wireless personal communications volume 40 issue 4 2007 [doi 10.1007%2fs11277-006-9114-x] jang-sub...

Wireless Personal Communications (2007) 40: 457–478

DOI: 10.1007/s11277-006-9114-x C© Springer 2006

A State-Space Approach to Multiuser Parameters Estimation UsingCentral Difference Filter for CDMA Systems

JANG-SUB KIM, SEOKHO YOON and DONG-RYEOL SHIN

School of Information and Communication Engineering, Sungkyunkwan University, 300 ChunChun-Dong,JangAn-Gu, Suwon, KoreaE-mails: [email protected], [email protected], [email protected]

Abstract. In this paper, it is shown that a state-space model applies to the code-division multiple-access (CDMA)

channel, and Central Difference Filter (CDF) produces channel estimates with the minimum mean-square error

(MMSE). This result may be used as compare to Extended Kalman Filter (EKF) which used as channel estimator in

CDMA system. The main purpose of this paper is to compare robustness of channel estimator for realistic rapidly

time-varying Rayleigh fading channels. To overcome the highly nonlinear nature of time delay estimation and also

improve the accuracy, consistency and efficiency of channel estimation, an iterative nonlinear filtering algorithm,

called the CDF has been applied in the field of CDMA System. The proposed channel estimator has a more near-far

resistant property than the conventional Extended Kalman Filter (EKF). Thus, it is believed that the proposed

estimator can replace well-known filters, such as the EKF. The Cramer-Rao lower bound (CRLB) is derived for

the estimator, and simulation result show that it is nearly near-far resistant and clearly outperforms the EKF.

Keywords: code division multiple access (CDMA), channel and delay estimation, nonlinear recursive filter

1. Introduction

The CDMA technology is well developed and currently used in many applications that requireanti-jamming, high resolution for position and velocity estimation, and in low detectable appli-cations as well as multiple access communications in cellular systems. In the cellular systems,common usage of wireless channel resources, i.e., frequency band and time slot, brings manybenefits; high capacity, ease of cell planning and development. As with all cellular systems,CDMA suffers from multiple-access interference (MAI) and the result of strongly-poweredusers masking weaker users is known as the near-far effect [1]. One means of combatingthe near-far problem is the use of strict power control which attempts to maintain equallyreceived power for the users [2]. A second approach is to design spreading codes with lowcross-correlation properties. A third, a better alternative, is to embrace ideas from multiusercommunication theory [3]. This theory is based on the idea that better performance is possibleif the receiver incorporates information about the users’ wave-forms. A significant amountof research from this idea has taken place to develop near-far resistant detectors and channelestimators.

Multiuser detection has the potential to reduce the MAI and solve the near-far problemin a CDMA channel [4]. The analysis of multi-user detectors for fading channels is oftenconducted under the ideal assumption of perfect channel estimation [5–8]. Imperfect channelestimation degrades performance of multiuser detectors since many multiuser detectors requirechannel estimates to cancel the MAI and/or to perform coherent reception. To improvement

458 J. -S. Kim et al.

performance of multiuser detectors for rapidly varying fading channels, we introduce the newnonlinear recursive filters that provide an improvement in the accuracy and stability. We focuson the accuracy of performance of channel estimator for combating the near-far effect.

The CDMA multiuser channels relates to the issue of estimating the states of a systemgiven a set of noisy or incomplete observations. This model is specifically on discrete-timenonlinear dynamic systems that can be described by a dynamic state-space model (DSSM)as shown in Equations (3) and (4) of Section 2. An optimal solution (state estimation) isgiven by the recursive Bayesian estimation algorithm which recursively updates the posteriordensity of the state (called parameter in this report) as new observations arrive on-line. Thisposterior density constitutes the complete solution to the probabilistic inference problem, andallows us to calculate the optimal estimation for the state. Unfortunately, for most real-worldproblems, the optimal Bayesian recursion is intractable, and other approximation solutionsmust be used. In an approximate solution, the EKF has become one of the most widely usedalgorithms. Kalman filtering is known as a recursive approximation to the optimum minimummean-square error (MMSE) solution [9].

A central and vital operation performed in the Kalman Filter is the propagation of a Gaus-sian random variable (GRV) through the state and observation models. In the EKF, the statedistribution is approximated by a GRV, which is then propagated analytically through the first-order linearization of the nonlinear system. This can introduce large errors in the true posteriormean and covariance of the transformed GRV, which may lead to sub-optimal performanceand sometimes divergence of the filter. A tracking-mode receiver for asynchronous direct-sequence CDMA was presented based on the EKF [10–13]. The EKF jointly estimates thedelays and multi-path coefficients of the received CDMA waveform, and provides a modifiedminimum mean-square error (MMSE) estimate of the user data. Extended Kalman filteringsolves much of the early problems of divergence.

