Anatoly Khina. Joint work with: Uri Erez, Ayal Hitron, Idan Livni TAU Yuval Kochman HUJI Gregory W. Wornell MIT

Transcription

1 Network Modulation: Transmission Technique for MIMO Networks Anatoly Khina Joint work with: Uri Erez, Ayal Hitron, Idan Livni TAU Yuval Kochman HUJI Gregory W. Wornell MIT ACC Workshop, Feder Family Award Ceremony February 27 th, 2012

2 Talk Outline Novel MIMO multicast scheme Two-user: via new joint decomposition of two matrices Multi-user: via algebraic space time coding structure Various applications New information-theoretic results

3 MIMO Multicast (Closed Loop): State of the Art Unicast Multicast Theory SISO MIMO

4 Single-Input Single-Output (SISO) Unicast SISO Unicast Unicast MIMO SISO Multicst MIMO Mlticst h Transmitter Receiver y = hx +z x Input of power 1 y i Output h Channel gain z White Gaussian noise CN(0,1) Optimal communication rate (capacity): C = log(1+ h 2 ) Good practical codes that approach capacity are known!

5 SISO Unicast Unicast MIMO SISO Multicst MIMO Mlticst MIMO Multicast (Closed Loop): State of the Art Unicast Multicast Theory SISO MIMO

6 SISO Unicast Unicast MIMO SISO Multicst MIMO Mlticst MIMO Multicast (Closed Loop): State of the Art Unicast Multicast Theory SISO MIMO

7 SISO Unicast Unicast MIMO SISO Multicst MIMO Mlticst Multiple-Input Multiple-Output (MIMO) Unicast Transmitter H x Input vector of power 1 N t y Output vector H Channel matrix Receiver y = Hx+z H kl Gain from transmit-antenna l to receive-antenna k. z White Gaussian noise CN(0,I) Optimal rate (capacity): C = max Cx log(1+hc xh ) log(1+hh )

8 SISO Unicast Unicast MIMO SISO Multicst MIMO Mlticst MIMO Multicast (Closed Loop): State of the Art Unicast Multicast Theory SISO MIMO

9 SISO Multicast SISO Unicast Unicast MIMO SISO Multicst MIMO Mlticst h 1 h 4 Receiver 1 x Input of power 1 y i Output of user i Receiver 2 h 2 y i = h i x +z i h i Channel gain to user i Transmitter z i White Gaussian noise CN(0,1) h 3 Receiver 3 i = 1,...,K Optimal rate (capacity): C = min i log(1+ h i 2 ) Receiver 4

10 SISO Unicast Unicast MIMO SISO Multicst MIMO Mlticst MIMO Multicast (Closed Loop): State of the Art Unicast Multicast Theory SISO MIMO

11 SISO Unicast Unicast MIMO SISO Multicst MIMO Mlticst MIMO Multicast (Closed Loop): State of the Art Unicast Multicast Theory SISO MIMO

12 Gaussian MIMO Multicast SISO Unicast Unicast MIMO SISO Multicst MIMO Mlticst H 1 H 4 Receiver 1 H 2 Transmitter H 3 Receiver 4 Receiver 2 y i = H i x+z i i = 1,...,K Receiver 3 x N t 1 input vector of power N t 1 y i Output vector of user i H i Channel matrix to user i z i White Gaussian noise vector CN(0,I) Closed loop (Full channel knowledge everywhere)

13 Optimal Achievable Rate (Capacity) SISO Unicast Unicast MIMO SISO Multicst MIMO Mlticst Multicasting capacity C = max C X { min log i=1,...,k ( det I +H i C X H i Optimization over covariance matrices C X satisfying the power constraint White Input / High SNR { ( )} C WI min log det I +H i H i=1,...,k i )}

14 SISO Unicast Unicast MIMO SISO Multicst MIMO Mlticst MIMO Multicast (Closed Loop): State of the Art Unicast Multicast Theory SISO MIMO?

15 SISO Unicast Unicast MIMO SISO Multicst MIMO Mlticst Summary: Multicast is (Almost) Everywhere... Multicast Rateless Codes Permuted Channels Full-Duplex Relay Network Modulation Parallel Relay Half-Duplex Relay Two-Way Relay Gaussian Source Multicast (JSCC)

16 Unicast SVD QR SIC &H1 &H3 &T1 &H2 Transmitter H &T3 &T2 Receiver y = Hx+z White-input capacity { C WI = log det (I +HH )} But how is this rate achieved?

