analysis and design of mappings for iterative decoding of bit-interleaved coded modulation*

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

Analysis and Design of Mappings for Iterative Decoding of

Bit-Interleaved Coded Modulation*

Frank SchreckenbachInstitute for Communications Engineering

Munich University of Technology, Germany

Norbert GörtzSchool of Engineering and Electronics,

University of Edinbrugh, UK

* This work was supported by NEWCOM and DoCoMo Communications Laboratories Europe GmbH


System model: BICM and BICM-ID

Encoder Interleaver

DecoderDe-

interleaver

data

data estimate

c Mapper

DemapperDetector/ Equalizer

Le(C)

InterleaverLa(C)

ChannelCode: Convolutional, Turbo, LDPC

e.g. QPSK, 16QAM

AWGN, OFDM, ISI, MIMO


Outline

• Consider mapping as coding entity: characterization with Euclidean distance spectrum EXIT charts

• Bit-Interleaved Coded Irregular Modulation (BICIM)

• Optimization of mapping: Quadratic Assignment Problem (QAP) Binary Switching Algorithm

• Future work - Open problems


Euclidean distance spectrum

Distance

Frequency λ1 λ2

Gray

Anti Gray

QPSK, no a priori information at the demapper.

1101

1000

1101

1000

1001

1100

1001

1100

Gray

Anti-Gray

1st bit 2nd bit

1 2d 2 2d



Distance

Frequency λ1 λ2

Gray 4Anti Gray


1101

1000

1101

1000

1001

1100

1001

1100

Gray

Anti-Gray

1st bit 2nd bit

1 2d 2 2d



Distance

Frequency λ1 λ2

Gray 4 4Anti Gray


1101

1000

1101

1000

1001

1100

1001

1100

Gray

Anti-Gray

1st bit 2nd bit

1 2d 2 2d



Distance

Frequency λ1 λ2

Gray 4 4

Anti Gray 6


1101

1000

1101

1000

1001

1100

1001

1100

Gray

Anti-Gray

1st bit 2nd bit

1 2d 2 2d



Distance

Frequency λ1 λ2

Gray 4 4

Anti Gray 6 2


1101

1000

1101

1000

1001

1100

1001

1100

Gray

Anti-Gray

1st bit 2nd bit

1 2d 2 2d

Note that without a priori information, the distances d2 might not be relevant. An expurgated distance spectrum would be more precise.



Distance

Frequency λ1 λ2

Gray 4 4

Anti Gray 6 2


1 2d 2 2d

Distance

Frequency λ1 λ2

Gray

Anti Gray

1 2d 2 2d

QPSK, ideal a priori information at the demapper : signal constellation is reduced to a symbol pair.

1101

1000

1101

1000

1001

1100

1001

1100

Gray

Anti-Gray

1st bit 2nd bit



Distance

Frequency λ1 λ2

Gray 4 4

Anti Gray 6 2


1 2d 2 2d

Distance

Frequency λ1 λ2

Gray 4 0Anti Gray

1 2d 2 2d


1101

1000

1101

1000

1001

1100

1001

1100

Gray

Anti-Gray

1st bit 2nd bit



Distance

Frequency λ1 λ2

Gray 4 4

Anti Gray 6 2


1 2d 2 2d

Distance

Frequency λ1 λ2

Gray 4 0

Anti Gray 2 2

1 2d 2 2d


1101

1000

1101

1000

1001

1100

1001

1100

Gray

Anti-Gray

1st bit 2nd bit


EXIT chart QPSK

Average mutual information between coded bits C at the transmitter and LLRs L at the receiver:

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

mutual information at input of demapper

mut

ual i

nfor

mat

ion

at o

utpu

t of d

emap

per

4-state conv. code

Gray

QPSK, AWGN channel


EXIT chart QPSK

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1


mut

ual i

nfor

mat

ion

at o

utpu

t of d

emap

per

4-state conv. code

GrayAnti-Gray

QPSK, AWGN channel

Average mutual information between coded bits C at the transmitter and LLRs L at the receiver:


Bit-wise EXIT chart QPSK

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1


mut

ual i

nfor

mat

ion

at o

utpu

t of d

emap

per

4-state conv. code

Anti-Gray

Anti-Gray,bit 1 Anti-Gray,

bit 2

Compare to multilevel codes!

