Partially Coherent Multiple-Antenna Constellations Mohammad Jaber Borran Nokia Research Center Ashutosh Sabharwal Rice University Behnaam Aazhang Rice University Prabodh Varshney Nokia Research Center November 11, 2003

System Model . sT1 … s21 s11 + xT1 … x21 x11 sT2 … s22 s12 wt1 wtN xT1 … x21 x11 h11 h12 h1N h21 h22 h2N hMN sT2 … s22 s12 sTM … s2M s1M xT2 … x22 x12 xTN … x2N x1N Entries of H and W are independent complex Gaussian rv’s with distribution CN(0, 1). At the receiver, is known (estimate of H), but is not. Furthermore, is independent from and .

Conditional Received Distribution The conditional pdf of the received signal The ML receiver

Design Criterion The exact expression and the Chernoff upper bound for the pairwise error probability are, in general, intractable. Inspired by the Stein’s Lemma, we propose to use the Kullback-Leibler (KL) distance between conditional distributions as the performance criterion. KL distance is an upper bound for the error exponent of the ML detector, and determines the rate of the exponential growth of the likelihood ratio.

Stein’s Lemma The best achievable error exponent for Pr(S2S1) with hypothesis testing and with the constraint that Pr(S1S2) < , is given by the KL (Kullback-Leibler) distance between p1(X) = p(X|S1) and p2(X) = p(X|S2): Only an upper bound for the error exponent of the ML detector

Lemma Let X1, X2, …, XN be i.i.d. ~ q. Consider two hypothesis tests between q = p0 and q = p1 between q = p0 and q = p2 where 0<D(p0||p2) < D(p0||p1) < . Let Then where

The KL Distance Using the expression for conditional received distribution The expected KL distance

Two Extreme Cases If (coherent) (results in the rank and determinant criteria) If (non-coherent) (which is the KL-based design criterion for non-coherent systems)

Multiple-Antenna Constellations For a fixed transmission rate, the constellation size grows exponentially with T (L = 2RT). We design constellations of 1M vectors. Use multilevel spherical constellations

Decoupled Optimization Define Solve

Recursive Spherical Constellations Sn(L): L-point n-dimensional real spherical constellations. Start from n = 2: Construct Sn(L) by using several n-dimensional constellations as its latitudes, and using S3(32) S2(14) S2(9)

Decoupled Optimization Assume P levels Define Solve

Performance Comparison (1)

Performance Comparison (2)

Performance Comparison (3)

Performance Comparison (4)

Performance Comparison (5)

Performance Comparison (6)

Performance Comparison (7)

Performance Comparison (8)