Exploitation of Retransmission Diversity in Large CDMA ... - CiteSeerX

Exploitation of Retransmission Diversity in Large CDMA Random Access Systems Yi Sun and Junmin Shi Department of Electrical Engineering The City College of City University of New York New York, NY 10031 Phone: (212)650-6621 E-mail: [email protected]; [email protected] Abstract 1 – This paper considers a large CDMA random access system where a number of users randomly access a basestation through a common CDMA channel. The number of users and spreading gain tend to infinity with their ratio convergent to a constant. An approach is proposed to exploitation of retransmission diversity. Performance of three systems is analyzed: (i) random spreading with exploitation of retransmission diversity; (ii) randomly selected but fixed spreading for each packet with exploitation of retransmission diversity; and (iii) random spreading without exploitation of retransmission diversity. Lower bounds on the limit SIRs of the first two systems and the exact limit SIR of the third system are obtained. User throughput, packet delay, transmission time, and waiting time as well as spectral efficiency in the limit system are obtained as functions of medium access control (MAC). The analytical results are confirmed by simulations. It is shown that out of many system settings, (i) is the most favorable in terms of maximization of spectral efficiency and simplicity of MAC. I. INTRODUCTION In a cellular system where multiple users randomly access a basestation through a common CDMA packet-switched channel, a user needs to send a packet a random number of times before it is successfully detected. The signals received in the multiple transmissions all contain information of the transmitted packet and provide retransmission diversity for detection of the packet. To exploit retransmission diversity, packet combining and multiuse detection were previously studied [1]-[4]. Different from the previous studies, the approach proposed in this paper first subtracts from the received signals the interference signals produced by the packets that are already successfully detected by the basestation, which significantly reduces the total interference power. Then the left signals related to the packet of interest and received in multiple transmissions are jointly used to detect the packet, which exploits retransmission diversity. By using the large CDMA system technique [5]-[8], the throughput, packet delay, and spectral efficiency for a large CDMA random access system are analyzed. It is demonstrated that the proposed approach can substantially improve throughput and spectral efficiency. Moreover, random 1

This work is in part prepared through collaborative participation in the Communications and Networks Consortium sponsored by the U. S. Army Research Laboratory under the Collaborative Technology Alliance Program, Cooperative Agreement DAAD19-01-2-0011. The U.S. Government is

authorized to reproduce and distribute reprints for Government purposes not withstanding any copyright notation thereon.

spreading sequence, can provide significantly more retransmission diversity gain and therefore outperform fixed spreading in terms of throughput and spectral efficiency. The rest of the paper is organized as follows. Section II presents the system and signal models and the approach to exploitation of retransmission diversity. Section III analyzes the limit SIR. Section IV addresses the throughput performance. Numerical examples are demonstrated in Section V, and conclusions are given in Section VI. II. SYSTEM AND SIGNAL MODELS A. System model We consider that a large number of K potential users access a basestation through a packet-slotted bit-synchronous CDMA channel. Packets of size L bits are transmitted in the first-in firstout order. The packet at the front end of the buffer is called the current packet of the user and the rest are waiting packets. Therefore, a user always transmits his current packet. New packet arrival rate at each user is assumed sufficiently high so that there will be always a packet in each user’s buffer waiting for transmission. Random access is employed in media access control (MAC) such that the probability that a user with signal amplitude A transmits a packet slot is equal to θ(A) ∈ (0,1], depending only on user’s signal amplitude A, and independent of other events. Each function θ(A) defines a MAC scheme of the system. Each packet is coded with an error detecting code. The receiver at the basestation is able to detect if the detected packet is in error. If the detected packet is erroneous, the basestation informs through another reliable channel the user the failure of transmission and then the user will retransmit the same packet. If the basestation successfully detects the packet, it sends an acknowledgement message informing the user the success. Then the user eliminates the current packet from his buffer and a waiting packet becomes the new current packet. B. Signal model The basestation first independently detects L bits of a transmitted packet by a MF detector and then checks error of the detected packet. For the bit-synchronous system, it is sufficient to consider the received signal in one bit period in detection of a bit. Consider the current slot n when the desired user indexed by one transmits his current packet. The chip matched filter at the basestation outputs a signal vector for a bit period during slot n as K

r (n) = A1s1 (n)b1 (n) + ∑ I i (n) Ai s i (n)bi (n) + w (n) .

