  This paper proposes an intelligent reflecting surface (IRS) assisted
integrated sensing and communication (ISAC) system operating at the
millimeter-wave (mmWave) band. Specifically, the ISAC system combines
communication and radar operations and performs, detecting and communicating
simultaneously with multiple targets and users. The IRS dynamically controls
the amplitude or phase of the radio signal via reflecting elements to
reconfigure the radio propagation environment and enhance the transmission rate
of the ISAC system. By jointly designing the radar signal covariance (RSC)
matrix, the beamforming vector of the communication system, and the IRS phase
shift, the ISAC system transmission rate can be improved while matching the
desired waveform for radar. The problem is non-convex due to multivariate
coupling, and thus we decompose it into two separate subproblems. First, a
closed-form solution of the RSC matrix is derived from the desired radar
waveform. Next, the quadratic transformation (QT) technique is applied to the
subproblem, and then alternating optimization (AO) is employed to determine the
communication beamforming vector and the IRS phase shift. For computing the IRS
phase shift, we adopt both the majorization minimization (MM) and the manifold
optimization (MO). Also, we derive a closed-form solution for the formulated
problem, effectively decreasing computational complexity. Furthermore, a
trade-off factor is introduced to balance the performance of communication and
sensing. Finally, the simulations verify the effectiveness of the algorithm and
demonstrate that the IRS can improve the performance of the ISAC system.
