On the Use of Division Algebras for Wireless Communication

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Nov 21, 2013 (3 years and 9 months ago)

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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