outline - shannon.cm.nctu.edu.twshannon.cm.nctu.edu.tw/html/thesis/jia-qin-s.pdf · pccc codes....

1

2001/6/14 1

An Efficient Technique for Bit Error Probability Estimation of MLSDA-Turbo Decoder

Presenter: Jia-Qin Pan (潘家斳)Advisory by Prof. Po-Ning Chen

Dept. of Communications EngineeringNational Chiao Tung University

2001/6/14 2

OutlineIntroduction and BackgroundPreliminaries

Fixed-Length Codes and Maximum Metric DecodingImportance SamplingMLSDA for PCCC

The Important Sampling Method on MLSDA-Turbo DecoderSimulation ResultsConclusions

2

2001/6/14 3

Introduction and BackgroundMLSDA (Maximum-likelihood soft-decision sequential decoding algorithm) was proposed by Chen and Han in 1999. Later, it was applied to the decoding of PCCC codes. Unlike the iterative decoding algorithm, the algorithm turns out to be a maximum-likelihood decoding algorithm for PCCC.Unfortunately, for this ML decoder, brute-force MC often requires a very large number of simulation trials in order to achieve a meaningful estimate of system performance.

2001/6/14 4

Introduction and BackgroundImportance Sampling(IS) is a modified MC technique which can efficiently reduce the number of simulation trials required to achieve a specified accuracy.

DecoderChannel (*)

Biased Noise

Decoded Codeword

Importance Sampling Importance Sampling Importance Sampling Importance Sampling SimulationSimulationSimulationSimulation

Codeword

Importance Sampling EstimatorImportance Sampling EstimatorImportance Sampling EstimatorImportance Sampling Estimator

)()|(1)|'(ˆ

.0'1

)(

)('

1

)(*

'

lx

L

l

lL

x

YIxYwL

xxP

otherwisedecodedx

yI

∑=

=

=

x y 'x

3

2001/6/14 5

Introduction and BackgroundIn recent years, there has been considerable interest in applying IS to digital communications systems. Some of them are applied to ML decoder, e.g., Sadowsky developed an error event simulation(EES) method for the simulation of the Viterbi decoders;Khaled and Khurram developed a modified error event simulation(MEES) method for the simulation of the ZJ decoders.Here, a new application of the important sampling method for estimating BER of MLSDA-Turbo Decoder is presented.

2001/6/14 6

Preliminaries Fixed-length Codes and Maximum Metric Decoding

System Model

DecoderStationary Stationary Stationary Stationary Memoryless Memoryless Memoryless Memoryless

ChannelDecoded Codeword

Monte-Carlo Monte-Carlo Monte-Carlo Monte-Carlo SimulationSimulationSimulationSimulation

Codeword

The joint densityThe joint densityThe joint densityThe joint density The decision regions of a maximum metric decoder The decision regions of a maximum metric decoder The decision regions of a maximum metric decoder The decision regions of a maximum metric decoder

∏=

=n

kkk xypxyf

1

)|()|(

== ∑∑

==),'(max),(:)(

11 ' k

n

kk

n

k xkk yxmyxmyxD

x y 'x

4

2001/6/14 7

Preliminaries Fixed-length Codes and Maximum Metric Decoding

System Model

The Indicator function of the decoding region The Indicator function of the decoding region The Indicator function of the decoding region The Indicator function of the decoding region

The specific decoding error probabilityThe specific decoding error probabilityThe specific decoding error probabilityThe specific decoding error probability

The average bit error probabilityThe average bit error probabilityThe average bit error probabilityThe average bit error probability

∉∈

=).'(0)'(1

)(' xDyxDy

yI x

∫ ∫==)'( ' )|()()|()|'(xD x ydxyfyIydxyfxxP

[ ] ∑=−='

)|'()',(1|1)(x

bbb xxPxxnm

dtransmittexNEm

xP

===

=

|)(|log)',(

2 codewordmxxn

N

b

b the number of bits errors that occer after decoding.

the number of information bits transmitted by a single codeword transmission.

the number of postdecoding information bit errors caused by decoding instead of . . . .

)'(xD

'x x

2001/6/14 8

Preliminaries Importance Sampling

How to apply importance sampling to coded communications systems?

