Phd Defence

Construction and Analysis of Non Systematic
Codes on Graphs for Redundant Data

Amira A LLOUM

Ecole Nationale Sup´ rieure des T´ l´ communications
e ee
Telecom Paris Tech

September 5th, 2008

Presentation Outline

Introduction and Motivations

Part I: Non Systematic LDPC Codes Constructions

Part II: Density Evolution Analysis for Split-LDPC Codes

Part III: Exit Chart Analysis for Split-LDPC Codes

Part IV : EM for Joint Source-Channel Estimation

Conclusions and Future Work

Introduction

The Non Uniform Assumption

0.5

0 1

P (s = 1) = 0.5

Source Encoder Channel Encoder Channel Channel Decoder Source Decoder

Amira A LLOUM Construction and Analysis of Non-Systematic Codes on Graphs for Redundant Data September 5th, 2008 1 / 35

Introduction


Source Encoder Source Decoder

0.9

0 Channel
0.1 Channel Encoder Channel Decoder
1
P (s = 1) = µ

The Uniform Assumption is not valid anymore When
1 It is not worth to compress (bad channel conditions)


Introduction



0.9

0 Channel
0.1 Channel Encoder Channel Decoder
1
P (s = 1) = µ


The Uniform Assumption is not valid anymore When
1 It is not worth to compress (bad channel conditions)
2 Using sub-optimal compression (highly redundant sources)


Introduction

Channel Coding Strategies for Non Uniform Sources

0.9
P (s = 1) = µ
0 0.1
1

Source Channel Encoder Channel Channel Decoder Sink

Shannon has intuited in his 1948 Paradigm
Any Redundancy in the source will usually help if it is utilized at the receiveing
...This redundancy will help to combat noise


Introduction


0.9
P (s = 1) = µ
0 0.1
1


SCCD
µ

Channel Coding for Redundant data follows the following strategies
1 Source Controlled Channel Coding (Hagenauer 1995).


Introduction


0.9
P (s = 1) = µ Systematic Coding
0 0.1
1 Info Parity


Parity
µ Non−Systematic Coding µ
Best Constructions

Channel Coding for Redundant data follows the following strategies
1 Source Controlled Channel Coding (Hagenauer 1995).
2 Non Systematic Encoding Structures (Shamai and Verdu 1997)


Introduction

Information Theoretical Limits wih Redandancy: AWGN Channel
Capacity limit Versus Source Entropy, Coding Rate = 0.5, AWGN Channel
2

0
Minimum Achievable Eb/N0 (dB)

-2

-4

-6

-8

-10
Systematic code, BPSK input
Non-Systematic code, BPSK input
Gaussian input
-12
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Source Entropy (bits)


Introduction

Information Theoretical Limits wih Redandancy: AWGN Channel
Capacity limit Versus Coding Rate , Source Entropy= 0.5, AWGN Channel
6
Systematic codes, BPSK input
5 Non-Systematic code, BPSK input
Gaussian input
4

3
Minimum Achievable Eb/N0 (dB)

2

1

0

-1

-2

-3

-4

-5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Coding Rate


Introduction

Achieving the Theoretical Limits 1

In the presence of Redundancy
The Theoretical Limits of Information Theory are mooving to better
regions.


Introduction

Achieving the Theoretical Limits 1

In the presence of Redundancy
The Theoretical Limits of Information Theory are mooving to better
regions.

To Attain these Challenging limits :
1 Building Non Systematic Capacity Achieving Encoding Structures In the
Codes On graphs Family.

2 Using Source Controlled Channel Decoding with Iterative Algorithms In
the Sum-Product Algorithms Family.


Introduction

Related Work and Motivation

Non−Systematic
Codes On Graph

Turbo Codes LDPC

Non−Systematic MN Codes for BSC
Alajaji Codes et.al. LDPC Codes


Introduction

Main System Assumptions

0.9
P (s = 1) = µ
0 0.1
1


Lossless source coding or no source coding.
Binary i.i.d. source with entropy Hs = H2 (µ), where µ = P (si = 1).
Source sequence s = (s1 , s2 , ..., sK ) encoded by a binary channel code of
rate Rc = K/N , dimension K, and length N .
x = (x1 , ..., xN ) denotes the codeword (channel input).
Transmitted information rate R = Hs × Rc bits per channel use.
Any symmetric binary-input channel can be considered, mainly BEC,
BSC, and AWGN.

