EURASIP Journal on Applied Signal Processing 2002:8, 797–806 c ○ 2002 Hindawi Publishing Corporation Performance of Reverse-Link Synchronous DS-CDMA System on a Frequency-Selective Multipath Fading Channel with Imperfect Power Control

We analyze the performance for reverse-link synchronous DS-CDMA system in a frequency-selective Rayleigh fading channel with an imperfect power control scheme. The performance degradation due to power control error (PCE), which is approximated by a log-normally distributed random variable, is estimated as a function of the standard deviation of the PCE. In addition, we investigate the impacts of the multipath intensity profile (MIP) shape and the number of resolvable paths on the performance. Finally, the coded bit error performance is evaluated in order to estimate the system capacity. Comparing with the conventional CDMA system, we show an achievable gain of from 59% to 23% for reverse-link synchronous transmission technique (RLSTT) in the presence of imperfect power control over asynchronous transmission for. As well, the effect of tradeoff between orthogonality and diversity can be seen according to the number of multipaths, and the tendency is kept even in the presence of PCE. We conclude that the capacity can be further improved via the RLSTT, because the DS-CDMA system is very sensitive to power control imperfections.


INTRODUCTION
Direct-sequence code-division multiple-access (DS-CDMA) has been considered as the most promising multiple-access scheme for the next generation mobile communications, because of its high flexibility in offering various services with variety of rates and its possibility of achieving greater capacity [1,2]. The capacity of DS-CDMA system is mainly limited by multiple-access interference (MAI), and thus techniques to reduce the MAI, such as multiuser detection or interference cancellation, are currently of great interest [3,4,5]. In particular, techniques for reverse links have attracted much attention, as the capacity of a reverse voice cellular network link is smaller than that of the forward link. One reason for this is that the code orthogonality is not maintained, because in the reverse link the arrival times of signals from mobile stations (MSs) at a cell site (CS) are different, given the random geographical distribution of MSs within the cell sector.
For terrestrial mobile systems, the reverse link synchronous transmission technique (hereafter, we denote it by RLSTT) has been proposed to reduce the interchannel interferences over a reverse link [6]. In RLSTT, a closed-form timing control based on a new parameter called the timing control bit is introduced. The DS-CDMA system considered uses an orthogonal reverse-link spreading sequence and timing control algorithm that allows the main paths to be synchronized. Analyses for a single-cell system have shown good performance, especially for an exponentially decaying multipath intensity profile with a large decay factor [6,7]. However, the previous analyses have assumed perfect power control, that is, all the users' transmissions arrive with the same power at the CS receiver. In a practical mobile radio environment, an adaptive power control (APC) scheme is always essential to compensate for the distance losses, shadowing, and fading effects. Such a scheme attempts to maintain a constant average performance among the users, and reduce the MAI effect. This results in a randomly varying power control error (PCE), which may be caused by the dynamic range of the APC, the spatial user distributions, and the propagation statistics [8,9,10,11]. Evaluation of the system capacity degradation due to PCE is the main focus of this paper. To investigate the overall effect of imperfect power control, PCE is considered in terms of the standard deviation of the lognormal distribution. We consider the capacity of a reverse-link synchronous DS-CDMA system over frequency selective Rayleigh fading channels in the presence of imperfect power control scheme. Using the results of [6], which are described in more detail in the appendices of this paper, the system performance degradation as a function of the standard deviation of the PCE is evaluated. We also investigate the impact of the multipath intensity profile (MIP) shape and the number of resolvable paths on the performance of RLSTT, because the extent of orthogonality destruction depends on the multipath channel power delay profile shape and number of resolvable paths. To estimate the system capacity, the coded bit error probability is evaluated and compared with conventional asynchronous CDMA.
The remainder of the paper is organized as follows. In Section 2, channel and system model are described. The performance is analytically derived and evaluated, assuming coherent binary phase shift keying (BPSK) data modulation and a Rake combiner using maximal ratio combining (MRC) in Section 3. Numerical results and conclusions are provided in Sections 4 and 5, respectively.

