Division algebras for coding in multiple antenna channels and wireless networks Frédérique Oggier frederique@systems.caltech.edu California Institute of Technology Cornell University, School of Electrical and Computer Engineering, March 13rd 2007
Outline Division algebras Motivation: coherent space-time coding Introducing division algebras Other space-time coding applications Coding for wireless relay networks Distributed space-time coding Noncoherent distributed space-time coding Conclusion Future work
The multiple antenna channel (I) 1. Time t = 1: 1st receive antenna:y 11 = h 11 x 11 + h 12 x 21 + v 11 2nd receive antenna:y 21 = h 21 x 11 + h 22 x 21 + v 21
The multiple antenna channel (I) 1. Time t = 1: 1st receive antenna:y 11 = h 11 x 11 + h 12 x 21 + v 11 2nd receive antenna:y 21 = h 21 x 11 + h 22 x 21 + v 21 2. Time t = 2: 1st receive antenna:y 12 = h 11 x 12 + h 12 x 22 + v 12 2nd receive antenna:y 22 = h 21 x 12 + h 22 x 22 + v 22
The multiple antenna channel (II) We get the matrix equation ( ) ( ) ( ) ( y11 y 12 h11 h = 12 x11 x 12 v11 v + 12 y 21 y 22 h 21 h 22 x 21 x 22 v 21 v 22 } {{ } space-time codeword X ).
Code design criteria (Coherent case) Reliability is modeled by the pairwise probability of error: P(X ˆX) const det(x ˆX) 2M. We assume the receiver knows the channel (coherent case).
Code design criteria (Coherent case) Reliability is modeled by the pairwise probability of error: P(X ˆX) const det(x ˆX) 2M. We assume the receiver knows the channel (coherent case). We need fully diverse codes, such that det(x X ) 0 X X.
Code design criteria (Coherent case) Reliability is modeled by the pairwise probability of error: P(X ˆX) const det(x ˆX) 2M. We assume the receiver knows the channel (coherent case). We need fully diverse codes, such that det(x X ) 0 X X. Diversity reflects the slope of the probability of error.
Code design criteria (Coherent case) Reliability is modeled by the pairwise probability of error: P(X ˆX) const det(x ˆX) 2M. We assume the receiver knows the channel (coherent case). We need fully diverse codes, such that det(x X ) 0 X X. Diversity reflects the slope of the probability of error. We attempt to maximize the minimum determinant min det(x X X X ) 2.
Previous work 1. E. Telatar,Capacity of multi-antenna Gaussian channels, 1999. 2. V. Tarokh and N. Seshadri and A. R. Calderbank, Space-time codes for high data rate wireless communications: Performance criterion and code construction, 1998. 3. B. Hassibi and B.M. Hochwald, High-Rate Codes That Are Linear in Space and Time, 2002. 4. H. El Gamal and M.O. Damen, Universal space-time coding, 2003.
The idea behind division algebras The difficulty in building C such that det(x i X j ) 0, X i X j C, comes from the non-linearity of the determinant.
The idea behind division algebras The difficulty in building C such that det(x i X j ) 0, X i X j C, comes from the non-linearity of the determinant. If C is taken inside an algebra of matrices, the problem simplifies to det(x) 0, 0 X C.
The idea behind division algebras The difficulty in building C such that det(x i X j ) 0, X i X j C, comes from the non-linearity of the determinant. If C is taken inside an algebra of matrices, the problem simplifies to det(x) 0, 0 X C. A division algebra is a non-commutative field.
The Hamiltonian Quaternions: the definition Let {1, i, j, k} be a basis for a vector space of dim 4 over R. We have the rule that i 2 = 1, j 2 = 1, and ij = ji. The Hamiltonian Quaternions is the set H defined by H = {x + yi + zj + wk x, y, z, w R}.
The Hamiltonian Quaternions: the definition Let {1, i, j, k} be a basis for a vector space of dim 4 over R. We have the rule that i 2 = 1, j 2 = 1, and ij = ji. The Hamiltonian Quaternions is the set H defined by H = {x + yi + zj + wk x, y, z, w R}. Hamiltonian Quaternions are a division algebra: q 1 = q q q, where q = x yi zj wk and q q = x 2 + y 2 + z 2 + w 2.
The Hamiltonian Quaternions: how to get matrices Any quaternion q = x + yi + zj + wk can be written as (x + yi) + j(z wi) = α + jβ, α, β C.
