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

2. 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.

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

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 KL distance 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.