CDMA Multiuser RadiolocationZahid Ali, Mohamad A. Deriche, Mohamed Adnan Landolsi

Electrical Engineering DepartmentKing Fahd University of Petroleum & Minerals

P.O.Box 8466,Dhahran 31261 Saudi ArabiaEmail: zalikhan@kfupm.edu.sa

Abstract—Wireless location refers to obtaining the positioninformation of a mobile subscriber in a cellular environment.Such positioning information is usually provided in terms ofgeographic coordinates of the mobile subscriber with respect to ageographic reference point. Wireless location finding has emergedas an essential feature of cellular systems and has many potentialapplications in areas such as location sensitive billing, assettracking, fraud protection, mobile yellow pages, fleet managementand the list continues to grow and so does the location basedservices market. Several location techniques utilizing terrestrialwireless network elements and radio signals have been proposedover the years. However, accurate mobile station (MS) position-ing in a terrestrial wireless system is impeded by multipathpropagation, low signal-to-noise ratios (SNRs), multiple accessinterference (MAI), and non-line-of-sight propagation. Whilesubscriber location has been previously studied for code divisionmultiple access (CDMA) networks, the effect of multiple accessinterference has not been fully explored especially for the case ofclosely spaced multipaths. Traditional location algorithms havederived location estimates usually assuming single user singlepath environment. However this assumption is not correct asmeasurement bias will be introduced due to MAI. We proposenon linear filtering approach based on unscented Kalman filter(UKF) for MS delay estimation that is subsequently used formultiuser radiolocation. We show through simulations that theproposed method provides MS position location quite accuratelyin the presence of MAI for closely spaced paths.

Index Terms—CDMA channel estimation, non-linear filters,closely spaced multipaths, multiple access interference, approxi-mate likelihood, radiolocation.

I. INTRODUCTION

Over the past decade, considerable attention has been givento wireless cellular mobile positioning systems, and a plethoraof new location-based applications have already started takingadvantage of location technology [1]. With the rapid evolutionof wireless technologies during the last decade or so, theemphasis on services offered by wireless carriers has shiftedfrom voice-based services to high data rate applications andvalue-added services. The widespread use of wireless phoneshas also boosted interest in the field of mobile positioning,especially for vital emergency and personal safety servicesand commercial applications such as location sensitive billing,fleet management and Intelligent Transportation Systems (ITS)[1,2,3]. Location based services (LBS) offer wireless devicemanufacturers and carriers opportunities to increase revenuesby offering users attractive services that are tailored to theirlocation. From the standpoint of subscribers, mobile locationoffers increased safety, access to emergency services, localized

information and higher quality of service [4]. The drivingforce behind the development of accurate wireless locationtechniques is the ever increasing revenue generated from lo-cation based services. It was estimated that the location basedindustry accounted for estimated revenue of about US$40billion in 2006 [5].

Numerous new and novel applications are being imple-mented to provide a range of services to the subscribers. Theaccuracy requirements for network-based and handset-basedtechnologies have been outlined in [1]. For network-basedtechnologies, these accuracies vary from 100m for 67% ofcalls as dictated for E-911 emergency services in the USA to25-125m in urban areas for E-112 in the EU. As the namesuggests, a hand-set based location system uses the mobilestation (MS) to measure certain signal characteristics andlocates itself while with network-based location, the locationdetermination is done at the base stations (BSs) [1,2]. Bothapproaches have their advantages and drawbacks [1], but thelatter takes advantage of the existing wireless communicationsinfrastructure without the need for supplementary technologysuch as dead reckoning. This paper examines the feasibilityand performance of radio location techniques in code divisionmultiple access (CDMA) cellular networks. CDMA is thechosen access scheme, since it is the leading candidate forthird generation cellular networks. The rest of the paper isorganized as follows. In Section III, the UKF approach andsignal model is discussed and time of arrival technique usingapproximate maximum likelihood (AML) is presented in sec-tions III. Computer simulations and performance discussionsare presented in section IV 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⌋,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 Gaussian

2010 IEEE 21st International Symposium on Personal, Indoor and Mobile Radio Communications Workshops

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Noise (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 [6],

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) = F𝜏 𝜏(𝑙) + v𝜏 (𝑙) (4)

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

E{v𝑐(𝑖)v𝑇𝑐 (𝑗)} = 𝛿𝑖𝑗Q𝑐,E{v𝜏 (𝑖)v

𝑇𝜏 (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 is written as

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 (1)

