Feedback Capacity of Finite-State Channels with Causal State Information Known at the Encoder Eli Shemuel Oron Sabag Haim Permuter Ben-Gurion University of the Negev, Israel 2020 IEEE International Symposium on Information Theory Los Angeles, California, USA June 2020 Shemuel, Sabag, Permuter (BGU) Los Angeles, California, USA ISIT 2020 1 / 23
Table of Contents The Setting 1 Main Results 2 Theorem- The Capacity Theorem- Q-Graphs Examples of solving specific known problems Open problem example - Energy Harvesting Model Conclusions 3 Shemuel, Sabag, Permuter (BGU) Los Angeles, California, USA ISIT 2020 2 / 23
Table of Contents The Setting 1 Main Results 2 Theorem- The Capacity Theorem- Q-Graphs Examples of solving specific known problems Open problem example - Energy Harvesting Model Conclusions 3 Shemuel, Sabag, Permuter (BGU) Los Angeles, California, USA ISIT 2020 3 / 23
FSC with Feedback and causal state information General Finite State Channel (FSC) p ( y i , s i | x i , s i − 1 , y i − 1 ) = P ( y i , s i | x i , s i − 1 ) Remarks There is no assumption that the state sequence ( S 1 , S 2 , ... ) is i.i.d. The state is input dependent. Shemuel, Sabag, Permuter (BGU) Los Angeles, California, USA ISIT 2020 4 / 23
Special Cases The general channel model: P ( y i , s i | x i , s i − 1 ) = P ( y i | x i , s i − 1 ) P ( s i | x i , s i − 1 , y i ) This setting covers the following specific cases: i.i.d state: p ( s i − 1 ) p ( y i | x i , s i − 1 ) Markovian state: p ( s i | s i − 1 ) p ( y i | x i , s i − 1 ), and look-ahead. (Can feedback increase the capacity?) Unifilar FSC: s i = f ( x i , s i − 1 , y i ) Energy Harvesting Shemuel, Sabag, Permuter (BGU) Los Angeles, California, USA ISIT 2020 5 / 23
Table of Contents The Setting 1 Main Results 2 Theorem- The Capacity Theorem- Q-Graphs Examples of solving specific known problems Open problem example - Energy Harvesting Model Conclusions 3 Shemuel, Sabag, Permuter (BGU) Los Angeles, California, USA ISIT 2020 6 / 23
The Capacity Theorem (Capacity) N 1 I ( U i , U i − 1 ; Y i | Y i − 1 ) � C = lim max (1) N N →∞ { p ( u i | u i − 1 , y i − 1 ) } N i =1 , i =1 x i = f ( u i , s i − 1 ) is the capacity for some f : U × S → X , with the joint distribution n p ( s i , y i | s i − 1 , x i ) 1 { x i = f ( u i , s i − 1 ) } p ( u i | y i − 1 , u i − 1 ) , � (2) i =1 Lack of cardinality bound of the auxiliary r.v. U . Multi-letter expression. For any fixed cardinality |U| and a fixed function f one can compute achievable rates by 2 main tools. Shemuel, Sabag, Permuter (BGU) Los Angeles, California, USA ISIT 2020 7 / 23
Achievability - Shannon Strategy Coding is based on a Shannon strategy-like argument, conversion to a new FSC P ( s i , y i | u i , s i − 1 ). P ( s i , y i | x i , u i , s i − 1 , y i − 1 ) = P ( s i , y i | x ( u i , s i − 1 ) , s i − 1 ) = P ( s i , y i | u i , s i − 1 ) Shemuel, Sabag, Permuter (BGU) Los Angeles, California, USA ISIT 2020 8 / 23
Converse - Sketch of Proof Any sequence of achievable block codes (2 nR , n ) has ( a ) ≤ I ( M ; Y n ) + n ǫ n nR where n ( b ) I ( M ; Y n ) � I ( M ; Y i | Y i − 1 ) = i =1 n ( c ) I ( U i ; Y i | Y i − 1 ) � = i =1 (a) follows from Fano’s inequality, where ǫ n → 0 as n → ∞ . (b) follows from the chain rule. (c) follows from defining U i � ( M , Y i − 1 ). Shemuel, Sabag, Permuter (BGU) Los Angeles, California, USA ISIT 2020 9 / 23
n n ( d ) � I ( U i ; Y i | Y i − 1 ) � I ( U i ; Y i | Y i − 1 ) ≤ max { p ( u i | u i − 1 , y i − 1 ) , p ( x i | u i , s i − 1 ) } n i =1 i =1 i =1 n ( e ) I ( U i ; Y i | Y i − 1 ) � = max { p ( u i | u i − 1 , y i − 1 ) , p ( v i ) , x i = f i ( u i , v i , s i − 1 ) } n i =1 i =1 (d) follows because the objective is determined by { p ( u i , y i , s i ) } n i =1 = { p ( u i , y i , s i ) } n i =1 due to the definition of U i � ( M , Y i − 1 ) and from the following lemma: Lemma p ( u k , y k , s k ) is determined by { p ( u i | u i − 1 , y i − 1 ) p ( x i | u i , s i − 1 ) } k i =1 . (e) follows from the Functional Representation Lemma, i.e. there exists a RV, V i , independent of ( U i , S i − 1 ), such that X i can be represented as a function of ( U i , S i − 1 ) and V i , and from a similar lemma. Shemuel, Sabag, Permuter (BGU) Los Angeles, California, USA ISIT 2020 10 / 23
