Sigma Point Kalman Filters for Multipath ChannelEstimation in CDMA Networks

Zahid Ali, Mohammed A. Deriche, Mohamed Adnan Landolsi

Electrical Engineering DepartmentKing Fahd University of Petroleum and Minerals

P.O.Box 8466Dhahran 31261 Saudi Arabiazalikhan@kfupm.edu.sa

Abstract— The paper investigates time delay and channelgain estimation for multipath fading Code Division MultipleAccess (CDMA) signals using the Sigma Point Kalman Filters(SPKF), namely Unscented Kalman Filter (UKF) and secondorder Divided Difference Filter (DDF). Given the nonlineardependency of the channel parameters on the received signals inmultiuser/multipath scenarios, we show that the SPKF achievebetter performance than its linear counterparts. This derivative-free Kalman filtering approach, avoids the errors associatedwith linearization in the conventional Extended Kalman Filter(EKF). The numerical results also show that the proposed SPKFalgorithms are simpler to implement, and more resilient to near-far interference in CDMA networks.

I. INTRODUCTION

Multiuser parameter estimation has an area of active re-search in the signal processing field. Time delay estimationtechniques are used in numerous applications such a radi-olocation, radar, sonar, seismology, geophysics, ultrasonic, tomention a few. In most applications, the estimated parametersare fed into subsequent processing blocks of communicationsystems to detect, identify, and locate radiating sources. Direct-sequence code-division multiple-access technology includeshigher bandwidth efficiency which translates into capacityincreases, speech privacy, immunity to multipath fading andinterference, and universal frequency reuse [1,2], over existingand other proposed technologies make it a popular choice.As with all cellular systems, CDMA suffers from multiple-access interference (MAI). In CDMA, however, the effectsof the MAI are more considerable since the frequency bandis being shared by all the users who are separated by theuse of the distinct pseudonoise (PN) spreading codes. ThesePN codes of the different users are non-orthogonal givingrise to the interference, which is considered to be the mainfactor limiting the capacity in DS-CDMA systems. Accuratechannel parameter estimation for CDMA signals impaired bymultipath fading and multi-user interference (MAI) is an activeresearch field that continues to draw attention in the CDMAliterature. In particular, the joint estimation of the arrivingmulti-path time delays and corresponding channel tap gainsfor closely-spaced (within a chip interval) delay profiles isquite challenging, and has led the development of severaljoint multiuser parameter estimators, e.g., [3,4]. These havebeen extended to the case of multipath channels with constant

channel taps and constant or slowly varying time delays [7].An attempt at extending subspace methods to tracking timedelays was given in [6], On the other hand, time delay trackersbased on the delay lock loop (DLL) combined with inter-ference cancellation techniques have also been developed formulti-user cases [10]. Near-far resistant time delay estimatorsare not only critical for accurate multi-user data detection,but also as a supporting technology for time-of-arrival basedradiolocation applications in CDMA cellular networks [5,8-10]. The maximum-likelihood-based technique has beenemployed in [11], and [12] for single-user channel and/ormultiuser channel estimation with training symbols or pilots.The Kalman filter framework based methods were consideredin [10, 13-16], where these methods have been applied toparameter estimations. Many of the algorithms presented inprevious work have focused on single-user and/or single-pathpropagation models. However, in practice, the arriving signaltypically consists of several epochs from different users, and itbecomes therefore necessary to consider multi-user/multi-pathchannel models. In this paper, we present a joint estimationalgorithm for channel coefficients and time delays in a multi-path CDMA environment using a non-linear filtering approachbased on the Sigma Point Kalman Filters (SPKF), with aparticular emphasis on closely spaced paths in a multipathfading channel. The rest of the paper is organized as follows.In Section II, the signal and channel models are presented.Section III provides a description of the nonlinear filteringmethod used for multiuser parameter estimation that utilizesUnscented Kalman Filter (UKF) and second order DividedDifference (DDF). Section IV describes computer simulationand performance discussion followed by the conclusion.

