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ACKNOWLEDGMENT The authors would like to thank the editor and anonymous reviewers. Their comments have significantly improved the quality of this paper. The corresponding author of this paper is Wonjun Lee. R EFERENCES

Fig. 4. Optimal binding-management-key refresh time. (a) Effect of μS . (b) Effect of δ.

a new BU procedure is performed. Therefore, the key exposal time and N decrease with the increase in μS . The effect of πON is also demonstrated in Fig. 3(a). When πON is high, the probability that the state is ON for a key exposal attack is also high. Hence, more packets can be affected by the key exposal when πON is high. In Fig. 3(a), it can be seen that the simulation results are consistent with the analytical results. When TR is high, the key exposal time can be long because the binding management key is not frequently refreshed. Therefore, as shown in Fig. 3(b), N increases as TR increases. It can be seen that N becomes fairly stable when TR exceeds a certain point. This is because the mobility rate μS is fixed at 1.0, and thus, the binding management key can be refreshed by a handoff even though TR is very high.

[1] D. Johnson, C. Perkins, and J. Arkko, Mobility Support in IPv6, Jun. 2004, IETF. RFC 3775. [2] P. Nikander, J. Arkko, T. Aura, and G. Montenegro, “Mobile IP version 6 (MIPv6) route optimization security design,” in Proc. IEEE VTC—Fall, Sep. 2003, pp. 2004–2008. [3] J. Kempf, J. Arkko, and P. Nikander, “Mobile IPv6 security,” Wirel. Pers. Commun., vol. 29, no. 3/4, pp. 389–414, Jun. 2004. [4] J. Hass and Y. Lin, “On optimizing the location update costs in the presence of database failures,” Wirel. Netw., vol. 4, no. 5, pp. 419–426, Aug. 1998. [5] L. Kleinrock, Queuing Systems Volume 1: Theory. New York: Wiley, 1975. [6] Y. Haung, “Determining the optimal buffer size for short message transfer in a heterogeneous GPRS/UMTS network,” IEEE Trans. Veh. Technol., vol. 52, no. 1, pp. 216–225, Jan. 2003. [7] X. Zhang, J. G. Castellanos, and A. T. Capbell, “P-MIP: Paging extensions for mobile IP,” ACM Mob. Netw. Appl., vol. 7, no. 2, pp. 127–141, Apr. 2002. [8] D. Staehle, K. Leibnitz, and K. Tsipotis, “QoS of Internet access with GPRS,” Wirel. Netw., vol. 9, no. 3, pp. 213–222, May 2003. [9] A. Downey, “The structural cause of file size distributions,” in Proc. IEEE MASCOT, Aug. 2001, pp. 361–370.

Performance of Multicarrier CDMA in the Presence of Correlated Fading

B. Optimal Binding-Management-Key Refresh Interval The optimal binding-management-key refresh interval in different μS is shown in Fig. 4(a). It can be seen that the optimal bindingmanagement-key refresh interval increases with the increase in μS . This can be explained as follows. As μS increases, BU procedures due to handoffs are more frequently performed. Therefore, when the bound for the number of packets influenced by the key exposal attack is given, the periodical binding-management-key refresh can be triggered with a long interval for a high μS . The impact of πON can also be found in Fig. 4(a). That is, a longer TR can be used for a low πON since the number of arrival packets is small. Fig. 4(b) demonstrates the effect of δ on the optimal bindingmanagement-key refresh interval. As shown in Fig. 4(b), a longer binding-management-key refresh interval can be permitted when δ is high. On the other hand, if δ is low, a shorter binding-management-key refresh interval should be employed to minimize the number of packets affected by the key exposal attack. In Fig. 4, it can be shown that the binding-management-key refresh interval should be determined by considering both the mobility rate and the traffic pattern. Furthermore, the security level for packets (i.e., δ) is an important consideration. V. C ONCLUSION Secure key negotiation is a challenging issue for the successful deployment of mobile IPv6. In this paper, we have derived the optimal binding-management-key refresh interval bounding the number of infected packets to a given threshold. From analytical and simulation results, the effects of the mobility rate and traffic intensity are demonstrated. By devising an algorithm, it is shown that the optimal binding-management-key refresh interval can be adaptively determined to the mobility rate and traffic load.

Siamak Sorooshyari and David G. Daut, Senior Member, IEEE

Abstract—Incorporating a framework from multivariate statistical theory, analytical results for the performance of the downlink and uplink of multicarrier code-division multiple access (MC-CDMA) in the presence of correlated Rayleigh fading are presented. Bit-error-probability expressions are derived for MC-CDMA diversity combining schemes, while statistical generality is maintained by assuming no specific form for the covariance matrix of the desired user and interferers. The scenarios of uncorrelated fading among subcarriers and equal-power users are considered as special cases. The analysis is exact in the sense that the error probability is derived without any assumptions on the distribution of the multiple-access interference encountered by a user. With the scenario of independent fading among subcarriers as a baseline, the performance degradation incurred by correlated fading is analyzed for an empirical wireless channel. Index Terms—Correlated fading, diversity combining, multicarrier code-division multiple access (MC-CDMA), multivariate analysis.

Manuscript received July 3, 2008; revised October 30, 2008 and December 25, 2008. First published February 6, 2009; current version published August 14, 2009. The review of this paper was coordinated by Prof. N. Arumugam. S. Sorooshyari is with Alcatel-Lucent, Whippany, NJ 07981 USA (e-mail: [email protected]). D. G. Daut is with the Department of Electrical and Computer Engineering, Rutgers University, Piscataway, NJ 08854 USA (e-mail: [email protected] ece.rutgers.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2009.2014635

0018-9545/$26.00 © 2009 IEEE

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I. I NTRODUCTION Of the numerous reports dedicated to multicarrier code-division multiple-access (MC-CDMA) performance analysis, the majority restrict attention solely to the scenario of independent identically distributed fading statistics among the received subcarriers (i.e., diversity branches). A few works identify the problem of statistical dependence, and even fewer actually take it into consideration when evaluating performance via either simulation or theoretical analysis. To the best of the authors’ knowledge, [1]–[7] are the only works that analytically assess the mathematical complication brought about by relaxing the unrealistic assumption of independent fading among subcarriers. The first two works provide an error floor while restricting attention to downlink transmission with maximal-ratio combining (MRC). In [3], a bit-error-probability (BEP) expression was derived, consisting of a double integral with infinite limits for the MC-CDMA uplink with MRC. Similarly, in [4], a BEP expression consisting of an integral with infinite limits is derived for the uplink with MRC. The result is an approximation since a Gaussian assumption was invoked on the distribution of a user’s multiple-access interference (MAI). In addition to not being exact, the Gaussian approximation (GA) on the distribution of the MC-CDMA signal has been reported as being particularly inaccurate for a small number of users [2], [8]. In [5] and [6], BEP expressions were derived for the MC-CDMA downlink with MRC. The numerical evaluation of the exact BEP expression presented in [5] requires the calculation of residues of a complicated ratio of polynomials and subsequent averaging over the data bits. In the case of [6], a GA is used on the MAI. To the best of the authors’ knowledge, [7] has been the only work to analyze the performance of MC-CDMA with equal-gain combining (EGC) in the presence of correlated fading. The analysis in [7] is restricted to the MC-CDMA uplink, equalpower users, and an exponential multipath intensity profile for the channel. Furthermore, a GA is invoked on the distribution of the MAI encountered by a user. Incorporating a framework from multivariate statistical theory, this paper presents a detailed performance analysis of the MC-CDMA system under non-independent fading conditions with arbitrary-power users.

channel gains experienced by the transmitted signal of the ith user along its N subcarriers. We have not considered pre-equalization [9] or the more sophisticated M&Q-modification technique [10] in our MC-CDMA model. The complex envelope of the signal present at each user’s receiver prior to combining is represented by r = APb + n

present at√the jth subcarrier, the diagonal where rj is the signal √ √ matrix P = diag( P1 , P2 , . . . , PM ), and the N × M matrix A Δ

