A Multipath Detection Scheme for CDMA Systems With Space–Time Spreading

146 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 1, JANUARY 2008

A Multipath Detection Scheme for CDMA SystemsWith Space–Time Spreading

Mohamed Abou-Khousa, Student Member, IEEE, Ali Ghrayeb, Senior Member, IEEE, andMohamed El-Tarhuni, Senior Member, IEEE

Abstract—Space–time spreading (STS) is an appealing open-loop transmit diversity scheme, which has recently been includedinto the cdma2000 standard. It has been shown that the perfor-mance of the STS scheme is highly sensitive to fading coefficientestimation errors, particularly when the channel is highly timedispersive. In practical systems, channel estimation is normallyperformed after the multipath components are resolved, whichsuggests that improving multipath detection reduces such esti-mation errors. Motivated by this, we address, in this paper, theproblem of multipath detection in STS-based code division mul-tiple access (CDMA) systems. We first extend the conventionalenergy-based multipath detection scheme (EMDS) to cope withthe spatial channel structure. We derive approximate expressionsfor the probability of detection and probability of false alarm. Itis shown that the errors produced by the conventional schemein detecting the potential multipath components severely impactthe performance of the receiver. To improve upon the EMDS, weintroduce and analyze an improved multipath detection scheme(IMDS) based on the estimation of the interference power in theindividual resolved multipath components. The efficacy of theproposed scheme stems from the fact that the interference in eachpotential path is estimated and subtracted before that path isdetected. We also present a simple and realizable version of theproposed IMDS detection scheme. Our results show that the pro-posed scheme not only improves the bit-error-rate performancesignificantly but also utilizes the pilot power much more efficiently.

Index Terms—Code division multiple access (CDMA) systems,multipath detection, space–time spreading (STS).

I. INTRODUCTION

T RANSMIT diversity techniques constitute promisingmeans to combat slow fading in third-generation code

division multiple access (CDMA) systems. Among all the tech-niques, open-loop transmit diversity techniques are particularlyappealing to the system designer, as they do not reduce the up-link capacity since no channel information feedback is needed.

Manuscript received March 20, 2006; revised January 28, 2007, April 16,2007, and May 2, 2007. The work of M. Abou-Khousa and A. Ghrayeb wassupported in part by the Natural Sciences and Engineering Research Councilof Canada under Grant N00858. This work was presented in part at the IEEEVehicular Technology Conference, Stockholm, Sweden, May–June 2005. Thereview of this paper was coordinated by Prof. T. J. Lim.

M. Abou-Khousa was with the Department of Electrical and ComputerEngineering, Concordia University, Montreal, QC H3G 1M8, Canada. He isnow with the Department of Electrical Engineering, Missouri University ofScience and Technology (Missouri S&T), Rolla, MO 65409 USA (e-mail:[email protected]).

A. Ghrayeb is with the Department of Electrical and Computer Engi-neering, Concordia University, Montreal, QC H3G 1M8, Canada (e-mail:[email protected]).

M. El-Tarhuni is with the Department of Electrical Engineering, AmericanUniversity of Sharjah, Sharjah, United Arab Emirates (e-mail: [email protected]).

Digital Object Identifier 10.1109/TVT.2007.905327

Recently, the so-called space–time spreading (STS) schemehas been included in the cdma2000 standard as an open-looptransmit diversity with two transmit antennas and one receiveantenna option. The STS scheme was first proposed in [1] andindependently in [2]. Due to its potential for enhancing thetransmission reliability and the capacity of the system withoutpenalizing the system resources, e.g., bandwidth and spreadingcodes, it has been subsequently adopted by the cdma2000standard [3]–[5].

Since its introduction, the STS scheme has been subjectedto extensive performance analysis and evaluation (see [3]and references therein). Moreover, various studies have beenconducted to compare the performance of the STS schemewith other open-loop transmit diversity schemes, includingspace–time transmit diversity [6], [7] and orthogonal transmitdiversity [3], [8]. In [9] and [10], the performance of a reverselink STS-based CDMA system is analyzed over frequency se-lective fading channels with imperfect estimation of the fadingcoefficients while assuming that the delays are perfectly known.Therein, it is shown that as the number of multipath componentsincreases, the estimation errors significantly impact the sys-tem performance. More recently, a comprehensive performanceevaluation of the STS scheme for the forward link has beendiscussed in [3].

The majority of the relevant work in the literature suggeststhat the STS scheme is rather sensitive to the fading coeffi-cient estimation errors, particularly when the channel is highlytime dispersive. In realistic systems, channel estimation errorsare normally composed of not only the errors in estimatingthe fading coefficients themselves but also the impairment indetecting the multipath components impinging at the receiverfront end [11]. The latter errors have a profound effect on theperformance of the system since the estimation of the fadingcoefficients of a certain multipath component is normally car-ried out after that path is identified. Hence, accurate multipathdetection is of great importance for STS-based CDMA sys-tems, as it impacts the overall performance of the receiver. Todate, neither the problem of multipath detection for STS-basedCDMA systems nor the effect of imperfect multipath detectionon the performance of the STS scheme has been investigated.Guenach et al. [12], [13] propose an iterative estimation schemefor the channel-fading coefficients and the path delays, withemphasis on the uplink channel. They also consider singletransmit and single receive antenna systems, i.e., no STS.This scheme can also be applied to the downlink channel, butgiven the associated high computational complexity, it may beprohibitive to implement this scheme in mobile terminals.

