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
SpaceTime Coding
Diﬀerential SpaceTime Coding
Distributed SpaceTime 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
SpaceTime 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
SpaceTime 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
SpaceTime 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
SpaceTime 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
spacetime 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
SpaceTime 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
spacetime 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
SpaceTime 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
fullydiverse
.
On the Use of Division Algebras for Wireless Communication
Fr´ed´erique Oggier
A few wireless coding problems
Division algebras
SpaceTime 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
fullydiverse
.
On the Use of Division Algebras for Wireless Communication
Fr´ed´erique Oggier
A few wireless coding problems
Division algebras
Diﬀerential SpaceTime Coding
The diﬀerential noncoherent MIMO channel
We assume
no channel knowledge
.
We use
diﬀerential unitary spacetime 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 SpaceTime Coding
The diﬀerential noncoherent MIMO channel
We assume
no channel knowledge
.
We use
diﬀerential unitary spacetime 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 SpaceTime Coding
The diﬀerential noncoherent MIMO channel
We assume
no channel knowledge
.
We use
diﬀerential unitary spacetime 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 SpaceTime 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 fullydiverse
matrices.
On the Use of Division Algebras for Wireless Communication
Fr´ed´erique Oggier
A few wireless coding problems
Division algebras
Diﬀerential SpaceTime 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 fullydiverse
matrices.
On the Use of Division Algebras for Wireless Communication
Fr´ed´erique Oggier
A few wireless coding problems
Division algebras
Diﬀerential SpaceTime 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 fullydiverse
matrices.
On the Use of Division Algebras for Wireless Communication
Fr´ed´erique Oggier
A few wireless coding problems
Division algebras
Diﬀerential SpaceTime 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 fullydiverse
matrices.
On the Use of Division Algebras for Wireless Communication
Fr´ed´erique Oggier
A few wireless coding problems
Division algebras
Distributed SpaceTime 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 SpaceTime 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 SpaceTime 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 SpaceTime 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 SpaceTime 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
SpaceTime Coding
Diﬀerential SpaceTime Coding
Distributed SpaceTime 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
nonlinearity
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 noncommutative 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
nonlinearity
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 noncommutative 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
nonlinearity
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 noncommutative 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
noncommutativity 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
noncommutativity 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
noncommutativity 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
noncommutativity 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
noncommutativity 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
spacetime coding
:use the underlying algebraic properties
to optimize the code (for example the discriminant of L/Q(i )).
2.
For
diﬀerential spacetime coding
:endowe the algebra with a
suitable
involution
,or use the
Cayley transform
.
3.
For
distributed spacetime 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
spacetime coding
:use the underlying algebraic properties
to optimize the code (for example the discriminant of L/Q(i )).
2.
For
diﬀerential spacetime coding
:endowe the algebra with a
suitable
involution
,or use the
Cayley transform
.
3.
For
distributed spacetime 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
spacetime coding
:use the underlying algebraic properties
to optimize the code (for example the discriminant of L/Q(i )).
2.
For
diﬀerential spacetime coding
:endowe the algebra with a
suitable
involution
,or use the
Cayley transform
.
3.
For
distributed spacetime 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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