Physical-Layer MIMO Relaying

Transcription

1 Model Gaussian SISO MIMO Gauss.-BC General. Physical-Layer MIMO Relaying Anatoly Khina, Tel Aviv University Joint work with: Yuval Kochman, MIT Uri Erez, Tel Aviv University August 5, 2011

2 Model Gaussian SISO MIMO Gauss.-BC General. Two-Way Relay Model General Nodes 1 and 2 want to convey messages to each other. No direct link have to use relay. Transmission is divided into two transmission phases: MAC phase: nodes transmit to relay. Relay y p(y x 1,x 2 ) x 1 x 2 Node 1 Node 2 Broadcast (BC) phase: Relay transmits to nodes. Relay x p(y 1,y 2 x) y 1 y 2 Node 1 Node 2

3 Model Gaussian SISO MIMO Gauss.-BC General. Model: MAC Phase in the Gaussian Case Model CS UB, Balanced PNC Unbalanced PNC DF, AF, CF y = h 1 x 1 +h 2 x 2 +z x i input of average power P. h i - channel gain of user i. y - channel output. z - Channel noise CN(0,1). Closed loop (Full channel knowledge everywhere). Balanced case Equal channel gains: h 1 = h 2 h. Equal rates: R 1 = R 2 R.

4 Model Gaussian SISO MIMO Gauss.-BC General. Model: Broadcast Phase Model CS UB, Balanced PNC Unbalanced PNC DF, AF, CF Broadcast (BC) phase can be general. Broadcast channel has common-message capacity C common.

5 Model Gaussian SISO MIMO Gauss.-BC General. Gaussian SISO Case Model CS UB, Balanced PNC Unbalanced PNC DF, AF, CF Cut-Set Upper-Bound R i < R CS = min { log ( 1+ h i 2 P ),C i } C i individual capacity of user i in BC phase. Balanced case: R < R CS = min { log ( 1+ h 2 P ),C common } Physical-layer network coding [Narayanan et al. 07] Assumes balanced gains: h = h 1 = h 2, R = R 1 = R 2. Extends XORing of Network-Coding to Physical-layer. Based on nested lattices. R PNC = min { log ( h 2 P ),C common }.

6 Model Gaussian SISO MIMO Gauss.-BC General. Gaussian SISO Case Model CS UB, Balanced PNC Unbalanced PNC DF, AF, CF Unbalanced PNC [Nam, Chung, Lee 08] h 1 h 2, R 1 R 2. { ( R 1 min log { R 2 min log h 1 2 P h 1 2 P + h 2 2 P + h 1 2 P ( h 2 2 P h 1 2 P + h 2 2 P + h 2 2 P ),C 1 } ),C 2 } BC phase uses random binning.

7 Model Gaussian SISO MIMO Gauss.-BC General. Gaussian SISO Case Model CS UB, Balanced PNC Unbalanced PNC DF, AF, CF Cut-Set Upper-Bound R < R CS = min { log ( 1+ h 2 P ),C common } Decode & Forward Decode both messages at relay. R DF = min { 1 2 log( 1+2 h 2 P ),C common }. Amplify & Forward / Compress & Forward Forward / Compress noisy sum of terminal transmissions. R AF = log(1+α h 2 P), where α = α(p h 2,C common ) < 1.

8 MIMO

9 H 1 H 2 Relay Terminal 1 Terminal 2 y = H 1 x 1 +H 2 x 2 +z, i = 1,2 x i n 1 input vector of power P. H i - n n channel matrix of user i. y - n 1 output vector. z - Channel noise CN(0,I n ). Closed loop (Full channel knowledge everywhere).

10 Model: Assumptions Balanced case R = R 1 = R 2. Equal link capacities For high SNR: H 1 = H 2. High SNR assumption. Optimum inputs are white. log I + P n H ih i log P n H ih i = 2log H i +nlog P n. Capacity determined by H i for high SNR.

11 Main Idea Problem How to apply scalar PNC in MIMO case? Idea Each antenna sees interference from other streams. Use joint matrix decomposition to reduce to scalar channels. Use unitary transformations at transmitters preserves power. Apply scalar PNC over each of these channels. Decomposition Joint diagonalization of both channel matrices not possible. Joint triangularization?

12 Triangularization Joint triangularization H 1 = UT 1 V 1 H 2 = UT 2 V 2 where U,V 1,V 2 Unitary. T 1,T 2 Triangular matrices. Application for MAC phase Unitary V i is applied at encoder i preserves power. U is applied at relay. T 1, T 2 Effective channels.

