  This paper studies a multiple-input multiple-output (MIMO) integrated sensing
and communication (ISAC) system, in which a multi-antenna base station (BS)
sends unified wireless signals to estimate an extended target and communicate
with a multi-antenna communication user (CU) at the same time. We investigate
the fundamental tradeoff between the estimation Cram\'er-Rao bound (CRB) for
sensing and the data rate for communication, by characterizing the Pareto
boundary of the achievable CRB-rate (C-R) region. Towards this end, we
formulate a new MIMO rate maximization problem by optimizing the transmit
covariance matrix at the BS, subject to a new form of maximum CRB constraint
together with a maximum transmit power constraint. We derive the optimal
transmit covariance solution in a semi-closed form, by first implementing the
singular-value decomposition (SVD) to diagonalize the communication channel and
then properly allocating the transmit power over these subchannels for
communication and other orthogonal subchannels (if any) for dedicated sensing.
It is shown that the optimal transmit covariance is of full rank, which unifies
the conventional rate maximization design with water-filling power allocation
and the CRB minimization design with isotropic transmission. Numerical results
are provided to validate the performance achieved by our proposed optimal
design, in comparison with other benchmark schemes.
