For subspace estimation with an unknown colored noise, Factor Analysis (FA) and its extensions, denoted as Extended FA (EFA), are good candidates for replacing the popular eigenvalue decomposition (EVD). Finding the unknowns in (E)FA can be done by solving a non-linear least square problem. For this type of optimization problems, the Gauss-Newton (GN) algorithm is a powerful and simple method. The most expensive part of the GN algorithm is finding the direction of descent by solving a system of equations at each iteration. In this paper we show that for (E)FA, the matrices involved in solving these systems of equations can be diagonalized in a closed form fashion and the solution can be found in a computationally efficient way. We show how the unknown parameters can be updated without actually constructing these matrices. The convergence performance of the algorithm is studied via numerical simulations. @en

Image formation using the data from an array of sensors is a familiar problem in many fields such as radio astronomy, biomedical and geodetic imaging. The problem can be formulated as a least squares (LS) estimation problem and becomes ill-posed at high resolutions, i.e. large number of image pixels. In this paper we propose two regularization methods, one based on weighted truncation of the eigenvalue decomposition of the image deconvolution matrix and the other based on the prior knowledge of the "dirty image" using the available data. The methods are evaluated by simulations as well as actual data from a phased-array radio telescope in the Netherlands, the Low Frequency Array Radio Telescope (LOFAR).@en

Many subspace-based array signal processing algorithms assume that the noise is spatially white. In this case, the noise covariance matrix is a multiple of the identity and the eigenvectors of the data covariance matrix are not affected by it. If the noise covariance is an unknown arbitrary diagonal (e.g., for an uncalibrated array), the eigenvalue decomposition leads to incorrect subspace estimates and it has to be replaced by a more general “factor analysis” decomposition (FA), which then reveals all relevant information. We consider this data model and several extensions where the noise covariance matrix has a more general structure, such as banded, sparse, block diagonal, and cases, where we have multiple data covariance matrices that share the same noise covariance matrix. Starting from a nonlinear weighted least squares formulation, we propose new estimation algorithms for both classical FA and its extensions. The optimization is based on Gauss–Newton gradient descent. Generally, this leads to an iteration involving the inversion of a very large matrix. Using the structure of the problem, we show how this can be reduced to the inversion of a matrix with dimension equal to the number of unknown noise covariance parameters. This results in new algorithms that have faster numerical convergence and lower complexity compared to several maximum-likelihood based algorithms that could be considered state of the art. The new algorithms scale well to large dimensions and can replace eigenvalue decompositions in many applications even if the noise can be assumed to be white.@en