Part I

Non Systematic LDPC Constructions

Encoding Structures Decoding Strategy Simulation Results

Scrambling-LDPC Encoding Structure

s u c(u,v)
Cs G

Scramble LDPC
c = G × Cs × s
v
(1−R).N β (1−R).N
v db
LDPC
Cs : sparse matrix of dimension
β K × K. In the regular case row and
u
db
R.N dc
u
column weight are ds
u: systematic bits for the inner
LDPC.
s α
ds v: parity bits for the inner LDPC.
R.N s Cs α R.N
ds
s: source bits.
s α
Scrambler


Splitting-LDPC Encoding Structure

s u c(u,v)
C−1
s G

Splitter LDPC
−1 × s
c = G × Cs
v
(1−R).N β (1−R).N

v
LDPC dc
db Cs : full rank sparse matrix of
u
β dimension K × K. In the regular
R.N case row and column weight are ds
u
Cs u: systematic bits for the inner
ds
ds
Splitter
LDPC.
s α
v: parity bits for the inner LDPC.
s α
R.N R.N s: source bits.
s α



Source Controlled Sum-Product Decoding

The bitnode Rule: v
β
v
LLRci →pcei =LLRtype + LLRpcej →ci LLR0+ Extrinsic Information

∗
pcej ∈Sc
i
β
u
LLR0 if bitnode ∈ {u, ϑ}
LLRtype = u
LLRs if bitnode ∈ {s}

s α
LLR0 is the channel observation LLR.
1−µ s α
LLRs = log( ) is the source LLR .
µ LLRs+ Extrinsic Information
s α




The bitnode Rule:
v
β
LLRci →pcei =LLRtype + LLRpcej →ci
∗
v
pcej ∈Sc
i

LLR0 if bitnode ∈ {u, ϑ} β
LLRtype = u Extrinsic Information
u

The checknode Rule:
s α
LLRcj →pcei
LLRpcei →ci = 2 tanh−1 tanh( ) s α
∗
2
cj ∈Spce
i
s α




The bitnode Rule:
v
β
LLRci →pcei =LLRtype + LLRpcej →ci
∗
v
pcej ∈Sc LLR0+ Extrinsic Information
i

LLR0 if bitnode ∈ {u, ϑ} β
LLRtype = u
u Extrinsic Information

The checknode Rule:
s α
LLRcj →pcei
LLRpcei →ci = 2 tanh−1 tanh( ) s α
∗
2 LLRs+ Extrinsic Information
cj ∈Spce
i
s α



Scrambling or Splitting: Information Theoretical Comparaison
Mutual information vs. Eb /N0 for Hs = 0.5 and coding rates Rc = 0.5 (AWGN Channel).
1.8

1.6
Gaussian input
1.4 Non-Systematic code, BPSK input
Scrambled ds=5
Scrambled ds=3
1.2 Systematic codes, BPSK input
Mutual information

1

0.8

0.6

0.4

0.2

0
-10 -5 0 5 10
Eb/N0(dB)


Finite-length Performance

1E-01 1E+00
(3,6) systematic LDPC without SCCD (3,6) systematic LDPC without SCCD
(3,6) systematic LDPC with SCCD (3,6) systematic LDPC with SCCD
scrambler ds=3 scrambler ds=3
MN code db=3 MN code db=3
splitter ds=4 splitter ds=4
1E-02 1E-01

Frame Error Rate
Bit Error Rate

1E-03 1E-02

1E-04 1E-03

1E-05 1E-04
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
SNR SNR

Bit (left) and word (right) error probabilities vs. signal to noise ratio for codes with rate
Rc = 1/2 length N = 2000 and non uniform source distribution µ = 0.1 : systematic (3,6)
LDPC with and without SCCD, split-LDPC with ds = 4, scramble-LDPC ds = 3 and MN
Codes db = 3.