Transmitted signal representation
We assume that the narrowband-modulated signal of each user is first spread by a short orthogonal sequence, and then randomized by a pseudonoise (PN) sequence. Assuming K active users (k = 1, 2, . . . , K), the kth transmitted signal is given by where P k is the average transmitted power, ω c is the common carrier frequency, φ (k) is the phase angle of the kth modulator, which is assumed to be uniformly distributed in [0, 2π), and a(t) is a PN randomization sequence, which is common to all the channels in a cell to maintain the CDMA orthogonality and is expressed as The orthogonal channelization sequence, W (k) (t), is given by and user k's data waveform, b (k) (t) is expressed as where T w is the chip duration in the orthogonal sequence and p T (t) is a rectangular pulse of unit height and duration T. The PN chip interval T c is related to the data bit interval T by the processing gain N = T/T c . We assume, for simplicity, that T w equals to T c . As well, w (k) j represents the sign of the jth chip for the kth user's orthogonal sequence, a j represents the sign of the jth chip for the PN sequence, and b (k) j is the sign of the jth transmitted symbol for the kth user.

Channel model
The low-pass impulse response of the band-pass channel for the kth user may be written as [12] h k (τ) = Each path is characterized by three variables: its strength β (k) l , its phase shift θ (k) l , and its propagation delay τ (k) l . A tapped delay line model describes the frequency selective channel with the lth multipath delay of the kth user given by τ (k) l = τ (k) 0 + lT c [13]. Assuming Rayleigh fading, the received signal strength of the kth user on the lth propagation path, l = 0, 1, . . . , L (k) − 1, has a probability density function (pdf) given by The parameter Ω (k) l is the second moment of β (k) l , that is, , with ∞ l=0 Ω 1 = 1, and we assume it to be related to the second moment of the initial path strength Ω (k) 0 for the MIP by The parameter δ reflects the decay rate of the average path strength as a function of path delay. Note that a more realistic profile model may be given by the exponential MIP, in which the main path occupies more than half of the total received signal power [12,14,15].

Rake combiner output
The receiver is a coherent Rake receiver, where the number of taps L r is a variable that is less than or equal to L (k) . The tap weights and phases are assumed to be perfect estimates of the channel parameters. The received signal is represented as where λ k corresponds to the PCE of the kth user, which is a random variable due to imperfect power control. We consider λ k to be log-normally distributed with standard deviation σ λk dB. In other words, λ k = 10 (x/10) where the variable x follows a normal distribution. As well, φ (k) l is the phase of the lth path of the carrier of the kth user, and n(t) is the additive white Gaussian noise (AWGN) with a two-sided spectral density η 0 /2. For the user of interest (k = 1), the output of the receiver is given by where where (1) n , and RW and RŴ are Walsh-PN continuous partial cross-correlation functions defined by From (9), we see that the output of the nth branch, n = 0, 1, . . . , L r − 1, consists of four terms. The first term represents the desired signal component to be detected. The second term represents the MAI from the (K − 1) other simultaneous users. The third term is the self-interference (SI) for the reference user. Finally, the last term is the noiseinterference (NI) caused by AWGN.

PERFORMANCE ANALYSIS
Suppose we have K transmitters able to adjust their timing clock of the main paths to be aligned at the CS by the timing control algorithm [6]. Therefore, in our analysis, the evaluation is carried out for the case in which the arrival time of paths is modeled as asynchronous in every branch (i.e., for multipaths) but as synchronous in the first branch (i.e., for main paths) exceptionally. We first estimate the uncoded bit error performance at different system parameter settings. Assuming perfect interleaving, we then evaluate an upper bound on the coded bit error performance of the system using convolutional codes with hard decision Viterbi decoding.

