Algebraic Methods for Wireless Coding

Transcription

1 Algebraic Methods for Wireless Coding Frédérique Oggier California Institute of Technology UC Davis, Mathematics Department, January 31st 2007

2 Outline The Rayleigh fading channel Algebraic lattices and minimum discriminant Coherent space-time codes Cyclic division algebras Other examples Current and future work

3 The Rayleigh fading channel y = xh + v y = (y 1,..., y n ) R n is the received signal x = (x 1,..., x n ) R n is the transmitted codeword v N (0, σ 2 I ) is the noise H = diag(h 1,..., h n ) is the channel fading matrix, h i Rayleigh distributed

4 Decoding and error probability Assuming that H is known at the receiver, the decoding rule is ˆx = arg min x S y xh 2.

5 Decoding and error probability Assuming that H is known at the receiver, the decoding rule is ˆx = arg min x S y xh 2. Reliability is modeled by the pairwise probability of error, bounded by P(x ˆx) 1 2 x i ˆx i 8σ2 (x i ˆx i ) 2 = 1 2 (8σ 2 ) l x i ˆx i x i ˆx i 2 when the two codewords differ in l components.

6 Code design To design a code with low error probability, we have to: 1. Maximize the modulation diversity L = min(l), ideally L = n.

7 Code design To design a code with low error probability, we have to: 1. Maximize the modulation diversity L = min(l), ideally L = n. 2. For a given L, maximize the minimum product distance d p,min = min x i x i x x x i x i under the constraint of bounded average energy.

8 Previous work K. Boullé and J.-C. Belfiore, Modulation schemes designed for the Rayleigh channel,1992. X. Giraud and J.-C. Belfiore, Constellations Matched to the Rayleigh fading channel, J. Boutros and E. Viterbo, Signal Space Diversity: a power and bandwidth efficient diversity technique for the Rayleigh fading channel, G. Taricco and E. Viterbo, Performance of High Diversity Multidimensional Constellations, 1998.

9 Algebraic lattices

10 Algebraic lattices Let K be a number field of degree n and signature (r 1, r 2 ). The canonical embedding σ : K R n is defined by σ(α) = (σ 1 (α),..., σ r1 (α), Rσ r1 +1(α), Iσ r1 +1(α),...)

11 Algebraic lattices Let K be a number field of degree n and signature (r 1, r 2 ). The canonical embedding σ : K R n is defined by σ(α) = (σ 1 (α),..., σ r1 (α), Rσ r1 +1(α), Iσ r1 +1(α),...) Let O K be the ring of integers of K with integral basis {ω 1,..., ω n }. An algebraic lattice Λ = σ(o K ) has generator matrix M = σ 1 (ω 1 )... σ r1 (ω 1 )... Iσ r1 +r 2 (ω 1 )... σ 1 (ω n )... σ n (ω r1 )... Iσ r1 +r 2 (ω n ).

12 The modulation diversity Let K be a number field of signature (r 1, r 2 ). Theorem. Algebraic lattices exhibit a diversity L = r 1 + r 2.

13 The modulation diversity Let K be a number field of signature (r 1, r 2 ). Theorem. Algebraic lattices exhibit a diversity L = r 1 + r 2. In order to guarantee maximal diversity, we consider totally real number fields.

14 The minimum product distance Let Λ = σ(o K ) be an algebraic lattice. Theorem: The minimum product distance of an algebraic lattice Λ is det(λ) d p,min =, d K where d K is the discriminant of K.

15 The minimum product distance Let Λ = σ(o K ) be an algebraic lattice. Theorem: The minimum product distance of an algebraic lattice Λ is det(λ) d p,min =, d K where d K is the discriminant of K. Since finding small discriminants is a difficult question, we use Odlyzko s bound: d 1/n K C n.

16 Summary In order to get good codes for the Rayleigh fading channel, we need 1. a number field K totally real 2. of small discriminant 3. on which we can build a suitable algebraic lattice.

17 Summary In order to get good codes for the Rayleigh fading channel, we need 1. a number field K totally real 2. of small discriminant 3. on which we can build a suitable algebraic lattice. Explicit constructions available in all dimensions, which are shown to be optimal. F. Oggier and E. Viterbo, Algebraic number theory and code design for Rayleigh fading channels.

18 The Rayleigh fading channel Algebraic lattices and minimum discriminant Coherent space-time codes Cyclic division algebras Other examples Current and future work

19 The multiple antenna channel (I)

20 The multiple antenna channel (I) 1. Time t = 1: 1st receive antenna:y 11 = h 11 x 11 + h 12 x 21 + v 11

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 Time t = 2:

22 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 Time t = 2: 1st receive antenna:y 12 = h 11 x 12 + h 12 x 22 + v 12

23 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 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

24 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 ).

