Shaping Low-Density Lattice Codes Using Voronoi Integers

Transcription

1 Shaping Low-Density Lattice Codes Using Voronoi Integers Nuwan S. Ferdinand Brian M. Kurkoski Behnaam Aazhang Matti Latva-aho University of Oulu, Finland Japan Advanced Institute of Science and Technology Rice University, USA University of Oulu, Finland Information Theory Workshop Hobart, Tasmania, Australia November 2014

2 Nested Lattice Codes Achieve Capacity Lattice codes can achieve the capacity of AWGN channel [Erez and Zamir 04] Nested lattice codes: Want which is simultaneously good for coding and shaping Other information theoretic results using lattices: Lattices for relay channel e.g. [Song-Devroye 13] Two-way (Bidirectional) relay channel e.g. [Wilson et al.] Compute-forward relaying [Nazer-Gastpar 11] How to move from information theory to practical lattice codes? 2/24

3 Capacity-Approaching Lattice Constructions Recent high-dimension lattice constructions approach capacity Construction A with LDPC codes Construction D with turbo codes, spatially coupled LDPC Lattices based on polar codes Low-Density Lattice Codes [Sommer et al. 2008] Common claim: within few tenth of db of unconstrained capacity: No assumption about the channel power constraint. 3/24

4 Satisfy Power Constraint with Nested Lattices Good for correcting errors Good for quantization (satisfy the power constraint) high complexity 4/24

5 1.53 db Shaping Gain of Sphere over Cube Separate lattice and shaping region B contribution to signal power: Average Power B x 2 dx Depends only on shape of B (normalized second moment) nv (B) 2 n +1 G(B) M n V ( ) Depends only on coding lattice Λ Shaping Gain B B lim n G(cube) G(n-sphere) = e 6 = 1.53 db lim n G(n-sphere) = 1 2 e G(cube) = /24

6 for Well-Known Lattices Z 1 A 2D D4 D E 7 E Λ 16 Λ lattice dimension n

7 Satisfy Power Constraint with Nested Lattices n = 1, 2 n = 8 n = 23 n = 1000 n = 10 4 n = 10 5 E8 Leech small n Well-known lattices Weak coding gain efficient shaping algorithms Good shaping gain (0.65~1.0 db) Large n BP-based lattices Strong coding gain Inefficient shaping algorithms Uncertain coding gains: two cases: 0.4 db shaping gain 7/24

8 It Would Be Great If Find a construction that: Has the capacity-approaching coding gain high-dimension lattices Has the shaping gains and implementation complexity of a well-known lattice like E8. Must overcome the problem of mismatch in dimensions 8/24

9 Outline Key result: a lattice construction technique for shaping LDLC lattices Elements of the technique: 1. Voronoi Integers Shape integers using small-dimension lattices 2. Systematic lattice encoding: lattice point is nearby corresponding integer Results Full 0.65 db shaping gain of the E8 lattice. (2.1 db from 1/2 log(snr+1) ) Competing nested LDLCs obtained only 0.4 db, using higher complexity First, review 1.53 shaping gain result and LDLC lattices 9/24

10 Low-Density Lattice Codes LDLC lattices introduced by Sommer, Shalvi and Feder [IT 2008] LDLC have a sparse inverse generator matrix H Gaussian Belief-propagation decoding High dimension, n = 100, 1000, 10000, Come within 0.6 db of unconstrained capacity LDLCs for the power-constrained channel [Sommer et al ITW 2009] H matrix in triangular form, use M algorithm for quantization Obtained 0.4 db gain over hypercube (out of 1.53 db) 10/24

11 LDLC Latin Square Construction Inverse generator H = G 1 has constant row and column weight d. Latin square: each row/column {h 1, h 2,,h d } with random ±, h 1 h 2 h d Choose h 1 = 1 Example: {1, 1/2, 1/3} (forces determinant to be 1) Random sign changes d = 7 gives good performance BP convergence condition: 11/24

12 Nested Lattice Codes With LDLCs Sommer [ITW 2009]: Triangular construction Construct dimension n = 10,000 Put 1's on main diagonal, make triangular 90% Latin square weight d: {h 1, h 2,,h d } Quantization/shaping using M-Algorithm Such a lower triangular matrix can be generated with simi Complexity is O(ndM), but M is large Shaping gain of 0.4 db over hypercube Modest shaping gain for high complexity 12/24

13 Proposed Construction u 1 c 1 u P Z n Splitter u 2 Z m { s c 2 Combine integers c P Z n Systematic shaping x H 1 pc kq Substract offset a x 1 x a u n{m c n{m Voronoi Integers Systematic lattice encoding Offset to reduce average power 13/24

14 Voronoi Integers Z m / shape 5 Under systematic shaping, if the integers are 4 shaped, then lattice code will be shaped. 3 0,3 2 3,3 0,2 1,1 L is a small-dimensional lattice 1 2,3 3,2 0,1 1,0 2,7 quantization, i.e. shaping, is easy c 2 0 1,3 2,2 3,1 0,0 1,7 2,6 3,5 1 1,2 2,1 3,0 0,7 1,6 2,5 3,4 Define Voronoi Integers Z m / shape 2 2,0 3,7 0,6 1,5 2,4 set of integers inside fundamental region 3 3,6 0,5 1,4 4 0, c 1 14/24

15 13 12 Voronoi Integers Example /24

16 13 12 Voronoi Integers Example /24

17 13 12 Voronoi Integers Example /24

18 Systematic Lattice Encoding Requirement Triangular H with 1 s on diagonal 18/24

19 Systematic Lattice Encoding Example using 4 c = (2,4) (3,4) (4,4) Recall: c = round(x) Note Voronoi volume det(h) = 1 3 (3,3) (4,3) and the integer grid also has vol. 1 (2,3) No gaps (4,2) 2 (3,2) (2,2) /24

20 Systematic Lattice Encoding Example using 4 c = (2,4) (3,4) (4,4) Recall: c = round(x) Note Voronoi volume det(h) = 1 3 (3,3) (4,3) and the integer grid also has vol. 1 (2,3) No gaps (4,2) 2 (3,2) (2,2) /24

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22 Average Transmit Power 0.7 E8 lattice alone No LDLC Transmit power, Gain over hypercube Gain over hypercube shaping a=a opt Z8/LE8 with LDLC systematic a=0 c (Z8/LE8) x (Z8/LE8 with LDLC) 0.2 E8 shaping gain bound log 2 (L) 21/24

23 Power-Constrained AWGN Channel AWGN channel with average power constraint 5 bits/dimension coding: LDLC lattice dimension n = 10,000 shaping: E8 lattice with m = 8 Compare with M-Algorithm LDLC shaping of Sommer et al 22/24

24 0.65 db Gain Over Hypercube Shaping Hypercube shaping [7] Nested lattice shaping 7] Proposed shaping Uniform input capacity AWGN capacity 0.15 db better than M algorithm, SER 10 3 and much lower complexity 0.4 db Average SNR in db 0.65dB 23/24

25 Conclusion Lattices are an alternative to finite-field codes for AWGN Shaping techniques to obtain 1.53 db are accessible Coset codes/nested lattice codes, high complexity We proposed: Voronoi integers using low-dimension lattices Systematic lattice shaping for LDLCs High coding gain of LDLCs, good shaping gain of E8 lattice 24/24