DecoderChannel (*)

Biased Noise

Decoded Codeword

Importance Sampling Importance Sampling Importance Sampling Importance Sampling SimulationSimulationSimulationSimulation

Codeword

Importance Sampling EstimatorImportance Sampling EstimatorImportance Sampling EstimatorImportance Sampling Estimator

)()|(1)|'(ˆ

)|(*)()|(*

)|()|'(

)('

1

)(*

'

lx

L

l

lL

x

YIxYwL

xxP

ydxyfyIxyfxyfxxP

∑

∫

==

=

=

=

.0'1

)(

)|(*)|()|(

' otherwisedecodedx

yI

xyfxyfxyw

x

Importance Sampling WeightImportance Sampling WeightImportance Sampling WeightImportance Sampling Weight

x y'x

5

2001/6/14 9


How to choose a good simulation channel?

The expectation operation for the simulation distributionThe expectation operation for the simulation distributionThe expectation operation for the simulation distributionThe expectation operation for the simulation distribution

The variance of the estimatorThe variance of the estimatorThe variance of the estimatorThe variance of the estimator

The optimal importance sampling densityThe optimal importance sampling densityThe optimal importance sampling densityThe optimal importance sampling density

)|'()|(*)|(*

)|()()|(*)|()()]()|([*)]|'(ˆ[* '')(

')(* xxPydxyf

xyfxyfyIydxyfxywyIYIxYwExxPE xx

lx

lL ==== ∫∫

−

== ∫ 22

)'('* )|'()|(*.

)|(*)|(1)]()|([var*1)]|'(ˆvar[ xxPydxyfxyfxyf

LYIxYw

LxxP

xDxL

)|'()()|(

)|( '*' xxP

yIxyfxyf xx =

2001/6/14 10


How to estimate the precision of importance sampling estimates?

The corresponding estimate of The corresponding estimate of The corresponding estimate of The corresponding estimate of

The relative precision estimates of The relative precision estimates of The relative precision estimates of The relative precision estimates of

)]|'(ˆvar[ * xxPL

−=== ∑

=

2*

1

)('

2)('

* ))|'(ˆ()()|(11ˆ)]()|([var*1)]|'(ˆvar[ xxPYIxYw

LLYIxYw

LxxPVar L

L

l

lx

lxL

)|'(ˆ *ˆ *

xxPVarRPE

LPL

=

)|'(ˆ * xxPL

The relative precision estimates of The relative precision estimates of The relative precision estimates of The relative precision estimates of )(ˆ * xPb

)(ˆ

])|'(ˆ)',(1var[

)(ˆ)](ˆvar[

*'

*

*

*

ˆ *

xP

xxPxxnm

xPxP

RPEb

xLb

b

bPb

∑==

6

2001/6/14 11

Preliminaries MLSDA for PCCC

PCCC with G1=37,G2=21 as its component code.

Block interleaver with N=36

π

Binary input1y

interleaver2y

⊕

⊕

⊕ ⊕

⊕

3y

⊕

⊕

⊕ ⊕

⊕

363534333231

302928272625

242322212019

181716151413

121110987

654321

2001/6/14 12


Tree-based Maximum-likelihood Soft-decision Sequential Decoding Algorithm on AWGN channels

The conventional sequential decoding algorithm with a new metric

where are respectively the ith bit of the jth

transmitted block and ith received bit at jth received block, and

( ) ( )

( )( )ji

ji

ji

ji

ji rxyxrM )(

)()(⊕

( ) ( )ji

ji randx

( )

<

=0

;0,1)(

otherwiserif

yj

iji

7

2001/6/14 13


ML decoding algorithmStep1

STACK

successor0

successor1

successor2

successor3

◆ Find the successors of the top path in the STACK

◆ There could be 1 or 2 or 4 successors.

Min metric

……

2001/6/14 14


Step 2

◆ Delete the top path.

◆ Insert the successors into the STACK.

◆ Reordering the STACK according to the metric.

successor1successor0

Min metric

……

.

STACK

x

reordering

8

2001/6/14 15


Step 3If the length of STACK exceeds the Threshold, delete the

path with Max metric in the stack.

