cooperative diversity coding strategiespaull/teaching/eece_496/coop_div_codes.pdf · 2006-07-28 ·...

Cooperative Diversity Coding Strategies

Summer School on Communications and InformationTheory,

August 1st, 2006

Paul Lusina, Lutz Lampe, and Robert Schober

{paull, lampe, rschober}@ece.ubc.ca

Communication Theory Group

Department of Electrical and Computer Engineering

University of British Columbia

Vancouver, Canada

Cooperative Diversity Code Design – p.1/31

’How to balance your codes’

αSD

αRD


Cooperative Diversity Topology

� ��

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� � ��

� � � � � � � � � � � � �� Destination

Relay

Source

hSR

hSD

hRD

• Diversity from independent paths hSD and hRD.• Diversity achieved without extra hardware on units.• Better use of system resources.


Simultaneous Decode and Forward

Tx. 1 Tx. 2

phase 1 phase 2 phase 1 phase 2

︸︷︷︸tail

︸︷︷︸head

sT

s′T

sH

s′T

sT

sHUnit 1

Unit 2

• Both the relay and the source transmit during phase 2.• Recovery techniques are needed at the receiver. (orthogo-

nal signalling, dirty paper codes, interference cancellation)


Time Division Decode and Forward

Tx. 1 Tx. 2

phase 1 phase 2 phase 3 phase 1

︸︷︷︸head

︸︷︷︸tail

Unit 1

Unit 2

sH sT

s′T

sH

• Relay transmits on an orthogonal channel in time in phase3.

• Avoids complexity caused by interfering signals by usingmore resources.


Old codes for the new channel.

Incremental redundancy type codes• The relay sends additional redundancy information over an inde-

pendent channel.

Iterative type codes• The relay sends symbols which give a-priori information.


Our focus

Low complexity encoders/decoders• Target simple low power and low complexity applications (i.e. Blue-

tooth encoding scheme).

Derive a theoretical performance bound• Try to identify new design parameters based on the channel model.


A new orthogonal design approach

TailHead

SD ChannelsH

RD Channel√

(1 − ζ) · sRD

√ζ · sSD

√ζ · sSD

√

(1 − ζ) · sRDsH

hSD︷︸︸︷

hRD︷︸︸︷

︸︷︷︸tail

︸︷︷︸head

d2(sH, s′H) ζ · d2(sSD, s′SD) (1 − ζ) · d2(sRD, s′RD)

• Derive the pairwise error probability based on the squared Eu-clidean distance d2 and the power distribution factor ζ.


Pairwise Error Probability (PEP) Derivation


αSD(ζ) =1

4

ˆd2(sH , s′

H) + ζd2(sSD, s′

SD)˜

| {z }

Distance over channel SD

αRD(ζ) =1

4(1 − ζ)d2(sRD, s′

RD)

| {z }

Distance over channel RD

„dist(s, s′, ζ,h)

2

«2

= |hSD|2 αSD(ζ) + |hRD|2 αRD(ζ)

P (s → s′|ζ) = E|h|2

2

4Q

0

@

s

γ ·

„dist(s, s′, ζ,h)

2

«2

1

A

3

5

γ: Signal to Noise Ratio s: Modulated codeword


Pairwise Error Probability Derivation cont.

Using Craigs form of the Q-Function and solving using partial fractionexpansion we get:

P (s → s′|ζ) =

1

2(αSD − αRD)

"

αSD

1 −

s

γαSD

1 + γαSD

!

− αRD

1 −

s

γαRD

1 + γαRD

!#

Using a Taylor series expansion for 1/γ → 0 (high SNR), gives:

P (s → s′|ζ) ≤

(3

16

)

·(

1

αSD · αRD

)

·(

1

γ

)2

P (s → s′) ≤ (4) ·

(1

∏r

i=1 λi

)

·(

1

γ

)Ntx


Design Criteria

P (s → s′|ζ) ≤

(316

)·(

1αSD·αRD

)

·(

1γ

)2

Diversity:• Full diversity is achieved when {αSD, αRD} 6= 0

• Non-zero α∗ is analogous to the full rank criterion for space-timecode diversity.

