This paper introduces an extension of minimum variance beamforming that explicitly takes into account variation or uncertainty in the array response. Sources of this uncertainty include imprecise knowledge of the angle of arrival and uncertainty in the array manifold. In our method, uncertainty in the array manifold is explicitly modeled via an ellipsoid that gives the possible values of the array for a particular look direction. We choose weights that minimize the total weighted power output of the array, subject to the constraint that the gain should exceed unity for all array responses in this ellipsoid. The robust weight selection process can be cast as a second-order cone program that can be solved efficiently using Lagrange multiplier techniques. If the ellipsoid reduces to a single point, the method coincides with Capon's method. We describe in detail several methods that can be used to derive an appropriate uncertainty ellipsoid for the array response. We form separate uncertainty ellipsoids for each component in the signal path (e.g., antenna, electronics) and then determine an aggregate uncertainty ellipsoid from these. We give new results for modeling the element-wise products of ellipsoids. We demonstrate the robust beamforming and the ellipsoidal modeling methods with several numerical examples.

Abstract. In this paper we give an overview, how to apply region based methods for the synthesis of Petri nets from languages to process mining.The research domain of process mining aims at constructing a process model from an event log, such that the process model can reproduce the log, and does not allow for much more behaviour than shown in the log. We here consider Petri nets to represent process models. Event logs can be interpreted as finite languages. Region based synthesis methods can be used to construct a Petri net from a language generating the minimal net behaviour including the given language. Therefore, it seems natural to apply such methods in the process mining domain. There are several different region based methods in literature yielding different Petri nets. We adapt these methods to the process mining domain and compare them concerning efficiency and usefulness of the resulting Petri net.

Abstract-This paper introduces an extension of minimum variance beamforming that explicitly takes into account variation or uncertainty in the array response. Sources of this uncertainty include imprecise knowledge of the angle of arrival and uncertainty in the array manifold.In our method, uncertainty in the array manifold is explicitly modeled via an ellipsoid that gives the possible values of the array for a particular look direction. We choose weights that minimize the total weighted power output of the array, subject to the constraint that the gain should exceed unity for all array responses in this ellipsoid. The robust weight selection process can be cast as a second-order cone program that can be solved efficiently using Lagrange multiplier techniques. If the ellipsoid reduces to a single point, the method coincides with Capon's method.We describe in detail several methods that can be used to derive an appropriate uncertainty ellipsoid for the array response. We form separate uncertainty ellipsoids for each component in the signal path (e.g., antenna, electronics) and then determine an aggregate uncertainty ellipsoid from these. We give new results for modeling the element-wise products of ellipsoids. We demonstrate the robust beamforming and the ellipsoidal modeling methods with several numerical examples.Index Terms-Ellipsoidal calculus, Hadamard product, robust beamforming, second-order cone programming.

Abstract-This paper introduces an extension of minimum variance beamforming that explicitly takes into account variation or uncertainty in the array response. Sources of this uncertainty include imprecise knowledge of the angle of arrival and uncertainty in the array manifold.In our method, uncertainty in the array manifold is explicitly modeled via an ellipsoid that gives the possible values of the array for a particular look direction. We choose weights that minimize the total weighted power output of the array, subject to the constraint that the gain should exceed unity for all array responses in this ellipsoid. The robust weight selection process can be cast as a second-order cone program that can be solved efficiently using Lagrange multiplier techniques. If the ellipsoid reduces to a single point, the method coincides with Capon's method.We describe in detail several methods that can be used to derive an appropriate uncertainty ellipsoid for the array response. We form separate uncertainty ellipsoids for each component in the signal path (e.g., antenna, electronics) and then determine an aggregate uncertainty ellipsoid from these. We give new results for modeling the element-wise products of ellipsoids. We demonstrate the robust beamforming and the ellipsoidal modeling methods with several numerical examples.Index Terms-Ellipsoidal calculus, Hadamard product, robust beamforming, second-order cone programming.

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