Finger Replacement Schemes for RAKE Receivers in the Soft...

2114 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 4, JULY 2008

Finger Replacement Schemes for RAKEReceivers in the Soft Handover Region

With Multiple Base StationsSeyeong Choi, Member, IEEE, Mohamed-Slim Alouini, Senior Member, IEEE,

Khalid A. Qaraqe, Senior Member, IEEE, and Hong-Chuan Yang, Senior Member, IEEE

Abstract—We propose and analyze new finger replacementtechniques for RAKE reception in the soft handover (SHO) regionwith multiple base stations. The proposed schemes are basicallybased on the block comparison among groups of resolvable pathsfrom different base stations and lead to the reduction of complex-ity while offering commensurate performance in comparison withpreviously proposed schemes. Relying on the newly derived ana-lytical expressions, we investigate not only the complexity in termsof the average number of required path estimations/comparisonsand the SHO overhead but the error performance of the proposedscheme over independent identically distributed (i.i.d.) fadingchannels as well. We also examine, via computer simulations, theeffect of path unbalance/correlation as well as outdated/imperfectchannel estimation and show the robustness of our proposedscheme to these practical limitations.

Index Terms—Diversity techniques, fading channels, perfor-mance analysis, RAKE receiver, soft handover (SHO).

I. INTRODUCTION

MULTIPATH fading is an unavoidable physical phenom-enon that affects considerably the performance of wide-

band wireless communication systems. While usually viewedas a deteriorating factor, multipath fading can also be exploitedto improve the performance by using RAKE-type receivers[1, Sec. 9.5.1]. These receivers use several baseband corre-lators, called fingers, to individually process multipath signalcomponents from different base stations (BSs) and, as such,facilitate soft handover (SHO). The outputs from the differentcorrelators are coherently combined to achieve improved reli-ability and performance. The effects of bandwidth on spreadspectrum systems have been investigated in [2] and [3] byquantifying the RAKE receiver output signal-to-noise ratio

Manuscript received January 8, 2007; revised July 6, 2007 and September 27,2007. This work was supported in part by the Qatar Foundation forEducation, Science, and Community Development, by Qatar Telecom, andby NSERC, Canada, under a Discovery Grant. The review of this paper wascoordinated by Dr. M. Stojanovic.

S. Choi, M.-S. Alouini, and K. A. Qaraqe are with the Department ofElectrical Engineering, Texas A&M University at Qatar, Doha 23874,Qatar (e-mail: [email protected]; [email protected]; [email protected]).

H.-C. Yang is with the Department of Electrical and Computer Engineering,University of Victoria, BC, V8W 3P6, Canada (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2007.912162

(SNR) and the error performance in terms of the numberof fingers, spreading bandwidth, and multipath spread of thechannel.

In the SHO region, however, due to the limited number offingers in the mobile unit, we are faced with a problem of howto judiciously select a subset of paths for the RAKE receptionto achieve the required performance while doing the following:1) maintaining a low complexity and low processing powerconsumption and 2) using a minimal amount of additional net-work resources. Recently, by considering macroscopic diversityschemes with two BSs, the authors have proposed and analyzednew finger assignment schemes that maintain a low complexityand reduce the usage of the network resources in the SHO re-gion [4], [5]. The main idea behind [4] and [5] is that in the SHOregion, the receiver uses the additional network resources onlyif needed. More specifically, with the scheme in [4], wheneverthe received signal from the currently used BS (serving BS) isunsatisfactory, the receiver scans the additional resolvable pathsfrom the neighboring BS (target BS) and selects the strongestpaths among the total available paths from both the servingand the target BSs. It has been shown that this scheme canreduce the unnecessary path SNR estimations and the SHOoverhead compared to the conventional generalized selectioncombining (GSC) scheme [6]–[10] in the SHO region. In [11],the results in [4] are generalized to the multi-BS situation bydeveloping two different path scanning schemes, which aredenoted as the full scanning scheme and the sequential scanningscheme. With the full scanning scheme, whenever the receivedsignal becomes unsatisfactory, the RAKE receiver scans all theavailable paths from all potential target BSs, whereas with thesequential scanning scheme, the RAKE receiver sequentiallyscans the target BSs until the combined SNR is satisfactory orall potential target BSs are scanned.

In [5], an alternative finger selection scheme for the SHOregion was proposed to further reduce the SHO overhead atthe expense of a certain degradation in performance. With thisscheme, when the output SNR falls below the target SNR,the receiver scans the additional resolvable paths from thetarget BS. However, unlike the scheme in [4], the receivercompares the sum of the SNRs of the strongest paths amongthe paths from the target BS with the sum of the weakestSNRs among the currently used paths from the serving BSand selects the better group. This scheme compares two blocksof equal size and, as such, avoids reordering all the paths,

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CHOI et al.: FINGER REPLACEMENT SCHEMES FOR RAKE RECEIVERS IN THE SOFT HANDOVER REGION 2115

which is required for the scheme in [4]. Therefore, a reductionin path estimations, SNR comparisons, and SHO overheadcan be obtained. The schemes proposed in [4] and [5] werefurther investigated in more practical fading environments in[12]. In this paper, we generalize the results in [5] to themulti-BS situation. Similar to [11], we propose two scanningschemes, which are denoted the full scanning scheme and thesequential scanning scheme. Both scanning schemes employblock comparison instead of the full GSC used in [11]. For thesake of clarity, we call the proposed scheme in this paper thereplacement scheme and the scheme in [11] the reassignmentscheme.

