This paper investigates the minimum energy required to transmit k
information bits with a given reliability over a multiple-antenna Rayleigh
block-fading channel, with and without channel state information (CSI) at the
receiver. No feedback is assumed. It is well known that the ratio between the
minimum energy per bit and the noise level converges to β1.59 dB as k goes
to infinity, regardless of whether CSI is available at the receiver or not.
This paper shows that lack of CSI at the receiver causes a slowdown in the
speed of convergence to β1.59 dB as kββ compared to the case of
perfect receiver CSI. Specifically, we show that, in the no-CSI case, the gap
to β1.59 dB is proportional to ((logk)/k)1/3, whereas when perfect
CSI is available at the receiver, this gap is proportional to 1/kβ. In
both cases, the gap to β1.59 dB is independent of the number of transmit
antennas and of the channel's coherence time. Numerically, we observe that,
when the receiver is equipped with a single antenna, to achieve an energy per
bit of β1.5 dB in the no-CSI case, one needs to transmit at least 7Γ107 information bits, whereas 6Γ104 bits suffice for the case of
perfect CSI at the receiver