The search for the answer to one of the most fundamental scientific questions, “How was the universe formed?”, requires us to study very weak radio signals from the early universe. In the last eighty years, radio astronomers have been able to use radio frequency observations for significant discoveries such as quasars, super massive Black Holes and the Cosmic Microwave Background radiation. Radio astronomers use a radio telescope to study the cosmos. A radio telescope usually consists of an array of radio receivers (antennas) and supporting hardware/software to produce synthesized images of the sky. While the earlier generation of the radio telescopes such as the Westerbork Synthesis Radio Telescope (WSRT), the Very Large Array (VLA) and the Giant Meter–wave Radio Telescope (GMRT) consisted of 14-45 receivers separated a few kilometers (3-25 km basedlines), the next generation of radio telescopes such as LOFAR and SKA have thousands of receivers which cover distances of over 1000 km. This massive increase in the number of receivers and the geometric dimensions is a consequence of the required (high) resolution and sensitivity for modern scientific studies and while it is necessary, it does not guarantee the desired results without the appropriate data and signal processing. The main challenges in radio astronomy can be divided in three closely related problems: mitigation of man–made radio frequency interference, calibration and image formation. The main goal of this thesis is to investigate how the signal processing formalism can be used to systematically model and analyze these three problems and what signal processing tools are needed for addressing them. The number of RFI free bands is diminishing rapidly as a consequence of the increased number of wireless services and applications. The shift towards wideband digital systems has created new problems which are not sufficiently addressed by currently implemented RFI detection and mitigation systems. For this class of continuously present wide–band RFI, the use of array processing techniques such as spatial filtering could provide access to frequency bands otherwise unusable by astronomers. Such a spatial filtering can be achieved by estimating and removing the subspace that the interfering signal is occupying. Many signal processing algorithms use the eigenvalue decomposition (EVD) for estimating the signal subspace. However the use of EVD is limited to systems where the noise is white or known from calibration. This requirement is a limiting factor for applying these techniques to uncalibrated arrays with unknown noise models. In these situations a more generic approaches which allows for combined RFI filtering and noise power calibration is preferred. In this thesis factor analysis (FA) is proposed as suitable substitution for EVD. FA is a technique that allows for the decomposition of the signal into a low–rank part corresponding to the signal and a diagonal part which represents the covariance of the noise on the receivers. Because the diagonal elements can be different this technique can be used when the noise is not white and forms a generalization of the EVD. In RFI mitigation applications the signal part of the data is dominated by RFI and changes more rapidly than the noise. Estimating the noise covariance which is shared by several measurements jointly allows for a more accurate estimation. As a result extensions to the classical FA are proposed to improve the estimates for the diagonal part of the decomposition in a joint fashion. Even a diagonal noise structure can be limiting in some applications. For example the contribution of the Milky Way affects the short baselines which can be modeled by using a non–diagonal covariance matrix. The FA model can be extended for this type of signals. An extension to FA called Extended FA (EFA) is used to allow for capturing such structures into the model. Similar to JFA we can also estimate the parameters in EFA jointly, and the resulting method is denoted by Joint EFA (JEFA). Using nonlinear optimization techniques combined with Krylov subspace based solvers an scalable algorithm is developed. The statistical efficiency of this algorithm is shown by comparing its results to the Cramér–Rao bound and its application in RFI mitigation has been demonstrated on measurements from the WSRT and LOFAR. Antenna gain calibration is an essential step in producing accurate images. Using common array processing datamodels, gain calibration is formulated as a nonlinear covariance matching problem. In this thesis we show that the matrices involved in this estimation problem are highly structured and that the system of equations involving these matrices can be efficiently solved using Krylov subspace based solvers (similar to JEFA). The resulting calibration algorithm is scalable and requires a low number of iterations in order to converge which makes it an attractive alternative to currently available techniques. Both classical and parametric based image formations consist of two steps. First a “dirty” image is constructed from the measurements and then an improved estimate is found by performing a deconvolution step. When the number of pixels on the image becomes large, the deconvolution step becomes an ill–posed problem. In this thesis we show that image values are bounded from below by a nonnegativity constraint and above by the dirty image. Using beamforming techniques, we show that tighter upper bounds can be constructed using the MVDR beamformer. These bounds allow us to regularize the deconvolution problem by a set of inequality constraints. Following a signal processing model, the image formation is then formulated as a parameter estimation problem with inequality constraints. This optimization problem can be solved using an active set algorithm. We show that, with the right initialization, the active set steps are very similar to sequential source removing techniques such as CLEAN. This connection between classical approaches and parametric imaging techniques provides the necessary theoretical basis for further analysis and allows for improving both methods. Based on the results presented in this thesis we can conclude that signal processing methodologies can provide new solutions to the radio astronomical problems and also shed light on the inner working of the classical techniques. Hence, a signal processing approach is extremely beneficial in tackling the problems that the next generation of radio telescopes will face.@en

Having an accurate calibration method is crucial for any scientific research done by a radio telescope. The next generation radio telescopes such as the Square Kilometre Array (SKA) will have a large number of receivers which will produce exabytes of data per day. In this paper we propose new direction-dependent and independent calibration algorithms that, while requiring much less storage during calibration, converge very fast. The calibration problem can be formulated as a non-linear least square optimization problem. We show that combining a block-LDU decomposition with Gauss-Newton iterations produces systems of equations with convergent matrices. This allows significant reduction in complexity per iteration and very fast converging algorithms. We also discuss extensions to direction-dependent calibration. The proposed algorithms are evaluated using simulations.@en

A simple and novel algorithm for source recovery based on array data measurements in radio astronomy is proposed. Considering that a radioastronomical image is composed of both point sources and extended emissions, prior information on the images, namely non-negativity and substantial black background are taken into account to choose source representation basis functions. Dirac delta functions are chosen to represent point sources and a Gaussian function approximated from the main beam of the antenna array is selected to capture the extended emissions. We apply the non-negative least squares (NNLS) algorithm to estimate the basis coefficients. It is shown that the sparsity promoted by the NNLS algorithm based on the chosen basis functions results in a super-resolution (finer resolution than prescribed by the main beam of the antenna array pattern) estimate for the point sources and smooth recovery for the extended emissions.@en

