The Additive Inverse Gaussian Noise Channel: an Attempt on Modeling Molecular Communication

Transcription

1 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

2 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

3 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 , July Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan 3

4 Channel Model (1) Brownian motion is modeled as Wiener process: for any time interval τ, the position increment is Gaussian: W N ( vτ,σ 2 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

5 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 f N (n) = ) λ ( 2πn 3 exp λ(n µ)2 2µ 2 I{n > 0} n average travel time: µ = d v Brownian motion parameter: λ = d2 σ 2 Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan 5

6 Channel Model (3) channel input: release time channel output: arrival time noise: input and noise independent: X Y = X + N N IG(µ,λ) X N = additive inverse Gaussian noise (AIGN) channel channel law: f Y X (y x) = λ 2π(y x) 3 exp ) λ(y x µ)2 ( 2µ 2 I{y > x} (y x) implicit nonnegativity constraint X 0 average-delay constraint E[X] m Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan 6

7 Properties of IG(µ, λ) (1) For N IG(µ,λ) we have E[N] = µ Var(N) = µ3 λ h(n) = 1 2 log 2πeµ3 λ e2λ µ Ei ( 2λ µ ) (Ei( ) exponential integral function) Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan 7

8 Properties of IG(µ, λ) (2) For N 1 IG(µ 1,λ 1 ) N 2 IG(µ 2,λ 2 ) N 1 N 2 we have N 1 +N 2 IG ( ( µ 1 +µ 2, λ1 + ) ) 2 λ 2 only if λ 1 µ 2 1 = λ 2 µ 2 2 Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan 8

9 First Lower Bound on Capacity (1) ( Lower Bound: choose X IG m, λm2 µ ): 2 C = max I(X;Y) P X : E[X] m { } = max h(y) h(y X) P X : E[X] m { } = max h(y) h(n) P X : E[X] m = max h(y) h(n) P X : E[X] m h(y) ( ) h(n) X IG m, λm2 µ 2 h(n) = 1 2 h(y) ( ) =? X IG m, λm2 µ 2 log 2πeµ3 λ e2λ µ Ei ( 2λ µ ) Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan 9

10 First Lower Bound on Capacity (2) ( Lower Bound: choose X IG m, λm2 µ 2 ): (continued) note: λm 2 µ 2 m 2 = λ µ 2 = Y = X +N ) IG (m, λm2 µ 2 +IG(µ,λ) ( ) 2 λm = IG m+µ, µ + λ ) = IG (m+µ, λ(m+µ)2 µ 2 Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan 10

11 First Lower Bound on Capacity (3) ( Lower Bound: choose X IG m, λm2 µ 2 ): (continued) C = max I(X;Y) P X : E[X] m { } = max h(y) h(y X) P X : E[X] m { } = max h(y) h(n) P X : E[X] m = max h(y) h(n) P X : E[X] m h(y) ( ) h(n) X IG m, λm2 µ 2 h(n) = 1 2 log 2πeµ3 λ h(y) ( ) = 1 X IG m, λm2 µ 2 2 log 2πeµ2 (m+µ) λ e2λ µ Ei ( 2λ µ ) e2λ(m+µ) µ 2 Ei ( 2λ(m+µ) µ 2 ) Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan 11

12 First Upper Bound on Capacity (1) Upper Bound: Note: C = max I(X;Y) P X : E[X] m { } = max h(y) h(y X) P X : E[X] m = max h(y) h(n) P X : E[X] m Under mean constraint: E[Y] = E[X]+E[N] m+µ h(y) maximized by exponential distribution Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan 12

13 First Upper Bound on Capacity (2) Upper Bound: h(y) h(exponential): Hence: h(y) loge(m+µ) C = max I(X;Y) P X : E[X] m { } = max h(y) h(y X) P X : E[X] m = max h(y) h(n) P X : E[X] m loge(m+µ) h(n) Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan 13

