On the Use of Division Algebras for Wireless Communication

Transcription

1 On the Use of Division Algebras for Wireless Communication California Institute of Technology AMS meeting, Davidson, March 3rd 2007

2 Outline A few wireless coding problems Space-Time Coding Differential Space-Time Coding Distributed Space-Time Coding Introducing Division Algebras Codewords from Division Algebras

3 Space-Time Coding Multiple antenna coding: the model

4 Space-Time Coding Multiple antenna coding: the model x 1 h 11 h 11 x 1 + h 12 x 3 + n 1 h 21 x 3 h 12 h 21 x 1 + h 22 x 3 + n 2 h 22

5 Space-Time Coding Multiple antenna coding: the model x 2 h 21 h 11 x 2 + h 12 x 4 + n 3 h 11 h 11 x 1 + h 12 x 3 + n 1 x 4 h 12 h 21 x 2 + h 22 x 4 + n 4 h 22 h 21 x 1 + h 22 x 3 + n 2

6 Space-Time Coding Multiple antenna coding: the coding problem We summarize the channel as ( ) ( ) h11 h Y = 12 x1 x 2 +W, W, H complex Gaussian h 21 h 22 x 3 x 4 }{{} space-time codeword The goal is the design of the codebook C: { ( ) } x1 x C = X = 2 x x 3 x 1, x 2, x 3, x 4 C 4 the x i are functions of the information symbols.

7 Space-Time Coding Multiple antenna coding: the coding problem We summarize the channel as ( ) ( ) h11 h Y = 12 x1 x 2 +W, W, H complex Gaussian h 21 h 22 x 3 x 4 }{{} space-time codeword The goal is the design of the codebook C: { ( ) } x1 x C = X = 2 x x 3 x 1, x 2, x 3, x 4 C 4 the x i are functions of the information symbols.

8 Space-Time Coding The code design The pairwise probability of error of sending X and decoding ˆX X is upper bounded by P(X ˆX) const det(x ˆX) 2M, where the receiver knows the channel (coherent case). Find a family C of M M matrices such that called fully-diverse. det(x i X j ) 0, X i X j C,

9 Space-Time Coding The code design The pairwise probability of error of sending X and decoding ˆX X is upper bounded by P(X ˆX) const det(x ˆX) 2M, where the receiver knows the channel (coherent case). Find a family C of M M matrices such that called fully-diverse. det(x i X j ) 0, X i X j C,

10 Differential Space-Time Coding The differential noncoherent MIMO channel We assume no channel knowledge. We use differential unitary space-time modulation. that is (assuming S 0 = I) S t = X zt S t 1, t = 1, 2,..., where z t {0,..., L 1} is the data to be transmitted, and C = {X 0,..., X L 1 } the constellation to be designed. The matrices X have to be unitary.

11 Differential Space-Time Coding The differential noncoherent MIMO channel We assume no channel knowledge. We use differential unitary space-time modulation. that is (assuming S 0 = I) S t = X zt S t 1, t = 1, 2,..., where z t {0,..., L 1} is the data to be transmitted, and C = {X 0,..., X L 1 } the constellation to be designed. The matrices X have to be unitary.

12 Differential Space-Time Coding The differential noncoherent MIMO channel We assume no channel knowledge. We use differential unitary space-time modulation. that is (assuming S 0 = I) S t = X zt S t 1, t = 1, 2,..., where z t {0,..., L 1} is the data to be transmitted, and C = {X 0,..., X L 1 } the constellation to be designed. The matrices X have to be unitary.

13 Differential Space-Time Coding Decoding and probability of error 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, H does not appear! The pairwise probability of error P e has the upper bound P e ( ) ( ) 1 8 MN 1 2 ρ det(x i X j ) 2N We need to design unitary fully-diverse matrices.

14 Differential Space-Time Coding Decoding and probability of error 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, H does not appear! The pairwise probability of error P e has the upper bound P e ( ) ( ) 1 8 MN 1 2 ρ det(x i X j ) 2N We need to design unitary fully-diverse matrices.

15 Differential Space-Time Coding Decoding and probability of error 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, H does not appear! The pairwise probability of error P e has the upper bound P e ( ) ( ) 1 8 MN 1 2 ρ det(x i X j ) 2N We need to design unitary fully-diverse matrices.

