Fundamental Limits of Molecular Communication Christopher Rose 1 I. - PowerPoint PPT Presentation

34 Wireless With Molecules Timing Channel Detail Review Mutual Information: I ( S ; T ) M tokens on an interval τ ( M ) I ( S ; T ) = h ( S ) − h ( S | T ) = h ( S ) − h ( D ) ≤ M ( h ( S ) − h ( D )) , (i.i.d. D ) Max h ( S ) , Done! School of Engineering@Brown University CTW 2016 C. Rose

34 Wireless With Molecules Timing Channel Detail Review Mutual Information: I ( S ; T ) M tokens on an interval τ ( M ) I ( S ; T ) = h ( S ) − h ( S | T ) = h ( S ) − h ( D ) ≤ M ( h ( S ) − h ( D )) , (i.i.d. D ) Max h ( S ) , Done! Easy, Right!?! School of Engineering@Brown University CTW 2016 C. Rose

34 Wireless With Molecules Timing Channel Detail Review Mutual Information: I ( S ; T ) M tokens on an interval τ ( M ) I ( S ; T ) = h ( S ) − h ( S | T ) = h ( S ) − h ( D ) ≤ M ( h ( S ) − h ( D )) , (i.i.d. D ) Max h ( S ) , Done! Easy, Right!?! S | T) = ? I ( � S; T) = h ( � S) − h ( � School of Engineering@Brown University CTW 2016 C. Rose

35 Wireless With Molecules Timing Channel Detail Review Hypersymmetries Departures S 1 T 1 S 2 T 2 S 3 T 3 T 4 S 4 S 5 T 5 S 1 S 2 S 3 S 4 S 5 t Arrivals School of Engineering@Brown University CTW 2016 C. Rose

36 Wireless With Molecules Timing Channel Detail Review Hypersymmetries Departures S 1 T 1 S 2 T 2 T 3 S 3 S 4 T 4 S 5 T 5 S 1 S 2 S 3 S 4 S 5 t Arrivals School of Engineering@Brown University CTW 2016 C. Rose

37 Wireless With Molecules Timing Channel Detail Review Hypersymmetry Buys You School of Engineering@Brown University CTW 2016 C. Rose

37 Wireless With Molecules Timing Channel Detail Review Hypersymmetry Buys You h ( � S) = h (S) − log M ! School of Engineering@Brown University CTW 2016 C. Rose

37 Wireless With Molecules Timing Channel Detail Review Hypersymmetry Buys You h ( � S) = h (S) − log M ! { � S , Ω } ↔ S School of Engineering@Brown University CTW 2016 C. Rose

37 Wireless With Molecules Timing Channel Detail Review Hypersymmetry Buys You h ( � S) = h (S) − log M ! { � S , Ω } ↔ S ⇓ h ( � S | T ) = H (Ω | � S , T ) − h ( S | T ) School of Engineering@Brown University CTW 2016 C. Rose

37 Wireless With Molecules Timing Channel Detail Review Hypersymmetry Buys You h ( � S) = h (S) − log M ! { � S , Ω } ↔ S ⇓ h ( � S | T ) = H (Ω | � S , T ) − h ( S | T ) I ( � S; T) = h (S) + H (Ω | � S , T) − (log M ! + h (D)) � �� � � �� � constant The Money! School of Engineering@Brown University CTW 2016 C. Rose

38 Wireless With Molecules Timing Channel Detail Review Channel Use Formalities School of Engineering@Brown University CTW 2016 C. Rose

38 Wireless With Molecules Timing Channel Detail Review Channel Use Formalities γ( Μ,ε ) k 1 2 ... τ( Μ ) Guard Interval: γ ( M, ǫ ) Overflow Probability: ǫ School of Engineering@Brown University CTW 2016 C. Rose

38 Wireless With Molecules Timing Channel Detail Review Channel Use Formalities γ( Μ,ε ) k 1 2 ... τ( Μ ) Guard Interval: γ ( M, ǫ ) Overflow Probability: ǫ Power Constraint (tokens cost energy): M ρ ≡ lim ǫ → 0 lim τ ( M ) + γ ( M, ǫ ) M →∞ School of Engineering@Brown University CTW 2016 C. Rose