13 Triangularization Interesting special cases Generalized SVD (GSVD). Joint Equi-diagonal Triangularization (JET). GSVD [Van Loan 76] T 1,T 2 have: Non-equal diagonals Proportional columns JET [Khina, Kochman, Erez 10] T 1,T 2 have: Equal diagonals Non-proportional columns

14 Example of GSVD and JET H 1 = ( det(h 1 ) = det(h 2 ) = 1. ), H 2 = GSVD gives rise to: ( ) 1/2 0 T 1 =, T = JET gives rise to: ( ) T 1 =, T = ( ( /4 ). ( ) ).

15 GSVD Scheme vs. JET Scheme GSVD-based [Yang et al. 10] n diagonal elements = n SISO sub-channels. diag(t 1 ) diag(t 2 ) unbalanced SISO sub-channels. Use unbalanced PNC. Proportional off-diagonal elements Subtract sum codewords. (GDFE/VBLAST) Relay must decode sum over the reals Can t achieve the 1 2. JET-based [New] n diagonal elements = n SISO sub-channels. diag(t 1 ) = diag(t 2 ) balanced SISO sub-channels. Use balanced PNC. Subtract off-diagonal elements in advance modulo-lattice. (dirty-paper coding (DPC)) Relay decodes sum modulo lattice achieves the 1 2.

16 Example - GSVD-based Scheme T 1 = ( 1/ ), T 2 = ( /4 Use unbalanced PNC over each sub-channel. ). Decode sum-codeword of first sub-channel: 2y 1 = x 1;1 +8x 2;1 +z 1 x sum = x 1;1 +8x 2;1. Second sub-channel: Interference {}}{ 4y 2 = 8x 1;2 +x 2;2 + 12(x 1;1 +8x 2;1 )+z 2. Subtract interference of first sub-channel: 12(x 1;1 +8x 2;1 ) = 12x sum.

17 Example - GSVD-based Scheme Remarks Needs to decode sum over the reals x sum (not modulo lattice). BC phase uses random binning scheme (as in [Nam, Chung, Lee 08]).

18 Example - JET-based Scheme T 1 = ( ), T 2 = Use balanced PNC over each sub-channel. ( ). Decode sum-codeword (modulo lattice) of first sub-channel. Second sub-channel: Interference ({}}{ y = (x 1;2+x 2;2 ) x 1; ) x 2;1 + z Use doubly dirty-paper coding (Philosof et al. 2007) for second sub-channel: x 1;2 = [ x 1; x ] 1;1 mod Λ x 2;2 = [ x 2; x ] 2;1 mod Λ Decode modulo lattice: y 2 = [ y ] mod Λ = [ x 1;2 + x 2;2 + z ] mod Λ

19 Example - JET-based Scheme Remarks Standard scalar balanced PNC strategy over each sub-channel is used. MMSE version of each sub-channel can be used achieves the 1 2 over each sub-channel. Any common-message scheme for BC phase.

20 GSVD Scheme vs. JET Scheme High SNR asymptotic optimality Both schemes achieve optimum for P : = P n H 2H 2 { {}}{ R min log P n H 1H 1,C common n = min log P j=1 n t 1;jj 2,C common } { } = min log P n T 1T 1,C common

21 GSVD Scheme vs. JET Scheme Achieving the 1 2 Achievable rate using GSVD specialized to balanced case n R = min log P n t 1;jj 2,C common j=1 Achievable rate using JET [New] n ( 1 R = min log 2 + P =t 2;jj {}}{ 2 ) t 1;jj,C common n j=1 Strictly better than GSVD-based scheme. Standard lattice coding (modulo lattice decoding at relay; no need of binning at BC stage)

22 The 1 2 MMSE Variant We saw how to achieve the 1 2 for each scalar channel. Can we achieve matrix variant of the 1 2 : { } R min minmaxlog 1 i Cx i 2 I +H icx i H i,c common? Currently under research (similar to BC treatment for general SNR [Khina, Kochman, Erez 2010]).

23 Model Gaussian SISO MIMO Gauss.-BC General. Gaussian BC Case G 1 G 2 Relay Terminal 1 Terminal 2 y i = G i x+z i, i = 1,2 Similar approach to MAC approach [Khina, Kochman, Erez 2010] Apply the JET to G 1 and G 2 (for augmented matrices if SNR not high). Achieves triangular matrices with equal diagonals. Use GDFE/VBLAST (instead of DPC). Optimal for any matrices and any SNR!

24 Model Gaussian SISO MIMO Gauss.-BC General. Generalizations MIMO Gaussian BC link can be treated in a similar manner (optimal for any SNRs [Khina, Kochman, Erez 10]). Better PNC-based scheme may be constructed for general SNR using JET. Time-sharing between balanced PNC and DF gives better results for intermediate SNR. Special case of Network Modulation : Joint decomposition of channel matrices for MIMO network problems. Generalization for more than 2 matrices/users exists (last talk in this session).