An Application Of The Divided Difference Filter to Multipath Channel Estimation in CDMA

Networks

Zahid Ali, Mohammad Deriche,

M. Andan Landolsi

King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

OUTLINE

Introduction Overview of DDF Algorithm Channel and Signal Model Application to Channel Estimation with

Multipath/ Multiuser model Simulation Results Conclusion

INTRODUCTION

Accurate channel parameter estimation for CDMA signals is challenging due to

Multipath fading Multiple Access interference (MAI)

Especially • Under near far environment• closely spaced multipath

CDMA multiuser parameter is the problem of estimating the states of a system given a set of noisy or incomplete measurements

INTRODUCTION

Advanced Signal Proc. techniques such asMaximum-LikelihoodJoint multiuser detection and

parametric channel estimation approaches

Subspace-based approachKalman filter framework

Kalman Filtering frameworkExtended Kalman Filter (EKF) for

nonlinear estimation and filteringSome Limitations of EKF

• First order terms of the Taylor series expansion

• Linearized approximation can be sometimes poor undermining the performance

• Jacobian matrix must exist

Divided Difference FilterDDF, unlike EKF, is a Sigma Point

Filter (SPF) where the filter linearizes the nonlinear dynamic and measurement functions by using an interpolation formula through systematically chosen sigma points.

DDF consistantly outperforms EKF.No analytic Jacobians or Hessians are

calculated.But DDF has same order of

computational complexity as the EKF

Channel and Signal Model Asynchronous CDMA system model where K users transmit over an M-path fading channel. The received baseband signal

, , ,1 1

( ) ( ) ( ( )) ( )l

K M

k i k m k l b k ik i

r l c l d a l mT l n l

, ( )k ic l complex channel coefficients

, lk md mth symbol transmitted by the kth user

( )ka l spreading waveform used by the kth user

, ( )k i l time delay associated with the ith path of the kth user

( )n l Additive White Gaussian Noise (AWGN) of zero mean and variance

2

State-Space Model Representation Unknown channel parameters (path delays

and gains) to be estimated are

2 1KM of[ ]x c;τ

11 12 1 21 2 1[ , ,..., , ,..., ,..., ,..., ]TM M K KMc c c c c c c c

with

11 12 1 21 2 1[ , ,..., , ,..., ,..., ,..., ]TM M K KM Dynamic Channel Model

( 1) ( ) ( )c cc l c l l F v

( 1) ( ) ( )l l l F v

The scalar measurement model

( ) ( ( )) ( )z l h l l x

, , ,1 1

( ( )) ( ) ( ( ))l

K M

k i k m k l b k ik i

h l c l d a l mT l

x

is a nonlinear function of the state ( )z l

DDF Algorithm

Consider a nonlinear function , with mean and covariance . If the function is analytic, then the multi-dimensional Taylor series expansion of a random variable about the mean is given by the following

)y = h(x x

xxP h

2 3 41 1 1) D D D D

2! 3! 4!x x x x y h(x x) h(x + h + h + h + h + .. .

1. Initialization Step:

ˆ ˆ ˆ, ( )( )Tk k k k k k kE E x x P x x x x

2. Square Cholesky factorizations

0T x xP S S

Tk w wQ S S

T v vR S S

(2)ˆ , ,

(2), ,

1ˆ ˆ ˆ( 1) ( , ) ( , ) 2 ( , )

2

1ˆ ˆ ˆ( 1) ( , ) ( , ) 2 ( , )

2

xx i k x j k i k x j k i k k

xw i k k w j i k k w j i k k

k h h

k h h

S f x s w f x s w f x w

S f x w s f x w s f x w

3.State and covariance Propagation:

1

, ,1

, ,1

( )ˆ ˆ( , )

1ˆ ˆ( , ) ( , )

2

1ˆ ˆ( , ) ( , )

2

x

x

x wk k k

n

k s p k i k s j kp

n

k k w p i k k s pp

n n

h h

h h

x f x w

f x s w f x s w

f x w s f x w s

(1) (1) (2) (2)ˆ ˆ( 1) ( 1) ( 1) ( 1) ( 1)xx xw xx xwk k k k k

-xS S S S S

(1) (1) (2) (2)ˆ ˆ( 1) ( 1) ( 1) ( 1) ( 1)

T

xx xw xx xwk k k k k -xS S S S S

1 ( 1)( ( 1))Tk k k - -

x xP S S4. Observation and Innovation Covariance

Propagation

1 1 1

1 , 1 1 , 11

1 1 , 1 1 ,1

1

( )ˆ ˆ( , )

1ˆ ˆ( , ) ( , )

2

1ˆ ˆ( , ) ( , )

2

( 1) ( 1)

x

x

x vk k k

n

k x p k k x p kp

n

k k v p k k v pp

vv Tk v v

n n

h h

h h

k k

y h x v

h x s v h x s v

h x v s h x v s

P S S

(1) (1)ˆ ˆ1 ( 1) ( 1)

Txyk x yxk k P S S

5. Update 1

1 1 1( )xy vvk k k

P P 1 1 1 1 1ˆ ˆ ˆk k k k k

x x y y

1 1 1 1 1vv T

k k k k k P P P

Application to Channel Estimation with Multipath/ Multiuser model

No. of users = 2, 5, 10 No of paths = 2 and 3 Near far ratio = 20 dB

Timing epoch estimation

100 200 300 400 500 600 700 800 900 100013

13.5

14

14.5

15

15.5

16

Number of Samples

Del

ay E

stim

atio

n-ch

ips

Timing poch of 1st arriving path

Estimated delay weaker user

Estimated delay stronger userTrue values

Timing epoch estimation for first arriving path with a five-user/ three-path channel model (with 1/2-chip path separation)

Timing epoch estimation

100 200 300 400 500 600 700 800 900 100013

13.5

14

14.5

15

15.5

16

Number of Samples

Del

ay E

stim

atio

n-ch

ips

Timing poch of 2nd arriving path

Estimated delay weaker user

Estimated delay stronger userTrue values

Timing epoch estimation for second arriving path with a five-user/ three-path channel model (with 1/2-chip path separation)

Timing epoch estimation

100 200 300 400 500 600 700 800 900 100013

13.5

14

14.5

15

15.5

16

Number of Samples

Del

ay E

stim

atio

n-ch

ips

Timing poch of 3rd arriving path

Estimated delay weaker user

Estimated delay stronger userTrue values

Timing epoch estimation for third arriving path with a five-user/ three-path channel model (with 1/2-chip path separation)

Channel Coefficients

50 100 150 200 250 300 350 400 450 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Samples

MS

E

MSE of the channel coefficients for first arriving path with a ten-user/ two-path channel model

100 200 300 400 500 600 700 800 900 100013

13.5

14

14.5

15

15.5

16

Number os samples

Del

ay E

stim

atio

n-ch

ips

Comparison of DDF with EKF

true value

EKFDDF

DDF vs. EKF

UKF vs. DDF

50 100 150 200 250 300 350 400 450 500 550 6000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of Samples

MS

EDDF

UKF

CONCLUSION

•DDF achieves better performance

•moderate complexity compared to the (linearized) EKF

•DDF is quite robust vis-a-vis near-far multiple-access interference

•Can be applied to track a given signal epoch even in the presence of other closely-spaced multipaths (within a fraction of a chip).