally good performance at moderately high signal to noise ratios the estimator is shown to be related to the linear prediction estimator this relationship is used to reveal why the linear prediction estimator does not attain the cramer rao bound even at high signal to noise ratios

low complexity and high resolution line spectral estimation using cyclic minimization the line spectral estimation problem has applications in radar wireless communications spectroscopy and power electronics among others the signal is modeled as a sparse linear combination of complex sinusoids and the problem target is to estimate the number of sinusoids in the mixture and respective parameters of each individual sinusoid such as magnitude phase and frequency in this work we first introduce a novel formulation of the rank function which involves the solution of a multi convex optimization problem using the multi convex reformulation of the rank function an over parameterization of the line estimation problem is proposed together with a cyclic minimization procedure to obtain a solution the cyclic method iterates between the optimization of the signal subspace and null space and stops when the two are orthogonal every limit point of the sequence of iterates is shown to be a stationary point of the original problem numerical experiments show that only a small number of iterations is required for convergence the signal subspace optimization is a semidefinite program sdp and the null space optimization has a closed form solution the cost per iteration of a general purpose interior point method ipm to solve the sdp is a quartic function of the problem dimension it is shown that by exploiting the problem structure the cost per iteration of the ipm may be reduced to a cubic function of the problem dimension comparable to the cost of the alternating direction method of multipliers admm however contrarily to the latter the former achieves the precision necessary for fine frequency localization a large set of numerical experiments show the effectiveness of the proposed approach

grid free compressive beamforming the direction of arrival doa estimation problem involves the localization of a few sources from a limited number of observations on an array of sensors thus it can be formulated as a sparse signal reconstruction problem and solved efficiently with compressive sensing cs to achieve high resolution imaging on a discrete angular grid the cs reconstruction degrades due to basis mismatch when the doas do not coincide with the angular directions on the grid to overcome this limitation a continuous formulation of the doa problem is employed and an optimization procedure is introduced which promotes sparsity on a continuous optimization variable the doa estimation problem with infinitely many unknowns i e source locations and amplitudes is solved over a few optimization variables with semidefinite programming the grid free cs reconstruction provides high resolution imaging even with non uniform arrays single snapshot data and under noisy conditions as demonstrated on experimental towed array data

off the grid line spectrum denoising and estimation with multiple measurement vectors compressed sensing suggests that the required number of samples for reconstructing a signal can be greatly reduced if it is sparse in a known discrete basis yet many real world signals are sparse in a continuous dictionary one example is the spectrally sparse signal which is composed of a small number of spectral atoms with arbitrary frequencies on the unit interval in this paper we study the problem of line spectrum denoising and estimation with an ensemble of spectrally sparse signals composed of the same set of continuous valued frequencies from their partial and noisy observations two approaches are developed based on atomic norm minimization and structured covariance estimation both of which can be solved efficiently via semidefinite programming the first approach aims to estimate and denoise the set of signals from their partial and noisy observations via atomic norm minimization and recover the frequencies via examining the dual polynomial of the convex program we characterize the optimality condition of the proposed algorithm and derive the expected error rate for denoising demonstrating the benefit of including multiple measurement vectors the second approach aims to recover the population covariance matrix from the partially observed sample covariance matrix by motivating its low rank toeplitz structure without recovering the signal ensemble performance guarantee is derived with a finite number of measurement vectors the frequencies can be recovered via conventional spectrum estimation methods such as music from the estimated covariance matrix finally numerical examples are provided to validate the favorable performance of the proposed algorithms with comparisons against several existing approaches

an algebraic model for free rational g spectra for connected compact lie groups g we show that the category of free rational g spectra for a connected compact lie group g is quillen equivalent to the category of torsion differential graded modules over the polynomial ring h bg the ingredients are the enriched morita equivalences of schwede and shipley topology 42 1 103 153 2003 the functors of shipley am j math 129 351 379 2007 making rational spectra algebraic koszul duality and thick subcategory arguments based on the simplicity of the derived category of a polynomial ring