The main contribution of this paper is to present a generalcomprehensive framework for the performance and the com-plexity assessment of the replacement scheme by providing notonly analytical results for independent identically distributed(i.i.d.) fading environments but computer simulation resultsfor non-i.i.d. fading channels and under outdated/imperfectchannel estimates as well. More specifically, in our derivations,we accurately quantify the complexity measures of interestand statistics of the receiver output SNR, which are used toanalyze the performance of the proposed scheme over i.i.d.fading channels. Through computer simulations, we show thatthe replacement scheme is also applicable to practical fad-ing conditions with correlated and/or nonidentical resolvablepaths. More importantly, the simulation results show that thereplacement scheme shows, in comparison to the reassign-ment scheme, a considerable robustness to channel estimationerrors.

The rest of this paper is organized as follows: In Section II,we present the channel and system model under considerationas well as the mode of operation of the proposed scheme.Based on this mode of operation, we illustrate in Section IIIthe complexity of the proposed schemes by quantifying theaverage number of path estimations, the average number ofSNR comparisons, and the SHO overhead. We then derivethe expressions for the statistics of the combined SNR inSection IV. These results are next applied to the performanceanalysis of the proposed systems in Section V. We consider theeffect of path unbalance/correlation in Section VI and evaluatethe performance in the presence of an outdated or imperfectchannel estimate in Section VII. Finally, Section VIII providessome concluding remarks.

II. SYSTEM MODEL

A. Channel and System Model

We assume that in the SHO region, N BSs are active,and there are a total of L(N) resolvable paths, where L(N) =∑N

n=1 Ln, and Ln is the number of resolvable paths fromthe nth BS. We also assume that the mobile unit in the SHOregion is at roughly the same distance from the serving and thetarget BSs, and therefore, the average signal strength on eachpath from the BSs is assumed to be the same. As such, we firstassume that the received signals on all the resolvable paths fromthe serving and the target BSs experience i.i.d. Rayleigh fadingin which the instantaneous received SNRs γ of all the available

resolvable paths follow the same exponential distribution witha common probability density function (PDF) and cumulativedistribution function (CDF) given by [1, eq. (6.5)]

fγ(x) =1γ

exp(−x

γ

), x ≥ 0 (1)

Fγ(x) = 1 − exp(−x

γ

), x ≥ 0 (2)

respectively, where γ is the common average faded SNR. Lateron, more practical fading environments will be considered.

Next, we consider that the mobile unit is equipped with anLc finger RAKE receiver and is capable of despreading sig-nals from different BSs using different fingers to facilitate theSHO process. The receiver operates over a “perfect” uniformpropagation delay profile provided by a multipath searcher ina way that the multipath components are correctly assigned tothe RAKE fingers. In the SHO region, according to the modeof operation described in Section II-B, only Lc out of L(n),1 ≤ n ≤ N paths are used for RAKE reception.

B. Mode of Operation

Without loss of generality, let L1 be the number of resolvablepaths from the serving BS and L2, L3, . . . , LN be those fromthe target BSs. In the SHO region, the receiver is assumed atfirst to rely only on L1 resolvable paths and, as such, startswith Lc/L1-GSC. The proposed schemes are based on thecomparison of blocks consisting of Ls(< Lc < L1) paths fromeach BS.

For simplicity, if we let

Y =Lc−Ls∑

i=1

γi:L1 (3)

Wn =

{∑Lc

i=Lc−Ls+1 γi:Ln, n = 1∑Ls

i=1 γi:Ln, n = 2, . . . , N

(4)

where γi:Lnis the ith order statistics out of Ln SNRs of paths

from the nth BS (see [8] for terminology), then, the receivedoutput SNR after GSC is given by Y + W1. At the beginningof every time slot, the receiver compares the GSC output SNRY + W1 with a certain target SNR, which is denoted by γT .If Y + W1 is greater than or equal to γT , a one-way SHOis used, and no finger replacement is needed. On the otherhand, whenever Y + W1 falls below γT , the receiver attemptsa two-way SHO by starting to scan additional paths from thetarget BSs. More specifically, we consider the two differentscanning schemes described below.Case I—Full Scanning: In this case, when Y + W1 < γT ,

the RAKE at once scans all target BSs and compares allWn, that is, the sum of the Ls smallest paths among the Lc

currently used paths from the serving BS (i.e., W1) and thesums of the Ls strongest paths from each target BS (i.e., Wi,2 ≤ i ≤ N ). Then, the receiver replaces W1 with the largest



one. Hence, the final combined SNR, which is denoted by γFull,is mathematically given by

γFull ={

Y + W1, Y + W1 ≥ γT

Y + max{W1, . . . ,Wn}, Y + W1 < γT .(5)

Note that even in the case of Y + W1 < γT , no replacementoccurs if W1 = max{W1, . . . ,WN}.