A new filtering called Scaled Unscented Filter (SUF) has been employed to tackle the non-linearity and shown its effectiveness in terms of the divergence reduction or error propagation[14]. The SUF addresses this problem by using a deterministic sampling approach. The statedistribution is again approximated by a GRV, but is now represented using a minimal set ofcarefully chosen sample points. All the iterative solutions including Kalman filters appearedin the literature necessarily assume that the unknown parameters are Gaussian distributed.These sample points completely capture the true mean and covariance of the GRV, and whenpropagated through the true nonlinear system, captures the posterior mean and covarianceaccurately up to the 3rd order (Taylor series expansion) for any nonlinearity. The EKF, incontrast, only achieves first-order accuracy. Remarkably, the computational complexity of theSUF is the same order as that of the EKF.

To relate this sample-based (sigma point) approach the EKF, we consider expanding thenonlinear function by polynomial approximations based on Sterling’s interpolating formulas.This filter is called central difference filter (CDF). This filter can interpret as a Taylor seriesexpansion where the derivatives are replaced by central differences which rely on functionalevaluations. The CDF and the SUF simply retain a different subset. It is shown in [15] and[16] that the CDF has marginally higher theoretical accuracy than the SUF in the higherterms of Taylor series expansion, but we’ve found in practice that both the CDF and SUFperform equally well with negligible difference in estimation accuracy. The CDF as proposedestimator does, however, have a smaller absolute error in the fourth order term and alsoguarantees positive semi-definiteness (PSD) of the posterior covariance. In contrast, the SUFmay result in a non-positive semi-definite covariance, which is compensated for using two

A State-Space Approach to Multiuser Parameters Estimation 459

additional heuristic scaling parameters [14]. Additional advantage of the CDF over the SUFis that it only uses a single scalar scaling parameter, the central difference half-step size h, asopposed to the three (α, β, κ) that the SUF uses. From this fact, the CDF has smaller degreeof freedom than the SUF. Thus the proposed estimator can easily control more than the SUF.Both filters generate estimates that are clearly superior to those calculated by a conventionalEKF.

In summary, it is the purpose of this paper to extend the development of multiuser channeland delay estimators to joint estimation of time-varying channel coefficients and time delays.Using the estimated channels and delays, an equalization method is applied to reduce boththe inter-symbol interference (ISI) and multiuser interference (MUI). The data decision isperformed on the resulting equalized signal. If necessary, the data decisions may be feed backto the channel and delay estimators. For a reliable data detection and position location, thereceiver algorithm has to estimate the users’ channel parameters. The CDF is using for thisCDMA channel estimation in here. And the channel model in this paper does not considermultipath propagation for simplicity of presentation and analysis, but the approach could beeasily extended to this case. We show that a proposed filter consistently outperform the forchannel estimation more than the conventional EKF, at an equal computational complexitywithout the need to analytical calculate Jacobians [17]. To ensure a validity of a proposedmethod, Monte Carlo simulations of the CDF to obtain the MSE (Mean Square Error) of thechannel estimator are used.

This paper is organized as follows. Section 2 introduces the signal and channel modelthat will be used throughout the paper and a description of problem formulation. In order tocompare to the performance of the proposed channel estimator, we derived the Cramer-Raolower bound (CRLB). Section 3 provides a description of the nonlinear filtering method (calledCDF) used for parameter estimation. The results of the performance from numerical analysisare given in Section 4. Finally, Section 5 provides concluding remarks.

2. Problem Formulation

2.1. SYSTEM AND CHANNEL MODEL

Assuming that each of K users transmits over an M-path fading channel, the received signalis given by

r (l) =K∑

k=1

M∑i=1

ck,i (l)dk,ml ak(l − ml Tb − τk,i (l)) + n(l) (1)

where the ck,i (l) represent complex channel coefficients, dk,ml is the mth symbol transmittedby the kth user, ml = �(l − τk(l))/Tb�, Tb is the symbol interval, ak(l) is the PN spreadingwaveform used by the k th user, τk,i (l) is the time delay introduced by the ith path of the kthuser, and n(l), AWGN (Additive White Gaussian Noise) is assumed to have a mean of zeroand variance of R = σ 2

n = E[|n(l)|2] = N0/Ts, where Ts is sampling time.

