Partially Information Coupled Duo-binary Turbo Codes Xiaowei Wu, Min Qiu, and Jinhong Yuan School of Electrical Engineering and Telecommunications University of New South Wales Sydney, Australia ISIT 2020 Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 1 / 28
Outline Introduction 1 Construction of partially information coupled duo-binary turbo codes (PIC-dTCs) 2 Performance analysis of PIC-dTCs via density evolution 3 Numerical results 4 Conclusion 5 Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 2 / 28
Table of Contents Introduction 1 Construction of partially information coupled duo-binary turbo codes (PIC-dTCs) 2 Performance analysis of PIC-dTCs via density evolution 3 Numerical results 4 Conclusion 5 Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 3 / 28
Introduction Spatial coupling: Connects a sequence of component codes to form a long codeword chain. Applied to: LDPC codes [1][2], turbo-like codes [3][4]. Spatially coupled codes provide close-to-capacity performance. Spatially coupled codes have threshold saturation phenomenon. [1] A. Jimenez Felstrom and K. S. Zigangirov, “Time-varying periodic convolutional codes with low-density parity-check matrix,” vol. 45, no. 6, Sep. 1999, pp. 2181–2191. [2] S. Kudekar, T. J. Richardson, and R. L. Urbanke, “Threshold saturation via spatial coupling: Why convolutional LDPC ensembles perform so well over the bec,” IEEE Trans. Inf. Theory , vol. 57, no. 2, pp. 803–834, Feb 2011. [3] S. Moloudi, M. Lentmaier, and A. Graell i Amat, “Spatially coupled turbo-like codes,” IEEE Trans. Inf. Theory , vol. 63, no. 10, pp. 6199–6215, Oct 2017. [4] W. Zhang, M. Lentmaier, K. S. Zigangirov, and D. J. Costello, “Braided convolutional codes: A new class of turbo-like codes,” IEEE Trans. Inf. Theory , vol. 56, no. 1, pp. 316–331, Jan 2010. Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 4 / 28
Introduction In [5][6], we proposed the partially information coupled turbo codes (PIC-TCs). Consecutive component turbo code blocks (CBs) are coupled by sharing a portion of information bits between each other. Achieve a large coupling gain without modifying the component encoder and decoder architecture. However, PIC-TCs have rate loss compared to its component turbo codes, i.e., R < 1 3 . e.g. coupling 1 2 the information bits results in R = 1 5 . [5] L. Yang, Y. Xie, X. Wu, J. Yuan, X. Cheng, and L. Wan, “Partially information-coupled turbo codes for LTE systems,” IEEE Trans. Commun. , pp. 1–1, 2018. [6] M. Qiu, X. Wu, Y. Xie, and J. Yuan, “Density evolution analysis of partially information coupled turbo codes on the erasure channel,” in 2019 IEEE Information Theory Workshop (ITW) , 2019, pp. 1–5. Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 5 / 28
Introduction Our contribution in this work: We proposed the partially information coupled duo-binary turbo codes (PIC-dTCs), which can have a consistent code rate R = 1 3 (no rate loss). We derive the density evolution (DE) equations for the PIC-dTCs, and show that the BP decoding threshold of PIC-dTCs is within a gap of 0.001 to the BEC capacity. Simulation results verify the DE analysis, and show that PIC-dTCs outperform PIC-TCs and the uncoupled turbo codes. Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 6 / 28
Review of PIC-TCs construction Component code: rate- 1 3 turbo code (TC1). [ u , v ] u ( t 1)-th CB u , t 1 t 1 t 1 t 1 t 1 CB t : component code block at time t . Encoding u 1, t t u t : information sequence at time t . [ u v , ] u u u t,t : uncoupled info. t -th CB t t t t , t Encoding u u t − 1 ,t : info. shared between CB t − 1 and CB t . t t , 1 u t,t +1 : info. shared between CB t and CB t +1 . [ u , v ] u u ( t+ 1)-th CB t 1 t 1 1 , 1 t t t 1 v t : parity bits. Encoding u t t , 1 � u t − 1 ,t � + � u t � = K . � u t − 1 ,t � = � u t,t +1 � = K c . Fig. 1: Block diagram of PIC-TC encoder with coupling memory m = 1 . Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 7 / 28
Review of PIC-TCs construction u t − i,t : info. shared between CB t − i and CB t , 1 ≤ i ≤ m . u t,t + i : info. shared between CB t and CB t + i , 1 ≤ i ≤ m . [ u , , u 1, ] � 1 ≤ i ≤ m � u t − i,t � + � u t � = K . , t m t t t [ u v , ] u u � u t − i,t � = � u t,t + i � = K c t -th CB m . t t t t , t Encoding Number of CBs: L . [ u , , u ] t t , 1 , t t m Coupling ratio: λ = K c K . Code rate: Fig. 2: Block diagram of PIC-TC encoder with R = KL − K c L − � m iK c 1 − λ coupling memory m ≥ 1 . i =1 L →∞ m = 3 − λ. 3 KL − K c L − � m iK c i =1 m ⊲ Rate loss due to coupling. Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 8 / 28
