Bandwidth Efficient and Rate-Matched Low-Density Parity-Check Coded Modulation

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1 Bandwidth Efficient and Rate-Matched Low-Density Parity-Check Coded Modulation Georg Böcherer, Patrick Schulte, Fabian Steiner Chair for Communications Engineering April 29, 2015 ITG Fachgruppentreffen Angewandte Informationstheorie 1 / 24

2 Outline Motivation: Bandwidth-Limited Communication Contribution: A New Coded Modulation Scheme Enabeling Technology: Fixed-Length Distribution Matching 2 / 24

3 Bandwidth-Limited Communication

4 Bandwidth-Limited Communication log 2(1 + SNR) bits/channel use SNR [db] 4 / 24

5 Higher Order-Modulation Equidistant 2 m -ASK constellation: X = {±1, ±3,..., ±(2 m 1)}. I/O-relation: Y = X + Z. Noise Z is zero mean, variance one. Input X with distribution P X on X. scales the constellation X. 5 / 24

6 Bandwidth-Limited Communication log 2(1 + SNR) {8, 16, 32, 64}ASK capacity bits/channel use SNR [db] 6 / 24

7 Bandwidth-Limited Communication log 2(1 + SNR) uniform {8, 16, 32, 64}ASK capacity bits/channel use SNR [db] 7 / 24

8 DVB-S2 LDPC Codes, Uniform Input, FER 10 3 bits/channel use log 2(1 + SNR) uniform {8, 16, 32, 64}ASK capacity uniform, {16, 32, 64}ASK, code rates 3 4, 4 5, 5 6, 8 9, SNR [db] 8 / 24

9 Probabilistic Amplitude Shaping

10 ASK Capacity ASK capacity: C ask (P) = Capacity-achieving distribution P X. max I(X ; X + Z).,P X : E[ X 2 ] P Construct a codebook with entries P X. 10 / 24

11 Capacity-Achieving Distribution P X P X is symmetric: P X (x) = P X ( x). P X factorizes: P X (x) = P S (sign(x)) P A ( x ) where A := X and S := sign(x ). The sign S is uniform: P S ( 1) = P S (1) = / 24

12 Shaping and Error Correction B k B k P R n k Binary systematic k n generator matrix G = [I P]. Binary information B k arbitrarily distributed. Uniform check bit assumption: R 1, R 2,... iid Bernoulli(1/2). 12 / 24

13 Probabilistic Amplitude Shaping P A A 1...A nc x X 1... X nc x b( ) b(a 1)...b(A nc ) P b(s 1)...b(S nc ) b 1 ( ) S 1...S nc n c number of channel uses. Represent amplitudes by (m 1) bits: A b(a). Represent signs by 1 bit: S b(s). Multiply amplitude labels by parity matrix P to get sign labels. Code rate c = (m 1)/m. P Xi = P X! 13 / 24

14 Rate Compatibility P A A 1...A nc x X 1... X nc x b( ) b(a 1)...b(A nc ) P b(s 1)...b(S nc ) b 1 ( ) S 1...S nc Control rate H(A) via P A. Control power E[ X 2 ] via. 14 / 24

15 Enabeling Technology: Distribution Matching Emulating discrete memoryless source P A : B k Matcher Ā n B i iid Bernoulli(1/2). Ā i approximately iid P A in the sense of D(PĀn P n A ) n n 0, k n n H(A) invertible: B k can be recovered from Ā n with zero error. 15 / 24

16 Constant Composition Codes The empirical distribution of a output sequence c of length n is P A,c (a) := n a(c) n, n a (c) = {i : c i = a} number of occurences of symbol a in c P A,c is the type of c T n P A is set of all n-type P A sequences Codebook C ccdm A n is called a constant composition code if all codewords are of the same type, i.e. C ccdm T n P A 16 / 24

17 Constant Composition Distribution Matching Idea Use codewords of the same type Choose sequence length of the matcher output n Choose n-type approximation PĀ of the target distribution P A Choose m = log 2 TPĀ n Construct a unique mapping {0, 1} m T n PĀ / 24

18 Bit-Metric Decoding (1) X Channel Y label(x ) = b(s)b(a) = B 1 B m. Demapper soft-decision: No iterative demapping! L i = log P B i Y (0 Y ), i = 1,..., m. P Bi Y (1 Y ) 18 / 24

19 Bit-Metric Decoding (2) To the receiver, the channel appears as B 1 P L1 B 1 L 1 B 2 P L2 B 2 L 2. B m P Lm B m L m 19 / 24

20 Achievable Rate (2) Achievable rate: 1 R BMD = H(B) = m H(B i L i ) i=1 m I(B i ; L i ) D(P B i P B i ). i=1 } {{ } ( ) ( ) known as BICM capacity, pragmatic capacity, G Böcherer Achievable Rates for Shaped Bit-Metric Decoding, 2 Guillen i Fabregas, Martinez Bit-interleaved coded modulation with shaping, ITW / 24

21 Achievable Rate (3) bits/channel use log 2(1 + SNR) 32-ASK capacity R BMD, dependent bits R BMD, independent bits SNR [db] 21 / 24

22 Probabilistic Amplitude Shaping with DVB-S2 Codes FER log 2(1 + SNR) 5 uniform {8, 16, 32, 64}ASK capacity 32ASK, code rate 5/6 64ASK, code rate 9/ bits/channel use SNR [db] 22 / 24

23 FER 10 3 within 0.69 db of Capacity bits/channel use log 2(1 + SNR) 32ASK, DVB-S2 5/6 64ASK, DVB-S2 9/10 Optimized Protograph SNR [db] 23 / 24

24 References G Böcherer, P. Schulte, F. Steiner Bandwidth Efficient and Rate-Matching Low-Density Parity-Check Coded Modulation, P.Schulte, G Böcherer: Constant Composition Distribution Matching, G Böcherer Achievable Rates for Shaped Bit-Metric Decoding, F. Steiner, G Böcherer, G. Liva Protograph-Based LDPC Code Design for Bit-Metric Decoding, ISIT / 24

Fixed Length Distribution Matching

Fixed Length Distribution Matching Patrick Schulte Chair for Communications Engineering patrick.schulte@tum.de May 26, 2015 Doktorandenseminar 1 / 15 Outline Motivation: Fixed Length Distribution Matching