Large Deviations for Channels with Memory

Transcription

1 Large Deviations for Channels with Memory Santhosh Kumar Jean-Francois Chamberland Henry Pfister Electrical and Computer Engineering Texas A&M University September 30, / 1

2 Digital Landscape Delay Tolerance and Rate Requirement Context Packetized system supporting delay-sensitive traffic Point-to-point communication with partial feedback 2/ 1

3 Overview Main Objective Analytic framework for delay sensitive systems Rigorous treatment of channels with memory Highlights Focus on delay sensitive systems Random linear codes Large deviations Potential Applications Optimal Code Rate Interface Selection 3/ 1

4 Preview of Result Main Result Large deviation principles on the delay and service rate for channels with memory 4/ 1

5 Preview of Result Main Result Large deviation principles on the delay and service rate for channels with memory Why Large deviations? 4/ 1

6 Preview of Result Main Result Large deviation principles on the delay and service rate for channels with memory Why Large deviations? Represents softer constraints on large buffers 4/ 1

7 Preview of Result Main Result Large deviation principles on the delay and service rate for channels with memory Why Large deviations? Represents softer constraints on large buffers Cumbersome to evaluate explicit probabilities 4/ 1

8 Preview of Result Main Result Large deviation principles on the delay and service rate for channels with memory Why Large deviations? Represents softer constraints on large buffers Cumbersome to evaluate explicit probabilities Provides foundation for other concepts like effective capacity 4/ 1

9 Channel Abstraction - Finite State Bit Erasure Channel At each instant channel can be in one of k states Channel transitions over time form a first-order Markov process In state i, transmitted bit is erased with probability ε i Transition matrix p 11 p 1k..... p k1 p kk Evolution at bit level 1 b 2 b 3 b 1 ε 1 ε 1 1 ε 2 ε 2 1 ε 3 ε 3 b e b e b e 5/ 1

10 Coding Scheme Code Length N Information Bits K Parity Bits N K Code Generation Generate a binary random parity check matrix H of size (N K) N Codebook is null space of H Code Properties Decoding failure probability (under ML) with e erasures P f (N K, e) = e 1 1 (1 2 i (N K)) i=0 6/ 1

11 Distribution of Erasures induced by Channel 7/ 1

12 Distribution of Erasures induced by Channel Interested in conditional probabilities of type ψ = Pr(e Erasures, C N+1 = j C }{{} 1 = i ) }{{} Channel exit state Channel beginning state 7/ 1

13 Distribution of Erasures induced by Channel Interested in conditional probabilities of type ψ = Pr(e Erasures, C N+1 = j C }{{} 1 = i ) }{{} Channel exit state Channel beginning state / 1

14 Distribution of Erasures induced by Channel Interested in conditional probabilities of type ψ = Pr(e Erasures, C N+1 = j C }{{} 1 = i ) }{{} Channel exit state Channel beginning state 3 2 e 3 1 e 7/ 1

15 Distribution of Erasures induced by Channel Interested in conditional probabilities of type ψ = Pr(e Erasures, C N+1 = j C }{{} 1 = i ) }{{} Channel exit state Channel beginning state 3 2 e 3 1 e (1 ε 3 ) p 32 ε 2 p 23 (1 ε 3 ) p 31 ε 1 7/ 1

16 Distribution of Erasures induced by Channel Interested in conditional probabilities of type ψ = Pr(e Erasures, C N+1 = j C }{{} 1 = i ) }{{} Channel exit state Channel beginning state 3 2 e 3 1 e (1 ε 3 ) p 32 ε 2 p 23 (1 ε 3 ) p 31 ε 1 7/ 1

17 Distribution of Erasures induced by Channel Interested in conditional probabilities of type ψ = Pr(e Erasures, C N+1 = j C }{{} 1 = i ) }{{} Channel exit state Channel beginning state 3 2 e 3 1 e (1 ε 3 ) p 32 ε 2 p 23 (1 ε 3 ) p 31 ε 1 7/ 1

18 Distribution of Erasures induced by Channel Interested in conditional probabilities of type ψ = Pr(e Erasures, C N+1 = j C }{{} 1 = i ) }{{} Channel exit state Channel beginning state 3 2 e 3 1 e (1 ε 3 ) p 32 ε 2 p 23 (1 ε 3 ) p 31 ε 1 ψ is the coefficient of x e in [P N x ] ij, where p 11 (1 ε 1 + ε 1 x) p 1k (1 ε 1 + ε 1 x) P x =..... p k1 (1 ε k + ε k x) p kk (1 ε k + ε k x) 7/ 1

19 Packetized Model & Acknowledgments Acknowledgements of successful transmissions are available When transmission fails, data segments are re-transmitted 8/ 1

