Constellations for Imperfect Channel State Information at the Receiver

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

2. wt1 h11 sT1 … s21 s11 xT1 … x21 x11 + + + h21 h12 sT2 … s22 s12 xT2 … x22 x12 h22 h2N . . . . . . h1N wtN hMN sTM … s2M s1M xTN … x2N x1N System Model • 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.

3. 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,

4. 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

5. The ML receiver • The conditional pdf of the received signal • The ML receiver

6. 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.

7. 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

8. 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  .

9. The KL Distance • Using the conditional distributions • The expected KL distance

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

11. Example, M = 1, T = 1 • KLdistance • Therefore, we use multilevel circular constellations

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

13. Performance Comparison (1)

14. Performance Comparison (2)

15. Performance Comparison (3)

16. Performance Comparison (4)

17. Performance Comparison (5)

18. 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.