This paper presents a comprehensive study on probing signal design for MIMO radar systems. The key contributions include the design of the covariance matrix of the probing signal vector to achieve desired transmit beampatterns and minimize cross-correlation between signals reflected from targets. The authors demonstrate that proper probing signal design can significantly improve the performance of adaptive MIMO radar techniques. They compare several MIMO transmit beampattern designs, including beampattern matching and minimum sidelobe designs, with their phased-array counterparts, showing that MIMO designs offer advantages in terms of performance and flexibility. The paper begins by introducing the MIMO radar system and its capabilities compared to phased-array radar. It then discusses the problem formulation, focusing on the design of the covariance matrix R to achieve specific beampattern goals. The authors propose a maximum power design for unknown target locations and a maximum power design for known target locations, both under uniform elemental power constraints. They also present a beampattern matching design that allows for the approximation of a desired beampattern while minimizing cross-correlation between signals. The paper further introduces a minimum sidelobe beampattern design, which aims to reduce the sidelobe levels in a specified region. The authors also discuss the conventional phased-array beampattern design problem and show that MIMO beampattern design problems are the semi-definite relaxations (SDR) of their phased-array counterparts. Numerical examples are provided to demonstrate the effectiveness of the proposed designs. These examples show that the MIMO radar with proper beampattern design can achieve better performance in terms of target localization and detection, especially in the presence of jammers. The results highlight the advantages of MIMO radar in terms of spatial resolution, robustness, and adaptability compared to phased-array radar systems. The paper concludes with a discussion on the practical implications of the proposed designs and their potential for future research.This paper presents a comprehensive study on probing signal design for MIMO radar systems. The key contributions include the design of the covariance matrix of the probing signal vector to achieve desired transmit beampatterns and minimize cross-correlation between signals reflected from targets. The authors demonstrate that proper probing signal design can significantly improve the performance of adaptive MIMO radar techniques. They compare several MIMO transmit beampattern designs, including beampattern matching and minimum sidelobe designs, with their phased-array counterparts, showing that MIMO designs offer advantages in terms of performance and flexibility. The paper begins by introducing the MIMO radar system and its capabilities compared to phased-array radar. It then discusses the problem formulation, focusing on the design of the covariance matrix R to achieve specific beampattern goals. The authors propose a maximum power design for unknown target locations and a maximum power design for known target locations, both under uniform elemental power constraints. They also present a beampattern matching design that allows for the approximation of a desired beampattern while minimizing cross-correlation between signals. The paper further introduces a minimum sidelobe beampattern design, which aims to reduce the sidelobe levels in a specified region. The authors also discuss the conventional phased-array beampattern design problem and show that MIMO beampattern design problems are the semi-definite relaxations (SDR) of their phased-array counterparts. Numerical examples are provided to demonstrate the effectiveness of the proposed designs. These examples show that the MIMO radar with proper beampattern design can achieve better performance in terms of target localization and detection, especially in the presence of jammers. The results highlight the advantages of MIMO radar in terms of spatial resolution, robustness, and adaptability compared to phased-array radar systems. The paper concludes with a discussion on the practical implications of the proposed designs and their potential for future research.