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Performance Analysis of Two-Branch Space-Time Block-Coded DS-CDMA Systems in Time-Varying Multipath Rayleigh Fading Channels

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[3] M. Zorzi and R. R. Rao, “Error control in multi-layered stacks,” in Proc. IEEE Global Telecommun. Conf., Nov. 1997, vol. 3, pp. 1413–1418. [4] S. Ramakrishna and J. M. Holtzman, “Interaction of TCP and data access

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Performance Analysis of Two-Branch Space-Time Block-Coded DS-CDMA Systems in Time-Varying

Multipath Rayleigh Fading Channels

Ping-Hung Chiang, Ding-Bing Lin, and Hsueh-Jyh Li

Abstract—For the two-branch space-time (ST) block-coded direct-sequence code-division multiple-access (DS-CDMA) systems, the impacts of a time-varying multipath channel on the downlink transmission are analyzed. By considering the systems using the random binary spreading code (RBSC) and deterministic binary spreading code (DBSC), the effects of the multipath interference and multiuser interference are included in the analyses of the bit-error rate and bit-error outage. Also, for the performance analysis of the system employing the decision-feedback (DF) detector, the effect of error propagation is taken into account. It is known that enlarging the spreading factor can enhance the interference-rejection ability of a DS-CDMA system and, hence, can improve the performance. However, it also lengthens the symbol duration and, thus, stiffens the diver-sity penalty resulting from the channel variation within an ST-code-word duration. Thus, a moderate spreading factor should be chosen. In this paper, for the RBSC system using the simple-maximum-likelihood (SML) detector, we derive an optimum spreading factor that is optimum in the minimum-error-probability sense. Numerical results have revealed that the derived optimum spreading factor is a good estimate of the ones for the DBSC systems using the SML, zero-forcing, and DF detectors. Therefore, it is very useful for system designers in determining the system parameters. Index Terms—Bit-error outage (BEO), direct-sequence code-division multiple access (DS-CDMA), multipath interference (MPI), multiuser interference (MUI), space-time block coding (STBC), transmit diversity.

I. INTRODUCTION

Space-time block coding (STBC), which is an effective transmit-diversity technique, was first proposed by Alamouti [1] for flat fading channels Assuming the channel being constant over an ST-code-word duration (i.e., two symbol durations) and the channel state information (CSI) being known at the receiver, he showed that the proposed two-input–single-output (2ISO) diversity system utilizing a simple-maximum-likelihood (SML) detector can obtain the full-diversity advantage. However, it suffers from a diversity penalty when the quasi-static (QS) channel assumption is dropped. Recently, the impacts of a time-varying channel on the Alamouti scheme were investigated [2]–[4]. Vielmon et al. [2] proposed three detectors to enhance the system robustness for rapid channel variation. These detectors are the zero-forcing (ZF), decision-feedback (DF), and joint-maximum-likelihood (JML) detectors. The ZF and DF detectors are of subop-timum performances but of moderate complexities, while the JML detector is of the optimum performance but of the highest complexity. Considering the CSI being not available at the receiver, Liu et al. [3] proposed a channel-tracking method using the Kalman filter to track

Manuscript received June 14, 2005; revised February 2, 2006 and April 20, 2006. This work was supported in part by the National Science Council, Republic of China, under Grant NSC 92-2219-E-002-010 and in part by the MOE Program for promoting academic excellence of universities un-der Grant 89E-FA06-2-4-7. The review of this paper was coordinated by Prof. L. H.-J. Lampe.

P.-H. Chiang and H.-J. Li are with the Graduate Institute of Communica-tion Engineering, NaCommunica-tional Taiwan University, Taipei 10617, Taiwan, R.O.C. (e-mail: d1942011@ee.ntu.edu.tw; hjli@ew.ee.ntu.edu.tw).

D.-B. Lin is with the Department of Electronic Engineering, National Taipei University of Technology, Taipei 10617, Taiwan, R.O.C. (e-mail: dblin@en. ntut.edu.tw).

Digital Object Identifier 10.1109/TVT.2007.891424 0018-9545/$25.00 © 2007 IEEE

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976 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007

Fig. 1. Discrete-time baseband-equivalent-system model.

the temporal channel variation. The proposed receiver introduced with the SML and JML detectors can perform the channel tracking and data detection iteratively. It is noteworthy that the extension of their work based on the DF detector was discussed in [4].

It is well-known that the interferences degrade the performance of a direct-sequence code-division multiple-access (DS-CDMA) system in multipath fading channels [5]. These interferences are 1) the multipath interferences (MPI), consisting of the intersymbol interference (ISI) and the interpath interference (IPI), due to multipath propagation and 2) the multiuser interference (MUI) from other active users. To improve the downlink transmission without drastically increasing the receiver complexity, the Alamouti scheme was adopted in the third-generation (3G) universal-mobile-telecommunication-system (UMTS) standard [6]. Please refer to [7] and [8] for the tutorial reviews of the applications of the Alamouti scheme and the other transmit-diversity schemes in the 3G-CDMA systems.

