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In this paper, we study an intelligent reflecting surface (IRS)-aided radar-communication (Radcom) system, where the IRS is leveraged to help Radcom base station (BS) transmit the joint of communication signals and radar signals for serving communication users and tracking targets simultaneously. The objective of this paper is to minimize the total transmit power at the Radcom BS by jointly optimizing the active beamformers, including communication beamformers and radar beamformers, at the Radcom BS and the phase shifts at the IRS, subject to the minimum signal-to-interference-plus-noise ratio (SINR) required by communication users, the minimum SINR required by the radar, and the cross-correlation pattern design. In particular, we consider two cases, namely, case I and case II, based on the presence or absence of the radar cross-correlation design and the interference introduced by the IRS on the …