A few wireless coding problems

Division algebras

On the Use of Division Algebras

for Wireless Communication

Fr´ed´erique Oggier

frederique@systems.caltech.edu

California Institute of Technology

AMS meeting,Davidson,March 3rd 2007

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

Outline

A few wireless coding problems

Space-Time Coding

Diﬀerential Space-Time Coding

Distributed Space-Time Coding

Division algebras

Introducing Division Algebras

Codewords from Division Algebras

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

Space-Time Coding

Multiple antenna coding:the model

1

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

Space-Time Coding

Multiple antenna coding:the model

x

1

x

3

h

11

h

11

x

1

+h

12

x

3

+n

1

h

21

x

1

+h

22

x

3

+n

2

h

21

h

12

h

22

1

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

Space-Time Coding

Multiple antenna coding:the model

x

2

x

4

h

11

h

11

x

2

+h

12

x

4

+n

3

h

21

x

2

+h

22

x

4

+n

4

h

21

h

12

h

22

h

11

x

1

+h

12

x

3

+n

1

h

21

x

1

+h

22

x

3

+n

2

1

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

Space-Time Coding

Multiple antenna coding:the coding problem

We summarize the channel as

Y =

h

11

h

12

h

21

h

22

x

1

x

2

x

3

x

4

space-time codeword

+W,W,H complex Gaussian

The goal is the design of the

codebook

C:

C =

X =

x

1

x

2

x

3

x

4

|x

1

,x

2

,x

3

,x

4

∈ C

the x

i

are functions of the

information symbols

.

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

Space-Time Coding

Multiple antenna coding:the coding problem

We summarize the channel as

Y =

h

11

h

12

h

21

h

22

x

1

x

2

x

3

x

4

space-time codeword

+W,W,H complex Gaussian

The goal is the design of the

codebook

C:

C =

X =

x

1

x

2

x

3

x

4

|x

1

,x

2

,x

3

,x

4

∈ C

the x

i

are functions of the

information symbols

.

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

Space-Time Coding

The code design

The

pairwise probability of error

of sending X and decoding

ˆ

X = X is upper bounded by

P(X →

ˆ

X) ≤

const

| det(X−

ˆ

X)|

2M

,

where the receiver knows the channel (

coherent case

).

Find a family C of M ×M matrices such that

det(X

i

−X

j

) = 0,X

i

= X

j

∈ C,

called

fully-diverse

.

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

Space-Time Coding

The code design

The

pairwise probability of error

of sending X and decoding

ˆ

X = X is upper bounded by

P(X →

ˆ

X) ≤

const

| det(X−

ˆ

X)|

2M

,

where the receiver knows the channel (

coherent case

).

Find a family C of M ×M matrices such that

det(X

i

−X

j

) = 0,X

i

= X

j

∈ C,

called

fully-diverse

.

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

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

z

t

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

.

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

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

z

t

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

.

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

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

z

t

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

.

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

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

z

t

S

t−1

H+W

t

= X

z

t

(Y

t−1

−W

t−1

) +W

t

= X

z

t

Y

t−1

+W

t

,H does

not

appear!

The

pairwise probability of error

P

e

has the upper bound

P

e

≤

1

2

8

ρ

MN

1

| det(X

i

−X

j

)|

2N

We need to design

unitary fully-diverse

matrices.

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

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

z

t

S

t−1

H+W

t

= X

z

t

(Y

t−1

−W

t−1

) +W

t

= X

z

t

Y

t−1

+W

t

,H does

not

appear!

The

pairwise probability of error

P

e

has the upper bound

P

e

≤

1

2

8

ρ

MN

1

| det(X

i

−X

j

)|

2N

We need to design

unitary fully-diverse

matrices.

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

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

z

t

S

t−1

H+W

t

= X

z

t

(Y

t−1

−W

t−1

) +W

t

= X

z

t

Y

t−1

+W

t

,H does

not

appear!