For the downlink transmission, the STBC-DS-CDMA system em-ploying the Alamouti scheme has been discussed by many researchers. Bjerke et al. [9] considered the combinations of the Alamouti scheme and various receive diversity schemes and derived their bit-error-rate (BER) expressions by assuming the channel being QS. Also, they assumed that the despreading operation is perfect, and hence, the effects of the MPI and MUI were not taken into account. In [10], the authors analyzed the effect of imperfect channel estimation with the QS channel assumption. Assuming the spreading code being random, they studied the effects of the IPI and MUI but ignored the one of ISI. In [11], the authors considered the SML and JML detectors and theoretically evaluated the performance of the system with noisy channel estimates in time-varying channels. However, as in [9], the effects of the MPI and MUI were not taken into consideration.

In this paper, assuming the CSI being available at the receiver, we analyze the impacts of a time-varying multipath channel on the downlink 2ISO STBC-DS-CDMA systems employing the SML, ZF, DF, and JML detectors. By considering the systems using the random binary spreading code (RBSC) and deterministic binary spreading code (DBSC), named the RBSC and DBSC systems, respectively, the effects of the MPI and MUI are included in the analyses of the BER and bit-error outage (BEO), i.e., outage probability [12]. It is known that enlarging the spreading factor (i.e., the number of chips per symbol) can enhance the interference-rejection ability of a DS-CDMA system and, hence, can improve the performance. However, it also lengthens the symbol duration and, thus, stiffens the diversity penalty resulting from the channel variation within an ST-code-word duration. Thereupon, a moderate spreading factor should be chosen according to a set of system and channel parameters, and our objective

is to develop a method for determining such a spreading factor. The major novelties of this paper are listed as follows.

1) For both RBSC and DBSC systems, the variances of the IPI, ISI, and MUI are derived. In the derivation, the channel is assumed to be constant within a chip duration only, and the ISI is assumed to be the contribution of the multiple previous symbols. These two assumptions make our derivation more practical and general than the ones in [7]–[11].

2) For the system employing the DF detector, the effect of error propagation is included in the derived BER expression. This was not considered in [2] and [13].

3) For the RBSC system employing the SML detector, an opti-mum spreading factor that is optiopti-mum in the miniopti-mum-error- minimum-error-probability sense is derived.

The rest of this paper is organized as follows. In Section II, the system and channel models are described. Then, various strategies for the receiver design are presented in Section III, and their BER expressions are given in Section IV. Also, the derivation of the optimum spreading factor is provided in Section V. Numerical results are shown in Section VI, while conclusions are drawn in Section VII.

II. SYSTEM ANDCHANNELMODELS

As shown in Fig. 1, we consider the STBC-DS-CDMA system, equipped with two transmit antennas at the base station and one receive antenna at the mobile terminal, in the synchronous downlink transmission.

A. Transmitter Model

Let Xk,idenote the information symbol of the kth user in the ith

symbol interval. Also, let TS and ES represent the symbol duration

and the symbol energy, respectively. At the transmitter, a pair of information symbols{Xk,i, i = 2q + 0, 2q + 1} for the qth

ST-code-word duration is first ST block-encoded via [1] Space Time a(0)k,2q+0 a(1)k,2q+0 a(0)k,2q+1 a(1)k,2q+1 = Xk,2q+0 Xk,2q+1 −X∗ k,2q+1 Xk,2q+0∗ (1)

where{a(g)k,i, i = 2q + 0, 2q + 1} is a pair of channel symbols to be transmitted from the gth transmit antenna. Let{bk,l, 0≤ l ≤ L − 1}

be the spreading sequence of the kth user, where L is the spreading factor. Also, bk,ltakes on the values of±1/

L with equal probability for RBSC or according to the assignment of the code book for DBSC. Considering short code spreading (i.e., TS= LTCand TCbeing the

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Fig. 2. Graphical illustration of the received signals forM = 5, L = 3, and N = λ = 2. chip duration or the reciprocal of the system bandwidth), we write the

transmitted sequence as ˜ s(g)i,l = K−1 k=0 a(g)k,ibk,l, for 0≤ l ≤ L − 1 (2)

where K is the number of users. Provided ˜s(g)i,l = 0 for l < 0 and l≥ L, the total transmitted baseband sequence from the gth transmit antenna is s(g)l =  i=−∞ ˜ s(g)i,l−iL. (3) B. Channel Model

In this paper, we employ the chip-spaced tapped-delay-line (TDL) channel model [14] (i.e., a TDL with a fixed tap spacing TC) and

assume that the channel from the gth transmit antenna to the receive antenna consists of M discrete paths, expressed as a set of tap coeffi-cients{h(g)m,l, 0≤ m ≤ M − 1}. It is assumed that the two channels, corresponding to two transmit antennas, experience independently and identically distributed (i.i.d.) Rayleigh fading and have an identical power-delay profile. More specifically, h(g)m,l∼ CN (0, σ2m) for g = 0, 1, where σm2 is the fading power of the mth path complying

with the constraint Mm=0−1σm2 = 1. Also, the channel is assumed

to be constant within a chip duration only. Subsequently, a temporal rearrangement of the tap coefficients is defined as h(g)m,i,l= h(g)m,iL+l, which will be used later in the receiver model (see Section II-C). For the wide-sense-stationary-uncorrelated-scattering (US) channel with classical Doppler spectrum, the correlation between the rearranged tap coefficients is expressed as [14] E  h(g)m,i,lh(gm),i∗,l  = σ2mJ0{2πfDTC[(i−i)L+(l−l)]} δmmδgg (4)

where J0(·) is the zeroth-order Bessel function of the first kind, fDis

the maximum Doppler frequency, and δijis the Kronecker delta.

