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 {c_n} 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 {c_n} 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 r_n from its past values. If the observation noise {v_n} 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 {v_n} 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.