0018-9545/$25.00 © 2008 IEEE

ABOU-KHOUSA et al.: MULTIPATH DETECTION SCHEME FOR CDMA SYSTEMS WITH SPACE–TIME SPREADING 147

Fig. 1. The (2, 1) STS scheme.

In this paper, we address the problem of multipath detec-tion in STS-based CDMA systems. We describe the multipathsearch algorithm, as it can be applied to the current optionof the STS scheme incorporated into the standard, i.e., twotransmit antennas and one receive antenna. Initially, we ex-tend the conventional energy-based multipath detection scheme(EMDS) to cope with the spatial structure of the channel.We derive expressions for the probability of detection andprobability of false alarm. It is shown that the EMDS exhibitspoor performance at low signal-to-interference-plus-noise ratio(SINR). To improve upon the EMDS, we introduce and analyzean improved multipath detection scheme (IMDS) based on theestimation of the interference power in the resolved multipathcomponents. Once the interference power per path is estimated,it is subtracted before that multipath component is detected.This leads to a significant improvement in multipath detectionprobability and, consequently, channel estimation. We deriveexpressions for the detection and false alarm probabilities forthe proposed scheme. To account for the hardware limitationsat the receiver, we present a simple and realizable version of theproposed detection scheme. We also provide a comprehensivecomparison between the conventional and proposed schemesin terms of performance. Our results show that the proposedscheme is superior to the conventional energy-based scheme.

The rest of this paper is organized as follows: The systemand channel models are described in Section II. In Section III,we present the search algorithm used to correlate the incomingsignal with locally generated versions of the intended user’sspreading code. In Section IV, we discuss the multipath de-tection schemes investigated in this paper. In Section V, wepresent a simple and realizable version of the IMDS scheme.Numerical and simulation results are presented and discussedin Section VI. Finally, Section VII concludes this paper.

II. SIGNAL AND CHANNEL MODEL

The system considered herein is similar to the proposeddownlink cdma2000 system, operating in the traffic mode,with the STS option. We assume that the base station (BS)transmitter is equipped with two antennas and the mobile

terminal receiver is equipped with one antenna, i.e., a (2, 1)STS scheme. To simplify the analysis, we assume binary phase-shift keying modulation and spreading. Each transmit antennacode-multiplexes a distinct pilot with the traffic signals. TheBS is assumed to serve M users, and each user is assigned twodifferent codes from a set of orthogonal codes.

A block diagram describing the spreading scheme is shownin Fig. 1. The mth user bit stream is split into two substreams,one corresponding to the even-indexed information bits andthe other corresponding to the odd-indexed information bits.The first substream is obtained by down sampling the originaluser bit stream by a factor of 2, whereas the other substreamis obtained by delaying the user bit stream by D = Tb (a bitduration) followed by down sampling by a factor of 2. Assuch, the baseband transmitted traffic and pilot signals fromthe first and second antennas during the nth bit time interval,respectively, are modeled as

s1(t) =M∑

m=1

[√Ebm

2(a1m(t)b1m(n) + a2m(t)b2m(n))

]

+

√Gp

2a1(M+1)(t) (1)

s2(t) =M∑

m=1

[√Ebm

2(a1m(t)b2m(n) − a2m(t)b1m(n))

]

+

√Gp

2a2(M+1)(t) (2)

where M is the number of users, Ebm is the average bit energyfor the mth user, and b1m(n) ∈ {±1} is the nth informationbit in the first substream for the mth user. Similarly, b2m(n) ∈{±1} is the nth information bit in the second substream for themth user, Gp is the pilot channel power gain compared to thetraffic channel, and ajq(t) is the signature waveform given as

ajq(t) =Nc−1∑i=0

cjq(i)g(t − iTc) (3)

where q ∈ {1, 2, . . . ,M + 1}, j ∈ {1, 2}, Nc is the spreadingfactor, cjq is the spreading code assigned for the qth user


Fig. 2. RAKE receiver structure for the (2, 1) STS scheme.

transmitted from the jth antenna, cjq(i) ∈ {±1}, g(t) isthe chip pulse shape, and Tc is the chip period. Note thata1(M+1)(t) and a2(M+1)(t) are the pilot channel signaturewaveforms for the first and second antennas, respectively. Thespreading code used for a particular user or pilot channel isthe multiplication of the channelization code assigned for thatchannel, i.e., Walsh code, and the BS-specific scrambling code.

The transmitted signal from the jth antenna traverses amultipath fading channel with L time-varying paths that arriveat the receiver front end with delays dc = {τ1, τ2, . . . , τL}.These paths are represented by the channel coefficients{αjl : l = 1, 2, . . . , L} modeled as independent and identicallydistributed complex Gaussian random variables with zero meanand 0.5φ(τl) variance per dimension, where φ(τ) is the channelpower delay profile (PDP), i.e., |αjl| is Rayleigh distributedand the phase is uniformly distributed in [0, 2π). The channelcoefficients are assumed to be constant during a bit duration,i.e., Nc chips. Furthermore, the in-phase and quadrature compo-nents of any channel coefficient are assumed to be independentof each other. It is also assumed that each pair of paths fromboth transmit antennas arrives at the receiver with the samedelay. This assumption is valid in practice since the propagationdelays between the two transmit antennas are typically on theorder of nanoseconds, whereas the multipath delays are on theorder of microseconds [4].

