Shaping Low-Density Lattice Codes Using Voronoi...

Shaping Low-Density Lattice Codes Using Voronoi Integers

Information Theory Workshop Hobart, Tasmania, Australia

November 2014

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

/24Brian Kurkoski, JAIST

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?

Nested Lattice Codes Achieve Capacity

2


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.

Capacity-Approaching Lattice Constructions

3


Satisfy Power Constraint with Nested Lattices

4

Good for correcting errors

Good for quantization (satisfy the power constraint)

high complexity!


1.53 dB Shaping Gain of Sphere over Cube

5

Depends only on shape of B (normalized second moment)

Depends only on coding lattice Λ

limn�⇥

G(n-sphere) =1

2�e G(cube) =112

limn�⇥

G(cube)G(n-sphere)

=�e

6= 1.53 dB

B B

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Separate lattice � and shaping region B contribution to signal power:

Average Power ⇥�

B ||x||2dxnV (B) 2

n +1

⌅ ⇤⇥ ⇧G(B)

·MnV (�)

Shaping Gain

1 2 3 4 5 7 8 16 240.055

0.06

0.065

0.07

0.075

0.08

0.085 Z1A2D3D4D5

E7E8Λ16

Λ24

for Well-Known Lattices

!6

lattice dimension n

/24Brian Kurkoski, JAIST 7

n = 1, 2 n = 8 n = 23 n = 105n = 104n = 1000

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

E8 Leech

Satisfy Power Constraint with Nested Lattices


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

It Would Be Great If…

8


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

Outline

9


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)

Low-Density Lattice Codes

10


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)

• Random sign changes • d = 7 gives good performance • BP convergence condition:

LDLC “Latin Square” Construction

11

Example: {1, 1/2, 1/3}


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

Nested Lattice Codes With LDLCs

12

that achieves the generalized capacity of the AWGN channel

without restrictions, also achieves the channel capacity of the

power constrained AWGN channel, with a properly chosen

spherical shaping region [1]. A shaping algorithm for LDLC,

based on the iterative LDLC decoding algorithm, was sug-

gested in [8], and was demonstrated for small dimensions. In

this paper, several shaping methods for LDLC are proposed.

The methods are demonstrated by simulations to give good

shaping gains for practical dimensions.

II. USING A LOWER TRIANGULAR PARITY CHECK

MATRIX

In [2], Latin square LDLC were defined. In a Latin square

LDLC, every row and column of the parity check matrix Hhas the same d nonzero values, except for a possible change

of order and random signs. The sorted sequence of these dvalues h1 ⌅ h2 ⌅ ... ⌅ hd > 0 is referred to as the generating

sequence of the Latin square LDLC.

We would like to use a simpler structure for H , which

will be more convenient for encoding and shaping. First, it

is assumed that all the values on the main diagonal of Hare 1. This can be assumed, without loss of generality, for

any Latin square LDLC that one of its generating sequence

elements is 1, by permuting rows and columns of H and by

multiplying whole rows by �1 as necessary. Also, we would

like H to have a lower triangular structure. Obviously, a parity

check matrix of an LDLC can not be lower triangular, since

the upper rows (as well as the rightmost columns) can not

contain d nonzero elements. Therefore, it is suggested that the

column degree of the rightmost column of H will start from

1 and gradually increase until d. In the same manner, the row

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 00 1.0 0 0 0 0 0 0

0.7 0 1.0 0 0 0 0 00 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 00 �0.5 0 0 0.7 0 1.0 00 0 �0.5 0 0 0.7 0 1.0

⌦

��

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

subset of the columns (starting at the leftmost column, ending

at the rightmost column where this element still appears).

As explained in the next section, the lower triangular struc-

ture is very convenient for encoding and shaping. However, it

has a drawback: the codeword components whose respective

H columns have low degree are less protected. For example,

an element whose column has a single nonzero element is

effectively uncoded, since it only takes place in a single check

equation. As a result, the information integers whose check

equations involve less protected codeword components should

contain a smaller amount of information, i.e. belong to a

smaller constellation. For example, we can use the following

constellations for the integers in the example matrix above.

