Efficient use of spatio-temporal resources, including sensor arrays and transmit waveforms, is a key challenge in modern MIMO active sensing systems. This paper studies the impact of array redundancy and WR on active sensing performance. Specifically, we show that parameter identifiability atreducedWR critically depends on subspace properties of the so-called array redundancy pattern. We show that array geometries with identical sum co-arrays can exhibit markedly different identifiability properties at low WR. We derive a novel necessary condition for maximizing identifiability at reduced WR, which reveals that the unfavorable redundancy patterns of certain redundant arrays fundamentally limits their performance. The results yield new insights into resource-efficient sensing systems, motivating redundancy-aware array and waveform design.

Sparse linear arrays (SLAs) have gained significant attention for automotive radar due to their enhanced aperture and angular resolution. A promising approach to Direction-of-Arrival (DOA) estimation with only a single temporal snapshot and SLAs is to interpolate the missing measurements by solving a low-rank Hankel matrix completion problem. Sampling geometry plays an important role in matrix completion, and random sampling techniques primarily provide non-uniform recovery guarantees, which can be undesirable. Using the unique geometry of nested-type SLAs, we formulate the Hankel matrix completion problem as a non-convex prediction problem which utilizes the latent structures of a low-rank Hankel matrix in order to reduce the number of unknowns. It also provides uniform recovery guarantees (provably in the absence of noise). Our approach can be interpreted as a self-supervised Untrained Neural Network (UNN) which utilizes the inner uniform linear array (ULA) of a nested array to perform prediction, and the outer array to curb error propagation. Our work establishes the first connection between array interpolation and untrained machine learning paradigms.

We consider the problem of adaptive sensing for multi-path channel estimation in millimeter-wave (mmWave) communications system with single RF chain. Current adaptive sensing approaches either focus on estimating the single dominant path or assume apriori knowledge of the number of multi-path components. A key challenge in Bayesian adaptive sensing is the choice of prior which captures the geometric structure of mmWave channels and leads to tractable adaptive sensing methods. In this work, we propose to learn a flexible empirical prior through supervised training that allows for test-time adaptations to the specific channel environment. We realize our framework using a novel recurrent neural network architecture which learns the prior implicitly as part of the recurrent update rule. At test time, our approach alternates between adaptively designing the beamformer to acquire measurement, estimating the channel, and refining the hyperparameters of the learned empirical prior based on history. We demonstrate our approach produces interpretable beamformers and generalizes to channel configurations with fewer multi-path components than it was trained upon.

We address the challenge of achieving angular super-resolution in multi-antenna radar systems that are widely used for localization, navigation, and automotive perception. A multi-antenna radar achieves very high resolution by computationally creating a large virtual sensing system using very few physical antennas. However, practical constraints imposed by hardware, noise, and a limited number of antennas can impede its performance. Conventional supervised learning models that rely on extensive pre-training with large datasets, often exhibit poor generalization in unseen environments. To overcome these limitations, we propose NEAR, an untrained implicit neural representation (INR) framework that predicts radar responses at unseen locations from sparse measurements, by leveraging latent harmonic structures inherent in radar wave propagation. We establish new theoretical results linking antenna array response to expressive power of INR architectures, and develop a novel physics-informed and latent geometry-aware regularizer. Our approach integrates classical signal representation with modern implicit neural learning, enabling high-resolution radar sensing that is both interpretable and generalizable. Extensive simulations and real-world experiments using radar platforms demonstrate NEAR's effectiveness and its ability to adapt to unseen environments.

This paper derives new non-asymptotic characterization of the Cramér-Rao Bound (CRB) of any sparse array as a function of the angular separation between two far-field narrowband sources in certain regimes characterized by a low Signal-to-Noise Ratio (SNR). The primary contribution is the derivation of matching upper and lower bounds on the CRB in a certain measurement-dependent SNR (MD-SNR) regime, where one can zoom into progressively lower SNR as the number of sensors increases. This tight characterization helps to establish that sparse arrays such as nested and coprime arrays provably exhibit lower CRB compared to Uniform Linear Arrays (ULAs) in the specified SNR regime.

