Precoding for Outage Probability Minimization on Block Fading Channels

Abstract

The outage probability limit is a fundamental and achievable lower bound on the word error rate of coded communication systems affected by fading. This limit is mainly determined by two parameters: the diversity order and the coding gain. With linear precoding, full diversity on a block fading channel can be achieved without error-correcting code. However, the effect of precoding on the coding gain is not well known, mainly due to the complicated expression of the outage probability. Using a geometric approach, this paper establishes simple upper bounds on the outage probability, the minimization of which yields to precoding matrices that achieve very good performance. For discrete alphabets, it is shown that the combination of constellation expansion and precoding is sufficient to closely approach the minimum possible outage achieved by an i.i.d. Gaussian input distribution, thus essentially maximizing the coding gain.Index Terms-Block fading channels, multidimensional mapping, precoding, outage probability.

Precoding for coded communication on block fading channels and cooperative communications

“…In [6], a boundary with a simple shape outer bounding B o (γ, P, Ω z , R) was determined, which is then much easier to optimize. A surface in the fading space, U (α) = 0, outer bounds B o (γ, P, Ω z , R) if and only if…”

“…By showing that for all α satisfying U (α) = 0, the constellation Ω t is distorted in such a way that I(T; Y|α, γ) ≥ BR, it is proved that U (α) = 0 outer bounds B o (γ, P, Ω z , R). For example, for general B and for high instantaneous SNR, it is proved in [6] that a B-hypersphere touching the outage boundary on the axes of the fading space, hence with radius α o , outer bounds B o (γ, P, Ω z , R). A B-hypersphere U (α) = 0 is a generalization of a sphere to B dimensions,…”

“…The optimization of the upper bound is covered in [6] and not repeated here. Summarizing, it consist of choosing P so that…”

“…[12] for MIMO, is not suitable for coded schemes because it optimizes the bit error rate of uncoded constellations transmitted on ergodic fading channels. Fortunately, a useful upper bound on the outage probability was established [6], [5], which can be optimized with a small computational effort (a few seconds with the current computer resources). The proof for this upper bound was given in [6] for high instantaneous SNR only.…”

“…The results can be applied to cooperative communications, which is shown in the last section. The work presented here is partly covered in [7], but new material such as the necessity of a new proof for low SNR and the application to cooperative communications is added.…”