C. Receiver Model

Provided that the receiver can achieve perfect chip-timing synchro-nization and power control, the received baseband sequence of the desired user, denoted by the zeroth user, can be expressed as

rl= 1  g=0 M−1 m=0 h(g)m,ls(g)l−m+ zl = 1  g=0  i=−∞ M−1 m=0 h(g)m,l˜s(g)i,l−m−iL+ zl (5)

where zl∼ CN (0, 2N0) is the additive white Gaussian noise. The

factor 2 is due to the sharing of the transmitting power for two antennas. To quantify the ISI due to multipath propagation, the number of previous symbols causing the ISI is defined as N =∆(M − 1)/L, where the functionx gives the smallest integer larger than or equal to x. Also, we define λ=∆M/L and

κ(n)=∆

L, if M = λL and 0≤ n ≤ λ − 1 L, if M= λL and 0 ≤ n ≤ λ − 2 ((M ))L, if M= λL and n = λ − 1

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where ((·))Ldenotes the modulo-L operation. It can be shown that λ

and N are related as λ =



N, if ((M ))L= 1

N + 1, if((M ))L= 1 . (7) Then, as shown in Fig. 2, rearranging rl symbol by symbol as

ri,l= riL+lyields ri,l= 1  g=0 λ−1  n=0

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978 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007

where zi,l

= ziL+lis the rearranged noise and y(g)i,l,nis the noise-free

received signal given by (9), shown at the bottom of the page. Without loss of generality, the zeroth path of the multipath channel is assumed to have the largest fading power. Thus, the code-matched filter per-forms despreading with respect to the zeroth path and produces the signal for the ith symbol duration as

Ri= L−1  l=0 ri,lb0,l= 1  g=0  Hi(g)a(g)0,i+Pi(g)+ S(g)i +Ui(g)  + Zi (10)

where Hi(g), Pi(g), S(g)i , Ui(g), and Ziare the multiplicative distortion

(MD), IPI, ISI, MUI, and noise, respectively. They are given by

Hi(g)=1 L L−1  l=0 h(g)0,i,l (11) Pi(g)= a(g)0,i κ(0)−1 m=1 L−1 l=m h(g)m,i,lb0,l−mb0,l (12) Si(g)= λ−1  n=1 a(g)0,i−n κ(n)−1 m=0 L−1  l=m h(g)m+nL,i,lb0,l−mb0,l + λ−1  n=0 a(g)0,i−n−1 κ(n)−1 m=1 m−1 l=0 h(g)m+nL,i,lb0,l−m+Lb0,l (13) Ui(g)= K−1 k=1 λ−1  n=0  a(g)k,i−n κ(n)−1 m=0 L−1  l=m h(g)m+nL,i,lbk,l−mb0,l + a(g)k,i−n−1 κ(n)−1 m=1 m−1 l=0 h(g)m+nL,i,lbk,l−m+Lb0,l  (14) Zi= L−1  l=0 zi,lb0,l (15)

where (12)–(14) are derived by changing the order of summations and rearranging the resultant terms.

Subsequently, some statistical properties regarding the random vari-ables (RVs) given in (11)–(15) are specified as follows.

1) The channel symbols {a(g)k,i, 0≤ k ≤ K − 1 and − ∞ ≤ i ≤ ∞} are i.i.d. and have zero mean and variance ES, provided

that{Xk,i, 0≤ k ≤ K − 1 and − ∞ ≤ i ≤ ∞} are i.i.d.

infor-mation symbols having zero mean and variance ES. This can be

verified from (1).

2) The tap coefficients {h(g)m,i,l, 0≤ m ≤ M − 1} are mutually uncorrelated due to the assumption of US.

3) The MD Hi(g) is distributed asCN (0, σH2), since it is a linear

combination of zero-mean complex Gaussian RVs. To char-acterize the channel variation, its normalized correlation of adjacent symbol durations, which is denoted by ρt, is used. See

Appendix A for the expressions of σH2 and ρt.

4) The noise Ziis distributed asCN (0, 2N0) for both RBSC and

DBSC systems.

5) The interferences, including Pi(g), S (g) i , and U

(g)

i , are

zero-mean RVs and mutually uncorrelated. Also, they are uncorre-lated to the MD Hi(g)and the noise Zi.