We study the problem of noisy sparse array interpolation, where a large virtual array is synthetically generated by interpolating missing sensors using matrix completion techniques that promote low rank. The current understanding is quite limited regarding the effect of the (sparse) array geometry on the angle estimation error (post interpolation) of these methods. In this paper, we make advances towards solidifying this understanding by revealing the role of the physical beampattern of the sparse array on the performance of low rank matrix completion techniques. When the beampattern is analytically tractable (such as for uniform linear arrays and nested arrays), our analysis provides concrete and interpretable bounds on the scaling of the angular error as a function of the number of sensors, and demonstrates the effectiveness of nested arrays in presence of noise and a single temporal snapshot.

A subspace code is defined as a collection of subspaces of an ambient vector space, where each information-encoding codeword is a subspace. This paper studies a class of spatial sensing problems, notably direction of arrival (DoA) estimation using multisensor arrays, from a novel subspace coding perspective. Specifically, we demonstrate how a canonical (passive) sensing model can be mapped into a subspace coding problem, with the sensing operation defining a unique structure for the subspace codewords. We introduce the concept of sensing subspace codes following this structure, and show how these codes can be controlled by judiciously designing the sensor array geometry. We further present a construction of sensing subspace codes leveraging a certain class of Golomb rulers that achieve near-optimal minimum codeword distance. These designs inspire novel noise-robust sparse array geometries achieving high angular resolution. We also prove that codes corresponding to conventional uniform linear arrays are suboptimal in this regard. This work is the first to establish connections between subspace coding and spatial sensing, with the aim of leveraging insights and methodologies in one field to tackle challenging problems in the other.

We study single-snapshot nested array interpolation with positive sources. The problem of sparse array interpolation is traditionally cast as a low-rank Toeplitz/Hankel matrix completion problem from partial observations. In recent work, we provided the first necessary and sufficient guarantees for nested array interpolation with real measurements in the rank minimization framework. In this work, we strengthen the sufficiency results by proving that in case of positive sources it is possible to interpolate the nested array by performing a simple convex feasibility search instead of solving a rank minimization problem. Simulations demonstrate that this framework is also effective for noisy measurements, and that noisy nested array interpolation outperforms ULA extrapolation.1

This paper proposes a novel method to focus or align array manifolds at different frequencies to a single reference frequency in wideband direction of arrival (DOA) estimation. Unlike existing methods, our focusing can be performed without explicitly constructing focusing matrices, or requiring any preliminary DOA estimates. Instead the focusing is done implicitly by obtaining focused measurements as the solution to a rank minimization procedure. This paper also provides theoretical guarantees for exact focusing via rank minimization. We call this procedure Initialization-Free Implicit-Focusing (IF2). Numerical simulations are provided to demonstrate the resilience of IF2in various SNR regimes compared to past and recent wideband DOA recovery methods, and its lack of error saturation in high SNR regimes1.

This paper considers the problem of recovering the joint support (of size K) of a set of unknown sparse vectors in ℝd, each of which can be sensed using a different measurement matrix. Such models have wide applicability ranging from communication to multi-task learning. We develop an adaptive strategy called Adaptive Joint Support Recovery (Ada- JSR) that enables exact support recovery in the extreme compression regime with only m = 1 measurement per unknown vector while requiring a total complexity of no more than K⌈log2(d)⌉ measurements. Unlike existing support recovery techniques which require suitable assumptions on the correlation structure or distribution of the unknown signals in order to operate in the regime m1

We study the problem of completing the virtual array of a nested array with a single snapshot. This involves synthesizing a virtual uniform linear array (ULA) with the same aperture as the nested array by estimating (or interpolating) the missing measurements. A popular approach for virtual array synthesis involves completing a certain Hankel/Toeplitz matrix from partial observations, by seeking low-rank solutions. However, existing theoretical guarantees for such structured rank minimization (which mostly provide sufficient conditions) do not readily extend to nested arrays. We provide the first necessary and sufficient conditions under which it is possible to exactly complete the virtual array of a nested array by minimizing the rank of a certain Toeplitz matrix constructed using a single temporal snapshot. Our results exploit the geometry of nested arrays and do not depend on the source configuration or on the separation between sources.