Case II—Sequential Scanning: In this case, when Y +W1 < γT , the RAKE receiver estimates L2 paths from the firsttarget BS and replaces W1 with W2. The receiver then checkswhether the combined SNR Y + W2 is above γT or not. Bysequentially scanning the remaining target BSs, this process isrepeated until either the combined SNR Y + Wi, 2 ≤ i ≤ N isabove γT or all the N BSs are examined. In the latter case,since the SNRs of all the paths are known, the receiver selectsthe largest Wn. Based on this mode of operation, we can seethat the final combined SNR, which is denoted by γSeq, ismathematically given by

γSeq =

Y + W1, Y + W1 ≥ γT

Y + Wi, Y + Wj < γT

Y + Wi ≥ γT

for j = 1, . . . , i − 1and i = 2, . . . , N

Y + max{W1, . . . ,Wn}, Y + Wn < γT

for n = 1, . . . , N .(6)

III. COMPLEXITY COMPARISONS

In this section, we look into the complexity of the proposedschemes by accurately quantifying the average number of pathestimations, the average number of SNR comparisons, and theSHO overhead, which are required during the SHO process ofthese schemes.

A. Average Number of Path Estimations

Case I—Full Scanning: With this scheme, the RAKE re-ceiver estimates SNRs of L1 paths in the case of Y + W1 ≥ γT

or L(N) in the case of Y + W1 < γT . Hence, we can easilyquantify the average number of path estimations, which isdenoted by EFull, as

EFull = L1 Pr[Y + W1 ≥ γT ] + L(N) Pr[Y + W1 < γT ](7)

which reduces to

EFull = L1 +(L(N) − L1

)FY +W1(γT ) (8)

where FY +W1(·) is the well-known CDF of the Lc/L1-GSCoutput SNR [13, eq. (9.440)].Case II—Sequential Scanning: In this scheme, we can write

the average number of path estimations, which is denoted byESeq, in the following summation form:

ESeq =N∑

n=1

L(n) · πn (9)

where πn is the probability that L(n) paths are estimated. Basedon the mode of operation in Section II-B2, we have

πn =

Pr[Y + W1 ≥ γT ], n = 1Pr[Y + W1 < γT , . . . ,

Y + Wn−1 < γT , Y + Wn ≥ γT ], 1 < n < NPr[Y + W1 < γT , . . . ,

Y +WN−1 <γT ], n = N .(10)

Note that Wn are independent, whereas Y + Wn are correlated.Hence, by conditioning on Y , the joint probabilities in (10) canbe calculated as

Pr[Y + W1 < γT , . . . , Y + Wn−1 < γT , Y + Wn ≥ γT ]

=

γT∫0

fY (y)FW1|Y =y(γT − y)

×(

n−1∏m=2

FWm(γT − y)

)(1 − FWn

(γT − y)) dy (11)

Pr[Y + W1 < γT , . . . , Y + WN−1 < γT ]

=

γT∫0

fY (y)FW1|Y =y(γT − y)N−1∏m=2

FWm(γT − y)dy (12)

respectively, where fY (·) is the PDF of (Lc − Ls)/L1-GSC output SNR [13, eq. (9.433)], FWm

(·) and FWn(·)

are the CDFs of Ls/Lm-GSC and Ls/Ln-GSC output SNR[13, eq. (9.440)], respectively, and

FW1|Y =y(x)=

∫ x

0fY,W1 (y,w1)dw1

fY (y) , 0 ≤ x< Ls

Lc−Lsy

1, x≥ Ls

Lc−Lsy.

(13)

In (13), fY,W1(·, ·) is the joint PDF of two adjacent par-tial sums Y and W1 of order statistics and can be found in[5, eq. (27)]. After successive substitutions from (13) to (9), wecan analytically obtain the average number of path estimations.

