Multiple-input multiple-output (MIMO) radars have many advantages over their phased-array counterparts: improved spatial resolution; better parametric identifiably and greater flexibility to design the transmit beampattern. The design of the transmit beampatterns generally requires the waveforms to have arbitrary auto- and cross-correlation properties. The correlation/covariance matrix, R, of the waveforms must be positive semidefinite, therefore synthesis of a desired beampattern is usually a constrained optimization problem. In this paper, to simplify the constrained optimization problem, two algorithms are proposed to synthesize the waveform covariance matrix for the desired beampattern. In the first proposed algorithm the elements of a square-root matrix of the covariance matrix R are parameterized using the coordinates of a hypersphere that implicitly fulfil the constraints. This yields an iterative algorithm, whose convergence speed can be increased significantly by providing good initial values. In the second algorithm the constraints and redundant information in the covariance matrix R are exploited to find a closed-form solution. The drawback of the second algorithm is that it may yield a pseudocovariance matrix (pseudo-CM) that is not guaranteed to be positive semidefinite. However, the pseudo-CM can be easily converted into a covariance matrix using eigenvalue decomposition and/or shrinkage methods. Moreover, a pseudo-CM can also be used to provide good initial values for the first algorithm to enable faster convergence.
Petillot, Y., Mulgrew, B., Ahmed, S., & Thompson, J. (2011). Unconstrained synthesis of covariance matrix for MIMO radar transmit beampattern. IEEE Transactions on Signal Processing, 59(8), 3837- 3849. https://doi.org/10.1109/TSP.2011.2153200