### 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:

Adding these equations gives

This is now in the form of the

*finite-difference time-domain*(FDTD) scheme analyzed in [222]:

#### 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

*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

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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Higher Order Terms

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Digital Filter Models of Damped Strings