20 Hidden Markovian System 9/ 1

21 Hidden Markovian System System is Hidden Markov 9/ 1

22 Hidden Markovian System System is Hidden Markov Append Channel State 9/ 1

23 Hidden Markovian System System is Hidden Markov Append Channel State Adding Channel State to Queue Length (C sn+1, q s ) 1, 0 1, 1 1, 2 1, 3 1, 4 2, 0 2, 1 2, 2 2, 3 2, 4 3, 0 3, 1 3, 2 3, 3 3, 4 9/ 1

24 Hidden Markovian System System is Hidden Markov Append Channel State Adding Channel State to Queue Length (C sn+1, q s ) 1, 0 1, 1 1, 2 1, 3 1, 4 2, 0 2, 1 2, 2 2, 3 2, 4 3, 0 3, 1 3, 2 3, 3 3, 4 The aggregate system is Markovian 9/ 1

25 Hidden Markovian System - System Evolution Level Transitions µ 11 µ 21 µ 12 µ 22 µ 32 µ 23 µ 33 κ 21 κ 32 1, l 2, l 3, l κ 12 κ 23 System Evolution Governed by Matrices: M (Success) and K (Failure) M = K = µ 11 µ 12 µ 13 µ 21 µ 22 µ 23 µ 31 µ 32 µ 33 κ 11 κ 12 κ 13 κ 21 κ 22 κ 23 κ 31 κ 32 κ 33 10/ 1

26 Delay & Service Rate - I D 1 = 0 D 2 = 0 D 3 = 0 D 4 = 1 D 5 = 0 D 6 = 0 D 7 = 1 T 1 = 4 T 2 = 3 Delay T q is the time for delivery of data segment q Average Time: Y m = T T m m Service Rate D s (0 or 1) indicates the success of transmission s Average Service Rate: Z n = D D n n 11/ 1

27 Delay & Service Rate - II Delay τ - delay requirement Minimizing Pr(Y m > τ) is a service guarantee on delay Service Rate γ - service requirement Minimizing Pr(Z n < γ) is a service guarantee on rate Basis for the large deviation principles Look at the asymptotic decay rates (as m, n ) of Pr(Y m > τ) and Pr(Z n < γ) For large buffers large deviations characterization is appropriate 12/ 1

28 Large Deviations - A Brief Overview From WLLN, Pr ( X1 + +X n n ) > δ 0, when δ > E(X ) Large deviations theory characterizes the exponential decay rate with which Pr 0 I (δ) = lim }{{} 1 ( ) X1 + + X n log Pr > δ n n n Rate function When {X i } are iid, Cramér s Theorem characterizes the large deviations principle (LDP) When {X i } are weakly dependent, invoke Gärtner-Ellis Theorem 13/ 1

29 Two Deviations Theorem Y m satisfies a large deviations principle with rate function I del ( ) where I del (x) = sup{λx log ρ((i Ke λ ) 1 Me λ )} λ R Theorem Z n satisfies a large deviations principle with rate function I ser ( ) where I ser (x) = sup{λx log ρ(k + Me λ )} λ R ρ( ) is the spectral radius of a Matrix 14/ 1

30 Relation between the deviations Set equivalence {T T m > n} = {D D n < m} Relation xi del (1/x) = I ser (x) This implies the means (in the asymptotic sense) of {D s } and {T q } have an inverse relation 15/ 1

31 Scaling for Commonality 12 bits N = 5, K = 3 N = 6, K = 4 Normalized Delay Ỹ m = NT 1+ +NT m/k m 1 K I del( K N τ) Normalized Service Rate Z n = KD 1+ +KD n/n n 1 N I ser ( N K γ) 16/ 1

32 GSM Example System Parameters Speech frame length 228 bits Physical layer 1 bit in 40µs Delay Tolerance 40ms Requires 228 bits in 1000 channel uses Choose τ = 1000/228 for the example 17/ 1

33 Numerical Analysis System Parameters Channel Gilbert-Elliott Channel Erasure Probabilities ε 1 = 1, ε 2 = 0 Channel Memory 0.9 Code length N=114 Channel Diagram e Average erasure rate is 20% K is the parameter to be optimized 18/ 1

34 Numerical Analysis ( K N τ) 1 K I del τ = τ = τ = τ = ( 1 N N Iser K γ) γ = γ = γ = γ = Information Bits (K) Information Bits (K) Optimal Code Rate Depends heavily on the underlying traffic Very conservative as the service requirements are lowered The perspective of delay and service rate are related 19/ 1

35 Concluding Remarks Main Contributions Provided a methodology for designing delay sensitive systems Rigorously modeled channels with memory Possible Extensions Channel Estimation Analyze variable block lengths 20/ 1