Cognitive radio networks (CRNs) and millimeter wave (mmWave) communications are two major technologies to enhance the spectrum efficiency (SE). Considering that the SE improvement in the CRNs is limited due to the interference temperature imposed on the primary user (PU), and the severe path loss and high directivity in mmWave communications make it vulnerable to blockage events, we introduce an intelligent reflecting surface (IRS) into mmWave CRNs. This paper investigates the robust secure beamforming (BF) design in the IRS-assisted mmWave CRNs. By using a uniform linear array (ULA) at the cognitive base station (CBS) and a uniform planar array (UPA) at the IRS, and supposing that the imperfect channel state information (CSI) of wiretap links is known, we formulate a constrained problem to maximize the worst-case achievable secrecy rate (ASR) of the secondary user (SU) by jointly designing the transmit BF at the CBS and reflect BF at the IRS. To solve the non-convex problem with coupled variables, an efficient alternating optimization algorithm is proposed. As for the transmit BF at the CBS, we propose a heuristic robust transmit BF algorithm to attain the BF vectors analytically. As for the reflect BF at the IRS, by means of an auxiliary variable, we transform the non-convex problem into a semi-definite programming (SDP) problem with rank-1 constraint, which is handled with the help of an iterative penalty function, and then obtain the optimal reflect BF through CVX. Finally, the simulation results indicate that the ASR performance of our proposed algorithm has a small gap with that of the optimal solution with perfect CSI compared with the other benchmarks.
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