Minimum Energy to Send $k$ Bits Over Multiple-Antenna Fading Channels

Abstract: 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 $-1.59$ 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 $-1.59$ dB as $k\to\infty$ compared to the case of perfect receiver CSI. Specifically, we show that, in the no-CSI case, the gap to $-1.59$ dB is proportional to $((\log k) /k)^{1/3}$, whereas when perfect CSI is available at the receiver, this gap is proportional to $1/\sqrt{k}$. In both cases, the gap to $-1.59$ 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 $ - 1.5$ dB in the no-CSI case, one needs to transmit at least $7\times 10^7$ information bits, whereas $6\times 10^4$ bits suffice for the case of perfect CSI at the receiver.