multilevel coding and iterative multistage decoding elec 599 project presentation
DESCRIPTION
Multilevel Coding and Iterative Multistage Decoding ELEC 599 Project Presentation. Mohammad Jaber Borran Rice University April 21, 2000. q 1 K 1. N x 1. M -way Partitioning of data. E 1 (rate R 1 ). Mapping (to 2 M -point constellation). data bits from the - PowerPoint PPT PresentationTRANSCRIPT
Multilevel Coding and Iterative Multistage Decoding
ELEC 599 Project Presentation
Mohammad Jaber Borran
Rice UniversityApril 21, 2000
Multilevel Coding
A number of parallel encoders
The outputs at each instant select one symbol
lbits/symbo 1
11
M
ii
M
ii K
NRR
M-wayPartitioning
of data
data bitsfrom theinformationsource
E1 (rate R1)
EM (rate RM)
E2 (rate R2)
q1 K1 N x1
Mapping(to 2M-point
constellation)
Signal Point
q2 K2
qM KM
N x2
N xM
• Minimum Hamming distance for encoder i: dHi ,
Minimum Hamming distance for symbol sequences
)(min
,,1Hi
MiH dd
• For TCM (because of the parallel transitions)
dH = 1
• MLC is a better candidate for coded modulation on fast fading channels
Distance Properties
Probability of error for Fading Channels
• Rayleigh fading with independent fading coefficients
Chernoff bound
L
dk
jikL
s
jie
k
dNEP
01
20
2
4)(
11)(
c,cc,c
L’: effective length of the error event (Hamming distance)
dk(ci,cj): distance between the kth symbols of the two sequences
• For a fast fading channel, or a slowly fading channel with interleaving/deinterleaving
Design criterion (Divsalar)
Design Criterion for Fading Channels
),(minmax,},,,{ 2
jiPji
dn
ccccc1
L
dk
jikjiP
k
dd
01
2
2
)()( c,cc,c
),(minmax,},,,{ 2
jiHji
dn
ccccc1
• For a slowly fading channel without interleaving/deinterleaving
Design criterion ),(minmax,},,,{ 2
jiEji
dn
ccccc1
• For a fast fading channel, or a slowly fading channel with interleaving/deinterleaving
Decoding Criterion
kkk
L
kikk
iyyd
~ where)~(||min1
22 c,y
k is the fading coefficient for kth symbol)
– Maximizes the likelihood function
• Optimum decoder: Maximum-Likelihood decoder
• If the encoder memories are 1, 2, …,M,
the total number of states is 2,
where = 1 + 2 + … + M.
• Complexity Need to look for suboptimum decoders
Decoding
• If A and Y denote the transmitted and received symbol sequences respectively, using the chain rule for mutual information:
),,,|;(
)|;();(
),,,X;();(
121
121
21
MM
M
XXXXYI
XXYIXYI
XXYIAYI
• Suggests a rule for a low-complexity staged decoding procedure
Multistage Decoding
• At stage i, decoder Di processes not only the sequence of received signal points, but also decisions of decoders Dj, for j = 1, 2, …, i-1.
Decoder D1
Decoder D2
Decoder DM
Y
1X
2X
MX
a
• The decoding (in stage i) is usually
done in two steps– Point in subset decoding
– Subset decoding
• This method is not optimal in maximum likelihood sense, but it is asymptotically optimal for high SNR.
Decoder DiY
1X 2X
...1
ˆiX
iX
Optimal Decoding
),ˆ,,ˆ(
|
)ˆ,,ˆ(
11
11
111
)|(}Pr{
}Pr{),ˆ,,ˆ(
iii
ii
xxxaAY
xxb
iii ayfb
axxxM
A
A
– Ai(x1,…, xi) is the subset determined by x1,…, xi
– fY|A(y|a) is the transition probability (determined by the channel)
ix
Rate Design Criterion
),,,|;(
)|;(
);(
121
122
11
MMM XXXXYIC
XXYIC
XYIC
then the rate of the code at level i, Ri, should satisfy
ii CR
Decoder D1
Decoder D2
Decoder DM
Y
1X
2X
MX
a
-5 0 5 10 15 200
0.5
1
1.5
2
2.5
3
SNR (dB)
Cap
acity
(bi
ts/s
ymbo
l)
C C1C2
Two-level, 8-ASK, AWGN channel
Rate Design Criterion
Using the multiaccess channel analogy, if optimal decoding is used,
);(),,;(
)}{|,;(
)}{|;(
1
,
AYIXXYIR
XXXYIRR
XXYIR
Mi
i
jikkjiji
ikkii
R1
R2
I(Y;X1)
I(Y;X2)
I(Y;X2|X1)
I(Y;X1|X2)
-5 0 5 10 15 200
0.5
1
1.5
2
2.5
3
SNR (dB)
Cap
acity
(bi
ts/s
ymbo
l)
C C1 C2 I(Y;X1|X2)
Two-level, 8-ASK, AWGN channel
Iterative Multistage Decoding
Assuming
)(
11
11
111
1
11
}Pr{
}Pr{}ˆ|Pr{
)}(|Pr{}ˆ|)(Pr{}ˆ|Pr{
xb
b
axx
xaxxxa
A
AA
This expression, then, can be used as a priori probability of point a for the second decoder.
