elec599
TRANSCRIPT
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Multilevel Coding andIterative Multistage Decoding
ELEC 599 Project Presentation
Mohammad Jaber Borran
Rice University
April 21, 2000
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Multilevel Coding
A number of parallel encoders
The outputs at each instant select one symbol
lbits/symbo1
11
!!
!!M
i
i
M
i
i KN
RR
M-way
Partitioning
of data
data bitsfrom the
information
source
E1 (rateR1)
EM (rateRM)
E2 (rateR2)
q1 K1 N x1
Mapping
(to 2M-point
constellation)
Signal
Point
q2 K2
qM KM
N x2
N xM
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Minimum Hamming distance for encoderi: dHi ,
Minimum Hamming distance for symbol
sequences
a_)(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
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Probability of error forFading Channels
Rayleigh fading with independent fading coefficients
Chernoff bound
{
!
d
e
L
d
k
jik
L
s
jie
k
dEP
0
1
2
0
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
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For a fast fading channel, or a slowly fading channel with
interleaving/deinterleaving
Design criterion (Divsalar)
Design Criterion forFading Channels
),(minmax,},,,{ 2
jiji
dn
ccccc1 -
{
!
!L
d
k
jikji
k
dd
0
1
2
2
)()( c,cc,c),(minmax ,},,,{ 2
jiHji dn ccccc1 -
For a slowly fading channel withoutinterleaving/deinterleaving
Design criterion ),(minmax,},,,{ 2
jiEji
dn
ccccc1 -
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For a fast fading channel, or a slowly fading channel with
interleaving/deinterleaving
Decoding Criterion
kkk
L
k
ikki
yyd EE !!
~where)~(||min1
22c,y
Ekis the fading coefficient forkth symbol)
Maximizes the likelihood function
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Optimum decoder: Maximum-Likelihood decoder
If the encoder memories are R1, R
2, ,R
M,
the total number of states is 2R,
where R=R1 + R2 + + RM.
Complexity Need to look for suboptimum
decoders
Decoding
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IfA
andY
denote the transmitted and receivedsymbol 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
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Multistage Decoding
At stage i, decoder Diprocesses not only thesequence of received signal points, but also
decisions of decoders Dj, forj = 1, 2, , i-1.
Decoder D1
Decoder D2
Decoder DM
Y
1X
2X
MX
a
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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
iX
iX
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Optimal Decoding
!
iii
ii
xxxa
AY
xxb
iii ayfb
axxxM
-
-
-
A
A
Ai(x1,,xi) is the subset determined byx1,,xi
fY|A(y|a) is the transition probability (determined by
the channel)
ix
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Rate Design Criterion
),,,;(
);(
);(
121
122
11
!
!
!
MMM XXXXYIC
XXYIC
XYIC
-
/
then the rate of the code atlevel i, i, should satisfy
ii Ce
Decoder D1
Decoder D2
Decoder DM
Y
1X
2X
MX
a
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-5 0 5 10 15 200
0. 5
1
1. 5
2
2. 5
3
SNR (dB)
Capacity(b
its/sym
bol)
C
C1
C2
Two-level, 8-ASK, AWGN channel
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-5 0 5 10 15 200
0. 5
1
1. 5
2
2. 5
3
SNR (dB)
Capacity(b
its/sym
bol)
C
C1
C2
I(Y;X1|X2)
Two-level, 8-ASK, AWGN channel
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Iterative Multistage Decoding
Assuming
!
!
r
rr
rrr
xb
bxx
xxxx
A
AA
This expression, then, can be used as a priori
probability of point a for the second decoder.
}|Pr
Two level Code
R e I(Y;X1|X2)
Decoder D1:
then the a posterioriprobabilities are
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Probability Mass Functions
Error free decoding Non-zero symbolerror probability
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-5 0 5 10 15 200
0. 5
1
1. 5
2
2. 5
3
SNR (dB)
Capacity(bits/sym
bol)
CC1
C2
I(Y;X1|X2)
I(Y;X2|partial X1)
Two-level, 8-ASK, AWGN channel
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8-PSK, 2-level, 4-state, uncoded, AWGN channel
0 1 2 3 4 5 6 71 0
-5
1 0-4
1 0-3
1 0-2
1 0-1
1 00
S N R e it
E
P
ilit
O ve
ll
E n
e
Un
e
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8-PSK, 2-level, 4-state, uncoded , fast Rayleigh fading channel
6 8 1 0 1 2 1 4 1 6 1 8 2 01 0
-5
1 0-4
1 0-3
1 0-2
1 0-1
S N R e it
E
!
P
!
"
#
"
ilit
$
O ve %
ll
E n& ' (
e (
Un& ' (
e (
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6 8 1 0 1 2 1 4 1 6 1 8 2 01 0
-5
1 0-4
1 0-3
1 0-2
1 0-1
1 00
S N R ) e 0 1 it
E
2
2
3
2P
2
3
4
5
4
ilit
6
O ve 0 7 llF i
0 8t Level
S e9 @
n A Level
8-PSK, 2-level, 4-state, zero-sum, fast Rayleigh fading channel
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6 8 1 0 1 2 1 4 16 1 81 0
-5
1 0-4
1 0-3
1 0-2
1 0-1
1 00
S N R B e C D it
E
E
E
F
EP
E
F
G
H
G
ilit
I
O ve C P llF i
C Qt Level
S eR S
n T Level
8-PSK, 2-level, 4-state, 2-state , fast Rayleigh fading channel
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6 8 1 0 1 2 1 4 1 6 1 8 2 010
-4
10-3
10-2
10-1
100
S NRU V W
B it
X
Y
Y
`
Y
a
Y
`
b
c
b
ilit
d
4- s t e t V , z V W f -s um4- s t e t V , 2- s t e t V , 1- it
V W et i
f g
4- s t e t V , 2- s t e t V , 2- it V W e t i f g
8-PSK, 2-level, fast Rayleigh fading
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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
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0 2 4 6 8 1 0 1 2 1 41 0
-5
1 0-4
1 0-3
1 0-2
1 0-1
1 00
S N R h e i p it
E
q
q
r
q
P
q
r
s
t
s
ilit
u
T v M , 8 -P S K
2 -level, 1 -it e i w t i x n, 16 -P S K
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14 1 5 1 6 1 7 1 8 19 2 01 0
-5
1 0-4
1 0-3
1 0-2
S N R y e it
E
P
ilit
T M , 8 -P S K
2 -level, 1 -it e
t i n, 1 6 -P S K
2 -level, 2 -it e t i n, 1 6 -P S K
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Conclusion
Using iterative MSD with updated a prioriprobabilities 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.
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6 8 1 0 1 2 1 4 1 6 1 8 2 01 0
-4
1 0-3
1 0-2
1 0-1
1 00
S N R e it
E
P
ilit
A
t i e, 1- it e
t i n
A
t i e, 2- it e
t i n
A f equen , 1 -it e t i n
A
f equen
, 2 -it e
t i n
8-PSK, 2-level, 4-state, 2-state