The first 2 integers can assume one of 2 possible integer

values (i.e. contain 1 bit of information). The next 2 integers

can assume one of 4 integer values and contain 2 bits of

information, where all the other integers can assume one of 8

possible values and contain 3 bits of information. As shown

in Section IV, the rate loss due to this selective constellation

reduction can be made relatively small, especially for large

codeword length n.

We note that for a lower triangular H , the LDLC shaping

problem becomes similar to shaping of signal codes [9], [10].

Signal codes are lattice codes for which encoding is done by

convolving the information integers with a fixed, finite-length

filter. The resulting lattice generator matrix has a Toeplitz

structure, which is close to being lower-triangular.

III. SHAPING METHODS FOR LDLC

For the methods in Sections III-A-III-C below, it is assumed

that H is lower triangular with ones on the diagonal, as

defined in Section II. In Section III-D, the methods of Sections

III-A and III-C are extended to an arbitrary H . We shall

assume that each information integer bi is drawn from the finite

constellation {0, 1, ..., L � 1}, where L is the constellation

size. As discussed above, we may want to use a different

constellation size for each integer, so the constellation size

of the ith integer bi will be denoted by Li.

A. Hypercube ShapingHypercube shaping finds b⇥ such that the components of

x⇥ = Gb⇥ are uniformly distributed. To achieve that, we shall

assume that

b⇥i = bi � Liki, (1)

where ki is an integer. The method starts from the first (top)

check equation, i.e., from i = 1, and continues to i = n. For

each equation, the value of ki is chosen such that |x⇥i| ⇤ Li/2,

where x⇥i is the resulting codeword element, i.e.

ki =

⌥1Li

⇧bi �

i�1⇣

l=1

Hi,lx⇥l

⌃�(2)

The modified integer b⇥i is then calculated according to (1),

and the codeword element x⇥i is then easily calculated as:

x⇥i = b⇥i �i�1⇣

l=1

Hi,lx⇥l (3)

239


Offset to reduce average power

Proposed Construction

13

u P Zn Splitter

u1

u2

un{m

Zm{⇤s

c1

c2

cn{m

Combineintegers

c P Zn Systematicshaping

x “ H´1pc ´ kq Substractoffset a

x1 “ x ´ a

(a) Encoder

noise (z)

y “ x1 ` z Addoffset a

x ` z LDLCDecoding

x Y

¨U c

Splitter

c1

c2

cn{m

⇤s Ñ Zm

u1

u2

un{m

Combinedintegers

u

(b) Decoder

Fig. 2. Encoder and Decoder using Zm{⇤s integers in LDLC.

E. Numerical Example

Fig. 1 shows an example of Z2{4D2, using the scaled D2

lattice:

G “ 4

„

1 0

1 2

⇢

. (23)

The figure shows the Voronoi region and the elements c, whichare integers inside the shaping region, labeled as u “ pu1, u2q.Note that for consistent tie-breaking, some elements on theVoronoi boundary and included, and some are not. In this case|detG| “ 32, so the rate is R “ 2.5 bits/dimension.

IV. PROPOSED CODE CONSTRUCTION AND DECODER

A. Code Construction

This section describes the code construction obtained byperforming systematic lattice encoding on the Voronoi inte-gers. Systematic lattice encoding does not change the structureof the lattice — it is only a mapping from integers to latticepoints, so the high coding gain properties of ⇤c are retained.By using systematic lattice shaping, the average power of xis similar to average power of c. The Voronoi integers givesthe same shaping gain observed in ⇤s lattice with much lesscomplexity. Thus, we can reach the goal of simultaneouslyhaving good coding gain along with some shaping gain.

The shaping lattice ⇤s has dimension m and the codinglattiice ⇤c has dimension n ° m. We require that n{m isan integer. The transmitter divides integer vector u to n{minteger vectors such that u “ ru1,u2, . . .un{ms where ui PZm. Then, these are encoded to Zm{⇤s to obtain ci,@i Pt1, . . . n{mu as described in Section III-B.

It is evident [13] that mean of c may not be zero, therefore,x also has non-zero mean. Hence, subtract the mean of Zm{⇤s

to obtain the transmitted codeword x1:

x1 “ x ´ a, (24)

where

a “ ras as . . . asslooooooomooooooon

repeat n{m

, (25)

and as P Rm is the mean of Zm{⇤s.