B. Average Number of SNR Comparisons

As another complexity measure, in this section, we evaluatethe average number of required SNR comparisons. Noting thatthe average number of SNR comparisons for i/j-GSC, whichis denoted by CGSC(i,j), can be obtained as

CGSC(i,j) =min[i,j−i]∑

k=1

(j − k) (14)

we can express the average number of SNR comparisons for thefull scanning scheme and the sequential scanning scheme as

CFull = CGSC(Lc,L1) Pr[Y + W1 ≥ γT ]

+

(CGSC(Lc,L1) + CGSC(Ls,Lc)

+N∑

n=2

CGSC(Ls,Ln) + CGSC(1,N)

)

× Pr[Y + W1 < γT ] (15)



CSeq =CGSC(Lc,L1) Pr[Y + W1 ≥ γT ]

+N∑

n=2


+n∑

i=2

CGSC(Ls,Li)

)· πn

+


+N∑

i=2

CGSC(Ls,Li)

)· φN

+ CGSC(1,N) · πN (16)

respectively, where πn is defined in (10), and

φN =Pr[Y +W1 <γT , . . . , Y +WN−1 <γT , Y +WN ≥ γT ].(17)

C. SHO Overhead

The SHO overhead, denoted by β, is commonly used toquantify the SHO activity in a network and is defined as[14, eq. (9.2)]

β =2∑

n=1

nPn − 1 (18)

where Pn is the average probability that the mobile unit usesn-way SHO. Note that in the proposed schemes, at most,two-way SHO is used, and both schemes have the same SHOoverhead since in both schemes, one-way SHO will be usedif Y +W1≥γT or Y +W1 <γT ,W1≥Wi for i=2, 3, . . . , N ;otherwise, two-way SHO is used. Therefore, we have

β= P2 =1−P1 (19)

P1 = Pr[Y +W1≥γT ]+ Pr [Y +W1 <γT ,W1 =max{W1, . . . ,Wn}] . (20)

Using the same conditioning method, we can express the jointprobability in (20) as

Pr [Y + W1 < γT ,W1 = max{W1, . . . ,WN}]= Pr[Y + W1 < γT ,W1 > W2, . . . ,W1 > WN ]

=

γT∫0

fW1(w1)FY |W1=w1(γT − w1)

×N∏

i=2

FWi(wi)dwi. (21)

Successive substitutions from (21) to (19) lead to the ana-lytical expression for the SHO overhead as

β = FY +W1(γT ) −γT∫0

fW1(w1)FY |W1=w1(γT − w1)

×N∏

i=2

FWi(wi)dwi (22)

where fW1(·) is the marginal density of W1, which can beobtained from the joint PDF fY,W1(·, ·) as

fW1(w1)

=

∞∫Lc−Ls

Lsw1

fY,W1(y, w1)dy (23)

FY |W1=w1(x)

=

0, 0 ≤ x < Lc−Ls

Lsw1∫ x

Lc−LsLs

w1fY,W1 (y,w1)dy

fW1 (w1), x ≥ Lc−Ls

Lsw1.

(24)

D. Numerical Examples1

In Fig. 1, we plot the following: 1) the average numberof path estimations; 2) the average number of SNR compar-isons; and 3) the SHO overhead versus the output thresholdγT of the proposed replacement scheme for various values ofLs over i.i.d. Rayleigh fading channels when N = 4, L1 =L2 = L3 = L4 = 5, Lc = 3, and γ = 0 dB. For comparisonpurposes, we also plot those for the reassignment scheme2

[11] and conventional Lc/L(N)-GSC. Recall from [11] thatthe reassignment scheme is acting as Lc/L(N)-GSC when theoutput threshold becomes large. Hence, we can observe from allthe subfigures that the reassignment scheme converges to GSCas γT increases.

Although both the replacement and reassignment schemeshave almost the same path estimation load [see Fig. 1(a)], itcan be seen in Fig. 1(b) that the replacement scheme leadsto a greater reduction of the SNR comparison load comparedto the reassignment scheme, and the amount of the reductionincreases as γT increases or the block size Ls decreases.

For the SHO overhead, it is clear in Fig. 1(c) that bothschemes have a better chance of relying on the target BSs asγT increases. Note that unlike the reassignment scheme, theproposed replacement scheme selects the acceptable paths fromat most two BSs in any case. Hence, the maximum value ofthe SHO overhead for the replacement scheme is one, whichessentially leads to a reduction of the SHO overhead comparedto the reassignment scheme. Also, we can see that in our pro-posed scheme, the smaller block size provides a slightly higherSHO overhead for large γT . This is because as Ls decreases,the sum of the Ls smallest paths among the Lc currently usedpaths from the serving BS has a higher probability of being lessthan the sums of the Ls strongest paths from the target BSs, andas such, we have a higher chance to replace groups.

1As a double check, all numerical evaluations obtained from the analyticalresults derived in this paper have been compared and verified by Monte Carlosimulations.

2Note that the analytical results in [11] are based on the approximationmethods used in [4]. Although these approximation methods have proved tobe precise, for a more accurate comparison in this paper, we use the simulationresults for the reassignment scheme.



Fig. 1. Complexity tradeoff versus the output threshold γT of the re-assignment (RA) and replacement (RP) schemes with the full scanning (FS)and sequential scanning (SS) and conventional GSC over i.i.d. Rayleighfading channels when N = 4, L1 = · · · = L4 = 5, Lc = 3, and γ = 0 dB.(a) Average number of path estimations. (b) Average number of SNR com-parisons. (c) SHO overhead.

