Polar Codes for Probabilistic Amplitude Shaping Tobias Prinz tobias.prinz@tum.de Second LNT & DLR Summer Workshop on Coding July 26, 2016 Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 1/16
Outline 1 Motivation 2 Polar Codes Polarization Polar Codes for BPSK Polar Codes for Higher-Order Modulation 3 Probabilistic Amplitude Shaping (PAS) 4 Numerical Results 5 Conclusions Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 2/16
Polar Codes Polar Codes achieve the symmetric capacity of binary-input memoryless channels. low-complexity encoding algorithms (O(N log N)) low-complexity decoding algorithms (O(N log N)) very good performance for finite lengths with some modifications (outer CRC-code) Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 3/16
Probabilistic Shaping 3.5 3 C(P) = 1 2 log 2(1 + P/1) 4-ASK uniform 8-ASK uniform Rate [bits/channel use] 2.5 2 1.5 1 0.5 6 8 10 12 14 16 18 20 22 SNR [db] Figure 1 : Rates for uniform and shaped ASK Constellations. Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 4/16
Probabilistic Shaping P Xuni (x) x Figure 1 : Rates for uniform and shaped ASK Constellations. Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 4/16
Probabilistic Shaping P Xuni (x) x P X (x) x Figure 1 : Rates for uniform and shaped ASK Constellations. Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 4/16
Probabilistic Shaping Rate [bits/channel use] 3.5 3 2.5 2 1.5 C(P) = 1 2 log 2(1 + P/1) 4-ASK shaped 4-ASK uniform 8-ASK shaped 8-ASK uniform 1 0.5 6 8 10 12 14 16 18 20 22 SNR [db] Figure 1 : Rates for uniform and shaped ASK Constellations. Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 4/16
Polarization 1 u 1 + + P Y X y 1 u 2 + P Y X y 2 u 3 + P Y X y 3 u 4 P Y X y 4 Figure 2 : Polar Transformation: N = 4. c 1 c 2 c 3 c 4 Polar Transformation: c = ug G = F m m = log 2 N [ ] 1 0 F = 1 1 Observation (Polarization) Most of the bit-channels U i (U1 i 1, Y1 N ) are either good channels I (U i ; Y1 N Ui 1 1 ) 1 or bad channels I (U i ; Y1 N Ui 1 1 ) 0. 1 E. Arıkan, Channel polarization: A method for constructing capacity-achieving codes, IEEE Trans. Inf. Theory, 2009 Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 5/16
Polar Codes Definition ((N,k) - Polar Code) An (N = 2 m, k) Polar Code is defined by the Polarization c = ug, where [ ] G = F m 1 0, m = log 2 N, F = 1 1 and a set of N k indices in u which correspond to frozen bits. The frozen bits are usually set to zero. The unfrozen bits of u contain the data to be transmitted. Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 6/16
Performance of Polar Codes 10 0 WiMax LDPC Code: 100 iterations LTE-Turbo (punctured) code: log-map: 5 iterations 10 1 10 2 FER 10 3 10 4 10 5 10 6 1 1.5 2 2.5 3 SNR [db] Figure 3 : rate 1/2. Performance of Polar Codes of length N = 1024 with overall Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 7/16
Performance of Polar Codes 10 0 10 1 WiMax LDPC Code: 100 iterations LTE-Turbo (punctured) code: log-map: 5 iterations Polar Code: SC Decoding 10 2 FER 10 3 10 4 10 5 10 6 1 1.5 2 2.5 3 SNR [db] Figure 3 : rate 1/2. Performance of Polar Codes of length N = 1024 with overall Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 7/16
Performance of Polar Codes 10 0 10 1 WiMax LDPC Code: 100 iterations LTE-Turbo (punctured) code: log-map: 5 iterations Polar Code: SC Decoding Polar Code: SCL Decoding L = 2 2 10 2 FER 10 3 10 4 10 5 10 6 1 1.5 2 2.5 3 SNR [db] Figure 3 : rate 1/2. Performance of Polar Codes of length N = 1024 with overall 2 I. Tal, A. Vardy, List decoding of polar codes, IEEE Trans. Inf. Theory,2015 Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 7/16