(1)

i =2

A1, s1(n), and b1(n) are the signal amplitude, spreading sequence, and transmitted bit of the desired user. Ai, si(n), and bi(n) for i ≥ 2 are of interference users. Ii(n) is the indicator function of user i’s

transmission of the current packet, i.e., Ii(n) = 1 if user i transmits his packet in slot n, and Ii(n) = 0 otherwise. The transmission probability of user i is Pr(Ii(n) = 1) = θ(Ai) and Pr(Ii(n) = 0) = 1−θ(Ai). w(n) is the AWGN vector with mean zero and covariance matrix σ2I, which is independent slot by slot. Signal amplitudes Ai are fixed in all slots. Bits from different users are independent and take ±1’s with equal probability. A spreading sequence is written as s i (n) = ( si1 (n), si 2 (n),L siN (n)) T where each chip sij(n) takes ± 1 N ’s independently and equally likely. The length N of spreading sequences is the spreading gain. At the basestation, the received signal r(n) passes through the MF s1(n) matching the desired signal and outputting an estimate of b1(n) K

y1 (n) = s1T (n)r (n) = A1b1 (n) + ∑ I i (n) Ai R1i (n)bi (n) + z1 (n) (2) i =2

where R1i (n) = s (n)s i (n) is the crosscorrelation between the spreading sequences of b1(n) and bi(n), and z1(n) ~ N(0, σ2) are Gaussian and independent in different slots. Two types of spreading sequences are considered. One is the random spreading that a user independently and equiprobably selected a spreading sequence to spread all bits of each packet in each transmission. The other is the fixed spreading that a user independently and equiprobably selects a spreading sequence to spread all bits of one packet. If the packet is retransmitted multiple times, the same spreading sequence is used. For another packet, a new spreading sequence will be randomly selected. A large CDMA system is considered where the number of users K and spreading gain N both tend to infinity and their ratio keeps a constant K/N = α > 0. Ak are upper bounded in N and their empirical distribution function converges to F. Moreover, the media access control (MAC) scheme θ(A) is a function of signal amplitude A such that E[A2θ(A)] exists with respect to F. T 1

We consider joint utilization of all received signals in packet detection. First, though r(j), j ≤ n, include all ever transmitted packets, the basestation does not know only the K current packets each for one user and knows all other previous packets. This is due to the fact that a user transmits packets in a first-in first-out order. Seeing this, we first eliminate all the interference signals produced by the known packets from the received signals y1(j), j ≤ n, thus reducing the total interference. Second, to exploit retransmission diversity we use a linear combiner to combine the signals that contain the desired packet. The linear combiner averages out an amount of interference and noise power in multiple transmissions and thus outputs a decision variable with increased signal to interference ratio (SIR). Consequently, throughput performance is improved. D. System state Let mk(n) ≥ 1 be the number of transmissions of the current packet for user k in slot n. If user k transmits the packet in slot n and is not successfully detected by the basestation at the end of slot n, the current packet will be held in the buffer to be transmitted in the next slot and the number of transmissions is increased by one, i.e. mk(n+1) = mk(n) + 1. If the current packet is successfully detected by the basestation, it will be eliminated from the buffer, a waiting packet will become a current packet and the number of transmissions for user k is renewed to mk(n) = 1. If the packet is not transmitted in slot n, then the number of transmissions is kept unchanged, i.e. mk(n+1) = mk(n). In general, m(n) = (m1(n), …, mK(n)) forms a Markov chain, which is nonhomogeneous because transition probabilities of mk(n)’s depend on m(n) and therefore they are coupled. III. SIR IN LIMITING SYSTEMS A. Traffic load A user in each slot is in one of two states: active in transmitting his packet, or idle. α in potential users per Hz per second is the demanded traffic. The traffic load in any slot n, measured in active users per Hz per second, is 1 K ρ ( n) = ∑ I k ( n ) (3) N k =1 which converges a.s. to a constant ρ (n) → αE[θ ( A)] as K, N → ∞ with K/N = α. Consider that at the end of the current slot n, user k has transmitted his current packet mk times and the number of slots from the first transmission of the packet to the current slot is Jk, k