Finite-length Performance (2/2)
1E+00
(3,30) systematic LDPC with SCCD
scrambler ds=5
splitter ds=7

1E-01

Frame Error Rate

1E-02

1E-03

1E-04
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4
SNR

Word error probabilities vs. signal to noise ratio Eb /N0 for codes with rate Rc = 0.9 length
N = 2000 and non uniform source distribution µ = 0.1 : systematic (3,30) LDPC with SCCD,
scramble-LDPC with ds = 5 and split-LDPC with ds = 7.

Part II

DE Analysis for Split-LDPC Codes

DE Motivation and Assumption DE statement Simulation Results Stability Analysis

Motivation and Contribution

The problem:
How close do Split-LDPC structures approach the Challenging
asymptotical limits ?

Exploring the Split-LDPC asymptotical convergence behaviour.



Motivation and Contribution

The problem:
How close do Split-LDPC structures approach the Challenging
asymptotical limits ?

Exploring the Split-LDPC asymptotical convergence behaviour.

The proposal: Density Evolution Analysis
1 Deriving a Density Evolution algorithm for Split-LDPC codes.

2 Investigating the stability issues related to the decoder convergence.



Density Evolution Assumptions

1 Concentration and the local tree Assumption.





3 types of message distribution =⇒ Local tree assumption over 3 types of trees

Message oriented Density Evolution.

db dc p3(x)
ϑ β ϑ β
3 dc
db
p2(x)
u ds u

2 ds
p1(x)
s
s
1 1 1
α
α
Messages Distributions




2 Symmetry Conditions (Channel, variable nodes, Checknodes).






3 The all-zero codeword restriction.






3 The all-zero codeword restriction.

0 0 0 0 0 /
∈B

A Typical Set
C[A]=2k
B
C(B)=2K×Hs



The DE statement: the Checknode level

u 0 0 0 0

{
0
s 0
s Averaging
p1(x) α 1
u 1 0 0 0

{
ds − 1 0

u u

pm (x) = Rc ps (x), (1 − µ) q1 (x) + µ q1 (−x)
α
m m

where:
1−µ
ps (x) = δ(x − log ) = δ(x − s)
µ

q1 (x) = ρα (pm (x))
m
1




β
R p2(x) + (1 − R) p3(x)

dc − 1

1
2
u + 1ϑ
2
1
2
u + 1ϑ
2

pm (x) = ρ Rc pm (x) + (1 − Rc ) pm (x)
β 2 3




s
β
α p1(x) R p2(x) + (1 − R) p3(x)

ds − 1 dc − 1

u u 1
u + 1ϑ 1
u + 1ϑ
2 2 2 2

pm (x) = Rc ps (x), (1 − µ) q1 (x) + µ q1 (−x)
α
m m

pm (x) = ρ Rc pm (x) + (1 − Rc ) pm (x)
β 2 3



The DE statement: the variable-node level
p1 α

u
ds − 1 db

η(x) xλ(x)

α α β β
p1 R p2 + (1 − R) p3

ds − 1 dc − 1

1 1 1
u u
2
u + 2
ϑ 2
u + 1ϑ
2

p1 (x) at the (m + 1)th iteration

pm+1 (x) = p0 (x) ⊗ λ1α pm (x)
1 α ⊗ λ1 pm (x)
β



β

p2

u

ds db − 1

xη(x) λ(x)

α α β β
p1 R p2 + (1 − R)p3

ds − 1 dc − 1

u u 1
+ 1 1
+ 1ϑ
2u 2ϑ 2u 2


pm+1 (x) = p0 (x) ⊗ λ2α pm (x)
2 α ⊗ λ pm (x)
β



β
p3

ϑ
db − 1

R p2 + (1 − R)p3 β β β

dc − 1

1 1 1
2u + 2ϑ 2u + 1ϑ
2


pm+1 (x) = p0 (x) ⊗ λ pm (x)
3 β



2
lambda(x)=0.32660x+0.11960x^2+0.18393x^3+0.36988x^4
rho(x)=0.78555x^5+0.21445x^6
0