2.2. A STATE-SPACE REPRESENTATIONS

Let the unknown parameters be represented by the following 2K × 1 vector.


x =[

cτ

](2)

where c = [c1, c2, . . . , cK ]T and τ = [τ1, τ2, . . . , τK ]T .From [10], we can write the state model as

x(l + 1) = F(l)x(l) + v(l) (3)

where F = diag{Fc, Fτ } is 2K × 2K augmented by the state transition matrix, v = [ vTc vT

τ ]is 2K × 1 process noise vector with mean of zero and covariance matrix Q = diag {Qc, Qτ } ,

and diag (•) is the diagonal matrix.The scalar measurement model follows from the received signal of (1) by

z(l) = h(x(l)) + n(l) (4)

where the measurement z (l) = r (l) , and

h(x(l)) =K∑

k=1

M∑i=1

ck,i (l)dk,ml ak(l − ml Tb − τk,i (l))

The scalar measurement z(l) is a nonlinear function of the state x(l). If it is assumed thatthe noise vectors v(l) and n(l) are individually and mutually uncorrelated with correlationmatrices

E[v(i)v( j)T ] = Qiδi j

E[n(i)n( j)T ] = Riδi j

E[v(i)n( j)T ] = 0

(5)

where δi j is the two-dimensional Kronecker delta function and the noise variance is assumedto R = σ 2

n .

Given measurement and the state-space model (3) and (4), find the optimal estimate ofx(1), x(l|l) = E{x(l)|zl} with 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)}.With (3) and (4), we now have the state-space model needed for the nonlinear adaptive

filters, which can therefore be used to iteratively determine the MMSE state estimates x. Inorder to detect the symbol d, assuming that channel and delay estimates cand τ are availableas depicted Figure 1. The recursive algorithm based on this description is given in next section(EKF or CDF), we also discuss how the proposed structure may be used for simultaneouschannel estimation.


Figure 1. The proposed receiver structure.

2.3. CR AMER-RAO LOWER BOUND (CRLB)

The CRLB is a lower bound on the covariance matrix of any unbiased estimator [17]. Supposex is an unbiased estimator of a vector of deterministic unknown parameters x (i.e., x = E[x])then the estimator’s covariance matrix is satisfies

J−1 ≤ E{(x − x)(x − x)T } (6)

where J is the 2K × 2K Fisher information matrix given by

J = E

{[∂

∂xln (zl)

][∂

∂xln (zl)

]T }

And (zl) is the log-likelihood function of the observed data zl with respect to x. In our case,the observed data is the received samples z(l) for l = 1, 2, . . . , L , hence, zl = [z(1) · · · z(l)]T .

Using (4), the expression for z(l) can be written in a more compact form as

z(l) = cT (l)D(l)a(l) + n(l) (7)

where


c =

⎡⎢⎣ c1(l)...

cK (l)

⎤⎥⎦ , D(l) =

⎡⎢⎣d1(l) 0. . .

0 dK (l)

⎤⎥⎦ , and a(l) =

⎡⎢⎣ a1(lTs − mk(l)Tb − τ1(l))...

aK (lTs − mk(l)Tb − τK (l))

⎤⎥⎦The CRLB presented here will still be a valid bound on the performance of estimators

based on the model (4). Since the noise is a Gaussian random variables [10], the likelihoodfunction of zl given x is

(zl) = 1(2πσ 2

n

)L/2exp

{− 1

σ 2n

L∑l=1

| z(l) − cT (l)D(l)a(l) | 2

}(8)

From which the log-likelihood function directly follows

ln (zl) = const. − 1

σ 2n

L∑l=1

| z(l) − cT (l)D(l)a(l) | 2 (9)

In order to calculate the Fisher information matrix J, the partial derivative with respect tothe parameter vector x is needed. From [17, 18], the Fisher information matrix J can be writtenas

J =[

Jcc Jcτ

JHτ c Jτ τ

](10)

where the matrices Jcc, Jcτ , Jτ c, Jττ are defined as

Jcc = 1

σ 2n

L∑l=1

D(l)a(l)aT (l)D(l)

Jcτ = 1

σ 2n

L∑l=1

D(l)a(l)aTd (l)D(l)

Jττ = 1

σ 2n

L∑l=1

C(l)D(l)ad(l)aTd (l)D(l)C(l)

The matrix C(l) = diag[c(l)], and ad(l) is the vector of derivatives ∂ a(l)/∂τk . Consequently,the CRLBs for the estimator of the amplitudes and delays is given by

CRLB(c) = [Jcc − Jcτ J−1

τ τ J Hcτ

]−1

CRLB(τ ) = [Jτ τ − JH

cτ J−1cc Jcr

]−1(11)

It is desirable that the performance of the developed estimator approach the CRLB.

3. Nonlinear Recursive Filters for CDMA Systems

The major shortcomings of the EKF have limited first-order accuracy of propagated meansand covariances resulting from a first-order truncated Taylor-series linearization method. But


the CDF also guarantees the same performance as the truncated second order filter, with thesame order of calculations as an EKF but without the need to calculate any approximationor derivatives [17]. These considerations motivate the development of the CDF in CDMAchannel estimation. In order to easily explain the CDF, first we present a brief review of theScaled Unscented Filter (SUF), and then we will extend the SUF to the CDF with Taylor seriesapproximation. We will compare to the performance of the EKF and the CDF.