percolation and cluster structure parameters the enhanced hoshen kopelman algorithm an enhanced hoshen kopelman ehk algorithm for calculating cluster structure parameters is presented for the site percolation problem the ehk algorithm enables efficient calculation of cluster spatial moments perimeters and bounding boxes for very large monte carlo simulated lattices the algorithm is used to compute the squared radius of gyration and internal perimeters of clusters in 3000 ifmmode times else texttimes fi 3000 square simulated lattices the squared radius of gyration was used to calculate the value of the correlation length exponent

two dimensional grid free compressive beamforming compressive beamforming realizes the direction of arrival doa estimation and strength quantification of acoustic sources by solving an underdetermined system of equations relating microphone pressures to a source distribution via compressive sensing the conventional method assumes doas of sources to lie on a grid its performance degrades due to basis mismatch when the assumption is not satisfied to overcome this limitation for the measurement with plane microphone arrays a two dimensional grid free compressive beamforming is developed first a continuum based atomic norm minimization is defined to denoise the measured pressure and thus obtain the pressure from sources next a positive semidefinite programming is formulated to approximate the atomic norm minimization subsequently a reasonably fast algorithm based on alternating direction method of multipliers is presented to solve the positive semidefinite programming finally the matrix enhancement and matrix pencil method is introduced to process the obtained pressure and reconstruct the source distribution both simulations and experiments demonstrate that under certain conditions the grid free compressive beamforming can provide high resolution and low contamination imaging allowing accurate and fast estimation of two dimensional doas and quantification of source strengths even with non uniform arrays and noisy measurements

estimating the frequency of a noisy sinusoid by linear regression corresp the estimation of the parameters of a sinusoid from observations of signal samples corrupted by additive noise is investigated at high signal to noise ratios the additive noise is viewed as an equivalent phase noise suggesting frequency and phase estimation by linear regression on the signal phase the variances of the regression estimates are shown to achieve the cramer rao bounds a formula for the variance of the regression frequency estimator is derived in terms of the noise power spectrum a simple formula for the variance with 1 f 2 phase noise is presented

s modules and symmetric spectra we study a symmetric monoidal adjoint functor pair between the category of s modules and the category of symmetric spectra the functors induce equivalences between the respective homotopy categories of spectra module spectra and ring spectra then at approximately the same time two categories of spectra with nice smash products were discov ered elmendorf kriz mandell and may constructed the category of s modules ekmm and je smith introduced symmetric spectra hss both categories are quillen model categories and have associated notions of ring and module spectra however these two categories arise in completely dierent ways and even though the homotopy categories are equivalent it is not a priori clear if both frameworks give rise to the same homotopy theory of rings and modules both categories have their merits described in detail in the introductions of ekmm and hss and it is desirable to be able to translate results obtained in one category into conclusions valid in the other the present paper describes an easy mechanism which facilitates such comparisons below we dene a lax symmetric monoidal functor ms sp from the category of s modules to the category of symmetric spectra the functor preserves homotopy groups and has a strong symmetric monoidal left adjoint we show that the two functors induce inverse equivalences of the homotopy categories of spectra ring spectra commutative ring spectra and module spectra main theorem the functor from the category ms of s modules to the category sp of symmetric spectra passes to a symmetric monoidal equivalence of homotopy categories

analysis of fisher information and the cramer rao bound for nonlinear parameter estimation after random compression in this paper we analyze the impact of compression with complex random matrices on fisher information and the cramer rao bound crb for estimating unknown parameters in the mean value function of a complex multivariate normal distribution we consider the class of random compression matrices whose distribution is right unitarily invariant the compression matrix whose elements are i i d standard complex normal random variables is one such matrix we show that for all such compression matrices the fisher information matrix has a complex matrix beta distribution we also derive the distribution of crb these distributions can be used to quantify the loss in crb as a function of the fisher information of the noncompressed data in our numerical examples we consider a direction of arrival estimation problem and discuss the use of these distributions as guidelines for choosing compression ratios based on the resulting loss in crb

