Non-stationary convolution and combination

There are several possible approaches to generalizing convolution described by equation () to deal with non-stationarity. The simplest approach Yilmaz (1987) is to apply multiple stationary filters and interpolate the results. This approach, however, gives incorrect spectral response in the interpolated areas Pann and Shin (1976).

Following Claerbout (1998a) and Margrave (1998), I extend the concept of a filter to that of a filter-bank, which is a set of N filters: one filter for every point in the input/output space. I identify ${\bf a}_j$ with the filter corresponding to the $j^{\rm th}$ location in the input/output vector, and the coefficient, a_i,j, with the $i^{\rm th}$ coefficient of the filter, ${\bf a}_j$.

Margrave (1998) describes two closely related alternatives which both reduce to normal convolution in the limit of stationarity. The first approach is to place the filters in the columns of the matrix, ${\bf A}$. This associates a single filter with a single point in the output space, and defines non-stationary convolution:  

$$y_k = x_k + \sum_{i=1}^{\min(N_a-1, k-1)} a_{i,(k-i)} \; x_{k-i}.$$ (116)

In contrast, the second approach is to place individual filters in the rows of the matrix, ${\bf A}$, associating a single filter with a single point in the input space. This defines what Margrave (1998) refers to as non-stationary combination:  

$$y_k = x_k + \sum_{i=1}^{\min(N_a-1, k-1)} a_{i,k} \; x_{k-i}.$$ (117)

The advantage of non-stationary convolution over non-stationary combination is that the response of equation () to an impulse at the $j^{\rm th}$ element of ${\bf x}$, is ${\bf a}_j$. A more general output is then a scaled superposition of filter-bank filters, which fits with Green's function theory for linear, constant coefficient, partial differential equations. Again, in contrast, the response of equation () to an impulse at the $j^{\rm th}$ element of ${\bf x}$, is the $j^{\rm th}$column of non-stationary combination matrix, which bears no direct relationship to the filter, ${\bf a}_j$, or any other individual filter for that matter.

As an illustration, consider the differences between matrices, ${\bf F}_{\rm conv}$ and ${\bf F}_{\rm comb}$ below, which represent, respectively, non-stationary convolution and combination with a causal three-point (N_f=3) filter-bank, ${\bf f}$, to vectors of length, N=5. The two are equivalent in the stationary limit; however, while the columns of ${\bf F}_{\rm conv}$ contain filters, ${\bf f}_j$, the columns of ${\bf F}_{\rm comb}$ do not.

$${\bf F}_{\rm conv} = \left[ \begin{array} {ccccccc} 1 & 0 & ... ...1 & 0 \\ 0 & 0 & 0 & f_{25} & f_{15} & 1 \\ \end{array} \right]$$

It is also clear that while ${\bf F}_{\rm conv}$ and ${\bf F}_{\rm comb}$ are related, they are not simply adjoint to each other.