The Concept of Soft Channel Encoding and its Applications ...sc.enseeiht.fr/doc/Seminar_Matz.pdf ·...

The Concept of Soft Channel Encoding andits Applications in Wireless Relay Networks

Gerald Matz

Institute of Telecommunications

Vienna University of Technology

institute oftelecommunications

Acknowledgements

People:

• Andreas Winkelbauer

• Clemens Novak

• Stefan Schwandter

Funding:

SISE - Information Networks (FWF Grant S10606)

2

Outline

1. Introduction

2. Soft channel encoding

3. Approximations

4. Applications

5. Conclusions and outlook

3

Outline

1. Introduction


3. Approximations

4. Applications


4

Introduction: Basic Idea

Soft information processing

• popular in modern receivers

• idea: use soft information also for transmission

Soft-input soft-output (SISO) encoding

• extension of hard-input hard-output (HIHO) encoding

• data only known in terms of probabilities

• how to (efficiently) encode “noisy” data?

Applications

• physical layer network coding

• distributed (turbo) coding

• joint source-channel coding

5

Introduction: Motivation (1)

Example: relay channel

D

xS

xS

xR

R

S

u′

R

πxR

xS

SR ch.

Encoder S

Encoder Ru

uS

• relay R assists S – D transmission

• decode-and-forward → distributed channel coding

• D can perform iterative (turbo) decoding

What if relay fails to decode?

• forwarding soft information might help

• how to transmit soft information?

6

Introduction: Motivation (2)

Soft information forwarding

• H. Sneessens and L. Vandendorpe, “Soft decode and forward improvescooperative communications,” in Proc. Workshop on Computational Advancesin Multi-Sensor Adaptive Processing, 2005.

• P. Weitkemper, D. Wubben, V. Kuhn, and K.D. Kammeyer, “Soft informationrelaying for wireless networks with errorprone source-relay link,” in Proc. ITGConference on Source and Channel Coding, 2008.

Transmission of soft information

• LLR quantization ⇒ information bottleneckN. Tishby, F. Pereira, and W. Bialek, “The information bottleneck method,” inProc. 37th Allerton Conference on Communication and Computation, 1999.

• quantizer labels & symbol mapping ⇒ binary switchingK. Zeger and A. Gersho, “Pseudo-gray coding,” IEEE Trans. Comm., vol. 38,no. 12, 1990.

7

Outline

1. Introduction


3. Approximations

4. Applications


8

Hard Encoding/Decoding Revisited

Notation

• information bit sequence: u = (u1 . . . uK)T ∈ {0, 1}K

• code bit sequence: c = (c1 . . . cN )T ∈ {0, 1}N

• assume N ≥ K

Linear binary channel code

• one-to-one mapping φ between data bits u and code bits c

• encoding: c = φ(u)

• codebook: C = φ({0, 1}K

)Hard decoding:

• observed code bit sequence c′ = c⊕ e = φ(u)⊕ e

• decoder: mapping ψ such that ψ(c′) is “close” to u

9

Soft Decoding Revisited

Word-level

• observations: code bit sequence probabilities pin(c′)

• enforce code constraint:

p′(c) =

{pin(c)∑

c′∈C pin(c′) , c ∈ C

0, else

• info bit sequence probabilities: pout(u) = p′(φ(u))• conceptually simple, computationally infeasible

Bit-level

• observed: code bit probabilities pin(cn) =∑∼cn

pin(c′)

• desired: info bit probabilities pout(uk) =∑∼uk

pout(φ(u))

• conceptually more difficult, computationally feasible

• example: equivalent to BCJR for convolutional codes

10

Soft Encoding: Basics

Word-level

• given: info bit sequence probabilities pin(u)• code bit sequence probabilities:

pout(c) =

{pin(φ

−1(c)), c ∈ C0, else

• conceptually simple, computationally infeasible

Bit-level

• given: info bit probabilities pin(uk) → pin(u) =K∏k=1

pin(uk)

• desired: code bit probabilities

pout(cn) =∑∼cn

pout(c) =∑

c∈C:cnpin(φ

−1(c))

• Main question: efficient implementation?11

Soft Encoding: LLRs

System model: SISO

encoder

binary

source

noisy binary source

u L(u) L(c|u)channel

noisy

Log-likelihood ratios (LLR)

• definition:

L(uk) = logpin(uk=0)

pin(uk=1), L(cn) = log

pout(cn=0)

pout(cn=1)

• encoder input: L(u) = (L(u1) . . . L(uK))T

• encoder output: L(c) = (L(c1) . . . L(cN ))T

12

HIHO versus SISO

HIHO is reversible:

SISO is irreversible:

• except if |L(uk)| =∞ for all k

• however: sign(L1(uk)) = sign(L2(uk))