The Hamiltonian Quaternions: how to get matrices Any quaternion q = x + yi + zj + wk can be written as (x + yi) + j(z wi) = α + jβ, α, β C. Now compute the multiplication by q: (α + jβ) (γ + jδ) } {{ } = αγ + jᾱδ + jβγ + j 2 βδ q = (αγ βδ) + j(ᾱδ + βγ)
The Hamiltonian Quaternions: how to get matrices Any quaternion q = x + yi + zj + wk can be written as (x + yi) + j(z wi) = α + jβ, α, β C. Now compute the multiplication by q: (α + jβ) (γ + jδ) } {{ } = αγ + jᾱδ + jβγ + j 2 βδ q = (αγ βδ) + j(ᾱδ + βγ) Write this equality in the basis {1, j}: ( ) ( ) ( ) α β γ αγ βδ = β ᾱ δ ᾱδ + βγ
Introducing cyclic algebras The Hamiltonian Quaternions gives the Alamouti code: ( ) α β q = α + jβ H β ᾱ
Introducing cyclic algebras The Hamiltonian Quaternions gives the Alamouti code: ( ) α β q = α + jβ H β ᾱ We similarly consider cyclic algebras: ( ) x0 γσ(x x = x 0 + ex 1 A 1 ) x 1 σ(x 0 )
Advantages of cyclic algebras 1. Yield full diversity and a practical encoding (of n 2 information symbols for an n n codeword), for any number of antennas.
Advantages of cyclic algebras 1. Yield full diversity and a practical encoding (of n 2 information symbols for an n n codeword), for any number of antennas. 2. Allow for a lower bound on the minimum determinant for constellations of arbitrary size.
Advantages of cyclic algebras 1. Yield full diversity and a practical encoding (of n 2 information symbols for an n n codeword), for any number of antennas. 2. Allow for a lower bound on the minimum determinant for constellations of arbitrary size. 3. Achieve the diversity-multiplexing tradeoff of Zheng and Tse, thanks to the non-vanishing determinant property.
Advantages of cyclic algebras 1. Yield full diversity and a practical encoding (of n 2 information symbols for an n n codeword), for any number of antennas. 2. Allow for a lower bound on the minimum determinant for constellations of arbitrary size. 3. Achieve the diversity-multiplexing tradeoff of Zheng and Tse, thanks to the non-vanishing determinant property. Constructions of codes are available where the algebraic structures are exploited to optimize the codes performance. F. E. Oggier, G. Rekaya, J.-C. Belfiore, E. Viterbo. Perfect Space-Time Block Codes. Trans. on IT.
Performances A 2 2 cyclic algebra based code is to be implemented in the future wireless standard 802.16e for wireless LANs.
Non-coherent unitary space-time coding We assume no channel knowledge.
Non-coherent unitary space-time coding We assume no channel knowledge. Use a cyclic division algebra endowed with an involution: A M n (L) x X α(x) X xα(x) = 1 XX = I F. Oggier, Cyclic Algebras for Noncoherent Differential Space-Time Coding. Trans. on IT.
Non-coherent unitary space-time coding We assume no channel knowledge. Use a cyclic division algebra endowed with an involution: A M n (L) x X α(x) X xα(x) = 1 XX = I F. Oggier, Cyclic Algebras for Noncoherent Differential Space-Time Coding. Trans. on IT. Use the Cayley transform of an Hermitian matrix A: X = (I + ia) 1 (I ia). F. Oggier, B. Hassibi, Algebraic Cayley differential Space-Time Codes. Trans. on IT.
Performances 10 1 algebraic orthogonal design cayley code 10 12 14 16 18 20 22 24 26 28 30 2 transmit/receive antennas, rate=6
Division algebras Motivation: coherent space-time coding Introducing division algebras Other space-time coding applications Coding for wireless relay networks Distributed space-time coding Noncoherent distributed space-time coding Conclusion Future work
Some references 1. J.N. Laneman and G. W. Wornell, Distributed space-time-coded protocols for exploiting cooperative diversity in wireless network, Oct. 2003. 2. K. Azarian, H. El Gamal and P. Schniter, On the achievable diversity-multiplexing tradeoff in half-duplex cooperative channels, Dec. 2005. 3. Y. Jing and B. Hassibi, Distributed space-time coding in Wireless Relay Networks, Dec. 2006. 4. S. Yang and J.-C. Belfiore, Optimal space-time codes for the MIMO Amplify-and-Forward cooperative channel, Feb. 2007.
