« Particle Filtering for Joint Data-Channel Estimation in Fast 

Fading Channels »

Tanya BERTOZZI

Didier Le Ruyet, Gilles Rigal and Han Vu-Thien

2

Outline   Problem statement

   Classical solutions to the problem: Why the PF (Particle Filtering)?

   Joint data-channel estimation applying the PF        Performance and computational complexity comparison      between the PF and the classical solutions

   Discussion:  When is it interesting to use the PF in digital

communications?   

Conclusion

3

Problem statementkn

kbMOD CHANNEL DEMOD DETECTOR

kb̂kr

bipodal modulation { 1}

i.i.d. bits organized into frames

Preamble Information bits Tail

Transmitted Signal Model:

4

kkk nFBr

1x2 1x(L+1) (L+1)x2 1x2

Received Signal Model:

Symbol-spaced FIR filter

L

llkf 0,

TT

Channel model:

Multipath fading channel

Problem Statement

5

Purpose of the receiver

Estimation of the transmitted sequence in the presence of an unknown channel

Classical MLSE solutions

Slow fading channels ( ):1.0BDTf

Channel Estimation

DataEstimation

Training sequence: LMS, RLS, Kalman filter

Classical solutions:Slow fading

6

Data Estimation: Discrete state space model

Complexity reduction solutions:

From one iteration to the next one, it retains only the M best paths, with M less than the total number of states.

M algorithm(Anderson and Mohan, 1984)

From one iteration to the next one, it retains a variable number of paths depending on T:

T algorithm(Simmons, 1990)

TLLbest

Viterbi algorithm

Optimal MLSE solution if the channel coefficients are known

Computational complexity L2

Classical solutions:Slow fading

7

The memory of the states in the Viterbi trellis is less than L and the terms of residual ISI are corrected along the survivor paths leading to each state.

PSP algorithm(Duel-Hallen and Heegard, 1989)

Fast fading channels ( ):1.0BDTf

Joint Data-Channel

Estimation

PSP approach:(Raheli and Polydoros, 1993)

Data-aided estimation of the channel(one estimation of the channel coefficients for each survivor path in the trellis)

Classical solutions:Fast fading

8

Data Estimation

Viterbi algorithm

Complexity reduction algorithms:

M algorithm T algorithm PSP algorithm

Data-aided Channel Estimation

LMS algorithm

RLS algorithm

Kalman filter algorithm

Better trade-off between Computational complexity – Performance:

Particle Filtering?

Classical solutions:Fast fadingPSP approach:

9

Joint data-channel estimation applying the Particle Filtering

MLSE Detector:

FRBpB KK

B

K

K

ˆ,maxargˆ111

1

Optimal solution

Viterbi algorithm

Data estimation: Estimation of the Posterior Probability Density (PPD) in a discrete state space

Particle Filtering

Suboptimal solution

Approximation of the PPD with particles

Exploration of a subset of the possible paths using the SISR algorithm

Complexity reduction algorithm

10

Observation model:

kkk nFBr

Lkkk bbb 1

Each state is represented by the L previous information bits because of the channel memory

State sequence:

Observations:

KkbB kK ,,1;1

KkrR kK ,,1;1

Initial distribution of the particles:

pi NiB ,,1;)(0 , where:

LbbB ,,10 L last bits of the preamble

Particle filtering:Joint data-channel estimation

11

Selection of the importance function:

Minimization of the variance of the importance weights , in order to limit degeneracy of the algorithm

ikk

ikkk

iik

kk FBrbpFBRb 111:01

ˆ,,ˆ,,

Particle filtering:Joint data-channel estimation

At time k-1, several particles are in the same position in the state space.

At time k, only two values are possible for : +1 and –1.

kb

The particles divide in two parts proportionally

to the importance function

Evolution of the particles in a discrete state space:

12

Tree-search algorithm

+1

-1

+1

+1

-1

-1

+1

+1

+1

-1

-1

-1The positions of the particles in the state space are seen as groups of particles.

Particle filtering:Joint data-channel estimation

13

The channel model

kkk WFF 1

L

00

00

001

0

Constant channel:

999.0i No a priori knowledge of the speed of the channel

variations:

1i

Particle filtering:Joint data-channel estimation

14

The channel estimation

Along each trajectory in the state space the channel is estimated by a Kalman filter.

