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