Coding gain:• αSD = αRD minimizes the PEP.• Equating the values for α∗ is analogous to equating the eigenval-

ues for space-time code coding gain.


PEP Coding Gain Design Balancing Act

4·αSD︷︸︸︷

4·αRD︷︸︸︷


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αSD αRD

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α′SD α′

RD


Balancing Technique 1: Optimize ζ

Setting the derivative of αSD · αRD with respect to ζ to zero gives:

ζ =

d2(sRD, s′RD) − d2(sH , s′H)

2 · d2(sRD, s′RD), d2(sRD, s′RD) − d2(sH , s′H) ≥ 0

0, otherwise

How do we select ζ for specific code and encoder?


Balancing Technique 1: Find ζ for the encoder

Orthogonal signalling (Alamouti) technique ζ = 0.5

• Source and relay must transmit at the same signal amplitude toensure orthogonal recovery of signals.

Minimum distance technique (MD)• The power factor is optimized based on the dominant error event

corresponding to the minimum distance codeword.

Power averaging technique (Avg)• ζ is found for each codeword, and a weighted average is taken

based on its multiplicity.


Balancing Technique 2: Control the Euclidean distance

d2(sH, s′H) ζ · d2(sSD, s′SD) (1 − ζ) · d2(sRD, s′RD)︸︷︷︸

4·αSD

︸︷︷︸

4·αRD

• The ideal code would have a weight distribution which wouldensure that αSD = αRD.

• Select an encoder that balances αSD and αRD.

Can ’good’ encoders be found based on this observation?


Balancing Technique 2: Finding ’good’ encoders

Control the information and transmission sequence mapping.

Selection of the finite field encoder polynomial• Controls the mapping between the information and codeword se-

quences.

Selection of the finite field primitive element• Controls the mapping between the information and codeword se-

quences.

Mapping to the signal constellation• Controls the mapping between the codeword and transmission se-

quences.


Challenging the theory:

’All encoders have the same performance’


Convolutional Code (1,7/5) and (1,5/7) Encoders

u(i)

c2(i)

c1(i)

u(i)

c2(i)

c1(i)

Code (1, 7/5) Code (1, 5/7)

• The Extended Weight Enumerator Polynomial gives theinformation and codeword weight distribution for all codewords.

• Assume the Relay repeats the redundancy bits of the source i.e.sSD = sRD


Minimum Distance Codeword Performance (high SNR)

Simultaneous decode and forward (Alamouti, ζ = 0.5)

Code dmin dheadmin dtail

min ζ αSD αRD

((αSDαRD)(1,7/5)

(αSDαRD)(1,5/7)

)

dB

(1, 7/5) 5 2 3 0.5 2.13 1.132.43

(1, 5/7) 5 3 2 0.5 2.75 0.50

Time division decode and forward (Min. Dist., MD)

Code dmin dheadmin dtail

min ζ αSD αRD

((αSDαRD)(1,7/5)

(αSDαRD)(1,5/7)

)

dB

(1, 7/5) 5 2 3 0.278 1.63 1.630.72

(1, 5/7) 5 3 2 0.000 2.25 1.00


Codeword Weight Distribution Ai,j

Encoder (1,7/5) Encoder (1,5/7)

12

34

56

78

910

12

34

56

78

910

0

10

20

30

40

50

Head Wt

Weight Distribution (1,7/5)

Tail Wt

Coe

f. (d

B)

12

34

56

78

910

12

34

56

78

910

0

10

20

30

40

50

Head Wt

Weight Distribution (1,5/7)

Tail WtC

oef.

(dB

)

• i → Head Weight j → Tail Weight• (1,7/5) encoder has more codewords dSD > dH .• Therefore a ζ exists for αSD = αRD.


