Constellations for Imperfect Channel State Information at the Receiver Mohammad Jaber Borran, Ashutosh Sabharwal, and Behnaam Aazhang Rice University October 2002

System Model . sT1 … s21 s11 + 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 known.

Channel Measurement at the Receiver Use a preamble based channel estimation The minimum number of required measurements is MN The minimum number of required preamble symbols is M If the preamble matrix to is where and Pp is the preamble power,

MMSE Channel Estimator The MMSE estimate for the entries of the H matrix is given by Linear MMSE estimate is orthogonal to the estimation error, hence . Since and are both zero-mean Gaussian, they are independent, and Define

The ML receiver The conditional pdf of the received signal

Design Criteria We use the maximum pairwise error probability as the performance 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 KL distance between conditional distributions as the performance criterion.

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 D(p0||p2) < D(p0||p1) < . Let Then Pr{L1N < L2N}  0 as N  , and Pr{L1N < 0 | L2N  0}  0 as N  .

The KL Distance Using the conditional distributions The expected KL distance

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

Example, M = 1, T = 1 KL distance Therefore, we use multilevel circular constellations

Examples (M=1, T=1) 16-point 8-point Pav= 10

Performance Comparison (1)

Performance Comparison (2)

Performance Comparison (3)

Performance Comparison (4)

Performance Comparison (5)

Conclusions We derived a design criterion for the partially coherent constellations based on the KL distance between distributions. The new design criterion reduces to the coherent and non-coherent design criteria for the two extreme values of the estimation error variance. The designed constellations show performance improvement compared to the commonly-used constellations of the same size. The performance improvement becomes more significant as the number of receive antennas increases.