- Sparsity and compressed sensing in mono-static and multi-static radar imaging
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A powerful constraint on matrices are constraints on the matrix rank. For example, in low-rank matrix recovery, the goal is to reconstruct a low-rank matrix given only a subset of its entries. Importantly, low-rank matrices also lie in a union of subspaces structure, although now, there are infinitely many subspaces though each of these is finite dimensional. Many other examples of union of subspaces signal models appear in applications, including sparse wavelet-tree structures which form a subset of the general sparse model and finite rate of innovations models, where we can have infinitely many infinite dimensional subspaces.

In this chapter, I will provide an introduction to these and related geometrical concepts and will show how they can be used to a develop algorithms to recover signals with given structures and b allow theoretical results that characterise the performance of these algorithmic approaches. The problem of sparse signal recovery from a relatively small number of noisy measurements has been studied extensively in the recent compressed sensing literature. From a statistical point of view, this problem is equivalent to maximum a posteriori probability MAP parameter estimation with Laplace prior on the vector of parameters i.

Classical results in compressed sensing e. A natural question to ask is whether one can accurately recover sparse signals under different noise assumptions. Optimization problems involving the minimization of the rank of a matrix subject to certain constraints are pervasive in a broad range of disciples, such as control theory [ 6 , 26 , 31 , 62 ], signal processing [ 25 ], and machine learning [ 3 , 77 , 89 ].

However, solving such rank minimization problems is usually very difficult as they are NP-hard in general [ 65 , 75 ]. The nuclear norm of a matrix, as the tightest convex surrogate of the matrix rank, has fueled much of the recent research and has proved to be a powerful tool in many areas. In this chapter, we aim to provide a brief review of some of the state-of-the-art in nuclear norm optimization algorithms as they relate to applications. We then propose a novel application of the nuclear norm to the linear model recovery problem, as well as a viable algorithm for solution of the recovery problem.

Preliminary numerical results presented here motivates further investigation of the proposed idea. It is more and more common to encounter applications where the collected data is most naturally stored or represented in a multi-dimensional array, known as a tensor. The goal is often to approximate this tensor as a sum of some type of combination of basic elements, where the notation of what is a basic element is specific to the type of factorization employed.

## Sparsity and compressed sensing in mono-static and multi-static radar imaging

If the number of terms in the combination is few, the tensor factorization gives implicitly a sparse approximate representation of the data. The terms e. This chapter highlights recent developments in the area of non-negative tensor factorization which admit such sparse representations.

Specifically, we consider the approximate factorization of third and fourth order tensors into non-negative sums of types of outer-products of objects with one dimension less using the so-called t-product.

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A demonstration on an application in facial recognition shows the potential promise of the overall approach. We discuss a number of algorithmic options for solving the resulting optimization problems, and modification of such algorithms for increasing the sparsity. Cognitive radio has become one of the most promising solutions for addressing the spectral under-utilization problem in wireless communication systems.

As a key technology, spectrum sensing enables cognitive radios to find spectrum holes and improve spectral utilization efficiency. To exploit more spectral opportunities, wideband spectrum sensing approaches should be adopted to search multiple frequency bands at a time. Sub-Nyquist sampling and compressed sensing play crucial roles in the efficient implementation of wideband spectrum sensing in cognitive radios. In this chapter, Sect. A literature review of spectrum sensing algorithms is given in Sect.

Wideband spectrum sensing algorithms are then discussed in Sect. Special attention is paid to the use of Sub-Nyquist sampling and compressed sensing techniques for realizing wideband spectrum sensing. Finally, Sect. In this chapter system identification algorithms for sparse nonlinear multi input multi output MIMO systems are developed. These algorithms are potentially useful in a variety of application areas including digital transmission systems incorporating power amplifier s along with multiple antennas, cognitive processing, adaptive control of nonlinear multivariable systems, and multivariable biological systems.

Sparsity is a key constraint imposed on the model. The presence of sparsity is often dictated by physical considerations as in wireless fading channel—estimation. In other cases it appears as a pragmatic modelling approach that seeks to cope with the curse of dimensionality, particularly acute in nonlinear systems like Volterra type series. Three identification approaches are discussed: conventional identification based on both input and output samples, Semi-Blind identification placing emphasis on minimal input resources and blind identification whereby only output samples are available plus a—priori information on input characteristics.