39 Wireless With Molecules Timing Channel Detail Review Limiting Details School of Engineering@Brown University CTW 2016 C. Rose

39 Wireless With Molecules Timing Channel Detail Review Limiting Details Set: γ ( M, ǫ ) = ǫτ ( M ) (convenience) School of Engineering@Brown University CTW 2016 C. Rose

39 Wireless With Molecules Timing Channel Detail Review Limiting Details Set: γ ( M, ǫ ) = ǫτ ( M ) (convenience) Require: lim M →∞ Prob { � S M ≤ τ ( M )(1 + ǫ ) } = 1 School of Engineering@Brown University CTW 2016 C. Rose

39 Wireless With Molecules Timing Channel Detail Review Limiting Details Set: γ ( M, ǫ ) = ǫτ ( M ) (convenience) Require: lim M →∞ Prob { � S M ≤ τ ( M )(1 + ǫ ) } = 1 Worst case: all tokens launched at time τ ( M ) School of Engineering@Brown University CTW 2016 C. Rose

39 Wireless With Molecules Timing Channel Detail Review Limiting Details Set: γ ( M, ǫ ) = ǫτ ( M ) (convenience) Require: lim M →∞ Prob { � S M ≤ τ ( M )(1 + ǫ ) } = 1 Worst case: all tokens launched at time τ ( M ) PUNCHLINE: all ok if E [ D ] exists School of Engineering@Brown University CTW 2016 C. Rose

40 Wireless With Molecules Timing Channel Detail Review Omitting the Details (or summary :) ) School of Engineering@Brown University CTW 2016 C. Rose

40 Wireless With Molecules Timing Channel Detail Review Omitting the Details (or summary :) ) Define: χ ≡ µ (first passage rate) M Set: ρ ≡ ρ (token launch rate) τ ( M ) � � I ( � Require: E [ D ] < ∞ C m ( M ) = max S ; T ) /M hypersymm f T () C m = lim M →∞ C m ( M ) C t = ρC m School of Engineering@Brown University CTW 2016 C. Rose

41 Wireless With Molecules Timing Channel Detail Review My Past Personal Struggles School of Engineering@Brown University CTW 2016 C. Rose

41 Wireless With Molecules Timing Channel Detail Review My Past Personal Struggles ∃ closed form results/bounds for H (Ω | � S , T ) School of Engineering@Brown University CTW 2016 C. Rose

41 Wireless With Molecules Timing Channel Detail Review My Past Personal Struggles ∃ closed form results/bounds for H (Ω | � S , T ) f T () h ( S ) + H (Ω | � max S , T ) ≥ ? (ISIT’13) School of Engineering@Brown University CTW 2016 C. Rose

41 Wireless With Molecules Timing Channel Detail Review My Past Personal Struggles ∃ closed form results/bounds for H (Ω | � S , T ) f T () h ( S ) + H (Ω | � max S , T ) ≥ ? (ISIT’13) f T () h ( S ) + H (Ω | � max S , T ) ≤ ? (ISIT’14) School of Engineering@Brown University CTW 2016 C. Rose

42 Wireless With Molecules Timing + Payload Timing + Payload School of Engineering@Brown University CTW 2016 C. Rose

42 Wireless With Molecules Timing + Payload Timing + Payload Identical tokens → timing info only School of Engineering@Brown University CTW 2016 C. Rose

42 Wireless With Molecules Timing + Payload Timing + Payload Identical tokens → timing info only Payloads → chop message into M B -bit pieces School of Engineering@Brown University CTW 2016 C. Rose

42 Wireless With Molecules Timing + Payload Timing + Payload Identical tokens → timing info only Payloads → chop message into M B -bit pieces BUT: Payloads can arrive out of order School of Engineering@Brown University CTW 2016 C. Rose

42 Wireless With Molecules Timing + Payload Timing + Payload Identical tokens → timing info only Payloads → chop message into M B -bit pieces BUT: Payloads can arrive out of order Add H (Ω | � S , T ) /M bits per token (for re-sequencing) School of Engineering@Brown University CTW 2016 C. Rose

43 Wireless With Molecules Timing + Payload Energy School of Engineering@Brown University CTW 2016 C. Rose

43 Wireless With Molecules Timing + Payload Energy Identical Tokens: c 0 joules per token School of Engineering@Brown University CTW 2016 C. Rose

