On transmit beamforming in MIMO radar with matrix completion

It was recently shown that MIMO radars with sparse sensing and matrix completion (MC) can significantly reduce the volume of data required by MIMO radars for accurate target detection and estimation. In MIMO-MC radars, the subsampled target returns are forwarded by the receive antennas to a fusion center, partially filling a matrix, referred to as the data matrix. The data matrix is first completed via MC techniques and then used to estimate the target parameters via standard array processing methods. This paper studies the applicability of MC theory on the data matrix arising in colocated MIMO radars using uniform linear arrays. It is shown that the data matrix coherence, and consequently the performance of MC, is directly related to the transmit waveforms. Among orthogonal waveforms, the optimum choices are those for which, any snapshot across the transmit array has a flat spectrum. The problem of waveform design is formulated as an optimization problem on the complex Stiefel manifold, and is solved via the modified steepest descent method, or the modified Newton algorithm with nonmonotone line search. Although the optimal waveforms are designed for the case of targets falling in the same range bin, sensitivity analysis is conducted to assess the performance degradation when those waveforms are used in scenarios in which the targets fall in different range bins.

It was recently shown that MIMO radars with matrix completion (MIMO-MC radars) can achieve the performance of MIMO radars but with significantly fewer samples. The matrix completion performance depends on the coherence of the matrix. In MIMO-MC radars and for the scenario in which the targets fall in the same range bin, the coherence of the received data matrix is asymptotically optimal, i.e., 1, when the transmit waveforms have flat spatial spectrum. In this paper, we study the performance of such waveforms when used to estimate targets that fall in different range bins. We show that, if such optimal waveforms are additionally designed to have good correlation properties, the matrix coherence is slightly larger than 1 but bounded by 2 as long as the maximal range delay is of the order of the pulse duration. Thus, in such cases the MC performance degrades only slightly.

In matrix completion based MIMO radars, the data matrix coherence, and consequently the performance of matrix completion depend on the transmit waveforms. It was recently shown that for uniform linear arrays and orthogonal waveforms, the optimal choice for waveforms are white-noise type functions. This paper deals with the design of optimal transmit waveforms. The design is formulated as an optimization problem on the complex Stiefel manifold, which is a non-convex set, and the optimal waveforms are found via a steepest descent algorithm with nonmonotone line search methods. Numerical results show that for large dimensional data matrices and for a large number of antennas, the objective function converges to its global minimum and the matrix coherence corresponding to the optimal waveforms is asymptotically optimal, thus resulting in very good target estimation performance.

Recently proposed multiple input multiple output radars based on matrix completion (MIMO-MC) employ sparse sampling to reduce the amount of data that need to be forwarded to the radar fusion center, and as such enable savings in communication power and bandwidth. This paper proposes designs that optimize the sharing of spectrum between a MIMO-MC radar and a communication system, so that the latter interferes minimally with the former. First, the communication system transmit covariance matrix is designed to minimize the effective interference power (EIP) to the radar receiver, while maintaining certain average capacity and transmit power for the communication system. Two approaches are proposed, namely a noncooperative and a cooperative approach, with the latter being applicable when the radar sampling scheme is known at the communication system. Second, a joint design of the communication transmit covariance matrix and the MIMO-MC radar sampling scheme is proposed, which achieves even further EIP reduction.

Wideband spectrum sensing is a popular topic in signal processing, especially for many radar and communication applications. What we face is a high sampling rate and a large volume of samples, in which demand of reducing the sampling rate without sacrificing the sensing resolution and quality. The generalized coprime sampling can break the limitation of the Nyquist sampling theorem with both characteristics of sparse sensing and coprime numbers. To fully utilize all the information received of the derived correlation matrix constructed by the different time delays, the matrix completion method is exploited. The theory of matrix completion is an extension of compressive sensing, though, which is not restrained by the sparsity and the restricted isometry property. The interpolation-based method presented via the convex framework of the nuclear norm minimization has no extra fine-tuned parameters, which different from techniques like compressive covariance sampling, positive definite Toeplitz matrix completion, and so on. Moreover, compared to the selection-based method under a continuous set, the proposed method improves the spectral resolution and estimation accuracy to avoid the information losing. The Simulation results indicate the performance of the algorithm.