The

pairwise probability of error

P

e

has the upper bound

P

e

≤

1

2

8

ρ

MN

1

| det(X

i

−X

j

)|

2N

We need to design

unitary fully-diverse

matrices.

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

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

z

t

S

t−1

H+W

t

= X

z

t

(Y

t−1

−W

t−1

) +W

t

= X

z

t

Y

t−1

+W

t

,H does

not

appear!

The

pairwise probability of error

P

e

has the upper bound

P

e

≤

1

2

8

ρ

MN

1

| det(X

i

−X

j

)|

2N

We need to design

unitary fully-diverse

matrices.

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

Distributed Space-Time Coding

Wireless relay network:model

A transmitter and a receiver node.

Relay nodes are small devices with

few

resources.

Tx

Rx

N antennas

M antennas

1

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

Distributed Space-Time Coding

Wireless relay network:phase 1

Tx

Rx

S

Sf

1

+v

1

= r

1

Sf

2

+v

2

= r

2

Sf

3

+v

3

= r

3

Sf

4

+v

4

= r

4

1

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

Distributed Space-Time Coding

Wireless relay network:phase 2

At each node:multiply by a

unitary

matrix.

Tx

Rx

S

Sf

1

+v

1

= r

1

Sf

2

+v

2

= r

2

Sf

3

+v

3

= r

3

Sf

4

+v

4

A

4

r

4

A

3

r

3

A

1

r

1

A

2

r

2

1

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

Distributed Space-Time Coding

Channel model

1.

At the receiver,

y

n

=

R

i =1

g

in

t

i

+w =

R

i =1

g

in

A

i

(Sf

i

+v

i

) +w

2.

So that ﬁnally

Y =

y

1

.

.

.

y

n

= [A

1

S∙ ∙ ∙ A

R

S]

X

f

1

g

1

.

.

.

f

n

g

n

H

+W

3.

Each relay encodes a set of columns,so that the encoding is

distributed

among the nodes.

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

Distributed Space-Time Coding

Channel model

1.

At the receiver,

y

n

=

R

i =1

g

in

t

i

+w =

R

i =1

g

in

A

i

(Sf

i

+v

i

) +w

2.

So that ﬁnally

Y =

y

1

.

.

.

y

n

= [A

1

S∙ ∙ ∙ A

R

S]

X

f

1

g

1

.

.

.

f

n

g

n

H

+W

3.

Each relay encodes a set of columns,so that the encoding is

distributed

among the nodes.

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

Introducing Division Algebras

A few wireless coding problems

Space-Time Coding

Diﬀerential Space-Time Coding

Distributed Space-Time Coding

Division algebras

Introducing Division Algebras

Codewords from Division Algebras

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

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.

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

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.

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

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.

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

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 ).

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

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 ).

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

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 ).

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

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 ).

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

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 ).

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

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

xy =

x

0

γσ(x

1

)

x

1

σ(x

0

)

y

0

y

1

.

3.

There is thus a correspondence between x and its

multiplication matrix

.

x = x

0

+ex

1

∈ A ↔

x

0

γσ(x

1

)

x

1

σ(x

0

)

.

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

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

xy =

x

0

γσ(x

1

)

x

1

σ(x

0

)

y

0

y

1

.

3.

There is thus a correspondence between x and its

multiplication matrix

.

x = x

0

+ex

1

∈ A ↔

x

0

γσ(x

1

)

x

1

σ(x

0

)

.

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

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

xy =

x

0

γσ(x

1

)

x

1

σ(x

0

)

y

0

y

1

.

3.

There is thus a correspondence between x and its

multiplication matrix

.

x = x

0

+ex

1

∈ A ↔

x

0

γσ(x

1

)

x

1

σ(x

0

)

.

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

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.

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

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.

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

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.

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

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.

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

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.

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

A few wireless coding problems

Division algebras

Codewords from Division Algebras

Thank you for your attention!

On the Use of Division Algebras for Wireless Communication

Fr´ed´erique Oggier

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