Soft Decision Decoding of RS Codes Using Adaptive Parity Check MatricesSoft Decision Decoding of RS Codes Using Adaptive Parity Check Matrices

Jing Jiang and Krishna R. Narayanan

Wireless Communication Group

Department of Electrical Engineering

Texas A&M University

Reed Solomon Codes

Consider an (n,k) RS code over GF(2m), n = 2m-1

Linear block code – e.g. (7,5) RS code over GF(8)

be a primitive element in GF(8)

Cyclic shift of any codeword is also a valid codeword

RS codes are MDS (dmin = n-k+1)

The dual code is also MDS

53642

65432

1

1

H

Introduction

DrawbackPerformance loss due to bounded distance decodingSoft input soft output (SISO) decoding is not easy!

AdvantagesGuaranteed minimum distance Efficient bounded distance hard decision decoder (HDD)Decoder can handle errors and erasures

RS Coded Turbo Equalization System

-

+

a priori

extrinsicinterleaving

a priori

extrinsic

ΠΣ

source

RS Encoder

interleaving

PR Encoder

sink

hard decision

+

AWGN+

RS Decoder

BCJR Equalizer

de-interleaving

Π

1Π

Σ

Presentation Outline

Existing soft decision decoding techniques

Iterative decoding based on adaptive parity check matricesVariations of the generic algorithm

Applications over various channels

Conclusion and future work

Existing Soft Decoding Techniques

Enhanced Algebraic Hard Decision Decoding

Generalized Minimum Distance (GMD) Decoding (Forney 1966):Basic Idea:

Erase some of the least reliable symbolsRun algebraic hard decision decoding several times

Drawback: GMD has a limited performance gainChase decoding (Chase 1972):

Exhaustively flip some of the least reliable symbolsRunning algebraic hard decision decoding several times

Drawback: Has an exponentially increasing complexity

Combined Chase & GMD(Tang et al. 2001).

Algebraic Soft Input Hard Output Decoding

Algebraic SIHO decoding:Algebraic interpolation based decoding (Koetter & Vardy 2003)Reduced complexity KV algorithm (Gross et al. submitted 2003)

Basic ideas:Based on Guruswami and Sudan’s algebraic list decodingConvert the reliability information into a set of interpolation pointsGenerate a list of candidate codewordsPick up the most likely codeword from the codeword list

Drawback:The complexity increases with , maximum number of multiplicity.

4maxm

Reliability based Ordered Statistic Decoding

Reliability based decoding: Ordered Statistic Decoding (OSD) (Fossorier & Lin 1995) Box & Match Algorithm(BMA) (Valembois & Fossorier to appear 2004)

Basic ideas: Order the received bits according to their reliabilities Make hard decisions on a set of independent reliable bits (MR Basis) Re encode to obtain a list of candidate codewords

Drawback: The complexity increases exponentially with the reprocessing order BMA must trade memory for complexity

Trellis based Decoding using the Binary Image Expansion

Maximum-likelihood decoding and variations Trellis based decoding using binary image expansion (Vardy & Be’ery ‘91) Reduced complexity version (Ponnampalam & Vucetic 2002)

Basic ideas:Binary image expansion of RSTrellis structure construction using the binary image expansion

Drawback:Exponentially increasing complexityWork only for very short codes or codes with very small distance