According to 5), it is reasonable to model the interferences as Gaussian RVs. Since independent Gaussian noise results in the smallest capacity, the performance bound can be achieved [15]. Indeed, (10) can be rewritten as Ri= 1  g=0 Hi(g)a(g)0,i+ Wi (16) where Wi∼ CN (0, σW2 ) is given by Wi= 1  g=0  Pi(g)+ Si(g)+ Ui(g)  + Zi. (17)

Here, σ2W = 2(σ2P+ σS2+ σU2 + N0), where σ2P, σ2S, and σU2

repre-sent the variances of Pi(g), S (g) i , and U

(g)

i , respectively. Please refer to

Appendices B and C for the derivations of σP2, σS2, and σ2U.

III. STRATEGIES FORRECEIVERDESIGN

In this section, assuming the CSI being known perfectly at the receiver, we introduce four strategies for the receiver design, including the JML, SML, ZF, and DF detectors and give a remark on their computational complexities. According to (1) and (16), the signal model for detecting the qth ST code word is

r = Hx + w R2q+0 R∗2q+1 = H2q+0(0) H2q+0(1) H2q+1(1) −H2q+1(0) X2q+0 X2q+1 + W2q+0 W2q+1 (18) where the subscript zero of the information symbol X0,iis omitted for

simplicity.

A. JML Detector

From (18), since the noise is white, the JML detector makes decision about x via [13, eq. (23)]

ˆ xJML = arg min x  r − Hx2 . (19)

Let the cascade of H and its matched filter be ˜ H = HHH = α0 β β∗ α1 (20) where α0=|H2q+0(0) |2+|H2q+1(1) |2, α1=|H2q+0(1) |2+|H2q+1(0) |2, and

β = H2q+0(0)∗H2q+0(1) − H2q+1(0)∗H2q+1(1) . Since ˜H is Hermitian, it has a

yi,l,n(g) =          l  m=0 h(g)m+nL,i,ls˜(g)i−n,l−m+ κ(n)−1 m=l+1 h(g)m+nL,i,l˜s(g)i−n−1,l−m+L, if 0≤ l ≤ κ(n) − 2 κ(n)−1 m=0 h(g)m+nL,i,l˜s(g)i−n,l−m, if κ(n)−1≤l≤L−1 (9)

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unique Cholesky factorization as ˜H = GHG, where G = ξα−1/21 0 β∗α−1/21 α1/21 (21) is a lower triangular matrix and ξ =|H2q+0(0) H2q+1(0) + H2q+0(1) H2q+1(1)∗|. Then, applying whiten-matched filtering on the received ST code word yields [13, eq. (30)] rW = CWr = Gx + wW (22) where rW= [RW,0 RW,1]T, CW = G−HHH, and wW = CWw

= [WW,0 WW,1]T. Since the noise wW is still white, the

JML detector is now equivalent to [13, eq. (31)] ˆ xJML = arg min x  rW− Gx2  . (23) B. SML Detector

Performing ST-matched filtering on the received ST code word results in [13, eq. (24)] rM = CMr = HMx + wM (24) where rM= [RM,0 RM,1]T, CM = AMHH, HM = AMH,˜ wM = CMw= [WM,0 WM,1]T, and AM = diag{α−1/20 , α−1/21 }.

From (24), without considering the correlation of the noise wM and

the crosstalks (i.e., the off-diagonal terms of HM), the SML detector

simply obtains the decisions about X2q+0and X2q+1via [13, eq. (32)]

ˆ X2q+pSML = arg min X  RM,p− α 1/2 P X 2 , for p = 0, 1. (25) C. ZF Detector

From (24), the ZF detector forces the crosstalks to zero through [13, eq. (33)] rZ= CZrM = AZx + wZ (26) where rZ= [RZ,0 RZ,1]T, CZ= AZH−1M, wZ= CZwM ∆ = [WZ,0WZ,1]T, and AZ= diag{ξα1−1/2, ξα0−1/2} ∆ = diag{AZ,0, AZ,1}.

Consequently, the ZF detector can separately make decisions about X2q+0and X2q+1via [13, eq. (35)]

ˆ X2q+pZF = arg min X  |RZ,p− AZ,pX|2  , for p = 0, 1. (27) D. DF Detector

From (22), the DF detector feeds back the decision about X2q+0to

help make a decision about X2q+1, namely [13, eq. (36)]

       ˆ X2q+0DF = arg min X  RW,0− ξα−1/21 X 2 ˆ X2q+1DF = arg min X  . RW,1−α1/21 X 2 (28) where .

RW,1= RW,1− β∗α−1/21 Xˆ2q+0DF is the decision statistic after

crosstalk cancellation.

IV. PERFORMANCEANALYSIS

For binary phase-shift keying (BPSK), we evaluate the theoretical BERs of the 2ISO STBC-DS-CDMA system in time-varying multipath Rayleigh fading channels. First, the effective signal-to-noise ratios (ESNRs), i.e., average signal-to-interference-plus-noise ratios, are de-termined according to the variances of the MD and the interferences. Then, the theoretical BERs of the JML, SML, and ZF detectors are obtained by substituting the ESNRs into the BER expressions given in [2]. Noteworthily, for the DF detector, we derive its BER expression being able to reflect the effect of error propagation. By using the derived ESNRs and BERs and following the same steps given in [13, Sec. VI-A], one can also evaluate the theoretical BEOs easily. Finally, a remark on computational complexities of all detectors is provided.

