interference calculus a general framework for interference management and network utility optimization

Foundations in Signal Processing,
Communications and Networking
Series Editors: W. Utschick, H. Boche, R. Mathar
Martin Schubert · Holger Boche
Interference Calculus
A General Framework for Interference
Management and Network Utility
Optimization
ABC
Series Editors:
Wolfgang Utschick
TU Munich
Associate Institute for Signal
Processing
Arcisstrasse 21
80290 Munich, Germany
Rudolf Mathar
RWTH Aachen University
Institute of Theoretical
Information Technology
52056 Aachen, Germany
Holger Boche
TU Munich
Institute of Theoretical Information Technology
Arcisstrasse 21 80290 Munich,
Germany
Authors:
Martin Schubert
Heinrich Hertz Institute for

Telecommunications HHI Einsteinufer 37
10587 Berlin
Germany
E-mail:
Holger Boche
TU Munich
Institute of Theoretical Information Technology
Arcisstrasse 21 80290 Munich,
Germany
E-mail:
ISSN 1863-8538 e-ISSN 1863-8546
ISBN 978-3-642-24620-3 e-ISBN 978-3-642-24621-0
DOI 10.1007/978-3-642-24621-0
Springer Dordrecht Heidelberg London New York
Library of Congress Control Number: 2011941485
c
 Springer-Verlag Berlin Heidelberg 2012
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M. Schubert, H. Boche, Interference Calculus, Foundations in Signal Processing,
Communications and Networking 7,
© Springer-Verlag Berlin Heidelberg 201 2
• R
+
R
++
R
K
Kg
•
• y y
l
= [y]
l
l A
mn
=
[A]
mn
A

• x > y x
k
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k
k
• y > 0
• y ≥ x y
l
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l
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y
l
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l
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• exp(y) log(y)
r p r K
r = p p K
u
K
u
I(r)
J(p)
p p = [p
1
, . . . , p
K
u
, σ

2
n
]
T
σ
2
n
σ
2
n
= 1
K K
u
K K
u
SINR
γ
Γ Γ = diag{γ}
V V = [v
1
, . . . , v
K
u
]
T
v
k
k
W
I : R

K
+
→ R
+
I
r > 0 I(r) > 0
I(αr) = αI(r) α ≥ 0
I(r) ≥ I(r

) r ≥ r

r
I(r)
I(r)
r
r
I(r) ≥ 0
r ≥ 0 I(r) < 0
0 < λ ≤ 1 0 > I(r) ≥ I(λr) = λI(r) λ → 0
r > 0 I(r) > 0
I(r) > 0 r > 0
I(r) = 0
r > 0
I : R
K
+
→ R
+
I R
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+
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+
I R
K
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f(s) s ∈ R
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log f(s)
f

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≤
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f(ˆs)

1−λ
·

f(ˇs)
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λ
, ˆs, ˇs ∈ R
K
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r = e

s
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f(ˇs)
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, ∀λ ∈ (0, 1); ˆs, ˇs ∈ R
K
,
s(λ) = (1 − λ)ˆs + λˇs , λ ∈ (0, 1) .
r(λ) = exp s(λ)
r(λ) = ˆr
(1−λ)
· ˇr
λ
r = exp{s}
I

k
(e
s
)
I
k

r(λ)
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I
k
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
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·

I
k
(ˇr)

λ
, λ ∈ (0, 1) .
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s
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1
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K
u
]
T
∈ R
K
u
+
.
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k
k
(p) =
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I
k
(p) k
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1
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K

u
p
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p
1
.
.
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1
=
p
1
I
1
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.
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SIR
K
=
p
L
I
K
(p)
interference
K
u

k
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k
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v
k
, k ∈ K
u
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v
k
∈ R
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u
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V = [v
1
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K
u
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T
,
I
k
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k

, k ∈ K
u
.
I
k
σ
2
n
p = [p
1
, . . . , p
K
u
, σ
2
n
]
T
∈ R
K
u
+1
+
I
k
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·

v

k
1

= p
T
v
k
+ σ
2
n
.
p
K
u
M S
1
, . . . , S
K
u
p
k
=
E
[|S
k
|
2
]
h
1

, . . . , h
K
u
∈ C
M
R
k
=
E
[h
k
h
H
k
]
u
1
, . . . , u
K
u
k
y
k
= u
H
k


l∈K
u

h
l
S
l
+ n

,
n ∈ C
M
E
[nn
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n
I k
SINR
k
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k
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E
[|u
H
k
h
k
S
k
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2
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E
[|

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2
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p
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k
R
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u
k
u

H
k
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l
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k
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l
R
l
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
u
k
u
H
k
R

k
u
k
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k
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2
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l=k
p
l
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H
k
R
l
u
k
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2
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k

2
u
H
k

R
k
u
k
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k
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2
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p
T
v
k
(u
k
) ,
v
k
(u
k
)
[v
k
(u
k
)]
l
=








u
H
k
R
l
u
k
u
H
k
R
k
u
k
1 ≤ l ≤ K
u
, l = k
u
k

2
u
H
k

R
k
u
k
l = K
u
+ 1,
0 l = k .
p > 0 u
k
h
1
, . . . , h
K
u
R
l
= h
l
h
H
l
I
k
(p) =
1
h
H
k


σ
2
n
I +

l=k
p
l
h
l
h
H
l

−1
h
k
.
k z
k
Z
k
z
k
v
k
(z
k
) ∈ R
K

u
+
z
k
v
k
z
k
p z
k
p
T
v
k
(z
k
)
I
k
(p) = min
z
k
∈Z
k
p
T
v
k
(z
k

) , ∀k ∈ K
u
.
I
k
(p) = min
z
k
∈Z
k
p
T
v
k
(z
k
) , ∀k ∈ K
u
.
u
k
u
k

2
= 1
Z
k
Z
k

I
k
(p) = [V p]
k
V
1
(p), . . . ,
K
u
(p)
γ
k
k
K
u
γ = [γ
1
, γ
2
, . . . , γ
K
u
]
T
∈ R
K
u
++
.
γ γ

ρ
V
(γ) = ρ(Γ V ) , Γ = diag{γ} .
ρ
V
ρ
V
(γ) ≤ 1 p > 0
k
(p) ≥ γ
k
k ∈ K
u
S = {γ > 0 : ρ
V
(γ) ≤ 1} .
ρ
V
(γ) γ
K
u
ρ
V
(γ)
S
ρ
V
(γ)
S
S

ρ
V
(γ)
ρ
V
(exp q) R
K
u
γ = exp q S
ρ
V
(exp q)
C(γ)
I
1
, . . . , I
K
u
p > 0
k
(p) ≥ γ
k
k ∈ K
u
max
k∈K
u
γ
k
k

(p)
= max
k∈K
u
γ
k
· I
k
(p)
p
k
≤ 1
C(γ) γ
C(γ) = inf
p>0

max
k∈K
u
γ
k
· I
k
(p)
p
k

.
γ > 0 C(γ) ≤ 1 C(γ) = 1
γ