II. CHANNEL AND SIGNAL MODEL

We consider a typical asynchronous CDMA system modelwhere K users transmit over an M-path fading channel. Thereceived baseband signal sampled at t = lTs is given by

r(l) =K∑

k=1

M∑i=1

ck,i(l)dk,mlak(l −mlTb − τk,i(l)) + n(l) (1)

where ck,i(l) represents the complex channel coefficients asso-ciated with the ith path of the kth user, dk,mk(l) is the mth sym-bol transmitted by the kth user, mk(l) = �(lTs − τk,i(l)/Tb�,

978-1-4244-3584-5/09/$25.00 © 2009 IEEE ISWCS 2009423

Tb is the symbol interval, ak(l) is the spreading waveformused by the kth user, τk,i(l) is the time delay of the ith pathof the kth user, and n(l) represents Additive White GaussianNoise (AWGN) assumed to have a zero mean and varianceσ2 = E[|n(l)|2] = N0/Ts where Ts is the sampling time.The state-space model representation with unknown channelparameters (path delays and gains) to be estimated are givenby the following 2KM × 1 vector [12],

x = [c ; τ ] (2)

with c = [c11, c12, ..., c1M , c21, ..., c2M , ..., cK1, ..., cKM ]T

and τ = [τ11, τ12, ..., τ1M , τ21, ..., τ2M , ..., τK1, ..., τKM ]T

The hidden system state xk evolves over time as an indirector partially observed first order Markov process. Thereforethe complex-valued channel amplitudes and real-valued timedelays of the K users are assumed to obey a Gauss-Markovdynamic channel model [6], i.e.

c(l + 1) = Fcc(l) + vc(l) (3)

τ(l + 1) = Fττ(l) + vτ (l) (4)

where Fc and Fτ are KM × KM state transition matricesfor the amplitudes and time delays respectively whereas vc(l)and vτ (l) are KM×1 mutually independent Gaussian randomvectors with zero mean and covariance given by

E{vc(i)vTc (j)} = δijQc,

E{vτ (i)vTτ (j)} = δijQτ ,

E{vc(i)vTτ (j)} = 0 ∀ i, j

with Qc = σ2c I and Qτ = σ2

τ I are the covariance matrices ofthe process noise vc and vτ respectively, and δij is the two-dimensional Kronecker delta function equal to 1 for i = j,and 0 otherwise. Using (1-2), the state model can be writtenas

x(l + 1) = Fx(l) + v(l) (5)

where F =[

Fc 00 Fτ

]are 2KM × 2KM state transition

matrix,v(l) is 2KM × 1 process noise vector with zero

mean and covariance matrix Q =[

Qc 00 Qτ

]. The scalar

measurement model follows from the received signal of (1)by

z(l) = h(x(l)) + n(l) (6)

The scalar measurement z(l) is a nonlinear function of the statex(l). The optimal estimate of x(l) is x(l|l) = E{x(l)|Zl},with the estimation error covariance

P = E{

[x(l) − x(l|l)] [x(l) − x(l|l)]T |Zl}

where Zl denotes the set of received samples up to time l,{z(l), z(l − 1), . . . , z(0)}.

III. PARAMETER ESTIMATION WITH KALMAN FILTERS

For the nonlinear dynamic system model such as above, theconventional Kalman algorithm can be invoked to obtain theparameter estimates [17, 18]. The most well known applicationof the Kalman filter framework to nonlinear systems is theExtended Kalman filter (EKF). Even though the EKF is oneof the most widely used approximate solutions for nonlinearestimation and filtering, it has some limitations [17]. The EKFonly uses the first order terms of the Taylor series expansionof the nonlinear functions introducing errors in the estimatedstatistics. Such transformations are only reliable if the errorpropagation can be well approximated by a linear function,otherwise the linearized approximation can be extremely poor.Also it requires the Jacobian matrix which in some cases isdifficult to calculate. UKF and DDF, unlike EKF, utilize min-imal set of deterministically chosen sample points to capturethe true mean and covariance of the state of the nonlinearsystem. Conceptually, the implementation principle resemblesthat of the EKF. However it is significantly simpler becauseit uses a finite number of functional evaluations instead ofanalytical derivatives. These SPKF outperform the EKF interms of estimation accuracy, filter robustness and ease ofimplementation.