Based on this taxonomy a variety of algorithms, existing and new, are studied and evaluated by simulations. In this chapter, we present the optimization formulation of the Kalman filtering and smoothing problems, and use this perspective to develop a variety of extensions and applications. We first formulate classic Kalman smoothing as a least squares problem, highlight special structure, and show that the classic filtering and smoothing algorithms are equivalent to a particular algorithm for solving this problem.

Once this equivalence is established, we present extensions of Kalman smoothing to systems with nonlinear process and measurement models, systems with linear and nonlinear inequality constraints, systems with outliers in the measurements or sudden changes in the state, and systems where the sparsity of the state sequence must be accounted for. The first part of this chapter presents a novel Kalman filtering-based method for estimating the coefficients of sparse, or more broadly, compressible autoregressive models using fewer observations than normally required.

By virtue of its unscented Kalman filter mechanism, the derived method essentially addresses the main difficulties attributed to the underlying estimation problem. In particular, it facilitates sequential processing of observations and is shown to attain a good recovery performance, particularly under substantial deviations from ideal conditions, those which are assumed to hold true by the theory of compressive sensing. In the remaining part of this chapter we derive a few information-theoretic bounds pertaining to the problem at hand.

The obtained bounds establish the relation between the complexity of the autoregressive process and the attainable estimation accuracy through the use of a novel measure of complexity. This measure is suggested herein as a substitute to the generally incomputable restricted isometric property. This chapter presents selective gossip which is an algorithm that applies the idea of iterative information exchange to vectors of data. Instead of communicating the entire vector and wasting network resources, our method adaptively focuses communication on the most significant entries of the vector.

We prove that nodes running selective gossip asymptotically reach consensus on these significant entries, and they simultaneously reach an agreement on the indices of entries which are insignificant. The results demonstrate that selective gossip provides significant communication savings in terms of the number of scalars transmitted. In the second part of the chapter we propose a distributed particle filter employing selective gossip. We show that distributed particle filters employing selective gossip provide comparable results to the centralized bootstrap particle filter while decreasing the communication overhead compared to using randomized gossip to distribute the filter computations.

In this chapter we describe our recent work on the design and analysis of recursive algorithms for causally reconstructing a time sequence of approximately sparse signals from a greatly reduced number of linear projection measurements. The signals are sparse in some transform domain referred to as the sparsity basis and their sparsity patterns support set of the sparsity basis coefficients can change with time. We also briefly summarize our exact reconstruction results for the noise-free case and our error bounds and error stability results conditions under which a time-invariant and small bound on the reconstruction error holds at all times for the noisy case.

A number of efforts have been devoted to improving the performance of channel estimation. The method is using coded auxiliary pilot symbols to eliminate the imaginary interference on each scatted pilot. However, these schemes have a higher computational complexity, and the phase ambiguity may occur and need longer observation data, which, to some extent, limits the availability. A more attractive approach to obtain well channel estimation performance is the recently researched compressive sensing CS method [ 22 , 23 , 24 , 25 , 26 , 27 ], where the wireless channels in practice tend to exhibit a sparse multipath structure.

Some channel estimation based on CS methods for OFDM systems have been studied in the past few years [ 28 , 29 , 30 , 31 ]. It is proved that the OMP [ 33 ] based method can get remarkable performance improvement compared with the conventional preamble based method. For the most greedy CS algorithms, such as OMP and compressive sampling matching pursuit CoSaMP [ 34 ], the sparsitylevel of the channel is given as a priori information. However, the sparsity of the channel is usually unknown in most practical application scenarios.

Furthermore, the proposed algorithm is based on the idea of regularization and the backtracking mechanism that attaches to CoSaMP algorithm, which removes the unreliable support and refines the current approximation iteratively. Simulations verify the proposed channel estimation scheme performs better than the conventional SAMP algorithm and the proposed algorithm can obtain an approximate performance compared with the CoSaMP algorithm.

The purpose of this paper is to propose an efficient sparse adaptive channel estimation method.

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We would like to convince the reader with the potential of the proposed method as a high performance channel estimator. The remainder of this paper is organized as follows. Section 3 reviews some conventional channel estimation methods including preamble-based methods and conventional CS recovery algorithms and presents the proposed scheme.

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In Section 4 , the performances of the proposed scheme associated with the conventional preamble-based and CS-based schemes are compared and simulation results are shown. Finally, Section 5 gives the concluding remarks. The design of pulse g enables the associated sub-carrier functions g m , n to be orthogonal in the real field,. We can find that, even in the distortion-free channel and with perfect time and frequency synchronization, some purely imaginary inter-carrier interference at the output still be existed, thus, we set interference weights.

