Constellation Shaping for LDPC-Coded APSK
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1 Constellation Shaping for LDPC-Coded APSK Matthew C. Valenti Lane Department of Computer Science and Electrical Engineering West Virginia University U.S.A. Mar. 14, 2013 ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013Virginia 1 / Univer 41
2 Acknowledgements I would like to thank: Xingyu Xiang. National Science Foundation. Army Research Lab. DirecTV. Hughes Network Systems. ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013Virginia 2 / Univer 41
3 Outline 1 Introduction 2 APSK Modulation 3 LDPC Coding 4 Iterative Reception 5 LDPC Degree Optimization 6 Constellation Shaping 7 Conclusion ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013Virginia 3 / Univer 41
4 Outline Introduction 1 Introduction 2 APSK Modulation 3 LDPC Coding 4 Iterative Reception 5 LDPC Degree Optimization 6 Constellation Shaping 7 Conclusion ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013Virginia 4 / Univer 41
5 Motivation for this Work Introduction DVB-S2 is a popular system for satellite broadcast and data transmission, and uses a combination of APSK modulation and LDPC coding. Goal of this work is to improve performance of LDPC-coded APSK by combining the following ideas: Iterative receiver implementation (a.k.a. BICM-ID). Constellation shaping. LDPC code optimization. ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013Virginia 5 / Univer 41
6 Preview of Our Results Introduction BER BICM Uniform BICM-ID Uniform Uniform with optimized 3/5 LDPC code (D=4) DVB-S2 2/3 LDPC and (4,2) shaping code optimized 9/14 LDPC in shaping system (D=4) (1) (2) (3) (4) E b /N 0 in db Baseline system: 32-APSK. R = 3 bits/symbol. AWGN channel. Performance improvements: 1 BICM-ID decoder: 0.3 db gain. 2 Optimized LDPC code s degree distribution: 0.3 db gain. 3 Constellation shaping: 0.5 db gain. 4 Both code optimization and constellation shaping: 0.9 db gain. ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013Virginia 6 / Univer 41
7 Outline APSK Modulation 1 Introduction 2 APSK Modulation 3 LDPC Coding 4 Iterative Reception 5 LDPC Degree Optimization 6 Constellation Shaping 7 Conclusion ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013Virginia 7 / Univer 41
8 APSK Modulation APSK vs. QAM for Nonlinear Channels Due to the use of TWTA, satellite channels are nonlinear. QAM constellations become highly distorted. Nonl linear Amplifier APSK maintains distinct rings despite nonlinearity. Nonlinear Amplifier ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013Virginia 8 / Univer 41
9 Phase noise APSK Modulation Amplitude Phase Shift Keying Non-linear magnitude and phase characteristics of a saturated transponder The fact that the transponder is power limited Group delay effects DVB-S2 uses the following APSK constellations: This work led to the defined constellations to be optimised for the above conditions. The constellations that were chosen are shown below: Q Q I I QPSK Q 8PSK Q I I 16APSK 32APSK Figure 1: DVB-S2 Constellations ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013Virginia 9 / Univer 41
10 Uncoded BER in AWGN APSK Modulation APSK 16APSK 8PSK QPSK 10 2 BER Es/No in db ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 10 / Univer 41
11 APSK Modulation Symmetric Information Rate of APSK Performance can be improved by using error control coding. Gains are limited by the modulation-constrained capacity. Capacity (bits per channel use) APSK 16APSK 8PSK QPSK LDPC codes are capable 0 of approaching capacity Es/No in db Symmetric information rate (assumes uniform input). ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 11 / Univer 41
12 Outline LDPC Coding 1 Introduction 2 APSK Modulation 3 LDPC Coding 4 Iterative Reception 5 LDPC Degree Optimization 6 Constellation Shaping 7 Conclusion ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 12 / Univer 41
13 LDPC Coding Single Parity-Check Codes Consider the following rate R = 5/6 single parity-check code: c = [ ] }{{}}{{} u parity bit One error in any position may be detected: c = [ 1 0 X ] Problem with using an SPC is that it can only detect a single error. ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 13 / Univer 41
