Causality in two-dimensions

In one dimension, most filters of interest have a short memory. Significant filter coefficients are concentrated shortly after t=0. The favorite example in Physics is the damped harmonic oscillator, all of which is packed into a two-lag filter (second order differential equation). The complete story is rich in mathematics and in concepts, but to sum up, filters fall into two categories according to the numerical values of their coefficients. There are filters for which equations (1) and (2) work as desired and expected. These filters are called ``minimum phase''. There are also filters for which (2) is a disaster numerically, the feedback process diverging to infinity.

Divergent cases correspond to physical processes that require boundary conditions. Equation (2) only allows for initial conditions. I oversimplify by trying to collapse an entire book (FGDP) into a few sentences by saying here that for any fixed spectrum there exist many filters. Of these, only one has stable polynomial division. That filter has its energy compacted as soon as possible after the ``1.0'' at zero lag.

Now let us turn to two dimensions. Filters of interest will correspond to energy concentrated near the end of a helix. Let us examine the end of a helix. At the very end, as in the 1-D case, is a coefficient with the numerical value 1.0. Keeping only coefficients within two mesh points in any direction from the 1.0, we copy the coefficients from near the end of the helix to a cartesian mesh like this:  

$$\begin{array} {ccccc} \begin{array} {ccc} h & c & 0 \\ p ... ...& {\rm variable} &\quad +\quad& {\rm constrained}\end{array}$$ (7)

where a,b,c,...,u are adjustable coefficients.

Which side of the little rectangular patch of coefficients we choose to place the 1.0 is rather arbitrary. The important matter is that as a matter of principle, the 1.0 is expected to lie along one side of the little patch. It is rarely (if ever) found at a corner of the patch. It is important that beyond the 1.0 (in whatever direction that may be) the filter coefficients must be zero because in one dimension, these coefficients lie before zero lag. Our foundations, the basic convolution-deconvolution pair (1) and (2) are applicable only to filters with all coefficients after zero lag.

Time-series analysis is rich with concepts that the helix now allows us to apply to many dimensions. First is the notion of an impulse function. Observe that an impulse function on the 2-D surface of the helical cylinder maps to an impulse function on the 1-D line of the unwound coil. An autocorrelation function that is an impulse corresponds both to a white (constant) spectrum in 1-D and to a white (constant) spectrum in 2-D. Next we look at a particularly important autocorrelation function and see how 2-D is the same as 1-D.