Minimum Achievable Eb/N0 (dB) -2

-4

-6

-8

Systematic code, BPSK input
-10
Non-Systematic code, BPSK input
Gaussian input
split-LDPC code, DE thresholds
-12
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Source Entropy (bits)
Minimum achievable Eb /N0 versus source entropy Hs for a regular ds = 3 splitter
concatenated to an irregular LDPC code of rate Rc = 1/2 over a BIAWGN channel.



The asymptotical behaviour of the decoder in the neighborhood of δ∞
1−R
ϑ ϑ β
CHANNEL P0

u u
Pu0
R

s BSC (µ δ−s + (1 − µ) δs )⊗ ds

µ
ds
s BSC

µ

s BSC

µ

Pu0 = p0 (x) ⊗ (µ δ−s + (1 − µ) δs )⊗ dS

Proposition 1
In the neighborhood of δ∞ , the message density given by the splitter to the core LDPC
(from node α to node u) is equivalent to the initial message density of ds parallel
concatenated BSC with a crossover probability µ.


1−R
ϑ ϑ β
CHANNEL P0

u u
Pu0
R

s BSC (µ δ−s + (1 − µ) δs )⊗ ds

µ
ds
s BSC

µ

s BSC

µ

Pu0 = p0 (x) ⊗ (µ δ−s + (1 − µ) δs )⊗ dS

Proposition 2
For non-uniform sources,type-1 message distribution (from node u to node α) shows a
permanent stability around the ﬁxed point δ∞ .


1−R
ϑ ϑ β
CHANNEL P0

u u
Pu0
R

s BSC (µ δ−s + (1 − µ) δs )⊗ ds

µ
ds
s BSC

µ

s BSC

µ

Pu0 = p0 (x) ⊗ (µ δ−s + (1 − µ) δs )⊗ dS

Proposition 3
When close to zero error rate, stability of the LDPC constituent is not disturbed by the
splitter.


1−R
ϑ ϑ β
CHANNEL P0

u u
Pu0
R

s BSC (µ δ−s + (1 − µ) δs )⊗ ds

µ
ds
s BSC

µ

s BSC

µ

Pu0 = p0 (x) ⊗ (µ δ−s + (1 − µ) δs )⊗ dS

If the embedded LDPC is stable the Split-LDPC would be so.
The inverse is not true !



Stability Condition for Split-LDPC

CHANNEL Systematic LDPC

p0(x)

The General Stability Condition for a systematicLDPC:

B(p0 )λ (0) ρ (1) ≤ 1

+∞
where B(p0 ) = p0 (x)e−x/2 dx is the Bhattacharyya constant of the channel.
−∞



Stability Condition for Split-LDPC

p0 (x) 1−R
ϑ CHANNEL ϑ β
Averaging
EQUIVALENT
u u
CHANNEL
R
ds
Pu0 = p0(x) ⊗ (µ δ−s + (1 − µ) δs )⊗

GLOBAL EQUIVALENT CHANNEL Systematic CORE−LDPC
Peq = R pu0 + (1 − R) p0(x)

The General Stability Condition for Split-LDPC:

B(Peq )λ (0) ρ (1) ≤ 1

where Peq = Rc Puo (x) + (1 − Rc ) p0 (x) is the initial message density of the global
equivalent channel; and B(Peq ) is the Bhattacharyya constant of the global equivalent
channel.



