  In this paper, we investigate a multiple-input multiple-output (MIMO)
integrated sensing and communication (ISAC) system under typical block-fading
channels. As a non-trivial extension to most existing works on ISAC, both the
training and transmission signals sent by the ISAC transmitter are exploited
for sensing. Specifically, we develop two training and transmission design
schemes to minimize a weighted sum of the mean-squared errors (MSEs) of data
transmission and radar target response matrix (TRM) estimation. For the former,
we first optimize the training signal for simultaneous communication channel
and radar TRM estimation. Then, based on the estimated instantaneous channel
state information (CSI), we propose an efficient majorization-minimization
(MM)-based robust ISAC transmission design, where a semi-closed form solution
is obtained in each iteration. For the second scheme, the ISAC transmitter is
assumed to have statistical CSI only for reducing the feedback overhead. With
CSI statistics available, we integrate the training and transmission design
into one single problem and propose an MM-based alternating algorithm to find a
high-quality solution. In addition, we provide alternative structured and
low-complexity solutions for both schemes under certain special cases. Finally,
simulation results demonstrate that the radar performance is significantly
improved compared to the existing scheme that integrates sensing into the
transmission stage only. Moreover, it is verified that the investigated two
schemes have advantages in terms of communication and sensing performances,
respectively.