The total received signal at the mth user’s receiver at time tis given by

r(t) =L∑

l=1

[α1l(n)s1(t − τl) + α2l(n)s2(t − τl)] + w(t) (4)

where αjl(n) is the complex channel coefficient seen during thenth bit duration in the lth path between the jth transmit antennaand the receive antenna, τl is the delay of the lth pair of paths,and w(t) is a complex additive white Gaussian noise (AWGN)process with zero mean and N0/2 variance per dimension. TheAWGN term herein models the receiver thermal noise and theintercell interference.

After multipath detection and channel estimation are per-formed, the received signal is processed by a RAKE receiverwith L fingers, as shown in Fig. 2. The fingers are time alignedwith the detected multipath components. After alignment andchip-matched filtering, each finger despreads the incoming sig-nal using the mth user’s codes for the first and second antennas.Each of the despread signals obtained using the jth antennacode in the lth finger, i.e., Sjl as shown in Fig. 2, is used bythe STS decoder/combiner to recover the transmitted bits as

b1m = sgn

(Re

{L∑

l=1

α∗1l(n)S1l − α∗

2l(n)S2l

})(5)

b2m = sgn

(Re

{L∑

l=1

α∗2l(n)S1l + α∗

1l(n)S2l

})(6)

where sgn(x) = −1 when x < 0 and 1 for x ≥ 0, and ∗ denotesthe complex conjugate.

As inferred from (5) and (6), there are two practical issuespertaining to the demodulation of the transmitted bits. First, theSTS decoder assumes the knowledge of the complex channelcoefficients at the receiver. Typically, the channel coefficientsare estimated in each finger using a moving average filter


operating over the pilot signal after proper alignment with thecorresponding path delay [14]. The performance of the STSscheme with imperfect channel coefficient estimation as drawnfrom the moving average filter has been addressed in [3] and[10] assuming perfect knowledge of the number of paths andtheir delays. The second issue is brought about by the factthat, before channel estimation is performed, the receiver hasno prior knowledge of the number of potential paths L andtheir delays. To estimate both sets of parameters, the receivertypically implements a search algorithm followed by a multi-path detection logic. Subsequently, a RAKE finger assignmentalgorithm (FAA) is used to allocate some/all of the detectedmultipath components to the available RAKE fingers [15].According to the applied multipath detection scheme and FAA,some of the actual potential paths might be missed, e.g., maskedby interference, and incorrect paths carrying interference poweronly might be detected and assigned to a RAKE finger [11].

The impact of the latter impairment in the multipath detec-tion is expected to have a profound effect on the performance ofthe STS scheme. This is reasonable since channel estimation isnormally carried out after the multipath delays in the channelare identified. For instance, if an incorrect path is detectedand assigned to a RAKE finger, forged channel estimates willbe obtained and used by the STS decoder. This essentiallyincreases the transmission errors, as it reduces the signal-to-noise ratio in the demodulation decision statistics.

III. SEARCH ALGORITHM

The search algorithm, henceforth the searcher, is essentiallyan integral part of the acquisition circuit used to perform thecorrelation between the received signal and delayed replicasof the pilot spreading code. Basically, the searcher despreadsthe incoming signal using different time-shifted versions of thepilot code. The searcher examines each delay in a window ofδ possible delays for a period of time Td (integration time),and the correlation results are stored for further processing.The search step size, which is denoted by S, is typically afraction of a chip, e.g., one-half chip. Hence, the delay uncer-tainty region can be represented by a set of K = �δ/S� delayoffsets as ds = {t1, t2, . . . , tK}, where �x� denotes the integerpart of x.

Without loss of generality, it is assumed that the searcherattempts to despread the incoming signal using one pilot codeat a time. Furthermore, the integration time Td is assumed tospan one bit duration, i.e., Nc chips. Hence, the search resultsobtained using the jth antenna’s pilot code are given by

hj(n, k)=Cj1(n, k)+Cj2(n, k)+Tj(n, k)+Nj(n, k) (7)

for k = 1, 2, . . . ,K, where the index k corresponds to the kthdelay offset within the search window, n is the search timeindex within a received data frame, Cj1(n, k) is the contributionof the first pilot signal, Cj2(n, k) is the contribution of thesecond pilot signal, Tj(n, k) is the interference coming from thetraffic channels transmitted from both antennas, and Nj(n, k) isthe contribution of the AWGN noise.

The first two components in (7) can be obtained from thefollowing expression:

Cji(n, k)=√

0.5Gp

L∑l=1

αil(n)Rji(M+1)(k, l), i=1, 2. (8)

The third and fourth components in (7), respectively, can beobtained as

Tj(n, k) =M∑

m=1

√Ebm

2

×L∑

l=1

(α1l(n)

[Rj

1m(k, l)b1m(n)]

+ α2l(n)

×[Rj

1m(k, l)b2m(n) − Rj2m(k, l)b1m(n)

]+ Rj

2m(k, l)b2m(n))

(9)

where

Rjim(k, l) =

1Tb

∫Tb

aj(M+1)(t − tk)aim(t − τl)dt (10)

Nj(n, k) =1Tb

∫Tb

aj(M+1)(t − tk)w(t)dt. (11)

The searcher should detect an effective path whenevertk = τl, where the spreading waveform’s autocorrelation func-tion will be at its maximum. In this case, (8) can be written as

Cjj(n, k)=

√Gp

2

L∑l=1,tk �=τl

αjl(n)Rjj(M+1)(k, l)+

√Gp

2αjk(n).