B. Channel and Decoding

The source transmits the codeword x1 via an AWGN chan-nel and the received signal is y:

y “ x1 ` z (26)

where z is the AWGN noise vector with variance �2z .

At the decoder, the receiver adds the offset a to obtain y1 “x`z and performs LDLC iterative decoding to obtain x. Next,it rounds to the nearest integer to find the integer vector c.Finally, it performs the inverse mapping for Voronoi integersas in Section III-C to obtain estimated information vector u.The encoding and decoding block diagram is shown in Fig. 2.

Remark 1: This shaping method significantly reduces thecomplexity compare to nested lattice shaping that uses of M -algorithm as proposed in [9]. Further, it has been found that forsmall dimensions, i.e., n § 1000, QR-decomposition methodgives better performance with hypercube/systematic shapingschemes, since, there is no rate penalty [9]. However, thenested lattice shaping is not possible with QR decompositionmethod due to high complexity, hence, there is no additionalshaping gain over hypercube. Interestingly, with the use ofproposed Voronoi integers with QR-decomposition, and sys-tematic shaping, we can obtain higher shaping gain with lowercomplexity.

V. NUMERICAL RESULTS

Efficient decoding and quantization schemes are availablefor E8 lattice [12], further it has a shaping gain of 0.65 dB[11]. Hence, it motivates us to use the E8 lattice to obtainVoronoi integers, i.e., Z8{ME8 in our simulations.

“Voronoi Integers” Systematic lattice encoding


Under systematic shaping, if the integers are “shaped,” then lattice code will be shaped. !

L is a small-dimensional lattice quantization, i.e. shaping, is easy !

Define “Voronoi Integers” set of integers inside fundamental region !

!

“Voronoi Integers”

