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____________2005 Conference on Information Sciences and Systems, The Johns Hopkins University, March 16-18, 2005

Exploitation of Historical Signals in Large Packet-Switched 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] Abstract1 – This paper proposes an approach to exploitation of retransmission diversity embedded in multiple packet transmissions of a large CDMA random access system. In the approach, the interference signals caused by packets that have been successfully demodulated previously are first subtracted thus reducing interference. Second, all signals received in slots when the desired packet was transmitted are jointed used thus obtaining retransmission diversity gain. Stationary distribution of the number of transmissions for a packet to succeed, throughput, delay, and spectral efficiency are analyzed by applying large CDMA techniques. The theoretical results are confirmed in simulations. 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 demodulated. The signals received in the multiple transmissions all contain information of the transmitted packet and provide retransmission diversity for demodulation of the packet. To exploit retransmission diversity, packet combining and multiuse detection were studied in the previous studies [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 demodulated 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 demodulate 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, long random spreading sequences, can provide significantly more retransmission diversity gain and therefore outperform short sequences in terms of throughput and spectral efficiency. II. SYSTEM AND SIGNAL MODELS A. System model 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.

We consider that a very large number of K potential users randomly access a basestation through a bit-synchronous CDMA channel. Each user has a sufficiently long buffer to store arrived packets. Packets are transmitted in a first-in-first-out order. At the front end, the packet that is currently served in transmission is called the current packet. All other packets in the buffer are packets waiting for transmissions. We assume that the new packet arrival rate at each user is sufficiently high so that there will be always a packet in the buffer waiting for transmission. Time is packet-slotted though the result can be extended to unslotted systems. In each slot, a user transmits a packet of L bits, which is coded with error detecting code. Therefore, the receiver at the basestation is able to detect if the demodulated packet is in error. If the demodulated packet is erroneous, the basestation informs through another reliable channel the user the failure of transmission and then the user will retransmit the packet. The retransmission continues until the basestation successfully demodulated the packet, followed with an acknowledgement message informing the user the success. Knowing the success of transmission, the user removes the successfully transmitted packet from the buffer and transmits another packet. While the error detecting code and the feedback channel take up certain spectrum, we will not take them into account in analysis of spectral efficiency. Two access schemes will be considered. One is the deterministic access that if a demodulated packet is erroneous in a slot, the user immediately sends the packet in the next slot. The other is the random access that the user sends a packet, either new or retransmitted, with a transmission probability θ ∈ (0,1] in the next slot. B. Signal model The basestation first independently detects L bits of a transmitted packet and then checks error of the demodulated packet. For the bit-synchronous system, we can consider the received signal in one bit period to detect a transmitted bit. Without loss of generality, the desired user is indexed by one. The chip matched filter at the basestation outputs a signal vector for a bit period as r (n) = A1s1 (n)b1 (n) + ∑ Ai s i (n)bi (n) + w (n) . (1) i∈I ( n )

where I(n) is the index set of active users in the current slot n. 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) are the signal amplitude, spreading sequence, and transmitted bit of

____________2005 Conference on Information Sciences and Systems, The Johns Hopkins University, March 16-18, 2005 interference users. w(n) is the AWGN vector with mean zero and covariance matrix σ2I. Noise vectors in different slots are mutually independent. Signal amplitudes Ai are time-invariant. Bits from different users are independent and equiprobably take ±1’s. A spreading sequence can be written as s i (n) = ( si1 (n), si 2 (n),L siN (n)) T where each chip sij(n) takes ± 1 N ’s. The length N of spreading sequences is also the spreading gain of the CDMA channel. Spreading sequences are known to the basestation. At the basestation, the received signal r(n) passes through the MF s1(n) matching the desired signal and outputting a decision variable y1 (n) = s1T (n)r (n) = A1b1 (n) +

∑ Ai R1i (n)bi (n) + z1 (n)

