EM based Multiuser detection in Fading Multipath Environments

Mohammad Jaber Borran, Željko akareski,

Ahmad Khoshnevis, and Vishwas Sundaramurthy

• Motivation

• Time-frequency representation

• Channel modeling

Outline

• Expectation Maximization algorithm

• EM algorithm based detector

• Performance comparison

• Conclusions and future work

Outline(continued)

Environment

• Noise

• Multipath

• Fading

• MAI

Time-Frequency RepresentationWhat is TFR?

• A 2-D signal representation

• Facilitates signaling by exploiting multipath and Doppler

• Identifies Doppler as another dimension for diversity

T

mtj

ckmlk elTtsts

2

)()(

Canonical basis corresponding to the uniform grid

Tc

1/T

Multipath

Dop

pler

M

-M

Canonical Coordinates

, / cm TTL TBM d

L

Channel Modeling Requirements

• Multipath environment

• Independent paths

• Rayleigh fast fading

Channel ModelingOur approach

• Jakes’ model for individual paths

• Independence assured by having:

Spacing >> Tcoh ( ~ )

• Random delays for different multipath

components

• Canonical representation

dB1

Channel ModelingCharacterization

• Linear time-varying system

h(t, ) +s(t) x(t)

n(t)

r(t)

• Represented by its impulse response h(t, )

Channel ModelingCharacterization

• The output r(t) determined as :

)()(),()( tndtsthtr

• Incorporate the canonical model into h(t, )

m d

d

T B

B

tj ddtseHtx0

2 )( ),()(

Channel ModelingCharacterization

• Spreading function H(, )

• Canonical finite-dimensional representation :

, / cm TTL where TBM d

TttslTT

mH

T

Ttx

L

l

M

Mm

mlkc

ck

0 ,)(),()(1

0

Channel ModelingCharacterization

• Bandlimited approximation of H(, )

m d

d

T B

B

CTj

C

ddTsincTsinceHT

TH

0

)'( '')/)'(())'((),(),(ˆ

TBM d

• In our case

mN

ii

C

i

C T

mE

Tlsinc

T

tlmH

1

)()(),(ˆ

Channel ModelingCharacterization

where

TBM d

))(()( tEFFTE ii

Ei(t) : Jakes’ model rep. for path i

Tt 0

cc TT

11

EM AlgorithmIntroduction

• Goal:

K

R rf

11b

b

, s.t.

);(log maximize

)(),,,( 21 ygyyygr K

• K-dim problem, direct approach is difficult.

• Define complete data, i.e. y, such that

and

K

f

11b

byY

, s.t.

);(log maximize

ybybyby YYYY drffrfE RR );|();(log|);(log ||

EM AlgorithmIntroduction (cnt’d)

ybybybb YY drffU R )';|();(log)',( |

• Since y is unavailable,

• b is unknown,

• Provides an iterative method for ML estimation:– E step: Compute U(b,b(n))

– M step:

EM AlgorithmIterative Nature, Decomposition

),(maxarg )(

1,1

)1( nn UK

bbbb

• K 1-dim problems (with suitable complete data)

• The value of b(0) is important.

• The log-likelihood function

New Multiuser Detection SchemeComplete Data

T K

kkkR dttxbtrArf

0

2

12

)()(2

1);(log

b

K

kk

kkkk

tytr

Kktntxbty

1

)()(then

,...,1 ,)()()(

• Define complete data, y(t) = (y1(t), …, yK(t)), as

• Defining

New Multiuser Detection SchemeIterative Expression, Special Cases

K

kjj

jkjHj

njk

Hk

kkkHk

knkk

nk bbb

1

)()()1( )1(Resgn hRhzhhRh

2

2

k

k

kHkkb zhResgn)0( • Assuming

– k=1 Multistage– k=0 Time-Frequency RAKE receiver

TFRAKE

+MRC

r(t)

sgn I-b(0)

MAI Estimation &Cancellation

+ I- +sgn sgn

HHz

b(1) b(n-1) b(n)

...

...

...

New Multiuser Detection SchemeBlock Diagram

MAI Estimation &Cancellation

Simulation Results(3 paths, Bd=100Hz, 5 users, User 4)

-10 -5 0 5 10 1510

-4

10-3

10-2

10-1

100

User 4, beta = 0.7

SNR in dB

Bit

Err

or R

ate

TF RAKE Multi-Stage 2-stageEM 2-stage Multi-Stage 3-stageEM 3-stage

Simulation Results(3 paths, Bd=100Hz, 5 users, User 3)

-10 -5 0 5 10 1510

-4

10-3

10-2

10-1

100

User 3, beta = 0.8

SNR in dB

Bit

Err

or R

ate

TF RAKE Multi-Stage 2-stageEM 2-stage Multi-Stage 3-stageEM 3-stage

Conclusion

• Canonical representation + EM algorithm

New Detector for Fast Fading Multipath Env.

