Direction of Arrival (DoA) estimation is extensively studied in the array signal processing with many applications areas including radar, sonar, medical diagnosis and radio astronomy. Since, in sparse target environments, Compressive Sensing (CS) provides comparable performance with the classical DoA estimation techniques by using fewer number of sensor outputs, there are a multitude of proposed techniques in the literature that focus on surveillance (detection) and tracking (estimation) of DoA in CS framework. Many of such works elaborate on recovery of compressed signal and employ random measurement matrices, such as Bernoulli or Gaussian matrices. Although random matrices satisfy Restricted Isometry Property (RIP) for reconstruction, the measurement matrices can be designed to provide improved performance in search sectors that they are designed for. In this thesis, a novel technique to design compressive measurement matrices is proposed in order to achieve enhanced DoA surveillance and tracking performance using sensor arrays. Measurement matrices are designed in order to minimize the Cramer-Rao Lower Bound (CRLB), which provides a lower bound for DoA estimation error. It is analytically shown that the proposed design technique attains the CRLB under mild conditions. Built upon the characteristics of proposed measurement design approach, a sequential surveillance technique using interference cancellation is introduced. A novel partitioning technique, which provides a greedy type solution to a minmax optimization problem, is also developed to ensure robust surveillance performance. In addition, an adaptive target tracking algorithm, which adaptively updates measurement matrices based on the available information of targets, is proposed. Via a comprehensive set of simulations, it is demonstrated that the proposed measurement design technique facilities significantly enhanced surveillance and tracking performance over the widely used random matrices in the compressive sensing literature.