Covariance matching techniques for radio astronomy calibration and imaging

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.