National Chiao
Tung University
Taiwan
Dept. of Electrical
and Computer
Engineering
2013 IEEE Taiwan/Hong Kong Joint Workshop
on Information Theory and Communications
The Additive Inverse Gaussian Noise Channel:
an Attempt on Modeling Molecular
Communication
Stefan M. Moser
(joint work with Chang Hui-Ting)
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
1
Nano Devices in Fluid Medium (1)
v
• nano devices communicate by exchange of molecular particles
• fluid medium flows with constant speed v
• particles suffer from Brownian motion
• information is encoded in release time
=⇒ fundamentally different channel behavior
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
2
Nano Devices in Fluid Medium (2)
• new channel model by Srinivas, Adve, and Eckford [1]
• simplifying assumptions:
– perfectly synchronized common clock
– no stray particles
– channel is memoryless, i.e., trajectories are independent
– once arrived at receiver, particle is absorbed
– receiver can arrange the arriving molecules in correct order of
release
– one-dimensional setup (generalization shouldn’t be too hard)
[1] K. V. Srinivas, R. S. Adve, and A. W. Eckford, “Molecular communication in
fluid media: The additive inverse Gaussian noise channel,” IEEE Transactions on
Information Theory, vol. 58, no. 7, pp. 4678–4692, July 2012.
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
3
Channel Model (1)
• Brownian motion is modeled as Wiener process: for any time
interval τ , the position increment is Gaussian:
2
∆W ∼ N vτ , σ τ
– σ 2 parameter depending on type of fluid, type of particle,
temperature, etc.
– v is constant drift velocity of fluid
– increments of nonoverlapping time intervals are independent
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
4
Channel Model (2)
• we need to invert problem:
– transmitter at fixed position w = 0
– receiver at fixed position w = d (normalized to d = 1)
– random travel time N
=⇒ “inverse Gaussian distribution”
N ∼ IG(µ, λ)
with PDF
fN (n) =
r
λ
λ(n − µ)
exp
−
2πn3
2µ2 n
– average travel time: µ =
2
I{n > 0}
d
v
d2
– Brownian motion parameter: λ = 2
σ
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
5
Channel Model (3)
• channel input: release time
X
• channel output: arrival time
Y =X +N
• noise:
N ∼ IG(µ, λ)
• input and noise independent:
X⊥
⊥N
=⇒ additive inverse Gaussian noise (AIGN) channel
• channel law:
fY |X (y|x) =
s
λ
λ(y − x − µ)
exp −
2π(y − x)3
2µ2 (y − x)
• implicit nonnegativity constraint
• average-delay constraint
2
I{y > x}
X≥0
E[X] ≤ m
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
6
Properties of IG(µ, λ) (1)
For
N ∼ IG(µ, λ)
we have
E[N ] = µ
µ3
Var(N ) =
λ
3
2πeµ
1
3 2λ
2λ
µ
h(N ) = log
+ e Ei −
2
λ
2
µ
(Ei(·) exponential integral function)
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
7
Properties of IG(µ, λ) (2)
For
N1 ∼ IG(µ1 , λ1 )
N2 ∼ IG(µ2 , λ2 )
N1 ⊥
⊥ N2
we have
N1 + N2 ∼ IG µ1 + µ2 ,
only if
p
λ1 +
p
λ2
2 λ1
λ2
= 2
µ21
µ2
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
8
First Lower Bound on Capacity (1)
• Lower Bound: choose X ∼ IG m, λm
µ2
C=
max
2
:
I(X; Y )
PX : E[X]≤m
=
max
h(Y ) − h(Y |X)
PX : E[X]≤m
=
max
h(Y ) − h(N )
PX : E[X]≤m
max
h(Y ) − h(N )
≥ h(Y )X∼IGm, λm2 − h(N )
=
PX : E[X]≤m
µ2
3
2πeµ
3
1
log
+ e
2
λ
2
=?
