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Chapters

Physical Audio Signal Processing
    Equivalence of Digital Waveguide and Finite Difference Schemes
       State Space Formulation
          DW State Space Model
             DW Displacement Inputs

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DW Displacement Inputs

We define general DW inputs as follows:

$\displaystyle y^{+}_{n,m}$	$\displaystyle =$	$\displaystyle y^{+}_{n-1,m-1} + (\underline{\gamma}^{+}_m)^T \underline{\upsilon}(n)$	(E.33)
$\displaystyle y^{-}_{n,m}$	$\displaystyle =$	$\displaystyle y^{-}_{n-1,m+1} + (\underline{\gamma}^{-}_m)^T \underline{\upsilon}(n)$	(E.34)

The

th $2q\times2$ block of the input matrix ${\mathbf{B}_W}$ driving state components $[y^{+}_{n+2,m},y^{-}_{n+2,m}]^T$ and multiplying $[\underline{\upsilon}(n+2)^T,\underline{\upsilon}(n+1)^T]^T$ is then given by

$\displaystyle \left({\mathbf{B}_W}\right)_m = \left[\! \begin{array}{cc} (\unde... ...ma}^{-}_m)^T & (\underline{\gamma}^{-}_{m+1})^T \end{array} \!\right]. \protect$

(E.35)

Typically, input signals are injected equally to the left and right along the string, in which case

$\displaystyle \underline{\gamma}^{+}_m = \underline{\gamma}^{-}_m \isdef \underline{\gamma}_m.$

Physically, this corresponds to applied forces at a single, non-moving, string position over time. The state update with this simplification appears as

$\begin{displaymath} \underbrace{\left[\! \begin{array}{c} \vdots\\ y^{+}_{n+2,m... ...ine{\upsilon}(n+1) \end{array}\!\right]}_{\underline{u}(n+2)}. \end{displaymath}$

Note that if there are no inputs driving the adjacent subgrid ( $\underline{\gamma}_{m-1}=\underline{\gamma}_{m+1}=0$ ), such as in a half-rate staggered grid scheme, the input reduces to

$\begin{displaymath} \underline{x}_W(n+2) = \mathbf{A}_W\underline{x}_W(n) + \un... ...d{array}\!\right]}_{{\mathbf{B}_W}} \underline{\upsilon}(n+2). \end{displaymath}$

To show that the directly obtained FDTD and DW state-space models correspond to the same dynamic system, it remains to verify that $\mathbf{A}_W=\mathbf{T}^{-1}\mathbf{A}_K\,\mathbf{T}$ . It is somewhat easier to show that

$\begin{eqnarray*} \mathbf{T}\,\mathbf{A}_W&=& \mathbf{A}_K\,\mathbf{T}\\ &=& \l... ...dots & \vdots & \vdots & \vdots & \vdots \end{array}\!\right]. \end{eqnarray*}$

A straightforward calculation verifies that the above identity holds, as expected. One can similarly verify $\mathbf{C}_W=\mathbf{C}_K\,\mathbf{T}$ , as expected. The relation ${\mathbf{B}_W}=\mathbf{T}^{-1}\,\mathbf{B}_K$ provides a recipe for translating any choice of input signals for the FDTD model to equivalent inputs for the DW model, or vice versa. For example, in the scalar input case (), the DW input-weights ${\mathbf{B}_W}$ become FDTD input-weights $\mathbf{B}_K$ according to

$\begin{displaymath} \left[\! \begin{array}{l} \qquad\vdots\\ y_{n+1,m-1}\\ y_{... ...psilon}(n+2)\\ \underline{\upsilon}(n+1) \end{array}\!\right] \end{displaymath}$

The left- and right-going input-weight superscripts indicate the origin of each coefficient. Setting $\gamma^{+}_m=\gamma^{-}_m$ results in

$\displaystyle \mathbf{B}_K= \left[\! \begin{array}{cc} \vdots & \vdots\\ \gamma... ...ma _{m+1}+\gamma _{m+3} \\ [5pt] \vdots & \vdots \end{array} \!\right] \protect$

(E.36)

Finally, when $\gamma _m=1$ and $\gamma _{\mu}=0$ for all $\mu\neq m$ , we obtain the result familiar from Eq. $\,$ (E.23):

$\begin{displaymath} \mathbf{B}_K= \left[\! \begin{array}{cc} \vdots & \vdots\\ ... ...0 \\ 2 & 0 \\ 1 & 0 \\ \vdots & \vdots \end{array}\!\right] \end{displaymath}$

Similarly, setting $\gamma^{\pm}_{\mu}=0$ for all $\mu\neq m+1$ , the weighting pattern

appears in the second column, shifted down one row. Thus, $\mathbf{B}_K$ in general (for physically stationary displacement inputs) can be seen as the superposition of weight patterns

in the left column for even

, and the right column for odd

(the other subgrid), where the

is aligned with the driven sample. This is the general collection of displacement inputs.

Previous: DW State Space Model
Next: DW Non-Displacement Inputs

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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.

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