Estimating principal components of covariance matrices using the Nystr\"{o}m method

Covariance matrix estimates are an essential part of many signal processing algorithms, and are often used to determine a low-dimensional principal subspace via their spectral decomposition. However, exact eigenanalysis is computationally intractable for sufficiently high-dimensional matrices, and in the case of small sample sizes, sample eigenvalues and eigenvectors are known to be poor estimators of their population counterparts. To address these issues, we propose a covariance estimator that is computationally efficient while also performing shrinkage on the sample eigenvalues. Our approach is based on the Nystr\"{o}m method, which uses a data-dependent orthogonal projection to obtain a fast low-rank approximation of a large positive semidefinite matrix. We provide a theoretical analysis of the error properties of our estimator as well as empirical results, including examples of its application to adaptive beamforming and image denoising

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

Assessment of Measurement Distortions in GNSS Antenna Array Space-Time Processing

Antenna array processing techniques are studied in GNSS as effective tools to mitigate interference in spatial and spatiotemporal domains. However, without specific considerations, the array processing results in biases and distortions in the cross-ambiguity function (CAF) of the ranging codes. In space-time processing (STP) the CAF misshaping can happen due to the combined effect of space-time processing and the unintentional signal attenuation by filtering. This paper focuses on characterizing these degradations for different controlled signal scenarios and for live data from an antenna array. The antenna array simulation method introduced in this paper enables one to perform accurate analyses in the field of STP. The effects of relative placement of the interference source with respect to the desired signal direction are shown using overall measurement errors and profile of the signal strength. Analyses of contributions from each source of distortion are conducted individually and collectively. Effects of distortions on GNSS pseudorange errors and position errors are compared for blind, semi-distortionless, and distortionless beamforming methods. The results from characterization can be useful for designing low distortion filters that are especially important for high accuracy GNSS applications in challenging environments