The Splitter Asymptotical Properties:

Proposition 4
For uniform sources, the threshold and the stability condition of the
split-LDPC code are the same for the CORE-LDPC




Proposition 4

Example
Split-LDPC Stability Condition for BEC Channel:
1 1
λ (0) ρ (1) < ×
ε [(1 − Rc ) + Rc (2 µ(1 − µ))ds ]




Proposition 4

Example
Split-LDPC Stability Condition for BSC Channel:
1 1
λ (0) ρ (1) < ×
2 λ(1 − λ) [(1 − Rc ) + Rc (2 µ(1 − µ))ds ]




Proposition 4

Example
Split-LDPC Stability Condition for AWGN Channel::
1 1
λ (0) ρ (1) < e 2σ2 ×
[(1 − Rc ) + Rc (2 µ(1 − µ))ds ]


Part III

EXIT Chart Analysis for Split-LDPC Codes

Motivation and Assumptions Split-LDPC EXIT Chart Statement Design of Irregular Split-LDPC Results

Motivation and Proposal

The problem
High computational complexity for Density Evolution
Low Complexity Approaches based on Gaussian Approximation
Lower Complexity and one-dimensional
More Insightful
Less Accurate
How to build capacity achieving Split-LDPC ?

The Proposal:
Message oriented Bi-dimensional low complexity approach based on a more
accurate Exit Chart method.



Assumptions and Notations:

Message-oriented local tree assumption.

p1 α

u
ds − 1 db

η(x) xλ(x)

α α β β
p1 R p2 + (1 − R) p3

ds − 1 dc − 1

1 1 1
u u
2
u + 2ϑ 2
u + 1ϑ
2




Consistency Condition is realized on all types of message distribution.

f (x) = f (−x) ex for all x ∈ R+



Gaussian approximation is applied at the bitnodes output based on equal
mutual information.
Uout

U0

Uout f
f −1 Uin

U0

Uin



Gaussian approximation is applied at the bitnodes output based on equal
mutual information.
We display the error probability as a measure of knowledge U .

Uout

U0

+∞
σ 1 t2
Perr = Q( ) = √ e− 2 dt
2 2π σ/2

Uout f
f −1 Uin

U0

Uin



The EXIT Chart Statement for Split-LDPC : Message Combining
β ∗
P2 = (1 − R)P3 + RP2 β

P2 P3

u
ϑ
ds db − 1
db − 1

xη(x) λ(x)

Pin2
∗ β β β
α α β β
Pin1
∗

dc − 1

ds − 1 dc − 1

1 1 1
u u 1
u + 1ϑ 1
u + 1ϑ 2u + 2ϑ 2u + 1ϑ
2
2 2 2 2

∗ ∗ ∗
Pout2 = G(Pin1 , Pin2 )



β ∗
P2 = (1 − R)P3 + RP2 β

P2 P3

u
ϑ
ds db − 1
db − 1

xη(x) λ(x)

Pin2
∗ β β β
α α β β
Pin1
∗

dc − 1

ds − 1 dc − 1

1 1 1
u u 1
u + 1ϑ 1
u + 1ϑ 2u + 2ϑ 2u + 1ϑ
2
2 2 2 2

db db
G(x, y) = Rc × λj fds +1,j (x, y) + (1 − Rc ) × λj gj (y)
j=2 j=2




∗ α
P1

u

ds − 1 db

η(x) xλ(x)

α α β β Pin2
∗
Pin1
∗

ds − 1 dc − 1

u u 1
u + 1ϑ 1
u + 1ϑ
2 2 2 2

∗ ∗ ∗
Pout1 = F (Pin1 , Pin2 )



∗ α
P1

u

ds − 1 db

η(x) xλ(x)

α α β β Pin2
∗
Pin1
∗

ds − 1 dc − 1

u u 1
u + 1ϑ 1
u + 1ϑ
2 2 2 2

db
F (x, y) = λj fds ,j+1 (x, y)
j=2



The validity Range for the Bidimensional Dynamical System
∗ ∗
The mixtures of Gaussian densities associated to inputs Pin1 and Pin2 must
satisfy the two equalities:

db
∗ 1 1
Pin1 = λj erf c m0 + (ds − 1)mα + jmβ
j=2
2 2
db
∗ 1 1
Pin2 = Rc × λj erf c m0 + ds mα + (j − 1)mβ
j=2
2 2
db
1 1
+ (1 − Rc ) × λj erf c m0 + (j − 1)mβ
j=2
2 2

m0 : Mean of messages from the channel.
mα : Mean of messages from the splitter checknodes α.
mβ : Mean of messages from the LDPC checknodes β.