(2)

i∈I ( n )

where R1i (n) = s1T (n)s i (n) is the crosscorrelation between the spreading sequences of b1(n) and bi(n), and z1(n)’s with different slots are mutually independent Gaussian random variables with mean zero and variance σ2. C. The approach Due to interference and noise, a packet must be transmitted a random number of times before it is successfully demodulated. At the end of current slot n, the desired bit b1(n) might have been transmitted several times. Suppose the system has been in a stationary state in the current slot n. Then, the basestation has received signals r(j), j ≤ n, all in the form of (1). The approach to exploiting all the received signals consists of two steps. First, though r(j), j ≤ n, include all ever transmitted packets, to the basestation only the current packets each for one user are unknown and all other earlier transmitted packets have been known. This is due to the fact that a user retransmits continuously an unsuccessful packet, and transmits a new packet only if the previously transmitted packet is successfully demodulated. In other words, only a total of K packets are unknown to the basestation and each is the one latest transmitted by one user. Seeing this, we first eliminate all the signals produced by the known packets from the received signals, thus considerably reducing interference signals. Second, to exploit retransmission diversity the left signals in multiple slots where the current packet of the desired user is transmitted are combined by a linear combiner. The linear combiner averages out a certain amount of interference and noise in multiple transmissions and thus outputs a decision variable with increased signal to interference ratio (SIR). Hence, the throughput performance is improved by exploiting the history signals properly. …… …… …… …… ……

r(n−6) b1(n−6) = b2(n−6) ≠ b3(n−6) = b4(n−6) =

r(n−5) b1(n−5) ≠ b2(n−5) = b3(n−5) = b4(n−5) =

r(n−4) b1(n−4) ≠ b2(n−4) = b3(n−4) ≠ b4(n−4) =

D. System state Let mk(n) ≥ 1 be the number of transmissions of the current packet for user k in slot n, that is, user k has transmitted his current packet mk(n) − 1 times before slot n. If user k does transmit the packet in slot n and the packet is not successfully demodulated by the basestation at the end of slot n, the current packet will be held in the buffer to be transmitted next time and the number of transmissions is increased by one. If the current packet is successfully demodulated 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. In general, m(n) = (m1(n), …, mK(n)) forms a nonhomogeneous Markov chain because mk(n)’s are coupled and their transition probabilities depend on m(n). We consider a large CDMA system where the number of users K and spreading gain N both tend to infinity and their ratio keeps a constant K/N = α > 0. It is observed that in such a large CDMA system [5], the SIR of matched filter output is a deterministic constant. We assume that the system is ergodic and is in a stationary state, thus mk(n) denoted by mk for notation brevity has a stationary probability distribution pk(m). Furthermore, the stationary state mk is absolutely integrable and squarely integrable, that is, E (mk ) and E (m k2 ) exist. III. PERFORMANCE ANALYSIS A. Probability of packet success In what follows, we analyze system performance of a deterministic access system where each use keeps transmitting his current packet until the packet is correctly demodulated, and then transmits another packet. Since all users are active in each slot, α is the traffic intensity in the number of active users per Hz per second. Suppose user k transmitted his current packet the mkth time in slot n. It means that the packet has been transmitted in slots n − mk + j, j = 1, 2, …, mk. Before slot n − mk + 1, other packets were transmitted and all were successfully demodulated by the basestation. All the signals caused by the previously successfully demodulated packets can be eliminated by reconstructing the signals. To detect the current packet of user 1, only the signals received in slots n − m1 + j, j = 1, 2, …, m1, are useful. Fig. 1 shows an example for packet retransmissions in a four-user system. The current packet of user 1 was transmitted in slots n − 4 + j, j = 1, 2, …, 4, where m1 = 4. As interference to user 1, user 2’s current packet was transmitted six times, user 3’s current packet was transmitted two times, and user 4’s current packet was transmitted three times.

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

r(n−2) b1(n−2) = b2(n−2) = b3(n−2) ≠ b4(n−2) =

r(n−1) b1(n−1) = b2(n−1) = b3(n−1) = b4(n−1) =

Fig. 1. Example of packet retransmissions in a four-user deterministic access system.

r(n) b1(n) b2(n) b3(n) b4(n)

____________2005 Conference on Information Sciences and Systems, The Johns Hopkins University, March 16-18, 2005 In the signals y(n−m1+j), j = 1, …, m1, only the K current bits each for one user are unknown, bk(n) = bk, k = 1, …, K. The m1 received signals with eliminating all the signals produced by the known bits are simply added to form a new decision variable * K mk −1

m1 −1

x(m1 ) = m1 A1b1 + ∑ ∑ R1k (n − i )Ak bk + ∑ z1 (n − i ) k = 2 i =0

(3)

i =0

where m k* = min(m1 , m k ) . The addition of the signals is equivalent to an equal weight combiner. The decision on b1 is bˆ = sgn( x(m )) . 1

1

In the following, we analyze the limit SIR of x(m1) with fixed m1 for three schemes of spreading sequences. The power of the desired signal is Ps = m12 A12 . The power of interference plus noise conditioned with m1 is, 2  K m −1  m −1     (4) Pin = E ∑ ∑ R1k (n − i )Ak bk + ∑ z1 (n − i ) m1    k =2 i =0  i =0     * k

1

2

  m −1 (5) = m1σ + ∑  ∑ R1k (n − i )Ak    k = 2  i =0  where the expectation is taken over the transmitted bits and noise. Scheme 1: Each user is assigned a fixed spreading sequence sk, which is initially randomly selected. Hence, the crosscorrelations R1k(j) = R1k for all j are fixed. Then the interference power is, 2

K

* k

K

Pin = m1σ 2 + ∑ m k*2 Ak2 R12k .