C. The proposed approach By the end of slot n, the desired bit b1(n) might have been transmitted several times. Then the basestation has received signals r(j), j ≤ n, all in the form of (1). In certain conventional ARQ protocols, previously received signals are discarded. A packet which was unsuccessful in previous detection is detected by using only the signal received in the current slot r(n). Without exploitation of retransmission diversity embedded in the previously received signals, system performance is poor. _____________________________________________________________________________________________________________ …… …… …… …… ……

r(n−6) b1(n−6) = b2(n−6) ≠ b3(n−6) =

r(n−5) b2(n−5) = b4(n−5) =

r(n−4) b1(n−4) ≠ b3(n−4) ≠ b4(n−4) =

r(n−3) b1(n−3) = b2(n−3) = b3(n−3) = b4(n−3) ≠

r(n−2) b1(n−2) =

r(n−1) b2(n−1) = b3(n−1) =

b4(n−2) =

r(n) b1(n) b3(n) b4(n)

Fig. 1. Example of packet retransmissions in a four-user random access system. A blank slot means that the user does not transmit in the slot.

= 1, 2, …, K. In particular, user 1 transmitted his current packet m1 times during the slots n−J1+j, j = 1, …, J1. To detect the current packet of user 1, the basestation uses only the signals received in the m1 slots when user 1 transmitted his current packet out of all received signals. Fig. 1 illustrates an example of

packet retransmissions in a four-user system. “≠” denotes correct detection of a packet in a slot and “=” denotes incorrect detection. User 1’s current packet was transmitted in slots n−3, n−2, and current slot n, thus m1 = 3 and J1 = 4. User 2’s current packet was transmitted in slots n−5, n−3, and n−1, and then m2 =

3 and J2 = 6. User 3’s current packet was transmitted in slots n−3, n−1, and n, and thus m3 = 3 and J3 = 4. User 4’s current packet was transmitted in slots n−2 and n, and so m4 = 2 and J4 = 3. To detect User 1’s current packet, the basestation utilizes the signals received in the slots n−3, n−2, and n. After eliminating all the known packets, the combiner in the basestation adds the signals received in the m1 slots, when user 1 transmitted his current packet, to form a new decision variable J1 −1

x(m1 ) = ∑ I 1 (n − i ) A1b1 i =0

* K J k −1

J1 −1

+ ∑ ∑ I 1 (n − i ) I k (n − i ) R1k (n − i ) Ak bk + ∑ I 1 (n − i ) z1 (n − i ) (4) k = 2 i =0

i =0

where J i* = min( J 1 , J i ) . We shall analyze the limit SIR of x(m1) with fixed m1 as K and N tend to infinity. Since b1 is transmitted m1 times in the J1 slots, the power of the desired signal equals Ps = m12 A12 . The power of interference plus noise conditioned with a fixed m1 equals  K J −1 Pin = E  ∑ ∑ I 1 (n − i ) I k (n − i ) R1k (n − i )Ak bk  k = 2 i =0 * k

2   + ∑ I 1 (n − i ) z1 (n − i )  m1   i =0   J1 −1

2

  J −1 (5) = m1σ + ∑ A  ∑ I1 (n − i ) I k (n − i ) R1k (n − i )  ,   k =2   i =0 where the expectation is taken with respect to transmitted bits and noise. The second term in (5) is the power of interference which is denoted by Pi. The SIR is defined as Ps/Pin. K

2

2 k

* k

B. Random spreading sequences Since all spreading sequences are random, R1k(n−i) for different i in (5) are not identical and then the interference power equals 2

