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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
Preliminaries Importance Sampling
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
Preliminaries Importance Sampling
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
Preliminaries MLSDA for PCCC
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
Preliminaries MLSDA for PCCC
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
Preliminaries MLSDA for PCCC
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
Preliminaries MLSDA for PCCC
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
The Important Sampling Method on MLSDA-Turbo Decoder
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)
n1=Gauss(0,Var1)n1=Gauss(0,Var1)n1=Gauss(0,Var1)n1=Gauss(0,Var1)
Var1Var1Var1Var1>>>>Var0Var0Var0Var0
2001/6/14 18
The Important Sampling Method on MLSDA-Turbo Decoder
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)
n1=Gauss(0,Var1)n1=Gauss(0,Var1)n1=Gauss(0,Var1)n1=Gauss(0,Var1)
Var1Var1Var1Var1>>>>Var0Var0Var0Var0
108)0...00(108)0...00(
36)01...00(108)0111...00(
10
2001/6/14 19
The Important Sampling Method on MLSDA-Turbo Decoder
Channel II
-1-1-1-1 1111
Var1Var1Var1Var1>>>>Var0Var0Var0Var0
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)
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)
n1=Gauss(0,Var1)n1=Gauss(0,Var1)n1=Gauss(0,Var1)n1=Gauss(0,Var1)
108)0...00(
36)01...00(108)0111...00(
2001/6/14 20
The Important Sampling Method on MLSDA-Turbo Decoder
Channel III
-1-1-1-1 1111
Var1Var1Var1Var1>>>>Var0Var0Var0Var0
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)
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)
n1=Gauss(0,Var1)n1=Gauss(0,Var1)n1=Gauss(0,Var1)n1=Gauss(0,Var1)
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
10-2 Es timate S NR=3.0dB with diffe rent s imula tion time s
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
100Estimate S NR=3.0dB with diffe rent s imula tion time s
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
1010 Es timate S NR=3.0dB with diffe rent s imula tion time s
S imula tion Time s
Met
ric C
ompu
tatio
n
Important S ampling S NR=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
10-5 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=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
100Estimate S NR=8.0dB with diffe rent s imula tion time s
S imula tion Time s
Rpe
%
2001/6/14 260 2 4 6 8 10 12 14 16
x 105
105
106
107
108
109 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=6dB Va r=0.3768Important S ampling S NR=3dB Va r=0.7518Monte Ca rlo (5.3e-06 5.80 161673235)
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
10-2 Es timate S NR=3.0dB with diffe rent s imula tion time s
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
100Estimate S NR=3.0dB with diffe rent s imula tion time s
S imula tion Time s
Rpe
%
2001/6/14 280 1000 2000 3000 4000 5000 6000 7000 8000
106
107
108
109 Es timate S NR=3.0dB with diffe rent s imula tion time s
S imula tion Time s
Met
ric C
ompu
tatio
n
Important S ampling S NR=3.0dB Var=0.7518Important S ampling S NR=1.5dB Var=1.0619Monte Ca rlo (1.2e-03 7.74 155713373)
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
10-5 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=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 second Biased Nonstationary Channels
0 2 4 6 8 10 12 14 16
x 105
0
10
20
30
40
50
60
70
80Estimate S NR=8.0dB with diffe rent s imula tion time s
S imula tion Time s
Rpe
%
2001/6/14 300 2 4 6 8 10 12 14 16
x 105
105
106
107
108
109 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=6dB Va r=0.3768Important S ampling S NR=3dB Va r=0.7518Monte Ca rlo (5.3e-06 5.80 161673235)
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
10-2 Es timate S NR=3.0dB with diffe rent s imula tion time s
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 second Biased Nonstationary Channels
0 1000 2000 3000 4000 5000 6000 7000 80000
5
10
15
20
25
30
35
40
45Estimate S NR=3.0dB with diffe rent s imula tion time s
S imula tion Time s
Rpe
%
2001/6/14 320 1000 2000 3000 4000 5000 6000 7000 8000
106
107
108
109 Es timate S NR=3.0dB with diffe rent s imula tion time s
S imula tion Time s
Met
ric C
ompu
tatio
n
Important S ampling S NR=3.0dB Var=0.7518Important S ampling S NR=1.5dB Var=1.0619Monte Ca rlo (1.2e-03 7.74 155713373)
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
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=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 third Biased Nonstationary 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
%
2001/6/14 340 2 4 6 8 10 12 14 16
x 105
105
106
107
108
109 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=6dB Va r=0.3768Important S ampling S NR=3dB Va r=0.7518Monte Ca rlo (5.3e-06 5.80 161673235)
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
10-2 Es timate S NR=3.0dB with diffe rent s imula tion time s
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 third Biased Nonstationary Channels
0 1000 2000 3000 4000 5000 6000 7000 80000
10
20
30
40
50
60
70Estimate S NR=3.0dB with diffe rent s imula tion time s
S imula tion Time s
Rpe
%
2001/6/14 360 1000 2000 3000 4000 5000 6000 7000 8000
106
107
108
109 Es timate S NR=3.0dB with diffe rent s imula tion time s
S imula tion Time s
Met
ric C
ompu
tatio
n
Important S ampling S NR=3.0dB Var=0.7518Important S ampling S NR=1.5dB Var=1.0619Monte Ca rlo (1.2e-03 7.74 155713373)
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
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 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
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 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.