25 Code design criteria (Coherent case) Reliability is modeled by the pairwise probability of error, bounded by P(X ˆX) const det(x ˆX) 2M. We assume the receiver knows the channel (coherent case).

26 Code design criteria (Coherent case) Reliability is modeled by the pairwise probability of error, bounded by P(X ˆX) const det(x ˆX) 2M. We assume the receiver knows the channel (coherent case). We need called fully diverse codes. det(x X ) 0 X X

27 Code design criteria (Coherent case) Reliability is modeled by the pairwise probability of error, bounded by P(X ˆX) const det(x ˆX) 2M. We assume the receiver knows the channel (coherent case). We need called fully diverse codes. det(x X ) 0 X X We attempt to maximize the minimum determinant min det(x X X X ) 2.

28 Previous work 1. E. Telatar,Capacity of multi-antenna Gaussian channels, V. Tarokh and N. Seshadri and A. R. Calderbank, Space-time codes for high data rate wireless communications: Performance criterion and code construction, B. Hassibi and B.M. Hochwald, High-Rate Codes That Are Linear in Space and Time, H. El Gamal and M.O. Damen, Universal space-time coding, 2003.

29 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.

30 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.

31 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.

32 An example: cyclic division algebras Let Q(i) = {a + ib, a, b Q} information symbols.

33 An example: cyclic division algebras Let Q(i) = {a + ib, a, b Q} information symbols. Let L/Q(i) be a cyclic number field of degree n. A cyclic algebra A is defined as follows A = {(x 0, x 1,..., x n 1 ) x i L}

34 An example: cyclic division algebras Let Q(i) = {a + ib, a, b Q} information symbols. Let L/Q(i) be a cyclic number field of degree n. A cyclic algebra A is defined as follows A = {(x 0, x 1,..., x n 1 ) x i L} with basis {1, e,..., e n 1 } and e n = γ Q(i).

35 An example: cyclic division algebras Let Q(i) = {a + ib, a, b Q} information symbols. Let L/Q(i) be a cyclic number field of degree n. A cyclic algebra A is defined as follows A = {(x 0, x 1,..., x n 1 ) x i L} with basis {1, e,..., e n 1 } and e n = γ Q(i). Think of i 2 = 1.

36 An example: cyclic division algebras Let Q(i) = {a + ib, a, b Q} information symbols. Let L/Q(i) be a cyclic number field of degree n. A cyclic algebra A is defined as follows A = {(x 0, x 1,..., x n 1 ) x i L} with basis {1, e,..., e n 1 } and e n = γ Q(i). Think of i 2 = 1. A non-commutativity rule: λe = eσ(λ), σ : L L the generator of the Galois group of L/Q(i).

37 Cyclic algebras: matrix formulation 1. For n = 2, compute the multiplication by x of any y A: xy = (x 0 + ex 1 )(y 0 + ey 1 ) = x 0 y 0 + eσ(x 0 )y 1 + ex 1 y 0 + γσ(x 1 )y 1 λe = eσ(λ) = [x 0 y 0 + γσ(x 1 )y 1 ] + e[σ(x 0 )y 1 + x 1 y 0 ] e 2 = γ

38 Cyclic algebras: matrix formulation 1. For n = 2, compute the multiplication by x of any y A: xy = (x 0 + ex 1 )(y 0 + ey 1 ) = x 0 y 0 + eσ(x 0 )y 1 + ex 1 y 0 + γσ(x 1 )y 1 λe = eσ(λ) = [x 0 y 0 + γσ(x 1 )y 1 ] + e[σ(x 0 )y 1 + x 1 y 0 ] e 2 = γ 2. In the basis {1, e}, this yields ( x0 γσ(x xy = 1 ) x 1 σ(x 0 ) ) ( y0 y 1 ).

39 Cyclic algebras: matrix formulation 1. For n = 2, compute the multiplication by x of any y A: xy = (x 0 + ex 1 )(y 0 + ey 1 ) = x 0 y 0 + eσ(x 0 )y 1 + ex 1 y 0 + γσ(x 1 )y 1 λe = eσ(λ) = [x 0 y 0 + γσ(x 1 )y 1 ] + e[σ(x 0 )y 1 + x 1 y 0 ] e 2 = γ 2. In the basis {1, e}, this yields ( x0 γσ(x xy = 1 ) x 1 σ(x 0 ) ) ( y0 y 1 ). 3. There is thus a correspondence between x and its multiplication matrix. ( ) x0 γσ(x x = x 0 + ex 1 A 1 ). x 1 σ(x 0 )

40 Cyclic division algebras and encoding Proposition. If γ and its powers γ 2,..., γ n 1 are not a norm, then the cyclic algebra A is a division algebra.

