Nonlinear Predictive Techniques
Linear predictive methods exploit the wideband nature of the
useful data signal to suppress the interference. In doing so, they are
exploiting only the spectral structure of the spread data signal, not its
further structure. These techniques can be improved upon in this application by
exploiting such further structure of the useful data signal as it manifests
itself in the sampled observations (7.5). In particular, on
examining (7.1), (7.2), (7.3), and (7.5), we see that for the
single-user case (i.e., K = 1), the discrete-time
data signal {cn} takes on values of
only 
. Although linear prediction would be optimal in the model of (7.5) in the case
in which all signals are Gaussian, this binary-valued direct-sequence data
signal {cn} is highly non-Gaussian.
So, even if the NBI and background noise are assumed to be Gaussian, the optimal
filter for performing the required prediction will, in general, be nonlinear
(e.g., [377]). This
non-Gaussian structure of direct-sequence signals can be exploited to obtain
nonlinear filters that exhibit significantly better suppression of narrowband
interference than do linear filters under conditions where this
non-Gaussian-ness is of sufficient import. In the following paragraphs we
elaborate on this idea, which was introduced in [522] and explored further in [133, 376, 387, 425, 535–538].
Consider again the state-space model of (7.10)–(7.11). The Kalman–Bucy
estimator discussed above is the best linear predictor of rn from its past values. If the observation
noise {vn} of (7.12) were a Gaussian process,
this filter would also give the global MMSE (or conditional mean) prediction of
the received signal (and hence of the interference). However, since {vn} is not Gaussian but rather is the sum of
two independent random variables, one of which is Gaussian and the other of
which is binary (), its probability density is the
weighted sum of two Gaussian densities. In this case, the exact conditional mean
estimator can be shown to have a complexity that increases exponentially in time
[452], which renders
it unsuitable for practical implementation.
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