Uncoded BER performance
Using the Gaussian approximation method, the MAI terms of the first branch and the rest of the branches are modeled as Gaussian processes with variances equal to the MAI variances for n = 0 and for n ≥ 1, respectively [6]. Using the random chip model for the Walsh-PN sequences [16] and performing some mathematical operations described in more detail in the appendices, we obtain the following results. The variance of MAI for n = 0, conditioned on β (1) n , is Similarly, the variance of MAI for n ≥ 1 is where E b = PT is the signal energy per bit. The conditional variance of σ 2 SI,n is approximated by [13] The variance of the AWGN term, conditioned on β (1) n , is given as Therefore, the output of the receiver U, conditioned on β (1) n , is a Gaussian random process with a mean given by (16) and the variance equal to the sum of the variances of all the interference terms. From (12), (13), (14), and (15), we have The variance of the total interference in (17) becomes then the received signal-to-noise ratio (SNR) at the output of the receiver may be written as σ 0 S, The pdf of and the pdf of where In addition to fast signal fluctuations caused by multipath reflections, slow signal fluctuations exist due to obstruction of the signal by hills and buildings. We can express the received signal power as P k = Pλ k , and λ k is the PCE, which is a log-normally distributed random variable [8,9,10,11] The CS receives the desired power of the reference user and the joint interference power. Fenton [17] showed that the pdf of λ I = K k=2 λ k for K − 1 users is approximately log-normal with the following logarithmic mean and logarithmic variance, which is valid for a logarithmic standard deviation σ less than 4 dB: where The average bit error probability is then P l e = P l e|X,Y,λ1,λI p x, y, λ 1 , λ I dx dy dλ 1 dλ I = P l e|X,Y,λ1,λI p(x)p(y)p λ 1 p λ I dx dy dλ 1 dλ I , where P l e|X,Y,λ1,λI = Q 2σ 0 S , From (20), (21), and (22), Using (24) and (29), we find that Assuming that z 1 = (ln λ 1 − m λ1 )/ √ 2σ λ1 , we can rewrite (30) as Therefore, the bit error probability is now given by Note that (32) can be calculated using the Hermite polynomial approach, which requires only summation and no integration [18].  upper-bounded by the well-known transfer function bound [12] p

Coded BER performance
where d f is the free distance of the code, and {γ x } are the coefficients in the expansion of the derivative of T (D, N), the transfer function of the code evaluated at N = 1 [19]. As well, P(x) is the probability of selecting an incorrect path, which can be bounded by the expression where p is the uncoded BER.

