Design Criteria and Construction of Non-coherent Space-Time Constellations Mohammad Jaber Borran, Ashutosh Sabharwal, and Behnaam Aazhang Rice University

Noncoherent System Model Entries of H and W are independent complex Gaussian rv’s from distribution CN (0, 1) s T1 … s 21 s w t1 w tN x T1 … x 21 x 11 h 11 h 12 h 1N h 21 h 22 h 2N h MN s T2 … s 22 s 12 s TM … s 2M s 1M x T2 … x 22 x 12 x TN … x 2N x 1N

Known Results Capacity achieving distribution: S =  V –  and V are independent –  : T  M isotropically distributed unitary matrix –V: M  M real, nonnegative, diagonal matrix Unitary space-time constellations –For T >> M, or at very high SNR, one can use a constellation of unitary matrices and still achieve capacity. Design criterion –Maximize the minimum of the chordal distance

Problems For small T, unitary constellations are not optimal. Performance degradation at high data rates. Optimal unitary constellations: –High detection complexity –Structured schemes Systematic unitary constellations [Hochwald, et. al.] Real PSK-type unitary constellations [Tarokh]

Design Criterion Average or pairwise error probability –Chernoff upper bound is intractable in the general non-unitary case. Stein’s lemma –The best achievable exponent for Pr(S 2  S 1 ), with the constraint that Pr(S 1  S 2 ) < , is given by the Kullback-Leibler distance D(p(X|S 1 )||p(X|S 2 )). The performance of the ML detector is also related to the above KL distance.

Design Criterion (cnt’d) The KL distance is given by –Linear in N (number of receive antennas) Design criterion:

Special Case 1: M = T = 1 KL distance For the L-point optimal signal set,, where  is the largest real root of

Special Case 1 (cnt’d)

Special Case 2: M = 1, T > 1, |S| 2 =P KL distance For T = 2, optimal real signal set (identical to “PSK constellation” [Tarokh])

General case for M = 1 KL distance Geometric interpretation

General case for M = 1 (cnt’d) Proposed constellation structure (multilevel unitary) Design criterion T-dimensional spheres

Examples (real, M=1, T=2) P av = 1P av = 10 Each coordinate corresponds to one symbol interval.

Performance Comparison

Summary General case M>1, with unitary constellations, if S ik · S jm = 0 for k  m, we will have –This means that we can use the previous design for the columns of the code matrices. Efficient multi-level unitary constellations are introduced By using structured unitary constellations at each level, the detection complexity is reduced (linear in the number of levels).