In summary, from the complexity point of view, our proposedscheme offers a big advantage over the reassignment schemeas well as conventional Lc/L(N)-GSC, particularly when theoutput threshold γT is relatively higher than the average SNRper path γ. In what follows, we obtain the analytical expressions

for the statistics of the output SNR, and based on this, wequantify the average bit error rate (BER) of the proposedscheme. As such, a comprehensive investigation of the tradeoffbetween complexity and performance is feasible.

IV. STATISTICS OF COMBINED SNR

Based on the mode of operation in Section II-B, we derive inthis section the statistics of the combined SNR of the proposedscheme.Case I—Full Scanning: From (5), the CDF of the combined

SNR γFull can be written as

FγFull(x) = Pr[γFull < x]= Pr[γT ≤ Y + W1 < x]

+ Pr [Y + W1 < γT

Y + max{W1, . . . ,WN} < x] . (25)

Following the same conditioning approach, we have

Pr [Y + W1 < γT , Y + max{W1, . . . ,WN} < x]

=

min{x,γT }∫0

fY (y)FW1|Y =y (min{x, γT } − y)

×N∏

i=2

FWi(x − y)dy. (26)

Substituting (26) into (25), we can obtain the analyticalexpression for the CDF of γFull in (27), shown at the bottomof the page.

By using Leibnitz’s rule [15, eq. (6.40)] and differentiating(27) with respect to x, we can obtain the generic expressionfor the PDF of the combined SNR γFull. Specifically, if weassume that the number of resolvable paths from each BS isthe same (i.e., L1 = · · · = LN ), then the PDF of γFull can begiven in (28), shown at the bottom of the page, where

fW1|Y =y(x) =

{fY,W1 (y,x)

fY (y) , 0 ≤ x < Ls

Lc−Lsy

0, x ≥ Ls

Lc−Lsy.

(29)

FγFull(x) =

x∫0

fY (y)FW1|Y =y(x − y)∏N

i=2 FWi(x − y)dy, 0 ≤ x < γT

FY +W1(x) − FY +W1(γT ) +γT∫0

fY (y)FW1|Y =y(γT − y)∏N

i=2 FWi(x − y)dy, x ≥ γT

(27)

fγFull(x) =

x∫0

fY (y)(fW1|Y =y(x − y) (FW2(x − y))N−1

+FW1|Y =y(x − y)(N − 1) (FW2(x − y))N−2 fW2(x − y))

dy, 0 ≤ x < γT

fY +W1(x) +γT∫0

fY (y)FW1|Y=y(γT −y)×(N−1) (FW2(x−y))N−2 fW2(x−y)dy, x ≥ γT

(28)



Case II—Sequential Scanning: From (6), the CDF of thecombined SNR γSeq can be calculated as

FγSeq(x)= Pr[γSeq < x]= Pr[γT ≤ Y + W1 < x]

+N∑

i=2

Pr[Y + W1 < γT , . . .

Y + Wi−1 < γT , γT ≤ Y + Wi < x]+ Pr [Y +W1 <γT , . . .

Y+WN <γT , Y+max{W1, . . . ,WN}<x] .(30)

All the joint probabilities in (30) can be expressed using thesame conditioning approach, leading to (31), shown at thebottom of the page. For L1 = · · · = LN , we can obtain the PDFof γSeq as in (32), shown at the bottom of the page.

V. AVERAGE BER COMPARISONS

In this section, we apply the results from Section IV tothe performance analysis of our proposed combining schemesover i.i.d. Rayleigh fading channels. More specifically, bypresenting some selected numerical examples, we examine itsaverage BER. The average BER can be calculated by findingthe expected value of the conditional probability of error. Forexample, the average BER for binary phase-shift keying(BPSK) can be expressed as

Pb(E) =

∞∫0

Q(√

2x)f(x)dx (33)

where Q(·) is the Gaussian Q-function, which is defined asQ(x) = (1/

√2π)

∫ ∞x e−t2/2dt, and f(x) is the PDF of the

combined SNR, which is obtained in (28) for the full scanningscheme and in (32) for the sequential scanning scheme.

Fig. 2. Average BER of BPSK versus the average SNR per path γ of theproposed scheme with the full scanning (FS) and sequential scanning (SS) forvarious values of γT and Ls over i.i.d. Rayleigh fading channels when N = 4,L1 = · · · = L4 = 5, and Lc = 3.