14 LDPC Coding Product Codes Place data into a k by k rectangular array. Encode each row with a SPC. Encode each column with a SPC. Result is a rate R = k 2 /(k + 1) 2 code. Example k = 2. c 1 = u 1 c 2 = u 2 c 3 = c 1 c 2 c 4 = u 3 c 5 = u 4 c 6 = c 4 c 5 = c 7 = c 1 c 4 c 8 = c 2 c 5 c 9 = c 3 c A single error can be corrected by detecting its row and column location ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 14 / Univer 41
15 LDPC Coding Linear Codes c 1 = u 1 c 2 = u 2 c 3 = c 1 c 2 c 4 = u 3 c 5 = u 4 c 6 = c 4 c 5 c 7 = c 1 c 4 c 8 = c 2 c 5 c 9 = c 3 c 6 The example product code is characterized by the set of five linearly-independent equations: c 3 = c 1 c 2 c 1 c 2 c 3 = 0 c 6 = c 4 c 5 c 4 c 5 c 6 = 0 c 7 = c 1 c 4 c 1 c 4 c 7 = 0 c 8 = c 2 c 5 c 2 c 5 c 8 = 0 c 9 = c 3 c 6 c 3 c 6 c 9 = 0 In general, it takes (n k) linearly-independent equations to specify a linear code. ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 15 / Univer 41
16 Parity-Check Matrix LDPC Coding The system of equations may be expressed in matrix form as: where H is a parity-check matrix. Example: c 1 c 2 c 3 = 0 c 4 c 5 c 6 = 0 c 1 c 4 c 7 = 0 c 2 c 5 c 8 = 0 c 3 c 6 c 9 = 0 System of equations ch T = H = Parity-check matrix ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 16 / Univer 41
17 LDPC Codes LDPC Coding An LDPC code is a code with a large, sparse H matrix. A code from MacKay and Neal (1996): H = The code called a (3, 4) regular code because: Each column has exactly 3 ones. Each row has exactly 4 ones. Irregular codes: An irregular LDPC code has columns with different Hamming weights. An irregular code can outperform a regular code. The DVB-S2 LDPC codes are irregular. ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 17 / Univer 41
18 Tanner Graphs LDPC Coding The parity-check matrix may be represented by a Tanner graph. Bipartite graph: Check nodes: Represent the n k parity-check equations. Variable nodes: Represent the n code bits. If H i,j = 1, then i th check node is connected to j th variable node. Example: For the parity-check matrix: H = The Tanner Graph is: ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 18 / Univer 41
19 LDPC Coding Degree Distribution Edge-perspective degree distributions: ρ i is the fraction of edges touching degree i check nodes. λ i is the fraction of edges touching degree i variable nodes. For example, consider the Tanner graph: 15 edges. All are connected to degree-3 check nodes, so ρ 3 = 15/15 = 1. Three are connected to degree-1 variable nodes, so λ 1 = 3/15 = 1/5. Twelve are connected to degree-2 variable nodes, so λ 2 = 12/15 = 4/5. ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 19 / Univer 41
20 LDPC Coding DVB-S2 standardized LDPC code Key features of the DVB-S2 LDPC code: Variable rate: R c = kc n c = { 1 4, 1 3, 1 2, 3 5, 2 3, 3 4, 4 5, 5 6, 8 9, 9 10 }. Two lengths: n c = 16, 200 (short) and n c = 64, 800 (long). Systematic encoding. Last m c = n c k c columns of H are a dual diagonal submatrix, making it an extended irregular repeat accumulate (eira) code 1. Constant row weight; i.e., check regular. Variable column weight, with D = 3 different values 2. 1 M. Yang, W. E. Ryan, and Y. Li, Design of efficiently encodable moderate-length high-rate irregular LDPC codes, IEEE Trans. Commun., vol. 52, pp , Apr Not including the last column, which has a weight of 1. ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 20 / Univer 41
21 Outline Iterative Reception 1 Introduction 2 APSK Modulation 3 LDPC Coding 4 Iterative Reception 5 LDPC Degree Optimization 6 Constellation Shaping 7 Conclusion ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 21 / Univer 41
22 Iterative Reception BICM-ID Iterative Demodulation and Decoding Conventional receivers first demodulation, then decode. LDPC Decoder Hard decision Performance is improved by iterating between the demodulator and decoder. y APSK demodulator Π 1-1 VND _ Π 3-1 CND BICM-ID: bit-interleaved modulation with iterative decoding. L a (z) Π 1 Feedback LLRs Π 3 ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 22 / Univer 41
23 BICM vs. BICM-ID Iterative Reception BICM-ID BER Rate 4 BICM (4by5 LDPC) Rate 4 BICM-ID (4by5 LDPC) Rate 3.75 BICM (3by4 LDPC) Rate 3.75 BICM-ID (3by4 LDPC) Rate 3 BICM (3by5 LDPC) Rate 3 BICM-ID (3by5 LDPC) E s /N 0 in db Curves show performance of 32APSK in AWGN. ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 23 / Univer 41
