Anti-crosstalk

Theoretical example

First let us consider an extremely simple theoretical example with an issue of ``cross talk''. Notice that word itself is absent from linear inverse theory. (Anyway, it's absent from the index of Tarantola's book.) Normally we think of ``high frequencies'' as being orthogonal to ``low frequencies''. In this example that will be true from a theoretical viewpoint, but no practical person would consider it to be true.

Consider a signal that is an impulse which is to be split into its low and high frequencies. We might write this as $0=l+h-d$, namely, zero equals low plus high minus data. Let us extract low frequencies from the data with a step function in the Fourier domain or convolving with a sinc function in the time domain. Clearly the high frequency component in the Fourier domain is the constant function minus the step. In the time domain it is a delta function minus a sinc (since high plus low is a delta function). Theoretically everything is fine. Usually a vanishing cross product means a sum of terms vanishes. Here every term vanishes in the Fourier domain when we multiply the step times one minus the step. That is powerful orthogonality! In the time domain the convolution of the two, $sinc *(1-sinc) = sinc - sinc*sinc = sinc -sinc =0$ vanishes not just at zero lag, but at all lags. That is powerfully strong orthogonality too. Theoretically, high and low frequency components of the data are orthogonal at every frequency and at every lag. What would the experimentalist have to say? The experimentalist would look in the time domain at the low frequency function and at the high frequency function and say, ``Everywhere I look at these two functions they are the same. They can't be orthogonal. They have a massive amount of crosstalk.'' Of course the two functions are not exactly the same. They have opposite polarities, and they are not the same at the origin point. But everywhere else they look the same. We don't like it. These signals should not be coherent but they are.

I first encountered this crosstalk issue in a serious geophysical application of a very simple nature. The lack of a good place for crosstalk in the theoretical framework was blatantly obvious.

Anti-crosstalk