(12)

Hence, the search result during the nth time instant is compactlywritten as

hj(n, k) =√

0.5Gpαjk(n) + Ij(n, k) (13)

where

Ij(n, k) =

√Gp

2

L∑l=1,tk �=τl

αjl(n)Rjj(M+1)(k, l)

+Cji(n, k) + Tj(n, k) + Nj(n, k), i �= j. (14)

Note that i, j ∈ {1, 2}. As such, since i �= j in (14), then i = 1when j = 2, and i = 2 when j = 1.

The term Ij(n, k) in (13) represents the total interferencethat is coming from other paths belonging to the desireduser, interference from other users along with their multipathcomponents, and the noise seen at the kth delay offset. Underthe standard Gaussian approximation, the interference termIj(n, k) can be approximated as AWGN [16]–[21]. This ap-proximation is justified by the central limit theorem and holdswhen the number of active users in the system is relatively largeand/or the spreading factor is large [10]. In addition to the fact


that this Gaussian approximation has been widely used in theliterature, simulation results for various combinations of num-ber of users and spreading factors are presented in Section VI tovalidate this approximation. As such, the search results can beapproximated as

hj(n, k) ≈√

p(k)αjk(n) +√

σ2I (k)Ij(n, k) (15)

where αjk(n) and Ij(n, k) are the normalized complexGaussian random variables with zero mean and 0.5 variance perdimension, p(k) = 0.5Gpφ(τk), and σ2

I (k) is the variance ofthe interference component. This model assumes equal powerallocation over both antennas.

To improve the probability of multipath detection, indepen-dent search results are obtained by repeating the search processNA times using each pilot at different time instants, e.g., bitsor search blocks within a data frame. These time instantsare usually chosen such that they are sufficiently spaced farapart from each other within a frame to decrease the correla-tion between the search results and, hence, enhance the timediversity.

IV. MULTIPATH DETECTION SCHEMES INVESTIGATED

A. EMDS

In the EMDS, the correlation energy is averaged over NA

independent search blocks at each delay offset and the resultsare compared to a threshold. If the average energy at a certaindelay offset exceeds the threshold, the path with that delayoffset is acquired. This process is repeated, in a serial or parallelfashion, for all delay offsets in the search window [14], [26].The conventional multipath detection scheme can be extendedto incorporate the STS schemes where two transmit antennasare employed at the BS. As we have shown previously, twosets of independent search results are obtained by despreadingthe incoming signal with the pilot code for each antenna. Foreach search result set, the multipath detection metric resultingfrom correlation with the jth antenna’s pilot code can beformulated as

Yj(k) =1

NA

NA∑n=1

|hj(n, k)|2

≈ pj(k) + σ2jI(k) (16)

where σ2jI(k) and pj(k) are the averages of the interference

power σ2I (k) and the user power p(k) obtained from NA

observations over the jth pilot code, respectively. In (16), it isassumed that the interference and the desired user’s signal areindependent. This is a valid assumption since each multipathcomponent fades independently.

It can be shown that both variables σ2jI(k) and pj(k)

have distributions that can be approximated by central chi-square distributions with 2NA degrees of freedom [22]. Theapproximation holds well when the time-bandwidth productNATd/Tc is large [23]. Since each pair of paths between thetransmit antennas and the receive antenna arrives at the receiverwith the same set of delay, the detection metrics resulting for

both antennas can be averaged out to yield the final detectionmetric as

Y (k) = 0.5 [Y1(k) + Y2(k)] . (17)

It is important to notice that averaging the search resultsbefore computing the energies provides no performance (diver-sity) gain over the 1-D EMDS. Averaging both detection met-rics, however, will have the effect of increasing the probabilityof correctly detecting the potential multipath components, asshown next.

Since the desired user’s signal arrives at the receiver from Ldifferent paths with a set of delays dc, any of the remainingdelay offsets in the search window ds other than the onesalready in dc may cause a false alarm. Hence, the false alarmevent occurs when tk /∈ dc, and yet, the average energy Y (k)exceeds the threshold. The probability of false alarm can becalculated from the cumulative distribution function (cdf) of thedecision metric of the kth delay offset when tk /∈ dc, i.e., whenthe received signal consists of noise and interference only. Inthis case, the decision metric is given as

Y (k) = σ21I(k) + σ2

2I(k)∆= σ2

I (k). (18)

It can be shown that Y (k) has a central chi-square distributionwith 4NA degrees of freedom. Basically, averaging over theindependent decision metrics for both antennas has the effectof doubling the degrees of freedom. Hence, better estimates ofthe interference variance are obtained. The cdf of Y (k) is thengiven as [22]

FYk(y) = 1 − e−2NAy/σ2

I (k)2NA−1∑

i=0

1i!

(2NAy

σ2I (k)

)i

, y ≥ 0.

(19)

Using (19), the probability that the kth delay offset producesa false alarm is

PEfa(k) = Pr [Y (k) ≥ η]

= 1 − FYk(η)

= e−2NAη/σ2I (k)

2NA−1∑i=0

1i!