This paper investigates the problem of recovering a binary-valued signal from compressed measurements of its convolution with a known finite impulse response filter. We show that it is possible to attain optimum sample complexity for exact recovery (in absence of noise) with a computationally efficient algorithm. We achieve this by adopting an algorithm-measurement co-design strategy where the measurement matrix is designed as a function of the filter, such that the recovery of binary signals with arbitrary sparsity is possible by using a sequential decoding algorithm. Such a filter-dependent sampler design can overcome the computational challenges associated with enforcing binary constraints, and enable us to operate in “extreme compression” regimes, where the number of measurements can be much smaller than the sparsity level.

This paper considers the problem of recovering a unipolar finite-valued signal from compressive measurements of its convolution with a known finite impulse response filter. We show that owing to the finite-value constraint the problem remains identifiable if the downsampling factor is smaller than the filter length. We develop a new computationally efficient decoding algorithm that can operate at the optimal downsampling factor under mild conditions on the filter. This allows us to explicitly impose the finite value constraint (no relaxation) without compromising on the computational tractability.1

This paper concerns understanding the fundamental performance benefits of co-array based super-resolution techniques in low SNR settings. Proper sensor placement in the form of a sparse nested array allows the generation of difference co-arrays with large virtual apertures that offer higher resolution. However, the performance of super-resolution aperture synthesis techniques is known to degrade in presence of large noise levels. The main contribution of this paper is to rigorously establish that nested arrays provide lower Cramér-Rao bounds than a ULA (with the same number of sensors) in the low SNR regime, and therefore can lead to better resolvability of closely spaced sources. Numerical experiments are performed to validate theoretical claims, including demonstration of MUSIC spectra with two closely-spaced sources using ULA and nested arrays in various SNR settings.1

We propose a new framework for leveraging the degrees of freedom in the difference set of a nested array with limited snapshots. Typically, this difference set (or coarray) is realized by first estimating a virtual coarray covariance matrix from the sample covariance matrix. However, with only a few snapshots, these techniques incur large estimation error, which saturates away from zero even as the signal-to-noise ratio (SNR) tends to infinity. We address this issue by moving away from estimating the coarray covariance matrix when snapshots are limited, and instead proposing a "proxy covariance matrix" (Prox-Cov) that provides an alternate estimate of the coarray subspace (but does not attempt to estimate the source powers). (Prox-Cov) is shown to outperform coarray MUSIC with limited snapshots when the number of sources exceeds the number of sensors. Moreover, when the number of sources is fewer than sensors, we prove that (Prox-Cov) leads to exact identification of the coarray subspace with very few snapshots in the absence of noise, while the error of coarray MUSIC provably saturates in this regime.

This paper considers the problem of lower bounding the mean-squared-error (MSE) of unbiased super-resolution estimates. In literature, only upper bounds on the MSE are available which scale with the so-called super-resolution factor (SRF). However, the upper bound does not indicate whether the MSE indeed exhibits noise amplification that increases with the target resolution. The main contribution of this paper is to derive the Cramér-Rao Bound for noisy super- resolution problem and understand its scaling as a function of the super-resolution factor. We compare our lower bound with the upper bound established in prior work and show that the dependency of MSE on SRF is fundamental. Our analysis can be applied to other unbiased estimates in the problem of super-resolution. Numerical experiments are conducted to demonstrate our theoretical claims.

This paper considers the problem of blind deconvolution where the input signal is non-negative and sparse, and the unknown convolutional kernel is a first order autoregressive filter. Our objective is to understand if it is possible to recover both the signal and the kernel from downsampled measurements of their convolution. This work is motivated by the problem of neural spike deconvolution from calcium imaging, where it is desirable to recover spikes at a higher rate from uniformly undersampled measurements. Assuming that the signals are generated according to a Bernoulli model, we show that it is possible to uniquely identify both the signal and the kernel with high probability using only O s measurements, where s is the expected sparsity. The key p qidea is to exploit non-negative constraints on the input signal as well as the parametric structure of the kernel1.

