Short Polar Codes Peihong Yuan Chair for Communications Engineering July 26, 2016 LNT & DLR Summer Workshop on Coding 1 / 23
Outline 1 Motivation 2 Improve the Distance Property 3 Simulation Results 4 sum-min Approx. 5 Rate Matching/IR-HARQ 6 Conclusions 2 / 23
Polar codes Provable Capacity-Achieving 1 Encoding: Precoding (k n) Polar transform (n n) 1 2 n log n (parallelizable) Successive Cancellation (SC) decoding (sequential): 1 n log n + 2 n log n 1 2 x y: 2 tanh 1 [tanh x 2 tanh y ] sign(x) sign(y) min( x, y ) 2 1 E. Arıkan. Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels. IEEE Trans. on Information Theory, 2009 3 / 23
Construction of Polar codes (Gaussian Approx. 2 with J-function 3 ) I (W (2i 1) N ) + I (W (i) N/2 ) I (W (2i) N ) I (W (i) N/2 ) I (W (1) 1 ) = J(2/σ n) I (W (2i 1) N ) = 1 J ( 2 ) ( )] 2 [J 1 1 I (W (i) N/2 ) ) ( )] 2 I (W (2i) N ( 2 [J ) = J 1 I (W (i) N/2 ) 2 S. ten Brink et al. Design of low-density parity-check codes for modulation and detection. IEEE Trans. on Communications, 2004 3 F. Brännström. Convergence analysis and design of multiple concatenated codes. Ph.D. dissertation, Chalmers Univ. Technol., Göteborg, Sweden, Mar. 2004. 4 / 23
Polar codes vs. LTE-Turbo codes (1024, 512) 10 0 Polar Codes, SC Polar Codes, SC (GA Est.) LTE-Turbo codes log-map, NoIt=10 10 1 10 2 FER 10 3 10 4 10 5 1 1.5 2 2.5 3 3.5 SNR in db 5 / 23
SC List Decoding 4 Time Complexity: O(Ln log n) Space Complexity: L(2n 1) float 2L(2n 1) boolean ML-achieving decoding (L 2 k ) 4 I. Tal and A. Vardy. List decoding of polar codes. IEEE Trans. on Information Theory, 2015 6 / 23
Polar codes vs. LTE-Turbo codes (1024, 512) 10 0 Polar Codes, SC Polar Codes, SCL, L=32 Polar ML (lower) bound 10 1 LTE-Turbo codes log-map, NoIt=10 10 2 FER 10 3 10 4 10 5 1 1.5 2 2.5 3 3.5 SNR in db 7 / 23
ML (lower) bound 5 Algorithm 1 Estimate ML bound 1: ĉ = decode(y) 2: if ĉ c then 3: error = error + 1; 4: if ĉy T > cy T then 5: error ml = error ml + 1; 6: end if 7: end if 5 I. Tal and A. Vardy. List decoding of polar codes. IEEE Trans. on Information Theory, 2015, Sec. 5 8 / 23
Distance Property of Polar codes F n = ( ) m 1 0 1 1 F n n n k row deletions G k n Minimum Hamming distance of (1024, 512) Polar codes: 16 (Design SNR < 5 db) 32 (Design SNR 5 db) 9 / 23
Outline 1 Motivation 2 Improve the Distance Property 3 Simulation Results 4 sum-min Approx. 5 Rate Matching/IR-HARQ 6 Conclusions 10 / 23
Outer codes + Polar codes CRC-aided (CA)-Polar codes 6 Flexible HARQ/Adaptive-decoding Minimum Distance? 6 K. Niu and K. Chen. CRC-aided decoding of polar codes. IEEE Communications Letters, 2012 11 / 23