  J −1 Pi = ∑ A  ∑ I1 (n − i ) I k (n − i ) R1k (n − i )  . (6)   k =2   i =0 In general, the limit interference power is difficult to obtain due to the random numbers J k* . However, a lower bound of the limit SIR is obtained in the following theorem. Theorem 1 (Random spreading): For the random access CDMA system where spreading sequences in all transmissions are random, as K → ∞, the SIR of the combiner output for a packet of m transmissions is almost surely lower bounded by mA 2 η rLB (m) = 2 . (7) σ + αE[ A 2θ ( A)] where the expectation is taken with respect to F. The lower bound equals the true limit SIR for m = 1. If no retransmission diversity is exploited, a packet is detected using only the signal received in the current slot regardless of previous transmissions. Consequently, the SIR for a packet of m transmission is equal to the SIR with one transmission. The exact limit SIR is obtained from Theorem 1 with m = 1. K

2 k

* k

Corollary 1: For the random access CDMA system where the spreading sequences are random, the SIR of combiner output for a packet of m transmissions without exploitation of retransmission diversity converges almost surely to A2 η w (m) = η rLB (m) = 2 . (8) σ + αE[ A 2θ ( A)] Compared with the system that does not exploit retransmission diversity, the random spreading sequences can provide a retransmission diversity gain higher than η rLB (m) = m, (9) η w ( m) which is 10log10m dB in SIR for a packet of m transmissions. The retransmission diversity gain is irrelevant to the MAC θ(A). C. Fixed spreading sequences With the fixed spreading, crosscorrelations R1k(j) = R1k for all j are identical. It follows from (5) that the interference power to user 1 conditioned with m1 transmissions equals 2

  J −1 Pi = ∑ A R  ∑ I1 (n − i ) I k (n − i )  . (10)   k =2   i =0 Theorem 2 (Fixed spreading sequences): For the random access CDMA system where a spreading sequence is randomly selected and then fixed for each packet, the SIR of combiner output for a packet of m transmissions is almost surely lower bounded by mA2 η LB (11) f ( m) = 2 σ + αE[(1 − θ ( A) + θ ( A)m) A 2θ ( A)] as K → ∞. The lower bound equals the true limit SIR for m = 1. K

2 k

2 1k

* k

D. Random spreading versus fixed spreading It is intuitively understandable that random spreading can provide more diversity gain than fixed spreading. The retransmission diversity gain obtained by random spreading over fixed spreading is also reflected on the lower bounds of the limit SIR for the two spreading types η rLB (m) (m − 1)αE[ A 2θ 2 ( A)] = 1+ , (12) LB η f (m) σ 2 + αE[ A 2θ ( A)] which is greater than one for all valid system where E[A2θ(A)] > 0. In the high SNR regime as σ → 0, η rLB (m) (m − 1) E[ A 2θ 2 ( A)] (13) = 1+ LB η f ( m) E[ A 2θ ( A)] which converges to m as θ(A) → 1. This implies, due to (9), that fixed spreading provides little retransmission diversity gain in the regime of high SNR and high transmission probabilities. When retransmission diversity is exploited, while random spreading can provide retransmission diversity gain more than 10log10m dB in SIR, retransmission diversity gain obtained from fixed spreading is lower and depends on MAC θ(A), η LB mσ 2 + mαE[ A 2θ ( A)] f ( m) = 2 . (14) η w (m) σ + αE[(1 − θ ( A) + θ ( A)m) A 2θ ( A)] In the regime of low traffic load as θ(A) → 0, η LB f ( m) (15) = m, η w ( m)

retransmission diversity gain achieves the maximum. The reason is that in high SNR regime, interference is dominated by noise and fixed spreading can acquire retransmission diversity gain equal to 10log10m dB in SIR from independent noises in different slots. In the regime as θ(A) → 1, η LB mσ 2 + mαE ( A 2 ) f ( m) = 2 (16) η w ( m) σ + mαE ( A 2 ) which converges to one as σ → 0. Thus, no retransmission diversity gain is obtained. E. Probability of packet success It is well-known [7][8] that the interference of MF output for one transmission in a large CDMA system is asymptotically Gaussian. Since the random access CDMA system in each slot is such a large CDMA system, the interference of combiner output is also asymptotically Gaussian. Therefore, the limit bit error rate (BER) for the hard decision of combiner output can be expressed by the Q function. For a packet of m transmissions with signal amplitude A, the limit BER equals

γ m ( A) = Q( η ( A, m) ) .