3.1. EXTENDED KALMAN FILTER (EKF)

Because CDMA measurement models are nonlinear, we cannot use the KF. The EKF hasprobably had the most widespread use in nonlinear estimation, and we briefly explain to theEKF. EKF has been the solution to the MMSE estimation in the nonlinear state transitionand/or the nonlinear measurement models [17].

Because CDMA measurement models are nonlinear, we cannot use the KF. The EKF hasprobably had the most widespread use in nonlinear estimation, and we briefly explain to theEKF. EKF has been the solution to the MMSE estimation in the nonlinear state transitionand/or the nonlinear measurement models. In order to retain the simple and strong predictor-corrector solution structure for the Kalman filter for the models which is obtained again fromthe linearity and Gaussian assumption, the EKF takes the linear approximation by the Taylorseries expansion of the nonlinear models. Taylor series expansion of the nonlinear functionsh around the estimates x(l + 1 | l) of the states x(l + 1) can be expressed as

z = g(x(l + 1)) = g(x(l + 1 | l)) + .∇g | x(l+1)=x(l+1 | l)(x(l + 1) − x(l + 1 | l)) + · · · (12)

Using only the linear expansion terms, it is easy to derive the following update equations.A full derivation of the EKF recursions is beyond the scope this paper, but time update andmeasurement update form based on the state-space Equations (3) and (4) is listed in Table 1[17, 18].

The EKF approximates the state distribution using a Gaussian random variable, which is

Table 1. Conventional extended kalman filter

Model x(l + 1) = F(l)x(l) + v(l)

z(l) = g(x(l)) + n(l)

Time Update x(l + 1 | l) = Fx(l)

Pxx(l + 1 | l) = FPxx(l)FT + QMeasurement Update K(l + 1) = Pxz(l + 1 | l)P−1

vv (l + 1 | l)

x(l + 1) = x(l + 1 | l) + K(l + 1)v(l + 1 | l)

Where

z(l + 1 | l) = g(x(l + 1 | l)),

v(l + 1 | l) = z(l + 1) − z(l + 1 | l),

Pvv(l + 1 | l) = HPxx(l + 1 | l)HH + σ 2n ,

and where K is known as the Kalman gain, Q is the variance of the

process noise, σ 2n is the variance of the measurement noise, G �=

.∇g | x(l+1)=x(l+1 | l) is the Jacobians of the measurement model, v is the

innovation.


then propagated analytically through the first-order linearization of the nonlinear system. TheEKF does not take into account the second and higher order terms in mean and fourth andhigher order terms in the covariance are negligible. These approximations can introduce largeerrors in the true posterior mean and covariance of the transformed random variable in manypractical situations, leading to suboptimal performance and divergence of the filter.

Weighted statistical linear regression (WSLR) have been proposed that address these issues.Our proposed filter is related through the implicit user of a technique called WSLR to calculatethe optimal terms in the Kalman update rule. This filter is called SUF or CDF according to amain approach method. The main idea of the WSLR is following as: Instead of linearizing thenonlinear function through a truncated Taylor-series expansion at a single point (mean valueof the random variable), we rather linearize the function through a linear regression betweenp points drawn from the prior distribution of the random variable, and the true nonlinearfunctional evaluations of those points. Since this approach takes into account the statisticalproperties of the prior random variable, the resulting linearization error tends to be smallerthan that of a truncated Taylor-series linearization. Refer to [16] for the detail derivation ofWSLR.

3.2. THE SCALED UNSCENTED FILTER (SUF)

The Unscented Filter (UF) derives the location of the sigma points as well as their correspond-ing weights according to the following rationale [19]: The Sigma Points should be chosenso that they capture the most important statistical properties of the prior random variable x.Transform builds on the principle that it is easier to approximate a probability distributionthan it is to approximate an arbitrary nonlinear function. A set of p + 1 weighted points whereS = {Wi , Xi } (such that

∑pi=0 Wi = 1) are chosen to reflect certain properties on x. In other

words, they obey a condition of the form

min〈W,X〉

g[S, px (x)] subject to g[S, px (x)] = 0 (13)

where g[•, •] specifies what information from x is to be matched by S [14]. The necessarystatistical information capture by the SUF is the first and second order moments of p(x). Thenumber of sigma points required is 2n + 1 where n is the dimension of x. See [19] for moredetails on how the sigma points are calculated as a solution to Equation (14). The resulting setof sigma points and weights utilized by the SUF are

X0(l | l) = x(l | l)