Fig. 2 presents the average BER of BPSK versus the averageSNR per path γ of the proposed replacement scheme with thefull scanning and the sequential scanning for various valuesof γT over i.i.d. Rayleigh fading channels when N = 4, L1 =L2 = L3 = L4 = 5, and Lc = 3. From this figure, it is clearthat the higher the threshold, the better the performance, asmore additional paths need to be used to meet the high thresholdrequirement. Both scanning schemes show almost the sameperformance when the threshold is small (i.e., γT = −5 dB)or large (i.e., γT = 15 dB). We also observe that when thethreshold is large, both schemes will benefit from larger Ls.This is because when the replacement is needed, we have little

FγSeq(x)

=

x∫0

fY (y)FW1|Y =y(x − y)∏N

i=2 FWi(x − y)dy, 0 ≤ x < γT

FY +W1(x) − FY +W1(γT )

+∑N

i=2

(γT∫0

fY (y)FW1|Y =y(γT − y)(∏i−1

j=2 FWj(γT − y)

)(FWi

(x − y) − FWi(γT − y))

)dy

+γT∫0

fY (y)FW1|Y =y(γT − y)∏N

i=2 FWi(γT − y)dy, x ≥ γT

(31)

fγSeq(x) =

x∫0

fY (y)(fW1|Y =y(x − y) (FW2(x − y))N−1

+FW1|Y =y(x − y)(N − 1) (FW2(x − y))N−2 fW2(x − y))

dy, 0 ≤ x < γT

fY +W1(x) +γT∫0

fY (y)FW1|Y =y(γT − y)fW2(x − y)1−(FW2 (γT −y))N−1

1−FW2 (γT −y) dy, x ≥ γT

(32)



Fig. 3. Average BER of BPSK versus the average SNR per path of thereassignment (RA) and replacement (RP) schemes for various values of γT

over i.i.d. Rayleigh fading channels when N = 4, L1 = · · ·=L4 =5, Lc =3,and Ls = 2. (a) Full scanning scheme. (b) Sequential scanning scheme.

benefit from the additional paths if Ls = 1 compared to the caseof Ls = 2, whereas for the low threshold, the variation of theblock size does not affect the performance since, in this case, noreplacement is needed. For the midrange of the output threshold(i.e., γT = 5 dB), the full scanning scheme has slightly betterperformance than the sequential scanning scheme over themedium SNR range. However, as shown in Fig. 1, with thisslight (negligible) performance loss, the sequential scanningscheme can reduce the unnecessary path estimations and SNRcomparisons, compared to the full scanning scheme.

In Fig. 3, we compare the error performance of the replace-ment scheme to that of the reassignment scheme when Ls = 2and other parameters are the same as in Fig. 2. For the fullscanning scheme [Fig. 3(a)], since the paths with the largestSNR values are selected whenever needed, the reassignmentscheme always provides better performance than the replace-ment scheme. However, with the sequential scanning scheme[Fig. 3(b)], we can observe that although the performance ofthe reassignment scheme is better than that of the replacementscheme for most cases, the replacement scheme performs betterfor the high-average-SNR region (γ > 0 dB) and mediumvalues of the threshold (γT = 5 dB). This can be interpretedas follows: In this case, to exceed the output threshold, thereplacement scheme has to scan more and more BSs, and as

such, there is a higher chance to acquire a block with betterquality, whereas the reassignment scheme needs fewer BSs tomeet that threshold requirement.

VI. EFFECT OF PATH UNBALANCE/CORRELATION

In practice, the i.i.d. fading scenario on the diversity paths isnot always realistic due to, for example, the different adjacentmultipath routes with the same path loss and the resulting un-balance and correlation among paths. In this section, we assessthrough some computer simulations the effect of nonidenticallydistributed paths with correlation on the performance of boththe reassignment and the replacement schemes. More specifi-cally, instead of the uniform power delay profile (PDP) consid-ered so far, we now consider an exponentially decaying PDP, forwhich γj = γ1e

−δ(j−1), where γj(1 ≤ j ≤ Ln, 1 ≤ n ≤ N) isthe average SNR of the jth path out of the total availableresolvable paths from each BS, and δ is the average fadingpower decaying factor. For the correlated paths, we considerthe exponential correlation models where an exponential powercorrelation coefficient ρ|j−j′|, for ρ ∈ [0, 1], between any pairof paths γj and γj′ from a certain BS is assumed. Note thatδ = 0 means identically distributed paths, and ρ = 0 meansindependent fading paths. When we set δ = 0 and ρ = 0, werevert to the i.i.d. fading channels.

In Fig. 4, we plot the average BER of BPSK versus γ1 ofthe replacement and the reassignment schemes over an expo-nentially decaying PDP with an exponential correlation acrossthe multipaths. These results show that the PDP and the corre-lation among paths induce a non-negligible degradation in theperformance and, therefore, must be taken into account for theaccurate prediction of the performance of proposed schemes. Inall cases, we can see the same relationship between the replace-ment and the reassignment schemes as observed in Fig. 3.