Δ

has ith column ai = Ci hi , with Ci = diag(ci ) and ci,j ∈ {−1, 1}. The complex-valued zero-mean additive-white-Gaussian-noise vector n has covariance matrix No I, where σ 2 = No /2 denotes the noise variance, and I is the identity matrix. The way in which we have specified the channel matrix H allows flexibility for both uplink and downlink analysis. In accordance with [11] and subsequent works, downlink transmission will lead to the signals received at each user to experience the same attenuation and phase offset at a given time instant via hi = h = [ρ1 ejφ1 , ρ2 ejφ2 , . . . , ρN ejφN ]T ∀i. The same scenario will not hold in the uplink, where the transmitted signal from each mobile is subject to a different channel response before its reception at the base station. Considering coherent detection and a linear receiver structure, the decision statistic at the output of the desired user’s combiner is given as

H H vd = Re tH d r = Re td APb + td n

r(t) =

M N

j 2πn TF t b

Pm Hm,n (t, τ )sm (t)cm,n e

Δ

fhd ,h1 ,...,hL (hd , h1 , . . . , hL ) = fhd (hd )

L

fhm (hm )

m=1

where L = M − 1 interferers are present. We shall restrict attention to Rayleigh fading at each diversity branch. Thus, the ith user’s channel propagation vector will follow a zero-mean complex Gaussian distribution denoted by hi ∼ CN (0, Rhi hi ), where Rhi hi ∈ CN ×N is the covariance matrix, and E[Rhi hi (j, j)] = E[ρ2i,j ] = Ωi,j . For the above processing to occur, we shall assume the channel state information available at the receiver of the desired user to be perfect. III. MC-CDMA W ITH MRC

+ n(t)

m=1 n=1

(1)

∞

with ˆbd = sgn{vd } and td = Cd wd . The complex vector wd represents the desired user’s gain vector, chosen in correspondence with performance and complexity criteria. By virtue of users independently transmitting signals, the N -dimensional propagation vectors present at a given receiver are assumed to be mutually independent. However, the individual vector elements, denoting the received signal present at different branches but originating from a common source, share a certain amount of correlation. Mathematically, for the uplink, this translates to the joint probability density function of the desired signal and interference as seen by desired user d from a set of M active users as being

II. MC-CDMA S YSTEM AND C HANNEL M ODEL We shall consider BPSK modulation, with bd ∈ {−1, 1} denoting the information bit of the desired user. We define the vector b = [b1 , b2 , . . . , bM ]T to be the transmitted bits of all M users at a given time instant, with P [bi = 1] = P [bi = −1] = 0.5∀i. A frequencyselective fading channel is considered with each transmitted subcarrier undergoing flat fading due to its narrow-band nature. Hence, the channel response witnessed by the nth transmitted subcarrier of the ith user is given as Hi,n (t, τ ) = ρi,n ejφi,n δ(t − τ ), where τ is the propagation delay. With synchronous MC-CDMA, the continuoustime baseband signal at the receiver is given as

(2)

where we have employed sm (t) = k=−∞ bm (k)u(t − kTb ), with one symbol being spread across the N subcarriers for each user. In (1), Pm is the transmit power of the mth user, Tb is the bit duration, cm,n is the nth component of the mth user’s spreading sequence, n(t) is the noise process, and F is a fixed integer that controls the frequency separation among subcarriers. The effect of the channel on the M users is described by a channel matrix H with the ith column, hi = [ρi,1 ejφi,1 , ρi,2 ejφi,2 , . . . , ρi,N ejφi,N ]T , denoting the complex

MRC, which is noted for mitigating the effects of multipath, requires that each subcarrier be scaled in proportion to its channel response via wdMRC = hd = [ρd,1 ejφd,1 , ρd,2 ejφd,2 , . . . , ρd,N ejφd,N ]T . We shall examine uplink and downlink performance separately for the scenarios of correlated and uncorrelated fading among received subcarriers. A. Uplink (Correlated Fading) With the received signal given by (2), then conditioning on hd , √ Δ setting bd = 1, and defining the intermediate vector xi = bi Pi Ci hi ,

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M

the assignment Δ

z = APb = x1 + x2 + · · · +

Pd Cd hd + · · · + xM

follows. Subsequently, by virtue of xi ∼ CN (0, Pi Ci Rhi hi Ci ) for M √ i = d, we have z ∼ CN ( Pd Cd hd , i=1,i=d Pi Ci Rhi hi Ci ). The portion of the decision statistic comprised of the signal√and interH ference αsig.+int. = Re{aH d APb} has distribution N ( Pd ad ad , M H (1/2)ad ( i=1,i=d Pi Ci Rhi hi Ci )ad ) since a linear transformation of a complex normal vector also yields a complex normal vector. With the noise portion of the decision statistic αnoise = 2 H Re{aH d n} ∼ N (0, σ ad ad ), the composite√decision statistic vd = H αsig.+int. + αnoise will have distribution N ( Pd aH d ad , ad ((1/2) × M 2 PCR C + σ I)ad ). By symmetry, the BEP condii=1,i=d i i hi hi i tioned on bd = 1 equals the BEP when bd = −1, leading to

P [e|hd ] = Q

⎛

μvd σvd

⎜ = Q⎝

π/2

=

1 π

exp

1 = π

aH d

2Pd hH d hd (M − 1)Pint Ωint + 2σ 2

2 Pd (aH d ad )

M 1 2

i=1,i=d

Pi Ci Rhi hi Ci +σ 2 I ad

π/2

−Pd hH d hd exp ((M −1)Pint Ωint +2σ 2 ) sin2 θ

2

≥Q

hH d Yhd

E

hH d hd

hH d Yhd

dt

k=0

1 − ν2 4

k (5)

with ν = c/(1 + c) and c = Pd Ωd /((M − 1)Pint Ωint + 2σ 2 ). In the derivation of (5), we have drawn upon the identity exp(−yH Ky)dy = π N / det(K), with K being Hermitian and y y being an N -dimensional complex vector, and used an integral relation from [13, eq. (5A.8)]. For the less general scenario of Pd = Pint and Ωd = Ωint , two different but numerically identical expressions have been derived in [8]. C. Downlink (Correlated Fading)

P1 C1 h + · · · +

Pd Cd h + · · · + bM

PM CM h

Pd Cd )h

with the random diagonal matrix X=

2

×t

dθ

N −1 2k 1 1−ν = k 2

= (X +

−1

d d

−3/2

N

0

z = APb

1 trace Rhd hd I − ln(t)Rhd hd Y × I − ln(t)Rh h Y 0

sin2 θ sin2 θ + c

= b1

M

bi

Pi Ci .

(6)

i=1,i=d

dθ

(4)

The decision statistic may now be expressed as

H vd = αsig.+int. + αnoise = Re aH d APb + ad n

is a result of the relation

H

x Ax 1 = E √ Γ(0.5) xH Bx

1

dθ.

With identical fading among users by virtue of hi = h∀i, we condition on bd = 1 to obtain

−1 2 (Γ(0.5) sin θ)2

0

1 π

(3)

Furthermore, for Rhd hd = Ωd I, the exact BEP can be derived as

⎟ ⎠

0

⎞

hH d hd

P [e|hd ] = Q

Pe =

hH d hd hH d Yhd

π

H where we have used the fact that aH i ai = hi hi and assigned Y = M 2 (1/2Pd )( i=1,i=d Pi Ci Rhi hi Ci + 2σ I), with Ci = Ci Cd = Cd Ci denoting another codeword. Unfortunately, a closed-form representation of the unconditional probability of error cannot be found. A lower bound on the BEP can be obtained via the calculation of a noninteger moment of ratios. The bound

(1/2Pd )( i=1,i=d Pi Rhi hi + 2σ 2 I) will be diagonal. A further simplification of Y = (1/2Pd )((M − 1)Pint Ωint + 2σ 2 )I exists if we assume equal-power interferers (i.e., Pi = Pint ∀i = d) and identically distributed fading statistics for each subcarrier, that is, Rhi hi = Ωint I ∀i = d. The conditional BEP given by (3) reduces to

=Q

Pe = E Q

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= hH (Cd X +

−1

trace RA (I − ln(t)RB) |I − ln(t)RB|

× t−3/2 dt

0

derived in [12] for x ∼ CN (0, R) and an alternate form of the Qfunction, which has finite limits. It is noteworthy that the above lower bound shall consist of two integrals with finite limits irrespective of either the diversity order N or the number of users M . B. Uplink (Uncorrelated Fading) With statistical independence among the N branches of the desired user, as well as those of the L interferers, the matrix Y =

Pd I)h + Re{hH Cd n}.