Through the channel, with an additive noise, the received signal can be expressed as. We first review the two classical preamble structures and CS theory for channel estimation. Then, we propose the new CS algorithm. Along with the algorithm process, we present numerical evidence showing that our proposed algorithm provides attractive results. CS theories [ 37 , 38 , 39 ] state that a sparse signal h can be recovered steadily from linear measurement.

The signal r t in Equation 5 can be given in matrix form as [ 31 ]. Rewrite Equation 10 as. Then, we can use the CS recovery algorithm to recover sparse signal h. A number of CS recovery algorithms have been proposed. One of the popular kinds of recovery algorithms is based on the iterative greedy pursuit.

According to whether the sparse K is known prior or not, this class of algorithms also can be divided into two types. For the OMP algorithm, in each iteration, the atom maximizes its inner product with the residual signal. However, the results of each iteration may be suboptimal. CoSaMP is proven to be the most advanced greedy algorithm. CoSaMP introduces the idea of backtracking that reduces the chance of error accumulation, selects 2 K coordinates and utilizes an iterative checking to refine them, and overcomes the defects of OMP, so that the atoms could not be changed once deposited in the candidate set.

In practical applications, the second type of algorithms has better prospects than the first.

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SAMP is the second type of algorithm. CoSaMP algorithm can reconstruct source signals with high efficiency. However, the algorithm requires the prior knowledge of sparsity. SAMP algorithm provides a way for blind sparse reconstruction. Motivated by the advantages in the two greedy algorithms and associated with a regularized process, we propose a new greedy algorithm, named sparse adaptive regularized compressive sampling matching pursuit ARCoSaMP.

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The proposed algorithm can automatically adjust the selected atoms to reconstruct the unknown sparsity signal in the iterative process. A similar backtracking theory of CoSaMP is utilized to reconstruct partial information of the target signal in the iterative process. An iterative process is divided into multiple stages, the proposed algorithm adaptively estimates the sparsity with steps through stage by stage and set it to the length of the initial support, then gets the accurate target signal by regularization screening of atoms in every stage.

The algorithm basic steps are shown below:. The deviation norm 2 is chosen as the basis of the iterative termination. The selection of the initial step is very important, and if the step size is too large, there may an overestimating problem. In the proposed algorithm, the initial step size is 1, which is less than the reality of the sparsity K until the final stage. Iteration loop follows the CoSaMP and regularization to identify support sets in the target signal. In the proposed algorithm, we trigger the stage switching between two consecutive iterations when the relevant residual improvement begins to disappear.

The estimation of multipath delay profile and percentage recovered of the algorithms are also given. The channel profile is shown in Table 1. The channel sparse K is 6. Figure 3 is a snapshot of the original and estimated delay profiles of IEEE The proposed scheme not only precisely estimates the multipath delay values but also exactly estimates the relative power of the multipath. Figure 4 depicts the probability curves of Gaussian sparse signal.

In Figure 5 a, it is obvious that CS based channel estimation methods can obtain significantly BER improvement compared with conventional least squares LS method. CoSaMP can obtain about 4. Figure 5 b plots the MSE performance comparisons. CoSaMP can obtain about 1. A new sparse adaptive regularized compressive sampling matching pursuit algorithm for channel estimation is proposed, which is associated with adaptive, regularized and CoSaMP.

The proposed algorithm can accurately estimate the multipath components. The proposed scheme outperforms SAMP for channel estimation with lesser time complexity and can provide approximate results than the state-of—the-art CoSaMP algorithm without a prior sparse knowledge of the channel. The authors would like to thank the editor and the anonymous reviewers for their valuable comments. Han Wang proposed the sparse adaptive regularized compressive sampling matching pursuit algorithm; Wencai Du conceived and designed the simulations and provided some comments on the paper organization; Lingwei Xu contributed towards the performance results and analytic evaluations.

National Center for Biotechnology Information , U. Journal List Sensors Basel v. Sensors Basel. Published online Jun Leonhard M. Reindl, Academic Editor. Author information Article notes Copyright and License information Disclaimer. Received Apr 17; Accepted Jun Introduction Filter bank multicarrier FBMC techniques have drawn increasing attention from many researchers [ 1 , 2 , 3 ]. Open in a separate window. Figure 1. Figure 2. Table 1 IEEE Figure 3.