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Constellations for Imperfect Channel State Information at the Receiver

Constellations for Imperfect Channel State Information at the Receiver. Mohammad Jaber Borran, Ashutosh Sabharwal, and Behnaam Aazhang Rice University October 2002. w t1. h 11. s T1 … s 21 s 11. x T1 … x 21 x 11. +. +. +. h 21. h 12. s T2 … s 22 s 12. x T2 … x 22 x 12. h 22.

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Constellations for Imperfect Channel State Information at the Receiver

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  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.

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