(7)

Conditioning on h and the interferers’ data bits bi ∀i = d, with 2 H αnoise = Re{aH d n} distributed as N (0, σ h h) and with the assign√ Δ ment W = Cd X + Pd I, we obtain vd ∼ N (hH Wh, σ 2 hH h). The conditional probability of error is then given as

P [e|h, {bint. }] = Q

hH Wh √ σ 2 hH h

(8)

with the designation {bint. } referring to the L interferers’ data bits (i.e., bi ∀i = d). Analogous to (3), a closed-form expression or more convenient representation of the BEP does not exist when unconditioning upon h. Additional complication results from the fact that

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unconditioning upon the interferers’ data bits must also be performed. Furthermore, not being able to simplify (8) in the case of a diagonal covariance matrix for h, the evaluation of downlink performance with uncorrelated fading would involve the simulation of (8) with the Rhh diagonal. Similar to the uplink analysis, a lower bound can be derived by evaluating a noninteger moment of ratios. We obtain

1 Pe ≥ E π

1 ×

π/2

exp

form. For the scenario of Rhi hi = Ωi I ∀i, however, an expression for the exact probability of error can be derived as a single integral with finite limits, which can be accurately and efficiently evaluated via the Gauss–Hermite method of integration [14]. First, M we obtain Y = ( i=1,i=d Pi Ωi )I, and consequently, wdH YwdH =

M

N ( i=1,i=d Pi Ωi ). Therefore, the conditional BEP in (10) can be rewritten as

⎛

−1 (Γ(0.5)σ sin θ)2

⎜

0

−1

trace Rhh W (I − ln(t)Rhh )

t−3/2 dt

|I − ln(t)Rhh |

P [e|{ρd,i }] = Q ⎜ ⎝

2 dθ

(9)

0

with the expectation used to remove the conditioning upon the interferers data sequence. IV. MC-CDMA W ITH EGC Noted for simplicity, EGC does not require channel information pertaining to the variation of each subcarrier’s envelope. The required cophasing is achieved through the assignment wdEGC = hd |ρd,1 =ρd,2 ,...,=ρd,N =1 = [ejφd,1 , ejφd,2 , . . . , ejφd,N ]T . We shall investigate the MC-CDMA uplink and downlink performance with correlated and uncorrelated fading among subcarriers.

N 2Pd

αsig.+int. = Re tH d APb ∼N

1 Pd wdH hd , tH 2 d

Pi Ci Rhi hi Ci

M

Pi Ci Rhi hi Ci td =

i=1,i=d

Pi Ωi + 2σ 2

⎟ ⎟ ⎠

ψv (jω) = ψnoise (jω) [ψsig.+int. (jω)]N d

1

= exp − ω η 2 2

1 F1

1 1 Ωd ω 2 − ; ;− 2 2 4

× exp −

= exp −

M

(11)

×

td .

Ωd ω 2 4

+jω

N

ω2 (2η + N Ωd ) 4

1 F1

1 1 Ωd ω 2 − ; ;− 2 2 4

πΩd 4

+ jω

N πΩd 4

(12)

i=1,i=d

Manipulating the quadratic form yields tH d

M i=1,i=d

ρd,i

i=1

and interpreted as having been based on the decision statistic N M vd = i=1 ρd,i + n, where n ∼ N (0, (N/2Pd )( i=1,i=d Pi Ωi + 2σ 2 )). It is well known that the distribution of a sum of independent Rayleigh random variables cannot be expressed in closed form. With this in mind, Zhang [15] attempted direct evaluation of the BEP by avoiding the intermediate step of determining the distribution of the Rayleigh sum. We shall incorporate Zhang’s general result in our application. The characteristic function (CF) of vd is given as

A. Uplink (Correlated Fading) With Msignal given by (2), maintaining z = APb ∼ √ the received CN ( Pd Cd hd , i=1,i=d Pi Ci Rhi hi Ci ) by conditioning on hd , and conditioning on bd = 1, it follows that

⎞

N

M

Pi tH d Ci Rhi hi (Ci td )

i=1,i=d

= wdH

M

where 1 F1 (.; .; .) is the confluent hypergeometric function, and η = M (N/2Pd )( i=1,i=d Pi Ωi + 2σ 2 ). The probability of error is obtained directly from the CF of the decision statistic via

Pi Ci Rhi hi Ci

1 1 Pe = − 2 2π

wd

" # ∞ Im ψv (jω) d ω

dω.

(13)

−∞

i=1,i=d

where, as before, Ci = Ci Cd = Cd Ci . Furthermore, αnoise = 2 H N (0, N σ 2 ) Re{tH d n} ∼ CN (0, 2σ td td ), or alternatively, αnoise ∼√ H H H since td td = wd Pd Pd wd = wd wd = N . Thus, vd ∼ N ( Pd wdH hd , M (1/2)wdH Ywd + N σ 2 ), with Y = i=1,i=d Pi Ci Rhi hi Ci being Hermitian. This leads to the conditional BEP of

⎛

P [e|hd ] = Q ⎝ !

Following the √ development in [15], we proceed with the change of variable z = ω 2η + N Ωd /2 and make the assignment

$ G(z) = z

−1

× Im

1 F1

Ωd z 2 1 1 − ; ; 2 2 2η + N Ωd

⎞

wdH hd 1 2Pd

⎠.

+ jz

(10)

N %

πΩd 2η + N Ωd

(14)

(wdH Ywd + 2N σ 2 )

In this case, the increased difficulty in the determination of the unconditional probability of error lies in the fact that wd is a function of hd via wd = hd |ρd,1 =ρd,2 ,...,=ρd,N =1 . Thus, the BEP must be evaluated numerically via the Monte Carlo simulation of (10). B. Uplink (Uncorrelated Fading) As a result of the matrix Y being diagonal, the conditional BEP expression given by (10) will maintain the same general

to express (13) in the following form: 1 1 Pe = − 2 π

∞ G(z) exp(−z 2 )dz ≈ 0

m/2 1 1 − vi G(zi ). 2 π

(15)

i=1

The mth-order Hermite polynomial Hm (.) is characterized as having zeros {zi } and corresponding weight factors {vi }. For various orders m, the zeros and weight factors of the Hermite polynomial are tabulated in [14, Table 25.10], with m = 20 being typically sufficient

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Fig. 1.

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Error performance of (left) MRC uplink and (right) MRC downlink, with N = 32 and SNR = 10 dB.

for excellent accuracy [13]. Since the integral is defined over the positive half axis, the calculation only requires the m/2 positive zeros and their corresponding weight factors. It should be noted that the Gauss–Hermite method of integration is only applicable because the CF of (12) is a smooth function. The evaluation of (14) can be facilitated by using the series relationship 1 F1

1 1

− ; ;k = − 2 2

∞ =0

k k ≈− (2 − 1) ! (2 − 1) ! b

(16)

=0

where for k < b + 2, the magnitude of the truncation error will be upper bounded by |Error| < ((b + 2)kb+1 e−k )/((2b + 1)(b + 2 − k)(b + 1)!). Thus, a desired and deterministic level of accuracy is attainable in (16) by choosing b to be large enough to satisfy b > k − 2. In the case of m = 20 in (15), the maximum absolute of zi is apvalue M proximately five [14]. With k = zi2 /((N/Pd Ωd )( i=1,i=d Pi Ωi + 2σ 2 ) + N ) from (14), the condition imposed on b is rather modest. Fig. 2. Error performance of EGC uplink and downlink, with N = 32 and SNR = 10 dB.