Coding for wireless relay network Relay nodes are small devices with few resources. Cooperation: we learnt diversity from space-time coding. Tx Rx
r 1 = f 1 s + v 1 r 2 = f 2 s + v 2 Wireless relay network: phase 1 s vector for one Tx antenna, matrix for several Tx antennas. s Tx r 3 = f 3 s + v 3 r 4 = f 4 s + v 4 Rx
t 1 t 2 A 1 r 1 A 2 r 2 Wireless relay network: phase 2 No decoding at the relays. The matrices A i are unitary. s Tx t 3 A 3 r 3 t 4 A 4 r 4 Rx
Channel model 1. At the receiver, y = R g i t i + w = i=1 R g i A i (sf i + v i ) + w i=1
Channel model 1. At the receiver, y = R g i t i + w = i=1 R g i A i (sf i + v i ) + w i=1 f 1 g 1 2. So that finally: y = [A 1 s A R s] } {{ }. +w X f n g n } {{ } H
Channel model 1. At the receiver, y = R g i t i + w = i=1 R g i A i (sf i + v i ) + w i=1 f 1 g 1 2. So that finally: y = [A 1 s A R s] } {{ }. +w X f n g n } {{ } H Each relay encodes a (set of) column(s), so that the encoding is distributed among the nodes (Jing-Hassibi). Need fully-diverse distributed space-time codes.
Distributed space-time codes from division algebras σ : L L, σ 4 = 1, 4 4 information symbols, does not work! } x 0 iσ(x 3 ) iσ 2 (x 2 ) iσ 3 (x 1 ) [A 1 s A 2 s A 3 s A 4 s] x 1 σ(x 0 ) iσ 2 (x 3 ) iσ 3 (x 2 ) [A 1 S A 2 S] x 2 σ(x 1 ) σ 2 (x 0 ) iσ 3 (x 3 ) x 3 σ(x 2 ) σ 2 (x 1 ) σ 3 (x 0 )
Distributed space-time codes from division algebras σ : L L, σ 4 = 1, 4 4 information symbols, does not work! } x 0 iσ(x 3 ) iσ 2 (x 2 ) iσ 3 (x 1 ) [A 1 s A 2 s A 3 s A 4 s] x 1 σ(x 0 ) iσ 2 (x 3 ) iσ 3 (x 2 ) [A 1 S A 2 S] x 2 σ(x 1 ) σ 2 (x 0 ) iσ 3 (x 3 ) x 3 σ(x 2 ) σ 2 (x 1 ) σ 3 (x 0 ) τ : K K, K L, τ 2 = 1 x 0 iτ(x 3 ) ix 2 iτ(x 1 ) x 1 τ(x 0 ) ix 3 iτ(x 2 ) x 2 τ(x 1 ) x 0 iτ(x 3 ) x 3 τ(x 2 ) x 1 τ(x 0 ), x i =information symbols x 0 ix 3 ix 2 ix 1 x 1 x 0 ix 3 ix 2 x 2 x 1 x 0 ix 3 x 3 x 2 x 1 x 0 F. Oggier, B. Hassibi, An Algebraic Coding Scheme for Wireless Relay Networks with Multiple-Antenna Nodes
Performances 10 0 M=N=R=2 algebraic random no coding 10 1 BLER 10 2 10 3 10 4 16 18 20 22 24 26 28 30 P(dB)
Division algebras Motivation: coherent space-time coding Introducing division algebras Other space-time coding applications Coding for wireless relay networks Distributed space-time coding Noncoherent distributed space-time coding Conclusion Future work
A Noncoherent Channel How to design a protocol to communicate over a wireless relay network with no channel information?
A Noncoherent Channel How to design a protocol to communicate over a wireless relay network with no channel information? f 1 g 1 Noncoherent network model: y = [A 1 s A R s] } {{ }. +w X unitary f n g n } {{ } H
A Noncoherent Channel How to design a protocol to communicate over a wireless relay network with no channel information? f 1 g 1 Noncoherent network model: y = [A 1 s A R s] } {{ }. +w X unitary f n g n } {{ } H Design s = 1 T (1,..., 1) t, s i = U i s, i = 1,..., L, where U i s are T T unitary matrices.
A Unitary Distributed Space-Time Code 1. Let M be a T T matrix M such that MM = T. Then A i = diag( M }{{} i ), i = 1,..., R. ith column
A Unitary Distributed Space-Time Code 1. Let M be a T T matrix M such that MM = T. Then A i = diag( M }{{} i ), i = 1,..., R. ith column 2. Choosing all U j diagonal (to commute with all A i ) [A 1 s j,..., A R s j ] = [U j A 1 s,..., U j A R s ] = U j M/ T,
A Unitary Distributed Space-Time Code 1. Let M be a T T matrix M such that MM = T. Then A i = diag( M }{{} i ), i = 1,..., R. ith column 2. Choosing all U j diagonal (to commute with all A i ) [A 1 s j,..., A R s j ] = [U j A 1 s,..., U j A R s ] = U j M/ T, 3. Let us keep in mind that the matrices A i have to be unitary.
