DW State Space Model

As discussed in §E.2, the traveling-wave decomposition Eq.(E.4) defines a linear transformation Eq.(E.10) from the DW state to the FDTD state:

Since is invertible, it qualifies as a linear transformation for performing a change of coordinates for the state space. Substituting Eq.(E.27) into the FDTD state space model Eq.(E.24) gives

Multiplying through Eq.(E.28) by gives a new state-space representation of the same dynamic system which we will show is in fact the DW model of Fig.E.2:

where

To verify that the DW model derived in this manner is the computation diagrammed in Fig.E.2, we may write down the state transition matrix for one subgrid from the figure to obtain the permutation matrix ,

and displacement output matrix :

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Subsections

• DW Displacement Inputs
• DW Non-Displacement Inputs
• Input Locality

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Order a Hardcopy of Physical Audio Signal Processing

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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.