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

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

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

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

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

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

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

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

Euclidean distance spectrum

Distance

Frequency λ1 λ2

Gray 4Anti 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

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

Euclidean distance spectrum

Distance

Frequency λ1 λ2

Gray 4 4Anti 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

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

Euclidean distance spectrum

Distance

Frequency λ1 λ2

Gray 4 4

Anti Gray 6

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

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

Euclidean distance spectrum

Distance

Frequency λ1 λ2

Gray 4 4

Anti Gray 6 2

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

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

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

Euclidean distance spectrum

Distance

Frequency λ1 λ2

Gray 4 4

Anti Gray 6 2

QPSK, no a priori information at the demapper.

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

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

Euclidean distance spectrum

Distance

Frequency λ1 λ2

Gray 4 4

Anti Gray 6 2

QPSK, no a priori information at the demapper.

1 2d 2 2d

Distance

Frequency λ1 λ2

Gray 4 0Anti 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

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

Euclidean distance spectrum

Distance

Frequency λ1 λ2

Gray 4 4

Anti Gray 6 2

QPSK, no a priori information at the demapper.

1 2d 2 2d

Distance

Frequency λ1 λ2

Gray 4 0

Anti Gray 2 2

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

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

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

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

EXIT chart QPSK

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

GrayAnti-Gray

QPSK, AWGN channel

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

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

Bit-wise EXIT chart QPSK

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

Anti-Gray

Anti-Gray,bit 1 Anti-Gray,

bit 2

Compare to multilevel codes!

QPSK, AWGN channel

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

Analytic EXIT chart QPSK

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

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.

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

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

mutual information at input of demapper

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.

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

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

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

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

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

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

• 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.

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

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• …

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

• 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

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

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

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

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

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

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

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

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

mutual information at input of demapper

rate 1/2, memory 2 convolutional code

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

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

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

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)

Frank Schreckenbach, Munich University of TechnologyNEWCOM 2005

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