Nonlinear Interpolating Filters
ACM Interpolator
Nonlinear interpolative interference suppression filters have
been developed in [425]. We next derive the
interpolating ACM filter. We consider the density of the current state
conditioned on previous and following states. We have
Equation 7.60
Equation 7.61
where in (7.60) we made
the approximation that, conditioned on in, 
and 
are
independent. The second term in (7.61) is
independent of in. If it is assumed
(analogously to what is done in the ACM filter) that the two densities in the
numerator of the first term in (7.61) are
Gaussian, the interpolated estimate is also Gaussian. Therefore, if we assume
that the densities are (where f indicates the
forward prediction and b indicates the backward
prediction)
the interpolated estimate is still Gaussian:
Equation 7.62
with
Equation 7.63
Equation 7.64
While the mean and variance of the interpolated estimate at
each sample n can be computed via the equations
above, recall that the forward and backward means and variances are determined
by the nonlinear ACM filter recursions.
Simulation Examples
The equations above can be used for both the linear Kalman
filter and the ACM filter to generate interpolative predictions from the forward
and backward predicted estimates. As in the ACM prediction filter, we have
approximated the conditional densities as being Gaussian, although the
observation noise is not Gaussian. The filters are run forward on a block of
data and then backward on the same data. The two results are combined to form
the interpolated prediction via (7.63)–(7.64).
Simulations were run on the same AR model for interference as
that given in Section 7.2. Figure 7.8 gives results for interpolative
filtering over predictive filtering for the known statistics case. The filters
were run forward and backward for all 1500 points in the block. Interpolator
SINR gain was calculated over the middle 500 points (when both forward and
backward predictors were in steady state).
Adaptive Nonlinear Block Interpolator
Recall that the ACM predictor uses the interference prediction
at time n, 
, to
generate a prediction of the observation less the spread spectrum signal 
.
This estimate is used in subsequent samples to generate new interference
predictions. Since the estimates of 
are not available for 
> 0 at time
n (i.e., samples that occur after the current
one), the ACM filter cannot be cast directly in the interpolator structure.
However, an approach similar to the one for the known-statistics ACM
interpolator can be used. In this approach the data are segmented into blocks
and run through a forward filter of length L to
give predictions 
and 
. The same
data are run through a backward adaptive ACM filter with a separate tap-weight
vector, also of length L, to generate estimates
and 
. After these calculations are made for the entire block, the
data are combined to form an interpolated prediction according to
Equation 7.65
Equation 7.66
The next block of data follows the same procedure. However,
when the next block is initialized, the previous tap weights are used to start
the forward predictor, and the interpolated predictions {} are used
to initialize the forward prediction. This "head start" on the adaptation can
only take place in the forward direction. We do not have any information on the
following block of data to give us insight into the backward prediction.
Therefore, the backward prediction is less reliable than the forward prediction.
To compensate for this effect, consecutive blocks are overlapped, with the
overlap being used to allow the backward predictor some startup time to begin
good predictions of the spread-spectrum signal [381, 425].
Simulation Examples
Results for the same simulation when the statistics are unknown
are given in Fig. 7.9. The adaptive
interpolator had a block length of 250 samples, with 100 samples being
overlapped. That is, for each block of 250 samples, 150 interpolated estimates
were made. For the case of known statistics, the ACM predictor already performs
well, and there is little margin for improvement via use of an interpolator. The
adaptive filter shows greater margin for improvement, on which the interpolator
capitalizes. However, in either case, the interpolator does offer improved phase
characteristics and some performance gain at the cost of additional complexity
and a delay in processing.
A number of further results have been developed using and
expanding the ideas discussed above. For example, performance analysis methods
have been developed for both predictive [537] and interpolative [538] nonlinear
suppression filters. Predictive filters for the further situation in which the
ambient noise {N(t)} has impulsive components have been developed in [133]. The multiuser
case, in which K > 1, has been considered in
[425]. Further
results can be found in [11, 15, 238, 535, 536].
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