1

Wireless Communication

Low Complexity Multiuser DetectionRami Abdallah

University of Illinois at Urbana Champaign

12/06/2007

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Outline

Multiuser Detectors

OptimalJoint-ML Suboptimal

LinearInterference Cancellation

Near-ML

Sphere Decoder

Semi-DefiniteRelaxation

ProbabilisticData

Association

Parallel

Succesive

Decision Feedback

MMSE

Decorrelator

PolynomialExpansion

3

Introduction

• Multiuser Detection (MUD): canceling or suppressing interfering users from the desired signals

• Benefits: – Capacity Improvement– Reduced requirement for power control

• Limitations:– Complexity– Intercell interference– Spreading – Coding tradeoff

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Problem Definition

• Optimum Multiuser Detection

– Search space exponential in number of users

K

llll tnNtgzaty

10 1,0CNn

K

zzyz

zzyz

z

z

z

z

HA

ARAA

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K

K

K

T K

llllML

2maxarg

2maxarg

maxarg2

0 1

KkKl

lkR:1:1 KaadiagA ,...,1

tgtg kllk ,

5

System Representation

• Matched Filter (MF)– Received Signal for user k:

– System Representation after MF:

• Noise Whitening– Cholesky Decomposition to decorrelate noise

– Enables layered decoding

kl

klkllkkk wzazay

Multiple-Access Interference (MAI)

wzy RA RNCN 0,0w

LLR nzyy

LAL1~ 0,0 NCNn

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Linear Detectors (1)

• Decorrelating Detector– Solve for z by inverting R– Independent User Decoding– Best near-far resistance– Noise enhancement

• Optimal Linear Detector (MMSE) – Trade-off between MAI elimination and noise enhancement

wzyx

kkkk wzAx

RAR~

11

120

ANRT

T

MMSE

MMSE yx

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Linear Detectors (2)

• Polynomial Expansion (PE) Detector :

– Weighted sum of MF output (R)– Weights (W) chosen depending on a performance

criterion and can be adaptively updated– Can approximate decorrelating and MMSE detector

(Cayley-Hamilton Theorem)– Regular architecture avoiding Matrix inversion

N

i

iiPE

PE

RwTwhere

T

0

yx

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Interference Cancellation

• Successive Interference Cancellation (SIC)

– Order users according to descending power

– Start detection with the highest power first and subtract its effect from the received signal

– Successive users benefits more for MAI cancellation

• Problems:

– Latency

– Decision error propagation

1y1y

Receiver 1

r2(t)

1z

r3(t)

2z

Receiver 2

Kz

Receiver K

rK(t)

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Interference Cancellation (2)

Stage 1

De

co

rre

lato

ro

r M

MS

E

)0(1z

)0(ˆ2z

)0(ˆKz

)1(ˆ1z

)1(ˆ2z

)1(ˆKzStage 2

)1(1 Nz

)1(2 Nz

)1(ˆ NzK

Stage N

r(t)

• Parallel Interference Cancellation (PIC) – Every stage use previous estimates to subtract

MAI for each user in parallel– Tradeoff between complexity and performance

z.yz iGAi ˆ1ˆ

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Performance Comparison

– PIC superior over SIC in well-power controlled environment

Power Controlled

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• Multistage decision feed-back detector: – In each stage use the already detected bits to improve detection of remaining bits in the same stage

• Partial interference cancellation– Decision is based on

– Partially cancel MAI with the amount being cancelled increasing with each stage

Variations of PIC

)1(),()1()( ,, iGAziGAziGAziGAzi fp

ipiGApi ii zz.yz ˆ1ˆsgn1ˆ

iiGAip zz.yz ˆ,ˆ,1ˆ

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• Decision feed-back detector:– User ordering in terms of descending power– Noise whitening – SIC to cancel MAI among user (F is lower triangular)

Decision Feedback MUD

r(t)

1z

2z

Kz

3z

1

1, ˆ~ˆ

k

iiikikk zAfyz

ZAzyy

FF ~1*

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• Sphere Decoders (SD) in AWGN Channel