Trajectory of error probability near the code threshold
Es
Right to the threshold: = −5.58dB, Threshold=−5.68dB. Final ﬁxed point is 0.
N0

Transfer Function F(x,y)
Trajectory of Pout1

Pout1 0.20

0.25 0.15

0.20
0.10
0.15
0.10 0.05

0.05
0.00
0.00

0.20
0.25
0.15
0.20
0.15 0.10 Pin1
0.10
0.05
Pin2 0.05
0.00
Illustration for an irregular Split-LDPC code.


Trajectory of error probability beyond the code threshold (2)
Es
Left to the threshold: = −5.78dB, Threshold=−5.68dB. Final ﬁxed point is non-zero.
N0

Trajectory of Pout1

Pout1 0.20

0.15
0.25
0.20 0.10
0.15
0.05
0.10
0.05 0.00
0.00

0.20

0.25 0.15
0.20
0.10 Pin1
0.15
0.10 0.05
Pin2 0.05
0.00
Illustration for an irregular Split-LDPC code.


Open Tunnel for the Bi-dimensional EXIT Chart
Open tunnel obtained by plotting the trajectory of error probability and its z = x plane
Es
reﬂection at = −5.58dB .
N0

Pout1 Trajectory of Pout1
Inverse of Trajectory of Pout1
Plan z=x
0.18
0.25
0.16
0.20 0.14
0.12
0.15 0.10
0.08
0.10 0.06
0.04
0.05 0.02
0.00
0.00

0.25
0.20
0.15
0.10 Pin2
0.00 0.05 0.10 Pin1 0.15 0.20 0.05



Design Irregular Split-LDPC codes

The Design linear Program:
1 Maximize λi /i
i≥2
2 subject to : λi ≥ 0, λi = 1
i≥2
∗ ∗
3 and ∀(P1in , P2in ) ∈ T (S)
db ∗ ∗
j=2 λj fds ,j+1 (P1in , P2in∗ ) < P1in
db ∗ ∗ ∗ ∗
j=2 λj [Rc × fds +1,j (P1in , P2in ) + (1 − Rc ) × gj (P2in )] < P2in




The Design Process:
Select a regular code with the desired coding rate.

∗ ∗
Use the regular code degree sequence for mapping (P1in , P2in ) to the
appropriate input Gaussian mixture densities.

Find the EXIT charts surfaces for different variable degrees.

Find a linear combination with an open EXIT chart that maximizes the
rate and meets all the required design criterion.




The Design Process:
Select a regular code with the desired coding rate.

∗ ∗
Use the regular code degree sequence for mapping (P1in , P2in ) to the
appropriate input Gaussian mixture densities.

Find the EXIT charts surfaces for different variable degrees.

Find a linear combination with an open EXIT chart that maximizes the
rate and meets all the required design criterion.

Best Approach to Shannon limits is within 0.1 dB !



Bi-dimensional EXIT Chart Accuracy

Error in approximation of threshold (dB) using EXIT chart analysis for various split-LDPC
codes of Rate one-half with ds = 3 , Hs = 0.5. ∆ is the log-ratio quantization step.

Eb /N0 ∗ (dB) Eb /N0 ∗ (dB) Error ∆Eb /N0 ∗
db dc Rate DE ∆ = 0.005 EC ∆ = 0.005 for ∆ = 0.005
3 6 0.5 −2.22 −2.199 0.020
4 8 0.5 −1.12 −1.129 0.009
5 10 0.5 −0.32 −0.369 0.049


Part IV

EM Source-Channel Estimation

Motivation and Proposal Expectation-Maximization Complex BIAWGN Channel Simultations Results


Motivation
Full utilization of systematic and non-systematic LDPC codes requires:
1 Knowledge of the source probability distribution (SSI) at the decoder side.
2 Knowledge of channel parameters (CSI) at the decoder side.




