|The helical coordinate|
Mathematics sometimes seems a mundane subject, like when it does the ``accounting'' for an engineer. Other times it brings unexpected amazing new concepts into our lives. This is the case with the study of causality and spectral factorization. There are many little-known, amazing, fundamental ideas here I would like to tell you about. We won't get to the bottom of any of them but it's fun and useful to see what they are and how to use them.
Start with an example. Consider a mechanical object. We can strain it and watch it stress or we can stress it and watch it strain. We feel knowledge of the present and past stress history is all we need to determine the present value of strain. Likewise, the converse, history of strain should tell us the stress. We could say there is a filter that takes us from stress to strain; likewise another filter takes us from strain to stress. What we have here is a pair of filters that are mutually inverse under convolution. In the Fourier domain, one is literally the inverse of the other. What is remarkable is that in the time domain, both are causal. They both vanish before zero lag .
Not all causal filters have a causal inverse. The best known name for one that does is ``minimum-phase filter.'' Unfortunately, this name is not suggestive of the fundamental property of interest, ``causal with a causal (convolutional) inverse.'' I could call it CwCI. An example of a causal filter without a causal inverse is the unit delay operator -- with -transforms, the operator itself. If you delay something, you can't get it back without seeing into the future, which you are not allowed to do. Mathematically, cannot be expressed as a polynomial (actually, a convergent infinite series) in positive powers of .
Physics books don't tell us where to expect to find transfer functions that are CwCI. I think I know why they don't. Any causal filter has a ``sharp edge'' at zero time lag where it switches from nonresponsiveness to responsiveness. The sharp edge might cause the spectrum to be large at infinite frequency. If so, the inverse filter is small at infinite frequency. Either way, one of the two filters is unmanageable with Fourier transform theory which (you might have noticed in the mathematical fine print) requires signals (and spectra) to have finite energy which means the function must get real small in that immense space on the -axis and the axis. It is impossible for a function to be small and its inverse be small. These imponderables get more managable in the world of Time Series Analysis (discretized time axis).
|The helical coordinate|