Band Limited Signals Observed Over Finite Spatial and Temporal Windows: An Upper Bound to Signal Degrees of Freedom

The study of degrees of freedom of signals observed within spatially diverse broadband multipath fields is an area of ongoing investigation and has a wide range of applications, including characterising broadband MIMO and cooperative networks. However, a fundamental question arises: given a size limitation on the observation region, what is the upper bound on the degrees of freedom of signals observed within a broadband multipath field over a finite time window? In order to address this question, we characterize the multipath field as a sum of a finite number of orthogonal waveforms or spatial modes. We show that (i) the "effective observation time" is independent of spatial modes and different from actual observation time, (ii) in wideband transmission regimes, the "effective bandwidth" is spatial mode dependent and varies from the given frequency bandwidth. These findings clearly indicate the strong coupling between space and time as well as space and frequency in spatially diverse wideband multipath fields. As a result, signal degrees of freedom does not agree with the well-established degrees of freedom result as a product of spatial degrees of freedom and time-frequency degrees of freedom. Instead, analogous to Shannon's communication model where signals are encoded in only one spatial mode, the available signal degrees of freedom in spatially diverse wideband multipath fields is the time-bandwidth product result extended from one spatial mode to finite modes. We also show that the degrees of freedom is affected by the acceptable signal to noise ratio (SNR) in each spatial mode.Comment: Submitted to IEEE Transactions on Signal Processin

Sparse Covariance Fitting Method for Direction of Arrival Estimation of Uncorrelated Wideband Signals

We propose a new direction of arrival estimation method for wideband uncorrelated signals. The wideband signals are first decomposed into narrowband signals. A group sparse Lasso formulation is proposed that jointly fits the powers of the signals using overcomplete dictionaries of the directions of arrival, into the estimated covariance matrices for all narrowband signals. Then, we propose a new formulation that determines the regularization parameters and becomes the group Lasso formulation. Additionally, we propose a modified algorithm for direction of arrival estimation with lower complexity that uses conventional based methods in the preprocessing stage to reduce the number of variables in the optimization task. We compare the performance of the proposed method to the conventional methods for a circular antenna array

Wide-band maximum likelihood direction finding by using the tree-structured EM algorithm

Ankara : The Department of Electrical and Electronics Engineering and the Institute of Engineering and Sciences of Bilkent Univ., 1996.Thesis (Master's) -- Bilkent University, 1996.Includes bibliographical references leaves 38-43A thorough derivation of the Expectation Maximization (EM) algorithm, which is an iterative numerical method of Maximum Likelihood (ML) estimation, is presented for the case of estimating direction of arrivals of unknown deterministic wide-band signals incident from different directions onto a passive array. For the rec^uired signal estimation, alternative regularized least squares estimation techniques are proposed with significant improvement over the standard least squares techniques. Also, for the angle of arrival estimation of a large number of signals, a tree structured EM algorithm is proposed and compared with the conventional EM approach. Extensive simulation results are presented for comparison of the proposed algorithms with the current high-resolution methods of wide-band direction finding. In order to handle efficiently the case of available parametric prior models on the received waveforms, the required modifications are also given.Çadallı, NailM.S

Signal Subspace Processing in the Beam Space of a True Time Delay Beamformer Bank

A number of techniques for Radio Frequency (RF) source location for wide bandwidth signals have been described that utilize coherent signal subspace processing, but often suffer from limitations such as the requirement for preliminary source location estimation, the need to apply the technique iteratively, computational expense or others. This dissertation examines a method that performs subspace processing of the data from a bank of true time delay beamformers. The spatial diversity of the beamformer bank alleviates the need for a preliminary estimate while simultaneously reducing the dimensionality of subsequent signal subspace processing resulting in computational efficiency. The pointing direction of the true time delay beams is independent of frequency, which results in a mapping from element space to beam space that is wide bandwidth in nature. This dissertation reviews previous methods, introduces the present method, presents simulation results that demonstrate the assertions, discusses an analysis of performance in relation to the Cramer-Rao Lower Bound (CRLB) with various levels of noise in the system, and discusses computational efficiency. One limitation of the method is that in practice it may be appropriate for systems that can tolerate a limited field of view. The application of Electronic Intelligence is one such application. This application is discussed as one that is appropriate for a method exhibiting high resolution of very wide bandwidth closely spaced sources and often does not require a wide field of view. In relation to system applications, this dissertation also discusses practical employment of the novel method in terms of antenna elements, arrays, platforms, engagement geometries, and other parameters. The true time delay beam space method is shown through modeling and simulation to be capable of resolving closely spaced very wideband sources over a relevant field of view in a single algorithmic pass, requiring no course preliminary estimation, and exhibiting low computational expense superior to many previous wideband coherent integration techniques