h(N ) =
h(Y )
X∼IG
2
m, λm
2
µ
2λ
µ
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
Ei −
2λ
µ
9
First Lower Bound on Capacity (2)
• Lower Bound: choose X ∼ IG m,
note:
=⇒
λm2
µ2
m2
=
λm2
µ2
: (continued)
λ
µ2
Y =X+
N
2
λm
∼ IG m, 2
+ IG(µ, λ)
µ

!2 
√
λm √

+ λ 
= IG m + µ,
µ
2
λ(m + µ)
= IG m + µ,
µ2
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
10
First Lower Bound on Capacity (3)
• Lower Bound: choose X ∼ IG m,
C=
max
λm2
µ2
: (continued)
I(X; Y )
=
max
h(Y ) − h(Y |X)
PX : E[X]≤m
=
max
h(Y ) − h(N )
PX : E[X]≤m
PX : E[X]≤m
max
h(Y ) − h(N )
PX : E[X]≤m
≥ h(Y )X∼IGm, λm2 − h(N )
=
µ2
3
X∼IG
2
m, λm
2
µ
2λ
1
2πeµ
3 2λ
log
+ e µ Ei −
2
λ
2
µ
2
2λ(m+µ)
2πeµ (m + µ) 3
1
2λ(m + µ)
2
µ
+ e
= log
Ei −
2
λ
2
µ2
h(N ) =
h(Y )
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
11
First Upper Bound on Capacity (1)
• Upper Bound:
C=
=
=
max
PX : E[X]≤m
max
PX : E[X]≤m
max
PX : E[X]≤m
I(X; Y )
h(Y ) − h(Y |X)
h(Y ) − h(N )
Note:
E[Y ] = E[X] + E[N ] ≤ m + µ
Under mean constraint:
h(Y ) maximized by exponential distribution
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
12
First Upper Bound on Capacity (2)
• Upper Bound: h(Y ) ≤ h(exponential):
h(Y ) ≤ log e(m + µ)
Hence:
C=
max
I(X; Y )
PX : E[X]≤m
=
max
h(Y ) − h(Y |X)
PX : E[X]≤m
=
max
PX : E[X]≤m
h(Y ) − h(N )
≤ log e(m + µ) − h(N )
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
13
Summary First Bounds (1)
• Upper Bound: [1]:
2
1
λe(m + µ)
3 2λ
2λ
µ
C ≤ log
− e Ei −
2
2πµ3
2
µ
• Lower Bound: [1]:
C≥
m+µ 3
1
log
+ e
2
µ
2
2λ(m+µ)
µ2
Ei −
2λ(m + µ)
µ2
3 2λ
2λ
− e µ Ei −
2
µ
[1] K. V. Srinivas, R. S. Adve, and A. W. Eckford, “Molecular communication in
fluid media: The additive inverse Gaussian noise channel,” IEEE Transactions on
Information Theory, vol. 58, no. 7, pp. 4678–4692, July 2012.
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
14
Summary First Bounds (2)
6
Capacity C [bits]
5
4
3
2
1
ZOOM IN
0
0
1
2
3
4
5
6
7
8
9
10
Delay m
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
15
Summary First Bounds (2)
ZOOMED
0.5
0.45
Capacity C [bits]
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Delay m
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
16
Summary First Bounds (3)
6
5
Capacity C [bits]
ZOOM IN
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
Drift velocity v
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
17
Summary First Bounds (3)
ZOOMED
5
4.5
Capacity C [bits]
4
3.5
3
2.5
2
1.5
1
0.5
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Drift velocity v
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
18
Idea for New Upper Bounds
• Duality-based upper bound:
C ≤ EQ∗ D fY |X (·|X) R(·)
for arbitrary choice of R(·):
– R(·) exponential: (α > 0)
R(y) = αe−αy
=⇒ yields first upper bound
– R(·) power inverse Gaussian: (α, β > 0, η 6= 0)


!
