Twisted-Pair Superposition Transmission for Low Latency Communications Suihua Cai and Xiao Ma School of Data and Computer Science Sun Yat-sen University, Guangzhou 510006, China June, 2020 S. Cai and X. Ma (SYSU) TPST Codes June, 2020 1 / 17
Outline Motivation 1 Twisted-Pair Superposition Transmission 2 Encoding Decoding Performance Analysis and Examples 3 Genie-Aided lower bounds Superposition Fraction Optimization Early Termination Constructions with different rates Conclusions 4 S. Cai and X. Ma (SYSU) TPST Codes June, 2020 2 / 17
Coding for URLLC To meet the stringent requirements of low latency, the channel codes for URLLC should be designed for short block length. Performance bounds on the channel capacity in the finite length regime. Efficient short code designs BCH Tail-biting convolutional codes (TBCC), TBCC+CRC polar, polar+CRC LDPC . . . When short blocks are transmitted, the concatenation of CRC incurs considerable rate loss. Y. Polyanskiy, H. V. Poor, and S. Verd´ u, “Channel coding rate in the finite blocklength regime,” IEEE Transactions on Information Theory , vol. 56, no. 5, pp. 2307–2359, May 2010. S. Cai and X. Ma (SYSU) TPST Codes June, 2020 3 / 17
Encoding of TPST u (0) v (0) c (0) C LAYER 0 w (1) R S w (0) u (1) v (1) c (1) C LAYER 1 Basic Forward Backward Encoding Superposition Superposition Figure: Encoding structure of TPST codes. (1) Basic Encoding : Encode u ( i ) into v ( i ) ∈ F n 2 , for i = 0 , 1 , by the encoding algorithm of the basic code C [ n , k ] . We assume that C [ n , k ] has an efficient list decoding algorithm. The coderate of TPST is determined by the basic code. S. Cai and X. Ma (SYSU) TPST Codes June, 2020 4 / 17
Encoding of TPST u (0) v (0) c (0) C LAYER 0 w (1) R S w (0) v (1) c (1) u (1) C LAYER 1 Basic Forward Backward Encoding Superposition Superposition Figure: Encoding structure of TPST codes. (2) Forward Superposition : Compute w (0) = v (0) R and c (1) = v (1) + w (0) . R is an n × n binary random matrix. Any difference in Layer 0 will result in a random flipping effect on Layer 1. Intuitive perspective: We can identify the correctness of Layer 0 by looking at the decoding behavior of Layer 1. S. Cai and X. Ma (SYSU) TPST Codes June, 2020 5 / 17
Encoding of TPST u (0) v (0) c (0) C LAYER 0 w (1) R S w (0) v (1) c (1) u (1) C LAYER 1 Basic Forward Backward Encoding Superposition Superposition Figure: Encoding structure of TPST codes. (3) Backward Superposition : Compute w (1) = c (1) S and c (0) = v (0) + w (1) . S is a binary diagonal matrix, referred to as a selection matrix . The fraction of ones in the diagonal entries of S is denoted by α . Intuitive perspective: We balance the channels by a“ partial one-step polarization ” , which improves the Layer 1 and worsens the Layer 0. S. Cai and X. Ma (SYSU) TPST Codes June, 2020 6 / 17
Decoding of TPST C LAYER 0 R S C LAYER 1 Figure: Decoding of TPST codes. (1) Compute Λ( v (0) ) , the LLR of Layer 0, from the“bad”channel similar to polar codes. (2) Perform the list decoding of the basic code by taking Λ( v (0) ) as input, v (0) resulting in a list of candidate codewords ˆ ℓ , ℓ = 0 , 1 , . . . , ℓ max − 1 . S. Cai and X. Ma (SYSU) TPST Codes June, 2020 7 / 17
Decoding of TPST C LAYER 0 R S C LAYER 1 Figure: Decoding of TPST codes. v (0) v (0) ℓ , compute Λ ℓ ( v (1) ) by treating ˆ (3) For each candidate codeword ˆ as ℓ correct. v (1) by taking Λ ℓ ( v (1) ) as input to the basic decoder. (4) Estimate ˆ ℓ v (0) v (1) (5) Output the basic codewords (ˆ ℓ ∗ , ˆ ℓ ∗ ) such that P ( y | ˆ c ℓ ∗ ) = max P ( y | ˆ c ℓ ) . ℓ S. Cai and X. Ma (SYSU) TPST Codes June, 2020 8 / 17
Decoding of TPST Intuitive perspective: S C LAYER 1 Figure: Decoding of TPST codes. If the ℓ -th candidate codeword is correct, i.e. ∆ v (0) = ˆ + v (0) = 0 , then v (0) ℓ v (1) is equivalently transmitted twice (one is with partial masking) over the channel. Otherwise, ∆ v (0) � = 0 , v (1) is equivalent to be flipped randomly (since R is a random matrix) before the transmission. We expect these two cases to have distinguishable decoding performance of Layer 1. S. Cai and X. Ma (SYSU) TPST Codes June, 2020 9 / 17