1) Overview of UKF Algorithm : The unscented Kalmanfilter was introduced by Julier and Ulmann [18] and uses anonlinear transformation, called the unscented transformation(UT) in which a given state probability distribution is repre-sented by a minimal set of sampled points that can be usedto parameterize the true mean and covariance of the statedistribution. Since for a fixed number of parameters it is easierto approximate a Gaussian distribution rather than an arbitrarynonlinear function [17]. Such a parameterization captures themean and covariance information allowing the propagation ofthe information through an arbitrary set of nonlinear equations.This is accomplished by generating a discrete distributioncomposed of the minimum number of ”sigma” points whichhave the same first and second (and possibly higher) moments,where each point in the discrete approximation can be directlytransformed. Given an n-dimensional Gaussian distributionhaving covariance P, we generate a set of points having thesame covariance from the columns (or rows) of the matrices(the positive and negative roots). This set of 2n points have thedesired mean and covariance. Because the set is symmetric itsodd central moments are zero, so its first three moments arethe same as the original Gaussian distribution. The algorithmis shown in table I.

2) Overview of DDF Algorithm : The linearization is basedon polynomial approximations of the nonlinear transforma-tions that are obtained by Stirlings interpolation formula, ratherthan the derivative-based Taylor series approximation.In caseof DDf we have a single scaling parameter which controls thespread of the sigma points around the prior mean [10].DDFalgorithm is shown in Table II where γ = 3 for a Gaussiandistribution.

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TABLE I

DDF ALGORITHM

1. Initialization: xk = E [xk]Pk = E[(xk − xk)(xk − xk)T

]2. Square Cholesky factorizations:P0 = SxST

x ,Qk = SwSTw,R = SvST

v

S(2)xx (k + 1) =√γ−12γ

{fi(xk + hsx,j , wk) + fi(xk − hsx,j , wk) − 2fi(xk, wk)}S

(2)xw(k + 1) =√γ−12γ

{fi(xk, wk + hsw,j) + fi(xk, wk − hsw,j) − 2fi(xk, wk)}3.State and covariance Propagation:

x−k+1 =

γ−(nx+nw)γ

f(xk, wk)

+ 12γ

nx∑p=1

{f(xk + hss,p, wk) + fi(xk − hss,j , wk)}

+ 12γ

nx∑p=1

{f(xk, wk + hsw,p) + fi(xk, wk − hss,p)}

S−x (k + 1) =

[S

(1)xx (k + 1)S

(1)xw(k + 1)S

(2)xx (k + 1)S

(2)xw(k + 1)

]P−

k+1 = S−x (k + 1)(S−

x (k + 1))T

4. Observation and Innovation Covariance Propagation:

y−k+1 =

γ−(nx+nv)γ h(x−

k+1, vk+1)

+ 12γ

nx∑p=1

{h(x−

k+1 + hs−x,p, vk+1) + h(x−k+1 − hs−x,p, vk+1)

}

+ 12γ

nx∑p=1

{h(x−

k+1, vk+1 + hsv,p) + h(x−k+1, vk+1 − hsv,p)

}

Pvvk+1 = Sv(k + 1)ST

v (k + 1)

Pxyk+1 = S

(1)x (k + 1)

(S

(1)yx (k + 1)

)T

5. Update:Kk+1 = Pxy

k+1(Pvvk+1)−1

x+k+1 = x−

k+1 + Kk+1 (yk+1 − yk+1)

P+k+1 = P−

k+1 − Kk+1Pvvk+1KT

k+1

IV. SIMULATION RESULTS

We have simulated the system varying number of userswith multipaths. The delays have been assumed to be constantduring one measurement. The spreading codes are of 31 inlength with oversampling factor of 2. For the state space modelwe have taken state transition matrix to be and the processnoise covariance matrix as . We have simulated 2, 5 and 10user scenario with SNR at the receiver of the weak user is10 dB. The near far ratio of 20 dB has been taken and thepower of the strong user is P1 = 1 and that of the weak useris P1/10. The scaling paramter γ = 3 for the DDF algorithmand α = 0.01, β = 2, κ = 0. We have considered the case ofclosely spaced multipaths.