VII. EFFECT OF OUTDATED OR IMPERFECT

CHANNEL ESTIMATION

In general, diversity-combining techniques rely, to a largeextent, on accurate channel estimation. As a typical first stepin performance analysis, perfect estimation was assumed so far.However, in practice, these estimates must be obtained in thepresence of noise and time delay. Hence, the effects of channelestimation error or channel decorrelation on the performanceof diversity systems are of interest. In this section, we studythe effect of outdated or imperfect channel estimates on theperformance. To isolate the effects from other factors, all thediversity paths are assumed to be i.i.d. Let γτ be the estimatedreceived signal power. Due to imperfect or outdated channelestimates, γτ may or may not be the same as γ. Hence, wecan assume that γτ is the correlated sample from γ with apower correlation factor ρτ ∈ [0, 1] between γ and γτ . Here,ρτ can be viewed as a measure of the channel fluctuation rateand a measure of the channel estimation quality as well. As anexample, from the well-known Clark’s model, we know thatρτ = J2

0 (2πfDτ) [1, Sec. 2.1.1], where J0(·) is the zero-orderBessel function of the first kind, τ is the time delay, and fD isthe maximum Doppler frequency shift. Note that ρτ = 0 means



Fig. 4. Average BER of BPSK versus the average SNR of first path γ1of the reassignment (RA) and replacement (RP) schemes over nonidentical/exponentially correlated Rayleigh fading channels when N = 4, L1 = · · · =L4 = 5, Lc = 3, Ls = 2, and γT = 5 dB. (a) Full scanning scheme.(b) Sequential scanning scheme.

completely outdated channel estimates, whereas ρτ = 1 meansup-to-date and perfect channel estimates.

Fig. 5 compares the effect of the correlation factor ρτ on theaverage BER of BPSK of the replacement and the reassignmentschemes with full scanning for several values of the outputthreshold γT over i.i.d. Rayleigh fading channels. We can seefrom these curves that in all cases, the diversity gain offered bythe proposed schemes decreases as ρτ decreases, as expected.It is very notable that contrary to the analysis over perfect chan-nel estimation, the replacement scheme shows a lower errorprobability than the reassignment scheme when ρτ = 0 and0.5 for γT = 5 and 15 dB. Recall that the replacement schemecompares two sums of paths from the serving and target BSs,whereas the reassignment scheme relies on the SNR of eachpath. Therefore, the replacement scheme is more robust to thechannel estimation errors, particularly when the comparisonsof the two sums are needed. In other words, the more oftenthe combined SNR is below the threshold, the less sensitive tothe channel estimation error the replacement scheme is, and themore sensitive the reassignment scheme is.

VIII. CONCLUSION

In this paper, we proposed new finger replacement schemesthat are applicable for RAKE receivers in the SHO region. We

Fig. 5. Average BER of BPSK versus the average SNR per path γ of thereassignment (RA) and replacement (RP) schemes for the full scanning withoutdated channel estimation over i.i.d. Rayleigh fading channels when N = 4,L1 = · · · = L4 = 5, Lc = 3, and Ls = 2. (a) γT = −5 dB. (b) γT = 5 dB.(c) γT = −15 dB.

considered two path scanning schemes: a full scanning schemeand a sequential scanning scheme. For both schemes, we pro-vided a general comprehensive framework for the assessment ofthese proposed schemes by offering not only analytical resultsfor i.i.d. fading environments but also computer simulationresults for non-i.i.d. fading channels and an outdated channelestimation. We showed through numerical examples that theproposed schemes can save a certain amount of complexitywith negligible performance loss compared to the previouslyproposed schemes. Computer simulation results showed thatour proposed scheme is also applicable to non-i.i.d. fadingchannels and offers, moreover, a great advantage in the presenceof channel estimation errors.

REFERENCES

[1] G. L. Stüber, Principles of Mobile Communication, 2nd ed. Norwell,MA: Kluwer, 2001.

[2] M. Z. Win and Z. A. Kostic, “Impact of spreading bandwidth on Rakereception in dense multipath channels,” IEEE J. Sel. Areas Commun.,vol. 17, no. 10, pp. 1794–1806, Oct. 1999.



[3] M. Z. Win, G. Chrisikos, and N. R. Sollenberger, “Performance of RAKEreception in dense multipath channels: Implication of spreading band-width and selection diversity order,” IEEE J. Sel. Areas Commun., vol. 18,no. 8, pp. 1516–1525, Aug. 2000.

[4] S. Choi, M.-S. Alouini, K. A. Qaraqe, and H.-C. Yang, “Soft handoveroverhead reduction by RAKE reception with finger reassignment,” IEEETrans. Commun., vol. 56, no. 2, pp. 213–221, Feb. 2008.

[5] S. Choi, M.-S. Alouini, K. A. Qaraqe, and H.-C. Yang, “Finger replace-ment method for RAKE receivers in the soft handover region,” IEEETrans. Wireless Commun., vol. 7, no. 4, Apr. 2008. to be published.

[6] T. Eng, N. Kong, and L. B. Milstein, “Comparison of diversity combiningtechniques for Rayleigh-fading channels,” IEEE Trans. Commun., vol. 44,no. 9, pp. 1117–1129, Sep. 1996.

