Resistive Terminations

Let's begin with simple ``resistive'' terminations at the string endpoints, resulting in the reflection coefficient $g$ at each end of the string, where $\vert g\vert\leq 1$ corresponds to nonnegative (passive) termination resistances [450]. Inspection of Fig.E.2 makes it clear that terminating the left endpoint may be accomplished by setting

$$y^{+}_{n,0} = g_ly^{-}_{n,0},$$

and the right termination corresponds to

$$y^{-}_{n,6} = g_ry^{+}_{n,6}.$$

By allowing an additional two samples of round-trip delay in each endpoint reflectance (one sample in the chosen subgrid), we can implement these reflections within the state-transition matrix:

$$\tilde{\mathbf{A}}_W= \left[\! \begin{array}{ccccccccccc} 0 & g_l & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & g_r & 0 \end{array} \!\right]$$ (E.38)

The simplest choice of state transformation matrix $\mathbf{T}$ is obtained by cropping it to size $M\times M$ :

$$\mathbf{T}\isdef \left[\! \begin{array}{ccccccccccc} 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{array}\!\right]$$

An advantage of this choice is that its inverse $\mathbf{T}^{-1}$ is similarly a simple cropping of the $M=\infty$ case. However, the corresponding FDTD system is not so elegant:

$$\begin{eqnarray*} \tilde{\mathbf{A}}_K&\isdef & \mathbf{T}\tilde{\mathbf{A}}_W\mathbf{T}^{-1}\\ & = & \left[\! \begin{array}{ccccccccccc} 0 & g_l & -g_l & h_l & -h_l & h_l & -h_l & h_l \\ 1 & -1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & -1 & 1 & -1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1 & 1 & -1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & -1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & -1 & h_r & -h_r \\ 0 & 0 & 0 & 0 & 0 & 0 & g_r & -g_r \end{array}\!\right], \end{eqnarray*}$$

where $h_l\isdef 1+g_l$ and $h_r\isdef 1+g_r$ . We see that the left FDTD termination is non-local for $g\neq -1$ , while the right termination is local (to two adjacent spatial samples) for all $g$ . This can be viewed as a consequence of having ordered the FDTD state variables as $[y_{n,m},y_{n-1,m+1},\ldots]$ instead of $[y_{n-1,m},y_{n,m+1},\ldots]$ . Choosing the other ordering interchanges the endpoint behavior. Call these orderings Type I and Type II, respectively. Then $\mathbf{T}_{II}=\mathbf{T}_I^T$ ; that is, the similarity transformation matrix $\mathbf{T}$ is transposed when converting from Type I to Type II or vice versa. By anechoically coupling a Type I FDTD simulation on the right with a Type II simulation on the left, general resistive terminations may be obtained on both ends which are localized to two spatial samples.

In nearly all musical sound synthesis applications, at least one of the string endpoints is modeled as rigidly clamped at the ``nut''. Therefore, since the FDTD, as defined here, most naturally provides a clamped endpoint on the left, with more general localized terminations possible on the right, we will proceed with this case for simplicity in what follows. Thus, we set $g_l=-1$ and obtain

$$\begin{eqnarray*} \mbox{$\stackrel{{\scriptscriptstyle \vdash}}{\mathbf{A}}$}_K&\isdef & \left[\! \begin{array}{ccccccccccc} 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & -1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & -1 & 1 & -1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1 & 1 & -1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & -1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & -1 & 1+g_r & -1-g_r \\ 0 & 0 & 0 & 0 & 0 & 0 & g_r & -g_r \end{array}\!\right] \end{eqnarray*}$$