Performance of Polar Codes 10 0 10 1 WiMax LDPC Code: 100 iterations LTE-Turbo (punctured) code: log-map: 5 iterations Polar Code: SC Decoding Polar Code: SCL Decoding L = 2 2 Polar Code: SCL Decoding L = 4 10 2 FER 10 3 10 4 10 5 10 6 1 1.5 2 2.5 3 SNR [db] Figure 3 : rate 1/2. Performance of Polar Codes of length N = 1024 with overall 2 I. Tal, A. Vardy, List decoding of polar codes, IEEE Trans. Inf. Theory,2015 Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 7/16
Performance of Polar Codes 10 0 10 1 10 2 WiMax LDPC Code: 100 iterations LTE-Turbo (punctured) code: log-map: 5 iterations Polar Code: SC Decoding Polar Code: SCL Decoding L = 2 2 Polar Code: SCL Decoding L = 4 Polar Code: SCL Decoding L = 8 FER 10 3 10 4 10 5 10 6 1 1.5 2 2.5 3 SNR [db] Figure 3 : rate 1/2. Performance of Polar Codes of length N = 1024 with overall 2 I. Tal, A. Vardy, List decoding of polar codes, IEEE Trans. Inf. Theory,2015 Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 7/16
Performance of Polar Codes 10 0 10 1 10 2 WiMax LDPC Code: 100 iterations LTE-Turbo (punctured) code: log-map: 5 iterations Polar Code: SC Decoding Polar Code: SCL Decoding L = 2 2 Polar Code: SCL Decoding L = 4 Polar Code: SCL Decoding L = 8 Polar Code: SCL Decoding L = 32 FER 10 3 10 4 10 5 10 6 1 1.5 2 2.5 3 SNR [db] Figure 3 : rate 1/2. Performance of Polar Codes of length N = 1024 with overall 2 I. Tal, A. Vardy, List decoding of polar codes, IEEE Trans. Inf. Theory,2015 Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 7/16
Performance of Polar Codes 10 0 10 1 10 2 WiMax LDPC Code: 100 iterations LTE-Turbo (punctured) code: log-map: 5 iterations Polar Code: SC Decoding Polar Code: SCL Decoding L = 2 2 Polar Code: SCL Decoding L = 4 Polar Code: SCL Decoding L = 8 Polar Code: SCL Decoding L = 32 Polar Code: ML lower bound FER 10 3 10 4 10 5 10 6 1 1.5 2 2.5 3 SNR [db] Figure 3 : rate 1/2. Performance of Polar Codes of length N = 1024 with overall 2 I. Tal, A. Vardy, List decoding of polar codes, IEEE Trans. Inf. Theory,2015 Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 7/16
Performance of Polar Codes 10 0 10 1 10 2 WiMax LDPC Code: 100 iterations LTE-Turbo (punctured) code: log-map: 5 iterations Polar Code: SC Decoding Polar Code: SCL Decoding L = 2 2 Polar Code: SCL Decoding L = 4 Polar Code: SCL Decoding L = 8 Polar Code: SCL Decoding L = 32 Polar Code: ML lower bound Polar Code with CRC: L = 32, 16-CRC FER 10 3 10 4 10 5 10 6 1 1.5 2 2.5 3 SNR [db] Figure 3 : rate 1/2. Performance of Polar Codes of length N = 1024 with overall 2 I. Tal, A. Vardy, List decoding of polar codes, IEEE Trans. Inf. Theory,2015 Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 7/16
Compound Polar Codes for Higher-Order Modulation 3 u nc 1 Arıkan Polar Transformation + c 1 + c 2 + c 3. Bit Level One u u 2nc nc+1 Arıkan Polar Transformation + c nc+1 + c nc+2 + c nc+3. Bit Level Two Mapper x 1,..., x nc u 3nc 2nc+1 Arıkan Polar Transformation c 2nc+1 c 2nc+2 c 2nc+3. c 3nc Bit Level Three Figure 4 : Compound Polar Code for 8-ASK 3 H. Mahdavifar et al., Polar coding for bit-interleaved coded modulation, IEEE Trans. Veh. Technol., 2015 Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 8/16
Probabilistic Amplitude Shaping (PAS) 4 2 m -ASK constellation: label X by B 1 B 2... B m (Gray labeling) PX = PB1B2B3 000 001 011 010 110 111 101 100 X = B1B2B3 4 G. Böcherer et al., Bandwidth efficient and rate-matched low-density parity-check coded modulation, IEEE Trans. Commun., 2015 Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 9/16
