Precoding for coded communication on block fading channels and cooperative communications

Abstract

Abstract-We study precoding for the outage probability minimization of block fading (BF) channels and BF relay channels. Recently, an upper bound on the outage probability with precoding was established for BF channels, but only for high instantaneous SNR. This upper bound is much easier to minimize than the actual outage probability, so that optimal precoding matrices can be determined without much computational effort. Here, we provide a proof for the upper bound on the outage probability at low instantaneous SNR. Next, the structure of the precoding matrix is simplified so that it can be easily constructed for an arbitrary number of blocks in the BF channel. Finally, we apply this technique to cooperative communications.

Precoding for Outage Probability Minimization on Block Fading Channels

“…The mutual information is [14], [12] see (11) at the bottom of the page where . However, more insight can be gained when considering .…”

“…As a unitary transformation preserves distance, it follows from (11) with that the value is insensitive to orthogonal transformations. Hence, the selection of the precoding matrix affects only the radius of the outer region .…”

“…It is easy to show that the fading points at the intersection of the sphere and the axes are the only fading points achieving the minimum distance except when the outage boundary is a sphere itself. 11 Hence, according to the relation between minimal distance and mutual information given in (35), with on a -hypersphere is minimized on the coordinate axes at high SNR. Indeed, for any other fading point on the sphere, the increase in minimum distance is greater than zero, so that its mutual information is larger at large SNR.…”

“…, we obtain , which is minimized to when 11 Indeed, (36) represents an ellipsoid which can only intersect the hypersphere on the coordinate axes, except when the ellipsoid is a hypersphere itself. For constellations where all the pairs and at minimum distance satisfy that is constant for all , it is easy to show that the outage boundary is a hypersphere itself.…”