14−5 −4 −3 −2 −1 0 1 2 3 4 5−5

−4

−3

−2

−1

0

1

2

3

4

5

1,7

2,4

2,5

0,6

3,0

3,6

3,1 3,5

0,2

3,6

0,3

0,7 1,6

1,7

0,6

0,5

2,3

0,6

0,3

0,1

1,6

0,5

2,0

1,7

0,5

2,4

1,4

1,0

1,4

2,7

3,0

2,4

1,2

1,5

3,0 3,4 3,0

1,0

2,6

1,1

1,3

2,4

1,7 3,5

0,4

1,7

1,5

2,1

2,0

0,0

3,6

0,3

0,4

3,2 3,2

0,5

3,4

0,2

0,1

0,7

3,2

3,6

2,3 1,0

1,4

1,5

2,6

0,3

2,1

2,0

2,6 1,3 0,0

3,6

1,0

1,6

3,5

3,7

0,5

1,0

3,3

3,0

1,5

0,5

1,6

2,3

1,2

3,3

1,3 3,1

3,0 1,6

1,4

1,0

2,4

2,2

1,5

3,0

0,5

3,7

0,2

2,3 3,2

0,5

0,6

3,3

0,1

1,6

2,2 1,3

3,7

2,5

2,0 2,0

1,3 0,0

3,4 1,2

0,1

1,2

3,3 1,1

3,5 3,5

3,7

1,4

2,3

3,1 0,0

0,4

2,3 1,0

3,5

1,1

3,6

3,3

3,6

0,1

3,3

1,7

3,6

2,7

1,1

1,2

2,0

0,1

3,6

2,7

3,7

0,7

2,0

1,7

3,0

2,6 0,0

1,0

1,3 1,7

1,2

0,1

0,0

2,0

2,3

0,5

1,0

2,1

3,5

3,6

3,5

2,5

1,4

3,0

2,3

3,5

3,4

3,3

2,1

2,7 2,3

3,0 3,0 2,5

2,3

1,4

3,5

3,0

2,0

2,5

0,6

1,7

1,0

0,0

3,0

0,0 0,0

2,4

0,1

0,2

0,4

1,2

2,6

1,4

1,2

3,7

2,1

0,4

1,2 2,1

2,0

0,2

0,7

3,3

0,3

1,5

1,7

3,7

1,2

3,1

0,4

3,3

0,0

3,4

1,5

3,2

2,1

3,5

0,3

2,7

2,5

2,7

0,3

1,5

3,3

3,2 2,3

3,5

3,0 1,6

0,0 1,7

1,4

1,0

2,6

1,0

0,5

1,5

0,5

1,0

1,5 3,7 3,7

3,6

2,3

0,5

3,2

3,0 1,2

0,5

1,7

1,5

0,4

2,6

2,7

3,1

0,6

0,3

1,5 0,6

3,4

1,0

2,0

0,1

1,5

3,3 0,2

2,6 1,3

3,4

2,2

1,6

1,3

1,5

2,5

1,3

1,0

1,7 3,5

3,6 3,6

3,0

1,3

1,0

3,5

0,1 2,3

0,3

0,6

3,0 2,5 1,6

1,0

0,6

2,6 2,6

0,3 0,3

3,2

2,0 0,6 0,6 1,5

0,5

1,7 1,3

0,7

1,0

2,4

1,6

3,2 2,3

0,4

3,0 3,4

1,1

0,3

2,2

3,7

2,3

2,5

0,3

1,4

3,3

0,7

3,2

1,6

2,4 2,0 3,7 0,6

3,0 2,5

1,5

3,4

2,2

3,3

2,6 1,3

2,3

0,5

0,3

0,7

2,7

1,7 1,7

3,3

2,4

3,1

0,7

0,1

2,2

3,7

3,3

3,1

2,5

0,4

3,2

0,6

1,1

0,5

2,2

2,5

1,3

1,6 0,7

1,3 3,5

0,5

0,7 3,4

2,7

0,6

3,6

0,3

0,2

0,3

2,6

1,4

3,7

2,7

0,6

1,1

3,2

0,0 3,5

3,7

2,7

1,7

3,2

0,6

0,5

1,5

0,4

2,0

1,7

0,3

2,2 3,5

0,4

3,4

3,7

3,1

0,1

0,0

2,4

3,5 1,3

1,5

0,5

3,5

1,4

1,0

3,5 1,7

0,2 1,1

0,5

0,4

1,3 3,1

0,7

2,4 2,0

3,2 2,3

0,4

2,1

0,6

0,2

3,7

2,6

2,5

2,7 2,3 2,3

0,3

2,3

1,6

2,3

0,2

1,5

2,7

3,0

3,1

3,4

0,0

1,6

0,2

0,1

0,5

0,4

0,5

1,2

3,6

3,0

2,0

0,0

1,1 1,1

1,3 2,6

1,6

2,6 0,0

2,5

0,0 2,6

3,0

2,7 1,0

0,5

0,3

2,7

0,0

2,0

3,1 2,6

3,3

3,7

3,1 2,6

0,5 1,4

1,0

1,7

1,2

3,6

0,7

3,2

2,2

3,2

c1

c 2

Zm/�shape

Zm/�shape


Voronoi Integers Example

15−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

/24Brian Kurkoski, JAIST 16−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

7

8

9

10

11

12

13


/24Brian Kurkoski, JAIST 17−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

7

8

9

10

11

12

13



Systematic Lattice Encoding

18

Triangular H with 1’s on diagonal

Requirement


c =

2 3 4

2

3

4

(2,2)

(2,3)

(2,4)

(3,2)

(3,3)

(3,4)

(4,2)

(4,3)

(4,4)!

Example using !

Recall: c = round(x) !

Note Voronoi volume det(H) = 1 and the integer grid also has vol. 1 No “gaps”

Systematic Lattice Encoding

19


2 2.5 3 3.5 4 4.5 5

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

log2(L)

Gai

n o

ver

hy

per

cub

e sh

apin

g

c (Z8/LE8)

x’ (Z8/LE8 with LDLC)

E8 shaping gain bound

a=aopt

a=0

Z8/LE8 with LDLC systematic

No LDLC

Transmit power,Gain over hypercube

E8 lattice alone

Average Transmit Power


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

Power-Constrained AWGN Channel

22


0.65 dB Gain Over Hypercube Shaping!

23

23.5 24 24.5 25 25.5 26 26.5

10−5

10−4

10−3

10−2

10−1

Average SNR in dB

SE

R

Hypercube shaping [7]

Nested lattice shaping |7]

Proposed shaping

Uniform input capacity

AWGN capacity

0.4 dB

0.65dB

0.15 dB better than M algorithm, and much lower complexity


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

Conclusion

24

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