41 Cyclic division algebras and encoding Proposition. If γ and its powers γ 2,..., γ n 1 are not a norm, then the cyclic algebra A is a division algebra. In general x x 0 γσ(x n 1 ) γσ 2 (x n 2 )... γσ n 1 (x 1 ) x 1 σ(x 0 ) γσ 2 (x n 1 )... γσ n 1 (x 2 )... x n 1 σ(x n 2 ) σ 2 (x n 3 )... σ n 1 (x 0 ) Each x i L encodes n information symbols..

42 Summary In order to get good space-time codes for the coherent multiple antenna channel, we follow those steps: 1. Take a cyclic number field L/Q(i) of degree n (# antennas). 2. Construct a cyclic division algebra.

43 Summary In order to get good space-time codes for the coherent multiple antenna channel, we follow those steps: 1. Take a cyclic number field L/Q(i) of degree n (# antennas). 2. Construct a cyclic division algebra. 3. This yields full diversity and a practical encoding, for any n.

44 Summary In order to get good space-time codes for the coherent multiple antenna channel, we follow those steps: 1. Take a cyclic number field L/Q(i) of degree n (# antennas). 2. Construct a cyclic division algebra. 3. This yields full diversity and a practical encoding, for any n. Constructions of codes are available where the algebraic structures are exploited to optimize the codes performance. J.-C. Belfiore, F. E. Oggier, E. Viterbo. Cyclic division algebras: a tool for space-time coding.

45 Summary In order to get good space-time codes for the coherent multiple antenna channel, we follow those steps: 1. Take a cyclic number field L/Q(i) of degree n (# antennas). 2. Construct a cyclic division algebra. 3. This yields full diversity and a practical encoding, for any n. Constructions of codes are available where the algebraic structures are exploited to optimize the codes performance. J.-C. Belfiore, F. E. Oggier, E. Viterbo. Cyclic division algebras: a tool for space-time coding. A 2 2 cyclic algebra based code is to be implemented in the future wireless standard e for wireless LANs.

46 The Rayleigh fading channel Algebraic lattices and minimum discriminant Coherent space-time codes Cyclic division algebras Other examples Current and future work

47 Differential unitary space-time codes (I) Consider the same scenario but with no channel information. How to decode?

48 Differential unitary space-time codes (I) Consider the same scenario but with no channel information. How to decode? We use differential unitary space-time modulation, i.e, S t = X zt S t 1, t = 1, 2,... where {X 0,..., X L 1 } are unitary and S 0 = I.

49 Differential unitary space-time codes (I) Consider the same scenario but with no channel information. How to decode? We use differential unitary space-time modulation, i.e, S t = X zt S t 1, t = 1, 2,... where {X 0,..., X L 1 } are unitary and S 0 = I. If we assume the channel is roughly constant, we have Y t = S t H + W t = X zt S t 1 H + W t = X zt (Y t 1 W t 1 ) + W t = X zt Y t 1 + W t.

50 Differential unitary space-time codes (I) Consider the same scenario but with no channel information. How to decode? We use differential unitary space-time modulation, i.e, S t = X zt S t 1, t = 1, 2,... where {X 0,..., X L 1 } are unitary and S 0 = I. If we assume the channel is roughly constant, we have Y t = S t H + W t = X zt S t 1 H + W t = X zt (Y t 1 W t 1 ) + W t = X zt Y t 1 + W t. The matrix H does not appear in the last equation.

51 Differential unitary space-time codes (II) 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.

52 Differential unitary space-time codes (II) 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. 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.

53 Wireless networks (I) Y = [A 1 s,..., A R s]h + W The relays and the transmitter cooperate to encode the signal.

54 Wireless networks (II) Distributed space-time coding requires: 1. fully-diverse codewords [A 1 s,..., A R s] 2. where the matrices A i are unitary.

55 Wireless networks (II) Distributed space-time coding requires: 1. fully-diverse codewords [A 1 s,..., A R s] 2. where the matrices A i are unitary. The first algebraic strategies have been recently proposed, for one or multiple antennas.

56 Some open questions 1. Coherent space-time coding: Find large families of fully-diverse matrices with large determinant. Investigate crossed product algebras.

57 Some open questions 1. Coherent space-time coding: Find large families of fully-diverse matrices with large determinant. Investigate crossed product algebras. 2. Differential space-time coding: Find large families of unitary fully-diverse matrices. Investigate Lie algebras. 3. Wireless networks, coherent and non-coherent cases

58 Thank you for your attention!