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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 - PowerPoint PPT Presentation
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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).