Talk Overview
LDPC Codes Message Passing Decoding Analysis of Message Passing Decoding
(Density Evolution) Approximations to Density Evolution Design of LDPC Codes using D.E.
The Channel Coding Strategy
Encoder chooses the mth codeword in codebook C and transmits it across the channel
Decoder observes the channel output y and generates m’ based on the knowledge of the codebook C and the channel statistics.
Decoder
Encoder
Channel
k}1,0{'m nYy
nXC xk}1,0{m
Linear Codes
A linear code C (over a finite field) can be defined in terms of either a generator matrix or parity-check matrix.
Generator matrix G (k×n)
Parity-check matrix H (n-k×n)
}{mGC
}':{ 0cHc C
LDPC Codes
LDPC Codes -- linear codes defined in terms of H.
H has a small average number of non-zero elements per row
1011001001
0100111010
0111010011
1100001101
0000100110
H
Graph Representation of LDPC Codes
H is represented by a bipartite graph.
There is an edge from v to c if and only if:
A codeword is an assignment of v's s.t.:
cxcv
v 0|
. . . . .
.
Variable nodes
Check nodes
0),( cvH
Regular (λ,ρ) LDPC codes
Every variable node has degree λ, every check node has degree ρ.
Ensemble is defined by matching left edge "stubs" with right edge "stubs via a random permutation
. . . . .
.
Variable nodes
Check nodes
π
Message-Passing Decoding of LDPC Codes
Message Passing (or Belief Propagation) decoding is a low-complexity algorithm which approximately answers the question “what is the most likely x given y?”
MP recursively defines messages mv,c(i) and
mc,v(i) from each node variable node v to each
adjacent check node c, for iteration i=0,1,...
Two Types of Messages...
Liklihood Ratio
For y1,...yn independent conditionally on x:
Probability Difference
For x1,...xn independent:
)0|(
)1|(,
xyp
xypyx )|0()|1(, yxpyxpyx
i
yxyx in ,, 1
i
yxyi
x ii ,,
...Related by the Biliniear Transform
Definition:
Properties:
x
xxB
1
1)(
yx
yx
yxpyxp
yp
ypyxpypyxp
yp
xypxyp
xypxyp
xypxyp
xyp
xypBB
,
,
)|1()|0(
)(2
)()|1(2)()|0(2
)(2
)1|()0|(
)1|()0|(
)1|()0|(
))0|(
)1|(()(
yxyx
yxyx
B
B
xxBB
,,
,,
)(
)(
))((
Message Domains
Likelihood Ratio
Log Likelihood Ratio
Log Prob. Difference
Probability Difference
)1|(
)0|(
xyP
xyP)|1()|0( yxPyxP
)(B
)(' B
Variable to Check Messages
On any iteration i, the message from v to c is:
It is computed like:
)()1(
'|,',
)(,
i
cvcvcyx
icv mBm
vv
. . . . . .
v
c
)0|(
)1|(
xeP
xeP
Check to Variable Messages
On any iteration, the message from c to v is:
It is computed like:
Assumption: Incoming messages are indep.
)()(
'|,'
)(,
i
vcvcv
ivc mBm
. . . . . .
v
c)|1()|0( evPevP
Decision Rule
After sufficiently many iterations, return the likelihood ratio:
otherwise ,1
0)( if ,0ˆ
)1(
|,,
i
vcvcyx mB
xvv
Theorem about MP Algorithm
If the algorithm stops after r iterations, then the algorithm returns the maximum a posteriori probability estimate of xv given y within radius r of v.
However, the variables within a radius r of v must be dependent only by the equations within radius r of v,
v
r
...
...
...
Analysis of Message Passing Decoding (Density Evolution)
in Density Evolution we keep track of message densities, rather than the densities themselves.
At each iteration, we average over all of the edges which are connected by a permutation.
We assume that the all-zeros codeword was transmitted (which requires that the channel be symmetric).
D.E. Update Rule
The update rule for Density Evolution is defined in the additive domain of each type of node.
Whereas in B.P, we add (log) messages:
In D.E, we convolve message densities:
vcv
icv
ivc mBm
'|
)(,'
)(, )('
vcv
icv
ivc mDBmD
'|
)(,'
)(, ))(('*)(
Familiar Example:
If one die has density function given by:
The density function for the sum of two dice is given by the convolution:
1 3 6542
2 4 7653 8 10 12119
D.E. Threshold
Fixing the channel message densities, the message densities will either "converge" to minus infinity, or they won't.
For the gaussian channel, the smallest (infimum) SNR for which the densities converge is called the density evolution threshold.
Regular (λ,ρ) LDPC codes
Every variable node has degree λ, every check node has degree ρ.
Best rate 1/2 code is (3,6), with threshold 1.09 dB.
This code had been invented by 1962 by Robert Gallager.
D.E. Simulation of (3,6) codes
Set SNR to 1.12 dB (.03 above threshold)
Watch fraction of "erroneous messages" from check to variable.
(note that this fraction does not characterize the distribution fully)
Improvement vs. current error fraction for Regular (3,6)
Improvement per iteration is plotted against current error fraction.
Note there is a single bottleneck which took most of the decoding iterations.
Irregular (λ, ρ) LDPC codes
a fraction λi of variable nodes have degree i. ρi of check nodes have degree i.
Edges are connected by a single random permutation.
Nodes have become specialized.
. . . . .
.
Variable nodes
Check nodes
πλ3
λn
ρ4
λ2
ρm
D.E. Simulation of Irregular Codes (Maximum degree 10)
Set SNR to 0.42 dB (~.03 above threshold)
Watch fraction of erroneous check to variable messages.
This Code was designed by Richardson et. al.
Comparison of Regular and Irregular codes
Notice that the Irregular graph is much flatter.
Note: Capacity achieving LDPC codes for the erasure channel were designed by making this line exactly flat.
Multi-edge-type construction
Edges of a particular "color" are connected through a permutation.
Edges become specialized. Each edge type has a different message distribution each iteration.
D.E. of MET codes.
For Multi-edge-type codes, Density evolution tracks the density of each type of message separately.
Comparison was made to real decoding, next slide (courtesy of Ken Andrews).
Regular, Irregular, MET comparison
Multi-edge-type LDPC codes improve gradually through most of the decoding.
MET codes have a threshold below the more "complex" irregular codes.
Design of Codes using D.E.
Density evolution provides a moderately fast way to evaluate single- and multi- edge type degree distributions (hypothesis-testing).
Much faster approximations to Density Evolution can easily be put into an outer loop which performs function minimization.
Semi-Analytic techniques exist as well.
Review
Iterative Message Passing can be used to decode LDPC codes.
Density Evolution can be used to predict the performance of MP for infinitely large codes.
More sophisticated classes of codes can be designed to close the gap to the Shannon limit.
Beyond MET Codes (future work)
Traditional LDPC codes are designed in two stages: design of the degree distribution and design of the specific graph.
Using very fast and accurate approximations to density evolution, we can evaluate the effect of placing or removing a single edge in the graph.
Using an evolutionary algorithm like Simulated Annealing, we can optimize the graph directly, changing the degree profile as needed.