This letter provides a Cramér-Rao Bound (CRB) based analysis of compressive arrays with applications in mmWave channel sensing, where the measurements at the output of an antenna array are further compressed using a complex-valued compression matrix, in order to reduce the system complexity and power consumption. While necessary conditions for the existence of CRB for compressed arrays have been recently derived, currently no sufficient conditions exist that can guarantee the existence of the CRB in different compressive regimes, and therefore can be used to guide the design of the overall system. We overcome this drawback by deriving tight sufficient conditions (that agree with the necessary conditions) for almost all choices of the compression matrix. Our results decisively demonstrate the additional benefit gained by using sparse arrays (such as nested array) even when a compression matrix is deployed.

This paper considers the problem of super-resolution reconstruction by casting it as an optimization problem with positive constraints and non-convex objective function. Enforcing the solution to be simultaneously sparse and non-negative naturally leads to a non-convex l1/2quasinorm minimization problem. A reweighted l1norm minimization algorithm is proposed to solve this problem, which is tailored for l1/2quasinorm minimization using the idea of Majorization-Minimization. Although the problem is non-convex and non-smooth, and the measurement matrix does not satisfy restricted isometry conditions, we are able to obtain deterministic stable reconstruction guarantees in presence of bounded noise by using the structure of the measurement matrix and non-negativity of the signal. Numerical results demonstrate that l1/2minimization promotes sparser solution and outperforms l1minimization.

This paper considers the problem of sparse support recovery in Multiple Measurement Vector (MMV) models, where the support size (K) can exceed the dimension (M) of individual measurement vectors. Existing results in this regime mostly establish asymptotic performance guarantees, where the number of measurement vectors L → ∞. In this paper, we develop non-asymptotic guarantees (finite L), and demonstrate that it is possible to recover supports of size K = O(M2) provided the sparse signals are statistically uncorrelated. In particular, the probability of detecting a wrong support is shown to approach zero exponentially fast in L even when K > M, for appropriately designed measurement matrices. Our analysis is based on a simple least squares estimation of signal powers, followed by hard thresholding to detect the support.

This paper considers the problem of super-resolution with positive constraints. By utilizing the concept of Modulus of Continuity (MC), we propose a unified framework for analyzing the robustness of super-resolution reconstruction in presence of noise, which is algorithm-independent and emphasizes the role of signal structures. In contrast to earlier works, we show that incorporation of positive constraints improves the scaling factor of MC and provides tighter upper bound on the estimation error of any algorithm that exploits such structure. The unified framework is further applied to analyze convex algorithms for positive super-resolution, and the theoretical results are validated by numerical experiments.

This paper analyzes the Cramér-Rao Bound associated with the estimation of certain sparse hyper-parameters in the Sparse Bayesian Learning (SBL) framework, that crucially control the sparsity of the desired signal. The CRB is shown to exhibit saturation with respect to the number of measurements, i.e., it can be lower bounded by a non-negative quantity that does not go to zero even when the number of measurements tends to infinity. Moreover, the CRB corresponding to the nonzero and zero elements of the sparse hyper-parameter can exhibit different behaviors. While the CRB for the non-zero elements always saturate regardless of the type of dictionary, saturation of the CRB for zero elements provably happens when the dictionary has normalized columns. For an unnormalized dictionary, singular values of certain sub-dictionaries determine if saturation can happen, prompting future research into this interesting phenomenon.

