Multilevel Coding and
Iterative Multistage Decoding
ELEC 599 Project Presentation
Mohammad Jaber Borran
Rice University
April 21, 2000
Multilevel Coding
A number of parallel encoders
The outputs at each instant select one symbol
data bits
from the
information
source
q1
K1
q2
K2
M-way
Partitioning
of data
qM
M
KM
1
R   Ri 
N
i 1
E1 (rate R1)
E2 (rate R2)
EM (rate RM)
N
x1
N
x2
Mapping
(to 2M-point
constellation)
N
xM
M
K
i 1
i
bits/symbo l
Signal
Point
Distance Properties
• Minimum Hamming distance for encoder i: dHi ,
Minimum Hamming distance for symbol
sequences
d H  min (d Hi )
i1,, M 
• For TCM (because of the parallel transitions)
dH = 1
• MLC is a better candidate for coded modulation
on fast fading channels
Probability of error for Fading Channels
• Rayleigh fading with independent fading coefficients
Chernoff bound
1
1
Pe (ci , c j ) 
 L
L
 d k2 (ci , c j )
Es N 0 


4
k 1 
2
d k 0



L’: effective length of the error event (Hamming distance)
dk(ci,cj): distance between the kth symbols of the two
sequences
Design Criterion for Fading Channels
• For a fast fading channel, or a slowly fading channel with
interleaving/deinterleaving
Design criterion
(Divsalar)
max min d H (ci , c j )
{c1 ,c 2 ,,c n } i , j
max min d P (ci , c j )
{c1 ,c 2 ,,c n } i , j
L
d P (ci , c j )   d k2 (ci , c j )
k 1
d k2  0
• For a slowly fading channel without
interleaving/deinterleaving
Design criterion
max min d E (ci , c j )
{c1 ,c 2 ,,c n } i , j
Decoding Criterion
• For a fast fading channel, or a slowly fading channel with
interleaving/deinterleaving
L
min
i
2
2 ~
|

|
d
 k k (y, ci ) where
k 1
~
yk  yk  k
k is the fading coefficient for kth symbol)
– Maximizes the likelihood function
Decoding
• Optimum decoder: Maximum-Likelihood decoder
• If the encoder memories are n1, n2, …,nM,
the total number of states is 2n,
where n = n1 + n2 + … + nM.
• Complexity  Need to look for suboptimum
decoders
• If A and Y denote the transmitted and received
symbol sequences respectively, using the chain
rule for mutual information:
I (Y ; A)  I (Y ; X1 , X 2 ,, X M )
 I (Y ; X 1 )  I (Y ; X 2 | X 1 )  
 I (Y ; X M | X 1 , X 2 ,, X M 1 )
• Suggests a rule for a low-complexity staged
decoding procedure
Multistage Decoding
X̂1
Decoder D1
X̂ 2
Y
Decoder D2
â
X̂ M
Decoder DM
• 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.
ˆ
X
i 1
X̂1 X̂ 2
...
Y
Decoder Di
X̂i
• 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.
Optimal Decoding
xi
M i ( xˆ1 ,, xˆi 1 , xi ) 

aA i ( xˆ1 ,, xˆi 1 , xi )
Pr{a}
fY | A ( y | a)
 Pr{b}
bA i1 ( xˆ1 ,, xˆi1 )
– Ai(x1,…, xi) is the subset determined by x1,…, xi
– fY|A(y|a) is the transition probability (determined by
the channel)
Rate Design Criterion
X̂1
Decoder D1
X̂ 2
Y
Decoder D2
â
X̂ M
Decoder DM
C1  I (Y ; X 1 )
C2  I (Y ; X 2 | X 1 )

CM  I (Y ; X M | X 1 , X 2 ,, X M 1 )
then the rate of the code at
level i, Ri, should satisfy
Ri  Ci
3
Capacity (bits/symbol)
2.5
C
C1
C2
2
1.5
1
0.5
0
-5
0
5
10
15
SNR (dB)
Two-level, 8-ASK, AWGN channel
20
Rate Design Criterion
Using the multiaccess channel analogy, if optimal
decoding is used,
Ri  I (Y ; X i | { X k }k i )
Ri  R j  I (Y ; X i , X j | { X k }k i , j )