Xi (l | l) = x(l | l) + (η√

P(l | l))i i = 1, . . . , n

Xi+n(l | l) = x(l | l) − (η√

P(l | l))i i = n + 1, . . . , 2n

W (m)0 = λ/(n + λ)

W (c)0 = λ/(n + λ) + (1 − α2 + β)

W (m)i = W (c)

i = 1/{2(n + λ)} i = 1, . . . , 2n

(14)

where, κ ∈ �, (√

(n + κ)P(l | l))i is the ith row or column of the matrix square root of(n + κ)P(l | l) and Wi is the weight that associated with the i th point. λ = α2(n + κ) − n isa scaling parameter and η = √

(n + λ). α is a positive scaling parameter which can be made


arbitrarily small to minimize higher order effects (e.g. 1e − 2 ≤ α ≤ 1). κ is a secondaryscaling parameter which is usually set to either 0 or 3 − n. β is an extra degree of freedomscalar parameter used to incorporate any extra prior knowledge of the distribution of x (forGuassian distributions, β = 2 is optimal). These sigma vectors are propagated through thenonlinear function

Zi = h(Xi ) i = 0, . . . , 2n (15)

The mean and covariance for z are approximated using a weighted sample mean andcovariance of the posterior sigma points

z ≈2n∑

i=0

W (m)i Zi

(16)

Pz ≈2n∑

i=0

W (c)i {Zi − z}{Zi − z}T

The deceptively simple approach taken with the SUT (Scaled Unscented Transform) resultsin approximations that are accurate to the third order for Gaussian inputs for all nonlinearities.For non-Gaussian inputs, approximations are accurate to at least the second-order, with theaccuracy of third and higher order moments determined by the choice of α and β.

It can be shown that matching the moments of x accurately up to the second order meansthat Equation (16) capture z and Pz accurately up the second order as well. The sigma pointscalculated in here make use of to calculate the optimal estimation using the Kalman updaterule (see Table 1).

For precise implementation of the SUF in CDMA environments given by [20]:Step. 1 The sigma point is calculated as using Equation (14)

X(l) = [x(l | l) x(l | l) + η√

P(l | l) + Q x(l | l) − η√

P(l | l) + Q]

Step. 2 The SUF time updates as follows

• The transformed set is given by instantiating each point through the process model

Xi (l + 1 | l) = FXi (l | l)

• The predicted mean is computed as

x(l + 1 | l) =2n∑

i=0

W (m)i Xi (l + 1 | l)

• The predicted covariance is computed as

P(l + 1 | l) =2n∑

i=0

W (c)i [Xi (l + 1 | l) − x(l + 1 | l)][Xi (l + 1 | l) − x(l + 1 | l)]T


• Instantiate each of the prediction points through the observation model

Z (l + 1 | l) = g(X(l + 1 | l))

• The predicted observation is calculated by

z (l + 1 | l) =n+1∑i=0

Wi Zi (l + 1 | l)

Step. 3 The SUF measurement updates as follows

• The innovation covariance is given by

Pzz(l + 1) =2n∑

i=0

W (c)i [Zi (l + 1 | l) − z(l + 1 | l)][Zi (l + 1 | l) − z(l + 1 | l)]T

• Since the observation noise is additive and independent, the innovation covariance is

Pvv(l + 1) = Pzz(l + 1) + σ 2n

• The cross-covariance matrix of x and z, is determined by

Pxz(l + 1) =2n∑

i=0

W (c)i [Xi (l + 1 | l) − x(l + 1 | l)][Zi (l + 1 | l) − z(l + 1 | l)]T

• The Kalman gain matrix can be found according to

K(l + 1) = Pxz(l + 1)/Pvv(l + 1)

• The update mean (parameter estimated) is calculated

x(l + 1) = x(l + 1 | l) + K(l + 1)v(l + 1)

v (l + 1) = z (l + 1) − z (l + 1 | l)

• The update covariance (error covariance matrix are updated) is also provided by

P(l + 1) = P(l + 1 | l) − K(l + 1)Pzz(l + 1)KT (l + 1)


In the SUF where the SUT is employed in the prediction stages follows the given nonlinearfunction, no harmful loss in the above is expected. It is not necessary to calculate the Jacobian(and Hessian if 2nd order approximation in the Taylor series) and the prediction stage onlyconsists of standard linear algebra operations (matrix square root, etc). The superior perfor-mance of the SUF over that of the EKF has been reported in numerous publications including[14, 19].

