Mansoor Ahmed and Noor M. KhanDepartment of Electronic Engineering

Muhammad Ali Jinnah University Islamabad, Pakistan

mansur.abbasi@gmail.com, noor@ieee.org

Abstract—This paper deals with the estimation of rapidly time varying non isotropic Rayleigh fading channels in synchronous direct sequence spread spectrum (DS-CDMA) systems. The effects of non isotropic scattering can be captured by using Von Mises distribution for the angle of arrival (AOA) of the scattered waves at the receiver. Kalman Filter based on higher order autoregressive (AR) model is used for tracking and estimation of the non isotropic Rayleigh channel. The proposed algorithm works much better than AR(1) based algorithms in tracking time varying channel especially on high Doppler spreads in directional scenarios. However, it is observed, the computational cost of the proposed algorithm increases when the AR model order increases.

Index Terms—CDMA, Non isotropic Rayleigh fading, Autoregressive models, Kalman filters.

I. INTRODUCTION

The use of spread spectrum multiple access schemes is well established in the wireless communication arena. In particular, DS-CDMA has been widely studied in the wireless literature and has been implemented in several commercial systems as well [1]. DS CDMA is based on the principle of Multi-User Detection (MUD), as various users transmit their information simultaneously on a common channel utilizing the complete available bandwidth resource of the system, each using a unique code sequence, known as signatures. If the signatures remain orthogonal, than the receiver is able to optimally demodulate the received signal by using a bank of matched filters followed by threshold detectors. However, at high vehicle speeds the channel conditions change significantly and the signatures of different users get highly correlated. This severely degrades the performance of the system. To combat this dynamic nature of the system, state space approach has been proposed as a good technique [2]. Autoregressive models have been used to accurately predict fading channel dynamics with Kalman filter based channel estimation [3]. In [4] authors have used higher order AR model to estimate MC-CDMAfading channels based on Kalman filtering. In [5] the problem of channel tracking for multi-input multi-output (MIMO) time-varying frequency-selective channels is addressed by using AR(1) and AR(2) models.

All of the above mentioned papers assumed isotropicscattering, while the scattering encountered in many practical environments is non isotropic [6], [7]. For isotropic scattering, the probability of AOA at the receiver is assumed to be uniform. In [8] authors have used Von Mises distribution for AOA at the receiver and showed that, Von Mises distribution can be used to accurately model the effects of directionality.

In this paper, we consider the estimation of rapidly time varying DS-CDMA channels in non isotropic Rayleigh fading based on Kalman filtering. Moreover, we have also used higher order AR model in Kalman filter based channelestimator.

The remainder of the paper is organized as follows. In section II we present proposed system model and thedirectional channel model. In section III Kalman filter based Channel estimation algorithm is introduced. Simulation results are presented in section IV, while section V concludes the paper.

II. SYSTEM MODEL

A downlink synchronous DS-CDMA system with binary Phase shift Keying (BPSK) modulation is considered with K users and processing gain N, the received vector is given by [2]

)()()()(1

inihsibAirK

kkkkk +=∑

=

(1)

where )(ibk is the ith symbol transmitted by the kth user

with 1])([2

=ibE k , ks is the signature vector and kA is the

amplitude of the kth user. )(ihk is the complex, non isotropic

Rayleigh fading process and )(in is the white Gaussian noise

vector with covariance 2

nσ .

A. Channel Model A Rayleigh characterization of the land mobile radio

channel follows from the Gaussian WSS uncorrelated scattering fading model, where the fading process is modeled as a complex Gaussian process. In this model, the variability

Model Based Approach for Time-Varying Channel Estimation in DS-CDMA Systems Using Non-

Isotropic Scattering Environment

2009 International Conference on Emerging Technologies

978-1-4244-5632-1/09/$26.00 ©2009 IEEE 62

of the wireless channel over time is reflected in its autocorrelation function (ACF). This second order statistic generally depends on the propagation geometry, the velocity and the antenna characteristics [9]. The theoretical PSD associated with either in phase or quadrature portion of the

fading process )(ihk is band limited and U-shaped as

( )⎪⎩

⎪⎨

⎧≤

−=

elsewhere

ffff

d

ff

dhhd

0

,1

1

)( 2πψ (2)

The corresponding normalized (zero variance) discrete time autocorrelation function hence satisfies

)2()( 0 iTfJiR bdhh π= (3)

where (.)0J denotes the zero-order Bessel function and

bdTf denotes the Doppler rate. In the above case a common

assumption is that the propagation path consists of two dimensional isotropic scattering with a vertical monopole antenna at the receiver so the probability of angle of arrival

(AOA) is uniformly distributed on ( )ππ ,− .

However, in order to capture the effects of directional

scattering on )(iRhh , we use the Von Mises distribution for

the probability angle of arrival [8]:

( )ππθκπ

θ µθκ ,,)(2

1)( )cos(

0

−∈= −eI

p

(4)

where )(κoI is the 0th order modified Bessel function,

µ represents the mean direction of the AOA, and κ controls

the beamwidth.