C. Downlink (Correlated Fading) Downlink analysis starts by noting that z = APb = (X + √ bd Pd Cd )h, with X as defined in (6). Subsequent conditioning upon bd = 1 leads to the decision statistic

H vd = αsig.+int. + αnoise = Re tH d APb + td n

= wdH (Cd X +

Pd I)h + Re wdH Cd n .

P [e|{ρi }, {bint. }] = Q

N

wρ i=1 i i √ N σ2

⎡

(17)

The term αsig.+int. = wdH Wh may be simplified to qT diag(W), where q = abs{h} = [ρ1 , ρ2 , . . . , ρN ]T . With αnoise ∼ N (0, N σ 2 ), it follows that

for the EGC uplink with uncorrelated fading. Henceforth, incorporating the same notation and approach as in the uplink, the exact BEP is represented as

1 1 Pe = − M −1 ⎣ 2 π2

∞

(18)

,...,

,...,

bi=d ∈{−1,1}

⎤

G (z, {bint. }) exp(−z 2 )dz ⎦

(19)

bM ∈{−1,1} 0

with the modification G (z, {bint. }) = Im

$N i=1

D. Downlink (Uncorrelated Fading) Independence among branches results in the conditional BEP expression of (18), taking the same general form as that of (11) derived

b1 ∈{−1,1} b2 ∈{−1,1}

M √ where wi = cd,i ( j=1,j=d bj Pj cj,i ) + Pd in connection with √ W = diag(w1 , w2 , . . . , wN ) = Cd X + Pd I. With correlated fading, the exact BEP would be computed via the simulation of (18).

+ jz

1 F1

* Ω

Ωw2 z 2 1 1 − ; ; N i 2 2 Ω w2 +2N σ 2 i=1 i

N

πΩwi2

i=1

wi2 + 2N σ 2

%

× z −1

(20)

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Fig. 3. Error performance of (left) MRC uplink and (right) MRC downlink for lightly loaded (L = 5) and heavily loaded (L = 20) systems, with N = 32.

where the dependence of the BEP on the interferers’ data sequence is expressed through the dependence of G(z) on {bint. }. The integral in (19) can be readily evaluated via (15); however, it is obvious that the computation of the BEP in this case will be more tedious due to the necessity of removing the conditioning on the interferers’ data sequence. V. S IMULATION R ESULTS AND D ISCUSSION We shall empirically assess the impact of correlated fading on a user’s performance and investigate the tightness of the lower bounds derived for MRC. With the scenario of mutually independent fading among subcarriers as a baseline, we specify a channel model for which the impact of correlated fading on a user’s performance is to be evaluated. Incorporating the notation and definitions of [16] for an isotropic 2 ) scattering environment, we specify u1ij = Jo (2πfm τij )/(1 + kij and u2ij = −kij u1ij with the elements of the desired user’s covariance matrix determined as

$ Rhd hd (i, j) =

Ωd u1ij + jΩd u2ij , Ωd u1ij − jΩd u2ij , Ωd ,

i>j i 11, performance improvements of up to a factor of two are apparent. Although counterintuitive, such unexpected improvement can be attributed to the correlation between subcarriers reducing the MAI for high interference levels. A similar result was obtained in [1] for the MRC downlink where the effects of thermal noise were neglected. The derived lower bounds are shown as having little deviation from the exact BEP in Fig. 1. Analogous to the MRC

1 This is frequently referred to as Monte Carlo integration since the unconditioning (integration) is, in effect, performed by averaging over a large number of realizations for the random quantity. 2 In Figs. 1–4, we have M/N ∈ [1/16, 1]. The same general trends were also verified for N = 64 with L = 1 to 63 for which M/N ∈ [1/32, 1], wherein 1010 samples were used for the Monte Carlo simulation of the conditional BEP expressions.

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Fig. 4.

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Error performance of (left) EGC uplink and (right) EGC downlink for lightly loaded (L = 5) and heavily loaded (L = 20) systems, with N = 32.

downlink, EGC performance is severely degraded with the loss of diversity for a small number of interferers, with the degradation diminishing for increasing levels of MAI, and correlation is actually beneficial when the system is interference dominated. Specifically, Fig. 2 shows that for the uplink, the degradation brought on by correlated fading is most pronounced for L < 5, where a degradation of up to an order of magnitude is apparent. The degradation gradually diminishes until L > 14, where correlated fading improves performance. For the EGC downlink with L < 5, a minimal performance loss of a factor of three is apparent with correlated fading, with the degradation exceeding an order of magnitude with low interference levels. For L > 15, correlated fading improves performance. The system-level simulation of the MC-CDMA system is included in Figs. 1 and 2, with the correlated Rayleigh variates generated according to the methodology described in [17]. The primary results shown in Figs. 3 and 4 are summarized as follows. In Fig. 3, we note that for the MRC uplink, in a heavily loaded system, there is little degradation in BEP brought about by correlated fading. This is untrue for a lightly loaded system, where the degradation is apparent and increases with larger SNR values. For the MRC downlink, Fig. 3 indicates that correlated fading is either beneficial or nondetrimental for both loading conditions. In the case of the EGC uplink and downlink, Fig. 4 illustrates that for lightly loaded systems, correlated fading introduces a significant deterioration in BEP with increasing SNR values, while little deviation is apparent in heavily loaded systems.

VI. C ONCLUSION The performance of MC-CDMA in the presence of correlated fading has been investigated. The analysis has been exact with respect to not invoking the traditional GA on the MAI. The MRC and EGC schemes were considered with variable-power interferers, correlated Rayleigh fading among subcarriers, and thermal background noise. In general, the results show that the effects of correlated fading are not negligible, particularly for lightly loaded communication systems.

ACKNOWLEDGMENT The authors wish to thank R. Berk, X.-L. Meng, and L. Shepp for helpful discussions. We also thank the anonymous reviewers for constructive comments, which have improved the quality of the paper.

R EFERENCES [1] Q. Shi and M. Latva-aho, “An exact error floor for downlink MC-CDMA in correlated Rayleigh fading channels,” IEEE Commun. Lett., vol. 6, no. 5, pp. 196–198, May 2002. [2] Q. Shi and M. Latva-aho, “Accurate bit-error rate evaluation for synchronous MC-CDMA over Nakagami-m fading channels using moment generating functions,” IEEE Trans. Wireless Commun., vol. 4, no. 2, pp. 422–433, Mar. 2005. [3] T. Kim, Y. Kim, J. Park, K. Ko, S. Choi, C. Kang, and D. Hong, “Performance of an MC-CDMA system with frequency offsets in correlated fading,” in Proc. IEEE ICC, New Orleans, LA, Jun. 2000, pp. 1095–1099. [4] J. Park, J. Kim, S. Choi, N. Cho, and D. Hong, “Performance of MC-CDMA systems in non-independent Rayleigh fading,” in Proc. IEEE ICC, Vancouver, Canada, Jun. 1999, pp. 506–510. [5] Z. Hou and V. K. Dubey, “Exact analysis for downlink MC-CDMA in Rayleigh fading channels,” IEEE Commun. Lett., vol. 8, no. 2, pp. 90–92, Feb. 2004. [6] Z. Li and M. Latva-aho, “Error probability of interleaved MCCDMA systems with MRC receiver and correlated Nakagami-m fading channels,” IEEE Trans. Commun., vol. 53, no. 6, pp. 919–923, Jun. 2005. [7] Y. Feng and J. Qin, “BER of MC-CDMA systems with EGC in correlated Nakagami-m fading,” IEEE Commun. Lett., vol. 10, no. 10, pp. 689–691, Oct. 2006. [8] Q. Shi and M. Latva-aho, “Exact bit error rate calculations for synchronous MC-CDMA over a Rayleigh fading channel,” IEEE Commun. Lett., vol. 6, no. 7, pp. 276–278, Jul. 2002. [9] I. Cosovic and S. Kaiser, “A unified analysis of diversity exploitation in multicarrier CDMA,” IEEE Trans. Veh. Technol., vol. 56, no. 4, pp. 2051–2062, Jul. 2007. [10] K. Fazel and S. Kaiser, Multi-Carrier and Spread Spectrum Systems. Hoboken, NJ: Wiley, 2003. [11] N. Yee, J. P. Linnartz, and G. Fettweis, “Multi-carrier CDMA in indoor wireless radio networks,” IEICE Trans. Commun., vol. E77-B, no. 7, pp. 900–904, Jul. 1994. [12] X. Meng, “From unit root to Stein’s estimator to Fisher’s k statistics: If you have a moment, I can tell you more,” Stat. Sci., vol. 20, no. 2, pp. 141–162, 2005. [13] M. K. Simon and M. S. Alouini, Digital Communication Over Fading Channels. New York: Wiley, 2000. [14] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. [15] Q. T. Zhang, “A simple approach to probability distribution of error for equal gain combiners over Rayleigh channels,” IEEE Trans. Veh. Technol., vol. 48, no. 4, pp. 1151–1154, Jul. 1999. [16] W. C. Jakes, Ed., Microwave Mobile Communications. New York: IEEE Press, 1974. [17] S. Sorooshyari and D. G. Daut, “On the generation of correlated Rayleigh fading envelopes for accurate simulation of diversity channels,” IEEE Trans. Commun., vol. 54, no. 8, pp. 1381–1386, Aug. 2006.