3.3. THE CENTRAL DIFFERENCE FILTER (CDF)

To relate this sample-based approach to the EKF, we consider expanding the nonlinear functionby polynomial approximations based on interpolating formulas. One such formula is Sterling’sinterpolation formula, which, if we are restricted to first and second-order polynomial approx-imations gives the following approximation

z = g(x) (17)

≈ g(x) + �xg′C D(x) | x=x + 1/2 • �2

xg′′C DC D

(x) | x=x

where

g′C D(x) = g(x + h) − g(x − h)

2h

g′C DC D

(x) = g(x + h) + g(x − h) − 2g(x)

h2

One can thus interpret above equation as a second order Taylor series expansion where thederivatives are replaced by central differences which only rely on functional evaluations. Thisformulation was the basis of Norgaard’s [21] recent derivation of the divided difference filter aswell a Ito and Xiong’s [22] central difference filter. These two filters are essentially identical.The CDF was developed in this manner, careful analysis of the Taylor series expansion of boththe CDF and the SUF approximations, show that both approaches are essentially the same[14]. Even though this approach is not explicitly derived starting from the statistical linearregression rationale, it can be shown [16] that the resulting Kalman filter again implicitlyemploys WSLR-based linearization. The resulting set of sigma-points for the CDF is onceagain 2n + 1 points deterministically drawn from the prior statistics of x, i.e.

X0(l | l) = x(l | l)Xi (l | l) = x(l | l) + (

√h2P(l | l))i i = 1, . . . , n

Xi+n(l | l) = x(l | l) − (√

h2P(l | l))i i = n + 1, . . . , 2nW (m)

0 = (h2 − n)/h2

W (c)0 = 1/(4h2)

W (m)i = 1/(2h2) i = 1, . . . , 2n

W (c)i = (h2 − 1)/(4h2) i = 1, . . . , 2n

(18)

It is shown in [16] that the CDF has marginally higher theoretical accuracy than the SUFin higher order terms of the Taylor series expansion, but we’ve found in practice that both theCDF and the SUF perform equally well with negligible difference in estimation accuracy. One


advantage of the CDF over the SUF is that it uses only a single scalar scaling parameter, thecentral difference half-step size h, as opposed to the three (α, β, κ) that SUF uses. Once againthis parameter determines the spread of the sigma-points around the prior mean. For Gaussianpriors, its optimal value is

√3.

The complete CDF algorithm that updates the mean x and P of the Gaussian approximationto the posterior distribution of the states is given by subsection B but only difference is thatthe sigma point is calculated as

X(l) = [x(l | l) x(l | l) +√

h2(P + Q) x(l | l) −√

h2(P + Q)]

The remaining procedure is a repeated implementation of subsection B. For precise imple-mentation of the CDF in CDMA environment given by :

Step. 1 The sigma point is calculated as using Equation (18)

X(l) = [x(l | l) x(l | l) +√

h2(P + Q) x(l | l) −√

h2(P + Q)]

Step. 2 The CDF time updates as follows

• The transformed set is given by instantiating each point through the process model

Xi (l + 1 | l) = FXi (l | l)

• The predicted mean is computed as

x(l + 1 | l) =2n∑

i=0

W (m)i Xi (l + 1 | l)

• The predicted covariance is computed as

P(l + 1 | l) =2n∑

i=0

W (c)i [Xi (l + 1 | l) − x(l + 1 | l)][Xi (l + 1 | l) − x(l + 1 | l)]T

• Instantiate each of the prediction points through the observation model

Z (l + 1 | l) = g(X(l + 1 | l))

• The predicted observation is calculated by

z(l + 1 | l) =n+1∑i=0

Wi Zi (l + 1 | l)


Step. 3 The CDF measurement updates as follows

• The innovation covariance is given by

Pzz(l + 1) =2n∑

i=0

W (c)i [Zi (l + 1 | l) − z(l + 1 | l)][Zi (l + 1 | l) − z(l + 1 | l)]T

• Since the observation noise is additive and independent, the innovation covariance is

Pνν (l + 1) = Pzz (l + 1) + σ 2n

• The cross-covariance matrix of x and z, is determined by

Pxz(l + 1) =2n∑

i=0

W (c)i [Xi (l + 1 | l) − x(l + 1 | l)][Zi (l + 1 | l) − z(l + 1 | l)]T

• The Kalman gain matrix can be found according to

K(l + 1) = Pxz(l + 1)/Pvv(l + 1)

• The update mean (parameter estimated) is calculated

x(l + 1) = x(l + 1 | l) + K(l + 1)v(l + 1)

ν (l + 1) = z (l + 1) − z (l + 1 | l)

• The update covariance (error covariance matrix are updated) is also provided by

P(l + 1) = P(l + 1 | l) − K(l + 1)Pzz(l + 1)KT (l + 1)

Steps 2 and 3 stages are time update and measurement update in the Kalman update rule(Table 1), respectively. Strictly speaking, we say that the main difference between EKF andCDF is whether use sigma points or not. And SUF and CDF perform equally well in estimationaccuracy. Because both filters are derived by analytically derived 1st and 2nd order derivativesin the Taylor series expansion. But the EKF is to the nature of the 1st order Taylor serieslinearization. Thus, SUF and CDF are clearly superior to the EKF. And one advantage of theCDF over the SUF is that it only uses a single scalar scaling parameter, as opposed to the three.