Genie-aided Lower Bounds For Layer 0, P ( E 0 ) ≥ P (0) genie , The genie-aided decoder outputs the transmitted codeword if it is in the list (told by a genie). P (0) genie is obtained by taking the LLRs from the“bad”channel as input and employing list decoding of the basic code. For Layer 1, P ( E 1 ) ≥ P (1) genie , The genie-aided decoder correctly remove the effect of Layer 0 and employ decoding of the basic code. P (1) genie is obtained by transmitting the codeword of Layer 1 twice (once with partial masking) over the channel. FER = P ( E 0 ∪ E 1 ) ≥ max { P ( E 0 ) , P ( E 1 ) } ≥ max { P (0) genie , P (1) genie } . S. Cai and X. Ma (SYSU) TPST Codes June, 2020 10 / 17
Genie-aided Lower Bounds 10 0 10 -1 10 -2 FER 10 -3 sim, list=512 (0) P genie , list=512 sim, list=1024 10 -4 (0) P genie , list=1024 sim, list=2048 10 -5 (0) P genie , list=2048 (1) P genie 10 -6 1 1.5 2 2.5 3 3.5 SNR (dB) Figure: Decoding of TPST-TBCCs with different list sizes. The basic code is (2 , 1 , 4) TBCC with information length k = 32 ( n = 64 ) and α = 1 . S. Cai and X. Ma (SYSU) TPST Codes June, 2020 11 / 17
Superposition Fraction Optimization To narrow down the gap between the two layers, we introduce partial superposition characterized by the so-called superposition fraction α . From Layer 0, LAYER 0 C v (0) is transmitted with binary Binary interference c (1) S . R S α ↓ ⇒ P (0) Interfrence genie ↓ C LAYER 1 From Layer 1, LAYER 0 C v (1) is transmitted partially twice Partial if v (0) is known. R S Retransmission α ↓ ⇒ P (1) genie ↑ C LAYER 1 S. Cai and X. Ma (SYSU) TPST Codes June, 2020 12 / 17
Superposition Fraction Optimization 10 0 10 -1 sim, α =1 10 -2 (0) P genie , α =1 FER (1) P genie , α =1 sim, α =0.75 10 -3 (0) P genie , α =0.75 (1) P genie , α =0.75 sim, α =0.5 10 -4 (0) P genie , α =0.5 (1) P genie , α =0.5 1 1.5 2 2.5 3 3.5 SNR (dB) Figure: Decoding of TPST-TBCCs with different superposition fractions. The basic code is (2 , 1 , 4) TBCC with information length k = 32 ( n = 64 ) and ℓ max = 2048 . S. Cai and X. Ma (SYSU) TPST Codes June, 2020 13 / 17
Complexity Analysis and Early Termination We use the metric of empirical divergence function (EDF) for the early termination, which is given as D ( y , v ) = 1 P ( y | c ) 2 n log 2 P ( y ) . Note that an erroneous candidate can cause a significant change on the joint c (0) c (1) ℓ ) and ( y (0) , y (1) ) . typicality between (ˆ ℓ , ˆ v (0) v (1) A candidate ˆ v ℓ = (ˆ ℓ , ˆ ℓ ) is treated as correct if D ( y , ˆ v ℓ ) > T . Table: Average list size needed for different threshold T SNR 1.0 1.4 1.8 2.2 2.6 3.0 3.4 3.6 T = 0 . 4 12.1 18.1 18.8 13.1 9.1 3.7 1.8 1.4 T = 0 . 5 459 275 132 55.3 22.9 7.5 2.7 1.9 T = 0 . 6 1412 1042 685 396 199 86.6 33.4 20.1 The basic code is (2 , 1 , 4) TBCC with information length k = 32 ( n = 64 ), α = 0 . 75 and ℓ max = 2048 . S. Cai and X. Ma (SYSU) TPST Codes June, 2020 14 / 17
Complexity Analysis and Early Termination 10 0 MC bound RCU bound TBCC, m=8 10 -1 TBCC, m=11 CRC-TBCC, m=11, known state CRC-TBCC, m=11, unknown state TPST-TBCC, T=0.4 10 -2 TPST-TBCC, T=0.5 TPST-TBCC, T=0.6 FER TPST-TBCC, GA bound 10 -3 10 -4 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 SNR (dB) Figure: Decoding of the TPST-TBCCs with different thresholds. M. C. Co¸ skun, G. Durisi, T. Jerkovits, G. Liva, W. Ryan, B. Stein, and F. Steiner, “Efficient error-correcting codes in the short blocklength regime,” Physical Communication , vol. 34, pp. 66 – 79, Mar. 2019. S. Cai and X. Ma (SYSU) TPST Codes June, 2020 15 / 17
Constructions with different rates 10 0 MC bounds RCU bounds rate-1/4 TPST-TBCC 10 -1 rate-1/3 TPST-TBCC rate-1/2 TPST-TBCC rate-3/4 TPST-TBCC 10 -2 FER 10 -3 10 -4 -2 -1 0 1 2 3 4 5 6 SNR (dB) Figure: Decoding of the TPST-TBCCs with different coding rates. We take TBCCs with constraint length m = 4 as basic codes, α = 0 . 75 and ℓ max = 2048 . S. Cai and X. Ma (SYSU) TPST Codes June, 2020 16 / 17
Conclusions We proposed the TPST codes with capacity-approaching performance in the finite length regime. The basic idea is to use the decoding behavior of the next layer to identify the correctness of the candidate codeword instead of CRC. We presented the genie-aided bounds for the two layers of TPST, providing an intuitive perspective how the performance of the two layer are effected by the partial superposition. We also presented thresholds for early termination of list decoding, which can significantly reduce the decoding complexity. We showed by numerical results that the proposed TPST codes have near-capacity performance in a wide range of coding rates. Thank You! S. Cai and X. Ma (SYSU) TPST Codes June, 2020 17 / 17
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