24 Outline LDPC Degree Optimization 1 Introduction 2 APSK Modulation 3 LDPC Coding 4 Iterative Reception 5 LDPC Degree Optimization 6 Constellation Shaping 7 Conclusion ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 24 / Univer 41
25 EXIT charts LDPC Degree Optimization The convergence threshold is the SNR value in which the bit error rate of an LDPC-coded system starts dropping sharply. The value of the threshold depends on the degree distribution. EXIT charts 3 Predict the convergence threshold. Can be used to identify good candidate degree distributions. 0.6 However, because it is just a 0.5 prediction, the candidate codes still VND, DVB-S2 standard LDPC code with rate 3/5 0.4 CND, all nodes degree 11 need to be simulated to determine which is best. I E,VND, I A,CND I A,VND, I E,CND Figure : EXIT chart for the uniform system at E b /N 0 = 4.93 db. 3 S. ten Brink, G. Kramer, and A. Ashikhmin, Design of low-density parity-check codes for modulation and detection, IEEE Trans. Commun., vol. 52, pp , Apr ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 25 / Univer 41
26 LDPC Degree Optimization Optimal Degree Distributions Degree distributions for uniform 32-APSK. The DVB-S2 standard rate R c = 3/5 LDPC code has degree distributions: λ 2 = λ 3 = λ 12 = The optimized degree distributions with D = 3 are: λ 2 = λ 4 = λ 19 = The optimized degree distributions with D = 4 are: λ 2 = λ 3 = λ 4 = λ 25 = All codes are check regular with ρ 11 = 1. ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 26 / Univer 41
27 LDPC Degree Optimization BER with Optimized Degree Distributions BICM-ID Uniform Uniform with optimized 3/5 LDPC code (D=3) Uniform with optimized 3/5 LDPC code (D=4) 10-3 BER E /N in db b 0 BER of 32-APSK in AWGN at rate R=3 bits/symbol. Comparison of standard vs. optimized LDPC codes. ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 27 / Univer 41
28 Outline Constellation Shaping 1 Introduction 2 APSK Modulation 3 LDPC Coding 4 Iterative Reception 5 LDPC Degree Optimization 6 Constellation Shaping 7 Conclusion ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 28 / Univer 41
29 Constellation Shaping Constellation Shaping The energy efficiency can be improved by transmitting lower-energy signals more frequently than higher-energy signals Figure : Uniform 32APSK vs. shaped 32APSK. Both constellations have the same energy. mutual information uniform shaping g= Es/No (db) Figure : The capacity of shaped 32APSK is about 0.3 db better than uniform 32APSK ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 29 / Univer 41
30 Constellation Shaping Shaping Through Signal Set Partitioning Partition the constellation into two equal-sized sub-constellations. Use a shaping bit to select between the two sub-constellations. The lower-energy sub-constellation is selected more frequently than the higher-energy sub-constellation. Requires the shaping bit to be encoded so that it is not uniform. The remaining bits select from among the M/2 symbols in the selected sub-constellation with equal probabability. ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 30 / Univer 41
31 Constellation Shaping Shaping Encoder The shaping encoder maps k s bits to a n s bit shaping codeword. Code is designed with the goal of having more zeros than ones. Example (k s = 3, n s = 5) code: 3 input data bits 5 output codeword bits p 0 = 31/40 is the probability of 0. p 1 = 9/40 is the probability of 1. ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 31 / Univer 41
32 Shaping Operation Constellation Shaping ENC Channel Encoder P/S th 5 bit (LSB) Here, the (5, 3) shaping code is used as an example. The P/S block segments groups of 23 bits. Three bits delivered to the shaping encoder. ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 32 / Univer 41
33 Shaping Operation Constellation Shaping Channel Encoder P/S ENC th 5 bit (LSB) Here, the (5, 3) shaping code is used as an example. The P/S block segments groups of 23 bits. Three bits delivered to the shaping encoder. ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 32 / Univer 41