Figure 2. Parameter estimation errors for (a) channel amplitudes and (b) time delay with User 1.

4. Numerical Analysis

We now examine the performance of the CDF for making parameter estimates for a multiuserdetector. We compare the CDF-based estimator with an estimator based on conventional EKF.We note that the form of channel amplitude corresponds to a Rayleigh uncorrelated scatteringmodel for the channel [10]. The Rayleigh fading of the channel coefficients for each user wasimplemented by IFFT ([23], Ch. 4) and normalized so that the average power was unity. Forsimplification purposes, we consider no multipath. For the state model, the augmented statetransition matrix of (11) was chosen to be F = 0.999I. Also the process noise covariancematrix was Q = 0.001I.


Figure 3. Parameter estimation error for (a) channel amplitude and (b) time delay with User 2.

We simulate a two-user scenario where the users’ PN spreading codes are chosen from theset of Gold codes of length 31 and generated by the polynomials x5 + x2 + 1 and x5 + x4 +x3 + x2 + 1. The SNR (Signal-to-Noise Ratio) at the receiver of the weaker user is 10 dB.The Near-Far Ratio is 20 dB. The oversampling factor (sample/chip) is 2. Because AWGN isGaussian, the parameter (h) using the CDF estimator is assume to h = √

3. One aspect aboutusing EKF or CDF is that they require proper initialization. Depending on the problem, theinitial guesses may need to be close to the correct value for convergence. For the simulationresults, we assume such an initial estimator is used to start the tracking algorithm fairly close tothe true values. Furthermore, we note that the data bits, dk,m, are not included in the estimationprocess, but are assumed to be unknown a priori. In the simulations, we assume that the data


Figure 4. Time-varying channel amplitude tracking with (a) User 1, (b) User 2.

bits are available from decision-directed adaptation, where the symbols dk,m are replaced bythe decisions dk,m shown in Figure 1.

The tracker for a two-user system is simulated for a fading channel where the channelcoefficients are time varying, but the delays remain constant. A simple channel model isassumed for each user with a single tap (flat fading). Furthermore we assume that User 1 and 2are moving with Doppler frequencies of 100 Hz and 200 Hz, respectively. The sampling timeis taken to be TS = 1/ (1.2288 Mbps × 2) and the bit rate is assumed to be 1/Tb = 9600 bpswith a processing gain of 31.

Figure 2(a) and (b) shows the estimation error for the channel coefficients and time delaysfor User 1 with imperfect power controlled using the CDF and the EKF, respectively.


Figure 5. Time-delay tracking with User 1 and User 2.

Figure 6. Instantaneous near-far ratio for the weaker user without power control.

As the figures indicate, the estimator/tracker is able to accurately track the time-varyingchannel coefficients of each user, even for fast fading rates. It can be seen that the user is capableof accurately converging to the correct delays and channel coefficient for both estimator. Andthe estimator is converge inner 100 samples (about 1 bit duration). The results for the sameset of assumptions, but with User 2 are shown in Figure 3. Again, the estimator is able toaccurately converge to the correct values of the parameters for both the CDF and the EKF.

The ability of the estimator to track time-varying parameters is shown in Figures 4 and 5where the time-delays are constant. Each user is assumed to move with a Doppler frequency of500 Hz for User 1 and 1000 Hz for User 2. Such great Doppler frequency are not likely to occurin practical situations. However, the estimator was able to converge and track the parameters


Figure 7. Estimation of the (a) User 1 (weaker user) and (b) User 2′ bits.

in such a harsh scenario. As the figure indicates, the estimator/tracker is able to accuratelytrack the time-varying channel coefficients of each user, even for fast fading rates of 500 Hzand 1000 Hz (Doppler). Also, although the average near-far ratio is 20 dB, the instantaneousratio varies drastically due to the fading of the channel since there is no power control. Asshown in Figure 6, the instantaneous near-far ratio with respect to the second user varies from−10 dB to +30 dB. In spite of the varying powers, the tracker is still able to display excellentperformance. But Monte Carlo simulations of the EKF often diverge in the same environment.

The simulation results assumed that the receiver operated in decision-directed mode, asillustrated in Figure 1. The ability of the estimator to estimate the bits of the User 1 (weakeruser) and User 2 in a 20 dB near-far ratio is illustrated in Figure 7(a) and (b), respectively.


Figure 8. The MSE of the (a) amplitude and (b) time delay estimates versus near-far ratio from the simulated

estimator and the CRLB.