Binary Image Expansion of RS Codes

)2( where, as expressed becan )2( )(1

0

)( GFcccGFc ib

m

i

iib

m

),2GF(in element primitive a be Let m )2GF( of basis a form ,...,,,1 12 mm

],...,,[ 1101 NN cccC ],...,,,...,,...,,[ )1(1

)1(1

)0(1

)1(0

)1(0

)0(0)1(

m

NNNm

Nmb ccccccC

NKNKNKN

N

NKN

HHH

HHH

H

),1(1),1(0),1(

1,01,00,0

)(

)1,1(1,1

)0,1(1,1

)1,1(0,1

)0,1(0,1

)1,0(1,1

)0,0(1,1

)1,0(0,1

)0,0(0,1

)1,1(1,0

)0,1(1,0

)1,1(0,0

)0,1(0,0

)1,0(1,0

)0,0(1,0

)1,0(0,0

)0,0(0,0

)(

mmNKN

mNKN

mmKN

mKN

mNKNNKN

mKNKN

mmN

mN

mmm

mNN

m

NmmKNb

hhhh

hhhh

hhhh

hhhh

H

bbb HCD HCD

)1,1()1,1()1,0(

)1,0()1,0()0,0(

mmmm

m

b

hhh

hhh

H

213 CCC 213 bbb CCC

Consider the (7,5) RS code

53642

65432

1

1

H

011110010001111101100

001111101100011110010

111101100011110010001

001011111110101010100

100001011111110101010

011111110101010100001

bH

Binary image expansion of the parity check matrix of RS(7, 5) over GF(23)

Recent Iterative Techniques

Sub-trellis based iterative decoding (Ungerboeck 2003)

Self-concatenation structure based on sub-trellis constructed from the parity check matrix

Drawbacks: Performance deteriorates due to large number of short cycles Work for short codes with small minimum distances Potential error floor problem in high SNR region

011110010001111101100

001111101100011110010

111101100011110010001

001011111110101010100

100001011111110101010

011111110101010100001

bH

Binary image expansion of the parity check matrix of RS(7, 5) over GF(23)

Recent Iterative Techniques (cont’d)

Stochastic shifting based iterative decoding (Jing & Narayanan, to appear 2004)

Due to the irregularity in the H matrix, iterative decoding favors some bits

Taking advantage of the cyclic structure of RS codes],,,,,,[ 4321065 rrrrrrr ],,,,,,[ 6543210 rrrrrrr

1011001

0110011

0001111

H

Stochastic shift prevent iterative procedure from getting stuck

Best result: RS(63,55) about 0.5dB gain from HDD

However, for long codes, this algorithm still doesnt provide good improvement

Shift by 2

Remarks on Existing Techniques

Most SIHO algorithms are either too complex to implement or having only marginal gain

Moreover, SIHO decoders cannot generate soft output directly Trellis-based decoders have exponentially increasing complexity

Iterative decoding algorithms do not work for long codes, since the parity check matrices of RS codes are not sparse

“Soft decoding of large RS codes as employed in many standard transmission systems, e.g., RS(255,239), with affordable complexity remains an open problem” (Ungerboeck, ISTC2003)

Questions

Q: Why doesn’t iterative decoding work for codes with non-sparse parity check matrices?

Q: Can we get some idea from the failure of iterative decoder?

How does standard message passing algorithm work?

bit nodes…………. ………..

. . . . . . . . . …………….

check nodes

…………….

erased bits

? If two or more of the incoming messages are erasures the check is erased Otherwise, check to bit message is the value of the bit that will satisfy the check

How does standard message passing algorithm work?

bit nodes…………. ………..

. . . . . . . . . …………….

check nodes

…………….

12 tanh tanh2j

kj k

vu

1 2 1, 1,..,| | min , ,..,k k k Ju v v v v v

Small values of vj can be thought of as erasures and hence more than two

edges with small vj’s saturate the check

1.15.01.08.02.09.06.01.02.09.05.01.03.04.08.01.10.17.06.19.08.0 r

A Few Unreliable Bits “Saturate” the Non-sparse Parity Check Matrix

000000000000000000000bc

Iterative decoding is stuck due to only a few unreliable bits “saturating” the whole non-sparse parity check matrix

011110010001111101100

001111101100011110010

111101100011110010001

001011111110101010100

100001011111110101010

011111110101010100001

bH

Binary image expansion of the parity check matrix of RS(7, 5) over GF(23)

Consider RS(7, 5) over GF(23)

Sparse Parity Check Matrices for RS Codes

Can we find an equivalent binary parity check matrix that is sparse?

For RS codes, this is not possible!

The H matrix is the G matrix of the dual code

The dual of an RS code is also an MDS Code

Every row has weight at least (N-K)!