(17)

Correspondingly, the probability of packet success for a packet of m transmissions without error correct coding equals q m ( A) = [1 − γ m ( A)] L . (18) IV. SYSTEM PERFORMANCE A. With retransmission diversity gain In the limit system, the probability of packet success qm(A) depends only on the user’s signal amplitude A. The probability that a packet is transmitted the mth time in slot n is stationary and depends only on the user’s signal amplitude A, which henceforth denoted by pm(A) can be determined by qm(A). The throughput of a user is defined as the average number of packets successfully transmitted per slot. A user of signal amplitude A transmits his packet with probability θ(A) and the transmitted packet, supposed to be in its mth transmission, has the probability of packet success qm(A). Then the probability for the user to successfully transmit a packet is equal to θ(A)qm(A). The throughput equals ∞

T ( A) = E[θ ( A)qm ( A)] = θ ( A)∑ q m ( A) p m ( A) .

(19)

m =1

The packet delay is defined as the average number of slots from the slot of first transmission to the slot of successful detection. Since T(A) packets are successfully detected on average, the packet delay is equal to the reciprocal of the throughput. Suppose that the packet is successfully detected after m transmissions. The probability distribution of m is equal to qm(A)(1−qm−1(A))…(1−q1(A)) that is the probability of m−1 unsuccessful detections followed by a success. The transmission time of the packet is defined as the average number of transmissions ∞

m −1

m =1

i =1

M ( A) = E (m) = ∑ mqm ( A)∏ [1 − qi ( A)]

(20)

which is irrelevant to the transmission probability θ(A). The waiting time is defined as the average number of slots where the

packet is not transmitted. It is clear that the waiting time is equal to W ( A) = D( A) − M ( A) . To measure the efficiency of spectrum usage in the entire system, we shall define the spectral efficiency as the average number of bits successfully transmitted per Hz per second in the entire system. Let W be the bandwidth and Tb the bit period and in general N = WTb. Since a user takes a share of 1/(αTb) in bandwidth to transmit LT(A) bits in LTb seconds, the spectral efficiency equals ∞

C = α ∫ T ( A)dF ( A)

(21)

0

bits per Hz per second. Theorem 3: Consider a limit random access system where the probability of packet success for a packet in its mth transmission with signal amplitude A equals qm(A). If θ > 0 and qm(A) > c for all m ≥ n with some c > 0 and some n ≥ 1, then the probability that a user of signal amplitude A transmits a packet with the mth time in a slot is stationary, unique, and equal to 1 , (22) p1 ( A) = i −1 ∞ 1 + ∑i =2 ∏ j =1 (1 − q j ( A))

∏ (1 − q ( A)) , m ≥ 2. ( A) = 1 + ∑ ∏ (1 − q ( A)) m −1

pm

j

j =1

∞

i −1

i=2

j =1

(23)

j

The throughput, packet delay, transmission time, and waiting time of the user equal, respectively, T ( A) = θ ( A) p1 ( A) , (24) 1 , (25) D( A) = θ ( A) p1 ( A) 1 , (26) M ( A) = p1 ( A) 1 − θ ( A) . (27) W ( A) = θ ( A) p1 ( A) The spectral efficiency equals ∞

C = α ∫ θ ( A) p1 ( A)dF ( A) .

(28)

0

It is worthwhile to mention that Theorem 3 is general and is applicable to any large random access system, provided that the packet success probability for a packet of m transmissions in the limit system is equal to a constant qm(A). B. Without retransmission diversity gain If packets are detected using only the signals received in the current slot without exploitation of retransmission diversity, the limit SIR is equal to ηw(1). Letting qm(A) = q1(A) in Theorem 3, we obtain the following corollary. Corollary 2 (No retransmission gain): For a random access system without exploitation of retransmission diversity, if θ > 0 and q1(A) > 0, the probability distribution that a user of signal amplitude A transmits a packet with the mth time in a slot is stationary, unique, and equal to p m ( A) = q1 ( A)(1 − q1 ( A)) m−1 , m ≥ 1. (29) The throughput, packet delay, transmission time, and waiting time of the user equal, respectively,

T ( A) = θ ( A)q1 ( A) , 1 , D( A) = θ ( A)q1 ( A) 1 , M ( A) = q1 ( A) 1 − θ ( A) . θ ( A)q1 ( A) The spectral efficiency equals W ( A) =

(30) (31) (32) (33)

∞

C = α ∫ θ ( A)q1 ( A)dF ( A) .