Max metric2nd Max Metric

2nd Min metricMin metric

……

Threshold

xxxx

STACK

2001/6/14 16

The Important Sampling Method on MLSDA-Turbo Decoder

Simulation Flow

NdD

1312

1311

2510

39

27

26

13

MLSDA-Turbo Decoder

Channel (*)

Biased Noise (default is AWGN channel)

Length 36 decoded data squenceCodeword

Simulation Channel ListSimulation Channel ListSimulation Channel ListSimulation Channel List

R=1/3 (37,21) Bi Turbo Eecoder

Channel Output Data sequence

I. stationary channelsI. stationary channelsI. stationary channelsI. stationary channelsII. nonstationary channelsII. nonstationary channelsII. nonstationary channelsII. nonstationary channels

)12~1( 36 −=nY

nError36)0...00( 108)1...11( −−−

9

2001/6/14 17


Stationary Channel

-1-1-1-1 1111

the default channelthe default channelthe default channelthe default channel(-1+n0) (-1+n0) ... (-1+n0) (-1+n0) (-1+n0) ... (-1+n0) (-1+n0) (-1+n0) ... (-1+n0) (-1+n0) (-1+n0) ... (-1+n0)

n0=Gauss(0,Var0)n0=Gauss(0,Var0)n0=Gauss(0,Var0)n0=Gauss(0,Var0)

the biased channelthe biased channelthe biased channelthe biased channel(-1+n1) (-1+n1) ... (-1+n1) (-1+n1) (-1+n1) ... (-1+n1) (-1+n1) (-1+n1) ... (-1+n1) (-1+n1) (-1+n1) ... (-1+n1)


Var1Var1Var1Var1>>>>Var0Var0Var0Var0

2001/6/14 18


Nonstationary channelsChannel I

-1-1-1-1 1111

transmit through transmit through transmit through transmit through the default channelthe default channelthe default channelthe default channel

(-1+(-1+(-1+(-1+n0n0n0n0)))) (-1+(-1+(-1+(-1+n0n0n0n0) ... (-1+) ... (-1+) ... (-1+) ... (-1+n0n0n0n0)))) n0=Gauss(0,Var0)n0=Gauss(0,Var0)n0=Gauss(0,Var0)n0=Gauss(0,Var0)

transmit through the biased channel for transmit through the biased channel for transmit through the biased channel for transmit through the biased channel for the error event the error event the error event the error event

the relative codeword the relative codeword the relative codeword the relative codeword (-1+n0) (-1+n0) ... (-1+n0)(-1+n0) (-1+n0) ... (-1+n0)(-1+n0) (-1+n0) ... (-1+n0)(-1+n0) (-1+n0) ... (-1+n0) (-1+n1) (-1+n1) (-1+n(-1+n1) (-1+n1) (-1+n(-1+n1) (-1+n1) (-1+n(-1+n1) (-1+n1) (-1+n1)1)1)1)



108)0...00(108)0...00(

36)01...00(108)0111...00(

10

2001/6/14 19


Channel II

-1-1-1-1 1111




108)0...00( transmit through the biased channel for transmit through the biased channel for transmit through the biased channel for transmit through the biased channel for the error event the error event the error event the error event

the relative codeword the relative codeword the relative codeword the relative codeword (-1+n0) (-1+n0) ... (-1+n0)(-1+n0) (-1+n0) ... (-1+n0)(-1+n0) (-1+n0) ... (-1+n0)(-1+n0) (-1+n0) ... (-1+n0) (0+n1) (0+n1) (0+n(0+n1) (0+n1) (0+n(0+n1) (0+n1) (0+n(0+n1) (0+n1) (0+n1)1)1)1)


108)0...00(

36)01...00(108)0111...00(

2001/6/14 20


Channel III

-1-1-1-1 1111




108)0...00( transmit through the biased channel for transmit through the biased channel for transmit through the biased channel for transmit through the biased channel for the error event the error event the error event the error event

the relative codeword the relative codeword the relative codeword the relative codeword (-1+n0) (-1+n0) ... (-1+n0)(-1+n0) (-1+n0) ... (-1+n0)(-1+n0) (-1+n0) ... (-1+n0)(-1+n0) (-1+n0) ... (-1+n0) (+1+n1) (+1+n1) (+1+n(+1+n1) (+1+n1) (+1+n(+1+n1) (+1+n1) (+1+n(+1+n1) (+1+n1) (+1+n1)1)1)1)


