Chapter 6

1. Chapter 6 Code Division Multiple Access

2. We can achieve code division multiple access (CDMA) based on spread spectrum techniques. • In particular, we can assign different spreading codes to different users in a DSSS system so that the users can share the communication channel. • For a DS-SS based CDMA (DSCDMA) system, multiple access interference (MAI) is the major factor limiting the performance and hence the capacity of the system. • Therefore, analyses of the effect of MAI on the system performance as well as ways to suppress MAI have been the major focus of CDMA research.

3. Roughly speaking, there are two different approaches to the problem. • The first approach is based on the concept of single-user detection. • In this approach, we identify one of the users in the system as the desired user and treat all signals from the other users as interference. • The receiver (for the desired user) detects only the desired user signal. • The second approach is called multiuser detection, in which all signals from all users are detected jointly and simultaneously by the receiver.

4. A common receiver based on the single-user detection approach is the matched filter receivermatched to the desired user’s spreading signal. • We note that the matched filter receiver is not optimal (in the sense of maximizing the likelihood function) in the presence of MAI. • Optimal receivers of such a kind are discussed in [1]. • However, due to their complexity, we will omit the optimal receivers here and focus on the simple matched filter receiver. • As a starting point, we will refine the crude analysis of the symbol error performance.

5. In general, it is difficult to conduct an exact symbol error performance analysis of a DS-CDMA system based on the matched filter receiver even in an AWGN channel. • Usually we have to resort to bounds and approximations. • We will discuss several common techniques to calculate the approximate symbol error probability of a DS-CDMA system. • These approximate analyses are important because they provide us simple ways to obtain the symbol error probability, which is crucial in determining the capacity of the system. • Towards the end of the chapter, we will also present some techniques based on the matched filter receiver to suppress MAI so that performance of the system can be improved.

6. 6.1 Asynchronous DS-CDMA model • In many cases, the transmissions from different users in a DS-CDMA system are not synchronized. • One of such examples is given by the uplink (reverse link) of IS-95 [2]. • In order to include these asynchronous cases, we consider a general asynchronous model of the DS-CDMA system here. • We assume that there are K actively transmitting users in the system. • We associate the kth user with a data signal bk(t) and a spreading signal ak(t), where

7. is the (transmitted) power of the kth user signal. • is the sequence of data symbols for the kth user. • is the spreading sequence assigned to the kth user. • For simplicity, we assume that BPSK modulation is employed, i.e., bi(k) are iid binary random variables taking values from the set {+1, -1} with equal probabilities. • We also assume that the spreading sequence is periodic with period N, where T = NTc, and the chip waveform is time-limited to [0; Tc) and is normalized such that • Extensions to other types of modulations and long sequences are straightforward.

8. The received signal at the receiver matched to the spreading signal of the mth user, for 1≦m ≦ K, is • n(t): AWGN with power spectral density N0 • : the amplitude response • : the phase response • : the delay (with respect to some time reference) of the channel from the transmitter of the kth user (we will call it the kth transmitter) to the receiver of the mth user (we will call it the mth receiver). • Also, where the term is the phase difference due to the delay

9. We employ to model the asynchronous nature of the system. • We assume that synchronization with the mth user’s signal has been achieved at the mth receiver. • Hence, we can assume without loss of generality. • For k ≠m, we model as independent uniform random variables on the intervals [0; T) and [0; 2π), respectively. • Moreover, we also make the assumption that all the random variables associated with different users are independent.

10. Now, let us look at the mth matched filter receiver which is designed to detect the mth user’s signal. • Without loss of generality, we consider the detection of the symbol b0(m). • Let be the received power of the kth user’s signal at the mth receiver.

11. The decision statistic for the 0th symbol is given by • ik,m is the interference component due to the kth signal. • ηis the component due to the AWGN that is a zero-mean Gaussian random variable with variance N0T.

12. On the other hand, is given by where is decomposed into

13. are the aperiodic, even, and odd cross correlation functions between the sequences respectively.