Aims. Image formation for radio astronomy can be defined as estimating the spatial intensity distribution of celestial sources throughout the sky, given an array of antennas. One of the challenges with image formation is that the problem becomes ill-posed as the number of pixels becomes large. The introduction of constraints that incorporate a priori knowledge is crucial. Methods. In this paper we show that in addition to non-negativity, the magnitude of each pixel in an image is also bounded from above. Indeed, the classical “dirty image” is an upper bound, but a much tighter upper bound can be formed from the data using array processing techniques. This formulates image formation as a least squares optimization problem with inequality constraints. We propose to solve this constrained least squares problem using active set techniques, and the steps needed to implement it are described. It is shown that the least squares part of the problem can be efficiently implemented with Krylov-subspace-based techniques. We also propose a method for correcting for the possible mismatch between source positions and the pixel grid. This correction improves both the detection of sources and their estimated intensities. The performance of these algorithms is evaluated using simulations. Results. Based on parametric modeling of the astronomical data, a new imaging algorithm based on convex optimization, active sets, and Krylov-subspace-based solvers is presented. The relation between the proposed algorithm and sequential source removing techniques is explained, and it gives a better mathematical framework for analyzing existing algorithms. We show that by using the structure of the algorithm, an efficient implementation that allows massive parallelism and storage reduction is feasible. Simulations are used to compare the new algorithm to classical CLEAN. Results illustrate that for a discrete point model, the proposed algorithm is capable of detecting the correct number of sources and producing highly accurate intensity estimates. @en

Aims. Image formation for radio astronomy can be defined as estimating the spatial intensity distribution of celestial sources throughout the sky, given an array of antennas. One of the challenges with image formation is that the problem becomes ill-posed as the number of pixels becomes large. The introduction of constraints that incorporate a priori knowledge is crucial. Methods. In this paper we show that in addition to non-negativity, the magnitude of each pixel in an image is also bounded from above. Indeed, the classical “dirty image” is an upper bound, but a much tighter upper bound can be formed from the data using array processing techniques. This formulates image formation as a least squares optimization problem with inequality constraints. We propose to solve this constrained least squares problem using active set techniques, and the steps needed to implement it are described. It is shown that the least squares part of the problem can be efficiently implemented with Krylov-subspace-based techniques. We also propose a method for correcting for the possible mismatch between source positions and the pixel grid. This correction improves both the detection of sources and their estimated intensities. The performance of these algorithms is evaluated using simulations. Results. Based on parametric modeling of the astronomical data, a new imaging algorithm based on convex optimization, active sets, and Krylov-subspace-based solvers is presented. The relation between the proposed algorithm and sequential source removing techniques is explained, and it gives a better mathematical framework for analyzing existing algorithms. We show that by using the structure of the algorithm, an efficient implementation that allows massive parallelism and storage reduction is feasible. Simulations are used to compare the new algorithm to classical CLEAN. Results illustrate that for a discrete point model, the proposed algorithm is capable of detecting the correct number of sources and producing highly accurate intensity estimates. @en

Radio astronomical observations are increasingly contaminated by RF interference. Assuming an array of telescopes, a previous technique considered spatial filtering based on projecting out the interferer array signature vector. A disadvantage is that this effectively reduces the array by one (expensive) telescope. In this paper, we consider extending the astronomical array with a reference antenna array, and develop spatial filtering algorithms for this situation. The information from the reference antennas improves the quality of the interferer signature vector estimation, hence more of the interference can be projected out. Moreover, since only the covariance data of the astronomical array has to be reconstructed, the conditioning of the problem improves as well. The algorithms are tested both on simulated and experimental data.@en

Radio astronomy is known for its very large telescope dishes but is currently making a transition towards the use of a large number of small antennas. For example, the Low Frequency Array, commissioned in 2010, uses about 50 stations each consisting of 96 low band antennas and 768 or 1536 high band antennas. The low-frequency receiving system for the future Square Kilometre Array is envisaged to initially consist of over 131,000 receiving elements and to be expanded later. These instruments pose interesting array signal processing challenges. To present some aspects, we start by describing how the measured correlation data is traditionally converted into an image, and translate this into an array signal processing framework. This paves the way to describe self-calibration and image reconstruction as estimation problems. Self-calibration of the instrument is required to handle instrumental effects such as the unknown, possibly direction dependent, response of the receiving elements, as well a unknown propagation conditions through the Earth’s troposphere and ionosphere. Array signal processing techniques seem well suited to handle these challenges. Interestingly, image reconstruction, calibration and interference mitigation are often intertwined in radio astronomy, turning this into an area with very challenging signal processing problems.@en