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
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?
Brian Kurkoski, JAIST
2/24
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.
Brian Kurkoski, JAIST
3/24
Satisfy Power Constraint with Nested Lattices
Good for correcting errors
Good for quantization
(satisfy the power constraint)
high complexity!
Brian Kurkoski, JAIST
4/24
1.53 dB Shaping Gain of Sphere over Cube
Separate lattice
and shaping region B contribution to signal power:
2
||x||
dx
B
Average Power ⇥
nV (B)
⌅
⇤⇥
Depends only on shape of B
(normalized second moment)
2
n +1
⇧
G(B)
·M V ( )
n
Depends only on coding lattice Λ
Shaping Gain
1
B
1
B
0.5
G(cube)
lim
n ⇥ G(n-sphere)
0.5
0
0
−0.5
−0.5
−1
−1
−1
−0.5
0
0.5
1
lim G(n-sphere) =
n
⇥
Brian Kurkoski, JAIST
e
=
6
= 1.53 dB
1
2 e
1
G(cube) =
12
−1
−0.5
0
0.5
1
5/24
for Well-Known Lattices
0.085
Z
1
A2
0.08
D
3
D4D
5
0.075
E
7E
8
Λ
0.07
16
Λ24
0.065
0.06
0.055
lattice dimension n
1
2
3
4
5
7
8
16
24
!6
Satisfy Power Constraint with Nested Lattices
n = 1, 2 n = 8
n = 23
n = 10
5
n = 10
h
ec
Le
E8
small n Well-known lattices
•Weak coding gain
•efficient shaping algorithms
•Good shaping gain (0.65~1.0 dB)
Brian Kurkoski, JAIST
n = 1000
4
Large n BP-based lattices
•Strong coding gain
• Inefficient shaping algorithms
•Uncertain coding gains:
two cases: 0.4 dB shaping gain
7/24
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
Brian Kurkoski, JAIST
8/24
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
Brian Kurkoski, JAIST
9/24
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, 100000
• 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)
Brian Kurkoski, JAIST
10/24
LDLC “Latin Square” Construction
Inverse generator H = G
–1
has constant row and column weight d.
Latin square: each row/column {h1, h2,…,hd} with random ±, h1 ≥ h2 ≥ … ≥hd
• Choose h1 = 1 (forces determinant to be 1)
Example: {1, 1/2, 1/3}
• Random sign changes
• d = 7 gives good performance
• BP convergence condition:
Brian Kurkoski, JAIST
11/24
degree of the top row will be 1, and it will gradually increase
until d.
The following matrix is an example of such a matrix with
dimension n = 8, degree d = 3 and generating sequence
(1, 0.7, 0.5). The two rightmost columns have a single nonzero
element, the next two have 2 nonzero element and the 4
leftmost columns have the final degree d = 3. The same is
true for the matrix rows, starting from the top.
⌦
1.0
0
0
0
0
0
0
0
0
1.0
0
0
0
0
0
0
0.7
0
1.0
0
0
0
0
0
0
0
0.7 1.0 0
0
0
0
0.5
0
0
0.7 1.0 0
0
0
0
0.7
0
0.5 0 1.0 0
0
↵ 0
0.5
0
0 0.7 0 1.0 0
0
0
0.5 0
0 0.7 0 1.0
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: {h1, h2,…,hd}
• Quantization/shaping using M-Algorithm
Complexity is O(ndM), but M is large
• Shaping gain of 0.4 dB over hypercube
Modest shaping gain for high complexity
Brian Kurkoski, JAIST
Such a lower triangular matrix can be generated with similar
methods to those described in [2] for Latin square LDLC,
where the location of each element of the generating sequence
in each row can be described by a permutation (since each
element appears once, and only once, in each row and column).
