1. 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

2. 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

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

4. 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.

5. 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

6. 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

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

8. 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)

9. 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

10. Decoupled Optimization
Define
Solve

11. 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)

12. Decoupled Optimization
Assume P levels
Define
Solve

13. Performance Comparison (1)

14. Performance Comparison (2)

15. Performance Comparison (3)

16. Performance Comparison (4)

17. Performance Comparison (5)

18. Performance Comparison (6)

19. Performance Comparison (7)

20. Performance Comparison (8)