Fundamental Imaging Limits of Radio Telescope Arrays

The fidelity of radio astronomical images is generally assessed by practical experience, i.e. using rules of thumb, although some aspects and cases have been treated rigorously. In this paper we present a mathematical framework capable of describing the fundamental limits of radio astronomical imaging problems. Although the data model assumes a single snapshot observation, i.e. variations in time and frequency are not considered, this framework is sufficiently general to allow extension to synthesis observations. Using tools from statistical signal processing and linear algebra, we discuss the tractability of the imaging and deconvolution problem, the redistribution of noise in the map by the imaging and deconvolution process, the covariance of the image values due to propagation of calibration errors and thermal noise and the upper limit on the number of sources tractable by self calibration. The combination of covariance of the image values and the number of tractable sources determines the effective noise floor achievable in the imaging process. The effective noise provides a better figure of merit than dynamic range since it includes the spatial variations of the noise. Our results provide handles for improving the imaging performance by design of the array.Comment: 12 pages, 8 figure

Multisource Self-calibration for Sensor Arrays

Calibration of a sensor array is more involved if the antennas have direction dependent gains and multiple calibrator sources are simultaneously present. We study this case for a sensor array with arbitrary geometry but identical elements, i.e. elements with the same direction dependent gain pattern. A weighted alternating least squares (WALS) algorithm is derived that iteratively solves for the direction independent complex gains of the array elements, their noise powers and their gains in the direction of the calibrator sources. An extension of the problem is the case where the apparent calibrator source locations are unknown, e.g., due to refractive propagation paths. For this case, the WALS method is supplemented with weighted subspace fitting (WSF) direction finding techniques. Using Monte Carlo simulations we demonstrate that both methods are asymptotically statistically efficient and converge within two iterations even in cases of low SNR.Comment: 11 pages, 8 figure

Die ekonomiese en kulturele selfstandingheid van die Afrikaner

Suid-Afrika het in die afgelope tien jaar ’n enorme ontwikkeling deur gemaak. Hierdie ontwikkeling het oor die algemeen groot voorspoed vir die Unie gebring, maar tegelykertyd is ’n aantal probleme geskep wat vir ’n deel altans, by ’n meer sistematiese beplanning vermy kon gewees het

Redundancy Calibration of Phased Array Stations

Our aim is to assess the benefits and limitations of using the redundant visibility information in regular phased array systems for improving the calibration. Regular arrays offer the possibility to use redundant visibility information to constrain the calibration of the array independent of a sky model and a beam models of the station elements. It requires a regular arrangement in the configuration of array elements and identical beam patterns. We revised a calibration method for phased array stations using the redundant visibility information in the system and applied it successfully to a LOFAR station. The performance and limitations of the method were demonstrated by comparing its use on real and simulated data. The main limitation is the mutual coupling between the station elements, which leads to non-identical beams and stronger baseline dependent noise. Comparing the variance of the estimated complex gains with the Cramer-Rao Bound (CRB) indicates that redundancy is a stable and optimum method for calibrating the complex gains of the system. Our study shows that the use of the redundant visibility does improve the quality of the calibration in phased array systems. In addition it provides a powerful tool for system diagnostics. Our results demonstrate that designing redundancy in both the station layout and the array configuration of future aperture arrays is strongly recommended. In particular in the case of the Square Kilometre Array with its dynamic range requirement which surpasses any existing array by an order of magnitude.Comment: 16 pages, 15 figures, accepted for publication in the A&A in Section 13, acceptance date: 1st May 2012. NOTE: Please contact the first author for high resolution figure

Fast gain calibration in radio astronomy using alternating direction implicit methods: Analysis and applications

Context. Modern radio astronomical arrays have (or will have) more than one order of magnitude more receivers than classical synthesis arrays, such as the VLA and the WSRT. This makes gain calibration a computationally demanding task. Several alternating direction implicit (ADI) approaches have therefore been proposed that reduce numerical complexity for this task from $\mathcal{O}(P^3)$ to $\mathcal{O}(P^2)$, where $P$ is the number of receive paths to be calibrated. Aims. We present an ADI method, show that it converges to the optimal solution, and assess its numerical, computational and statistical performance. We also discuss its suitability for application in self-calibration and report on its successful application in LOFAR standard pipelines. Methods. Convergence is proved by rigorous mathematical analysis using a contraction mapping. Its numerical, algorithmic, and statistical performance, as well as its suitability for application in self-calibration, are assessed using simulations. Results. Our simulations confirm the $\mathcal{O}(P^2)$ complexity and excellent numerical and computational properties of the algorithm. They also confirm that the algorithm performs at or close to the Cramer-Rao bound (CRB, lower bound on the variance of estimated parameters). We find that the algorithm is suitable for application in self-calibration and discuss how it can be included. We demonstrate an order-of-magnitude speed improvement in calibration over traditional methods on actual LOFAR data. Conclusions. In this paper, we demonstrate that ADI methods are a valid and computationally more efficient alternative to traditional gain calibration method and we report on its successful application in a number of actual data reduction pipelines.Comment: accepted for publication in Astronomy & Astrophysic

Calibration Challenges for Future Radio Telescopes

Instruments for radio astronomical observations have come a long way. While the first telescopes were based on very large dishes and 2-antenna interferometers, current instruments consist of dozens of steerable dishes, whereas future instruments will be even larger distributed sensor arrays with a hierarchy of phased array elements. For such arrays to provide meaningful output (images), accurate calibration is of critical importance. Calibration must solve for the unknown antenna gains and phases, as well as the unknown atmospheric and ionospheric disturbances. Future telescopes will have a large number of elements and a large field of view. In this case the parameters are strongly direction dependent, resulting in a large number of unknown parameters even if appropriately constrained physical or phenomenological descriptions are used. This makes calibration a daunting parameter estimation task, that is reviewed from a signal processing perspective in this article.Comment: 12 pages, 7 figures, 20 subfigures The title quoted in the meta-data is the title after release / final editing

Efficient Prediction of Array Element Patterns Using Physics-Based Expansions and a Single Far-Field Measurement

A method is proposed to predict the antenna array beam through employing a relatively small set of physics-based basis functions-called characteristic basis function patterns (CBFPs)-for modeling the embedded element patterns. The primary CBFP can be measured or extracted from numerical simulations, while additional (secondary) CBFPs are derived from the primary one. Furthermore, each numerically generated CBFP, which is typically simulated/measured for discrete directions only, can in turn be approximated by analytical basis functions with fixed expansion coefficients to evaluate the resulting array pattern at any angle through interpolation. This hierarchical basis reduces the number of unknown expansion coefficients significantly. Accordingly, the CBFP expansion coefficients can be determined through a single far-field measurement of only a few reference sources in the field of view. This is particularly important for multibeam array applications where only a limited number of reference sources are available for predicting the beam shape. Furthermore, this instantaneous beam calibration is fast, i.e., potentially capable to speed up the array calibration by one or two orders of magnitude, which is particularly important if the antenna radiation characteristics are subject to drifts