Background Ideas and Directions Single-Letter Lower Bound Applications Future Work Computable Lower Bounds for Capacities of Input-Driven Finite-State Channels V. Arvind Rameshwar Navin Kashyap Department of Electrical Communication Engineering Indian Institute of Science, Bengaluru ISIT 2020 V. Arvind Rameshwar, Navin Kashyap Computable Lower Bounds for Capacities of Input-Driven FSCs
Background Ideas and Directions Single-Letter Lower Bound Applications Future Work Outline Background 1 Motivation System Model Ideas and Directions 2 Single-Letter Lower Bound 3 Applications 4 Input-constrained BEC( ǫ ) Input-constrained BSC( p ) Future Work 5 V. Arvind Rameshwar, Navin Kashyap Computable Lower Bounds for Capacities of Input-Driven FSCs
Background Ideas and Directions Motivation Single-Letter Lower Bound System Model Applications Future Work FSCs and Capacity V. Arvind Rameshwar, Navin Kashyap Computable Lower Bounds for Capacities of Input-Driven FSCs
Background Ideas and Directions Motivation Single-Letter Lower Bound System Model Applications Future Work FSCs and Capacity Discrete Finite-State Channels (FSCs) are used to model: V. Arvind Rameshwar, Navin Kashyap Computable Lower Bounds for Capacities of Input-Driven FSCs
Background Ideas and Directions Motivation Single-Letter Lower Bound System Model Applications Future Work FSCs and Capacity Discrete Finite-State Channels (FSCs) are used to model: Inter-Symbol Interference in Magnetic and Optical Recording [e.g., Immink, Siegel, Wolf, ’98] Inter-Cell Interference in NAND Flash Memories [e.g., Li, Kavˇ ci´ c, Han, ’16] Fading in Mobile Radio Channels [e.g., Semmar, Lecours, Chouinard, Ahern, ’91] V. Arvind Rameshwar, Navin Kashyap Computable Lower Bounds for Capacities of Input-Driven FSCs
Background Ideas and Directions Motivation Single-Letter Lower Bound System Model Applications Future Work FSCs and Capacity Discrete Finite-State Channels (FSCs) are used to model: Inter-Symbol Interference in Magnetic and Optical Recording [e.g., Immink, Siegel, Wolf, ’98] Inter-Cell Interference in NAND Flash Memories [e.g., Li, Kavˇ ci´ c, Han, ’16] Fading in Mobile Radio Channels [e.g., Semmar, Lecours, Chouinard, Ahern, ’91] Computing Capacities of FSCs V. Arvind Rameshwar, Navin Kashyap Computable Lower Bounds for Capacities of Input-Driven FSCs
Background Ideas and Directions Motivation Single-Letter Lower Bound System Model Applications Future Work FSCs and Capacity Discrete Finite-State Channels (FSCs) are used to model: Inter-Symbol Interference in Magnetic and Optical Recording [e.g., Immink, Siegel, Wolf, ’98] Inter-Cell Interference in NAND Flash Memories [e.g., Li, Kavˇ ci´ c, Han, ’16] Fading in Mobile Radio Channels [e.g., Semmar, Lecours, Chouinard, Ahern, ’91] Computing Capacities of FSCs → Capacity-achieving coding schemes V. Arvind Rameshwar, Navin Kashyap Computable Lower Bounds for Capacities of Input-Driven FSCs
Background Ideas and Directions Motivation Single-Letter Lower Bound System Model Applications Future Work FSCs and Capacity Discrete Finite-State Channels (FSCs) are used to model: Inter-Symbol Interference in Magnetic and Optical Recording [e.g., Immink, Siegel, Wolf, ’98] Inter-Cell Interference in NAND Flash Memories [e.g., Li, Kavˇ ci´ c, Han, ’16] Fading in Mobile Radio Channels [e.g., Semmar, Lecours, Chouinard, Ahern, ’91] Computing Capacities of FSCs → Capacity-achieving coding schemes For even simple input-constrained DMCs, Computing Capacity ≡ Computing H ( Y ) V. Arvind Rameshwar, Navin Kashyap Computable Lower Bounds for Capacities of Input-Driven FSCs
Background Ideas and Directions Motivation Single-Letter Lower Bound System Model Applications Future Work FSCs and Capacity Discrete Finite-State Channels (FSCs) are used to model: Inter-Symbol Interference in Magnetic and Optical Recording [e.g., Immink, Siegel, Wolf, ’98] Inter-Cell Interference in NAND Flash Memories [e.g., Li, Kavˇ ci´ c, Han, ’16] Fading in Mobile Radio Channels [e.g., Semmar, Lecours, Chouinard, Ahern, ’91] Computing Capacities of FSCs → Capacity-achieving coding schemes For even simple input-constrained DMCs, Computing Capacity ≡ Computing H ( Y ) Hard! V. Arvind Rameshwar, Navin Kashyap Computable Lower Bounds for Capacities of Input-Driven FSCs