QPSK, AWGN channel


Analytic EXIT chart QPSK

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1


mut

ual i

nfor

mat

ion

at o

utpu

t of d

emap

per

simulationanalytic+numeric

4-state conv. code

Gray

Anti-Gray

Anti-Gray,bit 1 Anti-Gray,

bit 2

Analytic and numeric computation with BEC a priori information.


Bit Interleaved Coded Irregular Modulation (BICIM)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1


mut

ual i

nfor

mat

ion

at o

utpu

t of d

emap

per

QPSK16QAM50% QPSK, 50% 16QAM

4-state, rate 1/2 convolutional code

• Within one code block, use different signal constellations: fine adaptation of data rate to channel

characteristics with the modulation mappings: optimization of iterative decoding procedure

• Basic idea similar to irregular channel codes

• Low complexity, good performance with low and medium code rates

• EXIT chart: linear combination of EXIT functions.


Optimization of mapping

• Goal: find optimal assignment of binary indexes to signal points.• Optimization for:

• No a priori information at the demapper (Gray mapping)• Ideal a priori information at the demapper• Trade off no/ideal a priori• Optimization for bit positions





• Exhaustive search intractable for high order signal constellations: 2m! possible mappings. 16QAM: 2·1013 possible mappings





• Exhaustive search intractable for high order signal constellations: 2m! possible mappings. 16QAM: 2·1013 possible mappings

• Problem can be cast to a Quadratic Assignment Problem (QAP, Koopmans and Beckmann, 1957)• QAP is NP-hard, i.e. not solvable in polynomial time.• Famous applications are e.g. wirering in electronics or

assignment of facilities to locations.


QAP Algorithms

• Binary Switching Algorithm (Zeger, 1990): try to switch the symbol with highest costs, i.e. the strongest contribution to a bad performance, with an other symbol such that the total cost is minimized.

01100111 00110010

01000101 00010000

11001101 10011000

11101111 10111010

• Other possibilities:• Tabu search• Simulated annealing approaches• Integer Programming• …


• Cost function based on Euclidean distance spectrum

• AWGN channel:

• Fading channel:

• Optimized mapping:

Cost function

Possible distinctEuclidean distances

Frequency of distance dk in Euclidean distance spectrummapping


Euclidean distance spectrum 16QAM

Distance …Frequency λ1 λ2 λ3 … λ1 λ2 λ3 λ4 λ5 …

Gray 24 36 32 … 24 0 0 0 0 …SP 56 32 24 … 4 8 8 0 8 …

MSP 52 38 24 … 0 2 8 4 8 …M16a 56 42 40 … 0 0 0 16 4 …

I16 52 42 40 … 0 0 0 16 8 …

Gray M16a

no a priori ideal a priori2 21 Ed d 2 2

2 2 Ed d 2 23 4 Ed d 2 2

4 5 Ed d 2 25 8 Ed d2 2

1 Ed d 2 22 2 Ed d 2 2

3 4 Ed d

SP: Set Partitioning

MSP: Modified Set Partitioning

M16a: optimized for ideal a priori information in AWGN channels

I16: optimized for maximum sum of mutual info. without and with a priori


EXIT chart, 16QAM

• AWGN channel

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1GraySet Patrtitioning BICM-ID opt.

mut

ual i

nfor

mat

ion

at o

utpu

t of d

emap

per


rate 1/2, memory 2 convolutional code


Error rate, 16QAM

• BER for AWGN channel, 4-state, rate ½ conv. code, interleaver length 10000 bits

1 2 3 4 5 6

10-6

10-4

10-2

100

Eb/N

0 in dB

BE

RGraySet PartitioningBICM-ID opt.

10th iter.

1th iter.

analytical bounds for error free feedback


Conclusion

• Mapping has a big influence on the performance of iterative detection schemes.

• Consider mapping as coding entity: characterization with Euclidean distance spectrum EXIT chart

• Optimization of mapping: Quadratic Assignment Problem (QAP) Binary Switching Algorithm

• Bit-Interleaved Coded Irregular Modulation (BICIM)


Future work – Open problems

• Complexity:• trade-off “cheep” outer code vs. number of required iterations• Suboptimum demapping algorithms

• Combination of different (optimized) mappings with iterative MIMO detection, equalization, MU detection, …

• Further extensions:• Investigations on signal constellations• Multidimensional mappings: map a sequence of bits to a

sequence of symbols

analysis and design of mappings for iterative decoding of bit-interleaved coded modulation*

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