A. JML Detector

Since the JML detector is robust to the crosstalk resulting from the channel variation within an ST-code-word duration, we can treat its performance for the QS condition (i.e., H2q+0(g) = H2q+1(g) , for g = 0, 1)

as a lower bound. Under this condition, the first component of (22) is reduced to

RW,0= α1/20 X2q+0+ WW,0. (29)

According to (29), the BER of the JML detector is given as [2, eq. (16)] PbJML 1 4 1 ! ¯ γ 2 + ¯γ 2 2 + ! ¯ γ 2 + ¯γ= f1(¯γ) (30)

where the ESNR is ¯γ = 2σH2ES/σ2W.

B. SML Detector

The first component of (24) is

RM,0= α1/20 X2q+0+ βα−1/20 X2q+1+ WM,0. (31)

In [13, App.], we showed that the crosstalk βα−1/20 X2q+1is a

zero-mean RV and has variance σH2ES(1− ρ2t). By assuming that the

crosstalk is Gaussian distributed, the BER of the SML detector is derived as [13, eq. (43)]

PbSML= f1(¯γSML) (32)

where the ESNR is ¯γSML= 2σH2ES/[σH2ES(1− ρ2t) + σ2W].

C. ZF Detector

According to the first component of (26), which is given as RZ,0= ξα−1/21 X2q+0+ WZ,0 (33)

the BER of the ZF detector is derived as [2, eq. (30)]

PbZF=1− ρ2t 1 2 1 ! ¯ γ 2 + ¯γ + ρ2t " 1 4 1 ! ¯ γ 2 + ¯γ 2 2 + ! ¯ γ 2 + ¯γ # ∆ = f2(¯γ). (34)

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980 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007

D. DF Detector

Observing from (22) and (33), one can find that RW,0has the same form as RZ,0. Thus, we can write the BER of the DF detector for the

first symbol X2q+0as

Pb,0DF= PbZF= f2(¯γ). (35)

On the other hand, the BER of the DF detector for the second symbol X2q+1is computed via

Pb,1DF= Pb,DF1|CorrectPCorrect+ Pb,DF1|IncorrectPIncorrect (36) where Pb,DF1|Correct and Pb,DF1|Incorrect are the error probabilities for the second symbol given, respectively, correct and incorrect feedback decisions. Also, the probability of incorrect feedback decision is

PIncorrect= Pr

ˆ

X2q+0DF = X2q+0DF



= Pb,0DF= f2(¯γ) (37)

and hence, the probability of correct feedback decision is

PCorrect= 1− PIncorrect= 1− f2(¯γ). (38)

If the feedback decision ˆX2q+0DF is correct,

.

RW,1is of the same form

as (29), and thus, we have Pb,DF1|Correct= f1(¯γ). However, if ˆX2q+0DF is

incorrect,

.

RW,1becomes .

RW,1= α1/21 X2q+1DF + β∗α−1/21 ε2q+0+ WW,1 (39)

where ε2i+0= X2q+0− ˆX2q+0DF , for BPSK takes on the values of

±2√ES with equal probability. Since the error term β∗α−1/21 ε2q+0

has the same form as the crosstalk βα−1/21 X2q+0, we can assume that

its distribution isCN (0, 4σH2ES(1− ρ2t)). Then, since (39) is of the

same form as (31), one can obtain Pb,DF1|Incorrect= f1(¯γDF), where the

ESNR is ¯γDF= 2σ2HES/[4σH2ES(1− ρ2t) + σ 2

W]. Finally, the BER

of the DF detector can be calculated via PbDF= 1 2  Pb,0DF+ Pb,1DF  =1 2{f2(¯γ) + f1(¯γ) [1− f2(¯γ)] + f1(¯γDF)f2(¯γ)} . (40) E. Remark on Computational Complexity

As shown in (19), the JML detector detects two symbols jointly and, hence, has the highest computational complexity, especially when the constellation size of the information symbol is large. As compared to the SML detector, the ZF detector requires the additional computation for matrix inversion while the DF detector requires the ones for both Cholesky factorization and matrix inversion. Therefore, the ZF and DF detectors have higher computational complexities than the SML detector. Moreover, the DF detector has higher computational complexity than the ZF detector as a result of crosstalk cancellation. As compared to the JML and SML detectors, the ZF and DF detectors have suboptimum performances but moderate computational complex-ities. Indeed, we recommend the ZF and DF detectors for practical implementations.