This paper considers the problem of co-array interpolation for direction-of-arrival (DOA) estimation with sparse nonuniform arrays. By utilizing the much longer difference co-array associated with these arrays, it is possible to perform DOA estimation of more sources than sensors. Since the co-array may contain holes (or missing lags), interpolation algorithms have been proposed to fully utilize the remaining elements of the co-array beyond that captured in the contiguous ULA segment. However, the quality and stability of interpolation performed by such algorithms (especially in presence of modeling errors) have not been analyzed. This paper provides a unified analysis of co-array interpolation algorithms to bound the interpolation error in terms of modeling errors. The results are universal in the sense that they can be applied to analyze any algorithm that utilizes the positive semidefinite (PSD) structure of the interpolated covariance matrix. The general framework is then applied to analyze specific algorithms and simulations are conducted to study their interpolation errors.

This letter offers several new insights into the maximum-likelihood direction-of-arrival (DOA) estimation problem, when the number of sources exceeds the number of sensors. Two maximum-likelihood problems are studied: one for estimating the Toeplitz-structured coarray covariance matrix from the measurements, and the other for estimating the DOAs directly from the measurements. We establish the equivalence of both problems when the number of sources is assumed to be unknown and can potentially exceed the number of sensors. Additionally, it is shown that when the source waveforms satisfy certain orthogonality conditions, the Toeplitz-constrained maximum-likelihood covariance estimation framework provably produces the true DOAs without requiring to know the number of sources. When the number of sources exceeds the number of sensors, the maximum-likelihood algorithms studied in this letter outperform other recently studied methods, as demonstrated through numerical experiments.

The problem of sparse phase retrieval is considered where the goal is to recover a sparse complex valued vector (with s non zero elements) from the magnitudes of its linear measurements. Using a modified and partially randomized version of a newly proposed structured sampler, namely the Partial Nested Fourier Sampler (PNFS), it is shown to be possible to recover the unknown signal (up to a global phase ambiguity) from O(s log N) phaseless measurements where N is the dimension of the vector. The reconstruction is based on a novel idea of "decoupling" certain quadratic terms in the phaseless measurements acquired by the PNFS, leading to a simple l1-minimization-based recovery algorithm, without the need for "lifting" the unknown variable to a higher dimensional space. The proposed algorithm is also proved to be stable in presence of bounded noise.

Coprime arrays, consisting of two uniform linear arrays whose inter-element separations are coprime, can resolve O(MN) sources using only O(M + N) sensors. However, holes in the coarray prevent us from using the full coarray in the MUSIC algorithm for DOA estimation. Through interpolation, it may be possible to use the remaining elements of the coarray to increase the degrees of freedom beyond what is captured in the contiguous ULA section in the coarray. Techniques like positive definite Toeplitz completion, array interpolation, and sparse recovery, manage to include all the information in the coarray, but they demand extra fine-tuned parameters and have individual drawbacks. In this paper, a simple and tractable convex framework via nuclear norm minimization is presented. This approach has no extra tuning parameters and overcomes several undesired issues of other techniques. Numerical examples indicate that, in many instances, the proposed method not only increases the estimation accuracy but also distinguishes more sources than other methods1.

Although Cramér-Rao Bounds (CRB) for direction-of-arrival (DOA) estimation have been extensively studied for decades, existing results are mainly applicable when there are fewer sources than sensors. In contrast, this letter considers an underdetermined signal model (more sources than sensors) and investigates conditions under which CRB exist. Necessary and sufficient conditions are derived for the associated Fisher information matrix to be nonsingular, which in turn, leads to closed-form expressions for the CRBs for underdetermined DOA estimation. These conditions highlight crucial roles played by the array geometry, as well as the correlation between source signals. The CRB for different array geometries are numerically compared both in the overdetermined and underdetermined settings.

This paper considers the problem of compressively sampling wide sense stationary random vectors with a low rank Toeplitz correlation matrix. A new structured deterministic sampling method known as the “Generalized Nested Sampling” (GNS) is used to fully exploit the inherent redundancy of low rank Toeplitz matrices. For a Toeplitz matrix of size N ×N with rank r, this sampling scheme uses only O(√r) measurements and allows exact recovery from noiseless measurements. This compression factor is independent of N and is shown to be larger than that achieved by existing random sampling based techniques for compressing Toeplitz matrices. The recovery procedure exploits the connection between Toeplitz matrices and linear prediction.