R
i
i
 I (Y ; X 1 ,  , X M )  I (Y ; A)
R2
I(Y;X2|X1)
I(Y;X2)
I(Y;X1)
I(Y;X1|X2)
R1
3
Capacity (bits/symbol)
2.5
2
1.5
1
C
C1
C2
I(Y;X1|X2)
0.5
0
-5
0
5
10
15
SNR (dB)
Two-level, 8-ASK, AWGN channel
20
Iterative Multistage Decoding
Assuming
– Two level Code
– R1  I(Y;X1|X2)
– Decoder D1: Pr{x1 | xˆ1}
then the a posteriori probabilities are
Pr{a | xˆ1}  Pr{A1 ( x1 ) | xˆ1} Pr{a | A1 ( x1 )}
 Pr{ x1 | xˆ1}
Pr{a}
 Pr{b}
bA1 ( x1 )
This expression, then, can be used as a priori
probability of point a for the second decoder.
Probability Mass Functions
Error free decoding
Non-zero symbol
error probability
3
Capacity (bits/symbol)
2.5
C
C1
C2
I(Y;X1|X2)
I(Y;X2|partial X1)
2
1.5
1
0.5
0
-5
0
5
10
15
SNR (dB)
Two-level, 8-ASK, AWGN channel
20
3
C
C1
C2
I(Y;X1|X2)
I(Y;X2|partial X1)
Capacity (bits/symbol)
2.5
2
1.5
1
0.5
0
-5
0
5
10
15
SNR (dB)
20
25
30
35
Two-level, 8-ASK, Fast Rayleigh fading channel
0
10
Overall
Encoded
Uncoded
-1
Error Probability
10
-2
10
-3
10
-4
10
-5
10
0
1
2
3
4
SNR per Bit
5
6
7
8-PSK, 2-level, 4-state, uncoded, AWGN channel
-1
10
Overall
Encoded
Uncoded
-2
Error Probability
10
-3
10
-4
10
-5
10
6
8
10
12
14
SNR per Bit
16
18
20
8-PSK, 2-level, 4-state, uncoded , fast Rayleigh fading channel
0
10
Overall
First Level
Second Level
-1
Error Probability
10
-2
10
-3
10
-4
10
-5
10
6
8
10
12
14
SNR per Bit
16
18
20
8-PSK, 2-level, 4-state, zero-sum, fast Rayleigh fading channel
0
10
Overall
First Level
Second Level
-1
Error Probability
10
-2
10
-3
10
-4
10
-5
10
6
8
10
12
SNR per Bit
14
16
18
8-PSK, 2-level, 4-state, 2-state , fast Rayleigh fading channel
0
10
4-state, zero-sum
4-state, 2-state, 1-iteration
4-state, 2-state, 2-iteration
-1
Error Probability
10
-2
10
-3
10
-4
10
6
8
10
12
14
SNR per Bit
16
18
8-PSK, 2-level, fast Rayleigh fading
20
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
10
TCM, 8-PSK
2-level, 1-iteration, 16-PSK
-1
Error Probability
10
-2
10
-3
10
-4
10
-5
10
0
2
4
6
8
SNR per Bit
10
12
14
-2
10
-3
Error Probability
10
-4
10
TCM, 8-PSK
2-level, 1-iteration, 16-PSK
2-level, 2-iteration, 16-PSK
-5
10
14
15
16
17
SNR per Bit
18
19
20
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)
 Es h 2
Prc  cˆ | h  exp 
 4 N 0
c
Eh Prc  cˆ | h 
1
1
Es
4N0
n
 cˆn
n
 cn  cˆn
n
2
2



Coding Across Frequency Bins
• If coding is performed across frequency bins,
assuming independent fades for different
channels (interleaving/deinterleaving across
frequency bins is used)
 E
Prc  cˆ | h  exp  s
 4N0
2
2
ˆ
h
c

c
n n n n 

1
ˆ
Eh Prc  c | h  
Es
2
n 1
cn  cˆn
4N0
0
10
Accross
Accross
Accross
Accross
-1
time, 1-iteration
time, 2-iteration
frequency, 1-iteration
frequency, 2-iteration
Error Probability
10
-2
10
-3
10
-4
10
6
8
10
12
14
SNR per Bit
16
8-PSK, 2-level, 4-state, 2-state
18
20