1. Multilevel Codes and Iterative Multistage Decoding
M. Jaber Borran and Behnaam Aazhang
Rice University

2. Overview Motivation Multilevel codes Concatenated space-time codes
High rate communication in fading channels
Diversity gain and coding gain
Multilevel codes
Multistage decoding
Updated a priori probabilities
Concatenated space-time codes
Higher dimensional coded modulation

3. Motivation
Fast fading
environment

4. Multilevel Coding A number of parallel encoders
The outputs at each instant select one symbol 6 1 2 3 4 5 7
M-way
Partitioning
of data
data bits
from the
information
source
E1 (rate R1)
EM (rate RM)
E2 (rate R2)
q K1
N x1
Mapping
(to 2M-point
constellation)
Signal
Point
q K2
qM KM
N x2
N xM
1102=6 1 1

5. Distance Properties Minimum Hamming distance for encoder i: dHi ,
Minimum Hamming distance for symbol sequences
For TCM (because of the parallel transitions)
dH = 1
MLC is a better candidate for coded modulation over fast fading channels

6. 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

7. Multistage Decoding y 2 3 1 Pick the brown subset 4 7 5 6 Decoder D11 2 3 4 5 7
Pick the
brown subset
Decoder D1
Decoder D2
Decoder DM y

8. Rate Design Criterion (Multistage Decoding)
Decoder D1
Decoder D2
Decoder DM y
then the rate of the code at level i, Ri, should satisfy

9. Capacity Curves (Multistage Decoding)
Two-level, 8-ASK, AWGN channel

10. Rate Design Criterion (Optimal Decoding)
Using the multiaccess channel analogy, if optimal decoding is used,
Multistage decoding capacity region R2
I(Y;X2|X1)
I(Y;X2) R1
I(Y;X1)
I(Y;X1|X2)

11. Capacity Curves (Optimal Decoding)
Two-level, 8-ASK, AWGN channel

12. Iterative Multistage Decoding
Approximating the optimum decoding
Assuming
Two level Code
R1  I(Y;X1|X2)
Decoder D1:
then the a posteriori probabilities are
This expression, then, can be used as a priori probability of point a for the second decoder.

13. Probability Mass Functions
1/8 1 2 3 4 5 6 7
Decoder D1
Error free
decoding
Non-zero symbol error probability (Pe = 1/4)
1/4 1 2 3 4 5 6 7
3/16
1/16 1 2 3 4 5 6 7
Decoder D2
branch
metric

14. Capacity Curves (Iterative Decoding with Updated a priori)
Two-level, 8-ASK, Fast Rayleigh fading channel

15. Simulation Results
8-PSK, 2-level, fast Rayleigh fading

16. Design of MLC for OTD
Using Chernoff bounding technique, for orthogonal transmission over fast fading channel
Design criteria

17. Higher Dimensional Coded Modulation
(2D coordinate 2)
Design coded modulation scheme for a higher dimensional constellation.
Transmit the two-dimensional coordinates of the higher dimensional symbols in consecutive time intervals.
(4D point)
(2D coordinate 1)
Higher
Dimensional
Coded
Modulator
Projection
into 2D
Coordinates
Serial to Parallel
Alamouti
Encoder

18. Example (2-level code for 4D 256QAM)
M-way
Partitioning
of data
Mapping
to 256-point
constellation
(8 32-pint
subsets)
data bits
from the
information
source E1
(rate 2/3) 4D
Signal
Point E2
(rate 4/5)
E1 : 8-state convolutional code, dH1 = 2
E2 : 16-state convolutional code, dH2 = 2
dH = min{dH1, dH2} = 2

19. Simulation Results

20. Conclusions
Using iterative MSD with updated a priori probabilities, a broader sub-region of the capacity region of the MLC can be achieved.
MLC with iterative MSD and updated a priori probabilities can achieve better performance compared to regular MSD.
Higher dimensional constellations can be used to design MLC for OTD. As a result, temporal diversity can be traded off for coding gain.