Iterative Decoding Based on Adaptive Parity Check Matrix

transmitted codeword 0011010c

Idea: reduce the sub-matrix corresponding to the unreliable positions to a sparse nature.

For example, consider (7,4) Hamming code:

parity check matrix

received vector 1.01.02.14.11.06.01.1 r

1010110

1100101

1101010

H

1011001

0110011

0001111

H

1011001

0110011

0001111

H

After the adaptive update, iterative decoding can proceed.

Adaptive Decoding Procedure

bit nodes…………. ………..

. . . . . . . . . …………….

check nodes

…………….

unreliable bits

More Details about the Matrix Adaptive Scheme

transmitted codeword 0011010c

parity check matrix

1010110

1100101

1101010

H

0111100

1100101

1101010

H

1011001

1100101

1101010

H

1011001

0111100

0110011

H

1011001

1100101

1101010

H

4.03.02.14.12.01.01.1 rreceived vector

Consider the previous example: (7,4)Hamming code

We can guaranteed reduce some (n-k)m columns to degree 1

We attempt to chose these to be the least reliable independent bits

Least Reliable Basis

Interpretation as an Optimization Procedure

Standard iterative decoding procedure is interpreted as gradient descent optimization (Lucas et al. 1998).

Proposed algorithm is a generalization, two-stage optimization procedure:

The damping coefficient serves to control the convergent dynamics.

Parity check matrix update (change direction)

All bit-level reliabilities are sorted by their absolute valuesSystemize the sub-matrix corresponding to LRB in the parity check matrix

Bit reliabilities updating stage (gradient descent)

Iterative decoding is applied to generate extrinsic informationExtrinsic information is scaled by a damping coefficient and fed to update the bit-level reliabilities

A Hypothesis

Stuck at pseudo-equilibrium point

Adaption help gradient descent to converge

Complexity Analysis

Check Node Update

Overall Complexity

Variable Node Update

Matrix Adaption

Reliability Ordering

BinaryFloating PointOperation

)(log 2 NmNm

2})(,min{ mKNKmNm

2/)()( mkNKmmKN

)2/1()( KmmKN

2)(Nm 3)(Nm

Complexity can be even reduced when implemented in parallel

The complexity is in polynomial time with or N mind

Complexity Comparison

Metho

d

Dominant Complexity

GMD

Chase

KV

OSD

Trellis

ADP

)log2/( :operation )( 22

min NNdoqGF

exhausted be tosymbols ofnumber theis c ),log( :operation )( 22 NNqoqGF c

ty.multiplici maximum theis ),/1( :operation )( max4max

3 mmNRoqGF

})2,2(min{ :operationpoint Floating )( mKKNmo

processing-re oforder theis ),)(( :operationpoint Floating 1 iNmo i

processing-re oforder theis ),)(( :operationBinary 1 iNmo i

))(( :operationpoint Floating 2Nmo))(( :operationBinary 3Nmo

least reliable symbols

53642

65432

1

1

H

Variation1: Symbol-level Adaptive Scheme

Systemizing the sub-matrix involves undesirable Gaussian elimination.

4554

33

110

101

H

?????10

?????01H

We implement Symbol-level adaptive scheme.

This problem can be detoured via utilizing the structure of RS codes.

This step can be efficiently realized using Forney’s algorithm (Forney 1965)

111011011100111100000

011001001010011010000

110111111001110001000

011010110100110000100

100101111010111000010

100100101001101000001

bH

binary mapping

Variation2: Degree-2 sub-graph in the unreliable part

bit nodes…………. ………..

. . . . . . . . . …………….

check nodes

…………….

unreliable bits

weakly connected

Reduce the “unreliable” sub-matrix to a sparse sub-graph rather than an identity to improve the asymptotic performance.

