Using Noncoherent Modulation for Training

Transcription

1 EE8510 Project Using Noncoherent Modulation for Training Yingqun Yu May 5,

2 Noncoherent Channel Model X = ρt M ΦH + W Rayleigh flat block-fading, T: channel coherence interval Marzetta & Hochwald [IT, 99] Φ C T M : Transmitted signal matrix X C T N : Received signal matrix H C M N : Unknown channel matrix, i.i.d. CN(0, 1) W C T N : Additive noise matrix, i.i.d. CN(0, 1) Power Constraint: E{Tr (Φ Φ)} = M ρ: Average SNR per receive antenna 1

3 Known Results on Coherent Capacity Coherent Capacity (H known to Rx but not Tx) Foschini [Bell Labs. Tech. J, 96], Telatar [ETT, 99] Asymptotically (M = N): C coherent (ρ) = E{log 2 det(i M + ρ M HH )} = E{log 2 det(i N + ρ M H H)} lim lim M ρ [ Ccoherent (ρ) M log 2 ( ρ e ) ] = 0 2

4 Known Results on Noncoherent Capacity Noncoherent Capacity (H unknown to either Tx or Rx) Zheng & Tse [IT, 02] high SNR ρ 1, T 2M = 2N C M,M (ρ) = (1 M T )C coherent(ρ) + c(t, M) + o(1) Asymptotically (fix the ratio α = M/T): where C M,M (ρ) M (1 α) log 2 [( ρ ] e ) 2 k(α) (1 α) ln 2 k(α) = (1 α)2 2α ln(1 α) + α 2 lnα + 1 α 2 < 0 3

5 Unitary Space Time Modulation (USTM) Constellations of T M space-time signals Hochwald & Marzetta [IT, 00] {Φ l, l = 1,...,L} : Φ l Φ l = I M Capacity achieving when T M or ρ 1 with M min{n, T/2} Designed by numerical optimizations, no particular algebraic structure Exponential demodulation complexity Constellation size L: 2 RT for a given rate of R bits per symbol 4

6 Training-Based Schemes data Encoder modulator known pilot symbols Φ d Φ τ MUX De MUX X τ Φ τ X d channel estimator decoder Ĥ data Multiplexing known pilot symbols with data symbols A tight capacity lower bound (Φ τφ τ = I M ) Zheng & Tse [IT, 02], Hassibi & Hochwald [IT, 03] C L known (ρ) = (1 M T )C coherent(ρ eff ) Suffers SNR loss (due to estimation error) at high SNR: ρ eff < ρ Optimal for noncoherent channel when T is large Q: How large T is enough? 5

7 Asymptotic SNR Loss of Training with Known Symbols T 2M = 2N, ρ, but α = M/T fixed ρ loss (α) = [ ] α(1 α) 2 k(α) (1 α) ln 2 2 Asymptotical SNR loss (db) α=m/t ρ loss (0.5) = 2.17dB,ρ loss (10 1 ) = 1.598dB, ρ loss (10 2 ) = 0.698dB, ρ loss (10 3 ) = 0.252dB 6

8 Using Noncoherent Modulation for Training Modulators Decoders data Encoder Encoder coherent nonoherent Φ d Φ τ MUX De MUX X d X τ nonoherent data coherent Ĥ data T Reconstructer Φ τ channel estimator T τ T d noncoherent block coherent block Pilot symbols are unknown to the receiver, and can carry data information T τ is only a fraction of T, leading to less complexity The tradeoff between complexity and SNR loss can be obtained by selecting a suitable T τ 7

9 Using USTM as Training Symbols Choose Φ τ as USTM: Φ τφ τ = I M C L unknown(ρ) = I USTM(ρ) + (1 )C coherent (ρ eff ), = T τ /T: Time-Sharing factor! Asymptotically, T T τ 2M = 2N, ρ, but α = M/T, = T τ /T fixed Cunknown L (ρ) [ (1 α) log M 2 ( ρ e ) (1 + α ) ( 1+ 1 α ) ] k( α ) 2(1 α) ln 2 noncoherent capacity: C M,M (ρ) M (1 α) log 2 [( ρ ] e ) 2 k(α) (1 α) ln 2 8

10 Asymptotic SNR Loss of Training with USTM Symbols ρ loss (α, ) = (1 + α ) 1 α k(α) k( α ) 1 α1 1 α 2 (1 α) ln ρ loss (α) ρ loss (α, ) (db) =0.05 =0.1 =0.2 =0.3 =0.4 = =0.6 = =0.8 = α For most interested (α, ) combinations (α > 0.05,α > 0.1), ρ loss(α, ) < ρ loss (α) For sufficiently small α and, ρ loss(α, ) > ρ loss (α) benefit of power control > advantage of noncoherent training 9

11 Numerical Results 9 M=N=1, T=10, T τ =4 14 M=N=2, T=10, T τ = bits/symbol Trainng with USTM USTM Training with Known Symbols bits/symbol Trainng with USTM USTM Training with Known Symbols ρ (db) ρ (db) M = N = 1, T = 10, T τ = 4 simulation results: 1.5dB, 0.4dB, asymptotic results: ρ loss (0.1) = 1.598dB, ρ loss (0.1, 0.4) = 0.438dB M = N = 2, T = 10, T τ = 5 simulation results: 1.8dB, 0.55dB, asymptotic results: ρ loss (0.2) = 1.912dB, ρ loss (0.2, 0.5) = 0.580dB 10

12 Conclusion Training with known pilot symbols converges to the optimal very slowly Training with unknown USTM symbols provides a tradeoff between complexity and performance Thank You! 11