17 SVD QR SIC Practical (Capacity-Achieving) Unicast Approaches (Info. Theory) White-input capacity { C WI = log det (I +HH )} (Comm.) Achieving this rate with a practical scheme Singular value decomposition (SVD) QR decomposition (GDFE / V-BLAST) Geometric mean decomposition (GMD) Dirty-paper coding (DPC)

18 SVD QR SIC Practical (Capacity-Achieving) Unicast Approaches (Info. Theory) White-input capacity { C WI = log det (I +HH )} (Comm.) Achieving this rate with a practical scheme Singular value decomposition (SVD) QR decomposition (GDFE / V-BLAST) Geometric mean decomposition (GMD) Dirty-paper coding (DPC)

19 SVD QR SIC Singular Value Decomposition (SVD) Q,V Unitary H = QΛV Λ Diagonal Parallel AWGN SISO sub-channels off-the-shelf codes Diagonal of Λ = SISO channel gains R i = log ( 1+λ 2 ) i Generalization to Multicast? Precoding matrix V depends on channel matrix H Which V to take?? Bottleneck problem (Λ 1 Λ 2 ) SVD

20 SVD QR SIC QR Decomposition (GDFE/V-BLAST) Q Unitary H = QT T Upper-triangular matrix Successive interference cancellation Parallel AWGN SISO channels Diagonal of T SISO channel gains Remark Both zero-forcing and MMSE (capacity-achieving) solutions exist.

21 QR Based Scheme SVD QR SIC Scheme Channel: y = Hx+z = QTx+z Transmitter: x SISO codebooks Receiver: ỹ = Q y = Tx+Q z z = Q z CN(0,I N ) Example for 2 2 ỹ 1 = [T] 11 x 1 + Interference {}}{ [T] 12 x 2 + z 1 ỹ 2 = 0 x 1 + [T] 22 x 2 + z 2

22 QR Based Scheme QR-based Example JET Network Modulation Scheme Generalization of QR based solution to Multicast? T depends on H. For two channel matrices H 1 and H 2 : diag(t 1 ) diag(t 2 ) different sub-channel gains! Bottleneck problem: N t ( Info. Theory: log [T 1 ] jj 2) = j=1 Comm.: R j = log Can we have equal diagonals? N t j=1 ( min{[t 1 ] jj,[t 2 ] jj } 2) ( log [T 2 ] jj 2) X

23 QR Based Scheme QR-based Example JET Network Modulation Scheme Idea SVD uses both Q and V QR uses only Q Can V help in QR case to achieve equal diagonals? YES!

24 Illustrative Example QR-based Example JET Network Modulation Scheme H 1 H 2 H 1 = [ ], H 2 = [ 99 0 ] C1 WI = 2log ( 1+3 2) ( = log 1+ ( 99 ) ) 2 = C2 WI Send same signal over both antennas Losses half of the rate at High SNR! What if we add a precoding matrix V? How to choose V?

25 QR-based Example JET Network Modulation Scheme Joint Equi-Diagonal Triangularization (JET) Theorem [Kh., Kochman, Erez; Allerton2010, SP2012] H 1 and H 2 N N non-singular matrices det(h 1 ) = det(h 2 ) H 1 and H 2 can be jointly decomposed as: H 1 = Q 1 T 1 V H 2 = Q 2 T 2 V Q 1,Q 2,V unitary T 1 and T 2 are upper-triangular with equal diagonals For det(h 1 ) > det(h 2 ): H 1 = N det(h 1 )Q 1 T 1 V H 2 = N det(h 2 )Q 2 T 2 V

26 Illustrative Example QR-based Example JET Network Modulation Scheme Matrix V is applied to [ Hi I Nt ] (MMSE variant): [ H1 I 2 ] = = Q 1 {}}{ T 1 V {}}{{}}{ [ ][ ] [ H2 I 2 ] = Q 2 T {}}{ 2 V {}}{{}}{ [ ][ ] = [ ] Q Q1 = Q 2 Q2 = V V = 0 1 diag(t 1) = diag(t 2) = [ ] T Parallel SISO channels with equal gains for both users!

27 QR-based Example JET Network Modulation Scheme Network Modulation: MIMO Multicast Scheme Transmitter: info. bits Splitter b 1 Codebook 1 x 1 V x 1 b Nt x Nt x Nt Codebook N t

28 QR-based Example JET Network Modulation Scheme Network Modulation: MIMO Multicast Scheme Effective Channel: z i x x y i V H i + Q i ỹ i T i ỹ i = T i x i + z i z i = Q z i CN(0,I)

29 QR-based Example JET Network Modulation Scheme Network Modulation: MIMO Multicast Scheme Receiver: y i Q i ỹ i Decoder i ˆx j GDFE

30 QR-based Example JET Network Modulation Scheme Extensions

31 Non-white Capacity Multiple Users Extension: Optimal MMSE (Capacity-Achieving) Scheme Holds for channel matrices of any dimension and rank (not equal between H 1 and H 2 ) Based upon an extension of the decomposition to non-square matrices Similar to the extension of V-BLAST from zero-forcing to MMSE For non-white input covariance matrix Cx, decompose: [ ] H i Cx 1/2 I Nt Any number of codebooks number of Tx antennas

32 Multiple Users Non-white Capacity Multiple Users Problem We have used V to triangularize two matrices. What to do for more?? Is 2 just a bit more than 1? Or... Is 2 a simplified? How one buys more degrees of freedom? And and what price?