– ML: Search over all

– SD: Restrict search within a sphere of center s and radius R

• Complexity tradeoff in terms of choosing radius R

R

z~

Sphere (lattice) Decoder

zzzz

zzzyz

z

zz

~~minarg

~minargminarg22

HH

HHHML

nzy H

2~~ RHH zzzz

H: channel, n : AWGN

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Preprocessing for SD

• Triangularization in AWGN– QR Decomposition: a unitary matrix (Q) and an upper triangular matrix

• Triangularization in MUD – Noise Whitening

nzy

nznzy11

QRQ

QRH

New received vector

Still AWGN with equal variance

Channel Normalization

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+

1n

2n

1z

2z

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2222 nzry

nLAzyLy 1~

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• Layered/ Tree-based Decoding – Partial Euclidean Distance Accumulations by taking advantage of

channel triangularization

• Search Constraint: Radius or Best Candidates

Sphere Decoders

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,22,11,1 iiiiiiiL zlzlzlyzzCi

),...,,(....

),,(),()(

21

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321

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LLL

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Ld

K

zyz

16

Constrained SD Z1

C(z 1)

C(z

1,z

2)

C(z

1,z 2,

z 3)

Z2

Z3

1

1

1 1

1

11

0 0

0

0 0 00

d(z1,z2,z3)

• Depth First SD– Search the tree in downward and upward manner– Update the search radius after each pass

• Breadth First (K-best SD)– Search in downward direction only– K best candidates are retained at each level in the tree

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Performance Comparison

• 1000X reduction in complexity

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• SD limits search space• Relaxation increases search space by dropping certain constraints so that the search is

easier to implement• Unconstrained Relaxation (UR)

– Remove constraint on Alphabet

– Penalized UR:

Compare to MF, Decorrelator, MMSE

Relaxations and Heuristics

z

zzy z

z

z

J

HAz

K

Kopt

1,1

1,1

maxarg

2maxarg

yz yr

r rr

11 sgnˆ

maxarg

RAH

J

opt

optK

yz yr

rr r

11

r

2

2

sgnˆ

maxarg

ARAIH

J

opt

optK

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• Problem Setup:

• Semi-Definite Relaxation (SDR):

– Drop rank 1 constraint on X with X still symmetric positive semi definite:

– An efficient solution can be found in

Semi-Definite Relaxation

5.3K

1,.max

max1,1

iiK XtsQTrace

QK

,x xxX X

xx x

10.max iiXtsTrace ,X XQ

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• Approximate Boolean solution by randomization

– Randomize to approximate xi from vi

Semi-definite Relaxation (2)

K

i

K

jijji

KiX

QTrace

i 1 1,..1,12max

max

vv

X

v

K

i

K

jijji

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Q

i

K

1 1,..1,1

1,1

2max

max

xx

x KvvvV

VVX

,...,, 21

ii

i

xxx

uVsgnx -

u-

~~maxargˆ

~

1

,...,1Q

vector generate Randomly

M....,,ifor

randMiSDR

rand

21

SDR for MUD

SNR3=11dB

cA

AHc

HAccJJ

cz

cz

czz

K

KK

z

y

y z

zzyz z z

0max

2maxmaxmax

1,11,1

1,11,1

1,11,11,1

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• Problem Setup:

• PDA

– Order users in decreasing power

– Belief on the decision of user k at stage i

– Update this belief by treating MAI as AWGN:

– Stop when belief converges, Decide by comparing p to 0.5

Probabilistic Data Association

kl

klkllkkk wzazay

1)()(

ik

i zppkz

RNAApppayCN

pyzpp

oil

kl

il

il

kllk

kl

iiik

i

lk

*

)1,()()(

14,12~

,1

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Performance Comparison

Average BER with K=29 with gold codes

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Conclusions

• Multiuser Detection (MUD): canceling or suppressing interfering users from the desired signals

• Different techniques exist that trade-off complexity with performance

• Detection techniques can be applied to other detection problems (ex. MIMO)

• Viterbi Algorithm can be applied to MUD, How would low complexity “Viterbi algorithm” behave under MUD?