(6)

k =2

mk* is a random variable whose distribution depends on Ai’s. We consider that as K and N tends to infinity, the limit empirical distribution function FA of Ak’s exists and the limit expectation E(m*2A2) with respect to FA exists. We can obtain K

∑m k =2

*2 k

Ak2 R12k → αE (m *2 A 2 ) a.s.

(7)

Then the SIR satisfies Ps m12 A12 a.s. (8) → η (m1 ) = Pin m1σ 2 + αE (m *2 A 2 ) In the case that all users have the same power A2, mk have the identical probability distribution p(m). Thus, mk* = min(m1 , mk ) for k > 1 have the same distribution and we denote m * = mk* . Eq. (8) becomes m12 A 2 η (m1 ) = . (9) m1σ 2 + αA 2 E (m *2 ) With given m1, E(m*2) can be evaluated by the probability distribution function p(m). To see this, we note that the probability distribution function of m* can be expressed in terms of p(m) as  p (m), 1 ≤ m ≤ m1 − 1, m −1  ∞ * Pr(m = m) =  ∑ p(i ) = 1 − ∑ p (i ), m = m1 , (10) i =1 i =m 0, m > m1 .  Therefore, 1

1

m1

m1 −1

m1 −1

i =1

i =1

i =1

E (m *2 ) = ∑ i 2 Pr(m * = i ) = ∑ i 2 p(i ) + m12 [1 − ∑ p (i )] m1 −1

= m12 − ∑ (m12 − i 2 ) p (i ) .

(11)

i =1

Hence, the limit SIR for a packet of m transmissions equals m 2 A2 . (12) η ( m) = m −1 mσ 2 + αA 2 (m 2 − ∑i =1 (m 2 − i 2 ) p(i ))

When m = 1, there is no retransmission gain and A2 η (1) = 2 . (13) σ + αA 2 The retransmission diversity gain in SIR for a packet of m transmissions is equal to η ( m) σ 2 + αA 2 , (14) = η (1) σ 2 / m + αA2 (1 − ∑m−1 (1 − i 2 / m 2 ) p(i )) i =1

which is easy to see strictly greater than one. In many conventional ARQ protocols, previous signals were discarded and a packet is demodulated using only the signal received in the current transmission. The limit SIR is equal to η(1). Hence, η(m)/η(1) is the SIR gain of the proposed approach with exploitation of retransmission diversity over those conventional protocols. Scheme 2: Each packet is spread by a randomly equiprobably selected spreading sequence in its first transmission and all retransmission. If the packet is successfully demodulated, another packet will be assigned with a new randomly equiprobably selected spreading sequence. Since the combiner combines only the signals received in the slots when the desired packet is transmitted and the spreading sequence is fixed during these slots, the SIR for Scheme 2 is identical to that for Scheme 1. Thus, we have obtained the following theorem. Theorem 1 (Schemes 1 and 2): If the spreading sequences are random realization but fixed for each packet, the SIR of combiner output in the deterministic access system converges almost surely to (9) for an arbitrary power distribution and to (12) for an equal power distribution. Scheme 3: A spreading sequence is independently equiprobably selected for a packet in each transmission. That is, spreading sequences are completely random. Since spreading sequences are i.i.d., (4) cannot be further simplified. However, it can be shown that 2

  m −1 Ak2  ∑ R1k (n − i )  → αE (m * A 2 ) a.s. ∑   i =0 k =2   and then Ps m12 A12 → η (m1 ) = a.s. 2 Pin m1σ + αE (m* A 2 ) For an equal-power system, m12 A 2 η (m1 ) = . m1σ 2 + αA 2 E (m * ) Since K

* k

E (m * ) = m1 − ∑i =1 (m1 − i) p(i ) , m1 −1

(15)

(16)

(17)

(18)

____________2005 Conference on Information Sciences and Systems, The Johns Hopkins University, March 16-18, 2005

the limit SIR for a packet of m transmissions equals m 2 A2 . η ( m) = m−1 mσ 2 + αA2 (m − ∑i =1 (m − i ) p (i ))