The proof is in the pudding


Encoder Performance Evaluation

Simulation Environment

• Quasi-static independent Rayleigh block fading over all channels• Perfect avoidance of error propagation at the relay using a CRC code.• BPSK modulation with perfect receiver channel state information.

Theory Curves

• Calculation of the head and tail weight for codewords of length of 100.

• Union bound of the code averaged over n = 105 channel realizations.

Pbit ≤1

n

n∑

l

min

1

2,1

k

∑

i,j

Ai,j · Q

√

γ ·(

dist(si,j , s0, ζ,hl)

2

)2


Encoder comparison: Alamouti, ζ = 0.5

10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 1510

−3

10−2

Eb/No

BE

R

th. (1,7/5)th. (1,5/7)sim (1,7/5)sim (1,5/7)

15 15.5 16 16.5 17 17.5 18 18.5 19 19.5 2010

−4

10−3

Eb/No

BE

R

th. (1,7/5)th. (1,5/7)

• (1,7/5) has a theoretical advantage of 0.1 dB which is maintainedat high SNR.

• Simulation results show almost equal performance at lower SNR.


Encoder comparison: Min. Dist., ζ → MD

10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 1510

−3

10−2

Eb/No dB

BE

R

th. (1,7/5) 0.278th. (1,5/7) 0Sim. (1,7/5) 0.278Sim. (1,5/7) 0

� � � � ��

� � � ��

0.5 dB

15 15.5 16 16.5 17 17.5 18 18.5 19 19.5 2010

−4

10−3

Eb/No dB

BE

R

th. (1,7/5) 0.278th. (1,5/7) 0

• (1,5/7) has a simulation advantage of 0.5 dB at low SNR.• Theory predicts at high SNR, this advantage will decrease.


Power Factor Comparison, (1,7/5)

10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 1510

−3

10−2

Eb/No

BE

Rth. 0.5th. Min. Dist. (0.278)th. Avg. (0.124)th. Zero (0)sim. 0.5sim. Min. Dist. (0.278)sim. Avg. (0.124)sim. Zero (0)

� � � ��

� � � ��

1 dB


Power Factor Comparison, (1,5/7)

10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 1510

−3

10−2

Eb/No

BE

R

th. 0.5th. Min. Dist. (0)th. Avg. (0.0927)sim 0.5sim. Min. Dist. (0)sim. Avg. (0.0927)

� � � ��

� � � ��

1 dB� � � � � � �


Cooperative Diversity Protocol Design

Key Observation• For ζ = 0, both encoders have the best performance.• The protocol is greatly simplified by avoiding the need for orthogonal modu-

lation or signalling.Tx. 1

phase 1 phase 2

Tx. 1

phase 1 phase 2 phase 3

Tx. 1

phase 1 phase 2


Results

Encoder Comparison:• Theoretical results showed (1,7/5) is better for high SNR.• Simulation results showed the importance of encoder design and

the operating point at low SNR.

Power factor ζ comparison:• ζ can improve the overall code performance by up to 1 dB.• ζ = 0 is the best choice for both encoders considered.

Theory Results:• Theory predicted which encoder and ζ was qualitatively better at

high SNR.• Theory curves could not make quantitative predictions.


Conclusions

Construct low complexity cooperative diversity systems for moderatedata rates.

The Balancing Act:• Equate the codeword distance on the SD and RD channels.

Balancing based on ζ:• Choose ζ to equate the Euclidean head/tail weight distribution for

the codewords.

Balancing based on the encoder:• Choose the the information/transmission symbol mapping to

equate the Euclidean head/tail weight distribution for the code-words.


Future Work

Code Construction:• Other code classes (RS, BCH)• Probabilistic weight distributions for LDPC type codes.• Optimum modulation mappings.

Theory:• Tighter performance bounds• Find the optimum ζ for a given code.

Diversity:• Finding full diversity codes with minimum redundancy.


Thank you!

αSD αRD


cooperative diversity coding strategiespaull/teaching/eece_496/coop_div_codes.pdf · 2006-07-28 ·...

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