Post-sliced SISO identical to pre-scliced HIHO

SISO

encoder

L(u) c cHIHO

encoder

uL(u)≡

L(c)

13

Block Codes (1)

Binary (N,K) block code C with generator matrix G ∈ FN×K2

HIHO encoding: c = Gu, involves XOR/modulo 2 sum ⊕

Statistics of XOR:

p(uk⊕ul=0) = p(uk=0)p(ul=0) + p(uk=1)p(ul=1)

Boxplus: L(uk⊕ul) , L(uk)� L(ul) = 1 + eL(uk)+L(ul)

eL(uk) + eL(ul)

• � is associative and commutative

• |L1 � L2| ≤ min{|L1|, |L2|}

J. Hagenauer, E. Offer, and L. Papke, “Iterative decoding of binary block and

convolutional codes,” IEEE Trans. Inf. Theory, vol. 42, no. 2, Feb. 1996.

SISO encoder: replace “⊕” in HIHO encoder by “�”

14

Block Codes (2)

Example: systematic (5, 3) block codec1c2c3c4c5

︸︷︷︸

c

=

1 0 00 1 00 0 11 1 00 1 1

︸︷︷︸

G

u1u2u3

︸︷︷︸

u

=

u1u2u3

u1 ⊕ u2u2 ⊕ u3

L(u1) L(u3)L(u2)

�L(c4)

L(c3)

L(c2)

L(c1)

�L(c5)

u1 u3u2

⊕

⊕

c1

c2

c3

c4

c5

15

Soft Encoding: Convolutional Codes (1)

• Code C with given trellis: use BCJR algorithm

· · ·

· · ·

· · ·

· · ·(1, 1)

(1, 0)

(0, 1)

(0, 0)0

1

2

3

m(m(1),m(0)

)

k = 0 21 3 TT − 3 T − 2 T − 1

αk(m) βk(m)

γk(m′,m) = P

{Sk+1 = m|Sk = m′

}αk(m) =

∑M−1m′=0

αk−1(m′)γk−1(m

′,m)

βk(m) =∑M−1

m′=0βk+1(m

′)γk(m,m′)

pb(c(j)k

)=∑

(m′,m)∈A(j)b

αk(m′)γk(m

′,m)βk+1(m)

A. Winkelbauer and G. Matz, “On efficient soft-input soft-output encoding ofconvolutional codes,” in Proc. ICASSP 2011

16


• Observe: only 2 transition probabilites per time instantI Backward recursion βk(m) is rendered superfluous

· · ·

· · ·

· · ·

· · ·(1, 1)

(1, 0)

(0, 1)

(0, 0)0

1

2

3

m(m(1),m(0)

)

k = 0 21 3 TT − 3 T − 2 T − 1

αk(m) βk(m)

• Simplified forward recursion encoder (FRE), reducesI computational complexityI memory requirementsI encoding delay

sk+1(m) =∑

m′∈Bmsk(m

′)γk(m′,m)

pb(c(j)k

)=∑

(m′,m)∈A(j)b

sk(m′)γk(m

′,m)

17


• Observe: only 2 transition probabilites per time instantI Backward recursion βk(m) is rendered superfluous

· · ·

· · ·

· · ·

· · ·(1, 1)

(1, 0)

(0, 1)

(0, 0)0

1

2

3

m(m(1),m(0)

)

k = 0 21 3 TT − 3 T − 2 T − 1

αk(m) βk(m)

• Simplified forward recursion encoder (FRE), reducesI computational complexityI memory requirementsI encoding delay

sk+1(m) =∑

m′∈Bmsk(m

′)γk(m′,m)

pb(c(j)k

)=∑

(m′,m)∈A(j)b

sk(m′)γk(m

′,m)17


• Special case: non-recursive shift register encoder (SRE)I soft encoder with shift register implementationI linear complexity with minimal memory requirements

• Example: (7, 5)8 convolutional code

DD

L(c2k+1)

L(c2k)

L(uk)

BCJR

operationsper

unittime

constraint length ν

SRE (upper bound)

FRE

2 4 6 8 100

200

400

600

800

1000

18

Soft Encoding: LDPC Codes (1)

• Sparse parity check matrix H given: v ∈ C iff HTv = 0

• Graphical representation of H: Tanner graphI bipartite graph with variable nodes and check nodesI let V denote the set of variable nodes

• Encoding: iterative erasure fillingI matrix multiplication Gu is infeasible for large block lengthsI erasure pattern is known

19

Soft Encoding: First Approach

Consider systematic code: c = (uT pT )T

Erasure channel: L(c) = (L(u)T L(p)T )T 7→ (L(u)T 0T )T

deterministic

erasure channel

(L(u)TL(p)T )T

(L(u)TL(p)T )T (L(u)T 0T )T

SISO encoding = decoding ...

