  This paper investigates the energy efficiency of a multiple-input
multiple-output (MIMO) integrated sensing and communications (ISAC) system, in
which one multi-antenna base station (BS) transmits unified ISAC signals to a
multi-antenna communication user (CU) and at the same time use the echo signals
to estimate an extended target. We focus on one particular ISAC transmission
block and take into account the practical on-off non-transmission power at the
BS. Under this setup, we minimize the energy consumption at the BS while
ensuring a minimum average data rate requirement for communication and a
maximum Cram\'er-Rao bound (CRB) requirement for target estimation, by jointly
optimizing the transmit covariance matrix and the ``on'' duration for active
transmission. We obtain the optimal solution to the rate-and-CRB-constrained
energy minimization problem in a semi-closed form. Interestingly, the obtained
optimal solution is shown to unify the spectrum-efficient and energy-efficient
communications and sensing designs. In particular, for the special MIMO sensing
case with rate constraint inactive, the optimal solution follows the isotropic
transmission with shortest ``on'' duration, in which the BS radiates the
required sensing energy by using sufficiently high power over the shortest
duration. For the general ISAC case, the optimal transmit covariance solution
is of full rank and follows the eigenmode transmission based on the
communication channel, while the optimal ``on'' duration is determined based on
both the rate and CRB constraints. Numerical results show that the proposed
ISAC design achieves significantly reduced energy consumption as compared to
the benchmark schemes based on isotropic transmission, always-on transmission,
and sensing or communications only designs, especially when the rate and CRB
constraints become stringent.