Background Ideas and Directions Motivation Single-Letter Lower Bound System Model Applications Future Work FSCs and Capacity Discrete Finite-State Channels (FSCs) are used to model: Inter-Symbol Interference in Magnetic and Optical Recording [e.g., Immink, Siegel, Wolf, ’98] Inter-Cell Interference in NAND Flash Memories [e.g., Li, Kavˇ ci´ c, Han, ’16] Fading in Mobile Radio Channels [e.g., Semmar, Lecours, Chouinard, Ahern, ’91] Computing Capacities of FSCs → Capacity-achieving coding schemes For even simple input-constrained DMCs, Computing Capacity ≡ Computing H ( Y ) Hard! Goal : Get good bounds on capacity without feedback V. Arvind Rameshwar, Navin Kashyap Computable Lower Bounds for Capacities of Input-Driven FSCs
Background Ideas and Directions Motivation Single-Letter Lower Bound System Model Applications Future Work The Setup V. Arvind Rameshwar, Navin Kashyap Computable Lower Bounds for Capacities of Input-Driven FSCs
Background Ideas and Directions Motivation Single-Letter Lower Bound System Model Applications Future Work The Setup DMC: n � P ( y n | x n ) = P ( y i | x i ) i =1 V. Arvind Rameshwar, Navin Kashyap Computable Lower Bounds for Capacities of Input-Driven FSCs
Background Ideas and Directions Motivation Single-Letter Lower Bound System Model Applications Future Work The Setup Generic FSC: n P ( y n , s n | x n , s 0 ) = � P ( y i , s i | s i − 1 , x i ) i =1 V. Arvind Rameshwar, Navin Kashyap Computable Lower Bounds for Capacities of Input-Driven FSCs
Background Ideas and Directions Motivation Single-Letter Lower Bound System Model Applications Future Work The Setup Generic FSC: n � P ( y n , s n | x n , s 0 ) = P ( y i , s i | x i , s i − 1 ) i =1 Input-Driven FSC: n � P ( y n , s n | x n , s 0 ) = P ( y i | x i , s i − 1 ) ✶ { s i = f ( s i − 1 , x i ) } i =1 V. Arvind Rameshwar, Navin Kashyap Computable Lower Bounds for Capacities of Input-Driven FSCs
Background Ideas and Directions Motivation Single-Letter Lower Bound System Model Applications Future Work Examples V. Arvind Rameshwar, Navin Kashyap Computable Lower Bounds for Capacities of Input-Driven FSCs
Background Ideas and Directions Motivation Single-Letter Lower Bound System Model Applications Future Work Examples Input-Constrained DMCs: ( d , ∞ )-RLL input constraint: V. Arvind Rameshwar, Navin Kashyap Computable Lower Bounds for Capacities of Input-Driven FSCs
Background Ideas and Directions Motivation Single-Letter Lower Bound System Model Applications Future Work Examples Input-Constrained DMCs: ( d , ∞ )-RLL input constraint: ( d , k )-RLL input constraint: V. Arvind Rameshwar, Navin Kashyap Computable Lower Bounds for Capacities of Input-Driven FSCs
Background Ideas and Directions Motivation Single-Letter Lower Bound System Model Applications Future Work Examples Other Channels: Input-Constrained DMCs: ( d , ∞ )-RLL input constraint: ( d , k )-RLL input constraint: V. Arvind Rameshwar, Navin Kashyap Computable Lower Bounds for Capacities of Input-Driven FSCs
Background Ideas and Directions Motivation Single-Letter Lower Bound System Model Applications Future Work Examples Other Channels: Flash-Memory Channel Input-Constrained DMCs: (101 → 111 w.p. ǫ ): ( d , ∞ )-RLL input constraint: ( d , k )-RLL input s i = ( x i , x i − 1 ) constraint: V. Arvind Rameshwar, Navin Kashyap Computable Lower Bounds for Capacities of Input-Driven FSCs
Background Ideas and Directions Motivation Single-Letter Lower Bound System Model Applications Future Work Examples Other Channels: Flash-Memory Channel Input-Constrained DMCs: (101 → 111 w.p. ǫ ): ( d , ∞ )-RLL input constraint: ( d , k )-RLL input s i = ( x i , x i − 1 ) constraint: m ISI: y i = � h k x i − k + z i k =0 s i = ( x i − m , . . . , x i − 1 ) V. Arvind Rameshwar, Navin Kashyap Computable Lower Bounds for Capacities of Input-Driven FSCs
Background Ideas and Directions Motivation Single-Letter Lower Bound System Model Applications Future Work Examples Other Channels: Flash-Memory Channel Input-Constrained DMCs: (101 → 111 w.p. ǫ ): ( d , ∞ )-RLL input constraint: ( d , k )-RLL input s i = ( x i , x i − 1 ) constraint: m ISI: y i = � h k x i − k + z i k =0 s i = ( x i − m , . . . , x i − 1 ) Assumption: s 0 known to encoder and decoder V. Arvind Rameshwar, Navin Kashyap Computable Lower Bounds for Capacities of Input-Driven FSCs
Background Ideas and Directions Motivation Single-Letter Lower Bound System Model Applications Future Work Summary of Results V. Arvind Rameshwar, Navin Kashyap Computable Lower Bounds for Capacities of Input-Driven FSCs
Background Ideas and Directions Motivation Single-Letter Lower Bound System Model Applications Future Work Summary of Results 1 N I ( X N ; Y N | s 0 ) C = lim max N →∞ Q ( x N | s 0 ) V. Arvind Rameshwar, Navin Kashyap Computable Lower Bounds for Capacities of Input-Driven FSCs
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