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ACKNOWLEDGMENT The authors would like to thank the editor and anonymous reviewers. Their comments have significantly improved the quality of this paper. The corresponding author of this paper is Wonjun Lee. R EFERENCES

Fig. 4. Optimal binding-management-key refresh time. (a) Effect of μS . (b) Effect of δ.

a new BU procedure is performed. Therefore, the key exposal time and N decrease with the increase in μS . The effect of πON is also demonstrated in Fig. 3(a). When πON is high, the probability that the state is ON for a key exposal attack is also high. Hence, more packets can be affected by the key exposal when πON is high. In Fig. 3(a), it can be seen that the simulation results are consistent with the analytical results. When TR is high, the key exposal time can be long because the binding management key is not frequently refreshed. Therefore, as shown in Fig. 3(b), N increases as TR increases. It can be seen that N becomes fairly stable when TR exceeds a certain point. This is because the mobility rate μS is fixed at 1.0, and thus, the binding management key can be refreshed by a handoff even though TR is very high.

[1] D. Johnson, C. Perkins, and J. Arkko, Mobility Support in IPv6, Jun. 2004, IETF. RFC 3775. [2] P. Nikander, J. Arkko, T. Aura, and G. Montenegro, “Mobile IP version 6 (MIPv6) route optimization security design,” in Proc. IEEE VTC—Fall, Sep. 2003, pp. 2004–2008. [3] J. Kempf, J. Arkko, and P. Nikander, “Mobile IPv6 security,” Wirel. Pers. Commun., vol. 29, no. 3/4, pp. 389–414, Jun. 2004. [4] J. Hass and Y. Lin, “On optimizing the location update costs in the presence of database failures,” Wirel. Netw., vol. 4, no. 5, pp. 419–426, Aug. 1998. [5] L. Kleinrock, Queuing Systems Volume 1: Theory. New York: Wiley, 1975. [6] Y. Haung, “Determining the optimal buffer size for short message transfer in a heterogeneous GPRS/UMTS network,” IEEE Trans. Veh. Technol., vol. 52, no. 1, pp. 216–225, Jan. 2003. [7] X. Zhang, J. G. Castellanos, and A. T. Capbell, “P-MIP: Paging extensions for mobile IP,” ACM Mob. Netw. Appl., vol. 7, no. 2, pp. 127–141, Apr. 2002. [8] D. Staehle, K. Leibnitz, and K. Tsipotis, “QoS of Internet access with GPRS,” Wirel. Netw., vol. 9, no. 3, pp. 213–222, May 2003. [9] A. Downey, “The structural cause of file size distributions,” in Proc. IEEE MASCOT, Aug. 2001, pp. 361–370.

Performance of Multicarrier CDMA in the Presence of Correlated Fading

B. Optimal Binding-Management-Key Refresh Interval The optimal binding-management-key refresh interval in different μS is shown in Fig. 4(a). It can be seen that the optimal bindingmanagement-key refresh interval increases with the increase in μS . This can be explained as follows. As μS increases, BU procedures due to handoffs are more frequently performed. Therefore, when the bound for the number of packets influenced by the key exposal attack is given, the periodical binding-management-key refresh can be triggered with a long interval for a high μS . The impact of πON can also be found in Fig. 4(a). That is, a longer TR can be used for a low πON since the number of arrival packets is small. Fig. 4(b) demonstrates the effect of δ on the optimal bindingmanagement-key refresh interval. As shown in Fig. 4(b), a longer binding-management-key refresh interval can be permitted when δ is high. On the other hand, if δ is low, a shorter binding-management-key refresh interval should be employed to minimize the number of packets affected by the key exposal attack. In Fig. 4, it can be shown that the binding-management-key refresh interval should be determined by considering both the mobility rate and the traffic pattern. Furthermore, the security level for packets (i.e., δ) is an important consideration. V. C ONCLUSION Secure key negotiation is a challenging issue for the successful deployment of mobile IPv6. In this paper, we have derived the optimal binding-management-key refresh interval bounding the number of infected packets to a given threshold. From analytical and simulation results, the effects of the mobility rate and traffic intensity are demonstrated. By devising an algorithm, it is shown that the optimal binding-management-key refresh interval can be adaptively determined to the mobility rate and traffic load.

Siamak Sorooshyari and David G. Daut, Senior Member, IEEE

Abstract—Incorporating a framework from multivariate statistical theory, analytical results for the performance of the downlink and uplink of multicarrier code-division multiple access (MC-CDMA) in the presence of correlated Rayleigh fading are presented. Bit-error-probability expressions are derived for MC-CDMA diversity combining schemes, while statistical generality is maintained by assuming no specific form for the covariance matrix of the desired user and interferers. The scenarios of uncorrelated fading among subcarriers and equal-power users are considered as special cases. The analysis is exact in the sense that the error probability is derived without any assumptions on the distribution of the multiple-access interference encountered by a user. With the scenario of independent fading among subcarriers as a baseline, the performance degradation incurred by correlated fading is analyzed for an empirical wireless channel. Index Terms—Correlated fading, diversity combining, multicarrier code-division multiple access (MC-CDMA), multivariate analysis.

Manuscript received July 3, 2008; revised October 30, 2008 and December 25, 2008. First published February 6, 2009; current version published August 14, 2009. The review of this paper was coordinated by Prof. N. Arumugam. S. Sorooshyari is with Alcatel-Lucent, Whippany, NJ 07981 USA (e-mail: [email protected]). D. G. Daut is with the Department of Electrical and Computer Engineering, Rutgers University, Piscataway, NJ 08854 USA (e-mail: [email protected] ece.rutgers.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2009.2014635