14 Summary First Bounds (1) Upper Bound: [1]: C 1 2 λe(m+µ)2 log 2πµ 3 3 ( 2 e2λ µ Ei 2λ µ ) Lower Bound: [1]: C 1 2 log m+µ µ e2λ(m+µ) µ 2 Ei ( 2λ(m+µ) µ 2 ) 32 e2λµ Ei ( 2λ 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 , July Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan 14

15 Summary First Bounds (2) 6 5 Capacity C [bits] ZOOM IN Delay m Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan 15

16 0.5 Summary First Bounds (2) ZOOMED Capacity C [bits] Delay m Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan 16

17 Summary First Bounds (3) 6 5 ZOOM IN Capacity C [bits] Drift velocity v Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan 17

18 5 Summary First Bounds (3) ZOOMED Capacity C [bits] Drift velocity v Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan 18

19 Idea for New Upper Bounds Duality-based upper bound: [ ( C E Q D fy X ( X) )] R( ) for arbitrary choice of R( ): R( ) exponential: (α > 0) = yields first upper bound R(y) = αe αy R( ) power inverse Gaussian: (α,β > 0, η 0) ( ) 1+ η ( α β 2 (y R(y) = exp α )η ( ) )η 2 2 β 2 2πβ 3 y 2η 2 β β y (additional bounding required) Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan 19

20 New Upper Bounds on Capacity C 1 (1+ 2 log λm )+ 32 ( ( 1 µ(m+µ) log 1+m µ + 1 )) λ C 1 ( η 1 logλ+ (logµ+e 2λµ Ei 2λ )) 2 2 µ ( ) λ ) (m+µ) η log ( 2λ π µ η 1 2 e λ µ Kη+ 1 2 µ + η +2 ( ( 1 log 1+m 2 µ + 1 )) logη λ C 1 2 log λ µ + 1 ( 1 (1+m 2 log µ + 1 λ + 1 ( 2 log 1+ m µ λ ) µ+λ )) e 2λµ Ei ( 2λ µ (0 < η 1, K ν ( ) modified Bessel function of second kind) ) Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan 20

21 Idea for New Lower Bounds Choose exponential input: ( ) 1 X Exp m PDF of Y: convolution of exponential and inverse Gaussian [2]: Y 1 [ ( y m e m +λ µ e kλ Q ( )) ky 1 kλ ky ( ( ))] kλ ky +e kλ 1 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 , Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan 21

22 New Lower Bound on Capacity C log m λ + µ m λ µ +kλ+ 3 2 log λ µ 3 ( 2 e2λ µ Ei 2λ ) 1 µ 2 log 2π e ( log 1+ 1 ( ) m eλ µ λm 2λ 2+k 2 λm K 1 m +k2 λ ( )) 2m eλ µ λm λ +kλ 1+k 2 λm K 1 2 m +k2 λ 2 where k m 2µ2 λ 1 µ 2 2 mλ Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan 22

23 New Bounds on Capacity (1) 6 5 Capacity C [bits] known bounds: black Delay m Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan 23

24 New Bounds on Capacity (1) 6 5 Capacity C [bits] ZOOM IN known bounds: black our new bounds: colored Delay m Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan 24

25 0.5 New Bounds on Capacity (1) ZOOMED Capacity C [bits] Delay m Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan 25

26 New Bounds on Capacity (2) 6 5 Capacity C [bits] known bounds: black Drift velocity v Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan 26

27 Our Results: New Bounds on Capacity (2) 6 Capacity C [bits] ZOOM IN 1 known bounds: black our new bounds: colored Drift velocity v Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan 27

28 5 New Bounds on Capacity (2) ZOOMED Capacity C [bits] Drift velocity v Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan 28

29 Exact Asymptotic Capacity for m : C(m) = logm+ 1 2 λe log 2πµ 3 3 ( 2 e2λ µ Ei 2λ µ ) +o(1) for v : C(v) = 3 2 logv log λm2 e 2π +o(1) [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 , 2011, pp Stefan M. Moser, National Chiao Tung University (NCTU), Taiwan 29

30 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