}ˆ|Pr{ 11 xx
– Two level Code
– R1 I(Y;X1|X2)
– Decoder D1:
then the a posteriori probabilities are
Probability Mass Functions
Error free decoding Non-zero symbol error probability
-5 0 5 10 15 200
0.5
1
1.5
2
2.5
3
SNR (dB)
Cap
acity
(bi
ts/s
ymbo
l)
C C1 C2 I(Y;X1|X2) I(Y;X2|partial X1)
Two-level, 8-ASK, AWGN channel
-5 0 5 10 15 20 25 30 350
0.5
1
1.5
2
2.5
3
SNR (dB)
Cap
acity
(bi
ts/s
ymbo
l)
C C1 C2 I(Y;X1|X2) I(Y;X2|partial X1)
Two-level, 8-ASK, Fast Rayleigh fading channel
8-PSK, 2-level, 4-state, uncoded, AWGN channel
0 1 2 3 4 5 6 710
-5
10-4
10-3
10-2
10-1
100
SNR per Bit
Err
or P
roba
bilit
y
OverallEncodedUncoded
8-PSK, 2-level, 4-state, uncoded , fast Rayleigh fading channel
6 8 10 12 14 16 18 2010
-5
10-4
10-3
10-2
10-1
SNR per Bit
Err
or P
roba
bilit
y
OverallEncodedUncoded
6 8 10 12 14 16 18 2010
-5
10-4
10-3
10-2
10-1
100
SNR per Bit
Err
or P
roba
bilit
y
Overall First Level Second Level
8-PSK, 2-level, 4-state, zero-sum, fast Rayleigh fading channel
6 8 10 12 14 16 1810
-5
10-4
10-3
10-2
10-1
100
SNR per Bit
Err
or P
roba
bilit
y
Overall First Level Second Level
8-PSK, 2-level, 4-state, 2-state , fast Rayleigh fading channel
6 8 10 12 14 16 18 2010
-4
10-3
10-2
10-1
100
SNR per Bit
Err
or P
roba
bilit
y
4-state, zero-sum 4-state, 2-state, 1-iteration4-state, 2-state, 2-iteration
8-PSK, 2-level, fast Rayleigh fading
Higher Constellation Expansion Ratios
• For AWGN, CER is usually 2– Further expanding Smaller MSED
Reduced coding gain
• For fading channels, – Further expanding Smaller product distance
Reduced coding gain
– Further expanding Larger Hamming distance
Increased diversity gain
0 2 4 6 8 10 12 1410
-5
10-4
10-3
10-2
10-1
100
SNR per Bit
Err
or P
roba
bilit
y
TCM, 8-PSK 2-level, 1-iteration, 16-PSK
14 15 16 17 18 19 2010
-5
10-4
10-3
10-2
SNR per Bit
Err
or P
roba
bilit
y
TCM, 8-PSK 2-level, 1-iteration, 16-PSK2-level, 2-iteration, 16-PSK
Conclusion
• Using iterative MSD with updated a priori probabilities in the first iteration, a broader subregion of the capacity region of MLC scheme can be achieved.
• Lower complexity multilevel codes can be
designed to achieve the same performance.
• Coded modulation schemes with constellation expansion ratio greater than two can achieve better performance for fading channels.
Coding Across Time
• If channels are encoded separately, assuming– A slowly fading channel in each frequency bin, and
– Independent fades for different channels (interleaving/deinterleaving across frequency bins is used)
nnn
sh
nnn
s
ccN
EhccE
ccN
hEhcc
2
0
2
0
2
ˆ4
1
1|ˆPr
ˆ4
exp|ˆPr
Coding Across Frequency Bins
• If coding is performed across frequency bins, assuming independent fades for different channels (interleaving/deinterleaving across frequency bins is used)
nnn
s
nnnn
s
ccN
EccE
cchN
Ecc
2
0
22
0
ˆ4
1
1|ˆPr
ˆ4
exp|ˆPr
h
h
h
6 8 10 12 14 16 18 2010
-4
10-3
10-2
10-1
100
SNR per Bit
Err
or P
roba
bilit
yAccross time, 1-iteration Accross time, 2-iteration Accross frequency, 1-iterationAccross frequency, 2-iteration
8-PSK, 2-level, 4-state, 2-state