(34)

0

When system load α in the number of users per Hz per second is high, the probability of packet success q1(A) is small. Consequently, a packet needs a large number of transmissions before successfully detected. Without exploitation of retransmission diversity, the throughput and spectral efficiency are low, and the packet delay is high. On the other hand, if system load α is low, the probability of packet success is high. Since the probability for a packet to be retransmitted is low, the difference in throughput-delay performance per user between with and without exploitation of retransmission diversity is small. However, we point out that throughput and delay measure only the performance obtained on average for one user. It will be seen that when system load α is low, the spectral efficiency, which measures the efficiency of spectrum usage of the entire system, is low. Hence, to achieve a high efficiency of spectrum usage, retransmission diversity must be exploited. V. NUMERICAL AND SIMULATION RESULTS A. Optimization of MAC As the physical layer performance, given by the BER, depends on the MAC θ(A), the system performance also depends on the MAC θ(A). There must exist an optimal MAC θ(A) that attains the maximal spectral efficiency. The optimal MAC can be obtained by a standard numerical search algorithm for nonlinear optimization. For example, consider a two-class system where the users of the first and the second classes have SNR = 7 dB and 10 dB, respectively. The percentage of users of the first class is r1 = 0.75 and of the second class is r2 = 0.25. The demanded traffic is α = 1.1 users per Hz per second. Packet size is L = 16. Fig. 2 shows the spectral efficiency versus the MAC (θ1,θ2) where θi is the transmission probability of the ith class. From the top to the bottom, the systems are random spreading with exploitation of retransmission diversity, fixed spreading with exploitation of retransmission diversity, and random spreading without retransmission diversity. The lower bounds of limit SIR η rLB in

in (11) are used for the first two systems and the (7) and η LB f limit SIR ηw in (8) is employed for the third. For the first system, the optimal MAC is (θ1,θ2) = (1, 1) achieving the maximum of spectral efficiency equal to 0.30. The optimal medium access control is no control. Without control of medium access, the MAC becomes the simplest while the spectral efficiency achieves the maximum. In contrast, for the second system, the optimal MAC is about (θ1,θ2) = (0.45, 0.40) with the maximum of spectral efficiency equal to 0.16. For the third system, the optimal MAC is around (θ1,θ2) = (0, 0.90) attaining the maximum

spectral efficiency equal to 0.12. This MAC is unfair to the first class of users since they are not allowed to transmit their packets. Under all MAC, the system of random spreading with exploitation of retransmission diversity can achieve significantly higher spectral efficiency than the other two systems. B. Equal power systems In all simulations and numerical evaluations, packet size is L = 16. Fig. 3 shows throughput versus transmission probability θ with demanded traffic α = 1 and SNR = 9 dB. The traffic load is equal to θα = θ transmitted packets per Hz per second on average. Since α = 1, the spectral efficiency is equal to the throughput. As shown in the figure, the theoretically results predict the simulation. The proposed approach with random spreading sequences achieves higher throughput than with fixed spreading sequences and is much higher than without exploitation of retransmission diversity. The maximum throughput of 0.34 is attained at θ = 1, which is equivalent to the simplest MAC – no control of medium access. Fig. 4 and Fig. 5 show throughput and spectral efficiency versus demanded traffic α for a system with K = 64, SNR = 9 dB, and θ = 0.8. The traffic load is equal to θα = 0.8α transmitted packets per Hz per second. As observed in simulations, the throughput and spectral efficiency change slightly when K changes and thus only the simulation result of K = 64 is depicted in the figures. The system that employs random spreading and exploits retransmission diversity achieves higher throughput and spectral efficiency than the scheme that employs fixed spreading and exploits retransmission diversity, and much higher than the scheme that does not exploit retransmission diversity. For all systems, the throughput obtained by each user monotonically decreases as the traffic load increases. However, the spectral efficiencies for the system that employs random spreading and exploits retransmission diversity monotonically increase as the traffic load increases. In contrast, the spectral efficiencies for the system that employs fixed spreading and for the system that does not exploit retransmission diversity are convex down with respect to the traffic load and their maximums are much lower than the maximum of the scheme that employs random spreading and exploits retransmission diversity. VI. CONCLUSIONS* We propose an approach to exploitation of retransmission diversity in CDMA random access systems. The SIR, throughput, packet delay, and spectral efficiency in the limit large system are analyzed. In the approach, elimination of interference caused by known packets reduces total interference power. Joint use of signals received in multiple transmissions of a packet exploits retransmission diversity and further increases signal to interference ratio. In terms of maximization of spectral efficiency and simplicity of medium access control, the most favorable system out of many others is to employ random spreading, exploit retransmission diversity, and perform no access control. REFERENCES [1] A. Annamalai and V. K. Bhargava, “Mechanisms to ensure a reliable packet combining operation in DS/SSMA radio networks with retransmission diversity,” Proc. of IEEE VTC’98, Ottawa, pp. 1448–1452, May 1998.

