Division algebras for coding in multiple antenna channels and wireless networks

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Division algebras Coding for wireless relay networks Conclusion

Division algebras for coding in multiple antenna

channels and wireless networks

Fr´ed´erique Oggier

[email protected]

California Institute of Technology

Cornell University, School of Electrical and Computer Engineering, March 13rd 2007

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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

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The multiple antenna channel (I)

1. Time t = 1:

• 1st receive antenna:y11=h11x11+h12x21+v11 • 2nd receive antenna:y21=h21x11+h22x21+v21

2. Time t = 2:

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The multiple antenna channel (I)

1. Time t = 1:

• 1st receive antenna:y11=h11x11+h12x21+v11 • 2nd receive antenna:y21=h21x11+h22x21+v21 2. Time t = 2:

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The multiple antenna channel (II)

We get the matrix equation

y11 y12 y21 y22 = h11 h12 h21 h22 x11 x12 x21 x22 | {z } space-timecodeword X + v11 v12 v21 v22 .

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Code design criteria (Coherent case)

• Reliabilityis modeled by the pairwise probability of error:

P(X→Xˆ)≤ const |det(X−Xˆ)|2M.

• We assume the receiver knows the channel (coherent case).

• We need fully diversecodes, such that

det(X−X0)6= 0 ∀ X6=X0. • Diversity reflects the slope of the probability of error. • We attempt to maximize theminimum determinant

min

X6=X0|det(X−X

0

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Code design criteria (Coherent case)

• We assume the receiver knows the channel (coherent case). • We need fully diversecodes, such that

det(X−X0)6= 0 ∀ X6=X0.

• Diversity reflects the slope of the probability of error. • We attempt to maximize theminimum determinant

min

X6=X0|det(X−X

0

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Code design criteria (Coherent case)

det(X−X0)6= 0 ∀ X6=X0. • Diversity reflects the slope of the probability of error.

• We attempt to maximize theminimum determinant min

X6=X0|det(X−X

0

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Code design criteria (Coherent case)

det(X−X0)6= 0 ∀ X6=X0. • Diversity reflects the slope of the probability of error. • We attempt to maximize theminimum determinant

min

X6=X0|det(X−X

0

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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.

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The idea behind division algebras

• The difficulty in buildingC such that

det(Xi−Xj)= 0,6 Xi 6=Xj ∈ C,

comes from the non-linearityof the determinant.

• IfC is taken inside an algebraof matrices, the problem simplifies to

det(X)6= 0, 06=X∈ C. • A division algebrais a non-commutative field.

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The idea behind division algebras

comes from the non-linearityof the determinant. • IfC is taken inside an algebraof matrices, the problem

simplifies to

det(X)6= 0, 06=X∈ C.

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The idea behind division algebras

comes from the non-linearityof the determinant. • IfC is taken inside an algebraof matrices, the problem

simplifies to

det(X)6= 0, 06=X∈ C. • A division algebrais a non-commutative field.

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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 i2 =−1,j2 =−1, andij =−ji. • The Hamiltonian Quaternionsis the set Hdefined by

H={x+yi +zj +wk |x,y,z,w ∈R}.

• Hamiltonian Quaternions are a division algebra:

q−1 = ¯q

qq¯,

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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 i2 =−1,j2 =−1, andij =−ji. • The Hamiltonian Quaternionsis the set Hdefined by

H={x+yi +zj +wk |x,y,z,w ∈R}.

• Hamiltonian Quaternions are a division algebra:

q−1 = ¯q

qq¯,

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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 themultiplication byq: (α+jβ)

| {z }

q

(γ+jδ) = αγ+jαδ¯ +jβγ+j2βδ¯ = (αγ−βδ) +¯ j( ¯αδ+βγ)

• Write this equality in the basis {1,j}:

α −β¯ β α¯ γ δ = αγ−βδ¯ ¯ αδ+βγ

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The Hamiltonian Quaternions: how to get matrices

| {z }

q

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The Hamiltonian Quaternions: how to get matrices

| {z }

q

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Introducing cyclic algebras

• The Hamiltonian Quaternions gives the Alamouti code:

q =α+jβ ∈H↔

α −β¯ β α¯

• We similarly consider cyclic algebras:

x=x0+ex1 ∈ A ↔

x0 γσ(x1)

x1 σ(x0)

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Introducing cyclic algebras

• The Hamiltonian Quaternions gives the Alamouti code:

q =α+jβ ∈H↔

α −β¯ β α¯

• We similarly consider cyclic algebras:

x=x0+ex1 ∈ A ↔

x0 γσ(x1)

x1 σ(x0)

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Advantages of cyclic algebras

1. Yield full diversityand a practical encoding (of n2 information symbols for ann×n codeword), for anynumber of antennas.

