[Paper Review] Precoding for Outage Probability Minimization on Block Fading Channels

Dieter Duyck, Joseph J. Boutros+1 more|arXiv (Cornell University)|Mar 28, 2011

Advanced Wireless Communication Techniques24 references4 citations

TL;DR

This paper proposes a geometric approach to minimize outage probability in block fading channels using linear precoding, achieving full diversity and maximizing coding gain. By deriving tight upper bounds on outage probability, the method enables efficient optimization of precoding matrices without Monte Carlo simulations, showing that constellation expansion combined with precoding closely approaches the performance of i.i.d. Gaussian inputs.

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.

Motivation & Objective

• To understand the impact of linear precoding on coding gain in coded systems over block fading channels, where diversity order is well understood but coding gain is not.
• To establish a tractable optimization framework for precoding matrices that minimizes outage probability, a fundamental performance limit in fading channels.
• To show that combining constellation expansion with linear precoding can nearly achieve the outage performance of i.i.d. Gaussian inputs, thus maximizing coding gain.
• To provide a computationally efficient alternative to brute-force Monte Carlo optimization of precoding matrices by using geometric upper bounds on outage probability.

Proposed method

• A geometric approach is used to derive upper bounds on the outage probability, replacing the intractable expectation over fading distributions with a deterministic optimization problem.
• The method leverages the fact that the minimum pairwise distance in the transformed signal space determines the outage performance, and bounds the worst-case distance under fading.
• It proves that for any fading gain vector with fixed energy, the minimum transformed distance is bounded above by the minimum distance of the original constellation scaled by the energy per block.
• The outage probability upper bound is derived using the chi-square cumulative distribution function of the sum of squared fading gains, leading to a γ−B decay rate.
• The precoding matrix is optimized by minimizing this upper bound, which is computationally efficient and avoids repeated Monte Carlo simulations.
• The approach is validated for both discrete constellations and the theoretical limit of i.i.d. Gaussian inputs, showing near-optimality in coding gain.

Experimental results

Research questions

• RQ1 Can linear precoding be designed to minimize outage probability in block fading channels without relying on brute-force Monte Carlo optimization?
• RQ2 What is the effect of precoding on the coding gain of coded systems, especially when full diversity is already achieved?
• RQ3 Can discrete constellations with precoding approach the outage performance of i.i.d. Gaussian inputs, which are known to minimize outage probability?
• RQ4 Is there a tractable upper bound on outage probability that enables efficient precoding matrix optimization?

Key findings

• The proposed geometric upper bound on outage probability enables efficient optimization of precoding matrices without Monte Carlo simulations, significantly reducing computational complexity.
• For discrete constellations, the combination of constellation expansion and linear precoding achieves outage performance that is very close to the theoretical minimum achieved by i.i.d. Gaussian inputs.
• The method ensures full diversity order B on block fading channels, with outage probability decaying at least as fast as γ−B, matching the theoretical diversity limit.
• The upper bound on outage probability is derived using the chi-square CDF of the sum of squared fading gains, and it is shown to be proportional to γ−B, confirming the diversity order.
• The optimization of the precoding matrix via the upper bound yields performance close to the optimal, as verified through theoretical analysis and geometric arguments on signal point distances.