  This paper studies a near-field integrated sensing and communication (ISAC)
system with extremely large-scale antenna array (ELAA), in which a base station
(BS) deployed with enormous number of antennas transmits wireless signals to
communicate with multiple communication users (CUs) and simultaneously uses the
echo signals to localize multiple point targets in the three-dimension (3D)
space. To balance the performance tradeoff between communication and target
localization, we design the transmit covariance matrix at the BS to optimize
the localization performance while ensuring the
signal-to-interference-plus-noise ratio (SINR) constraints at individual CUs.
In particular, we formulate three design problems by considering different 3D
localization performance metrics, including minimizing the sum Cram\'er-Rao
bound (CRB), maximizing the minimum target illumination power, and maximizing
the minimum target echo signal power. Although the three design problems are
non-convex in general, we obtain their global optimal solutions via the
technique of semi-definite relaxation (SDR). It is shown that the three
problems have low-rank solution structures depending on the sensing and
communication channel matrices, helping reduce the complexity of the SDR-based
solutions. Interestingly, we find that in the special case with a single
collocated target/CU present towards the middle of a symmetric uniform planar
array (UPA), the optimal solutions to the three problems become identical to
the SINR-maximization design and have a closed form, while in other cases they
can be different in general. Besides, when the target/CU moves away from the
transmitter/receiver, the CRB may first decrease and then increase. These two
phenomena differ from those in the far-field. Numerical results show the
benefits of the proposed designs for near-field ISAC, by exploiting the beam
focusing capabilities of ELAA.
