Failure of independence assumption

The example in Figure 1 illustrates a pitfall of classical inversion theory. Had $\bold p$ not overlapped bolds, the crosstalk would have been removed perfectly. We were not interested in destroying $\bold p$ with $\bold s$,and vice versa. This result was just an accidental consequence of their overlap, which came to dominate the analysis because of the squaring in least squares. Our failure could be attributed to a tacit assumption that since $\bold p$ and $\bold s$ are somehow ``independent,'' they can be regarded as orthogonal, i.e., that $\bold p \cdot \bold s =0$.But the (potential) physical independence of $\bold p$ and $\bold s$does nothing to make a short sample of $\bold p$ and $\bold s$ orthogonal. Even vectors containing random numbers are unlikely to be orthogonal unless the vectors have an infinite number of components. Perhaps if the text were as long as the works of Shakespeare . . . .