First we consider the UKF implementation. Figure (1)displays the estimated delays for the three path model in a10 user scenario with the path separation of 1/2 a chip. Ifwe compare the UKF algorithm with the DDF algorithm, wesee that the performance of the two is nearly same. It is dueto the fact that DDF is based on the derivative approximationon Stirling formula whereas UKF is based on Taylor seriesapproximation for the nonlinear function. A comparison ofthe UKF algorithm with the EKF algorithm is also made andthe results shown in figure (5) and (6) clearly demonstrate thesuperior performance of the UKF and DDF over its counterpartEKF.

Fig. 1. Timing epoch for the first, second and third arriving paths with tenusers and a three path channel model (2nd path within 1/2-chip)(UKF)

Fig. 2. Timing epoch for the first, second and third arriving paths with tenusers and a three path channel model (2nd path within 1/2-chip)(UKF)

Fig. 3. MSE of the channel coefficients for first arriving path with a ten-user/two-path channel model(chips)(UKF)

425

Fig. 4. Timing epoch estimation for first arriving path with a five-user/three-path channel model (with 1/2-chip path separation)(DDF)

Fig. 5. Comparison of the MSE of timing epoch estimation for first arrivingpath for EKF and UKF ((2nd path within 1/2-chip delay)

Fig. 6. Comparison of the DDF with EKF in terms of timing epoch estimationfor first arriving path with a five-user/ three-path channel model (with 1/2-chippath separation)

TABLE II

UKF ALGORITHM

Step I: Sigma Points CalculationX0 = x i = 0

Xi = x +(√

(n + λ)P + Q)

i = 1, ..., n

Xi = x −(√

(n + λ)P + Q)

i = n + 1, ..., 2n

W(m)i =

{λ/(n + λ) i = 0

1/{2(n + λ)} i = 1, ..., 2n

W(c)i =

{λ/(n + λ) + (1 − α2 + β) i = 0

1/{2(n + λ)} i = 1, ..., 2n

Step II: Prediction 1. Prediction StateXi,l+1 = f(Xi,l, l)

x−l+1 =

4KM∑i=0

W(m)i Xi,l+1

P−l+1 =

4KM∑i=0

W(c)i

{Xi,l+1 − x−

l+1

} {Xi,l+1 − x−

l+1

}T

2. Observation PredictionZi,l+1 = h(Xi,l+1, l + 1)

z−l+1 =4KM∑i=0

W(m)i Zi,l+1

Pzzl+1 =

4KM∑i=0

W(c)i

{Zi,l+1 − z−l+1

} {Zi,l+1 − z−l+1

}T

Step III: Measurement Update1. Compute the innovation covariance and cross covariance asPvv

l+1 = Pzzl+1 + σ2

Pxzl+1 =

4KM∑i=0

W(c)i

{Xi,l+1 − x−

l+1

} {Xi,l+1 − x−

l+1

}T

2. Calculate Kalman GainKl+1 = Pxz

l+1(Pvvl+1)−1

3. Update state estimationxl+1 = x−

l+1 + Kl+1vl+1

vl+1 = zl+1 − zl+1

4. Update the covarianceP+

l+1 = P−l+1 − Kl+1P

zzl+1KT

l+1

V. CONCLUSION

This paper presented a SPKF approach for CDMA timedelay and channel gain estimation over multipath fading basedon the UKF and DDF. It was shown that the SPKF achievebetter performance and enjoys moderate complexity comparedto the (linearized) EKF algorithm because of the nonlineardependency of the channel parameters on the received signalsin multiuser/multipath scenarios. A general derivation theprocessing steps were presented, followed by a specializationto the epochs. The numerical results showed that the UKFand DDF are quite robust vis-a-vis near-far multiple-accessinterference, and can also track a case of time delay andchannel gain estimation for multipath CDMA signals, withparticular focus on closely-spaced multipath

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