2. Allow for alower bound on the minimum determinant for constellations of arbitrary size.

3. Achieve the diversity-multiplexing tradeoff of Zheng and Tse, thanks to thenon-vanishing determinant property.

• Constructions of codes are available where the algebraic structures are exploited tooptimize the codes performance. F. E. Oggier, G. Rekaya, J.-C. Belfiore, E. Viterbo. Perfect Space-Time

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Advantages of cyclic algebras

1. Yield full diversityand a practical encoding (of n2 information symbols for ann×n codeword), for anynumber of antennas. 2. Allow for alower bound on the minimum determinant for

constellations of arbitrary size.

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Advantages of cyclic algebras

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Advantages of cyclic algebras

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Performances

• A 2×2 cyclic algebra based code is to be implemented in the future wireless standard 802.16e for wireless LANs.

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Non-coherent unitary space-time coding

• We assume no channel knowledge.

• Use a cyclic division algebra endowed with an involution:

A Mn(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 transformof an Hermitian matrix A:

X= (I+iA)−1(I−iA).

F. Oggier, B. Hassibi,Algebraic Cayley differential Space-Time Codes. Trans. on IT.

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Non-coherent unitary space-time coding

A Mn(L)

x ↔ X

α(x) ↔ X†

xα(x) = 1 ↔ XX†=I

X= (I+iA)−1(I−iA).

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Non-coherent unitary space-time coding

A Mn(L)

x ↔ X

α(x) ↔ X†

xα(x) = 1 ↔ XX†=I

X= (I+iA)−1(I−iA).

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Performances

10 12 14 16 18 20 22 24 26 28 30 10−1 algebraic orthogonal design cayley code 1

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Division algebras

Other space-time coding applications

Coding for wireless relay networks Distributed space-time coding

Noncoherent distributed space-time coding

Conclusion Future work

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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.

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Coding for wireless relay network

• Relay nodes are small devices with few resources.

• Cooperation: we learntdiversityfrom space-time coding.

Tx

Rx

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Wireless relay network: phase 1

• s vector for one Tx antenna, matrix for several Tx antennas.

Tx Rx s r1=f1s+v1 r2=f2s+v2 r3=f3s+v3 r4=f4s+v4 1

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Wireless relay network: phase 2

• No decoding at the relays. • The matrices Ai are unitary.

Tx Rx s A1r1 A2r2 A3r3 A4r4 t1 t2 t3 t4 1

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Channel model

1. At the receiver, y= R X i=1 giti+w= R X i=1 giAi(sfi+vi) +w 2. So that finally: y= [A1s· · ·ARs] | {z } X    f1g1 .. . fngn    | {z } H +w0

• Each relay encodes a (set of) column(s), so that the encoding is distributedamong the nodes (Jing-Hassibi).

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Channel model

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Channel model

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Distributed space-time codes from division algebras

• σ :L→L,σ4 = 1, 4·4 information symbols, doesnotwork!

[A1s A2s A3s A4s] [A1S A2S] 6 =     x0 iσ(x3) iσ2(x2) iσ3(x1) x1 σ(x0) iσ2(x3) iσ3(x2) x2 σ(x1) σ2(x0) iσ3(x3) x3 σ(x2) σ2(x1) σ3(x0)     • τ :K →K,K ⊂L,τ2 = 1 xi=information symbols     x0 iτ(x3) ix2 iτ(x1) x1 τ(x0) ix3 iτ(x2) x2 τ(x1) x0 iτ(x3) x3 τ(x2) x1 τ(x0)     ,     x0 ix3 ix2 ix1 x1 x0 ix3 ix2 x2 x1 x0 ix3 x3 x2 x1 x0    

F. Oggier, B. Hassibi,An Algebraic Coding Scheme for Wireless Relay Networks with Multiple-Antenna Nodes

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Distributed space-time codes from division algebras

• σ :L→L,σ4 = 1, 4·4 information symbols, doesnotwork!

[A1s A2s A3s A4s] [A1S A2S] 6 =     x0 iσ(x3) iσ2(x2) iσ3(x1) x1 σ(x0) iσ2(x3) iσ3(x2) x2 σ(x1) σ2(x0) iσ3(x3) x3 σ(x2) σ2(x1) σ3(x0)     • τ :K →K,K ⊂L,τ2 = 1 xi=information symbols     x0 iτ(x3) ix2 iτ(x1) x1 τ(x0) ix3 iτ(x2) x2 τ(x1) x0 iτ(x3) x3 τ(x2) x1 τ(x0)     ,     x0 ix3 ix2 ix1 x1 x0 ix3 ix2 x2 x1 x0 ix3 x3 x2 x1 x0    

F. Oggier, B. Hassibi,An Algebraic Coding Scheme for Wireless Relay Networks with Multiple-Antenna Nodes

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Performances

16 18 20 22 24 26 28 30 10−4 10−3 10−2 10−1 100 P(dB) BLER M=N=R=2 algebraic random no coding 1

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Division algebras

Other space-time coding applications

Coding for wireless relay networks Distributed space-time coding

Noncoherent distributed space-time coding

Conclusion Future work

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A Noncoherent Channel

• How to design a protocol to communicate over a wireless relay network with no channel information?