108)0...00(

36)01...00(108)0111...00(

11

2001/6/14 21

0 2 4 6 8 10 12 14 16

x 105

10-9

10-8

10-7

10-6

10-5

10-4 Es timate S NR=8.0dB with diffe rent s imula tion time s

S imula tion Time s

Pb

Important S ampling S NR=8dB Va r=0.2377Important S ampling S NR=7dB Va r=0.2993Important S ampling S NR=6dB Va r=0.3768Important S ampling S NR=5dB Va r=0.4743Monte Ca rlo (5.3e-06 5.80 161673235)

Simulation ResultsEstimate the bit error probability of SNR=8.0dB by Biased Stationary Channels

0 2 4 6 8 10 12 14 16

x 105

0

10

20

30

40

50

60

70

80

90Estimate S NR=8.0dB with diffe rent s imula tion time s

S imula tion Time s

Rpe

%

2001/6/14 22

Simulation ResultsTime-spent during estimating the bit error probability of SNR=8.0dB by Biased Stationary Channels

0 2 4 6 8 10 12 14 16

x 105

105

106

107

108

109

1010 Es timate S NR=8.0dB with diffe rent s imula tion time s

S imula tion Time s

Met

ric C

ompu

tatio

n

Important S ampling S NR=8dB Va r=0.2377Important S ampling S NR=7dB Va r=0.2993Important S ampling S NR=6dB Va r=0.3768Important S ampling S NR=5dB Va r=0.4743Monte Ca rlo (5.3e-06 5.80 161673235)

12

2001/6/14 23

0 1000 2000 3000 4000 5000 6000 7000 800010

-4

10-3


S imula tion Time s

Pb

Important S ampling S NR=3.0dB Var=0.7518Important S ampling S NR=2.5dB Var=0.8435Important S ampling S NR=2.0dB Var=0.9464Important S ampling S NR=1.5dB Var=1.0619Monte Ca rlo (1.2e-03 7.74 155713373)

Simulation ResultsEstimate the bit error probability of SNR=3.0dB by Biased Stationary Channels

0 1000 2000 3000 4000 5000 6000 7000 80000

10

20

30

40

50

60

70

80

90


S imula tion Time s

Rpe

%

2001/6/14 24

Simulation ResultsTime-spent during estimating the bit error probability of SNR=3.0dB by Biased Stationary Channels

0 1000 2000 3000 4000 5000 6000 7000 800010

6

107

108

109


S imula tion Time s

Met

ric C

ompu

tatio

n

Important S ampling S NR=3.0dB Var=0.7518Important S ampling S NR=2.5dB Var=0.8435Important S ampling S NR=2.0dB Var=0.9464Important S ampling S NR=1.5dB Var=1.0619Monte Ca rlo (1.2e-03 7.74 155713373)

13

2001/6/14 25

0 2 4 6 8 10 12 14 16

x 105

10-6


S imula tion Time s

Pb

Important S ampling S NR=8dB Va r=0.2377Important S ampling S NR=6dB Va r=0.3768Important S ampling S NR=3dB Va r=0.7518Monte Ca rlo (5.3e-06 5.80 161673235)

Simulation ResultsEstimate the bit error probability of SNR=8.0dB by the first Biased Nonstationary Channels

0 2 4 6 8 10 12 14 16

x 105

0

10

20

30

40

50

60

70

80

90


S imula tion Time s

Rpe

%

2001/6/14 260 2 4 6 8 10 12 14 16

x 105

105

106

107

108


S imula tion Time s

Met

ric C

ompu

tatio

n


Simulation ResultsTime-spent during estimating the bit error probability of SNR=8.0dB by the first NonstationaryChannels

bP

0.29450.77531Ratio

47607830125344768161673235Com

5.845.755.80Rpe %

5.2e-066.2e-065.3e-06

SNR=3.0SNR=6.0MC

MC

IS

CComRatio =

14

2001/6/14 27

0 1000 2000 3000 4000 5000 6000 7000 800010

-4

10-3


S imula tion Time s

Pb

Important S ampling S NR=3.0dB Var=0.7518Important S ampling S NR=1.5dB Var=1.0619Monte Ca rlo (1.2e-03 7.74 155713373)