Signal Subspace Processing in the Beam Space of a True Time Delay Beamformer Bank

A number of techniques for Radio Frequency (RF) source location for wide bandwidth signals have been described that utilize coherent signal subspace processing, but often suffer from limitations such as the requirement for preliminary source location estimation, the need to apply the technique iteratively, computational expense or others. This dissertation examines a method that performs subspace processing of the data from a bank of true time delay beamformers. The spatial diversity of the beamformer bank alleviates the need for a preliminary estimate while simultaneously reducing the dimensionality of subsequent signal subspace processing resulting in computational efficiency. The pointing direction of the true time delay beams is independent of frequency, which results in a mapping from element space to beam space that is wide bandwidth in nature. This dissertation reviews previous methods, introduces the present method, presents simulation results that demonstrate the assertions, discusses an analysis of performance in relation to the Cramer-Rao Lower Bound (CRLB) with various levels of noise in the system, and discusses computational efficiency. One limitation of the method is that in practice it may be appropriate for systems that can tolerate a limited field of view. The application of Electronic Intelligence is one such application. This application is discussed as one that is appropriate for a method exhibiting high resolution of very wide bandwidth closely spaced sources and often does not require a wide field of view. In relation to system applications, this dissertation also discusses practical employment of the novel method in terms of antenna elements, arrays, platforms, engagement geometries, and other parameters. The true time delay beam space method is shown through modeling and simulation to be capable of resolving closely spaced very wideband sources over a relevant field of view in a single algorithmic pass, requiring no course preliminary estimation, and exhibiting low computational expense superior to many previous wideband coherent integration techniques

Sensor Array Processing with Manifold Uncertainty

<p>The spatial spectrum, also known as a field directionality map, is a description of the spatial distribution of energy in a wavefield. By sampling the wavefield at discrete locations in space, an estimate of the spatial spectrum can be derived using basic wave propagation models. The observable data space corresponding to physically realizable source locations for a given array configuration is referred to as the array manifold. In this thesis, array manifold ambiguities for linear arrays of omni-directional sensors in non-dispersive fields are considered. </p><p>First, the problem of underwater a hydrophone array towed behind a maneuvering platform is considered. The array consists of many hydrophones mounted to a flexible cable that is pulled behind a ship. The towed cable will bend or distort as the ship performs maneuvers. The motion of the cable through the turn can be used to resolve ambiguities that are inherent to nominally linear arrays. The first significant contribution is a method to estimate the spatial spectrum using a time-varying array shape in a dynamic field and broadband temporal data. Knowledge of the temporal spectral shape is shown to enhance detection performance. The field is approximated as a sum of uncorrelated planewaves located at uniform locations in angle, forming a gridded map on which a maximum likelihood estimate for broadband source power is derived. Uniform linear arrays also suffer from spatial aliasing when the inter-element spacing exceeds a half-wavelength. Broadband temporal knowledge is shown to significantly reduce aliasing and thus, in simulation, enhance target detection in interference dominated environments. </p><p>As an extension, the problem of towed array shape estimation is considered when the number and location of sources are unknown. A maximum likelihood estimate of the array shape using the field directionality map is derived. An acoustic-based array shape estimate that exploits the full 360$^\circ$ field via field directionality mapping is the second significant contribution. Towed hydrophone arrays have heading sensors in order to estimate array shape, but these sensors can malfunction during sharp turns. An array shape model is described that allows the heading sensor data to be statistically fused with heading sensor. The third significant contribution is method to exploit dynamical motion models for sharp turns for a robust array shape estimate that combines acoustic and heading data. The proposed array shape model works well for both acoustic and heading data and is valid for arbitrary continuous array shapes.</p><p>Finally, the problem of array manifold ambiguities for static under-sampled linear arrays is considered. Under-sampled arrays are non-uniformly sampled with average spacing greater than a half-wavelength. While spatial aliasing only occurs in uniformly sampled arrays with spacing greater than a half-wavelength, under-sampled arrays have increased spatial resolution at the cost of high sidelobes compared to half-wavelength sampled arrays with the same number of sensors. Additionally, non-uniformly sampled arrays suffer from rank deficient array manifolds that cause traditional subspace based techniques to fail. A class of fully agumentable arrays, minimally redundant linear arrays, is considered where the received data statistics of a uniformly spaced array of the same length can be reconstructed in wide sense stationary fields at the cost of increased variance. The forth significant contribution is a reduced rank processing method for fully augmentable arrays to reduce the variance from augmentation with limited snapshots. Array gain for reduced rank adaptive processing with diagonal loading for snapshot deficient scenarios is analytically derived using asymptotic results from random matrix theory for a set ratio of sensors to snapshots. Additionally, the problem of near-field sources is considered and a method to reduce the variance from augmentation is proposed. In simulation, these methods result in significant average and median array gains with limited snapshots.</p>Dissertatio