Probabilistic Amplitude Shaping (PAS) 4 2 m -ASK constellation: label X by B 1 B 2... B m (Gray labeling) P X is symmetric X = sign(x ) X = S A PX = PB1B2B3 000 001 011 010 110 111 101 100 X = B1B2B3 4 G. Böcherer et al., Bandwidth efficient and rate-matched low-density parity-check coded modulation, IEEE Trans. Commun., 2015 Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 9/16
Probabilistic Amplitude Shaping (PAS) 4 2 m -ASK constellation: label X by B 1 B 2... B m (Gray labeling) P X is symmetric X = sign(x ) X = S A P A ( x ) = 2P X ( x ) PA = PB2B3 10 11 01 00 A = B2B3 PX = PB1B2B3 000 001 011 010 110 111 101 100 X = B1B2B3 4 G. Böcherer et al., Bandwidth efficient and rate-matched low-density parity-check coded modulation, IEEE Trans. Commun., 2015 Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 9/16
Probabilistic Amplitude Shaping (PAS) 4 2 m -ASK constellation: label X by B 1 B 2... B m (Gray labeling) P X is symmetric X = sign(x ) X = S A P A ( x ) = 2P X ( x ) interpret B 1 as uniformly distributed sign PA = PB2B3 10 11 01 00 A = B2B3 B1 uniform PX = PB1B2B3 000 001 011 010 110 111 101 100 X = B1B2B3 4 G. Böcherer et al., Bandwidth efficient and rate-matched low-density parity-check coded modulation, IEEE Trans. Commun., 2015 Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 9/16
PAS encoding schemes 4 block transmission with n c symbols per block P A A nc = A 1... A nc X 1... X nc b(a 1)... b(a nc ) b(s 1)... b(s nc ) b( ) P b 1 ( ) S 1... S nc 4 G. Böcherer et al., Bandwidth efficient and rate-matched low-density parity-check coded modulation, IEEE Trans. Commun., 2015 Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 10/16
PAS encoding schemes 4 block transmission with n c symbols per block P A A nc = A 1... A nc X 1... X nc b(a 1)... b(a nc ) b(s 1)... b(s nc ) b( ) P b 1 ( ) S 1... S nc rate R = H(A) coderate c = m 1 m 4 G. Böcherer et al., Bandwidth efficient and rate-matched low-density parity-check coded modulation, IEEE Trans. Commun., 2015 Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 10/16
PAS encoding schemes 4 block transmission with n c symbols per block P A A nc = A 1... A nc X 1... X nc b(a 1)... b(a nc ) b(s γnc+1)... b(s nc ) b( ) P b 1 ( ) S 1... S nc P U b(s 1)... b(s γnc ) rate R = H(A) coderate c = m 1 m 4 G. Böcherer et al., Bandwidth efficient and rate-matched low-density parity-check coded modulation, IEEE Trans. Commun., 2015 Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 10/16
PAS encoding schemes 4 block transmission with n c symbols per block P A A nc = A 1... A nc X 1... X nc b(a 1)... b(a nc ) b(s γnc+1)... b(s nc ) b( ) P b 1 ( ) S 1... S nc P U b(s 1)... b(s γnc ) rate R = H(A) coderate c = m 1 m extended PAS scheme: rate R = H(A) + γ coderate c = m 1+γ m 4 G. Böcherer et al., Bandwidth efficient and rate-matched low-density parity-check coded modulation, IEEE Trans. Commun., 2015 Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 10/16
Bit-Metric Decoding y 1 demapper L 1,1. L 1,m y 2 demapper L 2,1. L 2,m SCL decoder ĉ y nc demapper L nc,1. L nc,m Figure 5 : Bit-Metric Decoding (BMD) L i,j = log 2 p Y Bj (y i b i,j =0) p Y Bj (y i b i,j =1) }{{} channel likelihood Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 11/16