I ) Prediction phase:

111ˆˆ

kkkk FF

QPP kkkk 111

~~

II ) Correction phase:

11ˆˆˆ

kkkkkkkk FBrGFF

11

~~~ kkkkkkk PBGPP

kkkk RFEF 1

ˆ

Estimate at time k

kkP~

Covariance of

kkkkk FFF ˆ~

Particle filtering:Joint data-channel estimation

15

Calculation of the importance function

1 / 2Bayes

ikk

ikkk FBrbp 11

ˆ,,1

kikk

ikk

ikk

ik

ikk

ikk

ikk

ik

bdFBbpFBrp

FBbpFBrp

111

111

ˆ,ˆ,

ˆ,1ˆ,

i

kki

kikk

ik

ikk

ik

FBrpFBrp

FBrp

11

1

ˆ,ˆ,

ˆ,

21

~n

Tiikk

i BPB

GaussianMean:

Variance:

ikk

i FB 1ˆ

Particle filtering:Joint data-channel estimation

16

Calculation of the importance weights

ikk

ikk

ik

ik FBrpww 11

*1

* ˆ,

ikk

ik

ikk

ik FBrpFBrp 11

ˆ,ˆ,

Normalisation of the importance weights

pN

j

jk

iki

k

w

ww

1

*

*~

Particle filtering:Joint data-channel estimation

17

Resampling

I ) Periodic every L bits:

II ) Uniformly according to the importance weights:

The particles with a weight < T are moved in the group with maximum weight.

thresN

i

ik

eff Nw

Np

1

2~1ˆIf the particles are distributed

uniformly according to the importance weights.

Particle filtering:Joint data-channel estimation

18

Alternative scheme(E. Punskaya, A. Doucet, W.J. Fitzgerald, EUSIPCO, September 2002)

+1

-1+1

-1

-1

-1

-1

+1

+1

+1

k-1 k k+1

+1

+1

+1

+1

+1

-1

-1

-1

-1

-1

At each time only the best M particles are retained

close to the M algorithm

Particle filtering:Joint data-channel estimation

19

Simulation results GSM system: the receiver detects only one slot for each

TDMA frame;

Preamble: 26 known bits for the channel initialisation; Information bits: 58;

First channel model:

07.0,14.0,21.0,28.0,35.0,42.0,49.0,56.070 aa

80,70,60,50,40,30,20,107,0, dd ff

memory L = 7;

sldllk Tkfaf ,, 2cosRe

sldllk Tkfaf ,, 2sinIm

Second channel model: HT240

20

Comparison PSP-Particle filteringFirst channel model: FER versus Eb/No

Simulation results

PSP: 8 states

PF: 8 particles

21

First channel model: Complexity versus Eb/No

Simulation results

PF

PSP

22

HT240: FER versus Eb/No

Simulation results

23

HT240: Complexity versus Eb/No

Simulation results

24

Comparison M-T-Particle filteringFirst channel model: FER versus Eb/No

Simulation results

M and T

PF

25

First channel model: Complexity versus Eb/No

Simulation results

M

T

PF

26

Preliminary conclusionIf the state space is discrete, the particle filtering technique is equivalent to the classical solutions.

When is it interesting to use the particle filteringin digital communications?

Joint estimation of discrete and continuous parameters

Example: Joint delay-channel-data estimation in DS-CDMA systems.

(The paper of Punskaya, Doucet and Fitzgerald reaches the same conclusion)

27

Joint delay-channel estimation in a DS-CDMA system

Data sequence: 1nd

Spreading sequence: 1kc

Chip duration: cT

Received signal:

,0

,

L

lklklkk nsfr

RX

cT2/1

LPF

cT/1

kr

kslk Tlkss

28

State model:

Channel lklkllk wff ,,1,

Delay kkk w 1Nearly constant channel coefficients and constant delay:

001.0,999.0 2 ll

001.0,999.0 2

Channel estimation

Delay estimation

Kalman filter

SISR algorithm

DS-CDMA:Joint delay-channel estimation

29

SISR algorithm for the delay estimation

Initial distribution of the particles:

Selection of the importance function: i

kkiki

kk pFr 111:1ˆ,,

uplow BB ,uniformly between

Calculation of the importance weights:

ikk

ikk

ik

ik Frpww 1

*1

* ˆ, Resampling:

uniformly according to the importance weights if

thresN

i

ik

eff Nw

Np

1

2~1ˆ

DS-CDMA:Joint delay-channel estimation

30

Simulation results

Time

31

Simulation results

Time

32

ConclusionPossible applications of the PF in digital communications:

Discrete state space equivalent to the classical solutions

(M and T algorithms)

More interesting:PF for the joint estimation of discrete

and continuous parameters

Example: Joint delay-channel estimation in a DS-CDMA system

The first results are encouraging; this approach can give better performance than the classical solutions.