Tonal Interference
For mathematical convenience, we assume that the narrowband interference signal consists of m complex sinusoids of the form
where Pl and fl are the power and normalized frequency of the lth sinusoid, and the {Fl} are independent random phases distributed uniformly on (0,2p). The covariance matrix Ri of the multitone interference signal i can be represented as
where
Denote 
,and 
. Then 
, and hence we have
where . According to (7.77), let 
. Then from (7.81) we can write
Assuming that the spread-spectrum user has a random signature sequence, we next derive expressions for the expected values of the output SINR with respect to the random signature vector s, for several special cases.
Case 1: m = 1. We have 
, and
where we have used 
and 
.
Substituting these into (7.82), we
obtain
Therefore, when N is large, the energy of a strong interferer is almost completely suppressed by the linear MMSE detector.
Case 2: m = 2. From (7.81) we have
where 
. Now using (7.82), we obtain
Using (7.85), we have
where 
, and
where 
. On the other hand, using (7.85), we can write
Substituting (4.55) and (7.89) into (7.86), we then have
Again we see that for large N, the interfering energy is almost completely suppressed. In general, it is difficult to obtain an explicit expression for E {SINRm} for m > 2. However, for the special case when the {gl} are mutually orthogonal, a closed-form expression for E {SINRm} can easily be found.
Case 3: Orthogonal {gl}. Assume that
This condition is met, for example, when fl - fK is a
multiple of 1/N for all l 
K. Under this
condition of orthogonality, it follows straightforwardly that
The expected value of the output SINR with respect to the random signature vector s is
where 
, and 
. Figure 7.10 shows some numerical examples of
tone suppression by the linear MMSE detector. In both plots the SINRs without
interference are 10dB (after despreading). Each curve in the plots corresponds
to one set of three-tone (or seven-tone) frequencies {fl} randomly chosen. (The vectors {gl} are
not necessarily orthogonal.) Interestingly, it is seen from Fig. 7.10 that the output SINRs are centered at the value
given by (7.93).