Source localization is the process of locating a source of electromagnetic waves or acoustic waves from passive measurements of the emitted fields made on distributed receivers. This is an important problem for both military and civilian applications. The Synthetic Aperture Passive Source Localization (SAPSL) algorithm combines statistical methods with techniques from Synthetic-Aperture Radar to attain a high-resolution image from two receivers. We focus on the case of wave propagation in a homogeneous medium, sources restricted to lie on a known surface, and slowly moving sensors. This paper contains resolution formulas for the SAPSL algorithm for two different sensor geometries. The first geometry consists of a stationary isotropic source located between one stationary sensor and one mobile sensor on a linear path. The second geometry consists of a stationary isotropic source and two mobile sensors following each other along a linear path. In both geometries, we make narrow-aperture assumptions in order to obtain simple resolution formulas. This paper also presents numerical simulations that verify the resolution predicted from the formulas.

Since most inverse problems arising in scientific and engineering applications are ill-posed, prior information about the solution space is incorporated, typically through regularization, to establish a well-posed problem with a unique solution. Often, this prior information is an assumed statistical distribution of the desired inverse problem solution. Recently, due to the unprecedented success of generative adversarial networks (GANs), the generative network from a GAN has been implemented as the prior information in imaging inverse problems. In this paper, we devise a novel iterative algorithm to solve inverse problems in imaging where a dual-structured prior is imposed by combining a GAN prior with the compound Gaussian (CG) class of distributions. A rigorous computational theory for the convergence of the proposed iterative algorithm, which is based upon the alternating direction method of multipliers, is established. Furthermore, elaborate empirical results for the proposed iterative algorithm are presented. By jointly exploiting the powerful CG and GAN classes of image priors, we find, in compressive sensing and tomographic imaging problems, our proposed algorithm outperforms and provides improved generalizability over competitive prior art approaches while avoiding performance saturation issues in previous GAN prior-based methods.

This paper follows a detection-theoretic approach for using synthetic-aperture measurements, made at multiple moving passive receivers, in order to form an image showing the locations of stationary sources that are radiating unknown electromagnetic or acoustic waves. The paper starts with a physics-based model for the propagating fields, and, following the general approach of McWhorter et al (2023 arXiv:2302.06816, IEEE Open J. Signal Process. 4 437-51), derives a detection statistic that is used for the image formation. This detection statistic is a quadratic function of the data. Each point in the scene is tested as a possible hypothesized location for a source, and the detection statistic is plotted as a function of location. Because this image formation process is nonlinear, the standard linear methods for determining resolution cannot be applied. This paper shows how to analyze the detection image by first writing the noiseless image as a coherent sum of shifted complex ambiguity functions of the source waveform. The paper then develops a technique for calculating image resolution; resolution is found to depend on the sensor-source geometry and also on the properties (bandwidth and temporal duration) of the source waveform. Optimal filtering of the image is given, but a simple example suggests that optimal filtering may have little effect. Analysis is also given for the case in which multiple sources are present.

Two key components in Synthetic Aperture Radar (SAR) are image resolution and scene coverage. To improve SAR image resolution, one may increase the bandwidth of the system (frequency diversity) or increase the range of aspect angles to view the target (geometric diversity). In most applications, the radar must operate within a fixed and specified frequency band, so increasing the frequency diversity is not feasible. To exploit geometric diversity and maximize scene coverage, one may fly around the target scene of interest in a non-straight path and steer the radar antenna towards the desired targets. This leads to a hybrid SAR mode that combines stripmap SAR and spotlight SAR modes. To investigate the trade-off between resolution and coverage, we develop a mathematical tool to quantify the attainable resolution of the target scene by incorporating the data-collection manifold (DCM). We use the DCM to build an objective function that is used as a means for determining optimal flight paths for desired coverage and resolution objectives by varying a SAR vehicle's heading, pitch, and antenna steering angles.

Algorithm unfolding or unrolling is the technique of constructing a deep neural network (DNN) from an iterative algorithm. Unrolled DNNs often provide better interpretability and superior empirical performance over standard DNNs in signal estimation tasks. An important theoretical question, which has only recently received attention, is the development of generalization error bounds for unrolled DNNs. These bounds deliver theoretical and practical insights into the performance of a DNN on empirical datasets that are distinct from, but sampled from, the probability density generating the DNN training data. In this paper, we develop novel generalization error bounds for a class of unrolled DNNs that are informed by a compound Gaussian prior. These compound Gaussian networks have been shown to outperform comparative standard and unfolded deep neural networks in compressive sensing and tomographic imaging problems. The generalization error bound is formulated by bounding the Rademacher complexity of the class of compound Gaussian network estimates with Dudley's integral. Under realistic conditions, we show that, at worst, the generalization error scales 𝒪(n√(ln(n))) in the signal dimension and 𝒪((Network Size)^3/2) in network size.

