Adaptive Nonlinear Predictor

Adaptive Nonlinear Predictor

It is seen that in the ACM filter, the predicted value of the state is obtained as a linear function of the previous estimate modified by a nonlinear function of the prediction error. We now use the same approach to modify the adaptive linear predictive filter described in Section 7.2.2. This technique was first developed in [387, 522]. To show the influence of the prediction error explicitly, using (7.34) we rewrite (7.25) as

Equation 7.50

Equation 7.51

By transforming the prediction error in (7.50) using the nonlinearity above, we get a nonlinear transversal filter for the prediction of r_n, namely,

Equation 7.52

Equation 7.53

The structure of this filter is shown in Fig. 7.6. To implement the filter of (7.52), an estimate of the parameter and an algorithm for updating the tap weights must be obtained. A useful estimate for is , where D_n is a sample estimate of the prediction error variance [e.g., ]. On the other hand, the tap-weight vector can be updated according to the modified LMS algorithm

Equation 7.54

where and p_n is given by (7.28). Note that the nonlinear prediction given by (7.52) is recursive in the sense that the prediction depends explicitly on the previous predicted values as well as on the previous input to the filter. This is in contrast to the linear prediction of (7.25), which depends explicitly only on the previous inputs to the filter, although it depends on the previous outputs implicitly through their influence on the tap-weight updates. The nonlinear prediction-based NBI suppression algorithm is summarized as follows.

Algorithm 7.4: [LMS nonlinear prediction-based NBI suppression] At time i, N received samples {r_iN, r_iN+1, ..., r_iN+N-1} are obtained at the chip-matched filter output (7.5).

• For n = iN, iN + 1, ..., iN + N - 1 perform the following steps:

  Equation 7.55

  

  
  

  Equation 7.56

  

  
  

  Equation 7.57

  

  
  

  Equation 7.58

  

  
  

• Detect the ith bit b₁[i] according to

  Equation 7.59

  

  
  

It is interesting to note that the predictor (7.52) can be viewed as a generalization of both linear and hard-decision-feedback (see, e.g., [107, 108, 254]) adaptive predictors, in which we use our knowledge of the prediction error statistics to make a soft decision about the binary signal, which is then fed back to the predictor. As noted above, introduction of this nonlinearity improves the prediction performance over the linear version. As discussed in [387], softening of this feedback nonlinearity improves the convergence properties of the adaptation over the use of hard-decision feedback.

To assess the nonlinear adaptive NBI suppression algorithm above, simulations were performed on the AR model for interference given in Section 7.2. The results are shown in Fig. 7.7. It is seen that, as in the case where the interference statistics are known, the nonlinear adaptive NBI suppression method significantly outperforms its linear counterpart.