(2NAη

σ2I (k)

)i

(20)

where the superscript E denotes the EMDS probability of falsealarm, and η denotes a precalculated threshold. The averageprobability of false alarm PE

FA is the probability that Y (k) atany delay offset in the search window, except for the offsets thatcorrespond to the actual delays, exceeds the threshold, which isgiven as

PEFA =

1K − L

K∑k=1,tk /∈dc

PEfa(k). (21)

It is clear from (20) that the probability of false alarm isproportional to the interference power. This is also the casewith the probability of detection, as will be shown next. Theprobability of detecting a multipath component in the desireduser’s channel is the probability that the average energy corre-sponding to that path is greater than the threshold (a detection


event occurs only when tk ∈ dc). When tk ∈ dc, the detectionmetric can be rewritten as

Y (k) = p1(k) + p2(k) + σ2I (k)

∆= p(k) + σ2I (k) (22)

where p(k) has distributions that can be approximated bycentral chi-square distributions with 4NA degrees of freedom.Since the decision metric is a function of two independentrandom variables, namely p(k) and σ2

I (k), we formulate itsconditional cdf as

FYk(y | p(k)) = 1 − e−2NAy/[σ2

I (k)+p(k)]

·2NA−1∑

i=0

1i!

(2NAy

p(k) + σ2I (k)

)i

, y ≥ 0. (23)

The probability of detecting the kth multipath componentwhen tk ∈ dc is given as

PED (k) =

∞∫0

e−2NAη/[σ2I (k)+p(k)]

·2NA−1∑

i=0

1i!

(2NAη

p(k) + σ2I (k)

)i

f (p(k)) dp(k) (24)

where f(p(k)) is the probability density function (pdf) of theaverage energy of the kth path given by the central chi-squaredistributions with 4NA degrees of freedom and is defined as

f (p(k)) =1

Γ(2NA)

(4NAp(k)Gpφ(τk)

)2NA−1

·e−4NAp(k)/Gpφ(τk), p(k) ≥ 0 (25)

where Γ(x) is the gamma function.Thus, for the EMDS scheme, the probability of detection

depends on the total received energy per path, i.e., p(k) +σ2

I (k). When the SINR is low, some of the actual multipathcomponents will be masked out by other delay offsets withstrong interference power, and consequently, wrong paths mightexceed the threshold test, resulting in a false alarm state. Ideally,the probability of detection should be solely dependent upon thereceived SINR per path, i.e., p(k)/σ2

I (k). For this purpose, anIMDS is developed next.

B. IMDS

An IMDS is developed by estimating the interference powerseen at each delay offset and thereupon subtracting this powerfrom the total received power at that delay. For the search resultsobtained from the jth antenna’s pilot channel, the detectionmetric of the IMDS is formulated by subtracting the estimatedinterference variance σ2

jI(k) from the total energy seen at thekth delay in the search window. Hence, the jth detection metricbecomes

Zj(k) = Yj(k) − σ2jI(k)

≈ pj(k) +[σ2

jI(k) − σ2jI(k)

]. (26)

A performance bound on the IMDS probability of detectionand probability of false alarm can be derived if we assume

that the interference power estimate σ2I (k) is produced by

the minimum variance unbiased (MVU) [24] estimator that isgiven as

σ2jI(k) =

1NA

NA∑n=1

∣∣∣hj(n, k) −√

p(k)αjk(n)∣∣∣2 . (27)

Similar to the EMDS, the detection metrics obtained from thesearch results using both pilots can be averaged as

Z(k) = 0.5 [Z1(k) + Z2(k)]

= p(k) + ζ(k) (28)

where p(k) is previously defined, and ζ(k) is the averageinterference power estimation error. Since the estimator in (27)is MVU, the estimation error is Gaussian distributed with zeromean and σ4

I (k)/2NA variance [25].When tk /∈ dc, the detection metric is equal to the es-

timation error, i.e., p(k) = 0, and consequently, Z(k) ∼N(0, σ4

I (k)/2NA). Hence, the probability of false alarm willbe the probability that the estimation error for the kth delayoffset exceeds a given threshold ε, which can be obtained as

P Ifa(k) = Q

(√2NAε

σ2I (k)

)(29)

where the superscript I denotes the IMDS probability of falsealarm, and Q(x) is the Q function. The average probability offalse alarm is then given by

P IFA =

1K − L

K∑k=1,tk /∈dc

Q

(√2NAε

σ2I (k)

). (30)

The probability of detection is derived following the samesteps used to obtain the probability of false alarm. The con-ditional probability of detecting the kth multipath componentwhen tk ∈ dc is given by

P ID (k | p(k)) = Q

(√2NA [ε − p(k)]

σ2I (k)

). (31)

Averaging over the pdf of p(k) results in the probability ofdetecting that component as given by

P ID(k) =

∞∫0

Q

(√2NA [ε − p(k)]

σ2I (k)

)f (p(k)) dp(k). (32)

From the foregoing discussion, it is clear that the IMDS pro-bability of detection is a function of the received SINR per path,i.e., p(k)/σ2

I (k), as given by (32). This feature of the proposedscheme allows one to more effectively take into considerationthe interference component in the multipath detection process.


Fig. 3. Low-complexity realization of the IMDS.