[2] Y. Sun, “Network diversity of random access slotted CDMA networks,” in Proc. 33rd Asilomar Conf. Signals, Systems, and Computers, Pacific Grove, CA, Oct. 24-27, 1999. [3] Y. Sun and X. Cai, “Multiuser detection for packet-switched CDMA networks with retransmission diversity,” IEEE Trans. Signal Processing, vol. 52, no. 3, pp. 826-832, March 2004. [4] X. Cai, Y. Sun, and A. N. Akansu, “Performance of CDMA random access systems with packet combining in fading channels,” IEEE Trans. Wireless Commun., vol. 2, no. 3, pp. 413419, May 2003. [5] D. N. C. Tse and S. V. Hanly, "Linear multiuser receivers: effective interference, effective bandwidth and user capacity,"

IEEE Trans. Inform. Theory, vol. 45, pp. 641-657, March 1999. [6] J. Zhang and X. Wang, “Large-system performance analysis of blind and group-blind multiuser receivers,” IEEE Trans. Inform. Theory, vol. 48, pp. 2507-2523, Sept. 2002. [7] D. Guo, S. Verdú, and L. K. Rasmussen, "Asymptotic normality of linear multiuser receiver outputs," IEEE Trans. Inform. Theory, vol. 48, no. 12, pp. 3080- 3095, Dec. 2002. [8] J. Zhang, E. K. P. Chong and D. N. C. Tse, "Output MAI distributions of linear MMSE multiuser receivers in DS-CDMA systems," IEEE Trans. Inform. Theory, vol. 47, pp. 1128-1144, Mar. 2001. 0.8

0.3

Random sequences Fixed seq. per pkt No retrans. gain, random seq. Simulation Limit

0.7 0.25

T (packets/slot/user)

C (bits/Hz/second)

0.6 0.2

0.15

0.1

0.5 0.4 0.3 0.2

0.05

0.1 0 0

0.5

θ1

1

0.4

0.2

0

1

0.8

0.6

0

θ2

Fig. 2. Spectral efficiency versus MAC scheme in a two-class system.

0.6

0.8

1

1.2

1.4

0.4

0.3

K = 16, 32, 64 →

0.2

↓ K = 16, 32, 64 ↓ K = 16, 32, 64

0.15

0.25 0.2 0.15

0.1

0.1

0.05

0.05

0.2

0.3

0.4

Random sequences Fixed seq. per pkt No retrans. gain, random seq. Simulation Limit

0.35

C (bits/Hz/second)

0.3 T (packets/slot/user)

0.4

Fig. 4. Throughput versus demanded traffic α in an equal power system with K = 64, SNR = 9 dB, packet size L = 16, and θ = 0.8.

Random sequences Fixed seq. per pkt No retr. gain, rand. seq. Simulation Limit

0.35

0 0.1

0.2

α (users/Hz/second)

0.4

0.25

0

0.5

0.6

0.7

0.8

0.9

0

1

θ

Fig. 3. Throughput versus transmission probability in an equal power system with α = 1, SNR = 9 dB, and packet size L = 16.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

α (users/Hz/second)

Fig. 5. Spectral efficiency versus demanded traffic α for the same system in Fig. 4.

*

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