(19)

We obtain the following theorem. Theorem 2 (Scheme 3): If the spreading sequences are completely random, the SIR of combiner output in the deterministic access system converges almost surely to (16) for an arbitrary power distribution and to (19) for an equal power distribution. Compared with Schemes 1 and 2, the random spreading sequences in Scheme 3 increase the limit SIR by a factor m1σ 2 + αE (m *2 A 2 ) >1. (20) m1σ 2 + αE (m * A 2 ) which is greater than one because the probability that a packet is retransmitted is greater than zero and so Pr{mk* > 1} > 0 . The random spreading sequences provide more diversity in packet retransmission and therefore produce the higher SIR. When traffic intensity is high, a packet is more likely to be transmitted a large number of times. The SIR gain produced by the random spreading sequences over the fixed spreading sequences is high. When m = 1, there is not retransmission diversity gain and the limit SIR becomes η(1) in (13). The retransmission diversity gain in the limit SIR obtained under Scheme 3 is equal to η ( m) m(σ 2 + αA 2 ) = , (21) η (1) σ 2 / m + αA2 (1 − ∑m−1 (1 − i / m) p(i )) i =1

which is greater than approximate m. It is well-known [7][8] that the interference of MF output for one transmission in a large CDMA system where K and N tends to infinity with their ratio kept a constant is asymptotically Gaussian. The result can be extended to the combiner output of the large random access CDMA system undertaken. Hence, as K and N tend to infinity, the limit bit error rate (BER) for the hard decision of the combiner output can be expressed by the Q function. Specifically, the limit BER for a bit of m transmissions equals

BER(m) = Q( η (m) ) .

(22)

Correspondingly, the probability of packet success for the packet of m transmissions equals q(m) = [1 − Q( η (m) )]L .

(23)

B. System performance In the analysis of system performance, we consider the equal power systems in limit. The throughput per user is defined as the average number of packets successfully transmitted per slot ∞

T = E[q(m)] = ∑ q(m) p(m) .

(24)

m =1

Let W be the bandwidth and Tb the bit period, and then in general N = WTb. The spectral efficiency C is defined as the average number of bits successfully transmitted per Hz per second in the system. Since each user takes a share 1/(αTb) in bandwidth to transmit LT bits in LTb seconds, the spectral efficiency equals C = αT (25) bits per Hz per second.

Suppose a packet is successfully demodulated after m transmissions. Then the probability distribution of m is equal to q(m)[1−q(m−1)]… [1−q(1)], that is, the probability of m−1 unsuccessful transmissions followed by a success. The packet delay is defined as the average number of transmissions taken by a packet until successfully demodulated, ∞

m −1

m =1

j =1

D = E (m) = ∑ mq(m)∏ [1 − q ( j )] .

(26)

Theorem 3: For the equal power and deterministic access system with any of the three schemes of spreading sequences, in the limit system the stationary probability distribution for a packet to be transmitted with the mth time in a slot is 1 , (27) p (1) = i −1 ∞ 1 + ∑i =2 ∏ j =1 (1 − q ( j ))

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



i −1

i =2

j =1

(28)

Moreover, the stationary throughput, the spectral efficiency, and the packet delay equal T = p (1) , (29) C = αp(1) , (30) 1 D= , (31) p(1) respectively. The results (29) and (31) are reasonable. First, each user keeps transmitting his packets and we see m = 1 if and only if one of his packets is successfully demodulated in a slot. Moreover, the system is ergodic. Hence, p(1) is the probability that a packet is successfully demodulated on average in a slot, which means the throughput in packets per slot per user in (29). Since a user continuously transmits his current packet once per slot, the packet delay must equal to the reciprocal of the throughput, which implies (31). Our analysis starts from the assumption that the system is in a stationary state in which the stationary probability distribution of state m, the number of transmissions of a user’s current packet in a slot, is given by p(m). It is shown in the preceding section that the probability of packet success q(m) is a function of p(m). Theorem 3 is significant in that it points out that p(m) in turn can be completely determined by q(m). In other words, we have obtained a group of equations for the unknowns p(m). By solving the equations, we can numerically evaluate p(m). Moreover, after obtaining p(1), we can get numerically the throughput per user, the spectral efficiency, and the packet delay.