• Noncoherent network model: y= [A1s· · ·ARs]

| {z } Xunitary    f1g1 .. . fngn    | {z } H +w0 • Design s0= √1 T(1, . . . ,1) t_, _s i =Uis0, i = 1, . . . ,L,where Ui’s areT ×T unitary matrices.

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A Noncoherent Channel

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A Noncoherent Channel

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A Unitary Distributed Space-Time Code

1. Let M be aT ×T matrixM such that MM†=T. Then

Ai =diag( Mi |{z} ith column

), i = 1, . . . ,R.

2. Choosing allUj diagonal (tocommute with allAi)

[A1sj, . . . ,ARsj] = [UjA1s0, . . . ,UjARs0] =UjM/

√ T,

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A Unitary Distributed Space-Time Code

), i = 1, . . . ,R.

√ T,

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A Unitary Distributed Space-Time Code

), i = 1, . . . ,R.

√ T,

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Butson-Hadamard Matrices

1. A Generalized Butson Hadamard(GBH) matrix is a T ×T

matrix M such that

MM∗ =M∗M =TIT

wherem_ij∗ =m−_ji1.

2. Choose the coefficients of M to beroots of unity. Furthermore all Ai are unitary.

3. Let ζ3 = exp(2iπ/3) be a primitive 3rd root of unity.

M =   1 1 1 1 ζ3 ζ32 1 ζ₃2 ζ3  

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Butson-Hadamard Matrices

matrix M such that

MM∗ =M∗M =TIT

M =   1 1 1 1 ζ3 ζ32 1 ζ₃2 ζ3  

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Butson-Hadamard Matrices

matrix M such that

MM∗ =M∗M =TIT

M =   1 1 1 1 ζ3 ζ32 1 ζ₃2 ζ3  

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A Differential Encoder

1. Transmitter: send st =s0 then st+T =U(zt+T)

| {z } data st. 2. Relays: ri(t) =fist+vi(t), ri(t+T) =fiU(zt+T)st+vi(t+T). 3. Receiver: y(t) =PR i=1gifiAist+w(t). y(t+T) =PR i=1gifiAiU(zt+T)st+w(t+T).

4. The differential channel:

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A Differential Encoder

1. Transmitter: send st =s0 then st+T =U(zt+T)

| {z } data st. 2. Relays: ri(t) =fist+vi(t), ri(t+T) =fiU(zt+T)st+vi(t+T). 3. Receiver: y(t) =PR i=1gifiAist+w(t). y(t+T) =PR i=1gifiAiU(zt+T)st+w(t+T).

4. The differential channel:

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A PEP computation

1. We consider a mismatched decoder.

2. We prove that

P(Uk →Ul)≤Egdet(IT +₈₍_c0 cρ

ρkgk2+1)D|g|(Uk −Ul)(Uk −Ul) †₎−1_.

3. So that, using a result by Jing-Hassibi, the diversity is

R 1−log logP logP , when (Uk−Ul)(Uk−Ul)† is full rank.

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A PEP computation

1. We consider a mismatched decoder. 2. We prove that

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A PEP computation

1. We consider a mismatched decoder. 2. We prove that

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Code Design

1. We want to design unitary diagonalmatricesU,independent of the matrices at the relays.

2. They have to satisfy

det(U(zt)−U(zt0))6= 0, t 6=t0.

3. Usecyclic codes(Hochwald and Sweldens):

   ζu1l L 0 0 . .. 0 0 ζuMl L   , l = 0, . . . ,L−1, ζL= exp(2iπ/L)

whereL andu = (u1, . . . ,uM) have to be designed.

F. Oggier, B. Hassibi,A Coding Strategy for Wireless Networks with no Channel Information

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Code Design

1. We want to design unitary diagonalmatricesU,independent of the matrices at the relays.

2. They have to satisfy

det(U(zt)−U(zt0))6= 0, t 6=t0.

3. Usecyclic codes(Hochwald and Sweldens):

   ζu1l L 0 0 . .. 0 0 ζuMl L   , l = 0, . . . ,L−1, ζL= exp(2iπ/L)

whereL andu = (u1, . . . ,uM) have to be designed.

F. Oggier, B. Hassibi,A Coding Strategy for Wireless Networks with no Channel Information

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Performances

12 14 16 18 20 22 24 26 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 P(dB) BLER R=3 R=6 R=9

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Conclusion...

• The problem of designing fully-diversematrices arise in a lot of wireless coding applications, and division algebras is a powerful tool to design such matrices.

• We thus could propose codes forspace-time coding as well as wireless relay networks.

• We also propose a strategy for communicate over a wireless relay network with no channel information.

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Conclusion...

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Conclusion...

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...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 securityin wireless communication.

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...and future work

• Can we do better for non-coherent wireless networks? • Synchronization is an issue to be dealt with.

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...and future work

• Can we do better for non-coherent wireless networks? • Synchronization is an issue to be dealt with.

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