Precoded Modulations for Outage Probability Minimization on Block Fading channels Dieter Duyck Marc Moeneclaey Joseph J. Boutros Ghent University Ghent University Texas A&M University Ghent, Belgium Ghent, Belgium Doha, Qatar dduyck@telin.ugent.be mm@telin.ugent.be boutros@tamu.edu Abstract We study precoded modulations for the outage probability minimization of block fading channels. This paper establishes interesting upper bounds on the outage probability, whose simple expressions allow to determine which precoding matrix minimizes these upper bounds. Through this optimization, the gap be- tween the outage probability corresponding with a discrete input alphabet and the best outage achieved by an i.i.d. Gaussian alphabet is almost closed. 1 Introduction A block fading (BF) channel [1] is a useful model for all channels that consist of parallel sub-channels (e.g. via time-interleaving, frequency hopping, OFDMA or cooperative communications). The outage probability limit is a fundamental and achievable lower bound on the average word error rate of coded systems [1], [10]. By choosing a well designed precoding matrix P , the outage probability can be minimized [5]. Because no closed form expression for the outage probability is known, only a brute force optimization could be applied. A unitary B × B precoding matrix has B 2 degrees of freedom, so that the optimization of the outage probability is multivariate. In a brute force optimization, Monte Carlo simulations are required to take into account the distribution of all fading gains when evaluating the outage probability, which is often intractable. In order to simplify the optimization of the outage probability, we establish upper bounds on the outage probability that can be optimized without performing Monte Carlo simulations. In [5], such upper bounds on the outage probability were established for BF channels with i.i.d. Gaussian input alphabets and discrete input alphabets without precoding. Also in [5], an illustration, without proof, of upper bounds on the outage probability of BF channels with B = 2 and discrete input alphabets with precoding was given. Here, we formally proof the latter case for high signal-to-noise ratio (SNR). The proof is assisted by a new channel model giving more insight. 2 A new channel equation for BF channels The transmitter output is a real or complex vector x =[x(1),... x(B)] where x(b)= [x(b) 1 ,...,x(b) N B ] is the b-th part of the transmitted vector. The received vector and the noise vector are similarly represented. The channel is memoryless with additive white Gaussian noise and multiplicative real fading (Rayleigh distributed). The fading coefficients are only known at the decoder side where the received signal vector is y(b)= α b x(b)+ w(b), b =1,...,B, where the fading coefficient α b is independent and identically distributed (i.i.d.) from block to block. The noise vector w(b) consists of N/B independent noise samples which are complex Gaussian distributed, w(b) n ∼ CN (0, 1 γ ), where γ is the average signal-to-noise ratio.