𝑧(𝑙) = ℎ(x(𝑙)) + 𝑛(𝑙) (6)

The scalar measurement z(l) is a nonlinear function of thestate x(𝑙). The optimal estimate of x(𝑙) denoted as 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. TIME-OF-ARRIVAL ESTIMATION WITH UKF

For the nonlinear dynamic system model such as givenabove, the conventional Kalman algorithm can be invoked toobtain the parameter estimates. The most well known appli-cation of the Kalman filter framework to nonlinear systems isthe EKF. Even though the EKF is one of the most widely usedapproximate solutions for nonlinear estimation and filtering, ithas some limitations such as limited accuracy due to truncated

Taylor series expansion, the Jacobian matrix calculation whichin some cases is difficult to calculate [6]. UKF, unlike EKF, uti-lizes a minimal set of deterministically chosen sample pointsto capture the true mean and covariance of the state of thenonlinear system. Conceptually, the implementation principleresembles that of the EKF. However it is significantly simplerbecause it uses a finite number of functional evaluationsinstead of analytical derivatives. The UKF outperforms theEKF in terms of estimation accuracy, filter robustness and easeof implementation.

1) Overview of UKF Algorithm: The unscented Kalmanfilter was introduced by Julier and Uhlmann [7] and usesa nonlinear transformation, called the unscented transforma-tion (UT) in which a given state probability distribution isrepresented by a minimal set of sampled sigma points thatcan be used to parameterize the true mean and covarianceof the state distribution. Given an n-dimensional Gaussiandistribution having covariance P, we generate 2𝑛 points havingthe desired mean and covariance. Because the set is symmetricits odd central moments are zero, so its first three momentsare the same as the original Gaussian distribution.

Suppose that we know the mean �� and covariance 𝑃of a vector 𝑥. We then find a set of deterministic vectorscalled sigma points whose ensemble mean and covarianceare equal to �� and 𝑃 . We next apply our known nonlinearfunction 𝑦 = ℎ(𝑥) to each deterministic vector to obtaintransformed vectors. The ensemble mean and covariance ofthe transformed vectors will give a good estimate of the truemean and covariance of 𝑦. This is the key to the unscentedtransformation.

We choose 2n sigma points such that

𝑥(𝑖) = ��+ �� 𝑖 = 1, ..., 2𝑛

��(𝑖) =(√

(𝑛+ 𝜆)P+Q)𝑇

𝑖𝑖 = 1, ..., 𝑛

��(𝑛+𝑖) = −(√

(𝑛+ 𝜆)P+Q)𝑇

𝑖𝑖 = 1, ..., 𝑛

(7)

The algorithm is shown in table I, where 𝛼 controls thespread of the sigma points and should be a small number (0 ≤𝛼 ≤ 1), 𝜆 = 𝛼2(𝑛 + 𝜅) − 𝑛, where 𝜅 is a scaling parameterused to describe the scaling direction of the sigma points and𝛽 incorporates the prior knowledge of the distribution of x.

IV. SIMULATION RESULTS

For the radio channel between the mobile and base stationBSi, we assume that the mobile signal is subject to attenuationincluding distance path loss and lognormal shadowing [8,9],

𝛼(d𝐵𝑆𝑖, 𝜉𝐵𝑆𝑖

) = 𝑝(d𝐵𝑆𝑖)10𝜉𝐵𝑆𝑖

/10

where 𝑝(𝑑) is the distance path loss, and 𝜉𝐵𝑆𝑖is the

shadowing variable. The path loss part follows a two-segmentmodel with breakpoint at 𝑑𝑜

𝑝(𝑑) = 10𝑛 log10(𝑑)

where 𝑛 is the path loss slope assumed to take two differentvalues, depending on whether the mobile is within or beyond

234

TABLE IUKF 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 − z−

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+1K

𝑇𝑙+1

the given breakpoint. In the subsequent numerical simulations,we use the slopes 𝑛=2, and a breakpoint at 200m, with acell radius of 2km. For a given mobile, shadowing vis-a-visthe different base stations is partially correlated, and givenby: 𝜉𝐵𝑆𝑖

= 𝑎𝜉𝑐 + 𝑏𝜉𝑖 where 𝜉𝑐 and 𝜉𝑖 are the commonand independent terms, respectively, and 𝑎2 + 𝑏2 = 1. Inthe numerical results, we assume the shadowing variablesare log-normal with standard deviation 𝜎𝑠ℎ = 8dB, and 50%correlation (a=b=1/

√2 ≈ 0.707).

