  Integrated sensing and communication (ISAC) has recently been considered as a
promising approach to save spectrum resources and reduce hardware cost.
Meanwhile, as information security becomes increasingly more critical issue,
government agencies urgently need to legitimately monitor suspicious
communications via proactive eavesdropping. Thus, in this paper, we investigate
a wireless legitimate surveillance system with radar function. We seek to
jointly optimize the receive and transmit beamforming vectors to maximize the
eavesdropping success probability which is transformed into the difference of
signal-to-interference-plus-noise ratios (SINRs) subject to the performance
requirements of radar and surveillance. The formulated problem is challenging
to solve. By employing the Rayleigh quotient and fully exploiting the structure
of the problem, we apply the divide-and-conquer principle to divide the
formulated problem into two subproblems for two different cases. For the first
case, we aim at minimizing the total transmit power, and for the second case we
focus on maximizing the jamming power. For both subproblems, with the aid of
orthogonal decomposition, we obtain the optimal solution of the receive and
transmit beamforming vectors in closed-form. Performance analysis and
discussion of some insightful results are also carried out. Finally, extensive
simulation results demonstrate the effectiveness of our proposed algorithm in
terms of eavesdropping success probability.
