Input Locality

The DW state-space model is given in terms of the FDTD state-space model by Eq.(P.31). The similarity transformation matrix is bidiagonal, so that and are both approximately diagonal when the output is string displacement for all . However, since given in Eq.(P.11) is upper triangular, the input matrix can replace sparse input matrices with only half-sparse , unless successive columns of are equally weighted, as discussed in §P.3. We can say that local K-variable excitations may correspond to non-local W-variable excitations. From Eq.(P.35) and Eq.(P.36), we see that displacement inputs are always local in both systems. Therefore, local FDTD and non-local DW excitations can only occur when a variable dual to displacement is being excited, such as velocity. If the external integrator Eq.(P.37) is used, all inputs are ultimately displacement inputs, and the distinction disappears.

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written by 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.