V. OPTIMUMSPREADINGFACTOR

In this section, the optimum spreading factor for RBSC system using the SML detector is derived. Here, we define the sum of the powers of the interferences and crosstalk as

σI,SML2 = 2σ2P+ σS2 + σU2+ σH2ES



1− ρ2t. (41)

From (31), the ESNR for the SML detector can then be rewritten as ¯

γSML= 2 HES

σI,SML2 + 2N0. (42)

Now, we derive the optimum spreading factor for the case of M≤ L first. The derivation for the case of M > L will be given later. For RBSC, according to (49) and (55), by using J0(x)≈ 1 − 0.25 x2 [16, eq. (8.441-1)] and L 1, (41) can be approximated as

σ2I,SML= ES " 2 L M−1  m=1 σ2m+K−1  +7L 2 3 (πfDTC) 2σ2 0 # (43)

of which the high-order terms are ignored. Since (43) is a convex function of L over (0,∞), an optimum spreading factor LSMLopt for

minimizing σ2I,SMLcan be found. From (32) and (42), it is obvious that PbSMLis a monotone increasing function of σ2I,SML. Thus, LSMLopt

is also optimum in the sense of minimizing PbSML. Since σ2I,SMLis convex, one can obtain LSMLopt by the following steps. First, solving

∂Lσ 2 I,SML   L=Lopt = 0 (44) yields Lopt=  3 7· M−1 m=1σm2 + K− 1 (πfDTC)2σ02 1/3 . (45)

Second, according to Lopt, one can determine L−opt and L+opt from

all possible spreading factors. For instance, if Lopt= 52.13 and

the set of possible spreading factors is{1, 3, 7, 15, 31, 63, . . .}, then L−opt= 31 and L+opt= 63. Finally, the optimum spreading factor

LSMLopt is determined via

LSMLopt = arg min L−opt,L+opt

σ2I,SML. (46)

For the case of M > L, the steps of finding the optimum spreading factor are the same as the ones for the case of M≤ L, except for modifying (43) and (45) as follows:

σ2I,SML=ES 2 L  M−L+L−1 M + K−1  +7L 2 3 (πfDTC) 2 σ20 (47) Lopt= 3 7· M− 1/M + K − 1 (πfDTC)2σ20 1/3 . (48)

The above modification is made by considering the worst case of M > L, i.e., M = λL and σ2m= 1/M for 0≤ m ≤ M − 1. Indeed, given the system and channel parameters, by using (45), (46), and (48), one can easily determine the optimum spreading factor LSMLopt for the

RBSC system employing the SML detector.

VI. NUMERICALRESULTS

The performances of the single-input–single-output (SISO) and 2ISO DS-CDMA systems are evaluated with the channel models specified in the UMTS standard [17]. For all numerical results, the carrier frequency and system bandwidth are 2.1 GHz and 3.84 MHz, respectively, and the modulation is BPSK. In addition, we consider the channel assumptions given in Section II-B and employ the typical urban (TU120) and hilly terrain (HT120) channel models [17] with the mobile speed of 120 km/h and the time resolution of TC= 0.26 µs.

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Fig. 3. BER versusL for K = 5 and ES/N0= ∞.

Fig. 4. BEO versusK for L = 511, ES/N0= 25 dB, and Pth= 3 × 10−2.

are nine and 70, respectively. Subsequently, the DBSC systems using Gold codes [14] are considered, and their averaged BERs and BEOs over all users are presented.

Fig. 3 shows the analytical error floors corresponding to a set of spreading factorsL = {31, 63, 127, 255, 511, 1023} for the case of K = 5. The SISO RBSC system and the 2ISO RBSC system using the SML detector are treated as the baseline systems, and only their theoretical BERs are illustrated. One should note that for the TU120 channel model, the number of previous symbols resulting in ISI is N = 1 if L∈ L. However, for the HT120 channel model, we have 1) N = 3 if L = 31; 2) N = 2 if L = 63; and 3) N = 1 if L∈ {127, 255, 511, 1023}. As discussed in Section IV, it is the MPI, MUI, and crosstalk that cause the error floors. For the DBSC systems, the JML detector, being robust to the crosstalk, obtains the best performance, while the SML detector, ignoring the crosstalk, obtains the worst performance. Moreover, increasing L has three impacts on the system performance. First, the power of the total interference

2(σ2P+ σS2+ σU2) decreases, and hence, all error floors are lower. This evidences that the interference-rejection ability of a DS-CDMA system can be enhanced by increasing L. Second, the power of the crosstalk σ2HES(1− ρ2t) increases, and thus, the error floors of

the 2ISO DS-CDMA systems without employing the JML detector become higher. Finally, the data rate 1/TS decreases. Therefore, a

moderate L should be chosen based on the specific channel condition and system requirements.

In the light of (45), (46), and (48), for both TU120 and HT120 channel models, the optimum spreading factor of the 2ISO RBSC system using the SML detector is LSMLopt = 511. This can be easily

verified from Fig. 3. In addition, one can find that LSMLopt is a good

estimate of the optimum spreading factors for DBSC systems utilizing the SML, ZF, and DF detectors. Indeed, LSMLopt is very useful in

determining system parameters.

To provide a more objective judgment on the transmission quality within a fading environment, Fig. 4 illustrates the theoretical BEOs of

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982 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007

DBSC systems corresponding to different number of users for L = 511, ES/N0= 25 dB, and the target BER being Pth= 3× 10−2.