33 Multiple Users Non-white Capacity Multiple Users Problem We have used V to triangularize two matrices. What to do for more?? Is 2 just a bit more than 1? Or... Is 2 a simplified? How one buys more degrees of freedom? And and what price? Space Time Coding to the Rescue!

34 Space Time Coding Structure Non-white Capacity Multiple Users H i = Q i T i V X Bunch two channel uses together: H i Q i { ( }} ) { ( {}} ) {{ ( }} ) { ({}} ){ Hi 0 Qi 0 Ti 0 V 0 = 0 H i 0 Q i 0 T i 0 V T i V X H i have a block-diagonal structure. Use general Q i, V (not block-diagonal): H i ( {}} ) { Hi 0 0 H i = ( Qi ) ( Ti ) ( V ) Exploiting off-diagonal 0s enables JET of more users!

35 Space Time Coding Structure Non-white Capacity Multiple Users Space Time Coding Structure [Kh., Hitron, Erez ISIT2011][Livni, Kh., Hitron, Erez ISIT2012] Any number of users K Any number of antennas at each node Joint constant-diagonal triangularization of K matrices Process jointly N K 1 t symbols Prefix suffix loss of N K 1 t symbols total Numerical evidence: Can be improved!

36 Perm. Rateless HD Relay 2-Way Relay Source Multicast Applications

37 Perm. Rateless HD Relay 2-Way Relay Source Multicast Gaussian Permuted Parallel Channels General channels: [Willems, Gorokhov][Hof, Sason, Shamai] x 1 α 1 z 1 x 2 z 2 Tx α 2 Rx x n. α n z n Gains {α i } are known Order of gains is not known at Tx, but known at Rx Equivalent Problem Be optimal for all permutation-orders simultaneously.

38 Perm. Rateless HD Relay 2-Way Relay Source Multicast Gaussian Permuted Parallel Channels Special case of MIMO multicasting problem! n! effective channel matrices: α πi (1) α πi (2) 0 H i α πi (n), π i S n i = 1,...,n! Optimal precoding matrices [Hitron, Kh., Erez ISIT2012] 2 gains: Hadamard/DFT; Single channel use 3 gains: DFT; Single channel use 4-6 gains: Quaternion-based matrices; Two channel uses

39 Perm. Rateless HD Relay 2-Way Relay Source Multicast Gaussian Rateless (Incremental Redundancy) Coding y = αx +z, α is unknown at Tx but is known at Rx Rx sends NACKS/ACKS until it is able to recover the message Assume α can take only a finite number of values: α 1,α 2,... Can be represented as a MIMO multicasting problem [Kh., Kochman, Erez, Wornell ITW2011] Example α {α 1,α 2 }, α 1 > α 2 C 1 = 2C 2 Effective matrices: H 1 = ( α 1 0 ) ( α2 0,H 2 = 0 α 2 Coincides with the solution of [Erez, Trott, Wornell] Works for MIMO channels H 1,H 2 (replacing α 1,α 2 ) )

40 Half-Duplex Relay Perm. Rateless HD Relay 2-Way Relay Source Multicast y rel Relay x rel h t,rel h rel,r Transmitter x t h t,r y r Receiver Half-duplex: Relay can receive or transmit but not both Decode-and-forward implementation: rateless relay Effective Matrices: [Kh., Kochman, Erez, Wornell ITW2011] H 1 = [ P 1 h t,rel 0 0 ],H 2 = P1 h t,r 0 0 P2 h 0 t,r + 0 P rel h rel,r P2 h t,r + P rel h rel,r

41 Perm. Rateless HD Relay 2-Way Relay Source Multicast MIMO Two-Way Relay (New Achievable) [Kh., Kochman, Erez ISIT2011] Two nodes want to exchange messages via a relay H 1 H 2 Relay G 1 G 2 Relay Node 1 Node 2 Node 1 Node 2 (a) MAC Phase (b) Broadcast Phase MAC Phase Apply JET to H 1 and H 2 (roles of V and Q switched) Use dirty-paper coding to pre-cancel off-diagonal elements (Replaces successive interference cancellation of broadcast) Broadcast (Multicast!) Phase Use previously discussed multicasting scheme

42 Perm. Rateless HD Relay 2-Way Relay Source Multicast MIMO Multicasting of a Gaussian Source [Kochman, Kh., Erez ICASSP2011][Kh., Kochman, Erez SP2012] Separation does not hold! Different triangularization is needed (N t 1) sub-channels with equal diagonal values (last gain may differ) Combine with hybrid digital analog scheme Decomposition possible under a generalized Weyl condition When decomposition is possible: New achievable distortion! For 2 transmit-antennas: Optimum performance!

43 Summary: Multicast is (Almost) Everywhere... Multicast More...? Rateless Codes Permuted Channels Full-Duplex Relay Network Modulation Parallel Relay Half-Duplex Relay Two-Way Relay Gaussian Source Multicast (JSCC) Even now, me talking to you...