Bit-Metric Decoding y 1 demapper L 1,1. L 1,m y 2 demapper L 2,1. L 2,m SCL decoder ĉ y nc demapper L nc,1. L nc,m Figure 5 : Bit-Metric Decoding (BMD) L i,j = log 2 p Y Bj (y i b i,j =0) p Y Bj (y i b i,j =1) }{{} channel likelihood + log 2 P Bj (b i,j = 0) P Bj (b i,j = 1) }{{} a priori information Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 11/16
Results for n c = 512 10 0 10 1 10 2 10 3 FER 10 4 10 5 10 6 uniform Compound Polar Code: c=2/3 shaped Compound Polar Code: c=3/4 shaped punctured LTE-Turbo code: 10 iterations shaped WiMax-LDPC Code: 100 iterations Shannon s cone-packing achievability bound 5 10 7 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 SNR [db] Figure 6 : Frame error rate of a shaped 8-ASK Compound Polar Code with rate R = 2 bits per channel use under SCL Decoding with L = 32 and 16 CRC. 5 C. Shannon, Probability of error for optimal codes in a Gaussian channel, Bell Systems Tech. Journal, 1959 Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 12/16
Results for n c = 512 4 3.5 3 C(P) 1 2 log 2(1 + P/1) Shannon s cone packing achievability bound meta-converse upper bound Uniform Compound Polar Code: 10-CRC Shaped Compound Polar Code (CCDM): 8-CRC bpcu 2.5 2 1.5 1 0.5 6 7 8 9 10 11 12 13 14 15 16 17 SNR [db] Figure 7 : Rate Curves for 8-ASK uniform and shaped Compound Polar Codes of blocklength n c = 512 with coderate c = 0.75 under SCL Decoding with L = 32. 6 Y. Polyanskiy et al., Channel coding rate in the finite blocklength regime, IEEE Trans. Inf. Theory, 2010 Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 13/16
Results for n c = 512 1.6 1.58 1.56 1.54 C(P) 1 2 log 2(1 + P/1) Shannon s cone packing achievability bound meta-converse upper bound 6 Uniform Compound Polar Code: 10-CRC Shaped Compound Polar Code: 8-CRC bpcu 1.52 1.5 0.8 db 1.48 1.46 1.44 1.42 1.4 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10 10.1 10.2 10.3 10.4 10.5 SNR [db] Figure 7 : Rate Curves for 8-ASK uniform and shaped Compound Polar Codes of blocklength n c = 512 with coderate c = 0.75 under SCL Decoding with L = 32. 6 Y. Polyanskiy et al., Channel coding rate in the finite blocklength regime, IEEE Trans. Inf. Theory, 2010 Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 13/16
Results for n c = 128 4 3.5 3 C(P) 1 2 log 2(1 + P/1) Shannon s cone packing achievability bound meta-converse upper bound Uniform Compound Polar Code: 8-CRC Compound Polar Code with PAS: 8-CRC bpcu 2.5 2 1.5 1 0.5 6 7 8 9 10 11 12 13 14 15 16 17 SNR [db] Figure 8 : Rate Curves for 8-ASK uniform and shaped Compound Polar Codes of blocklength n c = 128 with coderate c = 0.75 under SCL Decoding. Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 14/16
Results for n c = 128 1.6 1.58 1.56 1.54 C(P) 1 2 log 2(1 + P/1) Shannon s cone packing achievability bound meta-converse upper bound Uniform Compound Polar Code: 8-CRC Compound Polar Code with PAS: 8-CRC bpcu 1.52 1.5 0.5 db 1.48 1.46 1.44 1.42 1.4 9.8 9.9 10 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 11 SNR [db] Figure 8 : Rate Curves for 8-ASK uniform and shaped Compound Polar Codes of blocklength n c = 128 with coderate c = 0.75 under SCL Decoding. Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 14/16
Conclusions Polar Codes for PAS perform very close to the theoretical limits for intermediate lengths. Polar Codes for PAS can be constructed very efficiently. Future work: Code length constricted to powers of two combine Polar Codes for PAS with punctering schemes. Consider Polar Codes with multilevel coding and multistage decoding for PAS. Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 15/16
Thank you! Questions? Tobias Prinz Polar Codes for Probabilistic Amplitude Shaping 16/16