in the Finite Memory Length Regime Joint work of Yu-Chih Huang , - PowerPoint PPT Presentation

2020 IEEE International Symposium on Information Theory Error Rate Analysis for Random Linear Streaming Codes in the Finite Memory Length Regime Joint work of Yu-Chih Huang , National Chiao Tung University, Shih-Chun Lin , National Taiwan University of Science and Technology, I-Hsiang Wang , National Taiwan University, and Chih-Chun Wang , Pin-Wen Su , Purdue University 6/7/2020 Sponsored by NSF CCF-1422997, CCF-1618475 and CCF-1816013, and by MOST Taiwan 107-2628-E-011-003-MY3.

Outline  Motivation and Related Work  System Model  Main Contributions  Information-Debt Under Finite Memory  Exact Error Rate Analysis  A Provably-Tight Closed-Form Error Rate Approximation  Numerical Verification and Conclusion

5G Communication Systems  IMT-2020  Enhanced mobile broadband (eMBB)  Massive machine type communications (mMTC)  Ultra-reliable and low latency communications (URLLC)  Low Latency Requirements  End-to-end delay ≤ 1 ms  Streaming codes may be a possible solution. The figure is copied from International Telecommunication Union, “Setting the Scene for 5G: Opportunities and Challenges.” (2018) . https://www.itu.int/dms_pub/itu-d/opb/pref/D-PREF-BB.5G_01-2018-PDF-E.pdf

Streaming Codes  Very small queueing delay : Good for URLLC & mMTC  Streaming Codes:  Align the convolutional code structure with the actual transmission/encoding schedule.  There are other definitions of Streaming Codes, but we use this basic definition.  What we are interested: Error Rate of Streaming Codes with a given finite memory length.

Existing Results for Streaming Codes  Adversarial Channel Model: Optimal streaming code rate and code construction given a deterministic set of possible channel error patterns  Burst Channel Model [Martinian and Sundberg 2004] [Khisti and Singh 2009]  Burs t and Arbitrary Erasure Channel Model [Fong et al. 2019] [Krishnan et al. 2018] [Badr et al. 2017]  Variable-Size Arrivals [Rudow and Rashmi 2018]  Stochastic Channel Model: Error exponent analysis and finite-memor y code construction [Draper and Khisti 2011] Draper and Khisti 2011 Our Work Deadline Constraint (Δ) Finite Infinite Memory Length (𝛽) 𝛽 ≥ Δ Arbitrary Error Probability Error Exponent Analysis Exact Error Rate Analysis

Error Exponent Analysis for Convolutional-based Codes  Continue from previous slide Draper and Khisti 2011 Our Work Deadline Constraint (Δ) Finite Infinite Memory Length (𝛽) 𝛽 ≥ Δ Arbitrary Error Probability Error Exponent Analysis Exact Error Rate Analysis  Exponentially tight is not tight enough. Draper et al. 2011; Viterbi 1967 Our Work Asymptotic analysis, Exponential in addition to exact error rate analysis

Outline  Motivation and Related Work  System Model  Main Contributions  Information-Debt Under Finite Memory  Exact Error Rate Analysis  A Provably-Tight Closed-Form Error Rate Approximation  Numerical Verification and Conclusion

Slotted Coding System  𝛽 = 2 : In every time slot 𝑢 ≥ 1 ,  Encoder:  Receives 𝐿 packets:  𝑡 𝑙 (𝑢) in GF(2 𝑟 ) .  Stores 𝛽𝐿 packets in the previous 𝛽 slots  𝛽 is the memory length .  Encodes (𝛽 + 1)𝐿 packets and outputs 𝑂 coded packets:  Linear encoder : Define 𝐇 𝑢 as the 𝑂 -by- (min 𝛽 + 1, 𝑢 𝐿) generator matrix, we have  Random linear streaming codes (RLSCs): each entry of 𝐇 𝑢 is chosen uniformly randomly from GF(2 𝑟 ) , excluding 0.  Cumulative generator matrix:

Comparison to [Martinian 2004] Coding System  Our work: Finite memory  Martinian setting: Infinite memory 𝛽 = 2 finite 𝛽 𝛽 = ∞

Slotted Coding System  𝛽 = 2 : In every time slot 𝑢 ≥ 1 ,  Packet Erasure Channel:  Only a random subset of 𝑂 coded packets, denoted by , will arrive at the decoder perfectly.  is i.i.d. across 𝑢 . Define  is the probability of receiving 𝑗 packets successfully.

Slotted Coding System  𝛽 = 2 : In every time slot 𝑢 ≥ 1 ,  Packet Erasure Channel:  Only a random subset of 𝑂 coded packets, denoted by , will arrive at the decoder perfectly.  is i.i.d. across 𝑢 . Define  is the probability of receiving 𝑗 packets successfully.  Received Signal:  The received packets:  Denote 𝐈 𝑢 the projection of 𝐇 𝑢 onto the random set  Cumulative receiver matrix:

Slotted Coding System Random matrix depends on channel realization Definition 1. The vector 𝐭(𝑢) is decodable by time 𝑢 + Δ if all 𝑡 𝑙 𝑢 ∶ 𝑙 ∈ 1, 𝐿 are decodable by time 𝑢 + Δ . 𝑢+Δ , we  With optimal decoder on the received 𝐳 1 aim to solve the following:  Objective: Given any finite 𝑂 , 𝐿 , 𝛽 and 𝑄 𝑗 , Slot Error Rate Average Error Rate Infinite Deadline