• for the erasure channel (erasure filling) or, equivalently,

• w/o channel observation, but with prior soft information on u

Consider special problem structure

• yields efficient implementation

• (much) less complex than SISO decoding

• adjustable accuracy/complexity trade-off

20


• Iterative erasure filling algorithm1. find all check nodes involving a single erasure2. fill the erasures found in step 13. repeat steps 1-2 until there are no more (recoverable) erasures

• Encoding example:

• Problem: stopping sets ⇒ non-recoverable erasures

21


• Definition: S ⊆ V is a stopping set if all neighbors of S areconnected to S at least twice

• Example:

• Solution: modify H such that stopping sets vanishI constraints: code C remains unchanged and H remains sparse

M. Shaqfeh, N. Gortz, “Systematic Modification of Parity-Check Matrices forEfficient Encoding of LDPC Codes,” in Proc. ICC 2007

22

Outline

1. Introduction


3. Approximations

4. Applications


23

Approximations: Boxplus Operator

• Boxplus operator is used frequently in SISO encoding

I Recall a� b = 1 + ea+b

ea + eb

• Approximation: a� b ≈ a � b = sign(a)sign(b)min(|a|, |b|)I small error if

∣∣|a| − |b|∣∣ large

I overestimates true result: a � b ≥ a� bI suitable for hardware implementation

• Correction termsI a� b = a � b+ log(1 + e−|a+b|)− log(1 + e−|a−b|)︸︷︷︸

− log(2) ≤ additivecorrection ≤ 0

I a� b = a � b ·

(1− 1

min(|a|, |b|)log

1 + e−||a|−|b||

1 + e−||a|+|b||

)︸︷︷︸

0 ≤ multiplicativecorrection ≤ 1

I Store correction term in (small) lookup table

• Decrease lookup table size by LLR clipping

24

Approximations: Max-log Approximation

• FRE: perform computation in log-domain (f∗ = log f)

s∗k+1(m) = log∑

m′∈Bmexp(s∗k(m

′) + γ∗k(m′,m)

)p∗b(y(j)k

)= log

∑(m′,m)∈A(j)

b

exp(s∗k(m

′) + γ∗k(m′,m)

)• Approximation: log

∑k exp(ak) ≈ maxk ak , aM

• Correction term: log(ea + eb) = max(a, b) + log(1 + e−|a−b|)I nesting yields log

∑k exp(ak) = aM + log

∑k exp(ak − aM )

• Correction term depends only on |a− b| ⇒ lookup table

25

Outline

1. Introduction


3. Approximations

4. Applications


26

Applications: Soft Re-encoding (1)

• Soft network coding (NC) for the two-way relay channelI users A and B exchange independent messagesI relay R performs network coding with soft re-encoding

A B

RxR xRR

BA

xBxB

A B

R

xA

xA

• Transmission of quantized soft information is critical

A. Winkelbauer and G. Matz, “Soft-Information-Based Joint Network-Channel Codingfor the Two-Way Relay Channel,” submitted to NETCOD 2011

27


• Soft network coding (NC) for the two-way relay channelI users A and B exchange independent messagesI relay R performs network coding with soft re-encoding

A B

RxR xRR

BA

xBxB

A B

R

xA

xA

Puncturer

Puncturerπ

π

Demapperencoder

SISOencoder

Network

encoderModulatorQ(·)

SISOSISOdecoder

SISOdecoder

Demapper

yBR

yAR

xR

• Transmission of quantized soft information is critical

A. Winkelbauer and G. Matz, “Soft-Information-Based Joint Network-Channel Codingfor the Two-Way Relay Channel,” submitted to NETCOD 2011

27


• BER simulation resultsI sym. channel conditions, R halfway between A, B, rate 1 bpcu,

256 info bits, 4 level quantization, 1 decoder iteration

soft NC

hard NC

PCCC (5 it.)

BER

γAB [dB]

−8 −6 −4 −2 0 2 4 6

10−5

10−4

10−3

10−2

10−1

100

• SNR gain of ∼ 4.5dB at BER ≈ 10−3 over hard NC28

Applications: Convolutional Network Coding

• Physical layer NC for the multiple-access relay channel

xR

xA

xBB

A

R D

• NC at relay

SISO

encoder

L(cA)

L(cB)

LR

L(cA)

L(cB)

LRcR

convolutional NCsoft NChard NC

�⊕cB

cA

29

Outline

1. Introduction


3. Approximations

4. Applications


30

Conclusions and Outlook

Conclusions:

• Efficient methods for soft encoding

• Approximations facilitate practical implementation

• Applications show usefulness of soft encoding

Outlook:

• Frequency domain soft encoding

• Code and transceiver design for physical layer NC

• Performance analysis of physical layer NC schemes

Thank you!

31

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