Simulation ResultsEstimate the bit error probability of SNR=3.0dB by the first Biased Nonstationary Channels

0 1000 2000 3000 4000 5000 6000 7000 80000

10

20

30

40

50

60

70

80

90


S imula tion Time s

Rpe

%

2001/6/14 280 1000 2000 3000 4000 5000 6000 7000 8000

106

107

108


S imula tion Time s

Met

ric C

ompu

tatio

n


Simulation ResultsTime-spent during estimating the bit error probability of SNR=3.0dB by the first NonstationaryChannels

bP

0.72711.171Ratio

113213118182182136155713373Com

17.217.47.74Rpe %

1.0e-031.1e-031.2e-03

SNR=1.5SNR=3.0MC

MC

IS

CComRatio =

15

2001/6/14 29

0 2 4 6 8 10 12 14 16

x 105

10-6


S imula tion Time s

Pb


Simulation ResultsEstimate the bit error probability of SNR=8.0dB by the second Biased Nonstationary Channels

0 2 4 6 8 10 12 14 16

x 105

0

10

20

30

40

50

60

70


S imula tion Time s

Rpe

%

2001/6/14 300 2 4 6 8 10 12 14 16

x 105

105

106

107

108


S imula tion Time s

Met

ric C

ompu

tatio

n


Simulation ResultsTime-spent during estimating the bit error probability of SNR=8.0dB by the second NonstationaryChannels

bP

bP

0.025410.012051Ratio

41076721947871161673235Com

5.775.785.80Rpe %

5.4e-065.2e-065.3e-06

SNR=6.0SNR=8.0MC

0.11825Ratio

19117489Com

5.76Rpe %

5.2e-06

SNR=3.0

MC

IS

CComRatio =

16

2001/6/14 31

0 1000 2000 3000 4000 5000 6000 7000 800010

-4

10-3


S imula tion Time s

Pb


Simulation ResultsEstimate the bit error probability of SNR=3.0dB by the second Biased Nonstationary Channels

0 1000 2000 3000 4000 5000 6000 7000 80000

5

10

15

20

25

30

35

40


S imula tion Time s

Rpe

%

2001/6/14 320 1000 2000 3000 4000 5000 6000 7000 8000

106

107

108


S imula tion Time s

Met

ric C

ompu

tatio

n


Simulation ResultsTime-spent during estimating the bit error probability of SNR=3.0dB by the second NonstationaryChannels

bP

0.89810.48951Ratio

13984161576220279155713373Com

7.757.797.74Rpe %

9.8e-041.0e-031.2e-03

SNR=1.5SNR=3.0MC

MC

IS

CComRatio =

17

2001/6/14 33

0 2 4 6 8 10 12 14 16

x 105

10-6

10-5


S imula tion Time s

Pb


Simulation ResultsEstimate the bit error probability of SNR=8.0dB by the third Biased Nonstationary Channels

0 2 4 6 8 10 12 14 16

x 105

0

10

20

30

40

50

60

70

80


S imula tion Time s

Rpe

%

2001/6/14 340 2 4 6 8 10 12 14 16

x 105

105

106

107

108


S imula tion Time s

Met

ric C

ompu

tatio

n


Simulation ResultsTime-spent during estimating the bit error probability of SNR=8.0dB by the third NonstationaryChannels

bP

0.22780.47571Ratio

3682499576910196161673235Com

5.855.835.80Rpe %

5.4e-065.3e-065.3e-06

SNR=3.0SNR=6.0MC

MC

IS

CComRatio =

18

2001/6/14 35

0 1000 2000 3000 4000 5000 6000 7000 800010

-5

10-4

10-3


S imula tion Time s

Pb


Simulation ResultsEstimate the bit error probability of SNR=3.0dB by the third Biased Nonstationary Channels