Here, we define the system capacity as the number of users for which the BEO is smaller than 4%. Then, for the TU120 channel model, the system capacities are approximately four, five, six, and seven users, respectively, for the SML, ZF, DF, and JML detectors. On the other hand, for the HT120 channel model, the system capacities are ten, 12, 13, and 14 users, respectively, for the SML, ZF, DF, and JML detectors.

VII. CONCLUSION

The impacts of a time-varying multipath channel on the perfor-mance of the 2ISO STBC-DS-CDMA systems employing various detectors are addressed. The ZF and DF detectors have suboptimum performances but moderate computational complexities and, thus, are recommended for practical implementations. For both RBSC and DBSC systems, with the derived statistical properties of the MD and the interferences, we determine their ESNRs and give their theoretical BERs for time-varying multipath Rayleigh fading channels. Also, we present the theoretical BEOs and evaluate the system capacities. Furthermore, we derive an optimum spreading factor for the RBSC system using the SML detector. Numerical results have revealed that the derived optimum spreading factor is a good estimate of the ones for the DBSC systems using the SML, ZF, and DF detectors. Therefore, it is very useful for system designers in determining the system parameters.

APPENDIXA

CORRELATION ANDVARIANCE OFMD From (11), the correlation of the MD is derived as [13]

ρH(∆i) = E  Hi+∆i(g) Hi(g)∗  =σ 2 0 L2 L−1  l=−L+1 (L− |l|) J0[2πfDTC(∆i· L + l)] (49)

and its variance is σH2 = ρH(0). Also, its normalized correlation of

adjacent symbol durations is defined as ρt

= ρH(1)/σ02.

APPENDIXB

VARIANCES OF THEINTERFERENCES FORRBSC From (12) and (13), assuming that information symbols, tap coef-ficients, and random spreading sequences are mutually independent, one can calculate the variances of the IPI and ISI as follows:

σP2 = E  Pi(g) 2 =ES L2  L κ(0)−1 m=1 σ2m κ(0)−1 m=1 2m  (50) σ2S= E  Si(g) 2 = ES L2  L λ−1  n=1 κ(n) + κ(0)−1 m=1 m2  . (51)

Accordingly, summing up (50) and (51) yields the variance of the MPI as σ2P+ σ 2 S= ES L κ(0)−1  m=1 σm2 + λ−1  n=1 κ(n)  . (52)

By considering the worst case, namely, M λ = L and σ2m= 1/M for 0≤ m ≤ M − 1, its upper bound is obtained as

σ2P+ σS2 <  ES L, if M ≤ L ES L(M− L + 1), if M > L . (53)

From (14), the variance of the MUI can be derived as

σ2U= E  Ui(g) 2 =ES L K−1 k=1 λ−1  n=0 κ(n)−1 m=0 σm+nL2 =ES L (K−1). (54) Finally, the power of the total interference is

σ2P+ σS2+ σU2 =ES L κ(0)−1  m=1 σm2 + λ−1  n=1 κ(n) + K− 1  (55)

and hence, its upper bound is

σ2P+ σS2+ σU2 <  ES LK, if M≤ L ES L(M− L + K), if M > L . (56)

Thereupon, if M ≤ L, the total interference can be approximated as the MUI contributed by K users. This confirms the assumption in [5], where the case of M < L is considered. However, if M > L, the total interference should be approximated as the MUI contributed by M− L + K users. Indeed, from (56), it is evident that the number of paths M and the number of users K dominate the quantity of the total interference while the spreading factor L stands for the inherent interference-rejection ability of a DS-CDMA system.

APPENDIXC

VARIANCES OF THEINTERFERENCES FORDBSC From (12) and (13), for deterministic spreading, assuming that the information symbols and channel-tap coefficients are independent, one can derive the variances of the IPI and ISI as

σ2P = E  Pi(g) 2 = ES κ(0)−1 m=0 σm2 L−1  l=m L−1  l=m J0[2πfDTC(l− l)] × b0,l−mb0,l−mb0,lb0,l− ESσH2 (57) σS2 = E  Si(g) 2 = ES λ−1  n=1 κ(n)−1 m=0 σm+nL2 L−1  l=m L−1  l=m J0[2πfDTC(l− l)] × b0,l−mb0,l−mb0,lb0,l+ ES λ−1  n=0 κ(n)−1 m=1 σ2m+nL × m−1 l=0 m−1 l=0 J0[2πfDTC(l− l)] b0,l−m+Lb0,l−m+Lb0,lb0,l. (58)

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Thus, the variance of the MPI is σP2 + σ 2 S= ES λ−1  n=0 κ(n)−1 m=0 σm+nL2 × L−1  l=m L−1  l=m J0[2πfDTC(l− l)] b0,l−mb0,l−mb0,lb0,l + m−1 l=0 m−1 l=0 J0[2πfDTC(l− l)] × b0,l−m+Lb0,l−m+Lb0,lb0,l  − ESσH2. (59)

