Asymptotic-Information-Lossless Designs and the Diversity–Multiplexing Tradeoff

It is known that neither the Alamouti nor the V-BLAST scheme achieves the Zheng–Tse diversity–multiplexing tradeoff (DMT) of the multiple-input multiple-output (MIMO) channel. With respect to the DMT curve, the Alamouti scheme achieves the point corresponding to maximum diversity gain only, whereas V-BLAST meets only the point corresponding to maximum multiplexing gain. It is also known that D-BLAST achieves the optimal DMT for $n$ transmit and $n$ receive antennas, but only under the assumption that the leading and trailing zeros are ignored. When these zeros are taken into account, D-BLAST achieves the point corresponding to zero multiplexing gain, but not the point corresponding to zero diversity gain. The first scheme to achieve the DMT is the coding scheme of Yao and Wornell for the case of two transmit and two receive antennas. In this paper, we introduce the notion of an asymptotic-information-lossless (AILL) design and obtain a necessary and sufficient condition under which a design is AILL. Analogous to the result that full-rank designs achieve the point corresponding to the zero multiplexing gain of the optimal DMT curve, we show AILL to be a necessary and sufficient condition for a design to achieve the point on the DMT curve corresponding to zero diversity gain. We also derive a lower bound on the tradeoff achieved by designs from field extensions and show that the tradeoff is very close to the optimal tradeoff in the case of a single receive antenna. A lower bound to the tradeoff achieved by designs from division algebras is presented which indicates that these designs achieve both extreme points (corresponding to zero diversity and zero multiplexing gain) of the optimal DMT curve. Finally, we present simulations results for $n$ transmit and $n$ receive antennas, for $n=2,3,4$, which suggest that designs from division algebras are likely to have the property of being DMT achieving.