0 1000 2000 3000 4000 5000 6000 7000 80000

10

20

30

40

50

60


S imula tion Time s

Rpe

%

2001/6/14 360 1000 2000 3000 4000 5000 6000 7000 8000

106

107

108


S imula tion Time s

Met

ric C

ompu

tatio

n


Simulation ResultsTime-spent during estimating the bit error probability of SNR=3.0dB by the third NonstationaryChannels

bP

0.95091.10451Ratio

148068555171978318155713373Com

10.710.97.74Rpe %

9.8e-049.5e-041.2e-03

SNR=1.5SNR=3.0MC

MC

IS

CComRatio =

19

2001/6/14 37

Simulation ResultsEstimate the bit error probability by the Best Biased Nonstationary Channel in order to achieve the same estimates of system performance

Importance Sampling Simulation for recording at the same RPE

Monte Carlo Simulation for recording at 300 bit errors4.03.53.02.52.01.51.00.50.0SNR(dB)

4.5e-047.6e-041.2e-032.2e-034.8e-039.8e-032.4e-023.8e-026.5e-02Pb

6.266.877.749.8313.414.216.717.217.2Rpe

6306310793004594155713373185857455208319093215889806206555768192710434222276385Com

4.03.53.02.52.01.51.00.50.0SNR(dB)

4.3e-046.2e-041.0e-031.3e-032.8e-033.5e-034.2e-039.3e-031.2e-02Pb

6.266.897.799.7613.314.316.317.217.3Rpe

3207217350410610762202799830755489587250163045159181950128190742239206742239Com

0.508570.542020.489490.528940.430050. 755220.880880.989790.93011Gain

2001/6/14 38


Importance Sampling Simulation for recording at the same RPE

Monte Carlo Simulation for recording at 300 bit errors9.08.58.07.57.06.56.05.55.04.5

8.9e-072.2e-065.3e-061.0e-052.4e-053.9e-056.7e-051.1e-041.9e-042.9e-04

5.765.765.805.785.815.845.865.926.036.22

85676011735849031116167323594346705494440373971935933922090330179613558701947388940

9.08.58.07.57.06.56.05.55.04.5

9.7e-072.2e-065.2e-061.1e-052.3e-054.0e-056.8e-051.1e-041.8e-042.8e-04

5.795.775.785.815.815.845.805.966.096.22

167466719338221947871238452728869983867645531074080551171342090318694896

0.001950.005390.012050.025270.058390.097370.156560.243960.377130.39450

20

2001/6/14 39

Simulation ResultsMonte-Carlo VS Importance-Sampling

0 1 2 3 4 5 6 7 8 910

-7

10-6

10-5

10-4

10-3

10-2

10-1

100 Estima te Bit Error P robability

S NR(dB)

Pb

Monte Carlo Important S ampling Upper Bound(up to d=9)

2001/6/14 40


Importance Sampling Simulation for recording at RPE=6%

Importance Sampling Simulation for recording at RPE=6%12.512.011.511.010.510.09.5SNR(dB)

1402515139102513845071317392134857613891501397725Com

3.8e-112.7e-101.6e-097.9e-093.3e-081.1e-073.5e-07Pb

16.015.515.014.514.013.513.0SNR(dB)

6.016.056.096.086.106.096.05Rpe

7.3e-215.3e-192.8e-178.5e-161.9e-143.3e-133.9e-12Pb

6.066.076.046.036.006.046.10Rpe

1942095180547217528321700387161057815638251483599Com

21

2001/6/14 41

Simulation ResultsMonte-Carlo VS Importance-Sampling

8 9 10 11 12 13 14 15 1610

-25

10-20

10-15

10-10

10-5

100 Estima te Bit Error P robability

S NR(dB)

Pb

Monte Carlo Important S ampling Upper Bound(up to d=9)

2001/6/14 42

ConclusionsThere is no computational gain when we use the Importance Sampling Simulation by BiasdStationary Channels. In fact, the noisier channel we use the more metric computations we pay, and it brings out the larger relative precision because of its larger variation than MC Simulation.

22

2001/6/14 43

ConclusionsThe optimal distribution of the biased channel

can show why the second nonstationarychannel causes the best performance among all channels.

)|'()()|()|( '*

' xxPyIxyfxyf x

x =

2001/6/14 44

ConclusionsIn this Simulation, the Importance Sampling can work well in high SNR, but less effectiveness in low SNR when we use the second biased channel. In fact, We gained 10-500 time-saving when we estimate bit error probability between SNR=6dB to 9dB, and gained 0-10 time-saving for SNR=0dB to 5.5dB.

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