It is seen that the receiver correctly estimates the bit sequence and that convergence after bitchanges is fairly rapid.

To further quantify the performance of the estimator, the MSE from simulation of theestimator is presented. In near-far situations, the first user is the weaker user. A mean squareerror was computed for the estimates as follows:

MSEX(n) = 1

Ns

Ns∑i=1

∣∣X − X (i)∣∣2

(19)


where X(X = [x(1) x(2) . . . x (n)]) denote the MSE of the channel parameters the amplitudeand time delay, at iteration n of the estimator. Ns is the number of ensemble samples used toform the MSE and X are the estimates of channel parameters at time n. The index i correspondto the individual simulation runs over which the average is taken. The performance of theCDF estimator compared to the CRLB is first evaluated. Figure 8 compares the CDF mean-square error for the channel estimate and time delay to the corresponding CRLB and EKF.The number of simulation runs used in the average is Ns = 200. Since the nonlinear estimatoris not guaranteed to be efficient, the mean-square estimation error is above the CRLB. Thefigures indicate that three is a slight increase in the MSE as the near-far ratio is increased.We would expect a nearly near-far resistant estimator to be unaffected by the near-far ratio.But the EKF estimator is rapidly increasing in the MSE above near-far ratio of 20 dB, and itshow that a larger error happened (it is often a divergence). The near-far ratio means that theincreased power of the second user is treated as increased noise compare to the case of equalpower users. In this case, the EKF estimator is not near-far resistant, but the CDF estimator isnearly near-far resistant. Thus we expect that the CDF estimator with 2nd order accuracy hasa more near-far resistant and strong noise power immunity than the EKF estimator with 1storder accuracy.

5. Conclusions

In this paper we have applied a CDF-based algorithm to the estimation of multipath pathdelays and related coefficients in CDMA environments. It consistently performs better thanthe conventional EKF and has a more near-far resistant, with the added benefit of ease ofimplementation in that no analytical derivatives (Jacobians or Hessians) need to be calculated.And the CDF has a smaller scalar scaling parameter and guarantees positive semi-definitenessof the underlying state covariance compared to the SUF. We have constructed a parameterestimator based on the CDF that has a more near-far resistant and is capable of estimating thechannel coefficients and time delays in MAI for a near-far ratio of 40 dB, but the EKF divergeabove a near-far ratio of 20 dB. The CDF can provide a better alternative to nonlinear filteringthan the EKF and SUF since it requires much simpler numerical computations. Computersimulations also show that it provides a more viable means of tracking time-varying amplitudesand delays in CDMA communication systems than EKF.

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Jang Sub Kim was born June 15, 1974, in Yeongdeok, Korea. He received the M.S. de-gree in school of electrical and computer engineering from Sungkyunkwan University, Seoul,Korea. He is currently with the School of Information and Communication Engineering,Sungkyunkwan University, where he was a Ph. D. student since 1999. His research inter-ests include code-division multiple access, channel estimation, position location, and wirelesscommunications.


Seokho Yoon (S’99–M’1) received the B.S.E. (summa cum laude), M.S.E., and Ph.D. degreesin electrical engineering from KAIST, Daejeon, Korea, in 1997, 1999, and 2002, respectively.From April 2002 to June 2002, he was with the Department of Electrical Engineering andComputer Sciences, Massachusetts Institute of Technology, Cambridge, MA, and from July2002 to February 2003, he was with the Department of Electrical Engineering, Harvard Uni-versity, Cambridge, MA, as a Postdoctoral Research Fellow. In March 2003, he joined theSchool of Information and Communication Engineering, Sungkyunkwan University, Suwon,Korea, where he is currently an Assistant Professor. His research interests include spread spec-trum systems, mobile communications, detection and estimation theory, and statistical signalprocessing. Dr. Yoon is a member of the IEEK and KICS. He was the recipient of a BronzePrize at Samsung Humantech Paper Contest in 2000.

Dong-Ryeol Shin (M’97) was born in Seoul, Korea, in 1957. He received the B.S., M.S.and Ph.D degree in electrical engineering from the Sungkyunkwan University in 1980, andthe Korea Advanced Institute of Science and Technology (KAIST) in 1982 and the GeorgiaInstitute of Technology in 1992, respectively. During 1992-1994, he had worked for SamsungData Systems, Ltd., Korea. Since 1994, he has been with network research group at theSungkyunkwan University, Korea, as a professor. His current research interests include wirelesscommunications and ubiquitous computing.

wireless personal communications volume 40 issue 4 2007 [doi 10.1007%2fs11277-006-9114-x] jang-sub...

Documents

cdma channel

channel estimates

proposed channel estimator

nearfar problemin

nearfar effect

velocity estimation

multiple access communications

resistant detectors