[7] M. Z. Win and J. H. Winters, “Analysis of hybrid selection/maximal-ratiocombining in Rayleigh fading,” IEEE Trans. Commun., vol. 47, no. 12,pp. 1773–1776, Dec. 1999.

[8] M.-S. Alouini and M. K. Simon, “An MGF-based performance analysisof generalized selection combining over Rayleigh fading channels,” IEEETrans. Commun., vol. 48, no. 3, pp. 401–415, Mar. 2000.

[9] Y. Ma and C. C. Chai, “Unified error probability analysis for generalizedselection combining in Nakagami fading channels,” IEEE J. Sel. AreasCommun., vol. 18, no. 11, pp. 2198–2210, Nov. 2000.

[10] A. Annamalai and C. Tellambura, “Analysis of hybrid selection/maximal-ratio diversity combiner with Gaussian errors,” IEEE Trans. WirelessCommun., vol. 1, no. 3, pp. 498–512, Jul. 2002.

[11] S. Choi, M.-S. Alouini, K. A. Qaraqe, and H.-C. Yang, “Finger assignmentschemes for RAKE receivers with multiple-way soft handover,” IEEETrans. Wireless Commun., vol. 7, no. 2, pp. 495-499, Feb. 2008.

[12] S. Choi, M.-S. Alouini, K. A. Qaraqe, H.-C. Yang, and Y.-C. Ko, “Apractical study of adaptive finger reassignment and replacement in the softhandover region,” in Proc. IEEE ISSPA, Sharjah, U.A.E., Feb. 2007.

[13] M. K. Simon and M.-S. Alouini, Digital Communication Over FadingChannels, 2nd ed. New York: Wiley, 2005.

[14] H. Holma and A. Toskala, WCDMA for UMTS, 3rd ed. New York: Wiley,2004.

[15] A. Papoulis and S. U. Pillai, Probability, Random Variables, and Sto-chastic Processes, 4th ed. New York: McGraw-Hill, 2002.

Seyeong Choi (S’03–M’08) was born in Seoul,Korea. He received the B.S. and M.S. degrees fromHanyang University, Seoul, in 1996 and 1998, re-spectively, and the Ph.D. degree in electrical andcomputer engineering from Texas A&M University,College Station, in 2007.

From 1998 to 2001, he was a Researcher withLGTeleCom’s Network R&D Center, Seoul, oper-ating CDMA mobile communication networks. Heis currently with the Department of Electrical En-gineering, Texas A&M University at Qatar, Doha,

Qatar. His research interests include wireless communications, MIMO fadingchannels, diversity techniques, and system performance evaluation.

Mohamed-Slim Alouini (S’94–M’98–SM’03) wasborn in Tunis, Tunisia. He received the Ph.D. degreein electrical engineering from the California Instituteof Technology, Pasadena, in 1998.

He was an Associate Professor with the Depart-ment of Electrical and Computer Engineering,University of Minnesota, Minneapolis. Since July2005, he has been with the Department of Elec-trical Engineering, Texas A&M University at Qatar,Doha, Qatar. His current research interests includethe design and performance analysis of wirelesscommunication systems.

Khalid A. Qaraqe (M’97–S’00–SM’00) was bornin Bethlehem. He received the B.S. degree (withhonors) in electrical engineering from the Universityof Technology, Baghdad, Iraq, in 1986, the M.S.degree in electrical engineering from the Universityof Jordan, Amman, Jordan, in 1989, and the Ph.D.degree in electrical engineering from Texas A&MUniversity, College Station, in 1997.

From 1989 to 2004, he held a variety positionswith many companies. He has more than 12 yearsof experience in the telecommunication industry. He

was with Qualcomm, Enad Design Systems, Cadence Design Systems/TalityCorporation, STC, SBC, and Ericsson. He has worked on numerous GSM,CDMA, and WCDMA projects, and he has experience in product development,design, deployments, testing, and integration. He joined the Department ofElectrical Engineering, Texas A&M University at Qatar, Doha, Qatar, inJuly 2004, where he is currently a Visiting Associate Professor. His researchinterests include performance analysis of 3G UMTS wireless communication,WCDMA estimation theories, fading channels, frequency hopping, and STTDdiversity.

Hong-Chuan Yang (S’00–M’03–SM’07) was bornin Liaoning, China. He received the Ph.D. degreein electrical engineering from the University ofMinnesota, Minneapolis, in 2003.

He is currently an Assistant Professor with theDepartment of Electrical and Computer Engineering,University of Victoria, Victoria, BC, Canada. Hisresearch focuses on different aspects of wirelessand mobile communications, with special emphasison diversity techniques, cross-layer design, energy-efficient communications, and system performance

evaluation.Dr. Yang is a recipient of the Doctoral Dissertation Fellowship Award

from the Graduate School of the University of Minnesota for the 2002–2003academic year.


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