34 Shaping Operation Constellation Shaping ENC Channel Encoder P/S th 5 bit (LSB) Here, the (5, 3) shaping code is used as an example. The P/S block segments groups of 23 bits. Three bits delivered to the shaping encoder. ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 32 / Univer 41
35 Constellation Shaping Receiver Implementation Demodulator + Shaping Decoder LDPC Decoder Hard decision i y APSK demodulator S/P 2-1 shaping decoder P/S 1 _ VND -1 CND L a (z) P/S 2 S/P 1 3 Feed dback LLRs Additional complexity relative to BICM-ID due to shaping decoder. MAP shaping decoder compares against all 2 ks shaping codewords. ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 33 / Univer 41
36 Constellation Shaping EXIT Charts with Constellation Shaping When shaping is used, the variable-node decoder (VND) accounts for the effects of shaping. 1 Demodulator + (Shaping Decoder ) + VND Combined VND Hard decision y APSK demodulator shaping decoder IE,DET IA,DET VND IE,VND _ 3-1 CND CND I E,VND, I A,CND IA,VND VND, DVB-S2 standard LDPC code with rate 2/3 CND, all check node degree 10 Figure : Model of decoder used for constructing EXIT charts I A,VND, I E,CND Figure : EXIT chart for the shaped system at E b /N 0 = 4.53 db. ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 34 / Univer 41
37 Constellation Shaping Optimal Degree Distributions with Shaping Spectral efficiency of 3 bits per channel use. (3, 2) shaping code. rate r c = 9/14 LDPC code. Check regular with ρ 10 = 1. The optimized degree distributions with D = 3 are: λ 2 = λ 3 = λ 14 = The optimized degree distributions with D = 4 are: λ 2 = λ 3 = λ 5 = λ 13 = ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 35 / Univer 41
38 BER with Shaping Constellation Shaping BICM-ID Uniform Uniform with optimized 3/5 LDPC code (D=3) Uniform with optimized 3/5 LDPC code (D=4) DVB-S2 2/3 LDPC and (4,2) shaping code optimized 9/14 LDPC with D=3 in shaping system optimized 9/14 LDPC with D=4 in shaping system BER E /N in db b 0 BER of 32-APSK in AWGN at rate R=3 bits/symbol. ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 36 / Univer 41
39 Constellation Shaping Summary of Performance Gains Cumulative Gains BICM Uniform BICM-ID Uniform Uniform with optimized 3/5 LDPC code (D=4) DVB-S2 2/3 LDPC and (4,2) shaping code optimized 9/14 LDPC in shaping system (D=4) 10-3 BER 10-4 (2) (1) (3) 10-5 (4) E /N in db b 0 BICM-ID decoder: 0.3 db gain. Optimized LDPC degree distribution: 0.3 db gain. Constellation shaping: 0.5 db gain. Both code optimization and constellation shaping: 0.9 db gain. ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 37 / Univer 41
40 Outline Conclusion 1 Introduction 2 APSK Modulation 3 LDPC Coding 4 Iterative Reception 5 LDPC Degree Optimization 6 Constellation Shaping 7 Conclusion ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 38 / Univer 41
41 Conclusion Conclusion DVB-S2 is already a highly efficient system, thanks to APSK modulation. Capacity-approaching irregular LDPC codes. The performance of DVB-S2 can be improved by BICM-ID. Constellation shaping. Optimization of LDPC degree-distribution. The cumulative gain is 1 db with all of these. Future work: Application to 64APSK, 128APSK, and beyond. Improved symbol labeling map. ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 39 / Univer 41
42 References Conclusion 1 and X. Xiang, Constellation shaping for bit-interleaved LDPC coded APSK, IEEE Trans. Commun., vol. 60, no. 10, pp C. Nannapaneni,, and X. Xiang, Constellation shaping for communication channels with quantized outputs, in Proc. Conf. on Info. Sci. and Sys. (CISS), (Baltimore, MD), Mar and X. Xiang, Constellation shaping for bit-interleaved coded APSK, in Proc. IEEE Int. Conf. on Commun. (ICC), (Kyoto, Japan), June X. Xiang and, Improving DVB-S2 performance through constellation shaping and iterative demapping, in Proc. IEEE Military Commun. Conf. (MILCOM), (Baltimore, MD), Nov X. Xiang and, Closing the gap to the capacity of APSK: Constellation shaping and degree distributions, in Proc. Int. Conf. on Computing, Networking, and Commun. (ICNC), (San Diego, CA), Jan ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 40 / Univer 41
43 Conclusion Thank You. ( Lane Department LDPCof Codes Computer Science and Electrical Engineering Mar. 14, West 2013 Virginia 41 / Univer 41
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