V. PRACTICAL REALIZATION OF THE IMDS SCHEME

The MVU estimator given by (27) assumes perfect channelknowledge. However, this is not a reasonable assumption, par-ticularly at the delay detection stage where estimating the chan-nel seen at all the K delay offsets in ds becomes prohibitive. Toaccount for the hardware limitation of the receiver, a modifiedlow complexity version of the previous estimator should beimplemented at the detection stage. To estimate the interferencepower, coarse channel estimates at the search instants might beused instead. The channel estimates of the paths from the jthtransmit antenna to the receive antenna at the kth delay offset,i.e., αjk(n), are obtained by filtering the search results using afinite impulse response (FIR) filter with impulse response f(n)of length Kf as

αjk(n) =Kf−1∑i=0

f(i)hj(i − n, k). (33)

Once these channel estimates are obtained, the interferencevariance seen in the paths from the jth transmit antenna to thereceive antenna at the kth delay offset, i.e., σ2

jI(k), is estimatedusing the following estimator:

σ2jI(k) =

1NA

NA∑n=1

|hj(n, k) − αjk(n)|2 . (34)

In the low-complexity version of the IMDS, the estimator in(34) is used to compute the detection metric in accordancewith (26). Fig. 3 shows the structure for the low-complexityrealization of the IMDS used to produce the jth detectionmetric. As for the complexity associated with the added FIRfilter, multipath detection and finger assignment are typicallyperformed dynamically on a per-frame (or multiple frames)basis. In cdma2000, the search and assignment of fingers aretypically done every 20–80 ms (one to four frames). In ourwork, these processes are done every frame. Consequently, thissuggests that there is plenty of time to filter the search resultsand obtain the rough channel estimates needed by the IMDS.

The FIR filter can be systematically designed by referringto the optimum linear channel estimator working on the datamodel in (15), namely, the Wiener filter [25]. The shape ofthe Wiener filter is dictated by the covariance matrix of theinterference and the channel. Consequently, the Wiener filterimplementation requires very good estimates of the interferencevariance (per path) as well as the channel autocovariance func-tion. In order for the latter estimates to be accurate enough, long

TABLE IMSE IN ESTIMATING THE CHANNEL WITH DIFFERENT NUMBER OF TAPS

AND NORMALIZED DOPPLER RATES

records (compared to NA) should be used. We assume that suchlong despreading intervals are not available at the multipath de-tection stage where detection is done on a frame-by-frame ba-sis. Hence, the Wiener filter structure cannot provide us with arealistic filter shape to implement. Therefore, a normalized rec-tangular window FIR filter, i.e.,

∑Kf−1i=0 |f(i)|2 = 1, is adopted.

As we will see, this rectangular pulse shape is quite effective.The number of filter taps depends on the number of search

results NA and the normalized Doppler rate. Although adaptingthe number of filter taps according to the Doppler rate for agiven number of search results is a possible solution, it requiresestimating the Doppler rate at the acquisition stage, which, inturn, adds some extra complexity at the receiver side. To avoidany additional complexity, it is always desirable to have a fixeddesign for the FIR filter. To this end, the following steps aretaken to determine a reasonable filter length while taking intoconsideration the Wiener filter.

1) Use a normalized rectangular window FIR filter of lengthKf with impulse response, i.e.,

f(n) =1√Kf

Kf−1∑i=0

δ(n − iTb) (35)

where δ(t) is the Dirac delta function.2) Assuming that the channel is known, the number of

search results NA is fixed to yield a certain detectionperformance [as given by (30) and (32)]. NA, however,should not be selected arbitrarily large since that wouldincrease the acquisition time.

3) For a given NA, the number of taps in f(n) can beselected upon comparing the channel estimation perfor-mance, i.e., mean square error (MSE), with differentnumber of taps over a relatively wide Doppler bandwidth.For instance, when NA = 12 and SINR is 6 dB, Table Ishows the MSE in estimating the channel for one pathwith different number of taps and normalized Dopplerrates.

The residual noise-plus-interference power at the output ofthe filter f(n) is proportional to the noise-plus-interference


power at the input of the filter within the filter bandwidth.Consequently, the channel estimation performance deterioratesas the average SINR decreases.

VI. NUMERICAL AND SIMULATION RESULTS

A downlink CDMA system employing a (2, 1) STS schemeis considered for simulation. The relevant system parametersare listed:

• number of users in the sector: M = 20 with equal power(unless otherwise specified);

• random spreading sequence used as a BS-specific scram-bling code and a set of Walsh codes used as channelizationcodes with a spreading factor of 128 (unless otherwisespecified);

• number of paths: L = 3, with relative delays {0, 4, 8}chips and a uniform PDP, i.e., φ(τl) = 1/L ∀l;

• number of accumulated search results per pilot code:NA = 12;

• normalized Doppler rate: 10−3;• FIR filter length: Kf = 6 (from Table I, it is shown that

Kf = 6 provides relatively good performance);• pilot power gain: Gp = 10%, 15%, 20% of the total power

transmitted from the BS, which correspond to 3.5, 5.5, and7 dB, respectively, relative to a single traffic channel;

• fading coefficients estimated for the assigned RAKE fin-gers using a moving average filter with a length of 12 thatis operating over the pilot channel (after despreading);

• per-frame RAKE finger assignment for the EMDS andIMDS used whereby the detected paths corresponding tothe maximum three detection metrics in the search windoware assigned.

A. Validation of the Gaussian Approximation

In this section, we validate the use of the Gaussian approxi-mation in our derivation carried out in Section III. We use,in our simulations (only for this part of the results), an idealdownlink CDMA system with STS in a multiaccess frequency-selective fading channel. The receiver’s bit-error-rate (BER)performance under the conditions of perfect synchronizationand channel knowledge is computed for two cases. In the firstcase, no assumption is made about the statistical nature of themultiaccess interference (MAI) term in the decision statistics.For the second case, the MAI term is replaced by AWGN sourcewith a variance equal to the average power of the original MAIterm. The variance of the MAI is computed after despreadingthe signal corresponding to the MAI term alone, i.e., the desireduser’s signal is set to zero.

