Tonal Interference
For mathematical convenience, we assume that the narrowband
interference signal consists of m complex
sinusoids of the form
Equation 7.78
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
Equation 7.79
where
Equation 7.80
Denote  ,and  . Then  , and hence we have
Equation 7.81
where . According to (7.77), let  . Then from (7.81) we can write
Equation 7.82
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
Equation 7.83
where we have used  and  .
Substituting these into (7.82), we
obtain
Equation 7.84
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
Equation 7.85
where  . Now using (7.82), we obtain
Equation 7.86
Using (7.85), we
have
Equation 7.87
where  , and
Equation 7.88
where  . On the other hand, using (7.85), we can write
Equation 7.89
Substituting (4.55) and (7.89) into (7.86),
we then have
Equation 7.90
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
Equation 7.91
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
Equation 7.92
The expected value of the output SINR with respect to the
random signature vector s is
Equation 7.93
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).
| 
Sections
Syndication