0018-9545/$26.00 © 2009 IEEE

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I. I NTRODUCTION Of the numerous reports dedicated to multicarrier code-division multiple-access (MC-CDMA) performance analysis, the majority restrict attention solely to the scenario of independent identically distributed fading statistics among the received subcarriers (i.e., diversity branches). A few works identify the problem of statistical dependence, and even fewer actually take it into consideration when evaluating performance via either simulation or theoretical analysis. To the best of the authors’ knowledge, [1]–[7] are the only works that analytically assess the mathematical complication brought about by relaxing the unrealistic assumption of independent fading among subcarriers. The first two works provide an error floor while restricting attention to downlink transmission with maximal-ratio combining (MRC). In [3], a bit-error-probability (BEP) expression was derived, consisting of a double integral with infinite limits for the MC-CDMA uplink with MRC. Similarly, in [4], a BEP expression consisting of an integral with infinite limits is derived for the uplink with MRC. The result is an approximation since a Gaussian assumption was invoked on the distribution of a user’s multiple-access interference (MAI). In addition to not being exact, the Gaussian approximation (GA) on the distribution of the MC-CDMA signal has been reported as being particularly inaccurate for a small number of users [2], [8]. In [5] and [6], BEP expressions were derived for the MC-CDMA downlink with MRC. The numerical evaluation of the exact BEP expression presented in [5] requires the calculation of residues of a complicated ratio of polynomials and subsequent averaging over the data bits. In the case of [6], a GA is used on the MAI. To the best of the authors’ knowledge, [7] has been the only work to analyze the performance of MC-CDMA with equal-gain combining (EGC) in the presence of correlated fading. The analysis in [7] is restricted to the MC-CDMA uplink, equalpower users, and an exponential multipath intensity profile for the channel. Furthermore, a GA is invoked on the distribution of the MAI encountered by a user. Incorporating a framework from multivariate statistical theory, this paper presents a detailed performance analysis of the MC-CDMA system under non-independent fading conditions with arbitrary-power users.

channel gains experienced by the transmitted signal of the ith user along its N subcarriers. We have not considered pre-equalization [9] or the more sophisticated M&Q-modification technique [10] in our MC-CDMA model. The complex envelope of the signal present at each user’s receiver prior to combining is represented by r = APb + n

present at√the jth subcarrier, the diagonal where rj is the signal √ √ matrix P = diag( P1 , P2 , . . . , PM ), and the N × M matrix A Δ

Δ

has ith column ai = Ci hi , with Ci = diag(ci ) and ci,j ∈ {−1, 1}. The complex-valued zero-mean additive-white-Gaussian-noise vector n has covariance matrix No I, where σ 2 = No /2 denotes the noise variance, and I is the identity matrix. The way in which we have specified the channel matrix H allows flexibility for both uplink and downlink analysis. In accordance with [11] and subsequent works, downlink transmission will lead to the signals received at each user to experience the same attenuation and phase offset at a given time instant via hi = h = [ρ1 ejφ1 , ρ2 ejφ2 , . . . , ρN ejφN ]T ∀i. The same scenario will not hold in the uplink, where the transmitted signal from each mobile is subject to a different channel response before its reception at the base station. Considering coherent detection and a linear receiver structure, the decision statistic at the output of the desired user’s combiner is given as

H H vd = Re tH d r = Re td APb + td n

r(t) =

M N

j 2πn TF t b

Pm Hm,n (t, τ )sm (t)cm,n e

Δ

fhd ,h1 ,...,hL (hd , h1 , . . . , hL ) = fhd (hd )

L

fhm (hm )

m=1

where L = M − 1 interferers are present. We shall restrict attention to Rayleigh fading at each diversity branch. Thus, the ith user’s channel propagation vector will follow a zero-mean complex Gaussian distribution denoted by hi ∼ CN (0, Rhi hi ), where Rhi hi ∈ CN ×N is the covariance matrix, and E[Rhi hi (j, j)] = E[ρ2i,j ] = Ωi,j . For the above processing to occur, we shall assume the channel state information available at the receiver of the desired user to be perfect. III. MC-CDMA W ITH MRC

+ n(t)

m=1 n=1

(1)

∞

with ˆbd = sgn{vd } and td = Cd wd . The complex vector wd represents the desired user’s gain vector, chosen in correspondence with performance and complexity criteria. By virtue of users independently transmitting signals, the N -dimensional propagation vectors present at a given receiver are assumed to be mutually independent. However, the individual vector elements, denoting the received signal present at different branches but originating from a common source, share a certain amount of correlation. Mathematically, for the uplink, this translates to the joint probability density function of the desired signal and interference as seen by desired user d from a set of M active users as being

II. MC-CDMA S YSTEM AND C HANNEL M ODEL We shall consider BPSK modulation, with bd ∈ {−1, 1} denoting the information bit of the desired user. We define the vector b = [b1 , b2 , . . . , bM ]T to be the transmitted bits of all M users at a given time instant, with P [bi = 1] = P [bi = −1] = 0.5∀i. A frequencyselective fading channel is considered with each transmitted subcarrier undergoing flat fading due to its narrow-band nature. Hence, the channel response witnessed by the nth transmitted subcarrier of the ith user is given as Hi,n (t, τ ) = ρi,n ejφi,n δ(t − τ ), where τ is the propagation delay. With synchronous MC-CDMA, the continuoustime baseband signal at the receiver is given as

(2)

where we have employed sm (t) = k=−∞ bm (k)u(t − kTb ), with one symbol being spread across the N subcarriers for each user. In (1), Pm is the transmit power of the mth user, Tb is the bit duration, cm,n is the nth component of the mth user’s spreading sequence, n(t) is the noise process, and F is a fixed integer that controls the frequency separation among subcarriers. The effect of the channel on the M users is described by a channel matrix H with the ith column, hi = [ρi,1 ejφi,1 , ρi,2 ejφi,2 , . . . , ρi,N ejφi,N ]T , denoting the complex

MRC, which is noted for mitigating the effects of multipath, requires that each subcarrier be scaled in proportion to its channel response via wdMRC = hd = [ρd,1 ejφd,1 , ρd,2 ejφd,2 , . . . , ρd,N ejφd,N ]T . We shall examine uplink and downlink performance separately for the scenarios of correlated and uncorrelated fading among received subcarriers. A. Uplink (Correlated Fading) With the received signal given by (2), then conditioning on hd , √ Δ setting bd = 1, and defining the intermediate vector xi = bi Pi Ci hi ,

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 7, SEPTEMBER 2009

M

the assignment Δ

z = APb = x1 + x2 + · · · +

Pd Cd hd + · · · + xM

follows. Subsequently, by virtue of xi ∼ CN (0, Pi Ci Rhi hi Ci ) for M √ i = d, we have z ∼ CN ( Pd Cd hd , i=1,i=d Pi Ci Rhi hi Ci ). The portion of the decision statistic comprised of the signal√and interH ference αsig.+int. = Re{aH d APb} has distribution N ( Pd ad ad , M H (1/2)ad ( i=1,i=d Pi Ci Rhi hi Ci )ad ) since a linear transformation of a complex normal vector also yields a complex normal vector. With the noise portion of the decision statistic αnoise = 2 H Re{aH d n} ∼ N (0, σ ad ad ), the composite√decision statistic vd = H αsig.+int. + αnoise will have distribution N ( Pd aH d ad , ad ((1/2) × M 2 PCR C + σ I)ad ). By symmetry, the BEP condii=1,i=d i i hi hi i tioned on bd = 1 equals the BEP when bd = −1, leading to

P [e|hd ] = Q

⎛

μvd σvd

⎜ = Q⎝

π/2

=

1 π

exp

1 = π

aH d

2Pd hH d hd (M − 1)Pint Ωint + 2σ 2

2 Pd (aH d ad )

M 1 2

i=1,i=d

Pi Ci Rhi hi Ci +σ 2 I ad

π/2

−Pd hH d hd exp ((M −1)Pint Ωint +2σ 2 ) sin2 θ

2

≥Q

hH d Yhd

E

hH d hd

hH d Yhd

dt

k=0

1 − ν2 4

k (5)

with ν = c/(1 + c) and c = Pd Ωd /((M − 1)Pint Ωint + 2σ 2 ). In the derivation of (5), we have drawn upon the identity exp(−yH Ky)dy = π N / det(K), with K being Hermitian and y y being an N -dimensional complex vector, and used an integral relation from [13, eq. (5A.8)]. For the less general scenario of Pd = Pint and Ωd = Ωint , two different but numerically identical expressions have been derived in [8]. C. Downlink (Correlated Fading)

P1 C1 h + · · · +

Pd Cd h + · · · + bM

PM CM h

Pd Cd )h

with the random diagonal matrix X=

2

×t

dθ

N −1 2k 1 1−ν = k 2

= (X +

−1

d d

−3/2

N

0

z = APb

1 trace Rhd hd I − ln(t)Rhd hd Y × I − ln(t)Rh h Y 0

sin2 θ sin2 θ + c

= b1

M

bi

Pi Ci .