C. System without exploitation of retransmission diversity In several conventional ARQ protocols, retransmission diversity is not exploited and packets are demodulated using only the signals received in the current slot. In this case, the limit SIR η(1) regardless of scheme of spreading sequences is irrelevant to p(m), and so is the probability of packet success q(1). The later from (13) and (23) is

____________2005 Conference on Information Sciences and Systems, The Johns Hopkins University, March 16-18, 2005 L

   A  . q(1) = 1 − Q (32)  2 2   σ α A +   Letting p(1) = q(1) in (27)-(31) we obtain the following corollary. Corollary 1 (No retransmission gain): For an equal power and deterministic access system without exploitation of retransmission diversity, p(m) = q(1)(1 − q(1)) m−1 , m ≥ 1. (33) T = q(1) , (34) C = αq(1) , 1 . D= q (1)

(35) (36)

When traffic intensity α in the number of users per Hz per second is high, the probability of packet success q(1) is small. Consequently, a packet needs a large number of transmissions before successfully demodulated. Without exploitation of retransmission diversity, the throughput and spectral efficiency are low, and the packet delay is high. On the other hand, if traffic intensity 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 by one user on average. It will be seen that when traffic intensity α is low, the spectral efficiency, which measures the efficiency of spectrum usage of the entire system, is low. Hence, to achieve high efficiency of spectrum usage, retransmission diversity must be exploited. Since we have assumed that the arrival rate for each user is sufficiently high so that a user’s buffer always has a packet available for transmission. Consequently, the throughput per user T obtained above is actually the capacity that the system provides for the user. That is, the queue of packets in the user’s buffer is stable if the arrival rate is lower than T; and the queue is unstable if the arrival rate is higher than T. The analysis can be extended to the random access system. Due to limit of room, the analysis is omitted here. However, we point out that only a lower and an upper bound of throughput performance can be obtained for a random access system even with equal user power. IV. SIMULATION AND NUMERICAL RESULTS The systems considered in simulation and numerical evaluations have equal power. Fig. 2 shows theoretical and simulation throughputs versus SNR for a deterministic access system with random spreading sequences. The number of users is K = 20, spreading gain is N = 16, and packet size is L = 32. While the number of users is fairly small, the theoretical and simulation results are close to each other. Fig. 3 demonstrates theoretical and simulation throughputs versus traffic intensity for a deterministic access system with

random spreading sequences. Spreading gain is N = 32, SNR = 9 dB, and packet size is L = 16. Fig. 4 shows spectral efficiency for the same system. As we can see, while throughput per user monotonically decreases as traffic intensity increases, the spectral efficiency of the entire system monotonically increases. Fig. 5 compares the throughput difference between the long and short spreading sequences versus SNR for a system with K = 20, N = 16, and L = 16. The random (long) spreading sequences significantly increases throughput. V. CONCLUSIONS*

We propose an approach to exploitation of historical signals in CDMA random access systems. Stationary throughput, spectral efficiency, and packet delay are obtained analytically. 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 thus further increases signal to interference ratio. Thus, the proposed approach achieves a much higher throughput performance. Random (long) spreading sequences can significantly increase retransmission diversity gain and in turn improve system performance. Theoretical results are confirmed by simulations. 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 DSCDMA Systems," IEEE Trans. Inform. Theory, vol. 47, pp. 1128-1144, Mar. 2001.

____________2005 Conference on Information Sciences and Systems, The Johns Hopkins University, March 16-18, 2005

0.26

0.5

Spectral Efficiency (bits/Hz/Second)

Throughput (packets/slot/user)

0.24

0.22

0.2

0.18

0.16

0.14

0

2

4

6

8 10 SNR (dB)

12

14

16

0.46 0.44 0.42 0.4 0.38 0.36 0.34

Theoretical Simulation 0.12

Theoretical Simulation

0.48

0.32 0.8

18

Fig. 2. Throughputs in theory and simulation vs. SNR.

1

1.2

1.4

1.6

2

Fig. 4. Spectral efficiency in theory and simulation vs. traffic intensity.

0.36

Throughput versus SNR, while Alpha=5/4, L=16

0.4

Deterministic spreading Random spreading

Theoretical Simulation 0.34

0.35

0.32

0.3

0.3 Throughput

Throughput (packets/slot/user)

1.8

α (users/Hz/second)

0.28

0.25

0.2

0.26 0.15

0.24 0.1

0.22 0.8

1

1.2

1.4

1.6

1.8

2

α (users/Hz/second)

Fig. 3. Throughputs in theory and simulation versus traffic intensity α.

*

The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U. S. Government.

0.05 0

2

4

6

8

10

12

14

16

18

SNR

Fig. 5. Theoretical throughput vs. SNR for long and short spreading sequences.

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