State Transformations

In previous work, time-domain adaptors (digital filters) converting between K variables and W variables have been devised [223]. In this section, an alternative approach is proposed. Mapping Eq.(E.7) gives us an immediate conversion from W to K state variables, so all we need now is the inverse map for any time . This is complicated by the fact that non-local spatial dependencies can go indefinitely in one direction along the string, as we will see below. We will proceed by first writing down the conversion from W to K variables in matrix form, which is easy to do, and then invert that matrix. For simplicity, we will consider the case of an infinitely long string.

To initialize a K variable simulation for starting at time , we need initial spatial samples at all positions for two successive times and . From this state specification, the FDTD scheme Eq.(E.3) can compute for all , and so on for increasing . In the DW model, all state variables are defined as belonging to the same time , as shown in Fig.E.2.

From Eq.(E.6), and referring to the notation defined in Fig.E.2, we may write the conversion from W to K variables as

where the last equality follows from the traveling-wave behavior (see Fig.E.2).

Figure E.3 shows the so-called ``stencil'' of the FDTD scheme. The larger circles indicate the state at time which can be used to compute the state at time . The filled and unfilled circles indicate membership in one of two interleaved grids [55]. To see why there are two interleaved grids, note that when is even, the update for depends only on odd from time and even from time . Since the two W components of are converted to two W components at time in Eq.(E.8), we have that the update for depends only on W components from time and positions . Moving to the next position update, for , the state used is independent of that used for , and the W components used are from positions and . As a result of these observations, we see that we may write the state-variable transformation separately for even and odd , e.g.,

Denote the linear transformation operator by and the K and W state vectors by and , respectively. Then Eq.(E.9) can be restated as

The operator can be recognized as the Toeplitz operator associated with the linear, shift-invariant filter . While the present context is not a simple convolution since is not a simple time series, the inverse of corresponds to the Toeplitz operator associated with

Therefore, we may easily write down the inverted transformation:

The case of the finite string is identical to that of the infinite string when the matrix is simply ``cropped'' to a finite square size (leaving an isolated 1 in the lower right corner); in such cases, as given above is simply cropped to the same size, retaining its upper triangular structure. Another interesting set of cases is obtained by inserting a 1 in the lower-left corner of the cropped matrix to make it circulant; in these cases, the matrix contains in every position for even , and is singular for odd (when there is one zero eigenvalue).

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.