14. 6.2 Error analysis of matched filter receiver • As mentioned before, it is important to determine the error performance of a DS-CDMA system using the matched filter receiver with the presence of MAI. • Obviously, it would be most desirable if we could obtain the exact average symbol error probability. • Unfortunately, this task is exceedingly complex for most practical scenarios in which many users are actively transmitting. • Therefore, bounds and approximations are typically employed. • It is the goal of this section to give an introduction to some common bounding and approximation techniques. • We will focus on one of the receiver, namely the mth receiver. • To simplify notation, we write,

15. 6.2.1 Error bounds • Let us assume that the set of spreading sequences for the K users are given. • For convenience, we define a set of system parameters • Then the conditional symbol error probability given Sm and b0(m) = 1 and that given Sm and b0(m) = -1 are, respectively,

16. Where is the MAI component. • Hence the average symbol error probability is where the expectation is taken over all the random variables in the set Sm.

17. The second equality in (6.10) is due to the fact that the data symbols b0(k) and b-1(k) for k ≠m are symmetrically distributed about 0, i.e., the distribution of Imis symmetric about zero. • In general, the complexity of calculating of the expectation in (6.10) is exceedingly high even when there are only a moderate number of users. • In many cases, we have to resort to bounds which can be calculated with a practical level of computational complexity.

18. A simple bound based on (6.10) is that where the maximization is over all possible choices of values of the system parameters in the set Sm. • For example, with the rectangular chip waveform, i.e., and BPSK spreading, for k ≠m Where

19. Hence • Therefore where is the symbol energy of the mth user. • Although the bound in (6.11) or (6.15) can be calculated easily given the set of sequences used, this bound is often not tight and hence its usefulness is limited.

20. Another way to bound the average symbol error probability is to first determine and bound the distribution function of the MAI contribution Im and thenobtainbounds on the average symbol error probability by taking the expectation in (6.10) using the bounds on the distribution function of Im. • Using this approach, we can obtain very tight upper and lower bounds on the average symbol error probability with a complexity which increases linearly with K [3]. • In general, it is difficult to determine the distribution function of Im. • Only the distribution functions for some simple cases such as BPSK spreading and QPSK spreading, have been worked out. • Yet another way to bound the average symbol error probability is to make use of the idea of moment-space bounds [4].

21. 6.2.2 Gaussian approximations • Instead of obtaining bounds on the average symbol error probability, we can assume the MAI contribution Im as a Gaussian random variable and obtain an approximation to the average symbol error probability based on this assumption. • There are mainly two variations to this Gaussian-approximation approach. • Standard Gaussian approximation • Improved Gaussian approximation

22. Standard Gaussian approximation • The first method is to assume Im as a zero-mean Gaussian random variable. • This method is usually known as the standard Gaussian approximation(SGA) [5] and is applicable to situations in which there are a large number of users with similar received powers in the system. • Its validity is justified by the central limit theorem based on the fact that Im is a summation of independent random variables Re[ik,m].

23. When the number of users K is large, the distribution of Im approaches Gaussian. • With SGA, the approximate average symbol error probability is given by • Therefore, all we need is to calculate the mean and variance of Im

24. Hence, the problem reduces to the evaluation of the variance of Re[ik,m]. • To do this, we use the following simple identity • Then, for k ≠m, • Let us write

25. From (6.5), we have • The second and third equalities in (6.23) are due to the independence of the random variables involved. • Substituting (6.21) and (6.22) into (6.23) and making use of the fact that are independent,

26. Let us define

27. For rectangular chip waveform, i.e., as in BPSK or QPSK spreading, • Hence, (6.26) reduces to • Combining (6.17), (6.18), and (6.29),

28. Given the set of sequences, the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (6.15). • Sometimes, it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed. • One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (6.30) by their expectations.

30. Replacing the term in (6.30) by the expectation in (6.32), we have the approximation,

32. Figure 6.2 shows the plots of the approximate symbol error probability given by the SGA in (6.33) for the case in which all the users have equal received powers and N = 31. • Note that for K = 1, the symbol error probability is exact. • When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 6.2, we can further approximate (6.33) as

33. Comparing to the approximation of the average symbol error probability given by (2.49), The standard Gaussian approximation in (6.34) gives a more optimistic estimate on the system capacity. • For example, in order to achieve an average symbol error probability of 10-3, K≦N/3 approximately based on (6.34). • Hence, a DS-CDMA system with a processing gain N can accommodate N/3 users (compared to N/5 users predicted by (2.50)).