For the AOA given in (4) the corresponding autocorrelation function is given by:

( )( )κ

πµκπκ

0

220 )cos(4)2(

)(I

iTfjiTfIiR

bdbd

hh

+−=

(5) Autoregressive models can be used for the computer

simulation of correlated Rayleigh fading processes [9]. The complex AR process of order p [AR(p)] can be generated via the time domain recursion

∑=

+−−=p

kk ivkihih

1

)()()( α (6)

where )(iv is a zero mean, complex white Gaussian noise

process with uncorrelated real and imaginary components with

variance 2

vσ and p1,2,...,k } { =kα are the AR model

parameters. The relationship between the AR parameters and the fading

process ACF is given by

1,)()(1

≥−−= ∑=

kmkRkRp

mhhmhh α (7)

In matrix form (11) can be represented as

vR −=αr

hh (8)

or

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

+−

+−

)(

)2(

)1(

)0()2()1(

)2()0()1(

)1()1()0(

2

1

pR

R

R

RpRpR

pRRR

pRRR

hh

hh

hh

phhhhhh

hhhhhh

hhhhhh

MM

K

MOMM

K

K

α

α

α

and variance of driving process will become

∑=

−+=p

khhkhhp kRR

1

2 )()0( ασ (9)

By using )(iRhh given in (5), the AR parameters can be

obtained by solving the Yule Walker equations, i.e.

vR 1−−= hhαr

.

Due to band limited nature of Doppler spectrum, the YWE suffer from ill conditioning for all but very small AR model orders. This problem can be solved by using the method given by [9].

Figure 1: Von Mises probability density function (pdf) in polar

coordinates ( )0=µ .

B. Receiver Model The receiver consists of bank of matched filters followed by

MMSE detector. The output of matched filter for user k is given by [10]

∑≠

++=kj

kjjjkkkk inihibAihibAiy )()()()()()( (10)

where kk sinn )(= with variance 2

nσ as ks is normalize to

have unit energy. At this stage the output of matched filters is processed by

the MMSE detector, which eliminates the multiple access

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interference caused by other users. The MMSE detector can be written in following form for user k.

( )∑≠

−+=

kjkkjk sIRw

12σ (11)

where [ ] [ ]kH

k ssssssR ...... 2121= is the

cross correlation matrix of the spreading vectors and ijR ]1−

denotes the thji ),( element of the inverse of the matrix R .

The output MMSE detector is given by

)()()()(

),()(

iihibAiz

iywiz

kkk

kH

k

ξ+=

= (12)

where )(iξ is a zero mean Gaussian noise. The fading

process is then estimated and used to make the decision about the desired user data symbol

( )( ))()(Resgn)(ˆ izihib kk∗

= (13)

Since the fading process )(ihk is unknown, we propose their

estimation using Kalman filter with higher order AR models in the following section.

III. KALMAN FILTER BASED CHANNEL TRACKING

For estimating the fading process )(ihk , by using AR(p)

process and selecting the state vector

[ ]Tpihihihi )1()1()()( +−−= Kh , we obtain the

state equation as [4] )()()1( igvii +Ξ=+ hh (14)

where

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=Ξ

010

00121

K

MO

K

K pααα

,

and [ ]Tg 001 K= having dimension 1×p . Using

(12) the measurement equation is

)()()()( iiiiz T ξ+= hb (15)

where [ ]TibAi 001)()( 11 K=b .

The implementation of our algorithm starts with a training

mode that is used to acquire initial )(ihk estimates, after

which it reverts to decision directed mode. In the training mode, the receiver knows the transmitted symbols, whereas in the decision-directed mode, the decoded symbols

)(1̂ ib replace the information symbols )(1 ib .We will focus on

the decision-directed mode and assume that initial channel estimates are available.

Given the state space representation of system, we summarize our algorithm for the channel tracking in the following steps.

1. Obtain )1/1(h and )1/1(P from training.

2. Obtain )(1̂ ib using (13)

3. Obtain predictions )()/()/1( igviiii +Ξ=+ hh (16)

Tv

H ggiiii 2)/()/1( σ+ΞΞ=+ PP (17)

4. Obtain innovation process and its variance

)/1()()()( iiiizie T +−= hb (18) 2)()/1()()( ξσ++= iiiiiC T bPb (19)

5. Obtain Kalman Gain as

)()()/1()( 1 iCiiii −+Ξ= bPK (20)

6. Update the state vector and error covariance matrix)()()/1()1/1( ieiiiii Khh ++Ξ=++ (21)

)/1()()()()/1()/1()1/1( 1 iiiiCiiiiiii T ++−+=++ − PbbPPP(22)

The fading process will be estimated as

)1/1()1/1( ++=++ iigiih Th (23)

7. Use )1/1( ++ iih for the detection of )(1̂ ib and

repeat steps 3 to 7 for 1+i . It should be noted that during the training period the state

vector and error covariance matrix are initially assigned to zero vector and identity matrix respectively, i.e. 0h =)0/0(

and pIP =)0/0( .