– Two special cases: TF RAKE and MultiStage

• Outperforms TF RAKE and MultiStage

• For rapid convergence use appropriate k

Future work

• Theoretical error probability analysis

• Near-Far resistance analysis

• Optimum value for k

• Extension to asynchronous case

That’s all Folks!

,)()(

21

22221

11211

*

kkkk

k

k

T

RRR

RRR

RRR

dtttR

ss

Cross correlation matrix

TMLk

LMk

Mk

MLk

Mkk

Tlkkl

tststststst

dtttR

)](),...,(),...,(),(),...,([)(

,)()(

)1(0)1(0

*

s

ss

where

Signal model

• The new log-likelihood function

• It can be shown that

New Multiuser Detection SchemeExpectation Calculation Step

K

k

T

kkkk

dttxbtyAf1 0

2

2)()(

2

1);(log

byY

K

k

K

jjkj

Hj

njk

Hk

kkkk

Hk

nk

k

kn bbb

U1 1

)(2

2)(

2)( Re),( hRhzhhRhbb

• The coordinates for each symbol of a particular user are computed by:

Kk

dtttr kk

..., ,1

)()( *

sz

a

Canonical RAKE

Simulation Results(3 paths, Bd=100Hz, 5 users, User 1)

-10 -5 0 5 10 1510

-4

10-3

10-2

10-1

100

User 1, beta = 0.6

SNR in dB

Bit

Err

or R

ate

TF RAKE Multi-Stage 2-stageEM 2-stage Multi-Stage 3-stageEM 3-stage

Simulation Results(3 paths, Bd=100Hz, 5 users, User 2)

-10 -5 0 5 10 1510

-3

10-2

10-1

100

User 2, beta = 0.6

SNR in dB

Bit

Err

or R

ate

TF RAKE Multi-Stage 2-stageEM 2-stage Multi-Stage 3-stageEM 3-stage

Simulation Results(3 paths, Bd=100Hz, 5 users, User 5)

-10 -5 0 5 10 1510

-4

10-3

10-2

10-1

100

User 5, beta = 0.8

SNR in dB

Bit

Err

or R

ate

TF RAKE Multi-Stage 2-stageEM 2-stage Multi-Stage 3-stageEM 3-stage

Simulation Results(3 paths, Bd=200Hz, 5 users, User 1)

-10 -5 0 5 10 1510

-3

10-2

10-1

100

User 1, beta = 0.6

SNR in dB

Bit

Err

or R

ate

TF RAKE Multi-Stage 2-stageEM 2-stage Multi-Stage 3-stageEM 3-stage

Simulation Results(3 paths, Bd=200Hz, 5 users, User 2)

-10 -5 0 5 10 1510

-3

10-2

10-1

100

User 2, beta = 0.6

SNR in dB

Bit

Err

or R

ate

TF RAKE Multi-Stage 2-stageEM 2-stage Multi-Stage 3-stageEM 3-stage

Simulation Results(3 paths, Bd=200Hz, 5 users, User 3)

-10 -5 0 5 10 1510

-4

10-3

10-2

10-1

100

User 3, beta = 0.8

SNR in dB

Bit

Err

or R

ate

TF RAKE Multi-Stage 2-stageEM 2-stage Multi-Stage 3-stageEM 3-stage

Simulation Results(3 paths, Bd=200Hz, 5 users, User 4)

-10 -5 0 5 10 1510

-3

10-2

10-1

100

User 4, beta = 0.7

SNR in dB

Bit

Err

or R

ate

TF RAKE Multi-Stage 2-stageEM 2-stage Multi-Stage 3-stageEM 3-stage

Simulation Results(3 paths, Bd=200Hz, 5 users, User 5)

-10 -5 0 5 10 1510

-3

10-2

10-1

100

User 5, beta = 0.8

SNR in dB

Bit

Err

or R

ate

TF RAKE Multi-Stage 2-stageEM 2-stage Multi-Stage 3-stageEM 3-stage

Channel Modeling

TBM d

),(ˆ HVisualization of