This paper studies Frequency-Difference-of-Arrival (FDOA) curves for the 2-dimensional, 2-sensor case. The primary focus of this paper is to give a description of curves associated to the FDOA problem from the algebro-geometric point of view. To be more precise, the complex projective picture of the family of FDOA curves for all possible relative velocities is described.

In this paper we establish a general first-order statistical framework for the detection of a common signal impinging on spatially distributed receivers. We consider three types of channel models: 1) the propagation channel is completely known, 2) the propagation is known but channel gains are unknown, and 3) the propagation channel is unknown. For each problem, we address the cases of a) known noise variances, b) common but unknown noise variances, and c) different and unknown noise variances. For all 9 cases, we establish generalized-likelihood-ratio (GLR) detectors, and show that each one can be decomposed into two terms. The first term is a weighted combination of the GLR detectors that arise from considering each channel separately. This result is then modified by a fusion or cross-validation term, which expresses the level of confidence that the single-channel detectors have detected a common source. Of particular note are the constant false-alarm rate (CFAR) detectors that allow for scale-invariant detection in multiple channels with different noise powers.

For solving linear inverse problems, particularly of the type that appears in tomographic imaging and compressive sensing, this paper develops two new approaches. The first approach is an iterative algorithm that minimizes a regularized least squares objective function where the regularization is based on a compound Gaussian prior distribution. The compound Gaussian prior subsumes many of the commonly used priors in image reconstruction, including those of sparsity-based approaches. The developed iterative algorithm gives rise to the paper's second new approach, which is a deep neural network that corresponds to an “unrolling” or “unfolding” of the iterative algorithm. Unrolled deep neural networks have interpretable layers and outperform standard deep learning methods. This paper includes a detailed computational theory that provides insight into the construction and performance of both algorithms. The conclusion is that both algorithms outperform other state-of-the-art approaches to tomographic image formation and compressive sensing, especially in the difficult regime of low training.

The authors propose a nascent concept for an iterative time-reversal radar (ITRR) that shows promise for detecting and tracking (localizing) a target of interest by using multiple transmit-receive pairs (distributed radars) and iteratively applying time reversal (TR). Existing research suggests that the ITRR methodology rapidly converges to a waveform that is better suited (matched) to a target, because the waveform’s frequency profile is better aligned with a target’s resonances. Hence an ITRR may provide a more effective way of generating waveforms that respond more dynamically to targets. The fundamental premise is to replace the matched filter of standard radar methodology, which assumes that the received and transmitted signals are the same, with an iterative TR (ITR) process that allows the environment to do the matched filtering, thereby achieving better cross-correlation between the transmitted and received signals. Important issues and open problems are noted.

Matched-field processing for localizing an underwater acoustic source in range and depth from power-spectrum measurements obtained at a single hydrophone receiver often suffers from high side-lobes and ambiguities when the signal is narrowband or signal bandwidth is small. In this paper we review how source motion can be utilized to reduce side-lobes in depth and range via an incoherent synthetic-aperture-like approach and explain the mechanisms responsible for side-lobe reduction relative to the height of the main-lobe by using a normal-mode expansion for the pressure field. We also derive an approximation for the depth main-lobe width when the true source range is known for the ideal rigid-waveguide case. Numerical results are presented corroborating the analytical analysis along with some matched-field localization ambiguity surface examples for (a) an ideal shallow-water waveguide with a pressure-release top boundary and a rigid bottom boundary and (b) a more realistic shallow-water Pekeris environment, to demonstrate how side-lobes and ambiguities are reduced when source motion is exploited in the matched-field processing.

In this article a general first-order statistical framework is established for passive multi-channel detection and localization of an unknown radiated signal. This radiated signal is written in terms of a finite basis expansion, and the map between these basis coefficients and the measured data on a sensor is a channel that might be known, partially known, or unknown except for the dimension of the signal subspace. The noise at each sensor is assumed to be Gaussian and white; the noise variances at each sensor may be known, unknown and equal, or unknown and possibly unequal. This article develops detectors for all nine of these cases. These detectors are each generalized likelihood ratios, and typically decompose into locally computed detector statistics plus pairwise coherence, or cross-validation, statistics. Of particular note are the scale-invariant detectors that preserve a constant false alarm rate (CFAR) property with respect to noise power.

It is well-known that an iterative time-reversal (TR) process, applied via a distributed sensing system, can be used to produce a space-time waveform that maximizes the energy scattered from a stationary target back to the sensors. The TR process accomplishes this by automatically focusing energy, both spatially and spectrally, on the stationary target. When scatterers are moving, however, the TR focusing can break down. This article shows how to modify the TR process so that it automatically focuses energy on a moving target. This new TR process automatically generates a distributed beam that: 1) follows the target as it moves and 2) enhances the target resonances by concentrating the energy spectrally. This focusing occurs without a priori knowledge of the target's location or spectral response. The new TR algorithm is derived through a careful analysis of the idealized case of a single isotropic moving point scatterer. This article includes simulations that compare the TR focus both with and without the new modification. Although the simulations are carried out for the electromagnetic case, the theory applies equally well to the acoustic case when the target speed is significantly less than the ambient sound speed. Appendices are included with details of the calculations and with analysis of the range of relative velocities for which the modified TR process should be used.