(6)

i=1,i=d

dθ

(4)

The decision statistic may now be expressed as

H vd = αsig.+int. + αnoise = Re aH d APb + ad n

is a result of the relation

H

x Ax 1 = E √ Γ(0.5) xH Bx

1

dθ.

With identical fading among users by virtue of hi = h∀i, we condition on bd = 1 to obtain

−1 2 (Γ(0.5) sin θ)2

0

1 π

(3)

Furthermore, for Rhd hd = Ωd I, the exact BEP can be derived as

⎟ ⎠

0

⎞

hH d hd

P [e|hd ] = Q

Pe =

hH d hd hH d Yhd

π

H where we have used the fact that aH i ai = hi hi and assigned Y = M 2 (1/2Pd )( i=1,i=d Pi Ci Rhi hi Ci + 2σ I), with Ci = Ci Cd = Cd Ci denoting another codeword. Unfortunately, a closed-form representation of the unconditional probability of error cannot be found. A lower bound on the BEP can be obtained via the calculation of a noninteger moment of ratios. The bound

(1/2Pd )( i=1,i=d Pi Rhi hi + 2σ 2 I) will be diagonal. A further simplification of Y = (1/2Pd )((M − 1)Pint Ωint + 2σ 2 )I exists if we assume equal-power interferers (i.e., Pi = Pint ∀i = d) and identically distributed fading statistics for each subcarrier, that is, Rhi hi = Ωint I ∀i = d. The conditional BEP given by (3) reduces to

=Q

Pe = E Q

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= hH (Cd X +

−1

trace RA (I − ln(t)RB) |I − ln(t)RB|

× t−3/2 dt

0

derived in [12] for x ∼ CN (0, R) and an alternate form of the Qfunction, which has finite limits. It is noteworthy that the above lower bound shall consist of two integrals with finite limits irrespective of either the diversity order N or the number of users M . B. Uplink (Uncorrelated Fading) With statistical independence among the N branches of the desired user, as well as those of the L interferers, the matrix Y =

Pd I)h + Re{hH Cd n}.

(7)

Conditioning on h and the interferers’ data bits bi ∀i = d, with 2 H αnoise = Re{aH d n} distributed as N (0, σ h h) and with the assign√ Δ ment W = Cd X + Pd I, we obtain vd ∼ N (hH Wh, σ 2 hH h). The conditional probability of error is then given as

P [e|h, {bint. }] = Q

hH Wh √ σ 2 hH h

(8)

with the designation {bint. } referring to the L interferers’ data bits (i.e., bi ∀i = d). Analogous to (3), a closed-form expression or more convenient representation of the BEP does not exist when unconditioning upon h. Additional complication results from the fact that

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unconditioning upon the interferers’ data bits must also be performed. Furthermore, not being able to simplify (8) in the case of a diagonal covariance matrix for h, the evaluation of downlink performance with uncorrelated fading would involve the simulation of (8) with the Rhh diagonal. Similar to the uplink analysis, a lower bound can be derived by evaluating a noninteger moment of ratios. We obtain

1 Pe ≥ E π

1 ×

π/2

exp

form. For the scenario of Rhi hi = Ωi I ∀i, however, an expression for the exact probability of error can be derived as a single integral with finite limits, which can be accurately and efficiently evaluated via the Gauss–Hermite method of integration [14]. First, M we obtain Y = ( i=1,i=d Pi Ωi )I, and consequently, wdH YwdH =

M

N ( i=1,i=d Pi Ωi ). Therefore, the conditional BEP in (10) can be rewritten as

⎛

−1 (Γ(0.5)σ sin θ)2

⎜

0

−1

trace Rhh W (I − ln(t)Rhh )

t−3/2 dt

|I − ln(t)Rhh |

P [e|{ρd,i }] = Q ⎜ ⎝

2 dθ

(9)

0

with the expectation used to remove the conditioning upon the interferers data sequence. IV. MC-CDMA W ITH EGC Noted for simplicity, EGC does not require channel information pertaining to the variation of each subcarrier’s envelope. The required cophasing is achieved through the assignment wdEGC = hd |ρd,1 =ρd,2 ,...,=ρd,N =1 = [ejφd,1 , ejφd,2 , . . . , ejφd,N ]T . We shall investigate the MC-CDMA uplink and downlink performance with correlated and uncorrelated fading among subcarriers.

N 2Pd

αsig.+int. = Re tH d APb ∼N

1 Pd wdH hd , tH 2 d

Pi Ci Rhi hi Ci

M

Pi Ci Rhi hi Ci td =

i=1,i=d

Pi Ωi + 2σ 2

⎟ ⎟ ⎠

ψv (jω) = ψnoise (jω) [ψsig.+int. (jω)]N d

1

= exp − ω η 2 2

1 F1

1 1 Ωd ω 2 − ; ;− 2 2 4

× exp −

= exp −

M

(11)

×

td .

Ωd ω 2 4

+jω

N

ω2 (2η + N Ωd ) 4

1 F1

1 1 Ωd ω 2 − ; ;− 2 2 4

πΩd 4

+ jω

N πΩd 4

(12)

i=1,i=d

Manipulating the quadratic form yields tH d

M i=1,i=d

ρd,i

i=1

and interpreted as having been based on the decision statistic N M vd = i=1 ρd,i + n, where n ∼ N (0, (N/2Pd )( i=1,i=d Pi Ωi + 2σ 2 )). It is well known that the distribution of a sum of independent Rayleigh random variables cannot be expressed in closed form. With this in mind, Zhang [15] attempted direct evaluation of the BEP by avoiding the intermediate step of determining the distribution of the Rayleigh sum. We shall incorporate Zhang’s general result in our application. The characteristic function (CF) of vd is given as

A. Uplink (Correlated Fading) With Msignal given by (2), maintaining z = APb ∼ √ the received CN ( Pd Cd hd , i=1,i=d Pi Ci Rhi hi Ci ) by conditioning on hd , and conditioning on bd = 1, it follows that

⎞

N

M

Pi tH d Ci Rhi hi (Ci td )

i=1,i=d

= wdH

M

where 1 F1 (.; .; .) is the confluent hypergeometric function, and η = M (N/2Pd )( i=1,i=d Pi Ωi + 2σ 2 ). The probability of error is obtained directly from the CF of the decision statistic via

Pi Ci Rhi hi Ci

1 1 Pe = − 2 2π

wd

" # ∞ Im ψv (jω) d ω

dω.

(13)

−∞

i=1,i=d

where, as before, Ci = Ci Cd = Cd Ci . Furthermore, αnoise = 2 H N (0, N σ 2 ) Re{tH d n} ∼ CN (0, 2σ td td ), or alternatively, αnoise ∼√ H H H since td td = wd Pd Pd wd = wd wd = N . Thus, vd ∼ N ( Pd wdH hd , M (1/2)wdH Ywd + N σ 2 ), with Y = i=1,i=d Pi Ci Rhi hi Ci being Hermitian. This leads to the conditional BEP of

⎛

P [e|hd ] = Q ⎝ !

Following the √ development in [15], we proceed with the change of variable z = ω 2η + N Ωd /2 and make the assignment

$ G(z) = z

−1

× Im

1 F1

Ωd z 2 1 1 − ; ; 2 2 2η + N Ωd

⎞

wdH hd 1 2Pd

⎠.

