The parallel channel with ordered gains: A high snr analysis

A parallel channel with random but ordered gains is considered. A sufficient condition on the code-book (with unit block length codewords) for this channel is derived, so that the fundamental D-M tradeoff of this channel is achieved, irrespective of the distribution of the gains. This condition suggests to construct codes so that the magnitudes of elements of each codeword difference vector are ordered, and the minimum Euclidean distance is maximized. Moreover, an average power constraint on the codewords is sufficient, rather than a constraint on every codeword required for approximate universality [1] if the gains are not ordered. For a 2-parallel channel (i.e. having 2 sub-channels) with ordered gains, a simple explicit code, referred to as the Zig Zag code, is proposed, which achieves the highest diversity order at a fixed rate. For a multiple-input multiple-output (MIMO) block fading channel, a novel low-complexity D-M tradeoff optimal space-time (ST) architecture, referred to as the ordered D-BLAST-ZF, is proposed, which effectively decomposes the MIMO channel into a parallel channel with ordered gains. This architecture uses a few bits of feedback, containing information of the antenna strength order, as determined by a careful choice of mapping between channel matrices and permutation matrices. Numerical simulations show that, over a parallel channel obtained via the ordered D-BLAST-ZF in a 2 times 2 MIMO channel, a Zig Zag code gives comparable gain over the permutation code of [1] as the latter gives over ^{QAM code} used over only the stronger sub-channel.