The paper proposes a cooperative scheme for the coexistence of a multiple-input-multiple-output (MIMO) communication system and a matrix completion (MC) based, collocated MIMO (MIMO-MC) radar. To facilitate the coexistence, and also deal with clutter, both the radar and the communication systems use transmit precoding. For waveform flexibility, the radar uses a random unitary waveform matrix. We prove that for such waveforms and any precoding matrix, the error performance of MC is guaranteed. The radar transmit precoder, the radar subsampling scheme, and the communication transmit covariance matrix are jointly designed in order to maximize the radar SINR, while meeting certain communication rate and power constraints. The joint design is implemented at a control center, which is a node with whom both systems share physical layer information, and which also performs data fusion for the radar. We provide efficient algorithms for the proposed optimization problem, along with insight on the feasibility and properties of the proposed design. Simulation results show that the proposed scheme significantly improves the spectrum sharing performance in various scenarios.

With the wide deployment of digital image capturing equipment, the need of denoising to produce a crystal clear image from noisy capture environment has become indispensable. In this article, a novel type-2 fuzzy-based filter is proposed for denoising images corrupted by impulse noise, especially for the high density of salt-and-pepper noise. It operates two stages, namely, type-2 fuzzy identifier and matrix completion denoiser. In the proposed method, the type-2 fuzzy identifier is first employed to identify and trim the entries contaminated by impulse noise in the data matrix from fuzzy system. Then, the trimmed data matrix is utilized to retrieve the noiseless data matrix with the matrix completion technology. Herein, a novel matrix completion technique is developed without <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$a$</tex-math></inline-formula> <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$priori$</tex-math></inline-formula> rank information compared to its counterparts. Simulation results are presented, which vividly show the denoised images obtained by the proposed method can achieve crystal clear image with strong structural integrity, and are showing good performance in terms of peak signal-to-noise ratio.

Singular Value Decomposition (and Principal Component Analysis) is one of the most widely used techniques for dimensionality reduction: successful and efficiently computable, it is nevertheless plagued by a well-known, well-documented sensitivity to outliers. Recent work has considered the setting where each point has a few arbitrarily corrupted components. Yet, in applications of SVD or PCA such as robust collaborative filtering or bioinformatics, malicious agents, defective genes, or simply corrupted or contaminated experiments may effectively yield entire points that are completely corrupted. We present an efficient convex optimization-based algorithm we call Outlier Pursuit, that under some mild assumptions on the uncorrupted points (satisfied, e.g., by the standard generative assumption in PCA problems) recovers the exact optimal low-dimensional subspace, and identifies the corrupted points. Such identification of corrupted points that do not conform to the low-dimensional approximation, is of paramount interest in bioinformatics and financial applications, and beyond. Our techniques involve matrix decomposition using nuclear norm minimization, however, our results, setup, and approach, necessarily differ considerably from the existing line of work in matrix completion and matrix decomposition, since we develop an approach to recover the correct column space of the uncorrupted matrix, rather than the exact matrix itself. In any problem where one seeks to recover a structure rather than the exact initial matrices, techniques developed thus far relying on certificates of optimality, will fail. We present an important extension of these methods, that allows the treatment of such problems.

Compressed sensing based data gathering is in-efficient in small scale wireless sensor networks because of uncorrelated observations at the sink. Failure to exploit the low rank and suitable transform domain to explore the correlation structure of sensor data, results in significantly low recovery accuracy in such matrix completion algorithms for small scale networks. Targeting the spatio temporal correlation structure of the data, a novel data gathering and matrix completion scheme, which exploits the low rank property and compactness of sensor data which exists in wavelet transform domain, is proposed in this work. The compactness of the sensor data in wavelet transform domain is used for recovering the missing entries of the matrix. Experiments and simulations, for single-node and multi-node scenarios, prove the efficacy of the proposed approach over existing schemes significantly in terms of recovery accuracy even at extremely low sampling rate.