block sparse bayesian learning based nb iot interference elimination in lte advanced systems narrowband internet of things nb iot is one of the emerging 5g technologies but might introduce narrowband interference nbi to existing broadband systems such as long term evolution advanced lte a systems thus the mitigation of the nb iot interference to lte a is an important issue for the harmonic coexistence and compatibility between 4g and 5g in this paper a newly emerged sparse approximation technique block sparse bayesian learning bsbl is utilized to estimate the nb iot interference in lte a systems the block sparse representation of the nbi is constituted through the proposed temporal differential measuring approach and the bsbl theory is utilized to recover the practical block sparse nbi a bsbl based method partition estimated bsbl is proposed with the aid of the estimated block partition beforehand the bayesian parameters are obtained to yield the nbi estimation the intra block correlation ibc is considered to facilitate the recovery moreover exploiting the inherent structure of the identical ibc matrix another method of informative bsbl is proposed to further improve the accuracy which does not require prior estimation of the block partition reported simulation results demonstrate that the proposed methods are effective in canceling the nb iot interference in lte a systems and significantly outperform other conventional methods

estimating two dimensional frequencies by matrix enhancement and matrix pencil a new method called the matrix enhancement and matrix pencil memp method is presented for estimating two dimensional 2 d frequencies in the memp method an enhanced matrix is constructed from the data samples and then the matrix pencil approach is used to extract out the 2 d sinusoids from the principal eigenvectors of the enhanced matrix the memp method yields the estimates of the 2 d frequencies efficiently without solving the roots of a 2 d polynomial or searching in a 2 d space it is shown that the memp method can be faster than a 2 d fft method if the number of the 2 d sinusoids is much smaller than the data set simulation results are provided to show that the accuracy of the memp method can be very close to the cramer rao lower bound

compressed sensing off the grid this paper investigates the problem of estimating the frequency components of a mixture of s complex sinusoids from a random subset of n regularly spaced samples unlike previous work in compressed sensing the frequencies are not assumed to lie on a grid but can assume any values in the normalized frequency domain 0 1 an atomic norm minimization approach is proposed to exactly recover the unobserved samples and identify the unknown frequencies which is then reformulated as an exact semidefinite program even with this continuous dictionary it is shown that o slog s log n random samples are sufficient to guarantee exact frequency localization with high probability provided the frequencies are well separated extensive numerical experiments are performed to illustrate the effectiveness of the proposed method

hankel matrix nuclear norm regularized tensor completion for n dimensional exponential signals signals are generally modeled as a superposition of exponential functions in spectroscopy of chemistry biology and medical imaging for fast data acquisition or other inevitable reasons however only a small amount of samples may be acquired and thus how to recover the full signal becomes an active research topic but existing approaches cannot efficiently recover n dimensional exponential signals with n geq 3 in this paper we study the problem of recovering n dimensional particularly n geq 3 exponential signals from partial observations and formulate this problem as a low rank tensor completion problem with exponential factor vectors the full signal is reconstructed by simultaneously exploiting the candecomp parafac tensor structure and the exponential structure of the associated factor vectors the latter is promoted by minimizing an objective function involving the nuclear norm of hankel matrices experimental results on simulated and real magnetic resonance spectroscopy data show that the proposed approach can successfully recover full signals from very limited samples and is robust to the estimated tensor rank

towards a mathematical theory of super resolution this paper develops a mathematical theory of super resolution broadly speaking super resolution is the problem of recovering the fine details of an object the high end of its spectrum from coarse scale information only from samples at the low end of the spectrum suppose we have many point sources at unknown locations in 0 1 and with unknown complex valued amplitudes we only observe fourier samples of this object up to a frequency cutoff fc we show that one can super resolve these point sources with infinite precision i e recover the exact locations and amplitudes by solving a simple convex optimization problem which can essentially be reformulated as a semidefinite program this holds provided that the distance between sources is at least 2 fc this result extends to higher dimensions and other models in one dimension for instance it is possible to recover a piecewise smooth function by resolving the discontinuity points with infinite precision as well we also show that the theory and methods are robust to noise in particular in the discrete setting we develop some theoretical results explaining how the accuracy of the super resolved signal is expected to degrade when both the noise level and the super resolution factor vary 2014 wiley periodicals inc

monoidal uniqueness of stable homotopy theory we show that the monoidal product on the stable homotopy category of spectra is essentially unique this strengthens work of this author with schwede on the uniqueness of models of the stable homotopy theory of spectra also the equivalences constructed here give a unified construction of the known equivalences of the various symmetric monoidal categories of spectra s modules w spaces orthogonal spectra simplicial functors with symmetric spectra as an application we show that with an added assumption about underlying model structures margolis axioms uniquely determine the stable homotopy category of spectra up to monoidal equivalence

