Page 343 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

332 WORKED OUT EXAMPLES

def
2
2
The design parameters of the estimators are the SNR ¼ / , the
a
v
def
2
duration of echo generation T, the echo strength S r ¼ K , the number
d
of correlators J and the window size I.
Example
In the following example, the selected design parameters are SNR ¼
100, T ¼ 0:8 (ms), S r ¼ 0:2 and I ¼ 1000. Using (9.11) and (9.13), we
calculate C zjt , and from that the eigenvectors and corresponding eigen-
values and weights are obtained. Figure 9.10 shows the result. As
expected the response of the first filter/correlator is similar to the direct
response, and this part just implements the conventional matched filter.
From the seventh filter on, the weights decrease, and from the fifteenth
filter on the weights are near zero. Thus, the useful number of filters is
between 7 and 15. Figure 9.11 shows the results of application of the
filters to an observed waveform.

9.2.6 Optimization and evaluation

In order to find the estimator with the best performance the best par-
ameters of the estimators must be determined. The next step then is to
assess their performances.

Cross-validation

In order to prevent overfitting we apply a three-fold cross-validation
procedure to the data set consisting of 150 records of waveforms. The
corresponding MATLAB code is given in Listing 9.8. Here, it is assumed
that the estimator under test is realized in a MATLAB function called
ToF_estimator().
The optimization of the operator using a training set occurs according
to the procedure depicted in Figure 9.12. In Listing 9.8 this is imple-
mented by calling the function opt_ToF_estimator(). The actual
code for the optimization, given in Listing 9.9, uses the MATLAB function
fminsearch().
We assume here that the operator has a bias that can be compensated
for. The value of the compensation is just another parameter of the
estimator. However, for its optimization the use of the function
fminsearch() is not needed. This would unnecessarily increase the