The problem of parameter identifiability in linear underdetermined models is addressed, where the observed data vectors follow a multivariate Gaussian distribution. The problem is underdetermined because the dimension of parameters characterizing the distribution of the data is larger than the dimension of the observed vectors. Such models arise frequently in Bayesian Compressive sensing and Sparse Bayesian Learning problems, where the parameter vector to be estimated, is assumed to be sparse. We establish explicit conditions for parameter identifiability in such models, by relating the ambient dimension of the hyperparameter space and that of the data. We establish a crucial result that in such underdetermined models, even without requiring the parameter to be sparse, it is possible to guarantee unique identifiability of the parameters as long as these two dimensions satisfy a certain condition. When such a condition is violated, the unconstrained statistical model is no more identifiable and additional constraints in the form of sparsity need to be enforced to recover the true parameter.

A finite duration sequence exhibiting periodicities does not in general admit a sparse representation in terms of the DFT basis unless the period is a divisor of the duration. This paper develops a dictionary called the Farey dictionary for the efficient representation of such sequences. It is shown herein that this representation is especially useful for identifying hidden periodicities in a finite data record. The properties of the Farey dictionary are studied, and the dictionary is shown to be superior to the conventional DFT based uniform dictionary, from the view point of identifying hidden periods.

The problem of direction of arrival (DOA) estimation of narrowband sources using an antenna array is considered where the number of sources can potentially exceed the number of sensors. In earlier works, the authors showed that using a suitable antenna geometry, such as the nested and coprime arrays, it is possible to localize O(M2) sources using M sensors. To this end, two different approaches have been proposed. One is based on an extension of subspace based methods such as MUSIC to these sparse arrays, and the other employs l1 norm minimization based sparse estimation techniques by assuming an underlying grid. While the former requires the knowledge of number of sources, the latter suffers from basis mismatch effects. In this letter, a new approach is proposed which overcomes both these weaknesses. The method is hybrid in nature, using a low rank matrix denoising approach followed by a MUSIC-like subspace method to estimate the DOAs. The number of sources is revealed as a by-product of the low rank denoising stage. Moreover, it does not assume any underlying grid and thereby does not suffer from basis mismatch. Numerical examples validate the effectiveness of the proposed method when compared against existing techniques.

Consider a multiple measurement vector (MMV) model given by y[n] = Axs[n]; 1 ≤ n ≤ L where equation denote the L measurement vectors, A ∈ RM×Nis the measurement matrix and xs[n] ∈ RNare the unknown vectors with same sparsity support denoted by the set S0with |S0| = D. It has been shown in a recent paper by the authors that when the elements of xs[n] are uncorrelated from each other, one can recover sparsity levels as high as O(M2) for suitably designed measurement matrix. The recovery is exact when support recovery algorithms are applied on the ideal correlation matrix. When we only have estimates of the correlation, it is still possible to probabilistically argue the recovery of sparsity levels (using a coherence based argument) that is much higher than that guaranteed by existing coherence based results. However the lower bound on the probability of success is found to increase rather slowly with L (as 1-C/L for some constant C > 0) without any further assumption on the distribution of the source vectors. In this paper, we demonstrate that when the source vectors belong to a Gaussian distribution with diagonal covariance matrix, it is possible to guarantee the recovery of original support with overwhelming probability. We also provide numerical simulations to demonstrate the effectiveness of the proposed strategy by comparing it with other popular MMV based methods.

In this paper, we develop the theory of a new class of two dimensional arrays with sensors on lattice(s) which can be used to construct a virtual array of much larger size through passive processing. This structure is obtained by systematically nesting two arrays, one with sensors suitably placed on a sparse lattice and the other on an appropriately chosen dense lattice. The difference co-array of such an array with M and N elements respectively on the two lattices, is proved to be a larger two dimensional array with O(MN) sensors present contiguously (without holes) on the dense lattice. It will be shown that there is complete freedom to choose the sparse and dense arrays as long as they are related by an integer matrix (which can also be arbitrarily chosen). To exploit the increased degrees of freedom offered by the array for two dimensional DOA estimation of more sources than sensors, a novel algorithm based on the concept of two dimensional spatial smoothing is also proposed. The validity of the proposed methods is verified through numerical examples.