Fig. 4 shows the RAKE receiver BER performance for theaforementioned two cases for different combinations of numberof users and spreading factors. The simulated frequency selec-tive channel is composed of L = 3 paths with delays {0, 4, 8}.The normalized Doppler rate is set to 10−3. As can be seen fromthe figure, there is almost a perfect match between the two caseswhen the number of users is 20 (for both spreading factors). Theleast match is when the number of users is 5 and the spreadingfactor is 64, but this mismatch is not severe.

Fig. 4. Receiver BER performance with MAI approximated as AWGN com-pared to the actual system MAI for different number of users and spreadingfactors.

Fig. 5. Comparison between the simulation results and the derived approxi-mations for the probability of detection.

To further validate the Gaussian approximation used in theprobability of detection derivation for IMDS, we compare thisprobability as computed from the derived expressions withthose obtained via system simulations for 20 users. Fig. 5compares the approximate probability of detection expressionsin (24) and (32) with the results computed by simulating theactual system. In the figure, the probability of detecting the firstpath is plotted against the threshold value for both detectionschemes when the average bit-energy-to-noise power spectraldensity Eb/N0 = 0 dB and the pilot power gain Gp = 7 dB.As shown in the figure, there is an excellent agreement betweenthe simulation results and the approximations derived inthis paper.

The authors believe that the results presented in this paperalong with those published in the literature (for more recentconsiderations, see [16]–[18]) establish the point behind using


Fig. 6. Probability of detecting a certain multipath component as function ofthe pilot-to-interference-power ratio per antenna.

this approximation in the first place, e.g., to be able to, approxi-mately, predict the performance of the IMDS. In the following,the performance of the proposed approach is mostly assessedbased on system simulations only, i.e., no approximations.

B. Performance Evaluation

Fig. 6 shows the probability of detecting one of the mul-tipath components as a function of the pilot-to-interference-power ratio per antenna Gp/2σ2

I computed from the previousexpressions. The probability of detection has been computedwhile the false alarm rate is kept constant at 1% and NA = 12for both schemes. As shown in the figure, the IMDS providesa superior detection performance compared to the EMDS. Thepractical ramification of this improvement is a significant pilotpower saving (almost 3 dB), as will be further manifested in thefollowing results.

Fig. 7 shows the receiver operational characteristics (ROCs)for both detection schemes. The probability of detecting thefirst path is plotted against the probability of false alarm atEb/N0 = 8 dB for different pilot power gains. It is evidentfrom the figure that the IMDS considerably enhances the ROCscompared to the EMDS. In particular, at the same pilot powergain, e.g., Gp = 7 dB, the IMDS results in a higher probabilityof detection at any given practical probability of false alarm.For instance, at 1% probability of false alarm, while the EMDSresults in a probability of detection of around 77%, the IMDSresults in a probability of detection of around 95%. It is alsointeresting to observe that the IMDS with a pilot power gain of3.5 dB outperforms the EMDS operating at a pilot power gainof 7 dB. As far as multipath acquisition is concerned, this resultimplies that the IMDS with Gp = 3.5 dB is, at least, equivalentto the EMDS with Gp = 7 dB.

The probability of accurately assigning the first path to aRAKE finger as a function of Eb/N0 (in decibels) is investi-gated in Fig. 8, where the probability of false alarm is set to 1%.As shown in the figure, the IMDS is more capable of identifying

Fig. 7. ROCs produced by the EMDS and IMDS at Eb/N0 = 8 dB fordifferent pilot power gains.

Fig. 8. Probability of accurate assignment of the first path as a function ofEb/N0 at different pilot power gains for both schemes.

the first path and assigning it to one of the RAKE fingersthan the EMDS. Once again, the efficiency of the IMDS inutilizing the pilot power is illustrated by comparing the IMDSwith Gp = 3.5 dB to the EMDS with Gp = 7 dB.

In Fig. 9, the probability of accurately assigning all threepaths to the available RAKE fingers for both detection schemesis depicted as a function of Eb/N0 (in decibels) for different pi-lot power gains. It is clear that the IMDS consistently maintainsa superior assignment performance compared to the EMDS.This indicates that the IMDS provides better diversity to theRAKE system.

Fig. 10 shows the BER performance of the receiver as afunction of Eb/N0 (in decibels) for different pilot power gains.The ideal performance corresponding to the case when thechannel is perfectly known is also shown in the figure. It isevident from the figure that the deficiency of the EMDS indetecting the potential paths significantly deteriorates the BER


Fig. 9. Probability of accurate assignment of the three paths as a function ofEb/N0 at different pilot power gains for both schemes.