Variation2: Degree-2 sub-graph in the unreliable part (cont’d)

111011011100111100000

011001001010011010000

110111111001110001000

010010110100110000100

101101111010111000010

100100101001101000001

bH

100010010110100110000

101110110011101011000

100101001101000001100

111111001110001000110

001001010011010000011

100100101001101000001

bH

011111110101010100001

100010010110100110000

011010000011001001010

001011111110101010100

101101111010111000010

010011010000011001001

bH

Q: How to adapt the parity check matrix?

Variation3: Different grouping of unreliable bits (cont’d)

Some bits at the boundary part may also have the wrong sign.

]2.11.18.075.070.067.060.058.052.050.047.045.040.034.034.032.022.021.012.004.001.0[bL

…….Group1 Group2

A list of candidate codewords are generated using different groups. Pick up the most likely from the list.

Consider the received LLR of an RS(7,5) code:

Run the proposed algorithm several times, each time with an exchange of some “reliable” and “unreliable” bits at the boundary.

Variation4: Partial updating scheme (cont’d)

The main complexity comes from updating the bits in the high density part, however, only few bits at the boundary part will be affected.

In variable node updating stage: update only the “unreliable” bits in the sparse sub-matrix and a few “reliable” bits at the boundary part.

111011011100111100000

011001001010011010000

110111111001110001000

010010110100110000100

101101111010111000010

100100101001101000001

bH

ascending reliability

In check node updating stage: make an approximation of the check sum via taking advantage of the ordered reliabilities.

Complexity in floating point operation part is reduced to be .)(Nmo

Applications

Simulation setups:A “genie aided” HDD is assumed for AWGN and fading channel.In the TE system, all coded bits are interleaved at random. A “genie aided” stopping rule is applied.

Q: How do the proposed algorithm and its variations perform?

Simulation results:Proposed algorithm and variations over AWGN channelPerformance over symbol level fully interleaved slow fading channelRS coded turbo equalization (TE) system over EPR4 channelRS coded modulation over fast fading channel

Additive White Gaussian Noise (AWGN) Channel

AWGN Channels

AWGN Channels (cont’d)

Asymptotic performance is consistent with the ML upper-bound.

AWGN Channels (cont’d)

AWGN Channels (cont’d)

Remarks

Proposed scheme performs near ML for medium length codes.Symbol-level adaptive updating scheme provides non-trivial gain.Partial updating incurs little penalty with great reduction in complexity.For long codes, proposed scheme is still away from ML decoding.Q: How does it work over other channels?

Interleaved Slow Fading Channel

Fully Interleaved Slow Fading Channels

Fully Interleaved Slow Fading Channels (cont.)

Turbo Equalization Systems

Embed the Proposed Algorithm in the Turbo Equalization System

RS Coded Turbo Equalization System

-

+

a priori

extrinsicinterleaving

a priori

extrinsic

ΠΣ

source

RS Encoder

interleaving

PR Encoder

sink

hard decision

+

AWGN+

RS Decoder

BCJR Equalizer

de-interleaving

Π

1Π

Σ

Turbo Equalization over EPR4 Channels

Turbo Equalization over EPR4 Channels

RS Coded Modulation

RS Coded Modulation over Fast Rayleigh Fading Channels

RS Coded Modulation over Fast Rayleigh Fading Channels (cont’d)

Remarks

More noticeable gain is observed for fading channels, especially for symbol-level adaptive scheme.

In RS coded modulation scheme, utilizing bit-level soft information seems provide more gain.

The proposed TE scheme can combat ISI and performs almost identically as the performance over AWGN channels.

The proposed algorithm has a potential “error floor” problem.However, simulation down to even lower FER is impossible.Asymptotic performance analysis is still under investigation.

Conclusion and Future work

Iterative decoding of RS codes based on adaptive parity check matrix works favorably for practical codes over various channels.

The proposed algorithm and its variations provide a wide range of complexity-performance tradeoff for different applications.More works under investigation:

Asymptotic performance bound.Understanding how this algorithm works from an information theoretic perspective, e.g., entropy of ordered statistics.Improving the generic algorithm using more sophisticated optimization schemes, e.g., conjugate gradient method.

Thank you!