16 Differential Space-Time Coding Decoding and probability of error 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, H does not appear! The pairwise probability of error P e has the upper bound P e ( ) ( ) 1 8 MN 1 2 ρ det(x i X j ) 2N We need to design unitary fully-diverse matrices.

17 Distributed Space-Time Coding Wireless relay network: model A transmitter and a receiver node. Relay nodes are small devices with few resources. M antennas Tx Rx N antennas

18 Sf 1 + v 1 = r 1 Sf 2 + v 2 = r 2 Distributed Space-Time Coding Wireless relay network: phase 1 S Tx Sf 3 + v 3 = r 3 Rx Sf 4 + v 4 = r 4

19 Sf 1 + v 1 = r 1 Sf 2 + v 2 = r 2 Distributed Space-Time Coding Wireless relay network: phase 2 At each node: multiply by a unitary matrix. A 1 r 1 A 2 r 2 S Tx Sf 3 + v 3 = r 3 A 3 r 3 A 4 r 4 Rx Sf 4 + v 4

20 Distributed Space-Time Coding Channel model 1. At the receiver, y n = R g in t i + w = i=1 R g in A i (Sf i + v i ) + w i=1 2. So that finally Y = y 1. y n = [A 1 S A R S] }{{} X f 1 g 1. +W } f n g n {{ } H 3. Each relay encodes a set of columns, so that the encoding is distributed among the nodes.

21 Distributed Space-Time Coding Channel model 1. At the receiver, y n = R g in t i + w = i=1 R g in A i (Sf i + v i ) + w i=1 2. So that finally Y = y 1. y n = [A 1 S A R S] }{{} X f 1 g 1. +W } f n g n {{ } H 3. Each relay encodes a set of columns, so that the encoding is distributed among the nodes.

22 Introducing Division Algebras A few wireless coding problems Space-Time Coding Differential Space-Time Coding Distributed Space-Time Coding Introducing Division Algebras Codewords from Division Algebras

23 Introducing Division Algebras 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.

24 Introducing Division Algebras 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.

25 Introducing Division Algebras 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.

26 Introducing Division Algebras An example: cyclic division algebras Let Q(i) = {a + ib, a, b Q}. Let L be cyclic extension of degree n over Q(i). 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).

27 Introducing Division Algebras An example: cyclic division algebras Let Q(i) = {a + ib, a, b Q}. Let L be cyclic extension of degree n over Q(i). 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).

28 Introducing Division Algebras An example: cyclic division algebras Let Q(i) = {a + ib, a, b Q}. Let L be cyclic extension of degree n over Q(i). 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).

29 Introducing Division Algebras An example: cyclic division algebras Let Q(i) = {a + ib, a, b Q}. Let L be cyclic extension of degree n over Q(i). 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).

30 Introducing Division Algebras An example: cyclic division algebras Let Q(i) = {a + ib, a, b Q}. Let L be cyclic extension of degree n over Q(i). 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).

31 Codewords from Division Algebras 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 ) y0 x 1 σ(x 0 ) 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 )

32 Codewords from Division Algebras 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 ) y0 x 1 σ(x 0 ) 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 )

33 Codewords from Division Algebras 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 ) y0 x 1 σ(x 0 ) 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 )

34 Codewords from Division Algebras 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..

35 Codewords from Division Algebras 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..

36 Codewords from Division Algebras Solutions for the coding problems Start with a cyclic division algebra, and: 1. For space-time coding: use the underlying algebraic properties to optimize the code (for example the discriminant of L/Q(i)). 2. For differential space-time coding: endowe the algebra with a suitable involution, or use the Cayley transform. 3. For distributed space-time coding: work in a suitable subfield of L.

37 Codewords from Division Algebras Solutions for the coding problems Start with a cyclic division algebra, and: 1. For space-time coding: use the underlying algebraic properties to optimize the code (for example the discriminant of L/Q(i)). 2. For differential space-time coding: endowe the algebra with a suitable involution, or use the Cayley transform. 3. For distributed space-time coding: work in a suitable subfield of L.

38 Codewords from Division Algebras Solutions for the coding problems Start with a cyclic division algebra, and: 1. For space-time coding: use the underlying algebraic properties to optimize the code (for example the discriminant of L/Q(i)). 2. For differential space-time coding: endowe the algebra with a suitable involution, or use the Cayley transform. 3. For distributed space-time coding: work in a suitable subfield of L.

39 Codewords from Division Algebras Thank you for your attention!