The conventional answer to the crosstalk question is to choose so that has minimum power. The idea is that since adding one signal to an independent signal is likely to increase the power of ,removing as much power as possible may be a way to separate the independent components. The theory proceeds as follows. Minimize the dot product

(5) |

(6) |

In equation (5) the ``**fitting function**'' is ,because various amounts of can be subtracted to minimize
the power in the residual .Let us verify the well-known fact that
after the energy is minimized,
the **residual** is **orthogonal** to the **fitting function**.
Take the dot product of the fitting function and the residual ,and insert the optimum value of from equation (6):

Figure 1

At first it is hard to believe the result:
the crosstalk is *worse* on the output than on the input.
Our eyes are drawn to the weak signals in the open spaces,
which are obviously unwanted new crosstalk.
We do not
immediately notice that the new crosstalk
has a negative polarity.
Negative polarity results when we try to extinguish
the strong positive polarity of the main signal.
Since the residual misfit is *squared,*
our method tends to ignore small residuals
and focus attention on big ones:
hence the wide-scale growth of small residuals.

The least-squares method is easy to oversimplify,
and it is not unusual to see it give disappointing results.
Real-life data are generally more complicated than
artificial data like the data used in these examples.
It is always a good idea to test programs on such **synthetic data** since
the success or failure of a least-squares method may not be apparent
if the method is applied to real data without prior testing.

10/21/1998