43 Wireless With Molecules Timing + Payload Energy Identical Tokens: c 0 joules per token Inscribed Tokens: School of Engineering@Brown University CTW 2016 C. Rose

43 Wireless With Molecules Timing + Payload Energy Identical Tokens: c 0 joules per token Inscribed Tokens: substrate: c 1 joules per token payload bit B : B ∆ c 1 joules per token avg. sequence bits K : K ∆ c 1 joules per token, so School of Engineering@Brown University CTW 2016 C. Rose

43 Wireless With Molecules Timing + Payload Energy Identical Tokens: c 0 joules per token Inscribed Tokens: substrate: c 1 joules per token payload bit B : B ∆ c 1 joules per token avg. sequence bits K : K ∆ c 1 joules per token, so H (Ω | � S , T ) ≤ MK ≤ log M ! School of Engineering@Brown University CTW 2016 C. Rose

44 Wireless With Molecules Timing + Payload And Now ... School of Engineering@Brown University CTW 2016 C. Rose

44 Wireless With Molecules Timing + Payload And Now ... LOWER BOUNDS using exponential first passage (the timing channel’s “Gaussian”) School of Engineering@Brown University CTW 2016 C. Rose

45 Wireless With Molecules Bounds Timing-Only Bits/Joule Theorem 1.   ∞ � 1 � k C T ≥ 1  ( kχ − 1)log k !  �  log χ + e − 1   χ   k ! c 0 χ  k =2 � �� � H (Ω | � S , T) /M : average per-token order-uncertainty School of Engineering@Brown University CTW 2016 C. Rose

46 Wireless With Molecules Bounds Payload-Only Bits/Joule Theorem 2. B C P = � � M H (Ω | � B + min t 1 c 1 + ∆ c 1 S , t ) Theorem 3. B C P ≥    ∞  � 1 � k ( kχ − 1)log k ! �  B + e − 1  c 1 + ∆ c 1 χ    k !  χ  k =2  � �� � H (Ω | � S , T) /M : average per-token order-uncertainty School of Engineering@Brown University CTW 2016 C. Rose

47 Wireless With Molecules Bounds Payload + Timing Bits/Joule Lower Bound Theorem 4. � � 1 + χM log + B e R P + T ≈   ∞ � 1 � k  ( kχ − 1)log k !  �  B + e − 1   c 1 + ∆ c 1 χ   k ! χ  k =2 � �� � H (Ω | � S , T) /M : average per-token order-uncertainty where R P + T ≤ C P + T . School of Engineering@Brown University CTW 2016 C. Rose

48 Wireless With Molecules Bounds Info per Unit Energy χ ↔ passage rate per launch rate c 0 = 1 , c 1 = 0 , ∆ c 1 = 1 School of Engineering@Brown University CTW 2016 C. Rose

49 Wireless With Molecules Bounds Info per Passage per Unit Energy 1 χ ↔ launch rate per passage rate c 0 = 1 , c 1 = 0 , ∆ c 1 = 1 School of Engineering@Brown University CTW 2016 C. Rose

50 Wireless With Molecules Play Time And Now .... School of Engineering@Brown University CTW 2016 C. Rose

50 Wireless With Molecules Play Time And Now .... Numerical Play Time School of Engineering@Brown University CTW 2016 C. Rose

51 Wireless With Molecules Play Time Play Time Setup School of Engineering@Brown University CTW 2016 C. Rose

51 Wireless With Molecules Play Time Play Time Setup source sink R School of Engineering@Brown University CTW 2016 C. Rose

51 Wireless With Molecules Play Time Play Time Setup source sink R “Binary Protein” Token Construction 4 B ATP = 3 . 2 B × 10 − 19 J School of Engineering@Brown University CTW 2016 C. Rose

51 Wireless With Molecules Play Time Play Time Setup source sink R “Binary Protein” Token Construction 4 B ATP = 3 . 2 B × 10 − 19 J Diffusion Coefficient, D in air: ≈ 10 − 5 m 2 /s Mean First Passage Time, E [ D ] ≈ R 2 2 D School of Engineering@Brown University CTW 2016 C. Rose