From (14), the variance of the MUI is derived as

σ2U= E  Ui(g) 2 = ES K−1 k=1 λ−1  n=0 κ(n)−1 m=0 σ2m+nL × L−1  l=m L−1  l=m J0[2πfDTC(l− l)] bk,l−mbk,l−mb0,lb0,l + m−1 l=0 m−1 l=0 J0[2πfDTC(l− l)] × bk,l−m+Lbk,l−m+Lb0,lb0,l  . (60)

Finally, adding up (59) and (60) yields the power of the total inter-ference as σ2P+ σ2S+ σU2 = ES K−1 k=0 λ−1  n=0 κ(n)−1 m=0 σm+nL2 × L−1  l=m L−1  l=m J0[2πfDTC(l− l)]bk,l−mbk,l−mb0,lb0,l + m−1 l=0 m−1 l=0 J0[2πfDTC(l− l)] × bk,l−m+Lbk,l−m+Lb0,lb0,l  −ESσ2H. (61)

For the static channel, i.e., fD= 0, (61) is reduced to

σ2P+ σ 2 S+ σ 2 U= ES K−1 k=0 λ−1  n=0 κ(n)−1 m=0 σm+nL2 × "L−1  l=m bk,l−mb0,l 2 + m−1  l=0 bk,l−m+Lb0,l 2# − ESσ20 (62)

which is of much lower computational complexity. According to extensive computer simulations in which the well-designed pseudo-random codes, having good automatic and cross-correlation properties (e.g., Gold codes and Kasami codes [14]), were used as spreading sequences, we found that (62) is a good approximation of (61) as long as fDTC< 0.0025.

REFERENCES

[1] S. M. Alamouti, “A simple transmit diversity scheme for wireless commu-nications,” IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 1451–1458, Oct. 1998.

[2] A. Vielmon, Y. Li, and J. R. Barry, “Performance of Alamouti trans-mit diversity over time-varying Rayleigh-fading channels,” IEEE Trans. Wireless Commun., vol. 3, no. 5, pp. 1369–1373, Sep. 2004.

[3] Z. Liu, X. Ma, and G. B. Giannakis, “Space-time coding and Kalman fil-tering for time-selective fading channels,” IEEE Trans. Commun., vol. 50, no. 2, pp. 183–186, Feb. 2002.

[4] K. S. Ahn and H. K. Baik, “Performance improvement of space-time block codes in time-selective fading channels,” IEICE Trans. Commun., vol. E87-B, no. 2, pp. 364–368, Feb. 2004.

[5] T. Eng and L. Milstein, “Coherent DS-CDMA performance in Nakagami multipath fading,” IEEE Trans. Commun., vol. 43, no. 2–4, pp. 1134– 1143, Feb.–Apr. 1995.

[6] Physical Channels and Mapping of Transport Channels Onto Physical Channels (FDD) (3GPP TS 25.211), Dec. 2005.

[7] R. T. Derryberry, S. D. Gray, D. M. Ionescu, G. Mandyam, and B. Raghothaman, “Transmit diversity in 3G CDMA systems,” IEEE Commun. Mag., vol. 40, no. 4, pp. 68–75, Apr. 2002.

[8] R. A. Soni and R. M. Buehrer, “On the performance of open-loop transmit diversity techniques for IS-2000 systems: A comparative study,” IEEE Trans. Wireless Commun., vol. 3, no. 5, pp. 1602–1615, Sep. 2004. [9] B. A. Bjerke, Z. Zvonar, and J. G. Proakis, “Antenna diversity

combin-ing schemes for WCDMA systems in fadcombin-ing multipath channels,” IEEE Trans. Wireless Commun., vol. 3, no. 1, pp. 97–106, Jan. 2004.

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[11] J. Jootar, J. R. Zeidler, and J. G. Proakis, “Performance of Alamouti space-time code in time-varying channels with noisy channel estimates,” in Proc. IEEE Wireless Commun. Netw. Conf., Mar. 2005, pp. 498–503. [12] A. Conti, M. Z. Win, M. Chiani, and J. H. Winters, “Bit error outage

for diversity reception in shadowing environment,” IEEE Commun. Lett., vol. 7, no. 1, pp. 15–17, Jan. 2003.

[13] D. B. Lin, P. H. Chiang, and H. J. Li, “Performance analysis of two-branch transmit diversity block coded OFDM systems in time-varying multipath Rayleigh fading channels,” IEEE Trans. Veh. Technol., vol. 54, no. 1, pp. 136–148, Jan. 2005.

[14] G. L. Stüber, Principles of Mobile Communication, 2nd ed. London, U.K.: Kluwer, 2001.

[15] C. E. Shannon, “Communication in the presence of noise,” Proc. IRE, vol. 37, no. 1, pp. 10–21, Jan. 1949.

[16] I. S. Gradshteyn and I. M. Ryzhik, Table of Integral, Series, and Products, 6th ed. London, U.K.: Kluwer, 2000.

數據

Fig. 1. Discrete-time baseband-equivalent-system model.
Fig. 2. Graphical illustration of the received signals for M = 5, L = 3, and N = λ = 2.
Fig. 3. BER versus L for K = 5 and E S /N 0 = ∞.

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