Fig. 10. BER performance of the (2, 1) STS scheme as a function of Eb/N0

for different pilot power gains when the paths are perfectly known and whenthey are detected and assigned based on the EMDS and IMDS schemes.

performance. As the pilot power increases, the gap betweenthe ideal and practical performances reduces since better es-timates of the delays and the fading coefficients are used forcombining. For the EMDS, the performance gap is almost3.5 dB with Gp = 20% at high Eb/N0. Utilizing the IMDSwith the same pilot power, however, results in reducing the gapto almost 1.5 dB, suggesting that the IMDS provides a gainof around 2 dB compared to the EMDS at high Eb/N0. Wealso observe that IMDS with Gp = 10% draws the same BERperformance as that of the EMDS with Gp = 20%, suggestingagain that the IMDS saves at least 3.5 dB of the pilot power. Theobtained BER results are consistent with detection and assign-ment results shown in Figs. 7–9. The IMDS provides the RAKEwith accurate path delays most of the time, even when the pilotpower is low, and hence, accurate channel estimates are ob-tained most of the time. The EMDS, on the other hand, results

in assigning wrong path delays to the RAKE fingers, which, inturn, drastically increases the channel estimation errors.

Finally, we remark, as it has been demonstrated in [9] and[10], that there is a practical tradeoff between the diversityorder and the pilot power. This tradeoff is brought about bythe channel estimation errors, which increase as the pilot powerper path per antenna decreases. Therefore, the potential ofthe IMDS in reducing the required pilot power compared tothe EMDS is particularly appealing for the implementationof the STS schemes. Although the current standard does notinclude more than two transmit antennas, future versions mightincorporate this option. In this case, the IMDS would still be aprominent replacement of the EMDS.

VII. CONCLUSION

In this paper, the problem of multipath detection inCDMA systems employing space–time spreading has beeninvestigated. The conventional EMDS has been extended totake into consideration the use of multiple transmit antennas,and its performance has been derived in terms of detection andfalse alarm probabilities. Motivated by the poor performanceof the EMDS, we have proposed an IMDS, which is based onthe estimation and subtraction of the interference power perpath before that path is detected. Analytical expressions for thedetection and false alarm probabilities for the IMDS schemehave been presented. Analytical and simulation results showedthat the IMDS has a significant improvement over the EMDS interms of multipath detection capability and BER performanceof the RAKE receiver, with a gain of about 2 dB. We have alsoproposed a low-complexity version of the IMDS scheme.

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Mohamed Abou-Khousa (S’01) received the B.S.degree (magna cum laude) in electrical engineeringfrom the American University of Sharjah, Sharjah,United Arab Emirates, in 2003 and the M.S. degreein electrical engineering from Concordia University,Montreal, QC, Canada, in 2004. He is currentlyworking toward the Ph.D. degree in electrical engi-neering at Missouri University of Science and Tech-nology (Missouri S&T, formally UMR).

Since January 2005, he has been with the AppliedMicrowave Nondestructive Testing Laboratory,

Missouri S&T, as a Graduate Research Assistant. His current research interestsinclude millimeter-wave and microwave nondestructive testing, numericalelectromagnetic analysis, reconfigurable antennas, and wideband wirelesscommunication systems.

Ali Ghrayeb (S’97–M’00–SM’06) was born inPalestine. He received the B.Sc. degree in elec-trical engineering from the University of Jordan,Amman, Jordan, in 1994, the M.Sc. degree in electri-cal engineering from New Mexico State University,Las Cruces, in 1996, and the Ph.D. degree in elec-trical engineering from the University of Arizona,Tucson, in 2000.

He is currently an Associate Professor with theDepartment of Electrical and Computer Engineering,Concordia University, Montreal, QC, Canada. He has

been recently appointed as the Concordia University Research Chair of High-Speed Wireless Communications. He is the coauthor of the book Coding forMIMO Communication Systems (Wiley, 2007). His research interests includedigital and wireless communications, channel coding, turbo codes, space–timecodes, and signal processing and coding for data transmission and storage.

Dr. Ghrayeb has co-instructed technical tutorials on Coding for MIMOSystems and on Synchronization for WCDMA Systems at several major IEEEconferences, including the Global Telecommunications Conference and theInternational Conference on Communications. He serves as an Associate Editorof the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY. He also servesas an Associate Editor of the Wiley Wireless Communications and MobileComputing Journal.

Mohamed El-Tarhuni (S’93–M’97–SM’05) re-ceived the B.Sc. and M.Sc. degrees in electricalengineering from Garyounis University, Benghazi,Libya, in 1986 and 1990, respectively, and the Ph.D.degree in electrical engineering from Carleton Uni-versity, Ottawa, ON, Canada, in 1997.

From 1987 to 1993, he was with the GeneralElectric Company of Libya, Benghazi, Libya, as aTelecommunications Engineer and Manager of theCommunications Department. From 1997 to 2000,he was with Nortel Networks, Ottawa, ON, Canada,

as a member of Scientific Staff, working on third-generation wireless commu-nication systems using CDMA technology. Since 2000, he has been with theAmerican University of Sharjah (AUS), Sharjah, United Arab Emirates (UAE),where he is currently an Associate Professor and the Head of the Department ofElectrical Engineering. From 2001 to 2006, he was the Director for the CiscoRegional Networking Academy, AUS. He has published about 50 journal andconference papers. His current research interests include wireless and mobileradio systems, CDMA, OFDM, physical layer issues, and cross-layer design.

Dr. El-Tarhuni has served on the organizing and technical committeesof many international conferences, such as the International Symposium onPersonal, Indoor, and Mobile Radio Communications in 2006 and 2007,the International Conference on Communications in 2004, and the VehicularTechnology Conference in 2006. He served as the IEEE Industry RelationsOfficer for the UAE Section from 2002 to 2006.

A Multipath Detection Scheme for CDMA Systems With Space–Time Spreading

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