effects of sampling and quantization on single tone frequency estimation the effects of sampling and quantization on frequency estimation for a single sinusoid are investigated the cramer rao bound for 1 bit quantization is derived and compared with the limit of infinite quantization it is found that 1 bit quantization gives a slightly worse performance however with a dramatic increase of variance at certain frequencies this can be avoided by using four times oversampling the effect of sampling when using nonideal antialiasing lowpass filters is therefore investigated through derivation of the cramer rao lower bounds finally fast estimators for 1 bit quantization in particular correlation based estimators are derived and their performance is investigated the paper is concluded with simulation results for four times oversampled 1 bit quantization

towards generalized fri sampling with an application to source resolution in radioastronomy it is a classic problem to estimate continuous time sparse signals like point sources in a direction of arrival problem or pulses in a time of flight measurement the earliest occurrence is the estimation of sinusoids in time series using prony s method this is at the root of a substantial line of work on high resolution spectral estimation the estimation of continuous time sparse signals from discrete time samples is the goal of the sampling theory for finite rate of innovation fri signals both spectral estimation and fri sampling usually assume uniform sampling but not all measurements are obtained uniformly as exemplified by a concrete radioastronomy problem we set out to solve thus we develop the theory and algorithm to reconstruct sparse signals typically sum of sinusoids from nonuniform samples we achieve this by identifying a linear transformation that relates the unknown uniform samples of sinusoids to the given measurements these uniform samples are known to satisfy the annihilation equations a valid solution is then obtained by solving a constrained minimization such that the reconstructed signal is consistent with the given measurements and satisfies the annihilation constraint thanks to this new approach we unify a variety of fri based methods we demonstrate the versatility and robustness of the proposed approach with five fri reconstruction problems namely dirac reconstructions with irregular time or fourier domain samples fri curve reconstructions dirac reconstructions on the sphere and point source reconstructions in radioastronomy the proposed algorithm improves substantially over state of the art methods and is able to reconstruct point sources accurately from irregularly sampled fourier measurements under severe noise conditions

super resolution delay doppler estimation for ofdm passive radar in this paper we consider the problem of joint delay doppler estimation of moving targets in a passive radar that makes use of orthogonal frequency division multiplexing communication signals a compressed sensing algorithm is proposed to achieve supper resolution and better accuracy using both the atomic norm and the ell 1 norm the atomic norm is used to manifest the signal sparsity in the continuous domain unlike previous works that assume the demodulation to be error free we explicitly introduce the demodulation error signal whose sparsity is imposed by the ell 1 norm on this basis the delays and doppler frequencies are estimated by solving a semidefinite program sdp which is convex we also develop an iterative method for solving this sdp via the alternating direction method of multipliers where each iteration involves closed form computation simulation results are presented to illustrate the high performance of the proposed algorithm

recovery of sparse translation invariant signals with continuous basis pursuit we consider the problem of decomposing a signal into a linear combination of features each a continuously translated version of one of a small set of elementary features although these constituents are drawn from a continuous family most current signal decomposition methods rely on a finite dictionary of discrete examples selected from this family e g shifted copies of a set of basic waveforms and apply sparse optimization methods to select and solve for the relevant coefficients here we generate a dictionary that includes auxiliary interpolation functions that approximate translates of features via adjustment of their coefficients we formulate a constrained convex optimization problem in which the full set of dictionary coefficients represents a linear approximation of the signal the auxiliary coefficients are constrained so as to only represent translated features and sparsity is imposed on the primary coefficients using an l1 penalty the basis pursuit denoising bp method may be seen as a special case in which the auxiliary interpolation functions are omitted and we thus refer to our methodology as continuous basis pursuit cbp we develop two implementations of cbp for a one dimensional translation invariant source one using a first order taylor approximation and another using a form of trigonometric spline we examine the tradeoff between sparsity and signal reconstruction accuracy in these methods demonstrating empirically that trigonometric cbp substantially outperforms taylor cbp which in turn offers substantial gains over ordinary bp in addition the cbp bases can generally achieve equally good or better approximations with much coarser sampling than bp leading to a reduction in dictionary dimensionality