### Lossy Finite Difference Recursion

We will now derive a finite-difference model in terms of string displacement samples which correspond to the lossy digital waveguide model of Fig.C.5. This derivation generalizes the lossless case considered in §C.4.3.

Figure C.7 depicts a digital waveguide section once again in physical canonical form,'' as shown earlier in Fig.C.5, and introduces a doubly indexed notation for greater clarity in the derivation below [442,222,124,123].
Referring to Fig.C.7, we have the following time-update relations:

 (C.31)

This is now in the form of the finite-difference time-domain (FDTD) scheme analyzed in [222]:

with , and . In [124], it was shown by von Neumann analysisD.4) that these parameter choices give rise to a stable finite-difference schemeD.2.3), provided . In the present context, we expect stability to follow naturally from starting with a passive digital waveguide model.

#### Frequency-Dependent Losses

The preceding derivation generalizes immediately to frequency-dependent losses. First imagine each in Fig.C.7 to be replaced by , where for passivity we require

In the time domain, we interpret as the impulse response corresponding to . We may now derive the frequency-dependent counterpart of Eq.(C.31) as follows:

where denotes convolution (in the time dimension only). Define filtered node variables by

Then the frequency-dependent FDTD scheme is simply

We see that generalizing the FDTD scheme to frequency-dependent losses requires a simple filtering of each node variable by the per-sample propagation filter . For computational efficiency, two spatial lines should be stored in memory at time : and , for all . These state variables enable computation of , after which each sample of () is filtered by to produce for the next iteration, and is filtered by to produce for the next iteration. The frequency-dependent generalization of the FDTD scheme described in this section extends readily to the digital waveguide mesh. See §C.14.5 for the outline of the derivation.
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