Integrated sensing and communication (ISAC) is widely recognized as a fundamental enabler for future wireless communications. In this paper, we present a joint communication and radar beamforming framework for maximizing a sum spectral efficiency (SE) while guaranteeing desired radar performance with imperfect channel state information (CSI) in multi-user and multi-target ISAC systems. To this end, we adopt either a radar transmit beam mean square error (MSE) or receive signal-to-clutter-plus-noise ratio (SCNR) as a radar performance constraint of a sum SE maximization problem. To resolve inherent challenges such as non-convexity and imperfect CSI, we reformulate the problems and identify first-order optimality conditions for the joint radar and communication beamformer. Turning the condition to a nonlinear eigenvalue problem with eigenvector dependency (NEPv), we develop an alternating method which finds the joint beamformer through power iteration and a Lagrangian multiplier through binary search. The proposed framework encompasses both the radar metrics and is robust to channel estimation error with low complexity. Simulations validate the proposed methods. In particular, we observe that the MSE and SCNR constraints exhibit complementary performance depending on the operating environment, which manifests the importance of the proposed comprehensive and robust optimization framework.

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