In this thesis we study the problem of exact completion for 𝑚 × 𝑛 sized matrix of rank 𝑟 and the problem of low-rank estimation with the adaptive sampling method. We introduce a relation of the exact completion problem with the sparsest vector of column and row spaces. Using this relation, wepropose matrix completion algorithms that exactly recovers the target matrix. These algorithms are superior to previous works in two important ways. First, our algorithms exactly recovers 𝜇0-coherent column space matrices byprobability at least 1 − 𝜖 using much smaller observations complexity than - 𝒪(𝜇0𝑟𝑛log𝑟 𝜖 )—the state of art. Specifically, many of the previous adaptive sampling methods require to observe the entire matrix when the column space is highly coherent. However, we show that our method is still able to recover this type of matrices by observing a small fraction of entries under many scenarios. Second, we propose an exact completion algorithm, whichrequires minimal pre-information as either row or column space is not being highly coherent. We provide an extension of these algorithms that is robust to sparse random noise. Besides, we propose an additional low-rank estimationalgorithm that is robust to any small noise by adaptively studying the shape of column space. At the end of the thesis, we provide experimental results that illustrate the strength of the algorithms proposed here. This thesis have been written mainly based on the paper [12].

In this paper, we propose to use fuzzy fusion rules to improve the performances of the CA-CFAR and OS-CFAR detectors for non-coherent MIMO radars in homogenous and non-homogenous Pareto background. First, the membership function for each individual detector is computed. At the fusion center, the membership functions are combined using four fuzzy fusion rules, namely; the “MIN”, “MAX”, “algebraic product” and the “algebraic sum” to yield a binary decision after defuzzification. The obtained results show that for a number of nodes equal to four, the performance is the best for a high number of receivers. In homogeneous background, the “algebraic product” fusion rule gives the best result when SCR > 4 dB whereas the “algebraic sum” is the best when SCR < 4 dB for the CA-CFAR. For the OS-CFAR, it is clearly shown that the “algebraic product” is the best. In non-homogenous case, the “algebraic sum” fusion rule gives the best results for the CA-CFAR, and the “algebraic product” for the OS-CFAR.

To detect a potential moving target in presence of clutter, the statistical method such as the generalized likelihood ratio test (GLRT) may be computationally expensive, due to unknown target and clutter parameters. This paper exploits the potential of supervised learning, which can have the computational complexity under control, for multiple-input-multiple-output (MIMO) radar target detection in combating clutter. Compound Gaussian clutter with unknown parameters is considered for generating synthetic data. The measurements at all receivers are fed to a fully connected network (FCN) at the fusion center, where detection is viewed as a binary classification problem. The detection performance of the FCN-based detector is analyzed for scenarios with various MIMO radar settings under different clutter parameters. It is shown that the FCN-based detector can outperform the traditional GLRT detector in the tested cases.

Limited measurement availability at the distribution grid presents challenges for state estimation and situational awareness. This paper combines the advantages of two sparsity-based state estimation approaches (matrix completion and compressive sensing) that have been proposed recently to address the challenge of unobservability. The proposed approach exploits both the low rank structure and a suitable transform domain representation to leverage the correlation structure of the spatio-temporal data matrix while incorporating the powerflow constraints of the distribution grid. Simulations are carried out on three phase unbalanced IEEE 37 test system to verify the effectiveness of the proposed approach. The performance results reveal - (1) the superiority over traditional matrix completion and (2) very low state estimation errors for high compression ratios representing very low observability.

DOA estimation via shift-invariant matrix completion

We derive square-root based algorithms for structured matrices and discuss potential applications to interpolation and matrix completion problems. The mathematical machinery used here is based on a standard Gaussian elimination technique and on simple results from matrix and linear system theory. We show how to exploit the inherent displacement structure in order to construct a convenient transmission-line cascade that makes evident the required interpolation and completion conditions. We also introduce the concept of time-variant structured matrices and discuss its applications to matrix completions and time-variant interpolation problems.