Motivation
Full utilization of systematic and non-systematic LDPC codes requires:
1 Knowledge of the source probability distribution (SSI) at the decoder side.
2 Knowledge of channel parameters (CSI) at the decoder side.

Proposal
Joint Source-Channel Iterative Estimation and Decoding for non-uniform
sources based on the Expectation Maximization Algorithm (EM)

→ No performance loss → Negligible estimation complexity



Brief statement of the EM algorithm
κ : complete data, κ = (x, y)

0.9
x:missing data y: incomplete data (observed)
p(s = 1) = µ
0.1 0
1 x y


SSI = µ CSI
Θ=SSI+CSI
set of parameters to be estimated, SSI+CSI ˆ
Θ

E-step: Compute the Auxiliary function Q:

Q(θ|θi ) = E[log p(x, y|θ)|y, θi ]
= log[P (y|x, θ) P (x|θ)] AP Pi (x)
x




0.9
p(s = 1) = µ
0.1 0
1 x y


SSI = µ CSI
Θ=SSI+CSI
Θ


x
The SSI part in the auxiliary function is:
P (x|θ) ≡ P (s|θ) = µωH (s) (1 − µ)K−ωH (s)




0.9
p(s = 1) = µ
0.1 0
1 x y


SSI = µ CSI
Θ=SSI+CSI
Θ


x
M-step:

θ i+1 = arg max Q(θ|θ i )
θ


Joint Source-Channel Estimation on Complex BIAWGN

The Complex BIAWGN channel is deﬁned as
φ
√
yi = Ae xi + ηi with = −1

where the three CSI parameters are :
The amplitude A , which is real positive
The phase ambiguity φ, which is uniformly distributed between 0 and 2π.
The Gaussian noise variance σ 2 .




φ
√

E-step: Auxiliary Function

µ
Q(θ|θi ) = log[ ]˜ + K log[(1 − µ)] − N log[2πσ 2 ]
s
1−µ
N N N
1 2 A2 2 A
− yj − xj + R xj ∗ e−
˜ φ
yj
2σ 2 j=1
2σ 2 j=1
σ2 j=1




φ
√

M-step: Maximizing the auxiliary function for SSI
K
i+1 j=1 sj
˜
µ =
K




φ
√

M-step: Maximizing the auxiliary function for CSI
N φi
j=1 ˜ ∗
R xj yj e
Ai+1 = N 2
j=1 xj AP Pi (xj ) xj
N
2 (i+1) 1 φi x 2
σ = yj − Ai e j
2N j=1

N
φi+1 = − Arg ˜ ∗
xj yj
j=1



100
Systematic + Uniform
Systematic + Non Uniform (Perfect)
Systematic + Non Uniform (EM observation)
Systematic + Non Uniform (EM APP)
Non Systematic + Non Uniform (Perfect)
10-1 Non Systematic + Non Uniform (EM APP)
Frame Error Rate

10-2

10-3

10-4
-2.4 -2 -1.6 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 1.6 2 2.4
Eb/N0 (dB)

Systematic and non-systematic LDPC codes on AWGN, Rc = 1/2.


Conclusions and Open Issues

Conclusions

The theoretical limits for redundant data are mooving to better regions.



Conclusions


Split-LDPC codes are the best non systematic LDPC constructions.



Conclusions



Split-LDPC codes have good asymptotical performance and properties.



Conclusions




Designed irregular Split-LDPC codes are capacity achieving.



Conclusions





The 2-D EXIT Chart formalism is lower complexity and as accurate as the
DE.



Conclusions





The 2-D EXIT Chart formalism is lower complexity and as accurate as the
DE.

The performance using EM is as good as the perfect knowledge case.



Open Issues

Investigating more complex source and channel models.



Open Issues


Extending our study to irregular splitting and scrambling LDPC.



Open Issues



Performing Complexity and Finite-length Optimization.



Open Issues




Consider lossy sources assumption .



Open Issues




Consider lossy sources assumption .

Studying and incorporating Split-LDPC in practical communication
systems.


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