This exercise explores modifications of the example used to introduce an ICA. The sources used to generate the curves in Figure 9.8 are
1(
) = sin(
) and
2() = sin(1.7−5), and the recorded signals for Figure 9.9 are
1()=2
1() +
1() and
2()=3
1() − 2
1(). Also, 200 points were used over the interval 0 ≤
≤ 10.
(a) The area in Figure 9.12 comes from using the composite trapezoidal rule to compute
where
∗
1
and
∗
2
are the sources computed using (9.37). Note that the latter depend on the angle
used to evaluate
V, but the original sources do not depend on
. Plot a curve similar to the one in Figure 9.12, but use
Does the value of the optimal angle change much using this error function? Note that because the ICA can not determine which source is
1
or
2, you will need to create two plots. One with
∗
1
and
∗
2
as above, and a second with
∗
1
and
∗
2
interchanged in the formula. Also, the above formula is associated with the 2-norm, and is also the error measure associated with least squares.
(b) The kurtosis plotted in Figure 9.13 comes from evaluating (9.41). Suppose instead one uses the 2-norm, and takes
Plot this function and determine the locations of the local maxima. One of these, presumably, is close to the value obtained in part (a). Compare it with the value in part (a), as well as the solution obtained in the text.
The exercises to follow require synthetic data, which is either provided or you generate. For the latter, suppose that by using either the exact solution, or a numerical solution, you determine values
0,
1, ··· ,
n
of the solution at the time points
0,
1, ··· ,
n
. By appropriately modifying the MATLAB commands:
you can produce synthetic data. In doing this, the synthetic data values will satisfy (1−
) ≤
≤ (1 +), which means you should pick
so that 0
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