η
η
η
2
2 2
1+ 2
r
α
y
β
β
α


R(y) =
exp − 2
−
3
2πβ
y
2η β
β
y
(additional bounding required)
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
19
New Upper Bounds on Capacity
1
λm
3
1
1
log 1 +
+ log 1 + m
+
2
µ(m + µ)
2
µ λ
2λ
1
η−1
2λ
µ
C ≤ log λ +
log µ + e Ei −
2
2
µ
!
r
1
2λ −η− 1 µλ
λ
−η
2e K
1
+ log
µ
−
(m
+
µ)
η+ 2
2
π
µ
1
1
η+2
log 1 + m
+
− log η
+
2
µ λ
2λ
1
λ 1
1
1
2λ
C ≤ log + log 1 + m
+
− e µ Ei −
2
µ 2
µ λ
µ
m
λ
1
−
+ log 1 +
2
µ
µ+λ
C≤
(0 < η ≤ 1, Kν (·) modified Bessel function of second kind)
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
20
Idea for New Lower Bounds
• Choose exponential input:
1
X ∼ Exp
m
– PDF of Y : convolution of exponential and inverse Gaussian [2]:
p
√
1 − my + µλ −kλ
1
√
Y ∼ e
ky −
e
Q − kλ
m
ky
p
√
1
kλ
ky + √
+ ekλ Q
ky
[2] W. Schwarz, “On the convolution of inverse Gaussian and exponential random
variables,” Communications in Statistics — Theory and Methods, vol. 31,
no. 12, pp. 2113–2121, 2002.
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
21
New Lower Bound on Capacity
m
µ
λ
3
λ 3 2λ
1
2π
2λ
+
− + kλ + log − e µ Ei −
− log
λ
m µ
2
µ 2
µ
2
e
!
r
r
λm
2λ
1 µλ
2 λ2
K
+
k
− log 1 + e
1
m
2 + k 2 λm
m
!!
r
r
λm
λ
1 µλ +kλ
2 λ2
2
e
K
+
k
+
1
2m
1 + k 2 λm
m
C ≥ log
where
r
1
2
k,
−
2
µ
mλ
2µ2
m≥
λ
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
22
New Bounds on Capacity (1)
6
Capacity C [bits]
5
4
3
2
known bounds: black
1
0
0
1
2
3
4
5
6
7
8
9
10
Delay m
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
23
New Bounds on Capacity (1)
6
Capacity C [bits]
5
4
3
2
known bounds: black
our new bounds: colored
1
ZOOM IN
0
0
1
2
3
4
5
6
7
8
9
10
Delay m
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
24
New Bounds on Capacity (1)
ZOOMED
0.5
0.45
Capacity C [bits]
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Delay m
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
25
New Bounds on Capacity (2)
6
Capacity C [bits]
5
4
3
2
known bounds: black
1
0
0
1
2
3
4
5
6
7
8
9
10
Drift velocity v
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
26
Our Results: New Bounds on Capacity (2)
6
Capacity C [bits]
5
ZOOM IN
4
3
2
known bounds: black
our new bounds: colored
1
0
0
1
2
3
4
5
6
7
8
9
10
Drift velocity v
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
27
New Bounds on Capacity (2)
ZOOMED
5
4.5
Capacity C [bits]
4
3.5
3
2.5
2
1.5
1
0.5
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Drift velocity v
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
28
Exact Asymptotic Capacity
• for m → ∞:
3 2λ
1
λe
2λ
− e µ Ei −
+ o(1)
C(m) = log m + log
3
2
2πµ
2
µ
• for v → ∞:
3
1
λm2 e
+ o(1)
C(v) = log v + log
2
2
2π
[3] M. N. Khormuji, “On the Capacity of Molecular Communication over the AIGN
Channel,” in Proceedings 45th Annual Conference on Information Sciences and
Systems (CISS), Baltimore, MD, USA, Mar. 23–25, 2011, pp. 1–4.
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
29
Summary & Outlook
• interesting new channel model
• new bounds on capacity
• exact asymptotics
• find better bounds for small drift velocity or small delay
• improve channel model:
– include peak-delay constraints
– account for loss of particles
– account for mixup of particles
Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan
30