We currently witness that sensor array processing, more specifically direction-of-arrival (DoA) estimation, receives new momentum due to the emergence of new applications such as automotive radar, drone localization, and parametric channel estimation in Massive MIMO. This development is reinforced by the emergence of new powerful and affordable multiantenna hardware platforms. In this tutorial we provide a consistent overview of developments in four decades of sensor array processing techniques for DoA estimation, ranging from traditional super resolution techniques to modern sparse optimization based DoA estimation techniques. In our overview we take a modern optimization based view and retell the story of sensor array processing from the relaxation technique perspective. We will show, from that perspective, that constrained optimization problem formulations and problem relaxation techniques have always played an important role in the development of powerful DoA estimation methods. In this context we will introduce the partial relaxation technique that has been proposed as a new optimization based DoA estimation framework applying modern relaxation techniques to traditional multi-source estimation criteria to construct new estimators with excellent estimation performance at affordable computational complexity. In many senses it can be observed that the estimators designed under the partial relaxation framework admit new insights in existing methods of this well established field of research. 

automotive radar;drone localization;Array Signal Processing