In analyzing the impact of Multiple Access Interference in DS-CDMA systems, as described in Chapter 1, we must consider the detailed structure of the contribution of each interfering signal to the decision statistic, Z0. This is given by Ik in (1.32).
The relationship between bk(t − τk), ak(t − τk), and a0(t) is illustrated in Figure A-1. The quantities γk and Δk in Figure A-1 are defined from the delay of user k relative to user 0, τk, such that [Pur77]:
Based on Figure A-1, we rearrange (1.32)
It is convenient to rewrite this as
This may be rearranged to give
It is useful to define Zk, l as [Leh87a]
where each of the Zk,j are independent Bernoulli trials, equally distributed on {-1,+1}. Then, using (A.5) and the fact that a0,la0,l = 1, (A.4) may be expressed as
We define the set 
as the set of all integers in [0, N − 2] for which a0, l + ia0, l + i + 1 = 1. Similarly, the set 
is defined as the set of all integers in [0, N − 2] for which a0, l + ia0, l + i + 1 = −1. Then (A.6) is
We define
Note that {Zk, l} is a set of independent Bernoulli trials [Coo86], each with each outcome identically distributed on {-1,+1}.
As described in [Leh87a] and [Mor89]. the contribution from interfering user k to the decision statistic is given by
The term A is the number of integers i such that for a0, ia0,i+ 1 = 1 for 0 ≤ i≤ N − 2. Similarly, B is defined as the number of integers i such that for a0, ia0,i+ 1 = −1 for 0 ≤ i ≤ N − 2. The quantities Xk, Yk, Uk, and Vk, in (A.12) have distributions, conditioned on A and B which are given by
Thus the interference contributed to the decision statistic by a particular multiple access interferer is entirely defined by the quantities Pk, φk, Δk, A and B. Note that A and B are solely dependent on the sequence of user 0. Furthermore, A + B = N − 1 since sets and are disjoint and span the set of total possible signature sequences of length N in which there are a total of N − 1 possible locations for a chip level transition.
The use of the Gaussian Approximation to determine the bit error rate in CDMA multiple access communication systems is based on the argument that the decision statistic, Z0, given by (1.26), may be modeled as a Gaussian random variable [Pur77], [Mor89]. The first component in (1.26), I0, is deterministic and its value is given by (1.27). The other two components of Z0, ζ and η are assumed to be zero mean Gaussian random variables. If the additive receiver noise, n(t), is a Gaussian random process, then η, which is given by (1.28), is a zero-mean Gaussian random variable. In this section, the standard expression for the bit error rate is derived based on the assumption that the multiple access interference term, ζ, may be approximated by a Gaussian random variable.
We will first define a combined noise and interference term ξ, given by
such that the decision statistic, Z0, is given by
where Z0 is a Gaussian random variable with mean I0 and a variance which is equal to the variance, 
, of ξ.
The probability of error in determining the value of a received bit is equal to the probability that ξ <−I0 when I0 is positive and ξ > I0 when I0 is negative. Due to its structure, ξ is symmetrically distributed such that these two conditions occur with equal probability; therefore, we may say that the probability of error is equal to the probability that ξ > |I0|. If we may assume that ξ is a zero mean Gaussian random variable with variance, , the probability of a bit error is given by
Using (1.27), this may be rewritten as
Assuming that the multiple access interference contribution to the decision statistic, ζ, may be modeled as a zero mean Gaussian random variable with variance 
, and that the noise contribution to the decision statistic, η, may be modeled as a zero mean Gaussian noise process with variance 
, then, since the noise and the multiple access interference are independent, the variance of ξ is given by
All that remains is to justify the Gaussian Approximation for ζ, and to determine the value of .
The total multiple access interference contribution to the decision statistic for user 0, from (1.31), is
where Ik is the contribution from user k to the decision statistic for user 0. Using (A.12), we write Ik, as
and Wk is given by
The distributions for Xk and Yk are given by (A.13) and (A.14) and the distributions of Uk and Vk are given in (A.15) and (A.16) as derived by [Mor89] and shown in Appendix [Leh87a]. Note that Wk may take on only discrete values (in fact a maximum of (N − 1)B − B2 + 6 values) given Δk.
The Central Limit Theorem (C.L.T.), described in [Sta86] and [Coo86] is used to justify the approximation of ζ as a Gaussian random variable. According to the more general statement of the C.L.T. [Sta86], the sum, y,
of M independent random variables, xi (which are not necessarily identically distributed), each with mean μxi and variance 
, has a distribution which approaches a Gaussian distribution as M gets large, provided that
Furthermore, the mean and variance of the sum, y, are given by
In this application, the condition given in (A.26) is equivalent to specifying that no single user dominates the total multiple access interference.
The terms Ik in (A.22) are not actually independent. Since the distributions of Xk and Yk given in (A.13) and (A.14) are both dependent on B, which is itself a random variable, the random variables Xk and Yk are independent only when conditioned on B. Since the terms Ik are not independent, one of the conditions of the C.L.T. is violated. To continue development of the Gaussian Approximation, we will assume that the C.L.T. may be applied; however, this issue is addressed at length in [Lib95].
The variance of the multiple access interference is computed from
where the expected value is taken over all values of {φk}, {Δk}, and B. The phases, {φk}, are independent and uniformly distributed on [0, 2π]. Similarly, the fractional chip delays, {Δk} are independent and distributed on [0,Tc]. For random PN sequences, B is distributed as
The expected value of Ik is
Therefore (A.29) reduces to
To proceed with the derivation of the standard Gaussian Approximation, it is necessary to assume that the terms Ik are independent, which, as already stated, is not a strictly valid assumption. Assuming that the terms Ik are independent, and using (A.31), we may write
The variance of Ik is
where
and
As noted earlier, since distributions of random variables Xk and Yk both depend on random variable B, they are not strictly independent. In the standard derivation of the Gaussian Approximation, this fact is not taken into account. To proceed, we assume that Xk, Yk, Uk, and Vk are all independent with zero means. Then we may write
The term Xk is the sum of A independent and identically distributed Bernoulli trials, xi, each with equally likely outcomes from {-1,+1}. As described in [Pur77], A is the number of integers i such that first lag of the autocorrelation function for the chip sequence for the desired user is a0,ia0,i+1 = 1 for i ∈ [0,N − 2]. Note that A + B = N − 1 or A = N − B − 1. Therefore, when conditioned on A, the mean square value of Xk is
Then
Similarly, Yk, is the summation of the B independent Bernoulli trials yi, where B is the number of integers i such that a0,ia0,i+1 = −1 for i ϵ [0, N−2]. Note that A + B = N − 1 or A = N − B − 1.
Then
The random variables Uk and Vk take on values of {-1,+1} with equal probability, therefore 
and 
, Then we may express (A.37) as
The statistics of Δk are
and
Using these values and A = N − 1 − B in (A.42) we obtain
Finally, we find the expected value of B using the distribution given by (A.30),
Therefore,
Substituting (A.47) and (A.35) into (A.34) we obtain
If the K − 1 values of 
satisfy (A.26), we use the C.L.T. to model ζ as a zero mean Gaussian random variable as K − 1 gets large. The variance of ζ is given by
Therefore, we model the decision statistic Z0 as a Gaussian random variable with mean given by (1.27) and variance given by (A.21)
Using (A.20), the bit error rate is given by
This is the bit error rate for BPSK-DS-CDMA, using random, asynchronous, sequences and the Gaussian Approximation.