Equivalence of Digital Waveguide and Finite Difference Schemes
State Space Formulation
FDTD State Space ModelSearch Physical Audio Signal Processing
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Let
denote the FDTD state for one of the two subgrids at time
, as defined by Eq.
(P.10). The other subgrid is handled
identically and will not be considered explicitly. In fact, the other
subgrid can be dropped altogether to obtain a half-rate,
staggered grid scheme [55,149]. However, boundary
conditions and input signals will couple the subgrids, in general. To
land on the same subgrid after a state update, it is necessary to
advance time by two samples instead of one. The state-space model for
one subgrid of the FDTD model of the ideal string may then be written
as
When there is a general input signal vector
, it is necessary to
augment the input matrix
to accomodate contributions over both
time steps. This is because inputs to positions
at time
affect position
at time
. Henceforth, we assume
and
have been augmented in this way. Thus, if there are
input
signals