Since time-of-arrival estimation accuracy strongly dependson the received MAI levels, this issue can be a limiting factorin mobile radiolocation which typically requires TOA datafrom at least three base stations. For example, if we assumethat the mobile is served by the center base station BS1 andwill be radiolocated by the strongest seven base stations BS1,BS2,. . . , BS7 (sorted in a descending order from the basestation that receives the highest average received power, shownin table II), then we define the ratio of its average receivedpower at BS𝑖 compared to BS1 as [9]𝛽𝑖 = P𝑖/P1

where P𝑖 is the received power in BS𝑖 and 𝛽1 = 1 ≥ 𝛽2 ≥𝛽3 ≥ ⋅ ⋅ ⋅ ≥ 𝛽7

TABLE IIAVERAGES OF THE BETA FACTORS

𝛽1 𝛽2 𝛽3 𝛽4 𝛽5 𝛽6 𝛽7

Case 1 1 0.0216 0.0113 0.0069 0.0045 0.0031 0.0021Case 2 1 0.6982 0.2215 0.1202 0.0735 0.0485 0.0331Case 3 1 0.7922 0.6353 0.2993 0.1701 0.1065 0.0706Averages of 𝛽-factors for various soft-handover link conditionswhen shadowing st.dev. 𝜎𝑠ℎ = 8 dB and the cell radius is 2km [9]

It is found that this ratio can fluctuate widely depending onthe mobile position relative to the base stations of interest. Asan illustration, we present examples for four scenarios (cases1, 2 and 3) that will be used in the subsequent numericalresults. Case-1 refers to a mobile located in close proximityto its “serving” BS1, with a signal at least 10dB above thatat the other two base stations. Case-2 represents a two-waysoft handover scenario, with the mobile power at base station2 within 3dB (as an example) from that at BS1, and case-3denotes the 3-way soft handover situation where the mobilesignal is within 3dB at both BS2 and BS3 compared to BS1[9].

For the purpose of simulation we consider typical asyn-chronous CDMA model with K users and M-paths. TOAestimates have been obtained from UKF. We have simulatedthe system varying number of users with multipaths. We alsoassume slow frequency-selective Rayleigh fading with 2-pathlinks between mobile and base stations (the 2nd path is 3dBbelow the main one used for time tracking). The delays havebeen assumed to be constant during one measurement. Wenote that the data bits, 𝑑𝑘,𝑚, are not included in the estimationprocess, but are assumed unknown apriori. In simulations, weassume that the data bits are available from decision-directedadaptation, where the symbols 𝑑𝑘,𝑚 are replaced by the 𝑑𝑘,𝑚decisions as shown in Figure 1. For the state space modelwe have taken state transition matrix to be F = 0.999I andthe process noise covariance matrix as Q = 0.001I. The UKFscaling parameters are chosen as 𝛼 = 0.01, 𝜅 = 0 and 𝛽 = 2 .We have simulated multiuser scenario with SNR at the receiverof the weak user is 10 dB. The near far ratio is 20 dB. Wehave considered the case of closely spaced multipaths. Firstwe consider the UKF implementation. Figure 2 displays theestimated delays for the two path model in a 10 user scenariowith the path separation of 1

2 a chip. Whereas Figure 3 showsthe delay estimation error of the first arriving path vs usersfor 1

2 a chip apart three path scenario. We can see that theestimator attains an estimated value close to the true values.This demonstrates the robustness of the proposed estimator inthe near far environment.

In this paper, we are considering only line-of-sight (LOS)propagation, and base stations assumed synchronized. TheTOA measurements recorded at each BS is directly propor-tional to the mobile-base distance. We only consider threebase stations BS𝑖, 𝑖 = 1, 2, 3 for simplicity (those with themost favorable radio links), although statistical methods canuse data from more base stations. Assuming without lossof generality that BS1 has coordinates (0, 0), and the other

235

Fig. 1. Multiuser parameter estimation receiver

0 100 200 300 400 500 600 700 800

12

14

16

18

20

22

Samples

De

lay e

stim

atio

n(c

hip

s)