NUMERICAL RESULTS
In this section, we consider the capacity of a reverselink synchronous DS-CDMA system over frequency selective Rayleigh fading channels in the presence of imperfect power control scheme. The system performance degradation as a function of the standard deviation of the PCE is estimated. We also investigate the effects of the selection of system parameters on the performance of a coherent BPSK Rake receiver with RLSTT in terms of the average BER and the supportable number of users for exponential MIPs. The effect of different values of δ on the effective total capacity is given. The BER analysis of a conventional asynchronous CDMA system with a diversity technique utilizing maximal ratio combining (MRC) reception can be found in [12,13]. Figures 1 and 2 show the difference in the average uncoded BER performance as a function of E b /N 0 , when δ = 0.2 and δ = 1.0 are assumed for the different MIPs. Measurements made by Turin et al. [20] in an urban environment indicate that the MIP is exponential. For illustration, we have 30 25 20 15  chosen the arbitrary parameters K = 24, N = 128, and L (k) = L r = 3, but we vary the values of σ λ from 0 to 4 dB. From these figures, we observe that the bit error performance degrades with increased σ λ values, as expected. In particular, the degradation in performance is marginal when σ λ = 1 dB. However, for σ λ > 2 dB, significant degradation is observed. The analytical results derived for asynchronous with the PCE are found to closely match the simulation results obtained by Chockalingam and Milstein in [8]. It is noted that the average BER in RLSTT, when σ λ = 1 dB, is better than that in the asynchronous transmission when σ λ = 0 dB (i.e., perfect power control). It means that the RL-STT with imperfect power control has better performance, compared to non-RLSTT with perfect power control. In addition, as the decay constant δ of MIP increases, RLSTT in the DS-CDMA reverse link results in a significant BER improvement over the asynchronous transmission even in the presence of imperfect power control. For example, for E b /N 0 > 15 dB, the average BER of RLSTT when σ λ = 2 dB is better than that of the asynchronous transmission when σ λ = 0 dB. Figures 3 and 4 show the effect of having a large number of resolvable paths and coherently combining all of them at the Rake receiver, L (k) (= L r ) on the average BER. The average BER is plotted as a function of L (k) (= L r ) for various values of σ λ , when K = 24, N = 128, E b /N 0 = 20 dB, and exponential MIP are assumed. The figures illustrate the performance difference between RLSTT and the asynchronous case is smaller as L (k) (= L r ) increases due to the increased MAI resulting from multipath interference. Meanwhile, for large L (k) (= L r ), the potential improvement in performance exists due to the diversity gain from increased frequency selectivity. Therefore, we observe that there are tradeoff between orthogonality and diversity as a function of L (k) (= L r ), and the tendency is kept even in the presence of PCE. However, the BER Sync. Async.  RLSTT still offers better performance over the asynchronous transmission.
In Figures 5 and 6, the BER is plotted for the situation with perfect and imperfect power control. We see from Figures 5 and 6 that the BER increases with the increase in the imperfection of the power control scheme. This is due to the random nature of the received power in the case of imperfect power control, whereas in the case of perfect power control the received power can be assumed to be deterministic.
We consider the use of a rate 1/3 convolutional code of constraint length 9 on the reverse link [21]. The {γ x } coefficients for the corresponding code are taken from [19]. The upper bound on the coded BER performance of the system as   Tables 1 and 2. The system capacity is defined as the number of simultaneous users that can be supported while maintaining an acceptable BER performance needed by the specific application. When RLSTT is employed, a channel with exponential MIP of δ = 1.0 has more capacity than a channel with δ = 0.2 for the same BER value. For BER=10 −6 and δ = 1.0, when σ λ = 0 dB the capacity improvements are around 50%, while when σ λ = 2 dB the improvements increase to 59%. The results have shown that the capacity of DS-CDMA system can be improved by employing RLSTT even in the presence of imperfect power control.

CONCLUSIONS
In this paper, we have considered the effect of imperfect power control in the performance of reverse-link synchronous DS-CDMA system in Rayleigh multipath fading channel. Multiple-access and self-noise interference were modeled as additional Gaussian noise. Under these assumptions, the performance of the coherent system was derived in terms of the uncoded BER and the capacity from the coded BER. The results indicate that in Rayleigh fading with perfect power control, RLSTT shows capacity improvements from 50% to 22% for BER = 10 −6 . When RLSTT is employed in imperfect power control, it shows more capacity gains from 59% to 23%. It means that RLSTT with imperfect power control has better performance, compared to non-RLSTT with perfect power control. As well, the effect of tradeoff between orthogonality and diversity can be seen according to the number of multipaths, and the tendency is kept even in the presence of PCE. Finally, we conclude that the capacity can be somewhat further increased via the RLSTT, because the DS-CDMA system is very sensitive to power control imperfections.

A. THE AVERAGE CROSS-CORRELATION PARAMETER IN (12)
Let the discrete aperiodic cross-correlation CW k,1 (i) be Let I k be the correlator output resulting from the kth interferer. Then the variance of I k is given by The factor 1/(N − 1) and the summation from i = 1 to N − 1 in (A.2) arise from the existence of N − 1 chip intervals in time [T c , T], and τ (k) 0l can fall into any one of them with equal probability. Using the random chip model for the Walsh-PN sequences [17], making Var(I k ) a random variable, and taking the expectation of (A.2) k,1 (i + 1) + CW 2 k,1 (i) + CW k,1 (i)CW k,1 (i + 1) = PT 2 12N 2 (N − 1) where CW k,1 (0) = 0, E CW k,1 (i)CW k,1 (i + 1) = 0. (A.4) Thus, we find that