Sparse sampling with coprime lattice arrays was introduced recently in the literature. It has been shown that a dense coarray can be constructed from such a pair of arrays, and is useful in array processing and image processing applications. For example, the coarray allows one to identify many more sources than sensors. After a brief review of these fundamentals, this paper examines the case where the two arrays are generated by matrices that are adjugates of each other. In this case it is possible to obtain a dense rectangular tiling of the 2D frequency plane from a pair of coarse 2D DFT filter banks. The special case where the adjugate pairs are generated by skew circulant matrices has some advantages, which are examined in detail.

Commuting coprime integer matrices arise in signal processing in a number of contexts. This paper develops two ways, one nonlinear and the other linear, to generate new commuting coprime pairs from known pairs. The first method is based on computing powers of the initial pair of matrices, and the second method is based on appropriate types of linear combinations of the initial pair of matrices. This enriches the already known families of coprime matrices reported in recent literature. Several properties of the newly generated coprime pairs are also addressed.

A novel array structure for significantly increasing the degrees of freedom of linear arrays is proposed. This structure is obtained by systematically nesting two or more uniform linear arrays and can provide O(N2) degrees of freedom using only O(N) physical sensors. It is possible to provide closed form expressions for the sensor locations and the exact degrees of freedom obtainable from the proposed array as a function of the total number of sensors. This cannot be done for existing classes of arrays like minimum redundancy arrays which have been used earlier for detecting more sources than sensors. A novel spatial smoothing based technique is also proposed to exploit the increased degrees of freedom offered by the array to perform DOA estimation of more sources than sensors, using only second order statistics of the received data. This method does not suffer from inherent weaknesses of techniques employing higher order statistics or quasi stationarity of sources. The validity of all the proposed methods is verified through numerical examples.

A novel approach to beamforming using a new class of sensor arrays is proposed, which can increase the achievable degrees of freedom significantly beyond the conventional limits obtained from uniform linear arrays (ULA). This class of arrays is named as "nested arrays" since they are obtained by nesting two or more ULAs with increasing inter-sensor spacing. Using the second order statistics of the signal received by such an array in a novel way, it is possible to perform beamforming with O(N2) de grees of freedom using only O(N) physical elements. This kind of beamforming will be shown to be essentially non linear in nature and theoretically, it is capable of nulling the effect of noise provided enough snapshots are available.

Given a continuous time LTI system with impulse response hc(t), it is shown that the uniformly spaced samples hc(nT) can be identified for any chosen spacing T by using an impulse train input with an arbitrarily small rate 1/NT and sampling the system output with an arbitrarily small rate 1/MT , provided M and N are coprime. This idea, referred to here as the sparse coprime sensing method for system identification, is closely related to well known results in multirate signal processing. It is shown that the problem can be related to the identification of a decimation filter from input-output measurements. It is also shown that the problem is equivalent to the identification of a discrete time N × M LTI system from a knowledge of the full rate input and output vector sequences.

Frequency invariant beamforming with sensor arrays is generally achieved using filters in the form of tapped delay-lines following each sensor. However it has been recently shown that with the help of the rectangular smart antenna array, it is possible to generate frequency invariant beampattern without using filters. In this paper, this frequency invariant beamforming technique is utilized to perform MVDR beamforming in the beamspace by designing frequency invariant beams spanning the desired range of azimuthal angles and optimally combining them. However, the performance of the frequency invariant beamformer depends on the number of sensors which could be large for a rectangular array of size M times N. Making use of the virtual array concept used in MIMO radar, a novel method of producing the same frequency invariant beam, using only M transmitting and N receiving antennas, is proposed and a design example is provided to demonstrate the idea.