True valueEstimated value

weaker user

stronger user

Fig. 2. Timing epoch of the first arriving path of the weaker and strongeruser in a twelve-user three paths (paths 1/5 chip apart)

Fig. 3. Delay estimation error for first arriving path of the weak user withvarying number of users in a three path channel model (UKF)

two base stations are at (𝑥𝑖, 𝑦𝑖)𝑇 , 𝑖 = 2, 3, with the mobile

𝑥 = (𝑥𝑚, 𝑦𝑚)𝑇 . It is shown in [5] that the mobile coordinatessatisfy the set of nonlinear equations Hx = b, where

H =

[𝑥2 𝑦2𝑥3 𝑦3

],x =

[𝑥𝑚𝑦𝑚

],b =

1

2

[𝐾2

2 − 𝑟22 + 𝑟21𝐾2

3 − 𝑟23 + 𝑟21

]

(8)The least square solution of (8) is given by

x = (H𝑇H)

−1H𝑇b (9)

The mobile position estimation can be further improved byusing this LS solution as an initial guess for an “approximate”maximum likelihood algorithm (AML) [10], which solves alinearized version of the problem expressed by:

[ ∑𝑔𝑖𝑥𝑖

∑𝑔𝑖𝑦𝑖∑

ℎ𝑖𝑥𝑖∑

ℎ𝑖𝑦𝑖

] [𝑥𝑚𝑦𝑚

]=

[ ∑𝑔𝑖(𝑠+ 𝑘𝑖 − 𝛿2𝑖 )∑ℎ𝑖(𝑠+ 𝑘𝑖 − 𝛿2𝑖 )

]

(10)

𝑔𝑖 =𝑥𝑚 − 𝑥𝑖𝑑𝑖(𝑑𝑖 + 𝛿𝑖)

, ℎ𝑖 =𝑦𝑚 − 𝑦𝑖

𝑑𝑖(𝑑𝑖 + 𝛿𝑖), 𝑠 = 𝑥2𝑚 + 𝑦2𝑚 (11)

Here, 𝛿𝑖 is the measured (noisy) distance between the basestation BS𝑖 and the mobile, while d𝑖 is the true distance.Since the unknowns (𝑥𝑚, 𝑦𝑚) appear on the RHS of (11),the algorithm can be solved iteratively, starting with an initialguess from (9). For the purpose of radiolocation we considertypical cellular network with center cell and two tiers ofinterfering cells as shown in Figure 4. We assume a uniformlyloaded network, with 30 users per cell with a LOS path fromMS to BS in a two path model. Figure 5 shows the cumulativedistribution function (CDF) of the mobile position estimationerror for the three cases outlined in table II. It is clearlyseen that the case for 3-way soft handover gives the bestperformance, followed by 2-way soft handover one, and thecase when the mobile is closest to its own base station is worst.Figure 6 shows position location for the case of three BS. Wesee that the mean of the position location error increases asthe MS approaches its serving BS. This is because the MS,closer to the serving BS, needs to transmit at lower powerlevels to maintain the fixed received power at the serving BS.We may also observe that as the distance to the neighboringBSs increases, the signal experiences greater path loss. So thereceived signal power at the neighboring BSs reaches lowerlevels, making position location error to rise sharply. This inturn decreases the position location accuracy.

V. CONCLUSION

This paper dealt with the performance of time of arrivalbased techniques for mobile positioning in CDMA wirelesscellular networks. UKF has been used to estimate the delaysin a multiuser environment. As MAI has an impact on theprecision of mobile radiolocation, owing to the fact that, withpower control, the mobile signal at far-away base stations(not involved in soft handover) can be very weak, hence itsTOA estimation will be noisy. We showed that UKF can be

236

Fig. 4. Cellular network

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Positioning Error , e (in meters)

Cum

mula

tive D

istr

ibutio

n F

unct

ion, P

rob(E

rror

<=

e)

Case 1

Case 3

Case 2

Fig. 5. Cumulative distribution function (CDF) for the residual mobilepositioning error. Comparison for 3 cases

0 500 1000 1500 2000 2500 30000

20

40

60

80

100

120

140

Distance to serving BS, m

Mean P

L e

rror,

m

Fig. 6. Mean position location error vs distance from serving BS

employed for delay estimates in the presence of MAI evenin a closely spaced multipath scenario. We showed throughsimulations that the the proposed method provides locationestimates that are close to the true MS position in the presenceof MAI.

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