+ jz

(10)

N %

πΩd 2η + N Ωd

(14)

(wdH Ywd + 2N σ 2 )

In this case, the increased difficulty in the determination of the unconditional probability of error lies in the fact that wd is a function of hd via wd = hd |ρd,1 =ρd,2 ,...,=ρd,N =1 . Thus, the BEP must be evaluated numerically via the Monte Carlo simulation of (10). B. Uplink (Uncorrelated Fading) As a result of the matrix Y being diagonal, the conditional BEP expression given by (10) will maintain the same general

to express (13) in the following form: 1 1 Pe = − 2 π

∞ G(z) exp(−z 2 )dz ≈ 0

m/2 1 1 − vi G(zi ). 2 π

(15)

i=1

The mth-order Hermite polynomial Hm (.) is characterized as having zeros {zi } and corresponding weight factors {vi }. For various orders m, the zeros and weight factors of the Hermite polynomial are tabulated in [14, Table 25.10], with m = 20 being typically sufficient

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 7, SEPTEMBER 2009

Fig. 1.

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Error performance of (left) MRC uplink and (right) MRC downlink, with N = 32 and SNR = 10 dB.

for excellent accuracy [13]. Since the integral is defined over the positive half axis, the calculation only requires the m/2 positive zeros and their corresponding weight factors. It should be noted that the Gauss–Hermite method of integration is only applicable because the CF of (12) is a smooth function. The evaluation of (14) can be facilitated by using the series relationship 1 F1

1 1

− ; ;k = − 2 2

∞ =0

k k ≈− (2 − 1) ! (2 − 1) ! b

(16)

=0

where for k < b + 2, the magnitude of the truncation error will be upper bounded by |Error| < ((b + 2)kb+1 e−k )/((2b + 1)(b + 2 − k)(b + 1)!). Thus, a desired and deterministic level of accuracy is attainable in (16) by choosing b to be large enough to satisfy b > k − 2. In the case of m = 20 in (15), the maximum absolute of zi is apvalue M proximately five [14]. With k = zi2 /((N/Pd Ωd )( i=1,i=d Pi Ωi + 2σ 2 ) + N ) from (14), the condition imposed on b is rather modest. Fig. 2. Error performance of EGC uplink and downlink, with N = 32 and SNR = 10 dB.

C. Downlink (Correlated Fading) Downlink analysis starts by noting that z = APb = (X + √ bd Pd Cd )h, with X as defined in (6). Subsequent conditioning upon bd = 1 leads to the decision statistic

H vd = αsig.+int. + αnoise = Re tH d APb + td n

= wdH (Cd X +

Pd I)h + Re wdH Cd n .

P [e|{ρi }, {bint. }] = Q

N

wρ i=1 i i √ N σ2

⎡

(17)

The term αsig.+int. = wdH Wh may be simplified to qT diag(W), where q = abs{h} = [ρ1 , ρ2 , . . . , ρN ]T . With αnoise ∼ N (0, N σ 2 ), it follows that

for the EGC uplink with uncorrelated fading. Henceforth, incorporating the same notation and approach as in the uplink, the exact BEP is represented as

1 1 Pe = − M −1 ⎣ 2 π2

∞

(18)

,...,

,...,

bi=d ∈{−1,1}

⎤

G (z, {bint. }) exp(−z 2 )dz ⎦

(19)

bM ∈{−1,1} 0

with the modification G (z, {bint. }) = Im

$N i=1

D. Downlink (Uncorrelated Fading) Independence among branches results in the conditional BEP expression of (18), taking the same general form as that of (11) derived

b1 ∈{−1,1} b2 ∈{−1,1}

M √ where wi = cd,i ( j=1,j=d bj Pj cj,i ) + Pd in connection with √ W = diag(w1 , w2 , . . . , wN ) = Cd X + Pd I. With correlated fading, the exact BEP would be computed via the simulation of (18).

+ jz

1 F1

* Ω

Ωw2 z 2 1 1 − ; ; N i 2 2 Ω w2 +2N σ 2 i=1 i

N

πΩwi2

i=1

wi2 + 2N σ 2

%

× z −1

(20)

3842

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 7, SEPTEMBER 2009

Fig. 3. Error performance of (left) MRC uplink and (right) MRC downlink for lightly loaded (L = 5) and heavily loaded (L = 20) systems, with N = 32.

where the dependence of the BEP on the interferers’ data sequence is expressed through the dependence of G(z) on {bint. }. The integral in (19) can be readily evaluated via (15); however, it is obvious that the computation of the BEP in this case will be more tedious due to the necessity of removing the conditioning on the interferers’ data sequence. V. S IMULATION R ESULTS AND D ISCUSSION We shall empirically assess the impact of correlated fading on a user’s performance and investigate the tightness of the lower bounds derived for MRC. With the scenario of mutually independent fading among subcarriers as a baseline, we specify a channel model for which the impact of correlated fading on a user’s performance is to be evaluated. Incorporating the notation and definitions of [16] for an isotropic 2 ) scattering environment, we specify u1ij = Jo (2πfm τij )/(1 + kij and u2ij = −kij u1ij with the elements of the desired user’s covariance matrix determined as

$ Rhd hd (i, j) =

Ωd u1ij + jΩd u2ij , Ωd u1ij − jΩd u2ij , Ωd ,

i>j i 11, performance improvements of up to a factor of two are apparent. Although counterintuitive, such unexpected improvement can be attributed to the correlation between subcarriers reducing the MAI for high interference levels. A similar result was obtained in [1] for the MRC downlink where the effects of thermal noise were neglected. The derived lower bounds are shown as having little deviation from the exact BEP in Fig. 1. Analogous to the MRC

1 This is frequently referred to as Monte Carlo integration since the unconditioning (integration) is, in effect, performed by averaging over a large number of realizations for the random quantity. 2 In Figs. 1–4, we have M/N ∈ [1/16, 1]. The same general trends were also verified for N = 64 with L = 1 to 63 for which M/N ∈ [1/32, 1], wherein 1010 samples were used for the Monte Carlo simulation of the conditional BEP expressions.

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 7, SEPTEMBER 2009

Fig. 4.

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Error performance of (left) EGC uplink and (right) EGC downlink for lightly loaded (L = 5) and heavily loaded (L = 20) systems, with N = 32.

downlink, EGC performance is severely degraded with the loss of diversity for a small number of interferers, with the degradation diminishing for increasing levels of MAI, and correlation is actually beneficial when the system is interference dominated. Specifically, Fig. 2 shows that for the uplink, the degradation brought on by correlated fading is most pronounced for L < 5, where a degradation of up to an order of magnitude is apparent. The degradation gradually diminishes until L > 14, where correlated fading improves performance. For the EGC downlink with L < 5, a minimal performance loss of a factor of three is apparent with correlated fading, with the degradation exceeding an order of magnitude with low interference levels. For L > 15, correlated fading improves performance. The system-level simulation of the MC-CDMA system is included in Figs. 1 and 2, with the correlated Rayleigh variates generated according to the methodology described in [17]. The primary results shown in Figs. 3 and 4 are summarized as follows. In Fig. 3, we note that for the MRC uplink, in a heavily loaded system, there is little degradation in BEP brought about by correlated fading. This is untrue for a lightly loaded system, where the degradation is apparent and increases with larger SNR values. For the MRC downlink, Fig. 3 indicates that correlated fading is either beneficial or nondetrimental for both loading conditions. In the case of the EGC uplink and downlink, Fig. 4 illustrates that for lightly loaded systems, correlated fading introduces a significant deterioration in BEP with increasing SNR values, while little deviation is apparent in heavily loaded systems.

VI. C ONCLUSION The performance of MC-CDMA in the presence of correlated fading has been investigated. The analysis has been exact with respect to not invoking the traditional GA on the MAI. The MRC and EGC schemes were considered with variable-power interferers, correlated Rayleigh fading among subcarriers, and thermal background noise. In general, the results show that the effects of correlated fading are not negligible, particularly for lightly loaded communication systems.

ACKNOWLEDGMENT The authors wish to thank R. Berk, X.-L. Meng, and L. Shepp for helpful discussions. We also thank the anonymous reviewers for constructive comments, which have improved the quality of the paper.

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