34. Improved Gaussian approximation • In general, the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar. • However, when either the number of active users is not large or there are a few users with received powers much higher than those of the others, the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed. • We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable. • However, we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ik,m] now.

35. This difficulty can be circumvented by noting that each ik,m is a summation of a large number of terms involving the sequence elements (refer to (6.5) and (6.31)) when the processing gain N is large. • We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with • To aid our discussion and to conform to the notation in the literature, let us define to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (6.4) by the factor

36. Then, the real part of the normalized decision statistic becomes • is a zero-mean (real) Gaussian random variable with variance • It can be shown [6, 7, 8] that the normalized MAI terms are conditional independent given and the desired user’s spreading sequence. • Based on this, one can further show [6, 7, 8] that conditioning on the set the normalized MAI component in (6.36) approaches a zero-mean Gaussian random variable with varianceVmas N approaches infinity.

37. The conditional variance of the limiting Gaussian random variable is given by • where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7, 8]. • For example, with BPSK spreading,

38. While with QPSK spreading,

39. We note that are iid random variables given our model of the delays and phases. • The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large. • Hence, the approximate conditional symbol error probability given is

40. Averaging this over the delays and phases, we obtain an accurate approximation to the (unconditional) symbol error probability

41. The approximation in (6.41) is known as the improved Gaussian approximation (IGA). • We note that the IGA is accurate, regardless of the number of active users in the system, as long as the processing gain is large. • On the other hand, the SGA is accurate, regardless of the processing gain, when there are a large number of active users with equal received powers. • We notice that the conditional error probability depends on the delays and phases only through Vm. • Hence, an efficient way to calculate PIGA in (6.41) is to first obtain the probability density function pVm(v) of the random variable Vm and then evaluate the expectation by the integral

42. The density function pVm(v) of Vm, in turns, can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables • Compared to the SGA in (6.31), the computational complexity of (6.42) is still significant higher. • We can reduce the computational complexity of the IGA by further approximating the expectation in (6.41) based on a Taylor series approximation [7, 9] of the conditional symbol error probability as below:

43. where are the mean and variance of the random variable Vm, respectively. • From (6.37), since are iid. • For example, with BPSK spreading, while with QPSK spreading,

44. Putting these into (6.43), we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (6.30). • It is shown in [7, 9] that the approximated IGA in (6.43) is almost as accurate as the original IGA in (6.41) in many situations.

45. In summary, we point out that the IGA generally gives a more accurate approximation to the symbol error probabilitythan the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems. • To illustrate this point, let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading, respectively. • From (6.33), the SGA predicts that the symbol error probabilities of the two systems are the same. • On the other hand, the IGA (6.43) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading. • The latter is in fact true for randomly selected sequences.

46. 6.3 Near-far problem • Based on the SGA in (6.30), we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor as compared to the case in which only the user is active. • When the received powers of all users are the same and the set of spreading sequences are properly chosen, the degradation in SNR is relatively small if there are a moderate number of users. • However, when the received powers of some of the interferers are much larger than that of the desired user, the performance degradation is large.

47. In the context of wireless communications, this situation occurs when some of the interferers are located close to the base station while the desired user is far away. • This problem is known as the near-far problemin CDMA systems. • A common measure of the robustness of a receiver against the near-far problem is the near-far resistance, defined in [10], of the receiver. • For now, we argue the intuitive idea of the near-far resistance measure.

48. To understand the concept of near-far resistance, let us imagine that only one user, say the mth user, were active in the DS-CDMA system considered previously. • In this case, the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large. • We employ this as a benchmark to compare performance of different receivers in the multiuser scenario. • Going back to the realistic situation of multiple active users, the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI. • In fact, the larger the received powers of the interferers, the poorer is the performance.

49. We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values. • The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario. • If the received MAI power increases, this ratio will get smaller. • The ratio of the exponential decrease rate to in the limiting case of extremely large MAI power is the near-far resistance of the receiver. • A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done). • A near-far resistance of 0 indicates that the receiver will break down in a near-far situation.

50. For the matched filter receiver, we can employ (6.31) or (6.43) to conclude that the symbol error probability levels off when the SNR increase (for example, see Figure 6.2). • Hence the exponential decrease rate is 0 and the near-far resistance is 0.