Near–Far Resistance to Both NBI and MAI
Now suppose that the interference in the system includes (K–1) independent synchronous MAIs in addition to the NBI. Let the signature vector for the kth MAI be sk and the power be Pk. It is straightforward to generalize (7.77) to include the effect of MAI, and we obtain the output SINR of the MMSE detector as
Equation (7.135) suggests that when we consider the output SINR for the linear MMSE detector, the NBI signal can be viewed as being equivalent to N independent synchronous virtual MAIs. The lth virtual MAI has signature vector ul and power ll. Suppose that r = rank(Ri) and lr+1 = ... = lN = 0. Then using the results from [307], the near–far resistance (to both MAI and NBI) is nonzero if and only if s is not contained in the subspace span{s1,..., sK; u1,..., ur}.
Next we consider the effect of NBI on the near–far resistance to MAI, by fixing the power of the NBI and increasing the power of the MAIs. It is shown in [307] that the linear MMSE solution w is asymptotically orthogonal to the subspace spanned by s1,..., sK:
Such an asymptotic w can be found by solving the following constrained optimization problem:
where
It then follows from the method of Lagrange multipliers that
where 
. Let 
, where
span{si, ..., sK} and
s
span{s1,..., sK}.
The near–far resistance to MAI is nonzero if and only if sT w 
0.
Therefore, from (7.136)
and (7.138) it is easily seen that the
near–far resistance to MAI is nonzero if and only if s
0 and s
span
. Notice
that if there is no NBI (i.e., 
=s2 IN),
this condition for nonzero near–far resistance reduces to s
0 [307].
Simulation Examples
Figure 7.15 shows the output SINR of the linear MMSE detector in the presence of both MAI and NBI. The signal-to-noise ratio for the desired user in the absence of interference is fixed at 20 dB. The NBI is a second-order AR signal with both poles at 0.99. The MAIs are synchronous with the desired SS user, with random signature sequences and the same power. The processing gain is N = 31. Two cases are shown: three MAIs and six MAIs. For each case we vary the power of one type of interference (MAI or NBI) while keeping the power of the other fixed.
Figure 7.15. Output SINR of a linear MMSE detector in the presence of both MAI and NBI. The noise power is held constant at s2 = -20 dB relative to the SS signal after despreading. The NBI signal is a second-order AR signal with both poles at 0.99. The MAIs are synchronous with the SS user, with random signature sequences. The processing gain is N = 31.
It is seen from Fig. 7.15 that the effects of the MAI and the NBI on the output SINR are different. The output SINR is insensitive to the power of the MAI while it is more sensitive to the power of NBI. To see this, we consider a simple example where the CDMA system consists of the desired SS user signal, one MAI and one NBI, in the absence of background noise. Then by (7.135) the output SINR of the MMSE detector in this case is given by
where the second equality is obtained by using the matrix
inversion lemma. Now because of the pseudorandomness of the signature vectors
s and s1, 
. It is
seen from (7.139) that the power of the
MAI (P1) affects the SINR only through
the negligible second term in (7.139),
while the power of the NBI affects the SINR through the dominant first term in
(7.139).
Figure 7.16 is a plot of the probability of error of the MMSE detector, in the presence of strong MAI and NBI, in addition to ambient noise. The symbols o and + in this plot correspond to the data obtained from simulations, and the solid and dashed lines correspond to Gaussian approximations of the probability of error (i.e., BER). It was shown in [386] that in an environment of MAI and AWGN, the error probability for the MMSE detector can be well approximated by assuming that the output MAI plus noise is Gaussian. This plot seems to suggest that even in the presence of NBI, the output NBI plus MAI plus noise is still approximately Gaussian, as one would expect, since the NBI here is Gaussian.