Butson-Hadamard Matrices 1. A Generalized Butson Hadamard (GBH) matrix is a T T matrix M such that where m ij = m 1 ji. MM = M M = T I T
Butson-Hadamard Matrices 1. A Generalized Butson Hadamard (GBH) matrix is a T T matrix M such that MM = M M = T I T where m ij = m 1 ji. 2. Choose the coefficients of M to be roots of unity. Furthermore all A i are unitary.
Butson-Hadamard Matrices 1. A Generalized Butson Hadamard (GBH) matrix is a T T matrix M such that MM = M M = T I T where m ij = m 1 ji. 2. Choose the coefficients of M to be roots of unity. Furthermore all A i are unitary. 3. Let ζ 3 = exp(2iπ/3) be a primitive 3rd root of unity. M = 1 1 1 1 ζ 3 ζ 2 3 1 ζ 2 3 ζ 3
A Differential Encoder 1. Transmitter: send s t = s then s t+t = U(z t+t ) s } {{ } t. data 2. Relays: r i (t) = f i s t + v i (t), r i (t + T ) = f i U(z t+t )s t + v i (t + T ). 3. Receiver: y(t) = R i=1 g if i A i s t + w(t). y(t + T ) = R i=1 g if i A i U(z t+t )s t + w(t + T ).
A Differential Encoder 1. Transmitter: send s t = s then s t+t = U(z t+t ) s } {{ } t. data 2. Relays: r i (t) = f i s t + v i (t), r i (t + T ) = f i U(z t+t )s t + v i (t + T ). 3. Receiver: y(t) = R i=1 g if i A i s t + w(t). y(t + T ) = R i=1 g if i A i U(z t+t )s t + w(t + T ). 4. The differential channel: y(t + T ) = U(z t+t )y(t) + w(t, t + T ).
A PEP computation 1. We consider a mismatched decoder.
A PEP computation 1. We consider a mismatched decoder. 2. We prove that P(U k U l ) E g det(i T + c ρ 8(c ρ g 2 +1) D g (U k U l )(U k U l ) ) 1.
A PEP computation 1. We consider a mismatched decoder. 2. We prove that P(U k U l ) E g det(i T + c ρ 8(c ρ g 2 +1) D g (U k U l )(U k U l ) ) 1. 3. So that, using a result by Jing-Hassibi, the diversity is ( ) log log P R 1, log P when (U k U l )(U k U l ) is full rank.
Code Design 1. We want to design unitary diagonal matrices U, independent of the matrices at the relays. 2. They have to satisfy det(u(z t ) U(z t )) 0, t t.
Code Design 1. We want to design unitary diagonal matrices U, independent of the matrices at the relays. 2. They have to satisfy det(u(z t ) U(z t )) 0, t t. 3. Use cyclic codes (Hochwald and Sweldens): ζ u 1l L 0 0... 0 0 ζ u Ml L, l = 0,..., L 1, ζ L = exp(2iπ/l) where L and u = (u 1,..., u M ) have to be designed. F. Oggier, B. Hassibi, A Coding Strategy for Wireless Networks with no Channel Information
Performances 10 0 10 1 10 2 10 3 BLER 10 4 10 5 10 6 10 7 R=3 R=6 R=9 10 8 12 14 16 18 20 22 24 26 P(dB)
Conclusion... The problem of designing fully-diverse matrices arise in a lot of wireless coding applications, and division algebras is a powerful tool to design such matrices.
Conclusion... The problem of designing fully-diverse matrices arise in a lot of wireless coding applications, and division algebras is a powerful tool to design such matrices. We thus could propose codes for space-time coding as well as wireless relay networks.
Conclusion... The problem of designing fully-diverse matrices arise in a lot of wireless coding applications, and division algebras is a powerful tool to design such matrices. We thus could propose codes for space-time coding as well as wireless relay networks. We also propose a strategy for communicate over a wireless relay network with no channel information.
...and future work Can we do better for non-coherent wireless networks?
...and future work Can we do better for non-coherent wireless networks? Synchronization is an issue to be dealt with.
...and future work Can we do better for non-coherent wireless networks? Synchronization is an issue to be dealt with. Future work is also oriented towards including security in wireless communication.
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