The difference is that here, instead of a permutation, we shall
use a mapping from a subset of the rows (starting at the first
row where this element appears, ending at the bottom row) to a
12/24
Proposed Construction
uPZ
n
u1
c1
u2
c2
Splitter
Combine
integers
Z {⇤s
m
un{m
cPZ
Systematic
n
shaping
Systematic
lattice
(a) Encoder
encoding
noise (z)
Add
offset a
Brian Kurkoski, JAIST
Substract
offset a
x1 “ x ´ a
cn{m
“Voronoi Integers”
y “ x1 ` z
x “ H´1 pc ´ kq
x`z
LDLC
Decoding
x̃
Y U
¨
c̃
Offset to reduce
average power
c̃1
ũ1
c̃2
ũ2
Splitter
⇤s Ñ Z
c̃n{m
Combined
integers
m
ũ
ũn{m
13/24
m
“Voronoi Integers” Z / shape
5
Under systematic shaping, if the integers are
4
“shaped,” then lattice code will be shaped.
3
!
0,3
3,3
0,2
1,1
2,3
3,2
0,1
1,0
2,7
1,3
2,2
3,1
0,0
1,7
2,6
3,5
1,2
2,1
3,0
0,7
1,6
2,5
3,4
2,0
3,7
0,6
1,5
2,4
3,6
0,5
1,4
2
L is a small-dimensional lattice
quantization, i.e. shaping, is easy
!
Define “Voronoi Integers” Z /
m
!
Brian Kurkoski, JAIST
0
−1
shape
set of integers inside fundamental region
!
c2
1
−2
−3
0,4
−4
−5
−5
−4
−3
−2
−1
0
c1
1
2
3
4
5
14/24
13
Voronoi Integers Example
12
11
10
9
8
7
6
5
4
3
2
1
0
−1
−2
−3
−4
−5
Brian Kurkoski, JAIST
−5 −4 −3 −2 −1
0
1
2
3
4
5
15/24
13
Voronoi Integers Example
12
11
10
9
8
7
6
5
4
3
2
1
0
−1
−2
−3
−4
−5
Brian Kurkoski, JAIST
−5 −4 −3 −2 −1
0
1
2
3
4
5
16/24
13
Voronoi Integers Example
12
11
10
9
8
7
6
5
4
3
2
1
0
−1
−2
−3
−4
−5
Brian Kurkoski, JAIST
−5 −4 −3 −2 −1
0
1
2
3
4
5
17/24
Systematic Lattice Encoding
Requirement
Triangular H
with 1’s on
diagonal
Brian Kurkoski, JAIST
18/24
Systematic Lattice Encoding
!
Example using
(4,4)
4
!
c=
(2,4)
Recall: c = round(x)
!
Note Voronoi volume det(H) = 1
(4,3)
3
(3,3)
(2,3)
and the integer grid also has vol. 1
No “gaps”
(3,4)
(4,2)
2
(3,2)
(2,2)
2
Brian Kurkoski, JAIST
3
4
19/24
Systematic Lattice Encoding
!
Example using
(4,4)
4
!
c=
(2,4)
Recall: c = round(x)
!
Note Voronoi volume det(H) = 1
(4,3)
3
(3,3)
(2,3)
and the integer grid also has vol. 1
No “gaps”
(3,4)
(4,2)
2
(3,2)
(2,2)
2
Brian Kurkoski, JAIST
3
4
19/24
Brian Kurkoski, JAIST
20/24
Average Transmit Power
0.7
E8 lattice alone
0.65
No LDLC
Transmit power,
Gain over hypercube
Gain over hypercube shaping
0.6
Z8/LE8 with
LDLC systematic
0.55
0.5
0.45
0.4
a=aopt
0.35
a=0
0.3
c (Z8/LE8)
0.25
x’ (Z8/LE8 with LDLC)
0.2
E8 shaping gain bound
2
2.5
3
3.5
4
4.5
5
log (L)
2
Brian Kurkoski, JAIST
21/24
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
Brian Kurkoski, JAIST
22/24
0.65 dB Gain Over Hypercube Shaping!
−1
10
Hypercube shaping [7]
Nested lattice shaping |7]
−2
10
Proposed shaping
0.15 dB better than M algorithm,
and much lower complexity
Uniform input capacity
SER
AWGN capacity
−3
10
0.4 dB
−4
10
0.65dB
−5
10
23.5
24
24.5
25
25.5
26
26.5
Average SNR in dB
Brian Kurkoski, JAIST
23/24
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
Brian Kurkoski, JAIST
24/24