IV. SIMULATION RESULTS

In this section, we present the computer simulation results to illustrate the BER performance of the DS CDMA system with Kalman channel estimation using AR models of different order, under directional channel model.

These results were obtained using Monte-Carlo simulations. We considered a system with K = 4 multiple access active users, with equal power and orthogonal signature sequences. The length of signature sequences is 16. The non isotropic fading coefficients are generated in each trial using AR(50) model as described by [9]. Each user generates BPSKmodulated bit streams of data. The carrier frequency is taken as 2.4GHz and the mobile speed is considered as 90km/hr. So

the Doppler Spread will be 200Hz. The Doppler rate, Tfd , is

considered to be 0.05. Fig. 2 shows the BER performance of the proposed system using Kalman filter estimator and LMS estimator in non isotropic environment i.e. 2=κ . As shown in Fig. 1, for 2=κ highly non isotropic environment is encountered. The results show that Kalman filter based estimator can also be used to track the directional channels. From these figures one can notice that Kalman filter based estimator provides much better estimation than the LMS based one. In addition, increasing the order of AR model significantly increase the BER performance of the system.

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In decision directed mode there exists, a possibility of error propagation. By using higher order AR model we can make the probability of error propagation smaller. Also by periodically retraining the system the effect of error propagation can be reduced.

Figure 2: BER performance of the DS CDMA system in non isotropic fading environment with k=2.

V. CONCLUSIONS

This paper presents a method for accurately tracking directional Rayleigh fading channel using Kalman filter. This method is based on the calculation of higher order AR models with statistics closely matching those of the directional Rayleigh fading process. Simulation results show that Kalman filter with higher order AR models can accurately track the directional radio channel. It is also shown that Kalman filter higher order AR model gives much better performance than low order Kalman filters. However the computational cost of

the above Kalman estimation algorithm )( 3pO increases

much when the AR model order increases. Thus a compromise has to be found. Also, as the Doppler spread increases, higher order AR model based Kalman filters performs much better than low order Kalman estimation algorithm. Estimation algorithm based on AR(10) model is recommended, especially for higher Doppler rate scenarios.

REFERENCES

[1] R. L. Freeman, “Fundamentals of telecommunication”, 2nd ed. Hoboken, NJ: John Wiley & Sons, 2005.

[2] N. M. Khan and P. B. Rapajic, “Use of statespace approach and kalman filter estimation in channel modeling for multiuser detection in time-varying environment”, in Proc. International Workshop Ultra-Wideband Sys. (IWUWBS-2003), Oulu, Finland, June 2003, no. 1052.

[3]. H. Wu and A. Duel-Hallen, “Multiuser detectors with disjoint Kalman channel estimators for synchronous CDMA mobile radio channels,” IEEE Trans. Commun., vol. 48, no. 5, pp. 752–756, May 2000.

[4]. W. Hassasneh, A. Jamoos, E. Grivel and H. Abdel Nour, "Estimation of MC-DS-CDMA fading channels based on kalman filtering with high order autoregressive models," in Proc. 1st Mobile Comput. and Wireless Commun. International Conf. (IEEE MCWC2006), Amman, Jordan, September, 2006, pp. 145-149.

[5] C. Komninakis, C. Fragouli, A. H. Syed and R. D. Wesel, “Multi-Input Multi Output fading channel tracking and equalization using Kalman

estimation,” IEEE Trans. On Signal Processing, vol. 53, no. 5, pp. 1065-1076, May 2002.

[6] P. D. Teal, T. D. Abhayapala, and R. A. Kennedy, “Spatial correlation for general distributions of scatterers," IEEE Signal Processing Lett., vol. 9, no. 10, pp. 305-308, October 2002.

[7] N. M. Khan, M. T. Simsim and P. B. Rapajic, “A generalized model for the spatial characteristics of the cellular mobile channel", IEEE Trans. Veh. Technol., vol. 57, no. 1, pp. 22 – 37, January 2008.

[8]. A. Abdi, J. Barger, and M. Kaveh, “A parametric model for the distribution of the angle of arrival and the associated correlation function and power spectrum at the mobile station,” IEEE Trans. Veh. Technol., vol. 51, no. 1, pp. 425–434, May 2002.

[9]. K. E. Baddour and N. C. Beaulieu, “Autoregressive modeling for fading channel simulation,” IEEE Trans. On Wireless Commun., vol. 4, no. 4, pp. 1650-1662, July 2005.

[10]. S. Verdu, “Multiuser Detection”, Cambridge, U.K., Cambridge University Press, 1998.

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