The measurement of underwater acoustic noise radiated by a vessel using the vessel's own towed array

The work described in this thesis tested the feasibility of using a towed array of hydrophones to: 1. localise sources of underwater acoustic noise radiated by the towvessel, 2. determine the absolute amplitudes of these sources, and 3. determine the resulting far-field acoustic signature of the tow-vessel. The concept was for the towvessel to carry out a U-turn manoeuvre so as to bring the acoustic section of the array into a location suitable for beamforming along the length of the tow-vessel. All three of the above were shown to be feasible using both simulated and field data, although no independent field measurements were available to fully evaluate the accuracy of the far-field acoustic signature determinations. A computer program was written to simulate the acoustic signals received by moving hydrophones. This program had the ability to model a variety of acoustic sources and to deal with realistic acoustic propagation conditions, including shallow water propagation with significant bottom interactions. The latter was accomplished using both ray and wave methods and it was found that, for simple fluid half-space seabeds, a modified ray method gave results that were virtually identical to those obtained with a full wave method, even at very low frequencies, and with a substantial saving in execution time. A field experiment was carried out during which a tug towing a 60-hydrophone array carried out a series of U-turn manoeuvres. The signals received by the array included noise radiated by the tow-vessel, signals from acoustic tracking beacons mounted on the tow-vessel, and transient signals generated by imploding sources deployed from a second vessel.Algorithms were developed to obtain snapshots of the vertical plane and horizontal plane shapes of the array from the transient data and to use range data derived from the tracking beacon signals to track the hydrophones in the horizontal plane. The latter was complicated by a high proportion of dropouts and outliers in the range data caused by the directionality of the hydrophones at the high frequencies emitted by the beacons. Despite this, excellent tracking performance was obtained. Matched field inversion was used to determine the vertical plane array shapes at times when no transient signals were available, and to provide information about the geoacoustic properties of the seabed. There was very good agreement between the inversion results and array shapes determined using transient signals. During trial manoeuvres the array was moving rapidly relative to the vessel and changing shape. A number of different array-processing algorithms were developed to provide source localisation and amplitude estimates in this situation: a timedomain beamformer; two frequency-domain, data independent beamformers; an adaptive frequency-domain beamformer; and an array processor based on a regularised least-squares inversion. The relative performance of each of these algorithms was assessed using simulated and field data. Data from three different manoeuvres were processed and in each case a calibrated source was localised to within 1 m of its known position at the source's fundamental frequency of 112 Hz.Localisation was also successful in most instances at 336 Hz, 560 Hz and 784 Hz, although with somewhat reduced accuracy due to lower signal to noise ratios. Localisation results for vessel noise sources were also consistent with the positions of the corresponding items of machinery. The estimated levels of the calibrated source obtained during the three manoeuvres were all within 4.1 dB of the calibrated value, and varied by only 1.3 dB between manoeuvres. Results at the higher frequencies had larger errors, with a maximum variation of 3.8 dB between serials, and a maximum deviation from the calibrated value of 6.8 dB. An algorithm was also developed to predict the far-field signature of the tow-vessel from the measured data and results were produced. This algorithm performed well with simulated data but no independent measurements were available to compare with the field results