Technical Assumptions  Less-than-Capacity ( LC ) condition: Assume  Each slot: 𝑂 pkts 𝐿 pkts 𝐷 𝑢 pkts Encoder Channel  Generalized MDS Condition:  𝐇 (𝑢) : as full rank as possible (details in the paper)  MDS holds when 𝑟 → ∞  See Schwartz-Zippel Theorem  in [Ho et al. 2006, Theorems 3 and 4]  Avoid corner cases in the analysis

Outline  Motivation and Related Work  System Model  Main Contributions  Information-Debt Under Finite Memory  Exact Error Rate Analysis  A Provably-Tight Closed-Form Error Rate Approximation  Numerical Verification and Conclusion

Information-Debt Under Infinite Memory  Mutual information debt under infinite memory 𝛽 = ∞ [Martinian 2004] Definition. Initialize . For any , we iteratively comput e Debt is Nonnegative

Information-Debt Under Infinite Memory  Mutual information debt under infinite memory 𝛽 = ∞ [Martinian 2004] 3 1 Definition. Initialize . For any , 3 2 3 we iteratively comput e 7 Observation: wherever 𝐽 𝑒 (𝑢) hits 0, we can decode 𝐭(𝑢) backwards. 𝐿 < 𝐷 𝑢 : decrease 𝐿 > 𝐷 𝑢 : increase

Information-Debt Under Infinite Memory  Mutual information debt under infinite memory 𝛽 = ∞ [Martinian 2004] Definition. Initialize . For any , we iteratively comput e Observation: wherever 𝐽 𝑒 (𝑢) hits 0, we can decode 𝐭(𝑢) backwards. Q: What if 𝐈 (𝑢+Δ) is NOT full triangular? E.g. a 4-by-4 matrix which is not full rank

Information-Debt Under Finite Memory  New information debt definition under finite memory 𝛽 < ∞ Definition 2. Define a constant and initialize _ . For any , we iteratively comput e  Absolute “ceiling” ∵ Memory length 𝛽  Bankruptcy ∴ Maximum allowable debt one can carry forward is at most 𝛽𝐿 Bankruptcy 𝛽𝐿 Maximum Allowable Debt

Decodability Events Define , , and Proposition 1. For any fixed 𝑗 0 ≥ 0 , a) No 𝜐 𝑘 ∈ 𝑢 𝑗 0 , 𝑢 𝑗 0 +1 , then 𝐭(𝑢) is decodable by time 𝑢 𝑗 0 +1 for all 𝑢 ∈ (𝑢 𝑗 0 , 𝑢 𝑗 0 +1 ] . b) Exists 𝜐 𝑘 ∈ 𝑢 𝑗 0 , 𝑢 𝑗 0 +1 , define 𝜐 𝑘 ∗ the one with the as the 𝑗 -th time that 𝐽 𝑒 𝑢 hits 0 and , largest 𝑘 . Then 𝐭(𝑢) is decodable by time 𝑢 𝑗 0 +1 for all respectively. 𝑢 ∈ (𝜐 𝑘 ∗ − 𝛽 , 𝑢 𝑗 0 +1 ] . 𝐿 = 1, 𝛽 = 3 𝛽 − 1 o o o o o o o o o o o o o Decodable Decodable

Error Events Define , , and Proposition 2. None of 𝐭 𝑢 : 𝑢 ∈ (𝑢 𝑗 0 , 𝜐 𝑘 ∗ − 𝛽] is decodable by time 𝜐 𝑘 ∗ − 𝛽 + Δ , regardless how large we set the deadline Δ . as the 𝑗 -th time that 𝐽 𝑒 𝑢 hits 0 and , respectively. 𝐿 = 1, 𝛽 = 3 𝛽 − 1 o o o o o o o x o o o x x x x x o o o x x Decodable Error Decodable

Intuition Behind Define , , and  Enough linear equations  Start decoding from 𝐭(𝑢 𝑗 0 +1 ) , 𝐭(𝑢 𝑗 0 +1 − 1) , ⋯ , in a backward fashion as the 𝑗 -th time that 𝐽 𝑒 𝑢 hits 0 and , respectively. 𝐿 = 1, 𝛽 = 3 𝛽 − 1 o o o o o o o x o o o x x x x x o o o x x Decodable Error Decodable

Intuition Behind Define , , and  Coupling between 𝐭 𝑢 : 𝑢 ≤ 𝜐 𝑘 ∗ − 𝛽 and 𝐭 𝑢 : 𝑢 > 𝜐 𝑘 ∗ − 𝛽 is severed once 𝐽 𝑒 (𝑢) hits (bankruptcy) as the 𝑗 -th time that 𝐽 𝑒 𝑢 hits 0 and , respectively. 𝐿 = 1, 𝛽 = 3 𝛽 − 1 o o o o o o o x o o o x x x x x o o o x x Decodable Error Decodable

Exact Error Rate Analysis  𝐷 𝑢 is i.i.d. ⟹ 𝐽 𝑒 𝑢 is a renewal Markov process.  Information debt:  The state space: .  With 𝐽 𝑒 (𝑢) being renewal Markov process , the long term average error rate can be computed by Lemma 2. Assuming the LC and MDS conditions, we have  Not Stopping Time  More involved analysis  𝐽 𝑒 𝑢 from state-0 to state-0  Stopping Time for any fixed 𝑗 0 , where is the indicator function.