n I ( U i ; Y i | Y i − 1 ) � max { p ( u i | u i − 1 , y i − 1 ) , p ( v i ) , x i = f i ( u i , v i , s i − 1 ) } n i =1 i =1 n ( f ) I (˜ U i ; Y i | Y i − 1 ) � ≤ max u i , s i − 1 ) } n u i − 1 , y i − 1 ) , x i = f i (˜ { p (˜ u i | ˜ i =1 i =1 n ( g ) I (˜ U i ; Y i | Y i − 1 ) � = max u i , i ) } n u i − 1 , y i − 1 ) , x = f ( s i − 1 , ˜ { p (˜ u i | ˜ i =1 i =1 (f) follows from defining ˜ U i � ( U i , V i ), from the fact that u i − 1 , y i − 1 ) = p ( v i | u i − 1 , v i − 1 , y i − 1 ) p ( u i | v i , u i − 1 , v i − 1 , y i − 1 ). p ( v i ) and p (˜ u i | ˜ p ( u i | v i , u i − 1 , v i − 1 , y i − 1 ) are sub-domains of p ( v i | u i − 1 , v i − 1 , y i − 1 ) and p ( u i | u i − 1 , y i − 1 ) respectively; the mutual information increases since conditioning reduces entropy. (g) follows since there exist an invariant function f (˜ u , s , i ) = f i (˜ u i , s i − 1 ). Shemuel, Sabag, Permuter (BGU) Los Angeles, California, USA ISIT 2020 11 / 23
n I (˜ U i ; Y i | Y i − 1 ) � max u i , i ) } n { p (˜ u i | ˜ u i − 1 , y i − 1 ) , x = f ( s i − 1 , ˜ i =1 i =1 n ( h ) I ( U ′ i ; Y i | Y i − 1 ) � ≤ max i , s i − 1 ) } n { p ( u ′ i | u ′ i − 1 , y i − 1 ) , x = f ( u ′ i =1 i =1 n I ( U ′ i , U ′ i − 1 ; Y i | Y i − 1 ) � ≤ max i , s i − 1 ) n i − 1 , y i − 1 ) } , x = f ( u ′ { p ( u ′ i | u ′ i =1 i =1 (h) follows from defining U ′ = (˜ U i , T = i ) where T represents the time index. Finally, by dividing both sides by n we get n R ≤ 1 � I ( U ′ i , U ′ i − 1 ; Y i | Y i − 1 ) max n i , s i − 1 ) n i − 1 , y i − 1 ) } , x = f ( u ′ { p ( u ′ i | u ′ i =1 i =1 which completes the proof. Shemuel, Sabag, Permuter (BGU) Los Angeles, California, USA ISIT 2020 12 / 23
The Q-graph A Q-graph is an irreducible directed graph. Each node Q has |L| different labelled edges. E.g. L = { 0 , 1 , ? } The Q-graph defines a mapping: Φ i − 1 : L i − 1 → Q (or g : Q × L → Q ) Each outputs sequence is Q -uantized. Shemuel, Sabag, Permuter (BGU) Los Angeles, California, USA ISIT 2020 13 / 23
Q-Graph Lower Bound Theorem (Q-graph lower bound) The feedback capacity of an FSC with causal state at the encoder is lower bounded by C fb ≥ I ( U + , U ; Y | Q ) , ∀ Q-graph (3) for any mapping X = f ( U + , S ) and for all aperiodic inputs P U + | U , Q ∈ P π that are BCJR-invariant. The BCJR-invariant property: P S + , U + | Q , Y ( s + , u + | q , y ) = P S + , U + | Q + ( s + , u + | g ( q , y )) , (4) for all ( s + , u + , q , y ). Markov chain: ( S + , U + ) − Q + − ( Q , Y ). Evaluated with the stationary distribution. Shemuel, Sabag, Permuter (BGU) Los Angeles, California, USA ISIT 2020 14 / 23
Example - The Trapdoor channel State evolution: s + = s ⊕ x ⊕ y . The probability of the Channel p ( y t | x t , s t − 1 ): x t s t − 1 p ( y t = 0 | x t , s t − 1 ) 0 0 1 0 1 0 . 5 1 0 0 . 5 1 1 0 Lemma For the Trapdoor channel, there exists a policy that achieves √ 1 + 5 R = log 2 (5) , 2 with |U| = 2 and x = u ⊕ s. Shemuel, Sabag, Permuter (BGU) Los Angeles, California, USA ISIT 2020 15 / 23
Example - The input constrained BEC - (1 , ∞ )-RLL State evolution: s = x . The probability of the Channel p ( y t | x t , s t − 1 ): Binary Erasure Channel (BEC) with erasure probability parameter ǫ . Lemma For the input-constrained BEC, there exists a policy that achieves H 2 ( p ) R = max (6) , 1 p p + 1 − ǫ for all ǫ , with |U| = 2 and x = u ∧ ¯ s. Shemuel, Sabag, Permuter (BGU) Los Angeles, California, USA ISIT 2020 16 / 23
Example - The Energy Harvesting (EH) Model X is constrained by the available energy at each channel use. B max − the battery size. E i ∈ { 0 , 1 }− the arrival process. i.i.d. ∼ Bern( h ). If E i = 1 then the battery is charged after the transmission up to the battery size. The battery state evolves according to: S i = min { S i − 1 − X i + E i , B max } Shemuel, Sabag, Permuter (BGU) Los Angeles, California, USA ISIT 2020 17 / 23
The Binary Energy Harvesting Model (BEHM) X = { 0 , 1 } Y = { 0 , 1 } B max = 1, S = { 0 , 1 } If S i − 1 = 0 then X i = 0. If S i − 1 = 1 then X i ∈ { 0 , 1 } . Noiseless channel Y = X What is the capacity? Shemuel, Sabag, Permuter (BGU) Los Angeles, California, USA ISIT 2020 18 / 23
Graph suspected to be the optimal to solve the BEHM Shemuel, Sabag, Permuter (BGU) Los Angeles, California, USA ISIT 2020 19 / 23
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