To solve linear inverse problems in image estimation, this paper develops an iterative algorithm that solves an implicitly regularized least squares problem where the chosen regularization is formulated on a compound Gaussian prior. The compound Gaussian prior, which decomposes the sparse coefficients of an image into a scale vector and Gaussian vector, subsumes many of the commonly used priors, including those of sparsity-based approaches. An implicit regularization on the scale vector permits application-specific variations for the iterative algorithm by specifying a regularization, or scale vector density, that fits the problem at hand. This iterative algorithm is then recast into a deep neural network through the algorithm unrolling, or unfolding, technique. The resultant unfolded deep neural network learns the implicit regularization from application-specific data, which, in effect, learns a generalization of the compound Gaussian prior that best applies to the given data. The algorithm unfolding approach, in general, produces high-performance deep neural networks that have interpretable layers. Through experimental testing, we conclude that our unfolded deep neural network outperforms other state-of-the-art iterative approaches to tomographic image formation.

Passive localization of acoustic or radio-frequency sources is often performed using time-difference-of-arrival (TDOA) measurements and/or frequency-difference-of-arrival (FDOA) measurements. TDOA localization has been thoroughly studied, but FDOA less so. This is largely because the TDOA level surfaces are hyperboloids, which are well understood, whereas the FDOA level curves and surfaces are much more complicated. This article addresses the case of known sensor positions and velocities and a stationary source. This article shows examples of the FDOA level curves and surfaces, and shows that they simplify dramatically in the far field, i.e., when the source is much farther from the origin than the sensors. The far-field behavior is of two types, depending on whether the sensor velocities are equal or unequal. The far-field behavior gives insight into conditions needed for far-field TDOA–FDOA localization and FDOA-only localization. This article includes a characterization of feasible far-field TDOA and unequal-velocity FDOA data.

Many state-of-the-art methods in source localization require large numbers of sensors and perform poorly or require additional sensors when emitters of interest transmit highly correlated waveforms. We present a new source localization technique which employs a cross correlation measure of the time difference of arrival (TDOA) for signals recorded at two separate platforms, at least one of which is in motion. This data is backprojected through a process of synthetic aperture source localization (SASL) to form an image of the locations of the emitters in a region of interest (ROI) This method has the advantage of not requiring anya prioriknowledge of the number of emitters in the scene. Nor does it rest on an ability to identify regions of the data which come from individual emitters, though if this capability is present it may improve image quality. We demonstrate that this method is capable of localizing emitters which transmit highly correlated waveforms, though complications arise when several such emitters are present in the scene. We discuss these complications and strategies to mitigate them.

We develop a physics-based mathematical model for the signals received from a fixed radar that interrogates a moving target and a single rotating object located in the same range cell. From this mathematical model, we extract a statistical model, and use this model to develop a detector for a linearly moving target in "clutter" produced by the rotating object. In particular, we exploit the second-order correlation structure of wind-turbine clutter to derive a target detector that uses a multipulse coherence statistic consisting of an incoherent geometric average of coherently computed adaptive coherence scores. The detector compares favorably to a multipulse coherence detector that uses no modeling or estimation of the clutter. Although we focus on the case of rotating wind turbines, the analysis applies to other rotating objects such as propellers. Similar models may be used for acoustic signals.

This paper develops a new framework for jointly addressing the waveform design and imaging problems in multiple-input-multiple-output synthetic-aperture radar systems could be continuously transmitting wideband electromagnetic energy. Here the transmitters and receivers move along arbitrary trajectories, and the scene is assumed to be stationary. The paper begins with with the derivation of the forward model and uses this model to develop a matched-filter imaging method. Analysis of the resulting image shows that the point-spread function involves a certain wideband cross-ambiguity function, which forms the basis for addressing the waveform design problem.

Typical synthetic aperture radar imaging techniques neglect the dispersive nature of the so-called image "reflectivity" function over the bandwidth of the transmitted waveform. In this paper, we form an image of the complex scene reflectivity as it depends on (x, y, and frequency), or equivalently (x, y, and time delay), a technique we refer to as hyperspectral synthetic aperture radar (HSAR). Our approach is based on a signal model that allows arbitrary flight trajectories and arbitrary waveforms (including continuously transmitting signals such as noise waveforms), and incorporates the causal, dispersive nature of the scene reflectivity without resorting to resolution-degrading frequency-domain subbanding as others have previously proposed. We describe the resulting joint time-space resolution of HSAR in terms of the imaging point spread function for a selection of geometries and waveform bandwidths, and provide numerical examples to illustrate the approach.