Sign in

username:

password:



Not a member?

Search Online Books



Search tips

Free Online Books

Ads

Chapters

Physical Audio Signal Processing
    Equivalence of Digital Waveguide and Finite Difference Schemes
       State Space Formulation
          Boundary Conditions
             Resistive Terminations

Chapter Contents:

Search Physical Audio Signal Processing

  

Book Index | Global Index

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

  

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 [458]. Inspection of Fig.P.2 makes it clear that terminating the left endpoint may be accomplished by setting

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

and the right termination corresponds to

$\displaystyle 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:

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

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

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

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\m... ...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