Table of Contents Introduction 1 Construction of partially information coupled duo-binary turbo codes (PIC-dTCs) 2 Performance analysis of PIC-dTCs via density evolution 3 Numerical results 4 Conclusion 5 Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 9 / 28
Construction of PIC-dTCs Step 1: Construct a rate-2/3 RSC code (RSC2) with a given rate-1/2 RSC code (RSC1). � � 1 0 5 � 1 5 � , and G RSC 2 = . G RSC 1 = 7 7 0 1 3 7 When u ′ = 0 , the parity sequence from RSC2 is equal to that of RSC1. 𝐯′ 𝐯′ 𝐯 𝐯 𝐯 𝐯 𝐰 𝐰 D D D D (a) (b) Fig. 3: Encoder block diagram of (a) RSC1, (b) RSC2. Step 2: Construct a duo-binary turbo code (TC2) by concatenate RSC2 in parallel. [7] C. Berrou and M. Jezequel, “Non-binary convolutional codes for turbo coding,” Electronics Letters , vol. 35, no. 1, pp. 39–40, Jan 1999. [8] C. Douillard and C. Berrou, “Turbo codes with rate-m/(m+1) constituent convolutional codes,” IEEE Trans. Commun. , vol. 53, no. 10, pp. 1630–1638, Oct 2005. Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 10 / 28
Construction of PIC-dTCs u t 1 [ u , v ] 0 ( t 1)-th CB Step 3: Apply PIC to TC2 (coupling memory m = 1 ) u t 1 t 1 u t Encoding 1 t 1, t 1 u t : the 1st input sequence of TC2 at time t . u 1, t t u ′ t : the 2nd input sequence of TC2 at time t . t u [ u v , ] 0 u t,t : uncoupled info. t -th CB u t t u , Encoding t t t u t − 1 ,t : info. shared between CB t − 1 and CB t . u t t , 1 u t,t +1 : info. shared between CB t and CB t +1 . u t 1 [ u , v ] v t : parity bits. 0 ( t+ 1)-th CB t 1 t 1 u u Encoding 1, 1 t t t 1 � u t � = � u ′ t � = K . u 1, 2 t t � u t − 1 ,t � = � u t,t +1 � = K c . Fig. 4: Block diagram of PIC-dTC encoder with coupling memory m = 1 . Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 11 / 28
Construction of PIC-dTCs Step 3: Apply PIC to TC2 (coupling memory m ≥ 1 ) u t − i,t : info. shared between CB t − i and CB t , 1 ≤ i ≤ m . u t,t + i : info. shared between CB t and CB t + i , 1 ≤ i ≤ m . [ u , , u 1, ] , t m t t t t � u t � = � u ′ t � = K . u [ u v , ] 0 � u t − i,t � = � u t,t + i � = K c t -th CB m . t t u u Encoding , t t t Number of CBs: L . [ u , , u ] Coupling ratio: λ = K c t t , 1 , K . t t m Code rate: Fig. 5: Block diagram of PIC-dTC encoder with R = KL − � m iK c 1 L →∞ i =1 coupling memory m ≥ 1 . m 3 . = 3 KL − � m iK c i =1 m ⊲ no rate loss compared with TC1. Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 12 / 28
PIC-TC vs. PIC-dTC PIC-TC PIC-dTC TC1 TC2 Component code (parallel concatenation of rate- 1 (parallel concatenation of rate- 2 2 RSC) 3 RSC) 1 − λ 1 Code rate 3 − λ 3 Encoder [ u , , u 1, ] block , t m t t t [ u , , u 1, ] t diagram u , t m t t t 0 [ u v , ] u v [ , ] u t -th CB u t t t -th CB u , t t t t u t Encoding , t t Encoding t [ u , , u ] [ u , , u ] t t t t , 1 , , 1 t t m , t t m Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 13 / 28
Table of Contents Introduction 1 Construction of partially information coupled duo-binary turbo codes (PIC-dTCs) 2 Performance analysis of PIC-dTCs via density evolution 3 Numerical results 4 Conclusion 5 Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 14 / 28
Graph Model Representation 𝑉 𝑉 𝑉 𝐰 𝑢−1 𝐰 𝑢 𝐰 𝑢+1 𝑉 𝐰 𝑢 𝑔 𝑉 𝑔 𝑉 𝑔 𝑉 f U : upper decoder. 𝑔 𝑉 ′ ′ ′ 𝐯 𝑢−1 𝐯 𝑢−1 𝐯 𝑢 𝐯 𝑢 𝐯 𝑢+1 𝐯 𝑢+1 f L : lower decoder. 𝐯 t−1,t 𝐯 t,t+1 𝐯 t−2,t−1 𝐯 t+1,t+2 ′ 𝐯 𝑢 𝐯 𝑢 u : first information sequence. ⋮ ⋮ 𝟏 𝟏 𝟏 u ‘ : second information 𝐯 t−1,t−1 𝐯 t,t 𝐯 t+1,t+1 𝑔 𝑀 sequence. 𝑔 𝑀 𝑔 𝑀 𝑔 𝑀 v U , v L : parity sequence which 𝑀 𝐰 𝑢 enters upper or lower decoder. 𝑀 𝑀 𝑀 𝐰 𝑢−1 𝐰 𝑢 𝐰 𝑢+1 � : interleaver. (a) (b) Fig. 6: Compact factor graph of (a) uncoupled TC2, and (b) PIC-dTC with m = 1 . [9] B. M. Kurkoski, P. H. Siegel, and J. K. Wolf, “Exact probability of erasure and a decoding algorithm for convolutional codes on the binary erasure channel,” in IEEE Global Telecommunications Conference , vol. 3, Dec 2003, pp. 1741–1745 vol.3. Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 15 / 28
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