RM-Polar codes 7 RM codes and Polar codes are obtrained from same polarization matrix F m 2. Polar rule: freeze the unreliable bits RM rule: freeze the bits with low weight of their corresponding rows RM-Polar codes: semi-rm semi-polar rule SCL decodable Minimum Distance not Flexible 7 B. Li et al. A RM-polar codes. arxiv preprint arxiv:1407.5483, 2014 12 / 23
ebch-polar codes 8 Dynamic frozen bits (k, n, d) ebch codes (k > k) with H c = uf m 2, ch T = 0 uf m 2 H T = 0, let V n (n k ) = F m 2 H T (k, n, d) ebch-polar codes SCL decodable Minimum Distance Flexible 8 P. Trifonov and V. Miloslavskaya. Polar subcodes. IEEE Journal on Selected Areas in Communications, 2016 13 / 23
Outline 1 Motivation 2 Improve the Distance Property 3 Simulation Results 4 sum-min Approx. 5 Rate Matching/IR-HARQ 6 Conclusions 14 / 23
Simulation Results, R = 1/2, n = 1024 10 0 10 1 10 2 (1056,528) WiMax LDPC, BP, NoIt=100 LTE-Turbo, log-map, NoIt = 10 (1024,512) Polar SCL, L=32 (1024,512, 32) RM-Polar SCL, L = 32 (1024,512, 28) ebch-polar codes SCL, L=32 (1024,512) Polar CA-SCL, L=32, CRC-16 FER 10 3 10 4 10 5 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 SNR in db 15 / 23
Outline 1 Motivation 2 Improve the Distance Property 3 Simulation Results 4 sum-min Approx. 5 Rate Matching/IR-HARQ 6 Conclusions 16 / 23
sum-min Approximation Polar codes 5 4 1 (x + x ) 2 ln(1 + e x ) 3 2 1 0 5 4 3 2 1 0 1 2 3 4 5 x 2 tanh 1 [tanh x 2 tanh y 2 ] = sign(x) sign(y) min( x, y )+ ln(1 + e x+y ) ln(1 + e x y ) 17 / 23
sum-min Approximation for Polar codes 10 0 Polar Codes, SC (tanh rule) Polar Codes, SC (sum-min aprrox) Polar Codes, SCL, L=32 (tanh rule) 10 1 Polar Codes, SCL, L=32 (sum-min aprrox) Polar Codes+CRC, SCL, L=32 (tanh rule) Polar Codes+CRC, SCL, L=32 (sum-min aprrox) 10 2 FER 10 3 10 4 10 5 1 1.5 2 2.5 3 3.5 SNR in db 18 / 23
Outline 1 Motivation 2 Improve the Distance Property 3 Simulation Results 4 sum-min Approx. 5 Rate Matching/IR-HARQ 6 Conclusions 19 / 23
Rate Matching/IR-HARQ k is controlled via bit-freezing n is controlled via puncturing mother code length N = 2 log 2 n, the first N n bits will be punctured IR: equivalent puncture pattern I (1) 0 + I (0) 0 I (1) 0 = 1 J I (1) 1 = J I (1) 1 ( [ ( J 1 ( [ ( J 1 I (0) 0 1 I (0) 0 )] 2 + [J 1 ( I (0) 1 )] 2 + [J 1 ( I (0) 1 )] 2 ) 1 I (0) 1 )] 2 ) 20 / 23
Rate 0.2 Turbo & Polar Codes 10 0 (1600, 320) LTE-Turbo codes, log-map, NoIt=10 (1600, 320) Polar CA-SCL (L=32, CRC-16) 10 1 10 2 FER 10 3 10 4 10 5 10 6 4 3.5 3 2.5 2 1.5 1 SNR in db 21 / 23
Outline 1 Motivation 2 Improve the Distance Property 3 Simulation Results 4 sum-min Approx. 5 Rate Matching/IR-HARQ 6 Conclusions 22 / 23
Conclusions Pros: Cons: Good Performance Efficient Design Low Complexity Encoding/Decoding no High-Throughput VLSI Architecture no Adaptive Decoding 23 / 23